New University of Genoa Department of Civil, Chemical and … · 2018. 10. 19. · course, the reader who is not interested in the details of the algebra can skip them and focus her/his

1

University of GenoaDepartment of Civil, Chemical and Environmental Engineering

Copyright 2015All rights reserved. These notes, or parts thereof, may not be reproduced in any form or

by any means, electronic or mechanical, including photocopying, recording or any informationstorage and retrieval system now known or to be invented, without the written permission fromthe authors.

SEA WAVES, TIDAL CURRENTS ANDCOASTAL FORMS

Appunti del corso di

IDRAULICA MARITTIMA E COSTIERA

Paolo Blondeaux and Giovanna Vittori

19 ottobre 2018

1

PREFACE

During the last decades, the modelling of both the hydrodynamics and morphodynamicsof the coastal region has progressed significantly and nowadays sophisticated numerical modelsexist which allow to evaluate the propagation of tidal waves in the coastal region, the interactionof wind waves with the sea bottom, the generation of steady currents by the waves and themorphological changes of the sea bed and the coastline induced by waves and currents. However,the results provided by these models are not entirely satisfactory and further efforts should bemade to obtain reliable and accurate predictions.

These notes are aimed at introducing the reader to the dynamics of water motions in thecoastal region and at looking at the interaction of waves and currents with sediment motionand with the morphology of the sea bottom, the beach face and the coastline. The notes arewritten for master and PhD students but we think they can be useful also for young researcherswho want to improve their knowledge of the hydrodynamic and morphodynamic phenomenataking place in the coastal region.

The reader is supposed to have a basic knowledge of Calculus and Fluid Mechanics andattention is devoted to describe the dynamics of sea waves, tidal currents and bottom forms,starting from Navier-Stokes and continuity equations. Sediment motion is described by meansof sediment continuity equation and by quantifying its transport rate by means of empiricalrelationships, even though nowadays analyses describing the flow around sediment grains andthe forces/torques acting on them start to appear. However, direct numerical simulations ofwater and sediment motions in the large scale problems typical of the coastal region are far tobe made.

The amount of relevant research results produced during the last decades is enourmous andit is not possible to provide an exhaustive and detailed description of our actual knowledgeof this subject. Hence, some fundamental topics are treated providing both the mathematicaldetails and a physical explanation of the equation controlling a particular phenomenon. Ofcourse, the reader who is not interested in the details of the algebra can skip them and focusher/his attention on the physics of the phenomena and on the final results. On the other hand,other topics are dealt with an almost review style providing references to the reader who wishesto deepen her/his knowledge on the subject.

The notes are made available in the form of an e-book to make easier theupdating of its contents. The readers and in particular the students are invitedto send comments and suggestions for improvements of the book to the authors([email protected], [email protected]

2

Prefazione

Questi appunti sono stati scritti con un duplice scopo. In primo luogo essi vogliono essered’ausilio per la preparazione all’esame di Idraulica Marittima e Costiera e quindi in gran partetrattano gli argomenti illustrati durante il relativo corso. Ove possibile, sono esplicitati i dettaglidei passaggi matematici che conducono alle relazioni di interesse, per consentire allo studente lapiena comprensione delle relazioni riportate, anche se lo studente deve rivolgere la sua attenzioneal significato fisico di tali relazioni.

Alcuni capitoli sono invece dedicati a sintetizzare le piu recenti analisi di particolari fenome-ni, per stimolare il lettore ad approfondire alcuni temi. E per questo motivo che alcuni capitolisono scritti in inglese. Queste parti sono una prima stesura di un possibile libro che potrebbevedere la luce in un prossimo futuro.

Nell’ambito del corso si fara ampio uso dei cosiddetti metodi perturbativi. Lo studente/illettore, che non ha familiarita con tali metodi, e spronato a leggere la nota alla fine degli appuntied eventualmente ad approfondire l’argomento.

3

Indice capA.tex

Capitolo 1: Le onde di Stokes cap1.tex

IntroduzioneL’evoluzione di un’onda monocromatica: formulazione del problemaOnde di piccola ampiezza: la soluzione del problema lineareAlcune caratteristiche delle onde di piccola ampiezzaSovrapposizione di onde. Onde stazionarie

Capitolo 2: Gli effetti non lineari cap2.tex

IntroduzioneGli effetti non lineari su un’onda progressiva.Gli effetti non lineari su un’onda stazionaria. Diagramma di Saintflou

Capitolo 3: Fenomeni di rifrazione cap3.tex

L’energia per unita di superficie e il flusso di energiaLa velocita di gruppo: un’interpretazione cinematicaIl fenomeno dello shoalingL’equazione della wave actionRaggi d’onda e legge di Snell

Capitolo 4: Diffrazione cap4.tex

Equazione di HelmholtzCasi di interesse ingegneristico

Capitolo 5: Mild slope equation cap5.tex

Capitolo 6: Effetti viscosi: lo strato limite cap6.tex

L’analisi nello strato limite: l’approccio lineareGli effetti non lineari

Capitolo 7: Le onde su bassa profondita cap7.tex

Introduzione e l’analisi di BoussinesqL’onda solitariaLe onde cnoidali

4

Le equazioni di Korteweg - De VriesLe onde di bordo

Capitolo 8: Equazioni mediate sulla profondita per moti stazionari (le correnti) cap7bis.tex

Capitolo 9: Le equazioni mediate sulla profondita e nel tempo cap8.tex

IntroduzioneEquazione di continuitaEquazione del motoAlcune semplificazioni preliminariIl tensore delle tensioni radiative

Capitolo 10: La regione dei frangenti cap9.tex

Il fenomeno del frangimentoSet-down e set-upLa corrente litoranea

Capitolo 11: Le onde di mare cap10.tex

Capitolo 12: La previsione del clima ondoso cap11.tex

Capitolo 13: Ship waves cap17.tex

Capitolo 14: Tides cap15.tex

Capitolo 15: Il trasporto solido cap12.tex

Capitolo 16: Il trasporto di materiali coesivi cap16.tex

Capitolo 17: Morfologia costiera cap13.tex

Capitolo 18: Stability analyses and coastal forms cap14.tex

Capitolo 19: A few significant problems cap18.tex

Nota su metodi perturbativi capB.tex

Capitolo 1

LINEAR STOKES WAVES

1.1 Introduction

Figura 1.1: Waves observed along the coast of Genoa (courtesy of Giovanni Allievi)

Rarely, the free surface of a body of water is flat and has no waves. The waves, whichcharacterize the free surface of a water body, are generated by external forces which act on thefluid and tend to deform its free surface against the actions of gravity and surface tension whichtend to keep it flat and horizontal. Waves of different length can be observed on the surfaceof the sea depending on the kind and magnitude of the forces which generate them: a small

5

CAPITOLO 1. LINEAR STOKES WAVES 6

stone thrown by a child generates waves which are a few centimeters long while the action ofthe moon and the sun generates a wave characterized by a length of thousands of kilometres.

The waves which are observed on the sea surface do not repeat exactly the same and, if thewater elevation is recorded as function of time, the signal turns out to have a random behaviour.However, notwithstanding the randomness of the waves observed in the sea, the free surfacecan be well represented by the superposition of a large number of sinusoidal waves. Moreover,in shallow water and far from the generation area, the waves turn out to be more regular andit is reasonable to describe them as monochromatic waves. Finally, for some purposes, onesinusoidal wave provides a reasonably description of the sea surface and for this reason and theease of use of a linear approach, the theory of a monochromatic surface wave of small amplitudeis widely known and largely applied. In addition, the linear theory of surface waves is the firstnecessary step to address the study of sea waves of large amplitude. Therefore, the first part ofthe book is devoted to describe the linear theory of surface waves and the analysis of nonlinearphenomena is made later.

The surface waves observed in the sea can be classified according to their characteristicperiod T which ranges between values of order 10−1 s and values of order of 104 s and even larger.To give a qualitative idea of the relative importance of the different harmonic components, figure1.2 shows a schematic, qualitative plot of the distribution of the energy of ocean surface wavesas function of their frequency f = 1/T . The largest part of the energy is associated with the so-called wind waves which are characterized by a period of the order of magnitude of 10 s. Thesecomponents are responsible to a large extent of the hydrodynamic phenomena observed in thecoastal region and of the erosion and deposition processes of the sediment. For this reason, inthis chapter and in the following chapters, we focus our attention on wind waves, studying theircharacteristics, the process of generation by the wind, their propagation and their interactionwith the seabed and the coastline. The study of the tidal waves, which induce the presence ofthe two peaks which can be observed in figure 1.2 for f = 1/24 h−1 and f = 1/12 h−1, and ofthe morphological patterns they force, is postponed to chapters 13 and ??.

1.2 Vorticity dynamics

Since the flow induced by the propagation of wind waves is usually studied assuming the flowto be irrotational (i.e. the vorticity is assumed to vanish), before looking at the details of thevelocity field generated by a wave train, it is useful to analyse the dynamics of the vorticity

in order to understand why the assumption of an irrotational flow is reasonable and acceptableto study wind waves. The reader which is familar with vorticity dynamics can skip sections 1.2and 1.3.

The vorticity is defined as the curl of the velocity vector v ( = ∇× v) and is a mea-sure of the velocity of rotation of the fluid elements. In particular, it is crucial to understandthe mechanism which generates vorticity and makes it to pervade the whole space. The vor-ticity equation can be simply obtained applying the curl operator to Navier-Stokes equation


1e-06 0.0001 0.01 1 100

ener

gy [a

rbtr

ary

scal

e]

frequency [1/s]

gravity

surface tensionCoriolis force

wind wavesstorms, tsunami

sun, moon

tides

capillarywaves

ultragravitywaves

gravitywaves

infragravitywaves

longperiodwaves

transtidalwaves

Figura 1.2: Qualitative distribution of the energy of ocean surface waves as function of thetheir frequency (adapted from the Shore Protection Manual, Volume 1, 1984, Department ofthe Army, US Army Corps of Engineers). Both the disturbing (sun, moon, wind, storms,tsunami) and restoring (Coriolis force, gravity, surface tension) forces are indicated along withthe range of the frequency which they dominate.

(momentum equation)

ρdv

dt= ρ

[

∂v

∂t+ (v · ∇) v

]

= ρf −∇p+ µ∇2v (1.1)

where ρ and µ are the fluid density and dynamic viscosity, respectively, which in the followingare assumed to be given constants. This implies that the pressure and temperature variationsare assumed so small to neglect the variations of ρ and µ. Moreover, in (1.1) p is the pressureand the force field f is assumed to be conservative (f = ∇ϕ). Finally, let us point out that abold symbol denotes a vector.

By applying the curl operator to (1.1) and taking into account that

∇× (∇ϕ) = 0 (1.2)

and∇× (v · ∇)v = (v · ∇)(∇× v) − ((∇× v) · ∇)v = (v · ∇) − ( · ∇)v, (1.3)

vorticity equation is obtained

ρ

[

∂

∂t+ (v · ∇) − ( · ∇)v

]

= µ∇2 (1.4)


ord

dt= ( · ∇) v + ν∇2 (1.5)

where ν = µ/ρ is the kinematic viscosity of the fluid.Equation (1.5) can be physically interpreted by noting that:

• the term (v · ∇) represents advection of vorticity by the fluid as a result of an unevendistribution of vorticity itself;

• the term ν∇2 represents the variation of associated with the viscous (molecular)diffusion;

• the term ( · ∇)v, which has no equivalent term in the Navier-Stokes equations, iswhat gives peculiar characteristics to the dynamics of vorticity. In particular, this termcan intensify or damp the vorticity simply by deforming the fluid elements. The readercan understand the role of this term by considering an ice skater who can modify her/hisrotation speed in absence of external torque by simply spreading her/his arms or bringingthem closer to the body, i.e. through a ’deformation’ of her/his body.

1.3 The generation of vorticity

If a fluid initially at rest is considered, the initial vorticity vanishes everywhere. However, ifthe fluid is set into motion, the resulting flow field is characterized by non-vanishing valuesof vorticity, at least in particular regions. Therefore, it appears that the motion of the fluidgenerates vorticity. To understand the process which generates vorticity in a fluid of constantdensity subject to a conservative force, let us consider the component along the xi-axis of (1.5).By multiplying both the left and right hand sides of the equation by i and by adding thethree equations obtained by considering the three possible values of the index i, we obtain

∂

∂t

(

1

2ii

)

+ vj∂

∂xj

(

1

2ii

)

−ij∂vi∂xj

− νi∂2i

∂xj∂xj= 0 (1.6)

where Einstein notation is used (according to this notation, when an index appears twice onthe right or left hand side of an equation only, it implies summation over all the values of theindex). Moreover, the last term on the right hand side of (1.6) can be written in the form

ν

[

∂

∂xj∂xj

(

1

2ii

)

− ∂i

∂xj

∂i

∂xj

]

(1.7)

Introducing the modulus of the vorticity vector

=√ii


and integrating (1.6) over a fixed volume V0, it follows

∫

V0

d

dt

(

1

22

)

dV0 =

∫

V0

ij∂vi∂xj

dV0 + ν

∫

V0

[

∂

∂xj∂xj

(

1

22

)

− ∂i

∂xj

∂i

∂xj

]

dV0 (1.8)

Then, applying the kinematic transport theorem (ρ = constant), we obtain that the materialderivative of the square of the modulus of vorticity integrated on a mobile volume of fluid V (t)is given by the sum of three terms of simple physical interpretation

d

dt

∫

V

(

1

22

)

dV =

∫

V

ij∂vi∂xj

dV + ν

∫

V

∂

∂xj∂xj

(

1

22

)

dV − ν

∫

V

∂i

∂xj

∂i

∂xjdV (1.9)

The third term on the right hand side of (1.9) has a quadratic form and it is always negative: itrepresents the dissipation of vorticity which takes place because of viscous effects. The secondterm

ν

2

∫

V

∇22dV =ν

2

∫

V

∇ · (∇2)dV =ν

2

∫

S

n · ∇2dS

produces a variation of the content of vorticity in the volume V because of a flux of vorticitythrough the surface S which defines V . Consequently the second term neither produces vorticitynor it dissipates vorticity within the volume V . Finally, the first term can vary the content ofvorticity through the deformation of material elements. However, this term can either increaseor decrease the content of vorticity only if vorticity is already present in V , otherwise itscontribution vanishes. It is possible to conclude that the vorticity can not be generatedwithin the volume V , if we consider an incompressible fluid subject to a conservativeforce field.

Since, it is a common observation that in a motion starting from rest vorticity is usuallyfound, it appears obvious to ask what is the source of vorticity in an incompressible fluid subjectto a conservative field of body force. It is reasonable to expect that there is some mechanismthat gives rise to the generation of vorticity at the frontiers of the fluid domain.

When the fluid is bounded wholly or in part by solid walls while it is at rest at the infinite,vorticity is generated by the no-slip condition at the wall (similar mechanisms exist in the caseof free surface boundaries).

To better understand this mechanism, let us look at the flow generated by the impulsivemotion of a flat plate (Rayleigh boundary layer). Indeed, Rayleigh solution shows that vorticityis generated at the surface of the plate by the no-slip condition and spreads in the directionnormal to the plate because of viscous effects.

Consider an infinite flat plate which, at t = 0, is set into motion with velocity U in thex-direction parallel to the plate (this plate is coincident with the plane y = 0). The fluid thatoccupies the half-space y > 0 is set into motion by viscous effects.

Assuming the flow to be unidirectional, Navier-Stokes equation (1.1) leads to

∂u

∂t= ν

∂2u

∂y2(1.10)


which must be solved with the boundary conditions

u = U (y = 0, t ≥ 0) (1.11)

u→ 0 (y → ∞, t ≥ 0) (1.12)

and the initial conditionu(y) = 0 (t < 0, y ≥ 0) (1.13)

Independence of x and z of the initial condition (1.13) suggests that a similar property issatisfied by the solution. The introduction of the auxiliary variable η = y

2√νt

transforms (1.10)into the following ordinary differential equation

d2u

dη2+ 2η

du

dη= 0 (1.14)

(the algebra is simple and the reader can work out it). That the dimensionless variable from

0

2

4

6

8

10

0 0.02 0.04 0.06 0.08 0.1

y [m

]

u [m/s]

t=0 st=103 st=105 st=106 st=107 s

Figura 1.3: Velocity profiles in a Rayleigh boundary layer at different instants. The plate speedis equal to 0.1 m/s and the fluid is water.

which the phenomenon depends must involve only the quantities y, ν and t in the form (y/√νt)

follows from considerations of dimensional nature. Indeed, the characteristic length ℓ of the


phenomenon, which measures the distance at which the speed change propagates, can be derivedlooking at the order of magnitude of the terms appearing into (1.10) and it turns out to be

ℓ ∼√νt. (1.15)

Then, it can be easily verified that the solution of (1.10) subject to the boundary and initialconditions (1.11), (1.12), (1.13) reads

u = U

[

1 − 2√π

∫y

2√

νt

0

e−η2

dη

]

= U

[

1 − erf

(

y

2√νt

)]

(1.16)

The solution (1.16) provides the law with which the speed U forced at y = 0 and thevorticity generated at the wall by the no-slip condition tend, by viscous diffusion, to propagateinto the region occupied by the fluid. The velocity profiles, for different values of t, are shownin figure 1.3.

For sufficiently long times, vorticity reach quite large distances from the plate and tends topervade the whole fluid domain. This result seems to suggest that no irrotational flow doesexist.

However, let us investigate the flow generated by the oscillations of a flat plate which movesin its plane (y = 0) according to the law

u =U

2

(

eiωt + c.c.)

(y = 0) (1.17)

where c.c. denotes the complex conjugate of the previous quantity. If the half-space y > 0 isoccupied by an incompressible viscous fluid, the linearity of the problem suggests that, aftera short transient, the velocity field tends to be periodic in t and characterized by the sameangular frequency ω imposed by the oscillations of the plate.

Furthermore, since the velocity of the plate oscillates, the propagation speed of vorticityis inverted at each half period. This implies that diffusion cannot proceed beyond a certaindistance ℓ from the wall. Equation (1.15) allows the order of magnitude of the distance ℓ to beevaluated. Since t ∼ T ∼ (ω)−1, T being the period of oscillation, it follows ℓ ∼

√

ν/ω. Thisresult is immediately verified by determining the solution of the problem. The forcing (1.17)of the flow suggests to write the solution in the form

u = F (y)eiωt + c.c. (1.18)

Substitution of (1.18) into Navier-Stokes equation leads to

d2F

dy2−(

iω

ν

)

F = 0 (1.19)

Equation (1.19) can be solved to provide

u =

[

c1 exp

(

− y√

2ν/ω(1 + i)

)

+ c2 exp

(

y√

2ν/ω(1 + i)

)]

eiωt + c.c. (1.20)


0

0.005

0.01

0.015

0.02

-0.1 -0.05 0 0.05 0.1

y [m

]

u [m/s]

t=0.0 st=2.5 st=5.0 st=7.5 s

Figura 1.4: Velocity profiles in a Stokes boundary layer at phases of the cycle. The amplitudeof the velocity oscillations of the plate speed 0.1 m/s, the period of oscillation is 10 s and thefluid is water.

where c1 and c2 are constants and use is made of√i = 1

2(1 + i).

The boundary condition which forces the vanishing of the flow for y → ∞ implies thevanishing of c2. On the other hand, the boundary condition (1.17) provides c1 = U

2. Hence, the

velocity field is provided by

u =U

2e− y√

2ν/ω ei

„

ωt− y√2ν/ω

«

+ c.c. (1.21)

which can be written in the form

u = Ue− y√

2ν/ω cos

(

ωt− y√

2ν/ω

)

(1.22)

Again vorticity propagates in the form of a ’viscous wave’. However, the amplitude of thiswave is damped exponentially. The characteristic thickness of the fluid layer affected by thephenomenon is, not unexpectedly,

√

2ν/ω.It follows that vorticity keeps confined within a thin layer the thickness of which, if the

fluid is water and the period of the oscillations is of order 10 s, is of the order of millimetres.Outside this layer, the vorticity vanishes.


In the following chapters, it will be shown that the propagation of a monochromatic windwave of small amplitude generates, close to the sea bed, an oscillatory flow which, introducingan oscillating reference frame, is described by (1.22). Even when the laminar regime is unstableand the flow in the boundary layer becomes turbulent, the thickness of the layer where vorticityis significant is quite small and negligible when compared with the water depth. Hence, it willappear that the propagation of a monochromatic sea wave of small amplitude generates vorticitywhich, however, keeps confined in a very thin layer the presence of which can be safely neglectedto study the main wave characteristics.

Nonlinear effects modify these results, since make the wave to generate a steady velocitycomponent and the vorticity to diffuse over the entire water column. However, the irrotationalapproach is valid at the leading order of approximation and it will be used in the following.

1.4 The time development of a monochromatic surface

wave: formulation of the problem.

0.0001

0.001

0.01

0.001 0.01 0.1

H/(

gT2 )

h/(gT2)

H0/L0=0.14

H=0.78 h

H=HB/4

HL2/h3=26

streamfunction

Stokeslinear theory

Stokes

2nd order

Stokes3rd order

Stokes 4th order

shallowwaterwaves

intermediatedepthwaves

deepwaterwaves

cnoidal theory

Figura 1.5: Ranges of validity of the wave theories in the [h/(gT 2), H/(gT 2)]-plane as suggestedin Le Mehaute (1976).

As already pointed out, the surface of the sea can be represented by the superimpositionof different harmonic components, each characterized by its period T , or alternatively by itsangular frequency ω = 2π

T, and a well-defined amplitude a. Before determining the complex


interaction between the different harmonic components which are always present in a sea state,let us consider the evolution of a single harmonic component.

The study of the time development of a harmonic component can be addressed with differentapproaches depending on the value of the local water depth h, the length of the wave (which isrelated to its period T ) and its height H = 2a. Figure 1.5, that it is now difficult to understandbut which will become clearer later on, shows the most appropriate mathematical model forthe study of a harmonic component as function of the ratios h/gT 2 and H/gT 2. At this stage,it is useful to point out that the quantity gT 2 is strictly related to the wavelength of the waveand the ratios h/gT 2, H/gT 2 can be though to be the ratios between the local water depth andthe wave length and the wave height and the wave length, respectively.

If we do not consider the region closest to the coast, which is characterized by small waterdepths, and waves of large amplitude, it is appropriate to determine flow generated by thepropagation of a surface wave by means of the linear Stokes theory.

Let us consider a two-dimensional monochromatic surface wave propagating in a coastalregion characterized by a sea bottom described by z = −h(x, y), (x, y, z) being a Cartesiancoordinate system with the plane (x, y) coincident with the still water level and the z-axispointing upwards.

Using our knowledge of vorticity dynamics, the flow generated by the propagation of amonochromatic surface wave can be studied by considering an inviscid fluid and an irrotationalflow. Both the assumptions turn out to be reasonable if we do not consider thin boundarylayers close to the bottom and the free surface, where significant values of vorticity are presentand viscous effects are relevant.

Let us postpone the study of these boundary layers and let us focus our attention in theinviscid, irrotational core region.

The assumption of an irrotational flow allows the potential function φ to be introduced,such that

φ(x) = φ0 +

∫ x

x0

v · dx (1.23)

It follows that the velocity can be obtained by the knowledge of φ using

v = ∇φ (1.24)

The fluid dynamics is controlled by continuity equation

∇ · v = 0 (1.25)

which, using the velocity potential φ, becomes

∇ · v = ∇ · (∇φ) = ∇2φ = 0 (1.26)

and by momentum equation, which is Euler equation because of the assumption of an inviscidfluid

ρ

[

∂v

∂t+ (v · ∇) v

]

= ρdv

dt= ρf −∇p (1.27)


x

z

u

w

c

L

a

h0

Figura 1.6: Sketch of the problem

We remind to the reader that the viscous term of Navier-Stokes equation (1.1) µ∇2v can bewritten in the form

µ∇2v = µ [∇ (∇ · v) −∇× (∇× v)] (1.28)

Hence, µ∇2v vanishes because of continuity equation and the vanishing of the vorticity field.Since the fluid is subject to gravity force (f = −gk), the use of (1.24) along with the

integration of (1.27) leads to

p

ρ+ gz +

∂φ

∂t+

1

2∇φ · ∇φ = F(t) (1.29)

where the function F(t) can be merged in the function φ.Then, the problem is closed by appropriate boundary conditions. At the sea bed, the

velocity component orthogonal to the bottom should vanish

∂φ

∂x

∂h

∂x+∂φ

∂y

∂h

∂y+∂φ

∂z= 0 (1.30)

At the free surface, since both the surface tension effects and the viscous effects turn out tobe negligible, the fluid pressure should be equal to the atmospheric pressure. Introducing therelative pressure and (1.29), it is easy to obtain

gη +∂φ

∂t+

1

2|∇φ|2 = 0 at z = η (1.31)

where the relationshipF (x, y, z, t) = z − η(x, y, t) = 0 (1.32)


describes the free surface and the function F is assumed to vanish (see the appendix at theend of the chapter to see what are the effects of the surface tension). At last, the free surface,described by (1.32), should satisfy the kinematic boundary condition dF/dt = 0

∂η

∂t+∂φ

∂x

∂η

∂x+∂φ

∂y

∂η

∂y− ∂φ

∂z= 0 at z = η (1.33)

To solve the problem just formulated, it is necessary to introduce some simplifications. To thisgoal, let us consider dimensionless variables which are expected to be O(1). Let us consider aregion of order L (L being the length of the monochormatic wave we are considering) and letus define

(x′, y′, z′, h′) =2π(x, y, z, h)

L(1.34)

where the factor 2π is introduced for later convenience. Similarly, let us use the period T ofthe wave and its amplitude a as scales for the time and the free surface elevation, respectively

t′ =2πt

Tη′ =

η

a(1.35)

Since the fluid velocity can be scaled with a/T and the flow changes on a spatial scale L, thedimensionless velocity potential should be

φ′ =φ

(aL/T )(1.36)

Finally (1.29) suggests to introduce the variable

p′ =p

(2πρaL/T 2)(1.37)

If the problem is written using the dimensionless variables just introduced but dropping theapex for later convenience, the following equations and boundary conditions are obtained

∇2φ = 0 (1.38)

∂φ

∂x

∂h

∂x+∂φ

∂y

∂h

∂y+∂φ

∂z= 0 at z = −h (1.39)

gT 2

2πLη +

∂φ

∂t+

1

2ǫ|∇φ|2 = 0 at z = ǫη (1.40)

∂η

∂t− ∂φ

∂z+ ǫ

[

∂φ

∂x

∂η

∂x+∂φ

∂y

∂η

∂y

]

= 0 at z = ǫη (1.41)

where the parameter ǫ is defined by

ǫ =2πa

L(1.42)


Since the apex characterizing dimensionless variables is dropped, a bit of confusion might bearise, however, we think that the reader can easily distinguish dimensionless variables fromdimensional variables.

Typical periods of sea waves are of order 10 s, typical wavelengths are of order 100 m andtypical amplitudes are of order 1 m. It follows

gT 2

2πL= O(1); ǫ = O(10−1) (1.43)

Therefore, since the solution of the hydrodynamic problem is not known in closed form, let usexpand it using the parameter ǫ, which is assumed much smaller than 1

φ = φ0 + ǫφ1 + ǫ2φ2 + ... (1.44)

η = η0 + ǫη1 + ǫ2η2 + ... (1.45)

If (1.44) and (1.45) are plugged into (1.38)-(1.41), it follows

∇2φ0 + ǫ∇2φ1 +O(ǫ2) = 0 (1.46)

∂φ0

∂z+∂φ0

∂x

∂h

∂x+∂φ0

∂y

∂h

∂y+ ǫ

[

∂φ1

∂z+∂φ1

∂x

∂h

∂x+∂φ1

∂y

∂h

∂y

]

+O(ǫ2) = 0 for z = −h (1.47)

gT 2

2πLη0 +

∂φ0

∂t+ ǫ

[

gT 2

2πLη1 +

∂φ1

∂t+∂2φ0

∂t∂zη0 +

1

2|∇φ0|2

]

+O(ǫ2) = 0 for z = 0 (1.48)

∂η0

∂t− ∂φ0

∂z+ ǫ

[

∂η1

∂t− ∂φ1

∂z− ∂2φ0

∂z2η0 +

∂φ0

∂x

∂η0

∂x+∂φ0

∂y

∂η0

∂y

]

+O(ǫ2) = 0 for z = 0 (1.49)

where the boundary conditions on the free surface have been expanded around z = 0, usingthe assumption of small values of the parameter ǫ.

Since ǫ is much smaller than 1, the terms of order ǫn can be grouped and are requiredto satisfy the equations and the boundary conditions separately. Hence, a sequence of diffe-rential problems, which are described in the following, are obtained at the different orders ofapproximation

1.5 Waves of small amplitude: the solution of the linear

problem

At order ǫ0, i.e. at the leading order of approximation, we obtain


∇2φ0 = 0 (1.50)

∂φ0

∂z+∂φ0

∂x

∂h

∂x+∂φ0

∂y

∂h

∂y= 0 for z = −h (1.51)

gT 2

2πLη0 +

∂φ0

∂t= 0 for z = 0 (1.52)

∂η0

∂t− ∂φ0

∂z= 0 for z = 0 (1.53)

Equations and boundary conditions (1.50)-(1.53), written using dimensional variables andconsidering a horizontal bottom such that h = h0, lead to

∇2φ0 = 0 (1.54)

∂φ0

∂z= 0 for z = −h0 (1.55)

gη0 +∂φ0

∂t= 0 for z = 0 (1.56)

∂η0

∂t− ∂φ0

∂z= 0 for z = 0 (1.57)

Then, the boundary conditions (1.56) and (1.57) can be combined to obtain

∂2φ0

∂t2= −g∂φ0

∂zfor z = 0 (1.58)

Assuming the wave to be two-dimensional, i.e. assuming φ0 to depend only on (x, z, t), thesolution of the linear problem can be written in the form

φ0(x, z, t) = f(x, z)eiωt + c.c. (1.59)

Variable separation allows the function f to be written in the form

f(x, z) = P (x)Q(z) (1.60)

and Laplace equation leads tod2Pdx2

P= −

d2Qdz2

Q(1.61)

The left hand side of (1.61) depends only on x, while the right hand side depends only on z.It follows that both the terms should be equal to a constant which can be denoted with −k2


(positive values of the constant are not possible if the solution should be finite moving from−∞ to ∞ along the x-axis. Hence, it is easy to obtain

P (x) = a1e−ikx + a2e

ikx (1.62)

Q(z) = b1ekz + b2e

−kz (1.63)

The boundary condition (1.55) forces

b1 = b2e2kh0 (1.64)

Hence

Q(z) = b2(

e2kh0ekz + e−kz)

= 2b2ekh0

[

ek(z+h0) + e−k(z+h0)

2

]

= 2b2ekh0 cosh [k(z + h0)] (1.65)

and the solution of the problem turns out to be

φ0(x, z, t) = 2b2ekh0 cosh [k(z + h0)]

[

a1e−ikx + a2e

ikx]

eiωt + c.c. (1.66)

If the surface wave propagates in the positive direction of the x-axis, the constant a2 shouldvanish. Moreover, an appropriate choice of the origin of the x-axis, allows the function φ0 tobe written in the form

φ0 = C cosh [k(z + h0)] ei(ωt−kx) + c.c. (1.67)

The boundary condition on the free surface (1.58) forces a relationship between the angularfrequency ω and the wavenumber k, where the water depth h0 appears as a parameter

ω2 = gk tanh (kh0) (1.68)

The relationship (1.68), as discussed in the following, is called ’dispersion’ relatioship. If thewater depth h0 is fixed, the evaluation of the period T for a given value of L is straightforward.It is also possible to evaluate the wavelength L for a given value of T , even though a recursiveprocedure should be used (in the past it was common to use also graphical procedures as shownin figure 1.7). For a given value of ω, it is also possible to evaluate the wavenumber k by usingapproximate empirical relationships as those proposed by Fenton & Mckee (1990)

k =ω2

g

coth

(

ω

√

h0

g

)

32

23

(1.69)

and more recently by Guo (2002)

k =ω2

g

1 − e−

„

ωq

h0g

«52

− 25

(1.70)


0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4kh0

ω2 h0 /(gkh0) for ω2 h0 /g = 0.5

ω2 h0 /g = 1.0

ω2 h0 /g = 2.0

kh0 for ω2 h0 /g = 1.0

tanh(kh0)

Figura 1.7: Sketch of the graphic approach used to determine the length L of a wave given itsperiod T and the local water depth.

which provide the wavenumber k with an error smaller than 1.5% for all the wave periods.The constant C, which appears into (1.67) can be related to the amplitude of the wave by

means of (1.56). It is easy to obtain

η0 =a

2ei(ωt−kx) + c.c. = −1

giωC cosh(kh0)e

i(ωt−kx) + c.c. (1.71)

and, then,

C = iag

2ω cosh [kh0](1.72)

It follows

φ0 = iag

2ω

cosh [k(z + h0)]

cosh [kh0]ei(ωt−kx) + c.c. (1.73)

which can be written also in the form

φ0 = iaω

2k

cosh [k(z + h0)]

sinh [kh0]ei(ωt−kx) + c.c. (1.74)

by using the dispersion relationship (1.68).

1.6 The characteristics of the waves of small amplitude

As already pointed out, the dispersion relationship (1.68) relates the period T and the wave-length L, once the water depth h0 is given. It is easy to verify that, once the period of the wave


is fixed, the wavelength decreases if the water depth h0 decreases. Similarly, a decrease of h0

leads to a decrease of the wave speed c defined by

c =L

T=ω

k=

√

g

ktanh(kh0) (1.75)

Figure 1.8 shows the ratio(

c√gh0

)2

as function of 2πkh0

and shows that c tends to√gh0 as h0

tends to zero. On the other hand c√gh0

grows if h0 is increased and turns out to be proportional

to 1/√kh0 as h0 tends to infinity. Indeed

c2

gh0=

tanh(kh0)

kh0

Hence, c tends to√

g/k as h0 tends to infinity.

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50

c2 /gh 0

2 π/kh0

c2/gh0=1/kh0

c2/gh0=1

Figura 1.8: Dimensionless wave speed plotted versus the quantity kh0

Once the velocity potential φ0 is known, it is easy to obtain the velocity components uand w and the other kinematic quantities of interest such as the particle trajectories and thestreamlines. The velocity components can be obtained by (1.24),

u =∂φ0

∂x=agk

2ω

cosh [k(z + h0)]

cosh [kh0]ei(ωt−kx) + c.c. =

aω

2

cosh [k(z + h0)]


w =∂φ0

∂z= i

agk

2ω

sinh [k(z + h0)]

cosh [kh0]ei(ωt−kx) + c.c. = i

aω

2

sinh [k(z + h0)]



Figure 1.9 shows the qualitative behaviour of the velocity vector under a wave crest, a wavetrough and the locations where η vanishes. Under the crests and the troughs, the velocity ishorizontal and positive and negative, respectively. On the other hand, the velocity where ηvanishes is vertical. The trajectory of a fluid particle can be obtained by

x

z

L

Figura 1.9: Qualitative instantaneous velocity field under the crest, the trough and the sectionswhere the free surface elevation vanishes.

dxPdt

= uP ;dzPdt

= wP (1.78)

which, once integrated, give

xP (t) = x0 +

∫ t

t0

u(xP , zP , t)dt (1.79)

zP (t) = z0 +

∫ t

t0

w(xP , zP , t)dt (1.80)

where (x0, z0) is the position of the water particle at t = t0. To evaluate the integrals, it isnecessary to point out that, for small displacements of the water particles,

v(xP , t) ∼= v(x0) + ∇v · (xP − x0) (1.81)

The linearization of (1.79), (1.80) leads to

xP (t) = x0 +

∫ t

t0

u(x0, z0, t)dt (1.82)

zP (t) = z0 +

∫ t

t0

w(x0, z0, t)dt (1.83)


and, hence, to

xP (t) − x0 =agk

ω2

cosh [k(z0 + h0)]

cosh [kh0]sin(ωt− kx) = A sin(ωt− kx) (1.84)

zP (t) − z0 =agk

ω2

sinh [k(z0 + h0)]

cosh [kh0]cos(ωt− kx) = B cos(ωt− kx) (1.85)

Then, from the relationships above, it is easy to obtain the trajectory of the generic waterparticle

(xP − x0)2

A2+

(zP − z0)2

B2= 1 (1.86)

which is an ellipse characterized by semi-axes A and B. The relationships which provide A eB show that the semi-axes decrease exponentially as the water depth is increased (see figure1.10). The largest and smallest values are attained on the free surface and at the bottom,respectively, and are

A = a coth (kh0) =a

tanh (kh0); B = a at the free surface (1.87)

A =a

sinh (kh0); B = 0 at the bottom (1.88)

In particular, since coth(kh0) tends to 1 when kh0 tends to infinity, it appears that the

x

z

L

Figura 1.10: Trajectories of water particles and their location at the instant considered.

trajectories tend to be circular when the water depth is much larger than the length of thewaves. On the other hand, for small water depths, i.e. when kh0 → 0, the oscillations of thewater particles in the horizontal directions tend to be much larger that those in the vertical


direction. The streamlines can be easily obtained by considering the relationship between thevelocity potential φ and the stream function ψ

u =∂φ0

∂x=∂ψ0

∂z;w =

∂φ0

∂z= −∂ψ0

∂x(1.89)

From (1.89), it follows that

ψ0 =ag

2ω

sinh [k(z + h0)]


Hence, the streamlines turn out to be given by

ψ0 = C (1.91)

i.e.

x =1

k

[

ωt− arccos

(

C2ω cosh [kh0]

ag sinh [k(z + h0)]

)]

(1.92)

Figure 1.11 shows the streamlines for t = 0 in the region between two crests for kh0 equalto π/2 and when the quantity ag

2ωis equal to 7.8 m2/s which corresponds to a wave of period

T = 10 s and an amplitude a = 1 m.

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0

kx

kz

Figura 1.11: Streamlines computed for t = 0 and ag2ω

= 7.8 m2/s.

From the knowledge of the velocity potential φ, it is possible to obtain not only kinematicalquantities but also dynamical quantities, like the pressure p, by means of the Bernoulli theorem.The pressure field is provided by

p = −γz − ρ∂φ0

∂t(1.93)


If the static pressure ps = −γz is introduced along with the modified pressure pm = −ρ∂φ0

∂t, it

is easy to obtain

pm =ρag

2

cosh [k(z + h0)]


= ρagcosh [k(z + h0)]

cosh [kh0]cos (ωt− kx) = γKpη

Then, it is possible to simplify the relationships previously described when the water depthis much larger or much smaller than the length of the surface wave. In the former case (infinitewater depth), we have

φ0 = iag

2ωekzei(ωt−kx) + c.c. (1.95)

u =agk


w = iagk


or, using the dispersion relationship,

u =agk

2ωekzei(ωt−kx) + c.c. = aωekz cos(ωt− kx) (1.98)

w = iagk

2ωekzei(ωt−kx) + c.c. = −aωekz sin(ωt− kx) (1.99)

Moreover, as already pointed out, when the value of h0 is much larger than the wavelength,the quantity kh0 tends to infinity and the wave celerity tends to

c =

√

g

k(1.100)

Later, it is shown that the expansion of the velocity potential as a power series of theparameter ǫ does not converge when the water depth is much smaller than the length of thewave. For this reason, figure 1.5 shows that the linear Stokes theory cannot be applied whenthe ratio h0

gT 2 assumes small values. In any case, the relationship which provide the wave celeritywhen kh0 tends to zero is still valid and provides

c =√

gh0 (1.101)

Figure 1.8 shows the quantity c2/gh0 as function of L/h0 = 2π/(kh0). This figure can be usedto classify the waves as function of the ratio h0/L. Indeed, it is possible to verify that therelationships (1.100) are (1.101) are appropriate for h0 > L/2 and h0 < L/20, respectively,with an error which can be easily evaluated, as shown in figure 1.12. In the former case thequantity c2/gh0 is well approximated by a straight line with an angular coefficient equal to1/(2π), in the latter case by a horizontal line.


0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3 3.5

kh

intermediate depth waves

shallowwaterwaves

deepwaterwaves

tanh(kh)

cosh(kh)

sinh(kh)kh

ekh/2

h/L=1/20 h/L=1/2

Figura 1.12: Hyperbolic functions (continuous lines) compared with the linear and exponentialfunctions (broken lines).

1.7 The effect of currents on waves

In the coastal region, waves might coexist and interact with steady or slowly varying currents,which possibly change over horizontal extensions which are much larger than the length of thewave. A preliminary investigation of the wave-current interaction can be made assuming that1) the currents are not affected by the waves, 2) the effects of the currents on the waves canbe investigasted by considering a steady velocity field which does not depend on the horizontaland vertical coordinates.

Let us denote with U = (Ux, Uy) the constant steady velocity components of the current.Introducing a reference frame (x, y, z) = (x+Uxt, y+Uyt, z) which moves with the current, thewaves can be studied as described previously and the relative angular frequency ωr, which ismeasured by an observer who keeps a fixed position in the reference frame (x, y, z), is given by

ω2r = gk tanh(kh) (1.102)

The reader should notice that the wavelength as well as the wavenumber do not change whenmoving from a reference frame to the other.

In the fixed coordinate system (x, y, z) (absolute reference frame), the wave speed ca turnsout to be

ca = cr + Ux (1.103)

if the x-axis is aligned with the direction of wave propagation (the velocity component Uy,which is parallel to the wave crests does not contribute to the wave speed). Since the length of


the waves is the same in the two reference frames

ωa = ωr + kUx = ωr + kU cosα (1.104)

where α is the angle that the current direction forms with the direction of wave propagation.If the relationship (1.104), which describes Doppler effect and it is called Doppler relation, issubstituited into (1.102), it can be seen that

ωa − kU cosα = ±√

gk tanh(kh) (1.105)

orωa = kU cosα±

√

gk tanh(kh) (1.106)

0

2

4

6

8

10

0 1 2 3 4 5 6

ωr

k

A

B

C U=1.5 m/s

U=2.63 m/s

[g k tanh(kh)]1/2

ωa-k U cos α

ωa

Figura 1.13: Evaluation of the two terms appearing in (1.106) as functions of k (h = 3 m,T = 6.28 s (ωa = 1 s−1), α = 150), only for opposing currents.

Hence, it appears clearly that for negative values of cosα (currents which are directedagainst the waves) and strong currents (large values of U), no solution exists and waves cannotexist (negative values of ωa are not possible).

In order to quantify the limiting value Ucrit of U , let us follow Svendsen (2006) and considervalues of the parameters such that the curve ωa − kU cosα intersects

√

gk tanh(kh) at thepoints A and B (see figure 1.13). If the value of U is increased, the points A and B movetowards the point C and they become coincident with C when the curve ωa−kU cosα becomestangent to

√

gk tanh(kh), i.e. when

dωadk

= U cosα +dωrdk


i.e.cga = U cosα + cgr

where the velocities

cga =dωadk

and cgr =dωrdk

are called ’group velocities’. In the following chapters, the physical meaning of the groupvelocity is discussed and it is shown that cg is the velocity of propagation of the energy associatedto the wave. The limiting conditions, such that the waves can not propagate against the current,take place when the group velocity cgr = dωr

dkequals −U cosα (cosα is negative), i.e. the energy

can propagate no longer in the direction of wave propagation.Figure 1.13 shows also that for increasing but still weak currents, the wavenumber k becomes

larger since the point A moves towards C. In other words, opposing currents reduce the lengthof the waves. As already pointed out, when U cosα becomes large, i.e. the current becomesstrong, the waves can propagate no longer upstream and are blocked. The point C representswaves which are just able to keep their position against the current since their group velocity inthe absolute reference frame vanishes. A linear analysis shows that the amplitude of the wavestends to infinity when the blocking point conditions are attained. However, a nonlinear analysis(Peregrine, 1976) shows that the actual waves have large but finite amplitudes. Moreover, ifthe waves do not break, reflected waves are generated which are characterized by a differentwavenumber, i.e. that represented by the point B (see Madsen & Schaffer, 1999). In the field,blocking conditions are not common but, sometimes, they occur in inlets and estuaries whenstrong (usually tidally generated) currents are present.

1.8 Superposition of waves. Incident plus reflected wa-

ves: clapotis

A wave approaching the coast can be reflected by the beach or a coastal structure. Figure 1.14shows an almost regular wave which approaches Hamlin beach (NY) from North-East and ispartially reflected by two coastal structures.

Wave reflection is quite a complex phenomenon because it often coexists with other pheno-mena. However, in particular cases, it can be easily studied. The reflection of a monochromaticwave which approaches a near vertical coastal profile (like that of a seawall or breakwater) givesrise to a clapotis, i.e. a standing wave generated by the superposition of two monochromaticwaves characterized by the same angular frequency and amplitude but opposite directions ofpropagation. It follows that the free surface and the velocity potential are described by

η0 = 2a cos(ωt) cos(kx) (1.107)

φ0 = −2ag

ω

cosh [k(z + h0)]

cosh [kh0]sin(ωt) cos(kx) (1.108)


Figura 1.14: Hamlin beach (NY). The image, downloaded from Google Earth, shows an inco-ming wave train from North-East and waves reflected by the hard structures which are presentalong the coast.

Indeedcos (ωt− kx) + cos (ωt+ kx) =


cos (ωt) cos (kx) − sin (ωt) sin (kx) + cos (ωt) cos (kx) + sin (ωt) sin (kx) = 2 cos (ωt) cos (kx)

sin (ωt− kx) + sin (ωt+ kx) =

sin (ωt) cos (kx) − cos (ωt) sin (kx) + sin (ωt) cos (kx) + cos (ωt) sin (kx) = 2 sin (ωt) cos (kx)

The free surface elevation turns out to be still sinusoidal even though the sinusoid does notpropagate as for a progressive wave. Indeed the wave profile has fixed points (nodes) forkx = (2n + 1)π

2where the amplitude vanishes, and antinodes for kx = nπ where the free

surface oscillations are maximum. Figure 1.15 shows both the standing wave and the twopropagating waves at different phases of the wave cycle.

Following the procedure already shown for a progressive wave, it is possible to obtain thevertical and horizontal velocity components

u =2agk

ω

cosh [k(z + h0)]

cosh [kh0]sin(ωt) sin(kx) (1.109)

w = −2agk

ω

sinh [k(z + h0)]


and the water particle displacements along the x- and z-axes, when the initial position is givenby x0, z0

x− x0 = −2acosh [k(z0 + h0)]

sinh [kh0]cos(ωt) sin(kx0) (1.111)

z − z0 = 2asinh [k(z0 + h0)]

sinh [kh0]cos(ωt) cos(kx0) (1.112)

The trajectories are straight lines. Indeed, the slope α of the trajectories does not depend ont and is given by the relationship tanα = (z − z0)/(x− x0), i.e.

α(x0, z0) = arctan

[

−sinh [k(z0 + h0)] cos(kx0)

cosh [k(z0 + h0)] sin(kx0)

]

= arctan

[

−tanh [k(z0 + h0)]

tan(kx0)

]

(1.113)

The value of α vanishes for kx = (2n+1)π2

which correspond to the nodes of the standing wave,where the amplitude of the free surface oscillations is always equal to zero and the trajectoriesof the water particles are horizontal. On the other hand the value of α is equal to π/2 forkx = nπ which correspond to the antinodes where the trajectories of the water particles arevertical. The water particles oscillate harmonically with an amplitude D which depends on x0

and z0. In fact(x− x0)

2 + (z − z0)2 = D2 cos2(ωt) (1.114)

where

D(x0, z0) =2a

sinh (kh0)

√

cosh2 [k(z0 + h0)] sin2(kx0) + sinh2 [k(z0 + h0)] cos2(kx0) (1.115)


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

η/a

kx

ωt=0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

η/a

kx

ωt=π/8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

η/a

kx

ωt=π/4

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6η/

akx

ωt=3π/8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

η/a

kx

ωt=π/2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

η/a

kx

ωt=5π/8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

η/a

kx

ωt=3π/4

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

η/a

kx

ωt=7π/8

Figura 1.15: Free surface elevation at different phases for a standing wave (thick line) generatedby the superposition of an incoming wave and a reflected wave (thin broken lines).

=2a

sinh (kh0)

√

sin2(kx0) + sinh2 [k(z0 + h0)]

Relationship (1.115) can be obtained taking into account that sin2X+cos2X = 1 and cosh2X−


-6

-4

-2

0

2

4

6

0 2 4 6 8 10

y

x

k

kykx

ββ

wave ray (incoming wave)

wave ray (reflected wave)

Figura 1.16: Wave rays of an incoming wave which is reflected by a vertical wall located aty = 0.

0

1

2

3

4 0 2 4 6 8 10x/L

y/L

Figura 1.17: Wavefield generated by an incident wave (β = 20) fully reflected by a verticalwall located at y = 0.


sinh2X = 1.If the monochromatic wave propagates along a direction which is oblique with respect to a

vertical wall located at y = 0 and is reflected, the resulting wavefield is described by

η0 =a

2

[

ei(ωt−kxx+kyy) + ei(ωt−kxx−kyy)]

+ c.c. (1.116)

and the velocity potential by

φ0 = iag

2ω

cosh [k (z + h0)]

cosh [kh0]

[

ei(ωt−kxx+kyy) + ei(ωt−kxx−kyy)]

+ c.c. (1.117)

where kx and ky are the wavenumbers in the x- and y-directions, respectively, such that k2x+k

2y =

k2 and k is provided by the dispersion relationship (1.68). The angle β, which the direction ofpropagation of the reflected wave forms with the y-axis, is such that

tanβ =kxky

(1.118)

(see figure 1.16).Figure 1.17 shows the diamond shaped free surface, which is the result of the incoming

wave plus the reflected wave, for fixed values of t (namely t = 0) and β (namely β = 20). Themaxima of the free surface move in the x-direction with a celerity given by ω/kx, which is thecomponent of c along the x-direction (clapotis gaufre). Figure 1.18 shows a wave reflected bya vertical wall at Santander (El Sardinero) on 7 June 2013. However, the reader should takeinto account that the free surface is different from figure 1.17 because of the strong nonlineareffects and the small water depth characterizing the waves observed at Santander.

1.9 A note on capillary waves

If the effects of the surface tension are taken into account into (1.31), the boundary conditionswould be

gη +∂φ

∂t+

1

2|∇φ|2 ± σ

ρ

(

1

R1

+1

R2

)

= 0 at z = η (1.119)

where σ is the surface tension at air-water interface, equal to about 7.5 × 10−2 N/m, and R1,R2 are the principal radii of curvature of the free surface. The order of magnitude of the largestterms is aL/T 2 which is equal to ag because of the dispersion relation. The term due to thesurface tension is of order aσ/(ρL2) (let us remind that for a two-dimension surface 1

R1= 0

and 1R2

= ∂2η/∂x2

[1+(∂η/∂x)2](3/2) ). It follows that the ratio between the largest terms and the surface

tension term has an order of magnitude which is equal to the Weber number

aLρL2

T 2σa=c2ρL

σ= We where c =

L

T(1.120)


Figura 1.18: Incoming waves reflected by a vertical wall (Santander (El Sardinero) on 7 June2013).

Taking into account the value of the air-water surface tension and the typical values of L andT for sea waves,

We = O(104) (1.121)

Therefore the effects of the surface tension are negligible. They are relevant only for theso-called ’capillary waves’ which are characterized by wavelengths of O(10 cm) or smaller (seefigure 1.19). For such waves, it is possible to simplify the boundary condition (1.119), takinginto account that the wave amplitude is much smaller than the wavelength

gη +∂φ

∂t+

1

2|∇φ|2 − σ

ρ

∂2η

∂x2= 0 at z = η (1.122)

By introducing the dynamic boundary condition at the free surface, the velocity potentialis not modified but the dispersion relation changes and the wave celerity turns out to be

c =

√

g

k

(

1 + σk2

ρg

)

tanh(kh) (1.123)

The relation (1.123) shows that the effects of the surface tension are relevant only for wavescharacterized by wavelengths smaller than 10 cm (σk

2

ρgis equal to 1 when the wavelength is

equal to about 1.7 cm).


Figura 1.19: Capillary waves generated by wind blowing over the surface of a small lake (Ireland,August 2014).

Capitolo 2

NONLINEAR STOKES WAVES

2.1 Introduzione

In chapter 1, both the free surface elevation and the velocity potential generated by the pro-pagation of a monochromatic wave are expanded as a power series of the small parameter ǫand only the first term of the expansion is considered. Since the problem which is consideredin chapter 1 is linear, it was possible to consider different harmonic components and to supe-rimpose their effects to determine, for example, the flow generated by a wave which is fullyreflected by a coastal structure. With a similar procedure, it is possible to evaluate the flowfield generated by the partial reflection of an incoming wave.

To improve the linear solution and to include weak nonlinear effects, it is necessary todetermine the second term of the expansions. The problem which provides the second term isforced and its solution can be written as the sum of the solution of the homogeneous problemand a particular solution.

In the following, we focus our attention on the second contribution, which has a spatialand temporal structure fixed by the forcing terms. The structure of the forcing terms dependson the solution of the problem at the previous order of approximation and it is necessary toconsider separately a progressive wave, a fully reflected wave and a partially reflected wave.

2.2 Nonlinear effects on a progressive wave

The second part of chapter 1 considers dimensional variables which we use also in this chapterand in the following chapters, unless differently stated.

Let us remind that the velocity potential and the free surface elevation are expanded in theform

φ = φ0 + ǫφ1 + ǫ2φ2 + ... (2.1)

η = η0 + ǫη1 + ǫ2η2 + ... (2.2)

36

CAPITOLO 2. NONLINEAR STOKES WAVES 37

where ǫ = 2πa/L.The O(ǫ0)-problem is already solved and the solution reads

φ0 = iag

2ω

cosh [k(z + h0)]


η0 =a

2ei(ωt−kx) + c.c. (2.4)

Using (1.46)-(1.49), at order (ǫ), the problem is posed by

∇2φ1 = 0 (2.5)

with the boundary conditions

∂φ1

∂z= 0 per z = −h0 (2.6)

gη1 +∂φ1

∂t= −∂

2φ0

∂t∂zη0 −

1

2|∇φ0|2 for z = 0 (2.7)

∂η1

∂t− ∂φ1

∂z=∂2φ0

∂z2η0 −

∂φ0

∂x

∂η0

∂x− ∂φ0

∂y

∂η0

∂yper z = 0 (2.8)

The procedure which provides the solution is straightforward but tedious and long. Hence,we write only the final result

φ1 = i3

16

ag

ω

cosh [2k(z + h0)]

sinh3 [kh0] cosh [kh0]e2i(ωt−kx) + c.c. (2.9)

η1 =a

8

cosh [kh0] (2 + cosh [2kh0])

sinh3 [kh0]e2i(ωt−kx) + c.c. (2.10)

where the wavenumber k is provided by the relationship determined at O(ǫ0). Let us point outthat the right hand side of (2.8), which can be evaluated using (2.3) and (2.4), has no termindependent on x and t. On the other hand, it turns out that the right hand side of (2.7) hasa non-vanishing constant term, which can be balanced by a time dependent term in φ1 whichdoes not contribute to the flow field.

Figure 2.1 shows the free surface elevation for kh0 = π/2 and two different values of ǫ,namely ǫ = 0.1 and ǫ = 0.3 (the thin line describes a sinusoidal function and is drawn to makeeasier the evaluation of nonlinear effects which make sharper the crests and flatter the troughs).Similarly, figure 2.2 shows the free surface elevation for ǫ = 0.1 and two different values of kh0,namely kh0 = π/2 and kh0 = π/4. Obviously, nonlinear effects increase as ǫ is increased butthey become more relevant also when kh0 is decreased, i.e. when the water depth decreases.As already pointed out, it is possible to add a solution of the homogeneous problem describing


-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

η/a

kx

ε=0.1

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

η/a

kx

ε=0.3

Figura 2.1: Free surface elevation at t = 0 for kh0 = π/2 and two different values of ǫ

a free wave. Finally, let us point out that it is also possible to add the term Kx to the velocitypotentail, term which describes a weak current characterized by a constant velocity profile ofstrength ǫK.

Once the velocity field is known, simple (even though long) algebra allows the pressuredistributions to be obtained.

Here, we only mention how it is possible to fix the value of the constant K. The timeaveraged volume flux per unit width through a vertical surface, induced by the propagation of


-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

η/a

kx

kh0=π/2

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

η/a

kx

kh0=π/4

Figura 2.2: Free surface elevation at t = 0 for ǫ = 0.1 and two different values of kh0

a surface wave, is provided by

Q =1

T

∫ T

0

(∫ η

−hudz

)

dt (2.11)

where an overbar denotes the time averaged value, i.e. X = 1T

∫ T

0Xdt. However, the solution

previously described does not provide the value of u when z falls above the still water level,i.e. for z larger than 0. This problem can be solved by expanding the solution in a smallrange around z = 0 (u(x, z, t) = u(x, 0, t) + z ∂u

∂z(x, 0, t) + ...). Neglecting terms of order ǫ2 and


considering vanishing values of K, it is possible to obtain

Q =ga2

2c=

a2ω

2 tanh(kh)(2.12)

Indeed∫ η

−hudz =

∫ 0

−hudz +

∫ η

0

udz =

∫ η

0

udz (2.13)

=

∫ η

0

[

u(x, 0, t) + z∂u

∂z(x, 0, t) + ...

]

dz = u0(x, 0, t)η0(x, t)

=

[

agk

2ωei(ωt−kx) + c.c.

]

[a

2ei(ωt−kx) + c.c.

]

= 2agk

2ω

a

2

The presence of a current characterized by a constant velocity profile contributes to the volumeflux Q with a term equal to Kh, which can be introduced to force the vanishing of the totalvolume flux. This procedure leads to

K =a2ω

2 tanh(kh)(2.14)

The determination of the steady velocity component generated by the propagation of amonochomatic wave in the real case is a complex problem and requires the evaluation of thespatial distribution of vorticity which vanishes no longer. The interested reader can look at thepaper by Blondeaux et al. (2002).

Finally, let us point out that the propagation speed c of the waves is not affected by termsof order ǫ. The wave speed c turns out to depend on the wave amplitude only when terms oforder ǫ2 are taken into account.

Long and tedious algebra provides the functions φ2 and η2

φ2 = −iagω

[

(11 − 2 cosh [2kh0]) cosh [3k(z + h0)]

128 sinh6 [kh0] cosh [kh0]ei3(ωt−kx)

]

+ c.c.

η2 = a

[

3 + 24 cosh6 [kh0]

128 sinh6 [kh0]ei3(ωt−kx)

]

+ c.c.

At this order of approximation, the wave celerity turns out to depend on the wave amplitude

c =gT

2πtanh [kh0]

[

1 + (ak)2 8 + cosh [4kh0]

8 sinh4 [kh0]

]

Indeed,

ω2 = gk tanh [kh0]

[

1 + (ak)2 8 + cosh [4kh0]

8 sinh4 [kh0]

]


L =gT 2

2πtanh [kh0]

[

1 + (ak)2 8 + cosh [4kh0]

8 sinh4 [kh0]

]

and

H = 2a

[

1 + (ak)2 3 + 24 cosh6 [kh0]

64 sinh6 [kh0]

]

where the wavenumber k is that determined at the leading order of approximation.At this stage, it is worth pointing out that other phenomena can affect wave propagation.

In particular, the vorticity associated with a steady current can modify significantly the resultsprovided by the analysis described so far when terms of order ǫ or larger are considered.

Wave celerity It is also useful to point out that Stokes introduced two different definitionsof the wave celerity which are known as Stokes’ first and second wave celerity. Indeed, there issome degree of freedom in defining the celerity of a surface wave.

The first definition of the wave celerity is such that the time average of the horizontal com-ponent of the Eulerian velocity vanishes below the trough level. Because of i) the irrotationalityof flow, ii) the horizontal sea bed and iii) flow periodicity in the direction of wave propagation,the time average of the horizontal velocity component turns out to be a constant between thebed and the trough levels. Hence, Stokes’ first definition of wave celerity implies that the waveis considered from a frame of reference moving with the mean horizontal velocity.

The second definition of wave celerity is such that the mean horizontal mass transportvanishes. This definition is different from the former because of the mass transport in the so-called ’splash zone’, i.e. the region between the trough level and crest level. The mass transportin the splash zone is due the positive correlation between surface elevation and horizontalvelocity component. In the reference frame used in Stokes’ second definition, the wave-inducedmass transport in the splash zone is balanced by an opposing current (undertow) and the meanvalue of the horizontal Eulerian velocity is negative below the trough level.

2.3 Nonlinear effects on a fully reflected wave. Sainflou

diagram

Following a procedure similar to that previously described, it is possible to obtain the O(ǫ)terms for a stationary wave. Taking into account that the velocity potential and the freesurface at the leading order of approximation are described by

φ0 = −2ag

ω

cosh [k(z + h0)]


η0 = 2a cos(ωt) cos(kx) (2.16)


it is possible to obtain

φ1 = −2ag

ω

[

3 cosh [2k(z + h0)]

8 sinh3 [kh0] cosh [kh0]cos(2kx) sin(2ωt) (2.17)

−1 + 3 tanh2 [k(z + h0)]

8 tanh [kh0]sin(2ωt) +

1

2 sinh [2kh0]ωt

]

η1 = a

[

2 + cosh [2kh0]

2 sinh3 [kh0]cosh [kh0] cos(2kx) cos(2ωt) +

1

tanh [2kh0]cos(2kx)

]

(2.18)

The vertical distribution of the pressure field under the antinodes is quite important becauseit corresponds to the distribution of pressure on a vertical fully reflective wall. Hence, theintegral of this pressure distribution along the vertical direction provides the force generatedby a reflected wave on the wall (breakwater).

The linear approach provides

p = ρg

[

−z + 2acosh [k (z + h0)]

cosh [kh0]cos(kx) cos(ωt)

]

(2.19)

a value which vanishes if z = η0 (nonlinear terms are neglected in (2.19)). Once the value ofp is known, the horizontal force Fx per unit length in the y-direction acting on a vertical walllocated at x = 0, can be computed

Fx =

∫ η0

−h0

p(0, z)dz = ρg

∫ η0

−h0

[

−z + 2acosh [k (z + h0)]

cosh [kh0]cos(ωt)

]

dz (2.20)

It follows

Fx = ρg

[

h20

2+ 2a

sinh [kh0]

k cosh [kh0]cos(ωt)

]

= ρg

[

h20

2+

2a

ktanh [kh0] cos(ωt)

]

(2.21)

Of course the first term of (2.21) represents the hydrostatic force which is present in the stillwater case. When, on the other side of the wall, there is still water, the net force acting on thewall oscillates in time.

The force can be evaluated taking into account also nonlinear terms.Usually, the results which are used in practical applications are a simplified form which

is known as linear Sainflou diagram and is plotted in figure 2.3. Under a crest, above thestill water level, the modified pressure (the actual pressure minus the hydrostatic pressure) isassumed to be described by a triangular distribution (ABD) while below the still water levelis trapezoidal (DBFO) being the result of the subtraction of the triangle DEO, due to thehydrostatic pressure, from the trapezius DBCO. The point A is above the still water level ofthe quantity as + sη, where sη is provided by

sη =π4a2

L tanh(kh0)(2.22)


and as = 2a. The value b of CE=FO is given by

b =2ρga

cosh(kh0)(2.23)

Under a trough, the distribution of the modified pressure, which gives rise to a force which isnegative, is provided by the tringle DGI and the trapezius IGMO (the reader should noticethat the value of OM = LH is assumed equal to b).

Once the pressure distribution is know, the force can be easily determined.

incident waves

x

z

mean water level

crest

trough

seabottom

A

D

OFE

B

C

GI

M L H

as

as

sη

Figura 2.3: Sketch of the pressure distribution due to a monochromatic wave fully reflected bya vertical wall according to the Sainflou diagram

Capitolo 3

WAVE ENERGY ANDREFRACTION PHENOMENA

3.1 The wave energy (per unit surface) and the flux of

energy

Let us first evaluate the time average of the energy per unit surface which is associated withthe propagation of a monochromatic wave. Let us remind that an overbar denotes the timeaverage over the wave period of a generic quantity X(t)

X =1

T

∫ T

0

X(t)dt. (3.1)

The time average of the kinetic energy Ec over the water column (from the bottom up to thefree surface) is provided by

Ec =ρ

2

∫ η

−h0

(u2 + w2)dz (3.2)

At the leading order of approximation, the square of the modulus of the velocity is proportionalto a2 and the value of η is O(a). Hence, neglecting terms of order ǫ3, the integral can beevaluated from −h (which is assumed to be constant) and 0.

Ec =ρ

2

∫ 0

−h

[u(x, z)eiωt + c.c.]2 + [w(x, z)eiωt + c.c.]2

dz (3.3)

=ρ

2

∫ 0

−h2 [|u|2 + |w|2] dz

44

CAPITOLO 3. WAVE ENERGY AND REFRACTION PHENOMENA 45

where |X| indicates the modulus of the complex quantity X. Taking into account (1.76) and(1.77), the value of Ec is provided by

Ec = ρ

(

gk|a|2ω

)21

cosh2(kh)

∫ 0

−h

cosh2 [k(z + h)] + sinh2 [k(z + h)]

dz = (3.4)

= ρ

(

gk|a|2ω

)21

2k cosh2(kh)

∫ 2kh

0

cosh [2kζ ] d(2kζ) = ρ

(

gk|a|2ω

)2sinh [2kh]

2k cosh2 [kh]=

1

4ρg|a|2

(let us remind that i) cosh2(X) + sinh2(X) = cosh(2X), ii) sinh(2X) = 2 sinh(X) cosh(X)),iii) cosh2(X) − sinh2(X) = 1). Moreover, the time average of the potential energy of the fluidcolumn turns out to be

Ep =

∫ η

0

ρgzdz =1

2ρgη2 =

1

4ρg|a|2 (3.5)

The total energy Et = Ec + Ep is given by

Et = Ec + Ep =1

2ρg|a|2 (3.6)

Let us point out that the relationships (3.4), (3.5) and (3.6) are valid also when the amplitudea is a complex quantity and the time average of the kinetic energy is equal to the time averageof the potential energy. This results is known as equipartition of energy.

Let us consider a section of unit width along a wave crest. The theorem of the mechanicalpower, written for an inviscid fluid characterized by a constant density, shows that the timeaveraged energy flux F s equal to the work done by the pressure forces per unit time (see theappendix 1 at the end of the chapter)

F =

∫ η

−hρ(gz +

|v|22

)udz =

∫ η

−hpudz (3.7)

Moreover, by using the Bernoulli theorem (1.29) and neglecting the quadratic terms becauseof the small value of ǫ, it is possible to obtain

p = −γz − ρ∂φ

∂t(3.8)

and, noticing that the time average of γzu vanishes,

F = −ρ∫ 0

−h

∂φ

∂t

∂φ

∂xdz (3.9)

It is possible to verify that (see appendix 2)

F =1

2ρga2

[

1

2

ω

k

(

1 +2kh

sinh 2kh

)]

= Etcg (3.10)


where

cg =c

2

(

1 +2kh

sinh 2kh

)

(3.11)

The quantity cg is known as group velocity and is the velocity of propagation of the waveenergy.

To understand why cg is called group velocity, let us consider two monochromatic wavescharacterized by the same amplitude a and angular frequencies ω + dω and ω − dω. Becauseof the difference between the angular frequencies, also the wavenumbers are different and canbe denoted by k + dk and k − dk, respectively, k being the wavenumber of the monochromaticwave characterized by the angular frequency ω. The superimposition of the two waves leads to

a

2ei[(ω+dω)t−(k+dk)x] +

a

2ei[(ω−dω)t−(k−dk)x] + c.c. =

a

2

[

ei((dω)t−(dk)x)ei(ωt−kx) + e−i((dω)t−(dk)x)ei(ωt−kx) + c.c.]

=

a

2

[

ei((dω)t−(dk)x) + e−i((dω)t−(dk)x)]

ei(ωt−kx) + c.c. =

2a cos ((dω)t− (dk)x) cos (ωt− kx)

i.e. to a monochromatic wave of angular frequency ω and wavenumber k and an amplitudewhich is slowly modulated in time and space. The modulating function is a monochromaticwave characterized by a very long wavelength 2π

dkand a very long period 2π

dω, which travels

with a celerity dωdk

. The derivative of the function ω(k), provided by the dispersion relation ofmonochromatic waves propagating over a constant water depth h, reads

dω

dk=

1

2

ω

k

(

1 +2kh

sinh 2kh

)

=c

2

(

1 +2kh

sinh 2kh

)

= cg (3.12)

For large values of the water depth (kh >> 1),

cg =c

2

(

1 +2kh

sinh 2kh

)

∼= c

2∼= 1

2

√

g

k∼= 1

2

g

ω(3.13)

while for small values (kh << 1),

cg ∼= c ∼=√

gh (3.14)

Figure 3.1 shows the free surface elevation at a particular phase. Since the wave speedc turns out to be larger than the group velocity, the crests of the waves appearing in figure3.2 move from the tail towards the head of the group. Moreover, during the propagation, thevertical distance between a single crest and the previous/next trough first increases and thendecreases. The reader should consider that the nodes of the free surface envelop travel withthe group velocity and the wave energy which is trapped between the nodes should travel withthe same speed as alread shown by (3.10).


-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-100 -50 0 50 100

η

kx

Figura 3.1: Free surface generated at a fixed phase by the superimposition of two mo-nochromatic waves characterized by the same amplitude and slightly different angularfrequencies.

As shown by Mei (1989), it is possible to extent this simple argument by considering agroup of monochromatic waves characterized by a continuous distribution of the amplitude ina restricted small range of wavenumbers. In this case, the free surface is described by

η =

∫ k0+∆k

k0−∆k

a(k)

2ei(kx−ωt)dk + c.c. (3.15)

where ∆k/k0 is assumed to be a quantity much smaller than one, a(k) is the amplitude ofthe wave characterized by the wavenumber k and ω and k satisfy the dispersion relation ω =ω(k) = gk tanh(kh).

If (3.15) is expanded as a Taylor series, it is possible to write

ω(k) = ω(k0 + (k − k0)) = ω(k0) + (k − k0)

(

dω

dk

)

k0

+O(k − k0)2 (3.16)

At the leading order of approximation, using the following notation

k − k0

k0= ζ, ω0 = ω(k0),

(

dω

dk

)

k0

= cg (3.17)

for sufficiently regular function a(k), it is possible to obtain

η =a(k0)

2ei(k0x−ω0t)

∫ ∆k/k0

−∆k/k0

[eik0ζ(x−cgt)]k0dζ + c.c. (3.18)


Indeed,

e[i(kx−ωt)] = e[i((k−k0)x+k0x−ω0t−(k−k0)cgt)] = ei(k0x−ω0t)eik0

“

k−k0k0

”

(x−cgt)k0d

(

k − k0

k0

)

(3.19)

If we introduce the variable X = ik0ζ(x− cgt), it follows dX = ik0(x− cgt)dζ and

η =a(k0)

2

ei(k0x−ω0t)

i(x− cgt)

∫ i∆k(x−cgt)

−i∆k(x−cgt)eXdX + c.c. = (3.20)

=a(k0)

2

ei(k0x−ω0t)

x− cgt

[

−iei∆k(x−cgt) + ie−i∆k(x−cgt)]

+ c.c. =

= a(k0)sin(∆k(x− cgt))

(x− cgt)ei(k0x−ω0t) + c.c. =

a

2ei(k0x−ω0t) + c.c.

where

a = a(k0)2 sin(∆k(x− cgt))

(x− cgt)(3.21)

As in the previous case, because of the presence of the term ei(k0x−ω0t), equations (3.20),(3.21) can be thought to describe a sinusoidal wave characterized by an amplitude a whichslowly changes in space and time. In particular, the envelope defined by a is a group of waves(see figure 3.2) which moves with velocity cg. The distance between two adjacent nodes of theenvelope is π/∆k and turns out to be much larger than the length 2π/k0 of the single waveswhich originate the group.

3.2 The shoaling phenomenon

Some information on the behaviour of a two-dimensional wave propagating over a changingbottom described by h(x) (x being the direction of wave propagation) can be obtained assumingthat the dissipation of energy is negligible and the slope of the bottom so small to neglectenergy reflection and to allow the adaptation of the wave characteristics to the local conditions.Therefore, wave characteristics can be described by means of the relationships which are validfor a constant water depth h, of course using the local value of h.

Because of the assumptions just introduced, the flux of energy should be constant. Forcingthat the flux of energy for an infinite water depth is equal to that for a finite depth h, it follows

ρga2∞2cg∞ = ρg

a2

2cg (3.22)

Hence

a

a∞= Ks =

√

cg∞cg

=

√

1

2

g

ω

2k

ω

sinh(2kh)

2kh + sinh(2kh)= (3.23)


-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-100 -50 0 50 100

η

kx

Figura 3.2: Free surface generated at a fixed phase by the superimposition of a group of wavescharacterized by a continuous but narrow banded distribution of a(k).

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.0001 0.001 0.01 0.1 1

Ks

h/(gT2)

Figura 3.3: The ’shoaling’ coefficient as function of the parameter h/(gT 2)

=

√

cosh(hk)

sinh(kh)

2 sinh(kh) cosh(kh)

2kh+ sinh(2kh)=

√

2 cosh2 [kh]

2kh+ sinh [2kh]

where the coefficient Ks is called ’shoaling’ coefficient. This coefficient provides the local value


of the wave amplitude as function of the amplitude of the wave in the offshore, deep waterregion, region. Since the period of a wave does not change during its propagation, it is possibleto obtain

a

a∞= Ks = f

(

h

gT 2

)

(3.24)

which is drawn in figure 3.3. Moving from the deep water region towards the shallow waterregion (decreasing values of h/(gT 2)), the shoaling coefficient first decreases, it attains itsminimum value equal to 0.91 for h/(gT 2) = 2.5 × 10−2 and then becomes equal to 1 again forh/(gT 2) = 9.0 × 10−3 (isometric point) to become larger for smaller water depths.

Figura 3.4: Wave refraction


3.3 Wave action equation

Wave refraction is a conservative process due to the interaction of a propagating wave with acomplex sea bed morphology. If the water depth depends on the two horizontal coordinates(x, y), the direction of wave propagation changes as the wave moves from the deep water regiontowards the coast, along with its wavelength and amplitude. On the other hand, the periodof the wave is constant. In the previous analysis we have considered a changing water depthwhich depends only on the direction of wave propagation (h = h(x)) and we have seen that thewavelength of the waves decreases and the amplitude increases after a relatively initial smalldecrease. Let us consider the general case described by h = h(x, y), assuming that the changesof the water depth are ’slow’, i.e. that the parameter ǫ1 be much smaller than one

ǫ1 =∆h

h<< 1 (3.25)

where ∆h is the variation of the water depth taking place if a region of order L is considered.It is reasonable to assume that the potential function describing the wave propagation be

equal to the product of a local function which depends on (x, y, x, t) times a function whichdepends on much slower coordinates. Hence, let us introduce the variables

(x, y, t) = ǫ1(x, y, t) (3.26)

and let us assume the funtions φ and η to be given by

φ = eiθ(x,y,t)/ǫ1[

φ0

(

x, y, z, t)

+ ǫ1(−i)φ1

(

x, y, z, t)

+ ...]

+ c.c. (3.27)

η = eiθ(x,y,t)/ǫ1[

a0

(

x, y, t)

+ ...]

+ c.c. (3.28)

where θ(x, y, t) is the wave phase and the function eiθ/ǫ1 describes the dependence of the waveon the variables x and y. Indeed, taking into account that

∂

∂x= ǫ1

∂

∂x;

∂

∂y= ǫ1

∂

∂y;

∂

∂t= ǫ1

∂

∂t(3.29)

it follows, for example, that

kx =∂

∂x

(

θ

ǫ1

)

= ǫ11

ǫ1

∂θ

∂x=∂θ

∂x(3.30)

Moreover,∂2φ

∂x2= ǫ21

∂2φ

∂x2= ǫ21

∂(Eφ0)

∂x2= ǫ21

[

φ0∂2E

∂x2+ 2

∂E

∂x

∂φ0

∂x+ E

∂2φ0

∂x2

]

= (3.31)

ǫ21

[

E∂2φ0

∂x2+ 2

∂φ0

∂xEi

ǫ1

∂θ

∂x+ φ0E

(

i

ǫ1

∂θ

∂x

)2

+ φ0Ei

ǫ1

∂2θ

∂x2

]

∼= −φ0E

(

∂θ

∂x

)2

= −φ0Ek2x


E being ei θ

ǫ1 .The linearized problem which controls wave propagation (see equations (1.50)-(1.53)) is

ǫ21

(

∂2φ

∂x2+∂2φ

∂y2

)

+∂2φ

∂z2= 0 for − h(x, y) < z < 0 (3.32)

ǫ21∂2φ

∂t2+ g

∂φ

∂z= 0 for z = 0 (3.33)

∂φ

∂z+ ǫ21

(

∂φ

∂x

∂h

∂x+∂φ

∂y

∂h

∂y

)

for z = −h (3.34)

Substitution of (3.27) into (3.32)-(3.34) and neglecting terms of order ǫ1, it is possible to obtain

∂2φ0

∂z2− k2φ0 = 0 for − h(x, y) < z < 0 (3.35)


∂φ0

∂z− ω2

gφ0 = 0 for z = 0 (3.36)

∂φ0

∂z= 0 for z = −h (3.37)

where both

k =

(

∂θ

∂x,∂θ

∂y

)

= ∇θ (3.38)

and

ω = −∂θ∂t

(3.39)

depend on x, y and t. Using (3.35), (3.37), it is possible to obtain

φ0 = −iga2ω

cosh [k (z + h)]

cosh [kh](3.40)

where ω and k are related by the local dispersion relation

ω2 = gk tanh [kh] (3.41)

and a is an arbitrary function of x, y and t.The function θ can be easily determined using (3.38)-(3.39). Indeed, if we consider, for

example, a wave characterized by a fixed period, it is possible to obtain k as function of (x, y)using (3.41).


Then, equation (3.38) provides

k2 =

(

∂θ

∂x

)2

+

(

∂θ

∂y

)2

(3.42)

which is a nonlinear equation which is known as ’eikonal equation’ and can be solved withappropriate numerical approaches. In the following, other approaches will be described whichcan be used to determine θ(x, y, t).

Alternatively, a simple formulation that can describe the refraction of waves by a slowly-varying water depth is given by the Fermat priciple applied to our physical system. Fermat’sprinciple (also name the principle of least time) states that the path taken by a ray betweentwo points is the path that can be traversed in the least time. Since k = ∇θ, it follows that∇ × k = ∇ × ∇θ = 0, hence ∇ × k = 0. Therefore the wavenumber field is irrotational(conservation of wave crests)

At order ǫ1, it follows

∂2φ1

∂z2− k2φ1 = kx

∂φ0

∂x+ ky

∂φ0

∂y+∂(φ0kx)

∂x+∂(φ0ky)

∂yfor − h(x, y) < z < 0 (3.43)


∂φ1

∂z− ω2

gφ1 = −1

g

[

ω∂φ0

∂t+∂(ωφ0)

∂t

]

for z = 0 (3.44)

∂φ1

∂z= φ0

[

kx∂h

∂x+ ky

∂h

∂y

]

per z = −h (3.45)

The problem posed by (3.43)-(3.45) is a non-homegeneous problem, the homogensous partof which admits a non-trivial solution because of (3.41). Therefore, it is necessary to force asolvability condition which turns out to be (see appendix 3 at the end of the chapter)

∫ 0

−h

[

φ∗0

(

kx∂φ0

∂x+ ky

∂φ0

∂y+∂(kxφ0)

∂x+∂(kyφ0)

∂y

)]

dz (3.46)

= −1

g

[

φ∗0

(

ω∂φ0

∂t+∂(ωφ0)

∂t

)]

z=0

−[

φ∗0φ0

(

kx∂h

∂x+ ky

∂h

∂y

)]

z=−h

In (3.46), a complex quantity with an asterix denotes the complex conjugate of the quantity.Some algebra (see appendix 4 at the end of the chapter) shows that such solvability conditionleads to

∂

∂t

(

Et

ω

)

+∂

∂x

(

cgxEt

ω

)

+∂

∂y

(

cgyEt

ω

)

= 0 (3.47)

where the quantity Et

ωis named ’wave action’. Equation (3.47) allows the amplitude a of the

wave to be determined once the function θ is known. Using the definition of ω and k, it is easyto show that

∂k

∂t+ ∇ω = 0 (3.48)


(we remind that θ = ωt− k · x) Hence, if the characteristics of the incoming wavefield do notchange with time, it appears that the angular frequency ω does not depend on x, y and doesnot change during the propagation of the wave. Moreover, equation (3.47) becomes

∂(cgxEt)

∂x+∂(cgyEt)

∂y= ∇ · (Etcg) = 0 (3.49)

ω being a constant of the wave field and cg a vector which is orthogonal to the wave fronts witha modulus equal to the group velocity. Let us consider a volume delimited by two wave rays

y

x

k

k

d σ1

d σ2

Figura 3.5: Wave rays

(see figure 3.5), which are tangent to the vector k and separated by an infinitesimal distance.It is easy to verify that

∫

V

∇ · (Etcg)dV = 0 (3.50)

and∫

S

Etcg · ndS = 0 (3.51)

However, on the lateral surfaces (tangent to k), cg ·n vanishes and equation (3.51) implies thatthe energy flux through the infinitesimal surfaces dσ1 e dσ2 keeps constant, where dσ1 e dσ2

are the infinitesimal surfaces of the volume orthogonal to the wave rays. Since Et = 12γa2 and

the flux of energy is constant, it follows that

a2cgdσ = a2∞cg∞dσ∞ (3.52)


where dσ denotes the generic infinitesimal distance between two wave rays and dσ∞ its initialvalue. Then, it turns out that

a

a∞=

[

cg∞cg

dσ∞dσ

]1/2

(3.53)

dσ/dσ∞ being defined as the ’separation factor’ of the two wave rays. If the angular frequencyof the propagating waves does not change in time, equation (3.47) can be divided by theangular frequency ω and the wave action equation can be thought to be an energy balanceequation. This energy balance can be extended to consider waves superimposed to a uniformcurrent characterized by depth averaged velocity components U and V along the x- and y-axis,respectively,

∂Et

∂t+

∂

∂x

[

(cgx + U)Et

]

+∂

∂y

[

(cgy + V )Et

]

= 0 (3.54)

simply by considering a moving reference frame.Then, the dissipation of energy can be heuristically added in (3.54) by adding the term

−ǫd on the right hand side of (3.54) to take into account the energy dissipation due to wavebreaking and bottom friction. Moreover, an energy input can be also considered to take intoaccount the possible presence of a blowing wind (see the papers by Falques and coworkers).

3.4 The wave rays and Snell law

y

x

α

α

dα

ds

b

b+db

Figura 3.6: Variables characterizing a wave ray. The reader should take into account thatdb = ∂α

∂nbds, dα = ∂α

∂nb and a few lengths and angles are distorted to make their definition

clearer.

The wave rays can be determined by means of the procedure described in the following.


Figure 3.6 shows that

y′ =kykx

=∂θ/∂y

∂θ/∂x(3.55)

Let us remind to the reader that the wave phase θ is defined by θ = k · x − ωt. If a fixedvalue of t is considered θ = constant implies k · x = constant, i.e. θ = constant is a line inthe (x, y)-plane which can be thought as a crest or a line parallel to it. Hence ∇θ is a vectorparallel to k.

Since√

k2x + k2

y = k, it is easy to obtain

∂θ

∂x

√

1 + y′2 = k;∂θ

∂x=

k√

1 + y′2; kx =

k√

1 + y′2;

∂θ

∂xy′ =

∂θ

∂y=

ky′√

1 + y′2(3.56)

In particular the last relationship follows from

k2y + k2

x = k2, ky =√

k2 − k2x = k

√

1 − k2

(1 + y′2)k2= k

√

y′2

1 + y′2

d

dx

[

∂θ

∂y

]

=d

dx

[

ky′√

1 + y′2

]

=∂2θ

∂x∂y+∂2θ

∂y2y′ =

∂2θ

∂x∂y+∂2θ

∂y2

∂θ∂y

∂θ∂x

= (3.57)

(

∂θ∂x

∂2θ∂x∂y

+ ∂2θ∂y2

∂θ∂y

)

∂θ∂x

=

12∂∂y

(∇θ)2

∂θ∂x

=

12

∂(k2)∂y

∂θ∂x

=k ∂k∂y

∂θ∂x

=∂k

∂y

√

1 + y′2

where the last equation follows from the first relationship appearing in (3.56). Therefore, ageneric wave ray can be determined by numerically integrating the following ordinary differentialequation

d

dx

[

ky′√

1 + y′2

]

=∂k

∂y

√

1 + y′2 (3.58)

If the depth contours are parallel to the y-axis, i.e. h = h(x) and k = k(x), it follows

ky′√

1 + y′2= C1 (3.59)

Then, since

sinα =dy

√

dx2 + dy2=

y′√

1 + y′2(3.60)

equation (3.59) becomesk sinα = C1


0

10000

20000

30000

0 10000 20000 30000 40000

x [m

]

y [m]

Figura 3.7: Wave rays on a constant sloping beach (the slope is equal to 1% ) for three differentvalues of the wave period: T = 5 s (dotted line), T = 10 s (broken line) and T = 20 s(continuous line). The wave ray which first deviates from the straight line (initally the waverays form an angle of 45 with the coastline) is that of the wave characterized by the longestperiod. The wave characterized by the shortest period deviates from the straight trajectoryonly close to the coast which is located at x = 0.

By introducing the wave celerity c = ω/k and taking into account that ω does not changeduring wave propagation (if the characteristics of the wave climate are time invariant), it iseasy to obtain

sinα

c=

sinα0

c0

which is known as Snell law. Moving towards the coast (described by the line x =costant infigure 3.7), the water depth decreases and, hence, also the celerity decreases. It follows thatalso the value of α decreases and the wave rays tend to become orthogonal to the coastline.Moreover, equation (3.59) leads to

k2y′2 = C21

(

1 + y′2)

; y′2(

k2 − C21

)

= C21 (3.61)

y′ =±C1

√

k2 − C21

(3.62)

The function y(x) can be obtained by numerically integrating equation (3.62) where, to simplify


0

50

100

150

200

0 50 100 150 200

y [m

]

x [m]h=0.5 [m], h=1.0 [m],...

Figura 3.8: Wave rays (thick solid lines) over a sea bottom characterized by the presence ofcanyons orthogonal to the coast. The wave rays are computed by means of the numerical inte-gration of (3.58) using (1.70) to evaluate the wavenumber. The wave period is equal to 4 s andthe direction of wave propagation in the deep water region forms an angle of 153 with the x-axis.

Finally, the water depth h is provided by h(x, y) = 0.025x[

1 + 20 exp(

−3(

x20

)1/3)

sin10(

πy80

)

]

and the broken lines are constant values of the water depth h (This example is also shown in???).

the procedure, the value of k can be obtained by means of (1.69)

y′ =dy

dx= ±

√

√

√

√

√

√

√

√

√

√

coth

[

(

ω√h(x)

√g

) 32

]23

sinα0

2

− 1 (3.63)

and taking into account that k0 = ω2/g.To give an example, figure 3.7 shows wave rays for a beach topography which does not

depend on the longshore coordinate and is characterized by a constant slope equal to 1%. Theinitial value α0 of α is equal to 45 and different wave periods are considered. The wavescharacterized by the longer periods feel the bottom influence far from the coastline while thewave characterized by the shortest period is affected by the beach profile only close to thecoastline. A further example is shown in figure 3.8 for a bottom topography characterized bythe presence of canyons orthogonal to the coast (This case is also considered in ???.


For arbitrary bottom topography, to evalute the distance b between two wave rays, it isnecessary to integrate the following equation

∂2β

∂s2+ p(s)

∂β

∂s+ q(s)β = 0 (3.64)

where s is the curvilinear coordinate along the wave ray and the functions p(s) and q(s) aredefined by

p(s) = −cosα

c

∂c

∂x− sinα

c

∂c

∂y(3.65)

q(s) =sin2 α

c

∂2c

∂x2− 2 sinα cosα

c

∂2c

∂x∂y+

cos2 α

c

∂2c

∂y2(3.66)

In (3.111) β = bb0

is the ratio between b and b0, which is its initial value (the main steps of theprocedure which leads to (3.64) are summarized in appendix 5).

The tendency of the wave fronts to become parallel to the shore is clearly shown in figure3.9

Figura 3.9: Refraction effects around a small Caribbean island (the actual wave fronts feel alsothe diffraction effects which are considered in the following chapter.


3.5 Appendix 1

Assuming that the fluid has a constant density (ρ =costant) and behaves like an inviscid fluid(µ = 0), momentum equation becomes Euler equation

ρdvidt

= ρfi −∂p

∂xi(3.67)

Moreover, if the body force is due only to gravity (fi = gi), (3.67) becomes

ρdvidt

= ρgi −∂p

∂xi(3.68)

From (3.68), it is possible to derive an equation describing the spatial and temporal developmentof the kinetic energy of the fluid

ρvidvidt

= ρd

dt

(vjvj2

)

= ρgkvk − vk∂p

∂xk= ρgnvn −

∂ (vnp)

∂xn(3.69)

where, it has been taken into account that ∂vn

∂xn= 0 because of continuity equation. By integra-

ting (3.69) over the generic volume V and considering the transport theorem, it is possible toobtain

∫

V

ρd

dt

(vivi2

)

dV =d

dt

∫

V

ρ(vjvj

2

)

dV = (3.70)

∫

V

ρgnvndV −∫

V

∂ (vnp)

∂xndV =

∫

V

ρgkvkdV −∫

S

pvknkdS

S being the surface of V and nk the component along the xk-axis of the unit vector normal toS. Equation (3.70) shows that the temporal variation of the kinetic energy of a fluid volume Vis due to the power of the body forces and forces acting on the surface S. The term describingthe power of the body forces can be easily rewritten as minus the temporal derivative of thepotential energy.

∫

V

ρgkdxkdt

dV =

∫

V

ρd (gixi)

dtdV =

d

dt

∫

V

ρgjxjdV (3.71)

Henced

dt

∫

V

[

ρ(vivi

2

)

− ρgixi

]

dV = −∫

S

pvknkdS (3.72)

In other words, the sum of the temporal derivative of the kinetic energy plus the potentialenergy is balanced by the power of the forces (due to the pressure) acting on S.

Indeed, the application of (3.72) to a volume V0, which does not move, leads to

∫

V0

∂

∂t

[

ρ(vivi

2

)

− ρgixi

]

dV0 +

∫

S0

vlnl

[

ρ(vivi

2

)

− ρgixi

]

dS0 = −∫

S0

pvknkdS0 (3.73)


Then, taking into account that the leading order terms of the velocity field are time periodicfunctions characterized by vanishing time averaged values, relationship (3.7) follows from (3.73)applied to a volume which goes from z = −h to z = η and has a unit area in the (x, y)-plane.

It is worth pointing out that, if the flow does not depend on t, it is possible to write thepower of the forces (due to the pressure) acting on S as the temporal derivative of the pressureenergy d

dt

∫

VpdV . Indeed

d

dt

∫

V

pdV =

∫

V

∂p

∂tdV +

∫

S

pvknkdS (3.74)

Therefore, if the flow is steady, the power of the forces acting on the surface S of V is equalto minus the temporal derivative of the pressure energy. However, such a relationship does nothold if the pressure depends on time. Hence, the statement that the total head H = z+ p

γ+ v2

2g

is equal to the mechanical energy of the fluid per unit weight is correct only if the flow is steady.It follows that, if the flow field generated by the propagation of a sea wave is considered, thetemporal derivative of the kinetic energy plus the potential energy is equal tothe power of the forces acting on the surface of the fluid volume and H does notrepresent the mechanical energy of the fluid per unit weight.

3.6 Appendix 2

Let us compute the work done by the pressure forces per unit time

F = −ρ∫ 0

−h

∂φ

∂t

∂φ

∂xdz = (3.75)

−ρ∫ 0

−h

[

−ag2

cosh[k(z + h)]

cosh(kh)ei(ωt−kx) + c.c.

] [

agk

2ω

cosh[k(z + h)]

cosh(kh)ei(ωt−kx) + c.c.

]

dz =

−ρ∫ 0

−h−a

2g2k

2ω

cosh2[k(z + h)]

cosh2(kh)=

ρa2g2k

2ω cosh2(kh)

∫ 0

−hcosh2[k(z + h)]dz =

ρa2g2k

2ω cosh2(kh)

∫ 0

−h

e2k(z+h) + e−2k(z+h) + 2

4dz =

ρa2g2k

2ω cosh2(kh)

∫ 0

−h

(

1

2+

cosh[2k(z + h)]

2

)

dz =

ρa2g2k

2ω cosh2(kh)

[

1

2h+

1

2k

sinh[2k(z + h)]

2|0−h]

=ρa2g2

4ω cosh2(kh)

[

kh +sinh(2kh)

2

]

=

1

2ρga2 gk2hω

2ω2k cosh2(kh)

[

1 +sinh(2kh)

(kh)2

]

=1

2ρga21

2

ω

k

kh

sinh(kh) cosh(kh)

[

1 +2 sinh(kh) cosh(kh)

2kh

]

=

1

2ρga2

[

1

2

ω

k

(

2kh

sinh(2kh)+ 1

)]


3.7 Appendix 3

Introducing a coordinate system with the x-coordinate aligned with the direction of wavepropagation and taking into account that

∂

∂x

(

φ0∂φ1

∂x

)

= φ0∂2φ1

∂x2+∂φ0

∂x

∂φ1

∂x,

∂

∂z

(

φ0∂φ1

∂z

)

= φ0∂2φ1

∂z2+∂φ0

∂z

∂φ1

∂z(3.76)

∂

∂x

(

φ1∂φ0

∂x

)

= φ1∂2φ0

∂x2+∂φ1

∂x

∂φ0

∂x,

∂

∂z

(

φ1∂φ0

∂z

)

= φ1∂2φ0

∂z2+∂φ1

∂z

∂φ0

∂z(3.77)

it follows that

φ0

(

∂2φ1

∂x2+∂2φ1

∂z2

)

− φ1

(

∂2φ0

∂x2+∂2φ0

∂z2

)

= (3.78)

=∂

∂x

(

φ0∂φ1

∂x

)

+∂

∂z

(

φ0∂φ1

∂z

)

− ∂

∂x

(

φ1∂φ0

∂x

)

− ∂

∂z

(

φ1∂φ0

∂z

)

=

= φ0∇2φ1 − φ1∇2φ0 =∂

∂x

(

φ0∂φ1

∂x− φ1

∂φ0

∂x

)

+∂

∂z

(

φ0∂φ1

∂z− φ1

∂φ0

∂z

)

Hence, using Green’s theorem∫

S

(

∂M∂x

− ∂L∂z

)

dxdz =∫

cLdx+Mdz

∫

S

(

φ0∇2φ1 − φ1∇2φ0

)

dxdz =

∫

S

[

∂

∂x

(

φ0∂φ1

∂x− φ1

∂φ0

∂x

)

− ∂

∂z

(

φ1∂φ0

∂z− φ0

∂φ1

∂z

)]

dxdz =

(3.79)

=

∫

C

(

φ0∂φ1

∂x− φ1

∂φ0

∂x

)

dz +

(

φ1∂φ0

∂z− φ0

∂φ1

∂z

)

dx

Since ∇2φ0 = ∂2φ0

∂z2− k2φ0 and ∇2φ1 = ∂2φ1

∂z2− k2φ1, we obtain

∫ 0

−h

[

φ0

(

∂2φ1

∂z2− k2φ1

)

− φ1

(

∂2φ0

∂z2− k2φ1

)]

dz = (3.80)

∫

C

(φ0ikφ1 − φ1ikφ0) dz +

[

φ1∂φ0

∂z− φ0

∂φ1

∂z

]

−h+

[

−φ1∂φ0

∂z+ φ0

∂φ1

∂z

]

0

Using this relationship, the following equation

∂2φ1

∂z2− k2φ1 = k · ∇φ0 + ∇ · (kφ0) (3.81)

and the following boundary conditions

∂φ1

∂z− ω2φ1

g= −1

g

[

ω∂φ0

∂t+∂

∂t(ωφ0)

]

for z = 0 (3.82)


∂φ1

∂z= φ0k · ∇h for z = −h (3.83)

it is possible to obtain∫ 0

−hφ∗

0

[

∂2φ1

∂z2− k2φ1

]

dz =

∣

∣

∣

∣

φ1∂φ∗

0

∂z− φ∗

0

∂φ1

∂z

∣

∣

∣

∣

z=−h+

∣

∣

∣

∣

φ∗0

∂φ1

∂z− φ1

∂φ∗0

∂z

∣

∣

∣

∣

z=0

(3.84)

and∫ 0

−hφ∗

0 [k · ∇φ0 + ∇ · (kφ0)] dz = − φ∗0φ0k · ∇h|z=−h −

φ∗0

g

(

ω∂φ0

∂t+∂

∂t(ωφ0)

)

|0 (3.85)

It is also possible to verify that, because of Leibniz’s rule ( ∂∂α

∫ b

afdz =

∫ b

a∂f∂αdz+ ∂b

∂αf |b− ∂a

∂αf |a),

the previous equation leads to

∇ ·∫ 0

−hk |φ0|2 dz +

1

g

∂

∂t

(

ω |φ0|2)

z=0= 0 (3.86)

and then to (3.46). Indeed

∂

∂x

∫ 0

−hkx |φ0|2 dz +

∂

∂y

∫ 0

−hky |φ0|2 dz +

1

g

∂

∂t

(

ω |φ0|2)

z=0= 0 (3.87)

∫ 0

−h

∂

∂x

[

kx |φ0|2]

dz +

∫ 0

−h

∂

∂y

[

ky |φ0|2]

dz (3.88)

+∂h

∂x

[

kx |φ0|2]

z=−h dz +∂h

∂y

[

ky |φ0|2]

z=−h dz +1

g

∂

∂t

(

ω |φ0|2)

z=0= 0

∫ 0

−h

∂kx∂x

|φ0|2 dz +

∫ 0

−h2kx |φ0|

∂ |φ0|∂x

dz + ... (3.89)

∫ 0

−h

∂kx∂x

|φ0|2 dz +

∫ 0

−hkx |φ0|

|φ0|φ0

∂φ0

∂xdz +

∫ 0

−hkx |φ0|

|φ0|φ0

∂φ0

∂xdz + .... (3.90)

∫ 0

−h

∂kx∂x

|φ0|2 dz +

∫ 0

−hkxφ

∗0

∂φ0

∂xdz +

∫ 0

−hkxφ

∗0

∂φ0

∂xdz + .... (3.91)

∫ 0

−hφ∗

0

∂

∂x(kxφ0) dz +

∫ 0

−hkxφ

∗0

∂φ0

∂xdz + .... (3.92)

∫ 0

−hφ∗

0 [∇ · (kφ0) + k · ∇φ0] dz = − |∇hφ0φ∗0|z=−h−

1

g

[

∂ω

∂tφ0φ

∗0 + ω2 |φ0|

|φ0|φ0

∂φ0

∂t

]

z=−h(3.93)

∫ 0

−hφ∗

0 [∇ · (kφ0) + k · ∇φ0] dz = − |∇hφ0φ∗0|z=−h −

1

g

[

∂ω

∂tφ0φ

∗0 + ωφ∗

0

∂φ0

∂t+ ωφ∗

0

∂φ0

∂t

]

z=−h∫ 0

−hφ∗

0 [∇ · (kφ0) + k · ∇φ0] dz = − |∇hφ0φ∗0|z=−h −

1

gφ∗

0

[

ω∂φ0

∂t+∂ (ωφ0)

∂t

]

z=−h


3.8 Appendix 4

It is easy to verify that

|φ0|2 = φ0φ∗0 =

|a|2 g2

ω2

cosh2 [k (h+ z)]

cosh2(kh)

It followsω

g|φ0|2z=0 =

|a|2 gω

=2

ρ

1

2ρg |a|2 1

ω=

2

ρ

Etω

and∫ 0

−hk |φ0|2 dz =

k2 |a|2 g2ρ

ρω22 cosh2(kh)

∫ 0

−hcosh2 [k (h + z)] dz =

Etω

2

ρ

kg

ω cosh2(kh)

∫ 0

−hcosh2 [k (h + z)] dz =

Etω

2

ρ

kg

ω cosh2(kh)

∫ 0

−hcosh2 [k (h+ z)] dz =

Etω

2

ρ

ω

k sinh(kh) cosh(kh)

∫ kh

0

e2ζ + e−2ζ + 2

4dζ =

2

ρ

Etω

2ω

k sinh(2kh)

[

kh

2+

1

2

∫ kh

0

cosh(2ζ)dζ

]

=

2

ρ

Etω

ω

k sinh(2kh)

[

kh+1

2sinh(2kh)

]

=2

ρ

Etω

ω

2k

[

1 +2kh

sinh(2kh)

]

=2

ρ

Etω

c

2

[

1 +2kh

sinh(2kh)

]

=2

ρ

Etωcg

3.9 Appendix 5

Let us start to consider the following relationships

ds

dt= c ;

dx

dt= c cosα ;

dy

dt= c sinα (3.94)

Reminding that b0 is the initial value of b and β = bb0

is the ratio between b and b0, it is possibleto verify that

1

b

∂b

∂s=

1

β

∂β

∂s=∂α

∂n(3.95)

Indeed ∂β∂s

= 1b0∂b∂s

= 1bβ ∂b∂s

and 1β∂β∂s

= 1b∂b∂s

. Moreover, db = ∂b∂sds = ∂α

∂nbds. It follows that

1b∂b∂s

= ∂α∂n

. Taking into account c = L/T = ω/k and k = ω/c, it follows

1

k

∂k

∂n=c

ωω

(

− 1

c2

)

∂c

∂n= −1

c

∂c

∂n(3.96)

(kx = k cosα and ky = k sinα are the derivatives of the wave phase with respect to x and y,respectively).

It follows (because of ∂2θ∂x∂y

= ∂2θ∂y∂x

)

∂(k sinα)

∂x− ∂(k cosα)

∂y= 0 (3.97)


Indeed, the reader should notice that

kx =∂

∂x(k · x − ωt) ; ky =

∂

∂y(k · x − ωt) (3.98)

and∂2

∂y∂x(k · x − ωt) =

∂2

∂x∂y(k · x − ωt) (3.99)

Therefore

sinα∂k

∂x+ k cosα

∂α

∂x− cosα

∂k

∂y+ k sinα

∂α

∂y= 0 (3.100)

and

cosα∂α

∂x+ sinα

∂α

∂y=

1

k

(

− sinα∂k

∂x+ cosα

∂k

∂y

)

(3.101)

However, from figure 3.10, it appears that

dsy

dsx

dsdn

dnx

dny

α

y

x

Figura 3.10: Variables characterizing a wave ray

dx

ds= cosα ,

dy

ds= sinα ,

dx

dn= − sinα ,

dy

dn= cosα

hence∂α

∂s=

1

k

∂k

∂n

and∂α

∂s=

1

k

∂k

∂n= −1

c

∂c

∂n(3.102)


(consider (3.96)). Moreover,

∂α

∂s=∂α

∂x

∂x

∂s+∂α

∂y

∂y

∂s= cosα

∂α

∂x+ sinα

∂α

∂y(3.103)

and(

∂α

∂s

)2

= cos2 α

(

∂α

∂x

)2

+ sin2 α

(

∂α

∂y

)2

+ 2 cosα∂α

∂xsinα

∂α

∂y(3.104)

The evaluation of the mixed derivatives (taking into account that i) ∂∂s

= ∂∂x

∂x∂s

+ ∂∂y

∂y∂s

, ii)∂x∂s

= cosα, ∂y∂s

= sinα, iii) ∂∂n

= ∂∂x

∂x∂n

+ ∂∂y

∂y∂n

, iv) ∂x∂n

= sinα, ∂y∂n

= cosα, v) − sinα ∂∂x

(

cosα∂α∂x

)

=

sin2 α(

∂α∂x

)2 − sinα cosα∂2α∂x2 ) leads to

∂

∂n

∂α

∂s− ∂

∂s

∂α

∂n=

(

− sinα∂

∂x+ cosα

∂

∂y

)(

cosα∂α

∂x+ sinα

∂α

∂y

)

(3.105)

−(

cosα∂

∂x+ sinα

∂

∂y

)(

− sinα∂α

∂x+ cosα

∂α

∂y

)

=

= sin2 α

(

∂α

∂x

)2

+ cos2 α

(

∂α

∂y

)2

+ cos2 α

(

∂α

∂x

)2

+ sin2 α

(

∂α

∂y

)2

=

=

(

∂α

∂x

)2

+

(

∂α

∂y

)2

=

(

∂α

∂s

)2

+

(

∂α

∂n

)2

=1

c2

(

∂c

∂n

)2

+1

β2

(

∂β

∂s

)2

= 0

By considering (3.95) e (3.96) and derivating them, the following relationships are obtained

∂

∂n

∂α

∂s− ∂

∂s

∂α

∂n=

∂

∂n

(

−1

c

∂c

∂n

)

− ∂

∂s

(

1

β

∂β

∂s

)

=1

c2

(

∂c

∂n

)2

− 1

c

∂2c

∂n2+

1

β2

(

∂β

∂s

)2

− 1

β

∂2β

∂s2= 0

(3.106)Indeed,

−1

c

∂c

∂n=∂α

∂s

(see (3.102)) and1

β

∂β

∂s=∂α

∂n

(see (3.95))

By taking into account (3.105) which forces the vanishing of the term 1c2

(

∂c∂n

)2+ 1

β2

(

∂β∂s

)2,

the previous equation leads to∂2β

∂s2+β

c

∂2c

∂n2= 0 (3.107)

The second term can be rewritten in the form

β

c

∂2c

∂n2=β

c

(

− sinα∂

∂x+ cosα

∂

∂y

)(

− sinα∂c

∂x+ cosα

∂c

∂y

)

= (3.108)


=β

c

(

sinα cosα∂α

∂x

∂c

∂x+ sin2 α

∂2c

∂x2+ sin2 α

∂α

∂x

∂c

∂y− cos2 α

∂α

∂y

∂c

∂x

−2 sinα cosα∂2c

∂x∂y− sinα cosα

∂α

∂y

∂c

∂y+ cos2 α

∂2c

∂y2

)

=

=β

c

[

sin2 α∂2c


∂2c

∂x∂y+ cos2 α

∂2c

∂y2

+ sinα cosα∂α

∂x

∂c

∂x− sinα cosα

∂α

∂y

∂c

∂y+ sin2 α

∂α

∂x

∂c

∂y− cos2 α

∂α

∂y

∂c

∂x

]

=

=β

c

[

sin2 α∂2c


∂2c

∂x∂y+ cos2 α

∂2c

∂y2− ∂α

∂n

(

cosα∂c

∂x+ sinα

∂c

∂y

)]

From (3.108), it follows

∂2β

∂s2+β

c

∂2c

∂n2=∂2β

∂s2+ p

∂β

∂s+ qβ =

β

c2

(

∂c

∂n

)2

+1

β

(

∂β

∂s

)2

(3.109)

Then, taking into account (3.95)

β

c

∂2c

∂n2=

(

−1

ccosα

∂c

∂x− 1

csinα

∂c

∂y

)

∂β

∂s+ (3.110)

+

(

1

csin2 α

∂2c

∂x2− 2

csinα cosα

∂2c

∂x∂y+

1

ccos2 α

∂2c

∂y2

)

β

To conclude, we obtain∂2β

∂s2+ p

∂β

∂s+ qβ = 0 (3.111)

where

p(s) = −cosα

c

∂c

∂x− sinα

c

∂c

∂y(3.112)

q(s) =sin2 α

c

∂2c


c

∂2c

∂x∂y+

cos2 α

c

∂2c

∂y2(3.113)

To obtain some of the equations previously written, it is convenient to consider the followingrelationships, which can be easily derived also taking into account figure 3.10

∂k

∂n= ∇k · dn|dn| =

∂k

∂x

dnx|dn| +

∂k

∂y

dny|dn| = − sinα

∂k

∂x+ cosα

∂k

∂y(3.114)

The reader should notice that

∂α

∂s= ∇α · ds|ds| =

∂α

∂x

dsx|ds| +

∂α

∂y

dsy|ds| = cosα

∂α

∂x+ sinα

∂α

∂y(3.115)


Moreover from (3.100), it follows

k

(

cosα∂α

∂x+ sinα

∂α

∂y

)

= − sinα∂k

∂x+ cosα

∂k

∂y(3.116)

Then

cosα∂α

∂x+ sinα

∂α

∂y= − sinα

1

k

∂k

∂x+ cosα

1

k

∂k

∂y(3.117)

and∂α

∂x=

1

k

∂k

∂y;

∂α

∂y= −1

k

∂k

∂x

Capitolo 4

DIFFRACTION PHENOMENA

4.1 The diffraction problem

The method of wave ray tracing has the advantage to reduce a two-dimensional problem toa one-dimensional problem. However, when two wave rays tend to approach each other, theeffects in the transverse direction can be neglected no longer and, to find an accurate solution,a two-dimensional approach should be used. In these cases, it is necessary to solve the Eikonalequation. A two-dimensional approach should be used also when a wave train interacts witha coastal structure (e.g. a coastal breakwater). This typical problem is generically known asdiffraction problem.

The study of diffration phenomena around coastal structures built over a sea bed, which isslowly varying, is described hereinafter. In this chapter we consider a coastal region characte-rized by a constant value of the water depth such that the velocity potential can be written inthe form

φ = iag

2ωf(z)F (x, y)eiωt + c.c. (4.1)

where: (1) the function f(z), already introduced, is defined by

f(z) =cosh [k(z + h)]

cosh(kh), (4.2)

(2) the angular frequency ω depends on the wavenumber k according to the dispersion rela-tionship

ω2 = gk tanh(kh) (4.3)

and (3) a is the amplitude of the incoming wave. Let us remind that the free surface boundarycondition, at the leading order of approximation, reads η = −1

g∂φ∂t

∣

∣

z=0= a

2f |z=0 F (x, y)eiωt +

c.c.. It follows that the free surface elevation is described by

η =a

2F (x, y)eiωt + c.c.

69

CAPITOLO 4. DIFFRACTION PHENOMENA 70

and the function F (x, y) should tend to 1 in the coastal region which is not affected by thecoastal structure. If (4.1) is plugged into Laplace equation, it gives rise to the two-dimensionalHelmholtz equation

∇2F + k2F = 0 (4.4)

which describes the diffraction of a wave around a two-dimensional structure in the plane(x, y). The reader should notice that the operator ∇2 appearing into (4.4) is defined by ∇2 =∂2/∂x2 + ∂2/∂y2. Of course, appropriate boundary conditions should be considered both onthe surface of the coastal structure and far from it.

The solution of the problem depends on the geometry of the structure. In the followingsimple geometries are considered which allow an analytical solution to be found while complexgeometries require the use of numerical approaches.

4.2 Diffraction around simple coastal structures

4.2.1 A monochromatic sea wave approaching orthogonally a semi-

infinite breakwater

Let us consider an ideal fully reflecting breakwater which is located along the x-axis from x = 0to x = +∞. It follows that

∂F

∂y= 0 for 0 < x <∞, y = 0 (4.5)

the y-axis being along the direction of wave propagation. As already pointed out, because ofthe dynamic boundary condition at free surface, the function η describing the free surface canbe written in the form

η =a

2F (x, y)eiωt + c.c. (4.6)

Moreover, far from the breakwater for x < 0, the wave should be a progressive wave propagatingalong the y-axis

F (x, y) = e−iky for x→ −∞. (4.7)

The solution of the problem was found by Sommerfeld (1896) and can be written in the form

F (x, y) =1 + i

2

[

e−iky∫ β

−∞e−i

π2u2

du+ eiky∫ −β′

−∞e−i

π2u2

du

]

= e−ikyf(β) + eikyf(−β ′) (4.8)

where

β = ±√

4

L(r − y), β ′ = ±

√

4

L(r + y), r =

√

x2 + y2

The sign of β and β ′ depends on the quadrant and it can be verified that sgn(β, β ′) =(−,−), (+,−), (+,−), (+,+) in the first, second, third and fourth quadrants, respectively. Of


course, for x > 0 and y < 0, a stationary wave is generated by the interaction between theincoming wave and the reflected wave. On the other hand, for x < 0 the solution describes apure progressive wave which is not affected by the breakwater.

-6-4

-2 0

2 4

6 -2-1

0 1

2 3

4

x

y

Figura 4.1: Sketch of the free surface, at a fixed time, close to a semi-infinite barrier when amonochromatic wave approaches it in the normal direction.

Figure 4.1 shows the free surface elevation, at a fixed time, as function of x and y obtainedby means of (4.6),(4.8). It is possible to distinguish a protected area (called also shadow region),where the wave height is much smaller than the height of the incoming wave, a region whichis not affected by the breakwater and the region in front of the breakwater, where a standingwave is generated by the interaction of the incoming wave with the reflected wave.

To evaluate the function F (x, y), it is necessary to point out that∫ β

0

e−iπ2u2

du =

∫ β

0

cos(π

2u2)

du− i

∫ β

0

sin(π

2u2)

du = C(β) − iS(β) (4.9)

where C(β) and S(β), known as Fresnel integrals, can be obtained by numerical means. Bierensde Haan (1959) showed that

∫ ∞

−∞e−i

π2u2

du = 1 − i = 2

∫ ∞

0

e−iπ2u2

du = 2

∫ 0

−∞e−i

π2u2

du (4.10)


Figura 4.2: Wave height, close to a semi-infinite barrier, when a monochromatic waveapproaches it in the normal direction.

Hence,∫ β

−∞e−i

π2u2

du =

∫ 0

−∞e−i

π2u2

du+

∫ β

0

e−iπ2u2

du =1 − i

2+

∫ β

0

e−iπ2u2

du (4.11)

and

f(β) =1 + i

2

∫ β

−∞e−i

π2u2

du =1

2[1 + C(β) + S(β) − i (S(β) − C(β))] (4.12)

Moreover,

∫ −β′

−∞e−i

π2u2

du =

∫ 0

−∞e−i

π2u2

du+

∫ −β′

0

e−iπ2u2

du =1 − i

2−∫ β′

0

e−iπ2v2dv (4.13)

a relationship which can be easily obtained introducing the variable v = −u. Then, it follows

f(−β ′) =1 + i

2

∫ −β′

−∞e−i

π2u2

du =1

2[1 − C(β ′) − S(β ′) + i (S(β ′) − C(β ′))] (4.14)

Figura 4.2 shows the wave fronts and the ratio between the local wave height and the heightof the incoming wave as function of x e y obtained analitically and by means of a parabolicapproximation of Helmoltz equation (Penny & Price, 1952).


4.2.2 Monochromatic sea wave approaching obliquely a semi-infinitebreakwater

An example of the behaviour of the wave height generated by the reflection of a monochromaticsea wave propagating obliquely and approaching a semi-infinite breakwater is shown in figure4.3, when the angle of the direction of wave propagation with respect to the barrier is equalto θ0 = 30. The curves are characterized by a constant value of the diffraction coefficientK = H/Hi, H being the local wave height and Hi the height of the incoming wave. As in theprevious case, it is possible to distinguish a protected area (called also shadow region), wherethe wave height is much smaller than the height of the incoming wave, a region which is notaffected by the breakwater and the region in front of the breakwater, where the incoming waveinteracts with the reflected wave.

The results are obtained by writing the function which describes the free surface elevationin the form (4.6) and knowing that

F = e−ikr cos(θ−θ0)f(β) + e−ikr cos(θ+θ0)f(β ′) (4.15)

where

β = 2

√

kr

πsin

[

1

2(θ − θ0)

]

; β ′ = −2

√

kr

πsin

[

1

2(θ + θ0)

]

(4.16)

(r, θ) being polar coordinated centered in (x, y) = (0, 0).Similar figures are obtained for different values of the angle of incidence and are reported

in the Shore Protection Manual.

4.2.3 Monochromatic wave propagating normally to a finite openingin an infinite breakwater

Figura 4.4 shows the diffraction coefficient K behind an opening in an infinite breakwater,B = 2.5L being the width of the opening, when a monochromatic wave approaches ortogonally.The results were obtained by Penny & Price (1944, 1952) when the opening width is largerthan the wavelength L of the approaching wave.

To obtain the solution, it is convenient to introduce a Cartesian coordinate system (x, y, z)with the y-axis coincident with the breakwater and the x-axis orthogonal to it and aligned withthe direction of wave propagation. Moreover, the origin of the coordinate system is locatedat the centre of the opening which is characterized by the width B. The solution is writtenintroducing the functions

f1 = e−ikxf(β1), f2 = e−ikxf(β2), g1 = e−ikxf(β ′1), g2 = e−ikxf(β ′

2) (4.17)

where

β1 =

√

4 (r1 − x)

L, β2 =

√

4 (r2 − x)

L, β ′

1 =

√

4 (r1 + x)

L, β ′

2 =

√

4 (r2 + x)

L


Figura 4.3: Diffraction coefficient K = H/Hi for a monochromatic wave obliquelly approachinga semi-infinite barrier when the angle of incidence is equal to 30.

and r1 =√

x2 + (y −B/2)2 e r2 =√

x2 + (y +B/2)2 are the distances of the generic pointfrom the edges of the opening.

For x < 0 and y < −B/2, the function F , defined by (4.6), is written in the form

F = exp(−ikx) + exp(ikx) − f1 + g1 − f2 − g2 (4.18)

For x > 0 and y < −B/2 the solution is

F = −f1 + g1 + f2 + g2 (4.19)

For x < 0 and y > B/2 the solution turns out to be

F = exp(−ikx) + exp(ikx) − f1 − g1 − f2 + g2 (4.20)

For x > 0 and y > B/2 the solution is

F = f1 + g1 − f2 + g2 (4.21)


At last, for −B/2 < y < B/2 and arbitrary values of x, it can be shown that

F = exp(−ikx) − f1 + g1 − f2 + g2 (4.22)

Figura 4.4: Diffraction coefficient K behind an opening, characterized by a width B = 2.5L,in a breakwater.

4.2.4 Monochromatic sea wave propagating normally to a breakwa-ter of finite width

Figura 4.5 shows the diffraction coefficient K behind a single breakwater characterized by alength 10L, L being the length of a monochromatic wave which approaches the breakwater inthe direction orthogonal to it. Further results can be found in the Shore Protection Manual.

4.2.5 Diffraction around a vertical cylinder

Let us consider a wave which propagates towards a vertical cylinder of radius R, comparablewith the length L of the wave, which extends from the bottom up to the free surface andbeyond. In this case, the Froude-Krylov assumption can not be introduced, i.e. the effects thatthe cylinder has on the wavefield can not be neglected. The velocity potential describing the


Figura 4.5: Diffraction coefficient K behind a finite breakwater of width equal to 10L

incident wave far from the cylinder can be written in the form

φi =ag

ω

cosh [k(z + h)]

cosh [kh]sin [k(x− ct)] (4.23)

where the x-axis is aligned with the direction of wave propagation. This velocity potential canbe conveniently transformed using cylindrical coordinates (r, θ, z) in the form

φi = −i ag2ω

cosh [k(z + h)]

cosh [kh]e−iωt

∞∑

n=0

αnJn(kr) cos(nθ) + c.c. (4.24)

where α0 = 1, αn = 2in for n greater than or equal to 1 and Jn is the Bessel function of thefirst kind. Equation (4.24) can be obtained taking into account that

eikx = eikr cos(θ) = cos (kr cos(θ)) + i sin (kr cos(θ)) = J0(kr) +

∞∑

n=1

2inJn(kr) cos(nθ) (4.25)

The interaction of the incident wave with the cylinder generates a diffracted wave which, addedto the incident wave, should produce a vanishing velocity component normal to the cylinderon its surface. It can be shown that the velocity potential φi due to the incident wave plus thevelocity potential φd due to the diffracted wave is provided by

φi + φd = −i ag2ω

cosh [k(z + h)]

cosh [kh]e−iωt

∞∑

n=0

αn

Jn(kr) −dJn(x)dx

|x=kRdH

(1)n (x)dx

|x=kRH(1)n (kr)

cos(nθ) + c.c.

(4.26)


where H(1)n = Jn + iYn are the Hankel functions. Once the velocity potential is determined, it

is possible to evaluate all the quantities of interest using the relationship previously obtainedby means of the linear approach. To give an example, the free surface η is provided by −1

g∂φ∂t

evaluated at z = 0 and turns out to be

η = ηi + ηd =a

2e−iωt

∞∑

n=0

αn

Jn(kr) −dJn(x)dx

|x=kRdH

(1)n (x)dx

|x=kRH(1)n (kr)

cos(nθ) + c.c.

The reader should notice that the derivatives of the Bessel functions are related to the Besselfunctions by relationships provided, for example, by Abramowitz and Stegun (1965) Handbookof Mathematical Functions with Formulas, Graphs, and Mathematical Tables New York: Dover,ISBN 978-0486612720, MR 0167642. Figure 4.6 shows the amplitude (made dimensionlesswith the amplitude of the incoming wave) of the free surface oscillations around a cylinder ofradius R equal to the L/4 while figure 4.7 shows the run-up profile (the maximum level of thewater during the wave cycle around the surface of the cylinder), made dimensionless with theamplitude of the incoming wave, around a circular cylinder for kR = 1.

4.3 Harbour resonance (close basin)

When a monochromatic wave penetrates into a harbour, it might excite a free mode of oscillationof the free surface inside the protected area and cause a resonance phenomenon. Hence, it isimportant to determine the development of a small perturbation of the free surface inside aharbour, which can be evaluated by 1) using a linear approach, 2) assuming that the waterdepth is constant and equal to h0. In this case, the assumption of small amplitude of thesurface oscillations allows to study the generic time harmonic component and to write thevelocity potential in the form

φ = iag

2ωF (x, y)

cosh [k(z + h0)]

cosh [kh0]eiωt + c.c. (4.27)

Substitution of (4.27) into Laplace equation leads to

∇2F + k2F = 0 (4.28)

where, again, ∇ =(

∂∂x, ∂∂y

)

. Of course the solution of (4.28) depends on the harbour geometry.

For complex geometries, it is necessary to use numerical approaches but the problem can besolved by analytical means for simple geometries.


-10 -5 0 5 10

-10

-5

0

5

10

x/L

y/L

-10-5

0 5

10 -10-5

0 5

10

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

x/L

y/L

Figura 4.6: Wavefield, at t = 0, generated by a monochromatic wave interacting with a cylinderof radius comparable with its wavelength (R = L/4).

4.3.1 Rectangular geometry

If the protected surface has a rectangular geometry of sizes Lx and Ly in the x- and y-directionsrespectively, the function F (x, y) can be written in the form

F = cos

(

nπx

Lx

)

cos

(

mπy

Ly

)


0

0.5

1

1.5

2

0 20 40 60 80 100 120 140 160 180

run-

up

θ

Figura 4.7: Run-up profile, made dimensionless with the amplitude of the incoming wave,around a cylinder of radius R = 1/k.

(with n,m integers equal to or greater than 0) in such a way that the velocity componentnormal to the walls vanishes along the walls. By forcing Helmoltz equation, the value of k isobtained

k2 =

(

nπ

Lx

)2

+

(

mπ

Ly

)2

as function of n and m, i.e. of the different modes. Then, the dispersion relationship providesthe value of the period of oscillation which should be compared with that of the forcing wave.Resonance takes place when the angular frequency of the free modes is equal or close to thatof the forcing term.

4.3.2 Circular geometry

If the protected area has a circular geometry of radius R, the function F can be written in theform

F = Jn(kr) cos(nθ) (4.29)

where (r, θ) are polar coordinates and Jn is the Bessel function of the first kind of order n. Ifn = 0, F does not depend on θ while F is periodic of period 2π/n for n equal to or larger than1. The value of k follows by forcing the vanishing of the velocity component normal to thewalls of the harbour

dJn(kr)

d(kr)= 0 for r = R (4.30)


wheredJ0(x)

dx= −J1(x) and

dJn(x)

dx= Jn−1(x) −

n

xJn(x) for n ≥ 1

Since relationship (4.30) is satisfied for different values of k, different modes are possible.For example, for n = 0,

dJ0(kr)

d(kr)= −J1(kr)

which vanishes for kR = 3.832.., 7.016.., .... Hence, the values of k = 3.382..R

, 7.016..R

, ... give riseto different modes, the first two of which are shown in figure 4.8.

For n = 1, (4.30) leads to

J0(kr) −J1(kr)

kr= 0

which is satisfied for k = 1.841..R

, 5.33..R, .... Figure 4.9 shows the first two modes for n = 1.

Then, as for the rectangular geometry, the dispersion relationship provides the period T of theoscillations which should be compared with the period of the forcing wave to verify the possiblepresence of a resonance phenomenon.


-1-0.5

0 0.5

1-1

-0.5

0

0.5

1

-1

0

1

2

x/L

y/L

-1-0.5

0 0.5

1-1

-0.5

0

0.5

1

-1

0

1

2

x/L

y/L

Figura 4.8: (a) First mode and (b) second mode describing the free oscillations in a circularbasin for n = 0.


-1-0.5

0 0.5

1-1

-0.5

0

0.5

1

-1

0

1

2

x/L

y/L

-1-0.5

0 0.5

1-1

-0.5

0

0.5

1

-1

0

1

2

x/L

y/L

Figura 4.9: (a) First mode and (b) second mode describing the free oscillations in a circularbasin for n = 1.

Capitolo 5

MILD SLOPE EQUATION

It is rather common to face coastal problems characterized by geometries such that the propa-gation of a monochromatic wave feels both refraction and diffraction effects as, for example, theproblem shown in figure 5.1 where the breakwaters induce the diffraction of the approachingwaves and the depth variations cause their refraction. Under such circumstances, the hydrody-

Figura 5.1: Example of a refracted and diffracted wave.

namic problem can be tackled by means of the ’mild slope approximation’, which is describedin the following.

If we consider a coastal region where the water depth changes on a slow spatial scale, it isreasonable to assume that the propagation of a monochromatic wave can be described by the

83

CAPITOLO 5. MILD SLOPE EQUATION 84

following velocity potential

φ = −i gωη(x, y)f(x, y, z)eiωt + c.c. (5.1)

where the function f turns out to be

f(x, y, z) =cosh [k(x, y)(z + h(x, y))]

cosh [k(x, y)h(x, y)](5.2)

and the wavenumber k depends on the angular frequency ω through the dispersion relationship

ω2 = gk(x, y) tanh [k(x, y)h(x, y)] (5.3)

The reader should remind that the angular frequency ω of the wave is a constant because ofthe conservation of the number of waves which crosses any section per unit time. Of course, asexplicitly written, the values of h and k, beside the function η, depend on x and y, i.e. the twohorizontal coordinates, because of the presence of a bottom slope (h = h(x, y), k = k(x, y)).

Laplace equation, which provides the velocity potential, can be written in the form

∂2φ

∂z2+ ∇2φ = 0 (5.4)

where, herein, the operator ∇ is defined by

∇ =

(

∂

∂x,∂

∂y

)

(5.5)

The linearized boundary conditions at the free surface and at the bottom force

∂φ

∂z− ω2

gφ = 0 at z = 0 (5.6)

∂φ

∂z= −∇h · ∇φ at z = −h (5.7)

Since the function f , defined by (5.2), is a solution of the following problem

d2f

dz2− k2f = 0 (5.8)

df

dz− ω2

gf = 0 at z = 0 (5.9)

df

dz= 0 at z = −h (5.10)


Green’s formula, applied to φ and f , provides (see the note at the end of the chapter)

∫ 0

−h

(

k2φf + f∇2φ)

dz = − [f∇h · ∇φ]−h (5.11)

Then, by applying the gradient operator to (5.1) and making use of (5.4), it is possible to obtain

∇φ = −igω

[

f∇η + η∂f

∂h∇h]

(5.12)

and, then,

∇2φ = −igω

[

f∇2η + 2∂f

∂h∇η · ∇h+ η

∂2f

∂h2(∇h)2 + η

∂f

∂h∇2h

]

(5.13)

It follows that (5.11) can be written in the form

∫ 0

−h

[

f 2∇2η + 2f∂f

∂h∇η · ∇h+ ηf

∂2f

∂h2(∇h)2 + ηf

∂f

∂h∇2h+ k2ηf 2

]

dz = (5.14)

[

∇h · ∇ηf 2]

−h −[

η(∇h)2f∂f

∂h

]

−h

By using Leibniz’s rule, the first two terms on the left hand side of (5.14) can be combinedwith the first term on the right hand side leading to

∇ ·∫ 0

−hf 2∇ηdz +

∫ 0

−hk2f 2ηdz = (5.15)

−f[

∂f

∂h

]

−hη (∇h)2 −

∫ 0

−hηf∂2f

∂h2(∇h)2 dz −

∫ 0

−hηf∂f

∂h∇2hdz

Since ∇h/(kh) << 1 and ∇η/(kη) = O(1), the terms on the right hand side of (5.15) turn outto be much smaller than the terms on the left hand side and they can be neglected. Indeed theorder of magnitude of the terms on the left hand side of (5.15) is O(hk2f 2η), while the order

of magnitude of the terms on the right hand side is O(

f2

hη(∇hkh

)2h2k2

)

= O(

hk2f 2η(∇hkh

)2)

.

By integrating the left hand side of (5.15), it is possible to obtain

∇ · (b1∇η) + k2b1η = ∇ · (b1∇η) + ω2b2η = 0 (5.16)

where

b1 = ghtanh(kh)

kh

1

2

(

1 +2kh

sinh(2kh)

)

= ccg (5.17)

b2 =1

2

(

1 +2kh

sinh(2kh)

)

=cgc

(5.18)


When kh is constant, equation (5.16) becomes Helmoholtz equation. On the other hand,when kh << 1, (5.16) becomes

∇ · (h∇η) +ω2

gη = 0 (5.19)

which holds also when ∇h/(kh) = O(1) (Mei, 1989). Hence, (5.16) can be applied for arbitraywavelength provided the slope of the bottom is small. Such an equation is known as mild-slopeequation. The solution of (5.16) cannot be determined easily since it depends on the geometry ofthe problem. However, it is not difficult to find open source codes which provide the numericalsolution of the mild slope equation.

5.1 Note

Since both φ and f are solution of the Laplace equation, it is possible to write

∫ 0

−h

[

f

(

∇2φ+∂2φ

∂z2

)

− φ

(

d2f

dz2− k2f

)]

dz = 0 (5.20)

It follows∫ 0

−h

(

f∇2φ+ k2φf +∂2φ

∂z2f − φ

d2f

dz2

)

dz = 0 (5.21)

However,

∂2φ

∂z2f − φ

d2f

dz2dz =

∂

∂z

(

f∂φ

∂z

)

− ∂

∂z

(

φ∂f

∂z

)

=∂f

∂z

∂φ

∂z+ f

∂2φ

∂z2− ∂φ

∂z

∂f

∂z− φ

∂2f

∂z2(5.22)

Hence∫ 0

−h

[

f∇2φ+ k2φf +∂

∂z

(

f∂φ

∂z

)

− ∂

∂z

(

φ∂f

∂z

)]

dz = 0 (5.23)

∫ 0

−h

(

f∇2φ+ k2φf)

dz +

(

f∂φ

∂z

)0

−h−(

φ∂f

∂z

)0

−h= 0 (5.24)

Then, the boundary conditions, which should be forced to φ and f , lead to (5.11)

∫ 0

−h

(

k2φf + f∇2φ)

dz = − [f∇h · ∇φ]−h

Capitolo 6

VISCOUS EFFECTS: THE BOTTOMBOUNDARY LAYER

6.1 The bottom boundary layer: the linear approach

The inviscid, irrotational approach shows that the propagation of a surface gravity wave ina coastal region, characterized by a finite water depth, induces horizontal fluid oscillationsclose to the bottom which have an amplitude equal to a/ sinh(kh). Even though such result issignificant when the fluid is assumed to be inviscid, the oscillations of the fluid for z = −h arenot acceptable when a viscous fluid is considered. Indeed, for a real (viscous) fluid, both thehorizontal and vertical velocity components should vanish at the bottom. Hence, close to thebottom under actual surface waves, a viscous boundary layer is present where, even for verylarge values of the Reynolds number, viscous effects can not be neglected and the vorticity doesnot vanish.

However, even though to determine the flow within the bottom boundary layer it is necessaryto take into account viscous effects and to consider Navier-Stokes equations, it is possible tosimplify them by neglecting the terms which are much smaller than the others.

Let us consider the Cartesian coordinate system shown in figure 6.1, such that the x-axis iscoincident with the bottom and points in the direction of wave propagation and the z-axis isvertical and points upwards.

The knowledge of the irrotational flow induced by wave propagation provides the orderof magnitude U0 of the streamwise component u of the velocity close to the bottom (U0 =aω/ sinh[kh0]). The order of magnitude of the modified pressure follows from Bernoulli theoremand turns out to be O(P0) = ρωU0L. Moreover, it is reasonable to consider L and 1/ω, as spatialscale along the x-direction and temporal scale, respectively. Let us denote with δ the orderof magnitude of the thickness of the bottom boundary layer within which viscous effects arerelevant. Even though, at this stage, δ is not known, it is possible to assume that δ is muchsmaller than L because of the small values of the fluid viscosity (boundary layer approximation).

87

CAPITOLO 6. VISCOUS EFFECTS: THE BOTTOM BOUNDARY LAYER 88

x

z

u=u(x,z,t)

Figura 6.1: Sketch of the coordinate system used to investigate the flow in the oscillatoryboundary layer generated close to the bottom by the propagation of a surface wave.

Continuity equation, which is written below along with the order of magnitude of its terms,suggests that the order of magnitude of w is δU0/L

∂u

∂x+∂w

∂z= 0 (6.1)

O

(

U0

L

)

O

(

O(w)

δ

)

(6.2)

In fact, neither ∂u∂x

nor ∂w∂z

can be neglected since neglecting these terms would lead to aninconsistent solution. Indeed, if the term ∂u

∂xis neglected with respect to ∂w

∂z, continuity equation

implies ∂w∂z

= 0 and hence w =constant. However, the bottom boundary condition forces theconstant to vanish and the term ∂w

∂zturns out to be zero and cannot be much larger than ∂u

∂x.

Similarly, if ∂w∂z

is neglected, u can not depend on x and this result is inconsistent because ofthe matching with the external flow which depends on x.

Now, let us consider the component along the x-direction of Navier-Stokes equation and theorder of magnitude of its terms

∂u

∂t+ u

∂u

∂x+ w

∂u

∂z= −1

ρ

∂p

∂x+ ν

∂2u

∂x2+ ν

∂2u

∂z2(6.3)

O (ωU0) O

(

U20

L

)

O

(

δU0

L

U0

δ

)

O

(

1

ρ

ρωU0L

L

)

O

(

νU0

L2

)

O

(

νU0

δ2

)

(6.4)


The irrotational flow induced by wave propagation suggests that

O

(

U20

L

)

<< O (ωU0) (6.5)

IndeedU2

0

LωU0=U0

Lω=aω

Lω=a

L

and the ratio a/L is assumed to be much smaller than one. Then, it follows that the termsdescribing the convective acceleration are much smaller than the local acceleration term whichturns out to be of the same order of magnitude as the term related to the pressure gradient.Moreover, assuming that δ << L

O

(

νU0

L2

)

<< O

(

νU0

δ2

)

(6.6)

and the streamwise diffusion term can be neglected with respect to the vertical diffusion term.At last, by requiring that the largest viscous term has the same order of magnitude as both thelocal acceleration and pressure gradient terms, the thickness of the viscous bottom boundarylayer can be obtained

ωU0 ∼νU0

δ2(6.7)

δ ∼√

ν

ω(6.8)

A similar analysis of the order of magnitude of the terms appearing into the z-componentof Navier-Stokes equation shows that the pressure can be assumed to be constant within thebottom boundary layer. Indeed,

∂w

∂t+ u

∂w

∂x+ w

∂w

∂z= −1

ρ

∂p

∂z+ ν

∂2w

∂x2+ ν

∂2w

∂z2(6.9)

O

(

ωδ

LU0

)

O

(

δ

L2U2

0

)

O

(

(

δ

LU0

)21

δ

)

O

(

1

ρ

ρωLU0

δ

)

O

(

νδ

L

U0

L2

)

O

(

νδ

L

U0

δ2

)

(6.10)

To conclude, to determine the flow at the bottom of a propagating surface wave, it is necessaryto solve the following equation

∂u

∂t=∂ui∂t

+ ν∂2u

∂z2(6.11)

where

ui =U0(x)

2eiωt + c.c. =

aω

2 sinh[kh]e−ikxeiωt + c.c. (6.12)


0

2

4

6

8

10

-1.5 -1 -0.5 0 0.5 1 1.5

z/δ

u/U0

ω=π/4ω=π/2ω=3π/4ω=π

ω=5π/4 ω=3π/2

ω=7π/4ω=2π

-3

-2

-1

0

1

2

3

0 1 2 3 4 5 6

Rδ2

τ/ρU

2 0,

Ui/U

0

ωt

Figura 6.2: (a) Velocity profiles within the viscous boundary layer at the bottom of a mono-chromatic surface waves for different wave phases; (b) time development of the bottom shearstress τ properly scaled (continuous line) and the free stream irrotational velocity close to thebottom (broken line).

The relationship −1ρ∂p∂x

= ∂ui

∂tfollows from the limit of (6.11) for z → ∞ where the viscous

terms are negligible and the modified pressure does not depend on the vertical coordinate. If


the velocity is written in the form

u = f(z)eiωt + c.c. (6.13)

where the parametric dependence of the solution by the variable x is not explicitly written, itis easy to obtain

d2f

dz2− iω

νf = −iω

ν

U0

2(6.14)

which leads to

f(z) = c1 exp

[

(1 + i)z

√

2ν/ω

]

+ c2 exp

[

−(1 + i)z

√

2ν/ω

]

+U0

2(6.15)

The forcing of the no-slip condition at the bottom and the matching with the irrotational flowgive

c1 = 0, c2 = −U0

2It follows

u =U0

2

[

1 − e−(1+i)z

δ

]

eiωt + c.c. (6.16)

where

δ =

√

2ν

ωOnce the flow is known, it is possible to determine the bottom shear stress which, at the leadingorder, reads

τ = µ∂u

∂z|z=0 = µ

U0

2

(1 + i)

δeiωt + c.c. (6.17)

Indeed

τ = µ

(

∂u

∂z+∂w

∂x

)

|z=0

but w vanishes at z = 0 and ∂w∂x

at z = 0 identically vanishes. The term ∂w∂x

can be neglected

also for values of z different from zero since ∂w∂x

is of order δU0

L2 while the term ∂u∂z

is of order U0

δ.

The maximum value τmax of τ is provided by

τmax = µ

√2U0

δ(6.18)

and it takes place 45 before the maximum value of the free stream velocity. If the frictionfactor fw is defined as 2τmax/(ρU

20 ), it turns out that

fw =2√

2

Rδ(6.19)

where the Reynolds number Rδ is defined by

Rδ =U0δ

ν(6.20)


6.2 Nonlinear effects in the bottom boundary layer

At the leading order of approximation, a monochromatic surface gravity wave propagating overa flat bottom gives rise to a velocity field that close to the bottom oscillates harmonically. Atthe following order of approximation, a steady velocity component appears. Even though thesteady velocity component is small, it has a large influnce on sediment transport and on themorphological development of the bottom, because of its persistence.

To understand the mechanism which gives rise to this steady velocity component and toquantify it, let us consider Navier-Stokes equations within the bottom boundary layer.

As already pointed out, continuity equation leads to O(w) = O(δU0/L). Now, let us lookat the order of magnitude of the different terms appearing into the vertical component of theNavier-Stokes equation

∂w

∂t+ u

∂w

∂x+ w

∂w

∂z= −1

ρ

∂p

∂z+ ν

∂2w

∂x2+ ν

∂2w

∂z2(6.21)

O(ωδ

LU0) O(

δ

L2U2

0 ) O((δ

LU0)

21

δ) O(

1

ρ

ρωLU0

δ) O(ν

δ

L

U0

L2) O(ν

δ

L

U0

δ2) (6.22)

The ratio between the nonlinear acceleration terms and the local acceleration term has anorder of magnitude equal to O(U0/ωL). Since O(U0) = O(aω), it is easy to obtain O(U0/ωL) =O(a/L). Taking into account that

δ =

√

2ν

ω(6.23)

it appears thatO(∂w

∂t)

O(ν ∂2w∂z2

)= O(1) (6.24)

Hence, the local acceleration term has the same order of magnitude as the largest viscous termand both turn out to be larger than the convective terms since the ratio between them and thelocal acceleration term has an order of magnitude equal to L/a.

Because, the vertical pressure gradient turns out to be O(1ρρωLU0

δ) and the ratio between it

and the local acceleration term turns out to be

O(−1ρ∂p∂z

)

O(∂w∂t

)=O(1

ρρωLU0

δ)

O(ω δLU0)

= O(L2

δ2) (6.25)

it appears that, also taking into account terms of order a/L, the pressure within the bottomboundary layer keeps constant in the vertical direction.

However, the pressure gradient in the horizontal direction, which can be easily obtainedfrom the knowledge of the external irrotational flow, is equal to

−1

ρ

∂p

∂x=∂ui∂t

+ ui∂ui∂x

(6.26)


where

ui =U0(x)

2eiωt + c.c.+O

( a

L

)

=aω

2 sinh(kh0)e−ikxeiωt + c.c. +O

( a

L

)

(6.27)

Indeed the vertical velocity component wi of the irrotational flow vanishes at the bottom. Tobe rigorous, the function ui should include terms of order a/L. However, these terms do noaffect the steady velocity component within the bottom boundary layer and are not shown in(6.27). To conclude, the equation which should be solved in the bottom boundary layer is

∂u

∂t+ u

∂u

∂x+ w

∂u

∂z= −1

ρ

∂p

∂x+ ν

∂2u

∂z2(6.28)

or∂u

∂t+ u

∂u

∂x+ w

∂u

∂z− ν

∂2u

∂z2=∂ui∂t

+ ui∂ui∂x

(6.29)

Then, it is appropriate to expand the solution in the form

u = u0 +a

Lu1 + ... (6.30)

w = w0 +a

Lw1 + ... (6.31)

and to writeui = ui0 +

a

Lui1 + ... (6.32)

At the leading order of approximation, substitution of (6.30)-(6.32) into (6.28) leads to theequation for u0 already solved. Hence.

u0 =U0

2[1 − e−(1+i) z

δ ]eiωt + c.c. (6.33)

At the second order of approximation we have

∂u1

∂t− ν

∂2u1

∂z2= ui

∂ui∂x

− u0∂u0

∂x− w0

∂u0

∂z+ not relevant terms (6.34)

Taking into account that the functions ui, u0 and w0 are time periodic functions characte-rized by the angular frequency ω, the forcing terms have a contribution which is periodic withan angular frequency which is twice that of u0, i.e. equal to 2ω, and a contribution which isindependent of time. Hence

u1 = [u1e2iωt + c.c.] + u1 (6.35)

Looking at the time-independent contribution, it follows

−ν ∂2u1

∂z2= ui

∂ui∂x

− u0∂u0

∂x− w0

∂u0

∂z(6.36)


where an overbar denotes the time average of the underlying term. Taking into account thefunctions which describe ui and u0,

ui =U0(x)

2eiωt + c.c. =

aω

2 sinh(kh0)e−ikxeiωt + c.c. (6.37)

u0 =U0(x)

2f(z)eiωt + c.c. =

U0(x)

2

[

1 − e−(1+i)z

δ

]

eiωt + c.c (6.38)

and continuity equation∂u0

∂x+∂w0

∂z= 0 (6.39)

it is easy to obtain∂w0

∂z= −1

2

dU0

dxf(z)eiωt + c.c. = (6.40)

w0 = −1

2

dU0

dxeiωt

[

z +δ

1 + ie

−(1+i)zδ + c1

]

+ c.c. (6.41)

1

2

dU0

dxeiωt

[

−z +δ

1 + i

(

−e−(1+i)zδ + 1

)

]

+ c.c. =1

2

dU0

dxg(z)eiωt + c.c.

where the constant c1 should be determined by forcing the bottom boundary condition w0 = 0at the bed.

Introducing the variable ξ = z/δ, equation (6.36) becomes

− 1

δ2ν∂2u1

∂ξ2=

1

4U0

(

dU0

dx

)∗− 1

4U0f(ξ)f ∗(ξ)

(

dU0

dx

)∗− 1

4

dU0

dxU∗

0 g(ξ)

(

df

dξ

)∗1

δ+ c.c. (6.42)

=1

4U0

(

dU0

dx

)∗ [

1 − ff ∗ − 1

δg∗df

dξ

]

+ c.c.

=1

4U0

(

dU0

dx

)∗

1 −(

1 − e−(1+i)ξ) (

1 − e−(1−i)ξ)−(

1 + i

δe−(1+i)ξ

)

[

(

1 − e−(1−i)ξ) δ

1 − i− z

]

=1

4U0

(

dU0

dx

)∗

−e−2ξ(1 − i) + e−(1−i)ξ + e−(1+i)ξ(1 − i) + (1 + i)ξe−(1+i)ξ

+ c.c.

where a star denotes the complex conjugate of a complex quantity. From (6.42), it follows

−ω2u1 =

1

4U0

(

dU0

dx

)∗ [

−(1 − i)

4e−2ξ +

i

2e−(1−i)ξ − (1 + i)

2e−(1+i)ξ (6.43)

−ie−(1+i)ξ +(1 − i)

2ξe−(1+i)ξ + c1

]

+ c.c.


0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

z/δ

u1,steady/(a/L)U0

Figura 6.3: Velocity profile of the steady streaming at the bottom of a propagating wave

and

u1 =1

2ωU0

(

dU0

dx

)∗ [(1 + 3i)

2e−(1+i)ξ − i

2e−(1−i)ξ +

(1 − i)

4e−2ξ (6.44)

−(1 − i)

2ξe−(1+i)ξ − 3

4(1 + i)

]

+ c.c.

In particular, when ξ tends to infinity, i.e. moving far from the bottom, the velocity tends toa constant value equal to

u1 =3a2ωk

4 sinh2(kh0)

Such a results shows that, close to the bottom a monochromatic surface wave generates asteady current which points in the direction of wave propagation and it does not vanish movingtoward the irrotational, inviscid region. In order to determine the steady velocity profile farfrom the bottom, it is necessary to solve a complex problem and to consider the dynamics ofvorticity which is generated by the no-slip condition at the bottom and it is convented far fromit (Longuet-Higgins, 1953). However, such a steady current is much smaller than the oscillatoryflow which is of order aω. The vertical velocity profile of u1, within the bottom boundary layer,is shown in figure 6.3, where positive values of the velocity mean a velocity which is in thedirection of wave propagation.


Figura 6.4: Friction factor as function of the Reynolds number RE and different values of therelative bottom roughness.

6.3 The bottom boundary layer in the turbulent regime

The results described in the previous section are based on the assumption that the flow regimeis laminar. However, when the Reynolds number Rδ is larger than a critical value which fallsin the range (500, 600), the laminar flow becomes unstable and turbulence appears. The powerof actual computer allows the direct numerical simulation of Navier-Stokes and continuityequations which can provide different realizations of the turbulent flow field. However, it iscommon to consider ensemble average quantities and to analyse the phenomenon using Reynoldsaveraged equations which call for a turbulence model.

At the leading order of approximation, following the procedure outlined in the previoussection, it is possible to obtain

∂〈u〉∂t

=∂ui∂t

+∂

∂z

(

ν∂〈u〉∂z

− 〈u′w′〉)

where the velocity components are decomposed into an ensemble average part, denoted by 〈〉,and a random fluctuating component, denoted by an apex. For example

u = 〈u〉 + u′

It is not possible to give an exhaustive description of the different turbulence models which areemployed to close the problem. By introducing Boussinesq hypothesis

−〈u′w′〉 = νT∂〈u〉∂z


where νT is the (kinematic) eddy viscosity, the previous equation becomes

∂〈u〉∂t

=∂ui∂t

+∂

∂z

[

(ν + νT )∂〈u〉∂z

]

Reliable and accurate results can be obtained by means of two-equation turbulence models likethe k-ǫ model or the model of Saffman (1970) (Blondeaux, 1987). In the latter case the eddyviscosity is provided by the ratio between the turbulent pseudo-energy e and the turbulentpseudo-vorticity Ω, which are quantities satisfying nonlinear advection-diffusion equations

∂e

∂t= αee

∣

∣

∣

∣

∂〈u〉∂z

∣

∣

∣

∣

− βeeΩ +∂

∂z

[

(ν + σeνT )∂e

∂z

]

∂Ω2

∂t= αΩΩ2

∣

∣

∣

∣

∂〈u〉∂z

∣

∣

∣

∣

− βΩΩ3 +∂

∂z

[

(ν + σΩνT )∂Ω2

∂z

]

where αe, αΩ, βe, ... are constants given by Saffman (1970). Then, it is possible to solve theequations of the problem by numerical means, for example, by using a finite difference approach.Of course the flow in the boundary layer should be matched with the irrotation flow far fromthe bottom while at the bottom the no-slip condition 〈u〉 = 0 should be forced along with thevanishing of the pseudo-energy e = 0. Moreover, at the bottom the value of Ω depends onthe roughness according with a function introduced by Saffman (1970) and Wilcox & Saffman(1974) and later quantified Blondeaux & Colombini (1985).

For practical purpouses, it is often sufficient to evaluate the bottom shear stress and inparticular its maximum value τmax. The value of τmax can be obtained from the knowledge ofthe friction factor fw. Figure 6.4 shows the value of fw as function of the Reynolds number RE =R2δ/2 for different values of the relative roughness k = ksω/U0 (ks=dimensional roughness) as

computed by Blondeaux (1987) using the turbulence model of Saffman (1970).Of course, different empirical formulae exist to evaluate fw for both smooth and rough walls.

As described in the book of Soulsby (1997), Myrhaug (1989) proposed the implicit formula

0.32

fw= 1.64 +

ln

[

6.36

√fw

ks

]

− ln[

1 − exp(

−0.0262REks√

fw

)]

+4.71

REks√fw

2

(6.45)

where ks = ksωU0

and ks is the equivalent Nikuradse roughness which can be assumed to be equalto 3d90 or similar values (e.g. 2.5d50).

At last, it is necessary to point out that, quite often, the flat configuration of the sea bottomturns out to be unstable and ripples (periodic bottom forms characterized by a wavelength ofO(10 cm)) appear. The ripples induce flow separation at their crests and the generation ofcoherent vortex structures which increase the bottom shear stress and the energy dissipationand tend to pick-up the sediments from the bottom and carry them into suspension. Whenripples are present, the behaviour of the waves and the currents can be studied considering a flatbut rough bottom, characterized by a roughness size which is related to the size (amplitude and


length) of the ripples. For example, Grant & Madsen (1982) proposed to describe the effectsof the ripples by introducing a roughness characterized by a size equal to 27.7η2

r/λr where ηrand λr are the height and the length of the ripples, respectively. Soulsby (1997) suggested avalue of the roughness height equal to 30arη

2r/λr where ar is a value falling between 0.3 and 3.

Van Rijn (1991) suggested values of the roughness size falling between ηr and 3ηr. In chapter14, different approaches are presented to predict the size (height and length) of the ripples

Figura 6.5: Visualization of the suspended sediment over a rippled bed for different phases ofthe fluid oscillations.

Another quantity of practical interest is the energy Ediss which is dissipated per unit areaduring a wave cycle within the bottom boundary layer. In fact this quantity can be related tothe attenuation of the wave amplitude. The local dissipation rate per unit volume 2µ(D : D)(D is the strain rate tensor) in the boundary layer at the bottom of a linear Stokes wave isequal to that taking place in the flow field generated by the harmonic oscillation of a plate inits own plane. In fact, the two velocity fields are equal but for a uniform function of time. Sincethe mechanical energy theorem states that the energy dissipation taking place during a plateoscillation should be equal to the work made by the plate, it follows that

Ediss =

∫ T

0

[

2µ

∫ ∞

0

(D : D) dV

]

dt =

∫ T

0

τ(t)ui(t)dt

which can be easily evaluated once τ(t) (the shear stress acting on the bottom) is known.


Indeed, the kinetic energy is a periodic function of time and the body force does not makeany work on the fluid, because the body force is orthogonal to the fluid velocity. Even thoughthe maximum value of τ can be written as 1

2fwρU

20 , the evaluation of Ediss requires also the

knowledge of the time development of τ which is not known exactly. Usually, the mean rate ofenergy dissipation Ediss

Tcan be heuristically computed by means of

EdissT

=2

3πρfeU

30

fe being the energy dissipation factor which turns out to depend on the Reynolds number andthe relative roughness size and it is evaluated by means of empirical relationships. Then, asimple energy balance shows that

EdissT

= −d (Etcg)

dx

Capitolo 7

SHALLOW WATER WAVES

7.1 Introduction and Boussinesq analysis

Let us consider the amplitude of the spatial/temporal oscillations of the first term φ0 and thesecond term 2πa

Lφ1, appearing into Stokes expansion (2.1), which describes the velocity potential

of a monochromatic wave propagating over a constant water depth h and are equal to

φ0 =ag

ω

cosh [k(z + h)]

cosh [kh], (7.1)

and2πa

Lφ1 = ak

3

8

ag

ω

cosh [2k(z + h)]

sinh3 [kh] cosh [kh](7.2)

respectively.When the wave steepness a/L is much smaller than 1, the second term 2πa

Lφ1 turns out to

be much smaller than the first term φ0 both when the water depth h is much larger than thelength L of the surface wave (infinite water depths) and when h and L have the same order ofmagnitude (intermediate water depths). However, for small water depths, it might be that thesecond term becomes larger than the first term and Stokes expansion fails to describe the wavepropagation.

Indeed the hydrodynamic problem, which describes the propagation of a monochromaticwave in the region closest to the coast (where the water depth is small), is characterized by twodifferent parameters

µ = kh =2πh

Land ǫ =

a

h. (7.3)

both of which can be considered to be much smaller than one (the reader should notice thedifferent definitions of the parameter ǫ which, in this chapter, is equal to the ratio betweenthe wave amplitude and the water depth while in the previous chapter was equal to the ratiobetween the amplitude and the length of the wave). The first parameter in (7.3) is linked to the

100

CAPITOLO 7. SHALLOW WATER WAVES 101

ratio between the water depth and the length of the wave, the second parameter is equal to theratio between the amplitude of the wave and the local water depth. Ursell (1953) showed thatthe solution of the hydrodynamic problem depends crucially on the so-called Ursell number(parameter) Ur, which is defined by

Ur =a

h

1

(kh)2=

ǫ

µ2(7.4)

and is proportional to the ratio between the second term and first term of Stokes expansionevaluated at z = 0,

3ak

8

cosh [2kh]

sinh3 [kh] cosh [kh]∼ 3ak

8(kh)3∼ a

h(kh)2= Ur (7.5)

Hence, to have a meaningful Stokes expansion, it is necessary not only that 2πaL

be muchsmaller than one but also that the parameter Ur be much smaller than one (see figure 1.5). Toanalyse what happens for very small water depths, let us follow closely the analyses describedin Mei (1989) and Svendsen (2006) and let us consider the hydrodynamic problem writtenusing dimensionless variables. Let us introduce the following dimensionless variables which aredenoted by an apex

(x′, y′) =2π(x, y)

L= k(x, y), z′ =

z

h, t′ =

2π√gh

Lt = tk

√

gh (7.6)

η′ =η

a, φ′ = φ

2πh

aL√gh

where√gh is the wave celerity, when the water depth is much smaller than L, and 2π/T =

k√gh. The scale for the velocity potential is chosen in such a way that the scale for the hori-

zontal velocity component be aTLh

= a√ghL

Lh

as suggested by continuity equation. Therefore, thevelocity components in the horizontal and vertical directions are scaled with different quantities

(u, v) =

(

∂φ

∂x,∂φ

∂y

)

=a

h

√

gh

(

∂φ′

∂x′,∂φ′

∂y′

)

=a

h

√

gh (u′, v′) (7.7)

w =∂φ

∂z=

1

kh

a

h

√

gh∂φ′

∂z′=

1

kh

a

h

√

ghw′

The scale for the vertical velocity component turns out to be inappropriate. However, later, itis shown that ∂φ′/∂z′ is O(µ2) and not O(1).

The dimensionless form of Laplace equation is

µ2

(

∂2φ′

∂x′2+∂2φ′

∂y′2

)

+∂2φ′

∂z′2= 0 − 1 < z′ < ǫη′ (7.8)


while the boundary conditions at the free surface and at the bottom turn out to be

µ2

[

∂η′

∂t′+ ǫ

∂φ′

∂x′∂η′

∂x′+ ǫ

∂φ′

∂y′∂η′

∂y′

]

=∂φ′

∂z′z′ = ǫη′ (7.9)

µ2

[

∂φ′

∂t′+ η′

]

+1

2ǫ

[

µ2

(

(

∂φ′

∂x′

)2

+

(

∂φ′

∂y′

)2)

+

(

∂φ′

∂z′

)2]

= 0 z′ = ǫη′ (7.10)

∂φ′

∂z′= 0 z′ = −1 (7.11)

As already done previously, to simplify the notation, the apices are dropped and the reader isasked to distinguish between dimensional and dimensionless variables.

Since we are interested in the nearshore region where the water depth tends to vanish movingtowards the beach, let us consider small values of the parameter µ and arbitrary values of theparameter ǫ. Indeed, close to the beach the amplitude of the wave can attain large values.Since the function φ is an analytical function, it can be written as a power series of the verticalcoordinate

φ(x, y, z, t) =

∞∑

n=0

(z + 1)nφn (7.12)

where the terms φn depend on x, y, t (φn = φn(x, y, t)). If ∇ denotes the operator(

∂∂x, ∂∂y

)

, it

follows

∇φ =∞∑

n=0

(z + 1)n∇φn (7.13)

∇2φ =∞∑

n=0

(z + 1)n∇2φn (7.14)

∂φ

∂z=

∞∑

n=0

n(z + 1)(n−1)φn =

∞∑

n=0

(z + 1)n (n+ 1)φn+1 (7.15)

∂2φ

∂z2=

∞∑

n=0

n(n− 1)(z + 1)(n−2)φn =

∞∑

n=0

(z + 1)n(n+ 2)(n+ 1)φn+2 (7.16)

Substitution of (7.13)-(7.16) into Laplace equation leads to

µ2∇2φ+∂2φ

∂z2=

∞∑

n=0

(z + 1)n[

µ2∇2φn + (n+ 2)(n+ 1)φn+2

]

= 0 (7.17)

which should be satisfied for any value of z. Since z can take any value in the range (−1, ǫη),the coefficients of any power of (z + 1) should vanish. It follows

φn+2 =−µ2∇2φn

(n+ 2)(n+ 1), n = 0, 1, 2, ... (7.18)


Since the boundary condition at the bottom forces the vanishing of φ1, each coefficent φn, withn odd, vanishes. On the other hand, when n is even, equation (7.17) leads to

φ2 =−µ2∇2φ0

2 × 1=

−µ2∇2φ0

2!(7.19)

φ4 =−µ2∇2φ2

4 × 3=µ4∇2∇2φ0

4!

φ6 =−µ2∇2φ4

6 × 5=

−µ6∇2∇2∇2φ0

6!

Because φ0 is O(1), it is possible to conclude that φ2 = O(µ2), φ4 = O(µ4), ... . Therefore, thepotential function can be written in the form

φ = φ0 −µ2

2(z + 1)2∇2φ0 +

µ4

24(z + 1)4∇2∇2φ0 +O(µ6) (7.20)

and the kinematic and dynamic boundary conditions at the free surface read

µ2

[

1

ǫ

∂(1 + ǫη)

∂t+ ∇(1 + ǫη) ·

(

∇φ0 −µ2

2(1 + ǫη)2∇2∇φ0

)]

=

(7.21)

= −µ2(1 + ǫη)∇2φ0 +µ4

6(1 + ǫη)3∇2∇2φ0 +O(µ6)

µ2

[

∂φ0

∂t− µ2

2(1 + ǫη)2∇2

(

∂φ0

∂t

)

+ η

]

(7.22)

+1

2ǫµ2[

(∇φ0)2 − µ2(1 + ǫη)2∇φ0 · ∇2 (∇φ0)

]

+1

2ǫµ4(1 + ǫη)2

(

∇2φ0

)2= O(µ6),

respectively, where 1 + ǫη is the total dimensionless water depth. Moreover, by denoting withv0 the horizontal velocity evaluated at the bottom, i.e. for z = −1,

v0 = ∇φ0 (7.23)

equation (7.21) can be written in the form

1

ǫ

∂(1 + ǫη)

∂t+ ∇(1 + ǫη) ·

(

v0 −µ2

2(1 + ǫη)2∇2v0

)

(7.24)

+(1 + ǫη)∇ · v0 −µ2

6(1 + ǫη)3∇2(∇ · v0) = O(µ4)

By applying the gradient operator ∇ to (7.22), it follows

∂v0

∂t+ ǫv0 · ∇v0 +

∇(1 + ǫη)

ǫ(7.25)


+µ2∇[

− ǫ

2(1 + ǫη)2v0 · ∇2v0 +

ǫ

2(1 + ǫη)2 (∇ · v0)

2 − 1

2(1 + ǫη)2∇ · ∂v0

∂t

]

= O(µ4)

When the quantities η and u0 are known, the velocity components (u, v) can be easilyobtained and are

(u, v) = ∇φ = v0 −µ2

2(z + 1)2∇ (∇ · v0) +O(µ4) (7.26)

w =∂φ

∂z= −µ2(z + 1)∇2φ0 = µ2(z + 1)∇ · v0 +O(µ4) (7.27)

The pressure field can be obtained from Bernoulli theorem that in dimensionless form reads

P = −z − ǫ

[

∂φ

∂t+ǫ

2

(

(∇φ)2 +1

µ2

(

∂φ

∂z

)2)]

(7.28)

where the pressure is scaled with the quantity ρgh. The approximated pressure field turns outto be

P = −z − ǫ

[

∂φ0

∂t− µ2

2(z + 1)2∇ · ∂v0

∂t

]

(7.29)

−ǫ2

2

[

v20 − µ2(z + 1)2v0 · ∇2v0 + µ2(z + 1)2(∇ · v0)

2]

+O(µ4)

The relationship (7.22) can be used to eliminate ∂φ0/∂t from the previous equation and toobtain

P = −z + ǫη − ǫµ2

2

[

(1 + ǫη)2 − (z + 1)2]

[

∇ · ∂v0

∂t+ ǫv0 · ∇2v0 − (∇ · v0)

2

]

+O(µ4) (7.30)

In the study of the hydrodynamics of the coastal region, it is common to introduce thedepth averaged value of the velocity, which is defined by

V =1

1 + ǫη

∫ ǫη

−1

∇φdz (7.31)

By using (7.13), it follows

V =1

1 + ǫη

∫ ǫη

−1

[

v0 −µ2

2(z + 1)2∇∇ · v0 + ..

]

dz = v0 −µ2

6(1 + ǫη)2∇2v0 +O(µ4)

which provides

u0 = V +µ2

6(1 + ǫη)2∇2V +O(µ4) (7.32)

By substituting (7.32) into (7.24) and neglecting term of order µ4, it is easy to obtain

∂η

∂t+ ∇ · ((1 + ǫη)V) = 0 (7.33)


which describes the mass balance. Then, still neglecting terms of order µ4, it is possible toobtain

∂V

∂t+ ǫV · ∇V + ∇η +

µ2

6

∂ [(1 + ǫη)2∇2V]

∂t(7.34)

+µ2∇[

− ǫ

3(1 + ǫη)2V · ∇2V +

ǫ

2(1 + ǫη)2 (∇ · V)2 − (1 + ǫη)2

2∇ · ∂V

∂t

]

= 0

Of course the previous equations, which are written in dimensionless form, can be written alsoin a dimensional form

∂η

∂t+ ∇ · ((h+ η)V) = 0 (7.35)

∂V

∂t+V·∇V+g∇η+1

6

∂ [(h+ η)2∇2V]

∂t−∇

(h+ η)2

2

[

2

3V · ∇2V − (∇ · V)2 + ∇ · ∂V

∂t

]

= 0

(7.36)

P = ρg(η − z) − 1

2ρ[

(h+ η)2 − (z + h)2]

[

∇ · ∂V∂t

+(

V · ∇2V − (∇ · V)2)

]

(7.37)

Both the set of equations (7.33), (7.34) and (7.29), which are dimensionless, and the setof equations (7.35), (7.36) and (7.37), which are dimensional, describe the hydrodynamics ofthe coastal region close to the shore where the water depth is much smaller than the lengthof the incoming waves. However, the equations are approximated since terms of O(µ4), i.e.O((h/L)4), are neglected.

7.1.1 The theory for very long waves

In the region closest to the shore, where the parameter µ is quite small, it is reasonable toneglect also the terms of order µ2. Hence, considering the dimensional equations, (7.35)-(7.36)become

∂η

∂t+ ∇ · [(h+ η)V] = 0 (7.38)

∂V

∂t+ V · ∇V + g∇η = 0 (7.39)

and the pressure turns out to be hydrostatic

P = ρg(η − z) (7.40)

Moreover, it is worth pointing out that equations (7.38)-(7.39) are valid also when the waterdepth depends on x and y.

To obtain two important results, let us consider waves of small amplitude and linearizecontinuity and momentum equations. If the solution is written in the form of a propagatingwave

η =A

2ei(kx−ωt) + c.c., U =

U

2ei(kx−ωt) + c.c., V = 0, (7.41)


substitution of (7.41) into (7.38)-(7.39) leads to

−iωA+ ikhU = 0 (7.42)

ikgA− iωU = 0

which is a linear algebraic homogenous system where the unknowns are A e U . A non-trivialsolution exists if and only if the following eigenrelationship is satisfied, which follows from thevanishing of the determinant of the matrix of the coefficients

ω2 = ghk2 (7.43)

The eigenrelation (7.43) shows that the wave celerity c is equal to√gh and hence c does not

depend on the length of the wave. Moreover, it follows that

U =A

T

L

h(7.44)

i.e. the horizontal velocity component is much larger than the vertical velocity componentwhich is O(A/T ).

7.1.2 Boussinesq theory

For long waves in shallow water, when their amplitudes are not so small to neglect nonlineareffects, it is possible to consider terms of O(ǫ) and O(µ2) simultaneously. Assuming thatO(ǫ) ∼ O(µ2), continuity equation becomes

∂η

∂t+ ∇ · [(1 + ǫη)V] = 0, (7.45)

momentum equation reads

∂V

∂t+ ǫV · ∇V + ∇η − µ2

3∇(

∇ · ∂V∂t

)

= 0 (7.46)

and the pressure distribution is no longer hydrostatic

P = −z + ǫη +ǫµ2

2

(

z2 + 2z)

∇ · ∂V∂t

(7.47)

In dimensional form, we have

∂η

∂t+ ∇ · [(h+ η)V] = 0 (7.48)

∂V

∂t+ V · ∇V + g∇(h+ η) − h2

3∇(

∇ · ∂V∂t

)

= 0 (7.49)


P = ρg(η − z) − ρ

2

(

2zh + z2)

(

∇ · ∂V∂t

)

(7.50)

which are known as Boussinesq equations.The presence of terms of order ǫµ2 in (7.47) can be understood if the pressure is split

into the hydrostatic contribution −z and the contribution due to the presence of waves ǫη +ǫµ2

2(z2 + 2z)∇ · ∂V

∂t. The latter turns out to be proportional to ǫ simply because the pressure

is made dimensionless introducing the scaling factor ρgh. However, it is more appropriate toscale the modified pressure (i.e. the pressure due to the waves) with quantity ρga which makesǫ to dissapear.

Moreover, momentum equations differ from those for very long waves because of the termsproportional to µ2. To understand the relevance of these terms, let us consider waves of smallamplitude, such to linearize the problem and write the solution in the form

η =A

2ei(kx−ωt) + c.c., U =

U

2ei(kx−ωt) + c.c., v = 0 (7.51)

By substituting (7.51) into continuity and momentum equations, it is easy to obtain

−iωA+ ikhU = 0 (7.52)

−iωU + ikgA− h2

3(ik)2(−iω)U = 0

which is a homogenous linear algebraic system, characterized by the unknowns A and U . Anontrivial solution exists if and only if the determinant of the matrix of the coefficients vanishes

ω2 =ghk2

1 + 13k2h2

(7.53)

It follows that wave celerity is provided by

c =

√gh

√

1 + k2h2

3

(7.54)

which can be approximated by

c =√

gh

√

1 − k2h2

3(7.55)

if higher order terms are neglected.The term k2h2

3= 1

3µ2 is related to the ’dispersion frequency’ and is originated by the

term h2

3∂3u∂x2∂t

. This contribution makes waves of different wavelengths to travel with differentcelerities, thus causing the ’dispersion’ of the waves of a packet. Hence, it is possible to concludethat Boussinesq equations take into account both nonlinear effects and dispersion effects.


7.1.3 Variable depth

If the spatial scale which characterizes the depth changes is not much larger than the length ofthe wave, it is possible to extend the procedure previously described. In this case, the bottomboundary condition, written using dimensional variables, is

∂φ

∂z= −∂h

∂x

∂φ

∂x− ∂h

∂y

∂φ

∂y(7.56)

or, using dimensionless variables

∂φ

∂z= −µ2

(

∂h

∂x

∂φ

∂x+∂h

∂y

∂φ

∂y

)

(7.57)

where the water depth h is scaled with some value h0 representing the average water depth.Therefore, assuming O(ǫ) ∼ O(µ2) (where both ǫ and µ2 are parameters much smaller thanone), let us write

φ(x, y, z, t) =

∞∑

n=0

[z + h(x, y)]n φn(x, y) (7.58)

If the previous expansion is plugged into the boundary condition (7.57), recursive relationshipsfor φn are obtained. In particular, the functions φn for odd values of n vanish no longer.

A procedure similar to that previously outlined leads to

∂η

∂t+ ∇ · [(h+ η)u] = 0 (7.59)

∂V

∂t+ V · ∇V + g∇η =

h

2∇[

∇ ·(

h∂V

∂t

)]

− h2

6∇[

∇ ·(

∂V

∂t

)]

(7.60)

The algebra is simple but long and it is not written herein but described in details in Peregrine(1967).

7.2 ’Permanent waves’ of small amplitude and long wa-

velength

The dynamics of sea waves of small amplitude is often determined looking for waves proportionalto eik(x−ct), i.e. propagating waves. Therefore, let us look for solutions of the nonlinear shallowwater equations by assuming that O(ǫ) ∼ O(µ2) (where both ǫ and µ2 are parameters muchsmaller than one) and considering solutions characterized by a form such that the variables xand t appear in the combination (x − ct). Because these waves propagate without modifyingtheir shape, they are called ’permanent waves’.


If we introduce a potential function such that u = ∂φ∂x

, Boussinesq equations can be writtenin the form

∂η

∂t+

∂

∂x

[

(1 + ǫη)∂φ

∂x

]

= 0 (7.61)

∂2φ

∂t∂x+ ǫ

∂φ

∂x

∂2φ

∂x2+∂η

∂x− µ2

3

∂4φ

∂x∂t∂x2= 0 (7.62)

Integrating (7.62), it follows

∂φ

∂t+ǫ

2

(

∂φ

∂x

)2

+ η =µ2

3

∂3φ

∂t∂x2(7.63)

The spatial and temporal scales used to make dimensionless the equations are 1/k and 1/(k√gh),

respectively. By eliminating η from the previous equations, an equation for φ is obtained whichis also known as Boussinesq equation,

∂2φ

∂t2− ∂2φ

∂x2=µ2

3

∂4φ

∂x2∂t2− ǫ

∂

∂t

[

(

∂φ

∂x

)2

+1

2

(

∂φ

∂t

)2]

(7.64)

where the terms of order smaller than ǫ and/or µ2 are neglected.The reader should notice that, to obtain Boussinesq equation, use is made of the following

relationships

η = −∂φ∂t

+O(ǫ, µ2) (7.65)

(which comes from momentum equation (7.62))

∂η

∂x= − ∂2φ

∂x∂t+O(ǫ, µ2)

(which is obtained by deriving (7.65)). Moreover, from continuity equation (7.61), it follows

∂η

∂t= ǫ

∂2φ

∂t∂x

∂φ

∂x+ ǫ

∂φ

∂t

∂2φ

∂x2− ∂2φ

∂x2(7.66)

and and, from (7.65), it follows

∂η

∂t= −∂

2φ

∂t2+O(ǫ, µ2)

Finally, equation (7.66) shows that

∂η

∂t= −∂

2φ

∂x2+O(ǫ, µ2)

or∂2φ

∂x2=∂2φ

∂t2+O(ǫ, µ2)


and use is made ofǫ

2

∂

∂t

(

∂φ

∂x

)2

=∂φ

∂xǫ∂2φ

∂t∂x

Now, looking for a solution of Boussinesq equation (7.64) in the form φ = φ(ξ), withξ = x− ct, and knowing that

∂

∂x=

d

dξ

∂

∂t= −c d

dξ

it is easy to obtain

(

c2 − 1) d2φ

dξ2=µ2

3c2d4φ

dξ4+ ǫc

(

1 +c2

2

)

d

dξ

[

(

dφ

dξ

)2]

(7.67)

The previous equation implies that c be equal to 1 +O(ǫ, µ2). Hence, it is possible to considerc ∼ 1 on the right hand side of (7.67), thus obtaining

(

c2 − 1) dφ

dξ+ A1 =

µ2

3

d3φ

dξ3+

3ǫ

2(dφ

dξ)2 (7.68)

However, (7.65) provides

η = −∂φ∂t

=dφ

dξ+O(ǫ, µ2)

and, taking into account that c = 1 +O(ǫ, µ2), it follows

(

c2 − 1)

η + A1 =µ2

3

d2η

dξ2+

3ǫ

2η2 (7.69)

At last, by multiplying the previous equation for dηdξ

and integrating once more, it is possibleto obtain

− ǫ

2η3 + (c2 − 1)

η2

2+ A1η + A2 =

µ2

6

(

dη

dξ

)2

(7.70)

where the constants A1 and A2 turn out to be O(ǫ, µ2).

7.3 Solitary wave

A solitary wave is defined as a wave characterized by a single crest at ξ = 0, such that the freesurface elevation tends to vanish monotonically as |ξ| tends to infinity (see figure 7.1). Since


Figura 7.1: Solitary wave

η and its spatial derivatives tend to vanish far from the crest, the constants A1 and A2 shouldvanish and equation (7.70) becomes

(

dη

dξ

)2

=ǫ

µ23η2

(

c2 − 1

ǫ− η

)

(7.71)

In order to have a positive value on the right hand side, it is necessary that c > 1 (c >√gh if

dimensional variables are used). Since η < (c2 −1)/ǫ, it follows that (c2 −1)/ǫ is the maximumof the wave profile which should be equal to 1 because of the definition of the dimensionlessvariables. Hence, it follows

c2 = 1 + ǫ (7.72)


which, using dimensional variables, leads to (the reader should remind that the horizontalspatial scale is L

2π, the temporal scale is L

2π√gh

and the scale of c turns out to be√gh)

c =

√

gh(

1 +a

h

)

=√

g(h+ a) (7.73)

Therefore the wave celerity increases with the height of the wave. By using the relationshipc2 = 1 + ǫ, equation (7.71) leads to

dη

dξ=

√3ǫ

µη√

1 − η (7.74)

which can be integrated √3ǫ

µ(ξ − ξ0) = −2arctanh

√

1 − η (7.75)

to provide

η = sech2

[√3ǫ

2µ(ξ − ξ0)

]

(7.76)

where sech(x) = 1/ cosh(x) and the constant ξ0 can be set equal to zero. Indeed,

tanh

[

−√

3ǫ

2µ(ξ − ξ0)

]

=√

1 − η (7.77)

η = 1 − tanh2

[

−√

3ǫ

2µ(ξ − ξ0)

]

= 1 −sinh2

[

−√

3ǫ2µ

(ξ − ξ0)]

cosh2[

−√

3ǫ2µ

(ξ − ξ0)] (7.78)

=cosh2

[

−√

3ǫ2µ

(ξ − ξ0)]

− sinh2[

−√

3ǫ2µ

(ξ − ξ0)]

cosh2[

−√

3ǫ2µ

(ξ − ξ0)] =

1

cosh2[

−√

3ǫ2µ

(ξ − ξ0)]

In dimensional form

η = a sech2

[

√

3a

4h3(x− ct)

]

(7.79)

which shows that the wave is ’sharper’ as the height becomes larger. Figure 7.2 shows the freesurface elevation as function of x/h for t = 0 and different values of ǫ. The reader should keepin mind that the ’solitary wave’ solution is obtained by assuming that both ǫ and µ2 are smalland the two parameters have the same order of magnitude.


0

0.2

0.4

0.6

0.8

1

1.2

-20 -15 -10 -5 0 5 10 15 20

η/a

x/h

ε=0.05

ε=0.1

ε=0.2

Figura 7.2: Dimensionless free surface elevation of a solitary wave as function of x/h for differentvalues of ǫ, namely ǫ = 0.05, 0.1, 0.2.

Figura 7.3: Crossing swells, consisting of near-cnoidal wave trains. Photo taken from Pharesdes Baleines (Whale Lighthouse) at the western point of Ile de Re (Isle of Re), France, in theAtlantic Ocean.


Figura 7.4: US Army bombers flying over near-periodic swell in shallow water, close to thePanama coast (1933). The sharp crests and very flat troughs are characteristic for cnoidalwaves.

7.4 Cnoidal waves (see figures 7.3 and 7.4)

Beside the solitary wave, there are other ’permanent’ waves. To determine and to investigatethem, let us write (7.70) in the form

µ2

3ǫ

(

dη

dξ

)2

= −η3 +c2 − 1

ǫη2 +B1η +B2 = P3(η) = (η3 − η) (η − η2) (η − η1) (7.80)

where B1 = 2A1

ǫ, B2 = 2A2

ǫand η1 < η2 < η3 are the three zeros of the third-order polynomial

P3 appearing into (7.80). As shown by figure 7.5, since P3 should be positive (see the left handside of (7.80)), the value of η should fall between the two zeros η2 and η3 which corresponds tothe level of the troughs and the level of the crests, respectively. Then, their difference is thewave height H

H = η3 − η2

Equation (7.80) can be integrated by using the elliptic integrals. Let us introduce

η = η3 cos2 ψ + η2 sin2 ψ con ψ = ψ(η) (7.81)

and let us derive with respect to ξ, thus obtaining

dη

dξ= −2(η3 − η2)

dψ

dξsinψ cosψ (7.82)

Equations (7.81) and (7.82) can be inserted into (7.80) leading to

4

(

dψ

dξ

)2

(η3 − η2)2 sin2 ψ cos2 ψ =

3ǫ

µ2

[

η3(1 − cos2 ψ) − η2 sin2 ψ]

(7.83)


η1 η2 η3

( )ηP3

η

Figura 7.5:

[

−η3 cos2 ψ + η2(1 − sin2 ψ)] [

η1 − η3(1 − sin2 ψ) − η2 sin2 ψ]

=3ǫ

µ2(η3 − η2)

2 sin2 ψ cos2 ψ[

(η3 − η1) − (η3 − η2) sin2 ψ]

or(

dψ

dξ

)2

=3ǫ

4µ2(η3 − η1)

[

1 −m sin2 ψ]

(7.84)

where

m =η3 − η2

η3 − η1

η3 − η1 =H

m(7.85)

and 0 < m < 1 because(

dψdξ

)2

should be positive and (η3−η1) is positive. At last, by integrating

(7.84), it follows

∫ ψ

0

dψ√

1 −m sin2 ψ= F (ψ,m) = ±

√3ǫ

2µ

√η3 − η1(ξ − ξ0) (7.86)

The integral is the standard form of the incomplete elliptical integral of the first kind withmodulus m. The previous relationship can be considered as an implicit relationshipfor ψ as function of ξ. More explicitly, let us denote

cosψ = cn

[√3ǫ

2µ

√η3 − η1(ξ − ξ0) |m

]

(7.87)

sinψ = sn

[√3ǫ

2µ

√η3 − η1(ξ − ξ0) |m

]

(7.88)


where cn and sn are the elliptic sine and cosine functions (Abramowitz and Stegun (1965)Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables NewYork: Dover, ISBN 978-0486612720, MR 0167642). From (7.81), the wave surface turns out tobe

η = η2 + (η3 − η2)cn2

(√3ǫ

2µ

√η3 − η1(ξ − ξ0) | m

)

(7.89)

where η2 is the level of the trough with respect to the mean water level and it is negative. Usingdimensional physical variables, (7.89) can be written in the form

η = η2 + (η3 − η2)cn2

(√3

2µ

√η3 − η1

h3/2(x− ct− x0) |m

)

. (7.90)

Korteweg & De Vries (1895) introduced the term cnoidal to define the function cn and thewaves are known as cnoidal waves.

Since cosψ is a periodic function of period 2π, cn(z|m) is, by definition, a periodic functionof period 4K, where

K = F(π

2, m)

=

∫ π/2

0

dψ√

1 −m sin2 ψ(7.91)

is the standard symbol for the complete elliptical integral of the first kind. Sometimes, insteadof the parameter m, use is made of the parameter α such that m = sinα and 0 ≤ m ≤ 1 with0 ≤ α ≤ π/2. Since cn2 should have period 2K(m) (half of the period of cn), the dimensionallength λ of the cnoidal wave is provided by

λ

√3ǫ

2µ

√η3 − η1 = 2K(m) (7.92)

or

λ =4K(m)µ√

3ǫ

√

m

H(7.93)

where the value of η3 − η1 is obtained by means of (7.85).The wavelength of the cnoidal waves depends on the amplitude through the value of m.

Equation (7.89) can be written in the form

η = η2 + (η3 − η2)cn2

(

2K

λ(ξ − ξ0)|m

)

(7.94)

The wave celerity can be determined as function of η1, η2 and η3 by means of (7.80)

c =√

1 + ǫ(η1 + η2 + η3) (7.95)

Indeed,

P3(η) = (η3 − η) (η − η2) (η − η1) = (η3 − η)[

η2 − η (η1 + η2) + η1η2

]

=


−η3 + η2 (η1 + η2) − ηη1η2 + η2η3 − ηη3 (η1 + η2) + η1η2η3 =

−η3 + η2 (η1 + η2 + η3) − η (η1η2 + η2η3 − η1η3) + η1η2η3 =

and, because of (7.80), P3(η) should be equal to −η3 + c2−1ǫη2 +B1η +B2. It follows

c2 − 1

ǫ= η1 + η2 + η3 (7.96)

and, then, from (7.96), (7.95) can be demonstrated. The characteristics of a cnoidal wave areknown once the three parameters is η1, η2 and η3 are given. However, from an engineering pointof view, it is convenient to substitute η1, η2 and η3 with λ, the local water depth h and thewave height H . Let us define the coordinate in such a way that

∫ λ

0

ηdξ = 0

This implies that∫ π

0

(η3 cos2 ψ + η2 sin2 ψ)dξ

dψdψ = 0

In fact, because of (7.81), η = η3 cos2 ψ + η2 sin2 ψ. By using (7.85) and the root square of(7.84), the integral can be written in the form

∫ π/2

0

dψη1 + (η3 − η1)(1 −m sin2 ψ)

√

1 −m sin2 ψ= 0

where the multiplying constant is dropped and use has been made of the symmetry of sin2 ψwith respect to ψ = π

2. Therefore, by using (7.91) and the definition of the elliptical integral of

the second kind

E(m) =

∫ π/2

0

√

1 −m sin2 ψdψ (7.97)

we obtainη1K(m) + (η3 − η1)E(m) = 0

or, using (7.85)

η1 = −E

K(η3 − η1) = −η3 − η2

m

E

K= −H

m

E

K

It follows

η3 = −η1

(

K

E− 1

)

=H

m

(

1 − E

K

)

(7.98)

and, at last,

η2 = η3 −H =H

m

(

1 −m− E

K

)

(7.99)


By means of this procedure the three parameters η1, η2 and η3 are written as function of H andm, once the local water depth h is given. Then, these relationships can be substituited into therelationship which provides the dimensionless wave celerity

c2 = 1 + ǫH

m

(

−m+ 2 − 3E

K

)

(7.100)

and into the relationship which provide the dimensionless wavelength

λ =4Kµ√

3ǫ

√

m

H(7.101)

Finally, the period turns out to be

T =λ

c=

K 4µ√3ǫ

√

mH

√

1 + ǫHm

(

−m+ 2 − 3EK

)

(7.102)

If the dimensional variables are needed, it is necessary to remind that

x′ = kx, t′ = k√

ght, c′ =√

ghc, λ′ = λk H ′ = H/a (7.103)

Moreover, since µ = kh e ǫ = a/h, it is easy to obtain

c2 = gh

[

1 +H

h

1

m

(

−m+ 2 − 3E

K

)]

(7.104)

λ = 4Kh

√

m

3H/h(7.105)

T =

√

h

g

4K√

mh/3H√

1 + Hh

1m

(

−m+ 2 − 3EK

)

(7.106)

η = η2 +Hcn2

(

2K

λ(x− ct)|m

)

(7.107)

Of course, the parameter m could be eliminated from any pair of the relationships (7.104)-(7.106) thus obtaining relationships of the kind c = c(T,H), λ = λ(T,H) ... (sometimes thefunction cn(x|m) is indicated with cn(x|e) where e =

√m). However, it is simpler to consider

m as a parameter. Wiegel (1960) provides a plot of the surface profile for different values of min the range (0, 1) (see also figure 7.6).

To better understand the relationships previously obtained, let us analyse two limiting cases:

(i) m tending to 1


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(ζ-ζ

2)/H

x/λ

m=0.6m=0.7m=0.8m=0.9m=0.99m=0.9999

Figura 7.6: Surface profile of a cnoidal wave for different values of the parameter m.

In this case, η2 tends to η1, E(1) = 1 and K(1) tends to infinity. It follows that λ tends toinfinity and cn2u tends to sech2u. However, equation (7.101) shows that the ratio K/λ tendsto a finite limit. It follows that

η = Hsech2

[√3

2

√

H

h3(x− ct)

]

(7.108)

which is coincident with the profile of the solitary wave. Moreover, the wave celerity tends to

c =

√

gh

[

1 +H

h

]

(7.109)

It follows that the solitary wave is the limit of a cnoidal wave, when its wavelength tends toinfinity.

(ii) m tending to 0

In this case, η3 − η2 = H tends to 0, i.e. the waves are characterized by an infinitesimalamplitude. It is possible to verify that c2 tends to gh, cn(n|m) tends to cn(u|0) or cosu and K


0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

10 100

c/(g

h)1/

2

λ2 H/h3

H/h=0.8H/h=0.7H/h=0.6H/h=0.5

H/h=0.4H/h=0.3H/h=0.2H/h=0.1

Figura 7.7: Wave celerity for a cnoidal wave as function of its wavelength.

tends to π/2. Hence,

η = η2 +H cos2(π

λ(x− ct)

)

(7.110)

Since η2 = −a = −H/2, it follows

η = −H2

+H cos2(π

λ(x− ct)

)

= H

(

cos2(π

λ(x− ct)

)

− 1

2

)

and, knowing that

cos 2θ = 2 cos2 θ − 1 ;cos 2θ

2= cos2 θ − 1

2,


η =H

2cos

(

2π

λ(x− ct)

)

(7.111)

which is a sinusoidal wave.As pointed out by Fenton (1990) (Ocean Engineering Science, The Sea, Vol. 9), the cnoidal

wave theory is not widely employed because this theory uses Jacobian elliptic functions and


integrals which are thought to be difficult to evaluate. Alternative formulae are given by Fenton(1990) and are reported hereinafter.

The complete elliptic integral of the first kind K(m) can be approximated by

K(m) ≃ 2

(1 +m1/4)2 ln

[

2(

1 +m1/4)

1 −m1/4

]

(7.112)

and the complete elliptic integral of the second kind E(m) by

E(m) ≃ K(m)e(m) (7.113)

where

e(m) ≃ 2 −m

3+

π

2KK ′ + 2( π

K ′

)2[

− 1

24+

q21

(1 − q21)

2

]

(7.114)

K ′(m) ≃ 2π

(1 +m1/4)2 (7.115)

q1(m) ≃ e−πKK′ (7.116)

Finally the function cn(x|m) can be approximated by

cn(x|m) ≃ 1

2

(

1 −m

mq1

)1/41 − 2q1 cosh(2w)

cosh(w) + q21 cosh(3w)

(7.117)

wherew =

πx

2K ′

The approximations previously described give results which are accurate to five digits when mis larger than or equal to 0.5, i.e. when the cnoidal wave theory is more appropriate.

For example, by fixing h = 3 m, T = 9 s and H = 2 m, the solution of (7.106) which can befound by means of an iterative procedure, provides the valuem equal to about 0.9999776. Then,(7.105) can be used to compute the wavelength λ which turns out to be about 54 m. Finally,it can be verified that the value of c provided by (7.104) is about 6 m/s and equal to the ratioλ/T . Of course η2 can be evaluated by means of (7.99) and the free surface elevation followsfrom (7.107) while the velocity components can be evaluated by means of the relationshipswhich appear in the paper by Wiegel

U =√

gh

[

−5

4+

3yt2h

− y2t

h2+

(

3H

2h− Hyt

2h2

)

cn2 − H2

4h2cn4 (7.118)

−8HK(m)2

L2

(

h

3− y2

2h

)

(

−m sn2cn2 + cn2dn2 − sn2dn2)

]

V =√

ghy2HK(m)

Lhsn cn dn × (7.119)


[

1 +yth

+H

hcn2 +

32K2(m)

3L2

(

h2 − y2

2

)

(

m sn2 −m cn2 − dn2)

]

whereyth

= 1 − H

h+

16h2

3L2K(m) [K(m) − E(m)]

7.5 Korteweg-De Vries equation

The permanent waves considered so far depend on the following dimensionless variable

ξ = x− ct = x− t+O(ǫ)t

Hence, an observer, who travels with the linearized speed of the wave, observes a slow variationof the form of the wave. The temporal scale describing the slow variation is 1/ǫ and thissuggests to introduce the following variables

σ = x− t τ = ǫt (7.120)

With these variables, the spatial and temporal derivatives become

∂

∂x=

∂

∂σ

∂

∂t= − ∂

∂σ+ ǫ

∂

∂τ

Substitution of these relationships into Boussinesq equation (7.64) leads to

∂2φ

∂σ∂τ+

3

4

∂

∂σ

(

∂φ

∂σ

)2

+µ2

6ǫ

∂3φ

∂σ3= O(ǫ, µ2) (7.121)

At the leading order of approximation ∂φ/∂σ can be substituted with η and, hence,

∂η

∂τ+

3

2η∂η

∂σ+µ2

6ǫ

∂3φ

∂σ3= O(ǫ, µ2) (7.122)

which is commonly known as Korteweg-De Vries equation. Using dimensional variables

∂η

∂t+√

gh

(

1 +3

2

η

h

)

∂η

∂x+h2

6

√

gh∂3η

∂x3= 0 (7.123)

7.6 Edge waves

Let us consider surface waves characterized by a wavelength much longer than the water depthand described by Airy equations (7.38)-(7.40). Moreover, let us consider a beach characterizedby a constant slope such that the bottom be described by

z = −h = −sx (7.124)


s =constant being the slope of the bottom and the x-axis being orthogonal to the coast andpointing in the offshore direction (the y-axis being coincident with the coastline). If the am-plitude of the wave is small, it is possible to neglect nonlinear effects and the equations whichcontrol the phenomenon are

∂η

∂t+∂(hU)

∂x+∂(hV )

∂y= 0 (7.125)

∂U

∂t+ g

∂η

∂x= 0 (7.126)

∂V

∂t+ g

∂η

∂y= 0 (7.127)

Such equations can be combined to give rise to

∂2η

∂t2− g

dh

dx

∂η

∂x− gh

(

∂2η

∂x2+∂2η

∂y2

)

=∂2η

∂t2− gs

∂η

∂x− gh

(

∂2η

∂x2+∂2η

∂y2

)

= 0 (7.128)

Let us look for a solution in the form

η = η(x)ei(βy−ωt) + c.c. (7.129)

Subsitution of (7.129) into (7.128) leads to

xd2η

dx2+dη

dx+

(

ω2

sg− β2x

)

η = 0 (7.130)

By introducing the variable ζ = 2βx and the unknown η = e−ζ2 f(ζ), equation (7.130) becomes

ζd2f

dζ2+ (1 − ζ)

df

dζ+

(

ω2

2βsg− 1

2

)

f = 0 (7.131)

which is an equation belonging to the class of the hypergeometric confluent equations (Kummerequation) which admit two solutions. One of the solutions is singular for ζ = 0 and should bediscarded. Solutions exist which are not trivial and finite at ζ = 0 and tend to vanish when ζtends to infinity, only if

ω2

2βsg= n +

1

2with n = 0, 1, 2, 3, ... (7.132)

which is a dispersion relationship and relates the frequency with the wavenumber in the directionparallel to the coast. In this case (7.131) reads

ζd2f

dζ2+ (1 − ζ)

df

dζ+ nf = 0 (7.133)


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

η

2 β x

n=0

n=2

n=1

Figura 7.8: Function η plotted versus ξ = 2βx for different values of n.

The solutions are

Ln (ζ) =(−1)n

n!

[

ζn − n2

1!ζn−1 +

n2(n− 1)2

2!ζn−2 − n2(n− 1)2(n− 2)2

3!ζn−3 + ...

]

(7.134)

(L0 = 1, L1 = 1−ζ, L2 = 1−2ζ+ 12ζ2, ...). In figure 7.8, the function η(ξ) is plotted for different

values of n and it shows that these waves are characterized by an amplitude which is significantonly close to the beach. For this reason the waves are named ’edge waves’. A skecth of the freesurface at a fixed phase is shown in figure 7.9 for n = 0. There are different mechanisms whichcan generate edge waves. For example, very long edge waves can be generated by atmosphericperturbations travelling parallel to the coast. Edge waves characterized by a period which istwice that of the incident waves can be generated by sea waves approaching the coast througha subharmonic instability mechanism. Finally, let us point out that the presence of edge waveshas a significant influence on other important hydrodynamic phenomena like rip currents andmorphodynamic phenomena like beach cusps, coastal forms shown in figure 7.10


z

xy

z

Figura 7.9: Sketch of the free surface due to the presence of an edge wave (t = 0). The coastin in the left-bottom side of the figure.

Figura 7.10: Beach cuspis observed along the coast of Mexico.

Capitolo 8

STEADY CURRENTS IN THESHALLOW WATER REGION

8.1 Introduction

Before considering the hydrodynamic problem of the steady currents generated by the interac-tion of propagating surface waves with the sea bottom and the coastline (see chapter 9), let usconsider the dynamics of a current in a shallow water region, where the changes of the waterdepth take place over distances much larger than the depth itself.

8.2 The depth averaged equations

Since we are considering a steady current, the assumption of an irrotational flow is no longerjustified and it is necessary to consider the full momentum equations. The equations areaveraged according to the Reynolds procedure since the flow regime is assumed to be turbulentand only the averaged flow is of interest.

∂u

∂t+u

∂u

∂x+v

∂u

∂y+w

∂u

∂z= −∂p

∂x+

∂

∂x

[

2νT∂u

∂x

]

+∂

∂y

[

νT

(

∂u

∂y+∂v

∂x

)]

+∂

∂z

[

νT

(

∂u

∂z+∂w

∂x

)]

(8.1)∂v

∂t+u

∂v

∂x+ v

∂v

∂y+w

∂v

∂z= −∂p

∂y+

∂

∂x

[

νT

(

∂v

∂x+∂u

∂y

)]

+∂

∂y

[

2νT∂v

∂y

]

+∂

∂z

[

νT

(

∂v

∂z+∂w

∂y

)]

(8.2)∂w

∂t+u

∂w

∂x+v

∂w

∂y+w

∂w

∂z= −∂p

∂z−g+ ∂

∂x

[

νT

(

∂w

∂x+∂u

∂z

)]

+∂

∂y

[

νT

(

∂w

∂y+∂v

∂z

)]

+∂

∂z

[

2νT∂w

∂z

]

(8.3)In (8.1)-(8.3), the turbulent Reynolds stresses are quantified introducing an eddy viscosity νTwhich should be properly modelled and is assumed to be much larger than the viscosity of the

126

CAPITOLO 8. STEADY CURRENTS IN THE SHALLOW WATER REGION 127

fluid and such that viscous stresses can be safely neglected. Then, the hydrodynamic problemis closed by considering continuity equation

∂u

∂x+∂v

∂y+∂w

∂z= 0 (8.4)

and an appropriate turbulence model.However, the hydrodynamic problem can be simplified assuming that the order of magnitude

of the water depth h0 is much smaller than the length L which characterizes the bottom changes.When the ratio h0/L is much smaller than 1, the three velocity components have not the

same order of magnitude. In fact, the order of magnitude of the vertical velocity component wis much smaller that the order of magnitude of the horizontal components because of continuityequation. By denoting with U0 the order of magnitude of the horizontal velocity components,it follows

U0

L ≃ O(w)

h0(8.5)

and, hence, O(w) ≃ h0

L U0.Then, let us consider the order of magnitude of the different terms appearing into momentum

equation along the vertical axis. The advective terms have an order of magnitude equal toh0U

20 /L2 and they are negligible with respect to the gravitational term g. Indeed the ratio

between the former order of magnitude and the latter is

h0U20

gL2=h2

0

L2Fr2

and it turns out to be much smaller than one because the Froude number Fr = U0/√gh0 in

the coastal region is smaller than one and h0/L is assumed much smaller than one.The local time derivative can be also neglected either because of the velocity field is steady

or because the temporal variations of the current are quite slow and of order U0/(L/U0).In order to compare the order of magnitude of the terms related to the turbulent stresses

with the order of magnitude of the gravitational term, it is necessary to have an estimate ofthe eddy viscosity. It is reasonble to assume that O(νT ) ≃ uτh0, uτ being the friction velocity.Even considering the largest turbulent stress term

(

∂∂z

[

2νT∂w∂z

])

, it turns out that the ratiobetween the order of magnitude of the turbulent stress terms and the order of magnitude ofthe gravitational term is of order

Fr2 uτU0

h0

Lwhich can be considered much smaller than one.

It follows that the momentum equation along the vertical direction forces a balance betweenthe vertical pressure gradient and the gravity acceleration and the pressure distribution can beassumed to be hydrostatic.


Then, momentum equations in the x- and y-directions become

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z= −g ∂η

∂x+

∂

∂z

[

νT∂u

∂z

]

(8.6)

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z= −g∂η

∂y+

∂

∂z

[

νT∂v

∂z

]

(8.7)

since the derivatives along the vertical direction are much larger than the derivatives along thehorizontal directions. In (8.6), (8.7), η denotes the free surface elevation and the pressure termsare evaluated taking into account the hydrostatic pressure distribution.

Often, when a shallow water region is considered, the velocity profile is almost constant overthe depth but for a large gradient close to the bottom and only the depth averaged velocity isof interest. Hence, let us define the velocity components U and V

U =1

h+ η

∫ η

−hudz, V =

1

h+ η

∫ η

−hvdz (8.8)

and let us obtain the equations which describe their temporal and spatial distribution. Let usstart by integrating continuity equation from the bottom up to the free surface

∫ η

−h

∂u

∂xdz +

∫ η

−h

∂v

∂ydz + [w]η − [w]−h = 0 (8.9)

Then, using Leibniz’s rule (see the appendix of chapter 9)

∂

∂x

∫ b(x)

a(x)

f(z, x)dz =

∫ b(x)

a(x)

∂f

∂xdz + f(b(x), x)

∂b

∂x− f(a(x), x)

∂a

∂x(8.10)


∂

∂x

∫ η

−hudz +

∂

∂y

∫ η

−hvdz +

[

−u∂η∂x

− v∂η

∂y+ w

]

η

−[

u∂h

∂x+ v

∂h

∂y+ w

]

−h= 0 (8.11)

The kinematic boundary condition (dF/dt = 0) on the free surface, described by F = z −η(x, y, t) = 0, forces

∂η

∂t+

[

u∂η

∂x+ v

∂η

∂y− w

]

η

= 0 (8.12)

and the no-slip condition at the bottom forces

u = v = w = 0 (8.13)

It follows that∂η

∂t+

∂

∂x

∫ η

−hudz +

∂

∂y

∫ η

−hvdz = 0 (8.14)


i.e.∂η

∂t+

∂

∂x[U (h+ η)] +

∂

∂y[V (h+ η)] = 0 (8.15)

Then, let us integrate momentum equation along the x-direction from the bottom up to thefree surface and use Leibniz rule. The local and convective acceleration terms become∫ η

−h

[

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

]

dz =∂

∂t

∫ η

−hudz − [u]η

∂η

∂t− [u]−h

∂h

∂t+

∂

∂x

∫ η

−hu2dz (8.16)

−[

u2]

η

∂η

∂x−[

u2]

−h∂h

∂x+

∂

∂y

∫ η

−huvdz − [uv]η

∂η

∂y− [uv]−h

∂h

∂y+ [uw]η − [uw]−h

By using the kinematic boundary conditions at the free surface and at the bottom, which read

∂ (η,−h)∂t

+ [u]η,−h∂ (η,−h)

∂x+ [v]η,−h

∂ (η,−h)∂y

− [w]η,−h = 0 (8.17)

the quatity (8.16) becomes equal to

∂

∂t[(h+ η)U ]+

∂

∂x

[

(h+ η)U2]

+∂

∂y[(h+ η)UV ]+

∂

∂x

∫ η

−h(u− U)2 dz+

∂

∂y

∫ η

−h(u− U) (v − V ) dz

(8.18)To obtain (8.18), use is made of∫ η

−hu2dz =

∫ η

−h(u− U)2 dz + 2

∫ η

−huUdz −

∫ η

−hU2dz =

∫ η

−h(u− U)2 dz + (h + η)U2 (8.19)

A similar relationship can be obtained by considering momentum equation along the y-direction.The integration of the terms on the right hand side of the x-component of the momentumequation leads to

−g (h+ η)∂η

∂x+

[

νT∂u

∂z

]

η

−[

νT∂u

∂z

]

−h(8.20)

∂η∂x

being independent of z. Since, the z-derivative of any quantity is much larger than the x-and y-derivatives, it turns out that

[

νT∂u

∂z

]

η

=τ

(s)x

ρ,

[

νT∂u

∂z

]

−h=τ

(b)x

ρ(8.21)

where τ(s)x indicates the possible wind stress acting on the free surface and τ

(b)x indicates the

shear stress of the water on the bottom. Similarly, it is possible to obtain the right hand sideof the y-component of the momentum equation.

To conclude, the shallow water equations, which describe the dynamics of a current, are

∂

∂t(h+ η) +

∂

∂x[(h+ η)U ] +

∂

∂y[(h+ η)V ] = 0 (8.22)


∂

∂t[(h+ η)U ] +

∂

∂x

[

(h+ η)U2]

+∂

∂y[(h+ η)UV ] = (8.23)

−g (h+ η)∂η

∂x+

1

ρ

∂

∂x

[

(h+ η)T (disp)xx

]

+1

ρ

∂

∂y

[

(h+ η)T (disp)xy

]

+1

ρ

(

τ (s)x − τ (b)

x

)

∂

∂t[(h+ η)V ] +

∂

∂x[(h + η)UV ] +

∂

∂y

[

(h + η)V 2]

= (8.24)

−g (h+ η)∂η

∂y+

1

ρ

∂

∂x

[

(h + η)T (disp)yx

]

+1

ρ

∂

∂y

[

(h+ η)T (disp)yy

]

+1

ρ

(

τ (s)y − τ (b)

y

)

where we introduce the components of the tensor of the dispersive stresses

T (disp)xx = − ρ

h+ η

∫ η

−h(u− U)2 dz (8.25)

T (disp)xy = T (disp)

yx = − ρ

h + η

∫ η

−h(u− U) (v − V ) dz

T (disp)yy = − ρ

h + η

∫ η

−h(v − V )2 dz

which come from the convective terms and are due to the non uniform distribution of thevelocity components along the vertical direction.

Taking into account continuity equation, it is possible to obtain

∂

∂t(h+ η) +

∂

∂x[(h+ η)U ] +

∂

∂y[(h+ η)V ] = 0 (8.26)

∂U

∂t+U

∂U

∂x+V

∂U

∂y= −g ∂η

∂x+

1

ρ (h + η)

[

∂

∂x

(

(h+ η)T (disp)xx

)

+∂

∂y

(

(h+ η)T (disp)xy

)

]

+

(

τ(F )x − τ

(B)x

)

ρ (h + η)(8.27)

∂V

∂t+U

∂V

∂x+V

∂V

∂y= −g∂η

∂y+

1

ρ (h + η)

[

∂

∂x

(

(h+ η)T (disp)yx

)

+∂

∂y

(

(h+ η)T (disp)yy

)

]

+

(

τ(F )y − τ

(B)y

)

ρ (h + η)(8.28)

To quantify the shear stresses acting on the bottom, it is common to use empirical relation-ships which are strictly valid only when a steady flow is considered

τ (B)x = ρrbU

√U2 + V 2, τ (B)

y = ρrbV√U2 + V 2 (8.29)

where rb is a resistance coefficient which, in principle, depends on the flow Reynolds number andthe relative roughness but it can be assume to depend only on the bottom roughness becauseof the large values of the Reynolds number in the coastal region. The shear stress acting on


the free surface exists when the wind is blowing and can be quantified by means of a similarrelatioship

τ (F )x = ρairrfUwind

√

U2wind + V 2

wind, τ (F )y = ρairrfVwind

√

U2wind + V 2

wind (8.30)

Capitolo 9

TIME AND DEPTH AVERAGEDEQUATIONS

9.1 Introduction

To describe the steady currents generated by the interaction of the sea waves with the bottomand the coast, let us consider the time and depth average of the quantities describing thehydrodynamics of the coastal region. The procedure used to obtain the time and averagedequations follows a classical approach. Hereinafter, we introduce a modification to take intoaccount that the vertical profile of the actual steady component of the velocity is not describedby a constant value.

Let us introduce a Cartesian coordinate system such that x and y denote the horizontaldirections and z-axis is vertical and points upwards. Moreover, the vertical velocity componentis indicated by w while the horizontal velocity components by u and v. At last, let us remindthe definition of the time and depth averaged velocity components U, V , which are obtained byintegrating u, v over the instantaneous water depth and over the wave period T in such a waythat

(U, V ) =1

T

∫ T

0

[

1

η + h

∫ η

−h(u, v)dz

]

dt =1

η + h

∫ η

−h(u, v)dz (9.1)

where an overbar denotes the time average. In (9.1), η is the free surface elevation (η is itstime averaged value) and h is the local water depth. Of course, the quantities ρU(η + h) andρV (η + h) are the mass discharges through a vertical surface of unit width in the x- and y-directions, respectively. Hence, the vector (U, V ) is called average velocity and it depends onlyon (x, y) and possibly on a slow temporal variable such that (U, V ) change on a time scalemuch longer than the period of the waves. By denoting with (u, v) the difference between theinstantaneous local velocity and its average value (U, V ), we have

(u, v) = (U, V ) + (u, v) (9.2)

132

CAPITOLO 9. TIME AND DEPTH AVERAGED EQUATIONS 133

From the previous definitions, it follows that

∫ η

−hudz = 0,

∫ η

−hvdz = 0 (9.3)

Notwithstanding relationships (9.2) show that the depth and time averaged value of the velocitycomponents u, v vanishes, the reader should notice that the time averages of u and v might bedifferent from zero, their vertical profile being positive in some regions and negative in othersregions.

At this stage it is worth pointing out that (U, V ) are different from the depth averagedvalues of the steady velocity components because of different terms and in particular becauseof the contribution

∫ η

ηt

(u, v)dz

which, as discussed in chapter ??, contributes significantly to the steady volume flux per unitwidth of a vertical surface.

The procedure which leads to formulate the hydrodynamic problem by using U, V and ηto describe the flow field is standard. It is repeated here to clarify the roles of the turbulentstresses and the dispersive stresses, which have a different origin and, as explained in thefollowing, should be modelled with different models.

9.2 Continuity equation

Let us start by considering continuity equation, which comes from a mass balance. If continuityequation is integrated over the water depth, as explained in chapter 8 (see (8.15)), it follows

∂η

∂t+

∂

∂x

∫ η

−hudz +

∂

∂y

∫ η

−hvdz = 0 (9.4)

and, by considering the time average (see (9.1)),

∂η

∂t+

∂

∂x

[

U (η + h)]

+∂

∂y

[

V (η + h)]

= 0 (9.5)

where η changes only on a temporal scale much longer than the period of the waves.The reader should notice that equation (9.4) applies also to an inviscid fluid such that

the bottom boundary condition forces the vanishing of the velocity component normal to thebottom and reads

[

u∂h

∂x+ v

∂h

∂y+ w

]

−h= 0 (9.6)


9.3 Momentum equation

Now, let us consider momentum equations. Since we are interested in the dynamics of steadycurrents, it is not reasonable to assume the velocity field to be irrotational and it is necessaryto take into account the shear stresses (either viscous or turbulent)

ρ

[

∂u

∂t+∂(uu)

∂x+∂(uv)

∂y+∂(uw)

∂z

]

= −∂p∂x

+∂τxx∂x

+∂τyx∂y

+∂τzx∂z

(9.7)

ρ

[

∂v

∂t+∂(vu)

∂x+∂(vv)

∂y+∂(vw)

∂z

]

= −∂p∂y

+∂τxy∂x

+∂τyy∂y

+∂τzy∂z

(9.8)

In (9.7)-(9.8) τij denote the anisotropic part of the stress tensor and p is the pressure. Byintegrating the momentum equation (9.7) along the vertical direction and assuming that thedensity ρ is constant, it can be verified that the left hand side terms lead to

∫ η

−h

∂(ρu)

∂tdz =

∂

∂t

∫ η

−hρudz − ρ [u]η

∂η

∂t

∫ η

−h

(

∂(ρuu)

∂x+∂(ρvu)

∂y

)

dz =∂

∂x

∫ η

−hρuudz +

∂

∂y

∫ η

−hρvudz − ρ [uu]η

∂η

∂x− ρ [vu]η

∂η

∂y

−ρ [uu]−h∂h

∂x− ρ [vu]−h

∂h

∂y∫ η

−h

∂(ρwu)

∂zdz = ρ [wu]η − ρ [wu]−h

By using the kinematic boundary condition, the left hand side of (9.7) becomes

∂

∂t

[∫ η

−hρudz

]

+∂

∂x

[∫ η

−hρuudz

]

+∂

∂y

[∫ η

−hρvudz

]

(9.9)

Similarly, the right hand side leads to

∫ η

−h

(

−∂p∂x

+∂τxx∂x

+∂τyx∂y

+∂τzx∂z

)

dz = (9.10)

∂

∂x

∫ η

−h−pdz +

∂

∂x

∫ η

−hτxxdz +

∂

∂y

∫ η

−hτyxdz − [τxx]η

∂η

∂x− [τyx]η

∂η

∂y− [−p]−h

∂h

∂x

− [τxx]−h∂h

∂x− [τyx]−h

∂h

∂y+ [τzx]η − [τzx]−h

where it has been taken into account that the relative pressure on the free surface should vanishbecause the surface tension effects are neglected. At the free surface, the shear stress of the air,


which is characterized by the components τFx and τFy should be balanced by the shear stressdue to the water. Taking into account that the relative pressure vanishes, we have

− [τxx]η nx − [τxy]η ny − [τxz]ηnz + τFx = 0 (9.11)

− [τyx]η nx − [τyy ]η ny − [τyz]ηnz + τFy = 0 (9.12)

where τFx , τFy are the shear stresses applied to the free surface in the x- and y-directions respec-

tively and due, for example, to the blowing wind. Moreover, the unit vector orthogonal to thefree surface n is upward directed (-n is downward directed). On the free surface, described byz − η(x, y, t) = F (x, y, z, t) = 0, the unit vector orthogonal to the surface is ∇F/|∇F |, where∇F = (−∂η/∂x,−∂η/∂y, 1). Hence, the dynamic boundary condition leads to

− [τxx]η∂η

∂x− [τxy]η

∂η

∂y+ [τxz]η = τFx |∇F | (9.13)

− [τyx]η∂η

∂x− [τyy]η

∂η

∂y+ [τyz ]η = τFy |∇F | (9.14)

On the bottom, described by z+ h(x, y) = B(x, y, z) = 0, the unit vector orthogonal to thesea bed and pointing in the upward direction is ∇B/|∇B|, where ∇B = (∂h/∂x, ∂h/∂y, 1). Ifwe denote τBx and τBy the components of the shear stress on the bottom due to the fluid flow,along the x- and y-axis, respectively

[τxx]−hnx + [τxy]−hny + [τxz]−hnz = τBx

[τyx]−hnx + [τyy]−hny + [τyz]−hnz = τBy


[τxx]−h∂h

∂x+ [τxy]−h

∂h

∂y+ [τxz]−h = τBx |∇B| (9.15)

[τyx]−h∂h

∂x+ [τyy]−h

∂h

∂y+ [τyz]−h = τBy |∇B| (9.16)

By using these relationships, the right hand side of (9.7) becomes

− ∂

∂x

∫ η

−hpdz +

∂

∂x

∫ η

−hτxxdz +

∂

∂y

∫ η

−hτyxdz + [p]−h

∂h

∂x+ τFx |∇F | − τBx |∇B| (9.17)

Therefore, momentum equation in the x-direction reads

∂

∂t

∫ η

−hρudz +

∂

∂x

∫ η

−hρuudz +

∂

∂y

∫ η

−hρvudz = (9.18)

− ∂

∂x

∫ η

−hpdz +

∂

∂x

∫ η

−hτxxdz +

∂

∂y

∫ η

−hτyxdz + [p]−h

∂h

∂x+ τFx |∇F | − τBx |∇B|


The previous equation is the result of the momentum balance of a water column of unit widthalong the y-direction. On the left hand side, it is possible to recognize the acceleration termsand the terms due to the momentum flux. On the right hand side, the terms due to the pressureand the shear stress acting on the free surface and the bottom appear.

The temporal average of the left hand side, using continuity equation, leads to

ρ (η + h)

[

∂U

∂t+ U

∂U

∂x+ V

∂U

∂y

]

+ ρ∂

∂x

(∫ η

−huudz

)

+ ρ∂

∂y

(∫ η

−hvudz

)

(9.19)

If we define the modified average pressure [P ]−h

[P ]−h = [p]−h − ρg (η + h) , (9.20)


[p]−h∂h

∂x= [P ]−h

∂h

∂x+

∂

∂x

[

1

2ρg (η + h)2

]

− ρg (η + h)∂η

∂x(9.21)

and

ρ (η + h)

[

∂U

∂t+ U

∂U

∂x+ V

∂U

∂y

]

= [P ]−h∂h

∂x− ρg (η + h)

∂η

∂x(9.22)

+∂

∂x

[

−Sxx +

∫ η

−hτxxdz

]

+∂

∂y

[

−Sxy +

∫ η

−hτxydz

]

+ τFx |∇F | − τBx |∇B|

where the tensor components Sxx, Sxy are introduced

Sxx =

∫ η

−h[p+ ρuu] dz − ρg

2(η + h)2 , Sxy =

∫ η

−hρuvdz (9.23)

Since the quantityρg

2(η + h)2 =

∫ η

−hρg (η − z) dz

is the average value of the hydrostatic pressure, Sxx, Sxy can be written in the form

Sxx =

[

∫ η

−hpdz −

∫ η

−hρ (η − z) dz

]

+

∫ η

−hρuudz, Sxy =

∫ η

−hρuvdz (9.24)

By applying the same procedure to momentum equation (9.8) in the y-direction, it is easyto obtain

ρ (η + h)

[

∂V

∂t+ U

∂V

∂x+ V

∂V

∂y

]

= [P ]−h∂h

∂y− ρg (η + h)

∂η

∂y(9.25)

+∂

∂x

[

−Syx +

∫ η

−hτyxdz

]

+∂

∂y

[

−Syy +

∫ η

−hτyydz

]

+ τFy |∇F | − τBy |∇B|


where

Syx =

∫ η

−hρuvdz, Syy =

[

∫ η

−hpdz −

∫ η

−hρ (η − z) dz

]

+

∫ η

−hρvvdz, (9.26)

For practical applications, it is necessary to evaluate the time and depth averaged values ofthe anisotropic components of the stress tensor, the components of the tensor Sij and the valueof[

P]

−h. Hence, it is unavoidable to introduce some simplifications which require to specifythe order of magnitude of the steady velocity components and the order of magnitude of theamplitude of the velocity oscillations induced by wave propagation.

Let us now analyse the equations, previously introduced, assuming that the slope of thebottom |∇h| is much smaller than the wave steepness ka ≪ 1 which in turn is assumed muchsmaller than one (|∇h| ≪ ka≪ 1)

9.3.1 The shear stress

Assuming aω to be the order of magnitude of the velocity and considering a spatial scale equalto L, the integral of the viscous stresses turns out to be of order

∫ η

−hτijdz ∼ O(µkaωh) (9.27)

(the shear stress components τij which have to be taken into account are only those along thehorizontal axes x and y). Since the components of the tensor Sij are of order O(ρω2a2h), weobtain

∫ η

−h τijdz

Sij= O

(

kν

ωa

)

= O

(

1

Re

)

(9.28)

where Re is the Reynolds number defined with the amplitude of the velocity oscillations and thewavelength of the wave. Actual values of Re are much larger than one and make the viscousstresses negligible. However, even though the viscous stresses are negligible when comparedwith the radiation stresses, the bottom shear stress cannot be neglected.

Since, the bottom slope is small

|∇B| = 1 +O((∇h)2)

and thenτBi |∇B| = τBi +O((∇h)2) (9.29)

9.3.2 The modified pressure

In order to evaluate the modified pressure and to estimate[

P]

−h, it is necessary to considermomentum equation in the vertical direction

ρ

[

∂w

∂t+∂(uw)

∂x+∂(vw)

∂y+∂(ww)

∂z

]

= −∂ (p+ ρgz)

∂z+∂τxz∂x

+∂τyz∂y

+∂τzz∂z

(9.30)


Now, let us integrate (9.30) in the vertical direction and use Leibniz rule

ρ∂

∂t

∫ η

z

wdz−ρ[w]η∂η

∂t+ρ

∂

∂x

∫ η

z

uwdz+ρ∂

∂y

∫ η

z

vwdz−ρ[uw]η∂η

∂x−ρ[vw]η

∂η

∂y+ρ[w2]η−ρ[w2]z =

(9.31)

[p]z − [p]η + ρg(z − η) +∂

∂x

∫ η

z

τzxdz +∂

∂y

∫ η

z

τzydz − [τzx]η∂η

∂x− [τzy]η

∂η

∂y+ [τzz]η − [τzz]z

where the derivative of z with respect to time and the variables x and y vanish. It follows thatthe value [p]z of p at the generic level z is given by

[p]z = ρg (η − z)+ρ

[

∂

∂t

∫ η

z

wdz +∂

∂x

∫ η

z

uwdz +∂

∂y

∫ η

z

vwdz

]

−ρ[

w

(

∂η

∂t+ u

∂η

∂x+ v

∂η

∂y− w

)]

η

(9.32)

−ρ[

w2]

z− ∂

∂x

∫ η

z

τxzdz −∂

∂y

∫ η

z

τyzdz −[

−p + τzz − τxz∂η

∂x− τyz

∂η

∂y

]

η

+ [τzz]z

If we assume that the normal stress, which is equal to [−p + τzz − τxz∂η∂x

− τyz∂η∂y

]η if nonlinearterms in the wave steepness are neglected, vanishes on the free surface and we use the kinematicboundary condition ([∂η

∂t+ u ∂η

∂x+ v ∂η

∂y− w]η = 0), the time average of the previous equation

becomes

[p]z = ρg (η − z)+∂

∂x

∫ η

z

ρuwdz+∂

∂y

∫ η

z

ρvwdz−ρ[

w2]

z− ∂

∂x

∫ η

z

τxzdz−∂

∂y

∫ η

z

τyzdz+[τ zz]z

(9.33)The term due to the viscosity turns out to be of order 1

Rewhen compared with the other terms.

By using continuity equation, it is possible to write

[τ zz]z = µ∂w

∂z∼ O

(

µ∂u

∂x, µ∂v

∂y

)

which is negligible, too. It follows,

[p]z =

[

ρg (η − z) +∂

∂x

∫ η

z

ρuwdz +∂

∂y

∫ η

z

ρvwdz − ρ[

w2]

z

]

+O

(

1

Re

)

(9.34)

which implies that viscosity has no effect on the vertical distribution of [p]z and hence on [p]−h

[p]−h = ρg (η + h) +∂

∂x

∫ η

−hρuwdz +

∂

∂y

∫ η

−hρvwdz +O

(

1

Re

)

(9.35)

Combinining equations (9.35) and (9.20), we obtain

[P ]−h∂h

∂x=

[

∂h

∂x

(

∂

∂x

∫ η

−hρuwdz +

∂

∂y

∫ η

−hρvwdz

)]

+O

(

1

Re

)

(9.36)


At last, let us compare the previous term with the other terms of (9.22). By considering seawaves propagating on infinite depth, it is reasonable to assume that the spatial scale of thevariations of U, V , η, Sxx, Sxy be 1/k and the largest terms among

ρ(η + h)U∂U

∂x, ρ(η + h)V

∂U

∂y, ...., ρg(η + h)

∂η

∂x, ρg(η + h)

∂η

∂y,

∂Sxx∂x

,∂Sxy∂y

be O(khρω2a2) (it is necessary to consider that η, which is generated by nonlinear effects, is oforder (a2k)). It follows that, because of the factor ∇h, the term

[P ]−h∂h

∂x= O

(

|∇h|ρ(ωa)2kh)

(9.37)

can be neglected. Also for long waves, the average quantities change slowly moving along thehorizontal directions and [P ]−h

∂h∂x

can be neglected as well.A similar procedure can be followed to verify that

[P ]−h∂h

∂y= O

(

|∇h|ρ(ωa)2kh)

(9.38)

and this term can be neglected, too.

9.3.3 The dispersive stress tensor and the radiation stress tensor

Let us decompose u, v and η = η − η into three parts: a steady part which depends onthe vertical coordinate but it is characterized by a vanishing depth averaged value and twofluctuating parts due to the waves and turbulence, respectively.

u = us + u′ + u′′ v = vs + v′ + v′′ η = η − η = η′ + η′′

Of course ηs does not exist and η = O(a). If we assume that u′, v′, u′′ and v′′ do not correlate,it follows

u′u′′ = 0 u′v′′ = 0 v′u′′ = 0 v′v′′ = 0 η′η′′ = 0 (9.39)

which should be added to the trivial relationships

usu′ = 0 usu′′ = 0 usv′ = 0 usv′′ = 0 and so on (9.40)

The correlations of the oscillating parts of the velocity components can be approximated by

∫ η

−hρuudz =

∫ η

−hρuudz +O(ρω2a3) =

∫ η

−hρususdz +

∫ η

−hρu′u′dz +

∫ η

−hρu′′u′′dz +O(ρω2a3)

(9.41)∫ η

−hρuvdz =

∫ η

−hρuvdz +O(ρω2a3) =

∫ η

−hρusvsdz +

∫ η

−hρu′v′dz +

∫ η

−hρu′′v′′dz +O(ρω2a3)

(9.42)


and so on. Indeed

∫ η

−hXdz =

∫ η+η

−h

(

X + X)

dz =

∫ η

−hXdz +

∫ η

η

(

X + X)

dz (9.43)

To obtain an explicit formula for the other terms appearing into the definition of the radiationstress components, let us integrate (9.32) in the vertical direction and let us average the resultsover the wave period

∫ η

−hpdz = ρg

(η + h)2

2+ ρ

∫ η

−h

(

∂

∂t

∫ η

z

wdz′)

dz + ρ

∫ η

−h

(

∂

∂x

∫ η

z

uwdz′)

dz (9.44)

+ρ

∫ η

−h

(

∂

∂y

∫ η

z

vwdz′)

dz − ρ

∫ η

−hw2dz

In (9.44) the viscous terms are ignored and use has been made of the kinematic and dynamicboundary conditions on the free surface. Moreover, the first term on the right hand side of(9.44) can be written in the form

ρg(η + h)2

2=ρg

2

[

(η + h)2 + η′2 + η′′2]

(9.45)

The second term on the right hand side of (9.44) vanishes after the use of Leibniz’s rule

∫ η

−h

(

∂

∂t

∫ η

z

wdz′)

dz =∂

∂t

∫ η

−h

(∫ η

z

wdz′)

)

dz − ∂η

∂t

∫ η

z

wdz′ = 0 (9.46)

and because h does not depend on time. The average value of the velocity along the verticaldirection, because of its definition, is

W =1

η + h

∫ η

−hwdz =

1

η + h

(

∫ η

−hwdz +

∫ η

η

wdz

)

(9.47)

By assuming that the horizontal scale of the variations of h, U and V are comparable, both fora strong current

O(U) ∼ O(V ) =√

gh with∇hkh

< O(ka)

and for a weak current

O(U) ∼ O(V ) = ωa with∇hkh

< O(ka)

the value of W turns out to be much smaller than both w′ and w′′ by a factor O(∇h) and afactor O(ka), respectively. Indeed, the first term on the right hand side of (9.47) is proportional


to |∇h|, which is the slope of the bottom, and the second term is proportional to (kA)2, thevertical integral being from η to η. Hence,

∫ η

−hdz

∂

∂x

∫ η

z

uwdz ≃∫ η

−hdz

∂

∂x

∫ η

z

[

u′w′ + u′′w′′]

dz (9.48)

∫ η

−hdz

∂

∂y

∫ η

z

vwdz ≃∫ η

−hdz

∂

∂y

∫ η

z

[

v′w′ + v′′w′′]

dz (9.49)

and

−ρ∫ η

−hw2dz ≃ −ρ

∫ η

−h

[

w′2 + w′′2]

dz (9.50)

with a relative error of O(|∇h|, ka). By substituting (9.45)-(9.50) into (9.44) and into (9.24),it is possible to obtain

Sxx = S(d)xx + S ′

xx + S ′′xx, Syy = S(d)

yy + S ′yy + S ′′

yy (9.51)

Sxy = S(d)xy + S ′

xy + S ′′xy = Syx = S(d)

yx + S ′yx + S ′′

yx

The terms

S(d)xx = ρ

∫ η

−hususdz, S(d)

xy = S(d)yx = ρ

∫ η

−husvsdz, S(d)

yy = ρ

∫ η

−hvsvsdz (9.52)

are called dispersive stresses and describe the excess of momentum flux due to the non uniformdistribution along the vertical direction of the steady velocity components.

The terms

S ′xx = ρ

∫ η

−hu′u′dz + ρ

[

gη′2

2+

∫ η

−hdz

(

∂

∂x

∫ η

z

u′w′dz +∂

∂y

∫ η

z

v′w′dz

)

−∫ η

−hw′2dz

]

(9.53)

S ′xy = S ′

yx = ρ

∫ η

−hu′v′dz (9.54)

S ′yy = ρ

∫ η

−hv′v′dz + ρ

[

gη′2

2+

∫ η

−hdz

(

∂

∂x

∫ η

z

u′w′dz +∂

∂y

∫ η

z

v′w′dz

)

−∫ η

−hw′2dz

]

(9.55)

are the components of the radiation stress tensor. The relationships providing S ′′xx, S

′′xy, S

′′yx, S

′′yy

are equal to (9.53)-(9.55) with the quantities denoted by ′ substituted by the same quantitiesdenoted by ′′.

The components of S ′ij and S ′′

ij are the excess of momentum flux due to the waves and tothe turbulence, respectively. As already pointed out, the former terms were named radiationstress components by Longuet-Higgins & Stewart (1962, 1964) even though the reader shouldnote that they have not the dimensions of a stress. Indeed their dimensions are equal to MT−2.Far from the breaker region, S ′′

ij can be neglected and S ′ij can be evaluated using the linear

theory to described wave propagation.


9.3.4 Summary

Assuming that the wave climate does not change, the equations previously obtained are

∂

∂x

[

U (η + h)]

+∂

∂y

[

V (η + h)]

= 0 (9.56)

U∂U

∂x+V

∂U

∂y= −g ∂η

∂x− 1

ρ (η + h)

[

∂

∂x

(

S(d)xx + S ′

xx + S ′′xx

)

+∂

∂y

(

S(d)xy + S ′

xy + S ′′xy

)

]

− τBxρ(η + h)

(9.57)

U∂V

∂x+V

∂V

∂y= −g∂η

∂y− 1

ρ (η + h)

[

∂

∂x

(

S(d)yx + S ′

yx + S ′′yx

)

+∂

∂y

(

S(d)yy + S ′

yy + S ′′yy

)

]

−τBy

ρ(η + h)(9.58)

9.4 The radiation stress tensor

In many cases of practical interest, the steady current has the same order of magnitude as theamplitude of the velocity oscillations induced by the waves. It follows that the steady currentand the waves do no interact at the leading order of approximation and the linear wave theorycan be still used to determined the dynamics of the waves. It follows that the velocity potentialcan be written in the form

φ = iag

2ωf(z)η(x, y)eiωt + c.c.

where f(z) = cosh[k(z+h)]cosh[kh]

and the free surface assumes the form

η =1

2η(x, y)eiωt + c.c.

where the function η is provided by Helmholtz equation.By considering only terms of second order in the wave steepness, the upper boundary of the

integrals appearing into (9.53) can be substituted by 0. It follows

ρ

∫ 0

−hu′v′dz =

ρg

4

(

∂η

∂x

∂η∗

∂y+ c.c.

)

1

k2

[

1 +2kh

sinh(2kh)

]

(9.59)

1

2ρgη′2 =

1

4ρg|η|2 (9.60)

ρ

∫ 0

−hdz

∂

∂x

∫ 0

z

u′w′dz =ρg

4

∂

∂x

(

η∗∂η

∂x

)

1

2k2[2khcoth(2kh) − 1] (9.61)


ρ

∫ 0

−hdz

∂

∂y

∫ 0

z

v′w′dz =ρg

4

∂

∂y

(

η∗∂η

∂y

)

1

2k2[2khcoth(2kh) − 1] (9.62)

−ρ∫ 0

−hw′2dz =

ρg

4|η|2

(

2kh

sinh(2kh)− 1

)

(9.63)

where η∗ is the complex conjugate of η. The sum of these terms and the use of Helmholtzequation leads to

S ′xy = S ′

yx =ρg

4

[

∂η

∂x

∂η∗

∂y+ c.c.

](

1 +2kh

sinh(2kh)

)

(9.64)

S ′xx =

ρg

4

[

∂η

∂x

∂η∗

∂x+ c.c.

](

1 +2kh

sinh(2kh)

)

(9.65)

+

[

|η|2 2kh

sinh(2kh)+

2khcoth2kh− 1

2k2

(

∣

∣

∣

∣

∂η

∂x

∣

∣

∣

∣

2

+

∣

∣

∣

∣

∂η

∂y

∣

∣

∣

∣

2

− k2|η|2)]

S ′yy =

ρg

4

[

∂η

∂y

∂η∗

∂y+ c.c.

](

1 +2kh

sinh(2kh)

)

(9.66)

+

[

|η|2 2kh

sinh(2kh)+

2khcoth2kh− 1

2k2

(

∣

∣

∣

∣

∂η

∂x

∣

∣

∣

∣

2

+

∣

∣

∣

∣

∂η

∂y

∣

∣

∣

∣

2

− k2|η|2)]

It should be clear that the previous relationships simplify in the following cases.

1 - Infinite water depth

When kh >> 1, ω =√gk

Sxxh

=Syyh

=ρg

4

1

k

(

|∇η|2 − k2|η|2)

(9.67)

Sxyh

=Syxh

= 0 (9.68)

2 - Shallow water depth

When kh << 1, ω =√ghk

Sxx =ρg

2

(

1

k2

∣

∣

∣

∣

∂η

∂x

∣

∣

∣

∣

2

+ 2|η|2)

; Syy =ρg

2

(

1

k2

∣

∣

∣

∣

∂η

∂y

∣

∣

∣

∣

2

+ 2|η|2)

(9.69)

Sxy = Syx =ρg

2

1

k2

(

∂η

∂x

∂η∗

∂y+ c.c.

)

(9.70)

3 - Progressive wave which propagates along a direction forming an angle θ withthe x-axis


In this case we haveη = aeiψ with a real

whereψ = kxx+ kyy, kx = k cos θ, ky = k sin θ, x1 = x

and it is easy to show that the radiation stress components are

S ′xx =

ρga2

4

[

kxkxk2

(

1 +2kh

sinh 2kh

)

+2kh

sinh 2kh

]

(9.71)

=E

2

[

kxkxk2

2cgc

+

(

2cgc

− 1

)]

S ′yy =

ρga2

4

[

kykyk2

(

1 +2kh

sinh 2kh

)

+2kh

sinh 2kh

]

=E

2

[

kykyk2

2cgc

+

(

2cgc

− 1

)]

(9.72)

S ′xy =

ρga2

4

[

kxkyk2

(

1 +2kh

sinh 2kh

)]

=E

2

[

kxkyk2

2cgc

]

(9.73)

More explicitly

S ′xx =

E

2

[

2cgc

cos2 θ +

(

2cgc

− 1

)]

; S ′yy =

E

2

[

2cgc

sin2 θ +

(

2cgc

− 1

)]

(9.74)

S ′xy = S ′

yx = Ecgc

sin θ cos θ (9.75)

When the water depth is much larger than the length of the waves cg/c tends to 1/2 and

S ′xx =

E

2cos2 θ; S ′

yy =E

2sin2 θ; S ′

xy = S ′yx =

E

2sin θ cos θ (9.76)

On the other hand, for shallow water

S ′xx =

E

2

[

2 cos2 θ + 1]

; S ′yy =

E

2

[

2 sin2 θ + 1]

; S ′xy = S ′

yx = E sin θ cos θ (9.77)

9.5 Appendix 1 - Leibniz rule

It is simply to verify Leibniz rule

∂

∂x

∫ b(x)

a(x)

f(z, x)dz =

∫ b(x)

a(x)

∂f

∂xdz + f(b(x), x)

∂b

∂x− f(a(x), x)

∂a

∂x(9.78)

by considering the limit of the difference quotient for ∆x tending to zero

∂

∂x

∫ b(x)

a(x)

f(z, x)dz = lim∆x→0

1

∆x

[

∫ b(x)+∆b

a(x)+∆a

f(x+ ∆x, z)dz −∫ b(x)

a(x)

f(x, z)dz

]

(9.79)


where b(x+ ∆x) = b(x) + ∆b e a(x+ ∆x) = a(x) + ∆a. It follows

∂

∂x

∫ b(x)

a(x)

f(z, x)dz = (9.80)

lim∆x→0

[

∫ b(x)

a(x)

f(x+ ∆x, z) − f(x, z)

∆xdz −

∫ a(x)+∆a

a(x)

f(x+ ∆x, z)

∆xdz +

∫ b(x)+∆b

b(x)

f(x+ ∆x, z)

δxdz

]

By considering that ∆a = ∂a∂x

∆x e ∆b = ∂b∂x

∆x and using the mean value theorem, (9.78)follows.

Capitolo 10

THE BREAKING REGION

10.1 The breaking phenomenon

When the height of a wave increases, a limiting profile of the free surface is attained such thatthe wave breaks. The limiting wave height is named breaking height and is denoted by Hb.When the water depth h is infinite, the breaking height depends on the wave period T . Forintermediate values of the water depth, Hb depends also on h and for shallow water Hb dependsalso on the slope sb of the sea bottom.

The limiting value Hb of H is estimated by means on relationships which are obtained eitherusing Stokes’ criterion that the wave breaks when the horizontal velocity of the fluid at thecrest equals the celerity of the wave or on the basis of experimental observations.

Infinite water depth

When the water depth is much larger that the length of the wave, Michell (1893) proposed

(Hb)∞(Lb)∞

= 0.142 (10.1)

where the breaking wavelength (Lb)∞ turns out to be 20% larger than the wavelength of thelinear Stokes waves

(Lb)∞ = 1.2gT 2

2π(10.2)

By plugging (10.2) into (10.1), it is easy to obtain

(Hb)∞gT 2

= 0.027 (10.3)

Intermediate water depth

146

CAPITOLO 10. THE BREAKING REGION 147

Figura 10.1: A breaking wave

Assuming that the wave breaks when the fluid velocity at the crests equals the wave celerity,Miche (1944) proposed

Hb

Lb= 0.14 tanh

(

2πh

Lb

)

forh

Lb> 0.11 (10.4)

The relationship (10.4) is usually applied assuming that the value of Lb is provided by the linearStokes theory

Lb =gT 2

2πtanh

(

2πh

Lb

)

(10.5)

even though, using this value, the previous relationship does not matches relationship (10.3)for an infinite water depth.

Shallow water(

hL< 1

20; h√

ghT< 1

20; h

gT 2 <1

400

)


In a shallow water region, when h/L is smaller than about 1/20 (h/(gT 2) < 1/400), and fora horizontal bottom, in order to predict the breaking height, it is possible to use the relationshipsuggested by Scarsi & Stura (1980)

Hb

h= 0.727 − 1.12

√

h

gT 2(10.6)

When the sea bottom is characterized by a slope sb smaller than 0.05, Scarsi & Stura (1980)proposed to use the following empirical relationship

Hb

h=[

0.727 + (13sb)2]−

[

1.12 + (30sb)2]

√

h

gT 2for sb < 0.05 (10.7)

When the slope is larger than 0.05, Weggel (1972) suggested

Hb

h=

[

1.561+e−19.5sb

]

1 + 43.75 [1 − e−19.0sb] hgT 2

for sb > 0.05 (10.8)

Vanishing water depth

When the ratio hgT 2 tends to zero, relationships (10.6), (10.7) and (10.8) lead to

Hb

h= 0.727 (10.9)

Hb

h= 0.727 + (13sb)

2 for sb < 0.05 (10.10)

Hb

h=

1.56

1 + e−19.5sbfor sb > 0.05 (10.11)

which are appropriate also for solitary waves.

Breaking types

The breaking of the wave has different characteristics depending on the characteristics ofthe waves and the sea bottom. It is possible to identify four breaking types: i) spilling type(figure 10.2), ii) plunging type (figures 10.1 and 10.3), iii) surging type (figure 10.4), iv)collapsing type. The spilling breaking is characterized by a symmetric wave crest with foamin the direction of wave propagation and it takes place in deep water or when the sea bottomhas a very small slope. The plunging breaking is characterized by an asymmetric crest and bythe fluid particles on the crests which fall forward. Moreover, it is characteristic of mediumbottom slopes and offshore waves which are very steep. The surging breaking is characterizedby an asymmetric profile of the wave and the foam is found in the direction of wave propagationbut there is no jet of water in front of the crest. The surging breaking takes place when the


Figura 10.2: Spilling breaking

Figura 10.3: Plunging type

sea bottom is characterized by a large slope and the offshore waves have a mild steepness.The collapsing breaking has characteristics which fall between those of plunging breaking andsurging breaking. Figure 10.5 shows a sketch of the different breaking types.

There are different criteria to predict the type of breaking, and many of them assume that


Figura 10.4: Surging type

Figura 10.5: Sketches of the different breaking types: a) spilling breaker, b) plunging breaker,c) surging breaker, d) collapsing breaker (image from Wikipedia).

the breaking type depends on the Iribarren number as suggested by Batties (1974)

ζ =sb

√

2π Hb

gT 2

=sb√

Hb

L∞

(10.12)


For ζ < 0.4 the criterion predicts a spilling breaking, for 0.4 < ζ < 2 the predicted breaking isthe plunging breaking and a surging/collapsing breaking takes place for ζ > 2. Other authors(Galvin, 1968) consider the characteristics of the wave in the deep water region and use theparameter

ζ∞ =sb

√

H∞L∞

(10.13)

For ζ∞ < 0.5 the criterion predicts a spilling breaking, for 0.5 < ζ∞ < 3.3 the predicted breakingis of plunging type and for ζ∞ > 3.3 a surging/collapsing breaking should be observed.

10.2 Set-down and set-up

Let us consider a uniform beach such that its profile does not depend on the coordinate yparallel to the coastline and is characterized by a constant slope such that the still water depthh is given by h = sx, x being a horizontal coordinate othogonal to the coastline and pointingoffshore (see figure 10.6). Since ∂/∂y = 0, continuity equation becomes

0

2

4

6

8

10

0 1 2 3 4 5 6 7 8 9

xy

z

Figura 10.6: Sketch of the problem

∂[

U (η + h)]

∂x= 0 (10.14)

and implies that U = 0 for any value of x because of the vanishing of the flow rate at x = 0. Itfollows that also the value of τx vanishes, since τx is linked to the velocity. Therefore, momentum


equation along the x-axis leads to

−g ∂η∂x

− 1

ρ (η + h)

∂(

S(d)xx + S ′

xx + S ′′xx

)

∂x= 0 (10.15)

while momentum equation along the y-axis leads to

− 1

ρ (η + h)

∂(

S(d)xy + S ′

xy + S ′′xy

)

∂x+ τBy

= 0 (10.16)

where τBy is the bottom shear stress due to the fluid and acting on the bottom. These equationsare valid both in the breaking region and in the ’shoaling region’.

The shoaling region

In a large part of the shoaling region, the turbulence is very weak and S ′′xx << S ′

xx. Moreover,η turns out to be of second order in the wave steepness and it is negligible when compared withthe local water depth. Finally, the dispersive stresses can be neglected since their contributionto the momentum balance turns out to be quite small. Hence, equation (10.15) can be rewrittenin the form

−g ∂η∂x

− 1

ρh(x)

∂S ′xx

∂x= 0 (10.17)

where S ′xx can be approximated by

S ′xx =

ρga2

4

[

2cgc

cos2 α+

(

2cgc

− 1

)]

(10.18)

From a qualitatively point of view, S ′xx decreases when h increases, i.e. ∂S ′

xx/∂x < 0. It followsthat ∂η/∂x > 0 and, hence, η increases as x increases. This steady variation of the free surfaceposition is known as set-down, since the mean water level decreases when the wave approachesthe coast and the water depth decreases. Longuet-Higgins & Stewart (1962) integrated (10.17)when the direction of wave propagation is orthogonal to the coastline

η = − ka2

2 sinh(2kh)(10.19)

which, for kh tending to zero, leads to

η = − a2

4h(10.20)

Laboratory experiments supported these theoretical result. Moreover, in the shoaling region,S ′xy does not depend on x and τBy should vanish, thus forcing the vanishing of the longshore

current, too.


The breaking region

In the breaking region, the components of the radiation stress tensor cannot be determinedby means of the relationships obtained on the basis of the linearized monochromatic waveapproach. Hence, it is necessary to use empirical relationships. Bowen (1969), Longuet-Higgins(1970) and Thorton (1970) introduced the assumption that the relationships (9.69)-(9.70) arevalid even in the breaking region if the wave height is linked to the local water depth by meansof

a =c12

(η + h) (10.21)

where the constant c1 assumes values falling in the range (0.7, 1.2). Using these assumptionsand considering the shallow water approximation (cg ∼= c, α ∼= 0), it is possible to obtain

S ′xx =

3

16c21ρg(η + h)2 (10.22)

Moreover, neglecting the term S ′′xx which is linked to the turbulent stresses and without consi-

dering the contribution due to the dispersive stresses (see also Bowen (1969), Longuet-Higgins(1970) and Thorton(1970)) momentum equation along the x direction leads to

−∂η∂x

− 3

8c21∂

∂x(η + h) = 0 (10.23)

It follows

−∂η∂x

(

1 +3

8c21

)

− 3

8c21∂h

∂x= 0 (10.24)

and then∂η

∂x=

−38c21∂h∂x

1 + 38c21

(10.25)

By integrating the previous equation and matching the solution with the value ηb of η at thebreaker line, it is possible to obtain

η − ηb =38c21 (hb − h)

1 + 38c21

(10.26)

where the value of ηb can be evaluated by means of (10.20)(

η = − a2

4h

)

, assuming(

ah

)

b= 1

2c1.

It follows

η + h =h− hs1 + 3

8c21

(10.27)

where

hs = −[(

1 +3

8c21

)

ηb +3

8c21hb

]

(10.28)

is the value of h such that (η + h)s vanishes. A comparison between the model results and theexperimental measurements is shown in figure 10.7


Figura 10.7: Wave set-down and wave set-up: comparison between model results and laboratorymeasurements

10.3 The longshore current

The shoaling region

By using the assumptions discussed in the previous section, momentum equation, along thelongshore direction y and within in the shoaling region, enforces

∂S ′xy

∂x= 0 (10.29)

i.e. it enforces S ′xy to be a constant (note that S ′

xy = ρga2

2

cgc

sinα cosα)Equation (10.29) is verified becausei) sinα

cis constant along a wave ray (Snell law),

ii) the flux of energy between two wave rays is constant and equal to Ecgb, b being thedistance between two wave rays,

iii) the distance between two wave rays is equal to the inverse λ0 of the number of waverays per unit length in the longshore direction multiplied by cosα (see figures 10.8 and 10.9),


it appears that b = λ0 cosα and S ′xy = E cg

csinα cosα does not depend on x

∂

∂x

(

Ecgc

sinα cosα)

=∂

∂x

(

sinα

cEcg

b

λ0

)

= 0

and, hence,∂

∂x(Ecg cosα) =

∂

∂x

(

Ecgb

λ0

)

= 0 (10.30)

csinα

being constant. The relationship (10.30) implies that the energy flux between two waverays keeps constant (the term cosα takes into account the variation of the distance betweenthe two wave rays as the water depth changes).

0

2

4

6

8

10

0 2 4 6 8 10

y

x

wave rays

b(x)λ0

α(x)

Figura 10.8: Sketch of wave rays. The distance b between two wave rays depends on x andturns out to be proportional to cos[α(x)] (b(x) = λ0 cos[α(x)].)

The breaking region

In the breaking region, the water depth can be assumed to be much smaller than the lengthof the waves and cg = c =

√gh. Hence, the radiation stress component S ′

xy is given by

S ′xy = E sinα cosα (10.31)

and the relationship

a =c12

(η + h) (10.32)

leads to

S ′xy =

1

16ρgc21(η + h)2 sin(2α) (10.33)


y

x

λ0

α(x)

λ0

Figura 10.9: Sketch of wave rays. The distance b between two wave rays depends on x andturns out to be proportional to cos(α) (b(x) = λ0 cos[α(x)].)

The term related to S ′xy turns out to be the forcing term which is balanced only by the bottom

shear stress and the Reynolds stresses. The bottom shear stress is usually related to the depth-and time-averaged velocity by

τB =f

2ρ|V + v′|(V + v′) (10.34)

where f is a friction coefficient and V + v′ is the velocity induced by the propagating waves,which is split into a steady steady component (V) and in an oscillatory component (v′) ((V + v′)= (U + u′, V + v′) ). It follows that

τBy =f

2ρ|V + v′|(V + v′) (10.35)

Since, in the surf region, the angle formed by the direction of wave propagation and the x-axisis quite small, we can assume that v′ << u′. Moreover, V has only the y-component. It followsthat

V + v′ ≃ u′i + V j

Hence

τyB =

f

2ρ

√

u′2 + V2V

where it is reasonable to assume u′ = U0 cos[ω(t− t0)], U0 being the amplitude of the velocityoscillations induced by the waves close to the bottom. If the order of magnitude of V is assumedto be equal to that of U0, the quantity

√

U20 cos2[ω(t− t0)] + V

2=

1

T

∫ T

0

√

U20 cos2[ω(t− t0)] + V

2dt (10.36)


=1

T

∫ T

0

√

(V2+ U2

0 ) − U20 sin2[ω(t− t0)]dt

can be evaluated by using the complete elliptical integral of the second kind (Jonsson, Skovgaard& Jacobsen, 1974). However, by assuming that V << U0, it is possible to use the relationship

√

u′2 + V2 ∼ |u′| = U0| cos(ωt)| =

2

πU0 (10.37)

and the bottom shear stress becomes

τyB ∼ 1

πfρU0V (10.38)

Since the value of U0 can be estimated by means of

U0 ∼a

η + h

√

g(η + h) (10.39)

where η + h is the average water depth

a =c12

(η + h)


U0 ∼a

η + h

√

g(η + h) ∼ c12

√

g(η + h) (10.40)

The value of the average water depth can be obtained by

η + h = s′x′ (10.41)

wheres′ =

s

1 + 38c21

and

x′ = x− xs = x− hss

which is the distance in the off-shore direction of the generic location from the mean coastline.It follows

τyB =

1

πfρc12

√

gs′x′V (10.42)

Momentum equation in the y direction (10.17) (neglecting dispersive and turbulent stresses)reads

τBy = −∂S ′

xy

∂x(10.43)


s

x

hs < 0

xs

run-up

Figura 10.10: The value of hs turns out to be equal to −[(

1 + 38c21)

ηb + 38c21hb

]

.

and, using (10.31), it leads to

1

π

f

2ρc1√

gs′x′V = −ρg3/2c218

∂(s′x′)5/2

∂x′

(

sinα

c

)

∞(10.44)

where use is made of the following relationships/approximations already obtained/introduced

η + h = s′x′, x′ = x− xs, cosα ∼= 1,sinα

c=

(

sinα

c

)

∞, a =

c12

(η + h)

The velocity profile which is obtained assuming that the lateral diffusion due to turbulence anddispersive stresses vanishes is not realistic but it gives a qualitative behaviour which allows anestimate of the longshore current to be obtained

V =5π

8

c1s′

ρfgs′x′

(

sinα

c

)

b

for x′ < x′b

V = 0 for x′ > x′b

The velocity profile of the longshore current grows linearly from its vanishing value at thecoastline to its maximum value V max at the breakerline

V max =5π

8

c1s′

ρf

√

g(η + h)b sinαb

The current profile is that shown in figure 10.11 for the vanishing value of the parameter Pdefined by (10.48).


Now, let us introduce a relationship to take into account the effects of the turbulent stressesand the dispersive stresses, the latter being larger than the former because of the shallow waterapproximation.

−S(d)xy − S ′′

xy = (η + h) ρνT∂V

∂x(10.45)

where νT is the eddy viscosity. The eddy viscosity can be assumed equal to a length scale timesa velocity scale.

The former can be assumed to be equal to the distance x′ from the coastline (the mostenergetic vortices in the surf region are vortices with a vertical axis, the length scale of which isproportional to the distance from the shore, which are characterized by a velocity proportionalto Torricelli’s speed) while the latter to be proportional to

√

g(η + h). It follows

−S(d)xy − S ′′

xy = Nρ√gs′3/2x′5/2

dV

dx(10.46)

where N is an empirical coefficient. Then, momentum equation becomes

Nρ√gs′3/2

d

dx′

(

x′5/2dV

dx

)

− 1

π

f

2ρc1

√gs′1/2x′1/2V = (10.47)

= − 5

16g3/2s′5/2c21

(

sinα

c

)

∞x′3/2 for 0 < x′ < x′b

= 0 for x′ > x′b

which can be solved with a numerical approach.At this stage, it is convenient to introduce the dimensionless coordinate x′/x′b and the

dimensionless variable V /V max where

V max =5π

8

c1s′

ρf

√

g(η + h)b sinαb

is the maximum value assumed by the longshore current at the breakerline when the parameter

P = 2πs′N

c1f(10.48)

vanishes, i.e. when the effects of dispersive and turbulent stresses vanish. Equation (10.48)becomes

(

x′

x′b

)2 d2(

V /V max

)

d (x′/x′b)2 +

5

2

(

x′

x′b

)

d(

V /V max

)

d (x′/x′b)−(

V /V max

)

P(10.49)

=(x′/x′b)

Pfor 0 < (x′/x′b) ≤ 1

= 0 for (x′/x′b) > 1


which can be easily solved with a finite difference approach by adding a fictitious temporalderivative of V /V max and integrating (10.49) till a time independent solution in achieved.

The results depend on the parameter P which is linked to the relavance of the lateraldiffusion with respect to the bottom resistance. Figure 10.11 shows the numerical resultsobtained for different values of the parameter P .

0

0.25

0.5

0.75

1

0 0.5 1 1.5 2

V/V

max

x/xb

||

P=0P=0.001P=0.01P=0.1P=1

Figura 10.11: Longshore current obtained for different values of the parameter P .

Capitolo 11

LE ONDE DI MARE

11.1 Introduzione

Le onde di mare non sono ripetitive, ne nel tempo ne nello spazio, come mostrato in figura11.1 dove e riportato l’andamento della superficie libera rilevato durante la fase iniziale diuna mareggiata che ha avuto luogo, fra la fine del 2010 e l’inizio del 2011 (in particolare dal28/12/2009 al 04/01/2010), al largo della costa genovese.

-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140

Wav

e he

ight

[m]

Time [s]

Free Surface

Figura 11.1: Andamento temporale della superficie libera del mare misurato di fronte alla costagenovese il 28/12/2009 durante la fase iniziale di una mareggiata.

Tuttavia, la sovrapposizione di onde periodiche (precedentemente studiate), caratterizzateda diverse frequenze angolari, da luogo a un fenomeno apparentemente simile a quello osserva-bile in natura, naturalmente se l’ampiezza e la fase di ogni componente armonica sono scelte inmodo opportuno. Cio porta all’introduzione dello spettro delle onde che pero e associato non

161

CAPITOLO 11. LE ONDE DI MARE 162

all’ampiezza di ogni singola componente armonica ma alla sua energia (spettro d’energia). Lospettro delle onde ha una forma che dipende dal grado di non linearita del fenomeno e che, perle applicazioni pratiche, puo poi essere opportunamente parametrizzata sulla base di numeroseosservazioni sperimentali (usualmente lo spettro dell’energia ha un solo picco ma in situazioniparticolari e possibile individuare anche un secondo picco).

11.2 Rappresentazione di Fourier di un campo di moto

ondoso

L’elevazione della superficie libera in un posizione fissata puo essere pensata come sovrapposi-zione di molte componenti armoniche (onde lineari) ognuna delle quali puo essere scritta nellaforma

ηn(t) = An cos (ωnt− δn) con n = 1, 2, 3, ... (11.1)

e quindi si ha

η(t) =

∞∑

n=1

An cos (ωnt− δn) (11.2)

Alternativamente e piu convenientemente e possibile scrivere

η(t) =∞∑

n=1

[an cos (ωnt) + bn sin (ωnt)] (11.3)

dove

A2n = a2

n + b2n; tan δn =bnan

(11.4)

Naturalmente, l’espressione (11.3) corrisponde a un segnale osservato in un intervallo finito diosservazione di lunghezza TM . Si assume cioe che

1) l’onda piu lunga abbia un periodo pari a

TM =2π

ω1=

1

f1(11.5)

2) le altre onde abbiano periodi tali che

TM = T1 = 2T2 = ... = nTn o fn = nf1 (11.6)

3) il livello medio si annulli

η(t) =1

TM

∫ TM

0

η(t)dt = 0 (11.7)

e quindia0 = b0 = 0


E dunque possibile scrivere anche

η(t) =

∞∑

n=1

[

an cos

(

2πn

TMt

)

+ bn sin

(

2πn

TMt

)]

(11.8)

che rappresenta una funzione arbitraria η(t) purche periodica di periodo TM . I valori di an, bnpossono essere ottenuti dalla conoscenza di η(t) attraverso le relazioni

an =2

TM

∫ TM

0

η(t) cos

(

2πn

TMt

)

dt (11.9)

bn =2

TM

∫ TM

0

η(t) sin

(

2πn

TMt

)

dt (11.10)

Naturalmente, l’ipotesi che il segnale sia periodico di periodo TM non e realistica. Tuttavia enecessario notare che la conoscenza del segnale fra 0 e TM non ci permette di dire nulla sulcomportamento del segnale per t < 0 e per t > TM avendo ipotizzato il fenomeno stocastico.La conseguenza dell’ipotesi che il comportamento di η(t) sia stocastico e che una qualunquevariazione di TM , anche se arbitrariamente piccola, comporta significative variazioni di an e bn.In altre parole, i valori di an e bn non convergono per TM tendente a ∞.

11.3 Lo spettro d’energia

Tuttavia, per un processo stocastico e possibile definire lo spettro d’energia o lo spettro di poten-za. Per definire lo spettro d’energia consideriamo, per un assegnato valore di TM , l’espansione(11.8). Per questa espansione e facile verificare (teorema di Parseval) che

1

TM

∫ TM

0

η2(t)dt = η2 =1

2

∞∑

n=1

(

a2n + b2n

)

=1

2

∞∑

n=1

A2n (11.11)

Il valore di η2 puo essere poi messo in relazione con l’energia potenziale istantanea per unita disuperficie Ep(t) che e data da

Ep(t) =1

2ρgη2. (11.12)

Infatti un elemento infinitesimo di fluido di spessore dz ha un’energia potenziale pari a ρgzdze dunque

Ep(t) =

∫ η

0

ρgzdz = ρg1

2η2

Per un sufficientemente lungo intervallo temporale TM si ha

Epρg

=1

2η2 (11.13)


dove η2 e la varianza del moto ondoso nel periodo rilevato. Considerato che energia potenziale diuna singola componente armonica Epn e pari a ρgA2

n/4, l’energia totale dell’n-sima componentee 2Epn e quindi

En = 2Epn = ρgη2n (11.14)

E facile verificare cheE

ρg=

∞∑

n=1

η2n =

1

2

∞∑

n=1

(

a2n + b2n

)

=1

2

∞∑

n=1

A2n (11.15)

dove E e l’energia totale per m2. Possiamo quindi definire lo spettro d’energia S(fn) nell’inter-vallo fn − ∆f

2< f < fn + ∆f

2attraverso la relazione

S(fn)∆f =1

2

(

a2n + b2n

)

=1

2A2n (11.16)

con

∆f =1

TM(11.17)

Pertanto il valore di η2, che e definito come la varianza del moto ondoso complessivo, risulta

η2 =1

2

∞∑

n=1

A2n =

∞∑

n=1

S∆f (11.18)

S e uno spettro discreto (vedi figura 11.2) denominato anche sample spectrum o raw spectrum.

Figura 11.2: Esempio di spettro discreto.

Se si valuta lo spettro di energia S per ∆f tendente a zero si osserva la stessa mancanzadi convergenza dello sviluppo di Fourier. Tuttavia lo spettro di energia puo essere definito inmodo univoco con un approccio diverso che e basato sulla valutazione della funzione di auto-covarianza Cηη definita come l’autocorrelazione di η(t) (vedi il paragrafo ’Descrizione statistica


della superficie del mare’) in cui si assume nullo il valor medio di η(t)

Cηη(τ) = limTM→∞

1

TM

∫ TM

0

η(t)η(t+ τ)dt (11.19)

Infatti e stato mostrato (ad esempio da Jenkins & Watts (1968), ’ Spectral analysis and itsapplication’, Holden-Day) che i valori asintotici di Sηη e Cηη sono la trasformata di Fourier unodell’altro

Sηη(f) =

∫ ∞

−∞Cηη(τ)e

−i2πfτdτ (11.20)

Cηη(τ) =

∫ ∞

−∞Sηη(f)ei2πfτdf (11.21)

relazioni che hanno senso in quanto Cηη tende a zero al tendere di |τ | a infinito. Dal puntodi vista computazionale non e conveniente valutare Sηη utilizzando le (11.20)-(11.21), le qualituttavia consentono di definire in modo corretto Sηη attraverso la relazione

Cηη(0) = limTM→∞

1

TM

∫ TM

0

η2(t)dt = η2 =

∫ ∞

−∞Sηηdf (11.22)

Nei casi pratici lo spettro viene valutato calcolando lo spettro raw per un assegnato interval-lo temporale utilizzando la FFT. Diversi intervalli originano diversi spettri che poi, mediati,forniscono una buona approssimazione dello spettro corretto (Jenkins & Watts, 1968)

Si noti che e di impiego usuale lo spettro di frequenza a un solo lobo S(f), definito nelcampo positivo delle frequenze, tale cioe che

∫ ∞

0

S(f)df =

∫ ∞

−∞Sηη(f)df (11.23)

Inoltre si ha

Cηη(0) =

∫ ∞

0

S(f)df = limTM→∞

1

TM

∫ TM

0

η2(t)dt = η2 = σ2η (11.24)

dove σ2η e la cosidetta varianza del moto ondoso (si noti che la varianza non e σ).

11.4 La riproduzione di un’onda dallo spettro d’energia

In questa sezione viene affrontato il problema inverso cioe quello di generare una serie temporaleη(t) partendo da un assegnato spettro. Sulla base della (11.16), se abbiamo un piccolo intervallo∆f centrato intorno a fn, si ha

A2(fn) = 2S(fn)∆f (11.25)

(f solo positiva). Tuttavia lo spettro definisce solo la distribuzione di energia con la frequenza.Il valore di η puo essere ottenuto introducendo le diverse componenti armoniche con ampiezza


deterministica fornita dalla (11.25) e fase aleatoria che non puo essere determinata sulla basedella conoscenza dello spettro

η(t) =

N∑

n=1

√

2S(fn)∆f cos (2πfnt− δn) (11.26)

1 In generale si assume che δn sia una variabile stocastica distribuita uniformemente nell’inter-vallo (0, 2π). Su questo tema si consulti Kirby (2005).

Le formule precedenti affrontano il problema della determinazione della funzione η(t) in unpunto particolare, assegnato uno spettro. Tuttavia, se aggiungiamo l’ipotesi che le componentiindividuali dello spettro sono onde lineari progressive che non interagiscono fra di loro, e allorapossibile sostituire 2πfnt = ωnt con ωnt− knx e ottenere

η(x, t) =N∑

n=1

√

2S(fn)∆f cos (ωnt− knx− δn) (11.27)

dove il valore di kn e fornito dalla relazione di dispersione

ω2n = gkn tanh(knh) (11.28)

E necessario tuttavia notare che, nella realta, le interazioni non lineari fra componenti armonichefanno sı che la (11.27) fornisca una descrizione accettabile solo per brevi distanze o aree limitate.

11.5 Spettri direzionali

Nella realta le onde non hanno altezze costanti nella direzione trasversale ma sono chiaramentetridimensionali (short crested). In termini di componenti, la superficie del mare puo esserepensata come sovrapposizione di onde bidimensionali che si propagano in direzioni diverse. Cioporta a descrivere l’energia introducendo uno spettro direzionale che definisce la distribuzionedell’energia associata al moto ondoso sia nelle frequenze sia nelle direzioni.

Considerato che lo spettro d’energia S(f) non e in grado di distinguere fra le diverse direzionima contiene tutta l’energia della frequenza f , e conveniente distribuire l’energia contenuta inS(f) sulle diverse direzioni moltiplicando S(f) per una funzione distribuzione F (f, αw) chesoddisfa alla seguente relazione

∫ 2π

0

F (f, αw)dαw = 1 (11.29)

Per ogni frequenza abbiamo lo spettro direzionale S(f, αw) dato da

S(f, αw) = S(f)F (f, αw) (11.30)

Dato che la componente spettrale ha un’ampiezza An(f, αw) e un’energia En(f, αw) data da

1

ρgEn(f, αw) =

1

2A2n(f, αw) = 2S(f, αw)∆f (11.31)


si ottiene che l’energia totale nello spettro e data da

η2 =

∫ 2π

0

∫ ∞

0

S(f, αw)dαwdf =

∫ 2π

0

∫ ∞

0

S(f)F (f, αw)dαwdf (11.32)

Per misurare in campo la funzione F e necessario adottare complesse procedure che sono de-scritte in dettaglio in Kirby (2005) e necessitano almeno rilievi direzionali in tre posizionidistinte.

11.6 Parametrizzazione degli spettri

L’analisi di numerosi spettri rilevati in campo mostra che solitamente essi sono unimodalie presentano un massimo per una frequenza fp denominata frequenza di picco. Sono stateproposte diverse forme parametrizzate di spettri, ovvero formule matematiche per descrivereS(f).

Una delle formule piu usate per descrivere uno stato di mare completamente sviluppato,cioe il contenuto energetico massimo che si puo osservare per un’assegnata velocita del vento,e quella di Pierson-Moskowitz (1964)

S(f) = αg2(2π)−4f−5e− 5

4

f4p

f4 (11.33)

dove α e una costante empirica (costante di Phillips) e fp (frequenza di picco) e legata U ,essendo U e la velocita del vento a una quota di 19.5 m sulla superficie del mare. Per marecompletamente sviluppato si assume fp = 0.14 g

Uo anche fp = g

2πU, mentre un valore frequen-

temente usato per α e 8.1 · 10−3. In figura 11.3 e’ mostrato lo spettro di Pierson-Moskowitz(1964) per un valore di U pari a 15 m/s.

Un esempio piu recente e lo spettro JONSWAP (Joint North Sea Wave Project) propostosulla base di rilievi di mare prevalentemente stazionario per il quale il contenuto energeticodipende dal fectch. Lo spettro JONSWAP e ottenuto moltiplicando lo spettro di Pierson-Moskowitz per un fattore di appiccamento e ha la forma

S(f) = αg2(2π)−4f−5e− 5

4

f4p

f4 γr (11.34)

dove r = e−(f/fp−1)2

2σ2 e α, γ, σ sono costanti empiriche che determinano la forma dello spettro.Valori di σ usualmente utilizzati sono

σ = σa = 0.07 se f ≤ fp (11.35)

σ = σb = 0.09 se f > fp

Le altre costanti dipendono dalla situazione. Valori tipici di α sono ottenibili da

α = 0.076(gx

U2

)−0.22

= 0.033

(

fpU

g

)0.667

; fp = 3.50 ∗(

g2

Ux

)13


0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5

S(f

) [m

2 s]

f [1/s]

Figura 11.3: Spettro di Pierson-Moskowitz (U = 15 m/s.)

dove x e l’estensione del fetch, la velocita del vento U e qui valutata a una quota di 10 m eγ (compresa fra 1 e 7) ha un valore medio pari a circa 3.3. Quest’ultimo valore mostra che lospettro JONSWAP e piu ’piccato’ dello spettro di Pierson-Moskowitz.

Esistono poi altre forme di spettro parametrizzate che presentano diverse dipendenze fun-zionali da f descritte per esempio in Young (1999).

11.7 Descrizione statistica della superficie del mare

La superficie del mare e spesso interpretata come un processo stocastico le cui proprieta sono de-terminate a partire da diverse ’realizzazioni’ che costituiscono un insieme η1(t), η2(t), ..., ηn(t).Con riferimento a un certo istante temporale t, e possibile definire il valor medio µη e la funzionedi autocorrelazione Cηη(t, τ) gia introdotta. In termini discreti si ha

µη(t) = limN→∞

1

N

N∑

i=1

ηi(t) (11.36)

Cηη(t, τ) = limN→∞

1

N

N∑

i=1

(ηi(t) − µη(t)) (ηi(t+ τ) − µη(t)) (11.37)

Se le due grandezze cosı ottenute risultano indipendenti dal tempo, il processo si definiscedebolmente stazionario. Se invece l’indipendenza dal tempo e presente per tutte le proprietastatistiche, il processo si definisce stazionario. Infine se si considera una sola realizzazione


e la media e l’autocorrelazione di questa realizzazione non differiscono da quelle di un’altrarealizzazione, il processo e detto anche ergodico.

Nel caso di moto ondoso e usuale stimare le grandezze statistiche da una sola serie di datiassumendo l’ergodicita del processo. Quindi, considerando un intervallo finito di rilevamentoTSM , si introducono il valor medio

µη = η(t) =1

TSM

∫ TSM

0

η(t)dt (11.38)

e la funzione di autocovarianza

Cηη(τ) =1

TSM

∫ TSM

0

[η(t) − µη][η(t+ τ) − µη]dt (11.39)

In particolare Cηη(0) rappresenta la varianza σ2η che indica la dispersione dei dati attorno al

valor medio o il momento del secondo ordine rispetto alla media

σ2η =

1

TSM

∫ TSM

0

[η(t) − µη]2dt (11.40)

Si definiscono inoltre i momenti centrali Mr di ordine r che sono definiti da

Mr =1

TSM

∫ TSM

0

[η(t) − µη]r dt, (11.41)

il coefficiente di skewness γ3, che evidenzia la dissimetria rispetto al valor medio

γ3 =M3

M3/22

(11.42)

e il coefficiente di kurtosis, che indica la larghezza della distribuzione

γ4 =M4

M22

− 3 (11.43)

Si noti che la sottrazione del valore 3 e introdotta in modo tale che γ4 si annulli per unadistribuzione gaussiana.

11.8 Descrizione probabilistica della superficie del mare

La descrizione probabilistica viene fatta assumendo che η(t) sia un processo casuale stazionarioed ergodico, introducendo la funzione densita di probabilita p(η) in modo tale che il valoreatteso o valor medio E[η] sia fornito da

E[η] = µη =

∫ ∞

−∞ηp(η)dη (11.44)


Infatti p(η)dη e la probabilita che η cada fra η e η + dη e dunque∫∞−∞ p(η)dη = 1 cioe il 100

%. Quindi∫∞−∞ ηp(η)dη e la somma del valore di η moltiplicato per il numero di onde con quel

valore di η rapportato al numero di onde totale, cioe la definizione di valore atteso.La funzione densita di probabilita usualmente adottata e la gaussiana

p(η) =1

ση√

2πexp

[

−(η − µη)2

2σ2η

]

(11.45)

che e simmetrica rispetto al valor medio µη, con larghezza dipendente dal parametro ση.Se si considera il processo a media nulla e si adotta la variabile adimensionale η = η

σηla

funzione p(η) assume la forma universale

p(η) =1√2π

exp

[

− η2

2

]

(11.46)

la quale presenta il valore massimo pari a 1/√

2π per η = 0 e due flessi in corrispondenza diη = ±1 (figura 11.4).

0

0.1

0.2

0.3

0.4

0.5

-4 -2 0 2 4

p

η~

Figura 11.4: Forma universale della funzione densita di probabilita

La funzione probabilita totale o cumulata P (η), associata alla funzione densita di probabilitap e anche detta funzione distribuzione e rappresenta la probabilita che lo spostamento verticalesia compreso fra −∞ e η

P (η) =

∫ η

−∞p(η)dη (11.47)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4 -2 0 2 4

P

η~

Figura 11.5: Forma universale della funzione probabilita cumulata

Adottando la distribuzione gaussiana, la funzione cumulata e espressa dalla funzione errore(figura 11.5)

P (η) =1√2π

∫ η

−∞exp

[

− η2

2

]

dη (11.48)

Risulta

P (∞) =

∫ ∞

−∞p(η)dη = 1 (11.49)

La probabilita di superamento Ps(η) e invece definita da Ps = 1 − P .

11.9 Analisi statistica dell’altezza delle onde

Un altro modo di analizzare il moto ondoso e quello di considerare le altezze d’onda e i periodidelle onde irregolari (random) rilevate nel tempo. L’altezza d’onda e usualmente definita comel’altezza d’onda zero up-crossing e corrisponde alla distanza verticale fra il massimo e ilminimo della superficie fra due successivi istanti in cui l’elevazione attraversa lo zero versol’alto.

L’insieme delle altezze d’onda forma una distribuzione discreta che porta alla definizionedi una funzione densita di probabilita p(H) che indica la probabilita che un’onda abbia unassegnato valore. Le misure di campo hanno mostrato che p(H) e prossima a una distribuzionedi Rayleigh

p(H) = 2H

H2rms

e−( HHrms

)2

(11.50)


che e caratterizzata dal solo parametro H2rms = H2, dove Hrms e la root-mean-square wave-

height, cioe la radice del valore medio di H2 (la distribuzione di Rayleigh sarebbe corretta solonel limite di spettro infinitamente stretto).

La probabilita cumulata P (H) (la probabilita che l’altezza d’onda assuma valori minori diH) e quindi

P (H) = 1 − e−( HHrms

)2

(11.51)

Naturalmente la distribuzione delle altezze d’onda (11.50) definisce l’altezza d’onda media He l’altezza d’onda significativa Hs che e l’altezza d’onda media del terzo delle onde piu alte. Epossibile verificare che

H =

√

π

4Hrms

∼= 0.886Hrms

Hs ∼ 1.6H

e inoltre l’altezza d’onda piu probabile Hp e minore dell’altezza media ed e data da

Hp =

√

2

πH = 0.798H

Dalla sola conoscenza della distribuzione di Rayleigh per le altezze d’onda non e possibileottenere informazioni sui periodi zero up-crossing Tz che presentano una diversa distribuzione.

A uno stato di mare descritto nel dominio del tempo sono spesso associate onde caratteri-stiche che sono onde regolari definite a partire dalla media di una frazione 1/c assegnata delleonde piu alte dello stato di mare. Per la loro definizione si adotta la seguente procedura cheparte dall’individuazione di N onde individuali.

1) si attribuisce l’indice corrente i alle altezze delle onde2) si ordinano le altezze d’onda in senso crescente in modo tale che H1 corrisponda all’onda

di altezza piu piccola e HN all’onda di altezza piu alta.3) Si individua l’indice j associato alla prima altezza d’onda Hj da considerare per indivi-

duare le N/c altezze da prendere in esame

j = N

(

1 − 1

c

)

+ 1 (11.52)

4) si calcola l’altezza d’onda caratteristica H1/c valutando la media delle altezze delle ondeconsiderate

H1/c =c

N

N∑

i=j

Hj (11.53)

5) si calcola il periodo corrispondente TH,1/c come media dei periodi delle onde che concor-rono a originare l’altezza d’onda considerata

TH,1/c =c

N

N∑

i=j

TH,j (11.54)


L’onda media e definita dalla (11.53) con c = 1. Usualmente si considerano l’onda un terzo(c = 3) e l’onda massima relativa (c = 10). In particolare, l’onda significativa Hs introdottada Sverdrup & Munk (1943, 1944) come l’onda piu alta e ben formata di uno stato di mare, estata poi ridefinita da Longuet-Higgins come l’onda caratteristica con c = 3

11.10 Lo spettro infinitamente stretto e la funzione in-

viluppo

Talvolta la superficie del mare non presenta ne massimi negativi ne minimi positivi e le oscil-lazioni sono caratterizzate da periodi molto prossimi l’uno all’altro. In questi casi e possibilepensare la superficie libera come un processo a media nulla e varianza σ2

η, associato a uno spet-tro infinitamente stretto che origina un’oscillazione η caratterizzata da una frequenza angolareω0 e da una ampiezza modulata nel tempo. Si ha

η(t) = A(t) cos(ω0t+ δ(t)) = A(t) [cos(ω0t) cos(δ(t)) − sin(ω0t) sin(δ(t))] (11.55)

dove A(t) e δ(t) sono rispettivamente l’ampiezza e la fase, variabili nel tempo. L’andamento diη puo essere sviluppato utilizzando una serie di Fourier tale che

η(t) =

∞∑

n=0

[an cos(ωnt) + bn sin(ωnt)] (11.56)

Postoωnt = (ωn − ω0)t+ ω0t

e possibile esplicitare due distinte sommatorie

η(t) =

[ ∞∑

n=0

an cos[(ωn − ω0)t] + bn sin[(ωn − ω0)t]

]

cos(ω0t) (11.57)

−[ ∞∑

n=0

an sin[(ωn − ω0)t] − bn cos[(ωn − ω0)t]

]

sin(ω0t)

Confrontando quest’ultima equazione con la (11.55) si ottiene

A(t) cos δ(t) =∞∑

n=0

an cos[(ωn − ω0)t] + bn sin[(ωn − ω0)t] (11.58)

A(t) sin δ(t) =

∞∑

n=0

an sin[(ωn − ω0)t] − bn cos[(ωn − ω0)t] (11.59)


I termini a primo membro definiscono le grandezze ηC(t) e ηS(t)

ηC(t) = A(t) cos δ(t); ηS(t) = A(t) sin δ(t) (11.60)

che risultano grandezze aleatorie, statisticamente indipendenti, distribuite secondo una gaus-siana con media nulla e varianza σ2

η pari a quella del processo originario. La funzione A(t) edenominata funzione inviluppo in quanto si puo individuare una coppia di linee simmetricherispetto allo zero e definite da +A(t) e −A(t) che congiungono rispettivamente tutti i cavi etutte le creste. Si osserva che

A2(t) = η2C(t) + η2

S(t) (11.61)

E possibile mostrare che

p(H) =H

4σ2η

exp

(

−H2

8σ2η

)

(11.62)

(distribuzione di Rayleigh). Se quindi si accetta l’ipotesi di spettro infinitamente stretto, lealtezze d’onda sono distribuite secondo la distribuzione di Rayleigh e, introducendo l’altezzaadimensionale H = H

ση, tale che dH = σηdH, si ha

p(H) =H

4exp

[

−H2

8

]

(11.63)

P (H) = 1 − exp

[

−H2

8

]

(11.64)

E interessante osservare che

Ps(H∗) = 1 − P (H∗) =

∫ ∞

H∗

H

4exp

[

−H2

8

]

dH =1

3(11.65)

e verificata per H∗ = 2.97 e conduce a

H1/3 =

∫∞2.97

H2

4exp

[

− H2

8

]

dH

∫∞2.97

H4

exp[

− H2

8

]

dH= 4.0083 (11.66)

11.11 Parametri delle onde e caratteristiche dello spet-

tro

Sotto certe ipotesi semplificative i parametri caratterizzanti le onde possono essere legati diret-tamente alle caratteristiche dello spettro. A tal fine introduciamo i momenti mn di ordine ndello spettro definiti da

mn =

∫ ∞

0

fnS(f)df (11.67)


Il momento di ordine 0 e

m0 =

∫ ∞

0

S(f)df = η2 (11.68)

Un importante parametro e la larghezza dello spettro ǫ definita da

ǫ2 = 1 − m22

m0m4

(11.69)

E stato mostrato da Cartwright & Longuet-Higgins (1956) che per ǫ = 0, le ampiezze delleonde sono distribuite secondo la distribuzione di Rayleigh. Inoltre se si assume che H = 2aanche le altezze d’onda sono distribuite secondo la distribuzione di Rayleigh. Il valore nullo di ǫcorrisponde inoltre alla situazione in cui il periodo T e costante. Questo non e certamente il casonella realta. Tuttavia le misure mostrano che la distribuzione delle altezze d’onda si avvicinamolto a una distribuzione di Rayleigh e quindi assumere ǫ = 0 risulta comunque significativo.In questo caso

H2 = 8m0 = 8η2 = 8σ2η (11.70)

e quindi la distribuzione di Rayleigh per le altezze d’onda puo anche essere scritta nella forma

p(H) =H

4m0e− H2

8m0 (11.71)

e risulta

Hs =√m0

[√8 ln 3 + 3

√2π(

1 − erf(√

ln 3))]

= 4.0083√m0

∼= 4√m0 = Hm0

(si noti nella relazione precedente la definizione di Hm0 che risulta fornita da

Hm0 = 4√m0 = 4

√

∫ ∞

0

S(f)df (11.72)

ed e denominata altezza d’onda spettrale)Nel caso piu generale per cui ǫ 6= 0 si puo utilizzare la distribuzione di Rice. In questo caso

e possibile vedere cheHs = 4

(

1 − 0.092ǫ2)√

m0

11.12 L’onda piu alta di una stato di mare

La determinazione dell’altezza dell’onda piu alta di uno stato di mare Hmax non puo essereottenuta in maniera deterministica perche tale valore tende a crescere con il numero di ondeconsiderate. Hmax e quindi un valore aleatorio che puo essere descritto da una funzione densitadi probabilita.


Con riferimento all’altezza d’onda Hmax, adimensionalizzata con la deviazione standard σ diuno stato di mare, la funzione densita di probabilita g(Hmax) puo essere determinata a partiredalla relativa funzione cumulata G(Hmax), la quale a sua volta dipende dalla funzione cumulatadelle altezze d’onda P (H) e della numerosita del campione. Risulta

G(Hmax) =

[

P (H)]N

H=Hmax

(11.73)

da cui

g(Hmax) = N

p(H)[

P (H)]N−1

H=Hmax

(11.74)

Figura 11.6:

La funzione g(Hmax), rappresentata indicativamente in figura 11.6 insieme alla funzionep(H), al crescere del numero N di onde considerate si sposta verso valori piu elevati di Hmax.Adottando per le altezze d’onda la distribuzione di Rayleigh, il valor medio di Hmax risultamaggiore del valore piu probabile in quanto la distribuzione presenta un coefficiente di skewnessnegativo. A esempio per N = 300 il valore piu probabile risulta 6.75 mentre il valor mediorisulta pari a 7.09. Per valori molto grandi di N il valore piu probabile puo essere espresso dalla

(

Hmax

)

valorepiu′probabile=

√8 lnN (11.75)


Usualmente per valutare il valore medio si adotta una relazione approssimata, valida per unvalore realistico di onde, N > 100,

(

Hmax

)

valoremedio=

√8

[√lnN +

1

2

γ√lnN

]

(11.76)

dove γ = 0.5772 e la costante di Euler. Un valore orientativo adottato nella pratica, indipen-dentemente dal valore di N e

(

Hmax

)

valoremedio∼ 1.8Hm0 = 7.2 (11.77)

che corrisponde all’altezza dell’onda caratteristica con c = 250. Al fine di scegliere l’ondamassima di progetto si puo definire l’altezza massima che ha un’assegnata probabilita α diessere superata, espressa dalla

(

Hmax

)

valorediprogetto=

√

√

√

√8 ln

[

N

ln(

11−α)

]

(11.78)

11.13 Onde anomale/freak (rogue) waves

La natura statistica delle altezze d’onda implica che teoricamente non c’e un limite superiorea quanto alta puo essere una singola onda, essendo fissato il valore di H . Dato che ancheuna singola onda puo produrre danni devastanti a una struttura (a esempio una piattaforma’offshore’), si e sviluppata l’analisi delle onde anomale. Le misure di campo hanno mostrato chepossono esistere onde con altezze fino a 3H e anche maggiori. Il meccanismo che causa questeonde non e del tutto chiaro anche se si pensa che esse siano causate da effetti di focalizzazionedell’energia che spiegherebbero perche esse sembrano apparire dal nulla ed esistono solo perun brevissimo tempo. La figura 11.7 mostra la foto di un’onda anomala, foto la cui storiaraccontata di Paul Williams e qui riportata:

About 45 years ago, my father related a story to me about this photo that was taken byhim, with a Box Brownie, from the top deck stern of the troop ship Queen Mary during WorldWar 2, sometime in 1940. From memory, I think it was the HMAS Vidal, a mine sweeperescorting the Queen Mary from England to the Middle East. Knowing the height of the QueenMary (very high) which is sitting in the wave’s trough, and the angle of the shot (about 45from the deck of the Mary), you get a fair idea of the size of the wave that this little shipis sitting on. Apparently, this rogue wave came up so quickly that everyone was caught out,including the Vidal’s crew. It went up like an express elevator I remember my father tellingme of the white faces staring down at him when he took the photo. He could see them evenfrom that distance. All of the crew on his area of the deck were also grabbing onto anythingsolid. He realised this when he looked around to see that he was fully exposed on the deckand now completely alone. It did happen that both the Vidal and the Mary came through


unscathed but, for a few seconds, no one believed that they would. My father also told me thathe was later informed that the Vidal’s crew had never before seen the helmsman move so fast,spinning the helm to surf down the wave rather than try to run along the top and risk fallingoff. Unfortunately, my father could not wind the film on fast enough to catch what he saidwould have been the war’s most spectacular surfing photograph. When the Vidal bottomedout into the trough, the wave collapsed in on itself, rather than breaking, and the sea just wentalmost flat. When the two ships’ crews started breathing again, they said they were as amazedby the wave’s ending as they were by the wave itself. I am surprised at the residual qualityof this photo, considering that it has survived years of war in a soldier’s pack in the MiddleEast and New Guinea, followed by many more years of neglect, sitting forgotten with lots ofothers in a wooden box, before being rediscovered by my father and then sticky taped intoan album. I re-rediscovered them about five years ago and started scanning them in order to,at least, retain the images. My father took many hundreds of photos during the war. As abackground task, my wife and I are still scanning them with the intention of sending a DVD tothe War Memorial in Canberra. We are about half way through. It is a slow job, as they arenow incredibly fragile, as is the album that they are in. Removing the photos from the albumto scan them and then placing them back is a worrying thing to do. Old sticky tape and veryold photos just don’t like being separated.

Figura 11.7: HMAS Vidal sitting on a very high (freak wave) in 1940.

11.14 Appendix

The design of coastal works, offshore activities and operations related to navigation requires adetailed knowledge of the local wave climate. In Italy, the complex geographical configuration of


the coast leads to a wave climate which is extremely variable from site to site and the acquisitionof accurate and detailed data to locally forecast and hindcast wave climate is necessary.

Figura 11.8: Locations of the buoys of the Italian National Sea Wave Measurement Network.

To collect data representative for different sea areas, the Italian National Sea Wave Measu-rement Network (RON) has been set up by the Italian Ministry of Public Work and is currentlyrun by ISPRA (Istituto Superiore per la Ricerca Ambientale) which carries out an extensive ac-tivity of preparation, validation and retrieval of raw data from the RON stations and performsprimary analyses in order to create a reliable database for statistical analysis.

In 2004, some of the results were published in an Italian Wave Atlas. The Atlas contains1) a map of the location of the buoys (see figure 11.8 where the buoy of La Spezia is

highlighted),2) the diagrams of the related geographical fetches (see figure 11.9 where the fetches for La

Spezia site are shown),3) the characteristics of the most extreme recorded sea storms,4) the cumulative annual duration curve of significant wave height (see figure 11.10 for the

data at La Spezia site),5) the curve of the mean persistence of significant wave height overthreshold during storms

(see figure 11.11 for the data at La Spezia site),6) the mean directional annual and seasonal wave climate (see the annual wave climate at

La Spezia site in figure 11.12 for the spectral characterisitcs measured by La Spezia buoy ),


Figura 11.9: Diagram of the geographical fetches of La Spezia buoy.

Figura 11.10: Cumulative annual duration curve of significant wave height at La Spezia site.

7) the mean frequency spectrum shape with JONSWAP parametrization (see figure 11.13 forthe data at La Spezia site),

8) statistical analysis of extreme events both directional and omnidirectional (see figure


Figura 11.11: Mean persistence of significant wave height over threshold during storms at LaSpezia site.

11.14).It is interesting to clarify a few points of the results which appear in the Atlas.

Geographical and effective fetches

The geographical fetch is defined as the length of the area of the sea where the waves aregenerated. In limited sea basins, the geographical fetches are defined by pointing the upwindcoast along the mean wind direction. Moreover, it should be pointed out that, because of the sizeof typical storms in the Italian seas, the maximum length of the geographical fetch turns out tobe 500 km (see figure 11.9 for notations and the reference scheme). Moreover, it is commonto distinguish the fetches into primary and secondary sectors (settori di traversia primari esecondari. Il settore di traversia e dato dall’angolo comprendente le direzioni da cui possonogiungere i moti ondosi prodotti da venti foranei. Il settore dei moti ondosi piu violenti sidefinisce primario o principale) depending on the extension of areas potentially subjected towind action.

On the other hand, the effective fetch is defined as the portion of the sea area under the directaction of the wind, where waves are generated by taking into account also the lateral width ofthe area. Different approaches can be used to compute the value of the effective fetch. Usuallythe approach of Sverdrup, Munk & Bretcheneider (1947) (SMB approach), later updated bySaville (1954), Seymour (1977) and by the S.P.M. (1984), is employed. In the SMB method,


Figura 11.12: Mean directional annual wave climate at La Spezia site (let us point out that thepercentage of waves with a height larger than 3 m along the 17th sector is about 1%)

the length of the effective fetch Feff (θj), which is associated to the mean wind direction (θj),is computed from the mean geographical fetches along the i−sector averaged on an angle of 90

(Saville method) or 180 (Seymour method) centered on the wind direction. According to thelatter approach

Feff(θj) =

∑j+90i=j−90 Fi cosn+1 (θi − θj)∑j+90

i=j−90 cosn (θi − θj)(11.79)


Figura 11.13: JONSWAP parameterization of the mean frequency spectrum at La Spezia site.

where the value of n is usually fixed equal to 2 and the number of geographical fetches consideredin an angle of ±90 is usually 12.

Wave climates

Usually, to evaluate the wave climates, the joint-occurance frequency of wave height/direction(see figure 11.12) and the mean exceedance persistence of significant wave height above assignedthreshold are considered (see figure 11.11). Occurance frequencies can be calculated by dividingthe interval of all measurements into different classes. The frequency of each class is the ratiobetween the number of events falling within the considered class and the total number of data.

Similarly, exceedance frequencies can be calculated by dividing the interval of all measuredvalues into different classes. The frequency of each class is the ratio between the number ofevents above the minimum threshold and the total number of measured events.

To calculate the joint-occurance frequency of wave height/direction, sea states with signifi-cant wave heights below 0.50 m are defined as calm and are not considered. The interval ofmeasured wave height values has been divided into four classes (0.5 m ≤ Hm0 < 1.0 m; 1.0 m≤ Hm0 < 2.0 m; 2.0 m ≤ Hm0 < 3.0 m; 3.0 m ≤ Hm0), while each direction sector covers anangle of 15. The information is graphically represented with polar diagrams (see figure 11.12).

In order to calculate the wave height over-threshold persistence, the interval of measuredvalues is divided into classes each of 0.25 m. The information is graphically represented withhistograms which allow an immediate calculation of the mean annual duration of sea states abovea fixed wave height over-threshold. Moreover, the joint-occurrence of wave height/direction


Figura 11.14: Statistical analysis of omnidirectional extreme events at La Spezia site. Thefigure shows that about 13% of the waves come from a direction which forms an angle equal to225. Of these waves, a percentage slightly smaller than 1% has a spectral height Hm0 largerthan 3 m, a percentage of about 2% has a value of Hm0 falling between 2 and 3 m and so on.Moreover, a percentage around 50% is characterized by values of Hm0 smaller than 0.5 m (calmsea).

was computed on a quarterly/seasonal for winter (january, february, march), spring (april,may, june), summer (july, august, september) and autumn (october, november, december) andrepresented in frequency polar diagrams.

(E comune rilevare la superficie libera per 10-20 minuti con frequenza di circa 1 Hertz (nellarete ROM la frequenza e 1.28 Hertz), trasmettendo i dati relativi allo spettro rilevato ogni 3ore. Solo quando l’altezza d’onda significativa supera un pre-determinato valore soglia, i datimisurati vengono trasmessi, insieme all’andamento temporale della superficie libera, ogni 30minuti. )

Correlation between wave parameters

By analysing the wave records with the zero-upcrossing method, information on the corre-


lations between the statistical parameters of the wave spectra are obtained. In particular, thefollowing quantities are evaluated as function of the significant wave height Hm0 = Hs:

1) mean wave height Hm

2) one-tenth wave height H1/10

3) maximum wave height Hmax

4) mean wave period Tm5) significant wave period Ts6) spectral peak period Tp

Figura 11.15: Correlation between wave parameters.

The correlations were made by means of linear and nonlinear regression. A linear correlation(Hi = aHm0) was used for the wave height. For the La Spezia buoy, it turns out that the differentwave heights are correlated to Hm0 by

Hmax = 1.559Hm0,H1/10 = 1.195Hm0,Hm = 0.602Hm0,For the wave periods, the expression Ti = bHc

m0 was used, and for La Spezia site, thefollowing values of b, c were obtained

b = 6.604, c = 0.255 for Tp


b = 4.919, c = 0.348 for Tsb = 4.511, c = 0.324 for Tm(see figure 11.15)

Persistence of a sea storm over-threshold

To estimate the mean persistence of sea states above a fixed wave height, the followingapproach is used (Mathisen, 1993). First, the mean over-threshold duration is calculated byestimating the non-exceedance probability distribution function of significant wave height thre-shold (F (H)) and the ratio between Hm0 and the time variation rate of Hm0 (trends). Sea stormmean duration estimates over a given threshold are obtained only for the main direction sectorassuming that the trend series are stationary.

The mean duration τ over a given threshold H is given by

τ(H) =2F (H)

f(H)S(H)(11.80)

where F (H) is calculated by using the method for partial over-threshold duration series (POT)which is described in the paragraph on exterme waves, f(H) is the relative probability densityfunction and S(H) is the absolute trend expressed as the height Hm0 to the power r

S(H) = qHrm0 (11.81)

The coefficients q and r are determined by adapting (11.81) to the Hm0 data calculated with thefollowing relationship

Si(H) =∆Hi

∆ti(11.82)

where ∆Hi is defined as

∆Hi = Hm0,i+1 −Hm0,i with Hi =Hm0,i+1 +Hm0,i

2(11.83)

Index i varying between 1 and N − 1 and N being the total number of data. The value of ∆ti ifthe time interval of sea states characterized by H larger or equal to Hm0i minus the time intervalof sea states characterized by H larger or equal to Hm0i+1 (see figure 11.16). The least squaremethod is applied to find the values of q and r by dividing the data in N classes charaterizedby an interval of the height of 0.2 m.

It is observed that the duration τ obtained by means of the above procedure can be expressedas a parametrical function of the significant wave height in the form

τ(H) =

(

Hm0

ξ

)ψ

(11.84)

where ξ and ψ are calculated for each measuring station.


H

t

Hm0,i+1

ti+1

Hm0,i

ti

∆ti

∆Hi

Figura 11.16: Sketch of the approach used to compute Si(H).

For La Spezia, the following values are obtained: ψ = −1.72, ξ = 27.7. However, thesevalues appears somewhat questionable.

Wave frequency spectra

As known from previous paragraphs, the wave spectra are usually represented by simpleformula, the most used of which is the JONSWAP spectrum

SJ(f) = SPMγΘ(f) (11.85)

where 1) f is the frequency of the general component which gives a contribution to the energyequal to ρgS(f)df , 2) γ is the spectral peak enhancement parameter, 3) SPM is the spectrumproposed by Pierson & Moskovitz (1964) and 4) Θ(f) is a function having its maximum incorrespondence with the spectrum peak frequency

SPM(f) =αg2

(2π)4f 5exp

[

−5

4

(

fpf

)4]

(11.86)

Θ(f) = exp

[

− 1

2σ2

(

f − fpfp

)2]

(11.87)

where α is the Phillips constant (also called saturation or equailibrium parameter), fp is thepeak frequency and σ is equal to σb when f < fp and to σa for f > fp.


It is worth pointing out the possible presence of cross sea states which are generated bythe simultaneous presence by two wave systems usually originated by short wind waves andswell coming from areas far away. Fitting a JONSWAP model to an observed spectrum duringcross-sea states can originate serious mistakes.

For La Spezia the procedure leads to γ = 2.02, σb = 0.073, σa = 0.096

Statistical analysis of extreme waves

The risk that is associated to a design wave is generally associated to the return period Trof sea states and the occurence probability during the work’s lifetime (encounter probability).

The procedure to forecast the wave heights in a given time interval has the following steps:1) Selection of homogenous independent data from the available data2) Identification of the probability method that best represents the selected data3) Determination of wave heights within a given return period on the basis of the chosen

probability model4) Calculation of the confidence interval associated to the expected value.

Selection of homogenous independent data from the available data

A time serie of the free surface elevations measured by a buoy at an establish rate is re-presentative of the local meteo-marine conditions. Of course, the rate of measurement, calledseries aggregation scale, affects the accuracy with which the phenomenon is known. Increasingvalues of the aggregation scale lead to lower accuracy. The aggregation scale affects also theindependence of the data of the series. Indeed, the smaller is the aggregation scale the higheris the correlation degree.

It is important that the observed time series data form a homogeneous and independent setof samples. While, for wave data, it is relatively simple to assure the independency of the data,it is more difficult to assure that they are homogeneous since sea states might have a differentorign.

Some authors consider sea storms as the time series of sea states characterized by i) waveheight persistence over the threshold of 1.0 m for more than 12 consecutive hours, i) wave heightdecays below the threshold for less than 6 consecutive hours (in other words, it is accepted thatduring a small time interval the significant wave hirght of a storm becomes smaller than thethreshold value), iii) original direction belonging to a determined angular sector (±30 withrespect to a fixed direction) (figure 11.17 shows the values of Hs, Tm and the direction of theincoming waves during a sea storm in front of La Spezia city)

Identification of the probability method that best represents the selected data

The most common distributions in the analysis of extreme waves heights are of type I(Gumbel), type II (Fretchet) and type III (Weibull) and are given below

Gumbel distribution


0

1

2

3

4

5

6

7

8H

s [m

]

BuoyWW III

0

2

4

6

8

10

12

Tm

[s]

0

50

100

150

200

250

300

350

24/02 25/02 26/02 27/02 28/02 01/03 02/03 03/03 04/03 05/03 06/03

Dir

°

Figura 11.17: a) significant wave height, b) mean wave period, c) direction of wave propagationduring a storm recorded at La Spezia.

F (Hi ≤ H) = exp

[

− exp

(

−H − B

A

)]

with − x < H < x, −x < B < x, 0 < A < x

(11.88)

Fretchet distribution

F (Hi ≤ H) = exp

[

−(

H

A

)−k]

with 0 < H < x, 0 < k < x, 0 < A < x (11.89)


Weibull distribution

F (Hi ≤ H) = 1− exp

[

−(

H − B

A

)k]

with B < H < x, 0 < k < x, 0 < A < x (11.90)

where F (H) is the cumulated probability of not exceedence above the threshold H, namely theprobability that H is not exceeded by a random chosen value Hi (F (H) = F (Hi ≤ H)). Ofcourse the value of the function F is controlled by the scale factor A, the position factor B andthe shape factor k. To find the values of the paramters above, fitting methods are employed themost preferred being the least squares method. The procedure implies the issue of attributing anon-exceedance probability F to each element H of the selected sample and the issue to chosethe shape parameter k if the Weibull distribution is chosen. The latter problem is solved byfixing tentative values of k and then selecting the most appropriate one. To chose the value ofF to be attributed to H, the following procedure should be used

1) the wave height sample is ordered in decreasing order and H1 and HN are the maximumand minimum recorded values.

2) the non-exceedance frequency is computed for each element by using

Fm = 1 − m− α

N + β(11.91)

where α and β are constants which depend on the chosen distribution: for Gumbel distribution

α = 0.44, β = 0.12 (11.92)

for Weibull distribution

α = 0.20 +0.27√k, β = 0.20 +

0.23√k

(11.93)

for Fretchet distributionα = 0.0, β = 1.0 (11.94)

(a tentative value of k should be used).Then, a best fitting procedure should be used to select the values of the parameters of the

distribution. It is worth pointing out that the best fitting procedure provides also the correlationcoefficient r

r =

∑Ni=1 (yi − y) (xi − x)

√

(

∑Ni=1 (yi − y)2

)(

∑Ni=1 (xi − x)2

)

(11.95)

As discussed by Goda & Kobune (1990), the best choice is that characterized by the minimumvalue of

∆r

∆rm


Distribution Coefficient ǫ Coefficient δ Coefficient χ

Fretchet (k = 2.5) -2.470 + 0.015 (N/NT )3/2 -0.1530 -0.0052 (N/NT )5/2 0.0

Fretchet (k = 3.3) -2.462 - 0.009 (N/NT )2 -0.1933 - 0.0037 (N/NT )5/2 -0.007

Fretchet (k = 5.0) -2.463 -0.2110 - 0.0131 (N/NT )5/2 -0.019

Fretchet (k = 10.) -2.437 + 0.028 (N/NT )5/2 -0.2280 - 0.0300 (N/NT )5/2 -0.033

Gumbel -2.364 + 0.054 (N/NT )5/2 -0.2665 - 0.0457 (N/NT )5/2 -0.044

Weibull (k = 0.75) -2.435 -0.168 (N/NT )1/2 -0.2083 + 0.1074 (N/NT )1/2 -0.047Weibull (k = 1.0) -2.355 -0.2612 -0.043

Weibull (k = 1.4) -2.277 +0.056 (N/NT )1/2 -0.3169 - 0.0499 (N/NT ) -0.044Weibull (k = 2.0) -2.160 +0.133 (N/NT ) -0.3788 - 0.0979 (N/NT ) -0.041

Tabella 11.1: Coefficients of probability distribution

where ∆r is the complement of r to 1 and ∆rm is its mean value provided by

∆rm = exp[

ǫ+ δ lnN + χ (lnN)2]

where the constants ǫ, δ, χ depend on adopted distribution and are given in table 11.14For all the direction of the La Spezia site, it turns out that A = 1.03, B = 3.88, k = 1.40, c0 =

2.22, c4 = 0.40, c5 = 0.72λ = 4.26, σH = 0.67, n = 45, HT = 4.0, H1 = 4.6, Hmax = 7.1, H10 =6.6, H50 = 7.80, β50 = 0.7, γ50 = 1.18

Maximum wave heights within a given time interval

The final aim of analysing extreme events is to calculate the maximum wave height withina given time interval (to fix the useful lifetime of a maritime work reference can be made toCNR-GNCDI 1996: Technical instruction for the design of breakwaters).

The concept of the period TR is introduced, which is defined as the mean number of yearswithin which the general H value is not exceeded. In actual fact, the return period is subjectto statistical variability and the probability that an event with a given return period TR occursduring this interval is 63%. Therefore, it is more correct to define a probability level for theprediction which can be fixed by means of a cost-benefit analysis which is aimed at quantifyingthe tolarable risk for the service life τ of the structure.

The relation between the tolerable risk and the return period is supplied by the encouterprobability PI , i.e. the probability that an event with a given frequency occurs during the courseof τ years. The value of Pi can be obtained as follows.

The annual exceedance frequency of the threshold HTR(wave height value with return period

TR) is equal to 1year/TR. The corresponding non-exceedance annual probability is given by(1− 1/TR). Therefore, assuming that the occurrence of an event does not influence subsequentones, the probability of not exceeding the threshold HTR

in τ years is calculated as the product


of single annual frequancies (composed propability theorem).

PI (HTR) = 1 −

(

1 − 1year

TR

)τ

(11.96)

Once the value of the acceptable encounter probability is fixed and the return period is evaluatedby means of (11.96), the corresponding design wave height HTR

can be calculated.The probability that, in occurrence of a sea storm, the wave height exceeds the highest peak

HTRis given by 1year/TR. On the other hand, the annual probability that a sea storm with

a peak height exceeding the truncation threshold occurs is equivalent to 1/λ, λ being the meannumber of over-threshold sea storms observed in one year. Therefore, assuming an independencebetween the number of storms in one year and the peak values, the probability that another seastorm with a peak wave height H > HTR

occurs is equivalent to the product of probabilities ofthe two events

F (H > HTR) =

1year

TR

1

λ(11.97)

and the non-exceedance probability of the threshold HTRis

F (H < HTR) = 1 − 1year

TR

1

λ(11.98)

Once, the disaggregated sample probability and the probability of the aggregated time seriesare related, the wave height characterized by a given return period can be computed. The proba-bility is used to calculate the reduced variable value and, from this, HTR

can be evaluated. Forexample, for the Weibull distribution

XT =

[

− ln

(

1year

λTR

)]1/k

(11.99)

HTR= A

[

− ln

(

1year

λTR

)]1/k

+B (11.100)

It must be pointed out that the design values related to a return period must be interpreted asa statistical parameters the value of which is affected by an uncertainty which can be estimated(Goda, 1988).

Capitolo 12

LA PREVISIONE DEL CLIMAONDOSO

12.1 Premessa

I modelli di previsione del clima ondoso sono basati su metodi indiretti, che individuano le ca-ratteristiche delle onde attese a partire dai campi di vento, e su metodi diretti, che determinanotali caratteristiche a partire da rilievi effettuati in campo.

12.2 I metodi indiretti

Come gia accennato, i metodi indiretti permettono la predizione degli stati di mare partendodalla conoscenza i) dell’estensione della regione sul quale agisce il vento (fetch) indicata conx, ii) della velocita del vento U e iii) della sua durata t. Chiaramente anche la profondita hinfluisce sul clima ondoso ma nel seguito verra considerato solo il caso di profondita infinitaanche se le formule che seguono possono essere adottate anche su profondita finita a patto ditener conto dei limiti imposti dal frangimento.

Il metodo dell’onda spettrale

Il modello di previsione descritto nello Shore Protection Manual (1975) richiede che, perpoter ipotizzare il regime di generazione stazionario, siano soddisfatte le relazioni

gt

U10≥ gtcrU10

= 21.88

(

gx

U210

)0.797

(12.1)

gx

U210

<gxcrU2

10

= C1 (12.2)

dove U10 e la velocita del vento alla quota di riferimento pari a 10 m e il valore della costante C1

e specificato nel seguito. Se sono verificate le condizioni (12.1)-(12.2), i parametri adimensio-

193

CAPITOLO 12. LA PREVISIONE DEL CLIMA ONDOSO 194

nali che permettono di quantificare l’altezza d’onda spettrale (Hm0)0 e il suo periodo (THm0)0

risultano forniti dag(Hm0)0

U210

= 0.283 tanh

[

0.0125

(

gx

U210

)0.42]

(12.3)

g(THm0)0

U10= 7.92 tanh

[

0.077

(

gx

U210

)0.25]

(12.4)

La costante C1 e fornita dal valore del fetch che rende le tangenti iperboliche che appaiono nelle(12.3)-(12.4) prossime a uno con uno scostamento inferiore a 1 %.

Nelle (12.3)-(12.4), i valori di (Hm0)0 e (THm0)0 sono legati a (H1/3)0 e (TH,1/3)0 dalle re-lazioni (Hm0)0 = (H1/3)0; (THm0)0 = 1.05(TH,1/3)0 che derivano, la prima dall’adozione delladistribuzione di Rayleigh la seconda dall’osservazione che (THm0)0 coincide praticamente con ilperiodo di picco che risulta maggiore di (TH,1/3)0 mediamente del 5 %.

Il regime di mare completamente sviluppato richiede invece che siano soddisfatte le relazioni

gt

U10≥ gtcrU10

= 21.88

(

gx

U210

)0.797

(12.5)

gx

U210

≥ gxcrU2

10

= C1 (12.6)

In tal caso, l’altezza d’onda e il periodo sono fornite da

g(Hm0)0

U210

= 0.283 (12.7)

g(THm0)0

U10= 7.92 (12.8)

Infine, il regime di generazione transitorio si verifica quando

gt

U10<gtcrU10

= 21.88

(

gx

U210

)0.797

(12.9)

L’altezza d’onda e il suo periodo possono ancora essere determinate utilizzando le (12.3)-(12.4)sostituendo pero al valore di x il valore di xr (fetch ridotto), valore che rende stazionario ilregime con il parametro della durata effettiva gt/U10. E facile ottenere

gxrU2

10

= 0.0208

(

gt

U10

)1.255

(12.10)

Il modello precedentemente descritto e stato poi modificato dalla versione del 1984 delloShore Protection Manual. Infatti, per poter ipotizzare il regime di generazione stazionario, erichiesto che siano soddisfatte le relazioni

gt

UA≥ gtcr

UA= 68.80

(

gx

U2A

)0.667

(12.11)


gx

U2A

<gxcrU2A

= C2 (12.12)

La costante C2 e pari a 23000 quando si vuole calcolare l’altezza d’onda e a 25500 se si vuolevalutare il periodo. La velocita UA (adjusted wind speed) e legata alla velocita U10 dalla relazione.

UA = 0.71 (U10)1.23 (12.13)

Verificate le condizioni (12.11)-(12.12), l’altezza e il periodo dell’onda spettrale si ricavanoda

g(Hm0)0

U2A

= 0.0016

(

gx

U2A

)0.50

(12.14)

g(THm0)0

UA= 0.286

(

gx

U2A

)0.33

(12.15)

Il regime di mare completamente sviluppato richiede invece che siano soddisfatte le relazioni

gt

UA≥ gtcr

UA= 68.80

(

gx

U2A

)0.667

(12.16)

gx

U2A

≥ gxcrU2A

= C2 (12.17)

In questo caso, l’altezza d’onda e il periodo vengono valutate utilizzando le seguenti relazioni

g(Hm0)0

U2A

= 0.243 (12.18)

g(THm0)0

UA= 8.134 (12.19)

che sono ottenute dalle (12.14)-(12.15) ponendo i valori del parametro del fetch critico sopraindicati.

Infine, il regime di generazione transitorio si verifica quando

gt

UA<gtcrUA

= 68.80

(

gx

U2A

)0.667

(12.20)

L’altezza d’onda e il suo periodo possono ancora essere determinate utilizzando le (12.14)-(12.15) sostituendo pero al valore di x il valore di xr (fetch ridotto), valore che rende stazionariil regime con il parametro della durata effettiva gt/UA. E facile ottenere

gxrU2A

= 0.00175

(

gt

UA

)1.5

(12.21)

Il metodo dello spettro d’energia


Alternativamente a quanto esposto precedentemente, nota la velocita del vento U10, e pos-sibile predire lo spettro in frequenza. Assumendo che esso sia descritto utilizzando la formaJONSWAP e considerando il caso di regime di generazione stazionario, che e presente se sonosoddisfatte le condizioni

gt

U10

≥ gtcrU10

= 21.88

(

gx

U210

)0.797

(12.22)

gx

U210

<gxcrU2

10

= 19000, (12.23)

e possibile valutare i parametri dello spettro con le seguenti relazioni

α0 = 0.076

(

gx

U210

)−0.22

(12.24)

fp = 3.5

(

gx

U210

)−0.33g

U10

γ0 = 3.3

σ = 0.07, se f ≤ fp; oppure σ = 0.09 per f > fp

Nel caso invece di mare completamente sviluppato, che e presente se sono soddisfatte lecondizioni

gt

U10≥ gtcrU10

= 21.88

(

gx

U210

)0.797

(12.25)

gx

U210

≥ gxcrU2

10

= 19000, (12.26)

si haα0 = 0.0087 (12.27)

fp = 0.135g

U10

γ0 = 3.3

σ = 0.07, se f ≤ fp; oppure σ = 0.09 per f > fp

Infine, nel caso di regime di generazione transitorio, che si ha quando non e soddisfatta la

gt

U10≥ gtcrU10

= 21.88

(

gx

U210

)0.797

, (12.28)

si possono utilizzare ancora le (12.24) a patto di adottare il parametro di fetch ridotto il cuivalore e fornito dalla (12.10) che e nuovamente riportata

gxrU2

10

= 0.0208

(

gt

U10

)1.255

(12.29)


Ricordiamo cheHrms =

√

H2

e

H2 = H2rms = 8m0 = 8

∫ ∞

0

S(f)df = 8η2

eHs ∼ 4

√m0 = Hm0

12.3 I metodi diretti

I metodi diretti permettono di valutare le caratteristiche degli stati di mare previsti sullabase di rilievi di campo effettuati in modo sistematico. Nel passato venivano spesso utilizzateosservazioni visuali che conducevano a una stima diH1/3 e di T1/3. Oggigiorno sono piu comuni ecertamente piu affidabili rilievi strumentali che misurano il profilo η dell’onda per un intervallodi tempo ∆D appropriato, tipicamente ogni tre ore. Disponendo dei dati, si valutano poil’altezza d’onda H1/c e il periodo TH,1/c o l’onda spettrale calcolando l’altezza Hm0 = 4(m0)

0.5

e il periodo THm0 = Tp, ove Tp e il periodo di picco dello spettro.Gli spettri in frequenza sono poi generalmente ottenuti costruendo la funzione di autocorre-

lazione monodimensionale riferita al profilo d’onda e introducendo poi la trasformato di Fourierdi tale funzione.

12.4 Stima degli eventi estremi

La stima degli eventi ondosi estremi puo essere effettuata adottando un procedimento cheutilizza dati relativi agli stati di mare e ipotizza un’assegnata distribuzione probabilistica.Il procedimento richiede quindi la conoscenza di dati relativi agli stati di mare nell’area inconsiderazione, dedotti o con un metodo di previsione indiretto o da misure dirette, e puoessere articolato nella seguente sequenza operativa che puo essere adottata sia considerando lealtezze d’onda significative sia le altezze d’onda massime degli stati di mare previsti.

In primo luogo e necessario individuare i valori dell’altezza d’onda H (Hs oppure Hmax)disponibili, relativi a un certo intevallo temporale Γ, adottando quelli che superano una sogliaprefissata. I valori cosı selezionati devono essere ordinati con riferimanto a un indice m tale chem = 1 corrisponde all’onda piu alta e m = N all’onda piu bassa, essendo N il numero di valoridi H selezionati.

Successivamente, si devono costruire le probabilita di superamento Ps(Hm) e totale P (Hm)che, tenendo conto che gli intervalli in cui puo cadere Hm sono N + 1, risultano

Ps(Hm) =m

N + 1; P (Hm) = 1 − Ps(Hm) (12.30)


m Hm = Hsm [m] Ps(Hm) P (Hm)1 2.81 0.0909 0.90912 2.30 0.1818 0.81823 2.11 0.2727 0.72734 1.72 0.3636 0.63645 1.71 0.4545 0.54556 1.63 0.5455 0.45457 1.44 0.6364 0.36368 1.34 0.7273 0.27279 1.30 0.8182 0.181810 1.19 0.9091 0.0909

Tabella 12.1:

Inoltre si deve introdurre una particolare distribuzione per gli eventi estremi (a esempio la di-stribuzione di Gumbel), precisando i valori dei parametri che intervengono in tale distribuzione.Chiaramente, rimane da verificare che l’ipotizzata distribuzione sia appropriata, riportando nelpiano di questa distribuzione i valori dell’altezza d’onda Hm e della probabilita totale P (Hm).Infine e possibile valutare il tempo di ritorno R(H)

R(H) =Γ

Ps(H)N(12.31)

di un evento caratterizzato da un’altezza d’onda H , dopo aver calcolato la probabilita di su-peramento Ps(H). La relazione (12.31) permette di valutare, in termini di tempo di ritorno,eventi che possono verificarsi in intervalli di tempo maggiori di Γ.

Per chiarire la procedura, nel seguito si descrive un esempio relativo a Vado Ligure per cuisono disponibili dati per un intervallo Γ pari a un anno (dal 1 novembre 1971 al 31 ottobre1972) ottenuti con misure dirette. Anche se l’intervallo temporale pari a un anno e eccessi-vamente corto per avere una previsione accurata, si riporta nel seguito la procedura a titoloesemplificativo. Si considerano altezze d’onda significative Hs che superano la soglia di 1 m: intutto 10 dati (N = 10). La tabella (11.1) riporta i valori di Hs, Ps e P .

Adottando una distribuzione di Gumbel espressa da

P (Hs) = exp [− exp (−a(Hs − b))] (12.32)

dove i parametri a e b si ottengono dalle

µs = b+0.577

a

σ2s =

1.645

a2


con

µs =1

N

N∑

m=1r

Hm σ2s =

1

N

N∑

m=1

(Hm − µs)2

si ottiene µs = 1.755 m e σ2s = 0.235 m2 e quindi a = 2.643 m−1 e b = 1.5467 m.

Riferendosi alla distribuzione di Gumbel riferita al parametro W = a(Hs− b) = 2.643(Hs−1.5467) e tale che P (W ) = exp [− exp (−W )], e possibile costruire nel piano di Gumbel (Hs, P (Hs))la curva corrispondente alla (12.32) e confrontare tale curva con i dati originari. La figura 12.1mostra i risultati nel piano (Hs, P ) mentre la figura 12.2 considera il piano (Hs,− log(− log(P ))).

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

P

Hs

Figura 12.1:

Dopo aver verificato l’attendibilita della distribuzione adottata e possibile valutare i periodidi ritorno per un assegnato valore di Hs. A esempio per Hs = 3.5 m si ha W = 5.163, P (Hs) =0.994, Ps(Hs) = 0.006 e quindi R(Hs) = 16.7 anni. In altre parole il tempo di ritorno di unamareggiata caratterizzata da un’altezza d’onda significativa pari a 3.5 m risulta di circa 17 anni.

12.5 Il campo di vento

Al termine di questo capitolo puo essere utile riportare alcune nozioni sul campo di vento. Essopuo essere assegnato nella cosidetta regione geostrofica che inizia a una quota, rispetto al livellodi quiete del mare, pari a circa 1000 m o nello strato limite atmosferico che si estende dallaquota zero sino alla regione geostrofica. Lo strato limite atmosferico e usualmente suddivisoin due parti: i) una regione aderente alla superficie del mare che termina a una quota pari a


-3

-2

-1

0

1

2

3

4

5

0.5 1 1.5 2 2.5 3

-log(

-log(

P))

Hs

Figura 12.2:

100 m, ii) una parte che va da 100 m a 1000 m e che e denominata regione di Eckman. Nellaregione geostrofica, per analizzare la dinamica dell’aria, vengono introdotte alcune importantisemplificazioni. In primo luogo il moto e pensato come una successione di stati stazionari inquanto l’evoluzione temporale del vento e molto lenta. Inoltre la forza d’attrito puo esseretrascurata e in prima approssimazione si puo considerare solo la componente rotazionale, tra-scurando la componente divergente (ricordiamo che il vettore velocita puo essere scomposto,come ogni campo vettoriale, in una parte rotazionale (∇ × Ψ) e una divergente (∇φ). Talesemplificazioni danno origine al cosidetto vento geostrofico che e denotato con il pedice g. Essoe caratterizzato da una velocita orizzontale Ug che risulta parallela alle linee a uguale pressionee caratterizzata da una intensita fornita dalla

Ug =1

ρafC

∆p

∆n(12.33)

ove ρa e la densita dell’aria che e assunta pari a 1.25 kg/m3, ∆p e la differenza di pressione chesussiste tra due linee isobare, ∆n e la distanza fra tali linee e fC e il parametro di Coriolis paria 2Ω sinφ essendo φ la latitudine e Ω la velocita angolare di rotazione della terra.

Quando non esistono significative variazioni della densita dell’aria nella direzione verticale, lostrato limite atmosferico si dice in condizioni neutrali. In tali condizioni, nella regione aderentealla superficie del mare, la distribuzione della velocita puo essere assunta di tipo logaritmico

U

Uat=

1

κln

(

y

y0

)

(12.34)


dove κ e la costante di Von Karman e y0 e una lunghezza correlata alla scabrezza della paretelegata alla velocita d’attrito Uat dalla

y0 = 0.0144U2at

g(12.35)

In condizioni non neutrali le variazioni di densita influenzano la distribuzione di velocita. Inparticolare, la distribuzione di velocita in condizioni stabili (temperatura dell’aria maggioredi quella dell’acqua) e significativamente diversa da quella osservata in condizioni instabili(temperatura dell’aria minore di quella dell’acqua) (si consulti il libro di Scarsi).

Talvolta, la distribuzione di velocita logaritmica o quella modificata per condizioni nonneutrali viene adottata anche nella regione non adiacente alla superficie del mare (regione diEkman).

Gli adeguamenti della velocita del vento

Nei modelli di predizione del moto ondoso si rende necessario valutare la velocita del vento auna quota di 10 m. Operativamente, in condizioni neutrali, la velocita U10 si calcola utilizzandola (12.34) dopo aver ricavato per tentativi la velocita di attrito nota la velocita U a una genericaquota y. Si puo scrivere

U10 = RQU (12.36)

dove RQ e il cosidetto coefficiente di adeguamento per la quota. Qualora sia presente unasignificativa differenza di temperatura dell’aria Ta e quella dell’acqua T , e necessario porre

U10 = RQRTU (12.37)

dove RT e il coefficiente di adeguamento per la differenza di temperatura ∆T = Ta − T . Lafigura 12.3 mostra un esempio dell’andamento di RT in funzione di ∆T , andamento che perodipende anche da altre quantita come ad esempio la velocita del vento.

Infine e necessario tener conto dell’adeguamento per la localizzazione dell’anemometro cioeper trasferire a mare la velocita del vento misurata in una regione a terra.

U10 = RQRTRLU (12.38)

dove il valore di RL puo essere dedotto dalla figura 12.4 che mostra RL in funzione di RQU (illettore noti che i valori forniti dalla figura 12.4 devono essere utilizzati con cautela in quantoottenuti da misure effettuate su grandi laghi).


Figura 12.3:

Figura 12.4:

Capitolo 13

THE TIDES

Figura 13.1: Two boats touching the sea bottom because of the low tide (Aran Isle, Ireland,summer 2014).

203

CAPITOLO 13. THE TIDES 204

13.1 Introduction

Tides are observed all around the world (see figure 13.2) and the currents, they induce, playan important role in the hydrodynamics and morphodynamics of the coastal region.

To give just a few examples: when the length of a bay is close to a quarter of the lengthof the tidal wave or to a multiple, a resonance phenomenon occurs and the amplitude of thetidal excursion as well as the tidal currents become quite large, as it happens in the bay ofFundy (see figure 13.2), thus inducing large values of the sediment transport rate. When thetide propagates along a converging estuary with a decreasing water depth, a tidal bore mightform, originating the so-called mascaret. One of the most famous tidal bores was the mascaretobserved along the Seine river (France) which, near the river mouth, was a breaking bore andbecame an ondular bore further upstream (see figure 13.3). However, the mascaret almostdisappeared along the Seine river after the construction of the canal de Tancarville, which wascompleted in 1963. Large amplitude tidal bores can still be observed all around the world asshown in figure 13.4.

Even though it is a long time that the tides are associated to the motion of the Moon andthe Sun relative to the Earth, only recently a theoretical explanation of the tides has beenproposed on the basis of Newton and Laplace’s studies.

The main mechanism which generates the tides can be understood by using a very simplemodel (the static or equilibrium model) which neglects the presence of the continents andassumes the Earth to be a static sphere surrounded by a thin (relative to the radius of theEarth) layer of water in equilibrium under the gravitational forces. However, this equilibriummodel is unable to explain important aspects of the tides (e.g. the different water excursionswhich are observed at different locations) which can be explained only by introducing a dynamicmodel which considers the tide as a propagating wave.

13.2 The static (equilibrium) model

The tides are the results of the gravitational action of Moon, Sun and Earth on the water ofthe sea. The gravitational action can be quantified by Newton’s law of universal gravitation,which states that two bodies exert a force of attraction each on the other, the magnitude ofwhich is proportional to the product of the masses of the two bodies, inversely proportional tothe square of their distance and directed from the centre of one body to the centre of the other

|F | = kGm1m2

r2(13.1)

where |F | is the module of the attractive force, m1 and m2 are the masses of the two bodies,r is the distance between the centres of the two bodies and kG = 6.674 × 1011 Nm2/kg2 is theuniversal gravitational constant.

Earth and Moon can be considered to rotate around their common centre of gravity CM ,which lies on the line joining the centres of gravity of the Earth and the Moon and lies about


Figura 13.2: The Bay of Fundy in the Gulf of Maine is known for its large tidal range

1800 km below the Earth surface. The Earth describes a small ellipse around CM (the smallestbroken line in figure 13.5), while the Moon describes an ellipse, always around CM , which ismuch larger (the other broken line in figure 13.5). Both the Earth and the Moon take about27.5 days to complete one revolution. Similarly, the center of gravity of the Earth-Moon systemdescribes an elliptical orbit around the centre of gravity of the Sun-Earth system (located insidethe Sun) and takes about 365.25 days to complete one revolution. Because of the rotation ofthe Earth around the Sun, the Moon takes 29.5 day (lunar month) to complete her revolutionaround the Earth relative to the Sun.

To understand how tides are generated, let us consider a spherical Earth completely coveredby water and the plane where the relative motion of the Moon and the Earth takes place asshown in figure 13.6. As the Moon revolves around the Earth, a small rotation dθ of the Moonaround CM leads to a small displacement of all points of the Earth equal to bdθ, i.e. all thepoints move along an arc of size bdθ and are subject to the same centripetal acceleration directedaway from the Moon (the distance between the centre of mass of the Earth and CM is denotedwith b). This force is balanced by the Moon’s attraction which, however, is not uniform over


Figura 13.3: An old photo showing the mascaret along the Seine river at the beginning of thelast century. It is told that Leopoldine Hugo, the daughter of Victor, and her husband drownedin the Seine river when their boat capsized because of a tidal bore. A romantic version of thestory is that the husband, despite being a skilled swimmer, preferred to die with his wife ratherthan to save himself. However, this appears to be a legend because the day of the accidentwas during the neap phase and there was no tidal bore at the time of the accident. Moreover,the husband of Leopoldine Hugo was from a family of ship pilots who knew well the tidal borephenomenon.

the Earth’s surface since it is proportional to the square of the distance of a generic point on theEarth’s surface from the Moon’s centre. For a fluid element which faces the Moon (consider thepoint A), there is a gravitational force which is slightly larger than the apparent centrifugal forcepreviously described. The resulting acceleration δG is directed towards the Moon. Similarly,on the other face of the Earth (consider the point A1) the resulting acceleration is directedaway from the Moon and has the same modulus (−δG). The evaluation of |δG| can be madeeasily taking into account that the radius R of the Earth is much smaller than the distance ofMoon from Earth’s surface, denoted by D. Indeed, the centripetal acceleration is constant inall the points of the Earth and equal to that due to the Moon attraction

G = kGM2

(D +R)2 (13.2)

where M2 is the mass of the Moon. On the other hand, the gravity acceleration due to the


Figura 13.4: a) Photo of a tidal bore on the Salmon River near Truro (Nova Scotia) takenon August 23rd, 2004; b) surfers enjoying a tidal bore on La Gironde river near Saint Pardon(France).

Moon in A is

G = kGM2

D2(13.3)


M

CM

T

Figura 13.5: Sketch of the relative motion of the Earth and Moon relative to their commongravitational centre.

It follows that the difference is

δG =kGM2

D2

[

1 − D2

(D +R)2

]

≃ kGM2

D2

[

1 −(

1 − 2R

D

)]

= kGM22R

D3(13.4)

where terms of O(

(

RD

)2)

are neglected. Similarly, it is not difficult to show that in the point

B, the resulting difference between the gravitational acceleration and the apparent centrifugalacceleration is δG cosφ, φ being the zenith distance. Indeed, the distance of the point B fromthe Moon is equal to

√

(D − R cosφ)2 +R sin2 φ =√

D2 − 2DR cosφ+R2 ≃ D − R cosφD

. Onthe other hand, in the point B1, the gravitational acceleration is −δG cos φ. In figure 13.6 theaccelerations δG and δG cosφ appear parallel, since the angle they form is very small, i.e. oforder R/D. A careful analysis of the order of magnitude of the quantities considered in thefollowing shows that this angle can be safely neglected.

The resulting difference between the gravitational force and the apparent centrifugal forcehas two components: one which is normal to the Earth surface ((δG cosφ)V ) and the otherwhich is tangential ((δG cosφ)H). It can be verified that

(δG cosφ)V = kGM2R

D3

(

3 cos2 φ− 1)

(δG cosφ)H = kGM2R

D3

(

3

2sin(2φ)

)

Let us consider figure 13.7. A generic point B on the Earth surface, characterized by a zenithdistance φ, has a distance from the Moon equal to d. By considering the right triangles TMH ,


b

CM

d θb d θ

MOON

EARTH

δG cos φ

δG-δG

-δG cos φB

A

B1

A1

MOON

EARTH

R D

φ

Figura 13.6: Sketch of the motion of the Moon relative to the Earth and of the forces whichact on the water.

BMH and the triangle TBM , Carnot theorem states that

d2 = R2 +D2 − 2RD cosφ

while the law of sines states thatd

sin φ=

D

sinφ′

Moreover, it turns out thatR + d cosφ′ = D cos φ

Then, it is easy to obtain

d

D=

[

1 − 2R

Dcosφ+

(

R

D

)2]1/2

cosφ′ =D

d(cosφ)


H

MB

T

d

D

angle φ

angle φ,

Figura 13.7: Sketch of the distances among a generic point B, the Earth and the Moon.

sinφ′ =sinφ

[

1 − 2RD

cosφ+(

RD

)2]

Using these relationships, the vertical and horizontal components of δG cosφ are

(δG cosφ)V = kGM2

D2

cosφ− RD

h

1−2 RD

cosφ+(RD)

2i1/2

[

1 − 2RD

cosφ+(

RD

)2] − cosφ

(13.5)

= kGM2

D2

cosφ− RD

[

1 − 2RD

cos φ+(

RD

)2]3/2

− cosφ

(δG cosφ)H = kGM2

D2

sinφh

1−2 RD

cosφ+( RD )

2i1/2

[

1 − 2RD

cosφ+(

RD

)2] − sin φ

(13.6)

= kGM2

D2

sinφ[

1 − 2RD

cosφ+(

RD

)2]3/2

− sin φ

Taking into account that the ratio R/D is much smaller than one, Taylor expansions of (13.5)and (13.6) lead to

(δG cosφ)V = kGM2

D2

R

D

[

3 cos2 φ− 1]

(13.7)


(δG cosφ)H = kGM2

D2

R

D

3

2sin(2φ) (13.8)

The vertical component modifies the gravity acceleration of the Earth but of a negligibleamount. The horizontal component tends to cause a horizontal displacement of water par-ticles towards the sublunar point A and its antipode A1 till the resulting forces towards A andA1 are balanced by pressure gradients due to the the slope of the free surface. This mechanismgives rise to an ellipsoid of water with two bulges directed towards the Moon and away fromit. According to this mechanism, since the Moon takes 24 hours and 50 minutes (a lunar day)to return to its original position, there are two high tides each 24 hours and 50 minutes.

However, the case analysed so far considers the Moon on the equatorial plane. Figure 13.8

M

T

A

A1

δM

Earth,s spin axis

Figura 13.8: Sketch of the Earth-Moon system, taking into account the declination of the Moon.It clearly appears that the maximum of the tide at point A is higher than the maximum of thetide when the point takes the position A1.

simplifies the action of the declination δM of the Moon. The two maxima and two minima ofthe tide are different and this difference is known as diurnal inequality. Moreover, in the regionswhere the latitude is larger than 90−δM , only a diurnal tide is observed with a low and a hightide during a day. During the rotation of the Moon around the Earth (period of approximately27.3 days), the declination varies from a maximum to a minimum and this variation producesvariations of the tide range. The action of the differential tide force also varies depending theEarth-Moon distance, its effect is maximum when the distance is minimal (Moon at perigee)


and the tide is said perigeal. When the Moon is at her farthest distance from the Earth (theMoon apogee), its effect is minimal and the tides are said apogeal. The action of the moon atapogee may be lower by about 15 − 20% than that at perigee.

Even though this simple model shows that the fundamental period of the tide should beabout 12 hours and 25 minutes (12.42 h) and hence high tides take place about a hour later eachday, as observed in the field, it can not provide an accurate prediction of tide elevation. Indeed,to mantain their position relative to the Moon, the two water bulges should travel around theEarth with a speed which turns out to be larger than the speed obtained considering a longwave (the wave which has a wavelength equal to the distance between the two bulges) whichpropagates in shallow waters. A more complex model (dynamic model) should be used to havean accurate description of tide propagation.

Before considering the dynamic model, let us analyse the effects of the Sun. The Sun hasa mass which is much larger than the mass of the Moon (about 27 × 106 times larger) but itis much more distant than the Moon (about 387 times more distant). It turns out that thegravitational force due to the Sun is about 46 % of the gravitational force due to the Moon andgenerates an ellipsoid which is not aligned with the ellipsoid generated by the Moon becauseof Sun’s declination. Needless to write that the period of the solar tide is exactly 12 h.

The tides due to the Moon and the Sun are additive and the largest tidal excursions takeplace when the Moon and the Sun are almost aligned (spring tides). On the other hand, thesmallest excursions are observed when the Moon is about 90 behind the Sun (neap tides) (seefigure 13.9 where the oscillations of the free surface at Punta della Salute in the lagoon ofVenice are plotted versus time). Moreover, because the relative motion of the Earth, Moon andSun is quite complex and other incommensurate periods are present, the gravitational changesgenerate many other tide constituents (the main tide constituents are listed in table 13.3 alongwith their period).

13.3 The dynamic model

A comparison between the tides predicted by means of the equilibrium model and those observedin the field shows that the equilibrium model is not completely satisfactory. The discrepanciesare due to many reasons. First, as already pointed out, the average depth of the oceans issmaller than the depth required to make the tidal bulges travelling with the necessary speed(about 447 m/s). Secondly, the presence of the continents prevents the tidal wave to propagatefreely around the Earth. Then, the free modes of ocean oscillations, which have their ownfrequencies, nonlinearly interact with the tide. Moreover, water has its own inertia and it doesnot respond instantaneously to the gravitational force because of bottom resistence. Finally,the water motion is also affected by Earth rotation and its dynamics feels the effect of theCoriolis parameter.

Sophisticated numerical models are now available to predicts tides around the world. Howe-


Tide constituent Symbol Period [h]Principal lunar semi-diurnal M2 12.42Principal solar semi-diurnal S2 12.00

Larger lunar elliptic semidiurnal N2 12.66Lunar diurnal O1 25.82Solar diurnal K1 23.93

First overtide of M2 M4 6.21Second overtide of M2 M6

First overtide of S2 S4

Tabella 13.1: Principal tide constituents

ver, the predicted tidal excursion is still affected by significant errors and to have more accurateresults the history of measured field data is used to make predictions.

At different locations, the nature of the tide can be classified on the basis of the form numberF , which is defined as

F =O1 +K1

M2 + S2(13.9)

where O1, K1,M2, K2 indicate the amplitude of the major tide constituents (see table 13.3). IfF is smaller than 0.25 the tide is defined as semi-diurnal. If F is larger than 3 the tide is definedas diurnal. When F falls between 0.25 and 1.5 the tide is mixed but predominantly semi-diurnalwhile for values of F falling between 1.5 and 3 the tide is mixed but predominantly diurnal.An example of the tide oscillations observed at Punta della Salute-Canal Grande (lagoon ofVenice) is given in figure 13.9. The figure shows clearly the presence of the spring-neap cycleand the details of the tide oscillations during the neap and spring phases

13.4 The f-plane approximation

The study of tide propagation in the oceans all around the world is out of the scope of these noteswhich are focused on the coastal region. To study the currents generated by tide propagationin a relatively shallow region of limited extent L, it is convenient to introduce a Cartesiancoordinate system centered at some reference latitude and longitude that locally approximatesthe spherical metric in the chosen neighborhood of the Earth’s surface. This because thehydrodynamic problem in spherical coordinates is much more complex than that formulatedusing Cartesian coordinates. This approximation is known as the f -plane approximation andit was first introduced by Rossby (1939).

Let r, θ, φ denote a set of spherical polar coordinates (see figure 13.10) and let vr, vθ, vφdenote the corresponding velocity components in the direction of increasing radius, decreasinglatitude and increasing azimuth, respectively.

Let us denote with Ω the angular velocity of the Earth rotation, Ω(cos θ,− sin θ) being


-0.5

0

0.5

1

1600 1800 2000 2200 2400

wat

er le

vel [

m]

t [hours]

-0.5

0

0.5

1

2080 2100 2120 2140 2160 2180

wat

er le

vel [

m]

t [hours]

-0.5

0

0.5

1

460 480 500 520 540

wat

er le

vel [

m]

t [hours]

Figura 13.9: Tide oscillations observed at Punta della Salute-Canal Grande in 2004. The hoursare from 00:00 of 1/1/2004.

its radial (vertical) and latitudinal (tangential) components (the longitudinal components isassumed to vanish).


R

φ

θxy

z

Figura 13.10:

It is convenient to define a set of curvilinear coordinate (x, y, z)

x = R(θ − θ0) y = R sin θ0φ z = r − R (13.10)

such that∂

∂r=

∂

∂z

∂

∂θ= R

∂

∂x

∂

∂φ= R sin θ0

∂

∂y(13.11)

and new velocity components (u, v, w) such that

u = vθ v = vφ w = vr (13.12)

In (13.10), θ0 is the value of θ characteristic of the chosen latitude and R is the Earth radius.It follows that

r = z +R θ = θ0 +x

Rφ =

y

R sin θ0(13.13)

Then, continuity equation provides

1

r2

∂(r2vr)

∂r+

1

r sin θ

∂(sin θvθ)

∂θ+

1

r sin θ

∂vφ∂φ

= 0 (13.14)


or∂vr∂r

+2

rvr +

cos θ

r sin θvθ +

1

r

∂vθ∂θ

+1

r sin θ

∂vφ∂φ

= 0 (13.15)

The transformation (13.10)-(13.12) leads to

∂w

∂z+

2w

z +R+

cos(

θ0 + xR

)

(z +R) sin(

θ0 + xR

)u+R

(z +R)

∂u

∂x+

R sin θ0

(z +R) sin(

θ0 + xR

)

∂v

∂y= 0 (13.16)

By assuming that the ratios h0/R and L/R (h0 is the order of magnitude of z variations) aremuch smaller than one, at the leading order of approximation, (13.16) becomes

∂u

∂x+∂v

∂y+∂w

∂z= 0 (13.17)

(notice that θ0 should be different from zero and the study of fluid motion at high latitudes orin polar seas should be performed using a different approach).

Because of the high values of the Reynolds number characterizing the tidal currents, theflow regime turns out to be turbulent and it is convenient to consider the Reynolds averagedvelocity components and the Reynolds averaged Navier-Stokes equations. Moreover, the Rey-nolds stresses can be quantified introducing the so-called eddy viscosity. Then, momentumequation in the θ-direction reads

∂vθ∂t

+ vr∂vθ∂r

+vθr

∂vθ∂θ

+vφ

r sin θ

∂vθ∂φ

+vrvθr

−v2φ cot θ

r= − 1

r

∂p

∂θ+ 2Ω cos θvφ (13.18)

+1

r2 sin θ

∂

∂r

[

(µ+ µT )

r2 sin θ

(

r∂

∂r

(vθr

)

+1

r

∂vr∂θ

)]

+

+∂

∂θ

[

2(µ+ µT )

sin θ

(

∂vθ∂θ

+ vr

)

+∂

∂φ

[

(µ+ µT )

(

sin θ∂

∂θ

( vφsin θ

)

+1

sin θ

∂vθ∂φ

)]

−2(µ+ µT )

rcot θ

[

1

r sin θ

∂vφ∂φ

+vrr

+vφ cot θ

r

]

+(µ+ µT )

r

[

r∂

∂r

(vθr

)

+1

r

∂vr∂θ

]

By using (13.10)-(13.12), one readily finds

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z= −1

∂p

∂x+ 2Ω cos θv+ (13.19)

+∂

∂x

[

2(µ+ µT )

∂u

∂x

]

+∂

∂y

[

(µ+ µT )

(

∂u

∂y+∂v

∂x

)]

+∂

∂z

[

(µ+ µT )

(

∂u

∂z+∂w

∂x

)]

Similarly considering momentum equations along φ and r, it is possible to write

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z= −1

∂p

∂y− 2Ω(sin θw + cos θu) (13.20)


+∂

∂x

[

(µ+ µT )

(

∂u

∂y+∂v

∂x

)]

+∂

∂y

[

2(µ+ µT )

∂v

∂y

]

+∂

∂z

[

(µ+ µT )

(

∂v

∂z+∂w

∂y

)]

∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z= −1

∂p

∂z+ 2Ω sin θv − g (13.21)

+∂

∂x

[

(µ+ µT )

(

∂u

∂z+∂w

∂x

)]

+∂

∂y

[

(µ+ µT )

(

∂w

∂y+∂v

∂z

)]

+∂

∂z

[

2(µ+ µT )

∂w

∂z

]

Equations (13.17),(13.19)-(13.21) are the so-called f -plane equations which can be used as longas L cos Θ0/R is much smaller than 1, Θ = π/2 − θ being the local latitude and L the localtide wavelength as it results from a dynamic model which describes the tide propagation in ashallow sea. Since an estimate of L for a tidal wave can be obtained by L =

√gh0T (T is the

tidal period), the f -plane model can be applied when√gh0T cos Θ0/R ≪ 1. At mid-latitudes,

considering a diurnal tide, we have

h0 ≪(

R

T

)21

g∼ 103m (13.22)

a condition which is well satisfied in the coastal region.Moreover, since in many practical problems the water depth can be assumed to be much

smaller than the length of the tidal wave, the vertical velocity component is negligible withrespect to the horizontal components and the pressure distribution along the vertical coordinatecan be assumed to be hydrostatic. By introducing the depth averaged velocity components andfollowing the procedure described in chapter 8, section 8.2, it is possible to write

∂η

∂t+

∂

∂x[(h+ η)U ] +

∂

∂y[(h+ η)V ] = 0 (13.23)

∂U

∂t+ U

∂U

∂x+ V

∂U

∂y= fV − g

∂η

∂x− τxρ (h+ η)

(13.24)

∂V

∂t+ U

∂V

∂x+ V

∂V

∂y= −fU − g

∂η

∂y− τyρ (h + η)

(13.25)

where f = 2Ω sin(Θ) (Θ = π/2 − θ being the local latitude) is the Coriolis acceleration.

13.5 Tide propagation in a shallow sea

13.5.1 The inviscid approach

Many phenomena in shallow seas are affected by the presence of tidal currents and understan-ding tide propagation in coastal areas is quite important. Taylor (1922) solved the problem oftide propagation in a rectangular basin of width B, length L and constant depth h which canbe assumed to represents an enclosed sea (e.g. the Adriatic sea) (see figure 13.11).


L

BSEA

LAND

x

y

Figura 13.11:

In a simple idealized model, the water depth can be assumed to be much smaller than thelength of the tidal wave and simultaneously much larger than the amplitude of tidal excursion.If dissipative effects are neglected, the phenomenon is thus described by the following linearizedshallow water equations

∂η

∂t+ h

(

∂U

∂x+∂V

∂y

)

= 0 (13.26)

∂U

∂t− fV = −g ∂η

∂x(13.27)

∂V

∂t+ fU = −g∂η

∂y(13.28)

where the Coriolis parameter f can be assumed constant if the coastal region is not too large.Manipulation of (13.27), (13.28) gives

∂

∂t

(

∂U

∂x+∂V

∂y

)

− f

(

∂V

∂x− ∂U

∂y

)

= −g(

∂2η

∂x2+∂2η

∂y2

)

(13.29)

∂

∂t

(

∂V

∂x− ∂U

∂y

)

+ f

(

∂U

∂x+∂V

∂y

)

= −g(

∂2η

∂x∂y− ∂2η

∂y∂x

)

= 0 (13.30)


Then∂2

∂t2

(

∂U

∂x+∂V

∂y

)

+ f 2

(

∂U

∂x+∂V

∂y

)

= −g ∂∂t

(

∂2η

∂x2+∂2η

∂y2

)

(13.31)

and using (13.26)∂

∂t

[(

∂2

∂t2+ f 2

)

η − c20

(

∂2η

∂x2+∂2η

∂y2

)]

= 0 (13.32)

where c0 =√gh is the speed of long waves in shallow water.

The solution of (13.32) provides the surface elevation η and the depth averaged values Uand V of the velocity components can be obtained by means of

∂2U

∂t2− f

(

−fU − g∂η

∂y

)

= −g ∂2η

∂x∂t(13.33)

∂2V

∂t2+ f

(

fV − g∂η

∂x

)

= −g ∂2η

∂y∂t(13.34)

or equivalently by∂2U

∂t2+ f 2U = −g

(

∂2η

∂x∂t+ f

∂η

∂y

)

(13.35)

∂2V

∂t2+ f 2V = −g

(

∂2η

∂y∂t− f

∂η

∂x

)

(13.36)

Assuming that the forcing tide has an angular frequency ω (ω ∼ 2π12

hour−1 for a semidiurnaltide and ω ∼ 2π

24hour−1 for a diurnal tide), let us write the solution of (13.32) in the form

η = η(y)ei(ωt−kx) + c.c. (13.37)

which leads to−ω2 + f 2

c20η −

(

−k2η +d2η

dy2

)

= 0 (13.38)

d2η

dy2−(

f 2 − ω2

c20+ k2

)

η =d2η

dy2+ α2η = 0 (13.39)

where

α2 =ω2 − f 2

c20− k2 (13.40)

The boundary conditions at y = 0 and y = B force the vanishing of V , i.e.

dη

dy+fk

ωη = 0 at y = 0, B (13.41)

The general solution of (13.39) reads

η = a sin (αy) + b cos (αy) (13.42)


(we remind that negative values of α2 leads to solutions which decay or amplify in the y-direction) and the boundary conditions, i.e. the vanishing of the velocity component along they-axis for y = 0 and y = B, imply

αa+fk

ωb = 0 (13.43)

[

α sin(αB) cos(αB) +fk

ωsin2(αB)

]

a+

[

−α sin2(αB) +fk

ωsin(αB) cos(αB)

]

b = 0 (13.44)

A nontrivial solution can be found if and only if

−α2 sin2(αB) + αfk

ωsin(αB) cos(αB) − fk

ω

[

α sin(αB) cos(αB) +fk

ωsin2(αB)

]

= 0

i.e. when

sin2(αB)

[

−α2 −(

fk

ω

)2]

= 0 (13.45)

Using the definition of α2 (see (13.40)), it follows

(

ω2 − f 2) (

ω2 − c20k2)

sin2(αB) = 0 (13.46)

The eigenvalues are provided either by

α =nπ

Bfor n = 1, 2, 3, ... (13.47)

or byω2 = c20k

2 (13.48)

The case ω = ±f is not considered hereinafter since, when it exists, it originates inertial wavesequivalent to Kelvin waves (see the definition of Kelvin waves which is given later).

The first eigenrelationship gives rise to

α2 =ω2 − f 2

c20− k2 =

(nπ

B

)2

for n = 1, 2, 3, .. (13.49)

or

kn = ±√

ω2 − f 2

c20−(nπ

B

)2

for n = 1, 2, 3, .. (13.50)

where the sign of the eigenvalue should be chosen in such a way that the solution decays in thedirection of wave propagation. The second eigenrelationship leads to

k0 = ± ω

c0(13.51)

where sign of the eigenvalue should be chosen depending on the direction of wave propagation.


Then, the function η, which describes the free surface elevation, can be obtained by meansof (13.42) where use can be made of (13.43). If the first eigenvalue is considered, the free surfaceturns out to be described by

ηn(y) = an

[

cos(nπy

B

)

− fknB

nπωsin(nπy

B

)

]

for n = 1, 2, 3, ... (13.52)

which is called ’Poincare wave’. On the other hand, the second eigenvalue leads to

η0(y) = a0 exp

(

∓yfc0

)

when k0 = ±ωc0 respectively (13.53)

This second mode is called ’Kelvin wave’. The ratio c0f

=√ghf

= R0 is a length scale whichprovides the order of magnitude of the coastal region affected by the presence of the Kelvinwave and is called Rossby radius.

Incidentally, let us point out that the functions sin(z) and cos(z), z being a complex variable,have a periodic part and a growing/decaying part as the exponential function. Finally, thevelocity components can be obtained by means of (13.30) and (13.31).

Vn = − g

f 2 − ω2

[

iωdηndy

+ fiknηn

]

=giω

ω2 − f 2

[

dηndy

+fknωηn

]

(13.54)

Un = − g

f 2 − ω2

[

−iωiknηn + fdηndy

]

=gf

ω2 − f 2

[

dηndy

+ωknfηn

]

(13.55)

The main difference between Poincare waves and Kelvin waves is that the latter travel witha celerity equal to

√gh which does not depend on their wavelength. In other words, Kelvin

waves are not dispersive. On the other hand Poincare waves are dispersive since their propa-gation speed depend on their wavelength. Moreover, it can be easily verified that the velocitycomponent V0 orthogonal to the channel axis and induced by a Kelvin wave vanishes and thefluid moves only in the direction of the channel axis.

Because of the linearization, the eigenfunctions describing the surface elevation and thevelocity field which satisfy (13.26),(13.27) and (13.28) can be combined with arbitrary coeffi-cients. Since the eigenfunctions are orthogonal, it is easy to determine the coefficients in sucha way that the boundary condition at x = 0 (u = 0) and that at x = L (the presence of anincoming Kelvin wave) turn out to be satisfied

Figure 13.12 shows the amplitude of tide oscillations in the enclosed sea and indicates theexistence of the amphidromic points where there is no tidal excursion and around which thetidal wave circulates once per tidal period. The tidal wave rotates counter-clockwise in theNorthern Hemisphere and clockwise in the Southern Hemisphere. In real cases the geometryis much more complex and the amphidromic systems are much more complicated as shown infigures 13.13 and 13.14 where the amphidromic systems in the North Sea and in the AdriaticSea are drawn.


Note: if ω = ±f equation (13.39) becomes

d2η

dy2− k2η = 0

the solution of which isη = aeky + be−ky

Then, forcing the boundary condition (13.41), it is easy to obtain a = 0 and to verify that bcan assume an arbitrary value. Finally, the velocity field turns out to have only the velocitycomponent parallel to the channel asis

13.5.2 The dissipative effects

Let us now consider the effects induced by the presence of the bottom shear stresses. FollowingLorentz’s approach, we can linearize the quadratic friction law (τ = ρCDU |U |, where thecoefficient CD is equal to 1

C2 , C being the conductance coefficient which, for a rough bottom,can be evaluated as function of the roughness size zr by means of C = 5.75 log10 (11h/zr) )assuming

(τx, τy) = ρr(U, V ) (13.56)

If the power dissipated during a tide cycle, by assuming τ = ρCDU |U |, is forced to be equal tothat dissipated by assuming τ = ρrU , it follows

∫ T

0

τUdt =

∫ T

0

ρCDU |U |Udt =

∫ T

0

ρrU |U |Udt (13.57)

Then, by considering a sinusoidal tidal current U = U0 sin(2πt/T ), it is easy to obtain

r =8U0

3πCD (13.58)

where the order of magnitude of r is 10−3 times the order of magnitude of the amplitude of thevelocity oscillations induced by tide propagation (

∫ π

0sin2 tdt = π

2and

∫ π

0sin3 tdt = 4

3).

It follows that continuity and momentum equations become

∂η

∂t+ h

(

∂U

∂x+∂V

∂y

)

= 0 (13.59)

∂U

∂t− fV = −g ∂η

∂x− rU

h(13.60)

∂V

∂t+ fU = −g∂η

∂y− rV

h(13.61)


0 200 400 600 0

100

200x [km]

y [km]

0 500 1000 0 100 200

x [km]

y [km]

Figura 13.12: Amplitude of the free surface oscillations induced by the propagation of the M2

tide constituent (amphidromic system) (h = 50 m, latitude = 50).

Then, following the procedure already described, it is straightforward to obtain

∂

∂t

(

∂U

∂x+∂V

∂y

)

− f

(

∂V

∂x− ∂U

∂y

)

+r

h

(

∂U

∂x+∂V

∂y

)

= −g(

∂2η

∂x2+∂2η

∂y2

)

(13.62)

∂

∂t

(

∂V

∂x− ∂U

∂y

)

+ f

(

∂U

∂x+∂V

∂y

)

+r

h

(

∂V

∂x− ∂U

∂y

)

= −g(

∂2η

∂x∂y− ∂2η

∂y∂x

)

= 0 (13.63)


Figura 13.13: Amplitude of the free surface oscillations induced by the propagation of the tidaloscillations (amphidromic system) in the North Sea.

Hence, from (13.62)

∂

∂t

(

∂η

∂t

)

+ hf

(

∂V

∂x− ∂U

∂y

)

+r

h

(

∂η

∂t

)

= gh

(

∂2η

∂x2+∂2η

∂y2

)

and deriving with respect to time

∂3η

∂t3+ hf

∂

∂t

(

∂V

∂x− ∂U

∂y

)

+r

h

∂2η

∂t2= gh

∂

∂t

(

∂2η

∂x2+∂2η

∂y2

)

Taking into account that

∂

∂t

(

∂V

∂x− ∂U

∂y

)

= −f(

∂U

∂x+∂V

∂y

)

− r

h

(

∂V

∂x− ∂U

∂y

)

=f

h

∂η

∂t− r

h

(

∂V

∂x− ∂U

∂y

)

it follows

∂3η

∂t3+ hf

[

f

h

∂η

∂t− r

h

(

∂V

∂x− ∂U

∂y

)]

+r

h

∂2η

∂t2= gh

∂

∂t

(

∂2η

∂x2+∂2η

∂y2

)

(13.64)


Figura 13.14: Amplitude of the free surface oscillations induced by the propagation of the tidaloscillations (amphidromic system) in the Adriatic Sea.

∂3η

∂t3+ f 2∂η

∂t− rf

(

∂V

∂x− ∂U

∂y

)

+r

h

∂2η

∂t2= gh

∂

∂t

(

∂2η

∂x2+∂2η

∂y2

)

∂3η

∂t3+ f 2∂η

∂t− r

h

(

− r

h

∂η

∂t+ hg

(

∂2η

∂x2+∂2η

∂y2

)

− ∂2η

∂t2

]

+r

h

∂2η

∂t2= gh

∂

∂t

(

∂2η

∂x2+∂2η

∂y2

)

∂3η

∂t3+ f 2∂η

∂t+( r

h

)2 ∂η

∂t− rg

(

∂2η

∂x2+∂2η

∂y2

)

+ 2r

h

∂2η

∂t2= gh

∂

∂t

(

∂2η

∂x2+∂2η

∂y2

)

∂3η

∂t3+ 2

r

h

∂2η

∂t2+

[

f 2 +( r

h

)2]

∂η

∂t− rg

(

∂2η

∂x2+∂2η

∂y2

)

− hg∂

∂t

(

∂2η

∂x2+∂2η

∂y2

)

= 0 (13.65)

The solution of (13.65) provides the surface elevation η while the depth averaged velocitycomponent U can be obtained by solving

∂2U

∂t2− f

(

−fU − r

hV − g

∂η

∂y

)

+r

f

∂U

∂t= −g ∂

2η

∂x∂t


∂2U

∂t2+ f 2U +

r

h

∂U

∂t+ f

r

hV + gf

∂η

∂y= −g ∂

2η

∂x∂t

Since

V =1

f

(

∂U

∂t+r

hU + g

∂η

∂x

)

it follows∂2U

∂t2+ f 2U +

r

h

∂U

∂t+r

h

(

∂U

∂t+r

hU + g

∂η

∂x

)

+ fg∂η

∂y= −g ∂

2η

∂x∂t

∂2U

∂t2+ f 2U + 2

r

h

∂U

∂t+( r

h

)2

U = −rgh

∂η

∂x− g

∂2η

∂x∂t− gf

∂η

∂y(13.66)

Similarly, it is possible to obtain V

∂2V

∂t2+ f 2V + 2

r

h

∂V

∂t+( r

h

)2

V = −rgh

∂η

∂y− g

∂2η

∂y∂t+ gf

∂η

∂x(13.67)

If the free surface displacement is written in the form

η = η(y)ei(ωt−kx) + c.c. (13.68)

equation (13.65) leads tod2η

dy2+ α2η = 0 (13.69)

where now α is given by

α2 = −iωf2 − ω2 +

(

rh

)2+ 2iωr

h

c20(

rh

+ iω) − k2 = −iωh

2f 2 + (r + iωh)2

hc20 (r + iωh)− k2 (13.70)

and the boundary conditions at y = 0 and y = B, which force the vanishing of V , can bewritten in the form

dη

dy+

ifkh

r + ihωη = 0 at y = 0, B (13.71)

The general solution of (13.69) reads

η = a sin (αy) + b cos (αy) (13.72)

and the boundary conditions imply

αa+ifkh

r + iωhb = 0 (13.73)

[

α cos(αB) +ifkh

r + iωhsin(αB)

]

a+

[

−α sin(αB) +ifkh

r + iωhcos(αB)

]

b = 0


Hence, nontrivial solutions for a and b can be found if and only if[

( r

h+ iω

)2

+ f 2

] [

ω (−ωh+ ir)

gh2+ k2

]

sin(αB) = 0 (13.74)

The significant eigenvalues are provided either by

α =nπ

Bfor n = 1, 2, 3, ... (13.75)

or byω (−ωh+ ir)

gh2+ k2 = 0 (13.76)

The first eigenrelationship gives rise to

α2 = −iωh2f 2 + (r + ihω)2

hc20 (r + ihω)− k2 =

(nπ

B

)2

for n = 1, 2, 3, .. (13.77)

or

kn = ±

√

−iωh2f 2 + (r + ihω)2

hc20 (r + ihω)−(nπ

B

)2

for n = 1, 2, 3, .. (13.78)

The second eigenrelationship leads to

k0 = ±√

ω(ωh− ir)

gh2(13.79)

As in the inviscid case, the sign of the eigenvalue should be chosen in such a way that thesolution decays in the direction of wave propagation.

Then, the function η, which describes the free surface elevation, can be obtained by meansof (13.72) where use can be made of (13.73). It follows

ηn(y) = an

[

cos(nπy

B

)

+ifknhB

nπ(r + iωh)sin(nπy

B

)

]

for n = 1, 2, 3, ... (13.80)

for the first eigenfunction, which is called Poincare wave, and

η0(y) = a0 exp

(

∓ y√

ωhf 2

√

gh (ωh− ir)

)

(13.81)

for the second eigenfunction, which is called Kelvin wave. The sign ∓ should be chosen accro-

ding to k0 = ±√

ω(ωh−ir)gh2 as in the frictionless case. Finally, the veloticy components can be

obtained by (13.66) and (13.67) with straightforward algebra.As in the frictionless case, it is possible to combine the eigenmodes to satisfy the boundary

condition at x = 0 and thus to obtain the full solution. For reasonable values of the parameterr, the results do not change significantly as discussed by Roos & Schuttelaars (2010)


13.6 The velocity profile of a tidal current

Let us consider a water body of small depth h and introduce a Cartesian coordinate system(x, y, z) where x, y are two horizontal axes lying on the still water surface such that the x-axisis along the parallels pointing East, the y axis points North along the meridian line and thez-axis is vertical pointing upwards.

Since the flow regime is assumed to be turbulent and viscous effects are neglected, thehydrodynamic problem is posed by (13.17), (13.19)-(13.21) setting µ = 0. To solve the problemjust formulated, it is convenient to introduce the new pressure P such that

p = P − ρg(z − η) . (13.82)

When (13.82) is substituted into momentum equations, the pressure P replaces p, the extra-terms

−g ∂η∂x

; −g∂η∂y

appear in (13.19) and (13.20), respectively, while in (13.21) the gravitational term disappears.The solution of the problem for arbitrary functions h is a difficult task. However, as pointed

out in the introduction, locally it is possible to assume that the bottom configuration is flatand h = h0.

Since the length scale L =√gh0/ω of the variations in the horizontal directions can be

assumed to be much larger than the water depth h0, the shallow water approximation can beintroduced and P vanishes. However, since the vertical profile of the tidal current is of interesthereinafter, the depth average procedure is not introduced.

Because of the periodicity of the tidal flow, it is convenient to assume

(u, v, η) =

N∑

n=0

(

u(n), v(n), e(n))

e−iωnt + c.c. (13.83)

where ωn = nω = 2πn/T and all the variables defined by (13.83) depend on the vertical andhorizontal coordinates. However, it should be noted that the variations in the x and y directionstake place on the spatial scale L which is much larger than h0. If the dimensionless parametersh0/L and a/h0 are assumed much smaller than one

h0/L ≪ 1 ; a/h0 ≪ 1 (13.84)

by substituting (13.83) into continuity and momentum equations and keeping only the highestorder terms (Blondeaux & Vittori, ????), it is possible to obtain

∂

∂z

[

νT∂u(n)

∂z

]

+ 2Ω sinφ0v(n) + iωnu

(n)− =∂e(n)

∂x(13.85)

∂

∂z

[

νT∂v(n)

∂z

]

− 2Ω sin φ0u(n) + iωnv

(n) =∂e

(n)0

∂y(13.86)


∂u(n)

∂x+∂v(n)

∂y+∂w(n)

∂z= 0 (13.87)

The solution of (13.85)-(13.87) can be determined once the eddy viscosity is specified. Theturbulence structure is supposed to be in local equilibrium by assuming that the horizontallength scales of bottom variations are much larger than the water depth. Therefore the eddyviscosity νT is supposed to be provided by the relationship which holds for a constant depthspecified, of course, with the local values of the parameters.

By denoting with h0 the local water depth and considering a uniform current with a depth-averaged velocity U0, different models have been proposed in the past to evaluate the eddyviscosity and describe its vertical structure. The simplest model relates νT to the local shearvelocity uτ and to the local depth h0

νT = kuτh0F (13.88)

where k is the Von Karman constant. Very simple models assume F to be constant over thedepth. However, in this case, the no-slip condition at the bottom can be applied no longer andthe description of the flow close to the bottom is not accurate. More reliable models (see a.o.Nielsen, 1992) assume F to depend on z (e.g. F (ξ) = −ξ(1 + ξ) with ξ = z/h0). However, theabove relationship provides vanishing values of F and hence of νT at the free surface where therandom fluctuations of the horizontal velocity components are small but different from zero. Arefined relationship was proposed by Dean (1974) and it reads:

F (ξ) =−ξ(1 + ξ)

1 + 2A(1 + ξ)2 + 3B(1 + ξ)3with ξ =

z

h0, A = 1.84, B = −1.56 (13.89)

Relationship (13.89) provides the vertical structure for νT , but in order to compute the eddyviscosity, it is necessary to relate uτ , i.e. the shear velocity, to the depth-averaged velocity. Fora steady flow a conductivity coefficient C can be introduced such that

uτ =U0

C. (13.90)

Note that C depends on the flow Reynolds number and on the relative roughness of the bottom.However, the Reynolds number is tipically very large and C can be assumed to depend only onthe relative roughness according to

C = 5.75 log10

(

13.3ϕh0

zr

)

= 5.75 log10

(

10.9h0

zr

)

(13.91)

where zr is the size of the homogenous equivalent roughness and ϕ is a shape factor whichdepends on the form of the cross-section of the current and can be assumed equal to 0.82 in thehorizontally unbounded case. For a flat bottom of cohesionless sediments of uniform diameterd, the equivalent roughness zr is usually assumed to be equal to 2.5d. However, the sea bed is


seldom flat, but quite often is covered with ripples. In case of a rippled bed, zr can be assumedequal to the ripple height ∆r. For strong ambient flows, ripples are swashed out and a sheetflow appears. For these conditions, zr to be related to the thickness of the moving layer ofsediments.

When the current is not steady but oscillates in time, as it happens for tidal flows, theeddy viscosity is time dependent. Because the tidal period is very long, a slowly varyingapproach seems justified. In this case relationships (13.88), (13.89) can be used with theinstantaneous values of uτ . Even though hysteresis effects were detected in the field and inlaboratory experiments simulating tidal flows and although Anwar (1975), Bohlen (1976), Thorn(1975), Anwar & Atkins (1980), Knight & Ridgway (1977) observed that the flow structureand transport phenomena during the accelerating phase differ from those characterizing thedecelerating phase, the differences are small and can be neglected in a simplified approach.

In many studies a further simplifying assumption is introduced and νT is assumed to be time-independent and related to some ’average’ value of the shear velocity. Indeed the introductionof a time-dependent eddy viscosity greatly complicates the problem and does not lead to muchbetter results. Hence, a time-constant value of νT can be used where the shear velocity isprovided by:

uτ =aiU0

C. (13.92)

In (13.92), U0 is the maximum value of the flow rate divided by the local water depth duringthe cycle. For a pure tidal flow characterized by a uni-directional velocity, the coefficient aiturns out to be equal to 2/π and values of this order of magnitude can be employed in thegeneral case.

Because of tidal oscillations, the free surface is not coincident with z = 0 but it is describedby a time dependend function η, i.e. by the equation

z = η(x, y, t) (13.93)

Relationship (13.89) can still be used, even though the definition of ξ should be modified as:

ξ =z − η

h + η. (13.94)

and also the relationship providing C should be modified

C = 5.75 log10

[

10.9(h+ η)

zr

]

(13.95)

To conclude, it is possible to assume

νC = κai

5.75 log10

[

10.9(h+η)zr

] ·[∫ η

−h+zr/29.8

[

U2 + V 2]1/2

maxdz

]

· F (ξ) (13.96)


where 29.8 is a constant such that the velocity vanishes at distance from the bottom equal tozr/29.8.

At the bottom and at the free surface the following boundary conditions should be satisfied

u(n) = 0 ; v(n) = 0 ; w(n) = 0 for z = −h0 +zr

29.8(13.97)

∂u(n)

∂z= 0 ;

∂v(n)

∂z= 0 ; e

(n)0 = iωnw

(n)0 ; for z = 0 . (13.98)

Of course the solution of the hydrodynamic problem requires the boundaries in the horizontalplane to be specified along with appropriate conditions. However, the study of the tidal waveand of the horizontal distribution of the velocity field is not of interest here.

The vertical structure of the flow can then be determined from (13.85) and (13.86) once∂e(n)/∂x, ∂e(n)/∂y are given. The quantities ∂e(n)/∂x and ∂e(n)/∂y can be thought of as twoparameters which control the orientation and the form of the tidal ellipse. Since ∂e(n)/∂x and∂e(n)/∂y are complex, they can be written as

∂e(n)

∂x= 1e

iϑ1∂e(n)

∂y= 2e

iϑ2 . (13.99)

Then, the solution of the hydrodynamic problem can be divided by 1eiϑ1 and hence the only

parameters affecting u(1) and v(1), besides the physical ones, are

=2

1and ϑ = ϑ2 − ϑ1 . (13.100)

With a standard iterative procedure on and ϑ it is possible to determine the values of and ϑgiving rise to an assigned tidal ellipse. The solution of the ordinary differential problems posedby

∂

∂z

[

νT∂u(n)

∂z

]

+ 2Ω sinφ0v(n) + iωnu

(n) = 0 (13.101)

∂

∂z

[

νT∂v(n)

∂z

]

− 2Ω sin φ0u(n) + iωnv

(n) = eiϑ (13.102)

∂u(n)

∂z=∂v(n)

∂z= 0 at z = 0 (13.103)

u(n) = v(n) = 0 , at z = −h0 +zr

29.8(13.104)

is found with a standard shooting procedure based on a predictor-corrector method after theintroduction of a streched variable

η = ln

[

(z + h0)29.8

zr

]

(13.105)


Figure 13.15 shows the tidal ellipse, i.e. the dimensionless values of the depth averagedvelocity during the tidal cycle scaled with it maximum value. The Keulegan-Carpenter numberof the tide U0/(ωh) is equal to 530, the ratio between the bottom roughness and the waterdepth is equal to about 3 × 10−3, the viscous parameter µ defined by Blondeaux & Vittori(???) turns out to be about 266 and the tidal ellipticity is fixed equal to 0.12 and the locallatitude is 57 North (these values are representative of Trapergeer site on the continental shelfof Belgium). In figure 13.15, the dimensionless bottom shear stress scaled with its maximumvalue is also plotted to show the effects of Coriolis acceleration which makes the tidal ellipseto rotate when the depth is changed. Finally, in figure (13.16), the quantity

√

|u(1)|2 + |v(1)|2is plotted versus z showing the typical log-law close to the bottom.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1 -0.5 0 0.5 1

V0,

τ0y

U0, τ0x

depth averaged velocitybottom shear stress

Figura 13.15: Tidal ellipse


-1

-0.8

-0.6

-0.4

-0.2

0

0 0.1 0.2 0.3 0.4 0.5

z

[|u(1)|2+|v(1)|2]1/2

Figura 13.16: The quantity√

u|(1)|2 + |v(1)|2 plotted versus z for U0/(ωh) = 530, the ratiobetween the bottom roughness and the water depth is equal to about 3 × 10−3, the viscousparameter µ defined by Blondeaux & Vittori (???) equal to about 266, the tidal ellipticityequal to 0.12 and the local latitude equal to 57 North.

Capitolo 14

SEDIMENT TRANSPORT

14.1 Introduction

In the previous chapters, the flow field generated by the propagation of wind waves and tidalwaves is described. To determine the time development of both the sea bottom and the coastlineand to investigate the dynamics of the morphological patterns observed in the coastal region(see chapter 16), it is necessary to quantify the amount of sediment moved by the flowingwater. The evaluation of the sediment transport rate is also necessary to quantify the erosionand deposition processes which take place around coastal structures (see figures 14.1 and 14.2)and might affect their stability.

Figura 14.1: Example of scour around an offshore structure

Herein, we provide a brief description of the sediment characteristics, which have a largeinfluence on sediment dynamics, and of the empirical predictors used to evaluate the sedimenttransport rate. Even though attempts have been made to determine the dynamics of sedimentgrains evaluating the flow around them and the forces exerted by the fluid (see for example

234

CAPITOLO 14. SEDIMENT TRANSPORT 235

Figura 14.2: Example of scour at the bottom of an inlet

Mazzuoli et al. 2016) such sophisticated approaches are far to be applied to investigate thedynamics of large morphological patterns. The latter are usually studied by using quite simpleidealized models. Hence, in the following, we limit ourselves to give to the reader only somebasic elements of sediment transport, suggesting other books (e.g. ???) to the readers interestedon a more accurate modelling of sediment dynamics.

The sediments lying on the sea bed are moved by the surface waves and/or the currentswhen the hydrodynamic forces on the solid particles become larger than the forces which opposethe motion and are due to the particles at rest. The conditions, such that the stabilizing forcesbalance the destabilizing forces, are named ’critical conditions’ or conditions of incipient motion.Once the sediments start to move, they are transported according to two different mechanisms

- the bed load, such that the coarse fraction of the bed material rolls, slides and jumps butit keeps in strict contact with the resting particles;

- the suspended load, such that the fine fraction of the bed material is picked-up from thebed and transported for long distances (much larger that the size of the sediments themselves)before settling down again.

The so-called washed load, which is the portion of sediment that is carried by a fluid flowin near-permanent suspension with a negligible deposition, is not considered herein.

The sediment transport rate can be evaluated using different units, since it can be defined


in different ways. In the following we use the volumetric transport rate qt, which is thevolumetric flux of sediments per unit width. In this case, the units are m2/s. Other authorsuse the mass flux of sediments ρsqt and/or the immersed weight flux (ρs − ρ)gqb.

The conversion to the units usually used by engineers (tons and years) can be easily made byusing the following standard values of the sediment and bed characteristics, possibly modifiedto consider local peculiar situations: ρs = 2650 kg/m3 and porosity n = 0.4 (n is the ratiobetween the volume of voids and the total volume of the sample of sand). The value of theporosity should be considered when it is necessary to evaluate the volume of the material whichis deposited to or eroded from the bottom.

Before analysing the solid transport and the erosion/deposition processes, it is appropriateto discuss the main characteristics of the sediments. The very fine fraction (silt and clay)behaves differently from the fine and coarse fractions (sand and gravel), because the formeris largely affected by cohesion while the latter behaves like a cohesionless material. In thischapter, we consider the cohesionless sediment while the cohesive sediment is considered in thefollowing chapter.

14.2 The sediment characteristics

14.2.1 Sediment density

Most sands are mainly composed of quartz grains which can be assumed to have a density ρsclose to 2650 kg/m3 and an almost spherical shape such that the ratio between the major andminor axes of a sediment grain is usually smaller than 2. However, in coastal environments,particles originated by shells of marine creatures are also present and they have a smaller densityof about 2400 kg/m3 and a more irregular shape. In particular areas, sediments of other minerals(for example coal with a typical density of 1400 kg/m3) may be present. To determine sedimenttransport and to investigate morphodynamic phenomena, it is common to consider sedimentgrains as if they were quarts grains. In those parts of the world where sediments of volcanic orcoral origin are predominant, it is necessary to measure their characteristics and in particulartheir density in laboratory.

14.2.2 Grain size

The size of the sediment grains falls in wide range. Different shapes can be observed and theuse of the term particle ’diameter’ is somewhat ambiguous. The sieve diameter, which is thesize of the sieve through which the sediment particles just pass, is commonly used for sand andgravel while for silt and clay the equivalent fall diameter, i.e. the diameter of a sphere with thesame density and fall velocity, is used. Table 14.1 shows the classification of the sediment asproposed by the American Geophysical Union.

The sediment, which is found in natural environments, contains particles of different size.The probability density function p(d) of the grain size distribution is defined in such a way that


Class name millimetres ψ-scale micronsvery large boulders 4096,2048 12,11

large boulders 2048,1024 11,10medium boulders 1024,512 10,9small boulders 512,256 9,8large cobbles 256,128 8,7small cobbles 128,64 7,6

very coarse gravel 64,32 6,5coarse gravel 32,16 5,4

medium gravel 16,8 4,3fine gravel 8,4 3,2

very fine gravel 4,2 2,1very coarse sand 2,1 1,0

coarse sand 1,0.5 0,-1medium sand 0.5,0.25 -1,-2 500,250

fine sand 0.25,0.125 -2,-3 250,125very fine sand 0.125,0.0625 -3,-4 125, 62

coarse silt 1/16,1/32 -4,-5 62,31medium silt 1/32,1/64 -5,-6 31,16

fine silt 1/64,1/128 -6,-7 16,8very fine silt 1/128,1/256 -6,-7 8,4coarse clay 1/256,1/512 -7,-8 4,2

medium clay 1/512,1/1024 -8,-9 2,1fine clay 1/1024,1/2048 -9,-10 1,0.5

very fine clay 1/2048,1/4096 -10,-11 0.5,0.24

Tabella 14.1: Size classification of sediment particles


p(d)dd is the contribution (percentage) to the volume of the mixture of the sediments with thesize falling between d and d+dd. It is worth pointing out that the probability density functionis a dimensional function [p] = L−1 and that the cumulative distribution function F (d) is suchthat dF (d)/dd = f(d) and

∫

f(d)dd = F (d). Figure 14.3 shows the cumulative distributionfunction of three sediment samples taken at Hanford Site, close to a mostly decommissionednuclear production complex, operated by the United States federal government on the ColumbiaRiver in the U.S. state of Washington. The data have been obtained by the report ’ParticleSize Distribution Data From Existing Boreholes at the Immobilized Low-Activity Waste Site’prepared by M. M. Valenta, J. R. Moreno, M. B. Martin, R. E. Fern, D. G. Horton, S. P. Reidel(2000) for the U.S. Department of Energy under Contract DE-AC06-76RL01 830

0

20

40

60

80

100

0.01 0.1 1 10 100

perc

ent p

assi

ng [%

]

Grain size [mm]

sample 1sample 2sample 3

Figura 14.3: Cumulative distribution function of three sediment samples. Samples n.1 and n.3are slightly gravelly sand, sample n.2 is gravelly sand.

To characterize a sediment mixture, a characteristic size of the sediment is usually introdu-ced. Either the median diameter d50, i.e. the size such that 50 % of the mixture is finer/coarser,or the mean diameter dmean, defined by

dmean =

∫

dp(d)dd, (14.1)

are used. Moreover, to describe the grain size distribution, the standard deviation σ is alsointroduced

σ2 =

∫

(d− dmean)2 p(d)dd (14.2)

(σ2 is the variance) along with the skewness Cs and the curtosis Ck coefficients.

Cs =

∫

(d− dmean)3p(d)dd

[∫

(d− dmean)2p(d)dd]3/2

(14.3)


Ck =

∫

(d− dmean)4p(d)dd

[∫

(d− dmean)2p(d)dd]2 − 3 (14.4)

The value 3 is subtracted in (14.4) in such a way that the curtosis coefficient of the normaldistribution vanishes.

The normal distribution, which is often used in many practical problem, is described by

p(d) =1√2πσ

exp

[

−1

2

(

d− dmeanσ

)2]

(14.5)

and it is characterized only by its mean value dmean and its standard deviation σ. However, theactual sediment distribution turns out to be better approximated by a log-normal distribution,i.e. a normal distribution where the variable is ln (d/d50) (the reader should consider the rulesof the change of variable of an integral)

p(d) =1

d√

2πσexp

[

−1

2

(

ln (d/d50)

σ

)2]

(14.6)

Indeed, it is quite common to use the so called ψ- of φ-scales, such that

d

dref= 2ψ = 2−φ (14.7)

with dref = 1 mm, and to introduce a normal distribution in the ψ or φ variables. In this casethe mean grain size dmean turns out to be larger than d50. Once, the mean value ψmean of ψis computed, dg = dref2

ψmean is called the geometric mean size (dg is different from dmean) andσg = 2σ (σ being the standard deviation of the normal ψ distribution) is called the geometricstandard deviation which is larger than 1. A sediment mixture is defined to be well sorted ifσg < 1.6.

Assuming a Gaussian ψ-distribution, it is possible to verify that

dg =√

d84d16, σg =

√

d84

d16(14.8)

For example, considering a sediment mixture characterized by a mean value of ψ equal toψmean = 0.2 (dg ∼ 1.15 mm) and a standard deviation σ = 0.5, it turns out that d84 ≃ 1.62

mm, d16 ≃ 0.82 mm,√d84d16 ≃ 1.15 mm and 2σ ≃ 1.41 ≃

√

d84/d16.However, other grain size distributions are observed and in particular bi-modal distributions

are frequent which are originated by a mixture of sand and gravel.If the dynamics of a single sediment particle is analysed, also the shape of the grains plays

an important role. Different parameters can be introduced to define the shape of a particle,the most common being the shape factor

Sh =d1√d2d3

(14.9)


where d1, d2, d3 are the shortest, intermediate and longest sizes along mutually perpendicularaxes. Sometimes, the sphericity ψs of a particle is also used which is defined as the ratio betweenthe surface area of a sphere (with the same volume as the given particle) and the surface areaof the particle. Of course a sphere has a value of ψs equal to 1 while ψs is equal to 0.846 andto 0.8806 for a tetrahedron and a cube, respectively.

14.2.3 Fall (settling) velocity

As already pointed out the fall velocity ws is also used to classify the sediment particles. Thefall (settling) velocity is the equilibrium velocity attained by a falling sediment grain. If thesediment particle is approximated by a sphere, the fall velocity can be easily obtained by forcingthe balance between the drag force FD and the immersed weight P of the particle

FD = ρw2s

πd2

8cD, P = (ρs − ρ) g

πd3

6(14.10)

where the drag coefficient cD depends on the Reynolds number dws

ν. It follows

w2s =

4

3cD(s− 1) gd (14.11)

wheres =

ρsρ

is the relative density of the particle and cD depends also on the fall velocity. For a smallparticle, such that the Reynolds number is smaller than about 0.1 the drag coefficient can bederived by the Stokes law

cD =24ν

wsd(14.12)

and ws turns out to be

ws =(s− 1) gd2

18ν(14.13)

For larger values of the Reynolds number, empirical relationships providing cD as function ofthe Reynolds number are necessary to determine ws.

Of course the fall velocity of sediment grains characterized by different shape factors differsfrom that of a sphere. Figure 14.4a, similar to figure 3.2 of Sleath’s (1984) book, shows thefall velocity obtained from the data of the U.S. Inter-Agency Committee on Water Resources(1957) as function of the grain size for different values of the shape factor and T = 0 (ρs = 2650kg/m3). As pointed out by Sleath (1984), the fall velocity of natural sediments characterizedby a shape factor equal to one is different from that of a sphere, simply because the shapefactor does not unambiguously define the shape of a particle (for example both a cube and asphere have a shape factor equal to one but their shape is quite different). Since the value ofthe temperature affects the kinematic viscosity of the fluid, it has an indirect effect also on the


1

10

100

0.1 1 10

ws

[cm

/s]

d [mm]

Sh=0.3Sh=0.5Sh=0.7Sh=0.9sphere

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35 40

ws

[cm

/s]

T [o]

Sh=0.3Sh=0.5Sh=0.7Sh=0.9sphere

Figura 14.4: Fall velocity of sediment grains characterized by different shape factors plottedversus a) d for T = 0 C, b) T for d = 0.2 mm.

fall velocity. Hence it is necessary to point out that the results of figure 14.4a were obtainedfor a water temperature equal to 0. Figure 14.4b shows the fall velocity as function of thewater temperature for a grain size d equal to 0.2 mm for different values of the shape factor.Formulae exist to predict the fall velocity of natural sediments. Soulsby (1997) used a bestfitting procedure and experimental data and proposed

ws =ν

d

[√

10.362 + 1.049R2p − 10.36

]

(14.14)


which has a degree of accuracy similar to that of Van Rijn (1984)

ws =νR2

p

18dfor R2

p ≤ 16.187 (14.15)

ws =10ν

d

[√

1 + 0.01R2p − 1

]

16.187 < R2p ≤ 16187

ws =1.1νRp

d16187 < R2

p

but it turns out to be simpler. Dietrich (1982) proposed

ws√

(s− 1)gd= exp

−2.891394 + 0.95296 ln(Rp) − 0.056835 [ln(Rp)]2 (14.16)

−0.002892 [ln(Rp)]3 + 0.000245 [ln(Rp)]

4

In (14.14)-(14.16) the particle/sediment Reynolds number Rp is introduced such that

Rp =

√

(s− 1) gd3

ν(14.17)

10-4

10-3

10-2

10-1

100

101

10-2 10-1 100 101 102 103 104 105

ws/

[(s-

1)gd

]1/2

Rp

Soulsby (1997)Van Rijn (1984)Dietrich (1982)

Figura 14.5: Fall velocity plotted versus the sediment Reynolds number


Figure 14.5 shows a comparison of the results obtained using the empirical formulae descri-bed previously, which provide similar results for small and intermediate values of Rp and smalldifferences for large values of Rp.

The fall velocity is also affected by the sediment concentration and the values discussedso far are for an isolated grain in an infinite fluid domain. Since the downward motion of asediment grain induces an upward fluid motion elsewhere because of a volume balance, thefall velocity of a single grain in a sediment suspension is smaller than the values previouslydiscussed and it can be approximated by

wsws0

= (1 − c)a (14.18)

where ws0 is the fall velocity of a single particle in clear water and different values of theexponent a are suggested by different authors. For example Fredsoe & Deigaard (1992) suggest

a = 4.35

(

ws0d

ν

)−0.03

for 0.2 <ws0d

ν< 1 (14.19)

a = 4.45

(

ws0d

ν

)−0.1

for 1 <ws0d

ν< 500

a = 2.39 for 500 <ws0d

ν

On the other hand, Van Rijn (1991) suggests

ws = ws0(1 − c5)

while Soulsby (1997) proposes

ws =ν

d

[√

10.362 + 1.049(1 − c)4.7R2p − 10.36

]

Finally, a significant influence on the fall velocity is given by turbulence intensity. Turbu-lence fluctuations have no effects on the fall velocity for small sediment grains, since the dragforce is proportional to the relative velocity. However, for large particles the drag force turnsout to be proportional to the square of the relative velocity and the mean drag force is no longerproportional to the square of the mean velocity.

14.2.4 Angle of repose

It is well known that two angles of repose can be introduced: the angle of repose/rest andthe residual angle of shearing. The angle of repose (also angle of initial yield) φi is defined byengineers as the tilting angle of an open box filled with sediment grains such that the sedimentat the free surface becomes unstable and starts to fall.


The residual angle of shearing φr is the slope attained by the sediment after avalanching.According to Allen (1970), uniform spheres with diameter ranging between 0.08 mm and 3 mmare characterized by

φi = 33 φr = 23

For natural sediment, the value of φi ranges between 28 to 36 for large values of the porosityand between 45 to 53 for relatively small values of the porosity. Because of the large spreadof the experimental data, direct measurements of φi and φr should be made. The reader shouldnote that the angle of repose φr is usually thought to be coincident with the friction angle.

14.2.5 Bed porosity and permeability

As already pointed out, the sea bottom is also charaterized by other properties among whichthe porosity n and the permeability are the most relevant.

The porosity n is defined as the ratio between the volume of voids and the total volume.Spherical particles in a thetraedical arrangment (like cannon balls) have a very small value ofporosity (n = 0.26) while, when they are packed in a cubical arrangement, the porosity is 0.48.Typical values of the porosity n fall between 0.30 and 0.50, moving from gravel to clay, n = 0.3and n = 0.4 being the values for sand. When fine sand is mixed with gravels the value of n canbe as small as 0.2.

The bed permeability controls the flow inside the bed. For a sediment characterized by amean grain size smaller than 1 mm, the flow regime inside the bed is laminar and the apparentvelocity v is proportional to the pressure gradient through the specific permeability Kp (m2)

v =Kp

ρν∇p (14.20)

For grains larger than 1 mm, Darcy’s law (14.20) should be replaced by

∇p = apρν

d250

v + bpρ

d50v|v| (14.21)

where the dimensionless coefficients ap, bp depend on porosity, shape factor, packing, orientationand grading of the sediment mixture. Table 14.2 provides the values suggested by Soulsby(1997). These values, as well as the values suggested by other authors, gives a strong increaseof Kp/d

2 as the porosity is increased. However, as pointed out by Soulsby (1997), data bySleath (1970) and HR Wallingford show that Kp/d

2 is almost constant and equal to about1.1×10−3. Hence, if the bed porosity is uncertaint, the value of Kp can be computed by meansof

Kp = 0.0011d2


Kp ap bpn4.7d2

19.8(1−n)19.8(1−n)

n4.7

0.956(1−n)n4.7

Tabella 14.2: Values of the constants appearing into (14.21)

14.3 The inception of sediment transport

In order to predict erosion and deposition processes in coastal environments, it is necessary toquantify the amount of sediment moved by a given water flow which is often produced by thesimultaneous presence of waves and currents, which give rise to a steady velocity componentand an oscillatory velocity component.

When the flow is weak, the forces acting on sediment grains are small and unable to move thesediment particles. Only when the flow exceeds limiting conditions, the sediment is transported.Hence, a first step to quantify the sediment transport is to quantify the critical conditions whichgive rise to sediment motion. By considering steady conditions, a theoretical basis to determinethese conditions can be obtained looking at forces acting on a spherical sediment particle whichis going to be set in motion. When the forcing flow is oscillatory, as it happens the sedimentgrains are moved by surface waves, the destabilizing force can be evaluated by means of

FD =1

2ρU2cD

π

4d2 + ρ

dUdtcM

4

3πd3

8

and both the drag coefficient cD and the added mass coefficient cM depend on the Reynoldsnumber and the Keulegan-Carpenter number Kc of the phenomenon. However, if the ratiobetween the second term and the first term of the right hand side of the FD is evaluated

ρdUdtcM

43π d

3

812ρU2cD

π4d2

=4cM

dUdtd

3cDU2

its order of magnitude turns out to be inversely proportional to Kc = U0Td

where U0 and T arethe amplitude and the period of the velocity oscillations close to the bottom. Since the valueof the Keulegan-Carpenter number for sand under sea waves is usually large, it appears thatthe unsteadiness of the forcing flow play a minor role and a first approximation of the sedimenttransport under sea waves can be obtained by considering a sequence of steady flows.

In a simple model, which considers a steady flow, the destibilizing force due to the fluiddrag can be evaluated by means of

FD =1

2ρU2cD

π

4d2

where U is a characteristic velocity of the fluid near the bed. The stabilizing force is due to thefriction force which is proportional to the immersed weight decreased by the contribution dueto the lift force

FS = µs

[

ρg(s− 1)π

6d3 − 1

2ρU2cL

π

4d2

]


where µs (the static friction coefficient) is related to the internal friction angle. Forcing thebalance between these two forces, it is easy to obtain

U2

(s− 1)gd=

4

3

µs(cD + µscL)

Then, assuming that U is proportional to the shear velocity uτ =√

τ/ρ(

U2 = (αuτ )2 = α2τ/ρ

)

,it follows

θcrit =τcrit

(ρs − ρ) gd=

4

3

µsα2(cD + µscL)

The symbol θ denotes the so-called Shields parameter and the right hand side depends on theparticle Reynolds number Rp, since both cD and cL depend on the sediment Reynolds number.The critical value θcrit of the Shields paramater θ = τ

(ρs−ρ)gd obtained by means of the empirical

predictor (14.22) proposed by Soulsby & Whitehouse (1997) is plotted versus the Reynoldsnumber of the sediment particle Rp in figure 14.6.

θcrit =0.24

R3/2p

+ 0.055[

1 − exp(

−0.02R3/2p

)]

(14.22)

0.01

0.1

1

10

0.1 1 10 100 1000

θ crit

,θcr

it(s)

Rp

Soulsby & Whitehouse (1997)Soulsby (1997)

Engelund (????)Van Rijn (1984)

Figura 14.6: Critical values of the Shields parameter for the sediment to be transported (θcrit)

and put into suspension θ(s)crit.

However, Soulsby (1997) noticed that for small values of the particle Reynolds number, i.e.for small grain size, the value of θcrit never exceeds 0.3. To take into account this experimental


evidence, Soulsby (1997) proposed to use

θcrit =0.3

1 + 1.2R3/2p

+ 0.055[

1 − exp(

−0.02R3/2p

)]

(14.23)

which is also plotted in figure 14.6. To make the reader to realize that the predictions of thecritical value of the Shields parameter are affected by a large uncertainty, in figure 14.7 thevalue predicted by means of (14.23) is compared with that provided by

θcrit = 0.22R−0.6p + 0.06 exp

(

−17.77R−0.6p

)

(14.24)

which was proposed by Brownlie (1981) and later amended by Paker at al. (2003) dividing itsvalue by a factor 2.

0.01

0.1

1

10

0.1 1 10 100 1000

θ crit

Rp

Soulsby (1997)Parker et al. (2003)

Figura 14.7: Critical values of the Shields parameter for the sediment to be transported (θcrit)predicted by means of (14.23) and (14.24) divided by 2 as suggested by Parker et al. (2003).

If the bed is not horizontal but charaterized by a slope, the gravity force provides a furthercontribution which can be either stabilizing or destabilizing depending on the local slope. Assuggested by Soulsby (1997), the effects of the bottom slope can be taken into account relatingthe critical value of the Shields parameter for the sloping bed θcrit,α,β to the value for a horizontalbed θcrit. Let us denote with β the bed slope and with α the angle that the flow direction makeswith the upslope direction (see figure 14.8). Soulsby obtained

θcrit,α,βθcrit

=cosα sin β +

√

cos2 β tan2 φi − sin2 α sin2 β

tanφi(14.25)


x

zy

α

β

Figura 14.8:

where φi is the angle of repose of the sediment.Once the sediments are set into motion, they tend to roll, slide and jump on the sea bed

thus originating the so called bed load. However, when the flow is very strong, the sedimentparticles are entrained into the bulk flow and transported for long distance without any contactwith the bed, originating the so called suspended load. The suspended load can be empiricallyassumed to take place when the fall velocity of the sediment particles is smaller than the upwardcomponent of the turbulent fluctuations which are supposed to be proportional to the shearvelocity uτ . The critical condition for the inception of the suspended load is thus

u(s)τ,crit = ws (14.26)

Of course, if the sea bed is made up of a sediment mixture, (14.26) should be applied to eachgrain class and it is likely that only the finer fractions are put into suspension. Other authorssuggest criteria slightly different with respect to (14.26). For example Van Rijn suggests thatthe sediment is put into suspension when uτ is larger than a critical value given by

u(s)τ,crit

ws= 0.4 for Rp > 31.62 (14.27)

u(s)τ,crit

ws= 4R−2/3

p for Rp ≤ 31.62

Because uτ =√

τ/ρ and the dimensionless fall velocity ws/√

(s− 1)gd turns out to dependonly on Rp, the relationships (14.26),(14.27) can be transformed into relationships which give


a critical value θ(s)crit of θ for the initiation of the transport of the sediments in suspension. The

values of θ(s)crit given by (14.26),(14.27) are

θ(s)crit =

(

ws√

(s− 1)gd

)2

(14.28)

and

θ(s)crit = (0.4)2

(

ws√

(s− 1)gd

)2

for Rp > 31.62 (14.29)

θ(s)crit = 16R−4/3

p

(

ws√

(s− 1)gd

)2

for Rp ≤ 31.62

respectively and are plotted in figure 14.6 evaluating ws/√

(s− 1)gd by means of (14.14) .Of course, the slope of the bed affects also the critical value of the Shields parameter for theinitiation of the suspended sediment transport even though widely accepted approaches do notexist.

14.4 The sediment transport due to a current (the bed

load contribution)

The bed load takes place when i) the sea bottom is flat and the shear stress is relatively small,ii) when bedforms of small spatial scale (ripples) or large spatial scale (sand waves/dune) arepresent and the shear stress is relatively large, iii) for large values of the shear stress such thatthe bottom configuration is flat and the sediment transport very high (sheet flow condition).

Of course, the bed load provides the largest contribution to the sediment transport whenthe bottom shear stress is relatively small or when the sediment is characterized by a largediameter. When the bottom shear stress is relatively large and/or the sediment size quitesmall, the suspended load becomes larger than the bed load which in turn becomes negligible.

Fernandez-Luque (1974) analysed the bed load transport over a plane bed, close to thecritical conditions, looking at the dynamics of single particle and assuming that its motionreduces the fluid shear stress acting on the bed surface to the critical value. The reduction ofthe bed shear stress is due to the reaction force of the sediment particles on the fluid. If thesediment particles move with a steady velocity vP , the drag and lift forces acting on them canbe assumed to be

(FD, FL) =1

2ρ (cD, cL)π

d2

4(αuτ − vP )2 (14.30)

where uτ =√

τ/ρ. The stabilizing force can be evaluated by means of

FS = µd

[

π

6d3(s− 1)ρg − 1

2ρcLπ

d2

4(αuτ − vP )2

]

(14.31)


µd being the dynamic friction coefficient. A steady velocity is attained by a sediment particlewhen

1

2ρcD

πd2

4(αuτ − vP )2 = µd

[

π

6d3(s− 1)ρg − 1

2ρcLπ

d2

4(αuτ − vP )2

]

(14.32)

It follows1

2ρπd2

4(αuτ − vP )2 (cD + µdcL) = µd

π

6d3(s− 1)ρg (14.33)

(αuτ − vP )2 =4µdgd(s− 1)

3 (cD + µdcL)

vP = αuτ −√

4µdgd(s− 1)

3 (cD + µdcL)= αuτ

[

1 −√

4µdgd(ρs − ρ)

3 (cD + µdcL)α2τ

]

vP = αuτ

[

1 −√

θ0θ

]

where

θ0 =4µd

3α2 (cD + µdcL)

The dynamic friction coefficient µd turns out to be smaller than the static friction coefficientµs and it is possible to assume θ0 = 1

2θcrit. Moreover, assuming α = 10 as suggested by Fredsoe

& Deigaard (1992), it follows

vPuτ

= 10

(

1 − 0.7

√

θcritθ

)

(14.34)

Equation (14.34) is compared with the experimental measurements of Luque (1974) and Melond& Normann (1966) in figure 14.9. Then, the sediment transport rate Qb can be obtainedassuming that the bed load is originated by the transport of a fraction p of the particles whichare in the top (single) layer of the bed. Since the total number of grains on the bed surfacelayer per unit area is 1/d2, it follows

Qb =π

6d3vP

p

d2=π

6duτ10

(

1 − 0.7

√

θcritθ

)

p (14.35)

Hence, the dimensionless sediment transport rate Φ = Qb/√

(s− 1)gd3 is provided by

Φ =Qb

√

(s− 1) gd3=π

610(√

θ − 0.7√

θcrit

)

p (14.36)

An estimate of p can be obtained by assuming that the moving particles decrease the effectiveshear stress acting on the bed till the critical value is attained. In other words

τ = τcrit + nFD


Figura 14.9: Comparison between the particle velocity computed by means of (14.34) and thatmeasured during laboratory experiments.

Figura 14.10: Comparison between the probability p given by (14.37) and the data of FortCollins and Luque.

n being the moving particles per unit area and FD being approximated by

FD = ρg(s− 1)π

6d3µd or

FD(ρs − ρ) gd

=π

6d2µd

It followsθ = θcrit +

π

6µdnd

2 = θcrit +π

6µdp (14.37)

which is compared with experimental data in figure 14.10. Making use of the results obtained


so far and in particular ofπ

6µdp = θ − θcrit

it follows

Φ =10

µd(θ − θcrit)

(√θ − 0.7

√

θcrit

)

(14.38)

which is a formula valid only for small values of (θ − θcrit).For µd = 1, the relationship just obtained becomes close to the widely used formula of Meyer-

Peter & Muller (1948). However, both the formulae overestimate the sediment transport ratefor large values of θ.

Far from the critical conditions, the evaluation of the sediment transport rate induced bya steady current or by a slowly varying current, like a tidal current, can be made by means ofempirical formulae in the form

Φ = F (θ, θcrit) (14.39)

whereθ =

τ

(ρs − ρ)gd(14.40)

is the Shields parameter. When bottom forms cover the sea bed, it is almost always necessary (asin the formulae below) to use the ’skin friction’ component of θ. The ’skin friction’ componentis the bottom shear stress minus the contribution due to the presence of the bottom forms. Toevaluate the skin friction, there are empirical formulae proposed by different authors.

Some of the most common formulae used to evaluate the bed load rate are:

i) Formula of Meyer-Peter & Muller (1948)

Φ = 8 (θ − θcrit)3/2 (14.41)

with θcrit = 0.047

ii) Formula of Ashida & Michiue (1971)

Φ = 17(√

θ −√

θcrit

)

(θ − θcrit) (14.42)

iii) Formula of Nielsen (1992)

Φ = 12√θ (θ − θcrit) (14.43)

Needless to write that all the formulae provide the sediment transport rate only when θ islarger than θcrit while the sediment transport rate vanishes when θ ≤ θcrit.

Figure 14.11 shows a comparison of the different formulae while figure 14.12 shows theformula proposed by Nielsen (1992) along with some experimental measurements.


0.01

0.1

1

10

100

1000

0.1 1 10

Φ

θ

Nielsen (1992)Ashida & Michiue (1971)

Meyer-Petr & Muller (1948)

Figura 14.11:

The formulae described previously provide the sediment trasport rate for a horizontal bed.If a sloping bed is considered, it is possible to take into account the bed slope by modifying thecritical value of the bottom shear as suggested by by Fredsøe & Deigaard (1992).

14.5 The sediment transport induced by a current (the

suspended load contribution)

The most widely used model to evaluate the flux of sediment due to the suspended sedimentis the diffusive model which is based on the assumption the the concentration is so small toignore the effects of the sediments on the dynamics of the fluid. Moreover, the value of theconcentration should be so large to allow the use of the continuum approach to deal with thesolid phase: the smallest spatial scale which characterizes the fluid motion should be largeenough to contain a large number of particles such to define their average characteristics. Atlast, the motion of the sediment particles is assumed to differ from that of the water elementonly because of the settling velocity. In other words, the horizontal velocity of the sediment isassumed to be equal to the local velocity of the fluid while the vertical component is given bythe fluid velocity component plus the fall velocity.

To quantify the suspended load, it is necessary to determine the sediment concentrationwhich can be evaluated by means of an equation which is the result of a sediment balance andthe introduction of the flux of sediment q such that q ·n dS represent the volumetric sedimentflow rate through the surface dS.


Figura 14.12:

It is easy to show that∂c

∂t+∂qj∂xj

= 0 (14.44)

Since q is defined equal to cvP where vP denotes the velocity of the sediment particles andvP = v − wsk where k denotes the unit vector in the vertical direction (directed upwards), it


is possible to obtain∂c

∂t+ vj

∂c

∂xj− ws

∂c

∂x3

= 0 (14.45)

Moreover sediment particles are carried into suspension only when the flow regime is tur-bulent, hence, both the fluid velocity and the sediment concentration can be decomposed inan average part (which is denoted by an angle bracket) and an oscillatory random component(which is denoted by an apex). Using the decomposition and applying the average operator tothe concentration equation, it is possible to obtain

∂〈c〉∂t

+ 〈vj〉∂〈c〉∂xj

− ws∂〈c〉∂x3

= −∂〈v′kc

′〉∂xk

(14.46)

which poses a ’closure’ problem.The classical approach to overcome the closure problem assumes that

−∂〈v′kc

′〉∂xk

=∂

∂xk

(

DT∂〈c〉∂xk

)

(14.47)

where the diffusive coefficient DT is often assumed to be coincident with the eddy viscosity andit might depend on x1, x2, x3 and t.

It follows that the sediment concentration is usually obtained by solving

∂〈c〉∂t

+ 〈vj〉∂〈c〉∂xj

− ws∂〈c〉∂x3

=∂

∂xk

(

DT∂〈c〉∂xk

)

(14.48)

Then, it is necessary to introduce appropriate boundary conditions. On the free surface, theflux of sediment should vanish while at a distance za from the bottom the sediment concentrationis forced to be equal to 〈ca〉, a value which is assumed to depend on the local value of the Shieldsparameter, even though, for unsteady flow, it is necessary to modify such a boundary conditionto consider the possible presence of overloaded flow.

Different empirical formulae are proposed to evaluate za and 〈ca〉. For example Garcia &Parker (1991) proposed

〈ca〉 =AZ5

u

1 + A0.3Z5u

(14.49)

za = 0.05h (14.50)

where

Zu =u′τwsR0.6p A = 1.3 × 10−7 (14.51)

while Van Rijn (1984) suggested

〈ca〉 = 0.015d

aR0.2p

(

θ′ − θcritθcrit

)1.5

(14.52)


za = d90 if d90 > 0.01h (14.53)

za = 0.01h if d90 ≤ 0.01h

and Zyserman & Fredsoe (1994) propose

〈ca〉 =0.331 (θ′ − 0.045)1.75

1 + 0.3310.46

(θ′ − 0.045)1.75(14.54)

za = 2d50

Once the sediment concentration c is known, the sediment flux through any surface S can becomputed by means of

∫

S

[〈c〉 (v − wsk) +DT∇〈c〉] dS (14.55)

If a steady uniform current is considered, the concentration c can be determined by solving

−ws∂〈c〉∂x3

= −∂〈v′3c

′〉∂x3

=∂

∂x3

(

DT∂〈c〉∂x3

)

(14.56)

which leads to

DT∂〈c〉∂x3

+ ws〈c〉 = 0 (14.57)

where the boundary condition at the free surface is used which forces the vanishing of thevertical flux of sediment.

Equation (14.57) can be further integrated once the sediment diffusivity DT is fixed. Assu-ming

DT = νT = kuτx3

(

1 − x3

h

)

(14.58)

x3 being the vertical coordinate, it follows

d〈ca〉〈ca〉

= − wskuτ

dx3

x3

(

1 − x3

h

) (14.59)

Then

〈ca〉 = Costant

(

1 − x3/h

x3/h

)Z

(14.60)

whereZ =

wskuτ

(14.61)

is the so-called Rouse number.The contribution of the suspended sediments to the sediment transport rate can be easily

evaluated once the constant appearing into (14.60) is determined using the bottom boundarycondition

〈ca〉 = ca

[

za (h− x3)

x3 (h− za)

]Z

(14.62)


As already pointed out, assuming that the sediment trasport be in the direction of the current,the suspended load can be evaluated once the vertical distribution of sediment concentrationis known

Qs =

∫ η

za

[

v1(x3)c(x3) + νT∂c

∂x1

]

dx3 (14.63)

where v1(x3) and c(x3) describe the velocity profile and the distribution of sediment concentra-tion, za is the reference level where the bottom boundary condition for c is forced, and η is thefree surface elevation. Needless to write that in (14.63) x1 is the coordinate in the direction ofthe current. However, quite often, the diffusive contribution to Qs is neglected.

The total sediment transport rate Qt induced by a current is the sum of the bed load andthe suspended load which, sometimes, can be much larger than the former.

14.6 Slope effects on the sediment transport due to a

current

When a sloping bottom is considered, the sediment transport rate formulae should be correctedto take into account the effect of gravity which tends to drift particles in the downslope direction.

In the intrinsic orthogonal coordinate system (s, n), with s aligned with the bottom stressand n is the coordinate normal to it in a plane tangent to the bottom, simple dimensionalarguments coupled with linearization immediately lead to state that the sediment transportrate over a sloping bottom is equal to that characterizing the flat bottom case plus the followingcorrection term

Q(c,slo) = (Q(c,slo)s , Q(c,slo)

n ) = Qb(Gss∂zb∂s

,Gnn∂zb∂n

) (14.64)

where G is a dimensionless second order 2-D tensor.Experimental observations of various authors provide estimates for the components of G

which read

Gss = −θcritµd

dΦ

dθGnn = −Φ

r√θ

(14.65)

Gsn = Gns = 0 (14.66)

where Φ is the dimensionless sediment transport rate, µd is the dynamic friction coefficient ofthe sediment and r is an empirical factor with

µd = 0.4 ÷ 0.7 dynamic friction coefficient; r = 0.5 ÷ 0.6.

(Talmon et al., 1995).


14.7 The sediment transport due to waves

As already pointed out, the Stokes number of sediment grains at the bottom of sea waves ismuch smaller than one and it is reasonable to evaluate the instantaneous sediment transport ratedue to waves with the formulae used for steady currents even though this procedure providesonly an approximation. It follows that the sediment trasport due to waves, averaged over thewave period, vanishes if the wave is symmetric. However, the sediment trasport rate averagedover a semi-cycle might be of interest for some applications. Indeed, it is common to evaluatethe wave-averaged sediment transport rate as difference between the sediment transport rateaveraged over a semi-cycle and the sediment transport rate averaged over the following semi-cycle, both estimated by simple empirical formulae. To give an example, for low values of theShields parameter such that the sediment transport rate is mainly due to the bed load, thesediment transport rate averaged over a semi-cycle can be evaluated by means of the formulaproposed by Soulsby (1997)

Φ1/2 = 5.1 (θw − θcrit)3/2 (14.67)

where θw is the amplitude of the oscillation of θ due to the presence of waves. When ripplesare present, only the skin friction contribution should be considered. The value of θw can beobtained by using the friction factor fw defined by

τw = fw1

2ρU2

w (14.68)

where Uw is the maximum value of the velocity induced by the wave during the half periodconsidered. Of course the value of fw depends on the relative roughness and the Reynoldsnumber. Relationship (14.67) is obtained by integrating relationship (14.43) over half cycleand dividing by T/2.

Another relationship was proposed by Van Rijn (1991) and provides the total sedimenttransport due to the waves (bed load plus suspended load). In fact, even though the sedimentmay be picked up from the bottom and carried into suspension by the action of the waves,the major part of the sediment in suspension is confined to a region close to the bed andit is reasonable to compute the wave-related sediment transport rate by a simple formula inanalogy with the bed load. The sediment transport rate averaged over half-period qw,1/2 canbe computed, as proposed by Van Rijn (1991), by means of the following formula

Qw,1/2 =0.03

R0.2p

d50Uw

(

θ′w − θcritθcrit

)3/2

(14.69)

where Uw is the peak orbital velocity.The net time-average wave-induced sediment transport rate in asymmetrical conditions is

thus given byQw,net = Qw,max −Qw,min (14.70)


where the maximum Uw,max and minimum Uw,min values of Uw to be used in (14.70) can becomputed on the basis of the second order wave theory and turn out to be:

Uw,max =Hω

2 sinh(kh)

[

1 +3

16

kH

sinh3[kh]

]

Uw,min =Hω

2 sinh(kh)

[

1 − 3

16

kH

sinh3[kh]

]

where h,H, ω, k are the water depth, the wave height, the angular frequency and the wavenum-ber of the surface wave.

Even though the formulae just described provide an estimate of the amount of sedimenttransported by surface waves, its accurate evaluation requires a more detailed study of the flowgenerated close to the bottom by the propagation of surface waves. Recently, Blondeaux et al.(2012) determined the flow in the boundary layer at the bottom of a monochromatic propagatingwave, taking into account nonlinear effects and evaluated also the sediment transport rate.The analysis is able to predict the laboratory measurements of Ribberink et al. (????) and inparticular the direction of sediment transport which can be in the direction of wave propagationor in the opposite direction, depending on the parameters of the problem. The details of theanalysis are given in chapter ???

14.8 Sediment transport due to waves and currents

When the coastal region is characterized by the simultaneous presence of waves and currents,the oscillatory flow induced by the waves tends to mobilise the sediment which, then, aretransported by the steady component of the velocity field.

The flow generated by the interaction of a current with surface waves is quite complexbecause the oscillatory flow induced by the waves causes averaged momentum fluxes whichaffect the current and viceversa (see chapter ???). Hence, for engineering applications, simpleempirical formulae are used to estimate the sediment transport.

A formula which provides the sediment transport rate induced by waves and currents is thatproposed by Soulsby (1997), where the following quantities appear

φ = angle between the current direction and the direction of wave propagation

Qt = volumetric sediment transport rate per unit width due to both the bed load and the suspended load

Qtx = volumetric sediment transport rate per unit width in the direction of the current

Qty = volumetric sediment transport rate per unit width in the direction orthogonal

to the current

θm = value of θ averaged over the period of the wave


θw = amplitude of the oscillatory component of θ due to the wave

θmax =

√

(θm + θw cosφ)2 + (θw sinφ)2

The approach assumes that the oscillatory flow induced by the waves tends to mobilise thesediment which, then, are transported by the steady component of the velocity field.

The formula readsΦx1 = 12

√

θm (θm − θcrit) (14.71)

Φx2 = 12 (0.95 + 0.19 cos 2φ)√

θwθm (14.72)

Φx = maximum value of Φx1and Φx2 (14.73)

Φy =12 (0.19θmθ

2w sin 2φ)

θ3/2w + 1.5θ

3/2m

(14.74)

where Φx and Φy vanish if θmax is smaller than θcrit. Moreover, in the previous equation it isnecessary to use only the skin friction if ripples are present. Equation (14.71) is for a currentdominated case while (14.72) is for a wave dominated case.

There are many other empirical approaches, e.g. the formulae of Soulsby and Van Rijn(????) and Bailard (1981). Hereinafter, let us describe the approach by Van Rijn (1991) whichis used in chapter ?? to investigate the process which leads to the formation of large scalebedforms in coastal areas where tidel currents and wind waves coexist. Improvements of thisapproach are described in Van Rijn (2004, ????).

The formula of Van Rijn (1991)

The model proposed by Van Rijn (1991) suggests to evaluate the sediment transport inducedby currents and by waves, separately. When waves and current coexist, the total time-averagedsediment transport rate can be obtained by vector addition of the sediment transport rate dueto the current and that induced by the waves which point in the current direction and in thedirection of the wave propagation, respectively.

Current-related sediment transport

Bedload

In a coastal region characterized by 1) a water depth h, 2) a cohesionless bed materialwith a given value of d50 and 3) and by the simultaneous presence of waves and currents, thetime-averaged bed load rate Qb due to the current is modified by the presence of waves andcan be evaluated as

Qb =0.25

R0.2p

d50u′τ,c

(

θ′cw − θcritθcrit

)1.5

(14.75)

where θ′cw is the Shields parameter obtained by time-averaged effective bed shear stress whichcan be evaluated as:

τ ′cw = τ ′c + τ ′w


The value of the effective wave related bed shear stress τ ′w can be obtained by

τ ′w = µwτw (14.76)

where

τw =1

2ρfw (Uw)2 (14.77)

and the efficiency factor µw is provided by

µw =0.6

R23p

(14.78)

The friction factor fw can be estimated using

fw = exp

[

−6 + 5.2

(

Uwωks,w

)−0.19]

(14.79)

where ω is the angular frequency of the wave and ks,w is the wave related bed roughness providedby 3∆r in the ripple regime and by 30d90 in the sheet flow regime. In (14.45) θcrit is the criticalvalue of the Shields parameter which can be computed for example using the formula proposedby Soulsby & Whitehouse (1997)

Of course the direction of the current-related bedload is that of the current.The current-related grain bed shear velocity u′τ,c is provided by:

u′τ,c =

√

τ ′cρ

(14.80)

where τ ′c is the effective bed shear stress due to the current, given by

τ ′c = µcαcwτc (14.81)

with

µc =

(

C

C ′

)2

= efficiency factor

C = 5.75 log10

(

12h

ks

)

C ′ = 5.75 log10

(

12h

3d90

)

αcw =

[

ln (δw/ka)

ln (90δw/ks)

]2


δw = 0.072Aδ

(

Aδks

)−0.25

= thickness of the wave boundary layer,

Aδ = the peak value of the near bed orbital excursion

The bottom shear stress τc due to the current and appearing in (14.81) turns out to be:

τc = ρU

2

C2a

(14.82)

where

Ca = 5.75 log10

(

12h

ka

)

U = depth-averaged current velocity

The quantities ks and ka, current-related bed roughness height and apparent current-relatedbed roughness height respectively, are computed as written below.

In the rippled bed regimeks = 3∆r (14.83)

∆r being the ripple height. In the sheet flow regime

ks = 30d90 (14.84)

Then, the apparent current related bottom roughness ka can be computed using

ka = ks exp

(

Uw

U

)

(14.85)

with a maximum value of ka equal to 10ks. In (14.85) Uw is the amplitude of the velocityoscillations induced by the waves close to the bottom. Of course to estimate ∆r it is necessaryto use a ripple predictor (e.g. Soulsby & Whitehouse, 2005) or to have field measurements ofripple height.

Suspended load

The suspended load rate Qs due to currents alone can be evaluated by means of

Qs = (Fc + Fw)Uhca (14.86)

where

Fc =(a/h)ZC − (a/h)1.2

(1.2 − ZC) (1 − a/h)ZC(14.87)

Fw =(a/h)ZW − (a/h)1.2

(1.2 − ZW ) (1 − a/h)ZW(14.88)


ca =0.015

R0.2p

d50

aT 1.5 (14.89)

ZC =ws

βκuτ,ccurrent-related suspension number (14.90)

ZW = α

(

ws

U

)0.9(UTwHw

)1.05

wave-related suspension number

κ = 0.4 Von Karman constant

ws = fall velocity in clear water

uτ,c =√

τc/ρ

α = 7 if h > 100δs

α = 0.7

(

h

δs

)0.5

if h < 100δs

a = 0.5∆∗r if ripples are present

a = max20d∗50, 0.01m. sheet flow regime

δs being the thickness of the mixing layer near the bed, which can be evaluated by

δs = 3∆r in the ripple regime

δs = δw sheet flow regime

δw = 0.072Aδ

(

Aδkw

)−0.25

In the previous relationships, Hw and Tw are the significant wave height and peak wave period

respectively, β is provided by β = 1 + 2(

ws

uτ

)2

with a maximum value of β equal to 1.5. The

relationship which provides β is taken from Van Rijn (2007) since β is assumed to be about 1in Van Rijn (1991). Moreover, the reference level a is provided by:

Wave-related sediment transport

When a monochromatic sea wave is added to the current, the major part of the sedimentin suspension is confined to a region close to the bed, hence it seems reasonable to computethe wave-related sediment transport rate by a simple formula in analogy with the bed load.Considering the sediment transport rate averaged over half-period Qw, the following formulacan be used

Qw =0.03

R0.2p

d50Uw

(

θ′w − θcritθcrit

)1.5

(14.91)

where Uw is the peak orbital velocity.


The net time-average wave-induced sediment transport rate in asymmetrical conditions isthus given by

Qw,net = Qw,max −Qw,min (14.92)

where the maximum and minimum values of Uw, to be used in (14.92), can be computed onthe basis of the second order wave theory and turn out to be:

Umax =Hω

2 sinh(kh)

[

1 +3

16

kH

sinh3[kh]

]

Umin =Hω

2 sinh(kh)

[

1 − 3

16

kH

sinh3[kh]

]

Even though Van Rijn (1991) did not considered the effects of the bottom slope on sedimentdynamics, these effects play a fundamental role in the dynamics of the bottom forms like sandwaves and sand banks and should be properly taken into account to have a reliable descriptionof the dynamics of these morphological patterns.

The most common models assume that the bed slope affects only the sediment moving closeto the bed. Therefore, with reference to the transport formula by Van Rjin (1991) and usingthe same symbols, two further terms should be added to the total transport.

Needless to write that the terms described below should be considered only when the Shieldsparameters θ′c or θ′w are larger than the critical value θcrit, otherwise they vanish.

The bed slope effect on current-related sediment transportFollowing well established procedures used in channel flows, slope effects on current-related

sediment transport can be quantified as follows:

Q(c,pen) = (Q(c,pen)s , Q(c,pen)

n ) = Qb(Gss∂zb∂s

,Gnn∂zb∂n

) (14.93)

In (14.93) zb(x, y) indicates bed elevation, s is the coordinate in the direction of the current-related bed shear stress and n is the coordinate normal to it in a plane tangent to the bottomand the two components of q(c,pen) are in the s and n directions, respectively.

Moreover:

Gss = −θcritµd

0.25

R0.2p

(3θ′ + θ′w − θcrit)

2√θ′θcrit

Gnn = −r0.25

R0.2p

√θ′

θcrit(θ′ + θ′w − θcrit)

with

µd = 0.4 ÷ 0.7 dynamic friction coefficient; r = 0.5 ÷ 0.6.

The bed slope effects on wave-related sediment transport


There is no well accepted relationship in the literature to account for slope effects on wave-induced sediment transport. However, it seems reasonable to assume that such effects aresimilar to those previously quantified.

Q(w,pen) = (Q(w,pen)s , Q

(w,pen)n ) = Qw(Gss

∂zb∂s

, Gnn∂zb∂n

) (14.94)

In (14.94) s is the coordinate in the direction of the wave-related bed shear stress and n isthe coordinate normal to it in a plane tangent to the bottom. The two components of q(w,pen)

are in the s and n directions, respectively.Moreover

Gss = − 0.03

µR0.2p

√

2

µwfw

(3θ′w − θcrit)

2√

θ′w

Gnn = −0.03r

R0.2p

√

2

µwfw

(θ′w − θcrit)

θcrit

14.9 Sediment continuity equation and bottom time de-

velopment

When the water motion is strong enough to move the sediment, the sea bottom is subject toerosion/deposition processes Once the sediment transport rate is computed, the variations ofthe bottom elevation zb can be evaluated by mean of a balance of the solid phase:

∂zb∂t

= − 1

1 − n

(

∂Qtx

∂x+∂Qty

∂y

)

(14.95)

where:- zb(x, y, t) is the bottom elevation;- n is the porosity of the bottom material;- Qtx, Qty are the horizontal components of the total sediment transport rate per unit width.The physical interpretation of Exner equation is simple: when the sediment transport rate

increases in the streamwise direction, the bottom is eroded (in this case ∂zb/∂t turns out to benegative) and viceversa.

Let us consider the definition of the flux of sediment q per unit area and make a simplemass balance over an infinitesimal volume. It is straightforward to obtain

∂c

∂t+ ∇ · q = 0 (14.96)

By integranting (14.96) from the reference level where the reference concentration is forced upto the free surface, it is easy to obtain

∫ η

za

∂c

∂tdx3 +

∫ η

za

∂q1∂x1

dx3 +

∫ η

za

∂q2∂x2

dx3 + q3|ηza= 0 (14.97)


By using Leibniz rule, it follows

∂

∂t

∫ η

za

cdx3 + c|za

∂za∂t

− c|η∂η

∂t+

∂

∂x1

∫ η

za

q1dx3 + q1|za

∂za∂x1

− q1|η∂η

∂x1(14.98)

+∂

∂x2

∫ η

za

q2dx3 + q2|za

∂za∂x2

− q2|η∂η

∂x2− q3|za + q3|η = 0

The boundary conditions at the bottom and at the free surface force

q · n|x3=za = vnza (c− cM) |x3=za, q · n|x3=η = vnηc|x3=η, (14.99)

where vnza and vnη are the normal components of the velocity of the bottom and of the freesurface, respectively, and cM is the concentration of the sediments resting on the bottom. Itfollows that

vnza =dnza

dt, vnη =

dnηdt

Moreover, since

dFza =∂Fza

∂tdt+

∂Fza

∂nza

dnza = 0, dFη =∂Fη∂t

dt+∂Fη∂nη

dnη = 0 (14.100)

whereFza = x3 − za(x1, x2, t), Fη = x3 − η(x1, x2, t)

we obtain

vnza =∂za/∂t

√

1 + (∂za/∂x1)2 + (∂za/∂x2)2=∂za/∂t

|∇Fza|(14.101)

vnη =∂η/∂t

√

1 + (∂η/∂x1)2 + (∂η/∂x2)2=

∂η/∂t

|∇Fzη|Indeed from (14.100), it follows

dnzadt

= −∂Fza∂t

/

(

∂Fza∂nza

)

= −∂Fza∂t

/∇Fza·nza = −∂Fza∂t

/

(∇Fza · ∇Fza|∇Fza|

)

= −∂Fza∂t

/ |∇Fza|

and a similar relationship for dnη

dt

Making use of (14.100) and (14.101) and taking into account that

n|x3=za =

(

− ∂za

∂x1,− ∂za

∂x2, 1)

|∇Fza|, n|x3=η =

(

− ∂η∂x1,− ∂η

∂x2, 1)

|∇Fη|,


(c|x3=za − cM)∂za∂t

=

[

−q1∂za∂x1

− q2∂za∂x2

+ q3

]

x3=za

(14.102)


c|x3=η∂η

∂t=

[

−q1∂η

∂x1− q2

∂η

∂x2+ q3

]

x3=η

Hence, it follows

∂

∂t

∫ η

za

cdx3 + cM |za

∂za∂t

+∂

∂x1

∫ η

za

q1dx3 +∂

∂x2

∫ η

za

q2dx3 = 0 (14.103)

If the flux of sediment (Qs1, Qs2) from the bottom up to the free surface is introduced, equation(14.103) takes the form

∂

∂t

∫ η

za

cdx3 + (1 − n)∂za∂t

+∂Qs1

∂x1

+∂Qs2

∂x2

= 0 (14.104)

since cM = (1 − n). If the flux of sediment between the bed level and the reference level (thebed load) is taken into account, equation (14.104 becomes

∂

∂t

∫ η

za

cdx3 + (1 − n)∂za∂t

+∂Qt1

∂x1

+∂Qt2

∂x2

= 0 (14.105)

where Qt1, Qt2 are the components of the total load which is the sum of the suspended loadQs1, Qs2 and the bed load Qb1, Qb2 along the x1- and x2-axes. It is worth pointing out that for adilute suspension (values of c smaller than 10−2) the term ∂

∂t

∫ η

zacdx3 turns out to be negligible

and (14.104) becomes

(1 − n)∂za∂t

+∂Qt1

∂x1

+∂Qt2

∂x2

= 0 (14.106)

14.10 One line model of the time development of the

coastline

In one of the previous chapters, it is shown that propagating waves give rise to stresses (’ra-diation stresses’) which, in the breaking region give rise to a variation of the mean water level.Moreover, when the crests of the waves are not parallel to the coast, a longshore steady currentis generated which might induce a significant sediment transport. The total longshore sedimenttransport QL, integrated over the width of the surf zone, can be evaluated with the followingempirical formula

QL =0.023

√gH

5/2sb sin(2αb)

(s− 1)(14.107)

whereHsb = significant wave height at the breaker line

αb = angle between the wave crests at the breaker line and the coast line


This formula is relatively simple and improvements can be introduced to take into the sedimentsize (coarse/fine sediment), the beach slope and other effect which are neglected in (14.107).

Of course, when the value of QL is not constant along the coast, erosion and depositionprocesses take place and the coastline can advance or retreat.

A simple model to evaluate the time development of the coastline can be obtained by meansof a mass balance which neglects the cross-shore sediment transport and links the forward andbackward movements of the coastline to the sediment transport rate parallel to the coastline.By considering a section of the coastal region of infinitesimal width dx (see figure 14.13) anddenoting with QL the sediment transport rate in the longshore direction ([QL] =m3/s), a simplemass balance forces the volume of sediment stored within the coastal slice to be

dV = (QL(x, t) −QL(x+ dx, t)) dt = dV (14.108)

where dx, dt and dV are assumed to be infinitesimal quantities.Taking into account that dV can be written as

dV = Ddydx (14.109)

where D is a length scale such that Ddy is equal to the volume eroded from or deposited onthe beach (the value of D should take into account also the porosity of the bottom material).By combining (14.108) with (14.109) and expanding QL(x+ dx, t) as a Taylor expansion, it iseasy to obtain

−∂QL

∂xdxdt = Ddydx (14.110)

where terms of order dx2 are neglected. From (14.110), it follows

∂y

∂t= − 1

D

∂QL

∂x(14.111)

The previous equation, coupled with a formula which provides the longshore sediment trans-port rate as function of the angle which the waves for with the coastline (see equation (14.107)),allows to compute the time development of the coastline.

Let us now consider a few simple examples. In the first example, we consider the changesinduced in a straight beach by the construction of a series of groins. The values of the parametersare fixed to simulate the results of a test in a wave basin. In fact, the wave height at the breakerline is equal to 4 cm and the angle formed by the wave crests with the initial breaker line isequal to −4. Moreover, in the physical model the groin distance is equal to 30 cm. Figure14.14 shows the time development of the coast line computed by the numerical integration ofthe one line model performed with a finite difference approach such that the value of ∆y, takingplace at the generic coordinate x after a time interval ∆t, is given by

∆y = −QL(x+ ∆x, t) −QL(x− ∆x, t)

2∆xD∆t (14.112)


QL

D

xy

y

x

dd

Figura 14.13: Sketch of a slice of the nearshore region where a sediment balance is applied

In figure 14.14 also the experimental measurements of Price et al. (1973) are plotted and showthat the one line model can describe the gross features of the phenomenon. Since the wavecrests are not parallel to the shore but they form a small angle of −4, at the initial stage auniform longshore sediment transport is generated which is directed from the right to the left.However, the groin on the left blocks the sediment flux while the groin on the right does notallow any flux of sediment to enter the region between the groins. Hence, the sand is depositedon the left of the domain and erosion of the beach takes place on the right till the coastlineturns out to be parallel to the wave crests and the sediment transport vanishes everywhere.

In the second example, the time development of a strainght beach between two rocky head-lands is computed assuming that the wave refraction causes the breaker line to have a parabolicform and an initial minimum distance from the shore equal to 50 m. Figure 14.15 shows thecoastline position at different times considering a wave height at the breaker line equal to 2 mand a distance between the headlands equal to 1 km. Also in this case, the coastline reorientsitself till its shape becomes congruent with the shape of the breaker line and the sedimenttransport rate vanishes everywhere along the coastline. In both the examples, an appropriatevalue of D is chosen to have a correct morphodynamic temporal development.


-1

-0.5

0

0.5

1

0 5 10 15 20 25 30

y [m

]

x [m]

initial costline

after 1 hour

after 3 hours

after 6 hours

Figura 14.14: Beach reorientation next to a series of groins due to an oblique wave approachto an initially straight coastline. Comparison between the numerical predictions obtained bymeans of an one line method (continuous lines) and the laboratory observations of Price et al.(1973) (dots). The experimental data have been obtained by figure 10-6 of the book of Komar(1976)

-100

-50

0

50

100

150

200

250

300

0 200 400 600 800 1000

y [m

]

x [m]

initial costline

after 1 day

after 5 days

after 15 days breaker line

Figura 14.15: Time development of an initially straight beach between two rocky headlandswhen the breaker line has a curved shape.

Capitolo 15

COHESIVE SEDIMENTTRANSPORT

15.1 Introduction

The sediments, characterized by a size falling in the range of clay, have strong interparticle forcesbecause of the ionic charge of the surfaces of the particles. In fact the surface area per unitvolume (i.e. specific surface area) increases when the grain size decreases and the interparticleforce becomes as large as the gravitational force and even larger for the smallest sediments. Itfollows that the dynamics of sediment particles with a size smaller than 2µm (clay) is oftencontrolled by interparticle and hydrodynamic forces and they are considered cohesive sediments.On the other hand, sediments of size larger than 60µm behave like non-cohesive sediments sincetheir dynamics is dominated by gravitational and hydrodynamic forces. In practical problems,both silt and clay are considered to be cohesive sediments because silt (2µm - 60µm), whichfalls between cohesive and non-cohesive sediment, has cohesive characteristics which are mailyinduced by the presence of a small amount of clay but larger than about 10%. Often, cohesivesediments are a mixture of inorganic minerals (e.g. silica, kaolinite, quartz, carbonates, feldspar)and organic material (e.g. plant and animal detritus) (Hayter, 1983).

Because of the interparticle forces, the particles of cohesive sediment do not behave asindividual particles but tend to form aggregates which are named flocs, the size and settlingvelocity of which are much larger than those of the individual particles.

Cohesive sediments are a concern in many water bodies because they are closely relatedto water quality. Indeed, many pollutants, such as heavy metals, pesticides and nutrients areusually adsorbed by cohesive sediments. Moreover, cohesive sediments themselves are a concernfor water quality because the turbidity, caused by cohesive sediments in suspension, can limitthe penetration of light in water bodies and affect the aquatic life.

Before considering the modelling of the transport of cohesive sediments, let us describesome of the properties and phenomena which characterize the dynamics of cohesive sediments.However, let us point out that the description of the various properties and phenomena is largely

271

CAPITOLO 15. COHESIVE SEDIMENT TRANSPORT 272

empirical. Moreover, most information is based on laboratory experiments which, however, areoften not fully representative of actual phenomena. Indeed the biogenic mechanisms and theorganic materials are altered by the deposition and the consolidation history of the bed andthe phenomena observed and measured in laboratory are different from those taking place innature. Hence, in-situ measurements are essential to identify the basic parameters of the mostdominant phenomena at a given site and to reduce the overall modeling errors.

Cohesion

First, let us point out that a cohesive soil sample with a low water content can stand non-vanishing values of the shear stresses (τ) without failure. If a normal pressure (p) is applied,the value of the shear stress which leads to the sample failure turns out to be τ = τy + p tanφ,where τy is the yield stress and φ is the angle of internal friction (law of Coulomb).

The yield stress is generally interpreted as the cohesion of the sample. Thus, a cohesivesediment sample is able to withstand a finite shear stress even when p = 0 (no deformation). Onthe other hand, the angle of internal friction represents the mechanical resistance to deformationby friction and interlocking of the individual particles.

15.2 Cohesive Sediment Processes

15.2.1 Aggregation and settling velocity

Clay particles in close proximity fell both repulsive and attractive forces. The repulsive forcesare induced by clouds of positive ions which surround the sediment particles, negatively charged,and cause particles to repell each other. The attractive forces are due to the electrical fieldsformed by the dipoles of the individual elements. The final result can be either attraction orrepulsion depending on the number of positive ions in the water and the distance between theparticles. Usually, in fresh waters characterized by the presence of only a few positive ions, therepulsive forces between the negatively charged particles are larger than the attractive forcesand the sediment particles repel each other. In saline waters, the attractive forces are largerthan the repulsive forces because of the presence of a large number of positive sodium ionswhich form positive clouds of cations (positive ions) around the negatively charged sedimentparticles. It follows that the particle tend to aggregate and generate flocs. Other forces whichcontribute to the aggregation of sediment particles and to the generation of flocs are chemicalforces. Moreover, the presence of organic materials significantly intensifies the flocculationprocess, because of the binding properties of the organic materials.

Of course, the size of the flocs depends on the type of sediment, the type and concentrationof ions in the water and, more importantly, on the turbulence level (Mehta et al. 1989). Indeed,because of the turbulent stresses, the flocs break up and the size, density and strength of theflocs is the results of a dynamic equilibrium between aggregation and break-up. Cohesion makesthe sediments to form larger flocs, since the flocs which collide with other sediments or other


flocs adhere to them. On the other hand, the flocs become smaller when they are broken upby turbulent stresses.

Experimental observations show the existence of macroflocs, characterized by sizes fallingin the range (0.1, 1) mm, miniflocs with sizes falling in the range of (0.01, 0.1) mm and singleparticles, smaller than about 0.01 mm (see for example figure 15.1)

When the flocs become larger, their size increases but the density becomes smaller. Indivi-dual particles have a density of about 2600 kg/m3 but the density of large flocs (e.g. of about1 mm) falls in the range of (1, 10) kg/m3 in excess of the fluid density, because the flocs consistalso of (pore) fluid. The particle sedimentation is largely affected by the formation of flocs and

Figura 15.1: Particle images and displacement vectors from PTV analysis. Particle displace-ments are indicated from two superimposed image frames (separated by 0.375 sec). Particleimages are negative representations of the raw images with logarithmic intensity scaling. Par-ticle intensity from the first image is decreased by 25 percent to better indicate direction ofmotion. Vector lengths are scaled for display purposes and do not correspond to the lengthscale provided (from Fine sediment dynamics in dredge plumes, PhD Dissertation, presentedto the Faculty of the School of Marine Science, the College of William and Mary by StanleyJarrell Smith II).

the aggregation process. Indeed, the weight of an individual particle is not able to overcomethe forces which tend to keep the particle in suspension (turbulence forces) and to cause its


settling while the size and density of the flocs cause a settling velocity which is also affected byhinderance effects, which are due to the upward velocity induced by the other falling flocs.

It follows that an important parameter in the study of the dynamics of cohesive materialsis the settling velocity of the flocs. Laboratory and field data show that the settling velocityof the flocs is strongly related to the concentration of sediments and the salinity of the water,because these two quantities strongly affect the size of the flocs. In saline water, the settlingvelocity increases when the concentration increases up to about 1 g/l because of the flocculationphenomanon. When the sediment concentration becomes larger than about 10 g/l, the settlingvelocity starts to decrease with increasing concentrations because of the hindered settling effect.In fact the settling velocity of the flocs reduces because of the upward flow of fluid induced bythe falling flocs. When the concentration is very large, the upward flow can be so strong thatthe upward drag force on the flocs becomes equal to the downward gravity force and a dynamicequilibrium with no average displacement of the flocs. This state, which occurs close to bed, iscalled fluid mud. The increase of the settling velocity for increasing values of the concentration,for low concentrations, and then the decrease for high concentrations is confirmed by in-situmeasurements from all over the world (see figure 15.2 taken from Van Rijn (1993)).

Figura 15.2: The influence of sediment concentration on the settling velocity (source: Van Rijn,1993)


There are different empirical relationships to determine the settling velocity of the flocs.Nicholson and O’Connor (1986) used the following relationship to evaluate the settling velocitytaking into account hinderance effects

ws = A1cB1s for cs ≤ cH (15.1)

ws = A1cB1s [1 − A2 (cs − cH)]B2 for cs > cH

where the constants appearing in (15.1) should be experimentally determined and depend onthe sediment type and water salinity. Using cH = 25 g/l, as onset concentration of hinderedsettling, A1 = 6.0 × 10−4 m4/kg/s, A2 = 1 × 10−2 m3/kg, B1 = 1 and B2 = 5, the value of wsas function of cs is plotted figure 15.3. Burban et al. (1990) linked the settling velocity with

0.01

0.1

1

10

100

10 100 1000 10000 100000

ws

[mm

/s]

cs [mg/l]

cH

Figura 15.3: Settling velocity plotted versus sediment concentration (formula of Nicholson andO’Connor (1986).

the median floc diameter dm using laboratory data

ws = adbm (15.2)

with a = B1 (csτ)−0.85 , b = − [0.8 + 0.5 log (Csτ − B2)] where cs is the cohesive sediment

concentration (g/cm3) , τ is the fluid shear stress (dyne/cm2), dm is the median floc diameter


(cm) and B1 and B2 are experimentally determined constant equal to 9.6×10−4 and 7.5×10−6

respectively.As already pointed out, the settling velocity of the flocs is usually site-specific and should

be determined by in-situ experiments. A useful relationship which can be used to approximatethe experimental data is that plotted in figure 15.4, where the values of (csi, wsi) should beproperly chosen.

0.01

0.1

1

10

100

10 100 1000 10000 100000

ws

[mm

/s]

cs [mg/l]

(cs1,ws1)

(cs2,ws2) (cs3,ws3)

(cs4,ws4)

Figura 15.4: Sketch of the mean settling velocity versus sediment concentration.

15.2.2 Deposition

Deposition occurs when the bed shear stress falls below a critical value. Two different kinds ofsediment deposition can be considered: full and partial eposition.

Full deposition

Full deposition occurs when all the sediment particles and flocs settle down and it is observedwhen the bed shear stress becomes smaller than the critical shear stress for full deposition(τd,full). The deposition rate Qd can be evaluated by

Qd = Pdwscs (15.3)


where Pd is the deposition probability (Krone, 1962). Equation (15.3) is based on the observa-tion that only the aggregates with a shear strength sufficient to withstand the highly disruptiveshear stresses in the near bed region due to turbulence can deposit and adhere to the bed. Afraction of sediments settling to the near bed region can not withstand the high shear stres-ses at the sediment-water interface and are broken up and re-suspended. The probability ofdeposition can be evaluated by

Pd = 1 − τ

τd,fullfor τ ≤ τd,full (15.4)

where τ is the bottom shear stress and τd,full is the critical shear stress for full deposition. Manyexperiments were performed to determine the value of critical shear stress for full deposition ofcohesive sediments. It ranges between 0.06 and 1.1 N/m2 depending upon the sediment typeand concentration. In their study of the dynamics of the lagoon of Venice, Carniello et al.(????) assume τd,full to be equal to ???

Partial deposition

Partial deposition exists when the bed shear stress is larger than the critical shear stress forfull deposition but simultaneously smaller than a second critical value τd,part. In this range ofthe bed shear stress, relatively strong flocs are deposited while relatively weak flocs break-upand remain in suspension. The second critical value of the bed shear stress is named criticalshear stress for partial deposition. The deposition rate when partial deposition takes place canbe evaluated by

Qd = Pdws (cs − ceq) for τd,full < τ < τd,part (15.5)

where ceq is the equilibrium sediment concentration, i.e. the sediment concentration such thatrelatively weak flocs are broken before reaching the bed or are eroded immediately after beingdeposited. The probability of deposition is given by

Pd = 1 − τ

τd,partfor τd,full < τ < τd,part (15.6)

Of course there is no deposition when the bed shear stress is larger than the critical shear stressfor partial deposition, i.e.

Pd = 0 for τ ≥ τd,part (15.7)

QUANTO VALE ceq ?Formulae to evalute the critical shear stresses for full and partial depositions are not

available. Measurements indicate that τd,part falls around 1.5 and 2 N/m2

The accuracy of the deposition model mainly depends on the use of the correct values ofτd,full and τd,part, which become calibration parameters to determine the deposition rate whentheir values are uncertain.


15.2.3 Consolidation

The fluid-bed interface increases or decreases because of the deposition or erosion and theconsolidation of the bed. Consolidation is the process of floc compaction under the influence ofgravity and it takes places simultaneous with the expulsion of the water from the pores of thebed and the gain in strength of the bed material. The consolidation process is strongly affectedby:

- the initial thickness of the mud layer (h0),- the initial concentration of the mud layer (c0),- the permeability (k) of the mud layer (sediment composition and size, content of organic

material, salinity, water temperature).The consolidation of natural muds proceeds relatively fast in a thin layer and relatively slow

in a thick layer.Two types of consolidation are usually considered: primary and secondary consolidations

(Mehta et. al, 1989).

Primary consolidation

Primary consolidation is due to the self-weight of sediment and the deposition of additionalsediments. It begins when the weight of the sediment exceeds the seepage force induced by theupward flow of the pore water from the underlying sediment. During this stage, the weight ofthe particles expels the pore water and forces the particles closer together. The seepage forcelessens as the bed continues to undergo self-weight consolidation. Primary consolidation endswhen the seepage force has completely dissipated.

Secondary consolidation

Secondary consolidation is caused by the plastic deformation of the bed under a constantoverburden. It begins during the primary consolidation and may last for very long times (weeksor even months).

Numerical models take into account the consolidation process by representing the bed witha number of layers, each having a specific thickness, a consolidation time, and a critical shearstress. An idealized version of the consolidation process models the relationship by linking thelayer bulk density ρb to the consolidation time t (Nicholson and O’Connor, 1986)

ρb = ρf for t ≤ tf (15.8)

ρb = ρf + (ρ∞ − ρf ) 1 − exp [−A2 (t− tf )]B2 for tf < t < t∞

ρb = ρ∞ for t ≥ t∞

where ρb is dry bulk density, t is time and A2 and B2 are coefficients that account into accountthe mud characteristics and salinity of the water. The subscripts f and ∞ represent the freshlydeposited and the fully consolidated sediments, respectively.

Other similar consolidation models do exist


15.2.4 Erosion

Sediment particles or parts of the bed surface are eroded when the bed shear stress τ exceeds acritical value for erosion, which is denoted by τe. The value of τe depends on the characteristicsof the bed material (e.g. mineral composition, salinity, amount of organic material) and bedstructure. Experimental measurements show that the value of τe is also dependent on thehistory of the deposition and consolidation processes. The critical bed-shear stress for erosionis often found to be larger than the critical bed-shear stress for full deposition (τe > τd,full).Experimental results show also partial deposition and no erosion when and where the bedconsists of strong and dense flocs which are deposited during high-shear conditions for whichthe weak flocs are broken by turbulent stresses and do not settle. Hence, τe may be even largerthan τd,part).

In tidal conditions, with oscillating bed shear stresses, during slack tide weak flocs aredeposited on the top of the strong flocs deposited earlier when the velocity is higher. Duringthe next tidal cycle, these weak flocs are easily eroded at low velocities.

Different types of erosion can be distinguished: (i) surface erosion which is the one byone removal of particles and/or flocs and (ii) mass erosion which is the erosion of clusters orlumps of flocs due to failure of the bed structure. Many attempts have been made to relate thevalues of τe to basic soil parameters as plasticity index, voids ratio, water content, yield stressand others. However, no relationship is generally well accepted. Therefore, the determinationof τe is based on laboratory tests using natural mud or on in-situ field tests.

A three-layer system is usually observed in estuaries (see figure 15.5). Indeed, it is possibleto distinguish:

- a consolidated mud layer with concentrations larger than about 300 kg/m3. The particlesand the flocs are supported by the internal floc structure.

- a fluid mud suspension layer where the concentration falls in the range of 10 to 300 kg/m3.In the normal conditions, the thickness of this layer is of the order of 0.1 to 1 m but it can be upto 5 m in extreme conditions. In the fluid mud layer the particles and the flocs are supportedby the fluid drag exerted by the fluid which moves upward because of hinderance effect. Themud layer can be further subdivided into two layers: a turbulent upper layer and a laminar(viscous) lower layer. In the former layer, the sediment concentration ranges between about 10and 100 kg/m3 while in the latter layer the concentration ranges between about 100 and 300kg/m3.

- dilute mud suspension layer which is characterized by concentrations falling between about0 and 10 kg/m3 (see figure 15.5). Flocculation dominates in the dilute suspension layer andthe particles and the flocs are supported by the drag due to turbulent fluctuations.

The interfaces between the layers fluctuate during the accelerating and decelerating phasesbecause of the variations of the turbulence intensity.

Hereinafter, erosion is meant surface erosion, i.e. the removal from the bed of individualparticles or small aggregates because of the hydrodynamic forces (Millar & Quick, 1998) andmass erosion is not considered further. There is a difference between the shear stress originating


0.01

0.1

1

10

0.1 1 10 100 1000 10000

heig

ht a

bove

the

bed

[m]

mud concentration [kg/m3]

Figura 15.5: Sketch of the possible distributions of mud concentration which are generated bystrong or weak currents.

surface erosion and that causing mass erosion from one to three orders of magnitude (Kamphuis& Hall, 1983; Zreik et al., 1998; Hilldale, 2001). The ability of a cohesive soil to resist to surfaceerosion is defined as erosional strength (Zreik et al., 1998).

Due to the complex and widely varied nature of particle bonds, much less is known aboutthe properties influencing the bonding of cohesive soils.

Unlike cohesionless sediments, cohesive sediments can not be classified by their grain sizesince their behaviour depends mainly on the particle bonds and not on their size. The complexparticle bonds ask for a large number of paramaters to identify the behaviour of a mud. Forexample, Winterwerp et al. (1990) pointed out that twenty-eight parameters were employed byDelft Hydraulics to characterize a mud soil. Moreover, the soil behaviour is affected not onlyby physical and electro-chemical properties of the cohesive soil but also by biological factorswhich, sometimes, have stronger effects than the electrochemical effects (Paterson, 1994).

Saturation

In tidal environments, when the tidal current decreases, the sediments tend to settle andto form a layer of fluid mud close to the bed. In the mud layer, turbulence is damped and a


decrease of mixing is observed. In particular, when sediment concentration becomes larger thana certain value, turbulence collapses. The mud concentration prior to this collapse is named’saturation concentration’ and the flows with concentrations smaller/larger than the saturationconcentration are named subsaturation(underloading)/supersaturation(overloading) flows.

Experimental results show that turbulence collapses above the level where the flux Richard-son number exceeds a critical value which is about 0.2 and the corresponding concentration isthe saturation concentration. It might be worth to remind that the Richardson number cha-racterizing flows, in which the density differences are small, is defined as the ratio between thepotential energy and the kinetic energy but considering reduced gravity g′ = g∆ρ

ρ

Ri =g′h

U2(15.9)

When the Richardson number is much smaller than 1, buoyancy effects are small. On the otherhand, when Ri is large buoyancy effects are dominant and stratification is likely to be observed

Experimental methods to determine erosion parameters

Because of the large number of parameters which affect the erodibility of cohesive materials. Thecritical values of shear stress for erosion and deposition are usually determined by experimentaltests rather than determined from the knowledge of the soil properties.

Some experiments are carried out creating a sediment bed in a laboratory by letting apremixed slurry to settle down. This procedure generates a young soil deposits and has theadvantage to give control over some of the parameters characterizing the soil and the water(e.g. bulk density, salinity, pH). However, since this procedure uses a disturbed soil, somecritical characteristics of the actual soil are not reproduced. Therefore, it is better to testundisturbed soil deposits, even though this is a difficult task since obtaining an undisturbedsample of a cohesive soil is unlikely. However, it is possible to minimize the disturbances ofthe sample taking into account that also the water properties play a fundamental role in thecohesive properties of a soil.

In figure 15.6 an idealized plot of the erosion rate as function of the bed shear stress isshown. In general, a laboratory test should provide erosion rates and critical shear stressesfor deposition, surface erosion and mass erosion. This idealized plot shows how the values ofcritical shear stress for surface erosion (τ cse), the critical shear stress for mass erosion (τ cme) andthe subcritical condition of deposition (τd) can be determined along with the mass and surfaceerosion rates.

15.2.5 Critical shear stress and erosion rate formulae

A simple formula to predict the surface erosion rate is

Qse = Mseτ − τ cseτ cse

for τ ≥ τ cse (15.10)


Figura 15.6: Sketch of the behaviour of the erosion rate as function of the shear stress.

Qse = 0 for τ < τ cse

where Qse is the surface erosion rate, τ and τ cse are the bed shear stress and the critical sur-face erosion shear stress, respectively and Mse is the constant to quantify the surface erosionrate. The excess bed shear stress, defined as τ − τ cse, is a measure of the erosion force. Thecritical erosion shear stress depends on a number of factors including sediment composition,bed structure, chemical compositions of the pore and eroding fluids, deposition history, andthe organic matter and its state of oxidation (Ariathurai & Krone, 1976; Mehta et al., 1989).Usually, both Mse and τ cse change with the bed properties in depth and time. Field studiesor laboratory measurements must be made to obtain the critical shear stress and the erosionrate. Hwang & Mehta (1989) presented a relationship with the critical shear stress for surfaceerosion and the wet bulk density of the bed,

τ cse = ase (ρwb − ρl)bse + cse (15.11)


with ase, bse, cse and ρl having default values of 0.883, 0.2, 0.05 and 1.065, respectively for thestress in N/m2 and the bed bulk density in g/cm3. They are determined from field data.

Hwang & Mehta (1989) also presented a relationship for the erosion constant as a functionof the wet bulk density of the bed:

log10Mse = 0.23 exp

(

0.198

ρwb − 1.0023

)

(15.12)

where ρwb is the wet bulk density of the deposit in g/cm3 and Mse is the surfaceerosion rate constant in mg/cm2/hr.

According to Teisson & Latteux (1986) and Cormault (1971) the experimental formulationof Mse for the Gironde estuary is

Mse = 0.55( ρb

1000

)3

(15.13)

where ρb is the dry bulk density of the deposit in g/l and Mse is the surface erosion rate constantin kg/m2/s.

In the numerical model of Nicholson & O’Connor (1986), the critical erosive stress is assumedto depend on the bed density

τ cse = τef + A (ρb − ρf)B (15.14)

where τef is the critical shear stress of a freshly deposited bed and ρb, ρf are the dry bulkdensities of bed and freshly deposited bed, respectively, and A,B are constants. They usedτef = 0.8 × 10−1 N/m2, ρf = 80 kg/m3, A = 0.5 × 10−3 N m5/2/kg3/2, B = 1.5.

The rate of mass erosion over a time interval ∆t is given by Shrestha & Orlob (1996)

Qme =T

∆tρs

(

ρb − ρwρs − ρw

)

(15.15)

where T is the thickness of the erodible layer, which is also a function of the excess bed shearstress, Qme is the mass erosion rate and ρs, ρb, ρw are the densities of sediment material, drybulk density and density of the suspending medium, respectively.

15.3 Numerical models of cohesive sediment transport

Numerical models are becoming a useful tool To predict the transport of cohesive sediments,the equation describing the sediment concentration should be considered along with momentumequation which is necessary to determine the flow field.


15.3.1 Three-dimensional models

Three-dimensional models solve the full convection-diffusion equation

∂ci∂t

+∂ (uci)

∂x+∂ (vci)

∂y+∂ ((w − ws) ci)

∂z=

∂

∂x

(

Dx∂ci∂x

)

+∂

∂y

(

Dy∂ci∂y

)

+∂

∂z

(

Dz∂ci∂z

)

+ Si

(15.16)where ci is the sediment volume concentration (m3/m3) of constituent i, t is time, u, v, w are thevelocity components (m/s) in the x, y, z-directions, respectively, Dx,Dy,Dz are the dispersioncoefficients (m2/s) in the x, y, z-directions, respectively, and ws is the sediment fall velocity(m/s).

The bed boundary conditions usually state that(

Dz∂ci∂z

)

B

= Qe −Qd (15.17)

where Qe and Qd are the volume eroded and deposited per unit area and unit time (m/s),respectively, and the subscript B denotes the boundary condition at the bed.

The appropriate boundary condition at the water surface forces the upward diffusion rateto be equal to the downward settling rate

(

Dz∂ci∂z

)

S

= ((w − ws) ci)S (15.18)

where the subscript S denotes the boundary condition at the water surface.Of course, at the solid boundaries, the diffusion normal to the solid surface should vanish,

i.e.(

Dm∂ci∂nm

)

= 0 (15.19)

where nm is the unit normal vector of the boundary.

15.3.2 Two-dimensional models

Two-dimensional models solve the depth-averaged convection-diffusion equation for cohesivesediment transport

∂ (hci)

∂t+∂ (huci)

∂x+∂ (hvci)

∂y=

∂

∂x

(

hDx∂ci∂x

)

+∂

∂y

(

hDy∂ci∂y

)

+ Si (15.20)

where h is water depth (m), ci is the depth-averaged sediment volume concentration (m3/m3) ofconstituent i, t is time, u, v are the depth-averaged velocity components (m/s) in the streamwiseand transverse directions x and y respectively, Dx,Dy are the dispersion coefficients (m2/s)in the x and y-streamwise directions, respectively, and Si is the source (erosion) and sink(deposition) terms (m/s) for constituent i.


Different types of boundary conditions are forced in a two-dimensional model. At theinlet boundary, the sediment concentration is usually given while at the outlet boundary, noconcentration gradient is forced. At solid boundaries, it is assumed that there is no flux in thedirection orthogonal to the solid surface.

15.3.3 One-dimensional models

One-dimensional models consider cross-sectional average of the equations for the mass andmomentum balance. In particular, sediment concentration is determined by using

∂ (Aci)

∂t+∂ (Auci)

∂x=

∂

∂x

(

ADx∂ci∂x

)

+ Si (15.21)

where A is the cross-sectional area (m2), ci is the cross-sectionally averaged sediment volumeconcentration (m3/m3) of constituent i, t is time, u is the cross-sectionally averaged velocitycomponents (m/s) in the streamwise direction x, Q = uA is discharge (m3/s), Dx is thedispersion coefficient (m2/s) in the streamwise direction x and Si is the source (erosion) andsink (deposition) terms (m2/s) for constituent i.

15.4 Appendix A

Plastic limit

The procedure to determine the plastic limit (PL) is described in ASTM Standard D TestMethod 4318. A small portion of a soil is rolled out on a flat, non-porous surface till it becomes athin thread. If the soil is plastic, this thread can retain its shape down to a very narrow diameterwhen it starts to break. Then, the sample can be remoulded and the test repeated. Since themoisture content decreases because of evaporation, the thread breaks at larger diameters. Theplastic limit is defined as the moisture content such that the thread breaks apart at a diameterof 3.2 mm. If a thread cannot be rolled out down to 3.2 mm at any moisture, the soil isconsidered to be non-plastic.

Liquid limit

The liquid limit (LL) is the water content at which a soil changes from plastic to liquidbehavior. The procedure to determine the liquid limit (LL) is described in ASTM Standardtest method D 4318. A pat of soil is placed into the metal cup and a groove of 13.5 millimetreswidth is made with a standardized tool. The cup is repeatedly dropped from an height of 10mm onto a hard rubber base at a rate of 120 blows per minute. The number of blows for thegroove to close is recorded. The moisture content at which the groove closes after 25 dropsof the cup is defined as the liquid limit. Another method for measuring the liquid limit isthe fall cone test which is based on the measurement of penetration length into the soil of astandardized cone of specific mass.


PI behaviour0 Nonplastic

1-5 Slightly plastic5-10 Low plasticity10-20 Medium plasticity20-40 High plasticity> 40 Very high plasticity

Tabella 15.1: Plasticity Index

Plasticity index

As already pointed out, the plasticity index (PI), which is a measure of the plasticity ofa soil, is defined as the range of water contents such that the soil exhibits plastic properties.In other words, the PI is the difference between the liquid limit and the plastic limit (PI =LL-PL). Soils which are characterized by a high value of PI tend to be clay while those with asmall PI tend to be silt. The soil which is characterized by a PI equal to 0 (non-plastic) tendto have little or no silt or clay (see table 15.4).

Bulk density

Bulk density is a property of powders, granules, and other divided solids, especially usedin reference to mineral components (soil, gravel). It is defined as the mass of many particlesof the material divided by the total volume they occupy. The total volume includes particlevolume, inter-particle void volume and internal pore volume.

Bulk density is not an intrinsic property of a material. It can change depending on howthe material is handled. For example, a powder poured into a cylinder has a particular bulkdensity. If the cylinder is disturbed, the powder particles move and usually settle closer together,resulting in a higher bulk density. For this reason, the bulk density of powders is usuallyreported both as freely settled (or poured density) and tapped density (where the tappeddensity refers to the bulk density of the powder after a specified compaction process, usuallyinvolving vibration of the container.)

Plasticity

Plasticity is the property of a cohesive material to undergo substantial permanent deforma-tion without breaking. Dilute suspensions with concentrations smaller than 10 kg/m3 (noticethat 10 kg/m3= 10 g/l) generally show a Newtonian behavior (linear relationship between shearstress and velocity gradient). Deviations from this behaviour tend to occur at concentrationslarger than 10 kg/m3. High-concentration (> 50 kg/m3) suspensions of water, fine sand, silt,clay and organic material usually have a pseudo-plastic or a Bingham plastic shearing behaviour,which means that the relationship between shear-stress τ and shear rate is nonlinear.

Capitolo 16

COASTAL MORPHOLOGY

16.1 Introduction

This chapter is aimed at providing a qualitative description of the major characteristics of thesedimentary structures observed in the continental shelf and along the coast. The contents ofthe chapter are largely taken from Scarsi (2009).

16.2 Definitions

Tidal range: difference between the maximum and minimum average levels taking placeduring the tidal cycle. If the tidal range is small (less than 2 m), the tide is defined to be micro.If the tidal range falls between 2 and 4 m, the tide is defined to be meso. Finally, when thetidal range is larger than 4 m, the tide is defined to be macro.

Breaker region: region where the wave breaking takes place

Longshore bar: accumulation of sediments, parallel to the coast, which is usually submer-ged but it can emerge at low tide.

Surf region: region extending from the breaker-line up to the face of the beach. Theregion has an extension that largely depends on the slope of the seabed. The beaches, whichare characterized by the presence of fine sand, have a wide surf region. On the other hand thebeaches, which are characterized by the presence of gravel or pebbles, are steep and rarely havea significant surf region, since the waves break close to the beach face. The beaches, which arecharacterized by average slopes, have a significant surf zone at low tide but they can have nosurf region at high tide.

Swash region: region which is coincident with the beach face where the run-up and therun-down take place.

Lower beach: part of the beach face that remains submerged even at low tide.

287

CAPITOLO 16. COASTAL MORPHOLOGY 288

Shoreline: line of intersection between the beach face and the mean sea level

Upper beach: the region of the back of the beach, made up of sediments accumulated byprevious intense storms.

Foreshore or beach face: region of the beach where the run-up and run-down take place

Beach: the region that includes both the face of the beach (foreshore) and the backshore.

Berm: part of the beach, almost horizontal or characterized by a low slope, that is found inthe backshore. Under certain conditions, more than one berm exist or the berm can be absent.

Berm crest: edge of the first berm looking at the sea

Figura 16.1: Sketch of an open beach

Backshore: region that extends from the region where run-up takes, which is excluded, upto the region that has different characteristics from the beach itself, for example because of thepresence of vegetation.

Beach scarp: scarp of relatively significant slope that is often present in the backshoreand that is due to erosion of the storm surges

16.3 Planimetric geometries of beaches

The beaches have different planimetric geometries, which are briefly described in the following

Open beaches: The open beaches (see figure 16.1 and 16.2 are usually delimited by rockyborders. These beaches are generated by sediments deposited by strong coastal currents andthey may develop for long distances, exhibiting a straight trend or a concave modest curvature.


Figura 16.2: Open beach

Figura 16.3: Fleche with an end touching the land

Fleches: The fleches are not delimited by rocky borders and are anchored by one or bothends to a concave beach (see figures 16.3 and 16.4)

Tomboli: The tomboli (see figures 16.5 and 16.6) are ’islands’ which are connected tothe land by fleches that are generated because of the shelter of the islands themselves. Oftensedimentary structures similar to tomboli form because of the shelter of man-made structures(figure 16.7)

Barrier islands: The barrier islands consist of fleches which are not joined to the landand run parallel to the beach. Sometime more than one barrier island exist that are separatedeach from the other by openings (figures 16.8 and 16.9). These islands have sizes (length andwidth) which can vary considerably (the length ranging between hundred meters and tens of


Figura 16.4: Presque Isle (Pennsylvania) is a sand spit built into Lake Erie by wind and waveaction and longshore drift

Figura 16.5: Tombolo

kilometers).

16.4 Transverse beach profiles

The transverse profiles of the beaches, which are observed during summer time, are differentfrom those which are observed during winter time (see the sketches of figure 16.10). The sum-mer profiles are generated by swells or modest sea states characterized by a small significantwave height. These profiles have a significant backshore, generally important, and no longsho-re bar can be identified (see figure 16.10). Winter profiles are generated by intense storms,characterized by severe sea states with an high significant wave height. In general, these bea-


Figura 16.6: Tombolo: Cape Paximadhi, close to Euboea, Greece. Photo by Tim Bekaert (25Marzo 2005)

Figura 16.7: Coast of Emila-Romagna (Italy)

ch profiles have no significant backshore and have one or more longshore coastal bars. Thetransition from the winter profile to the summer profile or viceversa involves the movement of


Figura 16.8: Virginia’s Atlantic shoreline is home to some of the nation’s most pristine barrierislands and saltmarsh lagoons.

Figura 16.9: Barrier islands off the Louisiana coast.

significant volumes of sand that are towards shore during the formation of a summer profileand towards the sea during the formation of a winter profile. However, the total volume ofsediments of the beach does not change (see figure 16.11).

It follows that the average slope sb of the beach face of a summer profile is larger than thatof a winter profile and it can be predicted by using the following empirical relationship

sb = 0.25

(

H0

d50

)−0.25(H0

L0

)−0.15

(16.1)


Figura 16.10: Transverse profiles of the beach during summer and winter times.

where d50 is the median diameter of the sediments and H0 and L are the significant height andwavelength of the incoming wave evaluated for large water depths. Iwagaki & Noda (1962)

proposed to divide the plane(

H0

d50, H0

L0

)

into two regions (see figure 16.12): for low values of H0

d50

and H0

L0, summer profiles are usually observed while for high values of H0

d50and H0

L0, winter profile

are observed. Clearly, the curve in figure 16.12 identifies, for a given value of H0

d50, a critical value

of the wave steepness that allows to distinguish the summer profiles from the winter profiles.Recently, to quantify the critical value, the following relation has been proposed that is basedon experimental results

(

H0

L0

)

cr

= 115

(

πwsgT

)1.5

(16.2)

and improves a previous relationship proposed by Dean (1973)

(

H0

L0

)

cr

= 1.7πwsgT

(16.3)

where ws represents the falling velocity of the sediment. Finally, Dalrymple (1992) proposed

(

gH20

w3sT

)

cr

= 9000 (16.4)

The beach profiles are also defined to be dissipative or reflective depending on the ratiobetween the amplitude of the reflected wave and the amplitude of the incoming wave. Thedissipative profiles are generated by storms which dissipate much of their energy. Indeed,because of the small values of the water depth and the low slope of the bottom, wave breakingof plunging type takes place. These profiles have a geometry which is schematically sketchedin figure 16.13 and are observed when the condition

Hma,f

gT 2mas

2b

> 1 (16.5)


Figura 16.11: Photo of the same beach during summer time (top panel) and winter time (bottompanel)

where Hma,f is the annual averaged wave height at the breaker point, Tma is the period of thiswave and sb is the average bottom slope between the breaker point and the shoreline (Wrightet al., 1978).

The reflective profiles are observed when the bottom material is predominantly coarse and


Figura 16.12: Regions of existence of summer and winter profiles

Figura 16.13: Profilo di spiaggia dissipativo. Profile of a dissipative beach

when the energy of the waves is largely reflected, the remaining part being dissipated by thebreaking of the waves of surging type.

These profiles are schematically sketched in figure 16.14 and are observed when the condition

Hma,f

gT 2mas

2b

< 0.1 (16.6)

is satisfied (Wright et al. 1978).Finally, when

0.1 <Hma,f

gT 2mas

2b

< 1 (16.7)

intermediate beach profiles are observed (Wright et al. 1978).


Figura 16.14: Reflective beach profile.

At last, the beach profile can be an equilibrium profile or a non-equilibrium profile. Theequilibrium profiles are transverse profiles of the beach identified by the envelope of the profileswhich follow each other in time, within a year, but without the presence of erosion or depositionphenomena (see the sketches of figure 16.10). The equilibrium profile, characterized by a slopethat increases monotonically moving towards the shore (see figure 16.15) can be described bythe simple relationship

Figura 16.15: Equilibrium beach profile.

h = AXm (16.8)

where h is the local water depth, X is the distance from the shore line and A is a dimensionalparameter. Dean (1977) proposed a value of m equal to 2/3 that provides the profile knownas Bruun-Dean profile. In general, the parameter A depends on the diameter of the sedimentsand, in the international system, can be assumed to be

A = 7.35(d50)0.49 (16.9)

A = 0.067(100ws)0.44 (16.10)

A = 2.25

(

w2s

g

)1/3

(16.11)

In particular, relationshps (16.9) and (16.10) are more suitable for sediments with a diame-ter falling between 0.1 and 10 mm, while the relation (16.11) is more appropriate for coarsesediments.


16.5 The wave run-up and the beach berms

The maximum value of the run-up of the waves on the beach face (total run-up) can be quan-tified by means of the distance RT between the highest run-up and the mean still level, assketched in figure 16.16 The value of RT is usually split into two parts

Figura 16.16: Wave run-up and beach face.

RT = SU +RU

where the contribution SU is due to the set-up induced by the waves, the wind and the tidesand the contribution RU is due to the run-up. The value of SU is largely influenced by theset-up SW induced by the wave motion (in general, the set-up induced by the wind and/or bythe tides is of minor importance) that can be evaluated, as suggested by Guza & Thornton(1981), using the formula

SW = 0.17H0 (16.12)

where, as already pointed out, the wave height is the significant wave height in the deep waterregion. Later, Holman & Sallenger (1985) proposed

SW = 0.45(βIR)0H0 (16.13)

which uses the Irribarren number in the deep water region, defined by

(βIR) =sb

√

H0/L0

(16.14)

Using the dispersion relation, it is possible to obtain

SW = 0.18sbT0

√

gH0 (16.15)

The significant value of RU , defined as the average of the third largest values of the run-up,can be assumed equal to 0.7H0, as suggested by Guza & Thornton (1982), although there othermore recent estimates.


16.5.1 Two-dimensional beach topographies: the longshore bar sy-stem

Highly dissipative beaches are characterized by the presence of one or more longshore bars consi-sting of ridges of sediment running roughly parallel to the shore, which may have a considerablelongshore extent and an inter-bar spacing of the order of 100 m. The generation and mainte-nance of these bars are commonly associated with the shoaling and breaking of high-frequencywaves. The problem of predicting changes of the beach profile caused by changing hydrody-namic conditions, and in particular the formation of longshore bars, has been a challenge formany researchers over the past few decades. Since the complexity of the problem precludes anyclear-cut solution, a variety of modeling approaches have been developed to describe the timedevelopment of the beach profile. In process-based or deterministic profile models, the diffe-rent processes that contribute to the development of the beach profile are explicitly taken intoaccount. These models, which consist of different modules describing waves, steady currents,sediment transport, and bed level changes, are now able to mimic bar formation (Roevlink& Brker 1993). However, calibration and validation of these models is needed since the de-scription of both the hydrodynamics and the sediment transport is based mainly on empiricalapproaches. For example, wave breaking is described in a parametric fashion, and details ofthe vorticity field in the surf zone are ignored. This is in contrast with the recent experimentsof Zhang & Sunamura (1994) that have revealed that bar formation is controlled by vorticitydynamics and, in particular, that it takes place at the cross-shore locations where the largevortex structures, generated by wave breaking and by nonlinear vortex dynamics in the surfzone, touch the bottom. The experiments by Zhang & Sumamura (1994) along with previousinvestigations (see for example Nadaoka, 1989) show that two major vortex structures can beidentified: horizontal vortices associated with plunging breakers and oblique vortices associa-ted with spilling breakers. Moreover, the results by Zhang & Sunamura (1994) indicate that adifferent mechanism of bar formation is associated with each vortex type. The process leadingto the formation of horizontal vortices by plunging breakers was simulated by Pedersen et al(1995), who also investigated the interaction of these vortices with the cohesionless bottom.The flow domain is split into a core region, where the fluid is assumed to behave inviscidly, anda bottom boundary layer. In the core region the description of the hydrodynamics is based ona discrete vortex approach (cloud-in-cell) and the plunging wave is simulated by introducinga jet of water that falls down in front of the wave crest. Then, simple models are used todescribe the flow and sediment concentration close to the bed, whereas in the outer region themotion of suspended sediment is described by a Lagrangian simulation. In spite of its relativesimplicity, the model produces results indicating that the most relevant processes are capturedby the approach.

Some beaches have more than one longshore bar separated by troughs. Multiple bars mayresult from multiple wave breaking and the generation of multiple vortex structures. Thepossibility of the first mechanism is supported by the laboratory wave channel experimentsperformed by Sunamura & Maruyama (1987) and Zhang & Sumamura (1994) that show the


development of many bars, the outer bar associated with the first wave breaking and the innerbars resulting from the breaking of the reformed waves. In some multiple bar systems, however,the outer bars are in water too deep to be associated with breaking waves. Another theoryof multiple bar formation is associated with the presence of standing waves produced by thereflection of the incoming waves from the beach and the generation of steady streaming bynonlinear effects within the bottom boundary layer. A general formula to evaluate the steadyvelocity components generated within the bottom boundary layer of a three-dimensional waveof small amplitude was provided by Hunt & Johns (1963), who did not consider the flow farfrom the bottom. Carter et al (1973) applied these results to compute the velocity field atthe bottom of a sea wave that approaches the coastline and is partially reflected at the beach.The same problem was tackled by Lau & Travis (1973), who considered beaches characterizedby very gentle slopes such that the wave field can be described locally in terms of a linearStokes wave over a constant depth. As discussed clearly in the book by Mei (1989:59-66), theperturbation approach used by Lau & Travis (1973) is strictly valid when the ratio βL/h0 ismuch smaller than one. The quantity β denotes the beach slope, h0 is a characteristic value ofthe water depth, and L is the length of the incoming waves. For intermediate water depths,the assumption βL/h0 ≪ 1, is not particularly restrictive since actual values of the beach slopeare small. However, close to the beach, where the water depth turns out to be much smallerthan the length of the waves, the approach of Lau & Travis (1973) requires very small valuesof β. In fact from the assumption βL/h0 ≪ 1, it follows, β ≪ h0/L ≪ 1. Blondeaux etal. (1999b, 2000b) recently predicted the shoaling process of a wave propagating on a gentlysloping bottom and partially reflected at the coastline for larger beach slopes (β of order h0/L),and also determined the steady velocity components by solving an advection-diffusion equationfor the vorticity field. This equation contains some terms that result from changes in the waveamplitude induced by the sloping bottom, and hence the equation differs from that obtained byLonguet- Higgins (1953) and described by Mei (1989). As in Longuet-Higgins (1953) analyses,the solution depends on the ratio a/δ between the wave amplitude and the thickness of thebottom boundary layer. For realistic values of a/δ, when the surface waves are fully absorbedat the beach, vorticity tends to be confined within a thin boundary layer that remains attachedto the sea bottom. On the other hand, for a fully reflected wave, the vorticity generated at thebed leaves the bottom forming jets directed upward. Then vorticity is convected within the coreregion, and the steady flow field turns out to be very complicated. Knowledge of the steadycurrents generated far from the seabed, i.e. outside the bottom boundary layer, is particularlyrelevant for morphodynamics models. Indeed it is generally accepted (cfr. Fredsøe & Deigaard1992) that waves propagating over an erodible bed of fine sand generate a sediment suspensionwith large concentrations in the near-bed region. When currents (in this case wave-generatedcurrents) are present, additional mixing over the water depth occurs, resulting in an increasedsediment concentration in the upper layers. The basic mechanism of sediment transport incombined currents and waves is thus the entrainment of particles by the stirring action of wavesand their transport by the currents. Carter et al (1973) and Lau & Travis (1973), by using thehydrodynamic results and a simple sediment transport predictor, investigated the formation of


multiple bars generated by partially reflected waves. They found that the longshore bars formedby partially reflected waves have a spacing equal to a half-wavelength of the incoming wave.Hence a resonant Bragg-type interaction between the surface waves and the bedforms maytake place (Davies 1982a,b, Mei 1985, Davies et al 1989), thus increasing the wave reflection(Heathershaw & Davies 1985, OHare & Davies 1993). Moreover this increase in wave reflectioncan lead to the growth of new sandbars in the seaward direction. Despite the apparent successof the standing long wave mechanism for bar formation, a number of problems are still unsolved.In fact the mechanism depends on the existence of a well-defined, stable standing wave. Thisoccurs only when the long-wave frequency spectrum is narrow banded.

However, most field measurements reveal a broad banded spectrum for which nodal andantinodal positions would be continually changing. Another possible mechanism leading to theformation of the bars appearing in the nearshore region was suggested by Boczar-Karakiewiczet al (1987, 1995b). According to their analysis, in shallow waters there may be a periodic spa-tial pattern of a propagating short wave (i.e. a periodic variations of its amplitude) owing toan energy transfer from the primary short wave to long waves. Indeed sinusoidal waves of con-stant amplitude reaching shallow waters from the deep water region do not have the ideal formcorresponding to the shallow water physics of wave propagation. As the short waves propagate,they slowly change their form in a periodic way, which depends on the wave steepness and onthe ratio between the wave height and water depth. The periodic variation of the envelope ofthe wave amplitude leads to periodic variation of the peak orbital velocities and near-bed masstransport velocities inside the bottom boundary layer and, hence, to variations of the near-bedsediment flux, resulting in regions of sediment flux convergence and divergence. Consequentlybars will form in regions where the wave envelope decreases from a maximum toward a mini-mum. Boczar-Karakiewicz et al (1995a, 1995b) used the model to predict the numbers andspacings of longshore bars, starting from an initially planar slope, in many different sites. Theyfound that the predicted patterns correlate well with the observed bars along beaches withinthe Gulf of St. Lawrence (Canada) and for the Gold Coast (Australia). Even though examplesof stable bars are reported, longshore bars usually experience seasonally varying amounts ofchange in position and shape. Moreover varying changes in the bar profile are observed duringindividual storms. Nearshore bar behavior shows a rather consistent pattern of response towave energy input and is usually characterized by rapid straightening of the outer bar duringstorms and gradual development of a crescentic pattern during periods of low-energy exposure(e.g. Wright & Short 1984, Lippman & Holman 1990). There are also cyclic behaviors. Forexample along the Dutch coast, multiple systems of nearshore breaker bars exist such that allthe bars migrate seaward, with the outer bar decaying offshore and a new bar being generatednear the shoreline (Wijnberg & Terwind 1995, Wijnberg 1996). Remote sensing systems likethe video monitoring system introduced by Lippman & Holman (1989) are valuable tools inobtaining the required high resolution morphologic information on bar movement.

Capitolo 17

Appendice

Nell’ambito del corso si fara ampio uso dei cosiddetti metodi perturbativi, approcci che con-sentono di determinare la soluzione di problemi complessi, caratterizzati dalla presenza di unparametro molto maggiore o molto minore di 1, la cui soluzione non sarebbe altrimenti possi-bile determinare. Non e possibile illustrare estesamente e compiutamente i metodi perturbativiprima di affrontare lo studio dell’idraulica marittima. Conseguentemente, i diversi metodi per-turbativi verranno descritti negli appunti del corso la dove il loro uso sara necessario. In questabreve nota, con due semplici esempi, si cerca di dare un’idea, anche se necessariamente moltovaga, di questo tipo di approccio

Problema 1

Ipotizzando di ignorare l’espressione che fornisce la soluzione di un’equazione algebrica nonlineare del secondo ordine, determinare i valori di x che soddisfano la seguente equazione

x2 + ǫx− 1 = 0 (17.1)

ove ǫ≪ 1.

Soluzione

E evidente che i valori cercati di x sono prossimi a ±1, valori che si ottengono trascurandoil secondo termine, per ipotesi piccolo, nella (17.1). E possibile migliorare la conoscenza di xscrivendo

x = x0 + ǫx1 + ǫ2x2 + ... (17.2)

cioe espandendo la soluzione in serie di potenze di ǫ. Sostituendo la (17.2) nella (17.1), siottiene

[

x20 + 2ǫx0x1 + ǫ2

(

x21 + 2x0x2

)

+ ...]

+ ǫ[

x0 + ǫx1 + ǫ2x2 + ...]

− 1 = 0 (17.3)

e dunquex2

0 − 1 + ǫ [2x0x1 + x0] + ǫ2[

x21 + 2x0x2 + x1 + ...

]

+ ... = 0 (17.4)

301

CAPITOLO 17. APPENDICE 302

Considerato che l’equazione (17.4) deve essere soddisfatta per qualunque valore di ǫ, segue

x0 = ±1, x1 = −1

2, x2 = −x

21 + x1

2x0= ±1

8, ... (17.5)

Emerge quindi

x = ±1 − ǫ

2± ǫ2

8+ ... (17.6)

Concludendo, le due soluzioni cercate sono

x = 1 − ǫ

2+ǫ2

8+ ..., x = −1 − ǫ

2− ǫ2

8+ ... (17.7)

Si noti che le soluzioni (17.7) coincidono con le soluzioni fornite dall’espansione in serie dipotenze di ǫ delle due soluzioni esatte di (17.1). E infatti semplice ottenere, espandendo lesoluzioni esatte in serie di Taylor tenendo conto che ǫ ≪ 1

x =−ǫ±

√ǫ2 + 4

2=

−ǫ±[√

4 + 12√

4ǫ2 + ...

]

2(17.8)

=−ǫ±

[

2 + 14ǫ2 + ...

]

2= ±1 − ǫ

2± ǫ2

8+ ...

Problema 2

Si consideri ora la seguente equazione

ǫx2 + ax+ 1 = 0 (17.9)

dove ǫ≪ 1.Procedendo come nell’esempio precedente e ponendo

x = x0 + ǫx1 + ǫ2x2 + ... (17.10)

si ottiene

ǫ[

x20 + 2ǫx0x1 + ǫ2

(

x21 + 2x0x2

)

+ ...]

+ a[

x0 + ǫx1 + ǫ2x2 + ...]

+ 1 = 0 (17.11)

ax0 + 1 + ǫ[

x20 + ax1

]

+ ǫ2 [2x0x1 + ax2] + ... = 0 (17.12)

Da cui segue

x0 = −1

a, x1 = −x

20

a, x2 = −2x0x1

a, ... (17.13)

CAPITOLO 17. APPENDICE 303

Si osserva quindi che l’applicazione della procedura illustrata nel primo problema porta a de-terminare solo una soluzione dell’equazione. Per determinare l’altra soluzione e necessarioassumere la soluzione nella forma

x =1

ǫ

(

x0 + ǫx1 + ǫ2x2 + ...)

(17.14)

Si lascia allo studente il compito di svolgere i conti necessari per determinare x0, x1, ... in questoulteriore caso.

Da quanto illustrato nei due esempi, emerge che non sempre lo sviluppo della soluzione diun problema, caratterizzato dalla presenza di un parametro ǫ molto minore di uno, in terminidi ǫ e la determinazione della soluzione dei problemi ai diversi ordini di approssimazione condu-cono alla soluzione completa del problema originale. Infatti, talvolta, devono essere introdottiopportuni accorgimenti. Certamente, pero, i cosidetti metodi perturbativi, ampiamente uti-lizzati nel corso, costituiscono un valido strumento per lo studio di fenomeni complessi nonlineari caratterizzati dalla presenza di parametri che assumono valori molto minori di 1 o moltomaggiori di 1.

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New University of Genoa Department of Civil, Chemical and … · 2018. 10. 19. · course, the reader who is not interested in the details of the algebra can skip them and focus her/his