SHORELINE BOUNDARY CONDITIONS FOR WATER WAVE …host.uniroma3.it/docenti/bellotti/bellotti_file/BellottiPhDThesis.pdf · the nearshore hydrodynamics. Shoreline boundary conditions

SHORELINE BOUNDARY CONDITIONS FOR

WATER WAVE MODELS

by

Giorgio Bellotti

UNIVERSITA DEGLI STUDI DI ROMA “LA SAPIENZA”

Dipartimento di Idraulica, Trasporti e Strade

Tesi di Dottorato in Ingegneria Idraulica

XIV Ciclo

UNIVERSITA DEGLI STUDI DI NAPOLI “FEDERICO II”

Settembre 2001

Il candidato: Il Tutore:

Giorgio Bellotti Prof. Alberto Noli

Abstract

The aim of this work is to extend to include the swash zone the solution domain of

wave-resolving and wave-averaged depth-integrated numerical models used for simulating

the nearshore hydrodynamics. Shoreline boundary conditions are therefore derived and

verified for both the two categories of models.

For what concerns wave-resolving models, we discuss the applicability of Boussinesq-

type equations to model swash zone flows. Therefore the role of the dispersive-nonlinear

terms which characterize these equations, is studied on the basis of the Carrier and

Greenspan (1958) solution for periodic regular waves. Subsequently a technique suitable

to track the movements of the instantaneous shoreline and to impose the correct water

depth and velocity at the shoreward limit of the computational domain is introduced. This

technique is based on the use of a specific shock-capturing numerical scheme, capable of

treating the discontinuity characterizing the shoreline, where a transition between wet and

dry conditions occurs. A numerical model based on Boussinesq-type equations is therefore

coded and tested. The new boundary conditions are finally implemented in the numerical

model and verified against well-known analytical solutions for non-breaking waves. The

validation process suggests a very good performance both of the numerical model and of

the proposed boundary conditions.

As far as wave-averaged models are concerned, the problems related to averaging in

proximity and into the swash zone are illustrated. A solution strategy for these problems

is detailed, leading to avoiding direct simulation of the swash zone flows and prescribing

boundary conditions for the model at the seaward limit of this region. Boundary conditions

based on an integral model of the swash zone are therefore derived and finally verified by

means of full numerical solutions of the Nonlinear Shallow Water Equations. The results

obtained clearly indicate that such boundary conditions are suitable to be implemented

in wave-averaged circulation models.

Introduzione

La previsione delle trasformazioni che il moto ondoso subisce durante la propagazione

dal largo verso riva e delle correnti da questo generate e di estrema importanza per

la progettazione delle opere marittime e per la comprensione dei fenomeni fisici che

avvengono in prossimita’ della costa. Come e noto le caratteristiche delle onde variano

a causa dell’interazione con il fondale per fenomeni quali shoaling, rifrazione, riflessione,

diffrazione e frangimento; gli effetti non lineari in acqua bassa sono inoltre responsabili del

trasferimento di energia su componenti di frequenza differente da quella delle onde portanti

(triad interaction e bound long waves). Il trasporto di massa legato al frangimento genera

inoltre delle correnti che possono avere intensita notevoli e che influiscono in maniera

importante sulla morfodinamica costiera.

Negli ultimi anni, due tipi di modelli numerici si sono rivelati capaci di fornire una

accurata descrizione della evoluzione delle onde nelle regioni costiere e delle correnti da

queste generate. Il primo tipo di modelli e in grado di simulare contemporaneamente la

propagazione del moto ondoso e la generazione delle correnti; tale categoria di modelli viene

comunemente indicata nella letteratura internazionale con il termine wave-resolving (che

risolve l’onda) e si basa sulle cosiddette equazioni di Boussinesq (Peregrine 1967) o sulle

classiche equazioni di acque basse (Nonlinear Shallow Water Equations). Il secondo tipo di

modelli e in grado invece di simulare soltanto la generazione delle correnti: la propagazione

del moto ondoso deve essere studiata con un modello separato. Le equazioni sulle quali

si basano tali modelli sono simili alle equazioni delle acque basse ma, diversamente dai

modelli che risolvono l’onda, le variabili dipendenti sono mediate sul periodo del moto

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ondoso e dunque rappresentano esclusivamente l’idrodinamica relativa alle correnti. Come

forzante delle equazioni appaiono inoltre dei termini che rappresentano l’influenza del moto

ondoso sulle correnti e che vengono calcolati, come accennato in precedenza, tramite un

modello separato. Questi modelli vengono comunemente definiti con il termine wave-

averaged (mediati sull’onda).

L’obiettivo del presente lavoro e derivare delle nuove condizioni al contorno per le due

categorie di modelli sopra introdotte, atte in particolare a simulare l’idrodinamica della

zona di swash. Per quanto riguarda i modelli che risolvono l’onda dunque, e stata messa

a punto una tecnica che permette di simulare i movimenti della linea di riva, specificando

le opportune condizioni al contorno per il modello numerico. Per verificare le prestazioni

delle nuove condizioni e stato implementato e testato un modello numerico, basato sulle

equazioni di Boussinesq, tramite il quale sono state infine condotte delle prove numeriche

che hanno rivelato un ottimo comportamento della tecnica proposta.

Per quanto riguarda i modelli mediati sull’onda e stata messa a punto una tecnica che

permette di rappresentare la zona di swash tramite un modello integrale, evitandone la

simulazione diretta con il modello. La tecnica proposta e in grado di specificare le corrette

condizioni al contorno per il modello e di seguire i movimenti del limite inferiore della zona

di swash. Il buon funzionamento della tecnica proposta e stato verificato effettuando un

elevato numero di esperimenti numerici.

Il presente lavoro e motivato dal fatto che nel recente passato numerosi ricercatori

hanno impiegato i loro sforzi nel migliorare l’accuratezza dei modelli che risolvono

l’onda nelle aree a profondita intermedia, mentre poche indagini sono state rivolte a

raffinare la descrizione del moto del fluido in prossimita della riva dove, spesso, e

di fondamentale importanza una dettagliata descrizione dell’idrodinamica. Per quanto

riguarda i modelli mediati sull’onda e invece sorprendente notare come in passato il

problema della simulazione della zona di swash sia stato sempre affrontato con grande

leggerezza. Normalmente infatti le condizioni al contorno di riva utilizzate per tali modelli

sono identiche a quelle utilizzate per i modelli che risolvono l’onda, senza tenere conto che

in realta le variabili nelle quali sono scritte le equazioni, sono mediate sull’onda.

Contents

1 Introduction 15

1.1 Nearshore hydrodynamics phenomena and nearshore areas . . . . . . . . . . 16

1.2 A brief overview of the nonlinear numerical models for studying the

nearshore hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Depth-integrated wave-resolving models 24

2.1 The BTE and the NSWE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Scaling parameters, reference frame and variable used . . . . . . . . 25

2.1.2 Boundary conditions and preliminary assumptions . . . . . . . . . . 26

2.1.3 The depth-integrated continuity equation . . . . . . . . . . . . . . . 27

2.1.4 The depth-integrated momentum equation . . . . . . . . . . . . . . . 27

2.1.5 An approximate expression for the horizontal velocity . . . . . . . . 29

2.1.6 The final equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 A numerical model for solving the BTE . . . . . . . . . . . . . . . . . . . . 33

2.2.1 The Adam-Bashfort-Moulton time stepping scheme . . . . . . . . . . 35

2.2.2 Time differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.3 Spatial differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.4 Evaluation of u from the computed υ . . . . . . . . . . . . . . . . . 38

2.3 On using the Boussinesq equations in the swash zone: a note of caution . . 39

2.3.1 Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 41

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3 A new shoreline boundary condition for Boussinesq-type models 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 The shoreline Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Implementation of characteristic type SBCs . . . . . . . . . . . . . . . . . . 58

3.4.1 A WAF technique to move the shoreline . . . . . . . . . . . . . . . . 59

3.4.2 The basic steps of the proposed procedure . . . . . . . . . . . . . . . 64

3.5 Performance evaluation of the BTE model with the new SBCs . . . . . . . 65

3.5.1 The Carrier and Greenspan run-up solution . . . . . . . . . . . . . . 65

3.5.2 The Carrier and Greenspan standing wave solution . . . . . . . . . . 67

3.5.3 The Synolakis run-up solution . . . . . . . . . . . . . . . . . . . . . 70

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 A new shoreline boundary condition for wave-averaged models 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Wave-averaged models in proximity and into the swash zone . . . . . . . . . 75

4.3 The boundary conditions at the lower limit of the swash zone . . . . . . . . 76

4.3.1 The short-wave-averaged NSWE . . . . . . . . . . . . . . . . . . . . 76

4.3.2 NSWE integration over the swash zone . . . . . . . . . . . . . . . . 78

4.4 Evaluation of the chosen SBCs . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.1 The chosen numerical model . . . . . . . . . . . . . . . . . . . . . . 82

4.4.2 Choice of input data for the computations . . . . . . . . . . . . . . . 84

4.5 Features of the SBCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

The motion of a ‘rigid wall’ . . . . . . . . . . . . . . . . . . . . . . . 93

Steady state conditions . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.6 The SBCs at a moving ‘porous wall’ . . . . . . . . . . . . . . . . . . . . . . 94

4.7 Implementation of the SBCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Conclusions 102

List of Figures

1.1 Nonlinear numerical models classification . . . . . . . . . . . . . . . . . . . 22

2.1 Sketch of the typical problem geometry. . . . . . . . . . . . . . . . . . . . . 26

2.2 Free surface elevation plotted at different nondimensional times t∗ =

0, π/5, 3π/5, 4π/5 versus the x−coordinate. The straight line represents

the sloping beach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 For the same times of figure 2.2 contributions of O(µ2) to equation (2.2.23)

are plotted with dashed lines versus the x−coordinate. Superposed with a

solid line are the (I) contributions to equation (2.3.53). . . . . . . . . . . . 44

2.4 For the same times of figure 2.2 contributions of O(δµ2) to equation (2.2.23)

are plotted with dashed lines versus the x−coordinate. Superposed with a

solid line are the (II) contributions to equation (2.3.53). . . . . . . . . . . . 44

2.5 For the same times of figure 2.2 contributions of O(δ2µ2) + O(δ3µ2) to

equation (2.2.23) are plotted with dashed lines versus the x−coordinate.

Superposed with a solid line are the (III) contributions to equation (2.3.53). 45

3.1 The computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Sketch of typical problem geometry . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 The Riemann problem. Illustration of the initial data for: (a) a generic

Riemann problem, (b) the ‘shoreline Riemann problem’. . . . . . . . . . . . 52

3.4 The generic Riemann problem. . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Possible wave patterns in the solution of the Riemann problem. . . . . . . . 56

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3.6 C+ and C− characteristic patterns used to solve the Riemann problem at

the shoreline: (a) subcritical flow, (b) supercritical flow. . . . . . . . . . . . 56

3.7 Variables representation on a discretized domain: anticlockwise integration

of (3.4.18) on a discretized x− t space, (b) discrete solution behaviour. . . . 59

3.8 Example of solution of the ‘shoreline Riemann problem’. Only the problem

for the U1 = d component is illustrated. Top: the piece-wise initial

condition. Middle: the solution structure in the (x, t)−plane. Bottom:

solution of the Riemann problem in the physical space. . . . . . . . . . . . . 60

3.9 Riemann problem solution in the accelerating (a) and in the stationary (b)

reference frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.10 The ‘Carrier & Greenspan run-up test’ on a uniform plane beach.

Dimensionless, scaled, analytical (dotted lines) and numerical (solid lines)

profiles of water elevation ζ∗ are plotted versus the dimensionless onshore

coordinate x∗ at dimensionless times increasing of ∆t∗ = 0.05 from t∗ = 0.00

(bottom curves) to t∗ = 0.80 (top curves). . . . . . . . . . . . . . . . . . . . 66

3.11 The ‘Carrier & Greenspan standing wave test’ on a uniform plane beach:

envelope of surface elevations. Envelope of the dimensionless, analytical

solution by Carrier & Greenspan (dotted lines) and numerical (solid lines)

profiles of water elevation ζ∗ are plotted versus the dimensionless onshore

coordinate x∗. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.12 The ‘Carrier & Greenspan standing wave test’ on a uniform plane

beach: horizontal motion of the shoreline. Incident wave of dimensionless

amplitude A∗ = 0.6 and dimensionless frequency ω∗ = 1. Dimensionless

analytical shoreline path as from Carrier and Greenspan (1958) (dotted

line) and numerical shoreline path (shoreline line) in time. . . . . . . . . . . 69

3.13 Definition sketch for the initial condition of Synolakis’ run-up solution. . . . 70

7

3.14 The ‘Synolakis run-up solution’. Dimensionless free surface elevation ζ∗ as

functions of the dimensionless x∗ coordinate at different adimensional times

t∗ = 20, 30, 35, 40, 45, 50 (from left to right and from top to bottom). The

solid line represents computed data while solid circles are used for Synolakis

analytical solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1 Example of solution for a typical computation (beach slope α = 1/50,

test N. 12 of Table 4.1) performed with the depth-integrated, short-waves

resolving models. Top panel: instantaneous free surface elevation (solid

line) with envelopes of maxima and minima (dashed lines), mean water

level (dash-dotted line) and wave height (double-dotted-dashed line). Lower

panel: instantaneous velocity (solid line), mean velocity 〈u〉 (dashed line)

and mean velocity u (dashed-dotted line) of equation (4.5.22). In both

panels the thick, solid line represents the seabed. . . . . . . . . . . . . . . . 81

4.2 Instantaneous (solid lines) and wave-averaged (dashed lines) quantities are

plotted versus the computational time (s). Monochromatic SW. Beach slope

α = 1/10, Ts = 10s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


plotted versus the computational time (s). Monochromatic SW and LW.

Beach slope α = 1/10, Ts = 10s, Tl = 100s. . . . . . . . . . . . . . . . . . . 83


plotted versus the computational time (s). Monochromatic SW. Beach slope

α = 1/50, Ts = 10s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84


plotted versus the computational time (s). Monochromatic SW and LW.

Beach slope α = 1/50, Ts = 10s, Tl = 100s. . . . . . . . . . . . . . . . . . . 85


plotted versus the computational time (s). Random SW superposed to a

monochromatic LW. Beach slope α = 1/50, T ss = 10s, Tl = 100s. . . . . . . 86

8

4.7 Results of the validation procedure for a typical case with monochromatic

SW. Top panel: instantaneous (solid line) and mean (dashed line)

shorelines. Middle and lower panels: left hand side (solid line) and right

hand side (dots) of equations (4.3.18) and (4.3.19) respectively. . . . . . . . 87

4.8 Results of the validation procedure for a typical case with random SW. Top

panel: instantaneous (solid line) and mean (dashed line) shorelines. Middle

and lower panels: left hand side (solid line) and right hand side (dots) of

equations (4.3.18) and (4.3.19) respectively. . . . . . . . . . . . . . . . . . . 88

4.9 Top panels: comparison of the time series of dcomp (solid line) and deval

(dashed line) for the case ξm = 0.035, f = 0 (left) and ξm = 0.25, f = 0

(right). Lower panel: mean values of dcomp vs. deval for the cases f = 0

(diamonds), f = 0.5 (triangles) and f = 1 squares. The dashed line gives a

reference for the case of perfect matching. . . . . . . . . . . . . . . . . . . . 97

4.10 Fitting of SW properties. From left to right and from top to bottom are

the first four properties of equation (4.7.33): 〈V 〉,⟨Q

⟩,

⟨S

⟩and 〈Υ〉

respectively. In each plot the flow properties are plotted against the scaling

arguments defined by 4.7.33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

List of Tables

4.1 Short and long wave input parameters used for each value of the beach slope

and bottom friction parameter. Wave nonlinearity at the offshore boundary

of the domain, defined as the ratio of wave height and undisturbed water

depth, is indicated for both short (δs) and long waves (δl). . . . . . . . . . . 90

4.2 Synthetic results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9

Aknowledgments

I wish to thank my supervisors Prof. Alberto Noli (Universita degli Studi di Roma “La

Sapienza”), Dr. Maurizio Brocchini (Universita degli Studi di Genova) and Prof. Paolo

De Girolamo (Universita degli Studi di L’Aquila) for their continued support.

10

Declarations

Part of the work presented in this thesis is being published as:

G. Bellotti and M. Brocchini (2001). “On the shoreline boundary conditions for

Boussinesq Type models”. International Journal for Numerical Methods in Fluids. Vol.37

(4), pp. 479-500.

G. Bellotti and M. Brocchini (2001). “On using Boussinesq Type Equations near the

shoreline: a note of caution”. Accepted for publication in Ocean Engineering.

M. Brocchini and G. Bellotti (2001). “Integral flow properties of the swash zone

and averaging. Part 2. The shoreline boundary conditions for wave-averaged models”.

Submitted for publication to the Journal of Fluid Mechanics.

11

Abbreviations

2DH → Two Horizontal Dimensions resolved

ABM → Adam-Bashfort-Moulton

BP96→Brocchini and Peregrine (1996)

BTE → Boussinesq Type Equations

DNS → Direct Numerical Simulation

LES → Large Eddies simulation

LW → Long Waves

NSWE → Nonlinear Shallow Water Equations

RANS → Reynolds Averaged Navier-Stokes

SBCs→Shoreline Boundary Conditions

SW→Short Waves

SWFT→Short Waves Forcing Terms

12

List of symbols

a0 → wave amplitude

A, B, C → coefficients of the tridiagonal system

C+, C− → positive and negative characteristic curves

d → total water depth

D → dispersive and dispersive-nonlinear terms in BTE

δ → nondimensional parameter measuring the wave nonlinearity

∆x → grid spacing

∆t → time step

E, F → auxiliary variables used in the numerical model

F → flux term in the conservative form of the NSWE

h → undisturbed water depth

k0 → wavenumber

L → wavelength

λ → characteristic curves slope in the t− x plane

µ → nondimensional parameter measuring the wave dispersiveness

n, i → indexes used in the numerical model

νt → eddy viscosity

N → number of nodes in the computational grid

ω → angular wave frequency

p → pressure

R→ Riemann invariant

13

14

S → source term in the conservative form of the NSWE

t → time

T → number of time steps

τb → seabed friction

τxx → normal deviatoric stresses

τxz → turbulent stress

u → depth-integrated horizontal velocity component

u′ → horizontal component of the fluid velocity

U → vector of the unknowns in the conservative form of the NSWE

υ → auxiliary variable used in the numerical model

w′ → vertical component of the fluid velocity

x → horizontal coordinate

z → vertical coordinate

ζ → water elevation with respect to the undisturbed level

z → vertical coordinate

Chapter 1

Introduction

The design of structures to be built in the nearshore region generally involves the testing

of different possible layouts, under the effects of local wave and currents conditions, with

the aim of minimizing costs and maximizing the desired results. For example, the optimal

layout of a harbour, is the one that minimizes the total construction and maintenaince

costs, maximizes the protection of the internal areas from wave disturbances, minimizes

the negative effects on the littoral, etc.

The possible layout of the structures to be designed can be tested experimentally in

wave tanks and wave flumes using adequate scale models. An alternative and attractive

procedure is to employ suitable numerical and mathematical models. In principle, a

very advanced numerical model, able to correctly simulate all the nearshore phenomena

(turbulence, waves, currents, sediment transport, etc.) could be equivalent or even superior

to a physical model. In practice, the numerical models currently employed in engineering

activities, adopt several assumptions and simplifications: the phenomena that can be

simulated strictly rely on the governing equations solved by the model. Indeed, the great

advantage of numerical and mathematical models is that their application is usually much

less expensive than physical ones: it is certainly more economic to modify a computer file

describing the bathymetry of the area under investigation than rebuild a physical model

layout.

15

Introduction 16

The development of numerical models to be applied in the nearshore areas has therefore

received great attention in the last fourty years. The interest in numerical model increases

as much as computer speed and diffusion do. This work is aimed at improving depth-

integrated models, which belong to a specific category of numerical models applicable to

nearshore flows and are commonly used to simulate wave transformation and currents

generation. More specifically, the extension of these models up to the swash zone is

investigated.

This research is motivated by the fact that considerable efforts have been spent in

the recent past by several researchers in extending the applicability of these models to

the intermediate depths and/or in increasing their accuracy by developing advanced and

complicated theories. On the contrary, much less work has been done in studying the

problems that arise in very shallow water, in proximity and into the swash zone.

In order to make the reader familiar with the terminology used in this work, in section

1.1, both nearshore hydrodynamics phenomena and the different regions in which the area

of interest can be subdivided are briefly summarized. In section 1.2 a short overview of

the nonlinear numerical models used in maritime hydraulics for engineering and research

purposes is presented. Finally section 1.3 describes the organization of this work.

1.1 Nearshore hydrodynamics phenomena and nearshore

areas

Hydrodynamic phenomena occurring in the nearshore region can be subdivided into

three categories. Small scale turbulence (i.e. water movements with periods varying

between 10−2s and 10−1s) is the typical phenomena belonging to the first category and is

characterized by irregular fluctuations of the fluid velocity. In the second category fall those

oscillations of the free surface of the sea whose typical period is between 1s and 10s. These

oscillations are usually referred to as short waves and are typically generated by winds

blowing over large sea areas; as these waves propagate through coastal areas a number of

phenomena occur and dramatic changes of their characteristics are observed. Turbulence

Introduction 17

plays a fundamental role in the propagation of short waves: during wave-breaking a large

amount of waves energy is transformed into turbulence and is, subsequently, dissipated

at very small space and time scales leading to a strong decrease of the wave height. In

the third category of phenomena can be included those water motions with typical period

greater than 10s, commonly referred to as long waves and nearshore currents. Rip

currents, longshore currents, edge waves, surf-beat, set-up, set-down, undertow are typical

phenomena belonging to this third category.

The last two categories of phenomena introduced above are strictly linked. Short waves

are commonly responsible for generating long waves and currents. Winds, tides, large scale

circulation can also generate long waves and currents, but in this study we focus on the

methods that can be used to study and simulate short waves propagation and the related

long waves and current generation.

The nearshore area can be basically divided into three different regions, depending

on the type of phenomena occurring. The seaward region of the nearshore is commonly

referred to as the shoaling zone. In this region short waves generated in the deep sea start

feeling the decreasing water depth. Their height first decreases, then increases, on the

contrary their length first increases then decreases. Commonly this process leads to an

asymmetric wave steepening, until unstable conditions are reached and waves break. This

happens in the surf zone, which is characterized by a large amount of turbulence, vorticity

and sediments suspension. When waves finally reach the shore they move back and forth

the shoreline. This region is referred to as the swash zone and is characterized by very

peculiar hydrodynamic conditions: dry and wet conditions alternate and water depth is

in general extremely small.

Introduction 18

1.2 A brief overview of the nonlinear numerical models for

studying the nearshore hydrodynamics

In this section a short overview of the numerical models currently applied for engineering

and research purposes for studying nearshore hydrodynamics is presented. Emphasis

is focussed on an intuitive procedure for explaining the derivation process and the

characteristics of nonlinear models; only brief remarks are finally given on linear models.

Before discussing the many categories in which these nonlinear models can be divided, it

is necessary to define the dimensions of the areas under investigation and the time scales

of the phenomena. Coastal structures usually interest very large areas in comparison

to other engineering activities. Harbour breakwaters, beach defence structures such as

parallel or transversal (groins) barriers, can easily reach lengths of kilometers. It is clear

that some design problems can be simplified, for example investigating the performance of

a general cross-section of a breakwater, but more general problems like, for example, the

wave penetration in harbours or water flowing around structures, can hardly be reduced

to simpler configurations.

As introduced in the previous section the time scale of nearshore phenomena ranges

between 10−2s and 101s. In principle it is not possible to avoid simulating the shortest or

the longest period fluid motions: turbulence (10−2s - 10−1s) is extremely important for

a correct simulation of energy dissipations related to wave breaking and bottom friction;

long waves and currents (period of the order of 10s) are of fundamental importance in

several applications (harbour seiches, morphodynamics, etc.).

In this context a possible approach is to directly solve the Navier-Stokes equations

(DNS→Direct Numerical Simulation) over a wide area covering the region of interest.

These equations naturally take into account turbulence generation and energy dissipation

induced by wave breaking, adequately simulate short wave transformations and the relative

long waves and currents generation. Unfortunately it is to be noted that hydrodynamic

conditions induced by wave-breaking are characterized by very high Reynolds number: this

forces to use very fine grids and small time steps in order to resolve the scales at which

Introduction 19

energy dissipation occurs. It is well accepted that DNS are at present (and probably for

the next ten or twenty years) unsuited for application to full scale coastal hydrodynamics:

the range of time and space scale is too wide for this approach.

The natural solution is therefore to avoid simulating those component of the fluid

motion with very small periods and/or very small space scales. Averaging is therefore

introduced. This can be done in two different ways, averaging over time or averaging over

space. If the Navier-Stokes equations are averaged over the typical period of turbulence

the classical Reynolds equations are obtained. These are equations valid for the mean flow

(mean over the turbulence time scale) and are very similar to the original Navier-Stokes

equations but additional forcing terms (the Reynolds stresses), representing the effect of

unresolved turbulent fluid motion at smaller time scales, appear. Numerical solution of

the Reynolds equations (RANS) for modelling maritime hydraulics, is a very innovative

research topic. Very recently some researchers (Lin and Liu 1998) have shown that by

means of RANS it is possible to simulate breaking waves at a reasonable computational

cost. Neverthless it can be stated that simulations shown by these researcher are, at

the moment, limited to very special and simple cases. We expect that before this kind

of models can be applied to study large hydrodynamic patterns, much work has to be

done, both from coastal and from computer scientists. When averaging of the Navier-

Stokes equations is performed over space the method of Large Eddies Simulation (LES)

is obtained (Christensen and Deigaard 2001). Also in this case the unresolved scales of

fluid motion (the small eddies) are included in the resulting equations as contributions for

which closure is needed.

Further simplifications are therefore needed in order to obtain more easily and

conveniently applicable models. An effective procedure for simplifying the problem is

to integrate the Reynolds equations over the water depth assuming a given vertical

profile of the fluid motion (horizontal and vertical velocity components, pressure). The

equations obtained by applying this procedure are simpler because the vertical dimension

is eliminated. Such equations are therefore referred to as 2DH, i.e. two Horizontal

Dimensions are resolved. It is clear that the accuracy of the final equations relies on the

Introduction 20

chosen vertical profile. If, for example, the vertical velocity is assumed to be zero and the

horizontal component to be constant over the depth, integration over the depth leads to the

classical Nonlinear Shallow Water Equations (NSWE). Assuming that the vertical velocity

is zero coincides with assuming an hydrostatic distribution of the pressure and no frequency

dispersion can be simulated by the NSWE, the wave celerity only depending on total water

depth. By allowing depth-varying profiles of the horizontal and vertical components of

the velocity, the pressure distribution is no longer hydrostatic and a special category

of dispersive-nonlinear equations is obtained integrating over the depth: the Boussinesq

Type Equations (BTE). BTE (Peregrine 1967), can simulate frequency dispersion and in

the last ten years have received very much attention by coastal scientists: at the moment

numerical models based on BTE are probably the most advanced tools available for solving

engineering problems.

NSWE and BTE resolve short wave motion and are therefore referred to as wave-

resolving (we would probably better say short-wave-resolving). It is common experience

that, in order to obtain accurate solutions, at least 10-20 space points per wavelength

and 20 time-steps per period are needed by most of the numerical schemes. When very

large areas (dimensions of kilometers) are to be studied this category of 2DH models easily

becomes unapplicable. Furthermore the applicability of these models is limited to shallow

water areas, introducing additional restrictions to their use.

A practical alternative is to go back to the Reynolds equations, to average them over

the short wave period and to finally integrate over the water depth (Sancho and Svendsen

1997). As a result the wave-averaged 2DH model equations are obtained. These are

formally identical to the NSWE but now the dependent variable are short-wave-averaged

quantities and extra forcing terms appear, representing the unresolved short-waves motion

(these terms are hereinafter referred to as SWFT→Short Waves Forcing Terms). The

procedure is very similar to the process by which Reynolds equations are obtained from

the Navier-Stokes equations. As the turbulence is to be represented in the Reynolds

equations, the effect of short-waves (the gradients of the so-called radiation stress) is to

be provided to the wave-averaged NSWE. Numerical models based on this approach are

Introduction 21

referred to as wave-averaged (i.e. short-wave-averaged) and can simulate the generation of

long waves and currents provided that an appropriate (usually simple) model is employed

to calculate the short-wave quantities. As it happens for turbulence in the Reynolds

Equations, SWFT appearing in these equations depend in principle on the mean flow

characteristics: short-wave propagation is to some extent influenced by currents and long

waves. Short-wave propagation and mean flow cannot therefore be treated by separate,

decoupled models. Neverthless in many engineering applications the effect of currents on

short-waves is neglected and SWFT are preliminary evaluated by applying a dedicated

numerical model and then provided to the wave-averaged NSWE. In order to improve the

results it is possible to employ an iterative procedure, applying the short-waves model

taking into account the currents obtained by the first application of the NSWE and to

feed them again in the current model (e.g. Haas et al. 1998). After a few iterations the

process should converge towards a stable solution which take into account short-waves

interaction with currents.

Up to this point we only discussed nonlinear models (see Figure 1.1 for a classification

of these model). These can represent the generation of long waves and currents by short

waves. A further category of models, widely used for engineering purposes is to be

mentioned in this brief review. These are the linear models, which are based on very

strong assumptions about the characteristics of the waves and of the bottom variation.

More specifically the relative wave amplitude (compared with the water depth) is assumed

to be infinitesimal while the steepness (ratio of amplitude to wavelength) and the bottom

slope must be small. The vertical variation of the fluid motion and pressure is assumed to

coincide with the one derived by Airy for linear waves (from which the name of these

models). Similarly to the depth-integrated models only unknowns in two horizontal

dimensions are to be determined. In contrast to nonlinear models, linear ones cannot

simulate the generation of long waves and currents. The main reason rests in the fact

that Airy’s theory prescribes closed trajectories of water particle, these being circles in

very deep water, ellipses in intermediate waters and, at the limit, horizontal segments in

very shallow water. No mass transport, essential for the generation of long waves and

Introduction 22

Turbulence Short-waves Long-waves

Fully

3-D

Dep

th-int

egra

ted

(2D

H)

Resolved Times Scales and Phenomena

Res

olve

dSp

ace

Scal

es

BTE

NSWE

10−2s ∼ 10−1s 1s ∼ 10s > 10s

DNS

LES

RANS

Q-3D NSWE with SWFT

NSWE with SWFT

610−3

m∼

10−1

m10

0m

Figure 1.1: Nonlinear numerical models classification

currents, can therefore be simulated by linear models. Despite of this, linear models are

frequently used to calculate the SWFT used by wave-averaged NSWE and in several design

applications. It is to be mentioned that, for ten or twenty years, these models have been

the only available tool for simulating nearshore wave transformations.

1.3 Outline of the dissertation

The rest of this thesis is organized as follows.

In chapter 2 depth-integrated wave-resolving models are introduced. The procedure

for obtaining the NSWE and the BTE is illustrated. A specific numerical model, coded

by the Author for solving the BTE, is described. A detailed analysis of the problems

Introduction 23

related to using BTE in very shallow water is finally carried out. This chapter is a review

of already published works, except for section 2.3 which is innovative and that is being

published in Bellotti and Brocchini (2001b)

In chapter 3 new shoreline boundary conditions for Boussinesq-type models are

derived. These boundary conditions are then implemented in the numerical model

described in chapter 2, with the aim of verifying their capability in simulating swash

zone hydrodynamics. The comparison of the results obtained by means of the numerical

model against analytical solutions suggests a very good performance of the new shoreline

boundary conditions. The work presented in this chapter is innovative and has been

published in Bellotti and Brocchini (2001a).

In chapter 4 the problems related to using wave-averaged models in proximity and into

the swash zone are introduced. A solution strategy is indicated and boundary conditions

for incorporating the swash zone in this kind of model originally developed by Brocchini

and Peregrine (1996) are re-derived, verified and discussed. Several computations carried

out by means of an available numerical solver of the Nonlinear Shallow Water Equations,

allows to verify the ability of the proposed boundary conditions in predicting the motion of

a mean shoreline motion along with both wave-averaged water depth and wave-averaged

fluid velocity. The implementation of the boundary conditions in available models is

finally addressed. The work presented in this chapter is innovative and is being published

as Brocchini and Bellotti (2001).

Conclusions and some remarks on ongoing research are summarized in chapter 5.

Chapter 2

Depth-integrated wave-resolving

models

In this chapter the depth-integrated wave-resolving models for simulating the nearshore

hydrodynamics are introduced. In section 2.1 a derivation of the typical model equation

is presented, obtaining the NSWE and the BTE. Emphasis is posed on the differences

between the two formulations and on the advantages and disadvantages of each. A

numerical model, coded by the Author for solving the BTE is described in section 2.2. This

model was used to test the SBCs derived in chapter 3. In section 2.3 the applicability of

BTE is investigated on the basis of the analytical solution of Carrier and Greenspan (1958),

valid for periodic waves propagating on a sloping beach. Notice that in order to simplify

the derivation of the equations we limit our presentation to one horizontal dimension.

Therefore instead of obtaining 2DH equations, in this chapter and more generally in the

rest of this thesis we deal with 1DH equations.

The results presented in section 2.3 are innovative since before this work no careful

investigation was carried out in order to understand the role of dispersive-nonlinear terms

of BTE in very shallow water. The analysis shows that, if not properly handled, BTE

may become extremely unstable in the swash zone, leading to diverging solutions.

24

Depth-integrated wave-resolving models 25

2.1 The BTE and the NSWE

In this section a derivation of the NSWE and of the BTE is presented. Several possible

approaches have been followed in the past. We decided to follow the derivation by

Veeramony and Svendsen (1999), which is based on the direct integration over the water

depth of the Reynolds equations. The aim of this section is not to provide the reader with

a very detailed derivation of the equations, but to briefly illustrate the basic steps, the

simplifications and the assumptions that lead to NSWE and BTE.

2.1.1 Scaling parameters, reference frame and variable used

Two independent non-dimensional parameters are used during the derivation for

estimating the order of magnitude of each term appearing in the equations. These

parameters are obtained as ratios of the length scales associated with the wave motion:

the wave amplitude a0, the wave number k0 = 2π/L (L being the wavelength) and the

water depth h0. The first non-dimensional parameter δ = a0/h0 measures the nonlinearity

of the wave. The second parameter µ = k0h0 measures the frequency dispersiveness of

the waves. Large values of µ characterize the motion of waves in deep water, while small

values of µ are typical of long waves in very shallow water where wave celerity depends on

the water depth rather than on the wave frequency.

On figure 2.1 a typical problem geometry and the employed reference frame are shown.

The origin of the cartesian reference frame is at the undisturbed free surface, x is the

horizontal coordinate, positive shoreward; z is the vertical coordinate, measured positive

upwards. h is the undisturbed water depth, ζ the water elevation with respect to the

undisturbed level. d is the total water depth, given by the sum of h and ζ. In the

following sections u′ indicates the horizontal component of the velocity (measured positive

rightward in the adopted reference frame), w′ the vertical one (positive upwards).

Nondimensional variables are widely used in the following. These are defined by the


following relationships (superscript ∗ represents hereinafter non dimensional quantities)

x∗ = k0x, z∗ = zh0

, t∗ = k0√

gh0 t

u′∗ = δ√

gh0 u′, w′∗ = δµ√

gh0 w′(2.1.1)

x

z

d(x, t) h(x)

ζ(x, t)

Figure 2.1: Sketch of the typical problem geometry.

2.1.2 Boundary conditions and preliminary assumptions

In order to integrate the Reynolds equations over the water depth proper boundary

conditions should be imposed at the limits of the domain of interest. At the free surface

it is assumed that a particle originally on the surface stays at the surface during wave

propagation. From a mathematical point of view this kinetic boundary condition can be

expressed as

w′ (ζ) =∂ζ

∂t+ u′ (ζ)

∂ζ

∂x. (2.1.2)

At the free surface is, furthermore, assumed that atmospheric pressure is constant and

equal to zero:

p (ζ) = 0. (2.1.3)

An impermeable bottom is assumed in the derivation. Thus the kinematic condition at

the bottom can be expressed as

w′ (−h) = −u′ (−h)∂h

∂x. (2.1.4)


Several assumptions and simplifications are used during the derivation of the equations.

These are reported here below and briefly discussed. The fluid is assumed to be of constant

density and inviscid. The motion of the bulk fluid is non-rotational and a velocity potential

exists. The shear stress at the bottom and at the free surface is neglected. This coincides

with imposing a free slip condition at the bottom and to neglect any wind force acting on

the water.

2.1.3 The depth-integrated continuity equation

We start the derivation from the two-dimensional differential form of the continuity

equation∂u′

∂x+

∂w′

∂z= 0. (2.1.5)

Integration of (2.1.5) from the free surface (z = ζ) to the bottom (z = −h) gives∫ ζ

−h

∂u′

∂xdz + w′(ζ)− w′(−h) = 0. (2.1.6)

Use of the boundary conditions (2.1.2) and (2.1.4) allows to eliminate vertical velocities

at the boundaries and application of the Leibniz rule gives

∂ζ

∂t+

∂

∂x

∫ ζ

−hu′ dz = 0 (2.1.7)

that is the general, dimensional form of the depth-integrated continuity equation.

2.1.4 The depth-integrated momentum equation

We start from the horizontal and vertical momentum equation, which read respectively

∂u′

∂t+

∂u′2

∂x+

∂u′w′

∂z= −1

ρ

∂p

∂x+

1ρ

(∂τxx

∂x+

∂τxz

∂z

), (2.1.8)

∂w′

∂t+

∂u′w′

∂x+

∂w′2

∂z= g − 1

ρ

∂p

∂z+

1ρ

(∂τxz

∂x+

∂τzz

∂z

)(2.1.9)

where τxx and τzz are the normal deviatoric stresses and τxz is the turbulent stress.

Integration of (2.1.8) over the water depth, use of boundary conditions, Leibniz rule and

of assumptions about the forces acting on the fluid at the free surface give

∂

∂t

∫ ζ

−hu′ dz +

∂

∂x

∫ ζ

−hu′2 dz =

p (−h)ρ

∂h

∂x+

∂

∂x

∫ ζ

−h(−p + τxx) dz. (2.1.10)


The expression for the pressure p can be obtained by integrating (2.1.9) from the

surface to a generic level z, using both the boundary conditions and the Leibniz rule:

p (z)ρ

= g (ζ − z)− w′2 +∂

∂t

∫ ζ

zw′ dz +

∂

∂x

∫ ζ

z

(u′w′ − τxz

ρ

)dz. (2.1.11)

In order to write (2.1.11) in non-dimensional variables, a scaling for the turbulent stress

term is to be introduced. Veeramony and Svendsen (1999) proposed the following scaling

by assuming an eddy viscosity representation of τxz

τxz = δµρgh0νt

(∂u′

∂z+ µ2 ∂w′

∂x

)(2.1.12)

where νt is the eddy viscosity.

By replacing the dimensional variables with the non-dimensional ones given in section

2.1.1 and making use of (2.1.12), equation (2.1.11) becomes

p∗ (z∗) =(

ζ∗ − z∗

δ

)− δµ2w′∗2 + µ2 ∂

∂t∗

∫ δζ∗

z∗w′∗ dz∗

+δµ2 ∂

∂x∗

∫ δζ∗

z∗u′∗w′∗ dz∗ − µ2 ∂

∂x∗

∫ δζ∗

z∗νt

∂u′∗

∂z∗w′∗ dz∗ + O

(µ4

).

(2.1.13)

Equation (2.1.13) is suitable to illustrate the effects of frequency dispersiveness on

the pressure distribution over the water depth. The first term of the right hand side

of (2.1.13) represents the hydrostatic pressure component. The other terms, of order

O(µ2

)and O

(δµ2

)deviate the pressure distribution from the hydrostatic, thus allowing

for frequency dispersion. Neglecting all these smaller terms coincides with treating non

dispersive nonlinear waves, i.e. very long waves. As shown in the following sections

neglecting terms of order equal or smaller than O(µ2

)leads to obtain the NSWE, while

retaining these terms leads to the BTE, which can simulate frequency dispersion.

By inserting non-dimensional variables into (2.1.10), then using both the expression

for the pressure (2.1.13) and the following form of the continuity equation integrated from

−h to the generic level z

w′∗(z) = − ∂

∂x∗

∫ z∗

−h∗u′∗ dz∗ (2.1.14)


gives the following combined momentum equation in terms of horizontal velocity integrals

and water surface elevation

∂

∂t∗

∫ δζ∗

−h∗u′∗ dz∗ + (h∗ + δζ∗) ζ∗x∗ + δ

∂

∂x∗

∫ δζ∗

−h∗u′∗2 dz∗+

− µ2

∫ δζ∗

−h∗

∂2

∂x∗∂t∗

∫ δζ∗

z∗

∂

∂x∗

∫ z∗

−h∗u′∗ dz∗ dz∗ dz∗+

− µ2

∫ δζ∗

−h∗

∂2

∂x∗2

∫ δζ∗

z∗νt

∂u′∗

∂z∗dz∗ dz∗ +−δµ2

∫ δζ∗

−h∗

∂

∂x∗

(∂

∂x∗

∫ z∗

−h∗u′∗ dz∗

)2

dz∗+

− δµ2

∫ δζ∗

−h∗

∂2

∂x∗2

∫ δζ∗

z∗u′∗

∂

∂x∗

∫ z∗

−h∗u′∗ dz∗ dz∗ dz∗ = 0. (2.1.15)

2.1.5 An approximate expression for the horizontal velocity

The depth-integrated continuity and momentum equations (2.1.7) and (2.1.15) can be

solved analytically provided that a vertical profile of the horizontal velocity u′ is assumed.

By assuming irrotationality of the fluid motion such profile can be obtained by solving

the Laplace equation approximately, imposing the proper boundary conditions at the

free surface and at the bottom. In classical Boussinesq models this approximate solution

is achieved by expressing the velocity potential as an infinite polynomial series. In the

original work that is followed in this section (Veeramony and Svendsen 1999) a very original

approach was presented. Those Authors assumed that the wave-breaking process generates

vorticity. Therefore the fluid motion cannot be tread as irrotational and approximate

solution for the horizontal velocity was obtained by using the streamfunction instead

of a velocity potential. In this approach the final expression for the velocity contains

terms that represent the influence of the wave-breaking generated vorticity on the fluid

motion. The final Boussinesq equations obtained by using this expression for the velocity

can naturally simulate the dissipative effects of wave-breaking. Furthermore, since the

horizontal velocity depends on the vorticity generated by wave-breaking the vertical profile

of the velocity is not self-similar: it depends on both wave propagation and transformation,

while models based on the classical approach of irrotational flow present a self-similar

vertical profile. It is to be mentioned that very recently Rego et al. (2001) used the


same approach for simulating waves over flows with arbitrary vorticity, thus extending in

some sense the work of Veeramony and Svendsen (1999) which focussed on wave-breaking

generated vorticity.

In this section the final expression for the velocity u′ employed in classical weakly

nonlinear-dispersive Boussinesq models is reported. This expression can be also recovered

from that presented by Veeramony and Svendsen (1999) by neglecting the terms involving

vorticity. u′ can be expressed as a function of quantities measured at different levels in

the water column or in terms of averaged or integrated quantities. For example either

the horizontal velocity at the bottom u′0, or the velocity at a specific level zα, or the

depth-integrated velocity u∗ = 1(h∗+δζ∗)

∫ δζ−h∗ u′∗ dz can be used as variable. A widely used

expression relates u′∗ to u∗ and reads

u′∗ = u∗ + µ2

(∆1

2− z∗

)(h∗u∗)x∗x∗ +

µ2

2

(∆2

3− z∗2

)u∗x∗x∗ (2.1.16)

where ∆1 = δζ∗ − h∗ and ∆2 = δ2ζ∗2 − δζ∗h∗ + h∗2. Equation (2.1.16) represents the

classical parabolic velocity profile of BTE. It is evident that if wave dispersiveness is

neglected (i.e. µ = O (0)), the horizontal velocity is constant over the depth and, of

course, is equal to the depth-integrated velocity.

2.1.6 The final equations

The approximate expression for the velocity (2.1.16) can be used to evaluate the integrals

appearing in the non-dimensional form of (2.1.7) and in (2.1.15). During the algebraic

procedure terms of high order in δ and µ appear. Therefore a specified degree of accuracy

is chosen and only terms consistent with this accuracy are retained. Nevertheless the

depth-integrated continuity equation is not affected by the truncation in µ and δ. This

equation is, under the hypotesis summarized in section 2.1.2, exact and its final form reads

ζ∗t∗ + [(h∗ + ζ∗) u∗]x∗ = 0. (2.1.17)

Let us now discuss the influence of the scaling parameters on the final depth-integrated

momentum equations. If the final equations are to be used to simulate nonlinear waves in


very shallow water, the effect of frequency dispersion is neglected, since in these conditions

the order of magnitude of the scaling parameters is

δ = O (1) , µ2 = O (0) . (2.1.18)

The final form of the depth-integrated momentum equation, if all terms of order µ2 and

smaller are neglected, reads

u∗t∗ + δu∗u∗x∗ + ζ∗x∗ = 0 (2.1.19)

which is the well known momentum equation of the NSWE.

Extending the NSWE to the intermediate depths involves retaining terms of higher

order in µ. Weakly nonlinear waves in intermediate water can be simulated by chosing

the following order of magnitude of the parameters:

δ

µ2= O (1) , µ2 << O (1) . (2.1.20)

This choice leads to the classic weakly dispersive, weakly nonlinear BTE momentum

equations that read

u∗t∗ + δu∗u∗x∗ + ζ∗x∗ + µ2

[−1

3h∗2u∗x∗x∗t∗ −

12h∗h∗x∗x∗u∗t∗ − h∗h∗x∗u∗x∗t∗

]= 0. (2.1.21)

The NSWE (2.1.19) and the BTE (2.1.21) are the most widely used nonlinear depth-

integrated models for the nearshore hydrodynamics. Each of these equations presents

some advantages and some drawbacks, which we discuss in this final part of the section.

Any model application, aimed at studying nearshore waves and currents, consists of

simulating wave transformations from the offshore to the inshore, where knowledge of

flow conditions is usually required. Input wave conditions are usually known, with a

certain degree of accuracy, in deep water, where the influence of the bottom is small. This

information can be obtained for example by buoys, hindcasting or forecasting models, etc.

Neverthless, waves in very deep water are characterized by high values of the dispersiveness

parameter µ, and the assumptions (2.1.18) and (2.1.20) are clearly in contrast with this

feature of the waves to be simulated. It is therefore clear that a very important limitation


characterizes the application of the model discussed in this section: the offshore boundary

of the computational domain is to be carefully selected in order to impose, as boundary

conditions, waves consistent with the assumptions that are at the basis of the models.

In other words these models are shallow water models and can be applied in regions

where waves are strongly influenced by the bottom (shallow and intermediate areas). Wave

transformation from offshore areas to the intermediate or shallow areas, where accurate

modelling can be performed with NSWE and BTE, are to be taken into account using

alternative techniques. NSWE suffer greatly from this limitation if compared to BTE. Due

to their non-dispersive nature, NSWE cannot propagate waves of constant form. Being

purely amplitude-dispersive, NSWE simulate waveforms that evolve in time, since the

higher parts of the waves propagates faster than the lowers, inducing to steepening of

the wave profile. This leads to vertical wave fronts which, in some sense, model the

physical process of wave-breaking. The problem with NSWE is that the steepening

process is only partially influenced by the bottom: also very long waves, propagating

over a horizontal bottom would unrealistically break after a few wavelengths. Indeed it

is to be noted that once the shoreward part of the wave profile has become vertical, if a

suitable numerical method is employed to solve the equations, a bore-like solution would

be obtained, simulating the dissipative effects of wave-breaking. Furthermore NSWE are

usually conveniently solved by means of numerical methods that allow a very effective

treatment of the swash zone. It can be concluded that NSWE can be applied with good

results in the surf and in the swash zone, but they can not accurately predict where

the waves start breaking. Moreover wave-breaking effects are simulated by means of a

purely numerical treatment and no representation of the physical phenomena is generally

incorporated into the equations.

On the contrary BTE can accurately simulate wave propagation before wave-breaking

occurs: in the last ten years great efforts have been spent in order to extend to intermediate

and deep waters the BTE (Madsen and Schaffer 1998). Very recently Madsen et al. (2001)

presented a very advanced form of the equations that allows an accurate representation of

wave propagation from almost deep waters to the surf zone. In general these new form of


BTE are achieved relaxing the assumptions (2.1.20) and retaining high order terms both

in µ and δ. Wave-breaking in BTE models is simulated by introducing additional terms

in the equations, which appear when the wave front has reached a certain slope. BTE, if

compared to NSWE are a very complete and advanced model, but mainly two drawbacks

limit their application. First the computational times needed to solve the equations are

very large if compared to those required for solving the NSWE. By retaining high order

terms, very complicated governing equations are obtained. These are characterized by high

order derivatives of the dependent variables which, in order to be evaluated numerically,

need very fine space and time grids. Furthermore the flow in the swash zone is represented

using ad hoc techniques (see chapter 3), which limit the accuracy of the results.

2.2 A numerical model for solving the BTE

This section is dedicated to a brief description of the numerical model for solving the BTE

that was coded in order to verify the boundary conditions described in chapter 3. We have

chosen to adopt and code the model derived by Veeramony and Svendsen (1999), because

of its effectiveness in representing nearshore flows.

The basic model equations are both the mass conservation equation

ζt + [(h + ζ) u]x = 0 (2.2.22)

and the momentum conservation equation

ut + uux + gζx +(B − 1

3

)h2uxxt − 1

2hhxxut − hhxuxt + Bgh2ζxxx − 13h2uuxxx

+13h2uxuxx − 3

2hhxxuux − 12hhxxxu2 − hhxuuxx + Bh2 (uux)xx − 1

3hζuxuxx

−13huxx (ζu)x + h

(ζu2

x

)x− 2

3h (ζuuxx)x − ζxhxxu2 − ζhxuuxx − 12ζhxxxu2

−32ζhxxuux − ζxhxuux − 1

3ζ2uuxxx − ζζxuuxx + ζζxu2x + 1

3ζ2uxuxx − hζxutx

−23hζ (ut)xx + ζhx (ut)x − hxζxut − 1

2ζhxxut + 16ζ2 (ut)xx − 1

2

(ζ2 (ut)x

)x

= 0

(2.2.23)

with improved dispersion characteristics (here B = − 115). Good dispersion properties,

which make this model suitable for accurate flow predictions from the ‘intermediate’ to the

‘shallow waters’, have been obtained by retaining terms of order up to O(δ3µ2) inclusive.


Notice that, although the original form of equation (2.2.23) includes additional terms

which model energy dissipations caused by wave breaking, these are here neglected as our

present purpose is to investigate shoaling and run-up of non-breaking waves. They will

be re-introduced when modelling wave dynamics in the surf and swash zones. A second

note of caution is highlighted in section 2.3 and for the reason discussed there we coded

the equivalent (but written in terms of the water depth d instead of the surface elevation

ζ) momentum equation 2.3.53 which is reported here for convenience (notice that here Bhas been set to zero for simplicity sake):

ut + uux + gζx − 13d2uxxt + 1

2dhxxut + dhxuxt

−13d2uuxxx − 1

3d2uxuxx + 32dhxxuux + 1

2dhxxxu2 + dhxuuxx + dζxuxt + hxζxut

−dζxuuxx − dζxu2x + ζxhxxu2 + ζxhxuux = 0.

(2.2.24)

The governing equations (2.2.22) and (2.2.24) are solved adopting a 4th-order Adam-

Bashfort-Moulton scheme (ABM hereinafter) to step the model forward in time and a

five-point finite difference scheme to evaluate the spatial derivatives. The resulting model

scheme is widely adopted with good results to solve the Boussinesq equations (see for

example Wei et al. 1995) and the short-wave-averaged NSWE (Sancho and Svendsen 1997).

In order to apply the ABM scheme the governing equations are written in a more

convenient way:

ζt = E, (2.2.25)

υt = F (2.2.26)

where

E = − [(h + ζ) u]x , (2.2.27)

F = uux + gζx

−13d2uuxxx − 1

3d2uxuxx + 32dhxxuux + 1

2dhxxxu2 + dhxuuxx + dζxuxt + hxζxut

−dζxuuxx − dζxu2x + ζxhxxu2 + ζxhxuux = 0

(2.2.28)

and

υ =13d2uxx − 1

2dhxxu− dhxux. (2.2.29)


The independent variables x and t are discretized over an unstaggered grid by defining

xi = (i− 1)∆x, (i = 1, 2, ..., nx− 1, N) and tn = (n− 1)∆t, (n = 1, 2, ..., T − 1, T ), where

N is the number of nodes of the computational domain and T is the number of time-steps.

If initial conditions are specified, i.e. if the values of ζ and u at the time levels n, n−1,

n − 2 are available, the solution at the subsequent time level n + 1 can be obtained by

means of the following procedure:

1. evaluation of the right-hand sides of equations (2.2.25) and (2.2.26) at the time level

n, n− 1, n− 2;

2. integration in time of equations (2.2.25) and (2.2.26) by means of the predictor stage

of the ABM scheme;

3. evaluation of right-hand sides of equations (2.2.25) and (2.2.26) at the time level

n + 1;

4. integration in time of equations (2.2.25) and (2.2.26) by means of the corrector stage

of the ABM scheme;

5. evaluation of un+1i at all interior grid points by means of the solution of the

tridiagonal system resulting from the discretizing of (2.2.29).

Steps from 4 to 5 are iterated in order to improve the accuracy of the convergence.

The ABM time stepping scheme and the finite-differences expressions for the spatial

derivatives are detailed in the following sections.

2.2.1 The Adam-Bashfort-Moulton time stepping scheme

Once the right-hand sides of equations (2.2.25) and (2.2.26) are computed at time-steps

n, n− 1 and n− 2, estimates of quantities ζ and υ at the following time-step n + 1 can be

obtained by applying the ABM scheme which at the predictor stage reads:

ζn+1i = ζn

i +∆t

12[23En

i − 16En−1i + 5En−2

i

], (2.2.30)


υn+1i = υn

i +∆t

12[23Fn

i − 16Fn−1i + 5Fn−2

i

]. (2.2.31)

All values at the right hand sides of equations (2.2.30) and (2.2.31) are known from

previous calculations. The values of ζn+1i are thus straightforward to obtain. The

evaluation of horizontal velocities, u, at the new time level, however, requires solution

of the tridiagonal system resulting from the discretizing of (2.2.29) as detailed in section

2.2.4.

Once ζn+1i and un+1

i are estimated, the quantities E and F can be evaluated at the

time-step n + 1 and the corrector stage of the ABM scheme can be applied:

ζn+1i = ζn

i +∆t

24[9En+1

i + 19Eni − 5En−1

i + En−2i

], (2.2.32)

υn+1i = υn

i +∆t

24[9Fn+1

i + 19Fni − 5Fn−1

i + Fn−2i

]. (2.2.33)

The time-stepping scheme is accurate up to O(∆t)3 at the predictor stage and up to

O(∆t)4 at the corrector stage. As introduced before, by applying repeatidly the corrector

stage, very accurate estimates of the dependent variable can be obtained. More specifically,

the corrector step is iterated until the error between two successive results reaches a

required limit. The error is computed for each of the two dependent variables ζ and u and

is defined as:

∆f =

i=N∑

i=1

|f ′i − fi|

i=N∑

i=1

|f ′i |(2.2.34)

where f = {ζ, u}, while f ′, and f respectively denote the solution at successive iterations.

2.2.2 Time differencing

The quantity F , defined by (2.2.28) includes time derivatives of the dependent variable u.

These derivatives are evaluated employing time-differencing expressions consistent with

the accuracy of the selected ABM scheme. As far as the predictor stage is concerned, we

apply the following expressions

(ut)ni =

12∆t

[3un

i − 4un−1i + un−2

i

]+ O(∆t2), (2.2.35)


(ut)n−1i =

12∆t

[un

i − un−2i

]+ O(∆t2), (2.2.36)

(ut)n−2i = − 1

2∆t

[3un

i − 4un−1i + un−2

i

]+ O(∆t2). (2.2.37)

For the corrector stage, we evaluate ut according to

(ut)n+1i =

16∆t

[11un+1

i − 18uni + 9un−1

i − 2un−2i

]+ O(∆t2), (2.2.38)

(ut)ni =

16∆t

[2un+1

i − 3uni + 6un−1

i − un−2i

]+ O(∆t2), (2.2.39)

(ut)n−1i = − 1

6∆t

[2un+1

i − 3uni + 6un−1

i − un−2i

]+ O(∆t2), (2.2.40)

(ut)n−2i = − 1

6∆t

[11un+1

i − 18uni + 9un−1

i − 2un−2i

]+ O(∆t2). (2.2.41)

Notice that the expressions reported above use the value of u at the time level n + 1; this

implies that during the iterative application of the corrector stage, the value of ut, and

therefore F at the four time levels has to be computed repeatidily. In order to speed up

the computation we found useful storing at each time step the terms in F not containing

time derivatives of u and adding to them the terms containing ut at each iteration of the

corrector expression.

2.2.3 Spatial differencing

The spatial derivatives appearing in E and F are computed by means of high order

finite difference schemes in order to obtain estimates with truncation errors lower than

the highest order dispersive terms in the governing equations. In the interior region of

the domain central schemes can be applied while one-sided schemes are used to evaluate

derivatives at the boundaries.

These scheme reads, for first order derivatives with respect to x,

(wx)1 =1

12∆x(−25w1 + 48w2 − 36w3 + 16w4 − 3w5) , (2.2.42)

(wx)2 =1

12∆x(−3w1 − 10w2 + 18w3 − 6w4 + w5) , (2.2.43)

(wx)i = 112∆x [8 (wi+1 − wi−1)− (wi+2 − wi−2)] (i = 3, 4, ..., N − 2) , (2.2.44)


(wx)N−1 =1

12∆x(3wN + 10wN−1 − 18N−2 + 6wN−3 − wN−4) , (2.2.45)

(wx)N = − 112∆x

(25wN − 48wN−1 + 36N−2 − 16wN−3 + 3wN−4) (2.2.46)

where w is the variable to be differenced.

For second order derivatives, a three-point difference schemes is used:

(wxx)i =wi+1 − 2wi + wi−1

(∆x)2, (i = 2, 3, 4, ..., N − 1) (2.2.47)

2.2.4 Evaluation of u from the computed υ

Once the value of υn+1i has been determined at each grid node (i = 2, 3, ..., N − 2, N − 1),

a technique to solve the ordinary differential equation (2.2.29) is needed to compute the

water velocity.

Equation (2.2.29) can be discretized using a three-point finite difference scheme for the

second derivative of u and a simple two-point central finite difference scheme for the first

derivatives to give:

υn+1i = Aiu

n+1i−1 + Biu

n+1i + Ciu

n+1i+1 for i = 2, 3, ..., N − 2, N − 1. (2.2.48)

In which A,B, C are defined by:

Ai = − d2i

3∆x2+

di(hx)i

2∆x,

Bi =[1− 1

2di(hxx)i

]+

2d2i

3∆x2,

Ci = − d2i

3∆x2− di(hx)i

2∆x.

(2.2.49)

These N − 2 equations form a tridiagonal system that can be solved to obtain un+1i at

all the grid points if un+11 and un+1

N are specified. It is to be stressed that the velocity

at the boundaries at the time step n + 1 are requested by the numerical scheme and

are the boundary conditions needed to solve the governing system of partial differential

equations.


2.3 On using the Boussinesq equations in the swash zone:

a note of caution

The behaviour of BTE models at the shoreward boundary of the domain of interest i.e. in

the swash zone is discussed in this section. In order to illustrate problems and proposed

solutions we make use of the high-order BTE model given by equations (2.2.22) and (2.2.23)

(Veeramony and Svendsen 1999). As already mentioned we set to zero the parameter Band do not include additional terms which model energy dissipations caused by wave

breaking.

In our firsts attempts of using these equations from intermediate waters up to the

shoreline (see chapter 3) we run into numerical troubles when reaching the run-up region

i.e. x > 0. These problems were essentially related to numerical instabilities due

to the uncontrolled growth of the dispersive contributions (i.e. O(µ2)−terms). Such

contributions govern the three-points central difference scheme used to solve the tridiagonal

system obtained when adopting the numerical scheme described in the previous section, to

advance the solution at the i−th node from the instant n to the instant n + 1 (see section

2.2.4 for more details):

υn+1i = Ai−1u

n+1i−1 + Biu

n+1i + Ai+1u

n+1i+1 (2.3.50)

where, if the original formulation (2.2.23) is used instead of (2.2.24)

Ai = − h2i

3∆x2+

hi(hx)i

2∆x,

Bi =[1− 1

2hi(hxx)i

]+

2h2i

3∆x2,

Ci = − h2i

3∆x2− hi(hx)i

2∆x

(2.3.51)

are the equivalents of (2.2.49).

Since all the three coefficients contain hi, the solution of this tridiagonal system

depends on the size of hi which is shoreward decreasing until the still shoreline is reached.

Then, for x > 0 i.e. in the run-up region, hi grows to reach a maximum value at the

actual shoreline. This spurious behavior (we would expect pure dispersion to be zero


in very shallow waters) makes the solution unstable. It is correct mentioning we were

not the first to encounter difficulties in using BTE models in the swash. For example,

Madsen et al. (1997) report: “However, to make this technique (i.e. the ‘slot technique’

for moving the shoreline) operational in connection with Boussinesq type models a couple

of problems call for special attention.”. They, however, took the following very pragmatic

view: “Firstly the Boussinesq terms are switched off at the still water shoreline, where

their relative importance is extremely small anyway. Hence in this region the equations

simplify to the nonlinear shallow water equations...”.

On the other hand, we tried to solve the same problem in a different way i.e. forcing the

dispersive terms which appear in the tridiagonal system of (2.3.50) to identically vanish

at the shoreline. It is therefore obvious that the reference-dependent variable h had to be

replaced by the more physically important total water depth d. With this aim in mind we

re-wrote the model equations in the following slightly different but equivalent form:

ζt + [du]x = 0, (2.3.52)

ut +uux+gζx−[13d2uxxt +

12dhxxut + dhxuxt

]

︸︷︷︸(I)

−[13d2uuxxx− 1

3d2uxuxx+

32dhxxuux+

12dhxxxu2+dhxuuxx+dζxuxt+hxζxut

]

︸︷︷︸(II)

− [dζxuuxx − dζxu2

x + ζxhxxu2 + ζxhxuux

]︸︷︷︸

(III)

= 0. (2.3.53)

With this choice the coefficients Ai and Ci in (2.3.51) of the tridiagonal system (2.3.50),

which for these new equations are given by (2.2.49), identically vanish at the actual

shoreline hence allowing for a much more stable numerical solution while the coefficients

Bi reduces to unity. Some very small spurious oscillations were still detectable due to the

non-zero high-order dispersive-nonlinear terms. In order to better illustrate the role of

dispersive and dispersive-nonlinear contributions near the shoreline we have performed a

simple analysis which is presented in the next section.


2.3.1 Analysis and Discussion

The adopted BTE model contains contributions of order O(µ2), O(δµ2), O(δ2µ2) and

O(δ3µ2). In order to illustrate the role of each term near the shoreline we have computed

such contributions on the basis of a reference solution i.e. the only available analytical

solution for periodic waves in the swash zone. This is the solution of (Carrier and

Greenspan 1958). With this solution we do not have to rely on numerical computations

of each contribution of equations governing equations. Such computations would always

carry with them the uncertainties due to numerical errors and instabilities which could

obscure the results. Rather, the Carrier and Greenspan analytical solution allows to

directly and reliably estimate contributions to BTE equations under investigation.

We have performed a few computations but, to highlight what happens in the

swash zone, we choose to show only that concerning the motion of a wave of

dimensionless amplitude (hereafter stars characterize dimensionless quantities) a∗0 = 0.5

and dimensionless frequency ω∗ = 1. This non-breaking wave allows for a sufficiently wide

swash zone the width of which being of a∗0ω∗/2 = 0.25 (see Brocchini and Peregrine 1996).

Before discussing the results we want to clarify that, due to the dimensionless definition

of the total water depth:

d∗ = h∗ + δζ∗ (2.3.54)

the dimensionless form of the first bracket of equation (2.3.53), i.e. that denoted by

(I), not only contains O(µ2) terms but also O(δµ2). Similarly, the second bracket [i.e.

(II)] of the same equation contains both O(δµ2) and O(δ2µ2) terms, while the third [i.e.

(III)] contains both O(δ2µ2) and O(δ3µ2) contributions. Notwithstanding this mixture

of contributions we found it useful to compare the spatial distribution of the terms of

the three brackets of equation (2.3.53) respectively with that of the O(µ2), O(δµ2) and

O(δ2µ2) + O(δ3µ2) terms of equation (2.2.23). These are reported, with the same order,

in figures 2.3, 2.4 and 2.5, while figure 2.2 shows, for reference, the free surface position

(i.e. the phase of the wave) and the sloping beach. The wave run-up/run-down motion is

illustrated for the four following dimensionless times t∗ = 0, π/5, 3π/5, 4π/5 on the panels


(from left to right) of each figure. In each panel of figures 2.3, 2.4 and 2.5, the contributions

coming from equation (2.3.53) are plotted with solid lines while those relative to equation

(2.2.23) with dashed lines.

Notice that the comparison of the mixed-ordered brackets with the different order

contributions to (2.2.23) is even more reasonable when considering that δ and µ are built

with scales (still water depth, wave amplitude and wavelength) of the flow in the offshore

region of the domain of interest. Hence, they are typical of shallow but finite water depths

and are not suitable for describing the order of magnitude of the various contributions in

the swash zone which is characterized by very thin sheets of water. To be more precise

this inadequacy starts from the region in which h is a small contribution to d.

At a first glance (see figure 2.3) it is evident how the contributions of bracket (I)

of equation (2.3.53) are almost equivalent to the O(µ2) terms of equation (2.2.23). The

largest discrepancies occur inside the swash zone. At the shoreline the O(µ2) terms can

be positive or negative depending on run-up/run-down phase. On the contrary the (I)

contribution is always zero at the shoreline. As for the high-order contributions (see figures

2.4 and 2.5) the redistribution operated to get equation (2.3.53) is such that contributions

(II) and (III) are characterized by smaller oscillations than the O(δµ2) and O(δ2µ2) +

O(δ3µ2) terms. In order to fully illustrate the behaviour of these terms their spatial

distribution has been shown over a larger distance from the shoreline (about one and

a half wavelengths). It is clear that dispersive-nonlinear contributions are increasingly

important while approaching the shoreline. Hence, in principle, they cannot be neglected.

In particular the highest order term of figure 2.5 are almost zero at a distance of one

wavelength from the undisturbed shoreline and abruptly grow to reach their maximum

in the swash zone. This behaviour in the swash zone is such to require much care when

computing dispersive-nonlinear terms up to the shoreline: high-order spatial derivative

may easily introduce numerical instabilities and suitable techniques must be adopted to

handle them (see chapter 3).

In summary we have shown one significant practical result: using d instead of h in

the equations of BTE models leads to re-grouping terms in the form of equation (2.3.53).


This is equivalent to equation (2.2.23) but better tractable from a numerical point of view.

Notice that, though based on a specific BTE model, most of the observations we have made

are valid for a large number of high-order BTE models (also for the fully nonlinear models

like those of Wei et al. 1995 and Madsen and Schaffer 1998). In fact all of them have to face

the problem of treating high-order terms in the run-up region (i.e. for x > 0). Moreover,

most of the available and currently used BTE models are numerically solved by the ABM

scheme mentioned above (e.g. Wei et al. 1995; Skotner and Apelt 1999; Veeramony and

Svendsen 2000 and others) which carries with it the problem of dealing with the coefficients

of the tridiagonal system (2.3.50) for x > 0. One second result concerns quntification and

illustration of the importance of the various dispersive-nonlinear contributions in very

shallow waters on the basis of an analytical model solution.


−1.5 −1 −0.5 0

0

0.01

0.02

0.03

−1.5 −1 −0.5 0−0.05

0

0.05

0.1

−1.5 −1 −0.5 0

−0.04

−0.02

0

0.02

−1.5 −1 −0.5 0

−0.1

−0.05

0

0.05

Figure 2.2: Free surface elevation plotted at different nondimensional times t∗ =

0, π/5, 3π/5, 4π/5 versus the x−coordinate. The straight line represents the sloping beach.

−0.4 −0.2 0 0.2

−15

−10

−5

0

5

x 10−5

−0.4 −0.2 0 0.2

−3

−2

−1

0

x 10−4

−0.4 −0.2 0 0.2

−1

0

1

2

x 10−4

−0.4 −0.2 0 0.2

0

2

4

x 10−4

Figure 2.3: For the same times of figure 2.2 contributions of O(µ2) to equation (2.2.23)

are plotted with dashed lines versus the x−coordinate. Superposed with a solid line are

the (I) contributions to equation (2.3.53).

−1.5 −1 −0.5 0−6

−4

−2

0

x 10−5

−1.5 −1 −0.5 0

−10

−5

0

x 10−5

−1.5 −1 −0.5 0

0

2

4

6

x 10−5

−1.5 −1 −0.5 0

−4

−3

−2

−1

0

x 10−4

Figure 2.4: For the same times of figure 2.2 contributions of O(δµ2) to equation (2.2.23)

are plotted with dashed lines versus the x−coordinate. Superposed with a solid line are

the (II) contributions to equation (2.3.53).


−1.5 −1 −0.5 00

5

10

x 10−6

−1.5 −1 −0.5 0

−10

−5

0

x 10−6

−1.5 −1 −0.5 0

−15

−10

−5

0x 10

−6

−1.5 −1 −0.5 0

−3

−2

−1

0

x 10−4

Figure 2.5: For the same times of figure 2.2 contributions of O(δ2µ2)+O(δ3µ2) to equation

(2.2.23) are plotted with dashed lines versus the x−coordinate. Superposed with a solid

line are the (III) contributions to equation (2.3.53).

Chapter 3

A new shoreline boundary

condition for Boussinesq-type

models

3.1 Introduction

As already mentioned the most favoured approximate model equations for studying

nearshore hydrodynamics are both the NSWE and the many available BTE which all

stem from the work of Peregrine (1967).

BTE became very popular when it was proved they could model fairly well breaking

waves (Brocchini et al. 1992; Schaffer et al. 1993). Subsequently, in order to make such

equations more suitable for coastal engineering practice, dispersive characteristics were

greatly improved extending their seaward limit to reach the so-called ‘intermediate depths’

(see Madsen and Schaffer 1998 and references therein).

Notwithstanding these important improvements, which recently made BTE models

‘the models’ for coastal engineering, flow solvers based on those equations suffer of a

major problem. This is related to the mathematical/numerical treatment of both the

swash motions and the delicate shoreline boundary conditions (Brocchini and Peregrine

46

A new SBC for wave-resolving models 47

1996).

To our knowledge no available solver based on BTE correctly models the shoreline

motions and often ad-hoc artificial techniques are used to model wave run-up and run-

down (see for example the ‘slot technique’ used by Madsen et al. 1997). The quest for good

shoreline boundary conditions (SBCs hereinafter) to be implemented in BTE models is

currently being pushed in a number of different directions. Recently new SBCs are being

developed (Ozkan Haller and Kirby 1997a) with the use of coordinate transformations

which map the irregular shoreline to a straight line. Although a few examples are given

which testify good performances some doubts can be reasonably raised on the effectiveness

of such techniques in the case of heavily breaking waves which require strongly distorted

transformations. This is more true for breaking waves which interact in the swash zone

(e.g. backwash bores) as they generate cusps-like indentations at the shoreline which seem

hardly representable by a smooth coordinate transformation. No artificial techniques are

required when using the NSWE as model equations. NSWE are typically solved by means

of the method of characteristics and the shoreline is a characteristic itself!

It is now becoming clear that better modelling is required of the SBCs employed in

BTE models. To this purpose a number of methods can be applied a short list of which

is here given as reference.

BTE-NSWE matching

This method, currently applied by some researchers, does not directly address the real

problems concerning the definition of suitable SBCs. Rather, a pragmatic view is taken

according to which purely dispersive BTE (i.e. with no extra nonlinear contributions,

see section 2.3 for a discussion) reduce to NSWE in very shallow waters. Consequently

a matching is imposed (depending on the local Ursell number) between BTE and NSWE

solvers (Giarrusso 1998; Dodd and Giarrusso 2001). With this technique swash zone

motions are always modelled by the NSWE module which properly handles the motion of

the shoreline.

Extension of the NSWE to include dispersion


It is based on the view that NSWE are most suitable for modelling the swash zone

motions and track the shoreline position. In order to extend the range of validity of

the NSWE to the ‘intermediate depths’ suitable nonlinear-dispersive contributions can be

included either into the flux term Fx or into the source term S of the model equation:

Ut + Fx = S (3.1.1)

used to cast the 1DH-NSWE in a typical conservation form to be solved for the variable

U (Brocchini et al. 2001).

Characteristic-type SBCs for BTE

A third approach is here followed which is believed to both provide a close description

of what actually happens at the point (line) where the water meets the beach face and

to be easily implemented in any type of numerical models based on BTE. Analysis is

underway to define the most suitable form of the SBCs (1DH flow propagation):

dxs

dt= us, ds = 0 (3.1.2)

[xs being the shoreline position, ds and us respectively the water depth and the flow speed

at the shore].

This chapter is organized as follows. A description of the problem to be solved and of

the schematization adopted is given in section 3.2. A detailed analysis of the celerity at

which the shoreline moves is given in section 3.3, where the ‘shoreline Riemann problem’

is introduced. The description and the implementation of the new shoreline boundary

conditions in the model described in section 2.2 is detailed in section 3.4. Section 2.2

illustrates the specific BTE model which has been developed in order to evaluate the

performance of the new SBCs. In section 3.5 the performances of the new SBCs and

their implementation is verified by means of the comparison against well known analytical

solutions. Some concluding remarks are given in the final section 3.6 along with a short

description of ongoing research on this topic.


3.2 Problem statement

Our objective here is to derive a shoreline boundary condition suitable to be implemented

in most commonly used numerical schemes for solving the Boussinesq type equations, i.e.

finite differences schemes, working on fixed computational grids. In order to proceed a

careful definition of the problem to be solved is needed. With reference to Figure 3.1, it

can be noted that in the swash zone the water depth gradually decreases up to a point

where it becomes zero. This point is commonly referred to as the shoreline. Notice that

in the sketch reported in the figure it is assumed that a very smooth transition in the

solution, i.e. in water depth and velocity, verifies. This is not the case when a bore is

propagating on the beach or when a strong interaction between two waves, one running-up

and one running-down the slope, occurs. Nevertheless the essence of the problem is not

altered: it can always be recognized that there is a point where the transition between

wet and dry conditions occurs: that point is the shoreline.

x

1

ζ

wet

shoreline

dry

N NTOT

Figure 3.1: The computational domain.

If we look at the problem from a numerical point of view, a discrete representation of

the physical conditions described above results. The solution, i.e. the value of both water

depth and velocity, can be known only at the computational points and it is clear that an

exact definition of the shoreline is no longer available. The shoreline is hence somewhere

between the last wet node (N in the Figure) and the first dry node (N + 1). In this work


we assumed that if a small spacing between the nodes ∆x is adopted, the shoreline can

reasonably be considered to lay in the middle of the region [xN , xN+1], where xN and

xN+1 are the abscissa of the two nodes N and N + 1. The position of the shoreline xs is

hence defined as xs = xN + ∆x2 .

By taking this view the computational domain is divided into two sub-domains as

depicted in Figure 3.1. From node 1 to node N the nodes are wet and the governing

equations are solved by means of a finite differences scheme as detailed in section 2.2.

From node N + 1 to node Ntot the nodes are dry. At the first and last wet nodes (i.e. 1

and N), suitable boundary conditions are to be specified in order to solve the problem (see

section 2.2.4). Furthermore, by changing the value of N , i.e. by inundating and drying

the nodes, the movements of the shoreline can be simulated.

It is now becoming clear that any swash zone modelling based on the conceptual scheme

described above has to deal with two problems. The first one is the specification of water

depth and velocity at the last wet node. These two values are the boundary conditions

needed to solve the system of partial differential equations. The second problem deals

with the simulation of the shoreline movements: a technique is needed capable of deciding

whether the wet region of the computational domain is to be enlarged (thus increasing the

value of N simulating run-up) or restricted (decreasing N simulating run-down).

3.3 The shoreline Riemann problem

Different approaches can be used to model the motion of the shoreline on a beach.

Until recently one of the most used was the “thin film approach” in which the whole

computational domain is considered as “wet” but the thickness of the water is defined as

“very small” in the region of the beach not reached by the motion of the waves. However,

it can be demonstrated that such an approach leads both to a theoretically wrong solution

and to great numerical inaccuracies. Hence we prefer to define and solve the motion of a

genuine wet-dry interface.

In this section the ‘shoreline Riemann problem’ is introduced with the aim of


investigating the celerity at which, according to the Boussinesq equation, the shoreline

moves. As detailed in the previous section at the shoreline a transition occurs between a

finite and a null value of the water depth. Hence a discontinuity of the solution verifies.

A suitable theoretical approach for dealing with such a discontinuity originates from the

method of the characteristics and is based on the solution of the so called Riemann problem

(Toro 1997). In the following the conservative form of BTE, the generic and the shoreline

Riemann problems are introduced. Finally the solution in the BTE framework is derived.

Let us first of all recognize that for small enough water depth most dispersive-nonlinear

terms D which characterize BTE from NSWE become negligible. Hence near the shoreline

we can write the 1DH version of any BTEs as:

dt + (ud)x = 0 (3.3.3a)

ut + uux + gdx = ghx − τb + D, (3.3.3b)

where d = h + ζ (see figure 3.2) is the total water depth, u is a depth-averaged velocity,

τb is the seabed friction and subscripts are used to represent partial derivatives.

x

z

d(x, t) h(x)

ζ(x, t)

Figure 3.2: Sketch of typical problem geometry

These can be cast in suitable conservative vectorial form which is typically used in

shock-capturing numerical solvers:

Ut + F(U)x = S(U) (3.3.4)

U being the vector of the unknowns, F the flux term and S the source term

U =

d

ud

, F(U) =

ud

u2d + gd2

2

, S(U) =

0

gdhx − dτb + dD

(3.3.5)


which also includes all dispersive-nonlinear contributions D which characterize each

specific BTE.

The above mentioned Riemann problem is defined by equations (3.3.4), (3.3.5) and

constant initial conditions (see figure 3.3a) such that:

U(x, 0) = U0(x) =

UL if x < 0

UR if x > 0.(3.3.6)

A very specific Riemann problem is the one in which the right constant state is dry.

This helps to formulate and solve the transition which occurs at the shoreline (see figure

3.3b). A similar description was given by Stoker (1957) of the ‘retreating piston’ or

‘retreating wave paddle’ problem. We call the specific Riemann problem of figure 3.3b as

the ‘shoreline Riemann problem’.

U0(x) U0(x)

UL UL

UR

UR

x x(a) (b)

Figure 3.3: The Riemann problem. Illustration of the initial data for: (a) a generic

Riemann problem, (b) the ‘shoreline Riemann problem’.

In figure 3.4 the solution structure of a typical Riemann problem is shown. This is the

well known ‘dam break on a wet bed’. On the upper panel the initial configuration (time

t = 0) of the free surface is shown; it can be noted that an abrupt variation of the water

depth occurs at x = x0. The velocity of the fluid is in this case equal to zero. At a generic

time t† > 0 from the point x = x0 two wave families emanate (see lower panel). For the

specific case at hand the left wave family is a rarefaction wave and the right one is a shock

wave.

Rarefaction waves, also indicated as depressions, connect two data states through a


smooth transition. At any fixed time all flow quantities vary continuously across the

wave. As depicted in the figure the wave has a fan like structure, centered at the origin.

Rarefaction waves propagate in the deep water region (left in the figure) reducing the water

depth. Shock waves, also known as bores, connect, through a single jump discontinuity,

two constant data states. For the case reported in figure the shock wave propagates in the

shallow water region (right) rising the water depth abruptly.

A simple, intuitive method for determining what kind of waves (weather rarefaction

or shock) emanates from the original discontinuity is given here below, provided that an

exact description of this mathematical problem can be found in a number of books (see

for example Toro 1997). Given that the governing system of equations is hyperbolic,

it can always be recognized that some information on fluid motion travels along the

characteristics curves. Two characteristics originate from each point in the x − t space,

one is termed positive or advancing characteristic (C+) and one is termed negative or

receding (C−). In very shallow water the characteristic curves C− and C+ of (3.3.4) are:

dx

dt= λ1 = u− c (C−),

dx

dt= λ2 = u + c (C+) (3.3.7)

where c =√

gd.

Subcritical flow states are characterized by the fact that the signs of λ1 and λ2 are

discordant. The contrary occurs for supercritical flows, where the sign of λ1 and λ2 is

equal. Now, compare the magnitude of the quantities λL1 and λR

1 , where the superscripts

indicate if the quantity refers to left (L) or right (R) states. It can be stated that left

rarefaction waves are generated if λL1 < λR

1 and that left shock waves are generated if

λL1 > λR

1 . Furthermore right rarefaction waves are generated if λL2 < λR

2 and right shock

waves are generated if λL2 > λR

2 .

In the typical case shown in figure 3.4 the negative characteristics (notice that in this

case λL1 < λR

1 ) originating from the left and from the right of the initial discontinuity,

bound the left fan region. On the contrary the positive characteristics (λL2 > λR

2 ) collapse

into a right shock waves.

In principle there are four possible wave patterns for a generic Riemann problem.


These are depicted in the figure 3.5 where thick lines represent shock waves and the fans

represent rarefactions. Case (a) is characterized by a left rarefaction wave and a right

shock wave, case (b) by a left shock and a right rarefaction, case (c) by a left and right

rarefaction and finally case (d) by left and right shock waves. The waves that originates

from the initial discontinuity separate three constant states, indicated in figure 3.5 by UL,

U∗ and UR. The left (UL) and right (UR) states are known, being equal to the initial

left and right initial conditions. The region in the middle of the wave families is indicated

as the cross region. Water depth and velocity (d† and u†) can be calculated by means of

exact or approximate Riemann solvers (Toro 1997).

The ‘shoreline Riemann problem’ is in principle very similar. In this case (indicated in

the NSWE framework as ‘dry bed Riemann problem’) only the left wave family originates

from the initial discontinuity, being no medium in which the right wave can propagate. It

can be demonstrated that this left wave is a rarefaction and the rightmost wave of the fan

coincides with the instantaneous position of the shoreline (see figure 3.6).

The characteristic curves C+ and C− (Figure 3.6) are used to represent the solution

structure for the problem of figure 3.3b) meet at the shoreline which can be considered as

a C−-type characteristic such that:

dx

dt= λ1s = us − cs, (C−). (3.3.8)

In this case the (x, t)-plane is subdivided into three regions which characterize the

solution of the shoreline Riemann problem: region II is made of an expansion fan of C−-

type characteristics connecting conditions of region I of left constant conditions UL =

(dL, uLdL) with the dry conditions UR = (dR, uRdR) = (0, 0) of region III.

Notice that along the C− and C+ characteristics Riemann variables (R1,R2) = (u −2c, u + 2c) are not conserved (as in the case of inviscid NSWE) because of the presence

of non-zero source terms which also include dispersive-nonlinear contributions. On the

contrary the following is valid:

dR1

dt= S along C−,

dR2

dt= S along C+ (3.3.9)

where S = S2/d = ghx − τb + D.


water depth at time = 0

water depth at time t†

velocity at time t†

rarefaction shock

x

x

x

x

d

d

u

t

t†

x0

Figure 3.4: The generic Riemann problem.

It is, finally, essential to notice that SBCs are only a simplified version of:

dxs

dt= us, or xs =

∫us dt, (3.3.10a)

ds = 0 (3.3.10b)

and the purpose of any analyses dealing with SBCs is to suitably define us which appears

in (3.3.10a) by obeying the constraint (3.3.10b).

Following Brocchini et al. (2001) we compute us using conditions (3.3.9) in which


(a) (b)

(c) (d)

x

x

x

x

t

t

t

t

UL UR

U†

UL UR

U†

UL UR

U†

UL UR

U†

Figure 3.5: Possible wave patterns in the solution of the Riemann problem.

I

II

III I

II

III

(a) (b)x x

t tC− C−

C+

C− C−

C+

Figure 3.6: C+ and C− characteristic patterns used to solve the Riemann problem at the

shoreline: (a) subcritical flow, (b) supercritical flow.

ds = 0 =⇒ cs =√

gds = 0 is used on the C− characteristic which represents the shoreline:

dR1

dt= S along

dxs

dt= us, (3.3.11a)

dR2

dt= S along

dxs

dt= uL + cL. (3.3.11b)


Integration of these gives

R1(t + ∆t) = R1(t) +∫ t+∆t

tSdt along

dxs

dt= us, (3.3.12a)

R2(t + ∆t) = R2(t) +∫ t+∆t

tSdt along

dxs

dt= uL + cL (3.3.12b)

where in this case [R1(t),R2(t)] = [us − 2cs, uL + 2cL] = [us, u

L + 2cL].

Substitution into equations (3.3.12) and knowledge of the integration path gives

R1(t + ∆t) = us(t) +∫ (x+∆x)/us

x/us

Sus

dx, (3.3.13a)

R2(t + ∆t) = uL(t) + 2cL(t) +∫ (x+∆x)/(uL+2cL)

x/(uL+2cL)

SuL + 2cL

dx. (3.3.13b)

Notice that particular attention should be taken to evaluate integral contributions for

small velocity values. At the shoreline this only occurs when at the maximum run-up and

run-down locations.

At the shoreline the above conditions are simultaneously valid (see figure 3.6) hence

giving the final result

us(t) = uL(t) + 2cL(t) +∫ (x+∆x)/(uL+2cL)

x/(uL+2cL)

SuL + 2cL

dx−∫ (x+∆x)/us

x/us

Sus

dx (3.3.14)

which replaces the condition

us(t + ∆t) = uL(t) + 2cL(t) (3.3.15)

valid for NSWE.

In the case of inviscid BTEs (i.e. with no seabed friction included) with purely

dispersive extra contributions, D → 0 in very shallow depths and the source term reduces

to the acceleration due to the beach slope. Therefore (3.3.15) can suitably be used to

evolve the shoreline position xs in time through (3.3.10a) if either a splitting technique

is used for such term (Brocchini et al. 2001) or the coordinate transformation by Watson

et al. (1992) is adopted. On the other hand, if D also contains nonlinear-dispersive terms

(see section 2.3 for proper treatment of these terms) equation (3.3.14) replaces (3.3.15).


3.4 Implementation of characteristic type SBCs

In section 3.3 the fluid velocity un+1s at the interface between wet and dry states was

obtained by solving the ‘shoreline Riemann problem’. Now, in order to employ this solution

as a boundary condition for the BTE model, it should clarified what is the difference (if

any) between un+1s and un+1

N , that is the boundary condition for the BTE model.

We can state that by assuming un+1N = un+1

s unrealistic and numerically unstable

solutions are obtained by the BTE model. The reason is that us is the velocity of the

fluid at a specific point (the shoreline) of the computational domain, while un+1N should

be representative of flow conditions in the region [xN − ∆x2 , xN + ∆x

2 ].

A numerical technique to evaluate un+1N from un+1

s is therefore needed. The basic

assumption we start from is that u and ζ are piece-wise constant over the three regions

[xi − ∆x2 , xi + ∆x

2 ], i = N − 1, N, N + 1, hereinafter referred to as ‘computational cells’.

The quantities unN−1, un

N and unN+1 can therefore be viewed as integral averages of the

solution u(x)n, namely

uni =

1∆x

xi+1

2∫

xi− 1

2

u(x)ndx. (3.4.16)

Now a suitable numerical method is to be chosen in order to evaluate un+1N starting from

piece-wise constant initial conditions as depicted in figure 3.8. It is necessary that the

method can adequately deal with solution discontinuities (between cells N − 1 and N)

and treat the wet-dry interface between cells N and N + 1 by taking the most from the

accurate analysis performed in section 3.3. Brocchini and co-workers (Brocchini et al.

2001) showed that a NSWE nearshore flow solver based on the WAF (Weighted Averaged

Flux) method can accurately simulate swash zone flows and shoreline motions. The WAF

method is therefore adopted in the present study as the numerical tool to evaluate un+1N ,

i.e. the boundary condition of the BTE model. It is to be stressed that this method is

here merely used to convert the ‘real’ velocity value us into the ‘numerical’ value un+1N .


3.4.1 A WAF technique to move the shoreline

The WAF method is used to solve the conservative form of the NSWE. First, concentrate

on the homogenous form of equations (3.3.4) which is identical to the NSWE homogeneous

problem for horizontal bottom

Ut + F(U)x = 0. (3.4.17)

These equations can be integrated in a rectangular region of the x − t space (see figure

3.7) in order to obtain a weak form. Using Green’s theorem:∮

[Udx− F(U)dt] = 0. (3.4.18)

t

n+1

n

i−1 i i+1 x

Uk

i−1 i i+1 x(a) (b)

Figure 3.7: Variables representation on a discretized domain: anticlockwise integration of

(3.4.18) on a discretized x− t space, (b) discrete solution behaviour.

These equations can be solved on a staggered grid as depicted in figure 3.7 if written

in the following discrete form:

Un+1kN

= UnkN

+∆t

∆x

[F

n+ 12

kN− 1

2

− Fn+ 1

2k

N+12

], k = 1, 2 (3.4.19)

where Fn+ 1

2k

N− 12

and Fn+ 1

2k

i+12

are the intercell fluxes at the time level n + 12 .

un+1N can be obtained by time-stepping the solution applying equation (3.4.19) once

the fluxes between cells given by

Fn+ 1

2k

N− 12

=1

∆x

xN∫

xN−1

Fk

(Un+ 1

2 (x))dx, F

n+ 12

kN+1

2

=1

∆x

xN+1∫

xN

Fk

(Un+ 1

2 (x))dx, k = 1, 2

(3.4.20)


A B C D E F G H

rarefactionwave

rarefactionwave

shockwave

x

x

x

d0

t∆t2

d

dN−1

dN

dN+1 = 0

N − 1 N N + 1

dRitter

d†dN−1

dN

Figure 3.8: Example of solution of the ‘shoreline Riemann problem’. Only the problem

for the U1 = d component is illustrated. Top: the piece-wise initial condition. Middle:

the solution structure in the (x, t)−plane. Bottom: solution of the Riemann problem in

the physical space.

have been suitably evaluated (e.g. Toro 1992; Toro 1997). Notice that (3.4.20) coincide

with averages of the fluxes over the regions around the boundaries of each cell. From a

practical point of view, the fluxes are evaluated by performing a weighted average, from

which the name of the present method.

In order to evaluate the integrals in equation (3.4.20), a technique to estimate the value

of the variables at time level n + 1/2 is required. This technique is entirely based on the

solution of the Riemann problem, detailed in the previous section. In particular, at any


given time level the variables Uk have a piece-wise constant distribution and a Riemann

problem in which the initial data is made of a pair of constant states can be formulated.

The value of Uk(x, tn+1/2) over the whole computational cell is therefore preliminarily

calculated and then the fluxes between cells are obtained by means of equation (3.4.20).

Once the Riemann problems are solved, i.e. the state U† and the speed of the waves are

known, the fluxes between the computational cells can be evaluated by means of following

procedure (the WAF method). The first step is to decompose the two regions of width ∆x

between nodes N − 1 and N and between nodes N and N + 1 in a number of segments,

namely AB, BC, CD, DE, EF , FG and GH (refer to middle panel of figure 3.8). For the

specific case shown in figure 3.8, which consist of a left rarefaction wave and a right shock

wave emanating from the wet-wet interfaces, the length of the segments can be evaluated

by means of the following expressions:

AB = ∆x2 + ∆t

2 λlh,

BC = ∆t2

(λl

h − λlt

),

CD = ∆t2

(λr − λl

t

),

DE = ∆x2 − ∆t

2 λr,

EF = ∆x2 + ∆t

2 λ1,

FG = ∆t2 (λs − λs

t ) .

(3.4.21)

Where λlh and λl

t are respectively the celerity of the head and of the tail of the fan of the

left rarefaction wave. Since the right wave is a shock, a single speed λr ≡ λrh ≡ λr

t has

been used for the right wave, but the generalization to the case in which the right wave

is a rarefaction or the left one is a shock is straightforward. Notice that in the case under

investigation (see figure 3.8) the speeds λlh and λl

t are negative. Definitions (3.4.20) can

therefore be approximated by

Fn+ 1

2k

N− 12

=1

∆x

[Fk(UN−1)AB+Fk(Urarefact)BC+Fk

(U†

)CD+Fk(UN )DE

](3.4.22)

Fn+ 1

2k

N+12

=1

∆x

[Fk (UN ) EF + Fk (URitter) FG

](3.4.23)


where Urarefact is a state representative of the value of the variables within the head

and the tail of the rarefaction wave and URitter is Ritter’s solution (see Stoker 1957).

Equation (3.4.22) is found to be accurate even if a rough estimate of Urarefact is provided,

for example by assuming Urarefact = (UN−1 + U†)/2 (Toro 1997). This is due to the fact

that the segment BC is usually much smaller than AB, CD and DE.

Numerical tests have revealed that the flux given by equation (3.4.23) is very sensitive

to the definition of URitter. The integral average over the segment FG should therefore be

performed adopting a more accurate integration method on the basis of Ritter’s solution.

It turns out that the trapezoidal rule provides satisfactory estimate of the exact integral;

the final expression adopted for Fn+ 1

2k

N+12

, k = 1, 2 reads

Fn+ 1

2k

N+12

=1

∆x

[Fk (UN ) EF + F b

kFG]

(3.4.24)

where F bk is given by

F bk =

14Fk

(dL, uL

)+

12Fk (dr, ur) +

14Fk (0, us) (3.4.25)

in which dr and ur are the fluid depth and velocity at the center of the rarefaction fan. Note

that equation (3.4.25) is the expression of the trapezoidal rule adopted for the integration

in the region where Ritter’s solution holds.

In order to solve equations (3.3.4), which differ from (3.4.17) because of the presence of

the source terms, we follow the approach of Watson et al. (1992). These authors proposed

a technique based on the incorporation of the source terms into the Riemann problem.

The idea is to transform the problem into a reference frame with horizontal acceleration

equal to gα−D, where α is the bottom slope assumed to be constant in each cell and D are

the dispersive-nonlinears terms. This transformation gives a set of homogenous equations

that can be solved as described before. Then, by means of a reverse transformation, the

solution is obtained in the original reference frame. Note however that D, unlike gα, is not

constant over each cell since its value depends on both the water depth and the velocity.

To overcome this undetermination, D is assumed to be constant over ∆t, given that this

value is computed at the beginning of the time step.


The new variables in the accelerating reference frame are

ξ = x + 12 (gα−D) t2, τ = t,

v = u + (gα−D) t, ς = d.

(3.4.26)

If these new variables are substituted into (3.3.4) a set of homogenous equations, formally

identical to (3.4.17) is obtained. Once the solution is found in the accelerating frame,

the reverse transformation yields the following relations between (3.4.26) and the original

variablesu(x, t) = v

[x + 1

2 (gα−D) t2, t]− (gα−D) t,

d(x, t) = ς[x + 1

2 (gα−D) t2, t].

(3.4.27)

U† U†

UL ULUR UR

ξ x

τ t

(a) (b)

Figure 3.9: Riemann problem solution in the accelerating (a) and in the stationary (b)

reference frames.

The structure of the solution of the Riemann problem in the case of a left rarefaction

and a right shock wave is shown in figure 3.9. The solution in the accelerating (panel a)

reference frame is identical to the solution of equations (3.4.17) while in the stationary

frame (panel b) the trajectory of each wave is no longer a straight line but turns into a

parabola.

From a practical point of view, in order to apply the WAF method, the quantity12 (gα−D) ∆t must be subtracted from all the velocities and, in evaluating the weights

of each flux, it is to be considered that the solution is shifted in x by a constant amount12 (gα−D)

(12∆t

)2.


Equation (3.4.19), modified to take into account source terms effects reads

Un+1kN

= UnkN

+∆t

∆x

[F

n+ 12

kN− 1

2

− Fn+ 1

2k

N+12

]+ S

n+ 12

N ∆t, k = 1, 2 (3.4.28)

where

S(U) =

0

−gdα + dD

. (3.4.29)

[Notice that in the original work (Watson et al. 1992) because of a typographical error an

incorrect expression for S(U) is reported on equation 3.4.29.]

Finally, the technique to change the value of N during the run-up phase is based on

the volume of fluid entering the dry cell N + 1 at each time step. An estimate of this

volume can be obtained by applying the WAF method to the cell N + 1. The expression

(3.4.28) reads in this case:

Un+1kN+1

=∆t

∆x

[F

n+ 12

kN+1

2

]+ S

n+ 12

N+1∆t, k = 1, 2 (3.4.30)

since UnkN+1

= 0, k = 1, 2 and Fn+ 1

2k

N+32

= 0 , k = 1, 2.

If dn+1N+1∆x is greater than a threshold value the cell is inundated and at the following

time step the new value of N = N + 1 is employed.

During the run-down phase a simpler technique provides good results. This is based

on the use of the water depth at node N : if dn+1N is lower than a threshold value at the

following time step the new value of N = N − 1 is employed.

Note that in this work run-up and run-down phases were defined on the basis of flow

direction at node N−1 at the time level n: unN−1 > 0 defines run-up, un

N−1 < 0 run-down.

3.4.2 The basic steps of the proposed procedure

Let us now briefly summarize the basic steps required to apply the proposed procedure.

Assume that the dependent variable d and u are known over the computational grid at time

level n. The objective is to compute un+1N and dn+1

N which are the boundary conditions

for the wave solver. The first step is to solve the Riemann problems at the interfaces

between cells N − 1, N and N + 1. On the basis of the solution of the Riemann problems


the fluxes at the interfaces of the cell N are estimated by means of equations (3.4.22)

and (3.4.24). Once the fluxes are known, the solution at node N is updated by applying

equation (3.4.28). At this point, the model checks if at the subsequent time step the value

of N is to be changed, i.e. if the shoreline is to be moved. This can occur either in the form

of run-up when the volume of water in the cell N + 1 is larger than the chosen threshold

and the same cell is inundated becoming part of the computational domain or in form of

run-down when the depth in the cell N is lower than the threshold and the cell is removed

from the computational domain becoming a dry cell.

3.5 Performance evaluation of the BTE model with the new

SBCs

A number of tests are here reported to help the reader evaluate the performances of the

implementation of the new SBCs (section 3.4) in the BTE model described in section 2.2.

Analytical solutions are the most suitable for evaluating the performances of the

implemented SBCs as they represent an exact benchmark. We here consider three

important analytical solutions for waves propagating over a uniform sloping beach. They

respectively model the run-up due to a depression of the water level (the fluid held

motionless) which is suddenly released (the ‘Carrier and Greenspan’s run-up solution’,

Carrier and Greenspan 1958), the run-up and run-down characteristic of a periodic wave

travelling shoreward and being reflected out to sea (the ‘Carrier and Greenspan’s standing

wave solution’, Carrier and Greenspan 1958) and the run-up of a solitary wave (the

‘Synolakis run-up solution’, Synolakis 1987).

3.5.1 The Carrier and Greenspan run-up solution

This test corresponds to the physical problem in which the water level at the coastline of

a plane uniform beach is depressed, the fluid held motionless and then released. It also

represents the most classical test conditions for assessing the quality of any run-up solver.

Carrier & Greenspan (1958) used a hodograph transformation to solve the NSWE


and obtained an analytical solution of this problem. The transformation makes use of

two dimensionless variables (σ∗, λ∗) which are respectively a space-like and a time-like

coordinate). Dimensionless ordinary variables and flow properties are then related to the

hodograph coordinates as follows:

x∗ =14φ∗λ −

116

σ∗2 − 1

2u∗

2, t∗ =

12λ∗ − u∗, (3.5.31a)

η∗ =14φ∗λ −

12u∗

2, u∗ = φ∗σ∗/σ∗ (3.5.31b)

where φ∗ is a ‘potential function’ which depends on the specific propagation problem under

investigation.

−0.2 −0.1 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.10: The ‘Carrier & Greenspan run-up test’ on a uniform plane beach.

Dimensionless, scaled, analytical (dotted lines) and numerical (solid lines) profiles of water

elevation ζ∗ are plotted versus the dimensionless onshore coordinate x∗ at dimensionless

times increasing of ∆t∗ = 0.05 from t∗ = 0.00 (bottom curves) to t∗ = 0.80 (top curves).


The ‘run-up solution’ is specified by the following initial conditions at t∗ = 0

ζ∗ = ε

[1− 5

2a3

(a2 + σ∗2)3/2+

32

a5

(a2 + σ∗2)5/2

], (3.5.32a)

u∗ = 0, (3.5.32b)

x∗ = −σ∗

16+ ε

[1− 5

2a3

(a2 + σ∗2)3/2+

32

a5

(a2 + σ∗2)5/2

](3.5.32c)

where a = 3/2(1 + 0.9ε)1/2 and ε is a nonlinearity parameter.

Further details on both initial conditions and the analytical solution can be found in

the original work of Carrier & Greenspan.

In figure 3.10, which is the equivalent of figure 7 of (Carrier and Greenspan 1958), the

analytical solution ζ∗/ε versus the onshore coordinate x∗ is shown by means of dotted

lines for different adimensional times. On the other hand, solid lines pertain to the

numerical results while the thicker line represents the sloping seabed. It is evident that

an excellent matching exists between the analytical and the numerical solution. It is

also worth underlying that no spurious oscillations are present near the shoreline. Any

oscillatory behaviours would reveal two possible sources of errors:

• a bad implementation of the SBCs in the chosen BTE model;

• a bad implementation of the ‘wetting-drying’ procedure.

On the contrary, the smooth behaviour of the elevation profiles of figure 3.10 testifies the

absence of such problems.

3.5.2 The Carrier and Greenspan standing wave solution

This solution of the NSWE represents the motion of a wave of dimensionless amplitude

A∗ and dimensionless frequency ω∗ travelling shoreward and being reflected out to sea

generating a standing wave (Carrier and Greenspan 1958). In the past it has been widely

used to analyze the dynamics of water waves approaching a coast or a continental shelf

(Carrier 1966; Carrier 1971).

Such a solution can be specified by means of the following potential function:

φ∗(σ∗, λ∗) = A∗J0(ω∗σ∗) cos(ω∗λ∗) (3.5.33)


−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Figure 3.11: The ‘Carrier & Greenspan standing wave test’ on a uniform plane beach:

envelope of surface elevations. Envelope of the dimensionless, analytical solution by Carrier

& Greenspan (dotted lines) and numerical (solid lines) profiles of water elevation ζ∗ are

plotted versus the dimensionless onshore coordinate x∗.

where J0 is the Bessel function of the first kind.

Once (3.5.33) is substituted into (3.5.31) a solution can be found for all the flow

properties of interest in the ordinary (x∗, t∗)−space. Such a solution has been obtained

both analytically and numerically for the case A∗ = 0.6 and ω∗ = 1 (non-breaking wave).

In figure 3.11 both profiles of the numerically-computed free surface elevation and the

envelope of the analytically-derived surface elevations are reported. The figure reveals an

almost perfect agreement between analytical and numerical solutions. Again, the absence

of any oscillations in the numerical solution is particularly satisfying.

The comparison can also be pushed forward to analyze any possible differences in


2 3 4 5 6 7 8 9 10 11 12

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Figure 3.12: The ‘Carrier & Greenspan standing wave test’ on a uniform plane beach:

horizontal motion of the shoreline. Incident wave of dimensionless amplitude A∗ = 0.6

and dimensionless frequency ω∗ = 1. Dimensionless analytical shoreline path as from

Carrier and Greenspan (1958) (dotted line) and numerical shoreline path (shoreline line)

in time.

the horizontal motion of the shoreline. This is reported in figure 3.12 in which a dotted

line is used to represent the analytical solution while the solid line gives the numerical

shoreline. Apart from a very small underestimation at the peak of the run-up (which

could be fixed by increasing the spatial discretization) the numerical solution perfectly

matches the analytical one. This does not happen when employing artificial techniques

like the slot-technique (Madsen et al. 1997) which always introduce a loss of mass (revealed

by a reduced swash amplitude). The agreement is even more remarkable in view of the

structure of the proposed SBCs which does not depend on any calibration parameters.


3.5.3 The Synolakis run-up solution

Synolakis’ solution (Synolakis 1987) is one of the very few available analytical solutions

for the run-up of a solitary wave (a similar solution is also available for the interactions of

solitary waves in shallow waters, see Brocchini 1998). Such an equations has been obtained

in the framework of the NSWE but has been shown to model very well beach inundation

conditions caused by solitary waves.

In Synolakis’ solution a solitary wave of dimensionless height H∗ centered at a distance

X∗1 from the shore at time t∗ = 0:

ζ∗ =H∗

d∗sech2[γ(x∗ −X∗

1 )], where γ =√

3H∗/4d∗ (3.5.34)

is propagated over a combined topography made of a plateau of depth d∗ and a plane

sloping beach of slope β; matching of the two regions occurs at x∗ = X∗0 = cotβ (see

figure 3.13) .

x∗

y∗

H∗

d∗

X∗0X∗

1

β

Figure 3.13: Definition sketch for the initial condition of Synolakis’ run-up solution.

Propagation of the above signal by means of the NSWE is more easily modelled

if Carrier & Greenspan’s (Carrier and Greenspan 1958) hodograph transformation and

a Fourier transform technique are used in combination. This brings to the following

definition for φ∗:

φ∗(σ∗, λ∗) = −32i

3

∫ ∞

−∞cosech(ξk∗)

J0(k∗σ∗X∗0/2)eik∗θ

J0(2k∗X∗0 )− iJ1(2k∗X∗

0 )dk∗ (3.5.35)

where ξ = π/2γ and θ = X∗1 −X∗

0 + λ∗X∗0/2 is the pulse phase.

We refer the reader to Synolakis (1987) for a detailed description of the solution.


We used such a solution to illustrate the model performances to reproduce the run-up of

a solitary wave. More specifically we have tried to reproduce Synolakis’ results given in his

figure 6. This summarizes the comparison of the analytical solution and experimental data

in the case of solitary wave of H∗/d∗ = 0.019 climbing up a 1 : 19.85 beach. Cross-shore

profiles of the free surface elevation at different stages of the run-up process are reported

on figure 3.14. Notice that instead of centering the initial wave profile at X∗1 = 37.35 we

used X∗1 = 40. This only introduces a small shift in the origin of the times.

A very good matching exists between the numerical solution provided by the BTE

model (solid lines) and Synolakis’ analytical solution (dotted lines). The matching is

almost perfect during most of the run-up. However, when the wave is just to reach the

maximum run-up small discrepancies can be found far from the shore (i.e. x∗ > 10), the

numerical solution being slightly smaller than the analytical one. This discrepancy can be

ascribed to two connected reasons:

1. being obtained within the NSWE framework Synolakis’ solution best represents flow

conditions near the shoreline;

2. Synolakis’ solution was seen to slightly overestimate experimental data far from the

shore.

However, near the shoreline (i.e for x∗ < 4) matching of the two solutions is always

excellent, again suggesting a good implementation of the SBCs in the chosen BTE model.

3.6 Conclusions

A novel type of SBCs has been proposed for Boussinesq-type models. This is derived by

using the characteristic form of the NSWE and is shown to properly model the shoreline

motion. The methodology used to implement such SBCs in a specific BTE model is

illustrated and its effectiveness verified by means of three different analytical solutions.

The illustrated model represents an efficient tool for modelling nearshore flows and analysis

is underway to compare it with a shock-capturing version of the same BTE model in which

nonlinear-dispersive terms are regarded as forcings of the classical NSWE.


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0

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0

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0.1-2-101234567891011121314151617181920

Figure 3.14: The ‘Synolakis run-up solution’. Dimensionless free surface elevation ζ∗

as functions of the dimensionless x∗ coordinate at different adimensional times t∗ =

20, 30, 35, 40, 45, 50 (from left to right and from top to bottom). The solid line represents

computed data while solid circles are used for Synolakis analytical solution.

Chapter 4

A new shoreline boundary

condition for wave-averaged

models

4.1 Introduction

Nearshore water flows occur over a wide range of time scales. Because of this complexity

averaging over the typical period of wind waves (about 0−10 seconds) is often introduced

(see section 1.2). Wave-averaged models are used, for example, to resolve the vertical

velocity distribution and to predict both wave set-up and undertow. If the flow is non-

uniform in the longshore direction (i.e. horizontally bidimensional) circulation models are

chosen which can be further subdivided into 2DH models and quasi-3D models. 2DH

models employ depth-integrated, wave-averaged velocities and are used, for example,

to study the water circulation over longshore varying topographies or the instability of

longshore currents. In quasi-3D models the current velocity is split into a depth-uniform

and a depth-varying component. Then, solution (analytical) is found for the vertical

profiles of the horizontal currents in combination with a numerical solution of the depth-

integrated 2DH equations (e.g. Van Dongeren and Svendsen 2000). The most important

73

A new SBC for wave-averaged models 74

application of quasi-3D models is concerned with the proper modelling of nearshore

hydrodynamics and sediment transport at a time scale adequate for morphodynamic

purposes.

Although well developed both theoretically and numerically, wave-averaged models

use some assumptions which limit their capabilities of reproducing natural flow conditions.

One of the most crucial shortcomings concerns the treatment of the boundary between the

wet and dry domains. Since such boundary is taken as the intersection of the mean water

level with the beach face both theoretical and practical problems arise. For instance, swash

zone dynamics is incorrectly represented and flow properties are defined also in regions

which should be dry. Moreover, at the above mentioned mean shoreline the water depth

is usually taken to be zero, hence, serious computational troubles are faced in prescribing

the short-wave (SW hereinafter) forcing.

The aim of this chapter is to define new shoreline boundary conditions for wave-

averaged models. More specifically SBCs for unidirectional flow propagation derived

from the model equations described in Brocchini and Peregrine (1996) (BP96 hereinafter)

are verified, discussed and their implementation in a wave-averaged model is preliminary

addressed.

This chapter is organized as follows. In section 4.2 problems related with the use

of wave-averaged models in proximity and into the swash zone are illustrated and the

proposed solution is introduced. In section 4.3 the wave-averaged NSWE and the boundary

conditions of BP96 are re-derived. Hence, we first assess validity of boundary conditions

by means of both theoretical and numerical arguments (section 4.4). In section 4.5 we

rewrite the SBCs in terms of flow variables used in available wave-averaged models (Ozkan

Haller and Kirby 1997b; Van Dongeren and Svendsen 2000), simplify them on the basis

of numerical solutions of the NSWE and discuss their features through three special cases

(section 4.6). Section 4.7 is devoted to a description of the implementation of the SBCs

in available models and section 4.8 to some concluding remarks.


4.2 Wave-averaged models in proximity and into the swash

zone

Wave-averaged models are based on the principle that both fluid velocity and water depth

are averaged over a time greater than or equal to the typical period of SW (wind waves).

The resulting governing equations, in the case of depth-uniform currents, are the classical

NSWE with forcing terms representing the effect of SW.

Averaging over the SW period is a simple and intuitive process outside the swash

zone where the domain is always wet and averaging can be unambiguously performed

over the whole computational time. On the contrary, inside the swash zone, wet and dry

conditions alternate. Several problems arise when averaging in this region. First, it would

be probably more correct if the averaging was carried out only during the occurrence of wet

conditions. Second, it can be easily recognized that the wave-averaged shoreline coincides

with the shoreward limit of the swash zone. This implies that the wave-averaged shoreline

is reached by the water for a time negligible if compared to the wave period.

Previous methods for the treatment of the moving shoreline in wave-averaged models

do not seem to take into account the problems discussed above. The techniques currently

applied treat the shoreline without any difference with respect to wave-resolving models

and do not consider the particular hydrodynamic conditions of the SW swash zone.

A solution to the problems introduced above is here proposed which suggests to avoid

simulating the SW swash zone and to cut the computational domain at the lower (seaward)

limit of the swash zone xl. The work described in the following sections is therefore aimed

at deriving boundary conditions which hold at xl. More specifically our objective is to

prescribe the motion of the lower limit of the swash and water depth and velocity at that

point.


4.3 The boundary conditions at the lower limit of the swash

zone

The boundary conditions discussed and verified in this chapter were derived by BP96.

In this section the basic steps of the derivation are briefly reported for convenience. In

subsection 4.3.1 the NSWE are averaged over the SW periods in order to show the origin

of SW forcing terms such as the radiation stress and the local mass flux. In subsection

4.3.2 the NSWE are integrated over the swash zone and subsequently averaged over the

SW period. The resulting equations are the boundary conditions at the lower limit of the

swash chosen to be employed in wave-averaged models that are verified and discussed in

the subsequent sections of this chapter.

4.3.1 The short-wave-averaged NSWE

We start from the conservative nondimensional form of the inviscid NSWE which reads

∂d

∂t+

∂

∂x[ud] = 0, (4.3.1)

∂

∂t[ud] +

∂

∂x

[d

(u2 +

12d

)]+ d = 0. (4.3.2)

Now we decompose the main flow properties u and d into long-period motions 〈u〉 and 〈d〉(typical period greater than 10 s) and the short-period wave u and d (1-10s) contributions:

u = 〈u〉+ u, (4.3.3)

d = 〈d〉+ d (4.3.4)

where the operator 〈·〉 indicates averaging over the SW period. Notice that the above

decomposition has been here performed after depth integration of the governing equations

of fluid mechanics. An alternative procedure could be used in which flow decomposition

is made before depth integrations.

By substituting in the shallow water equations (4.3.1) and (4.3.2) for each flow variable

and by phase averaging, we obtain a set of equations for the long-period flow properties


〈u〉 and 〈d〉. These equations read:

∂ 〈d〉∂t

+∂

∂x

[〈u〉〈d〉+

⟨Q

⟩]= 0, (4.3.5)

∂

∂t

[〈u〉〈d〉+

⟨Q

⟩]+

∂

∂x

[〈d〉

(〈u〉2 +

⟨u2

⟩+

12〈d〉

)+ 2 〈u〉

⟨Q

⟩+

⟨S

⟩]+ 〈d〉 = 0

(4.3.6)

where Q = ud and S = u2d + 12 d2 are respectively the local mass flow and the local

momentum flux tensor due to SW (see table 2 in BP96). Equations (4.3.5) and (4.3.6)

can be rewritten as follows:

∂ 〈d〉∂t

+∂

∂x[〈u〉〈d〉] = −

∂⟨Q

⟩

∂x, (4.3.7)

∂

∂t[〈u〉〈d〉] + ∂

∂x

[〈d〉〈u〉2 +

12〈d〉2

]+ 〈d〉 = − ∂

∂x

[〈d〉 ⟨u2

⟩+ 2 〈u〉

⟨Q

⟩+

⟨S

⟩]−

∂⟨Q

⟩

∂t(4.3.8)

which are the classical NSWE written in the averaged variables 〈d〉 and 〈u〉; the right

hand sides contains extra terms which represent the SW forcings. Once the short-wave

quantities are evaluated, by means for example of a separate wave model (the so called

wave driver), long waves (LW hereinafter) and currents can be calculated by solving (4.3.7)

and (4.3.8).

Some more algebra is needed to obtain the SW averaged momentum equation of the

NSWE in non-conservative form, which is here written for reference. From (4.3.8) we get

〈d〉 ∂〈u〉∂t + 〈u〉 ∂〈d〉

∂t + 〈u〉2 ∂〈d〉∂x + 2 〈d〉〈u〉 ∂〈u〉

∂x + 〈d〉 ∂〈d〉∂x + 〈d〉 =

− ∂∂x

[〈d〉 ⟨u2

⟩+ 2 〈u〉

⟨Q

⟩+

⟨S

⟩]− ∂〈Q〉

∂t

(4.3.9)

and from (4.3.7)

∂ 〈d〉∂t

= −〈u〉 ∂ 〈d〉∂x

− 〈d〉 ∂ 〈u〉∂x

−∂

⟨Q

⟩

∂x(4.3.10)

By inserting (4.3.10) into (4.3.9) it follows

〈d〉 ∂〈u〉∂t − 〈u〉2 ∂〈d〉

∂x − 〈u〉〈d〉 ∂〈u〉∂x − 〈u〉 ∂〈Q〉

∂x + 〈u〉2 ∂〈d〉∂x + 2 〈d〉〈u〉 ∂〈u〉

∂x + 〈d〉 ∂〈d〉∂x + 〈d〉 =

− ∂∂x

[〈d〉 ⟨u2

⟩+ 2 〈u〉

⟨Q

⟩+

⟨S

⟩]− ∂〈Q〉

∂t .

(4.3.11)


Then dividing (4.3.11) by 〈d〉 the final form of the non conservative momentum equation

is obtained

∂〈u〉∂t + 〈u〉 ∂〈u〉

∂x + ∂〈d〉∂x + 1 =

− 1〈d〉

∂∂x

[〈d〉 ⟨u2

⟩+ 2 〈u〉

⟨Q

⟩+

⟨S

⟩]− 1

〈d〉∂〈Q〉

∂t + 〈u〉〈d〉

∂〈Q〉∂x .

(4.3.12)

4.3.2 NSWE integration over the swash zone

The NSWE in conservative form are here integrated over the swash zone and then averaged

over the SW period. We start from (4.3.1) and (4.3.2), which are integrated between xl

and the instantaneous position of the shoreline xh to give:xh∫

xl

dt dx = Q(t)|xl, (4.3.13)

xh∫

xl

(ud)t dx = S(t)|xl− V (t)−Υ (4.3.14)

where the following quantities have been introduced: Q = ud, S = u2d+ 12d2, V =

xh∫xl

d dx,

Υ =xh∫xl

τ dx. Q can be regarded as the local mass flow, S is the local momentum flux, V

is the volume of water in the swash zone and Υ is the friction force in the swash zone (see

also table 2 of BP96). If a slow variation of the limits of the integral appearing in (4.3.13)

and (4.3.14) is assumed, the equations simplify to

∂V

∂t= Q(t)|xl

, (4.3.15)

∂P

∂t= S(t)|xl

− V (t)−Υ (4.3.16)

where P =xh∫xl

ud dx =xh∫xl

Q dx is the momentum of water in the swash zone.

There is now an important choice to be made about how much of the swash motion

should be assigned to long time scales and how much to short time scales. We assume that

swash motion is almost entirely assigned to SW contributions and that the LW contribution

moves the lower limit of the swash zone. Specifically we assume that

d = d u =∂xl

∂t+ u (4.3.17)


where the simbol · pertains to SW contribution inside the swash zone rather than · which

pertains to SW contributions outside the swash zone.

Using a different basis for both SW and averaged quantities in the swash zone

compared with those outside the swash zone is justified by two reasons. It is in principle

possible to adopt two different types of solution for the SW outside and inside the swash

zone; for example, it is common practice to use linear theory to model SW but this is

clearly inadequate inside the swash zone where nonlinear solution like that of Carrier and

Greenspan (1958) are more appropriate. Moreover, on many sandy beaches there is also

a strong difference in the character of the bed in the swash zone compared with the bed

just outside the swash zone.

By averaging equation (4.3.15) and (4.3.16) and by using (4.3.17) the following

dimensional equations are obtained, which hold at the lower limit of the swash xl and

that can be used as boundary conditions for wave-averaged models:

d〈V 〉dt

=〈u〉〈d〉+〈ud〉− dxl

dt〈d〉 =⇒ dxl

dt=〈u〉+ 〈ud〉−d〈V 〉/dt

〈d〉 , (4.3.18)

d

dt

[〈Px〉+

dxl

dt〈V 〉

]+ gα〈V 〉+ 〈Υ〉 =

[〈u〉 − dxl

dt

]2

〈d〉+ 2〈ud〉[〈u〉 − dxl

dt

]+〈u2〉〈d〉+〈u2d〉+1

2g[〈d2〉+〈d〉2

](4.3.19)

in which α is the beach slope and g the acceleration of gravity. Notice that in this case

dimensional equations have been used in order to help individuating the origin and the

role of each term.

4.4 Evaluation of the chosen SBCs

Equations (4.3.18) and (4.3.19) express a relationship between fluid properties in the

swash zone and fluid properties at the lower limit of the swash xl. In order to verify

the performances of these equations we performed several computations by means of a

depth-integrated, SW resolving numerical model (briefly described and discussed later in

section 4.4.1), by means of which we could compute each flow property. More specifically


(see Figure 4.1, where typical results of a computation are shown) we obtained the

instantaneous water depth d and the depth-integrated fluid velocity u. By averaging

these variables over the typical SW period we therefore computed the mean values 〈u〉and 〈d〉 and, by subtracting average values from the instantaneous ones, we computed the

SW related variables d and u at each point of the computational domain. Furthermore the

lower limit of the swash zone was defined as follows. First we computed the time average

of the instantaneous position of the shoreline, thus obtaining a sort of LW related shoreline

(we employed a moving average with period equal to the SW one). Second, we translated

this curve so that it lies below the instantaneous shoreline position for the whole duration

of the computation. This curve therefore defines the lower limit of the swash zone xl.

Once obtained the position of xl at each time step, we computed the volume in the swash,

the radiation stress and the mass flux at xl, etc. In Figures from 4.2 to 4.6 typical results

of the computations are shown for some significant cases. On each figure we reported both

instantaneous (solid lines) and wave-averaged (dashed lines) quantities, the latter obtained

by means of a moving average. Notice that in Figures 4.2 and 4.4 monochromatic SW

without LW are simulated; these computations have been performed with the aim of

showing the difference with simulations in which LW are present.

On the basis of the calculations described above we could compute the left hand side

of equations (4.3.18) and (4.3.19) in order to verify, by comparison, if the two equations

are correct. Results of these computations illustrate a number of features of the chosen

equations and support the validity of the analytical derivation given in BP96 and in the

previous sections. Good agreement was found between the values of left and right sides of

the equations. Typical results are illustrated in Figures 4.7 and 4.8. Top panels illustrate

for reference the instantaneous motion of the shoreline and the motion of the lower limit of

the swash zone; middle and lower panels illustrate the comparison of the sides of equations

(4.3.18) and (4.3.19). In Figure 4.7 a bicromatic wave field is considered while in Figure

4.8 a random wave superposed to a long wave is shown.

In the following section a brief discussion and a description of the chosen numerical

model are reported. Since the results of our computations are also used in the following for


simplifying the SBCs and for deriving expressions for the SW properties, we describe in

the subsequent section 4.4.2 how the input parameters for the model were chosen and we

introduce a nondimensional parameter suitable to describe the wave conditions for each

computation.

Figure 4.1: Example of solution for a typical computation (beach slope α = 1/50, test

N. 12 of Table 4.1) performed with the depth-integrated, short-waves resolving models.

Top panel: instantaneous free surface elevation (solid line) with envelopes of maxima

and minima (dashed lines), mean water level (dash-dotted line) and wave height (double-

dotted-dashed line). Lower panel: instantaneous velocity (solid line), mean velocity 〈u〉(dashed line) and mean velocity u (dashed-dotted line) of equation (4.5.22). In both panels

the thick, solid line represents the seabed.


200 250 300 350 40050

55

60

65

70Shoreline position

m

200 250 300 350 4000

0.1

0.2

0.3

0.4Water depth at xl

m

200 250 300 350 400−5

0

5Velocity at xl

m/s

200 250 300 350 4000

0.2

0.4

0.6

0.8Volume in the swash

m3

200 250 300 350 400−2

0

2

4Radiation stress at xl

m3 /s

2

200 250 300 350 4000

0.1

0.2

0.3

0.4Mass flux at xl

m2 /s

Figure 4.2: Instantaneous (solid lines) and wave-averaged (dashed lines) quantities are

plotted versus the computational time (s). Monochromatic SW. Beach slope α = 1/10,

Ts = 10s.

4.4.1 The chosen numerical model

Although analytical solutions, like those used in BP96, are attractive and useful they are

not suited for describing wave breaking conditions which, on the contrary, are at present

most easily represented by numerical solutions of the NSWE. Furthermore wave-averaged

models are commonly used for simulating LW and currents generated by breaking waves.

In this work we therefore decided to use a shock-capturing solver of the NSWE (Watson

et al. 1992), which can adequately deal with breaking waves (allowing bore like solution

to propagate) and accurately simulate swash motions. It is clear that the chosen model


200 250 300 350 40050

55

60

65


m

200 250 300 350 4000

0.1

0.2

0.3


m

200 250 300 350 400−5

0

5Velocity at xl

m/s

200 250 300 350 4000

0.5

1Volume in the swash

m3

200 250 300 350 400−2

0

2

4Radiation stress at xl

m3 /s

2

200 250 300 350 400−0.2

0

0.2

0.4

0.6Mass flux at xl

m2 /s


plotted versus the computational time (s). Monochromatic SW and LW. Beach slope

α = 1/10, Ts = 10s, Tl = 100s.

is applicable only in very shallow water, since the equations on which it is based do not

take into account dispersiveness (see section 2.1.6 for a discussion). Nevertheless, in the

present study our objective is to study the shoreward part of the surf zone and the swash,

where the chosen model is expected to provide a good representation of the fluid motions.

The natural extension of this work is to use data collected during laboratory

experiments . At this preliminary stage we preferred numerical experiments, which are

much less expensive and allow for analysis of a wide range of flow conditions. Furthermore,

since the SBCs were derived on the basis of the NSWE, data obtained by solution of these


200 250 300 350 40053

54

55


m

200 250 300 350 4000

0.005

0.01

0.015


m

200 250 300 350 400−1

−0.5

0

0.5Velocity at xl

m/s

200 250 300 350 4000

0.005

0.01


m3

200 250 300 350 400−0.05

0

0.05

0.1

0.15Radiation stress at xl

m3 /s

2

200 250 300 350 400−2

0

2

4

6x 10

−3 Mass flux at xl

m2 /s


plotted versus the computational time (s). Monochromatic SW. Beach slope α = 1/50,

Ts = 10s.

equations allow for a very detailed comparison between expected and obtained data; when

using experimental data extra phenomena, not taken into account by NSWE could affect

the results leading to additional problems.

4.4.2 Choice of input data for the computations

In BP96 examples were only given for a steady mean motion (i.e. periodic SW motion) of

the mean shoreline. We here extend our analysis to the more complex and practically more

important case of an unsteady motion of xl. This means that the full solution must involve


200 250 300 350 40045

50

55


m

200 250 300 350 4000

0.01

0.02


m

200 250 300 350 400−2

−1

0

1Velocity at xl

m/s

200 250 300 350 4000

0.005

0.01

0.015


m3

200 250 300 350 400−0.05

0

0.05

0.1


m3 /s

2

200 250 300 350 400−5

0

5

10x 10

−3 Mass flux at xl

m2 /s


plotted versus the computational time (s). Monochromatic SW and LW. Beach slope

α = 1/50, Ts = 10s, Tl = 100s.

at least two wave modes, one related to breaking SW and one to LW. For what concerns

the SW we employed both monochromatic and random waves. The latter were generated

on the basis of JONSWAP type spectra. The SW mode is characterized by period, length

and height (Ts, Ls,Hs) or by a significant period, ‘significant length’ (associated with the

significant period) and a significant height (TSs , LS

s ,HSs ). As far as the LW is concerned

we used a single mode, superposed to the SW mode, characterized by (Tl, Ll, Hl). Notice

that wave height and wavelength considered above are measured at the offshore boundary

of the computational domain. The superposition of SW and LW mode is particularly easy


200 250 300 350 40040

45

50

55


m

200 250 300 350 4000

0.05


m

200 250 300 350 400−2

−1

0

1Velocity at xl

m/s

200 250 300 350 4000

0.05

0.1

0.15


m3

200 250 300 350 400−0.2

−0.1

0

0.1


m3 /s

2

200 250 300 350 400−0.01

0

0.01

0.02

0.03Mass flux at xl

m2 /s


plotted versus the computational time (s). Random SW superposed to a monochromatic

LW. Beach slope α = 1/50, T ss = 10s, Tl = 100s.

to perform in numerical computations and allows for a strict control of the properties of

the two modes. One second procedure, often used in experimental analysis, could have

been employed in which the LW is generated within the considered domain by nonlinear

wave-wave interaction of the SW. This, however, has one major drawback: the LW height

Hl is known with a lesser accuracy than with the former procedure which is therefore

chosen for clarity purposes. Also notice that the chosen procedure exactly reproduces

what occurs in standard applications of wave-averaged models in which the SW field is

represented through one single mode (‘representative’ or ‘significant’ wave).


150 200 250 300 350 400 45040

45

50

time (s)

m

150 200 250 300 350 400 450−0.4

−0.2

0

0.2

0.4

time (s)

m/s

150 200 250 300 350 400 4500.5

1

1.5

2x 10

−3

time (s)

m3 /s

2

Figure 4.7: Results of the validation procedure for a typical case with monochromatic

SW. Top panel: instantaneous (solid line) and mean (dashed line) shorelines. Middle and

lower panels: left hand side (solid line) and right hand side (dots) of equations (4.3.18)

and (4.3.19) respectively.

For what concerns the beach slope we used five values (α = 1/10, 1/20, 1/30, 1/40,

1/50), representative of steep to mildly sloping beaches. We did not consider very mildly

sloping beaches (α = 1/100) because we expect that the SW energy in the swash zone in

these conditions is almost completely dissipated by wave-breaking. Moreover experience

with numerical computations shows that for α < 1 : 100 breaking becomes intermittent,

condition which cannot be properly handled within the NSWE approximation. In Table 4.1

the selected input parameters used in our computations for each value of the beach slope


200 300 400 500 600 700 80040

45

50

55

60

m

time (s)

200 300 400 500 600 700 800−0.2

−0.1

0

0.1

0.2

m/s

time (s)

200 300 400 500 600 700 8000.01

0.015

0.02

0.025

m3 /s

2

time (s)

Figure 4.8: Results of the validation procedure for a typical case with random SW. Top

panel: instantaneous (solid line) and mean (dashed line) shorelines. Middle and lower

panels: left hand side (solid line) and right hand side (dots) of equations (4.3.18) and

(4.3.19) respectively.

and of the bottom friction parameter are summarized. More specifically we performed

16 computations for five different values of the beach slope and for three different values

of the bottom friction coefficient (f/α = 0, 0.5, 1). Furthermore we run the tests with

f/α = 0 twice, using monochromatic and random SW. This lead to perform a large number

of computations (320) that provided a wide data set on which results described in the

following sections are based. In view of the large number of computations to perform the

resolution of the computational grid, the duration of each computation and the extension


of the domain were optimized in order to guarantee both accurate results and reasonable

computational times. The final set of parameters used in all calculations was such that the

total computational time needed for each computations, including the time for elaboration

of synthetic results was approximately equal to 20 minutes on a 500Mhz Intel Processor

PC.

It is well known that the motion of water waves on beaches is suitably classified in

terms of the ratio between the beach slope and the square root of the wave steepness which

gives the so-called ‘Iribarren number’ typically used to classify the types of breaking. Our

results show (see section 4.6) that flow variables strongly depend both on α and on the

steepness Hs/Ls of the SW. A weaker influence is due to the LW. This is suitably expressed

in terms of the ratio Ll/Ls, the role of Hl being less important. Hence we choose to classify

our test conditions in terms of both the ‘modified Iribarren number’:

ξm =α√

(Hs/Ls)(Ll/Ls)(4.4.20)

and of the dimensionless, scaled friction parameter:

f = Cf/α (4.4.21)

used in the dimensionless form of the NSWE to express the seabed shear stress by a

Chezy-type formulation.

For the analyzed test conditions 0.01 ≤ ξm ≤ 0.40 and 0 ≤ f ≤ 1. The smallest values

of ξm characterize steep SW and relatively long LW while the largest values of ξm pertain

to less steep SW superposed to relatively short SW. Both ξm and f vary over a range

which is representative of typical wave conditions over most natural beaches.

4.5 Features of the SBCs

Before discussing the implementation of the SBCs (4.3.18) and (4.3.19) we prefer to write

them in a slightly different form, coherent with the flow description employed in available


Test N. Wave input data

[Ts(s), δs] [Tl(s), δl]

1 [5, 0.1] [100, 0.01]

2 [5, 0.1] [100, 0.02]

3 [5, 0.2] [100, 0.02]

4 [5, 0.2] [100, 0.04]

5 [5, 0.1] [300, 0.01]

6 [5, 0.1] [300, 0.02]

7 [5, 0.2] [300, 0.02]

8 [5, 0.2] [300, 0.04]

9 [10, 0.1] [100, 0.01]

10 [10, 0.1] [100, 0.02]

11 [10, 0.2] [100, 0.02]

12 [10, 0.2] [100, 0.04]

13 [10, 0.1] [300, 0.01]

14 [10, 0.1] [300, 0.02]

15 [10, 0.2] [300, 0.02]

16 [10, 0.2] [300, 0.04]

Table 4.1: Short and long wave input parameters used for each value of the beach slope and

bottom friction parameter. Wave nonlinearity at the offshore boundary of the domain,

defined as the ratio of wave height and undisturbed water depth, is indicated for both

short (δs) and long waves (δl).


wave-averaged models. These models are written in terms of variables for which splitting

between short- and long-waves is performed before averaging over the depth is applied

while, as shown in the previous section, the contrary occurs for wave-averaged models

which are directly derived from the NSWE. Notice that in very shallow waters, like that

of the swash zone, the two descriptions can be easily related (see for example Svendsen

and Putrevu 1995). Let (d, u) be respectively the mean water depth and onshore velocity

used in standard wave-averaged models. They are related to the variables used in the

classic description of the motion through the NSWE as follows:

d = 〈d〉, u = 〈u〉+〈ud〉〈d〉 . (4.5.22)

Moreover, it is common practice to use the following nonlinear SW variables:

⟨Q

⟩=〈ud〉=SW mass flux, (4.5.23a)

⟨S

⟩=

⟨u2

⟩d +

⟨u2d

⟩+

12g

⟨d2

⟩=SW momentum flux or ‘radiation stress’. (4.5.23b)

Hence, substitution of (4.5.22) into both (4.3.18) and (4.3.19) gives the desired form of

the SBCs:

dxl

dt= u− 1

d

d〈V 〉dt

, (4.5.24)

d

dt

[〈Px〉+ dxl

dt〈V 〉

]+gα〈V 〉+〈Υ〉=

[u2+

(dxl

dt

)2]

d−2uddxl

dt+

12gd2−

⟨Q

⟩

d+

⟨S

⟩.(4.5.25)

A third equation is required for completely solving for the mean flow i.e. prescribing

both the motion of the mean shoreline xl and the mean water depth and velocity at xl. This

must provide some information on ‘what is happening’ inside the computational domain.

Such an information is naturally carried by the positive Riemann variable R+ = u+2√

gd

which propagates from the interior towards the shoreline along positive characteristic

curves of the NSWE for the mean flow (e.g. Ozkan Haller and Kirby 1997b):

∂d

∂t+

∂(ud)∂x

= 0, (4.5.26)

∂u

∂t+

∂

∂x

(12u2 + gd

)= −gα−

⟨S

⟩

d. (4.5.27)


By writing these in characteristic form we find that:

dR+

dt= −gα−

⟨S

⟩

dalong

dx

dt= u +

√gd, (4.5.28a)

dR−dt

= −gα−

⟨S

⟩

dalong

dx

dt= u−

√gd (4.5.28b)

where R− = u − 2√

gd is the negative (outgoing) Riemann variable i.e. the signal

propagating along the receding characteristic curve of equation (4.5.28b). Notice such

a variable is propagated unaltered if the ‘source term’ −gα−⟨S

⟩/d is zero.

Now the three equations (4.5.24), (4.5.25) and (4.5.28a) can be solved for xl, u and d

once the positive (or incoming) Riemann variable R+ = u + 2√

gd is known at xl. These

equations can be simplified on the basis of the following analysis.

Within a run-up/run-down cycle the onshore velocity u is an antisymmetric function

of time (it changes sign at mid cycle) while the depth is given by a symmetric function.

Hence, the time-average of their product, i.e. the mean value of the onshore component of

water momentum in the swash zone 〈Px〉, vanishes. This intuitive and qualitative result

is also clearly supported by the more quantitative results illustrated both by figure 3b of

BP96, in which the space-time pattern of the onshore velocity is given as computed from

the analytical solution of Carrier and Greenspan (1958) and from results of a number of

computations which show that 〈Px〉 = 0 (see section 8 of BP96).

We also restrict our analysis to the case of beach slopes large enough that the

acceleration of the mean shoreline is negligible. Hence, for α >> (d2xl/dt2)/g we can

neglect d2xl/dt2 with respect to gα at the left hand side of (4.5.25). Our computations

suggest a maximum acceleration of the mean shoreline of about 0.001m/s2 which is about

ten times larger than measured accelerations (see Guza and Thornton 1985). For typical

wave conditions this means we restrict our attention to beaches of slope larger than about

1 : 1000 which is almost always the case for natural beaches.


A simple manipulation gives the final form of the SBCs:

u = R+ − 2√

gd, (4.5.29a)

dxl

dt= R+ − 2

√gd3 + d〈V 〉/dt

d, (4.5.29b)

gd3

︸︷︷︸(I)

+4√

gd〈V 〉dt

d3/2

︸︷︷︸(II)

−2

[gα〈V 〉+〈Υ〉+R+

d〈V 〉dt

−⟨S

⟩]d

︸︷︷︸(III)

+4

[d〈V 〉dt

]2

−2⟨Q

⟩2

︸︷︷︸(IV )

= 0.(4.5.29c)

Since SW properties in principle depend on LW characteristics equation (4.5.29c)

cannot be solved directly. However the above set of coupled equations can be solved

by an iterative procedure: a trial solution of (4.5.29c) can be substituted in the first

two equations to give both an approximate value of dxl/dt and of u. A similar iterative

procedure is sometimes used in wave-averaging solvers and is found to converge in few

iterations (e.g. Haas et al. 1998). However, a simpler solution strategy is most often used

which is based on the assumption of a weak dependence of SW properties on d and u.

In that case (4.5.29c) is an algebraic equation in d and an analytical solution for d can

be found. This is the approach we use in the following having in mind that an iterative

procedure can be easily implemented once found a good trial value for d. Much can be

learned on the SBCs (4.5.29) by the analysis of two special cases introduced here below

and one more general case described in section 4.6. The terms containing the four different

powers of d in (4.5.29c) have been labelled with Roman capital figures for ease of reference

in the following analysis.

The motion of a ‘rigid wall’

It is characterized by no water in the swash zone (i.e. 〈V 〉=0, 〈Υx〉=0) and simplify to:

dxl

dt= u = R+ − 2

√gd, (4.5.30a)

gd3 + 2⟨S

⟩d− 2

⟨Q

⟩2= 0. (4.5.30b)

These conditions are those of a rigid wall moving with velocity u (equation 4.5.30a) where

the water depth can change due to the change in the SW forcings⟨Q

⟩and

⟨S

⟩of equation

(4.5.23). This type of condition was already discussed in section 7 of BP96.


Steady state conditions

If steady state conditions are assumed equation (4.5.29c) simplifies to

gd3 − 2[gα〈V〉+ 〈Υ〉 −

⟨S

⟩]d− 2

⟨Q

⟩2= 0 (4.5.31)

and gives a depth of equilibrium at xl for given SW properties (〈V 〉,⟨Q

⟩,⟨S

⟩and 〈Υ〉).

4.6 The SBCs at a moving ‘porous wall’

These are the SBCs we suggest to implement in available wave-averaged models. They

are a slightly simplified version of (4.5.29) and allow both for the mean shoreline to move

(i.e. dxl/dt 6= 0) and for SW flows of mass and momentum to take place across the

boundary. The major simplification concerns the rate of change of the mean water volume

in the swash zone [d〈V 〉/dt appears in various terms of both (4.5.29b) and (4.5.29c)]. This

is found to be small and the mean water volume is almost constant in time. Thus the

situation described by such SBCs is comparable with that occurring if the swash zone was

to be replaced by an artificial, porous layer, hence the name of ‘porous wall conditions’.

We have evaluated the size of each contribution to the SBCs on the basis of sample

solutions described in section 4.4.2. One first important result is that regardless of the

values of the wave parameters, of α and f the ratio |d〈V 〉/dt|/|ud| ranges between 0.1 and

0.5. Thus the d〈V 〉/dt contribution to the right hand side of both (4.5.24) and (4.5.29b)

must be retained in the equation used to prescribe the motion of the shoreline. Other

synthetic results are summarized in table 4.2. In each column the largest value of ratios

between the size (absolute value of the mean over each wave cycle) of contributions to

(4.5.29b) and (4.5.29c) appears. Analysis of the results reveals that these can be grouped

into three classes depending on the value of ξm. In each class the values of the synthetic

results vary between 20% and 40% for all properties. As already mentioned in section 4.4.2

results are most sensitive to changes of the slope of the SW, less to changes of the ratio

Ll/Ls and of Hl, hence the use of ξm of equation (4.4.20). The first class (class ‘A’) is such

that ξm < 0.025 and approximately corresponds to that of the ‘Spilling breakers’. The


intermediate class ‘B’ is characterized by 0.025 ≤ ξm ≤ 0.15 and includes waves breaking

in the form of ‘Plungers’. For ξm > 0.15 the SW break very close to the shoreline (class

‘C’).

We gauge each contribution to (4.5.29c) by the value of the term (IV ) which does not

contain the unknown d. The fourth column of table 4.2 shows that for all flow conditions

this almost exactly coincides with 2⟨Q

⟩2. It is also clear that the contribution (II) is at

least ten times smaller than the leading terms for all the considered test conditions (see

sixth column of table 4.2). Moreover, we can also neglect the R+d〈V 〉/dt contribution to

the coefficient of d appearing in (4.5.29c) (see eighth column of table 4.2). Analysis of the

table also reveals the important role of the friction contribution. This is such to modify

the balance among the various terms of (4.5.29c). For f = 0 balance is such that only (II)

can be neglected the other three terms being of the same size, while for f = 0.5 the term

(IV ) becomes smaller than both (I) and (III). Finally, for f = 1 it would be possible to

neglect also (IV ) the approximate balance being between (I) and (III). Similar findings

are valid for irregular waves even in the case of f = 0. These results are useful as a guide to

simplifying of (4.5.29c) while retaining both unsteadiness of the motion of xl and allowing

for flows across the boundary. Retaining (IV ) even for f ≥ 0.5 and irregular waves:

dxl

dt= u = R+ − 2

√gd3 + d〈V 〉/dt

d, (4.6.32a)

gd3 − 2[gα〈V 〉+ 〈Υ〉 −

⟨S

⟩]d− 2

⟨Q

⟩2= 0. (4.6.32b)

In this case the coefficients of equation (4.6.32b) are only function of the SW motion.

One first important consequence of the above is that the second of equations (4.6.32)

is unaffected by the positive Riemann invariant and the depth at the mean shoreline is

only imposed by the SW properties. This is in good agreement with the fundamental

assumption (see also BP96) that the swash zone motion is entirely assigned to SW

contributions and that LW only force the motion of the mean shoreline. One practical

implication is that standard techniques can be used to model the motion of the mean

shoreline but a non-zero depth is attained at the mean shoreline which depends on the

local SW conditions and is found by solving (4.6.32b). Also this finding is not surprising


Wave type f Class[4(dV /dt)2]

(2⟨Q

⟩2)

(I)(IV )

(II)(IV )

(III)(IV )

R+dV /dt

gα〈V 〉+〈Υ〉−⟨S

⟩

Bichromatic 0 A 0.0280 2.50 0.31 4.90 0.09

Bichromatic 0 B 0.0080 0.64 0.07 0.35 0.04

Bichromatic 0 C 0.0002 0.60 0.01 0.58 0.01

Bichromatic 0.5 A 0.0480 6.00 0.74 16.00 0.10

Bichromatic 0.5 B 0.0230 3.40 0.42 5.20 0.15

Bichromatic 0.5 C 0.0140 3.20 0.29 3.30 0.15

Bichromatic 1 A 0.0670 11.00 1.10 32.00 0.10

Bichromatic 1 B 0.0350 6.00 0.65 9.90 0.15

Bichromatic 1 C 0.0170 5.10 0.36 6.90 0.15

Irregular 0 A 0.0520 46.20 2.20 52.00 0.12

Irregular 0 B 0.0470 33.00 2.00 35.90 0.14

Irregular 0 C 0.0780 49.10 3.26 51.20 0.15

Table 4.2: Synthetic results.


Figure 4.9: Top panels: comparison of the time series of dcomp (solid line) and deval (dashed

line) for the case ξm = 0.035, f = 0 (left) and ξm = 0.25, f = 0 (right). Lower panel: mean

values of dcomp vs. deval for the cases f = 0 (diamonds), f = 0.5 (triangles) and f = 1

squares. The dashed line gives a reference for the case of perfect matching.

in view of the analysis performed in BP96 on the characteristics of mean shorelines. For

example, figure 8 of BP96 shows the dependence on the wave amplitude of the mean

water depth at the location given by the phase-averaged waterline. However, that result

is specific to the analytical solution of Carrier and Greenspan (1958) while the result here

achieved holds at xl and is more general covering also the case of breaking waves.

Results of the comparison between the value of d determined from the numerical

solution of the NSWE (dcomp) and that evaluated by equation (4.6.32b) (deval) are reported

on figure 4.9. The top panels show the comparison of the time series of the functions for

two specific tests while the bottom panel collects the mean values of both signals for all

the cases here considered. The latter plot shows that deval overpredicts dcomp (symbols

are above the straight line of reference) of about 10% − 20% however a good overall

agreement exists between the two data sets. The overprediction weakly depends on f and

an improving agreement is achieved for increasing values of d i.e. for increasing ξm.


4.7 Implementation of the SBCs

Once a method is found to compute the positive Riemann variable at xl it is possible to

implement conditions (4.5.29) into available models like SHORECIRC (Van Dongeren and

Svendsen 2000). However, we prefer to discuss use of the simplified conditions (4.6.32) as

more amenable to analytical treatment and more suited for illustration purposes.

The main problem for implementing SBCs is to suitably predict d(xl) on the basis

of local (computed at xl) SW conditions. This requires a good prescription of the SW

coefficients of equation (4.6.32) in terms of the local wave parameters (H,T ), of α and f .

Dimensional arguments and analytical results based on the Carrier and Greenspan

(1958) solution (e.g. Mei 1989, pages 524-527) and the Shen and Meyer solution (e.g.

Peregrine and Williams 2001) suggest that the swash zone width scales with H/α while,

in very shallow water, the most suitable scale for the water depth is H. Hence according to

its definition (see section 4.3.2) 〈V 〉 scales with H2/α. Definition (4.5.23a) suggests that⟨Q

⟩scales with the product of the scale for the onshore velocity of the SW (in shallow

water√

gH) times the scale for the fluctuation of the water depth (H). In shallow water

the radiation stress⟨S

⟩is suitably scaled with gH2 and it is easy to show that 〈Υx〉 scales

with fgH2, hence:

〈V 〉=CVH2

α,

⟨Q

⟩=CQ

√gH3,

⟨S

⟩=CSgH2, 〈Υ〉=CΥgH2 and

d〈V 〉dt

=2CV H

α

dH

dt, (4.7.33)

with CV =0.615−0.201f, CQ=0.356−0.273√

f, CS=0.792−0.574√

f, CΥ=−0.034f. (4.7.34)

These values, obtained from fitting (with a regression coefficient ranging between

0.94 and 0.99 as shown in figure 4.10) a large number of solutions, give an excellent

representation of the SW properties of (4.7.33) over a wide range of H,T, α and f . Hence,

direct reference can be given of the coefficients of (4.5.29b) to the local height of the SW

by substitution of (4.7.33), into (4.6.32b):

d3 − 2 (CV + CΥ − CS) H2d− 2C2QH3 = 0. (4.7.35)

This equation is amenable to simple, analytical solution (see Abramowitz and Stegun


0 0.02 0.04 0.06 0.080

0.01

0.02

0.03

0.04

0.05

f=1.0

f=0.5

f=0.0

0 0.01 0.02 0.03 0.040

0.002

0.004

0.006

0.008

0.01

0.012

0.014f=0.0

f=0.5

f=1.0

0 0.005 0.01 0.015 0.02 0.0250

0.005

0.01

0.015

0.02

f=0.0

f=0.5

f=1.0

0 0.002 0.004 0.006 0.008 0.01−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−4

Figure 4.10: Fitting of SW properties. From left to right and from top to bottom are the

first four properties of equation (4.7.33): 〈V 〉,⟨Q

⟩,

⟨S

⟩and 〈Υ〉 respectively. In each

plot the flow properties are plotted against the scaling arguments defined by 4.7.33

1964). First of all the condition[C4

Q −827

(CV + CΥ − CS)3]

H6 > 0 (4.7.36)

valid for f < 0.5, states that equation (4.7.35) admits one real and two complex conjugate

roots. The real root is then computed to give:

d =[(1 + F )1/3 + (1− F )1/3

]C

2/3Q H where F =

√1− 8(CV + CΥ − CS)3

27C4Q

. (4.7.37)


Substitution for CV , CQ, CS and CΥ gives the desired solution:

0.45H ≤ d ≤ 0.55H for 0 ≤ f ≤ 0.5. (4.7.38)

This almost exactly coincides with the range 0.48H ≤ d ≤ 0.53H derived from the fully

numerical solution of the NSWE, demonstrating again both the validity of the chosen

SBCs and of the chosen simplifications. Both analysis of the previous section and equation

(4.7.36) suggest that for f > 0.5 the second order equation

d2 − 2 (CV + CΥ − CS) H2 = 0 (4.7.39)

can be used to get d =√

2(CV +CΥ−CS)H which approximately ranges between 0.48H

and 0.58H for 0.5 ≤ f ≤ 2. Larger values of f would be too far out of the range

considered in the computations and regressions would not be appropriate. However, all

evidence suggest that for 0≤f≤2 we can assume d∼=H/2 with good approximation.

4.8 Discussion

In summary a simple ‘recipe’ is suggested to prescribe the SBCs. The motion of the mean

shoreline is given by equation (4.6.32a) in which the rate of change of the volume in the

swash zone, related to the local SW height through equation (4.7.33), acts to decelerate

the shoreline motion during the run-up and accelerate it during run-down. At the mean

shoreline a mean water depth must be prescribed of about half of the local SW height.

Numerical investigation is currently underway to implement this ‘recipe’ in an available

wave-averaged solver and preliminary results reveal that an accurate computation of R+

at xl is the most crucial issue for correctly predicting the motion of the mean shoreline.

In order to confirm the validity of our analysis we are also planning to verify our findings

on the basis of experimental evidence. At that stage it will also be necessary to take into

account the different generation mechanisms (disregarded as unnecessary in the present

analysis) of the shoreline low frequency motion (‘surf beat’). Preliminary analysis of new

experimental data on surf beat generation by a time-varying breakpoint (Baldock et al.,

2000) suggests our analysis is likely to be valuable for modelling those flow conditions:


examples of experimental shoreline motion (e.g. figure 5b of Baldock et al. 2000) are

surprisingly similar to our computed curves (e.g. figure 4.7).

Chapter 5

Conclusions

In this thesis the solution domain of depth-integrated numerical models for studying the

nearshore hydrodynamics has been extended to cover the swash zone. Three innovative

results were achieved. These are here below summarized.

• The applicability of BTE in the swash zone and the role of dispersive and dispersive-

nonlinear terms was studied (section 2.3).

• A new SBC for BTE was derived and verified against well-known analytical solution

(chapter 3).

• A technique for avoiding the problems related to using wave-averaged models in

proximity and into the swash zone was introduced and verified; boundary conditions

for available circulation models, valid at the lower limit of the swash were finally

presented and discussed (chapter 4).

As far as the BTE are concerned, we have shown that if not properly handled, dispersive

and dispersive-nonlinear terms of these equations, can unrealistically grow in the swash

zone. A modified form of the BTE, written in terms of the total water depth instead of the

surface elevation respect to the undisturbed level was therefore introduced. According to

this new form of the equations, dispersive terms naturally tend to vanish at the shoreline,

where the water depth tends to zero.

102

103

A new technique for specifying SBC and tracking the movements of the instantaneous

shoreline was subsequently derived for numerical models based on the BTE. This technique

is based on the use of a specific shock-capturing method for dealing with the discontinuity

occurring at the shoreline, where a transition between wet and dry conditions occur. A

numerical model based on the scheme proposed by Wei and Kirby (1995) was coded in

order to implement and verify the new SBC. The comparison of the numerical model

results against well known analytical solutions suggested that using the new SBC very

accurate simulations of swash zone flows can be performed.

New SBCs were finally derived and verified for wave-averaged models. This specific

category of model cannot properly treat the swash zone, where alternating of dry and

wet conditions makes averaging problematic. In this work we proposed to avoid direct

simulation of the swash and to ‘cut’ the computational domain at the seaward (lower)

limit of the swash zone. A technique suitable to specify both wave-averaged fluid velocity

and water depth at that specific point was described and verified by means of full numerical

solution of the NSWE.

In the following brief remarks on ongoing research are given.

For what concern the numerical model based on BTE two different improvements are

being studied. On the one hand the model and the SBCs are being extended to two

horizontal dimensions (2DH). This would allow for simulation of complex hydrodynamics

phenomena. On the other hand a specific technique for representing wave-breaking effects

is being inserted in the 1DH model in order to test the capability of the new SBCs in

simulating breaking waves.

For what concerns the wave-averaged models, the SBC derived in chapter 4 are going

to be implemented in an available numerical model for the nearshore circulation. Moreover

further validation is underway on the basis of available experimental data.

Bibliography

Abramowitz, M. and I. A. Stegun. 1964. Handbook of mathematical functions. Dover

Publications.

Baldock, T.E., D.A. Huntley, P.A.D. Bird, T.J. O’Hare, and G.N. Bullock. 2000. “Surf

beat generation by a time-varying breakpoint.” Proc. of 27th ICCE, ASCE, Volume 2.

1398–1411.

Bellotti, G. and M. Brocchini. 2001a. “On the shoreline boundary conditions for

Boussinesq type models.” International Journal for Numerical Methods in Fluids

37 (4):479–500.

Bellotti, G. and M. Brocchini. 2001b. “On using Boussinesq Type Equations near the

shoreline: a note of caution.” Accepted for publication on Ocean Engineering.

Brocchini, M. 1998. “The run-up of weakly two-dimensional solitary pulses.” Nonlin.

Proc. In Geophys. 5:27–38.

Brocchini, M. and G Bellotti. 2001. “Integral flow properties of the swash zone and

averaging. Part 2. The shoreline boundary conditions for wave-averaged models.”

Submitted for publication to the Journal of Fluid Mechanics.

Brocchini, M. and D.H. Peregrine. 1996. “Integral flow properties of the swash zone and

averaging.” J. Fluid Mech. 317:241–273.

Brocchini, M, R. Bernetti, A. Mancinelli, and G. Albertini. 2001. “An efficient solver for

nearshore flows based on the WAF method.” Coastal Engng. 43:105–129.

Brocchini, M., M. Drago, and L. Iovenitti. 1992. “The modelling of short waves in shallow

104

105

waters. Comparison of numerical models based on Boussinesq and Serre equations.”

Proc. of 23rd ICCE, ASCE, Volume 23. 76–88.

Carrier, G.F. 1966. “Gravity waves on water of variable depth.” J. Fluid Mech. 24:641–

659.

Carrier, G.F. 1971. “Dynamics of tsunamis.” in Mathematical problems in the geophysical

sciences., vol. 1.

Carrier, G.F. and H. P. Greenspan. 1958. “Water waves of finite amplitude on a sloping

beach.” J. Fluid Mech. 4:97–109.

Christensen, E. D. and R. Deigaard. 2001. “Large eddy simulation of breaking waves.”

Coastal Engng. 42:53–86.

Dodd, N. and C. Giarrusso. 2001. “ANEMONE: Otto 1-d, A user manual.” Technical

Report 87, HR Wallingford.

Giarrusso, C. 1998. “Studio numerico del moto ondoso su fondali intermedi e bassi.”

Ph.D. diss., Universita degli studi della Calabria, Cosenza.

Guza, R. T. and E. B. Thornton. 1985. “Observation of surf beat.” J. Geohys. Res.

90:3161–3172.

Haas, K.A., I.A. Svendsen, and M.C. Haller. 1998. “Numerical modelling of nearshore

circulation on a barred beach with rip channels.” Proc. 26th Int. Conf. Coastal Eng.,

ASCE, Volume 1. 801–814.

Lin, P. and P. L. F. Liu. 1998. “A numerical study of breaking waves in the surf zone.”

J. Fluid Mech. 359:239–264.

Madsen, P. A. and H. A. Schaffer. 1998. “Higher order Boussinesq-type equations for

surface gravity waves: derivation and analysis.” Phil. Trans. R. Soc. Lond., Series

A. 356:3123–3184.

Madsen, P., B. Bingham, and H. Liu. 2001. “Nonlinear wave dynamics and breaking

waves modeled by the use of a highly accurate Boussinesq formulation.” Proc. of

WAVES 2001, ASCE.

106

Madsen, P. A., O. R. Sørensen, and H. A. Schaffer. 1997. “Surf zone dynamics simulated

by a Boussinesq type model. Part I: Model description and cross-shore motion of

regular waves.” Coastal Engng. 32:255–287.

Mei, C. C. 1989. The applied dynamics of ocean surface waves. World Scientific.

Ozkan Haller, H. and J. T. Kirby. 1997a. “A Fourier-Chebyshev collocation method

for the shallow water equations including shoreline runup.” Applied ocean research

19:21–34.

Ozkan Haller, H. and J. T. Kirby. 1997b. “Nonlinear evolution of shear instabilities of

the longshore current: A comparison of observations and computations.” J. Geophys.

Res. 104:25953–25984.

Peregrine, D. H. 1967. “Long waves on a beach.” J. Fluid Mech. 27:815–827.

Peregrine, D. H. and S. M. Williams. 2001. “Swash overtopping a trucated plane beach.”

J. Fluid Mech. 440:391–399.

Rego, V., J. Kirby, and D. Thompson. 2001. “Boussinesq Waves on Flows with Arbitrary

Vorticity.” Proceedings of Waves 2001-ASCE, in print.

Sancho, F. E. and I. A. Svendsen. 1997. “Unsteady nearshore currents on longshore

varying topographies.” Technical Report CACR-97-10, Center for Applied Coastal

Research, University of Delaware.

Schaffer, H. A., P. A. Madsen, and R. Deigaard. 1993. “A Boussinesq model for wave

breaking in shallow water.” Coastal Engng. 20:185–202.

Skotner, C. and CJ. Apelt. 1999. “Application of a Boussinesq model for the computation

of breaking waves Part 1: Development and verification.” Ocean Engng. 26 (10):905–

925.

Stoker, J. J. 1957. Water waves. Interscience: New York.

Svendsen, I. A. and U. Putrevu. 1995. Surf Zone Hydrodynamics. In Advances in Coastal

and Ocean Engineering. World Scientific.

Synolakis, C. E. 1987. “The runup of solitary waves.” J. Fluid Mech. 185:523–545.

107

Toro, E. F. 1992. “Riemann problems and the WAF method for solving twodimensaional

shallow water equations.” Phil. Trans. of the Roy. Soc. London A 338:43–68.

Toro, E. F. 1997. Riemann solvers and numerical methods for fluid dynamics. Springer,

Berlin.

Van Dongeren, A. R. and I. A. Svendsen. 2000. “Nonlinear and quasi 3-D effects in leaky

ingragravity waves.” Coastal Engng. 41:467–496.

Veeramony, J. and I. A. Svendsen. 1999. “Modeling the flow in surf zone waves.” Technical

Report CACR-99-04, Center for Applied Coastal Research, University of Delaware.

Veeramony, J. and I. A. Svendsen. 2000. “The flow in surf zone waves.” Coastal Engng.

39:93–122.

Watson, G., D. H. Peregrine, and E.F. Toro. 1992. “Numerical solution of the shallow-

water equations on a beach using the weighted average flux method.” Computational

Fluid Dynamics ’92.

Wei, G., J. T. Kirby, S. T. Grilli, and R. Subramanya. 1995. “A fully nonlinear Boussinesq

model for surface waves. I. Highly nonlinear, unsteady waves.” J. Fluid Mech. 294:71–

92.

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