Numerical modelling of barotropic tidal dynamics in the strait of Messina

Numerical modelling of barotropic tidal dynamics in the straitof Messina

A.A. Androsov, B.A. Kagan, D.A. Romanenkov, N.E. Voltzinger *

Institute of Oceanology, Russian Academy of Sciences, St. Petersburg Branch, 30, Pervaya Liniya, 199053 St. Petersburg, Russia

Accepted 22 January 2002

Abstract

A two-dimensional model for simulation of the intensive tidal dynamics in the Strait of Messina is presented. An initial

boundary-value problem for the viscous shallow-water equations in the complex domain representing the Strait is transformed to

the boundary-fitted curvilinear coordinates mapping the domain onto a rectangle with two opposite open boundaries. The trans-

formed equations in the form of contravariant fluxes are integrated by the semi-implicit difference method. The results of com-

putation include tidal maps of the M2- and M4-waves, currents induced by the M2-tide and the sum of the main harmonics, a tidal

energetic balance, main eddies, and residual circulation. � 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Boundary-value problem; Curvilinear coordinates; Numerical convergence; Tidal map; Spectra; Eddies; Residual

circulation

1. Introduction

The Strait of Messina separates the Italian Peninsulafrom the Island of Sicily and connects the Ionian Sea withthe Tyrrhenian Sea (Fig. 1). Despite its small extent ofabout 20 km the Strait is characterized by the strongvariability of its shoreline and bottom topography. In itssouthern and central parts, the Strait is oriented meridi-onally; in the northern part, orientation of its axischanges and has a SW–NE direction. At the place ofbending, with the area cross-section of about 0:3 km2, theStrait width and depth are minimal: the width decreasesdown to 3 km and the smallest depth is about 70 m. Atboth sides of the sill the depth rapidly increases reaching1200 m in the southern part. Here, at the Ionian Sea, thesteepMessina Canyon intrudes in the abyssal structure tothe south of Sicily. North of the sill, the coastal slopessharply diverge, the depth increases and the Strait ex-pands to the Tyrrhenian Sea in the form of an enormoussubmarine cone. Geometry of the Strait together with thelocal peculiarities of its tidal dynamics determines theexistence of intensive tidal currents exceeding 3 m/s.

1.1. Classical results

The tradition ascending to antiquity identifies thereal geography of the West Mediterranean with thefairyworld of Homer’s Odyssey. The striking featureof the Messina Strait dynamics gained a mythologicalinterpretation in this epic, and the powers of naturetaking the shape of artistic form went down into thetreasure-store of human culture. Two thousand yearsseparate Homer’s epic from the scientific descriptionof the Messina Strait dynamics. Data gathering andprocessing of all information available on that timewas done in 1825 by Ribaud, a French vice-consul inMessina. His description of the currents and navigat-ing charts remained unsurpassed for a long time. Atthe beginning of our century there appeared a hydro-graphic map with detailed navigator’s comments.However, all existing maps with instructions about thevortical circulation, ‘‘rema montante’’, and ‘‘rema scen-dente’’ (ascending and descending ‘‘violent tide’’), werevery inaccurate.A new stage in the study of the Strait dynamics began

by the work of Sterneck [30], in which a general patternof tidal oscillations generated by the semi-diurnal waveM2 in the Mediterranean Sea was presented. In thispattern, the tide was interpreted as a system of co-oscillations and the nodal zone located at the boundary

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Advances in Water Resources 25 (2002) 401–415

*Corresponding author. Fax: +7-812-3285-759.

E-mail addresses: [email protected] (B.A. Kagan), roman@

gk3103.spb.edu (D.A. Romanenkov), [email protected] (N.E.

Voltzinger).

0309-1708/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0309 -1708 (02)00007 -6

of the West Mediterranean Sea was depicted in the formof an amphidrome in the narrowest part of the Strait.Sterneck was the first who not only substantiated thegeneral scheme of oscillations, but also performed acomputation of the M2 tidal currents in the Strait usinga numerical solution of the one-dimensional linear tidaldynamics equations. As is clear now, this was an out-standing achievement for those times. By now, theseresults have been reproduced by the one-dimensionalmodel of the Triest Observatory [23] using new data.Creation of a clear and complete picture of the tides

in the Strait required qualitatively new and extensivedata. These were obtained in the course of the two-yearwork of the research vessel ‘‘Marsilli’’ organized byVercelli [33] in 1922–1923. The information assembledby Vercelli remains the most complete set of data todate. These materials allowed Mazzarelli [22] and De-fant [11,12] to give a description and explanation ofobserved dynamics.Defant’s analysis can be summarized as follows. The

dominating role in the formation of the barotropic tidesis played by the M2-wave. In the narrowest part of theStrait where the amphidrome is located, the phase oftidal velocity changes by 180� at a 3 km distance and thetidal elevation gradient acquires a minimal value of 1.7cm/km. The tidal elevations are out-of-phase, and tidal

velocities go up to 200 cm/s. Maximal velocities during‘‘rema montante’’ occur in the middle of the semi-diur-nal period, and during ‘‘rema scendente’’, at the end ofthe period. Periodic turbulent disturbances in the tidalvelocities determine the presence of intensive station-ary cyclonic gyres near the sill. These disturbances de-velop in the form of a bore (taglio). This phenomenonbecomes most distinctly apparent in the spring tide pe-riod.The tidal circulation is characterized by three main

cyclonic eddies: near Scylla, at Capo Peloro (Charybdiseddy), and at Messina. Some weaker anticyclonic eddieswith a smooth and seemingly oily surface exist also inthe center of the vortical domain (‘‘macchie d’oglio’’).

1.2. Present state

Subsequent works [8,10,17,23] confirmed the leadingrole of the semi-diurnal tide in determining the generalscheme of currents in the Strait. These works elucidatedthe elements of Strait dynamics and reflected the generalprogress of the methods and data interpretation. Un-expected implications of the activity of the main eddieswere analyzed by Alpers et al. [2]. On the basis of sat-ellite data and approximate estimates, these authors

(a) (b)

Fig. 1. (a) Geographical location of the Strait of Messina (after [17]); (b) geographical description of the Strait of Messina with bathymetry (after [33]).

402 A.A. Androsov et al. / Advances in Water Resources 25 (2002) 401–415

reported generation of internal waves near the sill.Subsequently this phenomenon was described within thescope of a two-layer numerical model [9]. Another es-sential feature of Messina dynamics involves verticaldisplacement of the interface between the Atlantic waterflowing southwards over the sill and the colder inter-mediate Levantine water flowing northwards. Numeri-cal computations of interfacial motions were performedby Del Ricco [13] using a width-averaged, two-dimen-sional model.Some essential features of the phenomenology of the

Strait such as the temporal evolution of the interfacebetween two water layers, origination and propagationof internal waves, coastal currents and columnar dis-turbances have been analyzed on the basis of simplebalance relations and new experimental observationsobtained in the expeditions of the Rome University,1980–1985 [14,26]. The totality of results forming thepresent-day concept of the dynamics and hydrology forthe Strait of Messina is presented in the overview paperby Bignami and Salusti [7].

1.3. Two-dimensional barotropic models

The complexity of the Strait geometry, the highvariability of the morphometric data and the strongnonlinearity of the phenomena complicate their simu-lation. This accounts for the lack of two-dimensionalmodels based on vertically averaged equations until re-cently. The first such model was offered for computationof the residual currents in the Strait [3], where thecomparative effects of physical and morphometric fac-tors on the generation and peculiarities of the residualcirculation were clarified. A method based on curvilin-ear coordinates fitted to the coastal line was used tosolve the initial boundary-value problem describing thetidal dynamics. This approach was used in the past de-cade to study some oceanographical problems in arbi-trary domains, such as tidal dynamics, storm surges,three-dimensional circulation and tsunami propagation[16,18,27,29,34].The formulation and numerical solution of the

boundary-value problem for the quasi-linear shallow-water equations in a domain with a partly open bound-ary are the subject of a number of papers [20,21,24,28]having primary significance for geophysical modelling.Formulation of the boundary-value problem for theshallow-water equations in the form of contravariantfluxes in a domain representing a strait mapped onto arectangle with two open boundaries and the accuracy ofnumerical realization of such problem was consideredby Androsov et al. [5]. Analysis of one of the most in-teresting features in Strait dynamics relating to the maineddy system was performed in Androsov et al. [6] wherethe quantitative role of different factors determining

the unusually strong tidal currents in the Strait wasascertained. These results enabled one to estimate in-tensity of the Scylla and Charybdis eddies (i.e., evaluatethe vertical component of the vorticity) in a hypotheti-cal situation referred to the remote past and to ver-ify the possibility of hydrodynamical interpretationof the Odysseus trajectories in the Strait of Messina[4,6].As has been pointed out by Vercelli [33] and analyzed

by Defant [11], the motion near the sill is mainly baro-tropic. Baroclinic effects related to the density fieldevolution, generation and propagation of the internalwaves, demand special consideration [9,13].In this paper, simulation results are assembled and

presented for the aspects of Messina Strait dynamicswhich can be reproduced within the framework of baro-tropic approximation to a satisfactorily degree of real-ism. They include a calculation of the vertically averagedcurrents due to individual harmonics and their totality,construction of tidal charts defining the spatial struc-ture of the tidal oscillations, calculation of the spectrarevealing the nature of nonlinear interaction of the tidalprocess components, and energy budget of the tidalwaves. We present new results comprising a compari-son with data of the calculated amplitudes and phasesof level oscillations for four main tidal harmonics;computation of current amplitudes in the semi-diurnalgroup of tidal waves; calculation of the level spectra andlong temporal series of the water level at some coastalpoints; construction and interpretation of tidal charts.At the same time, for the sake of completeness, we haveincluded earlier results relating to the Messina eddiesand residual circulation [3,4], basically in improvedform.In the next section a boundary-value problem for the

viscous shallow-water equations in the Cartesian coor-dinates is formulated, together with necessary informa-tion on data set, parameters, and numerical method.Convergence of the numerical solution is established byrepeated computations on a detailed grid. Formulationof this problem in the boundary-fitted coordinates isgiven in Appendix A. In Section 3, the barotropic modelresults are discussed. The accuracy of computations ischecked by fulfillment of the tidal energy balance and bycomparison with data. The final section contains con-clusions of the work.

2. Description of the model

2.1. Boundary-value problem in Cartesian coordinates

In the domain X�¼ fx; y � X; 06 t6 Tg, X is a plane

domain with boundary oX, we consider the verticallyaveraged equations of motion and continuity

A.A. Androsov et al. / Advances in Water Resources 25 (2002) 401–415 403

vt þ ðv � $Þvþ g$f ¼ U

fk� v� rH�1vjvjþ H�1$ KH$vð Þ; ð1Þ

ft þ $ � ðHvÞ ¼ 0; ð2Þwhere v ¼ ðu; vÞ is the fluid velocity, f is the sea surfacelevel, H ¼ hþ f, h is the water depth, $ ¼ ðo=ox; o=oyÞ isthe gradient operator, f is the Coriolis parameter, k isthe unit vector in the vertical direction, r is the bottomfriction coefficient, and K is the eddy viscosity coeffi-cient. The set of (1) and (2), called the viscous shallow-water equations, relates to the type of incompletelyparabolic equations [15].On the solid part of the contour, oX1, and on its open

part, oX2, we have

vnjoX1 ¼ 0; Cðv; fÞjoX2 ¼ W1; ð3Þ

where vn is the velocity normal to oX1, C is the operatorof the boundary conditions and W1 is the known vector-function determined by the boundary regime: at in-flow the three conditions must be posed, at outflow – thetwo ones [24]. In practice, when the necessary informa-tion on the open boundary is unavailable, level oscilla-tions, in place of the second condition (3) at the openboundary, are usually posed: f joX2¼ wðx; y; tÞ. Accuracyof the reduced boundary-value formulation when onlythe sea level is assigned at the open boundary regardlessof the boundary regime, was considered by Androsovet al. [5]. The problem (1)–(3) for the vector u ¼ ðv; fÞ issolved for given initial conditions: ujt¼0 ¼ u0. Arbitaryinitial conditions are admissible because we are con-cerned with only quasi-periodic regime dictated by theconditions at the open boundaries.The equation of energy for set (1) and (2) has the

form

oEot

þ $ � H gf��

þ 12vj j2

�v

�

¼ �r vj j3=2 þ v � $ðKH$vÞ; ð4Þ

where

E ¼ 1

2H vj j2

�þ gf2

�ð5Þ

is the total energy per unit area.

2.2. Method

Using the boundary-fitted curvilinear coordinates,the equations are transformed to the form of contra-variant fluxes presented in Appendix A. The boundary-value problem (A.4)–(A.7) of Appendix A with someinitial conditions is approximated by the finite-differenceboundary-value problem and integrated on a uniformrectangular grid. This grid is a map of the nonuniformcurvilinear grid generated in X by the elliptic method

[31]. Equations (A.4) and (A.5) are approximated by theCrank–Nicolson scheme realized by splitting [19]). Touse this algorithm, with the complex problem of theMessina Strait dynamics, requires some modifications asnoted in Androsov et al. [5].All model results presented below were obtained by

using this algorithm. A test of the accuracy, particu-larities and reliability of the numerical solutions wasconducted using another method. This, so-called pres-sure-correction method [32] consists in preliminarysolution of the Poisson equation for the level which isobtained from a linear combination of the continuityequation and the divergence of the momentum equa-tion. With the level thus specified, the velocity compo-nents are determined from the momentum equations.Comparison of the results using these two methodsreveals some differences. The second algorithm gives asomewhat better convergence of the numerical solutionboth in the local and integral norms, but it also demandsmore computer time. Furthermore, comparison esti-mate of the the model amplitudes and phases with datafor each case gives nonregular, although insignificant,scattering which confirms reliability of the results, butdoes not allow preference of any one of these methods.

2.3. Parameters

The bottom friction coefficient and Coriolis parame-ter were, correspondingly, r ¼ 2:6� 10�3 and f ¼ 0:895�10�4 s�1. For the coefficient of turbulent viscosity wetake the value K ¼ 1 m2=s; this moderate value does notdistort the physical results and is sufficient for suppres-sion of the high-frequency disturbances of numericalsolution.

2.4. Data

We have used the data set mainly from the systematicand detailed work of Moseti [23]. In this work the dataare presented by the records on several horizons andhave been averaged over depth. These mean data wereused for prescribing open-boundary conditions. Obser-vational data include also the harmonic constants (am-plitudes and phases) of the sea level oscillations at sevenshore stations and some characteristics of the tidalcurrents at nine points in the narrowness of the Strait.We consider the tidal harmonics of the diurnal group D1

(including K1, O1 and P1-waves with periods, corres-pondingly, T ¼ 23:93, 25.82 and 23.77 h), semi-diurnalgroup D2 (including M2, S2, N2 and K2-waves with pe-riods, correspondingly, T ¼ 12:42, 12.0, 12.66 and 11.96h), ter-diurnal group D3 (including S3 and MK3-waveswith periods, correspondingly, T ¼ 8:0 and 8.18 h) andquarto-diurnal group D4 (including S4, M4 and MS4-waves with periods, correspondingly, T ¼ 6:0, 6.21 and6.1 h).


The computation starts from a state of rest andcontinues until the periodic regime is achieved. Relax-ation of the solution is attained after 5–6 periods andafter this the evolution of the total energy (5) of oscil-lations in tidal cycle remains invariable. The computa-tional results of the last period have been subjected toFourier analysis to determine the amplitudes and phasesof the main and secondary harmonics at the grid points.

2.5. Boundary conditions

Conditions for the sea level elevation f were imposedat the open boundaries in the form:

fm ¼Xi

Ami cos

2ptTi

�� wm

i

�;

where the subscript i indicates a particular harmonicwith period Ti, the amplitude Ai and the phase wi;m ¼ 1; 2; the index m ¼ 1 relates to the southern bound-ary (Ionian Sea), m ¼ 2, to the northern boundary(Tyrrhenian Sea).The computations have been performed on a curvi-

linear grid 16� 27 (Fig. 2) with varying grid sizes fromDmin ’ 50 m to Dmax ’ 1000 m at Dt ¼ 240 s.

2.6. Comparative accuracy on a grid with double resolu-tion

The computations have been repeated on a detailedgrid with 31� 53 points, half of which coincided with

the nodes of the first grid. Fig. 3 presents a histogram of

the velocity difference di ¼ jvð1Þi j � jvð2Þi j for the M2-wave,where the upper indices relate to solutions on the cor-responding grids at the same points; P ¼ da=A, where da

is the number of points at which the difference d falls inthe certain interval ½�a; a� during the period T; A is thetotal number of the space–time grid points. The com-parison shows that in ’82% of the points, d falls in therange ½�5; 5� cm=s and only ’2% of the points containthe difference in d range 15 cm=s < jdj < 25 cm=s atjvmaxj ’ 150 cm=s). We deemed this convergence asquite satisfactory. Such an accuracy on a 16� 27 grid,which appears to be comparatively crude, confirms theeffectiveness of using the transformation to the bound-ary-fitted curvilinear coordinates.

3. Results

3.1. Tidal map

Spatial structure of amplitudes and phases of the leveloscillations of a single harmonic, i.e., the tidal map,gives some information about propagation of the tidalwave with a corresponding period. We shall begin theanalysis of the results with a discussion of a tidal mapof the M2-wave (Fig. 4). Here, we focus on an amphi-dromic system with cyclonic-rotation isophases and thecenter situated almost on the axis of the Strait. Struc-tures of this kind in a channel are usually interpretedin terms of Kelvin waves, i.e., long waves modified bythe influence of the earth’s rotation [12]. A combinationof two equal Kelvin waves with opposite direction ofpropagation produces an amphidrome in the middle of

Fig. 2. Curvilinear grid. Fig. 3. Histogram of the difference of solutions obtained on two grids.


the channel width. It is this situation precisely that isrealized in the Strait.

3.2. Comparison with observations [23]

From Table 1 one can see that the maximum differ-ences between predicted and observed values of the M2–tidal constants do not exceed 3.6 cm for amplitudes and20� for phases of the level oscillations, which is generallyacceptable for the amphidromic region. The most sig-nificant deviation of the computed value of the M2-waveamplitude from the observed one relates to Punto Pezzo.There is, however, some reason for rejecting this value: itis not compatible with site data at Villa S.G. situated at

a very short distance from Punto Pezzo. Table 1 alsoincludes a usual v-estimate which is the rms differenceover a tidal cycle

v ¼ f½A2o þ A2c � 2AoAc cosðwo � wcÞ�=2g1=2

; ð6Þwhere the indices ‘‘o’’ and ‘‘c’’ denote the observed andcomputed values, respectively.A comparison of the calculated and measured M2-

tide current vectors is presented in Figs. 5(a) and (b).Computed velocities refer to the same moments of tidalperiod as the measured ones. The model results are ingood agreement with observations, except for the pointsE. It is possible to assume that the data at this pointis erroneous. Thus, according to the Admiralty TideTables [1] the M2-wave velocity amplitude at a pointvery close to E is 1:6 m=s, which is much nearer to thecomputed value.

3.3. Evolution of the total energy and energy budget

For the M2-wave in the Strait this is given in Fig. 6.Fig. 6(a) reveals an important particularity of the totalenergy behavior within the tidal cycle, namely, the firsthalf of the tidal period with southward currents is morepronounced than the second one with northward cur-rents. Existence of a top of the underwater mountain inthe Strait narrows leads to intensification of the non-linear effects and this is accompanied by asymmetry inthe flows over the tidal period, with the southwardcurrents stronger than the northward currents. Com-ponents of the energy Eq. (A.8) are presented in Fig.6(b). Value of the energy budget discrepancy in Fig. 6(b)shows that the budget is fulfilled with acceptable accu-racy in numerical modelling. On the whole, there is abalance between the temporal change of energy andenergy fluxes through the open boundaries during thetidal cycle. The dissipation factors, especially the hori-zontal turbulent exchange, play a very appreciable rolein intensive Strait dynamics. In this connection, it isinteresting to note that the average (over the tidal cycle)tidal energy dissipation due to bottom friction in theStrait of Messina is found to be equal to 0:77� 107 W,i.e., about two to three orders of magnitude smallerthan, for example, in the English Channel. This is due tothe fact that strong tidal currents are only concentratedin the vicinity of the narrowest and shallowest part ofthe Strait, whereas within the rest of the Strait, wherethe depths vary from 200 to 1200 m, the tidal velocitiesare much weaker.

3.4. Higher harmonics

Because of nonlinearity of the advective transfer andbottom friction, interaction of the main tidal harmonicsgenerates harmonics with multiple frequencies. Theseoverharmonics, in practice, appear in the spectrum of

Table 1

Comparison of the observed and computed amplitudes and phases of

level oscillations for the M2-wave

Locality M2 v (cm)

Amplitudes (cm) Phases (�)

Record Calculate Record Calculate

Ganzirri 3.2 2.3 316 303 0.77

Faro 5.5 7.5 269 281 1.70

P. Pezzo 0.9 4.5 143 146 2.55

Reggio 6.2 6.6 95 75 1.60

Villa S.G. 3.3 3.8 116 124 0.50

Messina 5.3 4.2 31 38 0.89

Scylla 10.2 10.2 271 267 0.50

Fig. 4. Tidal map of the M2-wave: — isophases (in degrees); - - - iso-

amplitudes (in cm).


tidal oscillations everywhere, especially in the vicinity ofthe sill. Model spectra of the level oscillations inducedonly the M2-wave are presented in Fig. 7 at four coastallocations. In the spectra one can see peaks correspond-ing to main harmonic M2 and its overharmonics M4, M6

and M8. Thus, for Ganzirri and Punta Pezzo the am-plitude of the quarto-diurnal overharmonic M4 relatesto that of the main semi-diurnal harmonic M2 as 4=5approximately. The high nonlinearity at these pointsleads to frequency modulation and appears in thespectral peaks at zero frequency. Fig. 8 gives spatialdistribution of the level amplitude and phase for the M4-wave. The zones along the Faro–Ganzirri and P. Pezzo–

V.S. Giovanni shore-line with maximal amplitudes of theM4-wave are regions of expressed manifestation ofnonlinear effects and precisely here, as will be shown, thetwo main eddy systems Charybdis and Scylla are lo-cated. Phase distribution of the M4-harmonic in Fig. 8(their increase from northern to southern boundary)points out to the wave which propagates southwards (tothe Ionian Sea).

3.5. Summary tide

In this case, the boundary values of tidal elevationswere assigned in the form of superposition of four main

(a) (b)

Fig. 6. (a) Total energy; (b) energetic budget for the M2-wave in tidal cycle: 1 – energy change in time, 2 – flow through the open boundaries,

3 – bottom friction, 4 – turbulent viscosity, 5 – discrepancy from balance.

(a) (b)

Fig. 5. A comparison of calculated (—) and measured (- - -) velocity vectors for the M2-wave.


constituents corresponding to the M2, S2, K1 and O1 –tidal waves. The period of the summarized wave con-stitutes 29.5 days (synodic month). In addition to Table1, comparison with data for the last three harmonicsis presented in Table 2. Fig. 9 gives the time series ofmodeled and observed water level at locations withlargest oscillations. As may be seen the agreement be-tween the computation and the data is quite good. Wedraw attention to the following two features of thesolution obtained: first, enrichment of the tidal elevationspectrum and, second, considerable amplification oftidal velocities in the sill region. To confirm this, in Fig.10 we present the spectra of summary tidal elevationsat the same points of the coastline as in Fig. 7. We cansee in Fig. 10 that, apart from the spectral peaksat frequencies of the main semi-diurnal and diurnal

Fig. 7. Spectra of tidal oscillations for the M2-wave at stations: (a) Faro, (b) Ganzirri, (c) Punta Pezzo, (d) Reggio.

Fig. 8. Tidal map of the M4-wave: — isophases (in degrees); - - - iso-

amplitudes (in cm).


Table 2

Comparison of the observed and computed amplitudes and phases of level oscillations for the S2, K1 and O1 waves

Locality S2 K1 O1

Record Calculate Record Calculate Record Calculate

Amplitudes

(cm)

Ganzirri 1.6 1.3 1.4 0.9 0.5 0.3

Faro 2.2 2.6 2.1 2.5 0.8 0.9

P. Pezzo 0.6 0.8 0.9 2.4 1.6 0.6

Reggio 3.0 3.3 1.6 2.0 0.8 0.9

Villa S.G. 1.3 1.3 1.2 2.1 0.2 0.6

Messina 2.8 2.7 0.9 1.2 0.9 0.7

Scylla 3.2 3.5 2.8 3.2 0.9 0.7

Phases (�) Ganzirri 354 8 242 239 214 280

Faro 307 318 229 235 217 232

P. Pezzo 137 135 236 124 235 98

Reggio 100 95 57 67 50 52

Villa S.G. 104 115 48 113 – 84

Messina 98 69 290 8 65 12

Scylla 295 304 220 225 136 226

v (cm) Ganzirri 0.33 0.36 0.33

Faro 0.43 0.33 0.17

P. Pezzo 0.14 2.02 1.47

Reggio 0.29 0.36 0.07

Villa S.G. 0.18 1.36 –

Messina 0.98 0.95 0.57

Scylla 0.43 0.34 0.81

(a)

(b)

Fig. 9. Computed (solid line) and observed (dash line) water level time series for summary tide at Faro (a) and Scylla (b).


constituents and their overharmonics (M4 and S4), thespectra also include certain compound harmonics ofthe D3 and D4-groups due to nonlinear interaction ofthe main constituents. Compound tones appear mostsignificantly at Ganzirri and Punto Pezzo where nonlin-earity is especially pronounced. For the ter-diurnalharmonics D3 particularly significant is the S3-wave withfrequency 1/8 h�1 and the MK3-wave with frequency 1/8.18 h�1. Both spectra contain expressed quarto-diurnalharmonics among which the waves M4 with frequency 1/6.21 h�1 and MS4 with frequency 1/6.1 h

�1 are veryobvious. The S4-wave appears to be less significant. It isan interesting fact that energy redistribution at these

points is such that the D4 harmonics are amplified,whereas at Ganzirri the quarto-diurnal waves even pre-vail over the semi-diurnal oscillations. All spectra, espe-cially the ones at Ganzirri and Punta Pezzo, also containpeaks with frequency of the spring-neap tidal cycle, 1/14.75 day�1, due to interaction of the M2 and S2-waves.For the sum of the seven constituents M2, S2, N2, K2,

K1, O1 and P1 summary tidal currents are reproduced inFig. 11. The results refer to instances of maximal cur-rents in the southern and northern directions, respec-tively, within the synodic period. We can see that in thesill region the summary tidal velocities are approxi-mately twice as high as the M2-tidal velocities (see Fig.

Fig. 10. Spectra of summary tide at stations: (a) Faro; (b) Ganzirri; (c) Punta Pezzo; (d) Reggio.


5). The tidal currents are not only amplified, but theyalso slightly change their direction. Maximum velocityof the summary tidal current takes place near Ganzirriand near Punto Pezzo where it reaches 280 cm/s(according to the measurements [23] the maximum ve-locity of the summary tidal currents for 14 harmonics isequal to 289.5 cm/s). In Table 3 we present the ampli-tudes ratio for the semi-diurnal harmonics group incomparison with the Mosetti [23] data relating to thepoints in Fig. 5 (for the last column the data are un-available).

3.6. Gyres

The fields of vorticity generated by the summary tideM2, S2, N2, K2, K1, O1 and P1 are reproduced in Fig. 12.These fields correspond to the same stages of the tidalcycle as the velocity fields in Fig. 11. The computededdies, on the whole, agree with theoretical Defants’scheme and define it more precisely. An eddy near PuntaPezzo is added to the three main gyres: near Scylla, nearCapo Peloro (the so-called Charybdis gyre), and nearMessina. The direction of rotation of the eddynear Punta Pezzo coincides with that in front ofMessinaand is opposite to the other two main gyres Scylla andCharybdis. When the northward velocities in the Straitare maximal, the eddies near Punto Pezzo and nearMessina are cyclonic, whereas Scylla and Charybdis areanticyclonic (Fig. 12(b)). All gyres change their direction

of rotation to the moment when the southward veloci-ties are maximum (Fig. 12(a)). The gyre situated nearCapo Peloro and the gyre system near Scylla–Punta

Table 3

Ratio of the velocity amplitudes for the semi-diurnal harmonic group

Points M2=S2 M2=N2 M2=K2

A Record 2.49 5.50 –

Calculate 2.47 5.67 8.50

C Record 2.60 5.42 –

Calculate 2.54 5.81 9.38

D Record 2.93 5.87 –

Calculate 2.39 5.29 9.25

E Record 2.93 4.89 –

Calculate 2.39 5.41 8.11

F Record 2.47 5.27 –

Calculate 2.51 5.68 9.00

G Record 2.84 6.29 –

Calculate 2.45 5.40 9.82

H Record 2.53 5.38 –

Calculate 2.45 6.13 8.17

I Record 2.81 5.29 –

Calculate 2.43 5.23 9.71

L Record 2.71 4.22 –

Calculate 2.46 4.93 13.8

Mean ratio Record 2.70 5.35 –

Calculate 2.45 5.51 9.53

(a) (b)

Fig. 11. (a) Velocity field of the summary tidal currents at the moment when the southward current is maximum; (b) velocity field of the summary

tidal currents at the moment when the northern current is maximum.


Pezzo are the most obvious ones. The third main gyrenearMessina, which has been pointed out by Defant [11]for the M2-wave, is significantly weaker for the sum-mary tide. The field of vorticity in the Strait generatedonly by the semi-diurnal M2-wave retains the structuredescribed, but the intensity of each of the eddies falls bya factor of two, approximately.In additional experiments we have attempted to de-

termine the influence of the narrowing of the Strait inthe sill region on the vorticity amplification. This wasmotivated by the attempt to provide a hydrodynamicbasis of the Odysseus passage between the Scylla andCharybdis [4,6]. For this purpose, the Strait has beenreconstructed to a hypothetical form which might haveexisted at the time, before numerous earthquakes en-larged the Strait at its outlet to the Tyrrhenian Sea. Thewidth in this place was reduced by one-third via dis-placement of part at the western boundary. We find thatin this situation the currents were approximately inten-sified by a factor of two, whereas intensity of the eddiesincreased by one order of magnitude.

3.7. Residual circulation

We complete the analysis of the computation resultswith a discussion of the residual circulation. To deter-mine the long and stable transport of water with char-acteristic period T, the Euler residual transfer velocity v0

is used [25]

v0 ¼ vþ h�1vf; v ¼ 1

T

Z T

0

vðx; y; tÞdt:

In the case of summary tide the period T is the leastcommon multiple of the periods of tidal components.For the main four harmonics M2, S2, K1 and O1 thisperiod consists of 29.5 days (synodic month). Despitesome inaccuracy in this procedure, it gives a realistic

picture of the residual circulation. The main cause de-fining the residual currents is the nonlinear advectivetransfer. The effect of the bottom friction appears as atorque generating a fixed sign vorticity. The results showthat significant residual velocities only occur in the sillregion where their values are about 20 cm/s. In the deeppart of the domain they do not exceed 2 cm/s. Thegeneral residual transport is directed from the Tyrrhe-nian Sea to the Ionian Sea. This can be explained by themorphometric structure of the Strait and manifests itselfin significant asymmetry of the energy change within thetidal cycle so that the peaks of energy in each half of thecycle do not coincide (see Fig. 6(a)). Residual transportprevents penetration of the denser Ionian waters to theTyrrhenian Sea and causes a lowering of the mean sealevel in the narrow part of the Strait from Sicily toCalabria. The observations do confirm this conclusion[23].Computational results for vorticity of the residual

circulation reveal the presence of three cyclonic gyres: atCapo Peloro, at Scylla–Punta Pezzo and at Messina.These gyres prove to be stationary and do not changetheir positions during the synodic month.

4. Conclusions

For modelling the tidal dynamics in the Strait ofMessina the two-dimensional boundary-value problemis considered. This problem has been solved by trans-ferring to the curvilinear coordinates fitted to the con-figuration of the domain. Transformed equations in theform of contravariant components of fluxes are inte-grated by the semi-implicit method with the second-order accuracy in space and in time realized by splitting.Accuracy was checked both by repeated computationson a refined grid which show a high degree of conver-

(a) (b)

Fig. 12. (a) Field of vorticity ð10�4 c�1Þ induced by the summary currents of the southern direction; (b) field of vorticity ð10�4 c�1Þ induced by thesummary currents of the northern direction (sign ‘‘þ’’ indicates anticyclonic rotation; sign ‘‘�’’ indicates cyclonic rotation).


gence of the numerical solutions and by fulfillment ofthe energy balance.Comparison with data for four main harmonics M2,

S2, K1 and O1 of the tidal level gives in seven coastalstations the mean value of v-estimation (6), corre-spondingly, 1.35, 0.4, 0.82 and 0.57 cm. Comparisonwith data on the computed velocity amplitude ratio forthe semi-diurnal harmonics in Table 3 has also beenfound to be quite satisfactory. For the nine-point currentpolygon in Fig. 5 the differences between the recordedand computed values of the ratio M2=S2 and M2=N2 are,respectively, 0.1% and 0.03% of their recorded values.For verification of the results reliability some com-

putations have been repeated using another numericalmethod [32] which consists in preliminary determinationof the level by solution of the Poisson equation. Thelevel equation is solved iteratively at each time step usingsuccessive over-relaxation and after that the momentumequation is integrated by a two-step alternating-direc-tion procedure. The results obtained by the two methodsare in very close agreement. The difference between themean values of v for the M2-tide in seven points ob-tained by these two methods is 0.1 cm.Due to the nonlinear effects of the advective transport

of momentum and bottom friction, perceptible com-pound tones appear in the spectra of tidal oscillations ofthe level. Among the compound tones the most obviousare waves MK3 and MS4 with frequencies 1/8.18 and1=6:1 h�1.The computations reveal localization of three main

eddies in the Strait: an eddy at promontory Capo Peloro(Charybdis), eddy system Scylla–Punta Pezzo, and aneddy at Messina. The vorticity fields are presented forthe cases of maximal currents of both directions gener-ated by the summary tide. The estimation of the inten-sity and the position of the main eddies allow one tomake the classical notion of Defant [11] more precise.The high nonlinearity is also the cause of an unusu-

ally strong residual tidal circulation. The residual cur-rents in the throat of the Strait are directed from theTyrrhenian to the Ionian Sea and their velocity reaches20 cm/s. The existence of these currents explain theobserved lowering of the mean level from Sicily to Ca-labria. The solution reproduces three main stationarycyclonic eddies. Therefore, it is possible to say that for-mation of the legendary vortexes Scylla and Charybdisoriginates with immediate participation of vorticity ofthe residual tidal circulation.

Acknowledgements

The authors greatly appreciate the cooperation inmany ways with Dr. E. Salusti and are grateful to him

for helpful assistance. This research has been supportedby Russian Foundation for Basic Research, grant 00-05-64300.

Appendix A

Formulation of the boundary-value problem in bound-ary-fitted coordinates. Consider the transformation

n ¼ nðx; yÞ; g ¼ gðx; yÞ; t0 ¼ t ðA:1Þ

concordant with the configuration of the domain X withthe Jacobian J ¼ oðx; yÞ=oðn; gÞ, 0 6¼ J < 1 and withbasic vectors: ei ¼ rni , ei ¼ $ni; gik ¼ eiek; i; k ¼ 1; 2;n1 ¼ n; n2 ¼ g; r ¼ ðx; yÞ. We obtain equations forthe contravariant components of the flux pi ¼ JHUi;Ui ¼ v � ei are the contravariant components of velocity[31]. Multiplying the momentum Eq. (1) by the contra-variant vector ei we have

oUi

otþ UkUi

;k þ g$f � ei ¼ U � ei; i; k ¼ 1; 2; ðA:2Þ

where $f ¼ fniei and Ui

;k is the covariant derivative ofthe contravariant vector

Ui;k ¼

oUi

onj þ UkCijk ðA:3Þ

and Cijk ¼ eiðoej=onkÞ are the Christoffel symbols of the

second kind. The equation of continuity (2) takes theform

JoHot

þ o

oni pi ¼ 0: ðA:4Þ

By multiplying Eq. (A.4) by Ui, multiplying Eq. (A.2) byJH and adding the results we obtain a set of equations inconservative form for the fluxes pi; i ¼ 1; 2; p1 ¼p; p2 ¼ q:

pt þ ðUipÞni þ JHUjC1kjUk þ gHJg1ifni ¼ JHU � e1;

qt þ ðUiqÞni þ JHUjC2kjUk þ gHJg2ifni ¼ JHU � e2:

ðA:5Þ

In the n, g-plane, the domain X is now represented bythe rectangle X�. The transformation (A.1) is assumed tobe such that the solid part of the contour oX�

1, whichis the mapping of oX1, constitutes the side n ¼ constantof the rectangle X� and the open part oX�

2, which isthe mapping of oX2, has the sides g ¼ constant. Thecontravariant velocities Ui ði ¼ 1; 2Þ are the decompo-sition of the velocity vector into components along then coordinate lines, U 1 ¼ U and along the g coordinatelines, U 2 ¼ V . We assume that the boundary linesn ¼ constant contain only U-points and flux across theselines is zero. Let nn be the unit normal to the boundaryline n ¼ constant. Then on the solid boundary we have:


vn ¼ v � n ¼ v � $nj$nj ¼ v � e

1

je1j ¼ 0;

i.e.,

U joX�1¼ 0: ðA:6Þ

At the open boundary oX�2 as in the Cartesian case the

condition

fjoX�2¼ w ðA:7Þ

is posed which assumes that in the subdomain near theopen boundary, advection and viscosity in (A.5) areomitted and we have the reduced hyperbolic set ofequations in this region. The well-posed boundaryconditions in a general case, i.e., for the completeproblem considered, have been given by Androsov et al.[5].

Equation of energy. The notation for the covariantcomponents of velocity is Ui ¼ v � ei; U1 ¼ P , U2 ¼ Qand jvj2 ¼ UiUi ¼ G. Multiplying the first of Eq. (A.5)by P, the second one by Q and Eq. (A.4) by gf þ 1

2G and

adding the results, we have:

oEos

¼ �P þ U þ D; ðA:8Þ

where the integral energy is

E ¼ 1

2J HG

þ gf2��:

The terms on the right-hand side of Eq. (A.8) are: fluxesacross the open boundary, P; the rate of energy dissi-pation due to bottom friction, U; and energy dissipationdue to viscosity, D, respectively. The expressions foreach of these are as follows:

P ¼ o

oni gf��

þ 12G�pi�; U ¼ �rJG3=2;

D ¼ Uio

oni Kgikopi

onk

� �:

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Documents

Numerical modelling of barotropic tidal dynamics in the strait of Messina