NEARSHORE WAVES IN THE RHONE DELTA. IMPLICATIONS FOR ALONGSHORE DYNAMICS

Author: Oriol Serra Ribas Tutors: Jos Antonio Jimnez Quintana (Universitat Politcnica de Catalunya)

Franois Sabatier (Universit de Provence Aix-Marseille III) Marcel Stive (Technische Universiteit Delft)

NEARSHORE WAVES IN THE RHONE DELTA. IMPLICATIONS FOR ALONGSHORE DYNAMICS

Oriol Serra Ribas Author Jos Antonio Jimnez Quintana Tutor

Franois Sabatier- Tutor Marcel Stive - Tutor

ABSTRACT The Rhone Delta has seen a reversal of its growing behaviour during the last century. It is assumed to be caused by the trend towards a warmer climate which increases the sea-level and a diminution of the sediment input carried by the Rhone as a consequence of reforestation in its catchment basin coupled with damming of its flood-prone affluents. The previously existing equilibrium in the system has been broken as the erosive agents have not stopped working. Amongst them, sea waves are the most important factor driving sediment transport and with it ruling the changes in the shoreline. Focusing on a study area dominated by an accretive spit in an otherwise straight E-W sandy coast, the longshore dynamics have been addressed. The aims of this study are twofold. First, to characterise the wave climate at three different locations of the study area (offshore boundary, nearshore and on the verge of the breaker zone) based exclusively on a one-year data record from a buoy in front of the spit coupled with bathymetrical data. Secondly, to determine the waves effects on the longshore sediment transport in the coast. First a trimming of the original wave data has been performed. Afterwards, a simple backward refraction + shoaling propagation has provided tentative values for the offshore climate. This has been accurately determined using a forward propagation cycle involving the use of SWAN, whose output at the buoy is compared to the reference record. When the divergences amongst the two have shrinked to tolerable levels, SWANs input has been assumed to be the offshore climate and its nearshore output has been accepted. Results all along the coast following a -5 m bottom isoline have been found. From there, new forward refraction + shoaling propagations have brought the waves to the verge of the breaker zone, where a new set of wave climates has been established. The effects on them caused by the coasts morphology have been studied. Two formulae have been applied to determine the sediment transport due to the wave attack on the coast: those by Kamphuis and CERC. The former demands data regarding several beach parameters, amongst others, the grain size. However, these were not available but for its easternmost end, and therefore, the main study had to be performed in a unigrain fashion. The latter relies on a local calibration of its constant K in order to yield satisfactory results. A value previously obtained in the same delta was available and has been applied. It has been found that the offshore wave climate is similar to the reference one at the buoy, albeit with an increase in wave height and scattering of the main directions. Both nearshore and breaking zone wave climates show a strong sheltering effect in the gulf area, especially in the lee side of the spit. In some breaker zone locations the different wave directions merge into a single group fairly perpendicular to the coast. CERC and Kamphuis formulations yield qualitatively similar but quantitatively divergent results, with Kamphuis overtly underestimating both longshore transport and the erosive/accretive dynamics. In the latter, the differences between taking into account or not the variation of the grain size along the coast are minimal, though the scarcity of data impossibilities generalisations. Accretion is concentrated in the lee side of the spit, fed by erosion along nearby Faraman and (western) Beaduc beaches. A significant amount of sediment leaves the study area through its western end, where erosion is also present. These results are on line with those found in literature.

ONATGE LITORAL AL DELTA DEL ROINE. IMPLICACIONS PER LA DINMICA LONGITUDINAL COSTANERA

Oriol Serra Ribas Autor Jos Antonio Jimnez Quintana Tutor

Franois Sabatier- Tutor Marcel Stive - Tutor

RESUM Durant el darrer segle, el delta del Roine ha vist el seu comportament expansiu invertir-se. La tendncia cap a un clima ms clid que provoca un increment del nivell del mar i una disminuci de laportaci de sediments per part del Roine com a conseqncia de la reforestaci a la seva conca unida al represament de molts dels seus afluents en sn les causes sovint assumides. Lequilibri existent prviament ha estat trencat per tal com els agents erosius no shan aturat. Dentre ells, les ones del mar sn el factor ms important en el transport de sediments i amb aquest determinant els canvis a la lnia de la costa. Centrant-se en una rea destudi dominada per una barra acreciva en una costa altrament recta i dorientaci E-W, la dinmica longitudinal ha estat abordada. Els objectius daquest estudi sn dobles. Primerament, caracteritzar el clima donatge a tres llocs diferents de lrea destudi (lmit mar endins, prop de la costa i al llindar de la zona de rompents) basant-se exclusivament en un registre dun any duna boia de davant la barra unit a dades batimtriques. En segon lloc, determinar els efectes de les ones en el transport longitudinal de sediments a la costa. Primer sha dut a terme una purga de les dades donatge originals. Posteriorment, una simple propagaci enrere de refracci + somatge ha fornit les dades inicials pel clima a alta mar. La seva determinaci precisa ha estat aconseguida mitjanant una propagaci endavant implicant ls del SWAN, els resultats a la boia del qual han estat comparats amb el registre de referncia. Quan les divergncies entre ambds shan redut a nivells tolerables, les condicions de contorn del SWAN han estat assimilades al clima a alta mar, i els seus resultats prop de la costa acceptats. Hom ha trobat resultats seguint la batimtrica de -5 m. A partir dall una noves propagacions de refracci + somatge han dut les ones fins al llindar de la zona de trencants, on una nova tongada de climes donatge ha estat establerta. Els efectes que hi causa la morfologia costanera hi han estat estudiats. Dues frmules han estat aplicades a lhora de determinar el transport de sediments arran de latac de les ones sobre la costa: les de Kamphuis i CERC. La primera demana dades referents a diversos parmetres de la platja, entre daltres la mida de gra. Tanmateix, aquesta noms era disponible a lextrem oriental, i per tant, lestudi principal ha estat realitzat amb una mida uniforme de gra. El segon es basa en una calibratge local de la seva constant K per tal de proporcionar resultats satisfactoris. Hom comptava amb un valor del mateix delta obtingut prviament i ha estat aplicat. Sha trobat que el clima en alta mar s similar al de referncia a la boia, si b amb un increment de lalada dona i una dispersi de les direccions principals. Tant els climes a aiges intermdies com a la zona de trencants mostren un fort efecte darrecerament a lrea del golf, especialment al costat protegit de la punta. En alguns punts de la zona de rompents les diferents direccions dona sacaben unint en un nic grup fora perpendicular a la costa. Les formulacions de CERC i Kamphuis proporcionen resultats qualitativament similars per quantitativament divergents, amb Kamphuis infravalorant tant el transport longitudinal com la dinmica derosi/acreci. Les diferncies entre considerar-hi mida de gra variable o no a travs de la costa sn mnimes, si b lescassesa de dades en fa impossibles les generalitzacions. El costat arrecerat de la punta concentra lacreci, alimentat per lerosi de les platges properes de Faraman i (part occidental de) Beaduc. Una quantitat significativa de sediment abandona lrea destudi a travs del seu extrem occidental, on tamb hi ha erosi. Aquests resultats sn en la lnia dels trobats en la literatura.

Preface and acknowledgements

PREFACE AND ACKNOWLEDGEMENTS This study constitutes the last years compulsory thesis work for the degree in Civil Engineering at the Universitat Politcnica de Catalunya. It was begun while in Delfts University of Technology under the auspices of professors Franois Sabatier and Marcel Stive coordinated with Jos Jimnez in Barcelona. To them Im grateful for the chance they offered me, their trust and patience and for having introduced me to the Rhone Delta (not exclusively because of having an excuse for a detour while travelling back from the Netherlands!). My dearest gratitude to Jordina Boada, for her support, ideas, whit, and hours spent discussing this (and often some other) issue. To all other friends and family in Delft and Barcelona whove made this work possible either through their ideas, moral support or simply their happy presence: bedankt. Moltssimes grcies.

Contents

CONTENTS 1 INTRODUCTION...................................................................................................01

1.1 General introduction ......................................................................................01 1.2- Aim .................................................................................................................01 1.3- Problem approach ...........................................................................................02 1.4- Layout ............................................................................................................03

2- SWAN .....................................................................................................................04

2.1- Introduction ....................................................................................................04 2.2- General background .......................................................................................05

2.2.1- Units and coordinate systems ................................................................05 2.2.2- Grids and boundary conditions .............................................................06

2.3- Physical background ......................................................................................08 2.3.1- Wind generation ....................................................................................09 2.3.2- Dissipation .............................................................................................09 2.3.3- Non-linear wave-wave interactions .......................................................10

2.4- Implementation ..............................................................................................10 2.4.1- Propagation ...........................................................................................11

3- STUDY AREA AND AVAILABLE DATA...........................................................13

3.1- The Rhone river ..............................................................................................13 3.2- The Rhone Delta ............................................................................................14 3.3- Human use and settlement .............................................................................16 3.4- Available data .................................................................................................18

3.4.1- Bathymetry ............................................................................................18 3.4.2- Wave record ..........................................................................................19 3.4.3- Grain size ..............................................................................................19

4- RESULTS ................................................................................................................21

4.1- Buoy wave climate and basic assumptions ....................................................21 4.1.1- Assumptions ..........................................................................................21 4.1.2- Reference wave climate at buoy ............................................................22

4.2- Deep water wave climate ...............................................................................24 4.2.1- Backward propagation ..........................................................................25 4.2.2- SWAN grids ..........................................................................................26 4.2.3- Other parameters ...................................................................................31 4.2.4- Forward propagation .............................................................................31

4.3- Nearshore wave climate .................................................................................35 4.4- Breaking zone wave climate ..........................................................................38 4.5- Sediment transport ..........................................................................................41

4.5.1- CERC ....................................................................................................42 4.5.2- Kamphuis unigrain ................................................................................49 4.5.3- Kamphuis multigrain .............................................................................54

4.6- Comparison and implications .........................................................................59 5- SUMMARY AND CONCLUSIONS .....................................................................64

Contents

6- IMPROVEMENTS AND FURTHER DEVELOPMENTS ....................................66 REFERENCES ............................................................................................................68 LIST OF FIGURES .....................................................................................................71 LIST OF TABLES ......................................................................................................74 APPENDIX A: Source term formulation in SWAN ...................................................75 APPENDIX B: Wave data grouped into events ..........................................................79 APPENDIX C: Grain size at eastern end of study area ..............................................86 APPENDIX D: Tentative offshore values ...................................................................87 APPENDIX E: Definitive offshore values ..................................................................100 APPENDIX F: SWAN input files ...............................................................................105 APPENDIX G: Wave roses nearshore ........................................................................113 APPENDIX H: Breaker zone points ...........................................................................116 APPENDIX I: Wave roses at breaker zone ..................................................................118 APPENDIX J: Sediment transport with CERC formulation ......................................121 APPENDIX K: Sediment transport with Kamphuis unigrain formulation .................123 APPENDIX L: Sediment transport with Kamphuis multigrain formulation ..............125

1. Introduction

1

1 INTRODUCTION

1.1 General introduction The Rhone Delta is a relatively recent and extremely fragile geological feature intruding southwards into the north of the Gulf of Lyon. A reversal of its growing trend has been observed during the last century, mostly due to anthropological reasons (Sabatier, 2001). This poses a threat to the natural communities inhabiting the area and the human settlements alike. A global trend towards a warmer climate is increasing the sea-level (Suanez et al., 1996), while the construction of dams (both retaining sediment and laminating avenues, thus decreasing their peaks and sediment transport capabilities) and the increase in forest cover of the catchment basin have caused a diminution of the sediment input in the delta that it previously received from the river. These facts have made the previously existing equilibrium in the system no longer valid, since the two main agents eroding and reshaping the delta, eolian and maritime actions, have not changed and continue to move the sediment already there. In the context of the Rhone Delta, a third agent has to be taken into account: human actions. These need not be understood as general anthropological actions like the ones causing the dwindling sediment input, but as pure coastal engineering/delta management works, such as the opening of drainage canals from the deltaic lakes to the sea (Grau de Roustan) or the construction of dikes, breakwaters and harbours, specifically focused on driving the evolution of the delta in a particular direction, in general, preventing the retreat of the coast. These installations do not dot the whole shore involved, but, where present, their effect is profound and can be felt well beyond their immediate setting. Civil engineering works notwithstanding, the waves actions are the most important actor in the current evolution of the delta, being it a sandy area in a micro-tidal setting. Therefore, studying the waves conditions along its coast and evaluating their effects on the shore is of utter importance in order to predict the future evolution of the delta, key to adopt a correct management policy (if applicable), with the actuations needed. Ignoring the wind and human actions is certainly a simplification, but this is done in the spirit of efficiency, aiming at the main cause explaining a behaviour, so that valuable information can be extracted with minimal effort.

1.2- Aim The aim of this paper is to study the long-shore sediment transport at and around the Beaduc spit, in an area encompassing nearly 60 km of coastline, based exclusively on wave records and shore and sea-bottom data. In order to do so, two important partial goals are set:

1. Introduction

2

Estimate the wave climate both offshore and nearshore the study zone Estimate the effects of these waves on the coast, coupling their actions with its

given characteristics (geometry, material)

1.3- Problem approach In order to reach the goals stated above, several intermediate steps need to be taken, as shown below:

Figure 1.1: Schematic representation of the steps and processes to be followed. The reason for breaking the wave climate part into 3 different steps is that a direct propagation of the wave climate cannot be done from the buoy to the nearshore breaker zone points where the sediment transport formulae may be applied. First a backward propagation towards deep water conditions must be made, and from there, a forward propagation will return the shore conditions. This is in turn separated into two steps, because SWAN, the program used for the propagation, is deemed not precise enough nearby the breaker zone, and therefore a different method is used for its final stretch. On the first stretch of this forward propagation (which is performed by SWAN) results are also given at the buoy location, so that the calculated wave climate can be compared to the

Waves buoy (data record)

Waves offshore

Waves @ 5 m

Waves @ breaker zone

Waves @ buoy

C O M P A R I S ON

Backward propagation + Try & fail /eng. Know-how

Forward propagation: SWAN

Forward propagation: Iteration with =0.68

Sediment evolution. Time & space morphodynamics

Transport formulas: CERC & Kamphuis

PROCESS STEP / PARTIAL GOAL

1. Introduction

3

original record. This allows for a feedback mechanism to fine-tune the offshore boundary conditions that need to be fed to the program.

1.4- Layout After the present introduction, chapter 2 will offer a description of SWAN, the program used for the first part of the wave propagation. Chapter 3 physically describes the study area, enumerates its human settlements and presents the data available. Chapter 4 deals with the step by step work and the results this yielded. Chapter 5 summarises the work and outlines its main conclusions. Chapter 6 suggests possible improvements and future work lines. Finally, appendixes enclosed at the end of the report include all the data, formulation and results that, albeit relevant, were deemed too specific or tedious to be included in the main body of work, where they would have disrupted the continuity and easiness of read.

2. SWAN

4

2- SWAN

2.1- Introduction The following chapter provides some explanations on the program used to propagate the wave data from high sea to the nearshore output locations: SWAN, an acronym for Simulating WAves Nearshore. Its purpose is to simulate the evolution of random, short-crested wind-generated waves in estuaries, tidal inlets, lakes and coastal areas in general. SWAN is based on the discrete spectral action balance and is fully spectral, that is both in directions and frequencies. This enables random short-crested wave fields propagating simultaneously from widely different directions to be accommodated, and calculate their evolution through deep, intermediate and shallow waters, even including currents. The phenomena SWAN takes (can take) into account are wave generation by wind, dissipation due to white-capping, bottom friction and depth-induced wave braking, non-linear wave-wave interactions (quadruplets and triads) and wave blocking by currents. This is a third-generation program developed at Delft University of Technology (TUDelft), successor of the stationary, second generation model HISWA. Since SWAN has been commonly adopted as a standard for the above mentioned applications, WL|Delft Hydraulics has integrated it into the wider Delft3D model suite, where it can be used (under a different interface) as part of the Delft3D-WAVE. Delft3D is a software package targetting any process involving water in a free surface environment: flow, waves, water quality, ecology, sediment transport and bottom morphology; as well as the interactions among them. The capabilities of SWAN are widely broadened when coupled with the rest of Delft-3D (essentially the flow-module, which enables to study waves in a current). However, it ceases to be on the public domain, which is its original situation at TUDelft (WL / Delft Hydraulics, 2003). SWANs range of applications encompasses areas of up to more that 50 x 50 km and includes estuaries, tidal inlets, lakes, barrier islands with tidal flats, channels and coastal regions, making it suitable both for harbour or offshore design and coastal development and management projects. The second section takes a look at the general and practical matters that enable the use of the program. The third, briefly provides its physics background, whereas its numerical implementation is left for Section 2.2.4.

2. SWAN

5

2.2- General background

2.2.1- Units and coordinate systems The international system (S.I.) is the required manner of expressing quantities in Delft3D-Wave: m, kg, s, degrees, etc... The program doesnt take the curvature of Earth into account and it operates in a flat horizontal plane. Geographic locations and orientations (e.g. for the different grids) are defined in one common Cartesian coordinate system with a defined absolute origin (0,0). This can be seen in the following figure:

Figure 2.1: Nautical (left) and Cartesian (right) conventions of orientation.

Directions (of winds and waves) have to be introduced by either the Nautical or the Cartesian convention, always defined relative to the previously outlaid coordinate system, as presented below:

Figure 2.2: Grid location according to the conventions used by SWAN.

2. SWAN

6

2.2.2- Grids and boundary conditions Three different sorts of grids may co-exist while using Delft-3D: Input, computational and output grids. These need not coincide, but they must obviously encompass the same area. Therefore, each may have a different origin, orientation and resolution. The transition from one grid to another is done interpolating, which might cause some accuracy loss. Input grids Input grids provide the information the user has beforehand: bathymetrical -that is, the bottom features of the study area- as well as regarding the friction this bottom might exert, and about current and wind fields (if known). Preferably, they should be bigger than computational grids, so that all possible situations of the later can be covered. The resolution of the input grids (especially of the bottom one) should be fine enough so as to include all the relevant details in the sea bottom, especially sharp ridges. It is important that their minimal depth (shallowest part) is taken into account as points of the grid: otherwise, the calculations will be biased (wrong) because not all pertinent waves will have been clipped by surf breaking when attaining a minimal depth. Computational grids Computational grids are 4-dimensional: x-, y- and -, - space. For most of the calculations, where wave conditions are given at high sea (deep water), the grid in x-, y- space ought to be chosen in such a manner that the boundary up-wave sits in deep water or, at least, in a water deep enough so as not to have affected the wave field with refraction. If the boundary conditions, though, are given in such a manner that refraction and other processes have been taken into account (such as when a nested calculation is performed), then this caution needs not be taken. Similarly, if boundary conditions are only given in one of the ridges of the grid (the up-wave one), a lack of energy at the other lateral ridges will cripple the results at points near them, since there, energy comes not only from up-wave but also laterally, the more so the bigger the width of the directional energy distribution is. Therefore, older, swell, waves will have the least area contaminated in the grid.

2. SWAN

7

Figure 2.3: Disturbed lateral regions in the computational grid.

In order to resolve relevant details of the wave field the a certain spatial resolution of the computational grid is needed. Choosing it to be the same as those of the input grids usually suffices. There is a maximum number of nodes SWAN can operate with (under a standard configuration) The computational spectral grid has to be provided by the user too. A minimum and a maximum frequency coupled with the frequency resolution which is proportional to the frequency itself (e.g. f = 0.1f) define the frequency space. The user determines all these by choosing the lowest and highest frequencies as well as their total number. Advisable values for the extremes are a lowest frequency smaller than 0.6 times the value of the lowest peak frequency expected and a highest one of about 3 times the highest peak frequency expected (usually smaller than or equal to 1 Hz). The directional range is the full 360 unless specified otherwise. Doing so might save computer time and space, but this should only be performed when theres the certainty that waves approach the coast only from within a limited sector smaller than 180. The directional resolution is determined by the number of discrete directions choosen. This should be bigger for swell seas, since the directional spreading around the mean wave direction is smaller. Possible values might fit with a 2 resolution for sea and 10 for swell. Output grids The results of the calculations are presented via output grids. These results can be given at the points where they were performed (thus coinciding with those of the computational grids) or elsewhere, in which case they are obtained via spatial interpolation. However, no output grid has to be defined used per se. In a stationary mode, SWAN calculates the effects of any given wave situation (input/boundary condition) all over the study area (that is, the computational grid). It is up to the user to ask for results (a.k.a. output) at

2. SWAN

8

particular points. He might even ask for none, for example in case the computation at hand is being used barely to furnish with data a subsequent, nested computation. If this happens to be the situation, SWAN inputs the result of the general computation into the more particular one automatically. If output is asked for a point that belongs to the area covered by a nested grid, the results provided by Delft 3-D will be those resulting from the more detailed, nested computations.

2.3- Physical background The waves are described in SWAN with two-dimensional wave action density spectrums. This is done even in highly non-linear situations such as when dealing with the surf zone, because nevertheless enough accuracy is believed to be attained at calculating this spectral distribution of the second order moment of the waves (as opposed to sufficiently to a full statistical description of the waves). SWAN deals with the action density spectrum N(,) rather than the energy density one E(,) because in the presence of currents action density is conserved whereas energy density not (Whitman, 1974). Its independent variables are the relative frequency (as observed in a frame of reference moving with the action propagation velocity) and the wave direction , which is the direction normal to the wave crest of each spectral component). The action density is equal to the energy density divided by the relative frequency: N (,) = E (,) / . The spectral action balance equation in Cartesian coordinates that SWAN uses to describe the evolution of the wave spectrum is (Hasselmann et al.,1973) :

SNcNcNc

yNc

xN

t yx=

+

+

+

+ (eq. 2.1)

Where the first term at the left-hand side accounts for the local rate of change of action density in time. The second and third terms represent the propagation of action in geographical space (with cx and cy being propagation velocities in x- and y- space respectively). The fourth represents shifting of the relative frequency due to variations in depths and currents (with propagation velocity c in - space). The fifth one represents refraction, both depth and current induced (with propagation velocity c in - space). Linear wave theory provides for the expressions for these propagation speeds (e.g., Whitman, 1974; Mei, 1983; Dingemans, 1997). At the right-hand side S is the source term in terms of energy density representing the effects of generation (by wind Sin), dissipation (by white-capping Sds,w; bottom friction Sds,b and depth-induced breaking Sds,br) and non-linear wave-wave interactions (quadruplets Snl4 and triads Snl3), each of which will be shortly explained below. A complete formulation for these source terms can be found on appendix A at the end of this work.

2. SWAN

9

2.3.1- Wind generation (Sin) The transfer of energy from wind to waves is modelled with a resonance (Phillips, 1957) and feedback mechanisms (Miles,1957) , so that the source term can be described as the sum of linear and exponential growth:

( ) ( ) ,, BEASin += (eq. 2.2) where both A (linear growht) and BE (exponential growht) depend on wave frequency and direction and wind speed and direction. Current effects are taken into account thanks to the use of apparent local wind speed and direction.

2.3.2- Dissipation (Sds) The dissipation term is the summation of three different contributions: white-capping

( ) ,,wdsS , bottom friction ( ) ,,bdsS and depth-induced breaking ( ) ,,brdsS . Whitecapping The waves steepness controls whitecapping. In third-generation wave models such as SWAN, white-capping formulations rely on a pulse-bade model (Hasselmann, 1974), adapted by the WAMDI group (1988) as:

( ) ( ) ,~~,, EkkS wds = (eq. 2.3)

Where is a coefficient dependent on steepness, k is the wave number and ~ and k~ denote a mean frequency and a mean wave number (cf. the WAMDI group, 1988). Bottom friction Depth-induced dissipation may be caused by bottom friction, bottom motion, percolation or back-scattering on bottom irregularities (Shemdin et al., 1978). In continental shelf seas with sandy bottoms, however, bottom friction turns out to be the principal mechanism (Bertotti and Cavalieri, 1994). It can be formulated as:

( ) ( ) ( ) ,

sinh, 22

2

, EkdgCS bottombds = (eq. 2.4)

Where Cbottom is a friction coefficient dependent on the bottom orbital motion (Urms).

2. SWAN

10

Depth induced breaking The formulation of a spectral version of the bore model by Eldeberky and Battjes to acccount for the total dissipation (1995) is used in SWAN as a substitute for the process of depth-induced wave-breaking, which is still poorly understood (and little is known about its spectral modelling):

( ) ( ) ,,, EED

Stot

totbrds = (eq. 2.5)

Where Etot is the total energy and Dtot its rate of dissipation due to wave breaking (Battjes and Janssen, 1978).

2.3.3- Non-linear wave-wave interactions (Snl) Quadruplet wave-wave interactions dominate the evolution of the spectrum in deep water, transferring energy from the spectral peak to lower frequencies (thus lowering the peak frequency) and to higher frequencies (where whitecapping dissipates the energy). Triad wave-wave interactions in very shallow water transfer energy from lower to higher frequencies resulting often in higher harmonics (Beji and Battjes, 1993). Low frequency energy generation by triad wave-wave interactions is not considered here. The Lumped Triad Approximation (LTA) derived by Eldeberky (1996) is the model used in SWAN amongst many attempting to describe the triad wave-wave interaction, since it pictures fairly well the energy transfer from the primary peak of the spectrum to the harmonics.

2.4- Implementation The action balance equation has been implemented in SWAN with finite difference schemes in all five dimensions (time, geographical and spectral spaces). Time is however omitted from the equations in Delft-3D because SWAN is applied in a stationary mode. The geographical space is discretised with a rectangular grid with constant resolutions x and y in x- and y- directions respectively. As for the spectrum, it is discretised with a constant directional resolution and a constant relative frequency resolution / (logarithmic frequency distribution). The discrete frequencies are definided between fixed low and high cut-off values (the prognostic part of the spectrum), where spectral density is unconstrained. Below the low-frequency cut-off (typically fmin = 0.04 Hz for field conditions) the spectral densities are assumed to be zero. Above the high-frequency cut-off (usually 1 Hz) a diagnostic f--m tail is added (used to compute non-linear wave-wave interactions at the high frequencies and compute integral wave parameters). The reason for using this and not a dynamic cut-off frequency is that, in mixed sea states, the latter might fail to account for the characteristics of one of the seas composing it. For example, it might be too low to properly account for a local wind-generated sea state in a coastal region which is superimposed on a simultaneously occurring swell, albeit unrelated. The value of

2. SWAN

11

m should be between 4 and 5 (e.g., Phillips, 1985). If the Komen et al. (1984) formulation for wind input is chosen, SWAN uses m = 4.

2.4.1- Propagation Robustness, accuracy and economy are the basis for the numerical schemes in SWAN. Thus, for a basic equation like where the state in a grid point is determined by the state in the up-wave grid points, an implicit upwind scheme (both in geographical and spectral space) would be the ideal choice, since its the most robust scheme. With implicit it is meant that all derivatives of action density (x or y) are formulated at one computational level, ix, or iy, except the derivative in the integration dimension for which the previous or up-wave level is used too (x or y in stationary mode). For such a scheme the value of space steps, x and y would be mutually independent. An extra advantage of such a scheme in economical terms is that it is unconditionally stable, therefore allowing larger time steps in computations than with explicit schemes in shallow water. Thanks to the experience acquired with years of using the second-generation HISWA shallow water wave model (Holthuijsen et al., 1989) it is now known that a first-order upwind difference scheme I accurate enough for the geographical space, whereas not for the spectral. Thus, SWAN has implicit upwind schemes in both geographical space and spectral spaces, this last one supplemented with a second-order central approximation. In the geographical space the state in a point of the grid is determined by that of the up-wave points (as defined by the propagation direction): Thus, decomposing the spectral space in four quadrants is possible. In each one of them the computations can be carried out independently from the rest except for the interactions between them due to refraction and wave-wave interactions (correspondingly formulated as boundary conditions between quadrants). The wave components in SWAN are propagated in geographical space with the first-order upwind scheme in a four forward-marching sweeps sequence (one per quadrant). The computations are carried out iteratively at each time step so as to properly account for the boundary conditions between the four quadrants. The discretisation of the action balance equation is (for positive propagation speeds; including the computation of the source terms but ignoring their discretisation):

2. SWAN

12

[ ] [ ] [ ] [ ]

( )[ ] [ ] ( )[ ]

( )[ ] [ ] ( )[ ]

=

++

+

++

+

+

+

+

n

jjii

n

iii

iii

n

iii

iii

n

iii

iyiyn

iii

ixix

jxyx

yx

x

yy

x

xx

SNcNcNc

NcNcNc

y

NcNc

xNcNc

,,,,,

11

,,

11

,,

1

,,

1

2121

2121

(eq. 2.6)

With:

iiii yx ,,, : grid counters yx , : increments in geographic space , : increments in spectral space

n : iteration index n : iteration index for source terms (equal to n or n-1)

The degree to which the scheme in spectral space is central or upwind determined by the coefficients and : Values of = 0 or = 0 correspond to central schemes (which have the largest accuracy: numerical diffusion >> 0), whereas either or equalling 1 correspond to upwind schemes. These are somewhat more diffusive (and therefore, less accurate) but more robust. The propagation scheme is implicit as the derivatives of action density (in x or y) at the computation level (ix or iy respectively) are formulated at that level except in the integration dimension (x or y, depending on the direction of propagation) where the up-wave level is used too. The values of x and y are therefore still mutually independent. Boundary conditions for wave energy that is leaving the computational domain or crossing a coast line are fully absorbing in SWAN, both in the geographical and the spectral space. The user needs to prescribe the incoming wave energy along open geographical borders, although for coastal regions doing so only along the deep-water boundary may suffice. This implies that erroneous lateral boundary conditions are propagated into the computational area (see figure 2.3) from the apexes of the deep-water boundary conditions, and spreading towards the shore with the one sided width of the directional distribution of the incoming wave spectrum (that is, the spreading is smaller for swell conditions, and can reach up to 45 for wind sea). In order to avoid the propagation of such an error into the interest area, lateral boundaries should be sufficiently far away.

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3- STUDY AREA AND AVAILABLE DATA The Rhone Delta is located on the north of the Gulf of Lyon, on the western Mediterranean sea, at approximately 4330 N, 430E.

Figure 3.1: Situation of the Rhone Delta in the western Mediterranean (Wikipedia, 2005).

Figure 3.2: General features of the coast of the Rhone Delta (Suanez and Sabatier,1999) .

3.1- The Rhone river The Rhone river has its source in the Alps and flows into the Mediterranean after 812 km with an average discharge of 1,710 m3/s, which, given its basin area of 95,500 km2, makes it one of the European rivers with a highest relative discharge (17.9 l/s/km2). It is fed via three main mechanisms: oceanic fronts, snowmelt from the Alps and Mediterranean storms. This causes an irregular regime, with a marked summer low and spring and autumn peaks, with annual maximums topping at around 4,000 m3/s, and exceptional floods at more than 13,000 m3/s. It is precisely during floods when most of the sediment is transported (80% for Q > 3,000 m3/s), and especially if they are Mediterranean in origin, be it from the Cvennes or the Southern Alps. In more general terms, though, the sediment output has been diminishing for centuries due to 3 different phenomena: the natural diminishing hydrological evolution after the Little Ice Age, the reforestation in the catchment area as a result of a dwindling agricultural use and, especially, the damming of many of the Rhone affluents during the 20th century (retaining sediment, besides laminating floods). In this regard, the damming of the

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Durance in 1958 has had a deep impact, strongly curtailing the previously important sediment input from the Southern Alps. During its final stretch, and already inside the deltaic plain, the Rhone splits into two branches at Fourques: the Grand Rhone to the east, and the Petit Rhone to the west. The first one follows a straight south-east direction for 50 km and accounts for much of the discharge (85-90%, although the sediment output is somewhat smaller, especially during floods: 80%). The second one is older, shallower, has more meanders, and 70 km of length. Both of them have been heavily entrenched, with sediment spilling into the surrounding areas only during catastrophic floods.

Figure 3.3: The Petit Rhone and the Espiguette spit as seen from the air (Wikipedia, 2005).

3.2- The Rhone Delta The delta started to form some 7,000 years ago, with an initial growth pattern in its central-western side (corresponding to the current Petit Rhone, forming the Sant-Ferrol lobe) which was replaced with the formation of an eastern lobe (Bras de Fer) in the mouth of the Grand Rhone in recent times. Nowadays the delta covers around 1,700 km2, along 90 km of coast, in a context of a very broad continental shelf (50 km wide) with a slope ranging from 0.3% to 0.5%. The sea where it is located being microtidal (tidal range of 0.3 m; Provansal, 2003), waves and the human interference are the main factors dictating the evolution and modification of its shape (besides the dwindling river sediment input seen above). The reason why wind is not as important, is that it tends to blow in a south-east direction, both in terms of frequency and intensity. It therefore pushes sediment into the sea. In some particular areas of the delta, this sediment that has been blown seawards is later returned by wave-induced currents towards their original position, such as in the Pointe de lEspiguette (Sabatier, 2001), thus forming a cycle. However, the vegetation covering much of the emerged area does not really allow massive aeolian transport, which remains an order of magnitude smaller than maritime transport, and can, thus, be neglected when perfoming gross studies.

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Despite being its main erosive agent, waves are relatively non-energetic in the Rhone delta. The fact that the stronger winds blow in a seaward direction means that swell seas from the Gulf of Lyon are not usual, and low-energy, short-period, low-height, steep waves predominate. They tend to come from the southern quadrants: good weather waves from the SW are present 40% of the time. Stronger storm waves, on the contrary, issue more often from the SE-SSE or S-SSW sectors. Wave heights in the area average 0.75 m with periods slightly above the 5 s and steepness of 0.02 (CETMEF, 2005). Two sediment types are found in the delta: sands, the source for littoral budget (found along the beaches and in the Petite Camargue); and more cohesive silt (in the interior: lakes and flood plains covering much of the delta), which does not partake in the beaches budget. The sandy beaches are separated from the brackish lagoons by low and discontinuous dune formations. On the sea side, several bars parallel to the shore tend to be present, making the beaches fairly dissipative.

Figure 3.4:Morphologycal description of a typical profile of a beach in the delta (Sabatier, 2001).

West of the Grand Rhone-fed Gracieuse spit (Henrot, 1996) the delta can be divided into three main zones, with similar characteristics:

Grand Rhone to Beaduc spit: located at the ancient Bras the Fer system, most of the coast has been eroding for three centuries, allowing the formation and feeding of the Beaduc spit. Despite the rapid erosion (8m/year, Blanc 1979), a submerged shoal remains in front of the shore, modifying the wave behaviour.

Beaduc Gulf to Petit Rhone: correponds to the old Saint-Ferrol system, not even manifested in shoals due to millenia of erosion.The most sheltered area of the Delta, is on its east end.

Petit Rhone to Espiguette spit: pine-covered sandy stripes separated by lagoons become a source of sediment for the Espiguette spit during storms.

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Figure 3.5: Morpho-sedimentary heritage in the Rhone Delta (Vella, 1999).

In general, it can be said that most of the delta is receding, with only the three spits showing an accresive tendency: that of Gracieuse thanks to the Grand Rhone sediment input, Beauducs thanks to the erosion along Faraman and the western end of Beauducs Gulf and lEspiguette, which is fed by the erosion on the Petite Camargue. The western shore of the Grand Rhone rivemouth experiences some accretion as well. This is shown in the figure below, where the coast has been divided into sediment cells (Sabatier and Suanez, 2003):

Figure 3.6: Distribution of the alongshore sediment cells. Those in red are eroding; those in green accreting (Sabatier and Suanez, 2003).

3.3- Human use and settlement The Rhone delta has been inhabited for centuries, and, although the population density remains low (5 inhabitants/km2), it bears a huge human presence in the form of diverse land uses, ranging from traditional activities long established in the area (agriculture, fisheries) to newer, more powerful ones such as port activities.

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Fisheries and agriculture: Farming is especially significant in the northern and western (Petite Camargue) parts of the delta, with rice being its most important output and accounting for 64% of the agricultural area. The port of Saintes-Maries-la-Mer and Port-Camargue being focused on leisure, some dozens of families collect seashells along the coast (Provansal, 2003).

Habitation: Saintes-Maries-de-la-Mer boasts a population of around 2,500 permanent inhabitants, and Port-Saint-Louis-du-Rhne 8,200. Further inland, Aigues-Mortes, St. Gilles and Arles are home to several thousands more inhabitants.

Tourism and recreation: Sea-side and nautical activities are the main reasons that draw tourists to the Delta, numbering more than 60,000 per year. The sandy coasts consubstantial to the delta are its main asset and have been extensively equipped with coastal defence structures. Unregulated camping and hut construction exert further pressure on the dune area. Besides this traditional tourist activities, a new green tourism is emerging thanks to a remarkable heritage (traditional horse and bull breeding), and especially, the ecological richness of this wetland (flora and fauna, particularly birds), endowed by the Regional Natural Reserve, the National Reserve of the Camargue and the Coastal and Lake Environment Conservancy.

Salt Production: The Compagnie des Salins du Midi et Salins de lEst exploits two production sites (Petite Camargue, 10,000 Ha; Salin du Giraud, 12,000 Ha) with a total yearly output of 1.5 Mt. Protection schemes prevent the retreating coasts bordering them from giving way to the sea.

Commercial harbour activities: They are located at the eastern end of the Delta, where the Autonomous Port of Marseilles has ore and tanker terminals; petrol, gas and chemical docks; besides housing industrial activity per se.

Figure 3.7: The beach at Beauducs gulf as seen from the dunes.

Figure 3.8: The protected shore at Saintes-Maries-de-la-Mer.

The current receding tendency of the coastline constitutes a threat to all the above mentioned activities. Eighty-five percent of the coast is already equipped with defence equipment, which has so far prevented the village of Saintes-Marie-de-la-Mer from disappearing. However, the current trend of water level increase might turn it into an island separated from the mainland in a near future. Harder actions might protect the salt pans and industrial activities, but would depreciate the beaches and natural environment that entice the tourists (to come).

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Figure 3.9: Equipments at Saintes-Maries-de-la-Mer and year they were built (Sabatier, 2001).

Figure 3.10: Equipments at Faraman and year they were built (Sabatier, 2001).

3.4- Available data The data available consist on a bathymetry of the area around the Beaduc spit and a wave record from a buoy in front of it, and some average grain size measurements.

3.4.1- Bathymetry The bathymetry covers an area of 47.5 x 35.5 km (from east to west and south to north respectively), encompassing 59.5 km of coastline, which marks its northern limit. The Beaduc spit is located a little bit to the northeast of its centre, with still more than 10 km of the littoral by the Grau de Veran to its west covered. To the east, it stretches well into the Petite Camargue, after having mapped the whole Beaduc Gulf, and surpassed the fixed point of Saintes-Maries-de-la-Mer and the Petit Rhone river mouth. These data come in the form of contour lines, which means that already some resolution is lost by converting the original measured points into isolines. Given the context of this work, it would have been more convenient to use the original data, since the goal is not to map the sea bottom but to introduce its features into a computer program. The depth range goes from the 0 m isoline (which is taken as the coastline) to -90 m at the sides of the southern end. The bathymetrical survey was perfomed 30 years ago, and therefore it is old and prone to mismatches with the current state of the bottom. X and Y are given in the French Lambert III South coordinates.

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ORIGINAL BATHYMETRICAL POINTS

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Figure 3.11: Available bathymetry data.

3.4.2- Wave record A one-year wave record from a buoy located in front of the Beaduc spit is available. The depth at the buoy location was -15 m. The phenomena recorded involves hourly measurements of the Hmo (spectral estimate of the significant wave height, Hs, in metres), Tp (peak wave period, in seconds), (wave direction according to the nautical convention, in degrees) and directional dispersion. The series has some gaps, which, in total, amounts to less than 2 days, but on the other hand, it stretches slightly beyond the 365 day mark. In total, 8,996 waves. Theyre grouped by direction, period and frequency in appendix B. In order to simplify the notation, H will be used throughout this paper instead of Hs or Hmo, and T instead of Tp,. When subindexes do appear further on, theyll refer to other concepts, pertinently explained.

3.4.3- Grain size The grain size at the breaker zone ranges from 0.16 to 0.22 mm. This value is lower further into the sea (attaining 0.12 mm) whereas it grows towards the shore, surpassing the average values when already in the dune . All in all, a 0.2 mm is a good general estimation for the grain size (Sabatier, 2001). However, it is not transversal grain size variation that is important in order to calculate the longitudinal transport, but the longitudinal one. Unfortunately, no such data are available covering the whole study area. On its eastern side, and beginning from the

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south-eastern end of the Beaduc Gulf, samples were taken every kilometre all the way to the mouth of the Grand Rhone, covering 29 kilometres of coast in a study realised by Masselink (1992). Of these, only 23 km are inside the current study area, thus they encompass slightly more than the eastern third of the studied stretch of coast.

Figure 3.12: Locations where grain size data are available (Masselink, 1992).

In general the values are higher than the 0.20 mm taken as an average. There is however a tendency towards a reduction of the grain size from west to east, in such a manner that on the last (eastern) points of Masselinks study, the grain size coincides with the average considered above if not slightly smaller. The precise values of the data for the aforementioned stretch are provided in appendix C.

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4- RESULTS Once the background has been outlined, both regarding the study area and the program to be used, the different steps and partial results that will lead to the sediment transport along the coast can be given. These will consist of wave climates at different locations of the study area, each of which will constitute a section of this chapter. First of all, in section 4.1, the wave climate at the buoy will be presented. Furthermore, the basic assumptions whereupon this work is funded will be introduced and justified. These will lead to a trimming of the original wave data which will conclude with the establishment of a reference buoy wave climate. In section 4.2, the deep water wave climate will be established. A coarse backward propagation will provide a first approximation to its values, after which a cycle of forward propagations using SWAN will begin until the values it is fed with yield results similar enough to the reference buoy climate. Section 4.3 will present the wave climate nearshore, the result of the SWAN propagation previously described. A new forward propagation will bring the wave data from nearshore to the breaker zone, filling section 4.4. Finally, on section 4.5 CERC and Kamphuis sediment transport formulae will be applied, so that patterns and erosion/accretion zones can be found along the coast. The chapter will conclude on section 4.6 with a comparison between the results obtained at 4.5 and a look into its implications.

4.1- Buoy wave climate and basic assumptions

4.1.1- Assumptions In this fourth chapter, the different steps and methods used to determine the wave climate at the breaker zone are presented. As stated in the introduction, this goal is pursued using exclusively a wave record from one intermediate-water location in the study area and bottom/coast data, both regarding its morphology and grain size. Such an approach is clearly a simplification, done under the following assumptions:

The waves coming from the north are negligible compared to those from the south.

The waves coming from the south are essentially swell. The tides and the currents are negligible. The period is constant for each wave. The one year wave record at the buoy validly represents the wave climate at that

point.

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Given adequate boundary conditions, the propagation methods can find wave conditions at any point of the study area.

These assumptions allow to simplify the problem in the following ways:

The wind is not taken into account. Both the waves it generates from the north and the effect it may have in the waves coming from the south while running across the study area are ignored.

Only waves comprised between 135.1 and 254.9 degrees are taken into account (of the original wave record).

Tides and currents are not taken into account. The offshore wave climate is the main boundary condition to be determined.

The first two simplifications are derived from the fact that wind doesnt play a significant role in the problem: thus, the waves from the north, which are strictly wind-generated, can be skipped (not only are they very small having had such a short fetch run, they also tend to leave the study area), and those from the south can be modelled as swell that propagates from deep sea into the study area. Thus, the last points of both lists are the basis for finding the wave climate at different locations along the breaker zone: a propagation of the offshore climate will suffice, in the absence of any other factors seriously conditioning the waves evolution. Finally, a last assumption that is made but has not been explicitly stated yet, is the fact that the human equipment dotting the shores in the study area is ignored. At least, it is not taken into account directly (that is, as a modifier of the waves, or, later on, as a barrier to the sand transport), although, the protuberance in Saintes-Maries-de-la-Mer does appear in the bathymetry, and it is therefore an indirect acknowledgement of the presence and role of the breakwaters and walls in the area.

4.1.2- Reference wave climate at buoy The original wave climate record as registered in the buoy comprises 8996 wave data triads (H, T, ). Its average wave height is 0.79 m (ranging from 0.08 to 4.56), the average period is 5.36 s (ranging from 1.75 to 12.50 s) and the predominant directions are WSW, SSW, S and SSE, as can be seen in the figure below. In order not to have to deal with each one of these waves, these have been grouped into events, each characterised by a wave height (in a half meter gap), a period (with a one second spacing) and a direction (10 sectors). Thus, waves with periods ranging from 7.50 to 8.49 s were tagged as 8 s, and analogously, multiples of ten were taken for the directions. As for the wave height, an average of the maximum and minimum values entering the event was deemed not good enough because it would diminish the energy of the wave climate, and ultimately reduce the sediment transport calculated. Since many of the transport formulas are dependent on the square of the wave height (or higher), the reference value (Href) chosen for an event was that that would have the same energy as the whole set wave heights ranging from the minimum (Hmin) to the maximum (Hmax) values allowed to enter that group:

2

2min

2max HHH ref

= (eq. 4.1)

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Appendix B bears the classification of waves into events. In total, the waves were sorted into 443 events.

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Figure 4.1: Original wave rose at buoy.

The original wave climate record as registered in the buoy included some data which were physically impossible (e.g. H=0.5 and T=2s; H/L>>1/7), probably the result of fumbling with the buoy or the passage of boats nearby. Furthermore, these awkward values stood alone in a sea of perfectly plausible values both preceding and following them. They were removed straight away. More important has been the trimming due to the previous suppositions, which has meant reducing the number of waves to 7826 and that of events to 335. As it can be seen in the figure below, though, the associated wave rose hasnt changed much from the original. This confirms the adequacy of the assumptions made.

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BUOY WAVE CLIMATE (trimmed)

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Figure 4.2: Reference (trimmed) wave rose at buoy.

4.2- Deep water wave climate Determining the offshore wave climate is of paramount importance since its events will constitute the main boundary conditions to be fed into the propagation methods when calculating the wave climate at other locations. A backward propagation from the buoy, if perfect, would suffice, but this is plainly impossible because it is precisely the boundary conditions for SWAN (the most precise propagation method for deep and intermediate waters available) what is sought here, and thus SWAN itself cannot be used. The adopted procedure consists on performing a coarse backward propagation to have departure values from where an iterative method will be launched in the quest for appropriate offshore boundary conditions: an event (that has been propagated backwards) is fed into SWAN and its output at the buoy location is compared with the reference data record. If their difference is within a given range, the deep water boundary condition is accepted as correct, and the output values nearshore at intermediate depth can be adopted and the propagation can proceed. If not, the offshore values for the event are modified according to the deviation between their output at the buoy and the record and newly fed into SWAN. This cycle continues until the forward propagation performed by SWAN yields values within tolerance at the buoy output location, as seen in the figure below:

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Offshore dataBoundary cond:H , ,T

Amendment

Beyondtolerance

Comparisionwith reference

Calculatedbuoy data:

T,c H b , c b

o o

toleranceWithin

Lateral runs1-D SWAN

Lateralboundaryconditions

2-D SWANGeneral run

buoy data:Trimmed

Backwardpropagattion

refref T,,H

buoy data

Nearshoreconditions

nn T,,H

Figure 4.3: Forward propagation cycle. In 4.2.1 the backward propagation is explained. Sections 4.2.2 and 4.2.3 deal with the preparation and parameters choice of SWAN before the forward propagation, which occupies section 4.3.3 along with the tolerance range chosen.

4.2.1- Backward propagation A first estimate of the (tentative) offshore boundary conditions can be calculated performing a backward propagation of the (trimmed) wave climate available at the buoy. Only reflection and shoaling are taken into account, in a straight coast of parallel bottom isolines. Thus, for each event, the following formula will give an approximation for the deep water wave height:

sr

refo KK

HH

= (eq. 4.2)

Where: Ho : Offshore wave height Href : Reference wave height at the buoy

refg

ogr C

CK

,

,= : Refraction coefficient

ref

osK

coscos

= : Shoaling coefficient

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The direction is found with the following formula:

)sinarcsin( refref

oo L

L = (eq. 4.3)

With: o:Angle between coast and wave front offshore ref:Angle between coast and wave front at buoy

2

2gTLo = : Offshore wave length

ghTLref = : Wave length at buoy Since this is parallel coast is not the real situation (both at the buoy and offshore) a choice has to be made between idealising the coast as following an E-W direction (general trend for the study area except for the spit) or a NW-SE direction, which corresponds to the local orientation of the isolines at the buoy (deviated 37 from the horizontal). After some test runs with representative values, the first option was retained, although the differences amongst the two werent usually great, and in some occasions where the E-W didnt yield results the NW-SE were used instead. It has to be considered that those values would merely be used as an educated initial guess anyway. Analogously, the depth offshore is taken as 90 m, which is not strictly true but is deep enough so as to have no effect on the waves anyway. Details regarding this backward propagation and each of the events (tentative) values can be found in appendix D.

4.2.2- SWAN grids Having the same rectangular boundaries for bottom and computational grids was a general guideline adopted for the sake of simplicity. Therefore, many of the considerations presented below regarding the bottom grid also apply to the computational grid, with which it has been deliberately intertwined. Bottom grid Several options exist in order to convert the given bathymetrical points into a bottom grid that can be inputted into SWAN. These stem from the fact that the bathymetrical points available do not conform to a rectangular grid (and therefore, either some borders need to be trimmed or some points have to be sensibly added) and the shadowing effect that the lack of input in lateral boundaries may have on the study area. This last factor allows for an initial classification of the available options depending on whether this shadow is solved via extra computations or extra distance:

Option I: Small bottom grid covering only the study zone of interest. Extra points are limited to those strictly necessary to achieve a rectangular grid from the points available. Lateral boundary conditions will have to be found and input in order not to have energy shadows. Energy enters the study area from three different borders.

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Option II: Massive lateral extension of the grid both east and west. A bathymetry large enough so as to overcome the shielding effect due to the lack of input on the lateral boundaries on the study zone. Since wave directions might be nearly parallel to the coast, the bottom grid has to be more than six times (305 km) wider than the study zone.

Option III: Moderate lateral extension of the grid both east and west, shorter than that of II but feeding the lateral borders with the same boundary conditions as those applied at the south. This is obviously physically impossible (e.g. having a 2 m wave height constantly all along a cross-shore profile), but precisely because of this impossibility SWAN disperses the excess energy quickly when moving away from the borders until only physically sensible values exist in the study zone. Overall this last option is actually a small variation of option II, above which it bears the recommendation of the technicians who designed the programme as a way of taking into account waves energy that might escape II.

The advantage of the last two options is that they demand no extra calculations per run, and each of Swans runs can be done at once. Option I, on the contrary, bears the need, before the main SWAN calculation is performed, of lateral 1-D runs in both the eastern and western boundaries so that their state can be determined and used as input. Its advantages, though, are more fidelity to the known data and (when referring to the computational grid) a denser and therefore more precise distribution of the points of computation. In options II and III the disadvantage is obviously that the prolongation of the bathymetry to its sides (being unknown) has to be guessed. The most simple and at the same time most logical way of doing this is to translate the last bathymetrical profiles available (that is, those of the lateral boundaries of the study zone) all over the new zones. In theory, this should yield exactly the same result as option A, since this is what SWAN assumes when running 1-D lateral runs. Propagations performed on the same study area using option III have been proven to yield satisfying results (Boada, 2004). In this study the chosen option was the first one. This is for the sake of fidelity to the physical reality of the involved area and precision when computing. However, despite not needing a major extension, some minor adjustments are still needed in order to have the points ready for inputting. Not all the contour lines end at the same longitude. In order to have a perfectly rectangular grid, these have to be either prolonged manually or trimmed up to a longitude were data is available for all. Since the area is sandy and with no brusque variations, the latter option has been chosen under the assumption that no gross error would be made if all the contour lines were prolonged up to the latitudes of the furthest reaching point amongst them. Extra emerged points have also been added (again, to achieve a rectangular shape).

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Original points Newly added points Figure 4.4: Points used to define the bathymetry: original (black) and added (red). From these data, SWAN generates a terrain model triangulating:

Figure 4.5: Bathymetrical image generated by SWAN (inverted scale: red means deep).

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Computational grid SWAN will perform the computations over a general grid covering exactly the same area as the bathymetrical one and three smaller nested grids nearshore, where more precision is sought. The maximal number of nodes SWAN can operate with (under the standard configuration used) is 20,500. These have to be distributed in such a way that both the longitudinal (E-W) and transversal direction (N-S) have enough resolution. Along regular straight sandy coasts, variation on the waves parameters is sharper in a cross-shore direction than along-shore, thus a nodes need to be closer longitudinally than transversally. This is the case of our study area except for the zone around the spit, where the coast doesnt follow an E-W direction. It is because of this, and again for the sake of simplicity, that squared computational cells have been chosen for the general computational grid. In total, 164 stretches in the x direction and 123 in the y direction, each with a length of x = y = 291 m, totalising 20,460 nodes. Grid xmin xmax ymin ymax x nodes y nodes Step x Step y General 750,100 797,824 96,000 131,793 165 124 291 291 Nested 1 750,100 781,841 127,000 131,900 209 99 152,6 50 Nested 2 777,500 783,000 117,750 127,000 111 186 50 50 Nested 3 783,000 797,824 114,500 119,500 205 101 72,7 50 Table 4.1: Computational grids used in SWAN. In the nested grids, higher resolution is sought, reducing the area between nodes in a transversal direction to 50 m. Since two of the grids (those at the E and W) encompass essentially E-W stretches of coast, their resolution alongshore has been reduced. Only around the spit were the cells 50 x 50 m.

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Figure 4.6: Computational grids: general (red) and nested (brown, green and blue). Shoreline in black, nearshore output in grey and buoy in purple. Output The output is given at 78 points. Of these, one corresponds to the buoys location, so that the offshore climate can be calibrated with the feed-up mechanism by comparing the output with the reference climate. The other 77 points follow the -5 m depth isoline near the shore. This depth has been chosen as that which is as close as possible to the coast before bottom induced breaking has begun. The reason for doing this is that Swans not precise enough in the breaker zone, so that its use at determining each waves breaking position as well as its height and direction is not recommended, and will be done otherwise in section 4.4. In the case of many waves, this -5 m value is extremely conservative (the breaker depth being effectively located around 1 m depth), but higher waves do exist and, although not abundant, they account for most of the sediment transport, which makes the precise determination of their breaker depth crucial. All the requested output points lay on the area covered by the nested grids. Thus, the output given by SWAN will be the result of two computations: a general one covering the whole study area and a local one at the specific part of the coast where the given output point belongs. No regular output grid has been used, since the wave characteristics at other points of the study area are not needed.

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Figure 4.7:Shoreline, buoy and numbered output locations both nearshore (black) and breaking (blue).

4.2.3- Other parameters Besides the wave conditions at the boundaries of the study area, other parameters need to be specified in the model:

Directional spreading: An average amongst the directional spreading of all the waves composing an event has been made, and fed into SWAN as part of the boundary conditions. However, its output has not been controlled (and consequently tuned and refed).

Physical parameters: Most of these parameters have been kept at the default values. This means that the gravity was taken as 9.81 m2/s, the water density 1025 kg/m3, and the direction of north referring to the x-axis as 90 (as stated, all grids are rectangular, with the horizontal ridges in a E-W direction and the verticals N-S). No wind is taken into account: its input was a speed of 0 m/s. All of the processes were activated (3rd generation model, bottom friction, depth induced breaking and non-linear triad interactions) but the quadruplets are de-activated (due to lack of wind).

Numerical parameters: Again, for most of them, the default values have been adopted, (regarded as balanced when trying to limit time consumption while at the same time remaining accurate). The diffusion of the spectral space is 0.5 and the frequency space too. The percentage of wet grid points is 98% and the maximum number of iterations is 15.

4.2.4- Forward propagation After the first estimates of the offshore boundary conditions have been calculated, these can be input into SWAN and the forward propagation cycles can begin. However, since a small grid has been chosen, lateral boundary conditions E and W are needed as well (the N boundary needs no input because it is fully occupied by land). Since the values

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along these borders are not constant and equal to the offshore ones, they need to be calculated before the global, to which theyll be fed as boundary conditions. SWAN allows to run 1-D runs (that is, not across a surface but merely along a line). Only a bottom profile must be introduced and, in the absence of wind or currents, a boundary condition at one of its ends. The calculation will be done as if an infinitely wide coast with such a profile were present. Although the output can be given anywhere along the line, the 2-D run to be done afterwards limits it: only 11 points can be used as input along a border. Between them, SWAN interpolates linearly the values along the line. Therefore, in the 1-D run, eleven points of output were chosen. A quadratic distribution of points was chosen instead of a regular one because it is nearshore that more detail is needed.

-100

-80

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-40

-20

0

94000 99000 104000 109000 114000 119000 124000 129000 134000

Figure 4.8: 1-D calculation output points at the E boundary. Note that the quadratic distribution of output points is only applied on the wet part of the profile, not in the emerged section.

-100

-80

-60

-40

-20

0

94000 99000 104000 109000 114000 119000 124000 129000 134000

Figure 4.9: 1-D calculation output points at the W boundary. Each event needs, in order to undergo the SWAN propagation, three SWAN runs: two lateral ones and a global one (examples of their input can be found in appendix F). All three boundary conditions available, the global 2-D SWAN run is performed, and its output at the buoy is compared with the reference data. In order to decide if they are similar enough, maximum deviations between output and record data have to be established:

5 cm for the wave height 3 degrees for the direction

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A cycle involving lateral runs, general run, comparison with data record and correction is triggered for each event, and stops when the differences between calculated and reference values are small enough. If a given event has just one of these parameters within tolerance, yet another iteration is performed, and the results accepted if both fall within maximum deviation. In order to correct the offshore values for the next iteration, they are modified with the proportion that the output at the buoy deviated from the reference. For example, in the case of wave height, the input for a given iteration i would be:

1,1,,

=

icb

refioio H

HHH (eq. 4.4)

Where: Ho,i : Input offshore wave height for iteration i Ho,i-1 : Input offshore wave height for iteration i-1 Href : Reference wave height at the buoy Hcb,i-1 : Calculated wave height at buoy for iteration i-1 This goes on until both wave height and direction of an event are within tolerance, or it is decided that it is not possible to establish a satisfactory offshore conditions for an event (in no case more than 5 iterations were done). If an event cannot be correctly calibrated, it is not ignored. The number of waves it includes is transferred to the most similar event possible. That is, amongst the events classed in its same direction and wave height, the one with the nearest period (usually this means increasing the period). If this is not possible due to the absence of events with the same height, then the event is downgraded to the next wave height keeping the previous direction and period. The latter will be changed until a coincidence is found. This politic of trying to maintain the wave height rather than the period was chosen with the aim of not loosing much energy in the transfer, so that the effects of waves to the coast are as similar as possible. Finally, after all events have been propagated or moved to similar ones, both the wave climate offshore and at the buoy have been found. The waves offshore were found to be globally much higher than those at the buoy: their average was 1,40 m, ranging from 0.29 to 5,87 m. Their directions were, too, more scattered, and they comprised values from 100 to 313. (please see appendix E for the wave the definitive values of the offshore boundary conditions).

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BOUNDARY CONDITIONS

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Figure 4.10: Wave rose of the offshore wave climate (boundary conditions).

CALCULATED BUOY

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Figure 4.11: Wave rose of the calculated climate at buoy.

The moving of events that could not be calculated has meant reducing its number from 335 to 236 representing 6342 waves instead of the 7826 taken as reference. However, the total number of waves has not been lost, because when an event was moved its waves were added to the receiver.

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total trimmed reference offshore / buoy (good) offshore / buoy (moved) teta # waves # events # waves # events # waves # events # waves # events 0 9 4 - - - - - - 90 1 1 - - - - - - 100 3 3 - - - - - - 110 34 9 - - - - - - 120 86 13 - - - - - - 130 181 16 - - - - - - 140 375 22 375 22 230 8 145 14 150 483 27 483 27 244 8 239 19 160 590 28 590 28 429 15 161 13 170 584 39 584 39 538 28 46 11 180 722 40 722 40 700 36 22 4 190 964 40 964 40 951 35 13 5 200 853 46 853 46 847 43 6 3 210 257 28 257 28 251 26 6 2 220 175 16 175 16 155 11 20 5 230 439 15 439 15 381 10 58 5 240 1421 18 1421 18 1169 11 252 7 250 963 16 963 16 605 6 358 10 260 308 14 - - - - - - 270 91 11 - - - - - - 280 34 4 - - - - - - 290 22 6 - - - - - - 300 67 7 - - - - - - 310 150 8 - - - - - - 320 129 6 - - - - - - 330 47 5 - - - - - - 340 8 1 - - - - - -

total 8996 443 7826 335 6500 237 1326 98 Table 4.2: Number of waves and events: initially, after the trimming (and therefore, those taken into account), correctly calibrated and moved from other events into those correctly calibrated.

4.3- Nearshore wave climate The calculated wave climate at the buoy meeting the tolerance requirements, SWAN output (for the same calculation; that is, with the same boundary conditions) can be accepted at nearshore locations. These were chosen around the -5m contour line, a depth deemed big enough so as to safely assume that no depth-induced breaking has occurred yet, but, at the same time, as close as possible to the breaker zone (so that SWAN capabilities optimized). In total, 236 events were propagated for each of the 77 nearshore points, yielding an output of 18,172 wave conditions. Each location has its own wave climate, summarised in appendix G with wave roses at some representative locations (denser near the spit). The sheltering effect of the spit at the eastern end of Beaducs gulf (points 45 through 55) is clearly seen in the shape of the wave roses, where no waves from east are present. This is the case of point 54, as oppposed to transitional point 47 and points 2 and 66, already quite free from such an influence.

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NEARSHORE 2

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Figure 4.12: Wave rose at point 2 nearshore.

NEARSHORE 54

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Figure 4.14: Wave rose at point 54 nearshore.

NEARSHORE 47

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Figure 4.13: Wave rose at point 47 nearshore.

NEARSHORE 66

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Figure 4.15: Wave rose at point 47 nearshore

The sheltering effect also affects the wave height, as seen in the graphs below corresponding to waves from the south-east and the south-west respectively:

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Figure 4.16: Wave height in the study area according to SWAN for offshore conditions H = 1.40 m; T = 5

s; = 115.2. Buoy reference conditions: H = 0.76 m; T = 5 s; = 140.

Figure 4.17: Wave height in the study area according to SWAN for offshore conditions H =0.86 m; T = 5

s; = 232.9.Buoy reference conditions H = 0.76 m; T = 5 s; = 230. At the same time, these results also illustrate the limitation of not taking the wind into account: in both situations, the reference wave height at the buoy was 0.76 m, but when

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the waves had a south-eastern origin, the offshore height value was much bigger. Since no energy is entering the system, the waves keep loosing energy when moving through the study area, and the only way to compensate for this loss, especially when having an adverse origin (that is, closer to the east), is to increase the offshore height boundary condition. The same can be said about the direction: the first waves (e.g., from the south-east) undergo much more refraction than that those originally from the west. The position of the buoy, already under the underwater bulb of the spit, makes it more prone to this effect in oriental waves: the bathymetrical isolines there have an orientation of 137instead of the 90 common along the coast outside the spit.

4.4- Breaking zone wave climate Sediment transport formulas require the wave condition at the breaker zone. In order to get there from SWANs nearshore output locations, yet another forward propagation is needed. Just like in the backward propagation, a simple method taking refraction and shoaling into account will be used. In this case, however, the assumption of a straight coast with parallel isolines is fully acceptable in a local scale, from a nearshore location to its corresponding breaker point. The problem is though that no particular breaker point exists for each nearshore location, because each event, with a different height, direction and period, will brake at a different place. Since 236 events describe the wave climate at each location, a cloud of 236 breaker points are associated with one SWANs nearshore output location. In order to find them and the wave height and direction when breaking, the formulae seen in 4.2.4 are applied with the particularity that the breaker depth is unknown. This will be determined with the parameter:

b

b

hH

= (eq. 4.5)

Where: Hb : Wave height at breaking point hb : Water depth at breaking point Both numerator and denominator are initially unknown. An iterative process beginning at hb = 0 and with an increase of 0.01 m will be launched, in order to determine the first wave height that fulfills the condition that < 0.68. The formulae used will be again:

rsnb KKHH = . (eq. 4.6) Where: Hb : Wave height nearshore Ks : Shoaling coefficient Kb : Refraction coefficient Twenty of the waves could not be propagated till its breaking point due to the fact that their front angle nearshore was bigger than 90, and therefore unable to fit the formulation. They corresponded to events with a low energy (H< 0.35 m) and a low

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period (T 5 s) around and on the lee of the spit. Given their small number and their small height they were ignored assuming that the effect on the sediment transport would be negligible.

Figure 4.18: Detail of the wave energy evolution of a wave initially coming from an offshore direction of T = 5 s; = 132,2(withT=5s,H=0.39m..Buoy reference conditions H = 0.29 m; T = 5 s; = 150). Around the spit, the waves loose energy and are refracted, although