-
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)
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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.
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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.
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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.
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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
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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
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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:
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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
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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.
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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).
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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.
-
3. Study area and available data
13
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
-
3. Study area and available data
14
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.
-
3. Study area and available data
15
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.
-
3. Study area and available data
16
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.
-
3. Study area and available data
17
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).
-
3. Study area and available data
18
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.
-
3. Study area and available data
19
ORIGINAL BATHYMETRICAL POINTS
90000
95000
100000
105000
110000
115000
120000
125000
130000
135000
745000 750000 755000 760000 765000 770000 775000 780000 785000
790000 795000 800000
-90 m
-80 m
-70 m
-60 m
-50 m
-40 m
-30 m
-20 m
-15 m
-10 m
-5 m
-3 m
0 m
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
-
3. Study area and available data
20
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.
-
4. Results
21
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.
-
4. Results
22
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)
-
4. Results
23
Appendix B bears the classification of waves into events. In
total, the waves were sorted into 443 events.
BUOY WAVE CLIMATE (data record)
0
500
1000
1500
0
1020
30
40
50
60
70
80
90
100
110
120
130
140
150
160170
180
190200
210
220
230
240
250
260
270
280
290
300
310
320
330
340350
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.
-
4. Results
24
BUOY WAVE CLIMATE (trimmed)
0
500
1000
1500
0
1020
30
40
50
60
70
80
90
100
110
120
130
140
150
160170
180
190200
210
220
230
240
250
260
270
280
290
300
310
320
330
340350
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:
-
4. Results
25
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
-
4. Results
26
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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90000
95000
100000
105000
110000
115000
120000
125000
130000
135000
745000 750000 755000 760000 765000 770000 775000 780000 785000
790000 795000 800000
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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29
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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32
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
-60
-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
0
200
400
600
800
1000
1200
0
1020
30
40
50
60
70
80
90
100
110
120
130
140
150160
170
180
190200
210
220
230
240
250
260
270
280
290
300
310
320
330340
350
Figure 4.10: Wave rose of the offshore wave climate (boundary
conditions).
CALCULATED BUOY
0
500
1000
1500
0
1020
30
40
50
60
70
80
90
100
110
120
130
140
150160
170
180
190200
210
220
230
240
250
260
270
280
290
300
310
320
330340
350
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
0
500
1000
1500
2000
0
10 2030
40
50
60
70
80
90
100
110
120
130
140150
160170180
190200210
220
230
240
250
260
270
280
290
300
310
320330
340 350
Figure 4.12: Wave rose at point 2 nearshore.
NEARSHORE 54
0
1000
2000
3000
4000
0
10 2030
40
50
60
70
80
90
100
110
120
130
140150
160170
180
190200210
220
230
240
250
260
270
280
290
300
310
320330
340 350
Figure 4.14: Wave rose at point 54 nearshore.
NEARSHORE 47
0
1000
2000
3000
4000
0
1020
3040
50
60
70
80
90
100
110
120
130
140150
160170
180190
200210
220
230
240
250
260
270
280
290
300
310
320330
340350
Figure 4.13: Wave rose at point 47 nearshore.
NEARSHORE 66
0
500
1000
1500
2000
0
10 2030
40
50
60
70
80
90
100
110
120
130
140150
160170
180
190200210
220
230
240
250
260
270
280
290
300
310
320330
340 350
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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38
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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39
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