ch_model_coastal.tex

Many different approaches have been used and refined for coastal waves. Looking at models that solve for a single-valued field $\zeta(x,y,t)$ (i.e. the waves are not allowed to overturn) they can be phase resolving (with the individual waves resolved), spectral with random phases (like the models described in Chapter \ref{ch_model}), coupled spectral and bispectral (to keep some nonlinear properties specific of shallow water). There are also many fluid dynamics models for multi-phase flow (air, water and sediment) that have been applied to wave breaking and sediment transport. 

The choice of model is often dictated by the complexity that can be afforded and the space and time scales that must be resolved: a few minutes of a 100 by 100 m region can be studied with much more detail than century-scale shoreline evolution under climate change scenarios, even if only for a small bay. One very important aspect to keep in mind is that the shore is often a region of intense breaking and high fluid accelerations and velocities: in this context the waves, currents and even bathymetry evolution are tightly coupled. For this reason, many efforts have been made to develop and validate coupled wave-flow-sediment models \citep[e.g.][]{Reniers&al.2004b}. The first questions to ask are : What space and time scales do I need to resolve? Do I need an explicit resolution of these scales or some parameterization? What are the associated processes that need to be well represented? When in doubt, it is tempting to go for the "fullest" possible model. However, some very simple parameterizations (e.g. one-line models of shoreline change) can often capture most of the dynamics we are interested in. In short, all models are good, but some are better than others for one's specific needs.

 This chapter will thus not cover all possible modeling strategies  but explore the limits and usefulness of the same spectral approach that is used for marine weather forecasting offshore, and illustrate a few other possibilities. The quality of coastal wave models solution depends on many factors, and is more complex than open ocean situations. This is reviewed by \cite{Roland&Ardhuin2014}. 
In general, the important aspects are, 
\begin{itemize}
\item \emph{the forcing}: waves are generated by the wind, and are strongly influenced by currents, varying water depth, and any obstacles (small islands, sea ice, icebergs ...). All 
these forcing fields should thus be defined as accurately as possible, starting with the winds, but not forgetting currents that are generally stronger in coastal regions. 

\item  \emph{parameterizations of physical processes}: many errors, in particular in coastal areas are due to deficiencies in source term parameterizations. 
In particular, waves in coastal regions can be often fetch-limited, and the growth of waves when wind is oblique relative to the shoreline can reveal 
errors in the magnitude of the wind generation and whitecapping source terms \citep{Ardhuin&al.2007}. Also, bottom friction is a dominant factor 
when the water depth is less than half the dominant wavelength, and bottom properties can have a very large impact the dissipation rate of wave energy 
\citep{Roland&Ardhuin2014,Monismith2007}.

\item \emph{Numerics}: Although some numerical methods are clearly worse than others, there is no perfect method that would be fast and accurate enough to 
deal with the very localized depth-induced breaking. For resolutions coarser than 100~m, explicit triangle-based grids using residual distribution (RD) schemes 
\citep{Csik&al.2002,Ricchiuto&al.2005} may well be the most efficient \citep{Roland&Ardhuin2014}. At higher resolution, the CFL constraint of explicit schemes 
makes them prohibitively expensive, and one may have to live with low order schemes and their diffusion. 
\end{itemize}

%La qualité d'ensemble du résultat dépend donc des choix, explicites ou implicites, dans ces trois domaines: forçage, paramétrages, méthodes numériques, ces dernières peuvent aussi inclure les méthodes d'assimilation.  Dans tous ces domaines, les choix les plus complexes ne sont pas forcément les meilleurs, et le choix optimal dépend beaucoup de la configuration à traiter (simulation à grande échelle, propagation côtière avec ou sans courants ...). 

%Toutefois, on n'a souvent pas trop de choix: tel code est imposé parce qu'il fait partie d'une chaîne de modélisation ou bien parce qu'il est plus facile à mettre en oeuvre, ou bien parce qu'il donne un résultat plus rapidement. Cela n'est pas forcément très grave. Par contre il convient d'avoir conscience de ce que cela implique en terme de qualité du résultat.

%Par ailleurs, les paramétrages sont généralement ajustés aux forçages et aux schémas numériques. Changer le forçage, par exemple utiliser les vents fournis par le Centre Européen (CEPMMT ou ECMWF en anglais)au lieu de ceux du service météorologique des Etats-Unis (le National Center for Environmental Prediction - NCEP - , qui dépend de la National Ocean and Atmosphere Administration - NOAA) peut avoir des effets assez inattendus. En effet les vents du NCEP sont en moyenne 0.5 m/s plus forts que ceux du CEPMMT, et  les statistiques des vents extrêmes sont elles aussi très différentes.

%Enfin, les modèles ne sont généralement bons que pour les paramètres sur lesquels ils ont été validés et dans les intervales de valeurs où ils ont été validés. Ainsi prévoir que la hauteur significative centenaire au large de la Bretagne est de 18~m sur la base d'un modèle qui n'a été validé que jusqu'à 14~m est une  réelle extrapolation. Les phénomènes extrêmes sont très mal compris et les paramétrages et les forçages dans ces cas peuvent très bien se révéler insuffisants. Toutes ces questions sont l'objet de ce chapitre.

\section{Forcing fields}
\subsection{Winds, currents and water depth}
Since it takes a long fetch to develop waves, the primary concern is generally on the offshore winds, as discussed in chapter  and coastal winds only have a moderate impact on the sea state. These issues are discussed in chapter \ref{ch_model}. Recent operational weather forecasts are now available at resolutions around 1~km and can be compared to SAR-derived winds at resolutions around 500~m, for example from Sentinel-1, or meteorological Doppler radar data \citep[e.g.][]{Ahsbahs&al.2020}. 

Getting good currents can be much more problematic, except possibly for tidal current. Still, one should be careful that tidal models that are very good for sea level, thanks to satellite data assimilation, may not be that good for currents. Some regions have coastal High-Frequency radars that can provide a good validation dataset for surface currents, but the spatial resolution can be marginal for resolving some sharp current gradients \citep{Ardhuin&al.2009}. With small-scale ( $< 50$~km) details of currents important for the waves propagating to the coast \citep{Marechal&Ardhuin2021}, some regions are particularly prone to large model errors given the lack of detailed measurements of surface currents. 

Finally, water depth is the sum of the varying water level and of the (also varying but usually more slowly) bathymetry. Recent efforts to obtain bathymetry from remote sensing using wave dispersion or changes in wet-dry boundary associated with tides will certainly lead to a better knowledge in some of the poorly charted regions of the world. One possible exception are the last few kilometers to the coast, in particular for exposed sandy shorelines where wave-driven sand transport leads to large changes in water depths on the scale of storms and seasons. 


\section{Numerics}
Restricting ourselves to spectral methods, one of the first developed methods is the use ray-tracing to transform global-scale forecasts (or offshore bouy measurements) into coastal wave predictions. Whereas the early versions used forward ray tracing and considered a single or a few offshore wave periods and directions, later implementations used eq. (\ref{eq:ray_model}) with backward ray-tracing to transform the full wave spectrum. This is still used in some regions where local wind generation can be neglected and where time-varying currents do not require a frequency recomputation of wave rays \citep{OReilly&Guza1993,OReilly&Guza1998,Crosby&al.2017}. However, this method is not generally applicable due to wind generation and dissipation source terms. 

A more general approach is thus to apply the wave action equation in coastal areas, including shallow water effects such as bottom friction, and computing the evolution of the wave spectrum across a discretized model grid. A straightforward implementation of the global model in a coastal setting leads to a generally high cost due to the fine resolution required to resolve sharp gradients near the coast. This cost is particularly high if the model uses an explicit numerical scheme, with a time step that is not dictated by the true time scale of evolution of the wave field (driven by the forcing, i.e. one hour or so when taking into account tidal currents), but by the stability of the numerical scheme: explicit schemes requires time steps of a few minutes (dt around 50 s) at most because the wave energy cannot jump over more than one grid cell (imagine dx=1000 m) in a time step and the speed for the longest wave periods is still around 20 m/s. 
A possible solution is to use implicit schemes with issues related to non-linearity or non-monotonicity or numerical diffusion, as discussed in chapter \ref{ch_model}. 

Specific issues in coastal areas are the large range of time scales of different processes: wave breaking in the surf zone is a very "stiff" source term. 
Also, in order to optimize the model cost, grids that allow higher resolution where gradients are large (generally near the coast) can save a lot on the number of grid points, and for this reason many coastal models use grids with variable resolution, such as triangle-based meshes. Further details are discussed in \cite{Roland&Ardhuin2014}.

 Numerical effects are most visible on the directional spread because numerical diffusion directly increases the spread, as shown in Fig. \ref{Aaron_shelf}. We note that a finer spatial resolution does not necessarily reduce the error, as noted by \cite{Ardhuin&Herbers2005}: to converge to the true solution one needs finer resolution in both space and directions.


%Godunov. Dans le modèle SWAN ces valeurs négatives sont
%redistribuées suivant les directions afin de garder un énergie
%positive partout. Malheureusement cette procédure introduit une
%diffusion dans l'espace des directions, ce qui limite l'intérêt du
%schéma d'ordre 2 (figure \ref{Aaron_shelf})

%Par ailleurs, la résolution de l'ensemble du système d'un bloc
%alors que les vitesses d'advection sont très diverses dans les
%différentes dimensions, peut produire des erreurs numériques
%importantes, et la convergence vers la solution exacte peut être
%très lente.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure
\begin{figure}
\centerline{\includegraphics[width=0.8\textwidth]{FIGS_CH_MODELCOAST/Roland_diffusion.pdf}}
%\vspace{3.64in}
\caption{Test case with a wave spectrum propagating obliquely over a broad and shallow continental shelf, with a mean direction starting at 60 degrees from shore normal: the directional spread should decrease as the water depth decreases and all wave components are refracted to the shore-normal direction. In numerical models, this refraction is not exact. Here the test involves three different models: one uses ray-tracing \citep{Ardhuin&Herbers2005} and the others are SWAN with 2 different advection schemes and two different spatial resolutions in the top panel, and Contour Residual Distribution schemes \citep{Csik&al.2002} of different orders in the Wind Wave Model of \cite{Roland2008}, using triangle-based grids, in the bottom panel. Schemes  N1 and N2 are implicit while  N and FCT are explicit. Reproduced from \cite{Roland2008}.} \label{Aaron_shelf}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure
Although directional spread is not the first variable that people care about, it can have a significant impact on radiation stresses and infragravity wave amplitudes, and thus surf zone dynamics. 

At the same time, the numerical diffusion of some numerical schemes may compensate for the lack of diffraction and correct part of the Garden Sprinkler Effect that was discussed in chapter \ref{ch_model} and that is also present in coastal models.  The trade-off between these issues and the cost of running the model (CPU time) have led some of us to prefer the use of the "N" scheme (that is equivalent to a first order scheme for triangle-based grids) instead of the higher order "Flux Corrected  Transport" (FCT) scheme \citep{Cancouet2008,Ardhuin&al.2009}.


%Pour la deuxième approche, le problème est l'erreur liée à la séparation des modes. Elle tend vers zéro quand le pas de temps tend vers zéro, mais en pratique elle est finie, et d'autant plus importante que le terme source est une fonction non-linéaire variant rapidement en fonction du spectre ou des dimensions d'espace.

%Afin de ne pas utiliser des pas de temps trop petits, l'ensemble des méthodes utilise en général des limiteurs: le changement du  spectre lors de l'intégration est artificiellement limité. Ces limiteurs peuvent avoir un effet très important sur la solution numérique \citep{Tolman2002b}. Dans le code SWAN, comme l'ensemble de l'équation est résolue en même temps le limiteur peut avoir pour effet de limiter la réfraction lorsque le nombre de Courant devient trop élevé, c'est en particulier le cas pour des vagues sur de forts gradients de courant, il faut donc y prendre garde.

%Enfin, en pratique l'intégration du spectre est généralement limitée à une fréquence diagnostique $f_d$ au delà de laquelle on  suppose que le spectre décroît de façon régulière, comme $f^{-5}$ par exemple. Cette astuce évite d'avoir à trop réduire le pas de temps pour résoudre les temps d'évolution très rapides des vagues les plus courtes. Malheureusement, la forme du spectre n'est pas aussi simple, et il vaut mieux de pas prendre $f_d$ trop bas. Dans le modèle WAM du centre Européen, $f_d$ est fixé à 2.5 fois la fréquence moyenne $f_{m,0,-1}$ de la partie du spectre ou le terme de génération $S_{in}$ est positif. Ce facteur 2.5 est trop faible si on s'intéresse aux propriétés telles que la pente moyenne de la surface.


\section{Parameterizations of physical processes}

\subsection{Bottom friction}
Many models still use an empirical "JONSWAP" parameterization, presented in chapter \ref{ch_bbl}, that has very little connection with what happens in the bottom boundary layer, beyond the obvious transformation of the surface elevation spectrum into a bottom velocity spectrum.  Indeed, bottom friction is only significant where bottom velocities are large enough. The reason for this default empirical choice is that other choices require the use of some information about bottom sediments. In practice we found that using a ripple parameterization and assuming a grain size of 0.2~mm (coarse sand) everywhere was generally acceptable. Obviously, if information on the bottom is available, it should be used: friction over rock or mud can indeed be many times stronger than over sand \citep{Elgar&Raubenheimer2008,Monismith2007}.

\subsection{Triad wave-wave interactions: sum interactions}
Spectrum evolution around the surf zone is severely affected by the near-resonant triad interactions which can produce a net transfer of energy from the peak of the spectrum to harmonics, but also a flow of energy from the harmonics back to the peak. An accurate description of this process requires some knowlege of the bispectrum that is not available in spectral phase-averaged models. 
Even if results are not perfect, different parameterizations have been developed to mimic some of the energy transfers in phase-averaged models \citep{Eldeberky&Battjes1995,BecqGirard&al.1999}.


Solving this issue is one of the main reasons why specific models such as the bi-spectral model of \citep{Herbers&Burton1997} or the time domain model SWASH \citep{Zijlema&al.2011} have been developed specifically to model waves in surf zones. The other important benefit of time-domain models is that they can also solve for the wave-driven currents and sediment transport. 

\subsection{Triad wave-wave interactions: difference interactions and infragravity wave generation} 
Since infragravity (IG) motions can be very energetic in the surf zone, leading to changes in water level of 1~m or more over a time scale around 100~s, it is impossible to ignore IG waves near the shoreline: indeed the primary limitation of wave heigth due to the depth-induced breaking introduce a strong coupling between the IG water levels and the wave heights. IG waves can also transport sediment in the cross-shore. Resolving IG-waves while not resolving the shorter waves is the strategy that is used in the X-Beach model \citep{Reniers&al.2004b,Roelvink&al.2009}.

\section{Examples of wave model results and validation}
Our first example is a regional model that was designed for coastal applications and has been used, among other things for marine energy studies \citep{Boudiere&al.2013}. The model mesh is shown in Fig. \ref{fig:NORGASUG}. Triangles near the coast have a resoltion around 300~m, rapidly increasing to a few kilometers when moving offshore. The 300 m at the shoreline are not enough to resolve cross-shore gradients in the surf zone, but is sufficient to represent many coastal effects including refraction caused by depth and tidal currents and sheltering by headlands and islands that create "wave shadows". 

For example Fig. \ref{fig:Hs_storm} shows wave heights simulated for the severe storm "Johanna" that hit France in March 2008. The offshore wave orientation relative to the shoreline explains the lower values of wave heights in coastal bays, or behind islands.
One can also see the clear effect of the paleo-river in the middle of the Channel which causes a sharp refraction and local minimum in the wave height, because these very large storm waves also have long periods that are refracted at depths larger than 100~m. 

 Such a regional model can use satellite altimetry for validation. Fig. \ref{fig:NORGASUG_ALTI} shows a statistical validation over a full year. Recent altimeter retracking and denoising techniques \citep{Passaro&al.2014,Quilfen&al.2018,Dodet&al.2020} allow a more detailed validation at higher resolution. 

Thanks to their continuous time series it is often easier to use in situ measurements, such as performed by moored buoys to analyze particular events. Fig. \ref{fig:NGUG_buoys} highlights the different results obtained with different parameterizations of bottom friction. However, the only location where models are significantly different is buoy 62067 located inshore of Ile d'Yeu, across a shallow rock oucrop which is represented by a high roughness when using the "SHOWEX" parameterization. We also note that both models are not very good at the Cherbourg buoy (62059) for these February 2011 swells. All three selected location are strongly influenced by tidal currents. Unlike altimeter data, buoys say nothing about the spatial variability of the wave field, and it is often difficult to know how far a particular buoy is representative of a larger area. 
 
%%%%%%%%%%%%% figure
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\textwidth]{FIGS_CH_MODELCOAST/NORGASUG_mesh.pdf}}
  \caption{Map showing the North Sea - Channel - Biscay  mesh used for our hindcasts and forecasts. 
Magenta and green circles show location of permanent and temporary buoys used for calibration 
or validation in addition to satellite altimeter data. Inset are zooms of four grid areas, showing typical alongshore resolutions, 
with color bars displaying the elevation relative to mean sea level, in meters. 
The full mesh contains 110,000 wet nodes. Reproduced from \cite{Roland&Ardhuin2014}.
  \label{fig:NORGASUG}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% figure
\begin{figure}[htb]
\centerline{\includegraphics[width=\textwidth]{FIGS_CH_MODELCOAST/Johanna.jpg}}
  \caption{Wave heights simulated around Brittany and in the Channel the March 10, 2008 at 15h00 UTC. 
  \label{fig:Hs_storm}}
\end{figure}
%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%
\begin{figure}
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use

  \includegraphics[width=\textwidth]{FIGS_CH_MODELCOAST/NGUG_ALTI_v2.pdf}

% figure caption is below the figure
\caption{Validation of the bay of Biscay model grid for year 2011 using all available altimeter data. 
(a) bias in centimeters, and (b) normalized root mean square error (NRMSE). The satellite data was taken from the 
calibrated Ifremer database \citep{Queffeulou&CroizeFillon2010}. The along-track time series 
at 1~Hz sampling was averaged over 0.5 degree along the track. These 'super-observations' (SO) were then binned with 
latitude and longitude, with an average number of 35 SOs for one year in each 0.5 by 0.5 degree bin. 
Results are only shown for bins with at least 4 SOs for the bias and 6 SOs for the NRMSE.  Reproduced from \cite{Roland&Ardhuin2014}.}
 \label{fig:NORGASUG_ALTI}       % Give a unique label
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use

  \includegraphics[width=\textwidth]{FIGS_CH_MODELCOAST/NGUG_timeseries_v2.pdf}

% figure caption is below the figure
\caption{Time series of observed and modelled significant wave height at several buoys using the SHOWEX (blue diamonds) 
or JONSWAP (red triangles) 
parameterizations for bottom friction, compared to hourly buoy measurements (solid line).  Reproduced from \cite{Roland&Ardhuin2014}.}
 \label{fig:NGUG_buoys}       % Give a unique label
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We close this chapter with one example of modeled and observed time series at different locations across a surf zone, here in Duck, North Carolina, with data collected during the DUCK'94 experiment \citep{Elgar&al.1997,Gallagher&al.1998}. In this case the model used is also WAVEWATCH III with a spatial resolution of 5~m, and a ridiculously small time step of 0.2~s because here we use an explicit numerical scheme in a regular spatial grid.  The two model runs have different breaking parameterizations: the "Model 1" corresponds to \cite{Filipot&Ardhuin2012} and "Model 2" is \cite{Battjes&Janssen1978}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use

  \includegraphics[width=\textwidth]{FIGS_CH_MODELCOAST/Duck_validation.pdf}

% figure caption is below the figure
\caption{Left: depth profile and location $D_1$ to $D_5$ where waves were measured. Right: time series of the rms wave height (defined as $H_{\mathrm{rms}}=H_s / \sqrt{2}$) at the 5 measurement locations.  Reproduced from \cite{Filipot&Ardhuin2012}.}
 \label{fig:NGUG_buoys}       % Give a unique label
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%% end of figure
%La qualité des résultats en zone côtière est beaucoup plus
%difficile à obtenir et à interpréter. Outre les incertitudes sur
%les spectres au large, il faut rajouter des effets liées au
%courants, souvent plus forts près des côtes, le frottement sur le
%fond, etc. Pour des côtes assez découpées, les résultats à la côte
%sont très sensible aux directions au large (directions moyennes et
%étalement angulaire) du fait des effets d'abri et de forte
%réfraction. Ainsi sur la figure \ref{rayon_valid}, on peut
%constater que la tempête du 5 mai 2004 ($H_s=5.3$~m à Iroise) est
%bien atténuée aux Blancs Sablons
%  ($H_s=1.2$~m),
%  et complètement absente à Bertheaume ($H_s=0.4$~m). Cet effet est normal car il s'agit d'une tempête de nord-ouest.
%  Au contraire, le 25 avril, les vagues d'ouest
%  sont très peu atténuées à Bertheaume, et beaucoup plus faibles aux Blancs
%  Sablons. Différents modèles numériques peuvent donner des
%  résultats sensiblement différents. Tout d'abord on peut remarque
%  qu'un calcul de propagation seule (modèle de réfraction-diffraction), sans prise en compte de la
%  génération locale par le vent, est déjà assez précis. Ensuite,
%  il est probable que l'effet de des courants, négligés dans ce
%  calcul soit assez important.

%%%%%%%%%%%%% figure
%\begin{figure}
%\centerline{\includegraphics[width=1.0\textwidth]{FIGS_CH_MODELCOAST/bretagne_map.pdf}}
%  \caption{Exemples de carte de hauteur des vagues en mer d'Iroise, calculée avec la version
%  non-structurée du code WAVEWATCH III (schéma CRD-N de Csik), et, pour un autre jour, avec
%  un modèle de propagation linéaire (refraction-diffraction). On retrouve les mêmes structures. On
%  remarque que les conditions aux limites au large sont supposées homogènes dans le modèl%  \label{Hsmap}
%\end{figure}
%%%%%%%%%%%%%% end of figure

%%%%%%%%%%%%% figure
%\begin{figure}
%\centerline{\includegraphics[width=1.0\textwidth]{FIGS_CH_MODELCOAST/Hs_valid_iroise.pdf}}
%  \caption{Evaluation de modèles de propagation en mer d'Iroise}{En haut, résultats pour la bouée Iroise
%  en termes de hauteur (Hs), et direction moyenne. En violet le calcul est fait avec la version
%  non-structurée du code WAVEWATCH III utilisant le paramétrage décrit ici, en bleu par tracé de rayons
%  forcés par le spectre calculé en un point au large d'Ouessant avec le même paramétrage, en orange avec
%  WAVEWATCH III non-structuré et le paramétrage de Bidlot et coll. (2005) et en rouge par tracé de rayons
%  à partir d'un spectre calculé avec le paramétrage de Bidlot et coll. (2005).
%  On constate un assez bon accord général. En bas, comparaisons entre observations aux bouées Bertheaume et Blancs Sablons (voir figure \ref{Hsmap}).}
%  \label{rayon_valid}
%\end{figure}
%%%%%%%%%%%%% end of figur