Quentin Enard

Christine Louberry

Charles Consel

Xavier Blanc

An Experimental Study of A Design-driven, Tool-based Development Approach

Design-driven software development approaches have long been praised for their many benefits on the development process and the resulting software system. This paper discusses a step towards assessing these benefits by proposing an experimental study that involves a design-driven, tool-based development approach. This study raises various questions including whether a design-driven approach improves software quality and whether the tool-based approach improves productivity. In examining these questions, we explore specific issues such as the approaches that should be involved in the comparison, the metrics that should be used, and the experimental framework that is required.

I. Introduction

Design-driven software development approaches have been extensively studied in the past decades, as shown by the literature (e.g., [START_REF] Taylor | Software Architecture: Foundations, Theory, and Practice[END_REF]- [START_REF] Bass | Software Architecture in Practice[END_REF]). From the requirements specification, the design of a software system is described at a high level; it guides the various stages of the development process. There exists a variety of designdriven approaches, ranging from Architecture Description Languages (ADLs) (e.g., [START_REF] Abi-Antoun | Modeling and Implementing Software Architecture with ACME and ArchJava[END_REF]) to UML-based developments (e.g., [START_REF] Sheng | ContextUML: A UMLbased Modeling Language for Model-driven Development of Context-aware Web Services[END_REF], [START_REF] Lodderstedt | SecureUML: A UML-based Modeling Language for Model-driven Security[END_REF]). These approaches rely on a notion of design language that provides syntax (whether or not textual) and abstractions allowing one to express the design of a software system. Designing a software system prior to programming it is said to have a number of key benefits, in particular, regarding software quality [START_REF] Bass | Software Architecture in Practice[END_REF]. Although there are few case studies analyzing software architectures [START_REF] Babar | Assessment of a framework for comparing software architecture analysis methods[END_REF], [START_REF] Wermelinger | Assessing Architectural Evolution: a Case Study[END_REF], there is no experimental study to demonstrate that a design-driven development approach improves software quality [START_REF] Falessi | Applying Empirical Software Engineering to Software Architecture: Challenges and Lessons Learned[END_REF].

In this paper, we propose to conduct an experimental study to measure this assumption in the context of design artifacts used in a productive way. This study uses an existing design-driven approach, namely DiaSuite [START_REF] Cassou | Leveraging Software Architectures to Guide and Verify the Development of Sense/Compute/Control Applications[END_REF], [START_REF] Cassou | Towards a Tool-based Development Methodology for Pervasive Computing Applications[END_REF]. DiaSuite is a tool-based approach ensuring the conformance between the design and the implementation of a software system. The goal of this study is to address the following research questions:

• Q1: Does a design-driven software development approach improve software quality?

• Q2: Does such an approach improve productivity when it is tool-based?

The reminder of this paper is structured as follows. Section II presents DiaSuite, a tool-based, design-driven software development approach. Section III describes our experimental study: [START_REF] Taylor | Software Architecture: Foundations, Theory, and Practice[END_REF] we propose a framework for our study and a data collection process; (2) we discuss how to analyze the collected data to address our research questions. Finally, Section IV summarizes our work.

II. DiaSuite

A key aspect of software design is the software architecture. It is aimed at decomposing an application into components, characterizing them with properties, and defining relations among them. This process is said to improve software quality in general but this benefit can widely vary depending on the effectiveness of the design in guiding the implementation. A design artifact can play a variety of roles in the development process, ranging from contemplative to productive. When a design language is used in a contemplative way, its syntax and semantics may remain informal, making challenging, if not impossible, the process of rigorously tracing the design in the implementation code. As such, contemplative approaches to design-driven software development are difficult to assess.

To promote design traceability, we need a design language with a syntax and semantics that are precisely defined, if not formally, at least in the form of a compiler. To match these requirements, our study relies on a suite of tools, named DiaSuite, that supports the programming of a software system from its design artifact written in a design language, named DiaSpec. This language is specific to a design paradigm in that it offers dedicated abstractions and semantics for designing applications. A compiler automatically generates programming support from a DiaSpec artifact, guiding the development process. The generated programming support ensures the conformance between the design and the implementation, enabling traceability.

A. A paradigm-based approach

DiaSpec is dedicated to the Sense/Compute/Control (SCC) paradigm and provides a conceptual framework for the design of applications. An SCC application is one that interacts with an environment, whether or not physical. The SCC paradigm targets a large spectrum of applications, from avionics and automotive to assisted living and healthcare. This generality prompted us to consider this approach as a paradigm-based approach rather than a domain-specific one.

As depicted in Figure 1, our SCC paradigm distinguishes three layers of components. The first layer consists of entities, whether hardware or software, that interact with an environment. For example, a temperature sensor and a heater actuator are entities of an application that regulates the temperature of a room. The second layer consists of contexts that process raw data produced by the entities and compute high-level information. The designer of the temperature regulation application can specify a context for the computation of the average temperature. The third layer consists of controllers that use context information to compute orders for the control of the actuating entities. An example of controller is one that uses the average temperature of a room to compute the heating level and the duration required to control a heater in a room.

To enable the designer to refine the description of a software system and provide more support for the implementation, DiaSpec makes explicit the data and the control flow of the components of an application in the form of interaction contracts [START_REF] Cassou | Leveraging Software Architectures to Guide and Verify the Development of Sense/Compute/Control Applications[END_REF]. For example, the designer can specify that an action on the heater may be triggered whenever the temperature sensor produces a new value.

A DiaSpec design is processed by DiaSuite that comprises a number of tools to support testing, programming, and deployment.

B. A tool-based approach

DiaSuite provides support for each development stage as depicted in Figure 2. A designer describes an application using the DiaSpec language (stage ➀). From this description, the DiaSpec compiler generates a Java programming framework (stage ➁) that guides the developers to implement the application (stage ➂) and ensures the conformance with the design. Testing support is also provided (stage ➃). The generated framework abstracts over communication technologies (e.g., Web Services, OSGi) that can be selected for the deployment (stage ➄).

From a design, the generated programming framework consists of one abstract class per DiaSpec declaration. Implementing a DiaSpec declaration amounts to extending the corresponding abstract class. The generated abstract classes make the programming framework both prescriptive and proscriptive: they provide high-level methods to interact with other building blocks while constraining developers to implement abstract methods that ensure the conformance with the design.

Presumably, a tool-based approach to design-driven development should improve the productivity of developers because of the generated programming support. However, the paradigm-specific nature of the DiaSuite approach may raise a new question.

• Q3: Does a paradigm-specific approach to designing applications have an impact on software quality?

III. Experimental Study

In this section, we present the experimental study of DiaSuite. We first propose a framework of this experimental study. Then we present a process to collect data from the experiment. Finally, we discuss how to analyze these data to address our research questions.

A. A framework for our experimental study

As a first step, we plan to compare DiaSuite to only one other approach. As DiaSuite is dedicated to the SCC paradigm and is tool-based, a key question is:

What makes an approach pertinent to be used for a point of comparison with DiaSuite? Another question concerns a baseline for comparison:

What would make an adequate baseline for our comparative study? Furthermore, since DiaSuite is based on Java, Should we consider that developing a software system in Java is the baseline for our comparative study?

The objective is to evaluate the benefits of DiaSuite on both software quality and productivity. For software quality, we plan to evaluate it via the evolutivity dimension. For software productivity, we will require the participants of the study to develop an SCC application using either DiaSuite or a programming-only approach (i.e., Java); this development will be monitored. The characteristics of the application used for the experimental study has a key role. For example, using a design-driven approach for the development of a simple application may require unnecessary efforts from the developer. Alternatively, the development of a large application requires the participants to master the application domain. Besides, it is difficult to recruit participants for a long period of time.

Our proposed application is a simple avionics error monitor. This application is large enough to make the design stage relevant but it is simple enough to be developed in less than half a day.

To evaluate the evolutivity of the application, we propose to enrich it with additional features to be developed by the participants once the initial development is completed. An alternative would be to have participants extend each other's initial development.

B. Data collection process

To gather data about the participants and the application they develop, we propose to use a questionnaire and a versioning system.

1) Questionnaire: The questionnaire requires the participants to report on their level of expertise in Java. Indeed, the participants should be proficient in Java so that the language will not be an obstacle to the development.

To evaluate their productivity, we gather the participants' feedback about their productivity and any difficulty they might have had evolving the application.

2) Versioning system: To maximize our control on the experimental environment and minimize the overhead for the participants, the experimental infrastructure is running on a remote server. The versioning system tracks the evolution of the application developments: a script running on the server invokes a commit of the participant's workspace every minute. Our proposed setting allows to evaluate the participant's productivity by calculating the time he effectively spent on the application files. The effective time spent on the development is calculated by adding the time intervals between successive commits. Intervals that last more than a given time (e.g., 15 minutes) are discarded to take into account the breaks during the development.

This versioning system also allows to estimate the effort that is required for the participants to make the application evolve. This estimation is based on the time spent on the evolution and the number of modifications the participants had to apply.

C. Data analysis

To answer the questions Q1, Q2 and Q3, we propose to analyze the following data gathered by our data collection process:

• The information from the questionnaire about the participants and their development

• The log of the versioning system • The applications code. Analyzing these data with respect to our questions requires to select pertinent metrics. For each question, we discuss the metrics that could be used.

1) Question Q1: To evaluate whether the DiaSuite approach improves software quality, we propose to compare the evolution of applications developed using DiaSuite to applications developed using Java only. The comparison relies on the following metrics: the time spent making the application evolve and the number of modifications required. Combining these two metrics should provide an accurate measure of whether software quality depends on the approach. We are still investigating other metrics to assess the evolutivity of applications. For example, others techniques, such as the use of artifacts' shared content [START_REF] Arbuckle | Studying Software Evolution using Artefacts' shared Information Content[END_REF], could be used. Beyond the evolutivity, other aspects of software quality (e.g., modularity) could be measured by analyzing the application code with metrics such as complexity, cohesion and coupling [START_REF] Chidamber | A Metrics Suite for Object-oriented Design[END_REF], [START_REF] Brito E Abreu | Evaluating the Impact of Object-oriented Design on Software Quality[END_REF].

2) Question Q2: To evaluate whether our tool-based approach increases the productivity of developers, we propose to compare the time spent by the participants to develop their application using either DiaSuite or Java only. As for the analysis of the evolutivity, we use both the data from the questionnaire and the data from the versioning system. For a finer grained comparison, the time spent on the design and the time spent on the implementation by the DiaSuite participants could be distinguished. Another interesting metric could be the adequacy of the application to the specification. However, this metric is rather subjective.

3) Question Q3: Since we compare DiaSuite to a programming-only development approach, it could be difficult to distinguish between the impact of the toolbased approach and the paradigm on software quality. One step towards addressing this issue could be to evaluate to what extent our design language guides the participants for the design of their applications. A possible metric is the similarity between application design artifacts: if DiaSuite applications are more similar than the Java ones, then DiaSpec effectively guides the participants in their design. This similarity can be calculated by measuring the distance between the applications designs. Dedicated tools such as EMFCompare [START_REF] Toulmé | Presentation of EMF Compare Utility[END_REF], SiDiff [START_REF] Treude | Difference Computation of Large Models[END_REF] or DSMDiff [START_REF] Lin | DSMDiff: A Differentiation Tool for Domain-specific Models[END_REF] can be used to measure this distance. Nevertheless, to further investigate this question, additional studies are required. For example, we could require other participants to develop the same application, using another tool-based approach that is not based on the SCC paradigm. Considering Java as a baseline, the results of this new study could be compared to the existing results of the participants who used Java only.

IV. Summary

In this paper, we have presented a proposal for a study to assess the benefits of a design-driven software development approach, in terms of software quality and productivity. To conduct this study, we have chosen to compare the development of applications by participants using either Java or DiaSuite, a design-driven, toolbased software develop approach. Experimental data are gathered by questionnaires filled out by the participants and a versioning system that monitors the participants' activities. Finally, we discuss the relevance of metrics to be used to analyze the collected data.

The undergoing work consists of conducting the proposed study and analyzing the results. To address Question Q3, we aim to consider general-purpose approaches that are design-driven and tool-based, such as UML environments, to further our experimental study. We also plan to conduct other studies that target non-functional properties supported by DiaSuite such as quality of service and error handling [START_REF] Mercadal | A Domain-specific Approach to Architecturing Error Handling in Pervasive Computing[END_REF], [START_REF] Gatti | A Step-wise Approach for Integrating QoS throughout Software Development[END_REF].

Figure 2 :

 2 Figure 1: The Sense/Compute/Control paradigm


1

Data assimilation in glaciology 1.1 Introduction

In this short paper, we will give one example of an inverse problem in glaciology. This problem is fairly simple to state: how to infer a climatic scenario (i.e. how to reconstruct past polar temperature) from ice volume records?

Ice-sheet dynamics is quite complex. Ice is a non-newtonian fluid, its flow follows a highly non-linear Stokes equation. One key factor of ice dynamics is surface mass balance, that is the amount of ice accumulating at the surface (from snow precipitation) minus the amount of melting ice. This mass budget is related to temperature at the surface of the ice-sheet, depending in particular on temperature. See textbooks [START_REF] Greve | Dynamics of ice-sheets and glaciers[END_REF] and [START_REF] Paterson | The physics of glaciers[END_REF] for detailed modelling of ice-sheets.

Ice volume observations are available from oceanic records, that give information about sea level, and therefore about the amount of water stored in ice-sheets, ice-caps and glaciers.

As far as we know, there are very few past works on this subject. [START_REF] Bintanja | A new method to estimate ice age temperatures[END_REF] and [START_REF] Bintanja | Modelled atmospheric temperatures and global sea levels over the past million years[END_REF] use some a very simple correction method, equivalent to the nudging method, with quite good results. The idea of this work is to explore the ability of the adjoint method to solve the inverse problem of reconstructing past temperature given all available observations. We start here with a simplified ice-sheet model and perform twin experiments.

We present the ice-sheet model in section 2, the adjoint method in section 3, and give numerical illustrations in section 4.

Ice-sheet model

Ice dynamics processes

As the full Stokes model is very costful to run over paleoclimatology time scales (thousands of years), large scale ice-sheet models have been developed, using the so-called Shallow Ice Approximation (SIA). Such models are able to reproduce ice-sheet evolution over hundreds of thousands of years, for large ice-sheets such as Antarctica, Greenland, Laurentide and Fennoscandian ice-sheets. One such model, GRISLI, is developed by LGGE [START_REF] Ritz | Modeling the evolution of Antarctic ice sheet over the last 420, 000 years-Implications for altitude changes in the Vostok region[END_REF]. It is a three dimensional thermomechanically coupled model, whose main physical processes are summed up in figure 1.1.

Model equations

As GRISLI model is still costful to run, we will use a simplified, flowline, isothermal prototype of this model, on a flat bedrock topography, without floating shelves, which is faster to run, in order to study the paleoclimate inverse problem. Flowline means that the model is one-dimensional in space. The first equation is the mass balance equation:

∂H ∂t = ḃm - ∂ U H ∂x , H| t=0 = H 0 with • t time, x latitude • H(t, x) ice thickness, H 0 (x) initial ice thickness • U (t, x)
Eulerian velocity averaged over the ice thickness • ḃm (t, x) surface mass balance rate Vertically averaged ice velocity U (t) is a diagnostic variable, that is to say it depends only on geometry variables (S, H) at time t, and it is given by the following equation: • u B (t, x) basal velocity • S(t, x) ice surface elevation • B(t, x) ice base (bedrock elevation, constant in this study)

U = u B -a 1 ∂S ∂x H 2 3 -a 2 ∂S ∂x 3 H 4 3 S = B + H H ≥ 0 with Adjoint method and adjoint model ✸ -5 -4 -3 -2 -1 0 1 2 -2 -1.5 -1 -0.
• a 1 , a 2 given coefficients

Climatic scenario and surface mass balance

The surface mass balance is given as a function of the ice surface temperature: 1.2 to see how accumulation and ablation depend on surface temperature.

   ḃm = Acc + Abl Acc = f (T S ) Abl = g(T S ) with • Acc(t, x) accumulation rate • Abl(t, x) ablation rate • T S (t, x) surface temperature • f, g given functions See figure
And finally, the surface temperature itself depends on latitude x, on altitude S, and on the (poorly known) temperature at the northern point at sea-level height T clim (t):

T S (t, x) = T clim (t) + b x + c S(t, x)
with b, c given coefficients. The aim of the problem is to infer T clim (t) from volume measurements.

Adjoint method and adjoint model

We refer to main and specialized lectures regarding variational data assimilation and adjoint code construction. [START_REF] Raymo | The timing of major climate terminations[END_REF]; blue and red : different models

Observations and control variables

As we said before, we will use ice volume observations, available from sea level records, see figure 1.3 from [START_REF] Paillard | The Antarctic ice sheet and the triggering of deglaciations[END_REF] as an example. As we do twin experiments, we will generate ourselves our data, from a reference "true state". In the framework of four-dimensional variational assimilation, we write our inverse problem as a minimization problem: we want to find the "best" temperature T clim (t), sot that the misfit between modeled and observed volume is smallest. This means that our control variable is the climatic temperature T clim (t), and our cost function writes as follows:

J (T clim ) = J b + t f t0 Vol(t) -Vol obs (t) 2 dt, Vol(t) = x H(t, x) dx
where the background term J b is a classical regularising term, involving a background temperature and background error covariance matrix:

J b (T clim ) = T clim -T b clim 2 k
The norm . k is classicaly given by:

X 2 k = t f t=t0 t f s=t0 k(s, t)X(s)X(t) ds dt
with k a regularizing kernel. In discrete form, this would writes:

X 2 B -1 = X T B -1 X
where B is the background error covariance matrix, and X contains T clim (t i )-T b clim (t i ) for all discrete controle times t i .

Adjoint code validation

We minimize our cost function using quasi-Newton descent algorithm. This requires the gradient of J , obtained thanks to the adjoint model. As our flowline model is simple and in Matlab, we implemented by hand the adjoint code using classical recipes for adjoint code construction, see e.g. [START_REF] Giering | Recipes for adjoint code construction[END_REF]. To validate our gradient, we perform first-and second-order tests, that is we check that the 1st and 2nd order Taylor approximation formula

J (x + αz) = J (x) + α∇J (x)z + α 2 2 z T ∇ 2 J (x)z + o(α 2 )
is valid. To do so we form

δ = ∇J (x)z, τ = J (x + αz) -J (x) α , ε = τ (α, z) -δ(z) |δ(z)| , r = τ (α, z) -δ(z) α
and we check that ε tends to zero as α tends to zero, and r tends to a constant depending only on z, see figure 1.4 for an illustration with our model.

Numerical results on twin experiments

In order to validate our method and ajoint model, we present a first illustration on twin experiments (see figures 1.5 and 1.6). The principle of such experiment is as follows: first we choose a reference climatic scenario T clim , called the "true" temperature forcing. Second we perform a direct run of the model with the "true" forcing and we generate synthetic observations (figure 1.5). And finally, we forget that we know the true temperature, we start with a constant first guess for T clim , and we perform data assimilation thanks to the adjoint method, in order to get an assimilated T clim . We can then compare our result with the true T clim (figure 1.6).

Fig. 1 . 1

 11 Fig. 1.1 Physical processes impacting ice-sheet dynamics.

  Fig. 1.2 Accumulation, ablation and surface mass balance as a function of surface temperature.

Fig. 1 . 3

 13 Fig. 1.3 Past ice volume records. black: mesured data from (Raymo, 1997); blue and red : different models

  Fig. 1.4 First-(left) and second-(right) order gradient tests.

Fig. 1 . 5

 15 Fig. 1.5 Twin assimilation results: True climatic scenario T clim (top), generated volume data (bottom).

Fig. 1 . 6

 16 Fig. 1.6 Twin assimilation results: T clim recovered thanks to data assimilation.


Olivier Brivois
email: o.brivois@brgm.fr

Déborah Idier

Jérôme Thiébot

Bruno Castelle

Gonéri Le Cozannet

Daniel Calvete

Utilisation d'un modèle de stabilité linéaire pour caractériser le comportement morphodynamique d'un système à double barre. Application à la plage du Truc Vert

Keywords: Modélisation, modèle de stabilité linéaire, morphodynamique, barres en croissant, plage du Truc Vert, Aquitaine Modelling, linear stability model, morphodynamic, crescentic bar, Truc Vert Beach, Aquitania

France).

Introduction

Nearshore sandbars are often characterised by the presence of rhythmic or quasi-rhythmic features such as crescentic patterns and rip channels. The dynamics of these three-dimensional patterns is characterized by a wide range of temporal and spatial variability, undergoing different "states" depending on the wave energy, the short-term history of the bar, the beach slope and the sediment characteristics. From these parameters, three main beach states were identified as dissipative, intermediate and reflective states [START_REF] Wright | Morphodynamic variability of beaches and surf zones, a synthesis[END_REF]. Intermediate beaches were further divided into four sub-states. Immediately below the high energy dissipative state is the Longshore Bar and Trough (LBT), then the Rhythmic Bar and Beach (RBB), next Transverse Bar and Rip (TBR) and finally the Low Tide Terrace (LTT) or ridge and runnel system [START_REF] Wright | Morphodynamic variability of beaches and surf zones, a synthesis[END_REF]. These rhythmic features guide and constrain intense horizontal circulations driven by alongshore differential broken-wave energy dissipation [START_REF] Bonneton | Large-scale vorticity generation due to dissipating waves in the surf zone[END_REF] into strong offshore currents across the bays.

The so-called rip currents [START_REF] Macmahan | Rip current review[END_REF] play an important role during storm events, as they determine the place where localized beach and dune erosion occurs [START_REF] Thornton | Rip currents, mega-cusps, and eroding dunes[END_REF]. Therefore, understanding sandy pattern dynamics and their link with hydrodynamics forcing is of strong interest for coastal risk assessment.

The phenomena involved in 3D sandbar dynamics are complex. This is now established that the development of 3D patterns can be attributed to self-organization processes [START_REF] Coco | Patterns in the sand: From forcing templates to self-organisation[END_REF], that is, the positive feedback between bottom perturbations, sediment processes and hydrodynamics (waves and currents). The sandy patterns that arise through self-organisation are called morphodynamic instabilities. Their emergence is commonly analysed using either Linear Stability Analysis (LSA) or Non-Linear Stability Analysis (NLSA) models (e.g. [START_REF] Calvete | Morphological development of rip channel systems: normal and near normal wave incidence[END_REF]; Klein and Schuttelars, 2006;[START_REF] Smit | The morphological response of a nearshore double sandbar system to constant wave forcing[END_REF].

LSA models focus only on the initial development stage of instabilities (infinitesimal perturbations).

As they are exploratory models, they are commonly used for understanding the mechanisms governing sandbars dynamics rather than making quantitative comparisons with field data. Typical LSA studies commonly consist in making sensitivity analysis, for instance considering slight variations in wave conditions [START_REF] Deigaard | A morphological stability analysis for a long straight barred coast[END_REF][START_REF] Calvete | Morphological development of rip channel systems: normal and near normal wave incidence[END_REF] or on bed profiles [START_REF] Deigaard | A morphological stability analysis for a long straight barred coast[END_REF][START_REF] Calvete | Un)predictability in rip channel systems[END_REF][START_REF] Damgaard | Morphodynamic modelling of rip channel growth[END_REF]. These studies showed, among other findings, that wavelength of instabilities increases with increasing wave obliquity and that the effect of wave period can be different if shore-normal or oblique waves are considered. The emerging pattern characteristics were also found to be as sensitive to the initial beach profile as to wave conditions. More recently, In the present study, rather than in-depth investigating the physical processes involved in sandbar dynamics, the objective is to determine whether LSA can characterize the overall 3D beach response, considering a wide range of hydrodynamic and morphodynamic configurations. To that purpose, a key step is accurately characterizing both the wave climate and a basic state bathymetry. The considered site is Truc Vert Beach (French Aquitanian coast) which is a double bar system, i.e. a beach characterized by two alongshore submarine bars (one outer bar and one inner bar, see figure 4).

Typically, the outer bar (more seaward) and the inner bar (more shoreward) exhibits crescentic patterns and a Transverse Bar and Rip morphology, respectively [START_REF] Castelle | Double bar beach dynamics on the highenergy meso-macrotidal French Aquitanian Coast : A review[END_REF]).

The paper proceeds as follows: section 2 presents the site description and gives some background about Truc Vert beach. Section 3 describes briefly the LSA model Morfo60 [START_REF] Calvete | Morphological development of rip channel systems: normal and near normal wave incidence[END_REF]. Section 4 deals with the choice of scenarios for actual climate conditions. Section 5 is dedicated to the analyses of the numerical results for these conditions. Conclusions are drawn in section 6.

Description of the site

General settings

The Aquitanian Coast (SW France) is an approximately 250-km long straigth low coast bordered by high aeolian dunes. The tide is of meso-macro type, with a mean tidal range of 3.2 m and about a 5 m maximum tidal amplitude during spring tides. The sediments are composed of fine to medium quartz sand with mean grain sizes ranging from 200 to 400 μm [START_REF] Pedreros | Application of grain-size-trend analysis for the determination of sediment transport pathways in intertidal areas[END_REF]. This coast is exposed to high-energy Atlantic waves. Waves at the Aquitanian Coast are generated by W-E tracking subpolar deep low pressure systems over the North Atlantic Ocean. They are, therefore, strongly seasonally modulated. A significant interannual variability of the wave climate is related to the NAO Index [START_REF] Dupuis | Wave climate evolution in the bay of biscay over two decades[END_REF][START_REF] Cozannet | Teleconnection pattern influence in the Bay of Biscay[END_REF][START_REF] Charles | Present wave climate in the Bay of Biscay: spatiotemporal variability and trends from 1958 to 2001[END_REF]. Wave conditions vary from NW reasonably low-energy waves during summer and occasional storms with a mean NW direction to very high-energy W to WNW swells during winter. Since 1998, most of the field experiments along the Aquitanian Coast took place at Truc Vert Beach (see Figure 1) because it is both considered as being representative of most of the Aquitanian Coast beaches and non-exposed to human activity.

Beach morphology

Truc Vert Beach has received increasing attention over the last decade. This stretch of coastline has been studied through different approaches: hydrodynamic measurement analysis (e.g. [START_REF] Butel | Spatial variability of wave conditions on the French Atlantic Coast using in-situ data[END_REF], topographic surveys (e.g., De Melo [START_REF] De Melo Apoluceno | Morphodynamics of Ridge and Runnel Systems during Summer[END_REF][START_REF] Sénéchal | Morphodynamic response of a meso-macro-tidal intermediate beach based on a long-term data set[END_REF], satellite imagery analysis (e.g., [START_REF] Lafon | Morphodynamics of nearshore rhythmic sandbars in a mixed-energy environment (SW France): I. Mapping beach changes using visible satellite imagery[END_REF][START_REF] Lafon | Morphodynamics of nearshore rhythmic sandbars in a mixed-energy environment (SW France): 2. Physical forcing analysis[END_REF], numerical modelling (e.g., [START_REF] Castelle | Modelling of a rip current induced by waves over a ridge and runnel system on the Aquitanian Coast, France[END_REF] and time exposure imagery (e.g., [START_REF] Almar | Two-and three-dimensional double-sandbar system behaviour under intense wave forcing and a meso-macro tidal range[END_REF]. Truc Vert Beach has been observed to be systematically double barred.

The inner bar has been found to go through all the states within the intermediate classification of [START_REF] Wright | Morphodynamic variability of beaches and surf zones, a synthesis[END_REF] with the Transverse Bar and Rip (TBR) morphology observed most of the time (58% of occurence according to [START_REF] Sénéchal | Morphodynamic response of a meso-macro-tidal intermediate beach based on a long-term data set[END_REF]. This morphology becomes a Low Tide Terrace (LTT) morphology (16% of occurrence according to [START_REF] Sénéchal | Morphodynamic response of a meso-macro-tidal intermediate beach based on a long-term data set[END_REF] after at least 10 days of fair weather conditions [START_REF] Castelle | Double bar beach dynamics on the highenergy meso-macrotidal French Aquitanian Coast : A review[END_REF]. The other morphologies, i.e. the RBB (Rhythmic Bar and Beach) and the LBT (Longshore Bar and Trough) states have 14% and 12% of occurrence [START_REF] Sénéchal | Morphodynamic response of a meso-macro-tidal intermediate beach based on a long-term data set[END_REF], respectively. Based on satellite Spot imagery and considering the alongshore wavelength as the mean rip spacing, [START_REF] Lafon | Morphodynamics of nearshore rhythmic sandbars in a mixed-energy environment (SW France): I. Mapping beach changes using visible satellite imagery[END_REF] showed that the inner bar alongshore wavelength ranges from 370 to 463 m.

The outer bar exhibits quasi-persistent crescentic patterns (Figure 2), i.e. a RBB morphology. The average alongshore wavelength of the outer crescentic bar ranges from 579 to 818 m [START_REF] Lafon | Morphodynamics of nearshore rhythmic sandbars in a mixed-energy environment (SW France): I. Mapping beach changes using visible satellite imagery[END_REF]. The outer bar geometry vary from a typical crescent shape to a strong asymmetric shape with the landward part of the bar being linear and oriented NW-SE [START_REF] Lafon | Morphodynamics of nearshore rhythmic sandbars in a mixed-energy environment (SW France): I. Mapping beach changes using visible satellite imagery[END_REF]. Only during severe storms the outer bar was suspected to reshape into a shore-parallel linear bar [START_REF] Castelle | Double bar beach dynamics on the highenergy meso-macrotidal French Aquitanian Coast : A review[END_REF], resulting in a LBT morphology. This was recently corroborated by video observations during a storm event with significant wave height (Hs) of about 8 m [START_REF] Almar | Two-and three-dimensional double-sandbar system behaviour under intense wave forcing and a meso-macro tidal range[END_REF]. Of note, the inner bar rhythmicity sometimes appears to couple to that of the outer bar [START_REF] Castelle | Double bar beach dynamics on the highenergy meso-macrotidal French Aquitanian Coast : A review[END_REF], which is in line with previous observations of other double sandbar systems (e.g., [START_REF] Ruessink | Coupled and noncoupled behavior of three-dimensional morphological patterns in a double sandbar system[END_REF] and was recently depicted through linear [START_REF] Coco | The use of linear stability analysis to characterize the variability of multiple sandbar systems[END_REF] and nonlinear (e.g., [START_REF] Castelle | Coupling mechanisms in double sandbar systems, Part 1: Patterns and physical explanation[END_REF] Thiebot et al., 2012) stability analysis. Accordingly, coupling will not be addressed in this contribution.

Model description

The morphological model used in this study, Morfo60 [START_REF] Calvete | Morphological development of rip channel systems: normal and near normal wave incidence[END_REF], is based on linear stability analysis. It is used to identify the morphological instabilities that can arise from selforganization processes. This model is based on depth-and wave-averaged continuity and momentum equations, coupled with wave-energy, wave-phase and sediment conservation equations. Waves are assumed to dissipate their energy according to the model of [START_REF] Thornton | Transformation of wave height distribution[END_REF]. In the momentum equation, the Reynolds stresses are parameterized using a horizontal viscosity coefficient [START_REF] Battjes | Modelling of turbulence in the surfzone[END_REF] and the bed shear is described using a linear friction law. The sediment flux calculation is based on the formula of [START_REF] Soulsby | Dynamics of Marine Sands[END_REF]. It includes the contribution of bedload and suspended load, as well as gravitational downslope sediment transport (for more details about the model see [START_REF] Calvete | Morphological development of rip channel systems: normal and near normal wave incidence[END_REF].

For each simulation, the input parameters are an alongshore uniform bathymetry (the bed level depends only on the cross-shore distance to the coast, as for instance in Figure 3-a), the characteristics of the offshore waves (significant wave height: H S , peak period: T P and peak direction: D P ) and a water level which includes the contribution of both tide and storm surge (the level is given relatively to the Lowest Astronomical Tide level). This defines a basic state which is time-invariant, i.e. in equilibrium with all the alongshore-uniform dynamical variables. The LSA consists in adding periodic perturbations of different wavelength to the basic state solution, assuming a quasi-steady state. The response of the system to the perturbations is analysed with the relationship between the growth rate and the perturbations wavelength (Figure 3-b). Depending on their characteristics, the perturbations are amplified at different rate. The instability with the higher growth rate, called FGM (Fastest Growing Mode), is the one that is the more likely to emerge. wavelength of the instabilities λ; its growth rate σ or its e-folding time denoted by "Efold" ("Efold" = 1/σ), that gives an indication of the characteristic formation time in nature; and its migration velocity in the longshore direction "Vmig". Moreover the variable "Dist.-Pert." (abbreviation for distance between the maximum perturbations amplitude and the shore) indicates the distance from the shore where instabilities emerge (inner or outer bar). This distance is measured graphically, as shown in Figure 3-c. In the following, the term instability refers to the FGM.

In this study, simulations that give e-folding times longer than 50 days are not considered, as it is far larger than the typical time scale at which wave regimes change. The range of wavelength instabilities tested by Morfo60 in this study is between 100 m to 3000 m. The upper value (3000 m) has been chosen intentionally high to avoid testing values too close to the observed values in the field and to check the ability of the model to find realistic wavelengths from a large range of possibilities.

For some configurations, the relation between the wavelength and the growth rate is characterized by two maxima (i.e. two FGM), meaning that two modes of instabilities with different wavelength emerge. In this case, if the maxima correspond to instabilities located on different bar, both FGM are considered independently. This configuration will be hereafter referred to as a "double point". If the two maxima correspond to the same bar, only the instabilities with the higher growth rate will be kept.

Climate scenarios

To characterize the global beach dynamics, a limited number of configurations have been determined from field data. These configurations are combinations of (1) idealized bathymetries (cross-shore profiles), (2) wave classes (significant wave height H S , peak period T P , peak direction D P ), (3) tidal levels (given relatively to the Lowest Astronomical Tide level) and (4) storm surges. All these discrete combinations are representative of a large number of physical configurations commonly observed at this site. The method used to determine these configurations is detailed in the following section.

Bathymetries

Numerical studies show that the morphological response of a beach is strongly dependent on the initial beach profile, especially on the distance between bars and coast and on the depth of bars [START_REF] Damgaard | Morphodynamic modelling of rip channel growth[END_REF][START_REF] Calvete | Un)predictability in rip channel systems[END_REF]. Field observations show that beach bathymetries and their morphological responses are very variable depending on close past and present meteorological conditions [START_REF] Ruessink | Coupled and noncoupled behavior of three-dimensional morphological patterns in a double sandbar system[END_REF]. Thus, to take into account the different beach configurations and the effects of events chronology, two initial cross-shore profiles are chosen. The first one is representative of high energy conditions (Figure 4 For these two profiles, the distance of the bar crest to the shore has been chosen from typical configurations observed in the field (ECORS campaign, 2008; [START_REF] Parisot | Treatment of topographic and bathymetric data acquired at the Truc-Vert Beach during the ECORS Field Experiment[END_REF]. In the following, the high-and low-energy conditions bathymetries (alongshore-uniform bathymetries built from the two previous profiles) will be called the post-storm and quiet weather bathymetries, respectively.

Hydrodynamic conditions

Waves

The wave forcing has been synthesized into a limited number of wave conditions (H S , T P , D P ), thanks to a classification algorithm. The use of wave classification is relatively recent and aims at obtaining a synthetic description of complex wave climate, classifying waves in homogeneous classes based on defined resemblance criteria [START_REF] Butel | Spatial variability of wave conditions on the French Atlantic Coast using in-situ data[END_REF]. The classification algorithm used in this study is the K-Means algorithm, similarly to Le [START_REF] Cozannet | Teleconnection pattern influence in the Bay of Biscay[END_REF]. Each wave class is defined by its mean significant height (H S ), peak period (T P ) and peak direction (D P ), and an empirical probability of occurrence.

The representative wave classes chosen at Truc Vert beach are given in Table 1. They have been obtained following different stages: first a wave classification has been made from NOAA WaveWatchIII data [START_REF] Tolman | Effects of numerics on the physics in a third-generation wind-wave model[END_REF] over the years 1997 to 2008 at an offshore point relatively close to the beach (45°N, 2.5°W; depth: 1100m). Each of the 12 waves classes found following this first classification has then been propagated, with the code SWAN [START_REF] Booij | Swan Cycle III version 40.41[END_REF], at the offshore boundary of our calculation domain (44°44.65" N; 1°16.19"W; depth: 20m). Finally, considering each wave class occurrence and according to expert judgment, the number of representative wave classes has been reduced to four (Table 1), their cumulative occurrence representing approximately 60 % of time. The three first wave classes originate from WNW, the fourth originates from W (Incid.: the wave peak incidence is defined clockwise relatively to the shore normal direction).

Sea level: Tide and storm surge

To study the effect of the sea level, three tidal levels corresponding to a mean spring tide are considered in our simulations: one Low Tide (0.62 m), one Mid Tide (2.46 m) and one High Tide (4.1 m) above the Lowest Astronomical Tide level. Moreover, the effect of a possible storm surge is studied adding the maximum observed storm surge to the sea level considered. The value of this maximum observed storm surge is 0.84 m close to Truc Vert beach [START_REF] Pirazzoli | Données pour dImensionnement des Structures côtières et des Ouvrages de BOrd de mer à Longue Echéance[END_REF]. All these combinations of bathymetries, wave classes and sea levels represent 48 configurations to be performed by numerical simulation.

Results

In this section, the morphodynamic response associated to these configurations are presented and analysed. They are then compared with the available in situ observations.

Numerical results

The system behaviour is analysed here considering all the combinations of forcing (i.e. 12 configurations for each wave class). The results are presented in Figure 5 and Figure 6 for the quiet weather bathymetry and post-storm bathymetry, respectively.

Rather than looking each case separately, the following analysis focuses on the global response of the system, describing its main tendencies through the four output variables.

The instabilities wavelength globally increases with increasing offshore wave height (Figures 5-a and 6-a) except for one configuration (CL4 on post-storm bathymetry at High Tide plus storm surge, the right-hand point in Figure 6-a for CL4). When a double point is observed (see Section 3) (for CL3 and CL4 in Figures 5-a and6-a), the wavelength is systematically larger at the outer bar than at the inner bar (between 2.1 and 3.6 times the inner bar wavelength), which is in line with existing field observations of double sandbar systems (e.g., [START_REF] Van Enckevort | Observations of nearshore crescentic sandbars[END_REF]. The relationship between the water level and the wavelength of the instabilities is more complex and differs from one wave class to another. Non-surprisingly, the e-folding time strongly increases with sea level (Figures 5-b The longshore migration rate increases with increasing wave height (Figures 5-c and6-c). Maximum alongshore migration of inner-bar rip channels typically occurs around mid tide.

The distance from the shore where instabilities emerge (inner or outer bar) depends only on wave height and sea level (Figures 5-d and6-d): the higher is the wave height, the higher is the sea level for which instabilities emerge on the outer bar. This might be explained by the fact that instabilities appear where the waves break (Van Enckevort et al., 2004), and the higher is the wave height, the most they break on the outer bar for higher sea levels.

Overall, instability behaviour is similar for the two bathymetries. Some differences between them exist due to the different depth of their bars. Firstly, instabilities emerge on the outer bar at higher sea level on quiet weather bathymetry rather than on post-storm bathymetry. Secondly, wavelengths and longshore migration rates of the instabilities are larger for the post-storm bathymetry than for the quiet weather bathymetry.

From these numerical results, the main system tendencies are: (i) a significant wave height (Hs) of 1 m can leads to the emergence of rhythmic features for sea levels typically lower than Mid Tide; (ii) for Hs of 2 m, instabilities can arise at the inner bar during the whole tidal cycle; (iii) for Hs larger 3 m instabilities arise at the outer bar. This threshold value in Hs of 3 m is in agreement with observations at Truc Vert beach, as only offshore waves larger than 3 m were observed to drive a significant morphological evolution of the outer bar [START_REF] Castelle | Double bar beach dynamics on the highenergy meso-macrotidal French Aquitanian Coast : A review[END_REF].

Typical pattern formation

The previous results show a large variability in the instabilities characteristics depending on the configuration considered (bathymetry, wave class and water level). Nevertheless, except for one configuration, all the simulated morphodynamic patterns have a similar shape, that is, a bar and rip system on inner bar and a crescentic bar on outer bar. All these configurations do not have the same occurrence frequency as some combinations are more likely to occur. Our purpose is to relate a wave class to a particular instability. For a given wave class and bathymetry, the instability the most likely to emerge only depends on the water level and on its corresponding e-folding time (Figures 5-b and6-b). The histogram of a ten years period of water level measurements at St-Jean-de-Luz is given in Figure 7. The two most frequent water levels (1.50 m and 3.40 m) are readily apparent. As the e-folding time increases with water level (model results), instabilities are more likely to arise for the 1.50 m water level.

These morphodynamic patterns are presented for the most common bathymetry, namely the quiet weather bathymetry, in Figure 8, with their corresponding wavelengths and e-folding time given in Table 2. In Figures 8,a crescentic bar (similar to RBB morphology) and a bar and rip system, with slightly transverse pattern, is observed at the outer bar and the inner bar, respectively. These beach states are comparable to the most common beach states observed at Truc Vert beach (e.g., [START_REF] Sénéchal | Morphodynamic response of a meso-macro-tidal intermediate beach based on a long-term data set[END_REF][START_REF] Castelle | Double bar beach dynamics on the highenergy meso-macrotidal French Aquitanian Coast : A review[END_REF] and Figure 2).

The only configuration presenting a different morphodynamic pattern concerns the highest sea level (High Tide plus storm surge) for CL4 on post-storm bathymetry (Figure 6). The corresponding instabilities are presented in Figure 9-b, together with the instabilities obtained with the same conditions at High Tide, in Figure 9-a. At High Tide plus storm surge, the instabilities present a particular pattern. Firstly, the wavelength is much smaller. Secondly, the bar and rip system are strongly oblique and only emerge on the crest of the inner bar (plotted as dotted line on figures 9). This is different from every other patterns, for which instabilities emerge simultaneously on the crest and on the off-shore face of the inner bar (Figures 9-a and8). But this particular morphology has a very low probability to effectively emerge, as the probability of such level is weak and the corresponding e-folding time is long.

Discussion and comparison with observations

Morfo60 is an exploratory model dedicated to analyse the development of morphodynamic instabilities during their initial linear growth. Further nonlinear evolution and interaction between modes growing at different rates are not accounted for, while they can play important role in the morphodynamic evolution of the system. Moreover, the stability analysis is performed for stationary conditions (i.e. given water level and wave class) assuming initial alongshore uniform beach profile and wave field. In reality, pre-existing alongshore variations can exist and can significantly influence the subsequent development of bed patterns [START_REF] Tiessen | Development of crescentic bars for a periodically perturbated initial bathymetry[END_REF]. In addition, time-varying wave conditions have been found to lead to different spacing than for the time-invariant wave forcing as a result of nonlinear interactions [START_REF] Castelle | Modeling formation and subsequent nonlinear evolution of rip channels: time-varying versus time-invariant wave forcing[END_REF]. Accordingly, LSA results should be considered with care, keeping in mind that they give prediction for the initial development phase of instabilities for idealized conditions.

In this study, LSA is applied to characterize the global morphological behaviour of Truc Vert beach, using idealized initial bathymetries. As the forcing is steady in our numerical simulations, it is possible to relate a given wave condition with a preferred instabilities (Table 4). In the field, establishing this relation is difficult, as wave conditions persistently change with timescales smaller than typical morphodynamic time response [START_REF] Smit | The morphological response of a nearshore double sandbar system to constant wave forcing[END_REF]. Accordingly comparison of model results with field data is not straightforward.

As the beach morphologies predict by the model are consistent with observations, a qualitative comparison with available Truc Vert field observations is done below.

The instabilities wavelength values, presented in Figure 5-a and Figure 6-a, are summarized in Table 3; together with the observations in [START_REF] Lafon | Morphodynamics of nearshore rhythmic sandbars in a mixed-energy environment (SW France): I. Mapping beach changes using visible satellite imagery[END_REF]. The wavelengths of simulated rhythmic pattern are in fair agreement with the observations, except for CL4 (Hs = 4m) that overestimates the emerging wavelength. Concerning the e-folding time, predicted values are of the order of several days (between one and five days for most of the sea levels) which is realistic with respect to observations at Truc Vert beach. This is also consistent with Van Enckevort and [START_REF] Van Enckevort | Video observations of nearshore bar behaviour. Part 2: Alongshore non-uniform variability[END_REF] Overall, the numerical results of the LSA model are relatively consistent with field observations at Truc Vert beach, particularly in term of shapes and wavelengths of the rhythmic features. The formation timescales of the instabilities from the model are realistic, but their migration rates are overestimated.

Conclusion

Linear stability analysis model have been mainly used in earlier studies to examine the underlying physical mechanisms that control the development of crescentic bed patterns and to investigate the sensitivity to physical parameters, like the effect of different wave conditions or different bed profiles In this paper, a linear stability analysis model has been used to determine the characteristics of morphodynamic instabilities emerging for different hydrodynamic and bathymetric conditions with application to the wave dominated meso-macro-tidal Truc Vert Beach, which is double-barred. Model settings have been designed for two contrasting bathymetric profiles (representative of post-storm and quiet weather conditions), four wave classes, and a range of tide levels and maximum observed storm surge level representative of Truc Vert Beach. The model predictions, developing outer bar crescentic patterns and bar and rip morphology at the inner bar, show a reasonable agreement with field observations in term of patterns and wavelengths. This application of a LSA model to a double sandbar system with an appropriate knowledge of the field site (beach profile, wave climate and tidal range) suggests that LSA can provide quantitative information on the 3D surfzone sandbar patterns that can be typically found along a given beach.

Nevertheless our approach presents some limitation inherent to the use of linear stability analysis.

Using a non-linear model could be useful to estimate the possible differences in behaviour in the nonlinear regime. Concerning the input data, using wave classification gives a synthetic representation of the local wave regime. Thus, the analysis is restricted to a limited number of wave conditions. However, in the field the waves vary constantly. Discretizing the wave conditions with a higher sampling could improve our description of the system response. Dist.-Pert, distance to the coast of the instabilities (m) ("+" corresponds to inner bar; "x" to outer bar).

Légendes des figures

Légendes des tableaux

Tableau 1 : Classes de vagues représentatives à la frontière au large de notre domaine de calcul (coord. : 44°44.65" N; 1°16.19"W) avec H S la hauteur significative des vagues (en m), T P la période pic (en s), D P la direction pic (en °) mesurée dans le sens des aiguilles d"une montre par rapport au Nord et Incid. l"incidence correspondante par rapport à la normale à la plage (en °).

  Tiessen et al. (2010) used a LSA for predicting the development of observed morphological patterns of a real double-barred beach with the purpose to validate LSA as a potential coastal engineering tool. Most of these studies focused on single-barred beaches and the investigations of double-barred beaches dynamics with LSA are limited (Klein and Schuttelaars, 2006; Coco and Calvete, 2009).

Figure 3 -

 3 Figure 3-c and 3-d. The model output are the characteristics of the morphodynamic response: the

  , dashed line). It has two straight bars. The second one is representative of low energy conditions (Figure 4, solid line). It has the outer bar close to the inner.

  and 6b): as the bar is deeper, depth-induced wave breaking dissipation across the bar is less intense and therefore drives weaker flow circulations. So the time, or the e-folding time, necessary to create instabilities increases.

  who observed the development of crescentic patterns within three days after a storm. For the longshore migration rate, values predicted by the model (Figures 5-c and 6-c) are very strong (from 0.09 to 9.7 m/hour) and much larger than those reported at Truc Vert beach (from 0.04 to 0.2 m/hour, i.e. 1 to 5 m/day during summer, De Melo Apoluceno et al., 2002). The systematic overestimation of rip channel migration rates with nonlinear morphodynamic models has already been pointed out in Falqués et al. (2008).Yet, recent video observations from the ECORS"08 field experiment showed that the longshore migration of the outer bar crescentic patterns could exceed 20 m in a single day (0.8 m/hour), for Hs > 4 m with a high offshore wave angle to the shore[START_REF] Almar | Two-and three-dimensional double-sandbar system behaviour under intense wave forcing and a meso-macro tidal range[END_REF].

(

  see for instance: Deigaard et al., 1999; Damgaard et al. 2002; Calvete et al., 2005 and 2007; Coco and Calvete, 2009). These studies have mainly focused on single bar beach and have compared model prediction with generic field spatial and temporal characteristics. Only recently, Tiessen et al. (2010) have used a LSA for predicting the development of observed morphological patterns of a real doublebarred beach.

Figure 1 :

 1 Figure 1 : Emplacement du site d"étude.

Figure 1 :

 1 Figure 1: Location of the study area.

Figure 2 :

 2 Figure 2 : Morphologie typique de la plage du Truc Vert : Barre externe : Rhythmic Bar and Beach

Figure 2 :

 2 Figure 2: Typical morphology of the Truc Vert beach: Outer bar: Rhythmic Bar and Beach (RBB)

Figure 3 :

 3 Figure 3 : Principe de fonctionnement de Morfo60. (a) Bathymétrie initiale uniforme. (b) Taux de

Figure 3 :

 3 Figure 3: Illustrated principle of Morfo60. (a): Longshore uniform initial bathymetry. (b): Growth rate

Figure 4 :

 4 Figure 4 : Profils perpendiculaires à la côte des bathymétries idéalisées : bathymétrie tempête

Figure 4 :

 4 Figure 4: Cross-shore profiles of the two idealized bathymetries: post-storm (dashed line) and quite

Figure 5 :

 5 Figure 5 : Résultats pour les 4 classes de vagues, bathymétrie temps calme. (a) : λ, longueurs d"onde

Figure 5 :

 5 Figure 5: Results for all wave classes, quiet weather bathymetry; (a): λ, wavelength of instabilities

Figure 6 :

 6 Figure 6 : Résultats pour les 4 classes de vagues, bathymétrie tempête. (a) : λ, longueurs d"onde des

Figure 6 :

 6 Figure 6: Results for all wave classes, post-storm bathymetry; (a): λ, wavelength of instabilities (m);

Figure 7 :

 7 Figure 7 : Histogramme des niveaux marins mesurés (marée + éventuelle surcote) pendant 10 ans au

Figure 7 :

 7 Figure 7: Histogram of measured sea level values (tide + eventual storm surge) during 10 years at the

Figure 8 :

 8 Figure 8 : Exemple d"instabilités émergentes pour les quatre classes de vagues et la bathymétrie temps

Figure 8 :

 8 Figure 8: Example of emerging instabilities for the 4 wave classes and the quiet weather bathymetry

Figure 9 :

 9 Figure 9 : Instabilités créées par CL4 sur la bathymétrie tempête, (a) : morphologie TBR (Transverse

Figure 9 :

 9 Figure 9: CL4 instabilities on post-storm bathymetry, (a): Transverse Bar and Rip (TBR) morphology

Table 1 :

 1 Representative waves classes at the offshore boundary of our calculation domain (Coord.: 44°44.65" N; 1°16.19"W) with H S the significant wave height (in m), T P the peak period (in s), D P the peak direction (in °) measured clockwise from the North direction and Incid. the corresponding incidence evaluated from the normal direction of the beach (in °). Tableau 2: Longueur d"onde (λ en m) et temps caractéristique de formation (Efold. en jours) des instabilités pour chaque classe de vagues, obtenues avec la bathymétrie temps calme et le niveau d"eau de basse mer le plus fréquent.

Table 2 :Tableau 3 :

 23 Wavelength (λ in m) and characteristic formation time (Efod. in days) of emerging instabilities for each wave class, obtained on quiet weather bathymetry for the most frequent low tide level. Longueur d"onde des instabilités (λ en m) calculée sur chaque barre (inner/outer bar), pour les quatre classes de vagues et les deux bathymétries (quiet weather/post-storm bathy.) pour tous les niveaux d"eau considérés.

  From 370 m +/-146 m to 462 m +/-188 m (min: 85 m; max: 1480 m) From 579 m +/-200 m to 818 m +/-214 m (min: 248 m; max: 1569 m) Table 3

Table 3 :

 3 Wavelength instabilities (λ in m) calculated on each bar, for all wave classes and both bathymetries, for all considered water levels.

	Tableaux

Wave Class Number H S (m) T P (s) D P (°) Incid. (°)

		1	1	6	291	10
		2	2	10	286	5
		3	3	11	286	5
		4	4	13	271	-10
			Table 1
			Inner Bar Outer Bar
		CL1 λ	351 m	No instability
		Efold. 8.5 days
		CL2 λ	340 m	No instability
		Efold 3.25 days
		CL3 λ	345 m	740 m
		Efold. 2.7 days	2.8 days
		CL4 λ	419 m	1496 m
		Efold. 2.2 days	2.4 days
			Table 2
		Inner Bar			Outer Bar
		Quiet weather	Post-storm	Quiet weather	Post-storm
		bathy.	bathy.	bathy.	bathy.
	Wave class 1	331 m to 351 m	263 m to 403 m	No instability	No instability

Acknowledgments

This work has been supported by French research National Agency (ANR) through VMC 2006 program (project VULSACO, n° ANR-06-VUNL-009). Also we wish to thank Etienne Delvallée and Emmanuel Romieu.