Fabien Lotte
email: fabien.lotte@gmail.com

Aurélien Van Langhenhove

Fabrice Lamarche

Thomas Ernest

Yann Renard

Bruno Arnaldi

Anatole L Écuyer
email: anatole.lecuyer@irisa.fr

Exploring Large Virtual Environments by Thoughts using a Brain-Computer Interface based on Motor Imagery and High-Level Commands

Brain-computer interfaces (BCI) are interaction devices which enable users to send commands to a computer by using brain activity only. In this paper, we propose a new interaction technique to enable users to perform complex interaction tasks and to navigate within large virtual environments (VE) by using only a BCI based on imagined movements (motor imagery). This technique enables the user to send high-level mental commands, leaving the application in charge of most of the complex and tedious details of the interaction task. More precisely, it is based on points of interest and enables subjects to send only a few commands to the application in order to navigate from one point of interest to the other. Interestingly enough, the points of interest for a given VE can be generated automatically thanks to the processing of this VE geometry. As the navigation between two points of interest is also automatic, the proposed technique can be used to navigate efficiently by thoughts within any VE. The input of this interaction technique is a newlydesigned self-paced BCI which enables the user to send 3 different commands based on motor imagery. This BCI is based on a fuzzy inference system with reject options. In order to evaluate the efficiency of the proposed interaction technique, we compared it with the state-of-the-art method during a task of virtual museum exploration. The state-of-the-art method uses low-level commands, which means that each mental state of the user is associated to a simple command such as turning left or moving forwards in the VE. In

contrast, our method based on high-level commands enables the user to simply select its destination, leaving the application performing the necessary movements to reach this destination. Our results showed that with our interaction technique, users can navigate within a virtual museum almost twice as fast as with low-level commands, and with nearly twice less commands, meaning with less stress and more comfort for the user. This suggests that our technique enables to use efficiently the limited capacity of current motor imagery-based BCI in order to perform complex interaction tasks in VE, opening the way to promising new applications.

Introduction

BCI are communication devices which enable users to send commands to a computer application by using only their brain activity, this brain activity being measured and processed by the system [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. Current BCI systems mostly rely on electroencephalography (EEG) for the measurement of this brain activity [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. While BCI were initially designed as assistive devices for disabled people, they recently emerged as being also promising interaction devices for healthy users [START_REF] Allison | Why use a BCI if you are healthy ? In proceedings of brainplay 2007, playing with your brain[END_REF]. In this context, BCI have been notably used to interact successfully with virtual reality (VR) applications and video-games [START_REF] Lécuyer | Brain-computer interfaces, virtual reality and videogames[END_REF]. However, current techniques used for interacting with VE by thoughts remain relatively basic and do not compensate for the small number of commands provided by the BCI. For instance, navigation techniques proposed so far are mostly based on low-level commands, with generally a direct association between mental states and turning left or right, or moving forwards in the VE.

We believe that higher-level commands should be more exploited in order to provide a more flexible, convenient and efficient system to interact with VE and video games. In such a system, most of the complex tasks would be carried out by the system whereas the user would only have to provide a few high-level commands to accomplish the desired tasks. Such a principle is currently being applied to brain-computer interactions with robots [START_REF] Millán | Noninvasive brain-actuated control of a mobile robot by human EEG[END_REF][START_REF] Iturrate | Non-invasive brainactuated wheelchair based on a P300 neurophysiological protocol and automated navigation[END_REF]. Here, we rely on the fact that VE offer much more possibilities than the "real-life" conditions. For instance, it is very easy to add artificial stimuli or information in a VE and it is also much easier to extract information automatically from this VE, e.g., its topology, than from a real world environment. Therefore, we believe that a BCI-based interaction technique in VR should take advantage of these additional virtual possibilities.

In this paper, we present a novel interaction technique for exploring large VE by using a BCI based on motor imagery (i.e., imagined movements [START_REF] Pfurtscheller | Motor imagery and direct braincomputer communication[END_REF]), thanks to the use of high-level commands.

Our technique is based on an automatic processing pipeline which can be used to interact by thoughts with any VE. The input of this interaction technique is a newly designed self-paced BCI based on 3 motor imagery classes. We illustrate our interaction technique through the exploration of a virtual museum by thoughts, and compare it with the current technique used to explore VE with motor imagery-based BCI. Our results suggest that the use of highlevel commands enables users to navigate the VE faster than with the current technique, based on low-level commands.

This paper is organized as follows: Section 2 proposes a state-of-the art of the existing interaction techniques used to explore VE by thoughts while Section 3 presents the interaction technique we proposed. Then, Section 4 presents the newly designed BCI system used as the interaction device.

Section 5 describes the evaluation achieved and the obtained results. Finally, Sections 6 and 7 discuss the results and conclude, respectively.

Related work

Due to the appealing potential of BCI as a new input device, researchers have proposed an increasing number of 3D interaction techniques with VE based on BCI, with a special emphasis on navigation techniques [START_REF] Lécuyer | Brain-computer interfaces, virtual reality and videogames[END_REF]. In the work presented in this paper we mainly focus on navigation in VE by using a BCI. However, it should be mentioned that interesting works have been achieved on selection and manipulation of virtual objects with a BCI [START_REF] Lalor | Steady-state VEP-based brain-computer interface control in an immersive 3-D gaming environment[END_REF][START_REF] Bayliss | The use of the P3 evoked potential component for control in a virtual apartment[END_REF].

In order to navigate in a VE by thoughts, most current works rely on brain signals generated by Motor Imagery (MI), i.e., brain signals generated when the user imagines movements of his own limbs (e.g., hand or foot). Indeed, such mental tasks trigger variations of EEG signal power, in specific frequency bands, known as the µ (8-13Hz) and β (13-30 Hz) rhythms, over the motor cortex area of the brain [START_REF] Pfurtscheller | Motor imagery and direct braincomputer communication[END_REF]. These variations are well studied and described in the literature, which makes them relatively easy to identify within a BCI. Moreover, MI does not require the use of external stimulus and as such enables the user to generate commands in a completely spontaneous way. When using a BCI based on MI, the detection of a given imagined movement is associated to a given control command.

BCI have been used to rotate the VE camera left or right by using imagination of left or right hand movement respectively [START_REF] Friedman | Navigating virtual reality by thought: What is it like?[END_REF] or by using specific brain responses to 2 different visual stimuli [START_REF] Touyama | Advances in human computer interaction[END_REF]. It has also been used to walk and stop along a virtual street by using foot and right hand motor imagery [START_REF] Leeb | Walking by thinking: The brainwaves are crucial, not the muscles! Presence: Teleoperators and Virtual Environments[END_REF], to steer a virtual car left or right (Ron-Angevin & Diaz-Estrella, 2009) or to navigate in a maze (Ron-Angevin, Diaz-Estrella, & Velasco-Alvarez, 2009), still using MI.

Leeb et al. have also proposed an interaction technique that enables users

to explore a virtual apartment by thoughts [START_REF] Leeb | Brain-computer communication: Motivation, aim and impact of exploring a virtual apartment[END_REF]. In this application, the user had to select at each junction the corridor (among two) he wanted to walk next, by using two MI tasks. Once the corridor selected, the user was moved automatically along predefined paths towards the next junction. However, it should be noted that in this application, only two commands were provided to the user and that the trajectories from one junction to the other were manually defined. Moreover, if the user wanted to go to a distant junction, he could not select directly this location, but had to go through all the intermediate junctions. Recently, Guger et al. have also proposed an application to explore a virtual home using a P300-based BCI [START_REF] Guger | Brain-computer interface for virtual reality control[END_REF]. The P300 is a neurophysiological brain signal, known as an event related potential, that is triggered by a relevant and rare stimulus [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. In this work, in order to navigate the VE, multiple visual stimuli were displayed to the user, each stimulus corresponding to a particular location in the virtual home. The user should focus on the stimulus corresponding to the place he wanted to go to, in order to actually move there. However, this system was not general-purpose in the sense that the stimuli used and the possible navigation locations were manually defined. As such, this system could not be used to automatically navigate a different VE. Finally, this system, together with the other interaction techniques described so far, are based on a synchronous BCI. These BCI can be used only during specific time periods, imposed by the system. Thus, with such a BCI, the user cannot send commands any time, which is not really convenient nor natural.

Very recently, some groups have started to investigate self-paced BCI, i.e., BCI that can be used to send commands at any time, at will [START_REF] Leeb | Self-paced exploration of the austrian national library through thought[END_REF][START_REF] Leeb | Self-paced (asynchronous) BCI control of a wheelchair in virtual environments: A case study with a tetraplegic[END_REF][START_REF] Mason | Evaluating the performance of self-paced BCI technology[END_REF]. Self-paced BCI can also recognize the so-called "non-control" mental state, i.e., they can recognize when the user does not want to send a mental command [START_REF] Mason | Evaluating the performance of self-paced BCI technology[END_REF]. Using such self-paced BCI, researchers have enabled users to freely move forwards in a virtual street [START_REF] Leeb | Self-paced (asynchronous) BCI control of a wheelchair in virtual environments: A case study with a tetraplegic[END_REF] or to explore a virtual library by walking along predefined paths [START_REF] Leeb | Self-paced exploration of the austrian national library through thought[END_REF]. A self-paced BCI has also been used in an entertaining VR application in which the user could lift a virtual object up by using foot MI [START_REF] Lotte | Self-paced brain-computer interaction with virtual worlds: a qualitative and quantitative study 'out-of-the-lab[END_REF].

Most interaction techniques described so far provided only a small number of commands to the user: generally two commands for the synchronous BCI mentioned above and a single command for the self-paced ones. Indeed, when using more mental states, i.e., more commands, the performance of the BCI, in terms of rate of correct mental state recognition, drops rapidly, hence generally compromising the reliability of the system [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF][START_REF] Decker | Information transfer rate in a five-classes brain-computer interface[END_REF]. The most advanced technique for navigating a VE with a motor-imagery based BCI is currently the work of Scherer et al. In this work, the user could freely navigate in a virtual garden by using a self-paced BCI which provided 3 different commands [START_REF] Scherer | Towards self-paced brain-computer communication: Navigation through virtual worlds[END_REF]. In this application, users could turn left, turn right, or move forward in the VE by performing left hand, right hand or foot MI respectively. As an evaluation of this system, 3 subjects had to collect 3 coins in the VE, within a given time window. The results showed that 2 subjects out of 3 successfully completed the task [START_REF] Scherer | Towards self-paced brain-computer communication: Navigation through virtual worlds[END_REF].

As a summary, even though the existing VR applications based on BCI are already quite impressive, there is still room for improvement. In particular, most of the commands provided are low level commands, e.g., turn left or right. This leaves the user in charge of all the details of the navigation task, which can be tiring and uncomfortable. Moreover, with low level commands, each mental state used for control is associated to a fixed low-level command, hence restricting the number of interaction tasks offered to the user. The next section describes a new interaction technique based on high level commands, in order to overcome these current limitations. This technique uses as input a new self-paced motor imagery-based BCI providing the user with 3 different commands. The interaction technique is described here within the context of exploration of a virtual museum by thoughts.

A novel interaction technique to explore large VE based on a BCI using high-level commands

The targeted application for our interaction technique is an application in which a user could visit a large VE such as a virtual museum by using thoughts only. This application should enable the user to navigate in the virtual museum and to look at the artworks displayed. The targeted input device is a self-paced BCI which can provide its user with 3 commands, respectively associated to left hand, right hand and foot MI (see Section 4).

In order to provide the user with a flexible interface and several possible choices of tasks though using only 3 mental commands, we propose a new interaction technique which relies on a binary tree approach. This technique is described in the following.

Using binary decision trees for selecting interaction tasks

With our interaction technique, the tasks available to the user at a given instant are organized according to a binary tree structure. This means that the possible tasks are recursively divided into two subsets of tasks and are placed at the nodes and leaves of a binary tree. In order to select a specific task, the user should first select one of the two subsets of tasks displayed, by using either left hand MI (to select the first subset) or right hand MI (to select the second subset). This choice done, the selected subset of tasks is again divided into two subsets and displayed to the user who should take another choice. This means that the current node of the binary tree has been changed to the root node of the left or right subtree. This process is repeated until the selected subset contains only a single task (i.e., until a leaf of the binary tree is reached), which task is then carried out by the system. Contrary to existing BCI-based interaction techniques in VR, this principle enables the user to perform different kinds of interaction tasks, e.g., switching from a virtual object selection task to a navigation task. This is currently impossible with other existing techniques for MI-based BCI.

In order to cope with human or BCI mistakes, BCI mistakes being much more common due to the modest performances of current mental state classifiers [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF], we also provide the user with an "undo" command. At any time, the user can perform foot MI in order to cancel the last choice he took and go back in the binary tree.

Based on this task selection principle, two navigation modes are provided to the user: the free navigation mode and the assisted navigation mode. The user can select one mode or the other by using left or right MI at the top of the binary tree (see Figure 1). When the user leaves a given mode by using the undo option (foot MI), the other mode is automatically selected in order to save time. Figure 1 displays the structure of this binary tree. As illustrated by this figure, it should be noted that this binary tree is composed of two parts: (1) a fixed part, used to switch between the two modes and to manage the free navigation mode, and (2) a dynamic part, constructed according to the user's viewpoint in the VE, and corresponding to the assisted navigation mode. The principle of this mode and the construction of the corresponding dynamic part of the tree are described in the next section.

Assisted navigation mode: use of high-level commands

In the assisted navigation mode, the user can select points of interest by using the binary tree selection mechanism presented before. The points of interest that the user can select at a given instant depend on his position and field of view, as the user can only select visible points of interest. The points of interest located the farthest on the left of the user's field of view are placed on the left subtree of the dynamic part of the binary tree (see Figure 1) whereas the points located the farthest on the right are placed on the right subtree. This process is recursively repeated until all the points of interest are arranged in the binary selection tree. As such, the user can naturally use left hand MI to select the set of points on the left, and right hand MI to select the set of points on the right. Naturally, this dynamic subtree is constructed every time the user moves, as this changes his field of view. Thus, it should be stressed that the size of the binary tree does not depend directly on the size of the VE but only on the number of points of interest that are currently visible by the user. This means that, with appropriately defined points of interest, the binary tree will remain relatively small, even for very large VE.

For the museum application, these points of interest could be either artworks or navigation points. Artworks represent any object exposed in the museum that is worthy of interest, such as a painting, a picture or a statue.

Different kinds of interaction tasks could be proposed to the user according to the kind of artwork. For instance, concerning a painting, the user could need to focus and zoom on some parts of the painting in a 2D manner. On the other hand, concerning a statue, the user could need to turn around the statue to observe it from various points of view. While artworks can already be selected in our current system, it should be mentioned that artwork manipulation tasks are not implemented yet. However, the binary tree could be easily extended to handle them in the near future.

Navigation points are points that the user can navigate to. More precisely, in order to walk in the VE, the user just needs to select the next navigation point from the available navigation points, using the binary tree selection technique. The application automatically drives the walk from the current point to the selected one. This relieves the user from all the cumbersome details of the trajectory to go from one point to the other. During this assisted walk, the user can perform foot MI at any time (i.e., undo) in order to stop the walking where he is. This could be useful if the user changes his mind or if, during the walk, the user spots an interesting artwork, not visible previously. Interestingly enough, these navigation points can be generated automatically by our approach, by extracting topographical information from the geometry of the 3D model (see Section 3.5).

Free navigation mode: turning the view point left or right

This navigation mode is a simple mode which provides low-level commands to the user. It enables the user to rotate the camera towards the left or towards the right by using left or right hand MI respectively. This mode enables the user to look around its current position, in order to localize his next destination or to look for a given artwork in the case of the virtual museum application.

Graphical representation and visual feedback

Providing relevant feedback to any VR user is essential. This is particularly important for BCI-based applications as the user can only rely on the feedback to know whether the mental task was correctly detected by the BCI [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. Moreover, results of a previous study have suggested that, for self-paced BCI-based VR applications, providing a continuous and informative feedback at any time may reduce the user's frustation and improve his learning [START_REF] Lotte | Self-paced brain-computer interaction with virtual worlds: a qualitative and quantitative study 'out-of-the-lab[END_REF]. Consequently, we pay attention to provide such a feedback to the users.

In our application, various colored icons are displayed dynamically on the screen (see Figure 2). Among these icons, 3 are continuously displayed: these icons provide feedback on the mental states identified by the system. These 3 icons represent a left hand (blue), a right hand (yellow) and feet (green) and are naturally associated to left hand, right hand and foot MI, respectively. When the VR application receives a given mental state from the BCI classifier, the size of the corresponding icon increases. As long as the same mental state is being received, the size of the icon keeps increasing until the number of consecutive states required is reached. Indeed, to make the control of the application more robust, we require that the same mental state is received several times in a row in order to execute the corresponding command. In other words, we used a dwell time. Dynamically changing the icon size depending on the BCI output enables us to provide feedback to the user even when the non-control state (any mental state except the targeted MI states) is finally detected, as recommended by a previous study [START_REF] Lotte | Self-paced brain-computer interaction with virtual worlds: a qualitative and quantitative study 'out-of-the-lab[END_REF]. When the icon reaches its maximum size, the corresponding command is executed. This command is also represented on screen under the form of an icon placed next to the corresponding mental state icon, and displayed with the same color. This aims at informing the user of what will happen if he performs a given mental command. As the available commands depend on the mode used, the icons representing the commands are also dynamically changed. The visible points of interest are displayed in the 3D scene using colored pins (see Figure 2). When using the assisted navigation mode, the user can select these points to go automatically from one point to another. In this mode, the (set of) point(s) that can be selected using left hand MI are displayed in blue, i.e. with the same color as the left hand icon, whereas the point or the set of points that can be selected using right hand MI are displayed in yellow, i.e., with the same color as the right hand icon. The other points, which cannot be selected anymore, are displayed in red and black. Figure 3 displays an example of selection of navigation points, on which we can see these colored pins. When selecting these points of interest, no command icon is displayed on screen, as the points of interest are colored according to the mental command needed to select them.

Implementation

We have designed and implemented our application and interaction technique in such a way that it can be used with virtually any 3D model. In this section, we first present the algorithm used to generate the navigation points and to compute automatically trajectories between these points, i.e., to perform path planning. Then we present the general software architecture of the whole application.

Generation of navigation points and path planning

In order to generate navigation points and to automatically compute trajectories between these points, we used the TopoPlan (Topological Planner) Figure 3: Assisted navigation in the virtual museum. Picture 1: the user selects the assisted navigation mode by using left hand MI from the starting node of the binary tree. Picture 2: the user selects the set of two navigation points located on the left by using again left hand MI. Picture 3: from these two points, the user finally selects the one on the right (at the back) by using right hand MI. Picture 4: the application automatically drives the walk to the selected point.

algorithm proposed by Lamarche [START_REF] Lamarche | Topoplan: a topological path planner for real time human navigation under floor and ceiling constraints[END_REF]. Using only the 3D geometry of a VE, Topoplan can extract a 3D topology of this environment. This topology defines the VE areas that can be navigated, i.e navigation points, and their accessibility. As such, Topoplan can be used to define a roadmap (a graph of navigation points) that enables any virtual entity to go from one point of the environment to the other without encountering any obstacle. In other words, Topoplan can be used to perform automatic path planning [START_REF] Lamarche | Topoplan: a topological path planner for real time human navigation under floor and ceiling constraints[END_REF].

Concerning the assisted navigation mode of our application, the challenge was to find a small number of relevant navigation points that enable the user to navigate the VE by selecting these points. To this end, we propose to filter the initial roadmap of navigation points. To do so, all the points from the roadmap are first given a mark. This mark corresponds to the distance between the point and the nearest obstacle. In other words, the highest a point mark, the more this point maximizes the coverage of the environment. The different points are then analysed by decreasing order of their associated mark. Let us define S(p), a function which associates to each point p its successors in the graph representing the roadmap. A point s ∈ S(p) is removed from the roadmap if and only if for all points x ∈ S(s), the path (x, p) can be navigated. If a point s is removed, the paths (x, p) such that p ∈ S(s) are added to the roadmap. This process is repeated until the algorithm converges. This algorithm enables us to filter the roadmap by keeping the points which maximize the visibility on the 3D scene in priority.

As such, it can greatly simplify the roadmap while ensuring that there is at least another visible point which can be selected to navigate the VE. The navigation points we used within our interaction technique correspond to the points that have been selected after this filtering.

However, even if these points appeared as functional, some of them did not have an optimal position, e.g., some of these were located in the corners of the rooms. Thus, after this automatic generation of points, we can still perform a manual optimization of their positions and eventually remove/add some points. However, it is worth noting that Topoplan can still generate automatic trajectories from point to point, should the points be generated automatically or by hand. As a result, any 3D model can be processed and used as input by our application

General software architecture

In order to use our BCI-based interaction technique, some offline operations are required beforehand. The main requirement is to have a 3D model of the VE. For our evaluations, we used the Google SketchUp modeling tool [START_REF] Google | Google sketchup website[END_REF] in order to create the 3D model of a virtual museum. In addition, an XML file, which contains the pathname of the artwork 3D models (if any) and the positions and orientations of these models in the museum, should be written. From the museum 3D model and the XML file, our application uses Topoplan to automatically generate the topology, the roadmap and the points of interest of the virtual museum. Optionally, these points of interest can be manually optimized. The BCI was implemented using the OpenViBE platform (INRIA, 2009). Within OpenViBE, a subject-specific BCI was calibrated for each user by using training EEG data from this user.

The main component of the software architecture is called the interaction engine. This software module processes the commands received from the BCI in order to perform the corresponding interaction tasks. The interaction engine uses the Ogre 3D engine (Ogre3D, 2009) to render the VE and to display the visual feedback. Finally, the interaction engine uses Topoplan in order to perform the automatic navigation between two points of interest.

The general software architecture is summarized in Figure 4.

The BCI system

As mentioned earlier, our interaction technique is based on a self-paced BCI which can provide its user with 3 mental commands. As such, this BCI is a 4-state self-paced BCI. It can indeed recognize the 3 mental states associated to each command -the Intentional Control (IC) states -plus the Non-Control (NC) state, i.e., any mental state that does not correspond to a control command. The following sections describe the preprocessing, feature extraction and classification methods we used to design this self-paced BCI.

Preprocessing

In order to record the MI brain signals, we used 13 EEG electrodes, located over the motor cortex areas. These electrodes were FC3, FCz, FC4, C5, C3, C1, Cz, C2, C4, C6, CP3, CPz and CP4 according to the international 10-10 system (American Electroencephalographic Society, 1991) (see Figure 5). In order to preprocess the signals recorded by these electrodes, we first band-pass filtered the raw EEG signals in the 4-45 Hz frequency band as this frequency band is known to contain most of the neurophysiological signals generated by MI [START_REF] Pfurtscheller | Motor imagery and direct braincomputer communication[END_REF]. Moreover, performing such a filtering can reduce the influence of various undesired effects such as slow variations of the EEG signal (which can be due, for instance, to electrode polarization) or power-line interference. To achieve this filtering, we used a Butterworth filter of order 4. In order to enhance the brain signals of interest, we also used a Surface Laplacian (SL) spatial filter [START_REF] Mcfarland | Spatial filter selection for EEG-based communication[END_REF]) over C3, C4 and Cz, leading to three Laplacian channels C3', C4' and Cz'. Features were extracted from these three new channels.

Feature extraction

For feature extraction we used logarithmic Band Power (BP) features as in [START_REF] Pfurtscheller | Motor imagery and direct braincomputer communication[END_REF]. Extracting a band power feature consists in filtering the EEG signal of a given electrode in a given frequency band, squar-Figure 5: Placement of electrodes in the 10-10 international system. The 13 electrodes we used in our BCI are displayed in red.

ing the results and averaging it over a given time window, and finally taking its logarithm. Such a feature extraction is a popularly employed method which has been proved efficient for motor imagery-based BCI [START_REF] Pfurtscheller | Motor imagery and direct braincomputer communication[END_REF][START_REF] Zhong | Classifying EEG for brain computer interfaces using gaussian processes[END_REF]. To obtain a more efficient BCI, we extracted several band power features in different frequency bands for the different Laplacian channels and selected the most relevant ones using the Sequential Forward Floating Search (SFFS) feature selection algorithm [START_REF] Pudil | Floating search methods for feature selection with nonmonotonic criterion functions[END_REF]. This algorithm is indeed one of the most popular and efficient feature selection techniques [START_REF] Jain | Feature selection: evaluation, application, and small sample performance[END_REF]. More precisely, we investigated BP features extracted in 2 Hz wide frequency bands between 4 and 34 Hz, with a 1 Hz step, and selected the 12 most efficient features using the SFFS algorithm. We indeed observed that using more than 12 features did not increase significantly the performance whereas it increased the computational burden. As our BCI is self-paced, it requires continuous classification of EEG signals. Consequently, we extracted a BP feature vector 16 times per second, over the last 1 s time window (i.e., using a sliding window scheme).

Classification

For classification we used a Fuzzy Inference System (FIS) as described in [START_REF] Lotte | Studying the use of fuzzy inference systems for motor imagery classification[END_REF]. Such a classifier can learn fuzzy "if-then" rules from data, and use them to classify unseen data. This classifier has been reported to have many advantages for BCI, among which its efficiency for self-paced BCI design [START_REF] Lotte | Pattern rejection strategies for the design of self-paced eeg-based brain-computer interfaces[END_REF].

In order to design a self-paced BCI we relied on a pattern rejection approach. More precisely, we used the reject class technique, as this seemed to be the most efficient method according to results of a previous study [START_REF] Lotte | Pattern rejection strategies for the design of self-paced eeg-based brain-computer interfaces[END_REF]. With this technique, our FIS classifier will have to deal with 4 classes: one for each mental state used for control (Intentional Control states), and one class for all other mental states (Non Control state).

In order to make our self-paced BCI more robust, we also used a dwell time and a refractory period [START_REF] Townsend | Continuous EEG classification during motor imagery-simulation of an asynchronous BCI[END_REF]).

When using a dwell time, a given control command is generated only if the classification identifies the same class N D times in a row. Similarly, when using a refractory period, the N R classifications which immediately follow the identification of an IC state will be forced to the NC state. These two techniques lead to less False Positives (FP), i.e., to less identifications of an IC state instead of an NC state. In our system we used N D = N R = 7. These values were defined experimentally according to users' preferences.

This new BCI design enables the user to employ 3 different commands in a self-paced manner. So far, very few groups have proposed a self-paced BCI with that many commands, exceptions being [START_REF] Scherer | Towards self-paced brain-computer communication: Navigation through virtual worlds[END_REF] and [START_REF] Bashashati | Towards development of a 3-state self-paced brain-computer interface[END_REF]. Moreover, according to previous studies [START_REF] Lotte | Pattern rejection strategies for the design of self-paced eeg-based brain-computer interfaces[END_REF], this new design is expected to have better performances than these existing designs.

Evaluation

The aim of the evaluation was to identify whether the interaction technique we proposed was usable and efficient. Additionally, it was interesting to assess the performance of the BCI and the users' experience. To do so, we studied the performances of 3 participants who used our interaction technique to navigate from room to room in a virtual museum. As a comparison, participants also had to perform the same task by using the state-of-the-art method, i.e., turning left, turning right or moving forwards by using left hand, right hand and foot MI, respectively, as in [START_REF] Scherer | Towards self-paced brain-computer communication: Navigation through virtual worlds[END_REF]. This aimed at comparing our high-level approach to a low-level one. Naturally, in order to perform a fair comparison, we used the same BCI design (the BCI presented in Section 4) for both interaction techniques. This BCI design is different from the one used by Scherer et al. [START_REF] Scherer | Towards self-paced brain-computer communication: Navigation through virtual worlds[END_REF], as we used a different classification algorithm, different electrodes, different signal processing techniques, etc. However, according to previous studies [START_REF] Lotte | Pattern rejection strategies for the design of self-paced eeg-based brain-computer interfaces[END_REF], the algorithms we used should give better recognition performances than the BCI design used originally by Scherer et al. Moreover, in this paper we mainly aimed at comparing the interaction techniques, but not the BCI designs. It should also be mentioned that we did not use a refractory period with the low-level interaction technique. Indeed, this enabled the user to maintain the MI task in order to maintain the movement in the museum and as such to move continuously rather than by periods. When using the low-level technique, the user moved by about 25 cm for each classifier output labelled "foot MI" (forward movement), knowing that the classifier provides 16 outputs per second. In other words, with the low level approach, the user could move at up to 4 m per second (16 × 0.25).

When using the high-level technique, participants moved at the speed of about 2.5 m per second during the automatic navigation phases. Thus, it should be mentioned that the movement speed was set slightly lower with the high-level technique. Indeed, fast passive movements may be visually uncomfortable for the user.

As a measure of performance, we evaluated the time needed by each participant to perform different navigation tasks. As this evaluation was dedicated to navigation tasks, artworks were not considered during this experiment, which means users could not interact with them. However, artworks were still displayed in order to provide an engaging VE. The next sections describe the virtual museum used, the population and apparatus, the task participants had to do, the experimental procedure and finally the results obtained.

Virtual museum

For this evaluation, we used a fictional virtual museum. This museum was composed of 8 different rooms each one containing either several pictures or a statue (see Figure 6).

Population and apparatus

For the experiment, EEG signals were recorded using a Nexus 32b EEG machine from the Mind Media company, at a sampling frequency of 512 Hz.

Three participants took part in this experiment (S1,S2 and S3, all males, mean age: 25.7±2.11), two of which had a previous MI-based BCI experience (S1 and S2). The experimental setup is displayed on Figure 7.

Figure 6: 3D model of the virtual museum used in our experiment on which some paths followed by the participants (short, medium, long) are visible.

Navigation tasks

For this evaluation, participants had to navigate from one room to another as fast as they could. There were three possible distances to cover: short, medium and long paths. These tasks consisted in passing through 3, 4 or 5 rooms respectively (see Figure 6 for examples).

Procedure

Before taking part in the virtual museum experiment, each participant performed three to five sessions following the training protocol of the 3-class Graz BCI [START_REF] Pfurtscheller | Motor imagery and direct braincomputer communication[END_REF], in order to record training data for selecting the features and training the FIS classifier. These sessions were recorded on a different day than the virtual museum experiment. During a session, the participant had to perform 15 trials for each class (left hand, right hand and foot MI) i.e., 15 repetitions of each mental state for a duration of 4 seconds. As no training classifier was available for these first sessions, the participant was not provided with any feedback. The participant was also instructed not to perform any MI nor real movement during the inter trial periods as the data in these periods are used as examples of the NC state. Once these sessions were completed, the features were selected and the classifier was trained on these data.

Each participant had to perform each kind of navigation task twice for each of the two interaction techniques. The order of the tasks was arranged in blocks. Within a block, participants used a single interaction technique and had to perform once each kind of navigation task (short, medium and longthe actual order being randomized within the block). The order of the blocks was also randomized. These blocks were achieved over different days as the tasks were too tiring for the participants to be carried out on a single day.

Each day, the duration of the experiment was approximately 2 to 2.5 hours, including the time required for electrode montage. At the beginning of each day, the participant participated in three sessions of the training protocol of the 3-class Graz BCI (see above). During these sessions, the participant was provided with a feedback thanks to a classifier trained on data recorded during the previous day of experiment. It should be noted that the BCI used for these sessions was synchronous, which means that the NC state was not recognized by the BCI. However, participants were instructed not to perform any MI nor real movement during the inter trial periods as the data in these periods would be used as examples of the NC state for training the final self-paced BCI. Once the three sessions were completed, the classifier was re-trained on these new data, in order to adapt to the participant's current EEG signals and in order to design the self-paced BCI. Then the experiment with the virtual museum could begin. After the experiments were completed, subjects were asked to fill in a subjective questionnaire.

Results

As mentioned above, the main purpose of the evaluation was the efficiency of the interaction technique we proposed. In the mean time, we assessed the performance of the BCI system we designed and recorded the users' experience. The results of these different evaluations are reported below.

BCI performance

First, it should be mentioned that all participants managed to reach all the rooms they were instructed regardless of the interaction technique used. This suggests that participants could actually control the application and explore the virtual museum by using the proposed BCI system. As the BCI is selfpaced and as participants send commands at their free will, it is not possible to report on the online classification performances of the BCI. Indeed, we cannot compare the mental state recognized by the BCI with the mental state performed by the participants as we did not know what the participants intented to do at a given instant. However, in order to evaluate whether our BCI system works better than a randomly performing BCI (i.e., a BCI which is unable to properly identify any mental state), we simulated a random BCI.

More precisely, we sent randomly selected mental states to the application as if it was the mental state identified by the classifier. We performed these simulations twice for each interaction technique, the "objective" of the random classifier being to perform a short navigation task. For each task, even after 30 minutes of simulation (i.e., 1800 seconds), the random BCI was not able to reach the targeted room. By comparison, the average time required to go from one room to another (situated at a distance of 3 rooms or more) with our BCI, independently from the interaction technique or the navigation task, was 331.5 seconds, i.e., approximatly 5.5 minutes (see tables below). This suggests that our BCI indeed provided control to the user. These results show that, in this study, navigating from one room to another by using high-level commands is on average almost twice as fast as when using low-level commands. It should also be reminded that the movement speed was slightly lower with high-level commands than with low-level commands (see Section 5), which further emphasizes the difference in efficiency. This difference is logically smaller for short paths and larger for long paths. A paired t-test [START_REF] Norman | Biostatistics: The bare essentials[END_REF] com-paring the time needed to navigate using high-level commands and using low-level commands, over all blocks, navigation tasks, and participants (i.e.,

Interaction technique efficiency

with n = 3 participants × 2 blocks × 3 navigation tasks = 18 measures for each type of interaction technique), revealed that high-level commands were significantly faster (p < 0.001). In order to also investigate the presence of interactions between the technique used and the distance covered, we performed a two-way ANOVA [START_REF] Norman | Biostatistics: The bare essentials[END_REF] with factors technique (levels: low-level commands and high-level commands) and distance (levels: short, medium and long). The results revelead a significant effect for both factors technique and distance (p < 0.01), but no significant interaction technique × distance (p = 0.19). However, this lack of significant interaction might be due to the small sample size.

The time needed to navigate was of course related to the number of commands that participants had to send. Table 2 displays the average number of commands needed by each participant to accomplish the different navigation tasks, according to the interaction technique.

This table shows that participants sent on average 67 commands to go from the starting room to the destination room using low level commands, and only 36 commands to do the same thing using high-level commands.

Thus, these results suggested that by using high-level commands, users need to send to the application almost twice as few commands as with low-level commands, hence performing their navigation tasks faster. However, it should be mentioned that a paired t-test revealed no significant difference Again, this lack of significance might be due to the small sample size.

Users' experience

The analysis of the questionnaires shown that all subjects found navigation with points of interest less tiring than low-level navigation, as they could relax and rest during the automatic navigation periods from one point of interest to the other. Interestingly enough, a user reported that, with lowlevel commands, he sometimes stopped sending mental commands to rest a bit, as continuously performing MI was too tiring for him. In contrast, with high-level commands, this user reported that he used the automatic navigation phase to rest. In the questionnaire, participants also reported that the advantage of using an assisted navigation was particularly striking when an obstacle (e.g., a statue) was present in the path. Indeed, with the low-level technique, many commands were necessary to walk round the obstacle whereas when using high-level navigation in the same situation, the application avoided the obstacle on its own, without additional demand on the user.

Overall, the questionnaire revealed that participants have found navigation using points of interests more efficient and generally as much as or more comfortable than navigation based on low-level commands. On the other hand, questionnaires revealed that using points of interest was less simple and intuitive than using low-level commands. In other words, it seems that the increase in efficiency and comfort obtained by using highlevel commands was gain at the expense of a more complex and less natural navigation. However, these latter issues seemed to vanish after some pratice and training, as the participants finally got used to this high-level technique.

Moreover, participants reported that, between the two techniques, they all prefered using the one based on high-level commands.

Discussion

As expected, navigating using points of interest enables participants to send only a few high-level commands while leaving most of the work to the application itself. It should be stressed that this application is independent of the BCI. High-level commands enable users to navigate VE more rapidly by thoughts, especially when the VE is large and the path is long. Indeed, it takes approximatly the same time to select close and far points of interest, provided that they are both visible. Leaving the application in charge of performing the low-level parts of the interaction tasks also enables the user to perform only a limited number of mental tasks as well as to rest. This makes the whole technique more comfortable and less tiring for the user, as reported by the subjects during the experiments. Indeed, when using lowlevel techniques to navigate, the user generally has to send mental commands continuously, which can become rapidely exhausting.

Although only 3 participants participated in this experiment, we should stress that this is a typical number of participants for current BCI experiments in VR, as these experiments are tedious, long and demanding for the participants [START_REF] Leeb | Walking by thinking: The brainwaves are crucial, not the muscles! Presence: Teleoperators and Virtual Environments[END_REF][START_REF] Scherer | Towards self-paced brain-computer communication: Navigation through virtual worlds[END_REF][START_REF] Friedman | Navigating virtual reality by thought: What is it like?[END_REF].

A limitation of our approach is that it prevents users from going everywhere as they can only go from navigation point to navigation point. However, it is still possible to add more points of interest in order to increase the user's mobility. We could also grant the user with additional low level navigation techniques. Indeed, thanks to the binary tree selection principle, any number of new modes or tasks can be easily added to the interface. For instance, this may enable the use of two techniques successively: high-level commands to reach a far destination quickly and low-level commands for precise and final fine-grain positionning.

It is also worth noting that any other 3D scene could be used with our approach. Indeed, generating the navigation points and performing path planning can be done completely automatically using only the geometry of the VE, thanks to Topoplan [START_REF] Lamarche | Topoplan: a topological path planner for real time human navigation under floor and ceiling constraints[END_REF].

The proposed system is based on Motor Imagery (MI). Other brain signals could have been used to drive this application, such as the P300 [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. Indeed, some recent works have reported applications using a P300-based BCI to navigate in a virtual home [START_REF] Guger | Brain-computer interface for virtual reality control[END_REF] or in real environments (for wheelchair control) [START_REF] Rebsamen | Controlling a wheelchair indoors using thought[END_REF][START_REF] Iturrate | Non-invasive brainactuated wheelchair based on a P300 neurophysiological protocol and automated navigation[END_REF], by selecting a destination among several.

In our opinion, both P300 and MI have pros and cons. The P300 can be used without any (human) training and it can be used to select among many targets (typically more than 3) with high information transfer rates. On the other hand, the P300 relies on external stimuli, these stimuli being generally visual. As such, the resulting BCI is necessarily synchronous in the sense it is dependent on the stimuli presentation. Moreover, these visual stimuli can be annoying or uncomfortable for the user, as it usually consists of flashing stimuli. Finally, it requires the user's attention to be focused on these stimuli, which prevents him from drawing its attention on other parts of the real or virtual environment. For instance, in the works presented in [START_REF] Guger | Brain-computer interface for virtual reality control[END_REF][START_REF] Rebsamen | Controlling a wheelchair indoors using thought[END_REF][START_REF] Iturrate | Non-invasive brainactuated wheelchair based on a P300 neurophysiological protocol and automated navigation[END_REF], the user had to look at a separate screen which was in charge of providing the visual stimulus, hence preventing him from observing simultaneously the environment. This can be unconvenient when the environment itself is visually interesting or informative, e.g., as in our application with a virtual museum and artworks to observe.

On the contrary, MI-based BCI are not based on any stimulus. As such, they can be used to send command to the application at any time, when the user decides to. Moreover they leave the visual attention of the user available, hence allowing him to look at the environment he is exploring, possibly while sending mental commands at the same time. Finally, MI can be seen as a natural and intuitive mental task to use. However, when used to select among several targets, MI-based BCI are most probably slower than P300based BCI. MI-based BCI may also require training for the user, although it has be shown that with appropriate signal processing and classification this training can be reduced or suppressed [START_REF] Blankertz | The non-invasive Berlin brain-computer interface: Fast acquisition of effective performance in untrained subjects[END_REF].

In this paper, we have explored the use of MI-based BCI for exploring VE. However, we believe that a promising extension of the current work could be to use a hybrid BCI based on both MI and the P300. For instance, the part of our interaction technique based on visual information, such as the selection of points of interest, could be achieved using the P300. In that case, the points of interest could be randomly flashing, and the user would select one by focusing its visual attention on it. Other interaction tasks, e.g., rotating the camera towards the left or right, cancelling the last task performed, or interacting with an artwork (which requires the user's attention to be available) could be achieved using MI. This will be explored in a future work.

Conclusion

In this paper, we have presented a new interaction technique to explore large Virtual Environments (VE) by using a Brain-Computer Interface (BCI)

based on Motor Imagery (MI). This technique is based on points of interest and enables participants to send only a few high-level commands to the application in order to navigate from one point of interest to the other.

The actual navigation between two points of interest is carried out by the application itself (which is independent of the BCI), leaving the user free to rest or to observe the VE. Interestingly enough, the computation of these points and the navigation between these points can be achieved completely automatically. In order to select these points of interest, the user can employ a selection mechanism which relies on a binary tree and depends on the user's viewpoint. This binary tree mechanism provides a unified framework for BCI-based interaction in VR. Indeed, it enables, for instance, to select navigation commands or object manipulation commands in the same way. In addition, we also proposed a new design of a self-paced BCI which can issue three different commands, these commands being associated to left hand, right hand and foot MI, respectively. This BCI design is based on a fuzzy inference system with a reject option.

An evaluation of our system has been conducted within the framework of a virtual museum exploration. Results suggest that our method was efficient as, using our technique based on high-level commands, a user could navi-gate the museum almost twice as fast as when using low-level commands.

Moreover, our technique appeared as less tiring and more comfortable than the low-level approach. This suggests that our whole system may be used to navigate and interact with virtual worlds more efficiently and with more degrees of freedom than current approaches based on motor imagery. Thus, the proposed approach might enable both healthy and disabled users to interact seamlessly with the same virtual world, thanks to a BCI.

Future work could deal with evaluating this application with disabled people, in order to assess to which extent the application can enable them to visit virtually various museums or even other buildings which they might not be able to see for real. It should also be interesting to further improve the self-paced BCI in order to enable the user to navigate more easily and faster.

For instance, we could use more advanced feature extraction techniques such as Common Spatial Patterns (CSP) [START_REF] Blankertz | Optimizing spatial filters for robust eeg single-trial analysis[END_REF] or inverse solutions based algorithms [START_REF] Lotte | FuRIA: An inverse solution based feature extraction algorithm using fuzzy set theory for braincomputer interfaces[END_REF]. Finally, we could explore and add various interaction commands to our binary tree mechanism. For instance, interaction techniques such as Navidget [START_REF] Hachet | Navidget for easy 3D camera positioning from 2D inputs[END_REF]) could be adapted for BCI purposes in order to easily observe the museum virtual statues.

Figure 1 :

 1 Figure 1: Architecture of the binary tree used for task selection. It should be noted that the architecture of the subtree surrounded by a red line is dynamic. Indeed, the number of nodes in this subtree and their arrangement change automatically with the user's view point (see Section 3.2 for details).

Figure 2 :

 2 Figure 2: Graphical representation of the BCI-based virtual museum application and its associated interaction technique.

Figure 4 :

 4 Figure 4: General software architecture. Left: offline processing. Right: online processing.

Figure 7 :

 7 Figure 7: Setup of the experiment.

Table 1 :

 1 Table1displays the average time needed by each participant to accomplish the different navigation tasks (short, medium or long paths), according to the interaction technique. Average time (in seconds) needed by each participant to accomplish the different navigation tasks using the two interaction techniques.

	interaction navigation	S1	S2	S3	mean overall
	technique	task				mean
		long	158 381.9 157.6 232.5
	High-level	medium	204.5 640.6 146 330.37 232.01
	commands	short	135 151.4 113.1 133.17
		long	702 530.5 598.1 610.2
	Low level	medium	554.5 598.7 336.8 496.67 433.84
	commands	short	235	85.7 263.3 194.67

Table 2 :

 2 Average number of commands needed by each participant to accomplish the different navigation tasks using the two interaction techniques.

	interaction navigation S1	S2	S3 mean overall
	technique	task				mean
		long	32.5 43	24 33.17
	High-level	medium	33 76.5 44 51.17 36.11
	commands	short	20 26.5 25.5	24
		long	142	53 97.5 97.5
	Low-level	medium	131 32.5 63	75.5	66.89
	commands	short	19 10.5 53.5 27.67
	(p = 0.1) between the number of commands needed with each technique.

Acknowledgments

This work was supported by the French National Research Agency within the OpenViBE project and grant ANR05RNTL01601. During this work, Fabien Lotte was at INRIA Rennes, France. He now is at the Institute for Infocomm Research (I2R), Singapore.


Jean De Dieu

Carine Lucas
email: carine.lucas@univ-orleans.fr

Enrique Fern Ández-Nieto

AN ENERGETICALLY CONSISTENT VISCOUS SEDIMENTATION MODEL

Keywords: Sedimentation, shallow water, viscous models, energetic consistency, stability. AMS Subject Classification: 35Q30

In this paper we consider a two dimensional viscous sedimentation model which is a viscous Shallow-Water system coupled with a diffusive equation that describes the evolution of the bottom. For this model, we prove the stability of weak solutions for periodic domains and give some numerical experiments. We also discuss around various discharge quantity choices.

Introduction.

Phenomena related to sediment transport have a huge interest as they affect human life and earth morphology in a determinant way. Indeed, the geomorphological evolution of rivers under the effect of hydrodynamic transport of sediments constitutes a fundamental problem for rivers management, estimates of environmental risks and prevention of floods. The analysis of sediment transport is then important to predict and prevent natural disasters.

For this purpose, many physical and mathematical models are proposed in the literature in order to predict the bed evolution and the changes in water regime when such unsteady flows occur. Physical experiments are used in particular to calculate local scouring phenomena, such as the local erosion around bridge piers or the scour hole due to a jet issued from an underflow gate. However, when problems with large space or times scales have to be solved, a mathematical model is generally required. Among the mathematical models, the most often used is based on the Saint-Venant-Exner equations. This model, studied numerically in Refs. 10 and 19 for example, couples an hydrodynamic Saint-Venant (Shallow-Water) system to a morphodynamic bed-load transport sediment equation (similar to the one introduced in Ref. 21) as follows:

∂ t h + divq = 0, (1.1)
∂ t q + div q ⊗ q h + 1 F r 2 h∇(h + z b ) = 0, (1.2)
and ∂ t z b + ξdiv(q b (h, q)) = 0 (1.3) where F r is the Froude number (square root of the ratio between kinetic and gravitational energy), z b is the movable bed thickness, ξ = 1/(1ψ 0 ) with ψ 0 the porosity of the sediment layer and q b denotes the solid transport flux or sediment discharge. It depends on the height h of the fluid and the water discharge q = hu, where u is the velocity (see Fig. 1).

u(t, x): velocity For the solid transport flux q b , there exist several formulas in the literature: the Grass equation, [START_REF] Grass | Sediment transport by waves and currents[END_REF] the Meyer-Peter and Muller equation, [START_REF] Meyer-Peter | Formula for bed-load transport[END_REF] or the formulas of Fernández Luque and Van Beek, of Van Rijn, or of Nielsen [START_REF] Ferreiro | Development of post-process technics of hydrodynamics flux, modelization of sediment transport problems and numerical simulation through finite volume technics[END_REF][START_REF] Savary | Transcritical transient flow over mobile beds, boundary conditions treatment in a two-layer shallow-water model[END_REF] . All of them are obtained using empirical methods. The most basic sediment model is the Grass equation, where the sediment movement begins at the same time as the fluid motion. In this case, the solid transport flux is given by

q b (h, q) = A g q h q h mg = A g |u| mg u, 0 ≤ m g ≤ 3, (1.4)
where the constant A g includes the effects due to grain size and kinematic viscosity. However, system (1.1)- (1.3) does not take the viscosity into account. In the viscous case, we have to consider the viscous version of the Shallow-Water system. Several choices have been considered in the literature for the viscous term 14 : in Ref. 17, the author chooses the Laplacian and obtained an existence result, but this system is not energetically consistent. In Refs. 1 and 2, the viscous terms are div(h∇u) or div(hD(u)), which gives an energetically consistent system. In this case, the authors proved the existence of global weak solutions. The key point in those papers is to show that the structure of the diffusion term provides some extra regularity for the density thanks to a new mathematical entropy inequality named BD entropy. But note that the stability result is obtained using drag and turbulence terms or capillarity. Recently, keeping this choice of viscous terms but without any additional regularizing terms, Mellet and Vasseur 15 proved the stability of a class of barotropic compressible Navier-Stokes equations, which includes the case of the viscous energetically consistent Shallow-Water system. This paper also uses the new BD entropy with an extra key point which gives bounds on hu 2 in a better space than L ∞ (0, T ; L 1 (Ω)), thanks to new multipliers, namely |u| k u and u + u log(1 + |u| 2 ).

Let us recall now the existing results on the viscous sedimentation models, that are a viscous Shallow-Water system coupled with an evolution equation for the bottom. A recent work [START_REF] Toumbou | An existence theorem for a 2-D Coupled Sedimentation Shallow-Water Model[END_REF] has been done on a model that couples the Shallow-Water system studied by Orenga

∂ t h + div(hu) = 0 (1.5) ∂ t u + u • ∇u + 1 F r 2 ∇(h + z b ) -∆u = f, (1.6)
with a Grass equation (1.4) satisfying

ξq b = hu. (1.7)
As the authors assume small variations of the free surface around a fixed level (z = cst), they replace h by -z b in (1.7). Then they follow the lines given by Ref. 17: thanks to a Brower fixed point on the finite dimensional problem, they get global existence results assuming the data to be small enough.

In this paper, we propose a new viscous sedimentation model, stable and energetically consistent. It consists in coupling a viscous Shallow-Water system with a sediment diffusive equation in a bounded domain with periodic boundary conditions, that is Ω = T 2 . More precisely, if we denote by ν the non-dimensional viscosity (ν = 2/Re, where Re is the Reynolds number) and A a positive constant, we consider the following system

∂ t h + div(hu) = 0, (1.8) ∂ t (hu) + div(hu ⊗ u) + h∇(h + z b ) F r 2 -νdiv(hD(u)) = 0, (1.9)
∂ t z b + Adiv h|u| k u - ν 2 ∆z b = 0, (1.10)
with the initial conditions

h| t=0 = h 0 ≥ 0, z b | t=0 = z b0 , hu| t=0 = m 0 , (1.11)
where D(u) is the symmetric part of the gradient, D(u) = (∇u + t ∇u)/2, F r > 0 denotes the Froude number, k is a positive real number satisfying 0 < k < 1/2. The initial data are taken in such a way that

h 0 ∈ L 2 (Ω), z b0 ∈ L 2 (Ω), |q 0 | 2 h 0 ∈ L 1 (Ω), ∇ h 0 ∈ (L 2 (Ω)) 2 .
(1.12)

After stating the main results in Sec. 2, we establish, in Sec. 3 some energy and entropy relations that give us a priori estimates. These estimates are then used in Sec. 4 to prove the announced theorem. We also propose, in Sec. 5, two other models of sedimentation, inspired by the works mentioned above. More precisely, we first study the model considered in Ref. 21 but with the viscous Shallow-Water system (1.8)-(1.9) and, in a second part, we introduce one of the multipliers used by Mellet and Vasseur in the sediment equation. Lastly, we conclude, in Sec. 6, with numerical experiments on these new models.

Main results.

In this part, we first recall the definition that will be used in the following. We then give the main theorem of this paper that will be proved in Sec. 4.

Notion of weak solutions.

We shall say (h, q = hu, z b ) is a weak solution of (

√ hu ∈ L ∞ (0, T ; (L 2 (Ω)) 2 ), √ h∇u ∈ L 2 (0, T ; (L 2 (Ω)) 4 ), h 1/(k+2) u ∈ L ∞ (0, T ; (L k+2 (Ω)) 2 ), h + z b ∈ L ∞ (0, T ; L 2 (Ω)), ∇h + ∇z b ∈ L 2 (0, T ; (L 2 (Ω)) 2 ), ∇ √ h ∈ L ∞ (0, T ; (L 2 (Ω)) 2 ), h 1/k D(u) 2/k u ∈ L k (0, T ; (L k (Ω)) 2 ),
• h and z b are in C 0 (0, T ; H -s (Ω)) and hu is in C 0 0, T ; (H -s (Ω)) 2 for s large enough.

Main theorem.

The main result of this paper is the following:

Theorem 2.1. Let (h n , q n = h n u n , z bn ) be a sequence of weak solutions of (1.8)-(1.10) satisfying entropy inequalities (3.1), (3.4), with initial data

h n| t=0 = h n 0 (x), h n u n| t=0 = q n 0 (x) and z bn |t=0 = z n b0 (x),
where h n 0 , z n b0 and u n 0 verify

h n 0 ≥ 0, h n 0 → h 0 in L 1 (Ω), z n b0 → z b0 in L 1 (Ω), q n 0 → q 0 in L 1 (Ω), (2.1)
and satisfy the following bounds:

Ω h n 0 |u n 0 | 2 2 + |h n 0 + z n b0 | 2 2 + h n 0 |u n 0 | k+2 k + 2 < C, Ω ∇ h n 0 2 < C and Ω |h n 0 | < C. (2.2)
Then, up to a subsequence, h n , q n and z bn converge strongly in C 0 (0, T ; L 2p/(2+p) (Ω)), C 0 (0, T ; W -1,2p/(2+p) (Ω)) and C 0 (0, T ; L 2p/(2+p) (Ω)) respectively to a weak solution of (1.8)-(1.10) satisfying entropy inequalities (3.1) and (3.4).

Energy estimates and BD entropy.

In this section, we give some energy and entropy inequalities. These relations will be used in Sec. 4 where we prove Theorem 2.1. But let us first recall the energy inequality in the inviscid case, for system (1.1)-(1.3).

The case without viscosity.

Lemma 3.1. Let (h, q, z b ) be a smooth solution of the system

∂ t h + divq = 0, ∂ t q + div (hu ⊗ u) + 1 F r 2 h∇(h + z b ) = 0, ∂ t z b + Adiv h|u| k u = 0.
Then the following identity holds:

1 2 d dt Ω h|u| 2 + A k + 2 d dt Ω h|u| k+2 + 1 2F r 2 d dt Ω |h + z b | 2 = 0.
The proof of this lemma will be included in the viscous case.

The viscous case.

From now on, we consider the viscous system (1.8)-(1.10).

Proposition 3.1. Let (h, q, z b ) be a smooth solution of (1.8)-(1.10). Then the following energy inequality holds:

1 2 d dt Ω h|u| 2 + 1 2F r 2 d dt Ω |z b + h| 2 + A k + 2 d dt Ω h|u| k+2 + ν 2F r 2 Ω ∇h • ∇z b + ν 2F r 2 Ω |∇z b | 2 + ν 4 Ω h ∇u + t ∇u 2 + 1 -2k 4 Aν Ω h ∇u + t ∇u 2 |u| k ≤ 0. (3.1)
Proof. We multiply Eq. (1.9) by u, and integrate on Ω. This gives, using (1.8):

Ω h∂ t u • u + Ω (hu • ∇)u • u + 1 F r 2 Ω h∇(h + z b ) • u -ν Ω div (hD(u)) • u = 0.
Now let us simplify each term:

• Ω h∂ t u • u + Ω (hu • ∇)u • u = 1 2 d dt Ω h|u| 2 , • Ω h∇(h + z b ) • u = Ω (h + z b ) ∂ t h = 1 2 d dt Ω h 2 + Ω z b ∂ t h , • Ω div (hD(u)) • u = - Ω hD(u) : ∇u = - 1 4 Ω h ∇u + t ∇u 2 .
Substituting all these terms, we get:

1 2 d dt Ω h|u| 2 + 1 2F r 2 d dt Ω h 2 + 1 F r 2 Ω z b ∂ t h + ν 4 Ω h ∇u + t ∇u 2 = 0. (3.2)
Contrary to the study of the classical Shallow Water system, we cannot make any assumption on the regularity of the bottom z b : we have to use the energy relations to get such properties. That is the reason why we are led to carry on the calculation. We multiply Eq. (1.9) by |u| k u and we integrate on Ω:

Ω h∂ t u • |u| k u + Ω (hu • ∇)u • |u| k u + 1 F r 2 Ω h∇(h + z b ) • |u| k u -ν Ω div (hD(u)) • |u| k u = 0.
Here again, we study separately each term:

• Ω h∂ t u • |u| k u + Ω (hu • ∇)u • |u| k u = 1 k + 2 d dt Ω h|u| k+2 , • 1 F r 2 Ω h∇(h + z b ) • |u| k u = - 1 F r 2 Ω (h + z b )div h|u| k u .
Then we use Eq. (1.10) to write:

1 F r 2 Ω h∇(h + z b ) • |u| k u = - ν 2AF r 2 Ω (h + z b )∆z b + 1 AF r 2 Ω (h + z b )∂ t z b = ν 2AF r 2 Ω ∇h • ∇z b + ν 2AF r 2 Ω |∇z b | 2 + 1 AF r 2 Ω h∂ t z b + 1 2AF r 2 d dt Ω z b 2 , • Ω div (hD(u)) • |u| k u = - 1 4 Ω h ∇u + t ∇u 2 |u| k -k Ω (hD(u)u • ∇) u • u|u| k-2 ,
and

Ω (hD(u)u • ∇) u • u|u| k-2 ≤ 2 Ω h|D(u)| 2 |u| k .
Gathering all these results, we are led to:

1 k + 2 d dt Ω h|u| k+2 + ν 2AF r 2 Ω ∇h • ∇z b + ν 2AF r 2 Ω |∇z b | 2 + 1 AF r 2 Ω h∂ t z b + 1 2AF r 2 d dt Ω z b 2 + 1 -2k 4 ν Ω h ∇u + t ∇u 2 |u| k ≤ 0. (3.3)
Now we multiply Eq. (3.3) by A and we add Eq. (3.2): we find the proclaimed inequality.

However, we still do not know the sign of the integral of ∇h • ∇z b . To get more information, we study the BD entropy. Proposition 3.2. For (h, q, z b ) solution of the model (1.8)-(1.10), we show the following relation:

1 2 d dt Ω h|u + ν∇ log h| 2 + 1 F r 2 d dt Ω |z b + h| 2 + 2A k + 2 d dt Ω h|u| k+2 + 1 2 d dt Ω h|u| 2 + ν F r 2 Ω |∇(h + z b )| 2 + ν 4 Ω h ∇u + t ∇u 2 + ν 4 Ω h ∇u -t ∇u 2 + 1 -2k 4 Aν Ω h ∇u + t ∇u 2 |u| k ≤ 0. (3.4)
The proof relies on the following lemma:

Lemma 3.2. If (h, q, z b ) is a solution of the model (1.8)-(1.
10), we have the equal-ity:

ν 2 2 d dt Ω h|∇ log h| 2 + ν F r 2 Ω |∇h| 2 = -ν d dt Ω u • ∇h + ν Ω h∇u : t ∇u - ν F r 2 Ω ∇h • ∇z b . (3.5)
Proof. If we derive the mass equation (1.8) with respect to x i and multiply it by h∂ i log h, when we compute the sum over i and integrate on Ω (see Ref. 1), we get:

1 2 d dt Ω h|∇log h| 2 + Ω h∇div u • ∇ log h + Ω h∇u : ∇ log h ⊗ ∇ log h = 0. (3.6)
This relation will be used in the following. We multiply the momentum equation (1.9) by (ν/2)∇ log h:

ν 2 Ω (∂ t u + (u • ∇)u) • ∇h + ν 2 2 Ω D(u) : ∇∇h - ∇h ⊗ ∇h h + ν 2F r 2 Ω |∇h| 2 = - ν 2F r 2 Ω ∇z b • ∇h.
We simplify this expression using the following relations:

Ω h∇u : ∇ log h ⊗ ∇ log h = Ω D(u) : ∇h ⊗ ∇h h , Ω D(u) : ∇∇h + Ω ∇div u • ∇h = 0,
and add Eq. (3.6) multiplied by ν 2 /2. We get:

ν 2 4 d dt Ω h|∇ log h| 2 + ν 2F r 2 Ω |∇h| 2 = - ν 2 Ω (∂ t u + (u • ∇)u) • ∇h - ν 2F r 2 Ω ∇z b • ∇h, = - ν 2 d dt Ω u • ∇h + ν 2 Ω h∇u : t ∇u - ν 2F r 2 Ω ∇z b • ∇h,
which ends the proof of Lemma 3.2.

Proof. We come back to the proof of Proposition 3.2. Equation (3.5) gives us:

1 2 d dt Ω h|u + ν∇ log h| 2 + ν F r 2 Ω |∇h| 2 = 1 2 d dt Ω h|u| 2 + ν Ω h∇u : t ∇u - ν F r 2 Ω ∇h • ∇z b .
We add to this equality the energy inequality (3.1) multiplied by 2:

1 2

d dt Ω h|u + ν∇ log h| 2 + 1 F r 2 d dt Ω |z b + h| 2 + 2A k + 2 d dt Ω h|u| k+2 + 1 2 d dt Ω h|u| 2 + ν F r 2 Ω |∇(h + z b )| 2 + ν 2 Ω h ∇u + t ∇u 2 + 1 -2k 4 Aν Ω h ∇u + t ∇u 2 |u| k ≤ ν Ω h∇u : t ∇u,
which proves the proposition.

We then know that our system is dissipative. In addition, we can give a priori estimates:

Corollary 3.1. If (h, q, z b ) is solution of the model (1.8)-(1.10), then, thanks to Proposition 3.2, we have: √ hu L ∞ (0,T ;(L 2 (Ω)) 2 ) ≤ c ∈ R + , ∇ √ h L ∞ (0,T ;(L 2 (Ω)) 2 ) ≤ c, z b + h L ∞ (0,T ;L 2 (Ω)) ≤ c, √ h|u| (k+2)/2 L ∞ (0,T ;(L 2 (Ω)) 2 ) ≤ c, ∇(h + z b ) L 2 (0,T ;(L 2 (Ω)) 2 ) ≤ c, √ h∇u L 2 (0,T ;(L 2 (Ω)) 2 ) ≤ c, √ hD(u) |u| k/2 L 2 (0,T ;(L 2 (Ω)) 2 ) ≤ c.

Convergence theorem.

This section is devoted to the proof of Theorem 2.1, in four steps. Thanks to the previous estimates, we show the convergence of the different terms that compose the equation.

First step: Convergence of the sequences

√ h n n≥1 √ h n n≥1 √ h n n≥1 , (h n ) n≥1 (h n ) n≥1 (h n ) n≥1 and (z bn ) n≥1 (z bn ) n≥1 (z bn ) n≥1 .
First, we give the spaces in which ( √ h n ) n is bounded. If we integrate the mass equation, we directly get (

√ h n ) n in L ∞ (0, T ; L 2 (Ω)). Corollary 3.1 gives us ∇ √ h L ∞ (0,T ;(L 2 (Ω)) 2 )
≤ c, so we obtain:

( h n ) n is bounded in L ∞ (0, T ; H 1 (Ω)). (4.1)
Moreover, thanks to the mass equation again, we have the following equality:

∂ t h n = 1 2 h n div u n -div h n u n , which gives that (∂ t √ h n ) n is bounded in L 2 (0, T ; H -1 (Ω)
). Applying Aubin-Simon lemma, we can extract a subsequence, still denoted (h n ) n≥1 , such that √ h n strongly converges to √ h in C 0 (0, T ; L 2 (Ω)). Let us study now the subsequence (h n ) n . According to the property (4.1) and Sobolev embeddings, we know that, for all finite p, (

√ h n ) n is bounded in L ∞ (0, T ; L p (Ω)).
In the following, we will assume p ≥ 4 in order to simplify our expressions and ensure that (

h n ) n is in L ∞ (0, T ; L 2 (Ω)). The equality ∇h n = 2 √ h n ∇ √ h n enables us to bound the sequence (∇h n ) n in L ∞ (0, T ; (L 2p/(2+p) (Ω)) 2 ) and consequently the sequence (h n ) n is bounded in L ∞ (0, T ; W 1,2p/(2+p) (Ω)).
Moreover, we have some properties on the time derivative of h n ; actually the mass equation reads:

∂ t h n = -div(h n u n ). Splitting the product h n u n into h n u n = √ h n √ h n u n , we get (h n u n ) n in L ∞ (0, T ; (L 2p/(2+p) (Ω)) 2 ) and (∂ t h n ) n in L ∞ (0, T ; W -1,2p/(2+p) (Ω)).
Thanks to Aubin-Simon lemma again, we find:

h n → h in C 0 (0, T ; L 2p/(2+p) (Ω)).
Last, we consider the bottom term (z bn ) n : with Corollary 3.1 and the bound on

( √ h n ) n in L ∞ (0, T ; L p (Ω)), we know that the sequence (∇z bn ) n is bounded in L 2 (0, T ; (L 2p/(2+p) (Ω)) 2 ), which gives (z bn ) n is bounded in L ∞ (0, T ; W 1,2p/(2+p) (Ω)).
For the time derivative of z bn , we restart from Eq. (1.10). We have just shown that (∆z bn ) n is in L ∞ (0, T ; W -1,2p/(2+p) (Ω)). Let us come to the divergence term:

h n |u n | k u n = h (1-k)/2 n h 1/2 n |u n | k h 1/2 n u n (4.2)
where

• h (1-k)/2 n n is bounded in L ∞ (0, T ; L p/(1-k) (Ω)), • h 1/2 n |u n | k n is bounded in L ∞ (0, T ; L 2/k (Ω)), • h 1/2 n u n n is bounded in L ∞ (0, T ; (L 2 (Ω)) 2 ), that is to say (h n |u n | k u n ) n is bounded in L ∞ (0, T ; (L 2p/(2-2k+kp+p) (Ω)) 2
). As 0 < k < 1/2 and we assumed p ≥ 4, it leads us to:

(h n |u n | k u n ) n is bounded in L ∞ (0, T ; L 4p/(2+3p) (Ω)).
Since in our case 4p/(2 + 3p) ≤ 2p/(2 + p), we obtain:

(∂ t z bn ) n is bounded in L ∞ (0, T ; W -1,4p/(2+3p) (Ω)).
As we have the relations

W 1,2p/(2+p) (Ω) ⊂⊂ L 2p/(2+p) (Ω) ⊂ W -1,4p/(2+3p) (Ω),
with Aubin-Simon lemma we are able to assert that z bn strongly converges to z b in C 0 (0, T ; L 2p/(2+p) (Ω)).

Second step: Convergence of the water discharge

(q n ) n≥1 = (h n u n ) n≥1 (q n ) n≥1 = (h n u n ) n≥1 (q n ) n≥1 = (h n u n ) n≥1 .
In the previous part, we proved that the sequence (

h n u n ) n is bounded in L ∞ (0, T ; (L 2p/(2+p) (Ω)) 2 )
where p is an integer greater than four. Writing the gra-dient as follow:

∇(h n u n ) = 2 h n u n ∇ h n + h n h n ∇u n ,
since the first term is in L ∞ (0, T ; L 1 (Ω)) and the second one belongs to L 2 (0, T ; L 2p/(2+p) (Ω)), we have:

(h n u n ) n bounded in L 2 (0, T ; W 1,1 (Ω)).
Moreover, the momentum equation (1.9) enables us to write the time derivative of the water discharge:

∂ t (h n u n ) = -div(h n u n ⊗ u n ) - 1 F r 2 h n ∇(h n + z bn ) + ν div (h n D(u n )) .
We then study each term:

• div(h n u n ⊗ u n ) = div √ h n u n ⊗ √ h n u n which is in L ∞ (0, T ; W -1,1 (Ω)), • as h n is in L ∞ (0, T ; W 1,2p/(2+p) (Ω)), it is also in L ∞ (0, T ; L p (Ω)
) and we can write the following relation:

h n ∇(h n + z bn ) is in L 2 (0, T ; L 2p/(2+p) (Ω)) ⊂ L 2 (0, T ; W -1,2p/(2+p) (Ω)),
• remark that

h k ∇u k = ∇(h k u k ) -u k ⊗ ∇h k = ∇ h k h k u k -2 h k u k ∇ h k ; (4.3)
we know that the first term is in L ∞ (0, T ; W -1,2p/(2+p) (Ω)) and the second one in L ∞ (0, T ; L 1 (Ω)). So we have h n D(u n ) bounded in L 2 (0, T ; W -1,2p/(2+p) (Ω)).

Finally, note that these three terms are included in L 2 (0, T ; W -2,2p/(2+p) (Ω)), which means that ∂ t (h n u n ) is also in this space for all n ≥ 1.

Then, applying Aubin-Simon lemma, we obtain:

(h n u n ) n strongly converges to q in C 0 (0, T ; W -1,2p/(2+p) (Ω)).

Third step: Convergence of

√ h n u n n≥1 √ h n u n n≥1 √ h n u n n≥1 .
The product √ h n u n is nothing but the ratio q n / √ h n . For this term, we also want to prove a strong convergence. Compared with Ref. 15, the bound on √ hu (k+2)/2 simplifies the computation.

Before studying the convergence, let us develop some properties of the limit water discharge. We know that (q n / √ h n ) n is bounded in L ∞ (0, T ; L 2 (Ω)); consequently Fatou lemma reads:

Ω lim inf q 2 n h n ≤ lim inf Ω q 2 n h n < +∞.
In particular, q(t, x) is equal to zero for almost every x where h(t, x) vanishes. Then, we can define the limit velocity taking u(t, x) = q(t, x)/h(t, x) if h(t, x) = 0 or else u(t, x) = 0. So we have a link between the limits q(t, x) = h(t, x)u(t, x) and:

Ω q 2 h = Ω h|u| 2 < +∞.
Moreover, we can use Fatou lemma again to write

Ω h|u| k+2 ≤ Ω lim inf h n |u n | k+2 ≤ lim inf Ω h n |u n | k+2 ,
which gives √ h|u| (k+2)/2 in L 2 (0, T ; L 2 (Ω)). As (q n ) n and (h n ) n converge almost everywhere, the sequence of √ h n u n = q n / √ h n converges almost everywhere to √ hu = q/ √ h when h does not vanish. Moreover, for all M positive, ( √ h n u n 1 1 1 |un|≤M ) n converges almost everywhere to √ hu1 1 1 |u≤M (still assuming that h does not vanish). If h vanishes, we can write √ h n u n 1 1 1 |un|≤M ≤ M √ h n and then have convergence towards zero. Then, almost everywhere, we obtain the convergence of ( √ h n u n 1 1 1 |un|≤M ) n . Finally, let us consider the following norm:

Ω h k u k - √ hu 2 ≤ Ω h k u k 1 1 1 |u k |≤M - √ hu1 1 1 |u|≤M + h k u k 1 1 1 |u k |>M + √ hu1 1 1 |u|>M 2 ≤ 3 Ω h k u k 1 1 1 |u k |≤M - √ hu1 1 1 |u|≤M 2 + 3 Ω h k u k 1 1 1 |u k |>M 2 +3 Ω √ hu1 1 1 |u|>M 2 . Since ( √ h n ) n is in L ∞ (0, T ; L p (Ω)), ( √ h n u n 1 1 1 |un|≤M
) n is bounded in this space. So, as we have seen previously, the first integral tends to zero. Let us study the other two terms:

Ω h n u n 1 1 1 |un|>M 2 ≤ 1 M k Ω h n |u n | k+2 ≤ c M k , Ω √ Hu1 1 1 |u|>M 2 ≤ 1 M k Ω h|u| k+2 ≤ c ′ M k ,
for all M > 0. When M tends to the infinity, our two integrals tend to zero. Then ( h n u n ) n strongly converges to √ hu in L 2 (0, T ; L 2 (Ω)). Concerning the diffusion term, (∇(h n u n )) n converges to ∇(hu) in the sense of the distributions, in (D ′ ((0, T ) × Ω)) [START_REF] Bresch | On the compressible Navier-Stokes equations with density dependent viscosities in bounded domains[END_REF] . Since the sequence (∇ √ h n ) n weakly converges in L 2 (0, T ;(L 2 (Ω)) [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] ) and ( √ h n u n ) n strongly converges in this space, then (u n ⊗∇h n ) n weakly converges in L 1 (0, T ; (L 1 (Ω)) [START_REF] Bresch | On the compressible Navier-Stokes equations with density dependent viscosities in bounded domains[END_REF] ). So, using the relation (4.3) to write the product h n ∇u n , we have (h n ∇u n ) n that converges to h∇u in (D ′ ((0, T ) × Ω)) [START_REF] Bresch | On the compressible Navier-Stokes equations with density dependent viscosities in bounded domains[END_REF] . This gives the convergence of the complete diffusion term.

From Corollary 3.1, we know that (∇(h n + z bn )) n weakly converges to ∇(h + b) in L 2 (0, T ; (L 2 (Ω)) 2 ). In addition, the sequence (h n ) n strongly converges in C 0 (0, T ; L 2p/(2+p) (Ω)) so the product weakly converges to h∇(h

+ z b ) in L 2 (0, T ; (L p/(1+p) (Ω)) 2 ).
The last term is the term of solid transport flux: (h

(1-k)/2 n ) n strongly con- verges to h (1-k)/2 in C 0 (0, T ; L 2/(1-k) (Ω)) and ( √ h n u n ) n strongly converges to √ hu in L 2 (0, T ; (L 2 (Ω)) 2 ). Moreover, (h k/2 n |u n | k ) n strongly converges to h k/2 |u| k in L 2/k (0, T ; L 2/k (Ω)). Using Eq. (4.2), we obtain that the sequence (h n |u n | k u n ) n strongly converges to h|u| k u in the space L 2/(k+1) (0, T ; (L 1 (Ω)) 2 ).
This ends the proof of Theorem 2.1.

5.

Others sediment discharge choices.

Model coming from those studied above.

Let us consider the bed-load transport model

∂ t z b + div(hu) = 0. (5.1)
This model of sediment has been studied in Ref. 21 but, in this paper, the Shallow-Water system is taken as in Ref. 17, that is to say the viscous term is a Laplacian.

Here, we couple Eq. (5.1) with the Shallow-Water system used above, given by (1.8)-(1.9). We prove that this model can be studied as an usual Shallow-Water system. Indeed, combining (1.8) and (5.1) we get

∂ t (z b -h) = 0,
and, by an integration with respect to t, we obtain

z b (t, x) = h(t, x) -z b0 (x) + h 0 (x). Setting b(x) = h 0 (x) -z b0 (x), the expression of z b becomes z b (x, t) = h(x, t) -b(x). (5.2)
If we replace z b by this value in (1.9), we get:

∂ t (hu) + div(hu ⊗ u) + h∇(2h -b(x)) F r 2 -Adiv(hD(u)) = 0. (5.3)
Hence, the problem becomes no-coupled, which means we can determine h using Eqs. (1.8) and ( 5.3) and then use the relation (5.2) to deduce the value of z b , since b is given. The Shallow-Water system (1.8)-( 5.3) is studied in Ref. 5 where the authors proved an existence result, under the assumption b ≥ c > 0 and some assumptions on the regularity.

Another viscous sediment transport.

We propose here another viscous system. More precisely, we consider the Shallow-Water system

∂ t h + div(hu) = 0 (5.4) ∂ t (hu) + div(hu ⊗ u) + h∇(h + z b ) F r 2 -νdiv(hD(u)) = 0 (5.5)
with the bed-load equation

∂ t z b + Adiv hu 1 + log(1 + |u| 2 ) - ν 2 ∆z b = 0. (5.6)
As mentioned in the Introduction, we have modified the sediment equation. We deal here with the term u + log(1 + |u| 2 )u used in Ref. 15 to obtain a better bound on hu 2 . As for the previous model, multiplying the diffusion term by u + log(1 + |u| 2 )u gives some terms which are controllable. We get the existence of dissipative energy for this system.

Lemma 5.1. Let (h, q, z b ) be a smooth solution of (5.4) -(5.6). The following estimate holds 1 2

d dt Ω h|u| 2 + A d dt Ω h 1 + |u| 2 2 log(1 + |u| 2 ) + 1 2F r 2 d dt Ω |z b + h| 2 +ν(1 -3A) Ω h(D(u) : D(u)) + ν 2F r 2 Ω ∇h • ∇z b +Aν Ω h(D(u) : D(u)) log(1 + |u| 2 ) + ν 2F r 2 Ω |∇z b | 2 ≤ 0.
(5.7)

Lemma 5.2. Let (h, q, z b ) be a smooth solution of (5.4) -(5.6). We have

1 2 d dt Ω h|u + ν log h| 2 + A d dt Ω h 1 + |u| 2 2 log(1 + |u| 2 ) + 1 2F r 2 d dt Ω |z b + h| 2 + Aν Ω h(D(u) : D(u)) + ν Ω h(W (u) : W (u)) +Aν Ω h(D(u) : D(u)) log(1 + |u| 2 ) + ν F r 2 Ω |∇h| 2 + ν 2F r 2 Ω |∇z b | 2 + 3ν 2F r 2 Ω ∇h • ∇z b ≤ 4Aν Ω h(D(u) : D(u)) (5.8)
where W (u) is the skew-symmetric part of the gradient:

W (u) = ∇u -t ∇u 2 .
As in Sec. 3, if we sum the two estimates (5.7) and (5.8), we deduce some bounds on h, u and z b with the condition 0 < A < 1/6. These bounds allow us to prove the stability of the system (5.4)-(5.6).

6. Numerical experiments.

6.1. Numerical scheme.

The proposed model can be written under the structure of a 2D hyperbolic system with non-conservative terms plus the diffusion terms:

∂ t W + div(F (W )) + B 1 (W )∂ x W + B 2 (W )∂ y W -ν div(D(W )) = 0,
where

W =     h hu 1 hu 2 z b     , F = (F 1 , F 2 ), with F 1 (W ) =     h u 1 hu 2 1 + h 2 /(2F r 2 ) hu 1 u 2 Ah|u| k u 1     , F 2 (W ) =     h u 2 hu 1 u 2 hu 2 2 + h 2 /(2F r 2 ) Ah|u| k u 2     , B 1 (W ) =     0 0 0 0 0 0 0 h/F r 2 0 0 0 0 0 0 0 0     , B 2 (W ) =     0 0 0 0 0 0 0 0 0 0 0 h/F r 2 0 0 0 0     and D(W ) = hD(u) ∇z b /2 .
The discretization of the model has been done with a high order finite volume method for the hyperbolic system and a centered second order discretization of the diffusion terms.

The following notation is considered (see Fig. 2): We decompose the spatial domain in a mesh of cells, finite volumes, V i ⊂ R 2 for i = 1, . . . , N V . The area of the volume V i is denoted by |V i | and the center of the cell by x i . We consider that V i is a closed polygon and the boundary of V i is defined by the union of the segments E ij , where E ij is the common edge between the volumes V i and V j . The normal unit vector to E ij pointing towards V j is denoted by η ij . The length of the segment

E ij is |E ij |. The middle point of E ij is c ij . By b ij we denote the baricenter of V ij ,
where V ij is the triangle defined by E ij and x i . Its area is denoted by

|V ij |. K i is the set of indexes j such that V j is a neighbor of V i .
We use a second-order finite volume method for 2D non-conservative hyperbolic systems, [START_REF] Castro | High order extensions of Roe schemes for two dimensional nonconservative hyperbolic systems[END_REF] with a second order state reconstruction operator [START_REF] Castro | Two-dimensional sediment transport models in Shallow Water equations[END_REF][START_REF] Ferreiro | Development of post-process technics of hydrodynamics flux, modelization of sediment transport problems and numerical simulation through finite volume technics[END_REF] .

By W i (t) we denote the average value of W (x, t) over the volume V i . And we consider a state reconstruction operator over each volume P i (x, t), x ∈ V i , (P i (x, t) ≈ W (x, t) ∀x ∈ V i ); concretely we use a MUSCL second-order reconstruction operator [START_REF] Castro | Two-dimensional sediment transport models in Shallow Water equations[END_REF] . We denote W + ij (t) = P j (c ij , t) and W - ij (t) = P i (c ij , t).

V V E c x i ij ij b ij η ij V j i ij

Fig. 2. Finite volume mesh

We obtain the following numerical scheme,

W ′ i (t) = - 1 |V i | j∈Ki |E ij | G ij -B ij (W + ij -W - ij ) + j∈Ki |V ij | B 1 (P i (b ij ))∂ x P i (b ij ) + B 2 (P i (b ij ))∂ y P i (b ij ) + j∈Ki |E ij |D ij η ij ,
where

G ij = G(W - ij (t), W + ij (t), η ij
) is a numerical flux function, for example for Roe method [START_REF] Parés | On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems[END_REF] :

G(U, V, η) = F η (V ) + F η (U ) 2 - 1 2 |A η (U, V )|(V -U )
where

F η = F 1 η 1 + F 2 η 2 . |A η (U, V )| is the absolute value of matrix A η (U, V ) and A η (U, V ) = A η (U, V ) + B 1 ( U + V 2 )η 1 + B 2 ( U + V 2 )η 2 ,
where A η (U, V ) verifies

F η (V ) -F η (U ) = A η (U, V )(V -U ).
Moreover,

B ij = (B 1 η ij,1 + B 2 η ij,2 ) W + ij + W - ij 2 .
By D ij we denote a second order approximation of D(W (c ij )). The MUSCL operator reconstruction 7,10 uses a second-order approximation of the derivatives of the vector of unknowns, so the same computations can be used to define D ij . The discretization in time is done with a second order TVD Runge-Kutta method 20 .

Numerical test.

In this subsection we perform a test where we study the evolution of a sand conical dune in a channel. We compare the results for models defined by (1.8)-(1.9) and one of the considered sediment transport models:

(1) Grass model, given by (1.4), ( 2) the first proposed model, defined by (1.10). In what follows we denote it by MS1, (3) the last proposed model (5.6), denoted by MS2.

First, in this section, we study the results obtained with model MS1, with and without viscosity and for two different values of the constant of interaction between the fluid and the sediment. After we compare it with Grass model and model MS2.

In this test the sediment layer is deformed gradually towards a star shape, expanding along time with a certain angle [START_REF] Castro | Two-dimensional sediment transport models in Shallow Water equations[END_REF][START_REF] Ferreiro | Development of post-process technics of hydrodynamics flux, modelization of sediment transport problems and numerical simulation through finite volume technics[END_REF][START_REF] Hudson | Numerical technics for morphodynamic modelling[END_REF] .

De Vriend 8,9 determined a formula that relates the solid transport formula of the model with the spread angle.

Consider a given transport equation defined by

∂ t z b + ∂ x S x + ∂ y S y = 0 (6.1)
where the solid transport discharges S x , S y verifies

S x = u 1 u tot S tot , S y = u 2 u tot S tot ,
where u tot = |u|, and u = (u 1 , u 2 ). We denote by α the expansion angle of spread.

Under the hypothesis of a weak interaction between the fluid and the sediment layer, De Vriend 8,9 deduces that the angle of spread can be approximated by the following formula

tan α = 3 T u √ 3 9 T u -8 T h , (6.2)
where

T u = u tot S tot ∂S tot ∂u tot -1, T h = h S tot ∂S tot ∂h -1.
The proposed model MS1, defined by (1.8)-(1.10), without viscosity, corresponds to set

S tot = A h |u| k+1 = A h u k+1 tot . So, ∂S tot ∂u tot = A h (k + 1) u k tot , ∂S tot ∂h = A u k+1 tot .
Then, )). Applying (6.2), we also obtain for this model α = 30 o . Remark 6.1. Observe that for model MS1 we obtain that the angle of spread is independent of the value of k because S tot is not independent of h. Otherwise, if S tot is independent of h, we obtain ∂ h S tot = 0, then

T u = A h (k + 1) u k+1 tot S tot -1 = k, T h = A h u k+1 tot S tot -1 = 0. k 0.
T h = -1, thus tan α = 3 k √ 3 9 k √ 3 + 8 .
If we omit in our model the dependency of h we obtain the solid transport formula defined by

S tot = A|u| k+1 ,
and this definition of S tot corresponds to the definition obtained for Grass model (1.4). Nevertheless we remark that in our model the dependency of S tot with respect to h is crucial for the proof of the theoretical results. We present in Table 1 the different values obtained for α in function of different values of k for Grass model. The angle of spread of Grass model converge to 30 o with respect to k. That is, the angle of spread predicted for the proposed model (1.8)-(1.10) is the limit angle for Grass model.

The classical value of k used with Grass model is k = 2 (see for example Ref. 12), corresponding to α = 21.78 o . We observe in the numerical results that this angle corresponds to the angle of spread of internal level curves of the sand dune.

For the definition of the constant of interaction between the fluid and the sediment, A, observe that it depends on the porosity of the sediment layer,

A = ξ Ā, ξ = 1 1 -ψ 0
where ψ 0 is the porosity. In this test we consider ψ 0 = 0.4 and two different values of Ā: Ā = 0.001 and Ā = 0.01. For Ā = 0.001 corresponding to a very weak interaction between the fluid and the sediment we simulate until t = 360000 s. For Ā = 0.01, that can be considered as the limit of a weak interaction, we simulate until t = 36000 s.

We use an explicit finite volume method, then we impose a CFL condition. We set for this test the CFL condition to 0.8. We use a mesh of 7600 control volumes of edge type (see Fig. 3(a)). We impose a discharge q = (10, 0) and sediment layer thickness z b = 0.1 in boundary-line corresponding to x = 0 and free condition boundary-line corresponding to x = 1000. At lateral walls we impose sliding condition q • η = 0, if by η we denote the outward normal vector. Initial conditions are (see Fig. 3), h(x, y, 0) = 10.1z b (x, y, 0), q x (x, y, 0) = 10, q y (x, y, 0) = 0; and the initial sediment layer is a sand dune with a conical form,

z b (x, y, 0) =    0.1 + sin 2 π(x -300) 200 sin 2 π(y -400) 200 if 300 ≤ x ≤ 500, 400 ≤ y ≤ 600, 0.1 otherwise.
In Fig. 4 we present the evolution of the sand dune for Ā = 0.001. We superpose the level curves for t = 0, t = 180000 and t = 360000 s. Figure 4 In Fig. 5 we present the evolution of the sand dune for Ā = 0.01. We superpose the level cuves for t = 0, t = 18000 and t = 36000 s. Figure 5 We observe that for Ā = 0.001, when the interaction is weaker than for Ā = 0.01, the analytical solution corresponding to the spread angle of 30 o is better captured. This observation corresponds with the hypothesis under which De Vriend deduces formula (6.2); a weak interaction between the fluid and the sediment. By comparing the solutions for the model with and without viscosity, we observe that in both cases, Ā = 0.001 and Ā = 0.01, the angle of spread is slightly smaller in the case of the model with viscosity.

As we mentioned previously, Grass model is usually used with k = 2. In this case we obtain α = 21.78 o . We can observe in Figs. 4 and5 that the line corresponding to α = 21.78 o reproduces the angle of spread of an internal level curve (it is also better captured for Ā = 0.001 than for Ā = 0.01).

The results presented in Fig. 6 correspond to Ā = 0.01, without viscosity. In Fig. 6(a) we present the results obtained with Grass model, we observe that effectively the angle of spread approximates the predicted angle of α = 21.78 o (discontinuous line).

In Fig. 6(b) we study the angle of spread of model MS2. In this case the times of the superposed level curves correspond to t = 0, t = 7000 and t = 14000 s. We also observe that the predicted theoretical angle of spread for this model (α = 30 o ) is also well approximated.

Finally, by comparing Figs.

Fig. 1 .

 1 Fig. 1. Sediment and water heights

4. 4 .

 4 Fourth step: Convergence of the diffusion terms, the pressure and the solid transport flux.

  Fig. 3. Initial condition.

  (a) corresponds to the model without viscosity and Fig. 4(b) to the model with viscosity.

  (a) correspond to the model without viscosity and Fig. 5(b) to the model with viscosity. In these figures the continuous black line correspond with an angle of 30 o and the black dashed line with 21.78 o .

Fig. 4 .

 4 Fig. 4. MS1. Spread angle, Ā = 0.001

  5(a), 6(a) and 6(b), corresponding to set Ā = 0.01 in three cases, we can observe that: (i) Model MS1 and Grass model have different angles of spread. But the time evolution obtained with both models are nearly the same (see Fig. 7(a)). In both cases, Figs. 5(a) and 6(a) the final time is the same.

Fig. 5 .Fig. 6 .

 56 Fig. 5. MS1. Spread angle, Ā = 0.01

Fig. 7 .

 7 Fig. 7. Comparison of the level curves

Table 1 .

 1 16.99 o 21.78 o 24 o 25.28 o 26.11 o 27.93 o Values of α for different values of k for Grass model We obtain that the angle of spread is independent of the definition of the parameter k; α = 30 o for all values of k. For the numerical results presented in this section we have set k = 0.25.

	25	1	2	3	4	5	10
	α 7.22 o Then,		tan α =	√ 3 3

Model MS2 correspond to S tot = qhu tot (1 + log(1 + u 2 tot

Acknowledgments.

The first author was partially supported by'Réseau EDP-MC, ICTP grants Net 47 and SARIMA". The third author is partially supported by the Spanish Government Research project MTM2006-08075. The computations have been done in the 'Laboratorio de Métodos Numéricos de la Universidad de Málaga".

The authors are grateful to D. Bresch for initiating this work and for the fruitful discussions on the subject. The authors would also like to thank the three referees for their comments and suggestions.