Yannick Courbois
email: yannick.courbois@univ-lille3.fr

Emily K Farran

Mark Blades (c Axelle Lemahieu

Mark Blades

Hursula Mengue Topio

Pascal Sockeel

Hursula Mengue-Topio

Wayfinding behaviour in Down syndrome: a study with virtual environments

Introduction

Down's syndrome (DS), also referred to as Trisomy 21 (see [START_REF] Roubertoux | Trisomy 21: from chromosomes to mental retardation[END_REF], has an incidence of about 650-1000 live births and it is the most common genetic cause of intellectual disability [START_REF] Bittles | The four ages of Down syndrome[END_REF]. This genetic syndrome has been the subject of widespread cognitive research over the past two decades, highlighting a specific cognitive profile is in most individuals with DS. This profile involves relative weaknesses in expressive language and verbal short-term memory, contrasting with relative strengths in visuo-spatial tasks. Deficits in the language domain have been extensively studied. However, researchers have devoted less attention to spatial processes, which have mainly been investigated through visuo-spatial short-term memory. Convergent evidence showed that individuals with DS had a relatively good level of performance at the Corsi Block Task, a measure of short-term memory for a series of spatial location. Indeed, they performed as well as typically developing children with the same mental age (MA) [START_REF] Frenkel | Verbal, visual, and spatio-sequential short-term memory: assessment of the storage capacities of children and teenagers with Down's syndrome[END_REF][START_REF] Jarrold | Short-term memory for verbal and visuospatial information in Down's syndrome[END_REF][START_REF] Visu-Petra | Visual-spatial processing in children and adolescents with Down's syndrome: a computerized assessment of memory skills[END_REF] or individuals with intellectual disability with the same MA [START_REF] Numminen | Working memory and everyday cognition in adults with Down's syndrome[END_REF]. These results contrasted with their poor performances in verbal short-term memory, which were below their mental age level. [START_REF] Lanfranchi | Verbal and visuospatial working memory deficits in children with Down syndrome[END_REF] also found that the relative strength in visuo-spatial tasks disappeared when the control required by the task increased.

They used a battery of five visuo-spatial tasks requiring different level of control and executive resources. DS participants performed lower than the MA control group only when the visuo-spatial tasks involved high level of executive resources (see also [START_REF] Lanfranchi | Working memory in Down syndrome: is there a dual task deficit[END_REF].

A vast majority of experiments with individuals with Down syndrome have relied on small-scale spatial tasks. There have been very few attempts to assess visuo-spatial abilities in large-scale spatial task [START_REF] Pennington | The neuropsychology of Down syndrome: Evidence for hippocampal dysfunction[END_REF][START_REF] Uecker | Down syndrome in neurobiological perspective: an emphasis on spatial cognition[END_REF]. This is surprising since there is considerable evidence in cognitive psychology and in neuroscience showing that processing spatial information at different scale of space involves different processes and brain structures [START_REF] Hegarty | Spatial abilities at different scales: Individual differences in aptitude-test performance and spatial-layout learning[END_REF]. For example [START_REF] Previc | The neuropsychology of 3-D space[END_REF], identified different brain systems involved in spatial processing, including the peripersonal system and the action extrapersonal system. The former is mainly devoted to the near-body space and is involved in reaching and manipulative behaviours. It is predominantly located in the dorsolateral cortex. The later is used in spatial navigation and is located in the hippocampus and parahippocampal regions (see also [START_REF] Previc | Why your "head is in the clouds" during thinking: the relationship between cognition and upper space[END_REF]. Moreover, convergent evidences suggested that specific impairments in the hippocampal system are an important part of the cognitive phenotype in DS [START_REF] Nadel | Down's syndrome: a genetic disorder in biobehavioral perspective[END_REF]. Clearly, research on spatial navigation in DS is required. [START_REF] Mangan | Spatial memory abilities and abnormal development of the hippocampal formation in Down syndrome[END_REF] tested two groups of ambulatory children with DS (16 to 20 months and 26 to 30 months, n = 5 in both groups), and two groups of age-matched control children, on three spatial tasks. The response learning task, required the children to learn a specific motor response to locate a hidden toy (see also [START_REF] Uecker | Down syndrome in neurobiological perspective: an emphasis on spatial cognition[END_REF]. The cue learning task required them to learn an association between a single cue and the location of the goal (the participants found the toy when they approached a specific coloured cue). The place learning task required the utilisation of multiple cues to locate the goal. Successful performance on this task needed to rely on a representation of the spatial relationship between multiple cues and the toy location (a cognitive map), a spatial ability that depends on the hippocampal formation [START_REF] Mangan | Spatial memory abilities and abnormal development of the hippocampal formation in Down syndrome[END_REF][START_REF] Nadel | Down's syndrome: a genetic disorder in biobehavioral perspective[END_REF][START_REF] Uecker | Down syndrome in neurobiological perspective: an emphasis on spatial cognition[END_REF]. Each task began with training trials and, upon reaching criterion, a memory test was given (children were removed from the apparatus for a delay interval). The results showed that children with DS needed more trials to reach the criterion in all three tasks, but they managed to learn the toy's locations. On the memory test, they performed at the same level as CA children on the response and cue tasks but they were severely impaired on the place task. This pattern of result was consistent with the hippocampal dysfunction hypothesis [START_REF] Mangan | Spatial memory abilities and abnormal development of the hippocampal formation in Down syndrome[END_REF][START_REF] Nadel | Down's syndrome: a genetic disorder in biobehavioral perspective[END_REF][START_REF] Uecker | Down syndrome in neurobiological perspective: an emphasis on spatial cognition[END_REF]. [START_REF] Pennington | The neuropsychology of Down syndrome: Evidence for hippocampal dysfunction[END_REF] conducted a neuropsychological assessment of prefrontal and hippocampal functions in participants with DS (mean CA =14.7) who were compared with typically developing children individually matched on MA (mean MA = 4.9). The tasks included a computer generated virtual Morris water maze, which was designed to study place learning and memory in humans (this task was an adaptation from the water maze task used in animals to study cognitive maps). In this task, the participants were required to find an invisible platform located in a virtual circular arena, which was surrounded by four walls with distinctive features. The platform remained in the same place across trials and the participants had to learn its spatial location relative to the distal cues provided by the features on the walls.

The participants with Down syndrome performed lower than children with the same MA in this task, which was supposed to assess the generation and use of a cognitive map, a function of the hippocampus.

Thus, the existing research suggests individual with DS have difficulties in coding the spatial location of an object in term of distance and direction with respect to landmarks located in the local surround. These place learning difficulties may impact spatial navigation in more complex and ecologically valid tasks where the destination to reach is located in the neighbourhoods and is not visible from the current location. In these large-scale spatial tasks, space can be viewed only in segments, and information from multiple views must be integrated [START_REF] Mcdonald | Psychological perspectives on spatial cognition[END_REF]. Such tasks require wayfinding, which can be defined as a goal-directed and planned movement of one's body around an environment which is coordinated to distal as well to local surrounds (Montello, 2005, p. 259). An efficient wayfinding behaviour can take different forms. At a basic level, individuals simply have to know what action to take when they reach decisions points (or nodes). The sequence of the decision points, together with the sequences travelled from one decision point to another constitutes a route [START_REF] Hunt | Orientation and wayfinding: A review[END_REF]. At a higher level, individuals know the configuration of the environment and locations are defined in terms of distance and direction to landmarks or important places. Their wayfinding behaviour is flexible, they can navigate on novel paths and create shortcuts. A deficit in place learning may impair ability in individuals with DS to create a configurational representation of their environment. However, it would not impair their ability to learn routes, provided that proximal landmarks are placed along the path.

In the present experiment, we used virtual environments to study wayfinding abilities in individuals with DS. The methodology, adapted from Mengue-Topio, Courbois, Farran, and Sockeel ( 2011), allowed us to assess memory for landmarks, route learning abilities, and shortcut performance. Participants explored two routes (A  B) and (A  C) until they reached a learning criterion. Then they were placed at B and were asked to find the shortest way to C. We expected individuals with DS to be able to learn the two routes and to have a good memory for landmarks (There are evidences for a relative sparing in visual memory in Down syndrome [START_REF] Laws | Working memory in children and adolescents with Down syndrome: evidence from a colour memory experiment[END_REF]. Moreover, we expected individuals with DS to have a low level of performance in the shortcut task.

Method

Participants Three groups of participants were assessed: Individuals with Down's syndrome (DS), individuals without Down's syndrome matched on chronological age (CA), and typically developing children matched on mental age (MA). All the participants, except one child in the MA group, used computer several times per week. The Down syndrome group comprised 6 females and 4 males, ranging in age from 14.2 to 29.9 years (mean age: 22.22). Their mental age, assessed with the French scale NEMI-2 [START_REF] Cognet | Manuel de la NEMI-2[END_REF], ranged from 7 to 9 years (mean mental age: 7.68). The CA group comprised 10 participants who were individually matched on sex and age (+/-6 months; mean age: 22.11). The MA group was composed of 10 children individually matched on the basis of sex and mental age (+/-6 months; mean age: 7.73). We did not obtain authorizations for testing children in the MA group with an intelligence scale.

We assumed the chronological age was a good indicator of mental age in these children.

Consent for taking part in the study was obtained from the parents of the participants in DS and MA groups (and for under age children in the CA group). All the participants were informed regarding the nature of the study and gave their consent to take part in it. They were also informed that they were free to withdraw from the study at any time.

Materials

The study was conducted in a virtual environment using the 3D VIDIA VIRTOOLS software (Dassault Systèmes). It comprised a 4  4 regular grid of streets lined with high brick walls (see Figure 1 for a map of the VE). This space was surrounded by distant landscapes providing no distinctive cues. Three buildings and 17 landmarks were located in different places of the space. The buildings were a "railway station" (A), a "store" (B), and an "apartment building" (C) 1 . The three buildings were not visible from each other. The landmarks were a railing, a bus shelter, a streetlight, an old car, a fountain, a bench, a slide, a bin, a white car, a tree, a dog, a billboard, a statue, a road sign, a bicycle, a pedestrian, a traffic light.

1 These buildings were extracted from a virtual environment designed by [START_REF] Gyselinck | Construction of a spatial mental model from a verbal description or from navigation in a virtual environment[END_REF] During familiarization and the first two phases (memory for landmarks and route learning), the VE was presented such that the participants could not explore the whole space.

Barriers were used to signal the roads that were not available on a particular route. In one familiarization VE the shortest route between the station and the store (route AB) was demonstrated by using visible barriers that blocked all but the correct path. In the other familiarization VE, the barriers signified the shortest route between the station and the apartment building (route AC). During learning, the VE was presented in the same manner as at familiarization, except that the barriers were not visible. That is, when a participant attempted to walk down an incorrect path, the barrier appeared, blocking their way (the barriers were located two meters away from the intersection). During the test phase, the participant can walk along any street in the environment. The barriers no longer exist.

The VE was projected onto a 1.20  1.50 m screen. The distance between the screen and the participant was 2 meters. Participants navigated from a first person viewpoint, at a constant velocity. They controlled their movement using the keyboard and the mouse.

Pressing the backspace key effected forward movement and moving the mouse to the right or left controlled rotational movements.

Procedure

In a preliminary phase, participants were asked to practice moving along the VE using the backspace key and the mouse (route AB or route BC), counterbalanced across the participants). When they were proficient at controlling their movement, the experiment started. The test session was composed of three phases, each of them assessing a major component of spatial navigation: memory for landmarks, route learning and finding a shortcut.

Memory for landmarks.

Participants faced the station and were told to follow the route from the station to the store (or from the station to the apartment) and then to return to the station. The order of the routes was counterbalanced, with half the participants in each group walking from the station to the store first (route AB), and the other half walking from the station to the apartment building first (route AC). The experimenter asked the participants to look carefully at the surroundings. Each route was constrained by visible barriers that prevented participants from taking an incorrect path. The participants walked the route forwards and back twice. Then, they were presented with a sequence of eight slides showing landmarks. Across the eight slides, four showed landmarks located along the route the participants had just walked, and four showed landmarks located elsewhere in the town.

For each of the slides, the participants were simply asked to say if they saw this object along the route they had just walked.

Route learning.

Participants faced the station and were asked to find the route between the station and the store (or the apartment) without choosing a wrong path. The order of the routes was counterbalanced across participants. When participants entered an incorrect path, a barrier appeared, preventing the participant from going further. The trial was repeated until participants reached a criterion of walking the route forwards and back twice without any errors. The maximum number of learning trials was ten round trips. Participants who passed the criterion in routes (AB) and (AC) were given the test phase.

Finding a shortcut. This test phase began with the participants facing the store. They were told they could walk along any street, and no barriers would appear. They were also asked to find the shortest route between the store and the apartment. The trial was repeated until participants walked the shortest the route between the store and the apartment twice. No feedback was provided. The test was stopped after the tenth trials for participants who were not able to find the shortcut. The route explored by the participant was automatically recorded. The walked distance was also computed.

Results

All the participants were able to control their displacement within the maze after a short period of practice. As the data did not consistently meet the assumptions of normality (Shapiro-Wilk's test), it was analysed using non-parametric tests. We used the Kruskal-Wallis test to evaluate differences in medians among the three groups. When the test was significant (p<.05), we conducted pairwise comparisons using the Mann-Whitney U test.

Memory for landmarks

The numbers of correctly recognized landmarks in routes (AB) and (AC) were summed (max = 16). The Kruskal-Wallis test indicated there was a significant difference in the medians among the groups,  2 (2, N=30) = 10.99, p<.004. Pairwise comparisons indicated that the DS group recognised significantly less landmarks than the CA and the MA groups (see Table 1, Mann-Whitney U test, respectively: p<.003 and p<.01).

Route learning

All participants in the CA group reached the criterion of two consecutive trials without error in both routes. They were 9 in the MA group and 7 in the DS group. The numbers of trials to reach the criterion in routes (AB) and (AC) were summed in these participants.

The group effect was significant,  2 (2, N=26) = 14, p<.001. The number of trials was significantly higher in the DS group or the MA groups when they were compared with the CA group (see Table 1, respectively: p<.001 and p<.007).

Insert Table I about here

Test phase

The median walked distances were not significantly different among the groups during the first trial,  2 (2, N=26) = 3,57, p<.16. They became significantly different during the last trial  2 (2, N=26) = 13,02, p<.001. In this trial, the CA group walked significantly lower distances than the DS or the MA groups (see Table 1, respectively: p<.001 and p<.01).

Walked distances significantly decreased between the first and the last trial for the CA group and the MA group (Wilcoxon one-tailed test, respectively: p<.002 and p<.03). In the DS group, there was a trend for the walked distance to decrease between the first and the last trial (p<.07). All participants in the CA group found the shortcut when walking from the store to the apartment building (route B => C). They were 5 out of 9 in the MA group, and 2 out of 7 in the DS group. When the data of these participants was excluded from the statistical analyses, the trial effect on the walked distance was no more significant in both MA and DS groups (p>.50)

Discussion

The aim of this research was to study memory for landmarks, route learning and shortcut performance in individuals with DS. We expected these participants to have a good memory for landmarks since there is evidence for a relative sparing in visual memory in individuals with DS. Indeed, they performed at the same level as typically developing children of the same mental age in a colour memory task [START_REF] Laws | Working memory in children and adolescents with Down syndrome: evidence from a colour memory experiment[END_REF]. However, our DS group recognized significantly less landmarks than the CA and MA groups. This result is surprising since their performances in the learning phase suggested they could use the landmarks to learn new routes. Yet, the memory test took place before the learning phase. A plausible explanation may be that individuals with DS paid less attention to landmarks than the TD children at the beginning of the experiment since they simply had to follow the route. Moreover, it may be possible they were more concentrated in using the input device than the control participants during the first two trials of the experiment, even though they had experience with computers and were familiarized with the display prior the experimental session.

Seven out of ten participants with DS were able to learn the two routes, even though they needed more trials than the CA controls to reach the criterion. They might have achieved this performance by using different strategies. A first strategy may have involved the memorization of the ordered sequence of landmarks and actions to be taken. This strategy could not be based on a simple association of each landmark with a unique response (ie response learning), since the routes were walked and back. However, there were only four changes of heading in each route. Individuals with DS may have memorized this very simple sequence. Indeed, there is evidence that the spatial sequential working memory is relatively preserved in DS. [START_REF] Lanfranchi | A specific deficit in visuospatial simultaneous working memory in Down syndrome[END_REF] found they performed at the same level as TD children of the same mental age in a 2-D task where they had to recall pathways. A second strategy may have involved a beacon-following procedure [START_REF] Montello | Navigation[END_REF]. Individuals using this strategy may have looked for a known landmark in the local surround, moved toward it, and then looked for another known landmark. This undemanding strategy may be efficient in the very simple environment we used. Further research is needed to better understand route learning strategies in individual with DS.

Did individuals with DS encode the spatial layout of the environment during the learning phase? Results from the shortcut test suggested most of them did not elaborate a configurational representation of their environment. Only two participants with DS were able to find the shortest route between the store and the apartment whereas all the participants in the CA group found the shortcut. Moreover, distance did not decrease significantly during the ten trials of the test phase for the other participants. Qualitative data analysis in participants who did not find the shortcut revealed that most of them found a relatively long path to reach the destination and tended to follow it over trials. Indeed participants with DS used the same path in 48% of trials (35% for the MA group). Moreover, in 35% of trials, they combined the two known routes to reach the destination2 (15% in MA group). They did (B => A => C) instead of (B => C). These participants did not err. They were engaged in a goal directed and planned activity, however they lacked the configurational knowledge allowing them to find short paths.

The hippocampal dysfunction hypothesis would have predicted that individuals with DS would perform lower than TD children with similar mental age in the shortcut test [START_REF] Pennington | The neuropsychology of Down syndrome: Evidence for hippocampal dysfunction[END_REF]. Our results did not confirm this hypothesis. This null result is difficult to interpret and might be due to a lack of statistical power. Children in the MA group had slightly better performances than participants with DS, but the task seemed difficult for them too. Interestingly, in a research on spatial knowledge acquisition in a real and a virtual large-scale environment, [START_REF] Schmelter | Empirical evaluation of virtual environment technology as an experimental tool in developmental spatial cognition research[END_REF] found that 7-8 year old children showed less configurational knowledge than older children (11-12 year-olds) and adults.

Moreover, their results suggested that the developmental process in spatial knowledge was comparable in real and virtual environments.

The results suggested that most individuals with DS could acquire knowledge about specific routes without being able to integrate that knowledge into a configurational understanding. However, we also found individual differences in this group. Some participants were unable to learn the routes; others could learn the routes but their wayfinding behaviour was not flexible; and very few of them were able to find the shortcut. It may be interesting to check if the observed variations reflect individuals differences in wayfinding behaviour in real environments. If so, VEs would become promising tools for assessing wayfinding abilities in individual with DS.

Fig. 1 .

 1 Fig. 1. Map of the virtual environment (A = the railway station; B = the store; C = the

Table 1 .

 1 Results from the three experimental phases (DS = Down Syndrome, MA = mental age, CA = chronological age; interquartile ranges are in brackets).

	DS	MA	CA

They walked toward the station and, when they arrived at the junction of the second known route, they continued their way to reach the store.

Acknowledgements

This research was supported by the Fondation Jérôme Lejeune (France) and the ANR-ESRC research program (ANR-FRBR-035 ELSTRAD). Thanks go to Vincent Sockeel for programming the VE and to Hursula Mengue-Topio for her advice regarding the experimental procedure. We also wish to thank all the participants for taking part in the study.


V Génot

F Mottez

P Louarn

Particle acceleration linked to Alfvén wave propagation on small scale density gradients

We study how Alfvén waves propagate in the presence of sharp density gradients in the direction perpendicular to the ambient magnetic field. A fully electromagnetic electron guiding centre code is used for the simulation. During the propagation, initially parallel (

¢¡ ¤£ ¦¥

), transverse scales of the order of § © © are quickly reached which contributes to the creation of a significant parallel component of the electric field in the region of density inhomogeneity. The effects of this field on the velocity distribution functions are then discussed. In particular, we show that they can present a strong deviation from their initial Gaussian shape (global shift in energy) due to the action of the parallel electric field. Evidences are then given for a net energy gain of the electrons, to the expense of the wave, during this process. This energy transfer mechanism may be relevant in order to explain the particle acceleration in the auroral plasma cavities.

Introduction

Although the mean density of the auroral plasma is very small (a few cm ¢ ), it presents a high degree of inhomogeneity.

Following the results of Viking (e.g. [START_REF] Hilgers | Probe measurements of low plasma densities with application to the auroral acceleration regions and auroral kilometric radiation sources[END_REF], the recent measurements of the Fast satellite have shown that spatial scales of the density variation shorter than the electron inertial length ( § © © ) can be encountered (Strangeway et al., 1998). Such structures have been named auroral plasma cavities. A consistent literature has reported that they are involved in many auroral processes from Auroral Kilometric Radiation generation to electrostatic turbulence (e.g. [START_REF] Louarn | Trapped electrons as a free energy source for the auroral kilometric radiation[END_REF][START_REF] Hilgers | The auroral radiating plasma cavities[END_REF][START_REF] Carlson | Fast observations in the downward auroral current region: Energetic upgoing electron beams, parallel potential drops, and ion heating[END_REF]. Recently [START_REF] Génot | A study of the propagation of Alfvén waves in the auroral density cavities[END_REF] have shown that they could play a major role in the acceleration of particles (up to energy of about a few keV). We studied how the characteristics of an Alfvén wave are modified during its propagation in a density cavity; it was shown that a non vanishing component of the electric field parallel to the ambient magnetic field progressively Correspondence to: V. Génot (v.genot@qmw.ac.uk) forms, similarly to the Inertial Alfvén wave/Kinetic Alfvén wave model. This is briefly described in section 2 of the present article. The action of these waves on the particles is the main theme of the present study. Previously, using a model of parallel electric field (with a fixed ), [START_REF] Kletzing | Electron acceleration by kinetic Alfvén waves[END_REF] has studied the modification of electron distribution functions, whereas recently, [START_REF] Clark | Electron beam formation by small-scale oblique inertial Alfvén waves[END_REF] investigated the formation of electron beam consecutive to the non linear evolution of an Inertial Alfvén Wave (IAW), thanks to a 1D self-consistent simulation. These two imposed conditions (a priori parallel electric field model and initial obliqueness of the wave) are released in our model to be computed self-consistently. Here, we present 2D Particle In Cell (PIC) simulations of a situation equivalent to the one of [START_REF] Génot | A study of the propagation of Alfvén waves in the auroral density cavities[END_REF], and particularly focus on the velocity distribution functions of the electrons that are modified by the presence of the parallel electric field formed on the density gradients. This behaviour is studied in section 3. Finally, we present some preliminary results about the effective dissipation of the incident wave.

Parallel electric field formation

From the theoretical point of view, since pioneering works (e.g. [START_REF] Hasegawa | Kinetic process of plasma heating due to Alfvén wave excitation[END_REF][START_REF] Goertz | Kinetic Alfvén waves on auroral field lines[END_REF], it is well known that the role of small spatial scales is crucial for the formation of parallel electric fields. Depending on the plasma , retaining the transverse variation of the fields ( ) in comparison with either the electron inertial length ( § © ! © , for #" %$

), or the ion Larmor radius (& ' , for #( %$

), leads to a modification of the plasma dispersion relation. Note that in the second case, the characteristic length is in fact § 0) 21 43 ©65 87 ! ¢ © which equals & 9' for @ A' £ @ ¢ ; in both cases the process effectively relies on the electron dynamics. It appears then that the electric field of the Alfvén wave can exhibit a parallel component. For the collisionless and very magnetized auroral plasma, this corresponds to the case of the IAW. Auroral plasma values lead for an Alfvén wave to a typical ratio

¡ ¨ ¢¡ ¥ ¤£ ¥ 6¥ $
for the two components of the wave electric field. In their work, [START_REF] Génot | A study of the propagation of Alfvén waves in the auroral density cavities[END_REF] have shown that retaining the density (¥ ) variations in the direction perpendicular to the magnetic field (¦ ) leads to an additional term in the equations for the parallel electric field.

As soon as this term is preponderant (that is § ¨© ¨©

) the value of the ratio ¡ ¨ can be significantly larger than the one for the IAW. This was applied to the case of the auroral plasma cavities for which the typical density varies from a factor two to five on a few kilometers. An analytically simplified linear model has been established which describes the electric field behaviour when the Alfvén wave propagates into the cavity. The density variations with altitude come from a fit to in situ measurements, and the background magnetic field is dipolar. The convergence of magnetic field lines is effectively taken into account, as it is an important feature for auroral regions: the typical transverse scale indeed narrows from a factor 4 between 20000 and 2000 km. In the transverse direction, the density variations are modelled by an analytical expression which reproduce the variations inferred from Viking data. Thus, although the model is physically simplified (low frequency and cold plasma approximations), the typical quantities variations are realistic which enables to draw some relevant conclusions on the importance of plasma cavities in the acceleration mechanism. This model has been solved numerically for an initial parallel propagation (

£ ¥

). The main results is that when the density gradients are steep enough such that they scale on § © ! © , the parallel electric field can reach values up to 10 times the ones derived from the IAW model. These values maximizes on the locus of the density gradients where the wave front of the perpendicular component are also the more distorted (

£ ¥

). This is shown on Fig. 1 which presents a zoom on the simulation domain when the situation has evolved from the initial condition for which

¡ £ ¥

. The density contours have been superimposed on the parallel electric field plotted in colour code. The highest values are reached where these contours are the closest in the perpendicular direction (maximum gradient).

The physical mechanism is explain below. In the direction perpendicular to the magnetic field, the ions move following the polarization drift ( £ ! ¨ " $# © &% (' © 1 , which is negligible for the electrons). On the density gradient locus, this induces the formation of a space charge. To counter this deviation to electric neutrality, in the low plasma that characterizes the auroral plasma (and our simulations), a parallel component of the electric field forms, on large parallel distances; the electrons then react promptly by being accelerated.

Following this linear study, we considered a more complete simulation able to tackle the non-linear plasma response. The aim is mainly to study the strong dissipation which is believed to take place in an inhomogeneous medium, such as the auroral cavity. This has been achieved thanks to a new implicit 2D PIC code which treats the ions kinetically whereas the electron motion are averaged over the gyro-period (for further details on this code the reader should refer to [START_REF] Mottez | A new guiding centre PIC scheme for electromagnetic highly magnetized plasma simulation[END_REF]). The results, presented extensively in Fig. 1. Density and parallel electric field in a gradient region, computed from the analytical linear model of [START_REF] Génot | A study of the propagation of Alfvén waves in the auroral density cavities[END_REF], and as functions of both the altitude () ) and the transverse coordinate (0 ). The convergence of the magnetic field line is taken into account. The parallel electric field is shown when the Alfvén wave has propagated down to 1 4000 km in the cavity. The initial conditions would present the same density and no parallel electric field. Génot et al. (In press), confirm the first linear study about the formation of parallel electric field on the density gradients (see Fig. 3 and explanations in the following section). During these simulations, ratios ¡ ¨ of 0.01 have been reached. With such high values of the parallel component of the electric field, one can readily address the question about how the plasma distribution is affected, especially the light electrons which are very sensitive to the parallel electric field. The kinetic nature of the code enables to display the distribution functions which is done in the following section.

Parallel electric field effects on the particle distributions

The general conditions for the simulations described in this section, are the following: about 4.102 particles are contained in a simulation box of 2048 3 32 cells, which physi- cally represents about 1000 km in the direction parallel to the magnetic field (4 ) and 15 km in the perpendicular direction (¦ ). The boundary conditions are periodic in both directions. In the middle of the box the density is decreased by a factor of 4. It thus presents a channel-like configuration.

We study the temporal evolution of a linearly polarized wave (superposition of an Alfvén wave and a fast magnetosonic wave, whose wavelengths equal the size of the box) over a few Alfvén periods. The Alfvén speed is 0.4 § .

To have a first insight of the direct effect of the plasma inhomogeneities on the particles, we conducted a comparison test. The same wave was launched in two simulations with different density configuration: the first presented a channel as described above, whereas the second was uniform. We selected a set of particles, initially from the gradient region (for the inhomogeneous case) and followed their evolution in the phase space (4 © ) ¡

). The results are shown on Fig. 2.

As time is running, the particles move in the 4 direction depending on their initial parallel velocity. However, some of them encounter a region of parallel electric field and get accelerated. This only appears in the inhomogeneous case, as it can be seen on the third panel: the particles located in the zone (

¥ " " ¥ 6¥ © ¥ ¤£ " 5 ¡ " ¥ ¤£

) for this time only exist in the presence of the density gradient. These particles may thus be resonant particles interacting with the parallel component of the electric field which travels at the Alfvén velocity. Although the particles come initially from the density gradient region, they do not all encounter a parallel electric field. Indeed, the particles get dispersed from their original position to the entire simulation box due to their parallel velocity, but the parallel electric field forms locally, where the perpendicular component and thus the polarization drift is the largest. The consequence is that the distribution function, for this set of particles, integrated over all 4 presents a beam located at ) ¡ ¡ ¥ £ § superimposed to the Gaussian distribution of non accelerated particles.

On the contrary, distribution functions of particles strictly localized on density gradient locus where ¡ £ ¥ exhibit rather different characteristics. To compute such distributions we isolate, at a given time, a region with a parallel electric field extremum, typically a rectangle in the simulation box, and we retain all the particle phase space coordinates. This is done on Fig. 3: the computation region is superimposed on the colour level plot of the parallel electric field. The resulting distribution function is plotted on Fig. 4.

One notes a complete shift of the function (that now peaks at 0.05 § ) which denotes a global response of the electron bulk to the accelerating parallel electric field. Such a distortion of the distribution functions suggests that the light electrons immediately react to the parallel electric field applied. This quick time response may also explain why nothing significant appears on the ions, neither perpendicular heating nor 3). The deviation from the zero-centred shape denotes the bulk acceleration of the electrons. direct acceleration by the parallel electric field. This is confirmed by the energy diagnostics of the following section.

Energy transfer

In this section, we draw preliminary results about the significant energy transfer from the waves to the particles. This is an attempt to explain the strong dissipation of the Alfvén wave which is believed to take place in the auroral region. Fig. 5 presents the temporal variations of the parallel kinetic energy of both electrons and ions, as well as the temporal variation of the electromagnetic energy. The different curves correspond to different amplitudes of the initial perturbation with or without perpendicular density variation. We see that the more energy is injected into the system, the more it decreases with time whereas the parallel kinetic energies of the particles are enhanced.

This denotes a net transfer of energy from the wave to the particles. However, the non linearity of the process is a not yet well understood feature. The electrons see their parallel kinetic energy multiplied by a factor 2.5 (there is only a factor 1.12 for the ions). The point we want to emphasize here is that such a dissipation of the wave is directly linked to the existence of the density gradients. As pointed out in Fig. 5, case 4, there is neither dissipation nor acceleration in the case of a homogeneous density. The density gradients may thus be ideal locations for both processes of dissipation and particle acceleration by propagating Alfvén waves. This could be a suggested mechanism to explain the brightness of the auroral arc: the whole structure of the arc is related to the acceleration region (the density cavity) but the brightest part may be connected to the density gradient locus where particles are preferentially accelerated.

Conclusions

Improving the results of the linear model, these kinetic simulations show several key features of the physics of MHD waves in inhomogeneous plasmas. It is in particular relevant to the case of the auroral plasma cavities that exhibit strong density gradients in the direction perpendicular to the background magnetic field. The wave spectrum of an incoming Alfvén wave (initially in pure parallel propagation) cascades to small length scales ( § © © ) by a linear process (the pro- gressive distortion of the wavefront). Simultaneously large parallel electric fields arise on the density gradient locus. The electrons are particularly sensitive to this field, and in the region of its extremum, react globally and very dynamically, which results in a complete shift of their velocity distribution functions. The energy transfer is clearly in favour of the electrons that experience a doubling of their parallel kinetic energy over one Alfvén period. All these features are proved to be linked to the existence of plasma inhomogeneities, which emphasizes the importance of plasma cavity as regards of many auroral physics phenomena.

Fig. 2 .

 2 Fig. 2. Evolution of the electron velocity distribution function seen in PIC simulations. Symbols correspond to macro-electrons: crosses for the simulation with an uniform density, squares for the simulation with an inhomogeneous density. The distance and the velocity are normalized to 0.1 ¢¡ ¤£ ¦¥ ¨ § and respectively.

Fig. 3 .

 3 Fig. 3. Parallel electric field computed self-consistently with a PIC code (in unit of £ ¥ ! § ¢ #" $ § ¤¡ &% ). The electron density contours have been superimposed. The extremum arise on the density gradient locus. The white box bounds the region where the velocity distribution function shown on Fig. 4 is computed.

Fig. 4 .

 4 Fig.4. Electron velocity distribution function computed in a region with a parallel electric field extremum (see the white box on Fig.3). The deviation from the zero-centred shape denotes the bulk acceleration of the electrons.

Fig. 5 .

 5 Fig. 5. Electron and ion parallel kinetic energies and electromagnetic energy versus time. The curves correspond to simulations with different levels of the initial wave amplitude : 1) ¢¡ ¡ ¡ ¤£ ¦¥ 0.05, 2) ¢¡ ¡ ¡ §£ ¦¥ 0.02, 3) ¢¡ ¡ ¡ £ ¥ 0.01, 4) uniform density and ¢¡ ¡ ¡ £ ¥ 0.05.

Acknowledgements. The numerical computations have been supported by the CNRS and were made on the C98 at the IDRIS computation center (France).