River, Coastal and Estuarine Morphodynamics: RCEM 2005 – Parker & García (eds)© 2006 Taylor & Francis Group, London, ISBN 0 415 39270 5

Modelling of dune patterns by short range interactions

Clément Narteau, Eric Lajeunesse, Franc̨ois Métivier & Olivier RozierEquipe de Géomorphologie, Laboratoire de Dynamique des Systèmes Géologiques, Institut de Physiquedu Globe de Paris, Paris cedex, France

ABSTRACT: A 3D cellular automation is disclosed that enables modelling the dynamics of bedform. Theoverall mechanism can be regarded as a Markov chain, a discrete system with a finite number of configurationsand probabilities of transition between them. Physical processes such as erosion, deposition and transport aremodelled at the elementary scale by nearest neighbour interactions.At larger length scales, topographic structuresarise from internal relationships based upon these short range interactions. This article focuses on crescenticbarchan dunes that are used as a benchmark for our numerical model of bedforms. Length and time scales ofisolated barchan dunes are studied in order to constrain the parameters of the model. Then we discuss patternselection and the evolution of a population of dunes over a wide range of initial and boundary conditions. Weeventually show that our model can be generalized to bedforms through the increase of the sand availability.

1 INTRODUCTION

There is a considerable variation within bedforms asthey locally depend on the topography, the sedimentload, and the flow. Such variability might be expressedby different relationships between erosion and deposi-tion rates and the fluid velocity field. While empiricalrelationships estimate these quantities for a given bedunder particular conditions, theoretical relationshipssimplify the turbulence problem to make easier thedescription of the sediment capacity of the flow. Inboth cases, it remains extremely difficult to tacklethe impact of the size distribution of sediment par-ticles. Despite these limitations, the study of aeoliandunes has significantly filled the gap between observa-tions and models (Bagnold 1941; Pye and Tsoar 1990;Lancaster 1995).

Under dry conditions, the transport of sand grainsby the wind involves similar physical mechanisms thansediment transport in liquids. However the absenceof cohesion, dissolution and sedimentation limits thenumber of relations between fluid and solid ingredi-ents. Then, in order to investigate couplings betweenwind and topography, it is sufficient to formalize thewind velocity field with respect to the surface pro-file as well as the erosion and deposition responsesto shear stress (Jackson and Hunt 1975; Hunt et al.1988; Weng et al. 1991). In this framework, the prin-ciple of mass conservation is commonly ensured by acontinuity equation for the height profile

where q is defined as a volumic sand flux per unit oftime and per unit of length perpendicularly to the winddirection. The capacity of transport takes therefore theform of a saturated sand flux qs, and the only param-eters are those which are relevant for the magnitudeof qs according to the topography (Kroy et al. 2002a;Andreotti et al. 2002). Schematically, for a strongenough wind, deposition dominates if q approachesqs (i.e ∂xq < 0), else erosion occurs (i.e ∂xq > 0). In allcases, there are different ways the grains can be putinto movement. First, they can be dragged, lifted andaccelerated by the excess shear stress exerted by thefluid on the surface.This corresponds to saltation. Sec-ond, they can be released by impacts of falling grainsand crawl on the surface. This corresponds to repta-tion. These two transport modes are obviously relatedto one another, essentially because saltation impliesan irregular hopping process through the retroactionof transported grains on wind velocity.

In this paper, we concentrate on the formation andthe evolution of crescentic barchans dunes as a bench-mark for a new model of sediment transport. Barchandunes are isolated structures with horns extendingdownwind on both sides of an sand pile characterizedby a slip face and a windward face (Fig. 1). Saltationand reptation are active on the windward face and theslope angle may vary from 10◦ to 15◦; the slip faceis not submitted to the dominant wind and its geom-etry is controlled by avalanches of grains reachingthe dune crest. The angle of repose of the sand beingapproximately of 30◦, the dune profile is asymmet-ric in the direction of the wind. Barchans dunes havebeen observed in different geophysical environments

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Figure 1. (a) Comparison between crescentic barchan dunesin air and water, in desert and laboratory experiments respec-tively (courtesy of Physical Review Letters, Hersen et al.(2002)). (b) Tranverse view of a barchan dune in Morocco(picture taken by B. Andreotti).

on Earth (arid desert, icecap, deep water), on Mars,but also in laboratory experiments (Fig. 1). They arepropagating downwind, and independent relationshipscan be deduced from their dimensions, volumes andvelocities. Overall, the aim of this paper is to comparethe predictions of our model with these relationships.

Following Nishimori and Ouchi (1993a) andWerner (1995), our numerical approach is dedicated tothe analysis of emergence mechanisms in geomorphol-ogy. An emergence mechanism is met when one phe-nomenon leads to another, not in a direct cause/effectrelationship, but in a manner that involves patternof interactions between the elements of a systemover time. In other words, an emergent macroscopicbehavior can not be anticipated from the analysis ofthe constituent parts of the system alone, but canonly result from their capacity to produce complexbehaviours as a collective, through their mutual andrepeated interactions. Such a complexity is an intrin-sic property of cellular automata. Based on a discretestructure and a finite number of states at an elementaryscale, cellular automata (CA) are systems that evolveon a network according to local interaction rules.These rules determine how each element responds toinformation transmitted from other elements along thenetwork connections. Most of the time, these con-nections are simplified to include only interactions ata microscopic scale between nearest neighbors. CAare useful tools in physics, geophysics and biology toanalyze pattern formation because their output matchvery well what we observe in nature without beingdependent on a complete description of small scaleprocesses. Thus, the origin of macroscopic behaviorsas well as the emergence mechanism itself may beanalyzed from a limited set of parameters. We exploitthis property to implement a model conceptual enoughto be applied on different types of geomorphologicalenvironments from aeolian dunes to river beds.

2 THE MODEL

Sediment transport is modelled by a Markov chain, astochastic process characterized by a finite number ofconfigurations evolving from one another accordingto a set of actions with different transition rates.

2.1 Length and time scales

A three-dimensional regular lattice models an interfacebetween a turbulent fluid (air or water) and a layer oferodible sediment lying on a solid flat bedrock. Thisinterface is subject to a so-called fluid action constantin magnitude and direction. An elementary cell has theshape of a parallelepiped, with 90◦ angles, a squarebase of length l and a height h. We focus on sedi-ment flux rather than on individual particle motionsand h is therefore equal to ld , the distance for a grainto accelerate up to the average fluid velocity:

where ρs, ρf and d are the grain density, the fluiddensity and the characteristic length scale of a grainrespectively.The choice for such a length scale is moti-vated by observations in desert area and laboratoryexperiments (Bagnold 1941; Hersen et al. 2002) aswell as by analytical results together with numericalsimulations (Kroy et al. 2002b; Andreotti et al. 2002).The aspect ratio η = h/l corresponds to an upper limitof the slope angle θ upward in the direction of the flow(η = tan(θ) < 1). Finally, the characteristic time scaleτ is determined from the dimensions of an elementarycell and an arbitrary volumic sediment flux Q (seeEq. 1):

2.2 The discrete dynamic

Erosion does not affect the underlying solid bedrockand, at the bottom of the system, a layer of stable cellsforms a flat surface where the transport of particles cancreate a topography. Then, we consider 4 states, 2 solidand 2 fluid. This is the minimum number of states nec-essary to implement retroaction mechanisms betweena topography and a flow (Tab. 1). The two solid states,grains (G) and mobilized grains (M ), allow to imple-ment the action of fluid shear velocity on the surface.The two fluid states, fluid (F) and excess shear stress(S), allow to implement the action of topography onthe flow pattern.

The indices (i, j, k), i ∈ [1, L], j ∈ [1, L], k ∈ [1, H ]label the Cartesian coordinates and the cell ci,j,k iseither

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Table 1. 4 different states: 2 solid, 2 fluid. This is the min-imum number of states to explore feedback mechanismsbetween circulation patterns and evolving topography.

Action of fluid shearstress on topography grains (G)

mobilized grains (M )

Action of topographyon fluid motions fluid (F)

Excess shear stress (S)

Figure 2. First neighbors in a regular rectangular paral-lelepiped mesh and the transition rates associated with aci,j,k -cell.

• Grains (G): a G-cell represents a volume of sedi-ment of uniform particle size.

• Mobilized Grains (M ): a M -cell represents a volumeof sediment transported by the fluid. Transportationof grains involves two transport modes related toone another: saltation and reptation. Here we do notdifferentiate between those modes.

• Fluid (F): a F-cell represents a volume of fluidwhere the velocity is under a threshold of erosion.Furthermore, particles can not be transported andF-cells are agent of deposition.

• Excess Shear Stress (S): a S-cell represents a vol-ume of fluid where the shear stress exerted by theflow is above a threshold of erosion. S-cell are agentof erosion and transport by saltation and reptation.

NG , NM , NF , NS are the number of G-cells, M -cells,F-cells, S-cells respectively, and

We do not consider long range interaction like otherdiscrete approaches (Nishimori and Ouchi 1993b;Bishop et al. 2002), we only consider interactionsbetween two neighboring cells with a common side(Von Neumann neighborhood, Fig. 2). These doublets

of neighboring cells are noted (ci,j,k , ci±1, j,k ), (ci, j,k ,ci, j±1,k ) and (ci, j,k , ci, j,k±1). As detailed below, a cellmay change states only if it shares an edge with a neigh-boring cell in a different state. In addition, we makea distinction between the orientation of the doubletsaccording to gravity and the direction of the flow. Thischoice yields the lowest number of possible configu-rations and transitions while allowing for modellingof the physical processes involved in the formation ofbedforms.

The whole process is defined in terms of a Pois-son process with stationary transition rates betweenthe various possible states of the doublets of neigh-boring cells. Given a transition from state u to v, theprobability distribution of the waiting time until thenext transition is an exponential distribution with rateparameter λv

u. Then the probability that a pair of neigh-boring cells in state u undergoes a transition towardthe state v in the infinitesimal time interval dt is λv

udt.The practical way we proceed in the numerical simu-lations is detailed in appendix. The main point is thatat each iteration three random numbers determine thetime step, the doublet which undergoes a transitionand the transition kind itself. Therefore the model pos-sesses the Markov property as the next configuration(i.e. the future) is independent of the previous con-figurations (i.e. the past), given the knowledge of thepresent configuration. This probabilistic approach andthe physical processes represented by different set oftransitions distinguish our model from classical CA(Narteau et al. 2001).

2.3 The physical processes

Each of the physical processes that we will nowdescribe corresponds to a set of transitions. A tran-sition of a given set cannot be considered in isolationbecause only combined and repeated actions are capa-ble of reproducing these processes. For the samereason, transition rates are determined by referenceto characteristic times representative of the givenphysical processes.

2.3.1 Fluid flowAt a given altitude above a flat surface, the velocityfield might be considered constant in magnitude anddirection, and the velocity profile is known to increaseaccording to a logarithmic function (Landau andLifshitz 1963). This is not the case above a roughsurface, over which a flow might produce an highly tur-bulent circulation especially when it involves erosion,deposition and transport. Here, we adopt simplifyingassumptions to limit the scope of the model on fluidvelocity above topography.

As the streamlimes approach an obstacle, they con-verge and the velocity increases; after this obstacle, thestreamlines diverge and the velocity decreases. Suchobservations are put into practical effect through the

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Figure 3. Transition rates of the permutation between aS-cell located in ci, j,k and a F-cell located in ci±1, j,k , ci, j±1,k ,ci, j,k±1.

motions of S-cells in an ocean of F-cell by consideringthe following transitions:

Each of these transition can be characterized by a vec-tor according to the orientation of the doublet and themagnitude of the transition rate (Fig. 3). The resultingvector, i.e. the sum of the six vectors, determines thedirection of the flow as well as the intensity of the tur-bulent diffusion. λw being an estimate of the turbulentdiffusion, we takeλ

j−1w = λ

j+1w = λk−1

w = λk+1w = λw and

λi+1w > λw > λi−1

w . As a consequence, the flow is goingeastward (Fig. 3). On the other hand, the velocity of thisflow cannot be determined only by the transition prop-erties, but, as explained below, it can be approachedthrough the proportion of S-cells in the fluid.

2.3.2 ErosionGrains are lifted and dragged by the shear stressapplied by the moving fluid and set in motions inthe direction of the flow. This erosion process, whichdoes not discriminate between saltation and reptation,involves two types of transition:

Practically, S-cells in contact with G-cells produce M -cells upward and in the direction of the flow. Thesetransitions are the only source of transport in themodel. Taking the sediment flux Q as a control param-eter, the magnitude of the transition rate λs is deriveddirectly from τ (see Eq. 3).

2.3.3 TransportIf the fluid velocity is high, grains already in motioncan be transported upward and in the direction of theflow. Such a transportation involves two transitionsbased on the permutation of M -cells in contact withS-cells:

The magnitude of this sediment transport is propor-tional to Q and, by convention, the transition rates istaken equal to λs.

2.3.4 DepositionWhen the fluid velocity is not high enough to main-tain particles in suspension, deposition occurs. Thisdeposition of fluid-borne grains is enhanced by topo-graphic obstacles and occurs faster on slopes of exist-ing structures. This process involves the followingtransitions

where G-cells are created from M -cells if they arenot in contact with S-cells. a > 1 and, for the sake ofsimplicity, we assume that the deposition rate is equalto λs, the erosion rate.

2.3.5 DiffusionHorizontal diffusion disperses the grains and flattensthe topography. This process involve the followingpermutations between G-cells and F-cells:

where λd = λi−1d = λi+1

d = λj−1d = λ

j+1d is the inverse of

a characteristic time for diffusion.

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2.3.6 GravitySand grains fall under their own weight and exertpressure on all the other grains. Gravity prevents fur-ther motions and leads to deposition.We attribute thefollowing transition to these gravitational processes:

They include downward permutations between G-cellsand fluid-states as well as depositions of M -cellslocated under solid-states. λg is determined from theStokes velocity and it is generally few orders ofmagnitude larger than all the other transition rates.

3 NUMERICAL SIMULATIONS

The model involves 20 transitions characterized by alimited number of independent transition rates. Beforewe present the results of the numerical simulations,different aspects of the model can be addressed whenlooking at all the transitions together. First, the conser-vation of mass is ensured by the constant number ofsolid cells (NG + NM = cte). Second, S-cells are per-sistent in all transitions (NS = cte) in such a way as toensure the fluid forcing.As a consequence, we are deal-ing with an open system which relies on the balancebetween erosion and deposition to ensure the conser-vation of momentum at a macroscopic scale. For thisreason, the proportion of S-cells in the fluid has to below, β = NS /NF << 1, and the fluid velocity can onlybe derived from β.

Table 2 shows all the model parameters and theirnumerical values for all the simulations presentedin this work. In this case τ ≈ 3.5 day and ld /τ ≈0.13 m.day−1. Initial conditions and boundary condi-tions are essential aspects of the long-term develop-ment of topography and we focus here on two differentset of conditions in order to analyze the evolution ofsome physical quantities as well as different propertiesof pattern formation.

3.1 Evolution of a conical bump

First we are interested in stationary patterns and weanalyze the evolution of a conical bump of sedimentin a corridor (Fig. 4): L = 200, H = 40, β = 0.15 andtwo walls facing each other form a corridor L/2 wideparallel to the flow; the cone has a base radius of

Table 2. Parameters of the model and their values.

ld 0.44 m� 11.3◦Q 100 m2·yr−1

λsτ 6λwτ 6λdτ 0.06λgτ 6000a 10

Figure 4. Model for a conical pile of sediment in a corri-dor. Three layer of cells are represented without the aspectratio: (a) a horizontal layer just above the flat bedrock, (b) avertical layer parallel to the flow, (c) a vertical layer perpen-dicular to the flow. In each figures, black lines indicate layerintersections.

45 and a half summit angle equal to �. In addi-tion, we consider asymmetric boundary conditions inthe direction perpendicular to the flow in order tosatisfy simultaneously mass conservation and a homo-geneous injection of material. Practically, each G-cellejected from the system in the direction of the flow isreinjected randomly through the opposite boundary.

Under such conditions, the distribution of S-cellsrapidly changes on both sides of the obstacle as theyabandon slopes oriented in the direction of the flow andaccumulate on the slopes oriented against the directionof the flow.Where the density of S-cell increases, shearstress is higher and erosion and transport dominate (see

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the location of M -cells in Fig. 5). On opposite slopes,where the density of S-cell decreases, fluid velocity islower and deposition is more likely to occur. Then aflux of sediment results from the motion of grains (i.e.M -cells) from one side to another of the sediment pileand the symmetry of the cone shape is broken (Fig. 5).Interestingly, the slope is smaller on the face orientedagainst the direction of the flow than on the oppositeface. The height of the cone varies and, where it islower, particles propagate more rapidly in the direc-tion of the flow. This produces horns of both sides onthe cones (Fig. 5).As this evolution proceeds the struc-ture moves in the direction of the flow and convergestoward a stationary state which is commonly describesas a crescentic shape barchan dune.

Fig. 6 shows the evolution of the dimension of sucha barchan dune over long time. Width, length andheight reach a stable value despite important variationsinherent to our discrete approach. Stronger fluctua-tions of the length result from unstable behaviors alongthe horns where the density of G-cells is lower. Theemergence and persistence of a stationary crescenticshape (see contours in Fig. 6) demonstrate that erosionand deposition can balance each other under specific(unrealistic?) conditions. However, from a theoreticalpoint of view, this is a chance to analyze differentphysical quantities under a statistically-stable regime.

As the flux of grains on the crest and in the hornsstabilize as well, the barchan dune reaches a constantvelocity of v = 0.18 ld /τ (Fig. 7). In more conventionalunits, v = 8.25 m.yr−1 for an height of H ≈ 10.2 m.Over short time, the barchan tends to accelerate withrespect to a loss of volume associated to a redistribu-tion of grains in the entire system (see contour plotsin Fig. 7). This relationship between the volume of thedune and its velocity is of primary importance in dunefields because it results inevitably in dune interactionpatterns.

3.2 Evolution of randomly distributed sediment

We are now interested by these patterns of interactionbetween dunes and we analyze the evolution of homo-geneously distributed sediment in a corridor (Fig. 8).As before, L = 200, H = 40, β = 0.15, and two wallsfacing each other form a corridor L/2 wide parallelto the flow; on two layers just above the flat bedrock,G-cells are randomly distributed with the probabilityp = 0.65. In addition we consider periodic boundaryconditions.

Over short time, grains form clusters which deformin the direction of the flow (Fig. 9). These clus-ters coalesce to produce elongated structures thatstart to modify significantly the density of S-cells intheir neighborhood. All these structures propagates atdifferent velocity according to their geometry and vol-ume. Smaller structures being faster, they merge withlarger ones which in turn decelerate. Thus, a set of

Figure 5. Evolution of a conical bump over short time. Eachfigure is obtained by smoothing the topography obtained fromG-cells. Dots are the M -cells. They are essentially located onthe slope oriented against the direction of the flow.

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Figure 6. Evolution of the width, the length and the heightof the conical bump in a logarithmic scale over long time.Note that the dimensions of the barchan stabilize. Theinset shows the stationary crescentic shape where the heightbetween level-lines is 1.5 ld .

Figure 7. Position of the sediment pile with respect to time(solid line) in (a) linear and (b) logarithmic scales. Overlong time, the velocity of the barchan is constant (dottedlines: y ∼ 0.03x). Over short time, the sediment pile accel-erates during the transition from a conical shape to a barchanshape. It corresponds to a loss of volume associated with theredistribution of grain escaping from the horns (see contourplots).

Figure 8. Model for randomly distributed sediment in acorridor. Three layer of cells are represented without aspectratio: (a) a horizontal layer just above the flat bedrockwhere G-cells are randomly distributed with the probabilityp = 0.65, (b) a vertical layer parallel to the flow, (c) a verticallayer perpendicular to the flow. In each figures, black linesindicate layer intersections.

crescentic barchan dunes emerge. Each of them is sim-ilar to the one describe in the previous section but theirinteraction make their respective evolution much morecomplicated. Finally, in this particular simulation, thelast configuration of dunes (i.e t/τ = 1187) is not sub-ject to strong changes and the set of dunes of similarsize is extremely resilient. The main reason for such abehavior is the small size of the system and the peri-odic boundary conditions which impose that all dunespropagate within their own shadow.

Nevertheless, the formation and the evolution ofdune fields provides the opportunity to compare modelpredictions to field measurements. To reduce signif-icantly finite size effect, model statistics have beencomputed on much larger systems with L = 103 andH = 102, with similar initial conditions than in Fig. 8but without walls.

Structural properties are estimated from the out-put of the model at one point in time. For barchansobserved in arid deserts and in the numerical simula-tions, Fig. 10 shows the relationships between widthand length, Fig. 11 shows the relationships betweenwidth and height. These two independent morpho-logical relationships are well fit by lines and there

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Figure 9. Evolution of homogeneously distributed sedi-ment over long time. Figure are obtained by smoothing thetopography obtained from G-cells. Dots are the M -cells.

Figure 10. Relationships between barchan width and lengthin arid deserts (gray ◦) and in the model (black x ). Fieldmeasurements are averaged by ranges of height. Predic-tions of the model represent individual dune. The solid line(y = 1.40x − 1.27) and the dashed line (y = 1.39x − 6.5) fitthe synthetic and the real data respectively (Andreotti et al.(2002) compiled observations from Finkel (1959), Hastenrath(1967), Hastenrath (1987)).

Figure 11. Relationships between barchan width and heightin arid deserts (gray ◦) and in the model (black x). Fieldmeasurements are averaged by ranges of height. Predic-tions of the model represent individual dune. The solid line(y = 0.13x − 1.01) and dashed line (y = 0.14x − 2.1) fit thereal and the synthetic data respectively (Andreotti et al.(2002) compiled observations from Finkel (1959), Hastenrath(1967), Hastenrath (1987)).

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Figure 12. Relationships between barchan width and veloc-ity in arid deserts (gray symbols), and in the model (black x).Field measurements are averaged by ranges of height. Predic-tions of the model represent individual dune. The line fits thesynthetic data and follows v = 1.25/(H + 5.5). The diamondsymbol is for the barchan dune shown in Fig. 5. (Andreottiet al. (2002) compiled observations from Finkel (1959) (*),Hastenrath (1967) (◦), Hastenrath (1987) (◦), Slattery (1990)(square), and Long and Sharp (1964) (+)).

are good agreements between synthetic data and fieldmeasurements.

In the model, the dune measured in Figs. 10 and 11are studied over long time (t/τ = 335) to approximatetheir average velocity v. Fig. 12 shows the relation-ship between height and velocity for numerical andobserved barchan dunes. One more time, the output ofthe model and the natural data are in good agreementand it is possible to fit both set of measurements with

where Q1 is a volumic sand flux and H0 a characteris-tic height. Here, we have Q1 = 7l2

d ·τ−1 and H0 = 4.0ld .In more conventional units, Q1 = 140 m2·yr−1 andH0 = 1.7 m.

4 DISCUSSION AND CONCLUSION

Under specific conditions, our model reproduces cres-centic barchan dune patterns that compare well withobservations. Firm morphological and velocity con-straints are satisfied and validate the predictions of ourmodel based on short scales interactions. Flux mea-surements have now to be investigate as we are awarethat a full understanding of dune dynamics should

include estimation of transport capacity on differentparts of the topography.

In the last decades, different scientists have exploreda variety of CA approach in order to model dunepatterns under dry conditions (Nishimori and Ouchi1993a; Werner 1995; Nishimori et al. 1999). Modelsare based on a characteristic saltation length and onrules involving long range interactions. In all cases,despite a stochastic ingredient, the trajectory of an ele-mentary volume of grains is given from the presentconfiguration of cells, and the rules do not allow todissociate between transport and erosion-depositionprocesses. In addition, there is no treatment of aero-dynamics effect. Here, we present a CA approach inwhich erosion and deposition locally depend on flowpatterns that are affected in turn by the surface pro-file. Such an innovation will allow to concentrate onrelationships between shear stress and topography andto characterize turbulent flow patterns on an evolvingsurface (Wiggs 2001). This goal can be achieved byconsidering more realistic small scale interactions forthe dynamic of S-cells.

In this preliminary work, the transport is simpli-fied to the extreme in order to concentrate on thefirst requirement of this kind of models, the formationand the development of realistic structural patterns.Nevertheless, we have shown that anisotropic motionsof S-cells and an evolving topography are enough toreproduce the existence of a shadow zone downwind,the concentration of shear stress on the windward face,and, as said above, dynamical properties of barchandune fields. Such qualitative features of physical fluidbehaviour have now to be replaced by more quantita-tive analysis. In the field of turbulence, it is common toanalyze fluid dynamics from particle collisions (Frischet al. 1986). This lattice gas method converts dis-crete motions into physically meaningful quantitiesand dispenses with the need to solve the Navier-Stokesequations. The discrete nature of our system offersthe opportunity to develop such a lattice gas methodbased, as the actual model, on short range interactionsbetween a finite number of states. As a result of amore generalized set of transitions (i.e. erosion, depo-sition and transport occurring in all direction), realisticstructures and fluid circulations might appear together.

This model has been developed to be applied to dif-ferent type of systems, in particular for the analysis ofbedform. Fig. 13 shows a snapshot of a model simula-tion similar to the one presented on Fig. 8 but with 10times more sediment (i.e. G-cells). There is no indi-vidual structures and larger structures perpendicular tothe flow develop. These structures exhibit similaritieswith bars in river. The bar dynamic and the subsequentmorphology of river will be studied by going furtherin this direction.

More generally, our model can also be adapted todifferent types of geometry and boundary conditions.

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Figure 13. A snapshot of a model simulation with more sediment. This figure is obtained by smoothing the topographyobtained from G-cells. Dots are the M -cells.

In order to compare the predictions of the modelwith classical observations of physical geography, itis possible to concentrate on

1. the impact of the variability of wind direction aswell as the impact of sediment availability (com-pare Figs. 9 and 13).The primary objective will be toreproduce different types of dune (e.g. linear dunes,star dunes) and to locate them in a phase diagram.Another objective could be to estimate the orienta-tion of dune crests according to the magnitude of thewind fluctuation in order to analyze dune attractortrajectories (Anderson 1996) and defect behaviors(Werner and Kocurek 1997).

2. the role of vegetation on erosion and depositionrates, and the effect of a vegetal cover on structuraland dynamical dune patterns.

3. the interaction between dune patterns and structuresrelated to human activity (e.g. accumulation of sandaround constructions).

Note that in all cases there are reciprocal applica-tions in the analysis of bedforms underwater.

In geomorphology, as in many domains in sci-ence, description in terms of differential equationshave limits that can be related to a lack of theoreticalbackgrounds, the role of heterogeneities and numeri-cal limitations. Then, CA approach can be describedas an alternative which focus on self organization andthe emergence of structure without taking into accountall the diversity of the small scale physical processes.Through our model, we try to provide a link betweenclassical CA methods and continuum mechanics insuch the way that we will be able to constrain struc-tural complexity of geophysical system by a set ofwell-defined physical quantities. We believe that thediscontinuous nature of our model and the feedbackmechanisms between different types of states willallow this system to move between different basinsof attractions and therefore capture in detail someof the more distinctive features of the evolution ofbedforms.

ACKNOWLEDGMENTS

We thank B. Andreotti and P. Claudin for helpfuldiscussions, figures and fields measurements. A pre-liminary work of E. Sepulveda was of great help.This work was supported by E2C2, a Specific Tar-geted Research Project of the European Community.In the I.P.G.P., Clément Narteau benefit from a MarieCurie reintegration grant 510640-EVOROCK of theEuropean Community.

APPENDIX

Here we present in more detail how we combine aPoisson process with a Markov chain with station-ary transition rates. The algorithm is schematicallydescribe in Fig. 14.

As said above, we consider only first neighborsinteractions and transition of doublets of cells in aD-dimensional parallelepipedic mesh. Overall, all tran-sitions and doublets are independent of one another.Each cell can be in one of Ns states. Then, the numberof different doublets is

and the number of doublets in a L × W × H mesh is

For a given configuration at time t, we first deter-mine ni the number of doublets in state i:

A transition of doublet from state i to j is modelledby a Poisson process with a rate parameter λ

ji . Then,

the occurrence of such a transition in a population of

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Figure 14. The algorithm of the model. ni is the number ofdoublets in state i, π

ji the rate parameter of the transition of

doublet from state i to j, and π the rate parameter of the entiresystem. To respect the Markov property, only one doublet ofcells is changed at each iteration. The time step, the transitiontype and the doublet are successively chosen according tothe present configuration of the system and three randomnumbers.

doublets i follows also a Poisson process with rateparameter

Generalized to all doublets and transitions, the totaltransition rate in the entire system is

At each iteration, only one doublet makes a tran-sition from one state to another. The time step tis therefore variable, randomly chosen according tothe magnitude of π. Practically, we draw at randoma value R1 between 0 and 1, and we consider that thecharacteristic time necessary for a transition to occur is

During this time step, the type of transition is alsorandomly chosen with respect to a weighted probabil-ity determined from the π

ji -values

Numerically, we define a cumulative step functionranging from 0 to 1, where jumps are proportionalto the P j

i -values. Then we draw at random a valueR2 between 0 and 1. This value falls within a jumpof the cumulative step function which determines inturn the type of transition to occur. Thus, transitionwith the highest rates have more chance to be selectedbut transitions with small rates may also occur. Theserare events are an essential ingredient of the modellingdeveloped in this paper.

Finally, when the transition from i to j is selected,we draw at random an integer between 1 and ni. Thuswe identify the doublet which makes a transition.

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