Particle deposition

From a more general point of view, if we can extrapolate a maximum fluid ve-locity (depending onuentry, the landscape shape), we can therefore define a

parti-cle motion efficiencyǫ_p = 1/max_r,tu_f(r, t), and then takeξ_x/y/z =ǫ_p u_fx/y/z+u_{f allx/y/z} . The result of this method is to uniformly (over the simulation domain and time)

increase by a factor ǫp the moving particle flux, and to reach sooner the end of the simulation.

In this section, we have seen how to model aeolian transport of solid particles over a lattice, without splitting it among different phenomenon (creeping, saltation, suspension). However, we have not yet model deposition and erosion. Both algorithms presented in the folowing subsections seem rather simplistic, but they have shown to be enough to recover some interesting results.

3.1.3 Particle deposition

As defined in section 2.1.1, s(r, t) is a boolean value indicating if the site r, at time t, is solid. Therefore, if s(r+ci, t) is true, the particles directed from r in directionci cannot proceed as they would hit a wall. The solution is therefore to freeze them, and to increment consequently the local number of frozen particles pf rz(r). Once the number of frozen particles exceeds a given threshold θf rz, the site is solidified. This procedure is expressed in algorithm 3.1.3, and displayed in figure 3.7.

From a practical point a view, ifθf rz is too small, unrealistic deposit can build up (such as one site wide particle piles or arches). Meanwhile, setting a larger θ_{f rz} (&5) averages the deposit and naturally avoids these model artifacts. The

Algorithm 3.1.1 a sequential particle distribution. X([0,1]) is a random vari-able uniformly distributed in [0,1].

do i= 1, ρp(r, t)

Algorithm 3.1.2A faster particle re-direction process with binomial distributed drawings.

!ωi for remaining directions must sum to 1

¯

!not enough particle to extract B directly anymore

!the ρ remaining particles must be redirected individually do i= 1, ρ

000000000000

Figure 3.7: the deposition process, when two particles are leaving siter. The one directed rightwards can proceed normally as its destination is free. Meanwhile, the one directed downwards is heading towards a solid site; therefore, it will have to freeze on r, and increase the local frozen number of particles. Once this number reaches θf rz, the site r is solidified.

parameter θf rz is therefore a key to calibrate the “real” cell size (roughly, the more frozen particles are needed to solidify the cell, the wider it is), and ranges between 10 and 10⁴ in our simulations.

A toppling process can be incorporated in this model (as it has been, in a simple version, with the 3D model). Such a process can take into account critical slope angles, such as described in [Dupuiset Chopard 2000a].

3.1.4 Erosion

Many features can be incorporated in the erosion process: wind velocity pro-file characteristics [Sundsbo 1997], particle concentration in the fluid calming down vortices thus the erosion rate [Castelle 1995], landing particles shocking the deposit and ejecting (instantaneously or after a short delay) other parti-cles [Martinez 1996].

We did try some of the above ideas to recover this complex phenomenon, but the method finally kept is a large simplification of them. The idea is simply, to release frozen particles (see previous section) with a probability ζp, and to artificially place them on the above site if it is free; then, it follows the aeolian transport mechanism described in section 3.1.2, i.e. it will either fly away if the velocity is high enough, or fall back to its initial position (and thus changing nothing to the deposit).

From a practical point of view, there are two possible implementations:

• each frozen particle below a free site is ejected with probabilityζ_p,

• or, if nf rz, the number of frozen particles on a site below a free site, is large enough, B(n, ζp) gives in one evaluation the number of particles to be placed on the above site.

However, it is important to consider carefully the number of particle “candi-date” to erosion. If we consider the following situations:

000000000000

1111111111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 11111111111

Counting only the upper layer of frozen particles would make us pulling par-ticles among 100 of them in (a) (as θf rz = 100 and there is nothing above the solidified layer) and only among 1 in (b). This produces a severe discontinuity in the erosion process as a totally different stock of “erodable material” is available when situations are more or less the same in both cases.

Therefore, it appears more correct to consider as erodable all the first solidified layer (i.e. θf rz particles) plus the frozen particles in the underneath cell.

Hence, with this later option, we may consider differently situations (a) and (c), even if they are very close (indeed, freezing one more particle in the rightmost situation would freeze the above site and therefore return to the same situation as the leftmost one). The solution is therefore to consider at most θf rz particles as erodable ones (in fact we may consider less only if the particles are frozen directly above a solid cell which is not built from frozen particle, but from the initial ground configuration). This later choice is therefore equivalent to pull particles from a constant volume equivalent to a frozen site (θf rz particles).

Eroding particle with eddies

It is reasonnable to erode more particles where the wind eddies are the more de-velopped. Even if the eddies are smaller than the domain resolution scale, we have extrapolated theire influence with the fluid BGK subgrid model in section 2.2.3.

As the local non-equilibrium momentum flux tensor has been computed in the Smagorinski process, we may re-use it here: instead of being (straigthly) a con-stantζp, the local erosion probability can be multiplied by the maximum value of the non-equilibrium momentum flux tensor in the neighborhood. This idea has been implemented only on the 3D version of our code and gave better results.

3.1.5 Summary

In this section, we have introduced the different mechanisms, ruling locally the solid particles evolution. These mechanisms can be tuned with several parame-ters, mainly:

θf rz the number of particles to solidify a cell (frozen threshold ζp probality of erosion

uf all falling velocity

These parameters, together with fluid paramters and domain sizes are sum-marized for all the 3D eperiment in table 3.1.

Algorithm 3.1.3 Particle deposition: freezing and solidifying steps

do i= 1, q // q is the number of lattice directions // for the particle model

pf rz(r, t+ 1) ←pf rz(r, t) if s(r+ci, t) then

pf rz(r, t+ 1)←pf rz(r, t+ 1) +p⁽ⁱ⁾(r, t) p⁽ⁱ⁾(r, t)←0

endif enddo

if (pf rz(r, t+ 1)≥θf rz) s(r, t+ 1) =true