Satisfaction of the Chodura/Bohm condition

[mrhardman]

Using the branch https://github.com/mabarnes/moment_kinetics/tree/wall-boundary-condition-experiment I have experimented with changes to the wall boundary condition to test how sensitive the PDF at sheath entrance is to this condition.

The nominal wall boundary condition is that no particles should return from the wall, meaning that f = 0 for dzdt <0 at the lower wall plate, and f=0 for dzdt > 0 at the upper wall plate. However, due to the huge electric field generated in the steady state sheath solutions there are particle characteristics that would reverse direction before even reaching the first grid point in space away from the wall. One modification to the boundary condition (justified in a purely Lagrangian code) could be to set to zero the parts of the incoming f that should be filled by these characteristics. This is the experiment made in this branch.

Plots with the modified BC
modified_wall_BC.zip
Note that the Chodura integral is ~ 0.9, meaning that the Chodura/Bohm condition is over satisfied. The distribution function over vpa^2 is smooth and has the "exponentially" small region enforced to be zero near the origin.

Plots with the standard BC:
standard_wall_BC.zip
Note that the Chodura integral is ~ 1.20, meaning that the Chodura/Bohm condition is not satisfied. This is due to an unphysical spike near the origin.

Clearly the choice of where to set the incoming PDF to zero is somewhat arbitrary, which means that this experiment just shows how hard it is to resolve this physics in an Eulerian code. Note that the switch epsz that differentiates continuously from the standard BC (epsz = 0) and the modified BC (epsz = 1.0) is a hardcoded parameter is the current revision of the experiment branch. It may be there there is a tunable value of epsz that corresponds to zeroing out the part of the incoming f for particles that make it a certain fraction of the grid spacing towards the first point away from the wall.

@johnomotani Perhaps of interest as a quick project to scan over epsz if it can be made an input parameter?