Make tool for extrapolating profiles into the far SOL

[eldond]

Need to know some profiles in the region between the outer edge of the SOLPS mesh and the limiting surface.

• Neutral pressure is an example of an important one
• Need to simulate things like D_alpha emission out near the wall; this could be the dominant source of D_alpha light

[eldond]

My thinking so far:

1. Make up assumption for parameter (let's call the parameter x) values along the whole wall. They should meet with the on-mesh values at the divertor plates. The coordinate should be s, distance along the wall, and the functions should depend on s-s_outer (distance to the mesh edge at the outer divertor) and s-s_inner.
   • x, x', and x'' should be continuous at the mesh edges
2. Parameterize the volume of the neutral zone (I'm gonna call the space between the mesh and the wall the "neutral zone" because that sounds like fun) by
   • a : distance from the mesh edge, normal from the mesh (a is used for minor radius)
   • g : shortest distance to any wall other than the divertor plates (g stands for gap)
   • L_o : distance parallel to the magnetic field to the outer divertor plate
   • L_i : distance parallel to B to the inner divertor plate
   • I think this parameterization might work better than just doing x(R,Z) subject to boundary conditions.
3. Pick xmin and xmax. For some parameters, like T_e, these can just be the values at the mesh edge and at the wall. For others, like neutral density, it might be possible to have higher neutral density in a corner of the wall where kinetic neutrals bounce around and concentrate and near the plama due to a recycling process with lower density in between. So maybe dips would make sense.
4. Now fit x(a,g,L_o,L_i) subject to constraints:
   • x > xmin everywhere
   • x < xmax everywhere
   • x = xmesh at mesh edge
   • dx/da = d(xmesh)/da at mesh edge
   • x'' = xmesh'' at mesh edge
   • x = xwall at wall
5. And provide the following incentives in a cost function:
   • penalize dx/dg at the wall
6. I think x should be a polynomial function of a, g, L_o, L_i. Given the constraint of matching x'' and also not extrapolating out of bounds, a second order polynomial is insufficient. Maybe a third order solution would be good enough. Or maybe drop the x'' boundary condition and fit the neutral zone with second order polynomials.

[eldond]

This is wrong. Gradients at the plasma mesh boundary should probably be with respect to psi_N. Exponentially decaying profiles should probably be used for many parameters.

[eldond]

I need an extended mesh for this.

Needs ProjectTorreyPines/IMASggd.jl#8