Hyper-resistivity per level

[nicolasaunai]

We typically want the hyper-resistivity to be at the grid scale $\Delta x$ stronger than the Hall term:

This issue is about implementing a grid-scale hyper-resistive dissipation tuned per level.
Given the local dx, and the local B and n, Ohm should adjust the hyper-resistive coef to damp grid scale fluctuation efficiently.

Some notes:

$$ \nu\mathbf{\nabla}^2\mathbf{j} \sim \frac{\mathbf{j}\times\mathbf{B}}{n} $$

^equiv

At grid scale we have $j\sim B\Delta x^{-1}$ and $\nabla^2\sim \Delta x^{-2}$ so [[#^equiv]] becomes:

$$ \nu \sim \frac{B}{n}\Delta x^2 $$

So, given $\nu_0$ the hyper-resistive coefficient that is empirically convenient for $B=n=\Delta x=1$ we need to have:

$$ \nu = \nu_0\frac{B}{n}\Delta x^2 $$

We need estimates of B and n, we could imagine having a local average estimate of B and n per level taking the average for instance.