Sheath_boundary_simple needs an option for internal gamma setting

[mikekryjak]

Bohm speed depends on both ions and electrons:

$c_{bohm} = \sqrt{\frac{T_e + T_i}{m_i}}$

Our sheath speed is equal or greater than Bohm:
$c_s \geq c_{bohm}$

The sheath heat flux depends on $c_{s}$ for not just the particle flux, but also for the kinetic energy term:

$q^{i}_{sheath} = (\frac{5}{2} T_i+ \frac{1}{2} m_i c_s) n_e c_s$

The way the code is written is that the underlying equations of pressure and momentum advect out of the sheath, and therefore the above heat flux is always present in the code even without any sheath boundary component.

When $T_e = T_i$, this results simplifies to:

$q^{i}_{sheath} = \gamma_i n_e T_i c_s$

Where $\gamma_i = 3.5$. This is what's referred to as a "total gamma". In sheath_boundary_simple, the user can specify a total gamma and the component will add or take away heat to facilitate this.

However, $T_e != T_i$ then $\gamma_i != 3.5$ in the above picture. The heat flux can therefore go up or down depending on the ion/electron temperature ratio... so if we set our total gamma to be constant, then we are suppressing this effect.

I'm pretty sure this is why most other codes have a user setting for only the internal gamma, allowing the kinetic energy term to drift with the ion/electron temperature ratio:

$q^{i}_{sheath} = (\gamma_i \frac{5}{2} T_i+ \frac{1}{2} m_i c_s) n_e c_s$

I think this effect is small (10% - 20%) and will only matter if you are attached. However, I propose we add a setting to sheath_boundary_simple which changes the ion gamma to internal to be more accurate.