Single diffusive field with neutrals

Solves a system of \(1+S\) fields representing \(S\) ion species. This equation system is used by specifying

<I PROPERTY="EQTYPE" VALUE="SingleDiffusiveField"/>

in the <SOLVERINFO> field

Electrons are always present, and at least one ion species must be present. Explicitly the fields are:

\[\left[n_e, n_i, ...\right]\]

The continuity equation is:

\[\frac{\partial n}{\partial t} + \nabla \cdot \vec{\Gamma} = S_n\]

with the diffusive flux given by:

with D being the anisotropic diffusion tensor:

\[D = \vec{b}\vec{b} k_\parallel + (I - \vec{b}\vec{b}) k_\perp\]

where the \(k_\parallel\) and \(k_\perp\) have the option of being functions of fields, and \(\vec{b}\) is the local unit vector parallel to the magnetic field.

The source \(S_n\) is given by VANTAGE. The system is assumed to be isothermal, with all species at the same temperature.

Boundary Conditions

The available boundary conditions for this system of equations are as follows.

System Bohm

All ion species enter the sheath at the system sound velocity \(c_s\).

\[c_s = \sqrt{\sum_i \frac{n_{0i} k_B T_e}{n_{0e} m_i}}\]

The boundary condition should be set to Neumann in the directions perpendicular to the magnetic field, while the parallel component of the flux at the boundary

\[\vec{\Gamma}\cdot \vec{b} = n c_s\]

where \(c_s\) in this case is a constant (isothermal Bohm speed \(c_s = \sqrt{kT/m_i}\)).

\[\frac{\partial n}{\partial \mathbf{n}} = \mp \frac{n}{c_s} \frac{\partial v_{\parallel i}}{\partial \mathbf{n}}\]

Species Bohm

Dirichlet

Neumann