Multiple Ion Species MHD If I have a fully ionized plasma consisting of electrons and a mixture of different positive ion species of varying masses and charges...how can I use the MHD model to describe the plasma fluid motion, taking into account the coupling between different ion species and electrons?
 A: In a theoretical description you can use the general fluid equations to describe some properties of a multi-species plasma. For all quantities depending on space $\mathbf{x}$ and time $t$, you have the equation of continuity:
$$
\partial_t n_\alpha + \nabla \cdot (n_\alpha \mathbf{v_\alpha}) = 0,
$$
and the momentum equation:
$$
\partial_t \mathbf{v_\alpha} + \mathbf{v_\alpha} \cdot \nabla \mathbf{v_\alpha} = -\frac{1}{n_\alpha m_\alpha} \nabla p_\alpha + \frac{q_\alpha}{m_\alpha} \left(\mathbf{E} + \frac{1}{c} \mathbf{v_\alpha} \times \mathbf{B} \right),
$$
with mass $m$, charge $q$ and isotropic pressure $p_\alpha = n_\alpha T_\alpha$ (all equations given in CGS units). $\alpha$ stands here for the particle species, e.g. $\alpha = e^-, p^+, \dots$. For each species you have 1 equation. In addition to that the electric and magnetic fields have to satisfy the Maxwell equations (which I don't write here).
The coupling of the species happens via the charge and current densities in the Maxwell equations. These quantities are given by:
\begin{align}
\rho &= \sum_\alpha q_\alpha n_\alpha, \\
\mathbf{J} &= \sum_\alpha q_\alpha n_\alpha \mathbf{v}_\alpha.
\end{align}
On basis of this set of equations one can describe many properties of the plasma. Most commonly this is used to derive all kinds of electron and ion waves. E.g., Langmuir waves, upper hybrid waves, ion plasma waves, O/X modes, Whistler waves, etc.
You can find more on this in the book of D. R. Nicholson, Introduction to Plasma Theory.
