# 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?

• Typically one uses MHD assuming bulk properties of fluid (i.e., is a quasi-neutral plasma), no? Otherwise it's Particle-in-Cell methods that I'm aware of for modeling (at least numerically) – Kyle Kanos Aug 30 '19 at 19:13
• quasi-neutral yes...PIC would be too coding intensive not to mention the amount of computation power needed...are there any analytical methods? – Joeseph123 Aug 30 '19 at 19:25
• Well you could use two-fluid approx for the MHD, but that won't be any more work than a PIC code...it might help if you add more details to the question about what you are trying to do, as it is terribly vague (at least to me). – Kyle Kanos Aug 30 '19 at 20:05

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}