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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?

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    $\begingroup$ 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) $\endgroup$ – Kyle Kanos Aug 30 '19 at 19:13
  • $\begingroup$ quasi-neutral yes...PIC would be too coding intensive not to mention the amount of computation power needed...are there any analytical methods? $\endgroup$ – Joeseph123 Aug 30 '19 at 19:25
  • $\begingroup$ 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). $\endgroup$ – Kyle Kanos Aug 30 '19 at 20:05
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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.

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  • $\begingroup$ Could these equations be used to find “transient states?” Not necessarily oscillations, but to describe confinement phenomena like implosions in pinches? These don’t necessarily describe a steady-state system, but fully contain all information about the system, am I correct? $\endgroup$ – Joeseph123 Sep 7 '19 at 13:00
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    $\begingroup$ As long as these effects are describable by a fluid, then yes. If kinetic effects, such as collisions, etc., are important you have to go back to Klimontovich/Vlasov equation. Nevertheless if fluid model is valid you have to be careful with the pressure. I assumed it to be isotropic but maybe viscosity is of importance. Also the ideal gas law for the pressure is only valid for an isothermal case, i.e. slow time variations. For short time scale behaviour you have to use adiabatic gas law. $\endgroup$ – P. U. Sep 9 '19 at 6:23

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