MHD

Treatment of the plasma as on fluid with density ρ speed v and resistivity η.

ρ \frac{\partial v}{\partial t} = j \times B - \nabla p + ρ g

E + v \times B = η j

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ v) = 0

\frac{\partial σ}{\partial t} + \nabla \cdot j = 0

Coupled with Maxwell-Faraday Law and Ampere Circuital Law (without displacement current)

MHD waves

ω^{2} = k^{2} v_{A}^{2} c o s^{2} θ

ω^{2} = \frac{1}{2} k^{2} (v_{a}^{2} + c_{s}^{2} \pm \sqrt{(v_{a}^{2} + c_{s}^{2}) - 4 v_{A}^{2} c_{s}^{2} c o s^{2} θ})

Full Ohm's law (Hall term and electron inertia ?):

E + v \times B = η j + \frac{1}{e n} (j \times B - \nabla p_{e})

ω^{2} = \frac{k^{2} c^{2} ω_{c e}}{ω_{p e}^{2} + k^{2} c^{2}}