Introduction to Plasmas and Plasma Dynamics

Magnetohydrostatics

Analogous to hydrostatics in OFM, relating pressure and force, we are here interested in determining the interrelationship of force, pressure, and magnetic field (Boyd and Sanderson, 1969). Again we begin by writing the momentum equation:

$\begin{matrix} ρ \frac{D \vec{v}}{D t} = - \nabla p + \vec{J} \times \vec{B}, or : \\ ρ [\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla) \vec{v}] = - \nabla p + (\frac{\nabla \times \vec{B}}{μ}) \times \vec{B}, and for steady state; \\ ρ (\vec{v} \cdot \nabla) \vec{v} = - \nabla p + (\vec{B} \cdot \nabla) \frac{\vec{B}}{μ} - \frac{\nabla B^{2}}{2 μ} = - \nabla (p + \frac{B^{2}}{2 μ}) + (\vec{B} \cdot \nabla) \frac{\vec{B}}{μ}, \\ and : (\vec{v} \cdot \nabla) \vec{v} = - \frac{1}{ρ} \nabla (p + \frac{B^{2}}{2 μ}) + (\vec{B} \cdot \nabla) \frac{\vec{B}}{ρ μ} . \end{matrix}$

\vec{v}

= 0, ρ = constant, then

\nabla (p + \frac{B^{2}}{2 μ}) - ρ ({\vec{V}}_{A} \cdot \nabla) {\vec{V}}_{A} = 0

, where:

{(B^{2} / μ ρ)}^{1 / 2} \equiv {\vec{V}}_{A}

So, the fluid pressure term has an equivalence to (B²/2 ...

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