Nuclear Fusion Physics and Technology/Magnetohydrodynamics

Definition: Minimal MHD density equation
Minimal MHD density equation is equation $$ \frac{\partial \varrho}{\partial t} + \text{div} \varrho \vec{u} = 0 $$

Definition: Magnetic viskozity
Magnetic viskozity $$ \eta \in \mathbb{C} $$ is defined as $$ \eta = \frac{1}{\sigma \mu} $$ where $$ \sigma $$ is conductivity and $$ \mu $$ permeability.

Definition: Minimal MHD magnetic equation
Minimal MHD magnetic equation is equation $$ \frac{\partial \vec{B}}{\partial t} = \eta \Delta \vec{B} + \text{rot} ( \vec{u} \times \vec{B}) $$

Definition: Minimal MHD speed equation
Minimal MHD speed equation is equation $$ \varrho \frac{\partial \vec{u}}{\partial t} + \varrho ( \vec{u} \cdot \nabla ) \vec{u} = - \nabla p + \eta \Delta \vec{u} + ( \xi + \frac{\eta}{3} ) \nabla ( \text{div} \vec{u} ) + \vec{j} \times \vec{B} $$ where $$ \eta, \xi $$ are viskosity coefficients, $$ \varrho $$ is particle density.