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.