# Nuclear Fusion Physics and Technology/Magnetohydrodynamics

#### Definition: Minimal MHD density equation

Minimal MHD density equation is equation

${\displaystyle {\frac {\partial \varrho }{\partial t}}+{\text{div}}\varrho {\vec {u}}=0}$

#### Definition: Magnetic viskozity

Magnetic viskozity ${\displaystyle \eta \in \mathbb {C} }$  is defined as

${\displaystyle \eta ={\frac {1}{\sigma \mu }}}$

where ${\displaystyle \sigma }$  is conductivity and ${\displaystyle \mu }$  permeability.

#### Definition: Minimal MHD magnetic equation

Minimal MHD magnetic equation is equation

${\displaystyle {\frac {\partial {\vec {B}}}{\partial t}}=\eta \Delta {\vec {B}}+{\text{rot}}({\vec {u}}\times {\vec {B}})}$

#### Definition: Minimal MHD speed equation

Minimal MHD speed equation is equation

${\displaystyle \varrho {\frac {\partial {\vec {u}}}{\partial t}}+\varrho ({\vec {u}}\cdot \nabla ){\vec {u}}=-\nabla p+\eta \Delta {\vec {u}}+(\xi +{\frac {\eta }{3}})\nabla ({\text{div}}{\vec {u}})+{\vec {j}}\times {\vec {B}}}$

where ${\displaystyle \eta ,\xi }$  are viskosity coefficients, ${\displaystyle \varrho }$  is particle density.