Nuclear Fusion Physics and Technology/Magnetohydrodynamics

Contents

Definition: Minimal MHD density equationEdit

Minimal MHD density equation is equation

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

Definition: Magnetic viskozityEdit

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

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

where $\sigma$$\sigma$ is conductivity and $\mu$$\mu$ permeability.

Definition: Minimal MHD magnetic equationEdit

Minimal MHD magnetic equation is equation

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

Definition: Minimal MHD speed equationEdit

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