Basic Equations of Ideal MHD

Basic equations are as follows: The mass continuity as

$$\frac{\partial \rho}{\partial t}+\mbox{\boldmath${\nabla}$} \cdot (\rho \mbox{\boldmath${v}$})=0,$$ (B.7)

the equation of motion as

$$\rho\left(\frac{\partial \mbox{\boldmath${v}$}}{\partial t}+... ... + \frac{1}{4\pi}\mbox{\boldmath${(\nabla\times B)\times B}$},$$ (B.8)

the equation of thermal energy as

$$\frac{\partial \epsilon}{\partial t}+{\rm div}(\epsilon+p){\bf v}=\rho {\bf v}\cdot {\bf g},$$ (B.9)

or some barotropic relation $p=P(\rho)$ and the induction equation as

$$\frac{\partial \mbox{\boldmath${B}$}}{\partial t}=\mbox{\boldmath${\nabla \times (v\times B)}$}.$$ (B.10)