Summary of resistive MHD equations

For the convenience of reference, the MHD equations discussed above are summarized here. The time evolution of the four quantities, namely $\mathbf{B}$, $\mathbf{u}$, $p$, and $\rho$, are governed respectively by the following four equations:

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B}/{\textmu}_0,$$ (19)

$$\rho \left( \frac{\partial \mathbf{u}}{\partial t} +\mathbf{u} \c... ...ight) = - \nabla p + (\nabla \times \mathbf{B}) /{\textmu}_0 \times \mathbf{B},$$ (20)

$$\frac{\partial p}{\partial t} = - \gamma p \nabla \cdot \mathbf{u}-\mathbf{u} \cdot \nabla p,$$ (21)

$$\frac{\partial \rho}{\partial t} = - \rho \nabla \cdot \mathbf{u}-\mathbf{u} \cdot \nabla \rho .$$ (22)

Note that only Eq. (19) involves the resistivity $\eta$. When $\eta = 0$, the above system is called ideal MHD equations.