Summary

You must know the MHD equations.

$$\rho {D{\bf v}\over Dt} = -\nabla p + {\bf J}\times {\bf B} + \rho {\bf g},$$ (4.17)

$${\partial {\bf B}\over \partial t} = \nabla \times ({\bf v}\times {\bf B}),$$ (4.18)

$${\partial \rho \over \partial t} + \nabla \cdot (\rho {\bf v}) = 0,$$ (4.19)

$$\nabla \cdot {\bf B} = 0,$$ (4.20)

$${\bf J} = \nabla \times {\bf B}/\mu_{0},$$ (4.21)

$$p = \rho R T,$$ (4.22)

and an energy equation.

Simplifications to the MHD equations include looking for steady, static equilibrium solutions that satisfy force balance so that the equation of motion reduces to

$$\nabla p = {\nabla \times {\bf B}\over \mu_{0}}\times {\bf B} + \rho {\bf g}.$$ (4.23)

If gravitational effects are negligible, then we have

$$\nabla p = {\nabla \times {\bf B}\over \mu_{0}}\times {\bf B},$$ (4.24)

and finally, if the pressure gradient term is negligible (meaning that the gas pressure $p$ is much smaller than the magnetic pressure $B^{2}/2\mu_{0}$, the magnetic field is called force-free and

$${\nabla \times {\bf B}\over \mu_{0}}\times {\bf B} = 0.$$ (4.25)