Magnetohydrodynamics

Remarks

The model follows from Maxwell's equations by dropping the displacement current, which is valid for speeds much smaller than the speed of light. One adds Ohm's law $\mathbf{J} = \sigma (\mathbf{E} + \mathbf{u} \times \mathbf{B})$ in the moving fluid. Eliminating the electric field $\mathbf{E}$ gives the induction equation, after using $\nabla \times (\nabla \times \mathbf{B}) = -\Delta \mathbf{B}$ when $\nabla \cdot \mathbf{B} = 0$. Example: if $\mathbf{B} \equiv 0$, the equations reduce to the incompressible Navier–Stokes equations. Another example: a fluid at rest, $\mathbf{u} = 0$, in a uniform field $\mathbf{B} = B_0 \mathbf{e}_z$ with constant pressure is a solution, since $\nabla \times \mathbf{B} = 0$ and $\Delta \mathbf{B} = 0$. Non-example: a field $\mathbf{B}$ with $\nabla \cdot \mathbf{B} \ne 0$ is not admissible, even if it solves the induction equation. The magnetic Reynolds number $\mathrm{Rm} = U L / \eta$ compares advection of the field by the flow with its diffusion; the limit $\eta = 0$, $\mu = 0$ gives incompressible ideal MHD. The compressible version adds the continuity equation and an energy equation.