Ideal MHD equations

Remarks

They are the equations of magnetohydrodynamics with zero viscosity, zero magnetic diffusivity and adiabatic thermodynamics. Clause 5 is a constraint on the initial data: clause 3 gives $\partial_t (\nabla \cdot \mathbf{B}) = 0$, so it holds for all time once it holds at $t = 0$. The Lorentz force splits as $$\frac{1}{\mu_0} (\nabla \times \mathbf{B}) \times \mathbf{B} = -\nabla \frac{\abs{\mathbf{B}}^2}{2 \mu_0} + \frac{1}{\mu_0} (\mathbf{B} \cdot \nabla) \mathbf{B},$$ a magnetic pressure plus a magnetic tension along field lines. Example: constant $\rho$, $p$ and $\mathbf{B}$ with $\mathbf{u} = 0$ is a solution; every term with a derivative vanishes. Non-example: the resistive induction equation $\partial_t \mathbf{B} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \Delta \mathbf{B}$ with $\eta > 0$ is not ideal. With $\mathbf{B} = 0$ the system reduces to the compressible Euler equations. The system can be written in conservation form for mass, momentum, total energy and magnetic flux.