Past exam of the advanced fluid dynamics course of the University of Oxford / 2019 / 1 / c

With zero velocity and no time dependence, the magnetic second-moment tensor equation holds automatically, but the magnetohydrodynamic momentum equation still requiresThe magnetic tension must therefore be a pressure gradient. Locally, its necessary integrability condition ison a simply connected region this is also sufficient for a scalar $p_0$, subject to the global boundary conditions. In physical units the same balance is $\nabla P_0={\mathbf{J}}_0\times {\mathbf{B}}_{\mathrm{phys},0}$: the Lorentz force density is balanced by mechanical pressure, rather than necessarily vanishing. A zero mean magnetic field does not make the local magnetic tension zero, so an arbitrary tangle is not automatically a static equilibrium.