 Solution

The axial angular momentum density is $\rho R{u}_{\varphi }$. The second equation in part (b) writes its conservation law with magnetic axial angular momentum fluxThe nonmagnetic part of the energy flux is azimuthal. Using $(\mathbf{u}\times \mathbf{B})\times \mathbf{B}=\mathbf{B}(\mathbf{u}\cdot \mathbf{B})-\mathbf{u}{B}^2$, its poloidal part isThe ratio is therefore $\Omega$ wherever the compared component of the angular-momentum flux is nonzero; the proportionality remains meaningful at zero flux.

A fully steady magnetic configuration requires ${\partial }_t{B}_{\varphi }=0$, hence ${\mathbf{B}}_p\cdot \nabla {\Omega }_0=0$: angular velocity is constant along poloidal field lines. This is Ferraro's law of isorotation. The steady azimuthal force additionally requires ${\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })=0$, which holds, for example, if $B_{\varphi }=0$. The remaining meridional force balance is a separate equilibrium condition.

Since $\rho$ and ${\mathbf{B}}_p$ are time independent, differentiate the angular-momentum equation once more and substitute the induction equation:This is the variable-coefficient wave equation for the torsional Alfvén wave.