= Solution
The axial <angular momentum> density is $\rho Ru_\phi$. The second equation in part (b) writes its conservation law with <magnetic axial angular momentum flux>
$$
\mathbf F_{J,p}=-\frac{RB_\phi\mathbf B_p}{\mu_0}.
$$
The 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 is
$$
\boxed{\mathbf F_{E,p}=-\frac{u_\phi B_\phi\mathbf B_p}{\mu_0}=\Omega\mathbf F_{J,p}.}
$$
The 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 $\partial_tB_\phi=0$, 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 $\mathbf B_p\cdot\nabla(RB_\phi)=0$, which holds, for example, if $B_\phi=0$. 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:
$$
\boxed{\rho R^2\mu_0\partial_t^2\Omega=\mathbf B_p\cdot\nabla\!\left(R^2\mathbf B_p\cdot\nabla\Omega\right).}
$$
This is the variable-coefficient wave equation for the <torsional Alfvén wave>.
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