= Sub-Alfvénic isorotation in a steady axisymmetric wind
{title2=$\Omega\simeq\Omega_F\quad(M_{Ap}^2\ll1)$}
Assume zero toroidal loop voltage in smooth steady <ideal magnetohydrodynamics>, so the poloidal flow aligns with the <poloidal magnetic field>. Write $\rho\mathbf u_p=\eta\mathbf B_p$. The <continuity equation> makes $\eta$ constant along a magnetic line. The induction equation gives the conserved <field-line angular velocity> $\Omega_F=\Omega-\eta B_\phi/(\rho R)$. The azimuthal momentum equation is $\rho\mathbf u_p\cdot\nabla(R^2\Omega)=\mathbf B_p\cdot\nabla(RB_\phi)/\mu_0$; hence $L=R^2\Omega-RB_\phi/(\mu_0\eta)$ is also constant along the line.
With $M_{Ap}^2=\mu_0\eta^2/\rho=|\mathbf u_p|^2/v_{Ap}^2$, eliminating $B_\phi$ gives
$$
\Omega=\frac{\Omega_F-M_{Ap}^2L/R^2}{1-M_{Ap}^2}.
$$
For $M_{Ap}^2\ll1$ and nonsingular invariants, $\Omega=\Omega_F+O(M_{Ap}^2)$ and matter approximately corotates along a field line. With exactly zero poloidal flow, induction directly gives <Ferraro's law of isorotation>. The approximation requires the correction $M_{Ap}^2(\Omega_F-L/R^2)$ to be small on the segment in question.
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