Sub-Alfvénic isorotation in a steady axisymmetric wind ($\Omega \simeq {\Omega }_F(M_{Ap}^2\ll 1)$)

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 ${\Omega }_F=\Omega -\eta B_{\varphi }/(\rho R)$. The azimuthal momentum equation is $\rho {\mathbf{u}}_p\cdot \nabla (R^2\Omega )={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })/{\mu }_0$; hence $L=R^2\Omega -R{B}_{\varphi }/({\mu }_0\eta )$ is also constant along the line.

With $M_{Ap}^2={\mu }_0{\eta }^2/\rho =|{\mathbf{u}}_p{|}^2/v_{Ap}^2$, eliminating $B_{\varphi }$ givesFor $M_{Ap}^2\ll 1$ 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.