Field-line angular velocity of an axisymmetric wind ($\mathbf{u}=k\mathbf{B}/\rho +R\omega (\psi ){\mathbf{e}}_{\varphi }$)

In steady axisymmetric ideal magnetohydrodynamics, induction makes $\omega$ constant on regular connected poloidal flux surfaces. It describes field-line rotation. The material angular velocity is $u_{\varphi }/R=\omega +k{B}_{\varphi }/(\rho R)$ and differs when the toroidal magnetic field contributes. If the poloidal magnetic field vanishes, the flux-surface integration requires separate treatment.