= Solution
Use the aligned poloidal flow from part (i), and define the <magnetohydrodynamic mass loading> $\eta=\rho\alpha$, so $\rho\mathbf u_p=\eta\mathbf B_p$. Axisymmetric <mass conservation> and $\nabla\cdot\mathbf B_p=0$ give $\mathbf B_p\cdot\nabla\eta=0$. Introduce the <poloidal magnetic flux function> $\Psi$ by $\mathbf B_p=R^{-1}\nabla\Psi\times\mathbf e_\phi$, and write $u_\phi=R\Omega$. Then
$$
\mathbf u\times\mathbf B=(u_\phi-\alpha B_\phi)\mathbf e_\phi\times\mathbf B_p=\left(\Omega-\frac{\eta B_\phi}{\rho R}\right)\nabla\Psi.
$$
The steady induction equation says the curl of this expression is zero. Hence $\nabla\Omega_F\times\nabla\Psi=0$, where
$$
\Omega_F=\Omega-\frac{\eta B_\phi}{\rho R};
$$
the <field-line angular velocity> $\Omega_F$ is constant on each connected magnetic surface and along its field lines.
To control the difference between matter rotation and field-line rotation, use the azimuthal component of the <ideal magnetohydrodynamic momentum equation>. The inertial and magnetic terms are respectively $\rho R^{-1}\mathbf u_p\cdot\nabla(Ru_\phi)$ and $(\mu_0R)^{-1}\mathbf B_p\cdot\nabla(RB_\phi)$; <pressure> and gravity have no azimuthal derivative. Multiplying by $R$ and using constant mass loading gives
$$
\mathbf B_p\cdot\nabla\left(R^2\Omega-\frac{RB_\phi}{\mu_0\eta}\right)=0.
$$
Thus the <magnetohydrodynamic angular-momentum invariant> $L=R^2\Omega-RB_\phi/(\mu_0\eta)$ is constant along the line when $\eta\ne0$. Let $v_{Ap}=|\mathbf B_p|/\sqrt{\mu_0\rho}$ and define $M_{Ap}^2=|\mathbf u_p|^2/v_{Ap}^2=\mu_0\eta^2/\rho$. Eliminating $B_\phi$ between the two invariants gives
$$
\boxed{\Omega=\frac{\Omega_F-M_{Ap}^2L/R^2}{1-M_{Ap}^2},\qquad \Omega-\Omega_F=\frac{M_{Ap}^2}{1-M_{Ap}^2}\left(\Omega_F-\frac{L}{R^2}\right).}
$$
For a highly sub-Alfvénic poloidal flow, $M_{Ap}^2\ll1$, this is $\Omega\simeq\Omega_F$ along a nonsingular field-line segment. This proves <sub-Alfvénic isorotation in a steady axisymmetric wind>, with the correction explicitly displayed. If the poloidal flow is exactly zero, the induction equation directly gives $\mathbf B_p\cdot\nabla\Omega=0$, namely <Ferraro's law of isorotation>. The approximate claim presumes regular invariants: small $M_{Ap}$ alone is not a uniform relative bound if $L/R^2$ is allowed to become arbitrarily large.
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