= Solution
Use a steady <axisymmetric flow> governed by <ideal magnetohydrodynamics> and take the <Newtonian gravitational potential> to be axisymmetric. Work locally on connected regular <magnetic flux> surfaces with nonzero poloidal flow. As usual for a wind without an imposed toroidal loop voltage, $E_\phi=0$; regularity on an included rotation axis guarantees this condition. It is needed to exclude steady cross-field flow in an annular domain.
Since $\nabla\phi=\mathbf e_\phi/R$, the <poloidal magnetic flux function> gives
$$
\mathbf B_p=\frac1R\nabla\psi\times\mathbf e_\phi,
\qquad B_R=-\frac1R\partial_z\psi,\qquad
B_z=\frac1R\partial_R\psi,\qquad
\mathbf B_p\cdot\nabla\psi=0.
$$
The <poloidal magnetic field> is divergence-free. The zero toroidal <electric field>, with $\mathbf E=-\mathbf u\times\mathbf B$, gives $\mathbf u_p\parallel\mathbf B_p$. Define $k$ by $\rho\mathbf u_p=k\mathbf B_p$. Steady <mass conservation> then gives
$$
0=\nabla\cdot(\rho\mathbf u_p)=\nabla\cdot(k\mathbf B_p)=\mathbf B_p\cdot\nabla k.
$$
Thus $k=k(\psi)$ is the <magnetohydrodynamic mass loading>, the <mass flux> per unit poloidal <magnetic flux>. Subtracting $(k/\rho)\mathbf B$ from the full <velocity> leaves only an azimuthal component. The <cross product> is
$$
\mathbf u\times\mathbf B
=\left(u_\phi-\frac{kB_\phi}{\rho}\right)\frac{\nabla\psi}{R}
\equiv\omega\nabla\psi.
$$
The steady <ideal magnetohydrodynamic induction equation> requires $0=\nabla\times(\omega\nabla\psi)=\nabla\omega\times\nabla\psi$. Consequently the <field-line angular velocity> is $\omega=\omega(\psi)$ and
$$
\boxed{\mathbf u=\frac{k(\psi)\mathbf B}{\rho}+R\omega(\psi)\mathbf e_\phi}.
$$
Neither gas <pressure> nor axisymmetric gravity exerts an azimuthal <torque>. Multiplying the azimuthal momentum equation by $R$ combines the cylindrical curvature terms into angular-momentum transport:
$$
\rho\mathbf u_p\cdot\nabla(Ru_\phi)
=\frac1{\mu_0}\mathbf B_p\cdot\nabla(RB_\phi).
$$
Insert $\rho\mathbf u_p=k\mathbf B_p$ and use $\mathbf B_p\cdot\nabla k=0$. It follows that
$$
\mathbf B_p\cdot\nabla\left[R\left(u_\phi-\frac{B_\phi}{\mu_0k}\right)\right]=0,
$$
so the <magnetohydrodynamic angular-momentum invariant> is
$$
\boxed{u_\phi-\frac{B_\phi}{\mu_0k(\psi)}=\frac{\ell(\psi)}R}.
$$
The two terms account respectively for matter and <Maxwell stress tensor> transport of <angular momentum>.
Finally, subtracting $\gamma$ times the logarithmic <mass density> equation from the logarithmic <pressure> equation gives $D\ln(p/\rho^\gamma)/Dt=0$. The <specific entropy> of a fixed-composition <ideal gas> is a function of $p/\rho^\gamma$. Steadiness and <axisymmetry> reduce its advection to $\mathbf u_p\cdot\nabla s=0$. Hence
$$
\boxed{s=s(\psi),\qquad p=K(\psi)\rho^\gamma}.
$$
These flux-label functions are local on connected regular surfaces; disconnected components with the same numerical label need not share constants without an additional matching condition.
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