= Solution
Although the <Lorentz force> can exchange energy between poloidal and toroidal motion, it does no net work in the frame rotating with a magnetic surface. Dotting steady momentum with the poloidal field and using the mass-loading, isorotation, and angular-momentum invariants combines the magnetic work with the toroidal kinetic term to give
$$
\mathbf B_p\mathbin\cdot\nabla\widetilde\epsilon=0.
$$
Therefore
$$
\boxed{\widetilde\epsilon(\psi)
=\frac12u_p^2+\frac12(u_\phi-r\omega)^2
+\Phi_{cg}+h}
$$
is constant on each magnetic surface. Here
$$
\boxed{\Phi_{cg}=\Phi-\frac12\omega^2r^2}
$$
is the gravitational potential plus the centrifugal potential in the frame corotating at the field-line angular velocity. This is the modified <Bernoulli function> of the wind.
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