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The flux-function form solves $\nabla \cdot \mathbf{B}=0$ identically. With $k=k(\psi )$, the poloidal relation $\rho {\mathbf{u}}_p=k{\mathbf{B}}_p$ and axisymmetry solve steady mass conservation. The ideal induction equation is solved by the velocity representation with the field-line angular velocity $\omega =\omega (\psi )$. The azimuthal momentum equation integrates once to the magnetohydrodynamic angular-momentum invariant $\ell =\ell (\psi )$.

What remains is the pressure or entropy equation and the two poloidal components of momentum. Projecting poloidal momentum along a field line produces the Bernoulli invariant in part (c); projecting across magnetic surfaces produces the transfield or Grad-Shafranov equation that determines their shape. An equation of state and boundary conditions complete the problem. The stellar gravitational potential is prescribed, so no self-gravity Poisson equation remains to solve.

Solved by gpt-5.6-sol high.

Ideal flux freezing anchors a field line in the highly conducting disc, so its field-line angular velocity naturally equals the angular velocity of its footpoint. A circular orbit in the central potential is Keplerian:

Put $r=r_0+\delta r$ and expandThe linear radial term vanishes because $GM/r_0^2={\Omega }_0^2{r}_0$. The radial and vertical second derivatives are respectively $-3{\Omega }_0^2$ and ${\Omega }_0^2$, so

If the field line makes angle $\theta$ from the vertical, then locally $\delta r=z\tan \theta$. Its quadratic potential change iswhich is negative exactly whenThis is the local magnetocentrifugal acceleration launching criterion.

Solved by gpt-5.6-sol high.