In an axisymmetric magnetostatic state, pressure and gravitational forces are poloidal, so the azimuthal component of the <Lorentz force density> must vanish. Hence $f=0$ and
$$
\nabla(L\alpha)\times\nabla\beta=0.
$$
The two gradients are locally parallel, so $L\alpha$ is constant on each regular level surface of $\beta$. Therefore
$$
\boxed{L\alpha=F(\beta)}
$$
for an arbitrary flux function $F$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-314/3/c/solution