Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 4 / b / Solution

Introduce the squared poloidal Alfvén numberSolving the two linear azimuthal invariants givesAt an Alfvén surface, $M_A^2=1$. Smooth passage through the apparent singularity requires both numerators to vanish at the same cylindrical radius $r_A$, giving the Alfvén-surface regularity condition for an axisymmetric windThe finite limiting values of $u_{\varphi }$ and $B_{\varphi }$ then follow by l'Hopital's rule from the local variation of $M_A$ and $r$ along the field line; the algebraic invariants alone fix their combination rather than each value separately at the critical point.