Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 76 / 1 / iii / Solution

Let $\ell$ be the characteristic radius over which the differential rotation changes, with angular velocity of order $\Omega$. The winding source has magnitude $F\sim B_0\Omega$. For times too short for appreciable magnetic diffusion, the zero initial toroidal magnetic field therefore grows as $B\sim B_0\Omega t$. In the steady balance, $F$ is opposed by $\eta B/{\ell }^2$, giving $B_{\mathrm{steady}}\sim B_0\Omega {\ell }^2/\eta$. Comparing these amplitudes gives the diffusive establishment time of a wound toroidal field:With $R_m=\Omega {\ell }^2/\eta$, this is $R_m$ rotation times and the steady toroidal amplitude is of order $R_m{B}_0$. Thus high electrical conductivity makes establishment very slow even though winding begins immediately.

More precisely, if $\mathcal{L}={\nabla }^2-s^{-2}$ on regular toroidal magnetic fields, the solution with zero initial value satisfies $B(t)=B_{\mathrm{steady}}-e^{\eta t\mathcal{L}}{B}_{\mathrm{steady}}$. The residual relaxes diffusively. In the unbounded domain, ${\ell }^2/\eta$ estimates equilibration of the rotating region; exact establishment at every radius is only an asymptotic statement, with diffusion on scale $r$ taking order $r^2/\eta$. As $\eta \to 0$, the time diverges and the prescribed-flow ideal solution continues its secular winding. In a real fluid, magnetic tension and Lorentz force feedback can modify the differential rotation before the very large kinematic steady amplitude is attained.