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

Use the resistive induction equation with constant magnetic diffusivity,Seek ${\mathbf{B}}_{\mathrm{tot}}=B_0{\mathbf{e}}_z+B(s,z,t){\mathbf{e}}_{\varphi }$. The axisymmetric velocity $\mathbf{u}=s\omega (s,z){\mathbf{e}}_{\varphi }$ is incompressible, and the proposed field is solenoidal. The toroidal magnetic field is parallel to the velocity, so it does not contribute to their cross product. The remaining term isThe uniform axial field has zero vector Laplacian. For the azimuthal part, differentiation of the cylindrical basis gives ${\partial }_{\varphi }^2{\mathbf{e}}_{\varphi }=-{\mathbf{e}}_{\varphi }$. ConsequentlyThere is no source for a radial or axial perturbation, so zero initial values of those components remain zero. Equating the toroidal components provesThe last Laplacian is the scalar axisymmetric Laplacian. The additional $-B/s^2$ term is essential for a vector component in a rotating basis. The source is the Omega effect: rotation that varies along an axial field line winds it into a toroidal magnetic field.

With $s=r\sin \theta$ and $z=r\cos \theta$, radial differential rotation gives ${\partial }_z\omega =\cos \theta {\omega }^{\prime }(r)$. Thus the source is $B_{0}r{\omega }^{\prime }(r)\sin \theta \cos \theta$. Put $B=f(r)g(\theta )$ with $g=\sin \theta \cos \theta$. The toroidal diffusion operator isDirect differentiation of $g$ makes the bracket $-6g$. Therefore the steady radial equation isDefine $I(r)={\int }_0^r{x}^4\omega (x)dx$ and $h=I/r^3$. Since $I^{\prime }=r^4\omega$,so $h^{\prime \prime }+2h^{\prime }/r-6h/r^2=r{\omega }^{\prime }$. Taking $f=-B_{0}h/\eta$ proves the steady toroidal induction by spherical differential rotation:Near zero, $I(r)=\omega (0)r^5/5+O(r^6)$, hence $B=O(r^2)\sin \theta \cos \theta$ and is regular. Its leading vector field is proportional to $z(-y,x,0)$, so the apparent azimuthal-basis singularity on the axis is harmless. The rapid decay of $\omega$ makes $I_{\infty }={\int }_0^{\infty }{x}^4\omega (x)dx$ finite, giving $B\sim -(B_0{I}_{\infty }/\eta )r^{-3}\sin \theta \cos \theta$ at large radius. The homogeneous radial solutions are $r^2$ and $r^{-3}$; regularity at zero excludes the latter, and decay at infinity excludes the former. Thus these boundary requirements also single out the displayed solution in this angular sector.

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.

A field in a circulation plane has components crossing circular streamlines. Magnetic flux freezing carries those lines with fluid elements at different angular speeds: the lines become spirals, their azimuthal components are stretched, and neighboring turns acquire increasingly small separations. This is a nonaxisymmetric winding and phase-mixing problem.

For illustration, in the ideal limit take the initial horizontal field along $x$ and fix a horizontal plane. With $\chi =\varphi -\omega (s,z)t$, the induction equations giveThe first term describes advection around a circle; the term proportional to $t{\partial }_s\omega$ is the stretching by differential rotation. Spatial variation of $\omega$ also produces phase gradients proportional to $t\nabla \omega$. A rigidly rotating region merely turns the horizontal field, whereas a shearing region progressively winds it up.

Even small magnetic diffusivity becomes important once these gradients are large. Locally a phase mode of azimuthal order $m$ has damping rate of order $\eta m^2|\nabla \omega {|}^2{t}^2$, leading to attenuation of order $\exp [-\eta m^2|\nabla \omega {|}^2{t}^3/3]$. The associated magnetic phase mixing under differential rotation time isFor a horizontal uniform seed, $m=1$; if $|\nabla \omega |\sim \Omega /\ell$, this is ${\Omega }^{-1}{R}_m^{1/3}$, much shorter than the ordinary diffusion time at large $R_m$. Resistive smoothing cancels the closely spaced oppositely directed field and, in persistent circulation regions with an imposed surrounding field, drives flux expulsion from the interiors toward their boundaries. The detailed final distribution depends on the geometry and boundary forcing. Winding alone does not supply a self-sustaining poloidal–toroidal regeneration loop, and magnetic feedback can again limit the kinematic process.