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

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.