3 - Advection, Distortion and Diffusion

Summary

Alfvén’s theorem, the analogue of Kelvin’s circulation theorem, is proved. The concept of a ‘frozen-in’ field in a perfectly conducting fluid is described, and the associated conservation of magnetic helicity is proved. The analogy with vorticity in an ideal fluid under Euler evolution is presented. The evolution of a magnetic field subjected to uniform strain is considered, and the possibility of accelerated ohmic diffusion is described. Flux expulsion is introduced through the example of uniform shearing of a space-periodic field; magnetic instability associated with oscillating shear flow and with steadily rotating shear flow is described. Flux expulsion associated with differential rotation is then analysed, with focus on the initial distortion, the intermediate phase and the ultimate steady state. The law of isorotation and the generation of toroidal field by differential rotation are also analysed, with emphasis again on the initial phase and the ultimate steady state. The concept of topological pumping resulting from asymmetry between upward and downward convective flow is introduced, and the behaviour as a function of the magnetic Reynolds number is described.