6 - Laminar Dynamo Theory

Summary

The kinematic dynamo problem is defined, and a necessary condition for dynamo action in terms of the maximum rate-of-strain of the velocity field is obtained. It is shown that, under frozen-field conditions, the dipole moment of the current distribution in a sphere is bounded but that diffusion across the spherical surface, in conjunction with interior convection, can lead to its sustained increase. Cowling’s anti-dynamo theorem and variants are proved and some consequences described. Rotor dynamos, including Herzenberg’s two-sphere dynamo and Gailitis’s ring-vortex dynamo, are analysed. Helical dynamo action is illustrated by the Ponomarenko dynamo. The Riga dynamo experiment, involving a propellor-driven helical flow of liquid sodium, is described; in this, dynamo action is observed as the propellor speed is increased beyond a critical threshold. The Bullard–Gellman formalism is described, and early convergence issues are noted. It is noted that axisymmetric flows in a sphere can generate non-axisymmetric fields, of either steady or oscillatory form. Finally, the ‘stasis’ dynamo of Backus, involving three phases well separated in time, is described; this was an early model through which dynamo action in a spherical domain could be rigorously established.