Dynamical equilibration of a spatially periodic flow of conducting fluid with an embedded magnetic field: toward a self-consistent alpha-squared dynamo model

Abstract

Spatially periodic flows possessing helicity exhibit dynamo action. One such flow studied by G.O. Roberts serves as the model in this thesis. The first results detail the structure of the flow, including its symmetry properties as ascertained by group-theoretical methods. A kinematic dynamo model at low magnetic Reynolds number is developed in the third chapter. It is found that magnetic fields with axial wavenumbers O(R2M) are destabilized by the fluid motions, producing a dynamo effect. Roberts' cellular flow is next utilized to develop a simplified self-consistent dynamo model. The Navier-Stokes and induction equations are integrated over a cellular region in the x - y plane. Degrees of freedom are recovered by imposing time-dependent amplitudes on the axial and planar velocities. The resulting set of three scalar evolution equations for the mean magnetic field energy density and velocity amplitudes are made non-dimensional. The model thus represents a hydromagnetic analogue of the disc dynamo problem. Equilibria of the three variables are determined in terms of four dimensionless parameters arising from the analysis. The bifurcation structure of the system is analyzed, leading to a minimal criterion for dynamo action. Using linear stability analysis, mean magnetic field growth rates are determined in the regimes of the parameter space specified by the bifurcations. In a concluding chapter, implications for the geodynamo are discussed.