An approach to a dynamic theory of dynamo action in a rotating conducting fluid

Abstract

Dynamo action associated with the motion generated by a random body force f(x, t) in a conducting fluid rotating with uniform angular velocity ω is considered. It is supposed that, in the Fourier decomposition off, only waves having a phase velocity V satisfying V. ω > 0 are present and that the Fourier amplitudes of f are isotropically distributed. The resulting velocity field then lacks reflexional symmetry, and energy is transferred to a magnetic field h0(x, t) provided the scale L of h0 is sufficiently large. Attention is focused on a particular distribution of h0 (x, t) (a circularly polarized wave) for which this dynamo action is most efficient. Under these conditions, the mean stresses acting on the fluid are irrotational and no mean flow develops. It is supposed that \[ \lambda \ll \Omega l^2,\quad \lambda\ll h_0l\quad{\rm and}\quad\nu/\lambda = O(1)\quad\hbox{or smaller}, \] where l ([Lt ] L) is the scale of the f-field, and ν and λ are the kinematic viscosity and magnetic diffusivity of the fluid. The response to f is then dominated by resonant contributions near the natural frequencies of the free undamped system. As h0 grows in strength, these frequencies change, and the dynamo process is rendered less efficient. Ultimately the magnetic energy M (and also the kinetic energy E) asymptote to steady values. Expressions for these values are obtained for the particular situation when ν [Lt ]λ and when the frequency ω0 characteristic of the f-field is small compared with other relevant frequencies, notably ω and h0/l; under these conditions, it is shown that \[ \frac{M}{E}\sim C\left(\frac{\Omega}{\omega_0}\right)^{\frac{1}{2}}\left(\frac{\nu}{\lambda}\right)^{\frac{1}{2}}\frac{L}{l}, \] where C is a number of order unity determined by the spectral properties of the f-field. The implications for the terrestrial dynamo are discussed.