Statistical dynamo theory: Mode excitation

TL;DR

This paper develops a statistical framework for analyzing magnetic field generation in dynamos of arbitrary shape, focusing on mode excitation, energy distribution, and eigenmode properties using stochastic differential equations.

Contribution

It introduces a novel statistical approach to dynamo theory that accounts for arbitrary flow distributions and derives explicit expressions for mode excitation and energy distribution.

Findings

Eigenmodes are transiently excited with specific frequencies and coherence times.

The energy distribution among modes scales inversely with the number of convective cells.

Eigenvalues of the dynamo system have negative real parts, indicating stability.

Abstract

We compute statistical properties of the lowest-order multipole coefficients of the magnetic field generated by a dynamo of arbitrary shape. To this end we expand the field in a complete biorthogonal set of base functions, viz. B = sum_k a^k(t) b^k(r). We consider a linear problem and the statistical properties of the fluid flow are supposed to be given. The turbulent convection may have an arbitrary distribution of spatial scales. The time evolution of the expansion coefficients a^k(t) is governed by a stochastic differential equation from which we infer their averages <a^k>, autocorrelation functions <a^k(t) a^{k*}(t+tau)>, and an equation for the cross correlations <a^k a^l*>. The eigenfunctions of the dynamo equation (with eigenvalues lambda_k) turn out to be a preferred set in terms of which our results assume their simplest form. The magnetic field of the dynamo is shown to consist of transiently excited eigenmodes whose frequency and coherence time is given by Im(lambda_k) and -1/(Re lambda_k), respectively. The relative r.m.s. excitation level of the eigenmodes, and hence the distribution of magnetic energy over spatial scales, is determined by linear theory. An expression is derived for <|a^k|^2> / <|a^0|^2> in case the fundamental mode b^0 has a dominant amplitude, and we outline how this expression may be evaluated. It is estimated that <|a^k|^2>/<|a^0|^2> ~ 1/N where N is the number of convective cells in the dynamo. We show that the old problem of a short correlation time (or FOSA) has been partially eliminated. Finally we prove that for a simple statistically steady dynamo with finite resistivity all eigenvalues obey Re(lambda_k) < 0.