Statistical dynamo theory: Mode excitation

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= summation operator_{k}a;{k}(t)b;{k}(r) . The properties of these biorthogonal function sets are treated in detail. 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} 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 Ilambda_{k} and -1/Rlambda_{k} , respectively. The relative rms 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} approximately 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 first-order smoothing approximation) has been partially eliminated. Finally we prove that for a simple statistically steady dynamo with finite resistivity all eigenvalues obey Rlambda_{k}<0 .