Non-steady global magnetic fields in kinematic theory

Abstract

The dynamo equation for the mean field (B) contains a random forcing term of unknown magnitude, which is therefore always omitted. The influence of this term is potentially large. To evaluate its effect, we employ ensemble averaging. If an ensemble average is used, there is no random forcing term in the dynamo equation. The effect of fluctuations is that the ensemble members get out of phase, so that (B) → 0. The damping time of (B) can be found by requiring that the mean energy (BB) remains finite. The eigenvalues к of the dynamo equation then all have negative real parts. Imк determines the period, and -Reк/Imк the relative period stability of the dynamo. We have developed a code to solve the equation for (BB) in a spherical shell (the convection zone), assuming axisymmetry. We report our first results, which do not yet include differential rotation. Using spherically symmetric boundary conditions, we reproduce the well known α²-dynamo, whose behaviour is known analytically. For instance, for an α²-dynamo located in a shell with inner boundary at R/2, we find that (BB) remains finite for R²γ/β = 1.48, where 3 represents turbulent diffusion and .y turbulent vorticity. Taking α = 1/4(βy)^1/2 — a factor of four below maximum helicity — implies that we have a dynamo number C_a ≡ Rα/β = 0.30. Using this value we find a damping time of 6 X 10⁻² R ²/β for (B), which is a measure for the coherence time of B in a single ensemble member. This result implies that the large-scale field of this particular α²-dynamo reorganizes its structure completely on a time scale of only about one year (for solar values of R and β), and it shows the enormous influence of random forcing in general.