STOCHASTIC AND NONLINEAR FLUCTUATIONS IN A MEAN-FIELD DYNAMO

We study the effect of rapid stochastic fluctuations in the kinetic he licity in a plane parallel mean field dynamo model for the Sun. The al pha-parameter has a fluctuating component delta alpha = alpha - alpha( 0), which is modelled as a random forcing term. The fluctuations give rise to variations in the amplitude and phase of the dynamo wave, such that shorter cycles have higher amplitudes, as is observed in the sol ar cycle. By making a second order expansion close to the unperturbed marginally stable dynamo wave we are able to go beyond the weak forcin g limit studied by Hoyng (1993). We show that with increasing strength of the forcing the effective dynamo frequency decreases. We introduce a simple non-linearity to model alpha-quenching and derive a set of l inear equations for the mean field, valid in the weak forcing case. Wi th alpha-quenching, phase and amplitude fluctuations are bounded, but still correlated. The strength of the alpha-quenching is measured by a parameter q = -(T-e/alpha(0))(partial derivative alpha/partial deriva tive T)/(Te), where T-e is the equilibrium value of the toroidal field . We make a comparison with sunspot data, and conclude that these are well explained by the model if delta alpha/alpha(0) approximate to 2.2 and q approximate to 0.7. Finally we briefly consider the alternative possibility of fluctuations caused by nonlinear dynamics, without ext ernal forcing (delta alpha = 0). We show that the resulting phase-ampl itude diagram does not agree with observations. Although this is no pr oof that the phase-amplitude correlation cannot be reproduced by nonli near chaos, we conclude that stochastic noise provides a more natural explanation.