Stochastic dynamo model for subcritical transition.

The effects of stochastic perturbations in a nonlinear alpha Omega-dynamo model are investigated. By using transformation of variables we identify a "slow" variable that determines the global evolution of the non-normal alpha Omega-dynamo system in the subcritical case. We apply an adiabatic elimination procedure to derive a closed stochastic differential equation for the slow variable for which the dynamics is determined along one of the eigenvectors of the full system. We derive the corresponding Fokker-Planck equation and show that the generation of a large scale magnetic field can be regarded as a first-order phase transition. We show that the an advantage of the reduced system is that we have explicit expressions for both the stochastic and deterministic potentials. We also obtain the stationary solution of the Fokker-Planck equation and show that an increase in the intensity of the multiplicative noise leads to qualitative changes in the stationary probability density function. The latter can be interpreted as a noise-induced phase transition. By a numerical simulation of the stochastic galactic dynamo model, we show that the qualitative behavior of the "empirical" stationary pdf of the slow variable is accurately predicted by the stationary pdf of the reduced system.