Non-normal and Stochastic Amplification in Turbulent Dynamo: Subcritical Case

Abstract

Our attention focuses on the stochastic dynamo equation with non-normal operator that give an insight into the role of stochastics and non-normality in the galactic magnetic field generation. The main point of this Letter is a discussion of the generation of a large-scale magnetic field that cannot be explained by traditional linear eigenvalue analysis. We present a simple stochastic model for the thin-disk axisymmetric $\alpha \Omega $ dynamo involving three factors: (a) the non-normality generated by differential rotation, (b) the nonlinearity reflecting how the magnetic field affects the turbulent dynamo coefficients, and (c) stochastic perturbations. We show that even for \textit{subcritical case,} there are three possible mechanisms for the generation of magnetic field. The first mechanism is a deterministic one that describes an interplay between transient growth and nonlinear saturation of the turbulent $\alpha -$effect and diffusivity. It turns out that the trivial state is nonlinearity unstable to small but finite initial perturbations. The second and third are the stochastic mechanisms that account for the interaction of non-normal effect generated by differential rotation and random additive and multiplicative fluctuations. In particular, we show that in \textit{subcritical case}the average magnetic energy can grow exponentially with time due to the multiplicative noise associated with $\alpha -$effect.