Non-normal and stochastic amplification of magnetic energy in the turbulent dynamo: Subcritical case

Our attention focuses on the stochastic dynamo equation with non-normal operator that gives an insight into the role of stochastics and non-normality in magnetic field generation. The main point of this Brief Report is a discussion of the generation of a large-scale magnetic field that cannot be explained by traditional linear eigenvalue analysis. The main result is a discovery of nonlinear deterministic instability and growth of finite magnetic field fluctuations in $\alpha \beta$ dynamo theory. We present a simple stochastic model for the thin-disk axisymmetric $\alpha \Omega$ dynamo involving three factors: (a) non-normality generated by differential rotation, (b) nonlinearity reflecting how the magnetic field affects the turbulent dynamo coefficients, and (c) stochastic perturbations. We show that even for the subcritical case (all eigenvalues are negative), 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 nonlinearly unstable to small but finite initial perturbations. The second and third are stochastic mechanisms that account for the interaction of non-normal effect generated by differential rotation with random additive and multiplicative fluctuations.