Stochastic analysis of subcritical amplification of magnetic energy in a turbulent dynamo

Abstract

We present and analyze a simplified stochastic αΩ−dynamo model which is designed to assess the influence of additive and multiplicative noises, non-normality of dynamo equation, and nonlinearity of the α−effect and turbulent diffusivity, on the generation of a large-scale magnetic field in the subcritical case. Our model incorporates random fluctuations in the α−parameter and additive noise arising from the small-scale fluctuations of magnetic and turbulent velocity fields. We show that the noise effects along with non-normality can lead to the stochastic amplification of the magnetic field even in the subcritical case. The criteria for the stochastic instability during the early kinematic stage are established and the critical value for the intensity of multiplicative noise due to α−fluctuations is found. We obtain numerical solutions of nonlinear stochastic differential equations and find the series of phase transitions induced by random fluctuations in the α−parameter.

Triggering in a Thermoacoustic System with Stochastic Noise

“…[30], which studied a nonnormal model of magnetic energy in a turbulent dynamo, which exhibits a subcritical bifurcation. By including multiplicative noise in the model, their system triggers from the zero fixed point to the basin of attraction of a second fixed point.…”

Analytical description of the quasi-stationary solidification of ternary systems

“…Therefore, the displacement of the maximum toward the parent mushy zone is explained by the effect of the diffusion transport of impurity B(x) in a real ternary system. The theory proposed in this work describes the pulling of a crystal from a melt at a constant solidifica tion rate (Czochralski growth technique) and can also describe the slow (almost steady) solidification pro cesses at the interface of the internal (solid) Earth core that are responsible for geodynamo [54][55][56]. More over, the analytical solutions found in this work can be used to study the dynamic stability of the solidification of ternary systems (which is responsible for layer by layer impurity segregation) by analogy with the stabil ity theory developed in [57][58][59][60] for binary melts.…”

Stochastic stability analysis of a reduced galactic dynamo model with perturbed α-effect

“…To see why we work with Lyapunov exponents, consider that the work in [4,5,6] addresses the exponential mean-square stability of the linearised equation in the driftsubcritical regime, and the behaviour of the distribution of the slowly varying component of the nonlinear system. Exponential mean-square stability is sufficient but not necessary for stability in probability of the equilibrium of the nonlinear stochastic system.…”

“…A reduced αΩ-dynamo model. We select the following reduced 2-dimensional stochastic model of an αΩ-dynamo, which was used by Fedotov et al [5,6] to investigate the role of stochastic perturbation in the generation of large-scale magnetic fields in spiral galaxies. The radial (B r ) and azimuthal (B ϕ ) components of the magnetic field B = (B r , B ϕ ) T are governed by…”