Linear and non-linear dynamos associated with ABC flows

Abstract

Numerical simulations of dynamo action are presented for a fluid stirred by a body force such that in the absence of the magnetic field, an ABC flow (to be viewed as the superposition of three helical modes of same wavenumber k ₀), establishes for kinetic Reynolds numbers below a critical value. For the kinematic dynamo, we show that keeping a fixed magnetic Reynolds number and decreasing the period of the ABC flow with respect to the size of the system, generally amplifies the growth rate of the magnetic field. This effect is especially significant when changing k ₀ = 1 to k ₀ = 2, while for larger k ₀, it scales like the rate of strain of the flow. A correlation between the rate of growth of the dynamo and the largest Lyapunov exponent is also observed when the mode amplitudes of the ABC flow are varied. In the dynamical regime where the flow is subject to magnetic feed-back through the Lorentz force, we always observe that decreasing the spatial period of the stirring force strongly enhances the saturation level of the dynamo, as measured by the ratio of the magnetic to the kinetic energies in the long-time regime. It is noticeable that in the kinematic regime, the magnetic field develops predominantly at the scale of the flow, whereas in the long-time dynamical regime, the magnetic field is dominant at the largest available scale of the system.