Linear and nonlinear dynamo properties of time-dependent ABC flows

The linear and nonlinear dynamo properties of a class of periodically force d flows is considered. The forcing functions are chosen to drive, in the ab sence of magnetic effects (kinematic regime), a time-dependent version of t he ABC flow with A = B = C = 1. The time-dependence consists of a harmonic displacement of the origin along the line x = y = z = 1 with amplitude epsi lon and frequency Omega. The finite-time Lyapunov exponents are computed fo r several values of epsilon and Omega. It is found that for values of these parameters near unity chaotic streamlines occupy most of the volume. In th is parameter range, and for moderate kinetic and magnetic Reynolds numbers, the basic flow is both hydrodynamically and hydromagnetically unstable. Ho wever, the dynamo instability has a higher growth rate than the hydrodynami c one, so that the nonlinear regime can be reached with negligible departur es from the basic ABC flow. In the nonlinear regime, two distinct classes of behaviour are observed. In one, the exponential growth of the magnetic field saturates and the dynamo settles to a stationary state whereby the magnetic energy is maintained in definitely. In the other the velocity field evolves to a nondynamo state an d the magnetic field, following an initial amplification, decays to zero. T he transition from the dynamo to the nondynamo state can be mediated by the hydrodynamic instability or by magnetic perturbations. The properties of t he ensuing nonlinear dynamo states are investigated for different parameter values. The implications for a general theory of nonlinear dynamos are dis cussed. (C) 2001 Published by The Japan Society of Fluid Mechanics and Else vier Science B.V. All rights reserved.