Nonlinear dynamo with ABC forcing.
Abstract
We report results of a bifurcation study of the incompressible magnetohydrodynamic equations in three space dimensions with periodic boundary conditions and an external helical forcing which generates if sufficiently weak a stable ABC flow {vec}(v)_ABC_=(Asinz+Ccosy, Bsinx+Acosz, Csiny+Bcosx). Fourier representations of velocity, pressure and magnetic field have been used to transform the original partial differential equations into a system of ordinary differential equations, which then has been studied by means of numerical bifurcationanalysis techniques, supplemented by the simulative calculation of solutions for selected initial conditions. In a part of the calculations, in order to reduce the number of modes to be retained, the concept of approximate inertial manifolds has been applied. For increasing Reynolds number or strength of the imposed forcing, timeasymptotic states have been traced. There is no dynamo effect as long as the velocity field is stationary, but the nonmagnetic ABC flow loses in a Hopf bifurcation stability to a timeperiodic state with a nonvanishing magnetic field, showing the appearance of a generic dynamo effect. No gaps in dynamo action are then observed for further increased Reynolds number. The Hopf bifurcation is followed by secondary, symmetrybreaking bifurcations, leading first to torus or quasiperiodic solutions and finally via torus decay to chaos.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 October 1996
 Bibcode:
 1996A&A...314..693S
 Keywords:

 MAGNETIC FIELDS;
 MHD;
 INSTABILITIES;
 CHAOS