Dynamo onset as a first-order transition: Lessons from a shell model for magnetohydrodynamics

We carry out systematic and high-resolution studies of dynamo action in a shell model for magnetohydro-dynamic (MHD) turbulence over wide ranges of the magnetic Prandtl number Pr-M and the magnetic Reynolds number Re-M. Our study suggests that it is natural to think of dynamo onset as a nonequilibrium first-order phase transition between two different turbulent, but statistically steady, states. The ratio of the magnetic and kinetic energies is a convenient order parameter for this transition. By using this order parameter, we obtain the stability diagram (or nonequilibrium phase diagram) for dynamo formation in our MHD shell model in the (Pr-M(-1), Re-M) plane. The dynamo boundary, which separates dynamo and no-dynamo regions, appears to have a fractal character. We obtain a hysteretic behavior of the order parameter across this boundary and suggestions of nucleation-type phenomena.