Low-dimensional dynamo modelling and symmetry-breaking bifurcations

Abstract

Motivated by the successful Karlsruhe dynamo experiment, a relatively low-dimensional dynamo model is proposed. It is based on a strong truncation of the magnetohydrodynamic (MHD) equations with an external forcing of the Roberts type and the requirement that the model system satisfies the symmetries of the full MHD system, so that the first symmetry-breaking bifurcations can be captured. The backbone of the Roberts dynamo is formed by the Roberts flow, a helical mean magnetic field and another part of the magnetic field coupled to these two by triadic mode interactions. A minimum truncation model (MTM) containing only these energetically dominating primary mode triads is fully equivalent to the widely used first-order smoothing approximation. However, it is shown that this approach works only in the limit of small wave numbers of the excited magnetic field or small magnetic Reynolds numbers ( R m ≪ 1 ). To obtain dynamo action under more general conditions, secondary mode triads must be taken into account. Altogether a set of six primary and secondary mode types is found to be necessary for an optimum truncation model (OTM), corresponding to a system of 152 ordinary differential equations. In a second step, the OTM is used to study symmetry-breaking bifurcations on its route to chaos, with the Reynolds number or strength of the driving force as the control parameter. A decisive role in this scenario is played by a symmetry of the form of Z 2 × S 1 resulting from the Z 2 reflection symmetry of the magnetic field in the MHD equations in conjunction with a circle symmetry S 1 of the Roberts flow. Under its influence, in a secondary Hopf bifurcation from a circle of steady reflection-symmetric states a time-periodic solution branch of oscillating waves (OW) is generated retaining the reflection symmetry, however in a spatio-temporal manner. Finally, the subsequent bifurcations on the route to chaos are examined.