Physica D 223 (2006) 151–162 www.elsevier.com/locate/physd Low-dimensional dynamo modelling and symmetry-breaking bifurcations Reik Donner a , Norbert Seehafer a , Miguel A.F. Sanju´ an b , Fred Feudel a,∗ a Institut f¨ ur Physik, Universit¨ at Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany b Nonlinear Dynamics and Chaos Group, Departamento de Matem´ aticas y F´ ısica Aplicadas y Ciencias de la Naturaleza, Universidad Rey Juan Carlos, Tulip´ an s/n, 28933 M ´ ostoles, Spain Received 17 March 2006; received in revised form 30 August 2006; accepted 30 August 2006 Available online 11 October 2006 Communicated by A. Mikhailov 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 ( Rm ≪ 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. c  2006 Elsevier B.V. All rights reserved. Keywords: MHD equations; Symmetry-breaking bifurcations; Oscillating waves; Dynamo theory; Alpha-effect 1. Introduction The generation of magnetic fields due to the motion of electrically conducting fluids is a well studied phenomenon. Introducing an arbitrary weak seed field, one can observe either a weakening, a conservation or even an amplification of this initial field. In the latter two cases one speaks of a dynamo effect [1–7]. Since the pioneering suggestion of Larmor [8] it is believed to be the physical reason for the occurrence of magnetism in planetary and astrophysical objects found in observations. In the past few years homogeneous dynamos as expected to be working in cosmic bodies were realised in laboratory experiments carried out in Riga (Latvia) ∗ Corresponding author. Tel.: +49 331 9771227; fax: +49 331 9771142. E-mail address: fred@agnld.uni-potsdam.de (F. Feudel). [9–15] and Karlsruhe (Germany) [16–21]. Overviews on dynamo experiments in the laboratory can be found in [22–25]. The mathematical framework of dynamo theory is given in terms of a system of coupled nonlinear partial differential equations. In the case of an incompressible non-relativistic fluid, this set can be reduced to four equations, the resistive magnetohydrodynamic (MHD) equations, containing the Navier–Stokes equation (NSE) and the induction equation as coupled evolutionary equations for the fluid velocity  u (  x , t ) and the magnetic field  B (  x , t ) and two additional constraints. In dimensionless form, these equations read: ∂ t  u + (  u ·∇)  u = Re −1 ∇ 2  u −∇ P +   B ·∇   B +  f (1) ∂ t  B + (  u ·∇)  B = Rm −1 ∇ 2  B +   B ·∇   u (2) 0167-2789/$ - see front matter c  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2006.08.022