Bifurcations and dynamo action in a Taylor-Green flow
Résumé
We report successive bifurcations in direct numerical simulations (DNSs) of a Taylor–Green flow, in both a hydro- and a magneto-hydrodynamic case. Hydrodynamic bifurcations occur in between different metastable states with different dynamo action, and are triggered by the numerical noise. The various states encountered range from stationary to chaotic or turbulent through possible oscillatory states. The corresponding sequence of bifurcations is reminiscent of the sequence obtained in the von Karman (VK) flow, at aspect ratio Γ=2 (Nore et al 2003 J. Fluid Mech. 477 51). We then use kinematic simulations to compute the dynamo thresholds of the different metastable states. A more detailed study of the turbulent state reveals the existence of two windows of dynamo action. Stochastic numerical simulations are then used to mimic the influence of turbulence on the dynamo threshold of the turbulent state. We show that the dynamo threshold is increased (respectively decreased) by the presence of large scale (resp. small scale) turbulent velocity fluctuations. Finally, DNSs of the magneto-hydrodynamic equations are used to explore the linear and nonlinear stage of the dynamo instability. In the linear stage, we show that the magnetic field favours the bifurcation from the basic state directly towards the turbulent or chaotic stable state. The magnetic field can also temporarily stabilize a metastable state, resulting in cycles of dynamo action, with different Lyapunov exponents. The critical magnetic Reynolds number for dynamo action is found to increase strongly with the Reynolds number. Finally, we provide a preliminary study of the saturation regime above the dynamo threshold. At large magnetic Prandtl number, we have observed two main types of saturations, in agreement with an analytical prediction of Leprovost and Dubrulle (2005 Eur. Phys. J. B 44 395): (i) intermittent dynamo, with vanishing most probable value of the magnetic energy; (ii) dynamo with non vanishing mean value of the magnetic energy. We describe a sequence of bifurcation of the dynamo at low Reynolds number and at increasing magnetic Prandtl number, where the dynamo switches from stationary mode to chaotic modes in a complex manner, involving intermittency in a way reminiscent to what is observed in dynamical systems with low number of degrees of freedom.