Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

Abstract

We present numerical simulations of the kinematic induction equation in order to examine the dynamo efficiency of an axisymmetric von Kármán-like flow subject to time-dependent nonaxisymmetric velocity perturbations. The numerical model is based on the setup of the French von Kármán-sodium dynamo (VKS) and on the flow measurements from a water experiment conducted at the University of Navarra in Pamplona, Spain. The principal experimental observations that are modeled in our simulations are nonaxisymmetric vortexlike structures which perform an azimuthal drift motion in the equatorial plane. Our simulations show that the interactions of these periodic flow perturbations with the fundamental drift of the magnetic eigenmode (including the special case of nondrifting fields) essentially determine the temporal behavior of the dynamo state. We find two distinct regimes of dynamo action that depend on the (prescribed) drift frequency of an (m = 2) vortexlike flow perturbation. For comparatively slowly drifting vortices we observe a narrow window with enhanced growth rates and a drift of the magnetic eigenmode that is synchronized with the perturbation drift. The resonance-like enhancement of the growth rates takes place when the vortex drift frequency roughly equals the drift frequency of the magnetic eigenmode in the unperturbed system. Outside of this small window, the field generation is hampered compared to the unperturbed case, and the field amplitude of the magnetic eigenmode is modulated with approximately twice the vortex drift frequency. The abrupt transition between the resonant regime and the modulated regime is identified as a spectral exceptional point where eigenvalues (growth rates and frequencies) and eigenfunctions of two previously independent modes collapse. In the actual configuration the drift frequencies of the velocity perturbations that are observed in the water experiment are much larger than the fundamental drift frequency of the magnetic eigenmode that is obtained from our numerical simulations. Hence, we conclude that the fulfillment of the resonance condition might be unlikely in present day dynamo experiments. However, a possibility to increase the dynamo efficiency in the VKS experiment might be realized by an application of holes or fingers on the outer boundary in the equatorial plane. These mechanical distortions provoke an anchorage of the vortices at fixed positions thus allowing an adjustment of the temporal behavior of the nonaxisymmetric flow perturbations.

Parametric instability in periodically perturbed dynamos

“…The parametric resonances found in our simulations are characterized by a strong gain in the growth rate in a narrow range of excitation frequencies, a nonoscillating amplitude of the field and the linking of the magnetic phase to the phase of the perturbation. The resonances have also been found in previous simulations with a different spatial structure of the periodic disturbance (see also our previous study [23]) as well as in our analytical model in which the shape of the disturbance is not specified at all. Hence we suppose that the shape of the perturbations is not important for the occurrence of the observed resonances.…”

“…The vast regimes with parametric amplification have not been found in previous models, because either the higher magnetic field modes were not even considered (e.g., in the galactic dynamo models by Ref. [19]) or they decayed on a very fast time scale because the interaction triggered by the nonaxisymmetric perturbation has been too weak to become effective [23].…”

On the Synchronizability of Tayler–Spruit and Babcock–Leighton Type Dynamos

“…What is observed here is, therefore, a synchronization of order 1:2 (Pikovsky, Rosenblum, and Kurths, 2001). The typical parabolic shape of the curves is representative of the occurrence of parametric resonance (Giesecke, Stefani, and Burguete, 2012;Giesecke, Stefani, and Herault, 2017). Still, the amplitude of the perturbation must be large (around 0.4) in order to provide synchronization.…”

Inverse Cascades Sustained by the Transfer Rate of Angular Momentum in a 3D Turbulent Flow

“…These cascades, that have been observed both in spatial and temporal spectra, correspond to the conservation of the axial angular momentum. A similar behavior may be present in any situation where large coherent structures are relevant and appear on the top of very turbulent colliding flows, as, for example, in atmospheric circulations [13], large scale currents and vortices in oceans [14], MRI instabilities and accretion disks [15][16][17], dynamo action in MHD [18,19], mixing problems [20], and industrial applications, to name a few. A key characteristic of this setup is the high inertia of the propeller and motor set and the high stability of the propellers; i.e., the instantaneous fluctuations of each one of the propeller's velocities are well below one part in one thousand, 0.1%.…”