Chaotic dynamics of the magnetic field generated by dynamo action in a turbulent flow

Abstract

We present models related to the results of a recent experiment (the 'VKS experiment ') showing the generation of a magnetic field by a fully turbulent flow of liquid sodium. We first discuss the geometry of the mean magnetic field when the two coaxial impellers driving the flow counter-rotate at the same frequency. We then show how we expect this geometry to be modified when the impellers rotate at different frequencies. We also show that, in the latter case, dynamical regimes of the magnetic field can be easily understood from the interaction of modes with dipolar (respectively quadrupolar) symmetry. In particular, this interaction generates magnetic field reversals that have been observed in the experiment and display a hierarchy of timescales similar to the Earth's magnetic field: the duration of the steady phases is widely distributed, but is always much longer than the time needed to switch polarity. In addition to reversals, several other large scale features of the generated magnetic field are obtained when varying the governing parameters of the flow. These results are also understood in the framework of the same model.

MHD Dynamos and Turbulence

“…These limitations become extremely severe in the low P m regime. There is a generic expectation that for sufficiently low P m an asymptotic regime is reached where the level of saturation becomes independent of P m (Pétrélis & Fauve 2008). Some steps have been taken to move in the direction of simulating dynamos at low P m, but these are still in the P m ∼ O(1) regime, with all the associated dynamics (see e.g.…”

Mechanisms for magnetic field reversals

“…We have shown in Pétrélis & Fauve (2008) that this is related to the broken invariance under R p when F 1 = F 2 (rotation of an angle p along any axis in the mid-plane). In that case, symmetric and antisymmetric modes (under R p ) are coupled.…”

“…The coefficients of equation (4.1) can be chosen such that it has two stable dipolar solutions ±D and two unstable quadrupolar solutions ±Q when f = 0 (Pétrélis & Fauve 2008). When f is increased, these solutions become more and more mixed due to the increase of the strength of the coupling terms between dipolar and quadrupolar modes.…”

“…When f is increased further, a saddle-node bifurcation can occur, i.e. the stable and unstable solutions collide by pairs and disappear (Pétrélis & Fauve 2008). This generates a limit cycle that connects the collision point with its opposite.…”

“…The effect of turbulent fluctuations on the dynamics of the two magnetic modes governed by equation (4.1) can be easily modelled by adding some noisy component to the coefficients (Pétrélis & Fauve 2008). Random reversals are displayed in figure 3a.…”