Decay rates of magnetic modes below the threshold of a turbulent dynamo

We measure the decay rates of magnetic field modes in a turbulent flow of liquid sodium below the dynamo threshold. We observe that turbulent fluctuations induce energy transfers between modes with different symmetries (dipolar and quadrupolar). Using symmetry properties, we show how to measure the decay rate of each mode without being restricted to the one with the smallest damping rate. We observe that the respective values of the decay rates of these modes depend on the shape of the propellers driving the flow. Dynamical regimes, including field reversals, are observed only when the modes are both nearly marginal. This is in line with a recently proposed model.

Figure 1

Sketch of the experiment: two impellers counter rotate at frequencies $F_1$ and $F_2$. When $F_1=F_2$ the setup is invariant to ${\mathcal{R}}_{\pi }$ rotation. The vertical thick lines indicate the locations of the two arrays of three-axis Hall probes which measure the magnetic field close to each disk.

Figure 2

For an anti-Helmholtz applied magnetic field, time series of $\langle |B_1{|}^2+|B_2{|}^2\rangle /2$ (in black). Two behaviors are observed. The decay rate measured at initial time is the same as the one obtained from the quadrupolar projection (in red). At long time, it is the same as the one obtained from the dipolar projection (in blue). The gray curve represents the total amount of energy in the quadrupolar and dipolar modes. Top: lin-lin scale; bottom: log-lin scale, the long time value is subtracted so that exponential decays appear as straight lines.

Figure 3

Decay of the amplitude of the dipolar projection for $F=11$ Hz (applied field in the Helmholtz configuration). The thick black line is the average over the different realizations (displayed as thin gray lines). Inset: decay rate $1/\tau$ of the dipole as a function of the disk rotation frequency for an applied field in Helmholtz ($\square$) or anti-Helmholtz ($\blacksquare$) configuration.

Figure 4

Phase space for the induced field $(D,Q)$ for $F=11$ Hz. The gray set contains the trajectories when a field is applied in the anti-Helmholtz configuration. The (colored online, thin) continuous curves are individual trajectories that start when the field is set to zero. The thick black curve is the average over the realizations.

Figure 5

Decay rate of the dipolar mode (blue square) and of the quadrupolar mode (red circle) as a function of the rotation frequency. Upper panel: the flow is driven by disks with curved blades. Lower panel: the flow is driven by disks with straight blades.

Figure 6

Standard deviation of the decay rate of the quadrupolar (red circle) and dipolar mode (blue square). In the inset, standard deviation over the mean.