Dynamo threshold detection in the von Kármán sodium experiment

Predicting dynamo self-generation in liquid metal experiments has been an ongoing question for many years. In contrast to simple dynamical systems for which reliable techniques have been developed, the ability to predict the dynamo capacity of a flow and the estimate of the corresponding critical value of the magnetic Reynolds number (the control parameter of the instability) has been elusive, partly due to the high level of turbulent fluctuations of flows in such experiments (with kinetic Reynolds numbers in excess of ${10}^6$). We address these issues here, using the von Kármán sodium experiment and studying its response to an externally applied magnetic field. We first show that a dynamo threshold can be estimated from analysis related to critical slowing down and susceptibility divergence, in configurations for which dynamo action is indeed observed. These approaches are then applied to flow configurations that have failed to self-generate magnetic fields within operational limits, and we quantify the dynamo capacity of these configurations.

Figure 1

Schematic of the experimental setup, in flow configuration 2 (see text for details). Probe shafts represent the location of the magnetic field sensor arrays.

Figure 2

Relaxation dynamics at $\epsilon =-0.13$ for a dynamo run (run $R$) and applied $m=0$ dipole magnetic field: (a) time traces for a single realization: magnetic field at one location [full (blue) curve] and current in axial coils [dotted (green) curve], and (b) normalized realization-averaged global magnetic energy and coil current (30 realizations)—the inset shows the curves in lin-log scale; see text for details. (c) Individual probability distribution functions (dashed curves) of the decay time $\tau$ (normalized to the decay time with no sodium motion) and associated Gaussian distributions [full (red) curve]—dynamo run $U$.

Figure 3

Evolution of relaxation decay time from an applied Helmholtz axial magnetic field (open blue circles), having an $m=0$ dipolar structure, an applied anti-Helmholtz axial magnetic field (green squares), having an $m=0$ quadrupolar structure, and an applied Helmholtz transverse magnetic field (red stars), having an $m=1$ dipolar structure, as a function of $\epsilon$. The lines are the linear extrapolation of the $1/\tau$ evolution in the range $-0.3<\epsilon <0$.

Figure 4

Experimental (points) and imperfect supercritical model (lines) for the off and on states (with an axial applied dipolar magnetic field). Inset: Evolution of the decay time for the imperfect supercritical model, with $\alpha =0.005$.

Figure 5

$1/\chi$ as a function of $\epsilon$ for the dynamo run, Helmholtz axial, and anti-Helmholtz axial setups. Curves have been normalized to their values at $\epsilon =-0.4$; see text for detailed definitions of ${\chi }_1$ and ${\chi }_2$. Linear fits (in the range $-0.4<\epsilon <0$) shown as full and dashed lines.

Figure 6

Schematic of the eleven studied VKS configurations. Gray color stands for stainless steel (except for the outer copper cylinder, which is displayed in gray for clarity of the figure), yellow color for copper, and red for soft iron. Static soft-iron screens behind the impellers can clearly be seen for runs O, P, and $T$.

Figure 7

Decay time evolution with Rm for runs O, P, Q, R and applied Helmholtz axial magnetic field.

Figure 8

Evolution of $1/{\chi }_2$ with Rm for runs O, P, Q, R and applied Helmholtz axial magnetic field.

Figure 9

Normalized induced azimuthal magnetic field radial profile in probe shafts closer to one impeller as function of Rm for (a) run $P$, (b) run $Q$.

Figure 10

Evolution of $1/{\chi }_2^{\prime }$ as a function of Rm between different runs; see legend and text for details.