Robust estimate of dynamo thresholds in the von K\'arm\'an sodium experiment using the Extreme Value Theory

We apply a new threshold detection method based on the extreme value theory to the von K\'arm\'an sodium (VKS) experiment data. The VKS experiment is a successful attempt to get a dynamo magnetic field in a laboratory liquid-metal experiment. We first show that the dynamo threshold is associated to a change of the probability density function of the extreme values of the magnetic field. This method does not require the measurement of response functions from applied external perturbations, and thus provides a simple threshold estimate. We apply our method to different configurations in the VKS experiment showing that it yields a robust indication of the dynamo threshold as well as evidence of hysteretic behaviors. Moreover, for the experimental configurations in which a dynamo transition is not observed, the method provides a way to extrapolate an interval of possible threshold values.

the determination of the dynamo threshold. We briefly recall the basic intuition beyond the method referring to [14] for further discussions. Classical Extreme Value Theory (EVT) states that, under general assumptions, the statistics of maxima M m = max{X 0 , X 1 , ..., X m−1 } of independent and identically distributed (i.i.d.) variables X 0 , X 1 , . . . , X m−1 , with cumulative distribution function (cdf) F (x) in the form:

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The sign of κ discriminates the kind of tail decay of the parent distribution: When κ = 0, the distribution is of 52 Gumbel type (type 1). This is the asymptotic Extreme Value Law (EVL) to be expected when the parent distribution 53 shows an exponentially decaying tail. The Fréchet distribution (type 2), with κ > 0, is instead observed when the that a Weibull law is observed in these cases as well.

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The interest of the EVL statistics in bifurcation detection relies on the change of the nature of the fluctuations of 63 a given system, when going from a situation with one stable attractor to a situation with two competing attractors, 64 with jump between the two allowed either under the effect of external noise or due to internal chaotic fluctuations. In shape parameter κ then changes through the bifurcation from κ < 0 to κ > 0, wich enables a precise definition of the 74 threshold as the value at which the zero crossing of κ happens. Physical observables will display deviations of greater 75 amplitude in the direction of the state the system is doomed to tumble, than in the opposite direction, therefore one 76 expects to observe this switching either in the maxima or in the minima. Application to VKS data. We present the results for the detection of the dynamo threshold Rm * by using as 93 observable the modulus of the magnetic field | B(t)| measured by the 40 different detectors (Hall probes). 94 The method can be described as follows. First of all, the extremes of the magnetic field are extracted by using the so called block maxima approach which consists in dividing the series | B(t)|, t = 1, 2, ..., s into n bins each containing m observations (s = nm) and thus selecting the maximum (minimum) M j in each bin. The series of M j , j = 1, ..., n is then fitted to the GEV distribution via the L-moment procedure described in [23]. In order to sample proper extreme values one has to consider a bin length longer than the correlation time τ . For each of the sensors we have computed τ as the first zero of the autocorrelation function finding that 0.42 s < τ < 1 s lags depending on the cases considered. This value is similar to the magnetic diffusion time found in [24]. By choosing a bin duration longer than 1 s (or, equivalently, a number of samples m in each bin larger than 2000) and repeating the fit until the shape FIG. 2. Schematic representation of the studied VKS configurations. Gray colors stands for stainless steel, yellow color for copper and red for soft iron.
parameter κ is not changing in appreciable way, one can establish the convergence to the GEV model [18]. In our experiments we found that reliable estimates can be generally obtained for m > 4000. Being the length of each series 10 5 < s < 3 · 10 5 , for any choice of m > 4000 no more than n = 100 maxima can be extracted. Such a value of n is one order of magnitude smaller than the one prescribed in [25, 26] for avoiding biased fits to the GEV model. In order to overcome this problem we have grouped sensors located at the same radial position. The sensors of the four arrays of Hall effect sensors are not installed at the same radial distance (see Fig 1 for a visual explanation). However, an effective radial grouping can be obtained by adding to the n extremes of the sensor a l the ones of b l+2 , c l and d l+2 , l = 1, ..., 8, thus obtaining 8 different series with a sufficient number of maxima to perform the fit. The choice of grouping the sensors by their radial location is justified by checking that the shape of the distribution, which enters in the computation of the shape parameter κ does not change substantially for sensors located at the same radial position. In order to do so, we have computed the skewness and the kurtosis for the time series of the magnetic field, finding small variations for sensors located at the same radial position. We also checked that the maxima extracted by combining the series are independent by analyzing the cross-correlation function of different sensors. For example, for sensors a and b, the cross-correlation function is defined as: Here, the notation · j indicates the expectation value taken over the j index. The results of this analysis are shown and h = −5. The case h=0 corresponds to sensors located at the same radial position, which we grouped in our study.

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One can observe that although the correlation is non-zero, it is relatively small (about 0.5 for neighboring probes and 99 smaller than 0.2 for non-neighbouring probes), which validates our grouping of the sensors to increase our statistics.

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In addition, for sensors located at different radial positions the decorrelation is total (see the example at h = ±5 in    The same analysis has been carried out for all the configurations shown in Fig. 2 Rm. An example is shown for the Q' configuration in Fig. 7. An extrapolate threshold value Rm e can then be found  P  ------Q  -----200  Q'  ---85± 10 350 125  R  44  46  37  -51  56  S  ---150± 25  --T  ---100± 25 250 205  U  70  75  66  -58 100  V  66  67  45  -71  93   TABLE I. Dynamo threshold for various configuration in the VKS experiment, obtained through various technique: Rm || B| : from the increase of the magnetic field amplitude | B| [10, 11], Rm * f and Rm * b : forward and backward threshold obtained from the extreme value technique, with zero crossing detection (this paper); Rm e : from the extreme value technique, with cubic extrapolation to detect zero crossing (this paper); Rm d : from decay time divergency extrapolation [12]; Rm i from induction increase extrapolation. transitions.