An optimal scale separation for a dynamo experiment

Following [4], we assume that the power P is dissipated by turbulence, leading to P = ρL 2 HU 3 /l where ρ is the density and U the characteristic speed of the fluid. Defining the magnetic Reynolds number by R m = Ul/η, where η is the magnetic diffusivity of the fluid, we find after some simple algebra that

As a preliminary step, we assume that the first order smoothing approximation is valid Writing that the mean electromotive force Ub is equal to αB leads to the following relation

Then we can show that

, leading to

From this simple estimate we conclude that the power consumption increases with the number of cells which is not in favor of scale separation. This was found previously by

Fauve & Pétrélis [4] for a scale separation in the 3 cartesian directions (instead of only 2 in our case), leading to a different scaling N 5/6

c . Both estimates are based on the first order smoothing approximation which has been proved to be too simplistic in the theoretical predictions of the Karlsruhe experiment. Therefore we reconsider this problem below in the light of the subharmonic solutions as studied in [3].

The original Roberts [2] flow is defined by

and the relations between the dimensions defined above and those defined in [3]

where R * m = UL U /η is the magnetic Reynolds number defined in [3]. For a given value of N, we look for the subharmonic solution embedded in the box of size L × L × H and corresponding to the dimensionless wave numbers f = 1/N in the horizontal directions and k = 1 N L H in the vertical direction. Then (1) writes in the form

For a given value of N the critical R * m versus k has been plotted in Figure 4 L/H = 2 and N c = 50 for which we find P • H = 19 kW.m. For the same value of L/H but for N c = 18 cells, the power consumption is reduced roughly by a factor 2. In Figure 2, P • H is plotted versus N c for L/H = 2 (full curve). We see that there is indeed a minimum around N c = 20 and that at large N c , P •H increases with N c as predicted by (3). In a previous study [5], Tilgner calculated the critical magnetic Reynolds number for the Karlsruhe experiment geometry, varying the number of cells inside the device (Figure 4 of [5]). The resolution was made with a completely different method than the one used in [3] and it is then of interest to reconsider the results of [5] in terms of power consumption and see how they compare to our results. For that we need to make preliminary correspondance between our present notations and those used in [5]. In [5] the flow container is a cylinder. then the consumption power, instead of (1), writes in the form

where R is the cylinder radius. In [5] we have l = 8 4.1 R N T 97 where we call here N T 97 the parameter N of [5]. This leads to a number of cells N c = πR 2 l 2 = 0.825N 2 T 97 . Furthermore the magnetic Reynolds number in [5] is defined by R mT 97 = Ur 0 /η where r 0 = √ 1.25R is the radius of the conducting sphere in which the cylinder is embedded. This leads to R m = 1.745 N T 97 R mT 97 . Finally using the results from the Figure 4 of [5], the consumption power is plotted versus the cells number on Figure 2 (dashed curve). We find that there is again an optimal scale separation for which the dissipated power is minimum and again it corresponds to N c close to 20. Furthermore the levels of power are of the same order of magnitude. We could not expect better agreement as the geometries and boundary conditions of [3] and [5] are really different. Now considering the design of the Karlsruhe experiment, most of the dissipation power occurs in the pieces of pipes which redirect the flow into neighbouring cells at the end of each cell. The dissipation scaling in there is somewhat slower than U 3 but, most importantly, it is not proportional to the volume of the experiment. Therefore the scale separation of that experiment was guided by the characteristics of the available pumps [6] in order to minimize the critical R m , instead of minimizing the dissipated power, leading to N c = 52 (or alternatively to N T 97 = 8 corresponding to the minimum critical R m in [5]). Therefore the criterion that we derived here is relevant for propeller driven experiments, but the requirements are more complicated (and also less universal) for pump driven experiments. curve is derived from [3] (from [5]) for L/H = 2 (R/H = 1).