An optimal scale separation for a dynamo experiment

Scale separation between the flow and the magnetic field is a common feature of natural dynamos. It has also been used in the Karlsruhe sodium experiment in which the scale of the magnetic field is roughly 7 times larger than the scale of the flow [R. Stieglitz and U. M\"uller, Phys. Fluids 13, 561 (2001)]. Recently, Fauve & P\'etr\'elis ["Peyresq lectures on nonlinear phenomena", ed. J. Sepulchre, World Scientific, 1 (2003)] have shown that the power needed to reach the dynamo threshold in a dynamo experiment increases with the scale separation in the limit of large scale separation. With a more elaborate method based on subharmonic solutions [F. Plunian and K.-H. R\"adler, Geophys. Astrophys. Fluid Dynamics 96, 115 (2002)], we show, for the Roberts flow, the existence of an optimal scale separation for which this power is minimum. Previous results obtained by Tilgner [Phys. Lett. A 226, 75 (1997)] with a completely different numerical method are also reconsidered here. Again, we find an optimal scale separation in terms of minimum power for dynamo action. In addition we find that this scale separation compares very well with the one derived from the subharmonic solutions method.Comment: 6 pages, 2 figure

Cascades and dissipation ratio in rotating MHD turbulence at low magnetic Prandtl number

A phenomenology of isotropic magnetohydrodynamic turbulence subject to both rotation and applied magnetic field is presented. It is assumed that the triple correlations decay-time is the shortest between the eddy turn-over time and the ones associated to the rotating frequency and Alfv\'en wave period. For $Pm=1$ it leads to four kinds of piecewise spectra, depending on the four parameters, injection rate of energy, magnetic diffusivity, rotation rate and applied field. With a shell model of MHD turbulence (including rotation and applied magnetic field), spectra for $Pm \le 1$ are presented, together with the ratio between magnetic and viscous dissipation.Comment: 5 figures, 1 table, appear in PR

An optimal scale separation for a dynamo experiment.

Scale separation between the flow and the magnetic field is a common feature of natural dynamos. It has also been used in the Karlsruhe sodium experiment in which the scale of the magnetic field is roughly 7 times larger than the scale of the flow [1]. Recently, Fauve & P ́etr ́elis [2] have shown that the power needed to reach the dynamo threshold in a dynamo experiment increases with the scale separation in the limit of large scale separation. With a more elaborate method based on subharmonic solutions [3], we show, for the Roberts flow [4], the existence of an optimal scale separation for which this power is minimum. Previous results obtained by Tilgner [5] with a completely different numerical method are also reconsidered here. Again, we find an optimal scale separation in terms of minimum power for dynamo action. In addition we find that this scale separation compares very well with the one derived from the subharmonic solutions method [3]

Shell Models of Magnetohydrodynamic Turbulence

Shell models of hydrodynamic turbulence originated in the seventies. Their main aim was to describe the statistics of homogeneous and isotropic turbulence in spectral space, using a simple set of ordinary differential equations. In the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged based on the same principles as their hydrodynamic counter-part but also incorporating interactions between magnetic and velocity fields. In recent years, significant improvements have been made such as the inclusion of non-local interactions and appropriate definitions for helicities. Though shell models cannot account for the spatial complexity of MHD turbulence, their dynamics are not over simplified and do reflect those of real MHD turbulence including intermittency or chaotic reversals of large-scale modes. Furthermore, these models use realistic values for dimensionless parameters (high kinetic and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing extended inertial range and accurate dissipation rate. Using modern computers it is difficult to attain an inertial range of three decades with direct numerical simulations, whereas eight are possible using shell models. In this review we set up a general mathematical framework allowing the description of any MHD shell model. The variety of the latter, with their advantages and weaknesses, is introduced. Finally we consider a number of applications, dealing with free-decaying MHD turbulence, dynamo action, Alfven waves and the Hall effect.Comment: published in Physics Report

Intermittency in the homopolar disk-dynamo

We study a modified Bullard dynamo and show that this system is equivalent to a nonlinear oscillator subject to a multiplicative noise. The stability analysis of this oscillator is performed. Two bifurcations are identified, first towards an `` intermittent\rq\rq state where the absorbing (non-dynamo) state is no more stable but the most probable value of the amplitude of the oscillator is still zero and secondly towards a `` turbulent\rq\rq (dynamo) state where it is possible to define unambiguously a (non-zero) most probable value around which the amplitude of the oscillator fluctuates. The bifurcation diagram of this system exhibits three regions which are analytically characterized

Axisymmetric dynamo action produced by differential rotation, with anisotropic electrical conductivity and anisotropic magnetic permeability

The effect on dynamo action of an anisotropic electrical conductivity conjugated to an anisotropic magnetic permeability is considered. Not only is the dynamo fully axisymmetric, but it requires only a simple differential rotation, which twice challenges the well-established dynamo theory. Stability analysis is conducted entirely analytically, leading to an explicit expression of the dynamo threshold. The results show a competition between the anisotropy of electrical conductivity and that of magnetic permeability, the dynamo effect becoming impossible if the two anisotropies are identical. For isotropic electrical conductivity, Cowling's neutral point argument does imply the absence of an azimuthal component of current density, but does not prevent the dynamo effect as long as the magnetic permeability is anisotropic.Comment: 19 pages, 6 figure

Oscillating Ponomarenko dynamo in the highly conducting limit

This paper considers dynamo action in smooth helical flows in cylindrical geometry, otherwise known as Ponomarenko dynamos, with periodic time dependence. An asymptotic framework is developed that gives growth rates and frequencies in the highly conducting limit of large magnetic Reynolds number, when modes tend to be localized on resonant stream surfaces. This theory is validated by means of numerical simulations.Comment: 12 pages, 4 figure

Parametric instability of the helical dynamo

We study the dynamo threshold of a helical flow made of a mean (stationary) plus a fluctuating part. Two flow geometries are studied, either (i) solid body or (ii) smooth. Two well-known resonant dynamo conditions, elaborated for stationary helical flows in the limit of large magnetic Reynolds numbers, are tested against lower magnetic Reynolds numbers and for fluctuating flows (zero mean). For a flow made of a mean plus a fluctuating part the dynamo threshold depends on the frequency and the strength of the fluctuation. The resonant dynamo conditions applied on the fluctuating (resp. mean) part seems to be a good diagnostic to predict the existence of a dynamo threshold when the fluctuation level is high (resp. low).Comment: 37 pages, 8 figure

Influence of electromagnetic boundary conditions onto the onset of dynamo action in laboratory experiments

We study the onset of dynamo action of the Riga and Karlsruhe experiments with the addition of an external wall, the electro-magnetic properties of which being different from those of the fluid in motion. We consider a wall of different thickness, conductivity and permeability. We also consider the case of a ferro-fluid in motion.Comment: 9 pages, 9 figure