Franck Plunian

Impact of Impellers on the Axisymmetric Magnetic Mode in the VKS2 Dynamo Experiment

In the von Kármán Sodium 2 (VKS2) successful dynamo experiment of September 2006, the observed magnetic field showed a strong axisymmetric component, implying that nonaxisymmetric components of the flow field were acting. By modeling the induction effect of the spiraling flow between the blades of the impellers in a kinematic dynamo code, we find that the axisymmetric magnetic mode is excited. The control parameters are the magnetic Reynolds number of the mean flow, the coefficient measuring the induction effect alpha, and the type of boundary conditions. We show that using realistic values of alpha, the observed critical magnetic Reynolds number, Rm;{c} approximately 32, can be reached easily with ferromagnetic boundary conditions. We conjecture that the dynamo action achieved in this experiment may not be related to the turbulence in the bulk of the flow, but rather to the alpha effect induced by the impellers.

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 electromagnetic properties of which are 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 ferrofluid in motion.

Shell models of magnetohydrodynamic turbulence

Abstract 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.

Parametric instability of the helical dynamo

We study the dynamo threshold of a helical flow made of a mean plus a fluctuating part. Two flow geometries are studied: (i) solid body and (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 with 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 (respectively, mean) part seems to be a good diagnostic to predict the existence of a dynamo threshold when the fluctuation level is high (respectively, low).

A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence

We derive a new shell model of magnetohydrodynamic (MHD) turbulence in which the energy transfers are not necessarily local. Like the original MHD equations, the model conserves the total energy, magnetic helicity, cross-helicity and volume in phase space (Liouvilles theorem) apart from the effects of external forcing, viscous dissipation and magnetic diffusion. The model of hydrodynamic (HD) turbulence is derived from the MHD model setting the magnetic field to zero. In that case the conserved quantities are the kinetic energy and the kinetic helicity. In addition to a statistically stationary state with a Kolmogorov spectrum, the HD model exhibits multiscaling. The anomalous scaling exponents are found to depend on a free parameter α that measures the non-locality degree of the model. In freely decaying turbulence, the infra-red spectrum also depends on α. Comparison with theory suggests using α = −5/2. In MHD turbulence, we investigate the fully developed turbulent dynamo for a wide range of magnetic Prandtl numbers in both kinematic and dynamic cases. Both local and non-local energy transfers are clearly identified.

Fully developed turbulent dynamo at low magnetic Prandtl numbers

We investigate the dynamo problem in the limit of small magnetic Prandtl number (Pm) using a shell model of magnetohydrodynamic turbulence. The model is designed to satisfy conservation laws of total energy, cross helicity and magnetic helicity in the limit of inviscid fluid and null magnetic diffusivity. The forcing is chosen to have a constant injection rate of energy and no injection of kinetic helicity nor cross helicity. We find that the value of the critical magnetic Reynolds number (Rm) saturates in the limit of small Pm. Above the dynamo threshold we study the saturated regime versus Rm and Pm. In the case of equipartition, we find Kolmogorov spectra for both kinetic and magnetic energies except for wave numbers just below the resistive scale. Finally the ratio of both dissipation scales (viscous to resistive) evolves as Pm −3/4 for Pm < 1.

Transition from large-scale to small-scale dynamo.

The dynamo equations are solved numerically with a helical forcing corresponding to the Roberts flow. In the fully turbulent regime the flow behaves as a Roberts flow on long time scales, plus turbulent fluctuations at short time scales. The dynamo onset is controlled by the long time scales of the flow, in agreement with the former Karlsruhe experimental results. The dynamo mechanism is governed by a generalized α effect, which includes both the usual α effect and turbulent diffusion, plus all higher order effects. Beyond the onset we find that this generalized α effect scales as O(Rm(-1)), suggesting the takeover of small-scale dynamo action. This is confirmed by simulations in which dynamo occurs even if the large-scale field is artificially suppressed.

Homopolar oscillating-disc dynamo driven by parametric resonance

We use a simple model of Bullard-type disc dynamo, in which the disc rotation rate is subject to harmonic oscillations, to analyze the generation of magnetic field by the parametric resonance mechanism. The problem is governed by a damped Mathieu equation. The Floquet exponents, which define the magnetic field growth rates, are calculated depending on the amplitude and frequency of the oscillations. Firstly, we show that the dynamo can be excited at significantly subcritical disc rotation rate when the latter is subject to harmonic oscillations with a certain frequency. Secondly, at supercritical mean rotation rates, the dynamo can also be suppressed but only in narrow frequency bands and at sufficiently large oscillation amplitudes.

Cylindrical anisotropic α 2 dynamos

We explore the influence of geometry variations on the structure and the time-dependence of the magnetic field that is induced by kinematic α 2 dynamos in a finite cylinder. The dynamo action is due to an anisotropic α effect which can be derived from an underlying columnar flow. The investigated geometry variations concern, in particular, the aspect ratio of height to radius of the cylinder, and the thickness of the annular space to which the columnar flow is restricted. Motivated by the quest for laboratory dynamos which exhibit Earth-like features, we start with modifications of the Karlsruhe dynamo facility. Its dynamo action is reasonably described by an α 2 mechanism with anisotropic α tensor. We find a critical aspect ratio below which the dominant magnetic field structure changes from an equatorial dipole to an axial dipole. Similar results are found for α 2 dynamos working in an annular space when a radial dependence of α is assumed. Finally, we study the effect of varying aspect ratios of dynamos with an α tensor depending both on radial and axial coordinates. In this case only dominant equatorial dipoles are found and most of the solutions are oscillatory, contrary to all previous cases where the resulting fields are steady.

Non-Kolmogorov cascade of helicity-driven turbulence.

We solve the Navier-Stokes equations with two simultaneous forcings. One forcing is applied at a given large scale and it injects energy. The other forcing is applied at all scales belonging to the inertial range and it injects helicity. In this way we can vary the degree of turbulence helicity from nonhelical to maximally helical. We find that increasing the rate of helicity injection does not change the energy flux. On the other hand, the level of total energy is strongly increased and the energy spectrum gets steeper. The energy spectrum spans from a Kolmogorov scaling law k^{-5/3} for a nonhelical turbulence, to a non-Kolmogorov scaling law k^{-7/3} for a maximally helical turbulence. In the latter case we find that the characteristic time of the turbulence is not the turnover time but a time based on the helicity injection rate. We also analyze the results in terms of helical modes decomposition. For a maximally helical turbulence one type of helical mode is found to be much more energetic than the other one, by several orders of magnitude. The energy cascade of the most energetic type of helical mode results from the sum of two fluxes. One flux is negative and can be understood in terms of a decimated model. This negative flux, however, is not sufficient to lead an inverse energy cascade. Indeed, the other flux involving the least energetic type of helical mode is positive and the largest. The least energetic type of helical mode is then essential and cannot be neglected.