Yannick Ponty

Numerical study of dynamo action at low magnetic Prandtl numbers.

We present a three-pronged numerical approach to the dynamo problem at low magnetic Prandtl numbers P(M). The difficulty of resolving a large range of scales is circumvented by combining direct numerical simulations, a Lagrangian-averaged model and large-eddy simulations. The flow is generated by the Taylor-Green forcing; it combines a well defined structure at large scales and turbulent fluctuations at small scales. Our main findings are (i) dynamos are observed from P(M)=1 down to P(M)=10(-2), (ii) the critical magnetic Reynolds number increases sharply with P(M)(-1) as turbulence sets in and then it saturates, and (iii) in the linear growth phase, unstable magnetic modes move to smaller scales as P(M) is decreased. Then the dynamo grows at large scales and modifies the turbulent velocity fluctuations.

Simulation of induction at low magnetic Prandtl number.

We consider the induction of a magnetic field in flows of an electrically conducting fluid at low magnetic Prandtl number and large kinetic Reynolds number. Using the separation between the magnetic and kinetic diffusive length scales, we propose a new numerical approach. The coupled magnetic and fluid equations are solved using a mixed scheme, where the magnetic field fluctuations are fully resolved and the velocity fluctuations at small scale are modeled using a large eddy simulation (LES) scheme. We study the response of a forced Taylor-Green flow to an externally applied field: topology of the mean induction and time fluctuations at fixed locations. The results are in remarkable agreement with existing experimental data; a global 1/f behavior at long times is also evidenced.

Dynamo Regimes with a Nonhelical Forcing

A three-dimensional numerical computation of magnetohydrodynamic dynamo behavior is described. The dynamo is mechanically forced with a driving term of the Taylor-Green type. The magnetic field development is followed from negligibly small levels to saturated values that occur at magnetic energies comparable to the kinetic energies. Although there is locally a nonzero helicity density, there is no overall integrated helicity in the system. Persistent oscillations are observed in the saturated state for not-too-large mechanical Reynolds numbers, oscillations in which the kinetic and magnetic energies vary out of phase but with no reversal of the magnetic field. The flow pattern exhibits considerable geometrical structure in this regime. As the Reynolds number is increased, the oscillations disappear and the energies become more nearly stationary, but retain some unsystematically fluctuating turbulent time dependence. The regular geometrical structure of the fields gives way to a more spatially disordered distribution. The injection and dissipation scales are identified, and the different components of energy transfer in Fourier space are analyzed, particularly in the context of clarifying the role played by different flow scales in the amplification of the magnetic field. We observe that small and large scales interact and contribute to the dynamo process.

Dynamo action at low magnetic Prandtl numbers: mean flow versus fully turbulent motions

We compute numerically the threshold for dynamo action in Taylor- Green (TG) swirling flows. Kinematic dynamo calculations, for which the flow field is fixed to its time average, are compared to dynamical runs, with the Navier- Stokes and induction equations jointly solved. The dynamo instability for the kinematic calculations is found to have two branches. The dynamical dynamo threshold at low Reynolds numbers lies within the low branch, while at high Reynolds numbers it gets closer to the high branch. Based on these results, the effect of the mean flow and of the turbulent fluctuations in TG dynamos are discussed.

Spectral Modeling of Turbulent Flows and the Role of Helicity

We present a version of a dynamical spectral model for large eddy simulation based on the eddy damped quasinormal Markovian approximation [S. A. Orszag, in, edited by R. Balian, Proceedings of Les Houches Summer School, 1973 (Gordon and Breach, New York, 1977), p. 237; J. P. Chollet and M. Lesievr, J. Atmos. Sci. 38, 2747 (1981)]. Three distinct modifications are implemented and tested. On the one hand, whereas in current approaches, a Kolmogorov-like energy spectrum is usually assumed in order to evaluate the non-local transfer, in our method the energy spectrum of the subgrid scales adapts itself dynamically to the large-scale resolved spectrum; this first modification allows in particular for a better treatment of transient phases and instabilities, as shown on one specific example. Moreover, the model takes into account the phase relationships of the small scales, embodied, for example, in strongly localized structures such as vortex filaments. To that effect, phase information is implemented in the treatment of the so-called eddy noise in the closure model. Finally, we also consider the role that helical small scales may play in the evaluation of the transfer of energy and helicity, the two invariants of the primitive equations in the inviscid case; this leads as well to intrinsic variations in the development of helicity spectra. Therefore, our model allows for simulations of flows for a variety of circumstances and a priori at any given Reynolds number. Comparisons with direct numerical simulations of the three-dimensional Navier-Stokes equation are performed on fluids driven by an Arnold-Beltrami-Childress (ABC) flow which is a prototype of fully helical flows (velocity and vorticity fields are parallel). Good agreements are obtained for physical and spectral behavior of the large scales.

Subcritical Dynamo Bifurcation in the Taylor-Green Flow

We report direct numerical simulations of dynamo generation for flow generated using a Taylor-Green forcing. We find that the bifurcation is subcritical and show its bifurcation diagram. We connect the associated hysteretic behavior with hydrodynamics changes induced by the action of the Lorentz force. We show the geometry of the dynamo magnetic field and discuss how the dynamo transition can be induced when an external field is applied to the flow.

Kinematic dynamo action in large magnetic reynolds number flows driven by shear and convection

A numerical investigation is presented of kinematic dynamo action in a dynamically driven fluid flow. The model isolates basic dynamo processes relevant to eld generation in the Solar tachocline. The horizontal plane layer geometry adopted is chosen as the local representation of a dierentially rotating spherical fluid shell at co-latitude #; the unit vectors b, b and b point east, north and vertically upwards respectively. Relative to axes moving easterly with the local bulk motion of the fluid the rotation vector lies in the (y;z)-plane inclined at an angle # to the z-axis, while the base of the layer moves with constant velocity in the x-direction. An Ekman layer is formed on the lower boundary characterized by a strong localized spiralling shear flow. This basic state is destabilized by a convective instability through uniform heating at the base of the layer, or by a purely hydrodynamic instability of the Ekman layer shear flow. The onset of instability is characterized by a horizontal wave vector inclined at some angle to the x-axis. Such motion is two-dimensional, dependent only on two spatial coordinates together with time. It is supposed that this two-dimensionality persists into the various fully nonlinear regimes in which we study large magnetic Reynolds number kinematic dynamo action. When the Ekman layer flow is destabilized hydrodynamically, the fluid flow that results is steady in an appropriately chosen moving frame, and takes the form of a row of cat’s eyes. Kinematic magnetic eld growth is characterized by modes of two types. One is akin to the Ponomarenko dynamo mechanism and located close to some closed stream surface; the other appears to be associated with stagnation points and heteroclinic separatrices. When the Ekman layer flow is destabilized thermally, the well-developed convective instability far from onset is characterized by a flow that is intrinsically time-dependent in the sense that it is unsteady in any moving frame. The magnetic eld is concentrated in magnetic sheets situated around the convective cells in regions where chaotic particle paths are likely to exist; evidence for fast dynamo action is obtained. The presence of the Ekman layer close to the bottom boundary breaks the up{down symmetry of the layer and localizes the magnetic eld near the lower boundary.

Dynamos in weakly chaotic two-dimensional flows

Abstract The dynamo action of a time-periodic two-dimensional flow close to integrability is analyzed. At fixed Reynolds number R M and frequency ω, magnetic structures develop in the form of both eddies and filaments. The growth rate of the eddies appears to be the same for all frequencies and decreases with R M, while the growth rate of the filaments displays a strong co-dependence and, except in the limit of zero or infinite frequencies, converges to a non-zero value as R M → ∞. Magnetic filaments develop in the widest chaotic zones located near the homoclinic or heteroclinic tangles, and their growth rate is strongly influenced by the width of these zones which is estimated using Melnikov formalism. This study illustrates quantitatively that not only a local stretching but also a sizable chaotic zone is required for fast dynamo action.

Spectral modeling of magnetohydrodynamic turbulent flows

We present a dynamical spectral model for large-eddy simulation of the incompressible magnetohydrodynamic (MHD) equations based on the eddy damped quasinormal Markovian approximation. This model extends classical spectral large-eddy simulations for the Navier-Stokes equations to incorporate general (non-Kolmogorovian) spectra as well as eddy noise. We derive the model for MHD flows and show that the introduction of an eddy damping time for the dynamics of spectral tensors, in the absence of equipartition between the velocity and magnetic fields, leads to better agreement with direct numerical simulations, an important point for dynamo computations.

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.