Nicolas Leprovost

Influence of turbulence on the dynamo threshold.

We use direct and stochastic numerical simulations of the magnetohydrodynamic equations to explore the influence of turbulence on the dynamo threshold. In the spirit of the Kraichnan-Kazantsev model, we model the turbulence by a noise, with given amplitude, injection scale, and correlation time. The addition of a stochastic noise to the mean velocity significantly alters the dynamo threshold and increases it for any noise at large scale. For small-scale noise, the result depends on its correlation time and on the magnetic Prandtl number.

The turbulent dynamo as an instability in a noisy medium

Abstract.We study an example of instability in presence of a multiplicative noise, namely the spontaneous generation of a magnetic field in a turbulent medium. This so-called turbulent dynamo problem remains challenging, experimentally and theoretically. In this field, the prevailing theory is the Mean-Field Dynamo [1] where the dynamo effect is monitored by the mean magnetic field. In recent years, it has been shown on stochastic oscillators that this type of approach could be misleading. In this paper, we develop a stochastic description of the turbulent dynamo effect which enables us to define unambiguously a threshold for the dynamo effect, namely by globally analyzing the probability density function of the magnetic field instead of a given moment.

Dynamics and thermodynamics of axisymmetric flows: Theory

We develop variational principles to study the structure and the stability of equilibrium states of axisymmetric flows. We show that the axisymmetric Euler equations for inviscid flows admit an infinite number of steady state solutions. We find their general form and provide analytical solutions in some special cases. The system can be trapped in one of these steady states as a result of an inviscid violent relaxation. We show that the stable steady states maximize a (nonuniversal) function while conserving energy, helicity, circulation, and angular momentum (robust constraints). This can be viewed as a form of generalized selective decay principle. We derive relaxation equations which can be used as numerical algorithm to construct nonlinearly dynamically stable stationary solutions of axisymmetric flows. We also develop a thermodynamical approach to predict the equilibrium state at some fixed coarse-grained scale. We show that the resulting distribution can be divided in two parts: one universal coming from the conservation of robust invariants and one non-universal determined by the initial conditions through the fragile invariants (for freely evolving systems) or by a prior distribution encoding nonideal effects such as viscosity, small-scale forcing, and dissipation (for forced systems). Finally, we derive a parametrization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale. A conceptual interest of this axisymmetric model is to be intermediate between two-dimensional (2D) and 3D turbulence.

Self-consistent theory of turbulent transport in the solar tachocline. III. Gravity waves

Aims. To understand the fundamental physical processes important for the evolution of solar rotation and distribution of chemical species, we provide theoretical predictions for particle mixing and momentum transport in the stably stratified tachocline. Methods. By envisioning that turbulence is driven in the tachocline, we compute the amplitude of turbulent flow, turbulent particle diffusivities, and eddy viscosity, by incorporating the effect of a strong radial differential rotation and stable stratification. We identify the different roles that the shear flow and stable stratification play in turbulence regulation and transport. Results. Particle transport is found to be severely quenched due to stable stratification, as well as radial differential rotation, especially in the radial direction with an effectively more efficient horizontal transport. The eddy viscosity is shown to become negative for parameter values typical of the tachocline, suggesting that turbulence in the stably stratified tachocline leads to a non-uniform radial differential rotation. Similar results also hold in the radiative interiors of stars, in general.

Dynamo Quenching Due to Shear Flow

We provide a theory of dynamo (alpha effect) and momentum transport in three-dimensional magnetohydrodynamics. For the first time, we show that the alpha effect is reduced by the shear even in the absence of magnetic field. The alpha effect is further suppressed by magnetic fields well below equipartition (with the large-scale flow) with different scalings depending on the relative strength of shear and magnetic field. The turbulent viscosity is also found to be significantly reduced by shear and magnetic fields, with positive value. These results suggest a crucial effect of shear and magnetic field on dynamo quenching and momentum transport reduction, with important implications for laboratory and astrophysical plasmas, in particular, for the dynamics of the Sun.

On a long-term dynamics of the magnetised solar tachocline

We investigate the confinement and long-term dynamics of the magnetised solar tachocline. Starting from first principles, we derive the values of turbulent transport coefficients and then explore the implications for the confinement and long-term dynamics of the tachocline. For reasonable parameter values, the turbulent eddy viscosity is found to be negative, with turbulence enhancing the radial shear in the tachocline. Both magnetic diffusivity and thermal diffusivity are severely quenched, with the values much smaller than the magnitude of the eddy viscosity. The effect of the meridional circulation on momentum transport via the hyperviscosity becomes important when the radial shear becomes large (larger than the presently inferred value) due to the negative viscosity.

Effect of Rossby and Alfvén Waves on the Dynamics of the Tachocline

To understand magnetic diffusion, momentum transport, and mixing in the interior of the Sun, we consider an idealized model of the tachocline, namely, magnetohydrodynamic (MHD) turbulence on a β-plane subject to a large-scale shear (provided by the latitudinal differential rotation). This model enables us to self-consistently derive the influence of shear, Rossby, and Alfven waves on the transport properties of turbulence. In the strong magnetic field regime, we find that the turbulent viscosity and diffusivity are reduced by magnetic fields only, as in the two-dimensional MHD case (without Rossby waves). In the weak magnetic field regime, we find a crossover scale (LR) from a Alfven-dominated regime (on small scales) to a Rossby-dominated regime (on large scales). For parameter values typical of the tachocline, LR is larger than the solar radius so that Rossby waves are unlikely to play an important role in the transport of magnetic field and angular momentum. This is mainly due to the enhancement of magnetic back-reaction by shearing, which efficiently generates small scales, and thus strong currents.

A stochastic model of torques in von Karman swirling flow

Abstract.A stochastic model is derived to predict the turbulent torque produced by a swirling flow. It is a simple Langevin process, with a colored noise. Using the unified colored noise approximation, we derive analytically the PDF of the fluctuations of injected power in two forcing regimes: constant angular velocity or constant applied torque. In the limit of small velocity fluctuations and vanishing inertia, we predict that the injected power fluctuates twice less in the case of constant torque than in the case of constant angular velocity forcing. The model is further tested against experimental data in a von Karman device filled with water. It is shown to allow for a parameter-free prediction of the PDF of power fluctuations in the case where the forcing is made at constant torque. A physical interpretation of our model is finally given, using a quasi-linear model of turbulence.

Analytical theory of forced rotating sheared turbulence: the perpendicular case.

Rotation and shear flows are ubiquitous features of many astrophysical and geophysical bodies. To understand their origin and effect on turbulent transport in these systems, we consider a forced turbulence and investigate the combined effect of rotation and shear flow on the turbulence properties. Specifically, we study how rotation and flow shear influence the generation of shear flow (e.g., the direction of energy cascade), turbulence level, transport of particles and momentum, and the anisotropy in these quantities. In all the cases considered, turbulence amplitude is always quenched due to strong shear (xi=nuk_y2/A<<1 , where A is the shearing rate, nu is the molecular viscosity, and ky is a characteristic wave number of small-scale turbulence), with stronger reduction in the direction of the shear than those in the perpendicular directions. Specifically, in the large rotation limit (OmegaA) , they scale as A-1 and A-1|ln xi| , respectively, while in the weak rotation limit (Omega<<A) , they scale as A-1 and A-2/3 , respectively. Thus, flow shear always leads to weak turbulence with an effectively stronger turbulence in the plane perpendicular to shear than in the shear direction, regardless of rotation rate. The anisotropy in turbulence amplitude is, however, weaker by a factor of xi1/3|ln xi| ( proportional, variantA;-1/3|ln xi|) in the rapid rotation limit (OmegaA) than that in the weak rotation limit (OmegaA) since rotation favors almost-isotropic turbulence. Compared to turbulence amplitude, particle transport is found to crucially depend on whether rotation is stronger or weaker than flow shear. When rotation is stronger than flow shear (OmegaA) , the transport is inhibited by inertial waves, being quenched inversely proportional to the rotation rate (i.e., proportional, variantOmega;-1 ) while in the opposite case, it is reduced by shearing as A-1 . Furthermore, the anisotropy is found to be very weak in the strong rotation limit (by a factor of 2) while significant in the strong shear limit. The turbulent viscosity is found to be negative with inverse cascade of energy as long as rotation is sufficiently strong compared to flow shear (OmegaA) while positive in the opposite limit of weak rotation (OmegaA) . Even if the eddy viscosity is negative for strong rotation (OmegaA) , flow shear, which transfers energy to small scales, has an interesting effect by slowing down the rate of inverse cascade with the value of negative eddy viscosity decreasing as |nuT| proportional, variantA-2 for strong shear. Furthermore, the interaction between the shear and the rotation is shown to give rise to a nondiffusive flux of angular momentum ( Lambda effect), even in the absence of external sources of anisotropy. This effect provides a mechanism for the existence of shearing structures in astrophysical and geophysical systems.

Stability of a nonlinear oscillator with random damping

Abstract.A noisy damping parameter in the equation of motion of a nonlinear oscillator renders the fixed point of the system unstable when the amplitude of the noise is sufficiently large. However, the stability diagram of the system can not be predicted from the analysis of the moments of the linearized equation. In the case of a white noise, an exact formula for the Lyapunov exponent of the system is derived. We then calculate the critical damping for which the nonlinear system becomes unstable. We also characterize the intermittent structure of the bifurcated state above threshold and address the effect of temporal correlations of the noise by considering an Ornstein-Uhlenbeck noise.