Multi-scale theory of rotating turbulence

We consider turbulence induced by an arbitrary forcing and derive turbulence amplitude and turbulent transport coefficients, first by using a quasi-linear theory and then by using a multi-scale renormalisation analysis. With an isotropic forcing, the quasi-linear theory gives that the turbulent transport coefficients, both parallel and perpendicular to the rotation vector, have the asymptotic scaling $\Omega^{-1}$ for rapid rotation (i.e. when the rotation rate $\Omega$ is larger than the inverse of the correlation time of the forcing and the diffusion time), while the renormalisation analysis suggests a weaker dependence on $\Omega$, with $\Omega^{-1/2}$ scaling. The turbulence amplitude is found to scale as $\Omega^0 - \Omega^{-1}$ in the rapid rotation limit depending on the property of the forcing. In the case of an anisotropic forcing, we find that non-diffusive fluxes of angular momentum scale as $\Omega^{-2} - \Omega^{-1}$ for rapid rotation, depending on the temporal correlation of the forcing

Un modèle stochastique pour l'écoulement de von Kármán

URL: http://www-spht.cea.fr/articles/S03/015 , Paris, France, 13-14 mars 2003National audienceUn système d'équations stochastiques nous sert à décrire l'évolution de la vitesse de rotation d'un disque ainsi que le couple appliqué dans l'écoulement de Von-Karman. Ce dernier est étudié de façon analytique pour deux modes de forçage: vitesse angulaire ou couple constant. Le principal résultat est que l'on retrouve la relation expérimentale de Titon et Cadot : dans la limite de l'inertie du disque nulle, la puissance injecté dans la turbulence fluctue deux fois moins lorsque l'on force à couple constant comparé au forçage à vitesse angulaire constante. Ensuite, les distributions de probabilité de la vitesse angulaire et du couple sont comparées à des données expérimentales

Effect of rotation on the tachoclinic transport

We study the effect of rotation on sheared turbulence, due to differential rotation. By solving quasi-linear equations for the fluctuating fields, we derive turbulence amplitude and turbulent transport coefficients, taking into account the effects of shear and rotation on turbulence. We focus on the regions of the tachocline near the equator and the poles where the rotation and the shear are perpendicular and parallel, respectively. For parameter values typical of the tachocline, we show that the shear reduces both turbulence amplitude and transport, more strongly in the radial direction than in the horizontal one, resulting in an anisotropic turbulence. The rotation further reduces turbulence amplitude and transport at the equator whereas it does not have much effect near the pole. The interaction between the shear and the rotation is shown to give rise to a novel non diffusive flux of angular momentum, possibly offering a mechanism for the occurrence of a strong shear region in the solar interior

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. When the noise is at small (resp. large) scale, the dynamo threshold is decreased (resp. increased). For a large scale noise, a finite correlation time reinforces this effect

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

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. By envisioning that turbulence is driven externally in the tachocline (e.g. by plume penetration), 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. 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

Dynamics and thermodynamics of axisymmetric flows: I. Theory

We develop new variational principles to study stability and equilibrium of axisymmetric flows. We show that there is an infinite number of steady state solutions. We show that these steady states maximize a (non-universal) $H$-function. We derive relaxation equations which can be used as numerical algorithm to construct stable stationary solutions of axisymmetric flows. In a second part, we develop a thermodynamical approach to the equilibrium states at some fixed coarse-grained scale. We show that the resulting distribution can be divided in a universal part 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 non-ideal effects such as viscosity, small-scale forcing and dissipation (for forced systems). Finally, we derive a parameterization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale

The turbulent dynamo as an instability in a noisy medium

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 (Krause and R\"{a}dler, 1980) where the dynamo effect is monitored by the mean magnetic field (other possible choices would be the energy, flux,...). 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 permits 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.Comment: 6 page

A turbulent model of torque in von Karman swirling flow

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