On the effect of rotation on magnetohydrodynamic turbulence at high magnetic Reynolds number

This article is focused on the dynamics of a rotating electrically conducting fluid in a turbulent state. As inside the Earth's core or in various industrial processes, a flow is altered by the presence of both background rotation and a large scale magnetic field. In this context, we present a set of 3D direct numerical simulations of incompressible decaying turbulence. We focus on parameters similar to the ones encountered in geophysical and astrophysical flows, so that the Rossby number is small, the interaction parameter is large, but the Elsasser number, defining the ratio between Coriolis and Lorentz forces, is about unity. These simulations allow to quantify the effect of rotation and thus inertial waves on the growth of magnetic fluctuations due to Alfv\'en waves. Rotation prevents the occurrence of equipartition between kinetic and magnetic energies, with a reduction of magnetic energy at decreasing Elsasser number {\Lambda}. It also causes a decrease of energy transfer mediated by cubic correlations. In terms of flow structure, a decrease of {\Lambda} corresponds to an increase in the misalignment of velocity and magnetic field.Comment: 18 pages, 12 figure

Anisotropic Developments for Homogeneous Shear Flows

The general decomposition of the spectral correlation tensor R(sub ij)(k) by Cambon et al. (J. Fluid Mech., 202, 295; J. Fluid Mech., 337, 303) into directional and polarization components is applied to the representation of R(sub ij)(k) by spherically averaged quantities. The decomposition splits the deviatoric part H(sub ij)(k) of the spherical average of R(sub ij)(k) into directional and polarization components H(sub ij)(sup e)(k) and H(sub ij)(sup z)(k). A self-consistent representation of the spectral tensor in the limit of weak anisotropy is constructed in terms of these spherically averaged quantities. The directional polarization components must be treated independently: models that attempt the same representation of the spectral tensor using the spherical average H(sub ij)(k) alone prove to be inconsistent with Navier-Stokes dynamics. In particular, a spectral tensor consistent with a prescribed Reynolds stress is not unique. The degree of anisotropy permitted by this theory is restricted by realizability requirements. Since these requirements will be less severe in a more accurate theory, a preliminary account is given of how to generalize the formalism of spherical averages to higher expansion of the spectral tensor. Directionality is described by a conventional expansion in spherical harmonics, but polarization requires an expansion in tensorial spherical harmonics generated by irreducible representations of the spatial rotation group SO(exp 3). These expansions are considered in more detail in the special case of axial symmetry

The magneto-hydrodynamic instabilities in rotating and precessing sheared flows: An asymptotic analysis

International audienceLinear magnetohydrodynamic instabilities are studied analytically in the case of unbounded inviscid and electrically conducting flows that are submitted to both rotation and precession with shear in an external magnetic field. For given rotation and precession the possible configurations of the shear and of the magnetic field and their interplay are imposed by the "admissibility" condition (i.e., the base flow must be a solution of the magnetohydrodynamic Euler equations): we show that an "admissible" basic magnetic field must align with the basic absolute vorticity. For these flows with elliptical streamlines due to precession we undertake an analytical stability analysis for the corresponding Floquet system, by using an asymptotic expansion into the small parameter ε (ratio of precession to rotation frequencies) by a method first developed in the magnetoelliptical instabilities study by Lebovitz and Zweibel Astrophys. J. 609 301 (2004). The present stability analysis is performed into a suitable frame that is obtained by a systematic change of variables guided by symmetry and the existence of invariants of motion. The obtained Floquet system depends on three parameters: ε, η (ratio of the cyclotron frequency to the rotation frequency) and χ=cos α, with α being a characteristic angle which, for circular streamlines, ε=0, identifies with the angle between the wave vector and the axis of the solid body rotation. We look at the various (centrifugal or precessional) resonant couplings between the three present modes: hydrodynamical (inertial), magnetic (Alfvén), and mixed (magnetoinertial) modes by computing analytically to leading order in ε the instabilities by estimating their threshold, growth rate, and maximum growth rate and their bandwidths as functions of ε, η, and χ. We show that the subharmonic "magnetic" mode appears only for η>√5/2 and at large η (⪢1) the maximal growth rate of both the "hydrodynamic" and magnetic modes approaches ε/2, while the one of the subharmonic "mixed" mode approaches zero

Analyse paramétrique des échanges d'énergie en turbulence barocline

Le mécanisme barocline crée des structures turbulentes spécifiques via l'instabilité barocline, et contribue de manière importante aux transferts d'énergie dans la dynamique des écoulements géophysiques. Il est non seulement essentiel pour la prédiction du mouvement atmosphérique à grande échelle, mais aussi pour évaluer l'efficacité du mélange aux échelles plus petites. Le phénomène barocline résulte de la combinaison d'une rotation de solide rigide avec un cisaillement de vitesse moyenne et un gradient moyen de densité stabilisant. De nombreuses études combinent deux de ces mécanismes, mais moins fréquemment les trois ensemble, en raison d'une complexité accrue qui requiert l'étude d'un espace paramétrique vaste, et nécessite de démêler les dynamiques respectives multiéchelles des modes d'énergies cinétique et potentielle, et de la vorticité potentielle. Pour cette raison, les géophysiciens ont d'abord utilisé une dynamique réduite sous la forme de modèles à base des équations quasi-géostrophiques (Charney, 1947; Eady, 1949). À notre connaissance peu d'études récentes ont porté sur la simulation numérique de la turbulence barocline, en systématisant l'ouverture de l'espace paramétrique. Nous proposons ici d'étudier la turbulence homogène barocline par des simulations numériques directes (DNS), dans lesquelles l'écoulement de base tournant verticalement est constitué d'un cisaillement vertical de vitesse moyenne zonale, et d'un gradient stabilisant vertical de densité. Dans ce contexte, un gradient horizontal de densité est également créé pour satisfaire les équations du vent thermique. Le jeu minimal de nombres adimensionnels sont les nombres de Richardson Ri et de Rossby Ro -- ce dernier pouvant être remplacé par le paramètre de baroclinicité E -- qui combinent les paramètres de rotation, cisaillement, stratification (f,S,N). Nous présenterons les limites d'instabilité obtenues par les DNS en liaison avec les approches linéarisées, le role de la vorticité potentielle, qui est un invariant dans la théorie linéaire, et nous identifierons les différentes régions du domaine (Ri,E) dans lesquelles les mécanismes d'instabilité barocline ou inertielle de cisaillement sont prépondérants. Pour cela, nous nous appuierons sur les équations de bilan d'énergie

Magnetized stratified rotating shear waves

International audienceWe present a spectral linear analysis in terms of advected Fourier modes to describe the behavior of a fluid submitted to four constraints: shear (with rate S), rotation (with angular velocity Ω), stratification, and magnetic field within the linear spectral theory or the shearing box model in astrophysics. As a consequence of the fact that the base flow must be a solution of the Euler-Boussinesq equations, only radial and/or vertical density gradients can be taken into account. Ertel's theorem no longer is valid to show the conservation of potential vorticity, in the presence of the Lorentz force, but a similar theorem can be applied to a potential magnetic induction: The scalar product of the density gradient by the magnetic field is a Lagrangian invariant for an inviscid and nondiffusive fluid. The linear system with a minimal number of solenoidal components, two for both velocity and magnetic disturbance fields, is eventually expressed as a four-component inhomogeneous linear differential system in which the buoyancy scalar is a combination of solenoidal components (variables) and the (constant) potential magnetic induction. We study the stability of such a system for both an infinite streamwise wavelength (k1=0, axisymmetric disturbances) and a finite one (k1≠0, nonaxisymmetric disturbances). In the former case (k1=0), we recover and extend previous results characterizing the magnetorotational instability (MRI) for combined effects of radial and vertical magnetic fields and combined effects of radial and vertical density gradients. We derive an expression for the MRI growth rate in terms of the stratification strength, which indicates that purely radial stratification can inhibit the MRI instability, while purely vertical stratification cannot completely suppress the MRI instability. In the case of nonaxisymmetric disturbances (k1≠0), we only consider the effect of vertical stratification, and we use Levinson's theorem to demonstrate the stability of the solution at infinite vertical wavelength (k3=0): There is an oscillatory behavior for τ>1+∣∣K2/k1∣∣, where τ=St is a dimensionless time and K2 is the radial component of the wave vector at τ=0. The model is suitable to describe instabilities leading to turbulence by the bypass mechanism that can be relevant for the analysis of magnetized stratified Keplerian disks with a purely azimuthal field. For initial isotropic conditions, the time evolution of the spectral density of total energy (kinetic + magnetic + potential) is considered. At k3=0, the vertical motion is purely oscillatory, and the sum of the vertical (kinetic + magnetic) energy plus the potential energy does not evolve with time and remains equal to its initial value. The horizontal motion can induce a rapid transient growth provided K2/k1≫1. This rapid growth is due to the aperiodic velocity vortex mode that behaves like Kh/kh where kh(τ)=[k21+(K2−k1τ)2]1/2 and Kh=kh(0). After the leading phase (τ>K2/k1≫1), the horizontal magnetic energy and the horizontal kinetic energy exhibit a similar (oscillatory) behavior yielding a high level of total energy. The contribution to energies coming from the modes k1=0 and k3=0 is addressed by investigating the one-dimensional spectra for an initial Gaussian dense spectrum. For a magnetized Keplerian disk with a purely vertical field, it is found that an important contribution to magnetic and kinetic energies comes from the region near k1=0. The limit at k1=0 of the streamwise one-dimensional spectra of energies, or equivalently, the streamwise two-dimensional (2D) energy, is then computed. The comparison of the ratios of these 2D quantities with their three-dimensional counterparts provided by previous direct numerical simulations shows a quantitative agreement

From baroclinic instability to developed turbulence

The coupled effects of mean shear, density-stratification and system rotation are investigated in the context of strong turbulence, i.e. accounting for the baroclinic instability. Although there exists a large literature in the rotating shear case and the stratified shear case, with linear approaches, Direct or Large Eddy Simulations, very few studies consider the combined three ingredients in the context of distorted homogeneous turbulence

On the two-dimensionalization of quasistatic magnetohydrodynamic turbulence

We analyze the anisotropy of turbulence in an electrically conducting fluid in the presence of a uniform magnetic field, for low magnetic Reynolds number, using the quasi-static approximation. In the linear limit, the kinetic energy of velocity components normal to the magnetic field decays faster than the kinetic energy of component along the magnetic field [Moffatt, JFM 28, 1967]. However, numerous numerical studies predict a different behaviour, wherein the final state is characterized by dominant horizontal energy. We investigate the corresponding nonlinear phenomenon using Direct Numerical Simulations. The initial temporal evolution of the decaying flow indicates that the turbulence is very similar to the so-called "two-and-a-half-dimensional" flow [Montgomery & Turner, Phys. Fluids 25(2), 1982] and we offer an explanation for the dominance of horizontal kinetic energy.Comment: 17 pages, 8 figure

Passive Scalar and Scalar Flux in Homogeneous Anisotropic Turbulence

Le but de ce travail est l'étude théorique et la modélisation d'un champ scalaire passif dans une turbulence homogène anisotrope à l'aide d'une fermeture EDQNM (Eddy-Damped Quasi-Normal Markovian) adaptée à un tel contexte. Cette fermeture est une extension du modèle spectral pour le champ de vitesse de V. Mons et. al. (2015) à un champ scalaire passif et au flux scalaire, qui naît de l'interaction de gradients de vitesse et de scalaire. Des résultats généraux originaux de croissance et décroissance des énergies du scalaire et du flux en présence de gradients moyens sont ici présentés

Rotating shear-driven turbulent flows: Towards a spectral model with angle-dependent linear interactions.

Le modèle en cours fait suite à l'étude de Mons, Cambon et Sagaut [1] (MCS dans la suite), dans le cadre de la thèse de doctorat de Ying Zhu. Il s'agit de résoudre les équations des corrélations de vitesse en deux points pour une turbulence incompressible et (statistiquement) homogène, en présence de gradients de vitesse moyenne incluant déformation et rotation avec des taux uniformes dans l'espace. Les équations couplées, qui généralisent l'équation de Lin en turbulence isotrope, sont résolues avec les opérateurs linéaires exacts d'action du champ moyen, et donnent accès à l'anisotropie détaillée, de type directivité et polarisation, ainsi qu'à la dynamique non-linéaire induite par les corrélations triples (transferts généralisés inter-échelles). Ces derniers termes sont fermés par une procédure EDQNM (Eddy Damped Quasi-Normal Markovian) en ne retenant que les premiers harmoniques angulaires de la distribution anisotrope des corrélations doubles de vitesse en deux points, en accord avec le modèle MCS. En revanche, les termes linéaires ne sont pas tronqués au premier degré significatif des harmoniques angulaires, comme dans MCS, mais sont résolus avec un grand nombre de paramètres angulaires, de façon à rendre le modèle plus précis et fiable pour les plus grandes échelles du champ turbulent. En particulier, nous allons étudier la compétition entre les effets linéaires dits de ""distorsion rapide"" et ceux de transfert nonlinéaire avec ""backscatter"", qui contrôlent l'essentiel de la dynamique (cf. [2] dans le contexte différent de la stratification instable, mais avec des outils statistiques analogues.) Le nouveau modèle est en cours de validation pour retrouver la dynamique linéaire qui reflète les instabilités de base, exponentielles (configuration hyperbolique et elliptique) ou algébrique (cisaillement pur plan). L'activation des couplages non-linéaires permettra ensuite de comparer quantitativement ses prédictions spectrales anisotropes avec celles de simulations directes à haute résolution. Un effort particulier portera sur le cisaillement tournant, avec des résultats spectraux de simulation directe existants [3] et à poursuivre à plus haute résolution. [1] Vincent Mons, Claude Cambon & Pierre Sagaut (2016). A spectral model for homogeneous shear-driven turbulence, J Fluid Mech. 788, 147-182. [2] Alan Burlot, Benoît-Joseph Gréa, Fabien Godeferd, Claude Cambon & Olivier Soulard (2015). Large Reynolds number self-similar states of unstably stratified homogeneous turbulence, Phys. Fluids 27, 065114. [3] Aziz Salhi, Frank Jacobitz, Kai Schneider & Claude Cambon (2014). Nonlinear dynamics and anisotropic structure of rotating sheared turbulence, Phys. Rev. E 89, 013020