From zero-mode intermittency to hidden symmetry in random scalar advection

The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier-Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It ultimately identifies the scaling exponents as the eigenvalues of a Perron-Frobenius operator acting on invariant measures of the rescaled equations. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale & Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode scenario represents only one facet of intermittency, here prescribing the scaling exponents of even-order correlators. Besides, we argue that hidden symmetry provides a more generic mechanism, fully prescribing intermittency in terms of scaling anomalies, but also in terms of its multiplicative random nature and fusion rules required to explicitly compute zero-modes from first principles

A statistical mechanics framework for the large-scale structure of turbulent von K{\'a}rm{\'a}n flows

In the present paper, recent experimental results on large scale coherent steady states observed in experimental von K{\'a}rm{\'a}n flows are revisited from a statistical mechanics perspective. The latter is rooted on two levels of description. We first argue that the coherent steady states may be described as the equilibrium states of well-chosen lattice models, that can be used to define global properties of von K{\'a}rm{\'a}n flows, such as their temperatures. The equilibrium description is then enlarged, in order to reinterpret a series of results about the stability of those steady states, their susceptibility to symmetry breaking, in the light of a deep analogy with the statistical theory of Ferromagnetism. We call this analogy "Ferro-Turbulence

Anomalous spectral laws in differential models of turbulence

International audienceDifferential models for hydrodynamic, passive-scalar and wave turbulence given by nonlinear first- and second-order evolution equations for the energy spectrum in the $k$-space were analysed.Both types of models predict formation an anomalous transient power-law spectra.The second-order models were analysed in terms of self-similar solutions of the second kind, and a phenomenological formula for the anomalous spectrum exponent was constructed using numerics for a broad range of parameters covering all known physical examples. The first-order models were examined analytically, including finding an analytical prediction for the anomalous exponent of the transient spectrum and description of formation of the Kolmogorov-type spectrum as a reflection wave from the dissipative scale back into the inertial range. The latter behaviour was linked to pre-shock/shock singularities similar to the ones arising in the Burgers equation.Existence of the transient anomalous scaling and the reflection-wave scenario are argued to be a robust feature common to the finite-capacity turbulence systems. The anomalous exponent is independent of the initial conditions but varies for for different models of the same physical system

A statistical mechanics framework for the large-scale structure of turbulent von Kármán flows

In the present paper, recent experimental results on large scale coherent steady states observed in experimental von Kármán flows are revisited from a statistical mechanics perspective. The latter is rooted on two levels of description. We first argue that the coherent steady states may be described as the equilibrium states of well-chosen lattice models, that can be used to define global properties of von Kármán flows, such as their temperatures. The equilibrium description is then enlarged, in order to reinterpret a series of results about the stability of those steady states, their susceptibility to symmetry breaking, in the light of a deep analogy with the statistical theory of Ferromagnetism. We call this analogy ''Ferro-Turbulence'

Statistical mechanics of Beltrami flows in axisymmetric geometry: Equilibria and bifurcations

We characterize the thermodynamical equilibrium states of axisymmetric Euler-Beltrami flows. They have the form of coherent structures presenting one or several cells. We find the relevant control parameters and derive the corresponding equations of state. We prove the coexistence of several equilibrium states for a given value of the control parameter like in 2D turbulence [Chavanis and Sommeria, J. Fluid Mech. 314, 267 (1996)]. We explore the stability of these equilibrium states and show that all states are saddle points of entropy and can, in principle, be destabilized by a perturbation with a larger wavenumber, resulting in a structure at the smallest available scale. This mechanism is therefore reminiscent of the 3D Richardson energy cascade towards smaller and smaller scales. Therefore, our system is truly intermediate between 2D turbulence (coherent structures) and 3D turbulence (energy cascade). We further explore numerically the robustness of the equilibrium states with respect to random perturbations using a relaxation algorithm in both canonical and microcanonical ensembles. We show that saddle points of entropy can be very robust and therefore play a role in the dynamics. We evidence differences in the robustness of the solutions in the canonical and microcanonical ensembles. A scenario of bifurcation between two different equilibria (with one or two cells) is proposed and discussed in connection with a recent observation of a turbulent bifurcation in a von Karman experiment [Ravelet et al., Phys. Rev. Lett. 93, 164501 (2004)].Comment: 25 pages; 16 figure

Mécanique statistique d'écoulements idéaux à deux dimensions et demi

The present manuscript deals with the statistical mechanics of some inviscid fluidmodels which are possibly relevant in the context of geophysics and astrophysics. Weinvestigate the case of axially symmetric flows, two-dimensional Boussinesq flows, andtwo-dimensional magneto-hydro fluids. Those flows can be loosely referred to as twodimensionalflows with three components (“2D3C”). In addition to the two-dimensionalvelocity field, they describe the evolution of an additional field variable, which representseither a magnetic current, a salinity, a temperature or a swirl depending on the situation.In common with the dynamics of strictly two-dimensional hydrodynamical flows, thenon-linear dynamics of 2D3C flows is constrained by the presence of an infinite numberof Casimir invariants, which emerge as dynamical invariants in the limit of a vanishingforcing and a vanishing dissipation . In common with three-dimensional flows, the vorticityis not only mixed but also stretched by the dynamics. The additional field may actas a source or a sink of kinetic energy. It is commonly believed that such flows have thepropensity to develop large scale coherent structures. Whether those long lived structuresare equilibrium or metastable structures is however not so clear, nor are the exactconditions of their emergence. The role of the Casimir invariants in constraining those isnot so obvious either.Dans cette thèse, nous nous intéressons à la mécanique statistique d’une classe d’écoulements “quasi-bidimensionnels”. Nous nous penchons plus particulièrement sur le cas des écoulements tri-dimensionnels axisymétriques, bidimensionnels stratifiés et bidimensionnels magnéto hydrodynamiques. La dynamique de ces écoulements est génériquement décrite par les équations d’évolution d’un champ de vitesses incompressible bidimensionnel,couplées à une équation d’évolution d’un champ scalaire. Ce dernier représente tantôt une température, tantôt un courant électrique, tantôt un mouvement tourbillonnaire transverse. Ces écoulements ont un intérêt géophysique ou astrophysique : ils peuvent être utilisés pour modéliser grossièrement les ouragans, les courants océaniques à l’échelle planétaire, les taches solaires, etc. Ils ont aussi un intérêt plus fondamental.Malgré leur géométrie bidimensionnelle intrinsèque, les écoulements “2D3C” peuvent être en effet tri-dimensionnellement connotés. Dans les cas que l’on regarde, la vorticité n’est pas seulement transportée : elle est aussi étirée. Il n’est ainsi pas évident de savoir si la tendance naturelle des écoulements 2D3C est de s’organiser en structures cohérentes énergétiques à grande échelle comme en deux dimensions, ou plutôt de répartir leur énergie sur les petites échelles comme en trois dimensions. Il n’est a priori pas clair nonplus de savoir si une forme d’énergie (cinétique ou magnétique/tourbillonnaire) y est privilégiée aux dépends de l’autre.Pour répondre à ces questions de manière très générale, nous étudions et décrivons la mécanique statistique d’équilibre des écoulements 2D3C sus-mentionnés, en nous plaçant d’abord dans le cadre des “ensembles d’équilibre absolu” considérés par Robert Kraichnan à la fin des années 1960, puis dans le cadre plus moderne des “mesures microcanoniques stationnaires” introduites par Raoul Robert, Jonathan Miller et Joël Sommeria pour les fluides bidimensionnels au début des années 1990. Les équilibres 2D3C sont décrits dans la première partie de ce manuscript. La seconde partie du manuscript est plus pratique, et également plus spéculative. Nous nous servons d’ outils de la mécanique statistique d’équilibre pour interpréter des données turbulentes expérimentales provenant d’expériences de type Von Kármán . Nous utilisons ensuite des résultats récents de théorie de probabilité pour montrer que des régimes de turbulence quasi-bidimensionnelle (turbulence tri-dimensionnelle avec rotation,turbulence dans des couches savonneuses) ont des propriétés d’invariance conforme statistique, analogues à celles observées dans des systèmes de spins ferromagnétiques au point critique