I will describe explorative numerical studies of Navier-Stokes 3D turbulence under different decimation, either based on the helical properties or on the number of degrees of freedom. Decimation local in Fourier space, leading to non-local couplings in real space.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

I will start reviewing a basic mechanism of the Euler equations, namely the vorticity stretching phenomenon which is non local in nature. Then, from there, I will make some approximations and heuristics in order to build up a realistic random velocity field able to reproduce not only the intermittency phenomenon, but also energy transfers. If some time is left, I will finally present several recent mathematical progresses in this direction.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

We introduce an evolution equation, where one can tune the interaction to be local in real space or rather local in Fourier space. In the one extreme (locality in real space) we recover the Burgers equation with its high degree of anomalous scaling whereas in the other extreme (nearly local in Fourier space) we obtain nearly perfect scale invariant turbulence without any intermittency. We calculate the extreme statistics of rare events using the instant on formalism to clarify the role of the nonlocal interactions.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

Spontaneous symmetry breaking is a classical phenomenon in statistical or particle physics, where specific tools have been designed to characterize and study it. Spontaneous symmetry breaking is also present in out-of-equilibrium systems, but there is at the present time no general theory to describe it in these systems. To help developing such a theory, it is therefore interesting to study well-controlled laboratory model of out-of equilibrium spontaneous symmetry breaking. In that respect, the turbulent von Karman (VK) flow is an interesting example. In this system, the ow is forced by two counter-rotating impellers, providing the necessary energy injection to set the system out-of-equilibrium. This energy is naturally dissipated through molecular viscosity, so that, for well controlled forcing protocols, statistically states can be established, that may be seen as the out-of-equilibrium counterpart of the equilibria of classical ideal systems [1, 2]. Changing the forcing protocol for the VK flow leads to various transitions with associated symmetry breaking. In the sequel, we focus on the special case of O(2) symmetry breaking, that has been reported in [3]. For exact counter-rotation (zero relative rotation) of the impeller, the VK set up is exactly isomorphic to O(2) | which is the symmetry group of XY-models [4] |. Increasing the relative rotation between the two impellers, one induces an O(2) symmetry breaking, in analogy with an applied external magnetic field. Studying the flow response to this continuous symmetry breaking for a Reynolds number ranging from Re = 102 (laminar regime) to Re ' 106 (highly turbulent regime), Cortet et al. observe a divergence of the flow susceptibility around a critical Reynolds number Rec _ 40 000.This divergence coincides with intense fluctuations of the order parameter near Rec corresponding to time-wandering of the flow between states which spontaneously and dynamically break the forcing symmetry. In this talk, we suggest that the dynamical spontaneous symmetry breaking reported in a turbulent swirling flow at Re = 40 000 by Cortet et al., Phys. Rev. Lett., 105, 214501 (2010) can be described through a continuous one parameter family transformation (amounting to a phase shift) of steady states. We investigate a possible mechanism of emergence of such spontaneous symmetry breaking in a toy model of our out-equilibrium system, derived from its equilibrium counterpart. We show that the stationary states are solution of a linear differential equation. For a specific value of the Reynolds number, they are subject to a spontaneous symmetry breaking through a zero-mode mechanism. The associated susceptibility diverges at the transition, in a way similar to what is observed in the experimental turbulent flow. Overall, the susceptibility of the toy model reproduces quite well the features of the experimental one, meaning that the zero mode mechanism is a good candidate to explain the experimental symmetry breaking.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

We consider the regularity issues of the Navier-Stokes equations in the whole space, centering on self-similarity and criticality (scale-invariance). It is well-known that energy is critical in 2D, enstrophy in 4D and a "helicity-like integral" in 3D. By using the critical conditions, we first give shortened proofs of absence of self-similar blowup, i.e., of the fact that Leray equations have trivial solutions only. After deriving non-steady Leray equations by dynamic scaling transformations, we study how the long-time asymptotic behavior of their solutions can be consistent with absence of self-similar lowup. Finally, we compare time intervals in which blowup can possibly occur in 3D and 4D. We observe that i) the dangerous interval is smaller in size in 4D than in 3D and that ii) the median time, at which enstrophy is most seriously endangered, has the common scaling behavior.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

One of the fundamental questions in turbulence research is the one on the universal properties that the variety of flows, which are sustained in a statistically stationary state by various large scale driving mechanisms, have in common. Rather than focusing on statistical analysis of the velocity in the inertial cascade range we resolve the velocity gradients in the crossover range from the inertial to the viscous range by means of very high resolution direct numerical simulations. In detail, we investigate the high order moments of velocity derivatives. At Reynolds numbers of about 100 their statistics switches from sub-Gaussian or Gaussian regime to intermittent non-Gaussian behavior. Above this transition point derivative moments follow the same scaling laws with respect to the Reynolds number. The exponents of the moments are found to agree with predictions by a theoretical framework. We compare therefore three different turbulent flows with an increasing degree of complexity: homogeneous isotropic box turbulence with periodic boundary conditions in all three directions, shear flow turbulence in a channel and turbulent convection in a closed cylindrical cell.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

I am going to talk about our newest results from windtunnel measurements. I will summarize our results on the Eulerian velocity structure function and the decay of turbulence from passive grids up to Reë ~ 1200. I shall also present results from the active grid turbulence generated in an open windtunnel and on the dependence of the turbulence statistics on the correlation of the active grid structure.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

The nonlinearity in the Navier Stokes equations directly leads to the interaction of wave-number triads that couple dynamics on different length scales. In turbulence, these triads self-organize to produce a net transfer of energy from the scales at which it is injected into the flow to the scales at which it is dissipated. In two dimensions, this cascade drives energy from the forcing scale to larger length scales, where large scale friction damps the motion. Formally, the energy transfer between scales can be written as the inner product of a scale-dependent turbulent stress with a large scale rate of strain. I will present recent results from a quasi-two-dimensional laboratory experiment that explore the geometric alignment of these two quantities, and I will show that the turbulent stress tensor undergoes an ordering transition at the onset of the inverse cascade. Our results suggest potential ways of thinking about spectral nonlocality in turbulence in terms of the relative geometry of turbulent stresses and strain rates.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

Rapidly rotating turbulence is arguably the simplest example, in a neutral fluid, of a system that supports anisotropically propagating waves as well as nonlinear interactions. I will argue that the (anisotropic) structure of this turbulence can be understood in terms of a scale-by-scale balance between wave propagation and nonlinear decorrelation scales. What to an experimentalist looks like formation of Taylor columns, to an unreconstructed turbulence theoretician is an anisotropic energy cascade. I will show that within this framework, the isotropisation of the turbulence at the Zeman scale is a natural consequence of the way energy is transferred in and cross the direction of the axis of rotation [1]. Several existing experimental studies and very large numerical simulations suggest that these arguments are perhaps not without merit – and there is clear experimental opportunity and challenge to measure critical balance in the laboratory. I will argue that the principle of critical balance is universal to wave-supporting anisotropic systems and discuss the evidence for this claim from MHD and plasma turbulence systems [2,3,4] (even in messy environments like a tokamak [5]!). Time permitting, I will show some new MHD results that give critical balance a precise measurable statistical meaning [6] and also discuss the way a weakly turbulent system attains the critically balanced state [7] (here a degree of nonlocality will enter the otherwise unapologetically local picture).

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

I will present three examples of nonlocality arising in 2D MHD turbulence, geophysical beta-plane turbulence and in small-scale superfluid turbulence dominated by Kelvin waves. These are examples where nonlocality leads to three different types of behavior, from changing the turbulent scalings to suppressing turbulence altogether by large-scale shear.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

In magnetohydrodynamic (MHD) turbulence several phenomenological theories exist debating for the interpretation of the power law of the energy spectrum. Numerical simulations to date are unable to provide a definitive answer to this scaling. Some direct numerical simulations (DNS) obtained energy spectra with k-5/3 (Kolmogorov spectrum) while others k-3/2 (IroshnikovKraichnan spectrum) or k-2 (weak turbulence spectrum). Recently, simulations of zero flux MHD turbulence at 20483 resolution by Lee et al. 2010, Krstulovic et al. 2012 demonstrated all three exponents for different initial conditions/forcing functions of the magnetic field. The dependence of the scaling exponent on initial conditions suggests a possible lack of universality in MHD turbulence. Our work investigates this lack of universality. We focus on the origin of the k-2 spectrum that can be clearly distinguished from the other two proposed exponents. Using numerical simulations of the same resolution (2048^3) we demonstrate (a) that the origin of the k-2 spectrum is not weak turbulence, (b) the properties of the initial conditions that lead to such a spectrum, (c) its stability and (d) its final fate as the Reynolds number is increased. Thus, we determine if and at what Reynolds number the exponent becomes universal.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

Turbulence strongly influenced by a stable density stratification is dominated by horizontal motions and structured in very thin horizontal layers with a characteristic thickness of the order of the buoyancy length scale Lb = U/N, where U is the characteristic horizontal velocity and N the Brunt-Väisälä frequency. The effect of this strong anisotropy in terms of nonlocal transfers will be discussed on the basis of results of high resolution numerical simulations. We will first focus on the nonlinear evolution of a counter-rotating vortex pair in a stratified fluid. This flow has been extensively studied in particular because it is one of the simplest flow on which the zigzag instability develops and from which the buoyancy length scale naturally emerges as the vertical length. A spectral analysis shows that the transition to turbulence is dominated by two kinds of transfers: first, the shear instability induces a direct nonlocal transfer from the large scale towards horizontal wavelengths of the order of the buoyancy scale; second, the destabilization of the Kelvin-Helmholtz billows and the gravitational instability lead to small-scale weakly stratified turbulence. We will then present numerical results on forced stratified turbulence showing that such nonlocal transfers related to the anisotropy of the flow are also active in developed stratified turbulence.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

We report the results of high-resolution numerical experiments on decaying turbulence in rotating shallow water model, which is a proxy to the large-scale atmospheric and oceanic turbulence. We are using a new-generation well-balanced shock-resolving finite-volume numerical scheme which resolves both vortex and wave components of the flow very well. We find clear deviations from the universal decay predictions in the vortex sector, known in the 2D turbulence, which is a limit of rotating shallow water turbulence at small Rossby numbers. The evolution is dominated by interacting coherent structures. We also observe strong departures from all theoretical predictions in the wave sector. In both sectors the energy spectra are very steep.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

In collaboration with L. Bou_e, R. Dasgupta, J. Laurie, S. Nazarenko, I. Procaccia and O. Rudenko Kelvin waves propagating on quantum vortices play a crucial role in the energy dissipation of superfluid turbulence. The physics of interacting Kelvin waves is highly non-trivial and cannot be understood on the basis of pure dimensional reasoning only. A consistent theory of Kelvin waves turbulence in super-fluids should be based on explicit knowledge of the details of their interactions, presented in our Ref. [1]. In Ref. [2] we derive a type of kinetic equation for Kelvin waves on quantized vortex _laments with random large-scale curvature, that describes step-by-step (local) energy cascade over scales caused by 4-wave interactions. Resulting new energy spectrum ELN(k) ~ k-5/3 replaced in a theory of super-fluid turbulence the previously used Kosik-Svistunov spectrum EKS(k) ~ k-7/5, which is inconsistent due to nonlocality of the 6-wave energy cascade, as shown in [1]. We also show in Ref. [3] that the solution proposed in [2] enjoys existence, uniqueness and regularity of the pre-factor. Furthermore, we present numerical results of the Local Nonlinear Equation (LNE) for the description of Kelvin waves in quantum turbulence. The LNE was systematically derived from the Biot-Savart Equation in the limit of one long Kelvin wave - which was shown to be the main contribution to the Kelvin wave dynamics. We compare our results with the theoretical results from the proposed local and non-local theories for Kelvin wave dynamics and show an agreement with the non-local predictions. Previous theoretical studies have consistently focused on the zero-temperature limit of the statistical physics of Kelvin-wave turbulence. In Ref. [4] we go beyond this athermal limit by introducing a small but finite temperature in the form of non-zero mutual friction dissipative force; a situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length scale separating a quasi-inertial range of Kelvin-wave turbulence from a far-dissipation range. The Letter [4] culminates with analytical predictions for the energy spectrum of the Kelvin-wave turbulence in both of these regimes.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)

Quantum turbulence at finite temperature (within the framework of the two-fluid model) exhibits an “anormal” distribution of kinetic energy of its superfluid component at scales larger than the inter-vortex distance. This anormal behavior is consistent with a thermalization of superfluid excitations at small scales. An original phenomenological argument allows us to predict explicitly the extension of the thermalization range. It is predicted that this extension is independent of the Reynolds number, and scales as the inverse square root of the normal fluid fraction. The prediction is well supported by high-resolution pseudo-spectral simulations of the two fluid-model.

• Thematic program: Turbulence and Fusion (2013)
• Event: Workshop on "Nonlocality in turbulence" (2013)