We investigate and clarify the notion of locality as it pertains to the cascades of two-dimensional turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L'vov and Procaccia. As a point of departure we use a revised version of the system of hypotheses that was proposed by Frisch for three-dimensional turbulence. We show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non-perturbative statistical locality. We also investigate the stability conditions for both cascades. We have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade. For the enstrophy cascade, statistical stability requires large-scale dissipation and a vanishing downscale energy dissipation. A careful discussion of the subtle notion of locality is given at the end of the paper.
Using the fusion rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous non-perturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We shall call this generalization of non-perturbative locality to multiple applications of the nonlinear interactions operator "multilocality". The resulting cross-terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry. Our main resul…
Comment on "Superfluid turbulence from quantum Kelvin wave to classical Kolmogorov cascades" [arXiv:0905.0159] by J. Yepez, G. Vahala, L.Vahala and M. Soe
Visual manifestations of intermittency in computations of three dimensional Navier-Stokes fluid turbulence appear as the low-dimensional or `thin' filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. In order to study this phenomenon, the first task of this paper is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale wi…
This investigation presents evidence of the relation between the dynamics of intense events in small-scale turbulence and the energy cascade. We use the generalised (Hölder) means to track the temporal evolution of intense events of the enstrophy and the dissipation in direct numerical simulations of isotropic turbulence. We show that these events are modulated by large-scale fluctuations, and that their evolution is consistent with a local multiplicative cascade, as hypothesised by a broad class of intermittency models of turbulence.
We report on the observation of surface gravity wave turbulence at scales larger than the forcing ones in a large basin. In addition to the downscale transfer usually reported in gravity wave turbulence, an upscale transfer is observed, interpreted as the inverse cascade of weak turbulence theory. A steady state is achieved when the inverse cascade reaches a scale in between the forcing wavelength and the basin size, but far from the latter. This inverse cascade saturation, which depends on the wave steepness, is probably due to the emergence of nonlinear dissipative structures such as sharp-c…
Three-dimensional (3D) turbulence is characterized by a dual forward cascade of both kinetic energy and helicity, a second inviscid flow invariant, from the integral scale of motion to the viscous dissipative scale. In helical flows, however, such as strongly rotating flows with broken mirror symmetry, an inverse energy cascade can be observed analogous to that of two-dimensional turbulence (2D) where a second positive-definite flow invariant, enstrophy, unlike helicity in 3D, effectively blocks the forward cascade of energy. In the spectral-helical decomposition of the Navier-Stokes equation …
This paper analyses the turbulent energy cascade from the perspective of statistical mechanics, and relates inter-scale energy fluxes to statistical irreversibility and information-entropy production. The microscopical reversibility of the energy cascade is tested by constructing a reversible 3D turbulent system using a dynamic model for the sub-grid stresses. This system, when reversed in time, develops a sustained inverse cascade towards the large scales, evidencing that the characterisation of the inertial energy cascade must consider the possibility of an inverse regime. This experiment is…
To answer the question whether a cascade of energy exists or not in turbulence, we propose a set of correlation functions able to test if there is an irreversible transfert of energy, step by step, from large to small structures. These tests are applied to real Eulerian data of a turbulent velocity flow, taken in the wind grid tunnel of Modane, and also to a prototype model equation for wave turbulence. First we demonstrate the irreversible character of the flow by using multi-time correlation function at a given point of space. Moreover the unexpected behavior of the test function leads us to…
We investigate experimentally turbulence of surface gravity waves in the Coriolis facility in Grenoble by using both high sensitivity local probes and a time and space resolved stereoscopic reconstruction of the water surface. We show that the water deformation is made of the superposition of weakly nonlinear waves following the linear dispersion relation and of bound waves resulting from non resonant triadic interaction. Although the theory predicts a 4-wave resonant coupling supporting the presence of an inverse cascade of wave action, we do not observe such inverse cascade. We investigate 4…
Using the theory of wave turbulence for rapidly rotating incompressible fluids derived by Galtier (2003), we find the locality conditions that the solutions of the kinetic equation must satisfy. We show that the exact anisotropic Kolmogorov-Zakharov (KZ) spectrum satisfies these conditions, which justifies the existence of this constant (positive) energy flux solution. Although a direct cascade is predicted in the transverse ($\perp$) and parallel ($\parallel$) directions to the rotation axis, we show numerically that in the latter case some triadic interactions can have a negative contributio…
Comment on "Symmetry of Kelvin-wave dynamics and the Kelvin-wave cascade in the T = 0 superfluid turbulence" [arXiv:1201.3188]
本文对E. B. Sonin，PRB v 85，104516（2012）[arXiv：1201.3188]，我们发现自己严重不同意。我们用这个选项来阐明开尔文波湍流理论的一些重要问题，在E。B. Sonin的论文，特别是，在强的Vinen谱和L 'vov-Nazarenko谱的弱湍流的开尔文波之间的关系。我们还讨论了我们的明确计算的开尔文波相互作用的哈密顿量和“对称性参数”，应该解决之间的矛盾的Kozik-Svistunov和L 'vov-Nazarenko谱的弱湍流的开尔文波的作用。
Vortex annihilation and inverse cascades in two dimensional superfluid turbulence
我们研究了二维超流湍流，采用有效的描述有效的极限超流涡的密度是参数小。在足够低的温度下，有效的描述产生逆级联与柯尔莫哥洛夫能谱$E(k) \sim k^{-5/3}$。用$N(t)$表示涡的数目作为时间的函数，我们发现，涡的湮灭率的标度类似于$\dot N \sim N^{5/3}$的状态与反级联和$\dot N \sim N^2$的层流。
We study numerically a Constantin-Lax-Majda-De Gregorio model generalized by Okamoto, Sakajo and Wunsch, which is a model of fluid turbulence in one dimension with an inviscid conservation law. In the presence of the viscosity and two types of the large-scale forcings, we show that turbulent cascade of the inviscid invariant, which is not limited to quadratic quantity, occurs and that properties of this model's turbulent state are related to singularity of the inviscid case by adopting standard tools of analyzing fluid turbulence.
According to Richardson's cascade description of turbulence, large vortices break up to form smaller ones, thereby transferring kinetic energy towards smaller scales. Energy dissipation occurs at the smallest scales due to viscosity. We study this energy cascade in a phenomenological model of vortex breakdown. The model is a binary tree of decreasing masses connected by softening springs, with dampers acting on the lowest level. The masses and stiffnesses between levels change according to a power law. The different levels represent different scales, enabling the definition of "mass wavenumber…
Large-scale turbulence in fluid layers and other quasi-two-dimensional compressible systems consists of planar vortices and waves. Separately, wave turbulence usually produces a direct energy cascade, while solenoidal planar turbulence transports energy to large scales by an inverse cascade. Here, we consider turbulence at finite Mach numbers when the interaction between acoustic waves and vortices is substantial. We employ solenoidal pumping at intermediate scales and show how both direct and inverse energy cascades are formed starting from the pumping scale. We show that there is an inverse …
Fluid turbulence is a far-from-equilibrium phenomenon and remains one of the most challenging problems in physics. Two-dimensional, fully developed turbulence may possess the largest possible symmetry, the conformal symmetry. We focus on the steady-state solution of two-dimensional bounded turbulent flow and propose a $c=0$ boundary logarithmic conformal field theory for the inverse energy cascade and another bulk conformal field theory in the classical limit $c\rightarrow -\infty$ for the direct enstrophy cascade. We show that these theories give rise to the Kraichnan-Batchelor scaling $k^{-3…
