Surface quasi geostrophy (SQG) describes the two-dimensional active transport of a temperature field in a strongly stratified and rotating environment. Besides its relevance to geophysics, SQG bears formal resemblance with various flows of interest for turbulence studies, from passive scalar and Burgers to incompressible fluids in two and three dimensions. This analogy is here substantiated by considering the turbulent SQG regime emerging from deterministic and smooth initial data prescribed by the superposition of a few Fourier modes. While still unsettled in the inviscid case, the initial value problem is known to be mathematically well-posed when regularised by a small viscosity. In practice, numerics reveal that in the presence of viscosity, a turbulent regime appears in finite time, which features three of the distinctive anomalies usually observed in three-dimensional developed turbulence: (i) dissipative anomaly, (ii) multifractal scaling, and (iii) super-diffusive separation of fluid particles, both backward and forward in time. These three anomalies point towards three spontaneously broken symmetries in the vanishing viscosity limit: scale invariance, time reversal and uniqueness of the Lagrangian flow, a fascinating phenomenon that Krzysztof Gawedzki dubbed spontaneous stochasticity. In the light of Gawedzki's work on the passive scalar problem, we argue that spontaneous stochasticity and irreversibility are intertwined in SQG, and provide numerical evidence for this connection. Our numerics, though, reveal that the deterministic SQG setting only features a tempered version of spontaneous stochasticity, characterised in particular by non-universal statistics.
We analyze the phenomenon of spontaneous stochasticity in fluid dynamics formulated as the nonuniqueness of solutions resulting from viscosity at infinitesimal scales acting through intermediate on large scales of the flow. We study the finite-time onset of spontaneous stochasticity in a real version of the GOY shell model of turbulence. This model allows high-accuracy numerical simulations for a wide range of scales (up to ten orders of magnitude) and demonstrates non-chaotic dynamics, but leads to an infinite number of solutions in the vanishing viscosity limit after the blowup time. Thus, t…
We develop a theoretical approach to ``spontaneous stochasticity'' in classical dynamical systems that are nearly singular and weakly perturbed by noise. This phenomenon is associated to a breakdown in uniqueness of solutions for fixed initial data and underlies many fundamental effects of turbulence (unpredictability, anomalous dissipation, enhanced mixing). Based upon analogy with statistical-mechanical critical points at zero temperature, we elaborate a renormalization group (RG) theory that determines the universal statistics obtained for sufficiently long times after the precise initial d…
We develop a Lagrangian approach to conservation-law anomalies in weak solutions of inviscid Burgers equation, motivated by previous work on the Kraichnan model of turbulent scalar advection. We show that the entropy solutions of Burgers possess Markov stochastic processes of (generalized) Lagrangian trajectories backward in time for which the Burgers velocity is a backward martingale. This property is shown to guarantee dissipativity of conservation-law anomalies for general convex functions of the velocity. The backward stochastic Burgers flows with these properties are not unique, however. …
We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, like non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Be…
The quantum wave-function of a massive particle with small initial uncertainties (consistent with the uncertainty relation) is believed to spread very slowly, so that the dynamics is deterministic. This assumes that the classical motions for given initial data are unique. In fluid turbulence non-uniqueness due to "roughness" of the advecting velocity field is known to lead to stochastic motion of classical particles. Vanishingly small random perturbations are magnified by Richardson diffusion in a "nearly rough" velocity field so that motion remains stochastic as the noise disappears, or class…
A Measure-Theoretic Approach to Spontaneous Stochasticity
LPENSL, ENS de Lyon, CNRS（巴黎高等师范学院物理实验室，法国国家科学研究中心）；Institut de Physique de Nice, Université Côte d’Azur et CNRS（尼斯物理研究所，蔚蓝海岸大学及法国国家科学研究中心）；DAMTP, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心应用数学与理论物理系）；Inria, Centre at Université Côte d’Azur Team Calisto（法国国家信息与自动化研究所，蔚蓝海岸大学卡尔斯托团队中心）
The real butterfly effect: from the pop culture to mathematics and physics
University of Rome “Tor Vergata”（罗马托维加塔大学）；Università degli Studi di Torino（复杂系统研究所，国家研究委员会）；INFN Sezione Roma “Tor Vergata”（罗马托维加塔大学INFN分部）；Università di Roma “La Sapienza”（罗马第一大学）
An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This "Lagrangian fluctuation-dissipation relation" equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated to initial scalar values and internal scalar sources, as those are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of "Lagrangian spontaneous stochasticity" or persistent non-determin…
Statistical mechanics provides an elegant explanation to the appearance of coherent structures in two-dimensional inviscid turbulence: while the fine-grained vorticity field, described by the Euler equation, becomes more and more filamented through time, its dynamical evolution is constrained by some global conservation laws (energy, Casimir invariants). As a consequence, the coarse-grained vorticity field can be predicted through standard statistical mechanics arguments (relying on the Hamiltonian structure of the two-dimensional Euler flow), for any given set of the integral constraints. It …
We derive here Lagrangian fluctuation-dissipation relations for advected scalars in wall-bounded flows. The relations equate the dissipation rate for either passive or active scalars to the variance of scalar inputs from the initial values, boundary values, and internal sources, as those are sampled backward in time by stochastic Lagrangian trajectories. New probabilistic concepts are required to represent scalar boundary conditions at the walls: the boundary local-time density at points on the wall where scalar fluxes are imposed and the boundary first hitting-time at points where scalar valu…
The butterfly effect is today commonly identified with the sensitive dependence of deterministic chaotic systems upon initial conditions. However, this is only one facet of the notion of unpredictability pioneered by Lorenz, who actually predicted that multiscale fluid flows could spontaneously lose their deterministic nature and become intrinsically random. This effect, which is radically different from chaos, have remained out of reach for detailed physical observations. Here, we substantiate this scenario by showing that it is inherent to the elementary Kelvin--Helmholtz hydrodynamical inst…
In this paper, we study the inviscid limit of the Sabra shell model of turbulence, which is considered as a particular case of a viscous conservation law in one space dimension with a nonlocal quadratic flux function. We present a theoretical argument (with a detailed numerical confirmation) showing that a classical deterministic solution before a finite-time blowup, $t < t_b$, must be continued as a stochastic process after the blowup, $t > t_b$, representing a unique physically relevant description in the inviscid limit. This theory is based on the dynamical system formulation written for th…
We examine the role discrete symmetries, time-reversal and mirror symmetries, play in the context of geophysical waves and instabilities. By looking at three special cases from the two-layer quasi-geostrophic model as well as developing a general framework for translating real-space transformations to Fourier-space we are able to: 1) show that baroclinic instability is an example of spontaneous parity-time symmetry breaking; 2) show that pure parity symmetry for a fluid system is exactly analogous to charge-conjugation-parity symmetry in a condensed matter system; and 3) show that when a pure …
