arXiv subjects

Nicolas Valade

Publications and source records attributed to Nicolas Valade.

6 recordsLinked to original sources

arXiv2026

A Measure-Theoretic Approach to Spontaneous Stochasticity

Spontaneous stochasticity (SpSt), originating in Richardson's picture of turbulent dispersion and Lorenz's Eulerian view of finite-time loss of predictability, was later formulated under this name by Gaw\k{e}dzki and collaborators and developed in shell models by Mailybaev and collaborators. Whether it occurs in fully developed turbulence remains a major open question. Beyond a few specific classes of systems, however, SpSt has lacked a general mathematical definition. We introduce a measure-theoretic formalism in which it is understood as a measure-selection principle. Given an inviscid problem, a well-posed regularization, and an ambient measure, we study the pushforward of that measure by the regularized flow. Strong SpSt occurs when these pushforward measures converge to a non-Dirac probability law, replacing classical deterministic selection by statistical selection. For finite-dimensional systems, we establish several structural results. Our central attainability theorem shows that, whenever the inviscid problem is nonunique, any probability measure supported on the set of inviscid states can be selected as the limiting law of a suitable regularization. We also identify singular sets in the inviscid dynamics, detected through Dini-type directional growth, as necessary obstructions underlying nonuniqueness. We analyze the relation between SpSt and sensitivity to initial data, clarifying the scope and limitations of turbulence-inspired finite-time separation criteria. Finally, we develop a renormalization-(semi)group viewpoint in which limiting statistics arise as statistical attractors. Explicit examples illustrate how ambient measures, inviscid singularities, regularization scales, and initial-data sensitivity interact in the emergence of SpSt.

nlin.CD↗

arXiv2025

Spontaneous stochasticity in the Armstrong-Vicol passive scalar

Spontaneous stochasticity refers to the emergence of intrinsic randomness in deterministic systems under singular limits, a phenomenon conjectured to be fundamental in turbulence. Armstrong and Vicol recently constructed a deterministic, divergence-free multiscale vector field arbitrarily close to a weak Euler solution, proving that a passive scalar transported by this field exhibits anomalous dissipation and lacks a selection principle in the vanishing diffusivity limit. We show that this advection-diffusion PDE also selects a non-Dirac measure in the space of weak solutions in the inviscid limit, thereby exhibiting Eulerian spontaneous stochasticity. We further provide numerical evidence of Lagrangian spontaneous stochasticity, together with a numerical illustration of the Obukhov-Corrsin conjecture for this system. We formulate a general framework for spontaneous stochasticity in arbitrary finite dimensional systems under arbitrary regularizations, distinguishing two regimes: weak, where different probability measures may arise along subsequences of inviscid limits, and strong, where the limit measure is unique. The advection diffusion system of Armstrong and Vicol lies in the strong regime. We prove that the set of selected measures is compact and equals the closed convex hull of Dirac measures. Moreover, for any non-Dirac measure supported on the set of nonunique solutions of the inviscid system, there exists a regularization that produces strong spontaneous stochasticity. Finally, we relate this framework to renormalization-group methods \`a la Feigenbaum and examine how the underlying dynamical system influences the inviscid limit. The discussion is complemented by elementary finite-dimensional examples illustrating a variety of cases.

math-ph↗

arXiv2025

Surface quasigeostrophic turbulence: The refined study of an active scalar

SQG describes the 2D active transport of a scalar field, such as temperature, which -- when properly rescaled -- shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with 3D turbulence. In particular, the Kraichnan-Leith-Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling $\propto (\varepsilon \ell)^{1/3}$ for both scalar and velocity fields, and the presence of intermittency was pointed out by Sukhatme & Pierrehumbert ($Chaos$ $\boldsymbol{12}$, 439, 2002) in unforced settings. In this work, we refine these analogies using simulations up to $16,384^2$ collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, linking velocity increments with powers of scalar differences, exhibit clear scaling, revealing the role of anomalous fluxes of all the scalar moments. However, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires to resort to generalised notions of scale invariance, such as extended self-similarity and specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: They evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions, that have been repeatedly observed in SQG numerics for the past 20 years.

physics.flu-dyn↗

arXiv2022

Anomalous dissipation and spontaneous stochasticity in deterministic surface quasi-geostrophic flow

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.

physics.flu-dyn↗

arXiv2022

Modeling compressed turbulent plasma with rapid viscosity variations

We propose two-equations models in order to capture the dynamics of a turbulent plasma undergoing compression and experiencing large viscosity variations. The models account for possible relaminarization phases and rapid viscosity changes through closures dependent on the turbulent Reynolds and on the viscosity Froude numbers. These closures are determined from a data-driven approach using eddy-damped quasi normal markovian simulations. The best model is able to mimic the various self-similar regimes identified in \citet{Viciconte2018} and to recover the rapid transition limits identified by \citet{Coleman1991}.

physics.flu-dyn↗

arXiv2021

Folding instabilities in non-Newtonian viscous sheets: shear thinning and shear thickening effects

In this work, we extend the analyses devoted to Newtonian viscous fluids previously reported by Ribe [Physical Review E 68, 036305 (2003)], by investigating shear thickening (dilatant) and shear thinning (pseudoplastic) effects on the development of folding instabilities in non-Newtonian viscous sheets of which viscosity is given by a power-law constitutive equation. Such instabilities are trigged by compression stresses acting on viscous sheets that leave a channel at a very small initial velocity, fall, and then hit a solid surface or a fluid substrate. Our study is conducted through a mixed approach combining direct numerical simulations, energy budget analyses, scaling laws, and experiments. The numerical results are based on an adaptive variational multi-scale method for multiphase flows, while Carpobol gel sheets are considered for the conducted experiments. Two folding regimes are observed: (1) the viscous regime; and (2) the gravitational one. Interestingly, only the latter is affected by shear thinning/thickening manifestations within the material. In short, when gravity is balanced by viscous forces along the non-Newtonian viscous sheet, both the folding amplitude and the folding frequency are given by a power-law function of the sheet slenderness, the Galileo number (the ratio of the gravitational stress to the viscous one), and the flow behaviour index. Highly shear thickening materials develop large amplitude (and low frequency) instabilities, which, in contrast, tend to be suppressed by shear thinning effects, and eventually cease. Lastly, nonNewtonian effects on folding onset/cessation are also carefully explored. As a result, non-Newtonian folding onset and cessation criteria are presented.

physics.flu-dyn↗