arXiv subjects

Wandrille Ruffenach

Publications and source records attributed to Wandrille Ruffenach.

9 recordsLinked to original sources

arXiv2026

Spontaneous stochasticity and anomalous dissipation in collapsing wave turbulence

We study a focusing Majda--McLaughlin--Tabak type equation undergoing finite-time wave collapse. This singularity terminates the classical smooth solution and opens a post-blowup regime where infinitely many solutions may exist. To probe this nonunique regime, we regularize the dynamics either by viscous diffusion or by nonlinear saturation and study the corresponding vanishing-regularization limits. Both regularizations prevent blowup at fixed parameter and recover the same inviscid collapse as the parameter vanishes. Before collapse, they converge to the same smooth inviscid solution. After collapse, however, their limits differ. The viscous approximation undergoes anomalous mass dissipation whereas the saturating approximation remains conservative. Moreover, neither regularization selects a unique post-blowup solution. Vanishing perturbations of the regularization parameter or of the initial condition survive the singular limit and generate finite post-blowup uncertainty. This places collapsing wave turbulence in the setting of spontaneous stochasticity, where the inviscid limit is better described in terms probability law on inviscid solutions rather than deterministically. Scale-by-scale fluctuation budgets identify collapse events as localized sources of uncertainty production. While spontaneous stochasticity is usually associated with fluid turbulence, these results provide numerical evidence that it can be applied to a broader class of systems, including dispersive media in which experiments could be conducted.

physics.flu-dyn↗

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↗

arXiv2026

Numerical study of probabilistic well-posedness of one dimensional fractional nonlinear wave equations

The three dimensional cubic defocusing nonlinear wave equation is known to be ill-posed for general low regularity initial data. Well-posedness can however be recovered globally in time on a probabilistic level, provided the Fourier coefficients of the random initial data follow a sub-Gaussian law, an assumption on which the available proofs rely heavily. In this article we perform numerical simulations of the one dimensional fractional cubic defocusing wave equation in a periodic setting, with low regularity random initial data drawn either from a Gaussian or from a heavy tailed Student law. Besides illustrating probabilistic well-posedness and norm inflation in both the energy subcritical and supercritical regimes, our simulations display no qualitative difference between the heavy tailed and the sub-Gaussian cases. This leads us to conjecture that probabilistic well-posedness holds under the sole assumption that the Fourier coefficients are centered with finite variance.

math.AP↗

arXiv2026

The spatio-temporal statistical structure of the turbulent dissipation field and its stochastic representation as a Gaussian Multiplicative Chaos

The present article concerns the stochastic modeling of the turbulent dissipation field and in particular its temporal evolution. To do so, we will be calling for a random distribution, ubiquitous in several aspects of physics and probability theory, known as the Gaussian Multiplicative Chaos (GMC), that takes its roots in the phenomenology of fluid turbulence. Firstly introduced by Mandelbrot, shortly after Yaglom's discrete multiplicative cascade models, and rigorously studied by Kahane, the GMC appears as an appropriate statistically homogeneous model of the turbulent dissipation field. In this article, we will be recalling several ingredients of the associated turbulent phenomenology and its stochastic representation as a GMC, and propose a generalization to a spatio-temporal framework. All along the presentation of known properties in space, and in order to support new propositions concerning the temporal evolution, we will be calling for a comparison against Direct Numerical Simulations of the Navier-Stokes equations extracted from a publicly accessible database.

physics.flu-dyn↗

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

A spatio-temporal random synthetic turbulent velocity field: The underlying Gaussian structure

We develop, simulate and extend an initial proposition by Chaves et al. concerning a random incompressible vector field able to reproduce key ingredients of three-dimensional turbulence in both space and time. In this article, we focus on the important underlying Gaussian framework. Presently, the statistical spatial structure of this velocity field is consistent with a divergence-free fractional Gaussian vector field that encodes all known properties of homogeneous and isotropic fluid turbulence at a given finite Reynolds number, up to second-order statistics. The temporal structure of the velocity field is introduced through a stochastic evolution of the respective Fourier modes. In the simplest picture, Fourier modes evolve according to an Ornstein-Uhlenbeck process, where the characteristic time scale depends on the wave-vector amplitude. For consistency with direct numerical simulations (DNSs) of the Navier-Stokes equations, this time scale is inversely proportional to the wave vector amplitude. As a consequence, the characteristic velocity that governs the eddies is independent of their size and is related to the velocity standard deviation, which is consistent with some features of the so-called sweeping effect. To ensure differentiability in time while respecting the Markovian nature of the evolution, we use the methodology developed by Viggiano et al. to propose a fully consistent stochastic picture. We finally derive analytically all statistical quantities in a continuous setup and develop precise and efficient numerical schemes of the corresponding periodic framework. Both exact predictions and numerical estimations of the model are compared to DNSs provided by the Johns Hopkins database.

physics.flu-dyn↗

arXiv2025

A toy model of turbulent shear flow using vortons

We introduce a novel toy model for shear flows, exploiting the spatial intermittency and the scale separation between large-scale flows and small-scale structures. The model is highly sparse, focusing exclusively on the most intense structures, which are represented by vortons: dynamically regularized quasi-singularities that experience rapid distortion from the large-scale shear. The vortons, in turn, influence the large-scale flow through the sub-grid stress tensor. Despite its simplicity, the model displays an interesting transition between two distinct regimes: (i) a laminar regime, where dissipation is entirely attributed to the large-scale flow, and the vortons dynamics is essentially diffusive, and (ii) a turbulent regime, in which most of the dissipation arises from the vortons. These regimes correspond to different scalings of dissipation and the Grashof number as functions of the Reynolds number, with power-law relationships that resemble those observed in classical turbulence.

physics.flu-dyn↗

arXiv2024

Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: applications to fluid turbulence and generation of fractional Gaussian fields

Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics has been recently proposed able to generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear Partial Differential Equation (PDE) stirred by an additive random forcing term which is delta-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space which aims at transferring energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We here present novel simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudo-spectral simulations. We show that the novel algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two and three spatial dimensions, and we display the solutions both in physical and Fourier spaces. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities.

physics.flu-dyn↗

arXiv2022

Superfluid drain vortex

Drain vortices are among the most common vortices observed in everyday life, yet their physics is complex due to the competition of vorticity's transport and diffusion, and the presence of viscous layers and a free surface. Recently, it has become possible to study experimentally drain vortices in superfluid liquid helium, a fluid in which the physics is simplified by the absence of viscosity and the quantisation of the circulation. Using the Gross-Pitaevskii equation, we make a simple model of the problem which captures the essential physics ingredients, showing that the superfluid drain vortex consists of a bundle of vortex lines which twist, thus strengthening the axial flow into the drain.

physics.flu-dyn↗