ICTS

The rich multifractal properties of fluid turbulence illustrated by the work of Parisi and Frisch are related explicitly to Leray's weak solutions of the $3D$ Navier-Stokes equations. Directly from this correspondence it is found that the set on which energy dissipates, $\mathbb{F}_{m}$, has a range of dimensions $\Dim=3/m$ ($1 \leq m \leq \infty$), and a corresponding range of sub-Kolmogorov dissipation inverse length scales $L\eta_{m}^{-1} \leq Re^{3/(1+\Dim)}$ spanning $Re^{3/4}$ to $Re^{3}$. In joint work with Kiran, Padhan and Pandit (2022), the idea is extended to a $2D/3D$ model in the subject of active turbulence called the incompressible Toner-Tu equations (ITT), which has two Reynolds numbers.

Leonardo da Vinci had a strong interest in hydrodynamics. Around 1505 he got interested in "turbulence" (he was th first to use this name). Examining the "turbulences" (eddies) in the river Arno of Florence, he found that the amplitude of the turbulence was decreasing very slowly in time, until it would come to rest (within the surrounding river). In spite of Leonardo's strong interested in mathematics, at that time, it consisted basically of geometry and simple polynomial equations. There were no tools available to describe the very slowly temporal relaxation of turbulence.This topic would remain dormant for about 430 years, until in 1938 Karman, triggered by Taylor, established that the mean energy of the turbulence should decrease very slowly, indeed like an inverse power of the time elapsed. Three years later, Kolmogorov himself found another inverse power (10/7) of the time elapsed. This, likewise was wrong, because he was assuming a certain invariance property (Loitsiansky , proved later wrong by Proudman and Reid ), The main change in the last few decades is that fully developed turbulence is definitely not self-similar, not only is it fractal , but it can have infinitely many fractal scalings (multifractality), as proposed by Parisi and Frisch in the eighties. Furthermore, multifractality can manifest itself either at small scales or at large scales. The latter might change the law of energy decay. Not enough is understood for the 3D Euler equations, but large-scale multifractality for the Burgers is an interesting possibility, possibility, which is being explored by Frisch, Khanin, Pandit and Roy. A brief exploration of what happens to the energy decay-law will be presented.

Over the recent years, I have been concerned (with several friends E. Titi, To. Nguyen, Tr Nguyen, E. Wiedemann and D. Boutros and others) by the boundary effect for the 0 viscosity limit of solutions of the NavieStokes equations in presence of a no slip boundary condition.

After several interpretations of the Kato criteria which connects the convergence and the absence of anomalous energy dissipation the by now classical convergence result obtained by Caflisch and Sammartino for the half space is extended to domain with boundary.

The proof uses several basic ingredients.

1. A formulation of the no slip boundary condition in term of a Dirichlet-Neumann operator for vorticity in the half space following Mayekawa.
2. A representation of the problem near the boundary in terms of geodesic coordinates inspired by a previous contribution devoted to the Onsager conjecture for solutions of the Euler equa-
tion in presence of boundary.

3. The reduction to a Nash Moser theorem well adapted to this decomposition built on previous papers (Kukavica, Vicol Wang and To Nguyen and Tri. Nguyen.)

The above construction underlines the limitation of the time of convergence by the effect of the detachment points connecting this “abstract approach ” with the issue of the generation of G ̈ortler vortices.

We develop a numerical simulation of the construction due to Buckmaster, De Lellis, Szekelyhidi and Vicol (2019) of the weak solutions of the Euler equations, which can dissipate the energy. In this talk, we describe the construction and discuss insights obtainable from the simulation into the physics of turbulence. This is an on-going work in collaboration with U. Frisch and L. Szekelkyhidi.

We discuss data-driven approaches to two turbulent problems, one Eulerian and one Lagrangian: (i) the case of controlling the dispersion rate of two active particles advected by model 2d turbulent flows [1] and (ii) the case of data-assimilation of partial/noisy measurements in 3d rotating turbulence [2]. The first problem is approached by means of Multi Objective Reinforcement Learning (MORL), combining scalarization techniques together with a Q-learning algorithm, for Lagrangian drifters that have variable swimming velocity. For the latter, we compare linear approaches based on Extended-POD with fully non-linear methods using Generative Adversarial Networks.

[1] Taming Lagrangian Chaos with Multi-Objective Reinforcement Learning C Calascibetta, L Biferale, F Borra, A Celani, M Cencini arXiv preprint arXiv:2212.09612, submitted to EPJE (2022)

[2] Data reconstruction of turbulent flows with Gappy POD, Extended POD and Generative Adversarial Networks T Li, M Buzzicotti, L Biferale, F Bonaccorso, S Chen, M Wan arXiv preprint arXiv:2210.11921, submitte to JFM (2022)

We develop a theory of circulation statistics in strong turbulence ($\nu \rightarrow 0$ in the NS equation), treated as a degenerate fixed point of a Hopf equation. 
We use spherical Clebsch variables to parametrize vorticity in the stationary singular (weak) Euler flow, characterized by two winding numbers.
The singular vortex line is regularized by matching Burgers vortex.
We compute anomalous dissipation, helicity, and Hamiltonian for our singular flow in the limit of strong turbulence.
The Hamiltonian shows explicit dependence on the logarithm of the Reynolds number, suggesting some multifractal behavior.
As a result, we compute the probability distribution of velocity circulation $\Gamma$, which decays exponentially with pre-exponential factor $1/\sqrt{\Gamma}$ in perfect match with numerical simulations of conventional forced \NS{} equations on periodic lattice $8K^3$. 

The one-dimensional Galerkin-truncated Burgers equation, with both dissipation and noise terms included, is studied using spectral methods. When the truncation-scale Reynolds number is varied, from very small values to order 1 values, the scale-dependent correlation time is shown to follow the expected crossover from the short-distance Edwards–Wilkinson scaling to the universal long-distance Kardar–Parisi–Zhang scaling. In the inviscid limit we show that the system displays another crossover to the Galerkin-truncated inviscid-Burgers regime.

There is growing evidence that turbulence in simple fluids is governed by two fixed points arising in the statistical mechanical description of flows. The first controls the behaviour near the laminar-turbulence transition, while the second controls the behaviour at asymptotically large Reynolds numbers. In the first part of the talk, I review the phenomena associated with the sub-critical transition to turbulence, primarily in quasi-one-dimensional flows such as pipe or high aspect-ratio Taylor-Couette, and show how theory and experiment are converging on a description based on a non-equilibrium phase transition. In particular, I present a stochastic model that captures decay and splitting of localised regions of turbulence (puffs), and the way in which regions of turbulence grow at higher Reynolds number, through two modes of growth (weak and strong slugs). I also show how recent experimental measurements on puff dynamics, when combined with renormalization group and simulation methods, unequivocally supports the identification of laminar-turbulence transition of pipes in the directed percolation universality class. In the second part of the talk, I consider the problem of turbulent drag in pipes, as studied originally by Nikuradze and more recently at OIST and Bordeaux. I will discuss how simple theory and experiments suggest a relation between turbulent dissipation and velocity fluctuations, an area that is in much need of more detailed and systematic study.

If there is time, I will discuss very briefly the widely unappreciated role of thermal fluctuations in the far dissipation range of turbulence, and using shell models, show how these are amplified and propagated to large scales by spontaneous stochasticity, reaching the integral scale eddies in just a few eddy turnover times.

This work was partially supported by grants from the Simons Foundation through Targeted Grant “Revisiting the Turbulence Problem Using Statistical Mechanics" (Grant Nos. 663054 (G.E.), 662985 (N.G.) and 662960 (BH)).

Recent rigorous progress on Stochastic Partial Differential Equations opened the door to a new investigation of the classical topic of Boussinesq eddy viscosity hypothesis. In the case of classical white noise description of small scale turbulence, Boussinesq hypothesis emerges as a mean field limit, but establishing rigorously the validity of the mean field approximation is very difficult and, in general, presumably false. We show a few simple cases where we may give a positive answer.

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, 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 phenomenon dubbed spontaneous stochasticity. 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.

The phenomenon of active turbulence, a complex organization of matter driven at the scale of its constituent agents, is confounding. In particular, analogies with high Reynolds number, inertial (Kolmogorov) turbulence have remained moot. The lack of scale separation in these low Reynolds active flows breaks away from the familiar notions of the energy cascade and approximate scale-invariance of inertial turbulence. Now, using a generalized hydrodynamic model developed for bacterial turbulence, we provide compelling analytical and numerical evidence that, beyond a critical drive, active turbulence indeed attains universality akin to inertial turbulence.

In this talk I will discuss necessary and sufficient conditions on the regularity of the external force for energy balance to hold for weak solutions of the 2D incompressible Euler equations. The problem is motivated by turbulence modeling and the result should be contrasted with the existence of wild solutions in 3D.

This is joint work with Fabian Jin (ETH-Zurich), Samuel Lanthaler (CalTech), Milton C Lopes Filho (UFRJ) and Siddhartha Mishra (ETH-Zurich).

In this talk I will discuss several problems related to the random forced Burgers equation.
One of them is to describe universal properties of points associated with the global shocks. I will also discuss how the above problem is connected with the KPZ phenomenon.

I shall describe multi-mode correlations in wave turbulence and shell models.

Specifically, we discuss whether steady states can be isolated, wandering for solutions starting nearby certain steady states, singularity formation at infinite time, and finally some results/conjectures on the infinite-time limit near and far from equilibrium.

We will discuss some old and new results concerning the long-time behavior of solutions to the two-dimensional incompressible Euler equations. Specifically, we discuss whether steady states can be isolated, wandering for solutions starting nearby certain steady states, singularity formation at infinite time, and finally some results/conjectures on the infinite-time limit near and far from equilibrium.

In this talk we will report the results of a computational investigation of a new blow- up criterion for the 3D incompressible Euler equations, which does not rely on the seminal Beale-Kato Majda blow-up criterion. This criterion is based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler-Voigt equations. The new criterion is only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.

Notably, the Voigt inviscid regularization approach applies equally to other hydrody- namical models, and it can be shown that its solutions converge, as the regularization parameter α → 0, to the corresponding solutions of the underlying hydrodynamical model for as long as the latter exist.

In a viscous fluid, the energy dissipation is the signature of the breaking of the time-reversal symmetry (hereafter TSB) t-> -t, u-> -u, where u is the velocity. This symmetry of the Navier-Stokes equations is explicitly broken by viscosity. Yet, in the limit of large Reynolds numbers, when flow becomes turbulent, the non-dimensional energy dissipation per unit mass becomes independent of the viscosity, meaning that the time-reversal symmetry is spontaneously broken. Natural open questions related to such observation are: what is the mechanism of this spontaneous symmetry breaking? Can we associate the resulting time irreversibility to dynamical processes occurring in the flow? Can we devise tools to locally measure this time irreversibility?

In this talk, I first show that the TSB is indeed akin to a spontaneous phase transition in the Reversible Navier-Stokes equations, a modification of the Navier-Stokes equation suggested by G. Gallavotti to ensure energy conservation and relevance of statistical physics interpretation. I then discuss the mechanism of the TSB in Navier-Stokes via quasi-singularities that create a non-viscous dissipation and exhibit the tools to track them. I apply them to time and space-resolved Lagrangian and Eulerian velocity measurements in a turbulent von Karman flow. I finally compare Eulerian and Lagrangian signatures of irreversibility, and link them with peculiar properties of the local velocity field or trajectories.