Monday, 16 January 2023

I will talk about two problems: (1) A single thread in a time-periodic linear shear flow at zero Reynolds number. The thread shows spatio-temporally chaotic behaviour and the fluid stirred by the thread shows mixing. (2) Turbulence in polymeric turbulence with high Deborah number. We find that for small wavenumbers , the kinetic energy spectrum shows Kolmogorov--like behavior which crosses over at a larger to a novel, elastic scaling regime, , with , providing support to and extending the analysis of recent experimental results [Yi-Bao Zhang et. al., Science Advances 7, eabd3525 (2021)]. We uncover the mechanism of elastic scaling by studying the contribution of the polymers to the flux of kinetic energy through scales.The contribution can be decomposed into two parts: one, expected, increase in effective viscous dissipation of the flow and two, a purely elastic contribution that dominates over the nonlinear flux in the range of over which the elastic scaling is observed. The multiscale balance between the two fluxes determines the crossover wavenumber which, intriguingly, depends non-monotically on the Deborah number. Consistent with this picture, real space structure functions also show two scaling ranges, with intermittency present in both of them in equal measure.

The basic phenomenology of isotropic turbulence will be discussed with the focus primarily on the scaling laws for the velocity correlation functions. We will try to focus on the pitfalls and the efforts made to avoid them as much as possible.

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.

This proposes to be a pedagogical lecture on convex integration applied to fluid dynamics. I will revisit in some detail the original construction by De Lellis and Székelyhidi from 2009. Time permitting, in a second lecture if need be, I will discuss the historical progression in the context of the flexible side of the Onsager conjecture. I will also discuss the rigidity side of the Onsager conjecture and the particularities of 2D flows.

Tuesday, 17 January 2023

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.

We numerically study free decay of turbulence when the initial energy spectrum is localized around a high wavenumber k_c. We show that the energy spectrum in the wavenumber range, k < k_c, depelopes as k^3 and that the decay of the total kinetic energy takes a peculiar form with a possibly logarithmic correction.

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.

Wednesday, 18 January 2023

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.

Incompressible Euler equation has zero viscosity and no external forcing, hence, it can be treated as an isolated system. This equation has been studied widely. In particular, 3D Euler turbulence reaches equilibrium asymptotically, and follows the predictions of Kraichnan and Lee. This is the usual thermalization process.

However, there are surprises in 2D Euler turbulence. For ordered initial condition, two-dimensional (2D) Euler turbulence is out of equilibrium and it exhibits evolution from disorder to order, even though the system is an isolated one. The energy spectra for these cases deviate from Kraichnan's predictions. The small wavenumber modes exhibit nonzero energy energy transfers, hence, detailed balance is broken.

Since the constant thermodynamic entropy of Euler turbulence cannot capture the variable order of the flow, we propose “hydrodynamic entropy” for describing the disorder in Euler turbulence. The hydrodynamic entropy decreases with time for a significant period.

Ref: M. K. Verma and S. Chatterjee, Hydrodynamic entropy and emergence of order in two-dimensional Euler turbulence, Phys. Rev. Fluids, 7, 114608 (2022)); arXiv:2210.06445

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)

I will discuss measure of mixing in incompressible flows and give examples of flows that achieve the optimal rate of mixing. I will then give two applications, one to complete, instantaneous loss of regularity in linear transport equations, the other to enhanced dissipation and consequences. In particular, I will show that adding a linear advection term leads to global existence for the 2D Kuramoto-Sivashinsky equation, a model of front propagation in combustion.

Thursday, 19 January 2023

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 kinetic energy spectrum of turbulence in rotating fluids and in stratified fluids show a crossover from the Kolmogorov 5/3 law to a different spectrum. We believe these two crossovers reflect two different kinds of physical effect. In particular we would like to discuss why the stratified fluid crossover has been so difficult to observe.

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.

We develop a Fourier-Chebyshev pseudospectral direct numerical simulation (DNS) to examine a potentially singular solution of the radially bounded, three-dimensional (3D), axisymmetric Euler equations [cf., G. Luo and T.Y. Hou, Proc. Natl. Acad. Sci. USA 111, 12968 (2014)]. We demonstrate the following: (a) the time of singularity is preceded, in any spectrally truncated DNS, by the formation of oscillatory structures called tygers, first investigated in the one-dimensional (1D) inviscid Burgers and two-dimensional (2D) Euler equations by Ray, Frisch, Nazarenko, and Matsumoto; (b) the analyticity-strip method can be generalized to obtain an estimate for the (potential) singularity time. We also examine early-time resonances and singularities, which have been studied recently in the 1D inviscid Burgers equation by Rampf, Frisch, Hahn.

This work has been done with Sai Swetha Venkata Kolluru and Puneet Sharma [Ref.: PHYSICAL REVIEW E 105, 065107 (2022)].

Friday, 20 January 2023

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.

Monday, 23 January 2023

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.

Several open problems in fluid dynamics are related to the multi-scale nature of turbulence and the effects of boundary conditions. To overcome the limited resolution of direct numerical simulations, simplified toy-models are commonly employed in the study of large spatial range flows. In the construction of such models, one modifies equations of motion, preserving only certain parts believed to be important. In this talk, we propose a different approach. Instead of simplifying equations, one introduces a simplified configuration space: we define velocity fields and boundary effects on multi-dimensional logarithmic lattices in Fourier space. Operations upon these variables are provided in a rigorous mathematical framework, so equations of motion are written in their exact original form. As a consequence, the resulting models preserve the same symmetry groups, inviscid invariants and regularity properties. The strong reduction in degrees of freedom allows computational simulations of incredibly large spatial ranges. Using the new simplified models, we address two important open problems: the finite time singularities in ideal flow and the vanishing viscosity limit in the presence of solid boundaries. We observe strong robustness of the chaotic blowup scenario in the three-dimensional incompressible Euler equations and promising results towards the investigation of potentially singular behavior close to solid boundaries. This is a joint work with Alexei Mailybaev.

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.

I will give an overview of the results which can be obtained using the functional renormalisation group (FRG) on the statistical properties of homogeneous and isotropic turbulence. I will describe the RG fixed-point corresponding to stationary turbulence with large-scale forcing, and I will present some analytical results on the space-time dependence of generic n-point correlation functions of the turbulent velocity. I will compare these predictions with available results from direct numerical simulations and experiments. I will finally discuss the FRG analysis of shell models of 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).

Tuesday, 24 January 2023

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.

Polymers in a turbulent flow are stretched out by the fluctuating velocity gradient and exhibit a broad distribution of extensions R; the stationary probability distribution function of R has a power-law tail with an exponent that increases with the Weissenberg number Wi, a nondimensional measure of polymer elasticity. This talk addresses the following questions:

(i) What is the role of the non-Gaussian statistics of the turbulent velocity gradient on polymer stretching?

(ii) How does the probability distribution function of R evolve to its asymptotic stationary form?

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.

Wednesday, 25 January 2023

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.

We chart a singular landscape in the temporal domain of the inviscid Burgers equation in one space dimension for single-mode initial conditions. These so far undetected complex singularities are arranged in an eye shape entered around the origin in time. Interestingly, since the eye is squashed along the imaginary time axis, complex-time singularities can become physically relevant at times well before the first real singularity—the pre-shock. Indeed, employing a time-Taylor representation for the velocity around t=0, loss of convergence occurs roughly at 2/3 of the pre-shock time for the considered single- and multi-mode models. Furthermore, the loss of convergence is accompanied with the appearance of initially localized resonant behaviour which, as we claim, is a temporal manifestation of the so-called tyger phenomenon, reported in Galerkin-truncated implementations of inviscid fluids [Ray et al., Phys. Rev. E 84, 016301 (2011)]. We support our findings of such early-time-tygers by two complementary and independent means, namely by an asymptotic analysis of the time-Taylor series for the velocity, as well as by a novel singularity theory that employs Lagrangian coordinates.

Finally, we apply two methods that reduce the amplitude of early-time tygers, one is tyger purging which removes large Fourier modes from the velocity, and is a variant of a known procedure in the literature. The other method realizes an iterative UV completion, which, most interestingly, iteratively restores the conservation of energy once the Taylor series for the velocity diverges. Our techniques are straightforwardly adapted to higher dimensions and/or applied to other equations of hydrodynamics.

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.

In this talk we study the Hamiltonian dynamics of charged particles subject to a non self-consistent stochastic electric field, when the plasma is in the so-called weak

turbulent regime. We show that the asymptotic limit of the Vlasov equation is a diffusion equation in the velocity space, but homogeneous in the physical space.

We obtain a diffusion matrix, quadratic with respect to the electric field, which can be related to the diffusion matrix of the resonance broadening theory and of the quasilinear theory,

depending on whether the typical autocorrelation time of particles is finite or not. In the self-consistent deterministic case, using a convenient scaling, we show that the asymptotic distribution function is homogenized in the space variables, while the electric field converges weakly to zero. We also show that the lack of compactness in time for the electric field is necessary to obtain a genuine diffusion limit. By contrast, time compactness property leads to a “cheap” version of the Landau damping: the electric field converges strongly to zero, implying the vanishing of the diffusion matrix, while the distribution function relaxes, in a weak topology,

towards a spatially homogeneous stationary solution of the Vlasov-Poisson system. Finally, in the self-consistent case, without scaling, we prove the validity of the standard quasilinear approximation of the Vlasov-Poisson by a diffusion equation.