2023 Simons Collaboration on Wave Turbulence Annual Meeting
Organizer:
Jalal Shatah, New York University
Past Meetings:
Speakers:
Jacob Bedrosian
Lydia Bourouiba, MIT
Sebastien Galtier, Université ParisSaclay
Pierre Germain, Imperial College London
Sergey Nazarenko, Institute de Physique de Nice
Yulin Pan, University of Michigan
Natasa Pavlovic, University of Texas at Austin
Laure Zanna, Courant Institute, New York University
Meeting Goals:
The Simons Collaboration on Wave Turbulence has brought together scientists and mathematicians working in disparate areas, but with a common goal of understanding, verifying, and constructing a rigorous theory of wave turbulence. They have been meeting, discussing, and collaborating to achieve their goals. Scientists and mathematicians working within this collaboration have made substantial progress during the past four years. They have conducted experiments and have reported the first observation of weak inertial wave turbulence. In other experiments they have observed weak turbulence of internal waves.
Mathematically they have provided a rigorous proof of the validity of the wave kinetic equation, which is a central component of the theory. They have also analytically obtained solutions of these kinetic equations.
The Simons Collaboration on Wave Turbulence Annual Meeting provided the latest on our collaborative research, as well as highlighting important external research from related fields, with the scope of further developing the collaborative research approach started by the PIs of the grant.

Meeting Report
The 2023 annual meeting of the Collaboration on Wave Turbulence took place at the Simons Foundation on November 30th and December 1st. It was preceded by a threeday workshop on the same topic held at the Courant Institute, which focused on highlighting the work of the junior collaborators. Progress on a multitude of research directions were reported on, ranging from substantial breakthroughs in the theoretical aspects to novel applications in geophysical and other systems, which formed the basis for detailed conversations and lively discussions during the workshop. This set the scene for the annual meeting, which featured overview talks on emerging and related topics by senior members of the collaboration and invited guest speakers. The broad spectrum of talks included applications of machine learning to ocean turbulence modeling, wave turbulence in the solar wind, progress in surface wave turbulence, and the everincreasing scope of detailed solutions to the kinetic equations. The meeting closed with a talk on rotating turbulence simulations and fluid fragmentation in turbulent sprays of complex fluid systems.

Agenda
THURSDAY, NOVEMBER 30
9:30 AM Laure Zanna  (Machine) Learning Multiscale Ocean Interaction 11:00 AM Pierre Germain  The analysis of the kinetic wave equation 1:00 PM Sebastien Galtier  Wave turbulence in the solar wind 2:30 PM Sergey Nazarenko  Universal scalings in evolving and stationary wave turbulence 4:00 PM Nataša Pavlović  Two tales of a rigorous Derivation of the Hamiltonian Structure Friday
9:30 AM Jacob Bedrossian  Nonlinear dynamics in stochastic system 11:00 AM Yulin Pan  Topics in surface gravity wave and MMT turbulence 1:00 PM Lydia Bourouiba  Unsteadiness: instabilities, turbulence, and fragmentation 
Abstracts & Slides
Laure Zanna
(Machine) Learning Multiscale Ocean Interaction
Ocean mesoscale eddies (horizontal scale of 10 km100 km) are critical in setting the global ocean circulation (scales of 1000’s km) and for mixing of momentum and tracers in the ocean. Mesoscale eddies are a key player in the ocean energy cycle, as they form the bulk of the kinetic energy and can backscatter energy to the mean flow through an inverse energy cascade. Yet, most climate models only partially resolve them, and we must therefore find approximations to represent their effects on the largescale flow. Here, we will present results using data from numerical simulations and machine learning techniques to learn equations that represent the effects of mesoscale eddies on the largescale flow. We will discuss the physical and mathematical properties of the equations learned, and examine their role in improving coarseresolution simulations of the ocean.
Pierre Germain
The analysis of the kinetic wave equation
View Slides (PDF)The kinetic wave equation is at the heart of wave turbulence theory. As far as rigorous mathematics go, impressive progress has been made on its derivation but very little is known on its dynamics, or even more basic questions such as existence and uniqueness. A longterm aim is to build up the theory towards the application to turbulent spectra, ie KolmogorovZakharov spectra and their stability. I will review the state of the art and describe open problems.
Sebastien Galtier
Wave turbulence in the solar wind
View Slides (PDF)The Sun can be seen as a gigantic natural wind tunnel, producing a wind at speeds of 400 to 800km/s. This wind is made up of particles (mainly electrons and protons), and fields (magnetic and electric) that can be precisely measured by ESA and NASA spacecraft. The two main properties of the solar wind are waves and turbulence, so it is natural to think of wave turbulence as a relevant regime for describing this medium.
In this presentation, I will review the main properties of the solar wind observed from in situ measurements, and then present the models used, which are based on plasma physics. I will present the main properties of weak wave turbulence theory in compressible magnetohydrodynamics (MHD) and electron MHD, which are anisotropic problems. This will be illustrated by direct numerical simulations. A comparison will eventually be made with in situ data. In particular, I will introduce the notion of collisionless wave turbulence, particularly useful for the solar wind, which is a dilute plasma. Finally, I will show the link between this topic and rapidly rotating turbulence.
Sergey Nazarenko
Universal scalings in evolving and stationary wave turbulence
Using the NonlinearSchrodinger (NLS) equation as a master model, I will present analytical and numerical results concerning several types of universal scaling regimes in wave turbulence. In stationary turbulence, these will be concerned with a revised theory of the famous KolmogorovZakharov (KZ) spectra, both the direct and the inverse cascades. In evolving wave turbulence, the universal scalings manifest themselves in selfsimilar asymptotics (referred to as “nonthermal fixed points” in some recent papers). The latter behaviour comes in three flavours: selfsimilarity of the first, second and third kinds respectively. The selfsimilarity of the first kind appears as a large time asymptotic of the spectrum propagating toward high frequencies. Its scaling is fully determined by energy conservation. The selfsimilarity of the second kind appears as a finite time blowup of the wavekinetic equation (WKE) at the zero frequency: it is related to a physical phenomenon of the BoseEinstein condensation. The scaling of this selfsimilarity is nontrivial: it cannot be found from conservation laws, and it is determined by solving a “nonlinear eigenvalue problem”. The selfsimilarity of the third kind appears in the forceddissipated settings as a final stage of transition to the KZ spectrum and it takes the form of a frequencyspace wave reflected from the lowfrequency dissipative range. Its scaling is inherited from the previous (blowup) selfsimilar stage. I will present numerical results testing the analytical predictions arising from simulations of both the WKE and the 3D NLS equation.
Nataša Pavlović
Two tales of a rigorous Derivation of the Hamiltonian Structure
View Slides (PDF)Many mathematical works have focused on understanding the manner in which the dynamics of a nonlinear equation arises as an effective equation.
 For example, the cubic nonlinear Schrodinger equation (NLS) is an effective equation for a system of N bosons interacting pairwise via a delta or approximate delta potential. In this talk, we will advance a new perspective on deriving an effective equation, which focuses on structure. In particular, we will show how the Hamiltonian structure for the cubic NLS in any dimension arises from corresponding structure at the Nparticle level.
 On the other hand, the Vlasov equation in any spatial dimension has long been known to be an infinitedimensional Hamiltonian system whose bracket structure is of LiePoisson type. In parallel, it is classical that the Vlasov equation is a meanfield limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the manybody problem. This work settles a question of Marsden, Morrison, and Weinstein on providing a “statistical basis”’ for the bracket structure of the Vlasov equation.
Jacob Bedrosian
Nonlinear dynamics in stochastic systems
View Slides (PDF)In this overview talk we discuss several results regarding the longtime dynamics of dampeddriven Hamiltonian stochastic systems in the limit of weak damping. Examples of interest include dampeddriven cubic NLS or the 2D or 3D NavierStokes equations. In infinite dimensional systems one generally observes turbulent behavior and relatively little is known aside from some basic results characterizing dissipation anomalies. However, for high, but finite, dimensional approximations to such systems, such as Galerkin truncations of the NavierStokes equations, one can now say a lot more about the dynamics through a combination of ideas in random dynamical systems and PDE methods applied to stochastic dynamics. We discuss some of these results regarding quantitative estimates on positive Lyapunov exponents and estimates of the attractor dimension (i.e. almostsure exponential sensitivity to initial conditions), sharp estimates on the spectral gaps of the Markov semigroups (i.e. of the time required for timeaveraged statistics to be close to the ensemble averaged statistics), and some results regarding nonlinear energy transfer (i.e. if one leaves a subspace undamped, under what conditions does the nonlinearity transfer energy to the damped modes and give onaverage bounded trajectories vs unbounded growth due to energy piling up in the undamped subspace). This talk will mention joint works with Alex Blumenthal, Michele Coti Zelati, Sam PunshonSmith, Kyle Liss, and Franziska Weber.
Yulin Pan
Topics in surface gravity wave and MMT turbulence
View Slides (PDF)In this presentation, I will discuss several selected topics in surface gravity wave and MMT turbulence.
 Nonlocal surface gravity wave turbulence in the presence of a condensate: We develop new theory to explain the inversecascade spectral slope observed in experiments and simulations that differs from the KZ solution. The theory considers the condensate at large scales and describes the spectral evolution by a diffusion equation, with a stationary solution consistent with the observations. (joint work with Alexander Korotkevich, Sergey Nazarenko and Jalal Shatah)
 Role of threewave quasiresonances in surface gravity wave turbulence: In the standard derivation of the Hasselmann’s kinetic equation, the quadratic terms (threewave interactions) are removed from a Lee transformation which is supposed to be valid only at small nonlinearity level. We numerically study the role of these threewave quasiresonances at finite nonlinearity level and discuss the implication to Zakharov equation and kinetic equation. (joint work with Zhou Zhang)
 MMT turbulence at kinetic limit: We numerically study wave turbulence at the kinetic limit (small nonlinearity and large domain size simultaneously) in the context of 1D MMT equation, and examine the relation of the numerically obtained spectrum with the KZ solution focusing on the Kolmogorov constant (joint work with Alex Hrabski).
Lydia Bourouiba
Unsteadiness: instabilities, turbulence, and fragmentationUnsteady fragmentation of fluids into droplets is ubiquitous in industrial, energy, environmental, and healthrelated processes. More broadly, in a range of geophysical and health flow systems, unsteadiness contributes in shaping, often in unexpected ways, instabilities, emerging patterns, mixing, or final fragment sizes from multiscale fluid fragmentation processes, for example. We will discuss illustrative examples for which the theoretical or experimental frameworks are adapted to gain insights on the rich effects emerging from unsteadiness of complex fluid systems.