2022 Simons Collaboration on Wave Turbulence Annual Meeting

This year’s annual meeting was a great success. The two-day meeting at the Simons Foundation was preceded by a three-day PIs and postdocs meeting held at the Courant Institute.  The theme for the Courant meeting this year was to showcase the evolving work of the junior scientists, which are by far the most important human resources component of the collaboration.  Impressive progress on diverse  theoretical, experimental and computational aspects of wave turbulence was displayed, showing us that the young scientists are pushing the boundary of what is possible.  The subsequent two-day meeting at the Foundation combined larger-scale overview talks from several current PIs with presentations from new faculty in adjoining topical fields.  The overall success of the meeting was summarized well by one of the guest speakers from Princeton, who said that she had never been at any meeting that was as interactive and interdisciplinary as this one.

The week-long meeting in NY showcased the emergence of a new community out of this collaboration, a community that boosted enormously the scientific productivity in the field of wave turbulence, from experiments and numerics to applied and pure mathematics. The meeting clearly demonstrated that this community of mathematicians and physicists do not only understand and appreciate each other’s work, but they are able to establish fruitful exchanges of ideas and genuine collaboration across fields.

A few highlights of the collaborations achievements that were presented during the meeting are:

1. The rigorous derivation of the PDF for the wave turbulence theory for the nonlinear Schrödinger equation. This is a huge theoretical step forward. It is the analogue of Lanford’s theorem for the Boltzmann kinetic equation for a dilute gas.
2. For the first time, using numerical simulations (aided by supercomputers) of the non-linear Schrödinger equation and the associated wave-kinetic equation, we tested wave turbulence theory predictions for the long-time (several kinetic times) evolution of spectra.
3. We presented the resolution of a longstanding puzzle about a well-studied test model for wave turbulence (the MMT model), which in many cases did not show the inertial range scalings expected from kinetic wave theory. Ultimately, this is shown to be due to a lack of wave frequency bandwidth in the inertial range, which went undetected for a long time because frequency bandwidth is often different from wavenumber bandwidth for dispersive waves. This laid to rest a decades-long theoretical controversy in the field.
4. A new experimental setup was built to study wave turbulence of inertial waves. This led to the first observation of the inertial wave turbulence regime in a rotating fluid, which is a key ingredient of geo and astrophysical flows.
5. Experimental progress was made on turbulence in stratified fluids. These experiments were setup to understand turbulent transport and dissipation in the ocean interior. The outcome of these experiments are consistent with oceanic observations of the Garrett & Munk spectrum and suggests that it may be associated to a regime of strongly non linear wave turbulence rather than weak turbulence.

The Courant part of the meeting included three discussion periods. These discussions were,

1. The Wave Kinetic equations: The mathematical challenges of understanding forcing, damping, and anisotropy. This discussion was led by Sergey Nazarenko.
2. Wave Turbulence models that are used in the field: where do they come from and how dependable are they. This discussion was led by Miguel Onorato,
3. The use of PINN in PDEs and what they can tell us about wave turbulence. This discussion was led by Tristan Buckmaster, Ching-Yao Lai, and Eric Vanden-Eijnden.

The postdoc presentations at Courant can be found at: https://cims.nyu.edu/wave-turbulence/event/courant-pis-workshop/

Oliver Bühler
Pierre Philipe Cortet
Giovanni Dematteis
Gregory Eyink
Nigel Goldenfeld
Giorgio Krstulovic
Nicolas Mordant
Laure Saint Raymond

Nicolas Mordant
Université de Grenoble

Experimental investigation of stratified turbulence: Weak turbulence and beyond

Wave turbulence of internal waves is a major part of the global ocean dynamics as it contributes to large extent to dissipation of kinetic energy and mixing of temperature and salt by establishing a direct cascade in length scales. Although measurements of turbulence in the ocean interior, the famous Garrett–Munk spectrum, are attributed to internal inertial-gravity wave turbulence, clear evidence of such turbulence in the laboratory was missing. The large-scale Coriolis facility in Grenoble, France, is dedicated to geophysical fluid dynamics investigations of stratified and/or rotating flows. It consists of a 13-meter diameter pool that can be filled with water with a vertical density gradient (using varying salt concentration) and that can rotate to mimic the action of the rotation of the Earth. We use this facility to obtain a state of stratified turbulence forced by large-scale waves. We report a state of weak wave turbulence of internal waves at intermediate scales. The global flow is more complex, as very large scale vortical structures can be observed as well, together with small scale strongly nonlinear turbulence resulting from wave overturning. This complex flow is consistent with the expected phenomenology of such gravity dominated turbulence as observed in the ocean.
 

Pierre-Philippe Cortet
Laboratoire FAST, CNRS – Université Paris-Saclay

(Wave) Turbulence in rotating fluids

The influence of a global rotation on hydrodynamic turbulence is a key ingredient of geo and astrophysical flows. A consequence of rotation is the emergence of a specific class of waves, called inertial waves, which propagate in the volume of the fluid. Within rotating turbulence, these waves and the classical eddy structures of fluid dynamics can be entangled in different ways, leading to several possible regimes: among these, the wave turbulence regime. The theory of wave turbulence applied to the case of inertial waves led to analytical predictions in the early 2000s [1]. Since then, various attempts to discover experimentally this regime have been carried out, with limited success.

In this context, we recently built an original experimental setup that led to the first observation of the inertial wave turbulence regime in a rotating fluid [2]. I will show that the features of our experimental wave turbulence are in quantitative agreement with the theory. This achievement was made possible by the prior discovery of an instability affecting inertial waves, which we named the “quartetic instability.” I will show that the understanding of how to inhibit this instability has been crucial for the observation of the wave turbulence regime [3].

[1] S. Galtier, Weak inertial-wave turbulence theory, Physical Review E, 68, 015301(R) (2003).

[2] E. Monsalve, M. Brunet, B. Gallet & P.-P. Cortet, Quantitative Experimental Observation of Weak Inertial-Wave Turbulence, Physical Review Letters, 125, 254502 (2020).

[3] M. Brunet, B. Gallet & P.-P. Cortet, Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability, Physical Review Letters, 124, 124501 (2020).

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Giorgio Krstulovic
Université Côte d’Azur

Turbulent steady states in the nonlinear Schrodinger equation

The nonlinear Schrödinger (NLS) equation, also known as the Gross–Pitaevskii equation, is one of the most common equations in physics. Its applications go from the propagation of light in nonlinear media to the description of gravity waves and Bose–Einstein condensates. In general, the NLS equation describes the evolution of nonlinear waves. Such waves interact and transfer energy and other invariants along scales in a cascade process. This phenomenon is known as wave turbulence and is described by the (weak) wave turbulence theory (WWT). One of the most significant achievements of WWT is the complete analytical characterization of turbulent steady states obtained in the long time limit when the system contains forcing and dissipative terms acting on well-separated scales.

In the first part of my talk, I will present recent theoretical developments on the steady-state solutions of the 3D NLS equation in the four-wave regime. Those new predictions are validated using high-resolution numerical simulations of the NLS equation and its associated wave kinetic equation. In the second part, I will address the three-wave interaction regime of the NLS equation in which waves become non-dispersive. I will start by reviewing some of the mathematical issues of the WWT for acoustic waves in 2D and 3D. Then, I will present a new theory for 2D NLS in the acoustic regime, which is in excellent agreement with numerical simulations without adjustable parameters.

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Gregory Eyink
Johns Hopkins University

The Josephson–Anderson relation and the origin of turbulent drag

The Josephson–Anderson relation provides the modern understanding for energy dissipation in quantum superfluids and superconductors, relating drag to motion of quantized vortex lines across the background potential superflow. The same relation has been recently shown to apply widely to classical fluid flows described by the incompressible Navier–Stokes equation. Here it connects dissipation with flux of continuously distributed vorticity across the streamlines of the smooth potential Euler solution. We shall concisely review these concepts and explain how they give a new resolution of the classical d’Alembert paradox, connecting it with Onsager’s theory of high Reynolds number turbulence. The theory explains experimental puzzles about the wall conditions of solid bodies necessary for anomalous dissipation. The Josephson–Anderson relation is valid, however, at any Reynolds number and explains the origin of drag in flows ranging from low-Reynolds-number Stokes flow to transitional and fully developed turbulence. As such, it provides a sharp, precise tool with which to address the theoretical and practical issue of drag reduction. We shall review our initial efforts to apply the Josephson-Anderson relation to turbulent channel flow and to resolve the 75-year-old open problem of drag reduction by polymer additives.

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Laure Saint-Raymond
Institut des Hautes Études Scientifiques  – Université Paris-Saclay

Solutions of kinetic wave equations with constant fluxes

In this talk, I will present a new approach to Kolmogorov–Zakharov solutions of wave kinetic equations based on a work with F. Golse and M. Escobedo. This opens interesting perspectives regarding the anisotropic case.

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Oliver Bühler
Courant Institute of Mathematical Sciences – New York University

Modified inertial range power laws due to finite spectral bandwidth: Idealized modeling results and implications for waves in the ocean

Predictions for power laws from wave-kinetic theory are based on infinite spectral bandwidth, with arbitrarily large-scale separations assumed to hold between forcing and dissipation scales, for example. We show by way of very high resolution numerical simulations of an idealized model that large but finite spectral bandwidth has a strong and predictable impact on the observed power laws. This has practical implications for the interpretation of experimental results and field observations, and the relevance of this for the study of realistic ocean waves is discussed.

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Nigel Goldenfeld
University of California, San Diego

Stochasticity in transitional and fully developed turbulence

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 behavior near the laminar-turbulence transition, while the second controls the behavior 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 localized 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 discuss more 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.

Work performed in collaboration with: Hong-Yan Shih (UIUC/Academia Sinica, Taiwan), Xueying Wang (University of Illinois at Urbana-Champaign), Tsung-Lin Hsieh (UIUC/Princeton), Björn Hof (Institute of Science and Technology, Austria), Joachim Mathieson (Niels Bohr Institute, Denmark), Grégoire Lemoult (University of Le- Havre, France), Vasudevan Mukund (Institute of Science and Technology, Austria), Gaute Linga (Niels Bohr Institute), Dmytro Bandak (University of Illinois at Urbana-Champaign), Gregory Eyink (Johns Hopkins University) and Alexei Mailybaev (IMPA, Brazil).

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

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Giovanni Dematteis
with Yuri Lvov and Kurt Polzin
Rensselaer Polytechnic Institute

The Role of Internal Wave Turbulance
in the Oceanic Energy Pathways

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