The aim of this Summer School is to introduce graduate students to current questions of interest in Wave Turbulence. It will emphasize the interplay between physical and mathematical points of view, and will present different approaches to wave turbulence.

Due to sanitary conditions, it will be held online; the zoom link is

https://nyu.zoom.us/j/91998747859

and the schedule is as follows – All time indications are given for Paris!!!

Monday, Jul 12

15:00-16:00 (Paris time) Zaher Hani (U. Michigan) Some mathematical questions in wave turbulence

Abstract: Wave turbulence is the theory of nonequilibrium statistical physics for nonlinear waves. This physical theory inspired and motivated several mathematical endeavors over the past few years, particularly in terms of a) proving some of its conjectures, such as the  energy cascade phenomenon, and b) rigorously deriving the fundamental equations of wave turbulence. We will discuss such questions and some recent progress on them.

16:30-17:30 (Paris time) Herbert Spohn (TU Muenchen) Hydrodynamic equations for the discrete nonlinear Schroedinger equation in one space dimension.

Abstract: NLS in one dimension is an integrable PDE and the respective hydrodynamic description has to include all conserved fields. This goal can be achieved for the defocusing Ablowitz-Ladik discretization. We explain the construction and the resulting infinite set of coupled hyperbolic conservation laws. There is a curious connection to the circular unitary ensemble of random matrix theory.

Tuesday, Jul 13 – 0-homogeneous operators and wave turbulence in stratified flows

Laure Saint-Raymond (ENS de Lyon), Thierry Dauxois (CNRS and ENS de Lyon), Laurent Chevillard (CNRS and ENS de Lyon)

Abstract:

15:00 – 16:00 (Paris time) The first part of the afternoon will be devoted to internal waves and linear propagators homogeneous of degree 0. T. Dauxois will explain the generation of internal waves in stratified fluids, their experimental investigation and the phenomenon of wave attractors. L. Saint-Raymond will then introduce a mathematical setting to describe this phenomenology, which is actually very generic for linear propagators homogeneous of degree 0: because energy surfaces are unbounded in the frequency space, the dynamics transfers energy towards small scales along attractors. Based on this construction, L. Chevillard will show that, when forced by a stochastic smooth term, this peculiar operator is able to generate a random field that shares several statistical properties of spatial white noises.

16:30 – 17:30 (Paris time) The second part of the afternoon will then focus on nonlinear effects. L. Chevillard will go on with the modeling of the cascading process that eventually takes place in the statistically stationary regime of forced fluid turbulence. He will show how to introduce the fractional regularity demanded by the phenomenology of turbulence. Further intermittent, i.e. multifractal, corrections to this self-similar picture can be included by considering more general and nonlinear evolutions. Numerical simulations are then presented. T. Dauxois will carry on with the presentation of experimental observations obtained in the nonlinear regime, by considering a larger amplitude forcing term. To interpret the results, he will present the so-called triadic interaction, based on the interaction of three waves, which is shown to reproduce key experimental observations such as the appearance of additional harmonics. Spectra of kinetic energy and related power-law behaviors will be presented, and interpreted in the framework of wave turbulence. L. Saint Raymond will finally conclude this lecture by presenting some open problems from both a mathematical and physical point of view.

Wednesday, Jul 14 – National Holiday! Day off

Thursday, Jul 15 

15:00 – 16:00 (Paris time) Nicolas Mordant (U. Grenoble Alpes) Experimental evidence of weakly non linear wave turbulence

Abstract: I will present experimental evidence of states of weakly non linear wave turbulence in various physical systems such as capillary waves, surface gravity waves, internal gravity waves or elastic waves. In all these systems, wave turbulence develops in some range of scales. This observed turbulence is in qualitative agreement with the phenomenology of Weak Turbulence Theory but most often the observed Fourier spectra are not in quantitative agreement with the theory. I will discuss the impact of various features that can not really be avoided in experiments, including wide-band dissipation, finite size or finite strength of non linearity with the development of coherent structures or the excitation of other features beyond waves.

16:30 – 17:30 (Paris time) Gennady El (Northumbria University) Integrable turbulence and soliton gas in dispersive hydrodynamics

Friday, Jul 16 – Boundary layers and transition to turbulence

15:00 – 16:00 (Paris time) Anne-Laure Dalibard (U. Sorbonne) Recent advances in fluid boundary layer theory

Abstract: The Prandtl equation describes the behavior of a fluid with small viscosity in the vicinity of an obstacle. In the past few years, there has been important theoretical progress on the description of solutions of this equation: well-posedness and ill-posedness results for the time-dependent Prandtl equation and its variants, separation in time-dependent and stationary settings… In this talk, I will focus on the non-stationary equation, for which instabilities may be seen as a form of turbulent behavior. I will make a review of recent results, and I will present some potential next steps.

16:30 – 17:30 (Paris time) Bjorn Hof (IST Austria) The onset of turbulence in shear flows – a matter of life and death

Abstract: In pipe, channel and Couette flow turbulence arises despite the linear stability of the laminar state and the transition is caused by finite amplitude perturbations. Until recently and despite numerous experimental and theoretical studies it had not been possible to define a critical point, let alone to explain the nature of this transition. I will explain in the following that the onset of sustained turbulence is a nonequilibrium phase transition. The critical point can be measured by resolving the extremely long time scales of the underlying growth and decay processes. Exploiting analogies to statistical mechanics and by carrying out experiments in set ups of excessively large aspect ratios we can finally measure the critical exponents that determine this transition type and show that it falls into the directed percolation universality class.