## Spring School on Wave Turbulence

## Ecole de Physique des Houches

## May 24-29th, 2020

**Wave Turbulence is a general framework to describe a statistical state made of a large number of non linear waves. In the weakly nonlinear limit, Weak Turbulence Theory uses classical tools of out of equilibrium statistical physics to study waves and dispersive systems. A major concern is to understand the validity regimes of this theory and what are the main processes that restrict its validity.**

**This springs school aims a gathering Mathematicians and Physicists for an interdisciplinary meeting on the issue of Wave Turbulence, at Ecole de Physique des Houches, France. The courses will cover a vast panorama of issues from Mathematics, Physical theory, experiments and field measurements. The goal is also to confront theoretical and experimental points of view.**

**This school is organized by the Simons Collaboration on Wave Turbulence**

*The school is intended for graduate students as well as junior and senior researchers interested in the field of Wave Turbulence.*

The first equations of wave
turbulence were proposed by Peierls in 1928 as part of solid
state physics, and then by Hasselman in 1962 for surface waves.
The theory was then implemented systematically by Zakharov and
his school in the 70's. It predicts the evolution of the energy
distribution among the different modes of oscillation for a
forced system, and in particular in stationary regime, the
equilibrium spectrum of Kolmogorov-Zakharov, characterized by an
energy cascade between small and large scales (direct cascade
and/or inverse cascade). This theory is relevant for some
systems, but there are also a number of situations where it does
not correspond quantitatively to observations. The theoretical
approach is based on an idealization of the conditions that can
be obtained in nature or in the laboratory:

- dissipation is rarely confined to small scales. This impacts the shape of the spectrum, which often exhibits a stronger decay than that predicted by the theory;

- the finite size of the experiments induces effects that are not taken into account in the theory, in particular a discretization of the oscillation modes, with strong constraints on the discrete cascades.

- the amplitude of the waves is difficult to calibrate. If it is too weak, the finite size effects are dominant. If it is too large, the wave structure is destroyed by the strongly nonlinear couplings.

In attempting to establish a rigorous mathematical basis for Zakharov's theory (including quantitative estimates of small parameters), it should be possible to clarify the validity regimes, and propose corrections or alternative models when appropriate.

- dissipation is rarely confined to small scales. This impacts the shape of the spectrum, which often exhibits a stronger decay than that predicted by the theory;

- the finite size of the experiments induces effects that are not taken into account in the theory, in particular a discretization of the oscillation modes, with strong constraints on the discrete cascades.

- the amplitude of the waves is difficult to calibrate. If it is too weak, the finite size effects are dominant. If it is too large, the wave structure is destroyed by the strongly nonlinear couplings.

In attempting to establish a rigorous mathematical basis for Zakharov's theory (including quantitative estimates of small parameters), it should be possible to clarify the validity regimes, and propose corrections or alternative models when appropriate.

Recent technical developments in
ultra-fast digital imaging make it possible to obtain time- and
space-resolved measurements in various systems
(capillarity-gravity waves, internal waves and inertial waves,
elastic plates, optical waves, etc.). Such measurements make it
possible to obtain information on wave statistics beyond the
one-wave spectrum. Typically one can (at least in some
experiments):

- check if the waves are weakly non-linear by measuring the dispersion relation;

- measure correlations to detect non-linear effects;

- extract the characteristic times to check the separation of scales between wave period, non-linear time and viscous damping time;

- detect the presence of intermittency.

- check if the waves are weakly non-linear by measuring the dispersion relation;

- measure correlations to detect non-linear effects;

- extract the characteristic times to check the separation of scales between wave period, non-linear time and viscous damping time;

- detect the presence of intermittency.

Direct numerical simulations (DNS) of equations allow a complementary approach to theory and laboratory experiments, since they give great flexibility in the choice of parameters (wave velocity, size of nonlinearities or dissipation), and allow to test different scenarios of evolution for the establishment of the stationary regime. Nevertheless these simulations still represent a significant numerical challenge due to the presence of an extremely wide range of time scales.

We expect that gathering
lecturers from very distinct backgrounds — pure mathematics,
applied mathematics, statistical and nonlinear physics, fluid
mechanics, theoreticians and experimentalists — will trigger
fruitful interactions that are necessary to tackle this
challenging topic.

This school is funded by the Simons Foundation through the Simons Collaboration on Wave Turbulence.