Schedule for: 20w5025 - Mathematical Questions in Wave Turbulence (Online)

In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity κ should display a |k|−1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature and is a core prediction of passive scalar turbulence. Together with Alex Blumenthal and Sam Punshon-Smith, we have provided the first mathematically rigorous proof of this prediction for a scalar field evolving by advection-diffusion in a fluid governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in a periodic box subjected to stochastic forcing at arbitrary Reynolds number. As conjectured by physicists, we also show the results in fact hold for a variety of toy models, though Navier-Stokes at high Reynolds number is the most physically relevant and the most difficult mathematically that we have considered thus far. These results are proved by studying the Lagrangian flow map using extensions of ideas from random dynamical systems. We prove that the Lagrangian flow has a positive Lyapunov exponent (Lagrangian chaos) and show how this can be upgraded to almost sure exponential mixing of passive scalars at zero diffusivity and further to uniform-in-diffusivity mixing. This in turn is a sufficiently precise understanding of the low-to-high frequency cascade to deduce Batchelor's prediction.

I will talk about some recent work on the nonlinear asymptotic stability of point vortices and monotonic shear flows among solutions of the 2D Euler equations. This is joint work with Hao Jia.

In this talk we first give a quick background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic NLS in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. We then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence with uniqueness) for the periodic NLS in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call random averaging operators which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is joint work with Yu Deng (USC) and Haitian Yue (USC).

Usually in wave turbulence, one looks for a scaling stationary solution. However, the evolution preceeding the steady state is equally interesting and it may exhibit a nontrivial self-similar scalings. Problem of this kind naturally arises when we ask , for example, about the rate at which the condensate grows within the wave turbulence settings in the NLS model.

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

We consider the long-term dynamics of nonlinear dispersive waves in a finite periodic domain. The purpose of the work is to show, for the first time, that the statistical properties of the wave field rely critically on the structure of the discrete resonant manifold (DRM). To demonstrate this, we simulate the two-dimensional MMT equation on rational and irrational tori, resulting in remarkably different power-law spectra and energy cascades at low nonlinearity levels. The difference is explained in terms of different structures of the DRM, which makes use of the recent number theory results. The role of DRM will also be discussed in the context of physical wave systems and demonstrated for the case of capillary waves. Finally, we will discuss the implications of the findings to finite-domain wave turbulence in general.

Internal gravity waves play a primary role in geophysical fluids : they contribute significantly to mixing in the ocean and they redistribute energy and momentum in the middle atmosphere. In addition to their very interesting and very unusual theoretical properties, these waves are linked to one of the important questions in the dynamics of the oceans: the cascade of mechanical energy in the abyss and its contribution to mixing. I will discuss a setup that allows us to study experimentally the interaction of nonlinear internal waves in a stratified fluid confined in a trapezoidal tank. The set-up has been designed to produce internal wave turbulence from monochromatic and polychromatic forcing through three processes. The first is a linear transfer in wavelength obtained by wave reflection on inclined slopes, leading to an internal wave attractor which has a broad wavenumber spectrum. Second is the broad banded time-frequency spectrum of the trapezoidal geometry, as shown by the impulse response of the system. The third one is a nonlinear transfer in frequencies and wavevectors via triadic interactions, which results at large forcing amplitudes in a power law decay of the wavenumber power spectrum. This first experimental spectrum of internal wave turbulence displays a $k^{-3}$ behavior.