2024 Simons Collaboration on Wave Turbulence Annual Meeting

Laurent Chevillard
École Normale Superiéure de Lyon

A Linear Dynamics Leading to a Loss of Regularity: Generation of Fractional Gaussian Fields and Applications to Fluid Turbulence

We will begin with some reminders on the phenomenology of three-dimensional fluid turbulence and the observational aspects of the statistical behavior of solutions of the Navier-Stokes equations, forced by a smooth term in space. To account for the cascade phenomenon, which asymptotically leads to a velocity field that is continuous yet highly irregular (of Hölderian type), we propose a linear model dynamics capable of generating such rough fields from entirely smooth ingredients over infinite time. This will involve a theoretical and numerical study of a transport phenomenon, not in physical space but in Fourier space, allowing energy to be transferred (or cascaded) across scales. A scheme based on finite spectral volumes will then provide a coherent numerical representation of such dynamics. This is joint work with G. Apolinario, G. Beck, C.-E. Bréhier, I. Gallagher, R. Grande, J.-C. Mourrat and W. Ruffenach.
 

Vincenzo Vitelli
University of Chicago

Odd Turbulence

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which it is eventually arrested by dissipation. In this talk, we show how to harness turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state. By contrast, our fully nonlinear mechanism is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which it is usually dissipated. The tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems. Odd viscosity acts as a scale-dependent Coriolis-like force and leads to two-dimensionalization of the flow and suppression of intermittency. These effects are caused by parity-breaking waves that give rise to a regime of (odd) wave turbulence at small scales. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows, stellar plasma such as the solar wind, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades.
 

Léonie Canet
University Grenoble Alpes

Space-Time Dependence of Correlation Functions in Turbulence and Related Models

Calculating the statistical properties of homogeneous and isotropic fully developed turbulence, particularly intermittency effects, from the Navier-Stokes equations remains an open issue. The theoretical challenge lies in the need to close an infinite hierarchy of coupled equations that determine correlation functions. The functional renormalization group (FRG) offers a promising tool to tackle this problem, enabling a controlled closure in the large wavenumber limit. I will demonstrate how it allows for analytical results on the space-time dependence of generic multi-point correlation functions of turbulent velocity in this regime. I will compare these predictions with available results from direct numerical simulations and experiments. In the second part, I will examine related simplified models of turbulence, such as Kraichnan’s model for passive scalar turbulence and the stochastic Burgers equation, showing how a universal behavior emerges in the temporal decay of correlation functions. I will discuss how this originates from underlying extended symmetries common to these models.
 

Alex Ionescu
Princeton University

On the Wave Turbulence Theory of 2D Gravity Waves

I will discuss recent work on the rigorous study of wave turbulence in water wave systems. Wave turbulence has attracted significant attention in recent years, especially in the context of semilinear models, such as semilinear Schrödinger equations or multi-dimensional KdV-type equations. However, the scenario here is different, as the water wave equations are quasilinear, and the solutions cannot be obtained by iterating the Duhamel formula due to unavoidable derivative loss. I will discuss a new strategy to address this issue, combining deterministic energy estimates with a dispersive propagation of randomness argument. This is joint work with Yu Deng and Fabio Pusateri.
 

Anne-Sophie de Suzzoni
Ecole polytechnique

Propagation of Gaussianity and Application to the Incompressible Euler Dynamics

In this talk, I will present a result on the propagation of Gaussianity under the flow of Hamiltonian equations. Namely, I will explain why solutions to a Hamiltonian equation whose initial data follows a Gaussian law still satisfy the Wick formula at latter times in the large box regime. This process is connected to the propagation of chaos in weak turbulence. However, to emphasize the fact that it has nothing to do with dispersion, we illustrate it with the incompressible Euler dynamics. The result is twofold: a first part lies at the lever of quasisolutions and is relatively general, and the second part only applies to the Euler dynamics. We will explain how to improve this result in the case of a dispersive equation.
 

Ching-Yao Lai
Stanford University

Machine-Precision Neural Networks for Multiscale Dynamics

Deep-learning techniques are increasingly applied to scientific problems where the precision of networks is crucial. Despite being deemed universal function approximators, neural networks, in practice, struggle to reduce prediction errors below a certain threshold, even with large network sizes and extended training iterations. To address this issue, we developed multi-stage neural networks that divide the training process into different stages, with each stage utilizing a new network optimized to fit the residue from the previous stage. We demonstrate that the prediction error from multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine precision of double-floating point within a finite number of iterations. This advancement mitigates the longstanding accuracy limitations of neural network training and can be used to address the spectral bias in multiscale problems, including numerically finding blow-up solutions in fluid dynamics.
 

Vlad Vicol
New York University

Shock Formation and Maximal Hyperbolic Development in Multi-D Gas Dynamics

We consider the Cauchy problem for the multi-dimensional compressible Euler equations, evolving from an open set of compressive and generic smooth initial data. We construct unique solutions to the Euler equations which are as smooth as the initial data, in the maximal spacetime set characterized by: at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time until reaching the Cauchy data prescribed along the initial time-slice. This spacetime is sometimes referred to as the “maximal globally hyperbolic development” (MGHD) of the given Cauchy data. We prove that the future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-2 surface of “first singularities” called the pre-shock; second, a downstream co-dimension-1 surface emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream co-dimension-1 surface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. In order to establish this result, we develop a new geometric framework for the description of the acoustic characteristic surfaces, and combine this with a new type of differentiated Riemann-type variables which are linear combinations of gradients of velocity/sound speed and the curvature of the fast acoustic characteristic surfaces. This is a joint work with Steve Shkoller (University of California at Davis)
 

Yu Deng
University of Chicago

The Hilbert Sixth Problem: Particles and Waves

A major component of the Hilbert sixth problem concerns deriving macroscopic equations of motion, and the associated kinetic equations, from microscopic first principles. In the classical setting of Boltzmann’s kinetic theory, this corresponds to deriving the Boltzmann equation from particle systems governed by Newtonian dynamics. In the theory of wave turbulence, it corresponds to deriving the wave kinetic equation from nonlinear dispersive equations.

In this talk, I will present recent joint work with Zaher Hani and Xiao Ma, where we examine the hard sphere model in the particle setting and the cubic nonlinear Schrödinger equation in the wave setting. In both cases, we derive the corresponding kinetic equation for as long as the solution to this kinetic equation exists. This represents a crucial step toward resolving the Hilbert sixth problem.

The foundation will arrange and pay for round-trip air or train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Business-class or premium economy airfare will be booked for all flights over five hours when within budget.

Group A – Organizers and Speakers

• Economy Class: For flights that are three hours or less to your destination, the maximum allowable class of service is Economy class.
• Premium Economy Class: For flights where the total air travel time (excluding connection time) is more than three hours and less than seven hours per segment to your destination, the maximum allowable class of service is premium economy.
• Business Class: When traveling internationally (or to Hawaii/Alaska) travelers are permitted to travel in Business Class on those segments that are seven hours or more. If the routing is over budget, a premium economy or mixed-class ticket will be booked.

Group B – Funded Participants
The foundation will arrange and pay for round-trip air or train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Economy-class airfare will be booked for all flights.

Group C – Unfunded Participants
Individuals in Group C will not receive financial support, but are encouraged to enjoy all conference-hosted meals.

Group D – Remote Participants
Individuals in Group D will participate in the meeting remotely.

Air & Rail
For funded individuals, the foundation will arrange and pay for round-trip travel from their home city to the conference.

All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency.

Travel specifications, including preferred airline, will be accommodated provided that these specifications are reasonable and within budget.

Travel arrangements not booked through the preferred agency, including triangle trips and routing/preferred airlines outside budget, must be pre-approved by the Simons Foundation and a reimbursement quote must be obtained through the foundation’s travel agency.

All costs related to changes made to ticketed travel are to be paid for by the participant and are not reimbursable. Please contact the foundation’s travel agency for further assistance.

Personal & Rental Cars
Personal car and rental trips over 250 miles each way require prior approval from the Simons Foundation via email.

Rental cars must be pre-approved by the Simons Foundation.

The James NoMad Hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation.

Hotel
Funded individuals who require hotel accommodations are hosted by the foundation for a maximum of three nights at The James NoMad, arriving one day before the meeting and departing one day after the meeting.

Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link included in your invitation.

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Overview

In-person participants will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.

Expenses accrued as a result of meetings not directly related to the Simons Foundation-hosted meeting (a meeting held at another institution, for example) will not be reimbursed by the Simons Foundation and should be paid by other sources.

Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.

Meals

The daily meal limit is $125; itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses up the following amounts.

• Breakfast $20
• Lunch $30
• Dinner $75

Allowable Meal Expenses

• Meals taken on travel days (when you traveled by air or train).
• Meals not provided on a meeting day, dinner on Friday for example.
• Group dinners consisting of fellow meeting participants paid by a single person will be reimbursed up to $75 per person and the amount will count towards each individual’s $125 daily meal limit.

Unallowable Meal Expenses

• Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner).
• Meals taken on days not associated with Simons Foundation-coordinated events.
• Minibar expenses.
• Meal expenses for a non-foundation guest.

• Ubers, Lyfts, taxis, etc., taken to and from restaurants in Manhattan.

  • Accommodations will be made for those with mobility restrictions.

Ground Transportation

Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport or train station) as well as subway and bus fares while in Manhattan are reimbursable.

Transportation to/from satellite meetings are not reimbursable.

Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport or train station) as well as subway and bus fares while in Manhattan are reimbursable. Transportation to/from satellite meetings are not reimbursable.

Attendance
In-person participants and speakers are expected to attend all meeting days. Participants receiving hotel and travel support wishing to arrive on meeting days which conclude at 2:00 PM will be asked to attend remotely.

COVID-19 Vaccination
Individuals accessing Simons Foundation and Flatiron Institute buildings must be fully vaccinated against COVID-19.

Entry & Building Access
Upon arrival, guests will be required to show their photo ID to enter the Simons Foundation and Flatiron Institute buildings. After checking-in at the meeting reception desk, guests will be able to show their meeting name badge to re-enter the building. If you forget your name badge, you will need to provide your photo ID.

The Simons Foundation and Flatiron Institute buildings are not considered “open campuses” and meeting participants will only have access to the spaces in which the meeting will take place. All other areas are off limits without prior approval.

If you require a private space to conduct a phone call or remote meeting, please contact your meeting manager at least 48-hours ahead of time so that they may book a space for you within the foundation’s room reservation system.

Guests & Children
Meeting participants are required to give 24 hour advance notice of any guests meeting them at the Simons Foundation either before or after the meeting. Outside guests are discouraged from joining meeting activities, including meals.

With the exception of Simons Foundation and Flatiron Institute staff, ad hoc meeting participants who did not receive a meeting invitation directly from the Simons Foundation are not permitted.

Children under the age of 18 are not permitted to attend meetings at the Simons Foundation. Furthermore, the Simons Foundation does not provide childcare facilities or support of any kind. Special accommodations will be made for nursing parents.

Meeting & Policy Questions
Christina Darras
Event Manager
[email protected]

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