SCIENTIFIC PROGRAMS AND ACTIVITIES

May 13
2:00-4:00 p.m.

Roberto Camassa (North Carolina at Chapel Hill)
Asymptotic models of internal wave motion in layered fluids

Density variations in fluids can have interesting dynamical consequences. Of these, the case of internal waves is arguably one of the most important, especially in geophysical applications. In order to get some fundamental understanding of this motion it is useful to study what is possibly the simplest set-up capable of supporting internal wave propagation, that of a two-ayer incompressible Euler fluid under gravity. Density is assumed to be homogeneous in each layer, and waves evolve as deformations from an equilibrium position of the interface between the two fluids. As simple as this system is, it is still hard to get an analytical grip on the dynamics it supports. We will then develop asymptotic models, based on assumptions on classes of initial conditions, and identify several relevant régimes and solutions of practical relevance. Some insightful physical and mathematical properties of the models will be examined, and their consequences on more general set-ups of interest for applications will be discussed.

Denys Dutykh (University College Dublin),
Claudio Viotti (University College Dublin)
Numerical methods for fully nonlinear free surface water waves

In this short course we are going to review some of the main existing Eulerian approaches to the numerical simulation of the full water wave problem. More specifically, the lectures willl cover the boundary integral equation method, higher order spectral method, Dirichlet-to-Neumann operator approach and, finally, the conformal mapping technique. The advantages and shortcomings of different methods will be discussed as well.
The course does not assume any particular knowledge in hydrodynamics. It is therefore suitable for researchers and graduate students in applied mathematics and related fields.

Vladimir Zeitlin (Laboratoire de Meteorology Dynamique, ENS - Paris)
Modeling large-scale atmospheric and oceanic flows: from primitive to 2D Euler equations

I will be first deriving a hierarchy of models for large-scale atmospheric and oceanic phenomena, explaining approximations and how stochastic parameterizations may be introduced into them, and then showing how a simple model of the kind explained in the first part may be improved to include two-phase dynamics of the moist air.

Plan 1 (lecture slides) Introduction Review Workflow Crash Course in fluid dynamics Reminder: perfect fluid Molecular dissipation Primitive Equations Rotating frame. Spherical coordinates. Traditional approximation. Tangent plane "Primitive" equations (PE) Vertically averaged models Vertical averaging of PE Vortices and waves Vortex dynamics Vortex dynamics in 1-layer RSW 2-layer QG model QG dynamics by time averaging Summary

Plan 2 (lecture slides) Introduction Methodology Constructing the model Limiting equations and relation to the known models General properties of the model Conservation laws Characteristics and fronts Example: scattering of a simple wave on a moisture front Introducing evaporation Moist vs dry baroclinic instability (Dry) linear stability of the baroclinic jet Comparison of the evolution of dry and moist instability Conclusion Literature

Armen Shirikyan (Universite de Cergy - Pontoise)
Stationary measures of stochastic PDE’s in turbulent regime

The course is devoted to studying stationary measures for the Navier–Stokes system on the 2D torus and Burgers equation on the circle. Both equations are perturbed by a random force, white in time and smooth in the space variables. After recalling some results on the uniqueness and mixing of a stationary measure, we discuss the behaviour of stationary solutions as the viscosity goes to zero. In the case of the Navier–Stokes system, it is proved that, under suitable normalisation, any sequence of stationary measures converges to a limit, which is invariant under the dynamics of the Euler system. Some universal relations for stationary measures and the non-degeneracy of limits are also established. In the case of the Burgers equation, it is proved that the inviscid limit is uniquely defined and is the unique stationary measure for the limiting transport equation. We also derive two-sided estimates for averaged Sobolev norms of stationary solutions.

1. Ergodic theory for stochastic PDE's
1.1. Stochastic Navier–Stokes and Burgers equations
1.2. Stationary measures and a priori estimates
1.3. Uniqueness and mixing

2. Inviscid limit for the Navier–Stokes equations
2.1. Kuksin measures
2.2. Balance relations
2.3. Non-degeneracy of the limit
2.4. Physical interpretation of the results

3. Two-sided estimates for solutions of the Burgers equation
3.1. E–Khanin–Mazel–Sinai theorem
3.2. Upper bound for Sobolev norms
3.3. Lower bound for the time-average of Sobolev norms
3.4. Kolmogorov scales for the Burgers equation

Walter Craig (McMaster)
Hamiltonian and Water Waves

David Lannes (Ecole Normale Supereure)
The effects of vorticity on shallow water asymptotics of the water wave
equations

The Zakharov-Craig-Sulem (ZCS) formulation of the water waves equations has proved very useful in the water waves theory, and in particular to address the well-posedness issue and the derivation of asymptotic models providing simpler models in the so called shallow water regime. This formulations relies strongly on the assumption that the flow is irrotational. This is a very reasonable assumption in many situations, but in some cases, vorticity should be taken into account. In a joint work with Angel Castro, we propose an extension of the ZCS formulation in presence of vorticity; after proving the well posedness of this new formulation, we derive shallow water models for rotational flows and exhibit non trivial effects of the vorticity.