Schedule 
May 13
10:00 a.m.
12:00 p.m. 
Wooyoung Choi (NJIT)
Connecting the dots between asymptotic models: from deep to shallow
water waves 
May 13
2:004:00 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 setup capable of supporting internal wave propagation,
that of a twoayer 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
setups of interest for applications will be discussed.

May 15 &16
10:00 a.m.
12:00 p.m. 
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, DirichlettoNeumann
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.

Short Course on Stochastic
Fluid Dynamics

June 1718
10:00 a.m.
12:00 p.m. 
Vladimir Zeitlin (Laboratoire de Meteorology Dynamique, ENS
 Paris)
Modeling largescale atmospheric and oceanic flows: from primitive
to 2D Euler equations
I will be first deriving a hierarchy of models for largescale
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 twophase 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 1layer RSW
2layer 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


June 1920
10:00 a.m.
12:00 p.m. 
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 nondegeneracy 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 twosided estimates for averaged
Sobolev norms of stationary solutions.
Plan:
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. Nondegeneracy of the limit
2.4. Physical interpretation of the results
3. Twosided 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 timeaverage of Sobolev norms
3.4. Kolmogorov scales for the Burgers equation

Short Course on Hamiltonian
PDEs and water waves 
June 24, 25, 27
10:00  12:00 p.m 
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 ZakharovCraigSulem (ZCS) formulation of the water waves
equations has proved very useful in the water waves theory, and in particular
to address the wellposedness 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.
Catherine Sulem (Toronto)
Water Wave Scaling Regimes 