Schedule for: 16w5112  Theoretical and Computational Aspects of Nonlinear Surface Waves
Beginning on Sunday, October 30 and ending Friday November 4, 2016
All times in Banff, Alberta time, MDT (UTC6).
Sunday, October 30  

16:00  17:30  Checkin begins at 16:00 on Sunday and is open 24 hours (Front Desk  Professional Development Centre) 
17:30  19:30 
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) 
20:00  22:00  Informal gathering (Corbett Hall Lounge (CH 2110)) 
Monday, October 31  

07:00  08:45 
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) 
08:45  09:00  Introduction and Welcome by BIRS Station Manager (TCPL 201) 
09:00  09:30 
Jerry Bona: Higherorder Hamiltonian Models for Water Waves ↓ Higherorder models for unidirectional propagation of
longcrested water waves will be introduced and their
analysis sketched. (TCPL 201) 
09:30  10:00 
Daniel Ratliff: The Emergence of Higher Order Dispersion from Periodic Waves ↓ Following the method of Bridges (2013, PRSA), it is demonstrated how one may derive PDEs with fifth order dispersion from periodic waves (and, in general, relative equilibrium). Many of the coefficients of the emerging nonlinear approximations are directly related to the system's conservation laws, and those of the dispersive terms are tied to a Jordan chain analysis. Examples illustrating how the theory applies will also be discussed. (TCPL 201) 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00 
Walter Craig: Hamiltonians and normal forms for water waves ↓ It was shown by V. E. Zakharov that the equations for water waves can be posed as a Hamiltonian dynamical system, and that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context. Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions, including the function space mapping properties of these transformations. (TCPL 201) 
11:00  11:30 
Mariana Haragus: Counting unstable eigenvalues in Hamiltonian spectral problems via commuting operators ↓ We present a general counting result for the unstable eigenvalues of linear operators of the form JL in which J and L are skew and selfadjoint operators, respectively. Assuming that there exists a selfadjoint operator K such that the operators JL and JK commute, we prove that the number of unstable eigenvalues of JL is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of onedimensional periodic traveling waves in the classical KPII (KadomtsevPetviashvili) equation. (TCPL 201) 
11:30  13:00  Lunch (Vistas Dining Room) 
13:00  14:00 
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) 
14:00  14:20 
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL Foyer) 
14:30  15:00 
Alexander Korotkevich: Circular instability of a standing surface wave: numerical simulation and wavetank experiment ↓ It is shown that standing wave is unstable if we consider four wave nonlinear processes. We compare
numerical simulation of instability of weakly nonlinear standing waves on the surface of deep fluid
in the framework of the primordial dynamical equations and in a laboratory wave tank experiment. The instability offers a new approach for generation of nearly isotropic spectrum using parametric
excitation. Direct measurements of spacial Fourier spectrum confirm existence of the instability in
a real life conditions for gravitycapillary surface waves.
This is a joint work with Sergei Lukaschuk (The University of Hull, UK). (TCPL 201) 
15:00  15:30  Coffee Break (TCPL Foyer) 
15:30  16:00 
Emilian Parau: Stability of capillary waves on fluid sheets ↓ The linear stability of finiteamplitude capillary waves on inviscid sheets of fluid is investigated. Superharmonic and subharmonic perturbations are considered and a conformal mapping technique is used. The instability results are also checked by time integration of the fully nonlinear unsteady equations. This is joint work with Mark Blyth (UEA). (TCPL 201) 
16:00  16:30 
JeanMarc VandenBroeck: A numerical investigation of nonsymmetric nonlinear water waves ↓ Nonlinear waves travelling at a constant velocity at the surface of a fluid of finite
depth are considered. The fluid is assumed to be incompressible and inviscid and the flow
to be irrotational. Gravity and surface tension are taken into account. Both two and three
dimensional waves are studied. Classical solutions usually assume that the waves are symmetric.
We show that there are in addition an infinite number of branches of nonsymmetric waves.
These include periodic waves, solitary waves and generalised solitary waves. As time
permits extensions to hydroelastic waves will be presented. (TCPL 201) 
16:30  17:00 
Christian Klein: Multidomain spectral method for Schrödinger equations ↓ A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line or in higher dimensions. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied (TCPL 201) 
17:00  17:30 
Katie Oliveras: Relationships between pressure, bathymetry, and waveheight ↓ A new method is proposed to relate the pressure at the bottom of a fluid, the shape of the bathymetry, and the surface elevation of a wave for steady flow or traveling waves. Given a measurement of any one of these physical quantities (pressure, bathymetry, or surface elevation), a numerical representation of the other two quantities is obtained via a nonlocal nonlinear equation obtained from the Euler formulation of the waterwave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments. This is joint work with Vishal Vasan and Daniel Ferguson. (TCPL 201) 
17:30  19:30 
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) 
Tuesday, November 1  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  09:30 
Sijue Wu: On the Motion of a SelfGravitating Incompressible Fluid with Free Boundary ↓ We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid subject to selfgravitational force and neglecting surface tension in two space dimensions. We show that for smooth data which are size $\epsilon$ perturbations of an equilibrium state, the solution exists and remains smooth for time of at least $O(\epsilon^{2})$. This should be compared with the lifespan $O(\epsilon^{1})$ provided by local wellposedness. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate system has no quadratic nonlinear terms. This is a joint work with Lydia Bieri, Shuang Miao and Sohrab Shahshahani. (TCPL 201) 
09:30  10:00 
Mihaela Ifrim: Constant vorticity water waves ↓ The talk is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in positionvelocity potential holomorphic coordinates, and prove local wellposedness for large data, as well as cubic lifespan bounds for small data solutions. (TCPL 201) 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00 
Daniel Tataru: Long time solutions for finite bottom gravity waves ↓ I will provide an overview of recent work, joint with Benjamin HarropGriffiths and Mihaela Ifrim, concerning enhanced lifespan bounds for small data gravity waves in the presence of a flat bottom. This work is based on two key ideas, (i) the use of holomorphic coordinates, and (ii) our modified energy method, a quasilinear alternative to normal forms. (TCPL 201) 
11:00  11:30 
Steve Shkoller: Nonuniqueness of weak solutions to the SQG equation ↓ We prove that weak solutions of the inviscid surface quasigeostrophic (SQG) equation are not unique, thereby answering an open problem posed by De Lellis and Szekelihidi. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. This is joint work with T. Buckmaster and V. Vicol. (TCPL 201) 
11:30  13:30  Lunch (Vistas Dining Room) 
13:30  14:00 
Henrik Kalisch: On Existence and Uniqueness of Singular Solutions for Systems of Conservation Laws ↓ Existence and admissibility of deltashock solutions is discussed for hyperbolic systems of conservation laws. One of the systems discussed is fully nonlinear, and does not admit a classical Laxadmissible solution to certain Riemann problems.
By introducing complexvalued corrections in the framework of the weak asymptotic method, we show that a compressive deltashock wave solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued and the resulting distributions fit into a generalized concept of singular solutions [V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of deltashock waves in hyperbolic systems, J. Differential Equations 211 (2005), 333381]. In this framework, it can be shown that every 2x2 system of conservation laws admits deltashock solutions.
As an example, it is shown that the combination of discontinuous freesurface solutions and bottom
step transitions naturally leads to singular solutions featuring Dirac delta distributions in the context of shallowwater flows. (TCPL 201) 
14:00  14:30 
John Carter: Frequency Downshifting in a Viscous Fluid ↓ Frequency downshift, i.e. a shift in the spectral peak to a lower frequency, in a train of nearly monochromatic gravity waves was first reported by Lake et al. (1977). Even though it is generally agreed upon that frequency downshifting (FD) is related to the BenjaminFeir instability and many physical phenomena (including wave breaking and wind) have been proposed as mechanisms for FD, its precise cause remains an open question.
Dias et al. (2008) added a viscous correction to the Euler equations and derived the dissipative NLS equation (DNLS). In this talk, we introduce a higherorder generalization of the DNLS equation, which we call the viscous Dysthe equation. We outline the derivation of this new equation and present many of its properties. We establish that it predicts FD in both the spectral mean and spectral peak senses. Finally, we demonstrate that predictions obtained from the viscous Dysthe equation accurately model data from experiments in which frequency downshift occurred. (TCPL 201) 
14:30  15:00 
Vera Mikyoung Hur: Wave breaking and modulational instability in fulldispersion shallow water models ↓ In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) $\times$ (the undisturbed water depth) $> 1.363....$ In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simple approximate models to gain insights.
I will begin by Whitham's shallow water equation and the wave breaking conjecture, and move to the modulational instability index for smallamplitude periodic traveling waves, the effects of surface tension and constant vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and twodimensional surfaces. This is based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), Ashish Pandey (Illinois), and Leeds Tao (UC Riverside). (TCPL 201) 
15:00  15:30  Coffee Break (TCPL Foyer) 
15:30  16:00 
Rosa VargasMagana: A WhithamBoussinesq longwave model for variable topography ↓ We study the propagation of water waves in a channel of variable depth using the longwave asymptotic regime. We use the Hamiltonian formulation of the problem in which the nonlocal DirichletNeumann (DN) operator appears explicitly in the Hamiltonian and due to the complexity of the expressions of the asymptotic expansion associated with this operator in the presence of a nontrivial bottom topography. We perform an adhoc modification of these terms using a pseudo differential operator (PDO) associated with the bottom topography. In this talk we propose a WhithamBoussinesq model for bidirectional wave propagation in shallow water that involves a PDO that consider explicitly the expression for the depth profile. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth.
We will introduce an accurate and efficient numerical method that has been developed to compute this PDO. We present the results for the normal modes and eigenfrequencies of the linearized problem for families of different topographies. We also present some experiments of the evolution of some initial wave profiles over different topographies. Due to the adhoc nature of this simplified model we present some comparisons between the full expression of the first term of the asymptotic expansion of the DN operator given by Craig, Guyenne, Nicholls, and Sulem and our PDO approach for some specific topographies. (TCPL 201) 
16:00  16:30 
Mats Ehrnstrom: On waves of greatest height in fully dispersive equations ↓ Starting from the nonlocal Whitham equation with its fully dispersive linear operator, we consider the existence of periodic traveling waves that are not small, albeit connected to the line of vanishing solutions. Of particular interest is the existence of a highest, $C^{1/2}$cusped, traveling wave solution, which is obtained as a limiting case at the end of the main bifurcation branch of $P$periodic traveling wave solutions. We prove that this regularity is optimal.
Given that the Euler equations admits a highest wave that is not cusped, but Lipschitz continuous, it is an interesting question whether a bidirectional Whitham equation, which carries the full twoway dispersion relation from the Euler equations, could encompass a Lipschitz wave as well. Due to reasons to be explained in the talk, it however turns out that the highest wave for the bidirectional Whitham equation, which we prove to exist, is not Lipschitz  nor is its optimal regularity described by Hölder or Zygmund spaces. We characterize its behaviour near the wave crest.
At the end we outline the first steps towards a more general theory. Our interest is a somewhat general largeamplitude theory for nonlinear dispersive equations.
The talk is based on joint works with E. Wahlén, Lund, M. A. Johnson and K. M. Claassen, both Kansas. (TCPL 201) 
16:30  17:00 
Gabriele Bruell: On symmetry and decay of traveling wave solutions to the Whitham equation ↓ The Whitham equation is a nonlocal, nonlinear dispersive wave equation introduced by G. B. Whitham
as an alternative wave model equation to the Kortewegde Vries equation, describing the wave motion at the surface on shallow water. The existence of supercritical solitary wave solutions to the Whitham equation has been shown by Ehrnström, Groves, and Wahlén in 2012. We prove that any such solution decays exponentially, is symmetric and has exactly one crest. Moreover, the structure of the Whitham equation allows to conclude that conversely any classical, symmetric solution constitutes a traveling wave. In fact, the latter result holds true for a large class of partial differential equations sharing a certain structure. (TCPL 201) 
17:00  17:30 
Dag Nillson: Solitary waves of a class of GreenNaghdi type systems ↓ We consider a class of GreenNaghdi type systems and prove the existence of solitary wave
solutions. The solutions are identifed as critical points of a scalar functional and we are able to
show that there exist minimizers of this functional, under certain constraints. A key component
of the proof is the use of the concentration compactness principle. The talk is based upon a work
in progress with Erik Wahlen (Lund University) and Vincent Duchene (University of Rennes). (TCPL 201) 
17:30  19:30  Dinner (Vistas Dining Room) 
Wednesday, November 2  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  09:30 
John Grue: Ship generated tsunamis: linearity vs. nonlinearity ↓ Very long waves running ahead of ships is a recent phenomenon in the Oslofjord in Norway. The waves are triggered when the new very large and relatively fast cruiseferries are running across significant depth changes in the shallow fjord. An asymptotic linear analysis expressing the upstream wave field in terms of a pressure impulse at the depth change is complemented by fully dispersive calculations of the upstream waves. At very shallow positions the local ship speed becomes critical where the effect of nonlinearity is analyzed. A wave length of 700 m, wave height at the shore of 1 m, average depth of the fjord of 35 m, depth change similar to the average depth, ship length of 200 m, moving at subcritical speed, are typical characteristics. (TCPL 201) 
09:30  10:00 
Onno Bokhove: Variational coupling of nonlinear water wave and ship dynamics: continuum and finite element modelling ↓ We report on the mathematical and numerical modelling of (non)linear ship motion in (non)linear water waves. We derive a coupled model for the waveship dynamics following a variational methodology, in order to ensure zero numerical damping which is important for wave propagation. The final system of evolution equations comprises the classical waterwave equations for incompressible and irrotational waves, and a set of equations describing the dynamics of the ship. The novelty in our model is in the presence of a physical restriction on the water height under the ship, which is enforced through an inequality constraint via a Lagrange multiplier. The model is solved numerically using a variational (dis)continuous Galerkin finite element method with special, new and robust time integration methods. Here we aim to show numerical results for the dynamics of the coupled system in a hierarchy of increasing complexity: linear waterwave and linear ship dynamics, and potentially also fully coupled (non)linear waterwave and nonlinear ship dynamics. (TCPL 201) 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00 
David Lannes: On the dynamics of floating structures ↓ The goal of this talk is to derive some equations describing the interaction of a floating solid structure and the surface of a perfect fluid. This is a double free boundary problem since in addition to the water waves problem (determining the free boundary of the fluid region), one has to find the evolution of the contact line between the solid and the surface of the water. The socalled floating body problem has been studied so far as a threedimensional problem. Our first goal is to reduce it to a twodimensional problem that takes the form of a coupled compressibleincompressible system. We will also show that the hydrodynamic forces acting on the solid can be partly put under the form of an added massinertia matrix, which turns out to be affected by the dispersive terms of the equations. (TCPL 201) 
11:00  11:30 
Harry Bingham: Stable, highorder finite difference methods for nonlinear wavestructure interaction in a moving reference frame ↓ This talk will focus on highorder finite difference methods for solving potential
flow approximations of nonlinear surface waves interacting with marine structures. Of special interest
is the loading and waveinduced response of sailing ships, where it is convenient to work in a
reference frame which is translating at constant speed. This introduces nonlinear convective
terms into the freesurface boundary conditions which are found to require nonlinear numerical
schemes to achieve robust and stable solutions. The work builds on the basic numerical solution
strategy described in [1, 2]. Inspired by work reviewed for example by Shu [4], we have developed a
simplified version of the Weighted Essentially NonOscillatory (WENO) scheme which
is only slightly more diffusive than the equivalent order centered finite difference scheme. The
equations are then put into HamiltonJacobi form, and the WENO convective approximations
are combined using the RoeFix numerical flux proposed by Shu & Osher [5]. In contrast to
a simple upwinding strategy, this approach is found to give accurate and stable solutions for
all combinations of wave celerity and ship forward speed. To introduce the ship geometry into
the numerical solution, we have developed an Immersed Boundary Method based on Weighted
LeastSquares difference approximations. This scheme will be described and some preliminary
results will be presented. Challenges with respect to tracking the bodyfree surface intersection
line, and treating wavebreaking in a rational way will be raised for discussion.
(TCPL 201) References:

11:30  13:30  Lunch (Vistas Dining Room) 
13:30  17:30  Free Afternoon (Banff National Park) 
17:30  19:30  Dinner (Vistas Dining Room) 
Thursday, November 3  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  09:30 
David Ambrose: Sufficiently strong dispersion removes illposedness of truncated series models of water waves ↓ We have previously, in joint work with Jerry Bona, David Nicholls, and Michael Siegel, demonstrated that truncated series models of gravity water waves exhibit illposedness. In joint work with Shunlian Liu, we show that the addition of sufficiently strong dispersion makes such a system wellposed. Physically, this strong dispersion can be relevant, for instance, for hydroelastic waves. The proof uses techniques of paradifferential calculus. (TCPL 201) 
09:30  10:00 
Paul Milewski: Computational exploration of water waves in two acts. 1. Diving under rotational water waves and 2. Wave collapse for ripples. ↓ Act 1: We consider periodic traveling waves in a flow with constant vorticity and look in detail at the
internal structure of the flow, including particle paths, stagnation points and pressure fields. The results illustrate some of the known theoretical results and also point to new interesting features.
(TCPL 201) Act 2: Capillary gravity wave packets are described asymptotically by the focussing 2D NLS equation. This equation has solutions that blow up in finite time. We explore the dynamics of blowup initial data under the full Euler equations. 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00 
Mark Groves: Fully localised solitary gravitycapillary water waves ↓ We consider the classical gravitycapillary waterwave problem in its usual formulation as a threedimensional freeboundary problem for the Euler equations for a perfect fluid. A solitary wave is a solution representing a wave which moves in a fixed direction with constant speed and without change of shape; it is fully localised if its profile decays to the undisturbed state of the water in every horizontal direction.
The existence of fully localised solitary waves has been predicted on the basis of simpler model equations, namely the KadomtsevPetviashvili (KP) equation in the case of strong surface tension and the DaveyStewartson (DS) system in the case of weak surface tension. In this talk we confirm the existence of such waves as solutions to the full waterwave problem and give rigorous justification for the use of the model equations.
This is joint work with Boris Buffoni and Erik Wahlen. (TCPL 201) 
11:00  11:30 
Samuel Walsh: Existence and qualitative theory of stratified solitary water waves ↓ In this talk, we will report some recent results concerning twodimensional gravity solitary water waves with hereogeneous density. The fluid domain is assumed be bounded below by an impenetrable flat ocean bed, while the interface between the water and vacuum above is a free boundary. Our main existence result states that, for any smooth choice of upstream velocity and streamline density function, there exists a path connected set of such solutions that includes largeamplitude surface waves. Indeed, this solution set can be continued up to (but does not include) an ``extreme wave`` that possess a stagnation point.
We will also discuss a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the Froude number from above and below that are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores for stratified surface waves; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry. This is joint work with R. M. Chen and M. H. Wheeler. (TCPL 201) 
11:30  13:30  Lunch (Vistas Dining Room) 
13:30  14:00 
David Henry: Nonlinear water waves and wavecurrent interactions ↓ This talk will examine the nonlinear water waves, and wavecurrent interactions, which may be prescribed by Gerstnerlike exact and explicit solutions to the geophysical $\beta$plane equations in the equatorial region. In particular, I will present recent work which highlights the role played by the typicallyneglected centripetal force terms in describing such physical processes (TCPL 201) 
14:00  14:30 
Anna Geyer: On periodic traveling waves of the CamassaHolm equation ↓ In this talk we are concerned with the wave length $\lambda$ of smooth periodic traveling wave solutions of the CamassaHolm equation. The set of these solutions can be parametrized using the wave height $a$. Our main result establishes monotonicity properties of the map $a\mapsto \lambda(a)$ i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviours of $\lambda(a)$, namely monotonicity and unimodality. The key point is to relate $\lambda(a)$ to the period function of a planar differential system with a quadraticlike first integral, and to apply a criterion which bounds the number of critical periods for this type of systems. (TCPL 201) 
14:30  15:00 
Miles Wheeler: Global bifurcation of rotating vortex patches ↓ We construct continuous curves of rotating vortex patch solutions to the
twodimensional Euler equations. These curves are large in that, as the
parameter tends to infinity, the minimum value on the boundary of the
relative angular fluid velocity becomes arbitrarily small. This is joint
work with Zineb Hassainia and Nader Masmoudi. (TCPL 201) 
15:00  15:30  Coffee Break (TCPL Foyer) 
15:30  16:00 
Christopher Curtis: Surface and Interfacial Waves over Currents and PointVortices ↓ The computation of surface and interfacial waves is a central problem in fluid mechanics. While much has been done, the effect of vorticity on surface and internal wave propagation is still poorly understood. To address this, we first look at shallowwater propagation in density stratified fluids with piecewise linear shear profiles. We show that by allowing for jumps in the shear across the interface, strong nonlinear responses can be generated resulting in phenomena like dispersive shock waves. Thus depth varying currents could play a larger role in interface dynamics than is currently understood.
Second, we study the problem of collections of irrotational point vortices underneath a free fluid surface. We present a derivation of a model and numerical scheme which allows for arbitrary numbers of vortices in a shallowwater limit. While we are able to recreate much of the classical results for how surface waves form over two counterpropagating vortices, we go beyond this case and look at an example involving four vortices. Again, our approach allows for any number of vortices to be present, and this lets us provide some hint as to how underwater eddies might generate free surface waves. (TCPL 201) 
16:00  16:30 
Guyenne Philippe: Nonlinear waves in ice sheets ↓ This talk concerns the mathematical modeling and numerical simulation of waves in ice sheets as occurring, e.g., in polar regions. A threedimensional Hamiltonian formulation for ice sheets deforming on top of an ideal fluid of arbitrary depth is presented and nonlinear wave solutions are examined. In certain asymptotic regimes, analytical solutions are derived and compared with fully nonlinear solutions obtained numerically by a pseudospectral method. (TCPL 201) 
16:30  17:00 
Olga Trichtchenko: Computing ThreeDimensional FlexuralGravity Water Waves ↓ The goal of this work is to build on previous results by Parau et al. [1, 2, 3, 4]
and produce a more efficient and accurate method for computing solutions to
Euler's equations for water waves underneath an ice sheet in three dimensions.
As was previously done, we solve the equations via a numerically implemented
boundary integral equations method and utilize some high performance computing techniques. In this talk, we give details of the current methods and compare
solutions for different models of the ice sheet.
This is a joint work with Emilian Parau, JeanMarc VandenBroeck and Paul Milewski.
(TCPL 201) References:

17:00  17:30 
Erik Wahlen: Variational existence and stability theory for hydroelastic solitary waves ↓ I will present an existence and stability theory for solitary waves at the interface between a thin ice sheet and an ideal fluid, which is based on minimising the total energy subject to the constraint of fixed total horizontal momentum. The ice sheet is modelled using the Cosserat theory of hyperelastic shells. Since the energy functional is quadratic in the highest derivatives, stronger results are obtained than in the case of capillarygravity waves. This is joint work with Mark Groves and Benedikt Hewer. (TCPL 201) 
17:30  19:30  Dinner (Vistas Dining Room) 
Friday, November 4  

07:00  09:00  Breakfast (Vistas Dining Room) 
09:00  10:00  Free discussion and research in groups (TCPL) 
10:00  10:30  Coffee Break (TCPL Foyer) 
10:30  11:00  Free discussion and research in groups (TCPL) 
11:30  12:00 
Checkout by Noon ↓ 5day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon. (Front Desk  Professional Development Centre) 
12:00  13:30  Lunch from 11:30 to 13:30 (Vistas Dining Room) 