
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR

Workshop
on Ocean Wave Dynamics
May 6  10, 2013


Organizing
Committee:
Walter Craig (McMaster), Diane Henderson (Penn State),
Miguel Onorato (Universita di Torino),
Efim Pelinovsky (Institute of Applied Physics, Nizhniy
Novgorod)


Fields Institute program on the Mathematics of Oceans is to take place in
the year 2013 as a part of the initiative for the Mathematics of Planet Earth.
Workshop Schedule
Speaker &
Affiliation

Title
and Abstract 
Akhmediev, Nail
Australian National University, Canberra

Rogue waves  higher order structures
Peregrine breather being the lowest order rational solution of the
nonlinear Schroedinger equation is commonly considered as a prototype
of a rogue wave in the ocean. Higherorder rational solutions are
far from being as simple as the Peregrine breather itself. They are
not as simple as a nonlinear superposition of solitons either. Only
recently, the complexity of their spatiotemporal structures started
to be revealed.
Basic thoughts on their classification will be presented in this talk.

Alazard, Thomas
Ecole Normale Superieure, Paris

The Cauchy problem for the water waves equations, local and global
aspects
This talk will present recent results on the analysis of the
Cauchy problem for the water gravity waves. This includes firstly a
discussion of the regularity thresholds for the initial conditions :
the initial surfaces we consider turn out to have unbounded curvature
and no regularity is assumed on the bottom. An application is given
to 3D water waves in a canal or a basin. Secondly, normal form methods
will be discussed. This corresponds to the analysis of three and fourwave
interactions. These are joint works with Nicolas Burq and Claude Zuily,
and joint work with JeanMarc Delort.

Camassa, Roberto
University of North Carolina at Chapel Hill

Some fundamental issues in internal wave dynamics
One of the simplest physical setups supporting internal wave
motion is that of a stratified incompressible Euler fluid filling the
domain between two rigid horizontal plates. This talk will present asymptotic
models capable of describing large amplitude wave propagation in this
environment, and in particular of predicting the occurrence of selfinduced
shear instability in the waves' dynamics. Some curious properties of
the Euler setup for laterally unbounded domains revealed by the models
will be discussed.

Choi, Wooyoung
New Jersey Institute of Technology

Evolution of nonlinear wave packets with
and without wave breaking
We study both experimentally and numerically the evolution of nonlinear
wave packets. We solve numerically a system of nonlinear evolution
equations for the propagation of wave packets with various orders
of approximation and validate the numerical solutions with experimental
measurements. In the presence of wave breaking, a new parameterization
is introduced to account for energy dissipation due to wave breaking
and its capability to capture breaking wave characteristics is examined
in comparison with laboratory experiments.

Didenkulova, Ira
Tallinn University of Technology


Dutykh, Denys
University College Dublin

Relaxed Variational Principle for Water Wave
Modeling
A new method, based on a relaxed variational principle, is presented
for deriving approximate equations for water waves. It is particularly
suitable for the construction of approximations. The advantages will
be illustrated on numerous examples in shallow and deep water. Using
carefully chosen constraints in various combinations, several model
equations are derived, some being wellknown, others being new. These
models are studied analytically, exact travelling wave solutions are
constructed, and the Hamiltonian structure unveiled.
This is a joint work with Didier Clamond, University of Nice Sophia
Antipolis.

Gemmrich, Johannes
University of Victoria 
On the spectral shape of the source terms of the radiative transfer
equation
Currently, large domain wave forecast models are not phaseresolving
but are based on the socalled 3rd generation spectral models. The basis
of these models is the radiative transfer equation, which relates the
change of spectral energy to the sum of three source terms: energy input
from the wind, energy transfer between different wave scales and energy
dissipation, mainly due to wave breaking (in deep water) and bottom
friction (in shallow water). The net effect of these sources in a developing
sea is an increase of energy and a downshift of the spectral peak. However,
this does not provide sufficient constraints on the spectral shape of
the individual source terms.
I will present various field observations to shed light on the spectral
shape of the source terms, with emphasize on the contributions by
breaking waves.

Grimshaw, Roger
Loughborough University

Shoaling of nonlinear water waves
We review the classical theory for shoaling of a solitary
wave, and extend this to consider the propagation of an undular bore
over a gentle monotonic bottom slope connecting two regions of constant
depth, in the framework of the variablecoeficient Kortewegde Vries
equation. We show that, when the undular bore advances in the direction
of decreasing depth, its interaction with the slowly varying topography
results, apart from an adiabatic deformation of the bore itself, in
the generation of a sequence of isolated solitons, that is an expanding
largeamplitude modulated solitary wavetrain propagating ahead of the
bore. Using nonlinear modulation theory we construct an asymptotic solution
describing the formation and evolution of this solitary wavetrain. Our
analytical solution is supported by direct numerical simulations.

Guyenne, Philippe
University of Delaware

Surface signature of internal waves
Based on a Hamiltonian formulation of a twolayer ocean, we
consider the situation in which the internal waves are treated in the
longwave regime while the surface waves are described in the modulational
regime. Using Hamiltonian perturbation theory, we derive an asymptotic
model for surfaceinternal wave interactions, in which the nonlinear
internal waves evolve according to a KdV equation while the smalleramplitude
surface waves propagate at a resonant group velocity and their envelope
is described by a linear Schrodinger equation. In the case of an internal
soliton of depression for small depth and density ratios of the two
layers, the Schrodinger equation is shown to be in the semiclassical
regime and thus admits localized bound states. This leads to the phenomenon
of trapped surface modes which propagate as the signature of the internal
wave, and thus it is proposed as a possible explanation for bands of
surface roughness above internal waves in the ocean. Numerical simulations
taking oceanic parameters into account are also performed to illustrate
this phenomenon.
This is joint work with Walter Craig and Catherine Sulem.

Hara, Tetsu
University of Rhode Island

Wind turbulence over ocean waves and airsea momentum flux
We present our recent LES (large eddy simulation) results
of wind turbulence modified by ocean surface waves.
In the constant stress layer above a smooth nonbreaking surface wave
train, a waveinduced momentum flux (stress) reduces the turbulent stress
and the turbulent kinetic energy (TKE) dissipation rate inside a very
thin layer (inner layer), when the wind speed is much larger than the
wave phase speed. This leads to an increased equivalent surface roughness
(or the drag coefficient) for the wind. Since the inner layer height
is often smaller than the wave amplitude, it is necessary to introduce
a wavefollowing coordinate and redefine the waveinduced stress when
the LES results are analyzed.
When a surface waves breaks (or is sufficiently steep) airflow separates
and exerts a large force on the wave. The effects of breaking waves
on nearsurface wind turbulence and drag coefficient are investigated
using LES. The impact of intermittent and transient wave breaking events
is modeled as localized form drag, which generates airflow separation
bubbles downstream. The simulations are performed for very young sea
conditions under high winds, comparable to previous laboratory experiments
in hurricanestrength winds. In such conditions more than 90 percent
of the total airsea momentum flux is due to the form drag of breakers;
that is, the contributions of the nonbreaking wave form drag and the
surface viscous stress are small. Detailed analysis shows that the breaker
form drag impedes the shear production of the TKE near the surface and,
instead, produces a large amount of smallscale wake turbulence by transferring
energy from largescale motions (such as mean wind and gusts). This
process shortcuts the inertial energy cascade and results in large TKE
dissipation (integrated over the surface layer) normalized by friction
velocity cubed. Consequently, the large production of wake turbulence
by breakers in high winds results in the small drag coefficient obtained
in this study.

Helfrich, Karl
Woods Hole Oceanographic Institute

Breaking of the internal tide
Nonlinear steepening of lowmode internal tides and the subsequent
arrest of steepening by nonhydrostatic dispersion is a common mechanism
for the generation of internal solitary waves in the ocean. However,
it is known that the earth's rotation can retard the steepening process
and in some cases prevent the emergence of the solitary waves. The Ostrovsky
equation, the Kortewegde Vries equation with a nonlocal integral term
representing the effects of rotation, is introduced as model for these
processes. Recent work on a breaking criteria for the reduced Ostrovsky
equation (in which the linear nonhydrostatic dispersive term with a
thirdorder derivative is eliminated) is discussed. This equation is
integrable provided a certain curvature constraint is satisfied. It
is demonstrated, through theoretical analysis and numerical simulations,
that when this curvature constraint is not satisfied at the initial
time, then wave breaking inevitably occurs. The breaking criteria is
applied to several oceanic examples including internal tides in the
South China Sea and radiation of the internal tide from the Hawaiian
Island chain.

Henderson, Diane
Pennsylvania State University

Surface waves and dissipation
Surface waves at an airwater interface are usually modeled
using inviscid dynamics. However, water is a viscous fluid, and resulting
dissipative effects, though small, can play an important role in the
wave dynamics when the waves propagate over long distances. Previous
experiments have shown that the dissipation rate of waves is strongly
affected by conditions at the free surface. So to derive a model that
predicts dissipation rate, one usually allows for weak viscosity and
assumes one of three types of boundary conditions at the surface: (i)
the surface is shearfree, also referred to as ``clean'', (ii) the surface
admits no tangential velocities, also referred to as ``fully contaminated'',
or (iii) the surface is elastic. Here we discuss experiments within
the context of these three models in an effort to better understand
the boundary condition at the airwater interface and the ranges of
applicability of these models.

Josserand, Christophe
Institut D'Alembert

Wave turbulence in vibrating plates
The concept of wave turbulence that has been introduced originally
for ocean waves applies in fact in very different domains. We have recently
shown theoretically and numerically that wave turbulence could be observed
on elastic plates. Experiments performed by different groups have however
shown discrepancies with the theory. I will discuss here first the general
framework of wave turbulence on plates. Then I'll discuss how we can
explain the differences between theory and experiments. Finally, I will
show how inverse cascades could be present in the dynamicsm although
it is a priori not possible.

Kharif, Christian
Institut de Recherche sur les Phenomenes Hors Equilibre

Modulational instability of surface gravity
waves on water of finite depth with constant vorticity
Generally, in coastal and ocean waters, the velocity profiles are
typically established by bottom friction and by surface wind stress
and so are varying with depth. Currents generate shear at the bed
of the sea or of a river. For example ebb and flood currents due to
the tide may have an important effect on waves and wave packets. In
any region where the wind is blowing there is a surface drift of the
water and water waves are particularly sensitive to the velocity in
the surface layer. We consider the effect of constant non zero vorticity
on the BenjaminFeir instability of 2D surface gravity waves on arbitrary
depth.
Very recently, Thomas, Kharif & Manna [1] using the method of
multiple scales derived a nonlinear Schroedinger equation in finite
depth and in the presence of uniform vorticity. They demonstrated
that vorticity modifies significantly the modulational instability
properties of weakly nonlinear plane waves, namely the growth rate
and bandwidth. Furthermore, it was shown that these plane wave solutions
may be linearly stable to modulational instability for an opposite
shear current independently of the dimensionless parameter kh, where
k and h are the carrier wavenumber and depth respectively. Within
the framework of the fully nonlinear water wave equations, Francius
& Kharif, have recently investigated the modulational instability
of a uniform wave train on a shearing flow of constant vorticity and
extended to steeper waves the results of Thomas, Kharif & Manna
[1].

Lushnikov, Pavel
University of New Mexico

Logarithmic scaling of wave collapse
The dynamics of quasimonochromatic wave packet on the free
surface of infinite depth fluid is described by the focusing twodimensional
(2D) nonlinear Schrodinger equation (NLSE) for short enough wavelength
(when the capillary force is significant). The dynamics of similar wave
packet in the case of finite depth is given by 2D DaveyStewartson (BenneyRoskes)
equation (DSE). Both NLSE and DSE have generic solutions in the form
of finitetime singularity accompanied by the contraction of the spatial
scale of solution to zero which is called by wave collapse.These collapses
are responsible for the formation of the strongly nonlinear waves on
a fluid surface.
We study the universal selfsimilar behaviour near collapse time t_c,
i.e. the spatial and temporal structures near singularity. Collapses
in both NLSE and DSE share a strikingly common feature that the collapsing
solutions have a form of a rescaled soliton. The time dependence of
the rescaled soliton width L(t) determines also the solution amplitude
~1/L(t). At leading order L(t)~ (t_ct)^{1/2} for both NLSE and DSE.
Collapse of NLSE requires the modification of that scaling which has
a wellknown loglog form ~ (\ln\ln(t_ct))^{1/2}. Loglog scaling
for NLSE was first obtained asymptotically in 1980's and later proven
by Merle and Raphael in 2006. However, it remained a puzzle that this
scaling was never clearly observed in simulations or experiment. Here
solved that puzzle by developing a perturbation theory beyond the leading
order logarithmic corrections for NLSE. We found that the classical
loglog modification NLSE requires doubleexponentially large amplitudes
of the solution ~10^10^100, which is unrealistic to achieve in either
physical experiments or numerical simulations. In contrast, we found
that our new theory is valid starting from quite moderate (about 3 fold)
increase of the solution amplitude compare with the initial conditions.
New scaling is in excellent agreement with simulations.

Masmoudi, Nader
New York University

Nonlinear inviscid damping in 2D Euler
We prove the global asymptotic stability of shear flows close
to planar Couette flow in the 2D incompressible Euler equations. Specifically,
given an initial perturbation of the Couette flow which is small in
a suitable regularity class we show that the velocity converges strongly
in L2 to another shear flow which is not far from Couette. This strong
convergence is usually referred to as "inviscid damping" and
is roughly analogous to Landau damping in the Vlasov equations. Joint
work in progress with Jacob Bedrosian

Mei, Chiang
Massachusetts Institute of Technology

Nonlinear long waves over a muddy beach
Abstract

Melville, Ken
University of California, San Diego

The Equilibrium Dynamics and Statistics of WindDriven GravityCapillary
Waves
Recent field observations and modeling of breaking surface gravity
waves suggest that airentraining breaking is not sufficiently dissipative
of surface gravity waves to balance the dynamics of windwave growth,
nonlinear interactions and dissipation for the shorter gravity waves
of O(10) cm wavelength. Theories of parasitic capillary waves that
form at the crest and forward face of shorter steep gravity waves
have shown that the dissipative effects of these waves may be one
to two orders of magnitude greater than the viscous dissipation of
the underlying gravity waves. Thus the parasitic capillaries may provide
the required dissipation of the short windgenerated gravity waves.
This has been the subject of speculation and conjecture in the literature.
Using the nonlinear theory of Fedorov & Melville (1998), we show
that the dissipation due to the parasitic capillaries is sufficient
to balance the wind input over some range of wave ages and wave slopes.
The range of wavelengths over which these parasitic capillary waves
are dynamically significant approximately corresponds to the range
of wavelengths that are suppressed by oil on water, as measured by
Cox & Munk (1954), who also found that these waves contributed
significantly to the mean square slope of the ocean surface, which
they measured to be proportional to the wind speed. Here we show that
that the mean square slope predicted by the theory is proportional
to the square of the friction velocity of the wind, u*2, for small
wave slopes, and to u* for larger slopes.

Onorato, Miguel
Università di Torino

Modulational instability, wave breaking
and formation of large scale dipoles in the atmosphere
The Direct Numerical Simulation (DNS) of the NavierStokes equation
for a twophase flow (water and air) is used to study the dynamics
of the modulational instability of free surface waves and its contribution
to the interaction between ocean and atmosphere. If the steepness
of the initial wave is large enough, we observe a wave breaking and
the formation of large scale dipole structures in the air. Because
of the multiple steepening and breaking of the waves under unstable
wave packets, a train of dipoles is released and propagate in the
atmosphere at a height comparable with the wave length. The amount
of energy dissipated by the breaker in water and air is considered
and, contrary to expectations, we observe that the energy dissipation
in air is comparable to the one in the water.

Pelinovsky, Efim
Russian Academy of Sciences

Rogue Waves in Shallow Waters
An overview on the problem of rogue or freak wave formation
in shallow waters is given. A number of huge wave accidents, resulting
in damages, ship losses and people injuries and deaths, are known and
summarized in recent catalogues and books. This presentation addresses
the nature of the rogue wave problem from a general viewpoint based
on nondispersive and weakly dispersive wave process ideas. We start
by introducing some primitive elements of sea wave physics with the
purpose of paving the way for further discussion. We discuss linear
physical mechanisms which are responsible for high wave formation, at
first. Nonlinear effects which are able to cause rogue waves are emphasized.
In conclusion we briefly discuss the generality of the physical mechanisms
suggested for the rogue wave explanation; they are valid for rogue wave
phenomena in geophysics and plasma.
(in collaboration with Alexey Slunyaev, Ira Didenkulova, Christian.
Kharif, Irina Nikolkina, Anna Sergeeva, Tatiana Talipova and Ekaterina
Shurgalina)

Perrie, Will
Bedford Institute of Oceanography 
Nonlinear Energy Transfers in a Wind Wave Spectrum
Nonlinear wavewave interactions involving quadruplets constitute
the basis for modern wave modeling and wave forecasts. In most modern
operational wave models such as WAM, or WAVEWATCHIII, quadruplet wavewave
interactions are simulated by the Discrete Interaction Approximation,
commonly referred to as the DIA, as formulated by WAMD1 (1988). We give
a description of DIA, and we introduce a new approximation, the TwoScale
Approximation (TSA), based on the separation of a spectrum into a broadscale
component and a localscale (perturbation) component. TSA uses a parametric
representation of the lowerorder or "broadscale" spectral
structure, while preserving the degrees of freedom essential to a detailedbalance
source term formulation, by including the second order scale in the
approximation. We present tests using idealized wave spectra, including
JONSWAP spectra (Hasselmann et al., 1973) with selected wave hypothetical
peakednesses, and perturbation cases, as well as well as additional
tests for fetchlimited wave growth, and storm waves generated by hurricane
Juan (2003). Generally, TSA is shown to work well when its basic assumptions
are met, when its first order, broadscale term represents most of the
spectrum, and its second order term is a perturbationscale residual
term representing the rest of the spectrum. These conditions are easily
met for test cases involving idealized JONSWAPtype spectra and in timestepping
cases when winds are spatially and temporally constant. To some extent,
they also appear to be met in more demanding conditions, when storms
move through their life cycles, with winds that change in speed and
direction, and with complex wave spectra, involving swellwindsea interactions,
multiple peaks fp1, fp2, …and directional shears. In these cases,
we show that TSA can be generalized (e.g. double, or multiple TSAs)
and work reasonably well when the spectrum is partitioned according
to individual spectral peaks. In this situation, TSA's basic assumptions
are met in each segment of the spectrum (each spectral peak region),
in terms of its first order broadscale, and second order perturbationscale
terms. Comparisons will be made with integrations of the full Boltzmann
integral (FBI) for quadruplet wavewave interactions.

Sajjadi, Shahrdad G.
Embry Riddle Aeronautical University

Enhanced transfer of wind energy into surface waves
Asymptotic multilayer analyses and computation of solutions
for turbulent flows over steady and unsteady monochromatic surface wave
are reviewed, in the limits of low turbulent stresses and small wave
amplitude. The structure of the flow is defined in terms of asymptoticallymatched
thinlayers, namely the surface layer and a critical layer, whether
it is 'elevated' or 'immersed', corresponding to its location above
or within the surface layer. The results particularly demonstrate the
physical importance of the singular flow features and physical implications
of the elevated critical layer in the limit of the unsteadiness tending
to zero. These agree with the variational mathematical solution of Miles
[J. Fluid Mech., 3, 185204 (1957)] for small but finite growth rate,
but they are not consistent physically or mathematically with his analysis
in the limit of growth rate tending to zero. As this and other studies
conclude, in the limit of zero growth rate the effect of the elevated
critical layer is eliminated by finite turbulent diffusivity, so that
the perturbed flow and the drag force are determined by the asymmetric
or sheltering flow in the surface shear layer and its matched interaction
with the upper region. But for groups of waves, in which the individual
waves grow and decay, there is a net contribution of the elevated critical
layer to the wave growth. Critical layers, whether elevated or immersed,
affect this asymmetric sheltering mechanism, but in quite a different
way to their effect on growing waves. These asymptotic multilayer methods
lead to physical insight and suggest approximate methods for analyzing
higher amplitude and more complex flows, such as flow over wave groups.

Saut, JeanClaude
University of ParisSud

Long time existence for some water wave
systems
Most of approximate models for surface and internal waves are derived
from some asymptotic expansions (with respect to "small"
parameters) in various regimes of amplitudes, wavelenghts,.. Their
solutions are not supposed to be good approximates for all times but
only on some relevant "long"" time scales. Proving
such long time existence is not an easy task for most of water waves
systems. This talk will present such results for Boussinesq type systems
and for a "full dispersion" system.

Segur, Harvey
University of Colorado

The nonlinear Schrödinger equation, dissipation and ocean swell
The focus of this talk is less about how to solve a particular
mathematical model, and more about how to find the right model of a
physical problem. The nonlinear Schrödinger (NLS) equation was
discovered as an approximate model of wave propagation in several branches
of physics in the 1960s. It has become one of the most studied models
in mathematical physics, because of its interesting mathematical structure
and because of its wide applicability – it arises naturally as
an approximate model of surface water waves, nonlinear optics, BoseEinstein
condensates and plasma physics. In every physical application, the derivation
of NLS requires that one neglect the (small) dissipation that exists
in the physical problem. But our studies of water waves (including freely
propagating ocean waves, called “swell”) have shown that even
though dissipation is small, neglecting it can give qualitatively incorrect
results. This talk describes an ongoing quest to find an appropriate
generalization of NLS that correctly predicts experimental data for
ocean swell. As will be shown, adding a dissipative term to the usual
NLS model gives correct predictions in some situations. In other situations,
both NLS and dissipative NLS give incorrect predictions, and the “right
model” is still to be found.

Tataru, Daniel
University of California, Berkeley

Two dimensional water waves
TBA

Zakharov, Vladimir E.
University of Arizona

Coxeter Lecture 
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Confirmed Participants as of May 3, 2013
Full Name 
University/Affiliation 
Akhmediev, Nail 
Australian National University, Canberra 
Alazard, Thomas 
Ecole Normale Superieure, Paris 
Ayala, Diego 
McMaster University 
Bustamante, Miguel 
University College Dublin 
Camassa, Roberto 
University of North Carolina at Chapel Hill 
Castaing, M. Richard 
Ecole Polytechnique 
Chabchoub, Amin 
Hamburg University of Technology 
Chaudhary, Osman 
Boston University 
Choi, Wooyoung 
New Jersey Institute of Technology 
Craig, Walter 
McMaster University 
Dutykh, Denys 
University College Dublin 
Garcia, Carlos 
McMaster University 
Gemmrich, Johannes 
University of Victoria 
Goncalves, Iury Angelo 
National Institute for Space Research 
Grimshaw, Roger 
Loughborough University 
Guyenne, Philippe 
University of Delaware 
Hara, Tetsu 
University of Rhode Island 
He, Yangxin 
University Of Waterloo 
Helfrich, Karl 
Woods Hole Oceanographic Institute 
Henderson, Diane 
Pennsylvania State University 
Henry, Legena 
University of the West Indies 
Hoang, Tung 
University of Waterloo 
Josserand, Christophe 
CNRS & Université Pierre et Marie Curie (Paris VI) 
Kartashova, Elena 
Johannes Kepler University 
Kharif, Christian 
Institut de Recherche sur les Phenomenes Hors Equilibre 
Korotkevich, Alex 
University of New Mexico 
Lacave, Christophe 
l'université ParisDiderot (Paris 7) 
Lamb, Kevin 
University of Waterloo 
Lannes, David 
Ecole Normale Superieure  Paris 
Li, Yile 

Linares, Felipe 
IMPA 
Lushnikov, Pavel 
University of New Mexico 
Masmoudi, Nader 
Courant Institute of Mathematical Sciences, NYU 
Mei, Chiang 
Massachusetts Institute of Technology 
Melville, Kendall 
University of California, San Diego 
Oliveras, Katie 
Seattle University 
Onorato, Miguel 
Università di Torino 
Pelinovsky, Efim 
Russian Academy of Sciences 
Perrie, Will 
Bedford Institute of Oceanography 
Polnikov, Vladislav 
Obukhov Institute of Atmospheric Physics 
Quinn, Brenda 
University College Dublin 
Rabinovich, Alexander 
Institute of Ocean Sciences 
Rakhimov, Shokhrux 
McMaster University 
Restrepo, Juan 
University of Arizona 
Rowe, Kristopher 
University of Waterloo 
Sajjadi, Shahrdad 
EmbryRiddle Aeronautical University 
Saut, JeanClaude 
University of ParisSud 
Schober, Constance 
University of Central Florida 
Segur, Harvey 
University of Colorado 
Shirikyan, Armen 
Université de CergyPontoise 
Tataru, Daniel 
University of California, Berkeley 
Toledo, Yaron 
Wuppertal University 
Totz, Nathan 
Duke University 
Trichtchenko, Olga 
University of Washington 
Vasan, Vishal 
Pennsylvania State University 
Viotti, Claudio 
University college dublin 
Webb, Adrean 
University of Colorado at Boulder 
Wickramarachchi, Subasha 
University of Waterloo 
Yang, Chiru 
McMaster University 
Zakharov, Vladimir E. 
University of Arizona 
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