Nonlinear water waves
Monday 7th August 2017 to Thursday 10th August 2017
09:00 to 09:50  Registration  
09:50 to 10:00  Welcome from David Abrahams (INI Director)  
10:00 to 11:00 
Robin Johnson Physical oceanography: an applied mathematician's approach
In this talk we present the problem, based on the Euler
equation, which is at the heart of any mathematical description of the motion
of our oceans. This involves the prescription of a suitable model for the fluid
(e.g. inviscid but with vorticity), together with an appropriate set of
boundary conditions (and associated initial data that is consistent with our
solutions will be assumed to exist). The approach is based on classical ideas
of fluid mechanics, involving nondimensionalisation and scaling, leading to a
reasonable and suitable reduction of the system that is amenable to further
analysis. The aim is to show that such methods, which involve relatively little
conventional ‘physical oceanographic modelling’, can uncover some fundamental
processes that underpin many of the observed movements of the oceans
(typically, on the large scale). However, in addition to approximate (asymptotic)
versions of the various problems of interest, we find that this systematic
approach also enables some relevant and useful exact solutions of the system to
be developed.
In particular, we will show how problems can be
formulated that relate
to: linear waves in the presence of a thermocline over an
arbitrary flow (near the Pacific Equator); exact solutions associated with the
Antarctic Circumpolar Current and a corresponding equatorial flow; a
threedimensional flow, with a thermocline, appropriate to a neighbourhood of
the Pacific Equatorial Undercurrent; a representation of gyres of any size.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Gareth Thomas Conservation relations for twodimensional wavecurrent interactions
For a uniform current interacting with a monochromatic
wavetrain in water of locally constant depth, the resulting wavefield will be
irrotational.
The conservation relations required to link the
slowlyvarying properties of the wavefield to the local properties, such as
those induced by a change in water depth, can be obtained by an appropriate
method such as a variational approach of Whitham. There are relatively few
difficulties for linear waves but the problem is much more difficult for nonlinear
waves and this requires a careful description of the reference frame and the
free surface datum level. In this approach the Bernoulli constants, sometimes
referred to as the secondary triad, play an important role. When the current is
rotational, difficulties arise even for linear waves on a current possessing
uniform vorticity. The talk will identify and describe these difficulties in
the rotational case, for both linear and nonlinear waves, but will not promise
to provide a definitive formulation.

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Alexandru Ionescu On long term regularity of water wave models
We discuss the construction of long term and global
solutions for certain water wave models under the influence of surface tension
and/or gravity. This is based on joint work with Fabio Pusateri, Yu Deng, and
Benoit Pausader.

INI 1  
14:30 to 15:30 
Hisashi Okamoto Some thoughts on the role of the convection terms in the fluid mechanical PDEs.
Nonlinear PDEs appearing in fluid mechanics often have a convection term, whose role is not wellunderstood in my viewpoint. I want to state some problems which I cannot answer and wish to usher in young people who may solve them. A key theme is that the convection term prevents the solution from blowingup. I want to demonstrate this `proposition' by examples.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Andre Nachbin Capturing the flow structure beneath water waves
This talk addresses two problems, studied through
numerical models, for capturing the flow structure beneath surface water waves.
First we consider (nonlinear, traveling) periodic
rotational waves, the appearance of stagnation points and the possibility of
pressure anomalies. Secondly, we consider linear waves in the presence of quite
general topographies and the numerical construction of a threedimensional DirichlettoNeumann
operator. Simulations, restricted to a twodimensional free surface, illustrate
that the vertical structure of the 3D flow is captured accordingly even in the
presence of large depth variations.

INI 1  
17:00 to 18:00  Welcome Wine Reception at INI 
09:00 to 10:00 
John Grue Nonlinear surface waves at finite depth with and without surface cover
Results
of recent and ongoing sets of experiments on nonlinear surface waves at finite
depth are presented. We are particularly investigating the properties of the
maximum elevation and waveinduced velocity as well as particle drift. While
waves without a cover are realistic for ocean surface waves, the effect of a
cover is important for waves interacting with an ice sheet. Experiments are
connected to available measurements at moderately shallow reefs or shallow seas
where in the latter case winddriven waves are observed, obtaining a strong
limiting effect by wave breaking, on the amplitude. The vertically integrated
particle drift is zero. The particle drift is strongly enhanced below the wave
surface and above the bottom, due to the streaming effects, where also the
shear is enhanced, with implications to a magnified Stokes drift. The particle
drift velocity is zero at two vertical positions. Below a cover (ice) the drift
shear becomes negative. The investigations are motivated by requests from the
offshore wind industry as well as Polar research.

INI 1  
10:00 to 11:00 
Emilian I Parau Numerical study of solitary waves under continuous or fragmented ice plates
Nonlinear hydroelastic waves travelling at the surface of an ideal fluid covered by a thin ice plate are presented. The continuous iceplate model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchoff's hypothesis. Twodimensional solitary waves are computed using boundary integral methods and their evolution in time and stability is analysed using a pseudospectral method based on FFT and in the expansion of the DirichletNeuman operator. Extensions of this problem including internal waves and threedimensional waves will be considered. When the iceplate is fragmented, a new model is used by allowing the coefficient of the flexural rigidity to vary spatially. The attenuation of solitary waves is studied by using twodimensional simulations. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Onno Bokhove On linear and nonlinear waveship interactions
Dynamics of buoy and ship motion in linear and nonlinear water waves will be considered. The starting point will be a "preLuke" variational principle (see Cotter & B. [1] for the case without ship) for the combined waterwave motion and the dynamics of a rigid buoy/ship. This principle reduces to Luke's variational principle when a scaled density D is set to unity, in the case without buoy/ship dynamics. The coupling between waves and ship is straightforward: that the shape of the water surface equals the shape of the moving buoy/ship is imposed via a Lagrange multiplier. PreLuke's principle has the advantage that the boundary conditions on the Lagrange multiplier emerge directly from the variational principle due to the weak imposition of the density constraint, while in Luke's form of the variational principle of the coupled dynamics the boundary condition needs to be imposed on the Lagrange multiplier. The consequences for a completely spacetime variational numerical discretisation will be discussed. Attempts to find an exact/reduced test solution in the shallow water limit of buoyfluid motion may be considered as well. Movies of a waveenergy device and preliminary numerical solutions [2,3,4] will be used throughout the presentation. [1] C. Cotter and O.B. 2010: Water wave model with accurate dispersion and vertical vorticity. Peregrine Commemorative Issue J. Eng. Maths. 67, 3354. [2] Youtube channel Anna Kalogirou: Movies of simulations and scaled waveenergy device: https://www.youtube.com/user/anna9k [3] A. Kalogirou and O.B 2016: Mathematical and numerical modelling of wave impact on waveenergy buoys. Proc. ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2016), Busan, South Korea. [4] A. Kalogirou, O.B. and D. Ham 2017: Modelling of nonlinear wavebuoy dynamics using constrained variational methods. Proc. ASME 2017 36th Int. Conf. Ocean, Offshore and Arctic Engineering (OMAE 2017), Trondheim, Norway. 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Vladimir Vladimirov CraikLeibovich Equation, Distinguished Limits, Drifts, and PseudoDiffusion
The CraikLeibovich Equation (CLE) describes the Langmuir circulations in the upper layer of the oceans, lakes, etc. In this lecture we consider CLE and the related notions of drifts and pseudodiffusion. In our approach, CLE describes general vortex dynamics of oscillating flows, not only the Langmuir circulations. A number of new results on CLE are presented. An important elements of our presentation is the systematic deriving of boundary conditions, which represents a more difficult tusk than the deriving of averaged equations. We also consider a 'linearized’ version of CLE, which is different from that obtained by a straightforward linearization of an averaged equation. A possible treetiming procedure for the generalisation of CLE is proposed. The effects of viscosity and density stratification has been additionally reexamined. We also discuss two generalisations of CLE. First one is a MagnetoHydroDynamic version of CLE. It may have relation to the MHD dy namo and to the forming of various flows and phenomena in stars, galaxies etc. Second generalisation deals with a version of CLE for compressible fluid, where similar to CLE equations appears due to oscillations caused be acoustic waves. Mathematically, our consideration is based on a unified viewpoint. We use the twotiming method, Eulerian averaging, and the concept of distinguished limit. Such a consideration emphasises the generality, simplicity, and the rigour in all our derivations. We do not accept any additional assumption and suggestions except the presence of small parameters and the applicability of rigorous asymptotic procedure. Our approach allows us to obtain the classical results much simpler than it has been done before, and hence a noticeable progress in the obtaining of new results can be achieved. Key Words: CraikLeibovich Equation, Lamgmuir Circulations, TwoTiming Method, Distinguished Limits, Drift, PseudoDiffusion. 
INI 1  
14:30 to 15:30 
Michael Stiassnie Harnessing wavepower in open seas
The promise of wavedriven renewable energy is clear: with oceans covering more than 70% of the Earth's surface, and the ability of ocean waves, once created by the wind, to transport energy over large distances, it is natural to want to tap this vast resource. The rate of energy transfer from wind to waves is larger by a factor of 3 than the world's present power consumption. However, only 5 % of this power reaches coastal waters, the remainder being dissipated by wave breaking in open seas. This highlights the need to study wavepower harvesting in deep open seas: to estimate the global wavepower potential, develop devices suitable for deep water, and understand the geometry and layout of large arrays that will be needed to capture wave energy on a globally significant scale. This talk will report on recent progress in these various directions. A novel selfreacting twincylinder wave energy converter (WEC) is discussed, which extracts energy from three modes of motion without requiring a fixed bottom reference. This WEC is employed to investigate survivability in extreme seastates, and also as one element in a large array of energy converters. In order to overcome the computational complexities of multiplescattering theory, a new approximate framework for sparse WEC arrays is developed, which allows fast and flexible calculations of array absorption, transmission, and reflection for changing wave conditions, including directional seas. The tools developed are also applicable to other resource assessment and WEC design studies, and present a move towards understanding the third generation of waveenergy converters: large, deepwater devices deployed in farms throughout the open ocean. References D. Xu, R. Stuhlmeier and M. Stiassnie, Harnessing wave power in open seas II: very large arrays of waveenergy converters for 2D sea states, J. Ocean Eng. Marine Energy 3 (2017), 151160. D. Xu, R. Stuhlmeier and M. Stiassnie, Assessing the size of a twincylinder wave energy converter designed for real seastates, submitted (arXiv:1605.00428). M. Stiassnie, U. Kadri and R. Stuhlmeier, Harnessing wavepower in open seas, J. Ocean Eng. Marine Energy 2 (2016), 4757. 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Eugen Varvaruca Global bifurcation of steady gravity water waves with constant vorticity
We consider the problem of twodimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. By using a conformal mapping from a strip onto the fluid domain, the governing equations are recasted as a onedimensional pseudodifferential equation that generalizes Babenko's equation for irrotational waves of infinite depth. We show how an application of the theory of global bifurcation in the realanalytic setting leads to the existence of families of waves of large amplitude that may have critical layers and/or overhanging profiles. This is joint work with Adrian Constantin and Walter Strauss.

INI 1 
09:00 to 10:00 
Tatsuo Iguchi IsobeKakinuma model for water waves as a higher order shallow water approximation
We justify rigorously an IsobeKakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order $\delta^2$, where $\delta$ is a small nondimensional parameter defined as the ratio of the typical wavelength to the mean depth. The GreenNaghdi equations are known as higher order approximate equations to the water wave equations with an error of order $\delta^4$. In this paper we show that the IsobeKakinuma model is a much higher approximation to the water wave equations with an error of order $\delta^6$.

INI 1  
10:00 to 11:00 
Mayumi Shoji Numerical computation of water waves with discontinuous vorticity
We consider progressive water waves with a piecewise constant vorticity distribution. Both capillarygravity waves and gravity waves of finite depth are considered. This is a bifurcation problem of a complicated structure of solutions with many parameters. It is hard to classify the structures of solutions mathematically. We thus resort to a numerical method in order to see their bifurcating phenomena with systematic computations. Another concern of ours is to see whether and when stagnation points appear. The difficulties for numerical computations are that it is a free boundary problem and we need a formulation not to exclude stagnation points. We will show our numerical results with various values of vorticity, depth of fluid and traveling speed. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Yoshimasa Matsuno Twocomponent CamassaHolm system and its reductions
My talk is mainly concerned with an
integrable twocomponent CamassaHolm (CH2) system which describes the
propagation of nonlinear shallow water waves. After a brief review of strongly nonlinear
models for shallow water waves including the GreenNaghdi and related systems, I
develop a systematic procedure for constructing soliton solutions of the CH2
system. Specifically, using a direct
method combined with a reciprocal transformation, I obtain the parametric
representation of the multisoliton solutions, and investigate their properties.
Subsequently, I show that the CH2 system
reduces to the CH equation and the twocomponent HunterSaxton (HS2) system by
means of appropriate limiting procedures. The corresponding expressions of the
multisoliton solutions are presented in parametric forms, reproducing the
existing results for the reduced equations. Also, I discuss the reduction from
the HS2 system to the HS equation. Last, I comment on an interesting issue associated
with peaked wave (or peakon) solutions of the CH, DegasperisProcesi, Novikov
and modified CH equations.

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Vladimir Kozlov Smallamplitude steady water waves on flows with countercurrents.
The
twodimensional freeboundary problem describing steady gravity waves with vorticity is considered
for water of finite depth.
It is known
that the whole set of smallamplitude waves on unidirectional rotational flows
is exhausted by Stokes and solitary waves. In the case of flows with
countercurrents, there occur other patterns of behaviour. Two results of this
kind will be discussed. The first one concerns the existence of Nmodal steady
waves, whereas the second result deals with the following fact. If the number of roots of the dispersion
equation is greater than one, then the major part of the plethora of
smallamplitude waves is represented by nonsymmetric ones.
This is a
joint work with E. Lokharu, Lund University.

INI 1  
14:30 to 15:30 
Amin Chabchoub Rogue and Shock Waves within the Framework of Weakly Nonlinear Evolution Equations  Applicability and Limitations
Extreme ocean waves, also referred to as freak or rogue waves (RWs), are known to appear without warning and have a disastrous impact on ships and offshore structures as a consequence of the substantially large wave heights they can reach. Studies on RWs have recently attracted the scientific interest due to the interdisciplinary universal nature of the modulation instability (MI) of weakly nonlinear waves as well as for the sake of accurate modeling and prediction of these mysterious extremes. Indeed, solutions of the nonlinear Schrödinger equation provide advanced backbone models that can be used to describe the dynamics of RWs in time and space, providing therefore prototypes for a deterministic investigation of MI. Experiments as well as related numerical simulations on breathers will be presented. Furthermore, the existence of such localized structures in realistic sea state conditions will be discussed as well. Moreover, experiments on shock waves in shallow water will be also reported.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Michio Yamada Zonal flows and wave resonance.  Rossby wave case 
We discuss resonant and nonresonant interaction of waves and timedevelopment of energy distribution in a wave system. We take, not a water wave system, but a Rossby wave system on a rotating sphere, where spherical harmonic functions correspond to Rossby waves which are labelled by total wave number and longitudinal wavenumber. In the Rossby wave system, a zonal flow pattern is known to emerge even from random and isotropic initial conditions in the course of time development. We classify the Rossy waves into 4 groups according to whether it has resonant waves or not, and also whether it represents a zonal flow (i.e. with zero longitudinal wavenumber) or not. Numerical simulation shows that when there is no zonal energy initially, the energy goes into zonal resonant modes from nonzonal and nonresonant modes, which suggests that the nonresonant interaction between the nonzonal nonresonant modes and the zonal resonant modes plays a dominant role in the formation of the zonal flows. Introducing 'effective energy transfer', which enables us to talk about the energy transfer from one mode to another mode, we find that the nonresonant interaction transfers energy from the nonresonant mode mostly to the resonant modes.

INI 1  
19:30 to 22:00  Formal Dinner at Emmanuel College 
09:00 to 10:00 
Samuel Walsh Stability of traveling water waves with a point vortex
In this talk, we will present recent results on the (conditional) orbital stabillity of twodimensional steady capillarygravity water waves with a point vortex. One can think of
these waves as an idealization of traveling waves with compactly supported vorticity. The
governing equations have a Hamiltonian formulation, and the waves themselves can be
realized as minimizers of an energy subject to fixed momentum.
We are able to deduce stability by using an abstract framework that generalizes the
classical work of Grillakis, Shatah, and Strauss. In particular, our theory applies to systems
where the sympletic operator is state dependent and may fail to be surjective.
This is joint work with Kristoffer Varholm and Erik Wahlén

INI 1  
10:00 to 11:00 
Benjamin Harropgriffiths Long time dynamics of some dispersive models arising from the study of water waves
We discuss some recent work on the long time dynamics of some dispersive models arising from the study of water waves. In particular we discuss the existence and long time dynamics of solutions to some internal wave models derived by Camassa and Choi that are natural 2d generalisations of the BenjaminOno and Intermediate Long Wave Equations in the case of weak transverse effects. This is joint work with Jeremy L. Marzuola.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Walter Strauss Upper bound on the slope of a steady water wave
Consider the angle of inclination of the profile of a steady 2D (inviscid,
symmetric, periodic or solitary) water wave subject to gravity. Although
the angle may surpass 30 degrees for some irrotational waves close to the
extreme Stokes wave, Amick proved in 1987 that the angle must be less than
31.15 degrees if the wave is irrotational. However, for any wave that is
not irrotational, the question of whether there is any bound on the angle
has been completely open. An example is the extreme Gerstner wave, which
has adverse vorticity and vertical cusps. Moreover, numerical calculations
show that waves of finite depth with adverse vorticity can overturn,
so the angle can be 90 degrees. On the other hand, Miles Wheeler and I
prove that there is an upper bound of 45 degrees for a large class of
waves with favorable vorticity and finite depth. Seung Wook So and I
prove a similar bound for waves with small adverse vorticity.

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
David Henry Prediction of the freesurface elevation for rotational water waves using the recovery of pressure at the bed
In this talk we consider the pressurestreamfunction relationship for a train of regular water waves propagating on a steady current, which may possess an arbitrary distribution of vorticity, in two dimensions. The application of such work is to both nearshore and offshore environments, and in particular for linear waves we provide a description of the role which the pressure function on the seabed plays in determining the freesurface profile elevation. Our approach is shown to provide a good approximation for a range of current conditions, leading to the derivation of expressions for the pressure transfer function, and the related pressure amplification factor, which generalise the wellknown formulae for irrotational waves. An implementation of the moderate current approximation renders these expressions more tractable, leading to quite elegant and explicit formulae. This is joint work with Gareth Thomas. 
INI 1  
14:30 to 15:30 
Rossen Ivanov Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom
CoAuthors: Alan Compelli and Mihail Todorov A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the socalled DirichletNeumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth. 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Ton van den Bremer Experimental validations of various aspects of the waveinduced mean flow for surface gravity wave groups
For surface gravity wave groups, the wellknown Stokes
drift and transport are complemented by a setdown (or setup) of the free
surface and an Eulerian return flow first described by LonguetHiggins and
Stewart (1962). We present experimental validation of these classical
secondorder theoretical predictions for unidirectional wave groups in a
laboratory flume at the University of Plymouth COASTlab and for directionally
spread wave groups in the University of Edinburgh FloWave facility. Firstly, we
present detailed PTV (particle tracking
velocimetry) measurements of the Lagrangian transport and
trajectories of nearneutrally buoyant particles underneath twodimensional
surface gravity wave groups in a laboratory flume. By focussing our attention
on wave groups of moderate steepness, we confirm the predictions of standard
secondorder multichromatic wave theory, in which the body of fluid satisfies
the potential flow equations. Particles near the surface are transported
forwards and their motion is dominated by Stokes drift.
Particles at sufficient depth are transported backwards
by the Eulerian return current. Secondly, we present detailed measurements of
the surface elevation of the secondorder long boundwaves in directionally
spread groups. We demonstrate that the setdown becomes a setup for
sufficiently large degrees of directional spreading using measurements in the
newly built FloWave facility.

INI 1  
17:00 to 18:00 
Anatoly Abrashkin Vortex waves in deep water: Lagrange approach
A lecture consists of two parts. The first one deals with a theory of weakly nonlinear vortex waves. The vorticity is set in the series expansion in the small parameter of wave steepness. Each term of this row is an arbitrary function of the vertical Lagrange coordinate. We study different types of the waves: the stationary waves on shear flow, the standing vortex waves and the spatial vortex waves in the low vorticity fluid. The perturbation theory up to the third order is analyzed. The nonlinear Schrödinger (NLS) equation describing weakly nonlinear wave packets in an infinitydepth fluid with nonuniform vorticity is obtained. The vorticity is assumed to be an arbitrary function of both Lagrangian coordinates and quadratic in the small parameter proportional to the wave’s steepness. The effects of vorticity are manifested in a shift of the wavenumber of the carrier wave and a changing of the coefficient in nonlinear term of the NLS equation. The modulated Gouyon waves are studied. There is a special case of the vortex waves for which the resulting nonlinearity in the NLS equation vanishes. The Gerstner wave belongs among them. The second part of the lecture presents the theory of strongly nonlinear waves. A vortex model of a rogue wave formation at the background of uniform waves is proposed. It based on an exact analytical solution of equations of 2D hydrodynamics of an ideal incompressible fluid. A unique feature of flows of this class is the dependence of complex coordinate of liquid particle’s motion on two functions that may be arbitrary to a large extent. As a consequence the model may be used for the analysis of different forms of surface pressure as well as of liquid vorticity, i.e. when taking into account both these factors of air flow impact on the surface waves simultaneously. A process of formation of a rogue wave in the field of the Gerstner wave is studied. The physical parameters of the rogue wave and feasibility of the proposed scenario are disc.

INI 1 