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Timetable (NWWW01)

Nonlinear water waves

Monday 7th August 2017 to Thursday 10th August 2017

Monday 7th 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 non-dimensionalisation 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 three-dimensional flow, with a thermocline, appropriate to a neighbourhood of the Pacific Equatorial Undercurrent; a representation of gyres of any size.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Gareth Thomas
Conservation relations for two-dimensional wave-current 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 slowly-varying 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.
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.
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 well-understood 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 blowing-up. I want to demonstrate this `proposition' by examples.  
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 three-dimensional Dirichlet-to-Neumann operator. Simulations, restricted to a two-dimensional free surface, illustrate that the vertical structure of the 3D flow is captured accordingly even in the presence of large depth variations.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 8th August 2017
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 wave-induced 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 wind-driven 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.
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 ice-plate model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchoff's hypothesis. Two-dimensional 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 Dirichlet-Neuman operator. Extensions of this problem including internal waves and three-dimensional waves will be considered.
When the ice-plate 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 two-dimensional simulations.

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Onno Bokhove
On linear and nonlinear wave-ship interactions
Dynamics of buoy and ship motion in linear and nonlinear water waves will be considered. The starting point will be a "pre-Luke" variational principle (see Cotter & B. [1] for the case without ship) for the combined water-wave 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. Pre-Luke'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 space-time variational numerical discretisation will be discussed. Attempts to find an exact/reduced test solution in the shallow water limit of buoy-fluid motion may be considered as well. Movies of a wave-energy 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, 33-54.

[2] Youtube channel Anna Kalogirou: Movies of simulations and scaled wave-energy device:

[3] A. Kalogirou and O.B 2016: Mathematical and numerical modelling of wave impact on wave-energy 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 wave-buoy dynamics using constrained variational methods. Proc. ASME 2017 36th Int. Conf. Ocean, Offshore and Arctic Engineering (OMAE 2017), Trondheim, Norway.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Vladimir Vladimirov
Craik-Leibovich Equation, Distinguished Limits, Drifts, and Pseudo-Diffusion
The Craik-Leibovich 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 pseudo-diffusion. 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 tree-timing procedure for the generalisation of CLE is proposed. The effects of viscosity and density stratification has been additionally re-examined. We also discuss two generalisations of CLE. First one is a Magneto-Hydro-Dynamic 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 two-timing 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: Craik-Leibovich Equation, Lamgmuir Circulations, Two-Timing Method, Distinguished Limits, Drift, Pseudo-Diffusion.
14:30 to 15:30 Michael Stiassnie
Harnessing wave-power in open seas
The promise of wave-driven 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 wave-power harvesting in deep open seas: to estimate the global wave-power 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 self-reacting twin-cylinder 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 sea-states, and also as one element in a large array of energy converters. In order to overcome the computational complexities of multiple-scattering 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 wave-energy converters: large, deep-water devices deployed in farms throughout the open ocean.


D. Xu, R. Stuhlmeier and M. Stiassnie, Harnessing wave power in open seas II: very large arrays of wave-energy converters for 2D sea states, J. Ocean Eng. Marine Energy 3 (2017), 151-160.

D. Xu, R. Stuhlmeier and M. Stiassnie, Assessing the size of a twin-cylinder wave energy converter designed for real sea-states, submitted (arXiv:1605.00428).

M. Stiassnie, U. Kadri and R. Stuhlmeier, Harnessing wave-power in open seas, J. Ocean Eng. Marine Energy 2 (2016), 47-57.
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 two-dimensional 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 one-dimensional 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 real-analytic 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.
Wednesday 9th August 2017
09:00 to 10:00 Tatsuo Iguchi
Isobe-Kakinuma model for water waves as a higher order shallow water approximation
We justify rigorously an Isobe-Kakinuma 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 Green-Naghdi 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 Isobe-Kakinuma model is a much higher approximation to the water wave equations with an error of order $\delta^6$.
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 capillary-gravity 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.

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Yoshimasa Matsuno
Two-component Camassa-Holm system and its reductions
My talk is mainly concerned with an integrable two-component Camassa-Holm (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 Green-Naghdi 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 two-component Hunter-Saxton (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, Degasperis-Procesi, Novikov and modified CH equations.  
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Vladimir Kozlov
Small-amplitude steady water waves on flows with counter-currents.
The two-dimensional free-boundary problem describing steady  gravity waves with vorticity is considered for water of finite depth. It is known that the whole set of small-amplitude waves on unidirectional rotational flows is exhausted by Stokes and solitary waves. In the case of flows with counter-currents, there occur other patterns of behaviour. Two results of this kind will be discussed. The first one concerns the existence of N-modal 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 small-amplitude waves is represented by non-symmetric ones. This is a joint work with E. Lokharu, Lund University.
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.
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 time-development 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.  
19:30 to 22:00 Formal Dinner at Emmanuel College
Thursday 10th August 2017
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 two-dimensional steady capillary-gravity 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 
10:00 to 11:00 Benjamin Harrop-griffiths
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 Benjamin-Ono and Intermediate Long Wave Equations in the case of weak transverse effects. This is joint work with Jeremy L. Marzuola.
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.  
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 David Henry
Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed
In this talk we consider the pressure-streamfunction 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 sea-bed plays in determining the free-surface 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 well-known 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.
14:30 to 15:30 Rossen Ivanov
Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom
Co-Authors:  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 so-called Dirichlet-Neumann 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.

15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Ton van den Bremer
Experimental validations of various aspects of the wave-induced mean flow for surface gravity wave groups
For surface gravity wave groups, the well-known Stokes drift and transport are complemented by a set-down (or set-up) of the free surface and an Eulerian return flow first described by Longuet-Higgins and Stewart (1962). We present experimental validation of these classical second-order theoretical predictions for uni-directional wave groups in a laboratory flume at the University of Plymouth COAST-lab 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 near-neutrally buoyant particles underneath two-dimensional surface gravity wave groups in a laboratory flume. By focussing our attention on wave groups of moderate steepness, we confirm the predictions of standard second-order multi-chromatic 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 second-order long bound-waves in directionally spread groups. We demonstrate that the set-down becomes a set-up for sufficiently large degrees of directional spreading using measurements in the newly built FloWave facility.
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 infinity-depth fluid with non-uniform 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 non-linearity 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.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons