# Seminars (TWW)

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Event When Speaker Title Presentation Material
TWW 14th July 2014
10:00 to 11:00
P Janssen Random time series analysis and extreme wave events
While studying the problem of predicting freak waves it was realized that it would be advantageous to introduce a simple measure for such extreme events. Although it is customary to characterize extremes in terms of wave height or its maximum it is argued in this paper that wave height is an ill-defined quantity in contrast to, for example, the envelope of a wave train. Also, the last measure has physical relevance, because the square of the envelope is the potential energy of the wave train. The well-known representation of a narrow-band wave train is given in terms of an slowly varying envelope function and a slowly varying frequency $\omega = ?\partial \phi /\partial t$ where $\phi$ is the phase of the wave train. The key-point is now that the notion of a local frequency and envelope is generalized by applying the same definitions also for a wave train with a broad-banded spectrum. It turns out that this reduction of a complicated signal to only two parameters, namely envelope and frequency, still provides useful information on how to characterize extreme events in a time series. As an example, for a linear wave train the joint probability distribution of envelope height and period is obtained and is validated against results from a Monte Carlo simulation. The extension towards the nonlinear regime is, as will be seen, fairly straightforward.
TWW 14th July 2014
11:00 to 12:00
V Shrira Nonlinear dynamics of trapped waves on currents
The lecture gives an overview of a series works done with Alexey Slunyaev. We develop a new paradigm of how to describe linear and weakly nonlinear dynamics of waves on jet currents with a particular emphasis on new mechanisms of freak wave formation.

From numerous seamen accounts and insurer records it has been known for long time that rogue waves events are quite frequent on certain currents, e.g. on notorious in this respect Agulhas current. The theoretical explanation of this fact is still lacking, which is due to our overall poor understanding of wave nonlinear evolution on currents.

We develop a new systematic asymptotic theory of waves dynamics on jet currents which does not rely on the WKB approximation. First, solutions to the general linear 2D boundary-value problem for jet currents with arbitrary lateral profiles are found in terms of an asymptotic series in natural small parameters. In essence, we employ approximate separation of variables, which is always justified for the oceanic conditions.

There are three types of waves on the current: passing through, reflected and trapped. The key role in the new approach is played by the trapped modes. The trapped modes themselves (rather than comprising them harmonic components) participate in the nonlinear interactions. A general weakly nonlinear theory of trapped mode evolution is being developed. The properties of resonant interactions are qualitatively different from those between the waves in the absence of a current. In particular, three-wave interactions are always allowed in deep water and may play an important role in wave field evolution. There are three main advantages of the developed approach: (i) it is systematic and can identify and address the situations where the commonly adopted paradigm is not applicable; (ii) the current could be almost arbitrary, i.e. weak/strong, or smooth/with sharp edges; (iii) the initially 3-D problem is reduced to solving 1D evolution equations with the lateral and vertical dependence being prescribed by the corresponding modal structure. The modes can participate in both three and four- interactions. The prevalence of one type over another depends on parameters of waves and currents. A rich variety of possible regimes is being explored.

From the perspective of rogue wave occurrence we have identified several new mechanisms which have no analogues in the absence of currents.

TWW 14th July 2014
13:00 to 14:00
H Segur Tsunami
Tsunami have gained worldwide attention over the past decade, primarily because of the destruction caused by two tsunami: one that killed more than 200,000 people in coastal regions surrounding the Indian Ocean in December 2004; and another that killed 15,000 more and triggered a severe nuclear accident in Japan in March 2011. This talk has three parts. It begins with a description of how tsunami work: how they are created, how they propagate and why they are dangerous. This part involves almost no mathematics, and should be understandable by everyone. The second part of the talk is about the operational models now being used to provide tsunami warnings and forecasts. These models predict some features of tsunami accurately, and other features less accurately, as will be discussed. The last part of the talk is more subjective: what public policies could be enacted to mitigate some of the dangers of tsunami? Much of the material in this talk appeared in a paper by Arcas & Segur, Phil. Trans. Royal Soc. London, 370, 2012.
TWW 14th July 2014
14:00 to 15:00
Gravity - Capillary Solitary Waves and Lumps
For length scales of a few cm or less, where both gravity and surface tension are important, a rich variety of locally confined nonlinear waves of permanent form can propagate on the surface of shallow and deep water. Such gravity-capillary solitary waves (in one dimension) and lumps (in two dimensions) have been studied extensively in the last two decades theoretically, by analytical and numerical techniques, as well as in laboratory experiments. This body of work will be surveyed and open questions for future work will be highlighted.
TWW 14th July 2014
15:30 to 16:30
Birkhoff normal forms for the equations of water waves
A normal forms transformation for a dynamical system in a neighborhood of a stationary point retains only the significant nonlinearities, eliminating inessential terms. It is well known 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 the Birkhoff normal forms for this system of equations in the setting of spatially periodic solutions. Results include the regularity of the normal forms transformations, and the dynamical implications of the normal form. This is joint work with Catherine Sulem (University of Toronto).
TWW 14th July 2014
16:30 to 17:30
Standing and three-dimensional water waves
The talk will overview of results and open problems related to the Nash-Moser theory of periodic standing and spatial water waves. The topics include the Nash-Moser implicit function theorem, the descent method, and the paradifferential change of variables.
TWW 15th July 2014
09:30 to 10:30
Short Course: HOPS Methods: Boundary Value Problems: I
TWW 15th July 2014
11:00 to 12:00
T Fokas Short Course: The Unified Transform and Boundary Value Problems: I
TWW 15th July 2014
13:00 to 14:00
Nonlinear flexural and capillary-gravity waves
Different types of travelling waves which propagates at the surface of a fluid are presented. The influence of surface tension or of an elastic plate which cover the fluid is discussed. Stratified fluids and three-dimensional solutions will also be considered.
TWW 15th July 2014
14:00 to 15:00
Solitary waves with intrinsic wavelength
A solitary wave is a localized coherent structure that maintains its shape while it travels at constant speed. Solitary waves are caused by a cancellation of nonlinear and dispersive effects in the medium. In this talk, we discuss a special type of solitary wave: wavepacket solitary waves. In contrast to the celebrated KdV soliton, wavepacket solitary waves have intrinsic length scale and feature oscillatory decaying tails. Two examples, capillary-gravity waves and flexural-gravity waves, will be presented to illustrate the importance of wavepacket solitary waves, the numerical methods for constructing these waves and their dynamics.
TWW 15th July 2014
15:30 to 16:30
J-M Vanden-Broeck New nonsymmetric gravity-capillary solitary waves and related problems
Solitary waves with decaying tails propagating at the surface of a fluid are considered. Such waves are known to exist when both gravity and surface tension are included in the dynamic boundary condition. Other examples include hydroelastic waves. These solitary waves exist both in two and three dimensions. Three dimensional waves are characterised by decaying oscillations in the direction of propagation and monotonic decay in the direction perpendicular to the direction of propagation. Most of the waves previously calculated are symmetric.

In this talk we show that there are in addition many new families of nonsymmetric waves. These waves are computed for the full Euler equations in the two dimensional case. For three dimensional waves, a model equation is used.

TWW 16th July 2014
09:30 to 10:30
Short Course: High Order Perturbation of Surfaces (HOPS) Methods for Water Waves: II
TWW 16th July 2014
11:00 to 12:00
T Fokas Short Course: The Unified Transform and Boundary Value Problems: II
TWW 16th July 2014
13:00 to 14:00
Nearly time-periodic water waves
We compute new families of time-periodic and quasi-periodic solutions of the free-surface Euler equations involving standing waves, standing-traveling waves and collisions of solitary waves of various types. In the collision case, similarities notwithstanding, the new solutions are found to be well outside of the KdV and NLS regimes. A Floquet analysis shows that many of the new solutions are linearly stable to harmonic perturbations. Evolving such perturbations (nonlinearly) over tens of thousands of cycles suggests that the solutions remain nearly time-periodic forever. Parts of the talk will serve as an introduction to later talks in the session.
TWW 16th July 2014
14:00 to 15:00
Stability of periodic travelling waves of the water wave problem in one dimension
I will present recent results on the stability of periodic travelling wave solutions to the full water wave problem, mainly focusing on highfrequency instabilities. I will outline the numerical scheme used to compute the solutions for gravity-capillary waves as well as the numerical method to determine the spectral stability of these solutions.
TWW 16th July 2014
15:30 to 16:30
Stability of Traveling Waves with Vorticity
In this talk, I will discuss the spectral stability for the periodic traveling wave with constant vorticity. We will discuss relationships between stability of the traveling wave with respect to long-wave perturbations and the structure of the bifurcation curve for small amplitude solutions.
TWW 16th July 2014
16:30 to 17:30
B Deconinck Reconstructing the water surface from pressure measurements
A method is proposed to recover the water-wave surface elevation from pressure data obtained at the bottom of the fluid. The new method requires the numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of diff erent asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments.
TWW 17th July 2014
09:30 to 10:30
Short Course: High Order Perturbation of Surfaces (HOPS) Methods for Water Waves: III
TWW 17th July 2014
11:00 to 12:00
T Fokas Short Course: The Unified Transform and Boundary Value Problem: III
TWW 17th July 2014
15:00 to 16:00
Dynamics of critical layer in turbulent shear flows above unsteady water waves
Turbulent flow over unsteady monochromatic and Stokes waves has been investigated for various wave steepness ka and wave age $c_r/U_*$ using a Differntial Second-Moment (DSM) closure turbulence model. Also presented is DSM simulation of flow over steady monochromatic waves of different wave steepness for comparision with Direct Numerical Simulation (DNS) reported by Sullivan et al. (2000). It is shown that there is a good agreement between results obtained from DNS and DSM simulations. For the simulations of unsteady waves, the growth factor $kc_i$ is derived from a quadratic secular equation in complex phase speed c, obtained at the air-sea interface, whose solution leads to two wave speeds $c_1$ and $c_2$. The wave speed $c_1$ corresponds to free-surface waves damped by viscous stresses in the boundary layers at the interface, whilst the wave speed $c_2$ is associated with Tollmien-Schlichting instabilities in the shear airflow over the surface wave. It is shown that the wave growth through the complex phase speed $c_1$ is finite and waves do not become sharp-crested, therefore waves are unlikely to break. However, when the complex wave speed $c_2$ is adopted, for growth of Stokes waves, the waves rapidly peak and become sharp-crested. As the wave becomes relatively steep a secondary vortical motion is induced around the wave crest with a region of high pressure behind and low pressure ahead of the crest. This pressure asymmetry will very likely to cause the wave to break. It is also observed that as the wave grows and becomes steeper the critical layer elevates from the inner to the middle layer. Consequently, the cat's-eye become more and more asymmetrical and dramatically affects the flow field above the wave. Furthermore, at relatively high steepness, flow separation is observed in the lee of the wave. Also, in simulations reported here, the effect of non-separating sheltering is seen to be pronounced since the boundary layer is perturbed and thickeness on the leeside of the crest due to turbulent shear stress in the inner region. It is therefore argued that the wave growth is due to combined interactions between the elevated critical layer, the non-separated sheltering, and turbulence. From numerical results obtained, a new parametrization for the energy-transfer parameter is derived which agrees well with that obtained by Belcher & Hunt (1993) for slow moving waves and has the salient features of those evaluated numerically and asymptotically for intermidiate as well as fast moving waves. From the colective accounts of results of numerical simulations reported here, it is concluded that the critical layer has dynamical effect, and plays a crucial role, in shear flows over unsteady water waves.
TWW 17th July 2014
16:00 to 17:00
J Hunt New results on wind driven waves and negative tsunami waves at coastlines
Wind driven waves on the surfaces of water bodies have great practical importance on all scales (H. Jeffreys in 1926 illustrated his famous paper with photographs of ripples on the Newnham duck pond in Cambridge and huge waves on the Atlantic ocean). Despite 50 years of intense research, there is still disagreement about how idealised mathematical models apply to real air flow over real waves. In this lecture the model of Miles (1957) and Lighthill (1962) for inviscid shear flow over a growing monochromatic wave is explained in terms of critical layer dynamics, but is shown to be invalid for viscous, turbulent flow when the growth rate is asymptotically small. But analytical viscous turbulent shear flow models, also with critical layers, are valid in this limit and agree more closely with experimental wind profile data. But the former type of model (suitably adjusted) is widely used by oceanographers and meteorologists. However for real wind-driven waves both the critical-layer and sheltering mechanisms are significant and affect how waves travel in groups with characteristic asymmetry of the wave-shapes on the windward and leeward sides of the group. (Sajjadi et al. 2013).

The second part of the lecture concerns recent long tsunami-like waves, especially waves where the leading part of the wave is depressed, which was a characteristic feature of the tsunamis that approached the coast-lines of SEAsia in 2004 and Japan in 2011. As such waves travel from the source region, a non-linear Kortweg-de Vries model of R. Grimshaw, K.W. Lam and J.C.R. Hunt (2014) shows how when a depression wave is followed by an elevation (a 'breather) there is a transition at a location which can be estimated when the peak elevation catches up with the peak depression and nearly doubles in height before it then decreases and travels in front of the depression. In situations where the depression reaches the beach, recent modelling and laboratory studies show how the depression deepens, leading to a back flow and drying out of the beach, before there is a transition when the following much amplified elevation (in which the total momentum of the wave is maintained) surges up the beach and moves some kilometres inland, which corresponds with recent and past observations (Klettner et al. 2012).

TWW 18th July 2014
09:30 to 10:30
Short Course: High Order Perturbation of Surfaces (HOPS) Methods for Water Waves: IV
TWW 18th July 2014
13:00 to 14:00
P Guyenne Boundary element and spectral methods for water waves
TWW 18th July 2014
14:00 to 15:00
Variational principles for water waves beyond perturbations
An approximation method based on a 'relaxed' variational principle is presented. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters. A candidate model for tsunamis modelling over significant bathymetry variations is also presented.
TWW 18th July 2014
15:30 to 16:30
E Gagarina Finite element method for nonlinear free surfaces water waves
TWW 22nd July 2014
14:00 to 15:00
H Okamoto Trajectories of fluid particles in a water-wave
TWW 22nd July 2014
15:00 to 16:00
On the Benjamin-Feir instability
TWW 22nd July 2014
16:30 to 17:30
Waves over periodic topographies
TWWW02 23rd July 2014
11:15 to 12:00
G Schneider Validity and non-validity of the NLS approximation for the water wave problem - Recent developments and open problems
We consider the 2D water wave problem in case of finite depth with and without surface tension. We are interested in the validity of the NLS approximation for the description of surface water waves. After giving an overview about positive results we explain that in case of small surface tension there are situations where the NLS approximation fails to describe the water wave problem correctly.
TWWW02 23rd July 2014
14:00 to 14:45
Interface singularities for the Euler equations
In fluid dynamics, a "splash" singularity occurs when a locally smooth interface self-intersects in finite-time. It is now well-known that solutions to the water waves equations (and a host of other one-phase fluid interface models) has a finite-time splash singularity. By means of elementary arguments, we prove that such a singularity cannot occur in finite-time for vortex sheet evolution (or two-fluid interfaces). This means that the evolving interface must lose regularity prior to self-intersection. We give a proof by contradiction: we assume that such a singularity does indeed occur in finite-time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allows us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, giving the contradiction. This is joint work D. Coutand.
TWWW02 23rd July 2014
14:45 to 15:30
E Varvaruca Singularities of steady free surface water flows
We present some recent results, based on a geometric analysis approach, that provide a characterization of all possible singularities in two related free-boundary problems in hydrodynamics: that of steady two-dimensional gravity water waves and that of steady three-dimensional axisymmetric water flows under gravity. In the 2D problem, we outline a modern proof, using a blow-up analysis based on a monotonicity formula and a frequency formula, of the Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under some restrictive assumptions and by somewhat ad-hoc methods.) The new approach extends easily to the case when the effects of vorticity in the flow are included. Moreover, we explain how the methods can be adapted to the 3D axisymmetric problem, where several different types of singularities are possible, depending on whether one is dealing with a stagnation point, a point on the axis of symmetry, or both (in the case of the origin). For example, in the case of the origin, there are only two possible types of singular asymptotic behaviour: one is a conical singularity called Garabedian corner flow", and the other is a flat degenerate point; while in the case of points on the axis of symmetry different from the origin, cusps are the only possible singularities. These results were obtained in joint works with Georg Weiss (Dusseldorf).
TWWW02 23rd July 2014
16:15 to 17:00
Three-dimensional water waves
The existence of solitary-wave solutions to the three-dimensional water-wave problem with is predicted by the Kadomtsev-Petviashvili (KP) equation for strong surface tension and Davey-Stewartson (DS) equation for weak surface tension.The term solitary wave describes any solution which has a pulse-like profile in its direction of propagation, and these model equations admit three types of solitary waves. A line solitary wave is spatially homogeneous in the direction transverse to its direction of propagation, while a periodically modulated solitary wave is periodic in the transverse direction. A fully localised solitary wave on the other hand decays to zero in all spatial directions. In this talk I outline mathematical results which confirm the existence of all three types of solitary wave for the full gravity-capillary water-wave problem in its usual formulation as a free-boundary problem for the Euler equations. Both strong and weak surface tension are treated. The line solitary waves are found by establishing the existence of a low-dimensional invariant manifold containing homoclinic orbits. The periodically modulated solitary waves are created when a line solitary wave undergoes a dimension-breaking bifurcation in which it spontaneously loses its spatial homogeneity in the transverse direction; an infinite-dimensional version of the Lyapunov centre theorem is the main ingredient in the existence theorem. The fully localised solitary waves are obtained by finding critical points of a variational functional.
TWW 24th July 2014
11:00 to 12:00
M Wheeler The Froude number and solitary waves with vorticity
TWW 24th July 2014
12:00 to 12:30
M Cooker Violent wave motion due to impact
TWW 24th July 2014
14:00 to 15:00
Justification of the nonlinear Schrödinger equation as the modulation equation to the full water-wave problem
TWW 24th July 2014
15:00 to 16:00
On water waves with angled crests
TWW 28th July 2014
14:00 to 15:00
E Tobisch Dynamic energy cascades in the theory of water waves
The very notion of energy cascade in large bodies of water belongs to the English mathematician, physicist and meteorologist L.F. Richardson, who introduced it in 1922 and also suggested a qualitative heuristic model of this process. This conceptual model has been turned into a mathematical theory by A.N. Kolmogorov in 1941, for fully developed turbulence, and by V.E. Zakharov in 1967, for weak wave turbulence of weakly nonlinear amplitude. The model of a dynamic energy cascade presented in this talk covers a range of moderate nonlinearities not covered by any of the previous theories. The main physical mechanism generating a dynamical cascade is modulation instability. Analytical and numerical results will be presented, and also briefly compared with experimental observations.
TWW 29th July 2014
14:00 to 15:00
T Kataoka Resonance curves of finite-amplitude gravity waves in shallow-water limit
TWW 29th July 2014
15:00 to 16:00
J Grue Maximum finite depth waves: breaking, kinematics and particle drift
TWW 29th July 2014
16:30 to 17:30
H Bredmose Impacts from highly nonlinear waves on walls and substructures of wind turbines
The talk links together the paper of mine from 2010 on the flip-through impact and recent research on ringing loads on monopile substructures caused by near-breaking and breaking waves. We have made experiments in a collaborative project with the Danish Hydraulics Institute and computations with a free-surface Navier-Stokes solver coupled to a potential flow model of the outer wave field.
TWW 30th July 2014
13:30 to 13:40
J Leeks Welcome and Introduction
TWW 30th July 2014
13:40 to 14:10
A Success Story of Collaboration Between Academia and Industry in the Field of Wave Energy
TWW 30th July 2014
14:10 to 14:40
Ocean Wave Measurements: The Challenges and Consequences for the Wave Energy Industry
TWW 30th July 2014
14:40 to 15:10
Coastal Wave Modelling: General Engineering Usage and Areas for Improved Research
TWW 30th July 2014
15:30 to 16:00
H Jasak & I Gatin & V Vukcevic Numerical Simulation of Wave Loads on Static Offshore Structures
TWW 30th July 2014
16:00 to 16:30
R Rainey & J Colman 100-year and 10,000-year Extreme Significant Wave Heights - How Sure Can We Be of These Figures?
TWW 30th July 2014
16:30 to 17:00
Open Discussion
TWW 31st July 2014
14:00 to 15:00
Ill-posedness of truncated series models for water waves
Some numerical methods for water waves, such as the Craig-Sulem method, involve expanding terms in the water wave evolution equations as series, truncating those series, and then simulating the resulting equations. For one such scheme, we present analytical evidence that the truncated system is in fact ill-posed; this involves further reducing the evolution equations to a model for which we can prove ill-posedness. We then present numerical evidence that the full truncated system is ill-posed, showing that arbitrarily small data can lead to arbitrarily fast growth. We present this numerical evidence for multiple levels of truncation. We are able to prove that by adding a viscosity to the system, we instead arrive at a well-posed initial value problem. This is joint work with Jerry Bona and David Nicholls.
TWW 31st July 2014
15:00 to 16:00
E Varvaruca Global bifurcation for steady gravity water waves with constant vorticity and critical layers
TWW 31st July 2014
16:30 to 17:30
Hydrodynamic Surface Wave Analogues for Quantum Mechanics and Nonlinear Optics
TWW 5th August 2014
14:00 to 15:00
Changing forms and sudden smooth transitions of tsunami waves
In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small meter-scale displacement over a multi-kilometer horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a "down-up" or breather'' wave. Shallow water theory shows that the latter travels faster than the former and, according to the extended Korteweg-de Vries model presented here, the waves undergo a transition. Firstly, the two parts coincide at a given position and time producing a maximum elevation, whose amplitude depends on the shape of the approaching wave. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Secondly, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time.The implications for how these down-up'' tsunami waves reach beaches are considered. This is joint work with Chow & Hunt.
TWW 5th August 2014
15:00 to 16:00
A Brady Why bouncing droplets are a pretty good model of quantum mechanics
TWW 5th August 2014
16:30 to 17:30
E Wahlén Transverse instability of generalised solitary waves
TWWW04 6th August 2014
10:00 to 12:30
Initial Value Problem and Vortex Sheets: Analysis and Computation
In this lecture, we will discuss the irrotational water wave problem. We will place the problem in the larger context of vortex sheets; the vortex sheet is the interface between two irrotational fluids, allowing for a jump in the tangential components of velocity across the interface. We will present the equations of motion in both 2D and 3D, for the water wave problem both with and without surface tension. We will present the numerical method of Hou, Lowengrub, and Shelley (HLS) for computing solutions of the initial value problem in 2D, and how the HLS ideas can be used to prove short-time well-posedness of the initial value problem. The corresponding numerical method and well-posedness proofs for the three-dimensional problem will also be discussed. If time allows, we will go beyond the initial value problem, and discuss how the vortex sheet formulation with the HLS ideas can be extended to treat other problems, such as the traveling wave problem. This talk includes joint work with Nader Masmoudi, Michael Siegel, Svetlana Tlupova, and possibly others.
TWWW04 6th August 2014
15:30 to 16:30
Forward looking / open problems session continued
TWWW04 7th August 2014
10:00 to 12:30
Well Posedness and Singularities of Water Waves
A class of water wave problems concerns the dynamics of the free interface separating an inviscid, incompressible and irrotational fluid (water), under the influence of gravity, from a zero-density region (air). In these lectures I will present some recent methods and ideas developed concerning the local and global wellposedness of these problems, where the fluid region has either no fixed boundary, or has a fixed vertical wall. The emphasis will be on the understanding of the mathematical structure that leads to the results on global existence of small and smooth waves, local existence of arbitrary smooth waves, the persistence of water waves with angled crests, and in the case where there is a vertical wall, the interaction of the free interface with the wall. These lectures will be accessible to graduate students and post-doctors.
TWWW04 7th August 2014
14:00 to 15:00
A Nachbin Conformal Mapping and Complex Topography
Water waves propagating over non-smooth, large amplitude, disordered topography leads to novel asymptotic theory both at the level of equations (i.e. reduced models) as well as at the level of solutions (i.e. effective behavior). In the reduced modeling of two-dimensional flows, conformal mapping plays an important role. This lecture will introduce the use of conformal mapping together with nonlinear potential theory. The Schwarz-Christoffel Toolbox (by T. Driscoll) will also be introduced showing how to extract quantitative information from the conformal mapping in a specific flow domain. This is useful for computational applications. As time permits recent research examples will be presented such as pulse shaped waves over a disordered (random) topography or in branching channels
TWWW04 7th August 2014
15:30 to 16:30
A Nachbin Conformal Mapping and Complex Topography
TWWW04 8th August 2014
10:00 to 12:30
O Bokhove Variational Water Waves: on Continuum and Discrete Modelling, and Experimental Validation
1. Some variational mechanics will be introduced first before embarking on formulating continuum models for water waves in fluids. A forced-dissipative nonlinear oscillator will be used as an example, inspired by wave motion in a laboratory Hele-Shaw cell for (breaking) water waves and beach morphodynamics. Subsequently, variational principles for 3D water waves by Miles and Luke will be derived using constraints. 2. The simpler, depth-averaged, spatially 2D, shallow water analogue of Miles' variational principle will be discretized in space and subsequently in time, using space and time finite elements, to yield a compatible geometric algebraic variational principle. Likewise, Miles’ variational principle will be discretized in space and subsequently in time, using space and time finite elements, to yield a compatible geometric algebraic variational principle. The latter interim variational principle with continuous time will be essential to find the time discretization, ensuring that there is an underlying discrete boundary element structure and that the mesh movement is soundly integrated in the variational structure to ensure numerical stability. The simple oscillator will serve as an illustrative example for the discontinuous Galerkin finite element time discretization developed. The similarities between these discrete 2D shallow water and 3D water wave models at the fre e surface will be discussed. 3. The resulting discretizations will be validated against wave tank data from the Maritime Research Institute Netherlands (MARIN) and in-house Hele-Shaw wave tank experiments (shown live). 4. Time permitting, extensions of the above results will be discussed. These concern water wave models with a vertical component of vorticity, experiments of a bore-soliton rogue wave, modelling of breaking waves, or a wave-energy device using geometric rogue-wave focussing.
TWWW04 8th August 2014
14:00 to 15:00
T Bridges Modulation of Water Waves
The lectures will provide an introduction to the theory of "modulation" and its role in the derivation of model equations, such as the KdV equation, Boussinesq equation, KP equation, and Whitham modulation equations, and their role in the theory of water waves. The classical theory of modulation, such as Whitham modulation theory, will be reviewed, and a new approach will be introduced. The new approach is based on modulation of background flow. Methodology that is key to the theory is symmetry and conservation laws, relative equilibria, Hamiltonian and Lagrangian structures, multiple scale perturbation theory, and elementary differential geometry. By basing the theory on modulation of relative equilibria, new settings are discovered for the emergence of KdV and other modulation equations. For example, it is shown that the KdV equation can be a valid model for deep water as well as shallow water. The lectures are introductory, and no prior knowledge is assumed.
TWWW04 8th August 2014
15:30 to 16:30
T Bridges Modulation of Water Waves