Large Amplitude Internal Waves

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Large Amplitude Internal Waves

 01 - 04 Dec 2008
 ICMS, 14 India Street, Edinburgh

Scientific Organisers
Roger Grimshaw, Loughborough University
Jean-Claude Saut, Université de Paris Sud
Thierry Colin, Université Bordeaux 1

Keynote Speakers

Frederic Dias
Roger Grimshaw
Karl Helfrich
David Lannes

Report Introduction

Internal waves occur in density-stratified fluids. They are ubiquitous in the ocean and
atmosphere, and often have large amplitudes. Of particular concern in this workshop
were internal solitary waves, where nonlinearity is essential. In their simplest
representation internal solitary waves are nonlinear waves of quasi-permanent form
which owe their existence to a balance between nonlinear wave-steepening effects and
linear wave dispersion. They are a commonly occurring feature of coastal seas, straits,
fjords and lakes, and also can occur as spectacular roll clouds in the atmospheric
boundary layer. On the theoretical side, nonlinear evolution equations of the Korteweg-de
Vries (KdV) type are usually used as a first-order basis for qualitative modelling and
prediction.

These ubiquitous internal solitary waves are a major factor in the flow over the oceanic
continental shelf and slope. Their associated currents and pycnocline displacements have
profound implications for the design and placement of offshore structures, the safety of
submersibles, the coherence of underwater sound signals, the movement of nutrients and
biological matter, and sediment transport. Further, they can produce microstructure and
localized turbulent patches, and in general are a sink for the barotropic tidal energy.

Although these waves are now well documented, the associated theory and modelling are
not well developed, certainly by contrast to the corresponding state of affairs for water
waves. Although model equations of the KdV-type are widely used, validation has
largely been confined to comparisons with numerical simulations and laboratory
experiments, and there are very few rigorous theoretical results available, even for
steady-state solitary waves. Hence the aim of this workshop is to bring together physical
oceanographers and engineers concerned with the observation and measurements of
internal waves, applied mathematicians and physicists involved in modelling and
numerical simulations, with mathematicians concerned to develop a rigorous
understanding of the dynamics of large-amplitude internal waves. While the focus is on
internal waves, the general principles involved have relevance for the study of nonlinear
waves in other physical contexts.

PDF file of full report

Arrangements

Participation
This workshop is now full. The workshop will begin on the morning of Monday 1 December and finish with lunch on Thursday 4 December 2008.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will take place at the head-quarters of ICMS, 14 India Street, Edinburgh. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. For local information the finding ICMS page shows the location of ICMS and contains useful maps of the city centre.

The seminar room at ICMS has 4 whiteboards, 2 overhead projectors, a data projector and laptop. If you are giving a talk we ask that you use our ICMS laptop and bring your talk on a memory stick or CD. If you wish your talk to be pre-loaded onto our laptop, you may email it in advance to audrey.brown@icms.org.uk.

Wireless access is available throughout the ICMS building. There are also 4 public PCs which may be used at any time for internet access and to check email.

Accommodation
ICMS will arrange single en-suite rooms in local guest houses for those who require it. Accommodation is typically about 15 to 30 minutes walk from ICMS. Participants are also free to make their own arrangements and may claim back the cost (with receipts), up to a maximum of 60.00 GBP a night for 4 nights. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
Morning and afternoon refreshments and a light lunch will be provided each day of the workshop, Mon to Thurs at no cost to participants.

An informal evening meal will be arranged at a nearby restaurant at 19.00 on Mon 1 Dec at Ristorante Librizzi, 69 North Castle Street. This restaurant offers Sicilian/Italian cuisine (website currently being updated). On Tue 2 Dec a mezze style evening meal will be provided in Nargile Turkish Restaurant, 73 Hanover Street, at 19.00. The workshop dinner will take place at 19.00 at First Coast Restaurant 97-101 Dalry Road, on the evening of Wed 3 Dec. The workshop grant will cover the cost of these meals.

Registration
Registration will take place 09.00 - 09.45 on the morning of Mon 1 Dec at ICMS.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee and lunch Mon to Thurs, informal evening meals Mon and Tue and the workshop dinner on Wed evening.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world should supply their IBAN and SWIFT/BIC code. We cannot reimburse any item without a receipt.

Please note that, unless otherwise advised, all participants will be asked to pay a 30.00 GBP registration fee for this workshop. This may be paid at Registration in cash, by sterling cheque or by providing credit/debit card details. If you wish to pay this fee in advance please use this credit or debit card payment form. The form should be printed out, completed and faxed back (as email is not a secure way of sending credit card information). The fax number is on the form.


Programme

Session Chairs:

Mon 1 Dec 10.00 - 12.30 Jean-Claude Saut
Mon 1 Dec 14.00 - 17.00 Jerry Bona

Tues 2 Dec 09.00 - 12.30 Roberto Camassa
Tues 2 Dec 14.00 - 17.00 Karl Helfrich

Wed 3 Dec 09.00 - 12.30 David Lannes
Wed 3 Dec 14.00 - 15.30 Frederic Dias

Thur 4 Dec 09.00 - 12.00 Roger Grimshaw


Monday 1 December

09.00 - 09.45

Registration and coffee

09.45 - 10.00

Workshop Opening: Roger Grimshaw and Jean-Claude Saut

10.00 - 11.00

Roger Grimshaw (Loughborough University)
Keynote Lecture
Long wave models for internal solitary waves PDF file of presentation

11.00 - 11.30

Alexey Slunyaev (IAP, Russian Academy of Sciences)
Analytic solutions for long internal wave models with improved nonlinearity PPT file of presentation

11.30 - 12.00

Roberto Camassa (University of North Carolina)
Unstable internal waves in near two layer fluids

12.00 - 12.30

Jerry Bona (University of Illinois at Chicago)
Internal wave propagation and applications to sand ridge formation

12.30 - 14.00

Lunch

14.00 - 14.30

David Farmer (University of Rhode Island)
Observations of nonlinear internal wave generation and evolution

14.30 - 15.00

Steve Ramp (Monterey Bay Aquarium Research Institute)
Observations of large-amplitude nonlinear internal waves in the northeastern South China Sea

15.00 - 15.30

Alberto Scotti (University of North Carolina)
Nonlinear internal waves in Massachusetts Bay: using a model to make sense of observations

15.30 - 16.00

Tea/Coffee

16.00 - 16.30

Kevin Lamb (University of Waterloo) PDF file of presentation
Shoaling internal solitary waves: energetics, dissipation and reflection

16.30 - 17.00

Wooyoung Choi (New Jersey Institute of Technology)
Strongly nonlinear internal wave models and their applications

17.00 - 18.00

Discussion: Moderator, Jerry Bona

19.00

Informal group meal at Ristorante Librizzi, 69 North Castle Street, Edinburgh

 

Tuesday 2 December

09.00 - 10.00

David Lannes (École Normale Supérieure)
Keynote lecture
Asymptotic models for internal waves PDF file of presentation

10.00 - 10.30

Jean-Claude Saut (Université Paris Sud)
On a nonlocal system for large amplitude internal waves PDF file of presentation

10.30 - 11.00

Coffee

11.00 - 11.30

Mariana Haragus (Université de Franche-Comté)
Stability of periodic waves in dispersive models PDF file of presentation

11.30 - 12.00

Eugen Varvaruca (Imperial College London)
On the existence of extreme surface waves and the Stokes conjecture with vorticity PDF file of presentation

12.00 - 12.30

Gérard Iooss (IUF, Université de Nice)
Non-symmetric periodic patterns for surface gravity waves PDF file of presentation

12.30 - 14.00

Lunch

14.00 - 14.30

Magda Carr (University of St Andrews) PDF file of presentation
Stability characteristics of large amplitude internal solitary waves

14.30 - 15.00

John Grue (University of Oslo)
Run-up of very long internal waves

15.00 - 15.30

Alan Davies (Proudman Oceanographic Laboratory)
Internal wave generation, breaking, associated mixing and model validation PPT file of presentation

15.30 - 16.00

Tea/Coffee

16.00 - 16.30

Victor Shrira (University of Keele)
Exact fully nonlinear solutions for internal waves

16.30 - 17.00

Lev Ostrovsky (Zel Technologies/University of Colorado)
Modelling of 'genuinely strong' internal waves PDF file of presentation

17.00 - 18.00

Discussion: Moderator, Jean-Claude Saut

19.00

Informal group meal at Nargile (Turkish) Restaurant, 73 Hanover Street, Edinburgh

 

Wednesday 3 December

09.00 - 10.00

Frederic Dias (École Normale Supérieure de Cachan)
Keynote lecture
Large amplitude internal waves PDF file of presentation

10.00 - 10.30

Al Osborne (Universita' di Torino)
Modelling internal waves with the 2+1 Gardner Equation

10.30 - 11.00

Coffee

11.00 - 11.30

Tom Bridges (University of Surrey)
Degenerate conservation laws, bifurcation of solitary waves, and criticality of internal waves

11.30 - 12.00

Shu-Ming Sun (Virginia Polytechnic Institute and State University)
Linear and nonlinear stability of solitary waves

12.00 - 12.30

Florent Chazel (Université Paris-Est)
A double-layer Boussinesq-type model for fully nonlinear and highly dispersive water waves

12.30 - 14.00

Lunch

14.00 - 14.30

Vasily Vlasenko (University of Plymouth)
Amplification and suppression of internal waves in horizontally sheared currents

14.30 - 15.00

José da Silva (University of Lisbon)
Synthetic aperture radar observations of resonantly generated internal solitary waves at race point channel (Cape Cod)
PDF file of presentation

15.00 - 15.30

Karima Khusnutdinova (Loughborough University)
Oceanic subsurface bubble distributions and internal waves PDF file of presentation

15.30 - 16.00

Hai Yen Nguyen (Université Haute Bretagne)
Solitary wave interaction for the BBM equation

16.00 - 16.30

Tea/Coffee

16.30 - 17.30

Discussion: Moderator, Frederic Dias

19.00

Workshop dinner at First Coast Restaurant, 99-101 Dalry Road, Edinburgh

 

Thursday 4 December

09.00 - 10.00

Karl Helfrich (Woods Hole Oceanographic Institution)
Keynote lecture
Effects of rotation on large-amplitude internal waves

10.00 - 10.30

Natalia Stashchuk (University of Plymouth)
Evolution of large-amplitude internal waves over 3D topography

10.30 - 11.00

Coffee

11.00 - 11.30

Jean-Marc Vanden-Broeck (University College London)
Numerical studies of nonlinear two and three-dimensional interfacial waves

11.30 - 12.00

Tatiana Talipova (IAP, Russian Academy of Sciences)
Nonlinear interfacial wave transformation in basin of variable depth: analytical and numerical results

12.00 - 12.30

Triantaphyllos Akylas (Massachusetts Institute of Technology)
The effect of rotation on nonlinear stratified flow over topography

12.30 - 13.00

Closing Discussion: Moderator, Roger Grimshaw

13.00 - 14.00

Lunch

 

Presentations

Akylas, Triantaphyllos
The effect of rotation on nonlinear stratified flow over topography
View Abstract
Steady, two-dimensional, rotating stratified flow of large depth over topography is considered for the case of uniform flow speed and buoyancy frequency far upstream. Under these flow conditions and in the absence of rotation, the nonlinear response due to finite-amplitude topography can be found analytically via Long's model. Rotation, however, turns out to be a singular perturbation as it always becomes as important as stratification in the far field. We discuss the effects of rotation on Long's nonlinear flow state by a matched-asymptotics procedure in the weak-rotation (large Rossby number) limit. Nonlinear interactions over the topography induce propagating gravity-inertial waves that alter the nature of the linearized response far downstream. Extension of the theory to a two-layer flow configuration with different buoyancy frequencies, modelling the troposphere and stratosphere, is also presented. The theoretical predictions are compared against fully numerical flow simulations.
Bona, Jerry
Internal wave propagation and applications to sand ridge formation
View Abstract
Models for internal wave propagation can be used in conjunction with sediment transport models to provide a description of wave bottom interaction and the resulting prospect of sand ridge formation.
Bridges, Tom
Degenerate conservation laws, bifurcation of solitary waves, and criticality of internal waves
View Abstract
The abstract is available as a pdf file by clicking on this link:
Bridges' Abstract
Camassa, Roberto
Unstable internal waves in near two layer fluids
View Abstract
Recent advancements in observational techniques have revealed that internal gravity waves are an ubiquitous phenomena in the ocean and in the atmosphere. In particular, internal waves propagating in a stratified ocean have been observed and reported to have large amplitudes. Understanding the breaking mechanisms of these waves is crucial for explaining mixing and transport phenomena within the ocean. As experimental observations show, for near two layer stratification, waves become unstable in large amplitude regimes and the wave-breaking closely resembles Kelvin Helmholtz shear instability originating in the maximum displacement of the pycnocline region. The instability is modulated by the stream-wise variation of the shear. We simulate numerically the generation and propagation of solitary waves starting from a step function initial condition and monitor the wave-induced shear instabilities. A conservative projection method for the variable density Euler equations is implemented for this scope. The code is validated against experimental data as well as theoretical results. In an effort to elucidate whether the instabilities are an intrinsic property of the wave or they are induced by the experimental generation, we also study the time evolution of traveling wave solutions with initial conditions generated by an iterative code for the stationary problem.
Carr, Magda
Stability characteristics of large amplitude internal solitary waves
View Abstract
The stability characteristics of an internal solitary wave of depression in a shallow, two or three-layer fluid are investigated experimentally and theoretically. The initial back ground stratification is varied and it is found that the onset, type and intensity of breaking are dramatically effected by change in the background stratification. The three-layered stratification consists of linearly stratified pycnocline sandwiched between a deep homogeneous dense layer and a shallow homogeneous top layer. In this regime, shear instability is seen at the trough of the wave and develops throughout the tail. Experimental results are compared with fully nonlinear theory. Excellent agreement is found in stable cases.

In the unstable observations discrepancies between experiment and theory are used to explain the physics observed. Estimates of the wave properties (from experiment and theory) and evaluation of the Richardson number reveal significant new findings. The two-layered stratification consists of a homogeneous dense layer below a linearly stratified top layer. In this regime a combination of shear and convective instability is seen on the leading face of the wave. It is shown that there is interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. (Joint work with Dorian Fructus (University of Oslo), John Grue (University of Oslo), Atle Jensen (University of Oslo), Peter Davies (University of Dundee).
Chazel, Florent
A double-layer Boussinesq-type model for fully nonlinear and highly dispersive water waves
View Abstract
In this talk we present a new Boussinesq-type model based on a double-layer approach, where the fluid is artificially divided into two layers of equal density and separated by a static interface. The model derivation relies on two basic ideas. The first one is to look for Laplace equation solutions under the form of infinite Taylor series in the vertical variable, and to truncate them at an order motivated by an appropriate dimensional analysis. The second one is to take advantage of the double-layer framework by lowering the maximum derivatives order with the introduction of more unknowns. The final model includes only second order derivatives, and we finally show through linear analysis and numerical computations that it propagates waves accurately from shallow to deep water.
Choi, Wooyoung
Strongly nonlinear internal wave models and their applications
View Abstract
Strongly nonlinear asymptotic models for large amplitude internal waves and their stability characteristics will be first discussed and their improvements for more realistic oceanic applications will be introduced. Numerical solutions of the strongly nonlinear model for the shallow water configuration will be also presented for the evolution of internal solitary waves propagating over bottom topography and their surface signatures by integrating the internal wave model with the surface wave model.
da Silva, José
Synthetic aperture radar observations of resonantly generated internal solitary waves at race point channel (Cape Cod)
View Abstract
Synthetic Aperture Radar images revealed the two-dimensional propagation characteristics of short-period internal solitary waves in Race Point Channel in Massachusetts Bay. The images and in situ measurements of the flow in the channel are used to infer the likely generation mechanism of the waves. The solitary waves are generated during the ebb phase of the tide within the Channel. On some occasions, two trains of internal waves are generated presumably at the same location but at slightly different phases of the ebb tide. The main characteristics of the (two-layer) flow are described based on the criticality of the Froude Number. It is suggested that these two individual packets of waves result from flow passage through resonance (where the Froude number is one). One packet is generated as the flow passes through the transcritical regime during the acceleration phase of the (ebb) tidal current, and another packet is generated during the deceleration phase. Both packets propagate upstream when the tide slacks, but with slightly different propagation directions.
Davies, Alan
Internal wave generation, breaking, associated mixing and model validation
View Abstract
Processes producing internal waves will be presented and illustrated in terms of wind forced and tidally forced internal waves. For tidally forced waves the focus will be a non-hydrostatic model of the sill at the entrance to Loch Etive and the role of small scale topography , bottom friction and tidal turbulence upon internal wave generation and associated mixing .

The problem of model validation against field measurements will be considered particularly in terms of turbulence measurements and power specta of the internal wave field. The value of power spectral analysis as applied to model results as a means of separating long waves (internal tides) from short waves (lee waves) and as a tool for understanding processes and model validation will be examined.

The major difficulty of obtaining data sets of sufficient accuracy and intensity for model skill assesment will be briefly discussed. The question of up-scaling to generate parameterizations of small scale processes of sufficent accuracy to use in large scale primarily hydrostatic oceanographic models will be considered.

Joint work with Jiuxing Xing (POL) and Jarle Berntsen (Bergen University)
Dias, Frederic
Large amplitude internal waves
View Abstract
Internal waves have been less studied than water waves. In this talk we present some of the major differences between water waves and internal waves: trajectories, weakly nonlinear models, observations, existence of internal fronts, limiting configurations, tsunami generation. Then we review some results on internal waves in the presence of a free surface. Finally we present a new model which allows to study internal waves in the presence of a diffuse interface.
Farmer, David
Nonlinearity and rotational effects on internal waves in the South China Sea
View Abstract
Time series observations of first mode internal waves have been acquired in the South China Sea using inverted echo-sounders deployed on the sea floor. An internal tide generated by flow over topography in Luzon Strait propagates westward across the adjacent deep basin. Nonlinearity, rotation and nonhydrostatic dispersion shape the waves as they travel, providing an opportunity to compare observations with theory. Strongly mixed tides complicate the interpretation, but observations at one location can be used as initial conditions for model predictions, allowing comparison with observations downstream. Competition between the effects of rotation and nonlinearity are evident. Steep interfacial slopes lead to further steepening and subsequent nonhydrostatic dispersion into a nonlinear wave train; the breaking distance that appears consistent with weakly nonlinear theory. Less steep interfacial slopes are subject to the effects of rotational dispersion and form corner waves. Results are interpreted in terms of the balance between nonlinearity and rotational terms in the Ostrovsky (1978) equation. (Co-author, Li Qiang)
Grimshaw, Roger
Long wave models for internal solitary waves
View Abstract
We will review long wave models of the Korteweg-de Vries type, emphasizing both quadratic and cubic nonlinearity, and the effects of variable bottom topography.
Grue, John
Run-up of very long internal waves
View Abstract
Motivated by observations by offshore industry of very strong bottom currents along the shelf slope we investigate run-up of very long horizontal desplacements of the thermocline. Particular focus in on the dynamics of the level at rest of the intersection line between the thermocline and the shelf slope. The study is experimental. Velocity measurements are performed by Particle Image Velocimetry. Through a Froude number scaling, the experiements confirm the observations in the field. Results generalize previous publications on run-up of internal solitary waves.
Haragus, Mariana
Stability of periodic waves in dispersive models
View Abstract
We discuss the problem of stability of periodic travelling waves for a class of dispersive models, including the generalized KdV equation, the BBM equation and the nonlinear Schr"odinger equation. We focus on the spectral stability, which is concerned with the question of determining the spectrum for the linearization about the periodic traveling wave. All these models share the property that they have a Hamiltonian structure. Then, by using a Bloch-wave decomposition, we recast the problem as determining the point spectra for a family of operators of the form JL, where J is skew-symmetric with bounded inverse and L is symmetric with compact inverse. The main result relates the number of unstable eigenvalues of the operator JL to the number of negative eigenvalues of the symmetric operator L. The compactness of the resolvent operators allows to greatly simplify the proofs, as compared to those of similar results for linearizations about localized waves.
Helfrich, Karl
Effects of rotation on large-amplitude internal waves
View Abstract
The role of the Earth’s rotation on the evolution of internal solitary waves is generally considered to be small. There are, however, situations in which rotation leads to interesting and important effects.

Individual solitary waves are subject to radiation damping that drains energy from the wave, though not completely. In the oceanographically important problem of solitary wave emergence from the disintegration of internal tide, rotation may completely inhibit the production of the short solitary waves. Both situations involve a competition between high and low-frequency dispersion to balance nonlinearity. Weakly and, more recent, fully nonlinear models of these processes will be discussed as well as recent observational evidence.
Iooss, Gérard
Non-symmetric periodic patterns for surface gravity waves
View Abstract
This is a joint work with Pavel Plotnikov. We consider travelling water waves in a potential flow on an infinitely deep fluid layer, which form a bi-periodic horizontal pattern on the free surface, in absence of surface tension.

The pattern is not of diamond type, the two basic wave vectors K1 and K2 having different lengths. The waves may be considered as the nonlinear superposition of two plane waves, with corresponding wave numbers K1 and K2, and amplitudes \epsilon1 and \epsilon2, provided that at the bifurcation, K1,K2 and the bifurcation parameters \mu (built with a wave length, the acceleration of gravity and the velocity of the waves) and the direction of propagation u0, satisfy the dispersion relation.

We first show how to build the asymptotic expansion of bifurcating 3-dimensional waves defined up to a horizontal shift, as a power series of the amplitudes \epsilon1, \epsilon2, the direction of propagation being close to u0. All coefficients may be computed explicitly in terms of the two angles made by K1 and K2 with u0. Due to the occurrence of small divisors, the main difficulty for proving the existence of solutions, possessing the above asymptotic expansions, is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem.

We prove the existence of bifurcating doubly-periodic non-symmetric waves for any values of the parameters (\epsilon1, \epsilon2) in a product of one-dimensional Cantor sets having asymptotically a full measure as (\epsilon1, \epsilon2) tends towards 0.
Khusnutdinova, Karima
Oceanic subsurface bubble distributions and internal waves
View Abstract
There is observational evidence for the existence, for sufficiently high wind speeds, of highly structured bubble layers, extending to a depth of several meters beneath the surface. Our preliminary theoretical studies indicate that under some circumstances bubble distributions could be entirely responsible for the density stratification in the upper ocean, and hence could give rise to their own internal wave field. Two physical factors affect the buoyancy frequency in the mixture: the effective stratification due to the bubbles adds to the effect of stratification in the fluid, while the compressibility of the mixture due to the bubbles reduces the buoyancy frequency. In accordance with existing observational evidence that the void fraction profile in the ocean decays exponentially with depth, we obtain an explicit description of the normal modes for several simple models of the basic density profile and show that bubble distributions, when present, may considerably change the properties of the internal waveguide. We also present first numerical simulations of the experimentally observed change of the thickness of the bubble layer in accordance with the profile of the underlying internal waves.
Lamb, Kevin
Shoaling internal solitary waves: energetics, dissipation and reflection
View Abstract
Internal solitary waves (ISWs) are common, highly energetic features in the coastal ocean where they are predominately generated by tide-topography interaction. There are many unanswered questions about the generation and fate of these waves and a better understanding of these processes is necessary for developing parameterizations of their effects in large scale models. In this talk I will focus on the energetics and fate of shoaling ISWs as predicted by fully nonlinear, nonhydrostatic two-dimensional numerical simulations. The talk will first discuss the use of an available potential energy density in describing the energetics of shoaling waves. Then two scenarios will be considered: (1) waves propagating from deep water to shallow water where the bottom is flat in the shallow water and most of the stratification occurs above the bottom in the shallow water; and (2) waves propagating along a thin pycnocline which intersects a sloping bottom. In the first case most of the energy can be transmitted into the shallow water whereas in the second strong breaking occurs and all the energy is dissipated or reflected.

Waves propagating from deep to shallow water deform as the depth changes, in general fissioning into several smaller ISWs and a dispersive wave train. The presence of stratification below the top of the slope can result in significant reflection. For some stratifications significant energy can be transferred to higher-mode waves. Comparisons with two weakly-nonlinear shoaling models will be made. For the cases considered the weakly-nonlinear models agree well with the fully nonlinear simulations provided most of the energy is transmitted and remains in mode-one waves. Inclusion of cubic-nonlinearity in the weakly-nonlinear model is essential.

For waves propagating along a pycnocline intersecting the bottom we explore the ratio of reflected to incident energy, focussing on its dependence on bottom slope and pycnocline depth. The breaking process is also explored. This shows that much of the energy dissipation occurs beneath the pycnocline in the boundary layer, including a region of flow separation, associated with the strong jet of fluid squeezed out by the wave as it reaches the bottom slope.
Lannes, David
Asymptotic models for internal waves
View Abstract
We derive in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on Rd , d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators with respect to suitable small parameters that depend variously on the amplitude, wave-lengths and depth ratio of the two layers.
Nguyen, Hai Yen
Solitary wave interaction for the BBM equation
View Abstract
Interaction of solitary waves of the Benjamin-Bona-Mahony (BBM) equation is investigated numerically by using spectral discretization in space and an explicit fourth-stage Runge-Kutta scheme in time. The phase-shift and dispersive wavetrain, which occur due to the interaction between the waves of elevation, depression or mixed, are tabulated and compared for various cases. (Joint work with Henrik Kalisch and Nguyen Thanh Nguyet, University of Bergen, Norway)
Osborne, Alfred R
Modelling internal waves with the 2+1 Gardner Equation
View Abstract
I use the 2+1 Gardner Equation as a candidate equation for the study of large amplitude internal waves. My approach is to study the solutions of the equation using Riemann theta functions and in consequence one has (1) the full physics of the equation, (2) a hyperfast numerical model of the equation and (3) nonlinear space/time series analysis algorithms for analyzing data. I give in particular an overview of many of the solutions to the equation and discuss how to compute the Riemann matrix so that the theta functions provide a spectral decomposition with many types of basis functions which are solutions of sub equations within 2+1 Gardner, namely: KdV, modified KdV, KP, modified KP, the 1+1 Gardner equation. I also show how to derive the associated (integrable) 2+1 nonlinear Schroedinger equation associated with 2+1 Gardner. I then give an overview of a hyperfast model, using Riemann theta functions, for simulations of internal wave dynamics. An overview of time series analysis algorithms is also given.
Ostrovsky, Lev
Modelling of 'genuinely strong' internal waves
View Abstract
It is now a common knowledge that groups of tidally generated internal waves in coastal zones can often be “genuinely” strongly nonlinear in that the corresponding pycnocline displacements are comparable with or even significantly exceed its depth. This presentation includes, first, a brief overview of observational and theoretical results, and second, our recent attempts to construct effective evolution equations for strong internal waves based on a model Hamiltonian taking into account the effects of dissipation and sloping bottom. Some specific solutions are given and compared with in situ and numerical data.
Ramp, Steve
Observations of large-amplitude nonlinear internal waves in the northeastern South China Sea
View Abstract
Three field programs sponsored by the U. S. Office of Naval Research have now been completed to study the generation, propagation, shoaling, and transformation of very large amplitude non-linear internal waves in the northeastern South China Sea (SCS). A very large internal tide is generated by the interaction of the barotropic tide with abrupt topography in the vicinity of the Batan Islands in the Luzon Strait between Taiwan and the Philippines. Both observations and models show that the internal tide rapidly steepens and becomes more nonlinear until it spawns internal solitary waves (ISWs) in the deep basin between the generating ridges and the Chinese continental slope. The time/distance required for the ISWs to form is a direct function of the Froude Number in the generating strait. As the waves continue to propagate WNW across the basin in deep water, the ISWs become wave packets usually with 3-5 waves per packet. Once they start shoaling on the continental slope, many more waves form, and the large leading wave often splits into multiple smaller waves. In water depths less than about 120 m on the continental shelf, the depression waves often become elevation waves when the upper layer thickness exceeds that of the lower layer. The entire process can be linked to the tides in the Luzon Strait, which have both a very large fortnightly envelope and also a large diurnal variation. The largest waves form diurnally on the stronger beat near the spring tide. There is some evidence for hydraulic control in the straight at spring tide plus or minus about 3 days, and the flow appears to be sub- critical the rest of the time.
Saut, Jean-Claude
On a nonlocal system for large amplitude internal waves
View Abstract
We will prove the well-posedness of a new nonlocal system for
two-dimensional large amplitude internal waves introduced recently by J Bona,
D Lannes and the speaker. (This is is a joint work with David Lannes)
Scotti, Alberto
Nonlinear internal waves in Massachusetts Bay: using a model to make sense of observations
View Abstract
Massachusetts Bay is an excellent natural laboratory to study the life-cycle of nonlinear internal waves (NLIWs). NLIWs are generated by the barotropic tidal flow over Stellwagen Bank, propagate as undular bore across Stellwagen Basin and experience shoaling and polarity conversion over the shoaling section just offshore of Scituate, MA. Thus, within a few tens of kilometers we observe the entire life-cycle of these waves. In this talk, we use a nonlinear, nonhydrostatic model encompassing the region to provide an interpretative framework for field observations collected over several years across the area. In particular, the model shows that the unusual current patterns observed along the shoaling section of the Bay are in fact the signature of NLIWs strongly interacting with the bottom, a process that ultimately leads to the formation of waves of elevation in the shallow reach of the Bay, which have been observed in situ. Further, the model shows that both during generation and shoaling nonlinearity dominates the dynamics, with dispersion playing an ancillary role. Thus, a simple two-layer hydrostatic nonlinear model can be used to diagnose the onset of shoaling. Time permitting, results from a three-dimensional model will be presented, to highlight the importance of three-dimensional effects.
Shrira, Victor
Exact fully nonlinear solutions for internal waves
View Abstract
Several new classes of exact solutions of the fully nonlinear Euler and Navier-Stokes equations for the Boussinesq fluid with constant buoyancy frequency will be presented. The solutions describe nonlinear internal waves. The relationships between the Lagrangian and Eulerian description for the found solutions will be also discussed.
Slunyaev, Alexey
Analytic solutions for long internal wave models with improved nonlinearity
View Abstract
Exact solutions for the KdV equation with combined quadratic-cubic nonlinearity (the Gardner equation) are discussed, including the initial-value problem for initial disturbances of simple shapes. The fully-nonlinear dispersionless equations for two-layer fluid are formulated; its nonlinearity coincides with the one in the Gardner equation in a limiting case of equal-depth layers. The general case of a next-order Gardner equation is reduced to a simpler form in the manner similar to the proof of asymptotic integrability, what allows quantitative estimations of the solitary waves. (Joint work with E Pelinovsky, O Polukhina)
Stashchuk, Natalia
Evolution of large-amplitude internal waves over 3D bottom topography
View Abstract
The modelling of large amplitude internal waves (LAIW) propagating in the Strait of Gibraltar is carried out using a fully-nonlinear non-hydrostatic numerical model. The focus of the modelling efforts was on three-dimensional peculiarities of LAIW evolution, viz. cross-strait variability, interaction with lateral boundaries (including wave breaking and water mixing), radiation of secondary waves from orographic features and interaction of secondary scattered internal waves.

It was found that propagating along the channel packets of LAIWs reveal remarkable three-dimensional behaviour. Due to the Coriolis force and multiple reflections from the lateral boundaries, the largest leading LAIW loses its energy much faster than that in the attached wave tail. The latter, in turn, effectively adsorbs the scattered energy from the leading wave as it propagates and grows in amplitude. As a result of the energy transfer, the initially rank-ordered wave packet loses its regular structure transforming into a non-rank-ordered wave structure. The in-situ data collected in the eastern part of the Strait of Gibraltar confirm the idea that the non-rank-ordered structure is a common feature for the internal wave packets emerging from the strait into the Alboran Sea. (Joint work with Vlasenko, Sanchez Garrido, Garcia Lafuente, Losada)
Sun, Shu-Ming
Linear and nonlinear stability of solitary waves
View Abstract
The talk will discuss most recent development on the existence and stability theory of solitary waves in water using exact Euler equations.

It is known that the exact equations have traveling solitary-wave solutions and it is still an open question whether the solitary-wave solutions are stable when the exact equations are used. Here, various stability results will be discussed. In particular, spectral stability of 2D solitary waves will be presented, which gives the location of the spectrum of a linear operator obtained from the linearized Euler equations around the 2D solitary-wave solution of the exact equation with zero surface tension. (This is a joint project with R Pego)
Talipova, Tatiana
Nonlinear interfacial wave transformation in basin of variable depth: analytical and numerical results
View Abstract
The nonlinear wave propagation in a two-layer flow is studied for two limiting bottom profiles: step and “non-reflected” beach. The bottom step leads to the maximal reflection of wave energy, and it is studied analytically for a weakly nonlinear solitary waves. The theoretical approach includes: linear approximation for wave transformation at a step and the Korteweg – de Vries and Gardner equations for the wave propagation far from a step.

Transformation of large-amplitude solitary at a step is analyzed numerically in the framework of Euler equations. The process of the large-amplitude solitary wave transformation on a step differs from the prediction of the asymptotic weakly nonlinear theory. Details of the strongly nonlinear field in the vicinity of a step are described. The another case when the wave reflection is weak even the water depth is varied non-smoothly is “non-reflected” beach (special profile of bottom topography). The non-reflected bottom profiles are found for interfacial waves in two-layer flow in the framework of the shallow water theory, and some analytical solutions for the sign-variable N-pulse propagation are obtained for such geometry.
Vanden-Broeck, Jean-Marc
Numerical studies of nonlinear two and three-dimensional interfacial waves
View Abstract
Nonlinear waves traveling at the interface between two fluids of constant densities are considered. Fully nonlinear solutions are calculated. Both two-dimensional and three-dimensional waves are described. Periodic waves, solitary waves and the effects of surface tension are among the topics to be discussed.
Varvaruca, Eugen
On the existence of extreme surface waves and the Stokes conjecture with vorticity
View Abstract
We present some recent results on singular solutions of the problem of travelling gravity surface water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of large-amplitude regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a symmetric corner of 120 degrees or a horizontal tangent at any isolated stagnation point. Moreover, the profile necessarily has a symmetric corner of 120 degrees if the vorticity is nonnegative near the free surface.
Vlasenko, Vasily
Amplification and supression of internal waves in horizontally sheared currents
View Abstract
The energy exchange between internal waves and barotropic currents over inclined bottom topography is studied theoretically and in the framework of the numerical model. The energy balance equation derived for a continuously stratified fluid predicts that energy can either be transferred toward or away from the internal wave depending on the direction of propagation of both the wave and current. Four scenarios of wave-flow interaction over the inclined bottom were identified. Internal wave extracts energy from the background tidal flow during its propagation upslope-upstream or downslope-downstream and its amplitude grows.

The wave loses energy propagating downslope-upstream or upslope-downstream and reduces in amplitude. This mechanism of suppression or amplification of internal waves by a current over an inclined bottom is verified numerically.

When applied to the area of the Knight Inlet sill, a high resolution fully nonlinear, non-hydrostatic model reproduces the packets of internal waves generated by supercritical tidal flow over the sill.

Careful inspection of the wave fields revealed the presence of an irregular wave structure within wave packets - namely, internal waves are not arranged by amplitude. This phenomenon, obtained numerically and observed in-situ, is treated in terms of the mechanism of wave-flow interaction: the energy exchange between the tidal current and generated internal waves over the inclined bottom topography is the reason for the absence of traditional rank-ordered waves in the packet.

Participants

Name
Institution
Triantaphyllos, AkylasMassachusetts Institute of Technology
Jerry, BonaUniversity of Illinois at Chicago
Tom, BridgesUniversity of Surrey
Roberto, CamassaUniversity of North Carolina
Magda, CarrUniversity of St Andrews
Florent, ChazelUniversité Paris-Est
Wooyoung, ChoiNew Jersey Institute of Technology
Alex, CraikUniversity of St Andrews
José, da SilvaUniversity of Lisbon
Alan, DaviesProudman Oceanographic Laboratory
Frederic, DiasUniversity College Dublin
Vincent, DucheneÉcole Normale Supérieure
David, FarmerUniversity of Rhode Island
Roger, GrimshawUniversity College London
Mark, GrovesUniversität des Saarlandes
John, GrueUniversity of Oslo
Mariana, HaragusUniversité de Franche-Comté
Karl, HelfrichWoods Hole Oceanographic Institution
Gérard, IoossUniversité de Nice
Karima, KhusnutdinovaLoughborough University
Kevin, LambUniversity of Waterloo
David, LannesÉcole Normale Supérieure
Dimitris, MitsotakisUniversité de Paris-Sud
Hai Yen, NguyenUniversité Haute Bretagne
Alfred R, OsborneUniversita' di Torino
Lev, OstrovskyZel Technologies/University of Colorado
Steve, RampMonterey Bay Aquarium Research Institute
Jean-Claude, SautUniversité Paris Sud
Alberto, ScottiUniversity of North Carolina
Victor, ShriraUniversity of Keele
Alexey, SlunyaevIAP, Russian Academy of Sciences
Neil, StapletonDefence Science and Technology Laboratory (DSTL)
Natalia, StashchukUniversity of Plymouth
Shu-Ming, SunVirginia Polytechnic Institute and State University
Tatiana, TalipovaIAP, Russian Academy of Sciences
Steve, ThorpeBangor University (Retired)
John, TolandIsaac Newton Institute for Mathematical Sciences
Jean-Marc, Vanden-BroeckUniversity College London
Eugen, VarvarucaUniversity of Reading
Vasily, VlasenkoUniversity of Plymouth