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Radar detection and characterization of nonlinear waves in the ocean
Stuart Anderson, University of Adelaide, Australia
Theoretical models of wave phenomena in the ocean environment play a central role in our attempts to address the challenges associated with climate change, renewable energy, deep ocean marine engineering and many other scientific, technological and geopolitical domains. While the sophistication of these models continues to grow, along with the computational capacity of the computers we use to implement them for numerical evaluation, there is still an important place for model validation in the form of in situ measurements using state-of-the-art technologies. Moreover, it is not infrequently the case that experiments reveal secondary phenomena that have been overlooked in the prevailing formulations of the primary topics of investigation. Read more about this presentation.
Wave Breaking – Nonlinear Wave Phenomenon
Alexander V. Babanin, The University of Melbourne, Australia
Wave breaking is one of the most significant dynamic processes at the ocean interface, and it influences the lower atmospheric boundary layer as well as the upper-ocean mixing. It facilitates or moderates the air-sea interactions including exchanges of energy, momentum, heat, gas, moisture, aerosol, and its understanding is important across the full range of ocean and coastal engineering
applications, remote sensing of the ocean. Read more about this presentation.
Multidimensional solitons in dispersive complex media: structure and stability. Applications
Vasily Yu. Belashov, Institute of Physics, Kazan Federal University, Russia
This talk is devoted to a one of the most interesting and rapidly developing areas of modern nonlinear physics and mathematics - the analytical and advanced numerical study of the structure and stability of two- and three-dimensional solitons in dispersive complex media described by the generalized system which includes the Kadomtsev Petviashvili and derivative nonlinear Schrodinger classes of equations and takes into account the generalizations relevant to various complex physical media, associated with the effects of high-order dispersion corrections, influence of dissipation and instabilities. This is consistent representation of the both early known and new original results obtained by author and also some generalizations in theory and numerical simulation of nonlinear waves and solitons in complex dispersive media. The analysis of stability of solutions is based on study of transformational properties of system Hamiltonian. Read more about this presentation.
Water wave packet shoaling and dispersive shock waves
Amin Chabchoub, O. Kimmoun, H. Hsu and S. Trillo, Centre for Wind, Waves and Water, School of Civil Engineering, Faculty of Engineering & Information Technologies, The University of Sydney
Hydrodynamic dispersive shock waves (DSW) are well-known to occur in shallow water, for instance within the Korteweg–de Vries framework, and can describe the dynamics of undular or tidal bores, as reported by Zabusky and Kruskal (1965) as well as Trillo et al. (2016). DSW can be also observed in various nonlinear dispersive media, for instance in optics. Recently, Fatome et al. (2014) have experimentally confirmed that the propagation of optical pulses can lead to multiple optical dispersive shock events that interact as a result of four-wave mixing. Read more about this presentation.
Non-linearity in tsunami and tidal bores linked to multiphase flow
Hubert Chanson, Xinqian (Sophia) Leng, Youkai Li, University of Queensland, School of Civil Engineering, Australia
A tidal bore is an unsteady rapidly-varied open channel flow characterised by a rise in water surface elevation in estuarine zones, under spring tidal conditions. Related geophysical applications include the in-river tsunami bore and storm-surge induced bore. After formation, the bore is traditionally analysed as a hydraulic jump in translation and its leading edge is characterised by a breaking roller for Fr1 > 1.3-1.5. The roller is a key flow feature characterised by intense turbulence and air bubble entrainment, associated with intense turbulent stresses across the water column and bed
sediment motion. Read more about this presentation.
Computation of steady waves in shallow (and deep) water
Didier Clamond, Universit´e Cˆote d’Azur, Laboratoire J. A. Dieudonn´e, Parc Valrose, Nice cedex 2, France
Gravity waves with a characteristic wavelength significantly larger than the mean water depth are often encountered in coastal engineering problems. Several accurate algorithms exist for computing (fully nonlinear) steady surface waves. However, none of these algorithms can compute long (cnoidal) waves, as they fail when the length-over-depth ratio exceed about 30 − −60, depending on the method. Therefore, for long waves, one has to rely on shallow water approximations. Read more about this presentation.
The forced coupled KdV equations as a model for internal waves in the atmosphere
Igor Korostil and Simon Clarke, School of Mathematical Sciences, Monash University, Australia
The coupled Korteweg—de Vries (KdV) equations model the resonant interaction of two modes of long, weakly nonlinear waves. Such behaviour has been proposed as the generation mechanism for the Morning Glory roll cloud over Cape York. Here we derive a forced version of the coupled KdV equations to model internal waves propagating on two layer interfaces with velocity shear and uneven bottom topography. The characteristics of these equations are discussed for two simple configurations of velocity shear and stratification. An algorithm is then derived to obtain steady, solitary wave solutions of these equations. Finally, the various types of solitary wave solutions and their stability is discussed.
Simple exact solutions of one-diemnsional linear and nonlinear shallow water equations over sloping bottom.
Sergey Yu. Dobrokhotov, Ishlinsky Institute for Problems in Mechanics and Moscow Institute of Physics and Technology in cooperation with A. Aksenov, K. Druzhkov and B. Tirozzi.
First, we present a wide class of simple exact solutions to 2-D wave equation with constant velocity c describing the waves generated by special spatially localized sources. Far from the source and for large time t these solutions are localized near the circles (fronts) |x|=ct and have the simple effective asymptotics. Then we take these asymptotics as initial data for the 1-D linear shallow water equations over sloping bottom and show that this type of initial data implies wide class of solutions of this equation in simple algebraic form. The application of the Carrier-Greenspan transform gives the class of exact solutions in the parametric form of nonlinear shallow water equations over sloping bottom. In particular, these solutions describe the interaction of "solitary waves" and "smooth steps" with the sloping bottom. We discuss applications of these solutions to the run-up problem, their relationship with Pelinovskii–Masova solutions and asymptotic generalization.
This work was supported by Russian Scientific Foundation, project No 16-11-10282.
Interaction of Korteweg–de Vries solitons with external sources
Andrei Ermakov and Yury Stepanyants, School of Agricultural, Computational and Environmental Sciences, University of Southern Queensland, Australia
We revise the solutions of the forced Korteweg–de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous works where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg–de Vries equation can be obtained for the specific forcings of arbitrary amplitude. Read more about this presentation.
On the impossibility of solitary Rossby waves in meridionally unbounded domains
Georg Gottwald, The University of Sydney, Australia in cooperation with Dmitry Pelinovsky, McMaster University, Canada
Evolution of weakly nonlinear and slowly varying Rossby waves in planetary atmospheres and oceans is considered within the quasi-geostrophic equation on unbounded domains. When the mean flow profile has a jump in the ambient potential vorticity, localized eigenmodes are trapped by the mean flow with a non-resonant speed of propagation. We address amplitude equations for these modes. Whereas the linear problem is suggestive of a two-dimensional Zakharov–Kuznetsov equation, we found that the dynamics of Rossby waves is effectively linear and moreover confined to zonal waveguides of the mean flow. This eliminates even the ubiquitous Korteweg–de Vries equations as underlying models for spatially localized coherent structures in these geophysical flows.
Solitary wave trains and undular bores
Roger Grimshaw, University College London, London, UK
In the weakly nonlinear long wave regime many physical systems, notably nonlinear internal waves in the coastal oceans, can be modelled with the variable-coefficient Korteweg-de Vries equation: see the equation and remainder of the abstract.
Optimal shear instabilities of large-amplitude internal solitary waves
Karl R. Helfrich (Woods Hole Oceanographic Institution), P.-Y. Passaggia (Universite d’Orleans), and B. L. White (UNC-Chapel Hill)
The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) with thin interfaces is analyzed by means of linear optimal transient growth methods. Optimal perturbations are computed through direct-adjoint loop iterations of the Navier–Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin–Long (DJL) equation. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone of potentially unstable Richardson number, Ri < 0.25. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with ISW amplitude. A local WKB approximation for spatially growing Kelvin–Helmholtz (KH) waves through the Ri < 0.25 zone captures properties (e.g., carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. Read more about this presentation.
Finite-amplitude compact wavepackets in rotating flows
Edward R. Johnson, Department of Mathematics, University College London, UK
Steady solitary waves in non-rotating inviscid shallow water travel faster than any linear wave and so there is no mechanism through which they can lose energy. If however the undisturbed system is rotating as a whole, in solid body rotation about a vertical axis, then the resistance of the fluid to stretching along the axis of rotation adds an effective stiffness to the surface that becomes strongest for the longest waves. The phase velocity of linear waves thus increases without bound as their wavelength increases and so any localised steadily propagating disturbance generates a lee-wave wake, loses energy and disperses. Roger Grimshaw, working with J.-M. Read more about this presentation.
Martingale solution to stochastic Korteweg - de Vries equation
Anna Karczewska, Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Poland
The celebrated Korteweg-de Vries equation (KdV for short), derived from the set of Eulerian shallow water and long wavelength equations, becomes a paradigm in the field of nonlinear partial differential equations. KdV appears as the lowest approximations of wave motion in several fields of physics.
A natural continuation of the study of the KdV equation seems to be a consideration of stochastic versions of such an equation. The KdV equation driven by random noise can be a model of several kinds of waves (e.g., surface water waves, waves in plasma) influenced by random factors. Two cases of the stochastic KdV equation are possible - the case with an additive noise and the case with a multiplicative noise. Read more about this presentation.
On the Ostrovsky equation and related topics
Karima Khusnutdinova, Department of Mathematical Sciences, Loughborough University, Loughborough, UK
In this talk, I will overview some recent results related to the Ostrovsky equation [1]. Firstly, I will discuss the effects of the parallel shear flow on internal waves in a rotating ocean [2, 3]. We found first examples when the shear flow can change the sign of the rotation coefficient in the Ostrovsky equation, leading to unusual dynamics [3]. Secondly, I will discuss the dynamics of two distinct linear long wave modes with nearly coincident phase speeds, described by the system of coupled Ostrovsky equations. Interestingly, the dominant features of the complex dynamical behaviour observed in numerical simulations can be classified and interpreted in terms of the main features of the linear dispersion curves [3], resembling the qualitative theory of ODEs. Read more about this presentation and see the references.
The interaction of multi-lumps within the Kadomtsev–Petviashvili-1 equation: analytical and numerical results
Zhiming Lu, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, China
The interactions between the lumps and multi-lumps within the Kadomtsev–Petviashvili-1 (KP1) equation is studied both analytically and numerically. The dependence of the multi-lump structures on free parameters is discussed in details. Some interesting phenomena are obtained and demonstrated for the interactions of single lumps with each other and with more complex objects such as bi-lumps, as well as the interactions of bi-lumps with each other. Finally, the generation of these multi-lumps by a forced KP1 equation is discussed.
Generalised solitary gravity-capillary waves in the forced Korteweg-de Vries model
Montri Maleewong, Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok, Thailand
Free surface flow past an applied pressure distribution in water of finite depth is presented in this work. Gravity and surface tension effects are included in the problem. Applying the asymptoticanalysis, the resulting model is represented by the forced Korteweg-de Vries model (fKdV) where the Froude number and the Bond number are introduced to determine flow regime and capillary effects, respectively. At steady state, the fKdV model is solved numerically by the wavelet Galerkin method with Neumann boundaries in the far field. In the case of the capillary effect is dominated, a generalised solitary wave is found. This finding shows us more complete bifurcation diagram in the water wave theory.
Multidimensional solitons in optics and ultracold gases: Predictions and creation
Boris A. Malomed Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel also affiliated with
ITMO University, St. Petersburg 197101, Russia, School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
It is commonly known that the interplay of linear and nonlinear effects gives rise to solitons, i.e., self-trapped localized structures, in a wide range of physical settings, including optics, Bose–Einstein condensates (BECs), hydrodynamics, plasmas, condensed-matter physics, etc. Solitons are considered as an interdisciplinary class of modes, which feature diverse internal structures. While most experimental realizations and theoretical models of solitons have been elaborated in one-dimensional (1D) settings, a challenging issue is prediction of stable solitons in 2D and 3D media. In particular, multidimensional solitons may carry an intrinsic topological structure in the form of vorticity. In addition to the “simple” vortex solitons, fascinating objects featuring complex structures, such as hopfions, i.e., vortex rings with internal twist, have been predicted too. Read more about this presentation.
Dispersive shock waves governed by the Whitham equation and their stability
Timothy R. Marchant, School of Mathematics and Applied Statistics, University of Wollongong, Australia; X. An and Noel F. Smyth, School of Mathematics, University of Edinburgh, Scotland, U.K.
Dispersive shock waves (DSWs), also termed undular bores in uid mechanics, governed by the nonlocal Whitham equation are studied in order to investigate short wavelength e ects that lead to peaked and cusped waves within the DSW. This is done by combining the weak nonlinearity of the Korteweg-de Vries equation with full linear dispersion relations. The dispersion relations considered are those for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120o peaked Stokes wave of highest amplitude. A dispersive shock tting method is used to nd the leading (solitary wave) and trailing (linear wave) edges of the DSW. Read more about this presentation.
Linear and nonlinear surface wave patterns free-surface for flow past a submerged source or doublet
Scott W. McCue, Queensland University of Technology, Australia
This talk is concerned with steady free-surface flow past a submerged point source or doublet singularity. Behind the singularity, the wave pattern has a distinctive V-shape that is often characterised by the Kelvin wake angle. For the linearised regime, the problem is equivalent to flow past a submerged semi-infinite Rankine body with a rounded nose or submerged sphere. Interestingly, for both large and small Froude numbers, it turns out that the wake angle appears to be less than the Kelvin angle. For the nonlinear version of the problem, we apply a boundary integral-method based on Green's second formula. We are able to compute highly nonlinear solutions for which the waves have distinctive properties that are unlike their linear counterparts.
Nonlinear waves in rotating fluids
L. A. Ostrovsky, Department of Applied Mathematics, University of Colorado, USA; Y. A. Stepanyants, University of Southern Queensland, Australia
The non-trivial dynamics of nonlinear dispersive waves affected by the Coriolis force is discussed. Applications include surface and internal waves in the ocean, magnetic sound in plasma, and other phenomena. The corresponding model equation (rKdV equation) has the form: see the equation and remainder of the abstract.
Long traveling waves in a fluid with a variable depth
Efim Pelinovsky, Institute of Applied Physics of the Russian Academy of Sciences, Russia; Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Russia.
Long wave propagation in a one- or two-layer fluid with a variable depth is studied for the specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow-water theory and determined by a family of solutions of the 2nd order ODE with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the shore in non-stratified fluid or the point where the two-layer flow transforms into a uniform one. In the vicinity of this point the nonlinear shallow-water theory is used and the wave breaking criterion corresponding to the gradient catastrophe is found. Read more about this presentation.
Time-frequency analysis of nonlinear ship waves
Ravindra Pethiyagoda, Timothy J. Moroney, Scott W. McCue, Queensland University of Technology, Australia
A spectrogram is a useful way of applying short-time discrete Fourier Transforms to visualise surface height measurements taken of ship waves in the real world. Previous research had identified a number of components of a spectrogram of a ship wave pattern, only two of which could be explained via linear theory. We use computer simulations of nonlinear ship waves to further study this problem. For this purpose, we consider nonlinear potential flow past a pressure applied to a free surface. Spectrogram analysis is applied to the computed nonlinear ship waves. Key features of the spectrograms, such as the linear dispersion curve, primary and secondary modes are identified. The multiple modes observed in this study bear a striking resemblance to components identified on spectrograms taken from previous experimental measurements of a high-speed ferry in the Gulf of Finland. If time permits, the effects of acceleration and finite-depth will be discussed.
Interactions of vector solitons in the model of a particle chain
Nawin Raj and Yury Stepanyants, School of Agricultural, Computational and Environmental Sciences, University of Southern Queensland, Australia.
As has been shown in the paper [3], exural transverse long waves of a small amplitude in an anharmonic chain of atoms can be described by the following vector equation: see the equation and the remainder of the abstract.
Macroscale simulation of nonlinear waves via computation only on small staggered patches
Anthony Roberts, School of Mathematical Sciences, University of Adelaide, Australia
The multiscale gap-tooth scheme is built from given small microscale simulations of complicated physical processes to empower large macroscale simulations. By coupling small patches of simulations over un-simulated physical gaps, large savings in computational time are possible. Here we discuss generalising the gap-tooth scheme to the case of wave systems on staggered grids in both 1D and 2D. Classic macroscale interpolation provides a generic coupling between patches that achieves arbitrarily high order consistency between the emergent macroscale simulation and the underlying microscale dynamics. Eigen-analysis indicates that the resultant scheme empowers feasible computation of large macroscale simulations of wave systems even with complicated underlying physics. For examples, we simulate dam-breaking and turbulent oods by this scheme.
Exact soliton, periodic and superposition solutions to the fifth-order Korteweg-de Vries equation and derivation of a new equation for an uneven bottom
Piotr Rozmej, Faculty of Physics and Astronomy, University of Zielona Góra, Poland
In the presentation we discuss several kinds of the exact solutions to the extended Korteweg—de Vries equation (name given by Marchant and Smyth [1]). This equation is obtained in a perturbation approach of second order with respect to small parameters, whereas KdV results from the same perturbation approach but limited to fi rst order. That is why we call this equation KdV2 [2]. Alternatively, in the literature appears the name the fifth-order Korteweg—de Vries equation since it contains the fifth space derivative of the unknown function. Read more about this presentation.
Blow-up of vorticity waves in homogeneous and weakly stratified boundary layers
Victor Shrira, Keele University, United Kingdom
High Reynolds number boundary layers are ubiquitous in nature and engineering context. Often the velocity shear coexists with density stratification. For example, in the ocean the boundary layer in the water adjacent air-water interface plays a key role in ocean atmosphere interaction; the first 2.5 m have the heat capacity of the entire atmosphere above. This boundary layer might be stratified with a stable stratification caused by solar heating or by entrainment of air bubbles produced by breaking waves. The oceanic bottom boundary layers are often stratified due to sediment entrainment. Engineering provides ennumerous variety of laminar and turbulent boundary layers, which are often unstable. In the linear setting the nonstratified boundary layers support a single vorticity mode which might be stable or unstable, while in the presence of stratification there appears also an in nite number of modes corresponding to internal gravity waves. Read more about this presentation.
Scattering of surface waves by a periodic system of spikes at the bottom.
Alex Skvortsov, Maritime Division, Defence Science and Technology, Fishermans Bend, Australia
The reflection of small-amplitude surface waves from a sharp change of bottom morphology is analysed. A 2D comb-like surface is used as a model of ‘spiky’ bottom morphology. This surface can be characterised by two parameters, viz., height of the spikes and period of the comb, which can have arbitrary relative values. The problem is treated analytically by means of conformal mapping. The amplitude of transmitted and reflected waves is explicitly calculated in terms of the geometrical parameters of the comb-like surface. It is shown that the geometrical parameters of of bottom morphology can be combined in one aggregated parameter (‘effective’ height) that can be evaluated by taking far-field limit of the solution in transformed coordinates [1]. See the references.
Nematic liquid crystals, resonant undular bores and the fifth order Korteweg-de Vries equation
Noel Smyth, School of Mathematics, University of Edinburgh; Mark Hoefer and Pat Sprenger, Department of Applied Mathematics, University of Colorado Boulder
Undular bores are an unsteady waveform which is widely observed in nature, familiar examples being tidal bores and tsunamis. Standard undular bores are a continually evolving and spreading waveform consisting of solitary waves at one edge and linear dispersive waves at the other. While nonlinear, dispersive wave equations, such as the Korteweg-de Vries (KdV) and nonlinear chrodinger (NLS) equations, are well known and studied for their solitary wave solutions, these equations also possess undular bore solutions, also termed dispersive shock wave solutions. Undular bore solutions are typically found as simple wave solutions of the Whitham modulation equations of the governing equation when these equations are hyperbolic and so the underlying steady periodic wave solution is stable. The undular bore solution is thus found as a modulated periodic wave. This talk will look at non-standard undular bores which are not of this standard form. Read more about this presentation.
The effects of interplay between the rotation and shoaling for a solitary wave on variable topography
Yury Stepanyants, School of Agricultural, Computational and Environmental Sciences, University of Southern Queensland, Australia
The specific features of solitary wave dynamics within the framework of the Ostrovsky equation with variable coefficients are considered in relation to surface and internal waves in a rotating ocean with a variable bottom topography. It is shown that for solitary waves moving toward the beach, the terminal decay caused by the rotation effect can be suppressed by the shoaling effect. Two basic examples of a bottom profile are analysed in detail and supported by direct numerical modelling. One of them is a constant-slope bottom and the other is a specific bottom profile providing a constant amplitude solitary wave. Estimates with real oceanic parameters show that the predicted effects of stable soliton dynamics in a coastal zone can occur, in particular, for internal waves.
Breathers in the stratified fluid: generation, dynamics and stability
Tatyana G. Talipova and O.E. Kurkina, Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia and Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia.
Existence of internal wave breathers in a density stratified fluid has been predicted by the asymptotic theory within the framework of KdV-type equations. Observations of long-wave internal breather-like waves off the coast of New Jersey in the Yellow Sea near the Korean coast and in the Celtic Sea have been recently reported. Study of internal breather dynamics now is one of the topical problems in physical oceanography. The analytical description of long-wave breathers is based on the completely integrable Gardner equation with a positive coefficient of the cubic nonlinearity. Read more about this presentation.
Linear and nonlinear equatorial atmospheric waves coupled to moist convection
Vladimir Zeitlin, Laboratory of Dynamical Meteorology, Sorbonne University and Ecole Normale Superieure, France
Equatorial regions in the atmospheres and oceans of rotating planets are special, as they are home to a variety of specific waves trapped in the vicinity of equator. Some of these waves have unidirectional propagation and no dispersion (equatorial Kelvin waves), or unidirectional propagation and weak (strong) dispersion in the long(short)-wave limit (equatorial Rossby waves), others are bi-directional and weakly dispersive in the short-wave limit (inertia-gravity and Yanai waves). Combined with nonlinear effects, this variety of dispersion properties gives rise to a number of archetypal nonlinear wave behaviors for different types of waves: soliton formation, steepening and breaking, resonant interactions, etc, which I will illustrate. Read more about this presentation.