Dispersive shock waves (DSW) are a salient feature of long water waves often observed in tidal bores and tsunami/meteotsunami contexts. Their interaction with bathymetry is poorly understood. The shoreline hazard from tsunamis and meteotsunamis critically depends on the fraction of incoming energy flux transmitted across the shallow nearshore shelf. Here, by considering nonlinear dynamics of waves over variable depth within the framework of the Boussinesq equations we show that the transmitted energy flux fraction can strongly depend on the initial amplitude of the incoming wave and the distance it travels. The phenomenon is similar to self-induced transparency in nonlinear optics: small amplitude waves are reflected by bathymetry inhomogeneity, while a larger amplitude ones pass through. The mechanism of self-induced transparency of long water waves can be explained as follows. In linear setting a bathymetry inhomogeneity, of length comparable to that of the incident wavelength, by transmitting high wavenumber components acts as a high-pass filter. The DSW evolution efficiently transfers wave energy into high wavenumber band, where reflection is negligible. By examining an idealized model of bathymetry we show that this is an order one effect and explore its dependence on parameters in the range relevant for meteotsunamis. The role of wave energy transfer into high wavenumber band owing to the growth of bound harmonics unrelated to the DSW was found to be small.

The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg-de Vries (KdV) equation. Among the topics we focus upon are "radiating solitons," the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV equations with the modular and "sublinear" nonlinearities. Such equations have an interesting class of solutions, called compactons, solitary waves defined on a finite spatial interval. Both the properties of single solitons and the interactions between them are discussed. We show that even minor non-elastic effects in the soliton-soliton collisions can accumulate and result in a qualitatively different asymptotic behavior. A statistical description of soliton ensembles ("soliton gas"), which emerges as a major theme, has been discussed for several models. We briefly outline the recent progress in studies of ring solitons and lumps within the framework of the cylindrical KdV equation and its two-dimensional extension. Ring solitons and lumps (2D solitons) are of particular interest since they have many features in common with classical solitons and yet are qualitatively different. Particular attention is paid to interactions between the objects of different geometries, such as the interaction of ring solitons and shear flows, ring solitons and lumps, and lumps and line solitons. We conclude our review with views of the future developments of the selected lines of studies of localized wave structures in the theory of weakly nonlinear, weakly dispersive waves.

The shoreline hazard posed by ocean long waves such as tsunamis and meteotsunamis critically depends on the fraction of energy transmitted across the shallow near-shore shelf. In linear setting, bathymetric inhomogeneities of length comparable to the incident wavelength act as a protective high-pass filter, reflecting long waves and allowing only shorter waves to pass through. Here, we show that, for weakly nonlinear waves, the transmitted energy flux fraction can significantly depend on the amplitude of the incoming wave. The basis of this mechanism is the formation of dispersive shock waves (DSWs), a salient feature of nonlinear evolution of long water waves, often observed in tidal bores and tsunami/meteotsunami evolution. Within the framework of the Boussinesq equations, we show that the DSWs efficiently transfer wave energy into the high wavenumber band, where reflection is negligible. This is phenomenologically similar to self-induced transparency in nonlinear optics: small amplitude long waves are reflected by the bathymetric inhomogeneity, while larger amplitude waves that develop DSWs blueshift into the transparency regime and pass through. We investigate this mechanism in a simplified setting that retains only the key processes of DSW disintegration and reflection, while the effects such as bottom dissipation and breaking are ignored. The results suggests that the phenomenon is a robust, order-one effect. In contrast, the increased transmission due to the growth of bound harmonics associated with the steepening of the wave is weak. The results of the simplified modelling are validated by simulations with the FUNWAVE-TVD Boussinesq model.

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

The description of weakly nonlinear water-wave evolution over a horizontal bottom by the integro-differential Zakharov equation, because of utilising the underlying Hamiltonian structure, has many advantages over direct use of the Euler equations. However, its application to finite-depth situations is not straightforward since, in contrast to the deep-water case, the kernels governing the four-wave interactions are singular, as well as the kernels in the canonical transformation that removes non-resonant interactions from the original equations of motion. At the singularities, these kernels are finite but not unique. The issue of how to use the Zakharov equation for finite depth and whether it is possible at all was debated intensely in the literature for decades but remains outstanding. Here we show that the absence of a limit of the kernels at the singularities is inconsequential, since in the equations of motion it is only the integral that matters. By applying the definition of the Dirac- $\delta$ , we show that all the integrals involving a trivial manifold singularity are evaluated uniquely. Therefore, the Zakharov evolution equation and the nonlinear canonical transformation are only apparently singular. The findings are validated by application to examples where predictions based on the Zakharov equation are compared with known solutions obtained from the Euler equations.

Edge waves, the infragravity waves trapped by near-shore topography, are important in morphodynamics and flooding on mildly sloping beaches. Edge waves are usually generated by swell via triad interactions. Here, we examine the possibility that edge waves might be also generated directly by wind. By processing data from the SandyDuck’97 near-shore experiment, we show that pronounced directional asymmetry of edge waves does occur in nature, apparently unrelated to the direction of swells and along-shore currents. These observations exhibit edge waves propagating in the downwind direction under moderate wind against the along-shore currents, while swell is incident nearly normally to the shoreline, which strongly suggests generation of edge waves by wind. We examine theoretically possible mechanisms of edge-wave excitation by wind. We show that the ‘maser’ mechanism suggested by Longuet-Higgins (Proc. R. Soc. Lond. A, vol. 311, issue 1506, 1969b, pp. 371–389) in the context of excitation of free water waves is effective under favourable conditions: nonlinearly interacting random short wind-forced waves create a viscous shear stress on the water surface with the variation of stress being phase linked to edge waves, which allows self-excitation of a coherent edge wave. The model we put forward is based upon the kinetic equation for short wind waves propagating on the inhomogeneous current due to an edge wave. The model needs a dedicated experiment for validation. Analysis of plausible alternative mechanisms of generation via Miles’ critical layer and via the viscous shear stresses induced by the edge wave in the air revealed no instability in the consideration confined to the main mode and constant slope bathymetry.

We examine the long-term evolution of a random wind wave field generated by constant forcing, by comparing numerical simulations of the kinetic equation and direct numerical simulations (DNS) of the dynamical equations. While the integral characteristics of the spectra are in reasonably good agreement, the spectral shapes differ considerably at large times, the DNS spectral shape being in much better agreement with field observations. Varying the number of resonant and approximately resonant wave interactions in the DNS numerical scheme, we show that when the ratio of nonlinear and linear parts of the Hamiltonian tends to zero, the DNS spectral shape approaches the shape predicted by the kinetic equation. We attribute the discrepancies between the kinetic equation modeling, on one side, and the DNS and observations, on the other, to the neglect of non-Gaussianity in the derivation of the kinetic equation.

Mixing in the uppermost part of the water column is crucial for modelling air-sea interaction, yet it remains poorly understood, especially the processes under strong wind conditions. The Ekman boundary layers are a salient feature of the air-sea interface. In the the Ekman boundary layers the current velocity vectors always rotates, making two components of the basic flow vorticity comparable and, thus, the boundary layer three dimensional. Linear instabilities of the homogeneous steady Ekman layers were examined and found to occur for sufficiently large turbulent Reynolds numbers. Here, we derive a model of nonlinear instabilities of 3d Ekman layer in deep ocean taking into account also a possible weak stratification of the boundary layer caused by air entrainment due to wave breaking or solar heating. The model exploits the observation that the corresponding linearized boundary value problem always supports a “vorticity wave” mode which is often decaying. Employing an asymptotic procedure utilizing smallness of the boundary layer thickness to the characteristic wavelength of perturbations scaled as inverse Reynolds number squared we derive a novel nonlinear evolution equation with a pseudo-differential dispersion. We take into account viscosity and weak stratification in the boundary layer. Within the framework of this equation a wide class of initial conditions, which we a priori specify, leads to `collapses’ of localized perturbations, that is an initial perturbation becomes more and more localised and its amplitude becomes infinite in finite time forming a point singularity. We derived a self-similar solution describing these collapses. The mechanism of collapse is essentially nonlinear. A new insight into linear instabilities has been also obtained. The collapses are expected to result in intense mixing and even temporary destruction of the boundary layer.

Mixing in the uppermost part of the water column is crucial for modelling air-sea interaction, yet it remains poorly understood, especially the processes under strong wind conditions. The Ekman boundary layers are a salient feature of the air-sea interface. In the the Ekman boundary layers the current velocity vectors always rotates, making two components of the basic flow vorticity comparable and, thus, the boundary layer three dimensional. Linear instabilities of the homogeneous steady Ekman layers were examined and found to occur for sufficiently large turbulent Reynolds numbers. Here, we derive a model of nonlinear instabilities of 3d Ekman layer in deep ocean taking into account also a possible weak stratification of the boundary layer caused by air entrainment due to wave breaking or solar heating. The model exploits the observation that the corresponding linearized boundary value problem always supports a “vorticity wave” mode which is often decaying. Employing an asymptotic procedure utilizing smallness of the boundary layer thickness to the characteristic wavelength of perturbations scaled as inverse Reynolds number squared we derive a novel nonlinear evolution equation with a pseudo-differential dispersion. We take into account viscosity and weak stratification in the boundary layer. Within the framework of this equation a wide class of initial conditions, which we a priori specify, leads to `collapses’ of localized perturbations, that is an initial perturbation becomes more and more localised and its amplitude becomes infinite in finite time forming a point singularity. We derived a self-similar solution describing these collapses. The mechanism of collapse is essentially nonlinear. A new insight into linear instabilities has been also obtained. The collapses are expected to result in intense mixing and even temporary destruction of the boundary layer.

Edge waves (EW) are surface gravity waves topographically trapped near the highly reflective ocean shorelines. Over mildily sloping beaches, the high-reflectivity condition is only satisfied for infragravity waves (IGW, periods of a few minutes). Initially believed to drive alongshore-periodic shoreline features, EW have been shown to be important also for a variety of coastal ocean processes such as nonlinear shoaling of wind waves, coastal flooding, ice-shelf break up in polar oceans, and others. As IGW, on mildly sloping beaches EW are outside the wind-wave frequency range, which seems to exclude direct wind forcing as generating mechanism. It is generally agreed that IGW ove mildly sloping beaches are generated by nonlinear swell interaction. Wave-wave interactions can excite both alongshore progressive and standing EW, but EW directional symmetry should match swell directionalty. This simple rule is confirmed also by observations. Exceptions to thius rule are intriguing: if directionally-asymmetric edge waves fields that do not match the swell direction, occur, the implication is that wave-wave interactions are not the dominant IGW/EW generation mechanism. Direct wind forcing would then be the only conceivable candidate. The high correlation of swell and IG wave directionality, however, suggests that such occurrences must be rare, possibly associated with peculiar coastal weather conditions. We investigate data produced by the most comprehensive effort to date to study EW - the nearshore array deployed by Elgar, Herbers, O'Reilly and Guza during the SandyDuck'97 experiment - which recorded pressure and velocity continuously at 2 Hz from August to December 1997, at sensors distributed on six alongshore lines between approximately the 1-m and 6-m isobaths near the Duck NC pier. Estimates directional IGW/EW match well swell directionality. However, a few events exhibit strong IG/EW directional asymmetry matching wind direction, with nearly shorenormal offshore swells. In most of these cases, IGW propagate against the nearshore current. These events are consistent with a mechanism for direct generation of IGW/EW by wind. It is not clear whether their scarcity is due to intrinsic properties of the wind generation mechanism, or to the rather low-energy conditions of the SandyDuck'97 experiment. In general, both nonlinear wave-wave interactions and wind generation should be taken into account, and we expect the wind generation mechanism to play an increasingly important role in storms, for example, for modeling wave surges. An investigation into modeling EW generation by wind will be reported elsewhere.

We consider the evolution of directional spectra of waves generated by constant and changing wind, modelling it by direct numerical simulation (DNS), based on the Zakharov equation. Results are compared with numerical simulations performed with the Hasselmann kinetic equation and the generalised kinetic equation, and with airborne measurements of waves generated by offshore wind, collected during the GOTEX experiment off the coast of Mexico. Modelling is performed with wind measured during the experiment, and the initial conditions are taken as the observed spectrum at the moment when wind waves prevail over swell after the initial part of the evolution. Directional spreading is characterised by the second moment of the normalised angular distribution function, taken at selected wavenumbers relative to the spectral peak. We show that for scales longer than the spectral peak the angular spread predicted by the DNS is close to that predicted by both kinetic equations, but it underestimates the corresponding measured value, apparently due to the presence of swell. For the spectral peak and shorter waves, the DNS shows good agreement with the data. A notable feature is the steady growth of angular width at the spectral peak with time/fetch, in contrast to nearly constant width in the kinetic equations modelling. Dependence of angular width on wavenumber is shown to be much weaker than predicted by the kinetic equations. A more detailed consideration of the angular structure at the spectral peak at large fetches shows that the kinetic equations predict an angular distribution with a well-defined peak at the central angle, while the DNS reproduces the observed angular structure, with a flat peak over a range of angles. In order to study in detail the differences between the predictions of the DNS and the kinetic equations modelling under idealised conditions, we also perform numerical simulations for the case of constant wind forcing. As in the previous case of forcing by real wind, the most striking difference between the kinetic equations and the DNS is the steady growth with time of angular width at the spectral peak, which is demonstrated by the DNS, but is not present in the modelling with the kinetic equations. We show that while the kinetic theory, both in the case of the Hasselmann equation and the generalised kinetic equation, predicts a relatively simple shape of the spectral peak, the DNS shows a more complicated structure, with a flat top and dependence of the peak position on angle. We discuss the approximations employed in the derivation of the kinetic theory and the possible causes of the found differences of directional structure.

ABSTRACT The work is concerned with the theoretical examination of a new type of thermo-acoustic instability in combustors not reported in the literature. The instability results from linear coupling between the conventional acoustic mode and the recently discovered “flame intrinsic modes.” Within the framework of a 1D model of a quarter wave resonator with the standard model of flame heat release, intrinsic-acoustic mode coupling occurs when the real parts of the frequencies of neighboring acoustic and flame-intrinsic modes at small interaction index are close. While at small the Eigen-functions of close acoustic and flame intrinsic modes clearly exhibit their distinctive identities, with increase of the mode identities become blurred and the Eigen-functions of acoustic modes resemble more and more those of flame intrinsic modes and at a certain become indistinguishable. We refer them as coupled intrinsic-acoustic modes or coupled modes. When the “Rayleigh index” for a coupled mode behaving as an acoustic mode at small is negative, at a larger such a mode can nevertheless become unstable at one of the nearby intrinsic mode frequencies. We find analytically the instability domain due to coupling in the parameter space. Near the instability boundary, we reduce the transcendental dispersion relation to a quadratic or, if higher accuracy is desired, to a quartic equation. These models capture well all four possible coupling scenarios.

Abstract In nature, the wind waves of the gravity–capillary range are noticeably skewed forward. The salient feature of such waves is a characteristic pattern of capillary ripples on their crests. The train of these ‘parasitic capillaries’ is not symmetric with respect to the crest, it is localised on the front slope and decays towards the trough. Although understanding the gravity–capillary waves front–back asymmetry is important for remote sensing and, potentially, for wave–wind interaction, the physical mechanisms causing this asymmetry have not been identified. Here, we address this gap by extensive numerical simulations of the Euler equations employing the method of conformal mapping for two-dimensional potential flow and taking into account wave generation by wind and dissipation due to molecular viscosity. On examining the role of various factors contributing to the wave profile front–back asymmetry: wind forcing, viscous stresses and the Reynolds stresses caused by ripples, we found, in the absence of wave breaking, the latter to be by far the most important. It is the lopsided ripple distribution which leads to the noticeable fore–aft asymmetry of the mean wave profile. We also found how the asymmetry depends on wavelength, steepness, wind, viscosity and surface tension. The results of the model are discussed in the context of the available experimental data on asymmetry of gravity–capillary waves in both the breaking and non-breaking regimes. A reasonable agreement of the model with the data has been found for the regime without breaking or microbreaking.

Transformation of spectral shape during wind wave development and the transition from the spectrum of developing waves to the spectrum of fully developed waves are well documented in measurements, but have so far escaped all modelling, as well as theoretical explanation. Numerical models of long-term wind wave evolution are based on the Hasselmann kinetic equation (KE). The KE predicts strict self-similarity beyond the initial several thousand characteristic periods of wave development, and therefore cannot describe the subsequent change of spectral shape. Instead, it predicts that the self-similar spectral shape, with a steep front and an enhanced peak, holds at arbitrary fetch, notwithstanding the experimental evidence that mature waves are characterised by the much wider Pierson-Moskowitz spectral shape. To resolve the contradiction, we perform long-term modelling of wind wave evolution by direct numerical simulation (DNS), based on the Zakharov equation. We model a particular class of situations when the wave field at hand is generated by a strong quasi-stationary offshore wind jet, which is caused by pressure differences and accelerates passing through a valley into the sea. Examples of such phenomena are the Tehuano event off the Pacific coast of Mexico, and the Mistral in the northern Mediterranean. Modelling results are compared with the airborne observations of waves generated by these winds, collected during GOTEX and HYMEX experiments respectively. In parallel we also perform numerical simulations with the Hasselmann kinetic equation and the generalised kinetic equation. For modelling of waves off the Mexican coast, wind data are taken from measurements during the GOTEX experiment, and the initial conditions from the measured spectrum at the moment when wind waves prevail over swell after a short initial part of the evolution. Waves in the Mediterranean Sea are modelled with constant wind forcing and zero initial condition. We show that the evolution of integral characteristics, e.g. significant wave height and wave steepness, is reproduced reasonably well by all modelling approaches. However, the spectral shape of developed waves demonstrates a large discrepancy between, on the one hand, the measured spectra and the DNS modelling and, on the other hand, spectra modelled by both kinetic equations. At the intermediate and advanced stage of development, both measured spectra and the DNS spectra tend to Pierson-Moskowitz spectral shape, while the modelling based on the kinetic equations invariably predicts spectra with a higher, more pronounced peak. In terms of the parameter of spectral peakedness, a commonly convenient measure of spectral shape, there is a large (of order one) discrepancy. We propose a theoretical explanation of the discrepancy as being due to the neglect of non-gaussianity in the derivation of the kinetic equations, and provide a numerical confirmation of this hypothesis.

The work examines the Ekman current response to a steady wind within the Stokes-Ekman paradigm. Under constant wind in the classical Ekman model there is a single attractor corresponding to the Ekman (1905)steady solution. It is known that the account of wind waves strongly affects the Ekman current dynamics via the Stokes drift, which is described by the Stokes-Ekman model. Waves continue to evolve even under constant wind, which makes steady solutions of the Stokes-Ekman equation impossible. Since the dynamics of the Ekman response in the presence of evolving wave field have not been considered, the basic questions on how the Ekman current evolves and, especially, whether it grows or decays at large times, remain open. Here by employing the known self-similar laws of wave field evolution and solving analytically the the Stokes-Ekman equation we find and analyse evolution of the Ekman current. We show that the system has a single time dependent attractor which can be described asymptotically. The large time asymptotics of the Ekman current is found to be determined by the regime of wave field evolution: for the regimes typical of young waves the Ekman current grows with time to infinity, in contrast, for `old waves' the Ekman current asymptotically decays.

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter D. In the limits of large and small D, the 2d-ILW equation respectively tends to the 2d Benjamin-Ono and 2d Zakharov-Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude a and width σ, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane (a, σ) for various values of D. The amplitude threshold a increases as D and σ decrease and tends to infinity at D → 0. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all D except D = 0. But the equation itself has not been proved for small D. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.

The work examines upper-ocean response to time-varying winds within the Ekman paradigm. Here, in contrast to the earlier works we assume the eddy viscosity to be both time and depth dependent. For self-similar depth and time dependence of eddy viscosity and arbitrary time dependence of wind we find an exact general solution to the Navier-Stokes equations which describes the dynamics of the Ekman boundary layer in terms of the Green's function. Two basic scenarios (a periodic wind and an increase of wind ending up with a plateau) are examined in detail. We show that accounting for the time dependence of eddy viscosity is straightforward and that it substantially changes the ocean response, compared to the predictions of the models with constant-in-time viscosity. We also examine the Stokes-Ekman equations taking into account the Stokes drift created by surface waves with an arbitrary spectrum and derive the general solution for the case of a linearly varying with depth eddy viscosity. Stability of transient Ekman currents to small-scale perturbations has never been examined. We find that the Ekman currents evolving from rest quickly become unstable, which breaks down the assumed horizontal uniformity. These instabilities proved to be sensitive to the model of eddy viscosity, they have small (similar to 10(2) m) spatial scales and can be very fast compared to the inertial period, which suggests spikes of dramatically enhanced mixing localized in the vicinity of the water surface. This picture is incompatible with the Ekman paradigm and thus prompts radical revision of the Ekman-type models.

From in situ measurements of vertical velocity profiles in the marine upper layer, the eddy viscosity turbulent coefficient is estimated in reference to the unsteady Ekman model solution by using optimal control techniques based on Simultaneous Perturbation Stochastic Approximation Method.