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A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$ , compares to the basic-state amplitude parameter, $\epsilon \ll 1$ . For $\mu ={O}(1)$ , the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$ , this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

The parametric subharmonic instability (PSI) of finite-width internal gravity wave beams is revisited using a formal linear stability analysis based on Floquet theory. The Floquet stability eigenvalue problem is studied asymptotically in the limit where PSI arises, namely for a small-amplitude beam of frequency no longer dominate. By adopting a frame riding with the wave beam, this advection effect is factored out and it is shown that small-amplitude beams that are not generally susceptible to PSI may develop an essentially inviscid instability with broadband frequency spectrum.

The effect of a small background constant horizontal mean flow on the parametric subharmonic instability (PSI) of locally confined internal wave beams is discussed for the case where the beam frequency is close to twice the inertial frequency due to background rotation. Under this condition, PSI is particularly potent because of the vanishing of the group velocity at the inertial frequency, which prolongs contact of near-inertial subharmonic perturbations with the primary wave. The mean flow generally stabilizes the very short-scale limit of such perturbations. By contrast, the stability of longer-scale perturbations hinges on the strength and the direction of the mean flow; particularly, a negative mean flow (antiparallel to the horizontal projection of the beam group velocity) can extend the sub-inertial range of PSI. However, a large enough mean flow of either sign ultimately weakens PSI.

The mean flow induced by a three-dimensional propagating internal gravity wave beam in a uniformly stratified fluid is studied experimentally and theoretically. Previous related work concentrated on the early stage of mean-flow generation, dominated by the phenomenon of streaming – a horizontal mean flow that grows linearly in time – due to resonant production of mean potential vorticity in the vicinity of the beam. The focus here, by contrast, is on the long-time mean-flow evolution. Experimental observations in a stratified fluid tank for times up to is the beam period, reveal that the induced mean flow undergoes three distinct stages: (i) resonant growth of streaming in the beam vicinity; (ii) saturation of streaming and onset of horizontal advection; and (iii) establishment of a quasi-steady state where the mean flow is highly elongated and stretches in the along-tank horizontal direction. To capture (i)–(iii), the theoretical model of Fan et al. (J. Fluid Mech., vol. 838, 2018, R1) is extended by accounting for the effects of horizontal advection and viscous diffusion of mean potential vorticity. The predictions of the proposed model, over the entire mean-flow evolution, are in excellent agreement with the experimental observations as well as numerical simulations based on the full Navier–Stokes equations.

A joint experimental and theoretical study is made of instability mechanisms of locally confined internal gravity wave beams in a stratified fluid. Using as forcing a horizontal cylinder that is oscillated harmonically in the direction of beam propagation makes it possible to generate coherent finite-amplitude internal wave beams whose spatial profile comprises no more than a single wavelength. For forcing amplitude above a certain threshold depending on the driving frequency, such thin wave beams are observed to undergo an instability that involves two subharmonic perturbations with wavepacket-like spatial structure. Although it bears resemblance to the triadic resonant instability (TRI) of small-amplitude sinusoidal waves, the present instability cannot be predicted by TRI theory as the primary wave is not nearly monochromatic, but instead contains broadband wavenumber spectrum. In contrast, the experimental observations are in good agreement with the predictions of a formal linear stability analysis based on Floquet theory. Finally, experimental evidence is presented that transverse beam variations induce a horizontal mean flow of the streaming type and greatly subdue the instability.

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

An asymptotic model is developed for the parametric subharmonic instability (PSI) of finite-width nearly monochromatic internal gravity wave beams in the presence of a background constant horizontal mean flow. The subharmonic perturbations are taken to be short-scale wavepackets that may extract energy via resonant triad interactions while in contact with the underlying beam, and the mean flow is assumed to be small so that its advection effect on the perturbations is as important as dispersion, triad nonlinearity and viscous dissipation. In this ‘distinguished limit’, the perturbation dynamics are governed by the same evolution equations as those derived in Karimi & Akylas (J. Fluid Mech., vol. 757, 2014, pp. 381–402), except for a mean flow term that affects the group velocity of the perturbations and imposes an additional necessary condition for PSI, which stabilizes very short-scale perturbations. As a result, it is possible for a small amount of mean flow to weaken PSI dramatically.

The coupled nonlinear interaction of three-dimensional gravity–inertia internal wavepackets, in the form of beams with nearly monochromatic profile, with their induced mean flow is discussed. Unlike general three-dimensional wavepackets, such modulated nearly monochromatic beams are not susceptible to modulation instability from their inviscid, purely modulation-induced mean flow. However, streaming – the induced mean flow associated with the production of mean potential vorticity via the combined action of dissipation and nonlinearity – can cause cross-beam bending, transverse broadening and increased along-beam decay of the beam profile, in qualitative agreement with earlier laboratory experiments. For wavepackets with general three-dimensional modulations, by contrast, streaming does arise, but plays a less prominent role in the interaction dynamics.

Interactions between different spatial and temporal scales play a major role in determining the flow structure over the urban canopy in densely built agglomerations. Aiming to address the limitations which arise as a result of the physical disparities between the different modelling scales, a two-way scheme has been introduced for coupling the mesoscale model MEMO and the microscale model MIMO, utilising a collection of interpolating metamodels. The coupled system was used to simulate meteorological fields over the Greater Paris area during a multi-day wintertime case study. Its performance as a meteorological driver model was further evaluated by introducing the calculated meteorological fields as driving input in air quality calculations performed using the Eulerian chemical dispersion model MARS-aero.

The three-dimensional propagation of internal gravity wave beams in a uniformly stratified Boussinesq fluid is discussed, assuming that variations in the along-beam and transverse directions are of long length scale relative to the beam width. This situation applies, for instance, to the far-field behaviour of a wave beam generated by a horizontal line source with weak transverse dependence. In contrast to the two-dimensional case of purely along-beam variations, where nonlinear effects are minor even for beams of finite amplitude, three-dimensional nonlinear interactions trigger the transfer of energy to a circulating horizontal time-mean flow. This resonant beam–mean-flow coupling is analysed, and a system of two evolution equations is derived for the propagation of a small-amplitude beam along with the induced mean flow. This model explains the salient features of the experimental observations of Bordes et al. (Phys. Fluids, vol. 24, 2012, 086602).

The propagation of wave disturbances in water of uniform depth is discussed, accounting for both gravity and compressibility effects. In the linear theory, free-surface (gravity) waves are virtually decoupled from acoustic (compression) waves, because the speed of sound in water far exceeds the maximum phase speed of gravity waves. However, these two types of wave motion could exchange energy via resonant triad nonlinear interactions. This scenario is analysed for triads comprising a long-crested acoustic mode and two oppositely propagating subharmonic gravity waves. Owing to the disparity of the gravity and acoustic length scales, the interaction time scale is longer than that of a standard resonant triad, and the appropriate amplitude evolution equations, apart from the usual quadratic interaction terms, also involve certain cubic terms. Nevertheless, it is still possible for monochromatic wavetrains to form finely tuned triads, such that nearly all the energy initially in the gravity waves is transferred to the acoustic mode. This coupling mechanism, however, is far less effective for locally confined wavepackets., MIT-Technion Fellowship

Isotropic turbulence is typically studied numerically through the direct numerical simulations (DNS). The DNS flows are described by the Navier-Stokes equation in a 'box', defined through periodic boundary conditions. The DNS flows live in a compact space and they are not isotropic in their large scales. The investigation of important phenomena of isotropic turbulence, such as anomalous scaling, through the DNS is affected by large scale effects. In this work we put isotropic turbulence - or better, the associated formal theory - in a 'box', through imposing periodicity at the level of the correlations functions. We offer a framework where one may investigate isotropic theories/models through the data of DNS in a formally consistent manner. We work at the level of the Karman-Howarth equation. Unlike the Navier-Stokes equation, infinitely smooth periodicity is obstructed in this theory, a fact expressed by a sequence of relations obeyed by the normal modes of the Karman-Howarth equation. Similar relations are imparted to the two-point functions under the condition that the energy spectrum and energy transfer function are realizable. Naturally constructed closures scheme for the Karman-Howarth equation do not conform to such relations, thereby destroying realizability. A closure can be made to conform to a finite number of them by adding corrective terms, in a procedure which possesses certain analogies with the renormalization of quantum field theory. The spectrum becomes unphysical (through sign-changing oscillations) for infinitely large wavenumbers, but we can controllably extend the regime where the spectrum remains physical deep enough in the dissipation range so that to be realistically adequate. We show that one or two such 'regularity relations' are needed at most for comparisons of the predictions of the theory with the current resolution level results of the DNS., Comment: 57 pages, 3 figures

Internal gravity wavetrains in continuously stratified fluids are generally unstable as a result of resonant triad interactions which, in the inviscid limit, amplify short-scale perturbations with frequency equal to one half of that of the underlying wave. This so-called parametric subharmonic instability (PSI) has been studied extensively for spatially and temporally monochromatic waves. Here, an asymptotic analysis of PSI for time-harmonic plane waves with locally confined spatial profile is made, in an effort to understand how such wave beams differ, in regard to PSI, from monochromatic plane waves. The discussion centres upon a system of coupled evolution equations that govern the interaction of a small-amplitude wave beam with short-scale subharmonic wavepackets in a nearly inviscid uniformly stratified Boussinesq fluid. For beams with general localized profile, it is found that triad interactions are not strong enough to bring about instability in the limited time that subharmonic perturbations overlap with the beam. On the other hand, for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope, PSI is possible if the beam is wide enough. In this instance, a stability criterion is proposed which, under given flow conditions, provides the minimum number of carrier wavelengths a beam of small amplitude must comprise for instability to arise.

The linear stability of uniform, plane internal wave beams with locally confined spatial profile, in a stratified fluid of constant buoyancy frequency, is discussed. The associated eigenvalue problem is solved asymptotically, assuming perturbations of long wavelength relative to the beam width. In this limit, instability is found only for oblique disturbances which vary in the along-beam and the horizontal transverse directions. The mechanism of instability is a first-harmonic–mean resonant interaction between the underlying wave beam and three-dimensional perturbations that comprise a time-harmonic component, with the beam frequency, and a mean flow. Progressive beams which transport energy in one direction, in particular, are unstable if the beam steepness exceeds a certain threshold value, whereas purely standing beams are unstable even at infinitesimal steepness. A distinguishing feature of this three-dimensional modulational instability is the generation of circulating horizontal mean flows at large distances from the vicinity of the beam.

Quadratic nonlinear interactions between two colliding internal gravity wave beams in a uniformly stratified fluid, and the resulting radiation of secondary beams with frequencies equal to the sum and difference of those of the primary beams, are discussed. The analysis centres on oblique collisions, involving beams that propagate in different vertical planes. The propagation directions of generated secondary beams are deduced from kinematic considerations and the use of radiation conditions, thus extending to oblique collisions previously derived selection rules for plane collisions. Using small-amplitude expansions, radiated-beam profiles at steady state are also computed in terms of the characteristics of the colliding beams. It is pointed out that, for certain oblique collision configurations, radiated beams with frequency equal to the difference of the primary frequencies have unbounded steady-state amplitude. This resonance, which has no counterpart for plane collisions, is further analysed via the solution of an initial-value problem; ignoring dissipation, the transient resonant response grows in time like ${t}^{1/ 2} $, a behaviour akin to that of forced waves at cut-off frequencies.

The wave pattern generated by a pressure source moving over the free surface of deep water at speeds, U, below the minimum phase speed for linear gravity–capillary waves, cmin, was investigated experimentally using a combination of photographic measurement techniques. In similar experiments, using a single pressure amplitude, Diorio et al. (Phys. Rev. Lett., vol. 103, 2009, 214502) pointed out that the resulting surface response pattern exhibits remarkable nonlinear features as U approaches cmin, and three distinct response states were identified. Here, we present a set of measurements for four surface-pressure amplitudes and provide a detailed quantitative examination of the various behaviours. At low speeds, the pattern resembles the stationary state (U = 0), essentially a circular dimple located directly under the pressure source (called a state I response). At a critical speed, but still below cmin, there is an abrupt transition to a wave-like state (state II) that features a marked increase in the response amplitude and the formation of a localized solitary depression downstream of the pressure source. This solitary depression is steady, elongated in the cross-stream relative to the streamwise direction, and resembles freely propagating gravity–capillary ‘lump’ solutions of potential flow theory on deep water. Detailed measurements of the shape of this depression are presented and compared with computed lump profiles from the literature. The amplitude of the solitary depression decreases with increasing U (another known feature of lumps) and is independent of the surface pressure magnitude. The speed at which the transition from states I to II occurs decreases with increasing surface pressure. For speeds very close to the transition point, time-dependent oscillations are observed and their dependence on speed and pressure magnitude are reported. As the speed approaches cmin, a second transition is observed. Here, the steady solitary depression gives way to an unsteady state (state III), characterized by periodic shedding of lump-like disturbances from the tails of a V-shaped pattern.

A theoretical model is presented for the generation of waves by a localized pressure distribution moving on the surface of deep water with speed near the minimum gravity–capillary phase speed, cmin. The model employs a simple forced–damped nonlinear dispersive equation. Even though it is not formally derived from the full governing equations, the proposed model equation combines the main effects controlling the response and captures the salient features of the experimental results reported in Diorio et al. (J. Fluid Mech., vol. 672, 2011, pp. 268–287 – Part 1 of this work). Specifically, as the speed of the pressure disturbance is increased towards cmin, three distinct responses arise: state I is confined beneath the applied pressure and corresponds to the linear subcritical steady solution; state II is steady, too, but features a steep gravity–capillary lump downstream of the pressure source; and state III is time-periodic, involving continuous shedding of lumps downstream. The transitions from states I to II and from states II to III, observed experimentally, are associated with certain limit points in the steady-state response diagram computed via numerical continuation. Moreover, within the speed range that state II is reached, the maximum response amplitude turns out to be virtually independent of the strength of the pressure disturbance, in agreement with the experiment. The proposed model equation, while ad hoc, brings out the delicate interplay between dispersive, nonlinear and viscous effects that takes place near cmin, and may also prove useful in other physical settings where a phase-speed minimum at non-zero wavenumber occurs.

It is generally accepted that ocean internal solitary waves can arise from the interaction of the barotropic tide with the continental shelf, which generates an internal tide that in turn steepens and forms solitary waves as it propagates shorewards. Some field observations, however, reveal large-amplitude internal solitary waves in deep water, hundreds of kilometres away from the continental shelf, suggesting an alternative generation mechanism: tidal flow over steep topography forces a propagating beam of internal tidal wave energy which impacts the thermocline at a considerable distance from the forcing site and gives rise to internal solitary waves there. Motivated by this possibility, a simple nonlinear long-wave model is proposed for the interaction of a tidal wave beam with the thermocline and the ensuing local generation of solitary waves. The thermocline is modelled as a density jump across the interface of a shallow homogeneous fluid layer on top of a deep uniformly stratified fluid, and a finite-amplitude propagating internal wave beam of tidal frequency in the lower fluid is assumed to be incident and reflected at the interface. The induced weakly nonlinear long-wave disturbance on the interface is governed in the far field by an integral-differential equation which accounts for nonlinear and dispersive effects as well as energy loss owing to radiation into the lower fluid. Depending on the strength of the thermocline and the intensity of the incident beam, nonlinear wave steepening can overcome radiation damping so a series of solitary waves may arise in the thermocline. Sample numerical solutions of the governing evolution equation suggest that this mechanism is quite robust for typical oceanic conditions.

We generalize Batchelor's parameterization of the autocorrelation functions of isotropic turbulence in a form involving a product expansion with multiple small scales. The richer small scale structure acquired this way, compared to the usual Batchelor function, is necessary so that the associated energy spectrum approximate well actual spectra in the universal equilibrium range. We propose that the generalized function provides an approximation of arbitrary accuracy for actual spectra of isotropic turbulence over the universal equilibrium range. The degree of accuracy depends on the number of higher moments which are determinable and it is reflected in the number of small scales involved. The energy spectrum of the generalized function is derived, and for the case of two small scales is compared with data from high-resolution direct numerical simulations. We show that the compensated spectra (which illustrate the bottleneck effect) and dissipation spectra are encapsulated excellently, in accordance with our proposal.

Using small-amplitude expansions, we discuss nonlinear effects in the reflection from a sloping wall of a time-harmonic (frequency $\omega$ ) plane-wave beam of finite cross-section in a uniformly stratified Boussinesq fluid with constant buoyancy frequency $N_{0}$ . The linear solution features the incident and a reflected beam, also of frequency $\omega$ , that is found on the same (opposite) side to the vertical as the incident beam if the angle of incidence relative to the horizontal is less (greater) than the wall inclination. As each of these beams is an exact nonlinear solution, nonlinear interactions are confined solely in the vicinity of the wall where the two beams meet. At higher orders, this interaction region acts as a source of a mean and higher-harmonic disturbances with frequencies $n\omega$ ( $n\,{=}\,2,3,\ldots$ ); for $n\omega\,{ the latter radiate in the far field, forming additional reflected beams along $\sin^{-1}(n\omega/N_{0})$ to the horizontal. Depending on the flow geometry, higher-harmonic beams can be found on the opposite side of the vertical from the primary reflected beam. Using the same approach, we also discuss collisions of two beams propagating in different directions. Nonlinear interactions in the vicinity of the collision region induce secondary beams with frequencies equal to the sum and difference of those of the colliding beams. The predictions of the steady-state theory are illustrated by specific examples and compared against unsteady numerical simulations.

A theoretical study is made of continuously stratified flow of large depth over topography when small periodic vertical fluctuations are present in the Brunt-Vaisala frequency, the background flow conditions being otherwise uniform. It is known from Phillips (1968) that, owing to nonlinear interactions with such fluctuations, internal gravity waves with vertical wavelength twice that of the background variations become trapped along the vertical, suggesting a waveguide-like behaviour. Using the asymptotic theory of Kantzios & Akylas (1993), we explore the role that this interaction-trapping mechanism plays in the generation of finite-amplitude long-wave disturbances near the hydrostatic limit. As a result of vertical trapping, a resonance phenomenon occurs and the linear hydrostatic response grows unbounded when the flow speed coincides with the long-wave speed of a free propagation mode that is trapped close to the ground. Near this critical flow speed, according to weakly nonlinear analysis, the wave evolution along the streamwise direction is governed by a forced extended Korteweg-de Vries equation, which predicts upstream-propagating solitary waves and bores similar to those obtained in resonant stratified flow of finite depth. The finite-amplitude response is then studied numerically and in some cases features strong upstream influence in the form of vertically trapped solitary waves and bores. On the other hand, incipient wave breaking is often encountered during the evolution of the nonlinear resonant response, and this flow feature, which is beyond the reach of weakly nonlinear theory, arises at topography amplitudes significantly below the critical value for overturning predicted by the classical model of Long (1953) for uniformly stratified steady flow.

Based on linear inviscid theory, a two-dimensional source oscillating with frequency , however, the transient evolution of nearly vertical beams takes place on a slower time scale than that of oblique beams; this is shown to account for the discrepancies between the steady-state similarity solution of Gordon & Stevenson (1972) and their experimental observations. Finally, the present asymptotic theory is used to study the refraction of nearly vertical nonlinear beams in the presence of background shear and variations in the Brunt–Vaisala frequency.

Symmetric solitary waves on the interface between two fluid layers are discussed for the case when the upper fluid is bounded above and the lower fluid is infinitely deep. In this flow geometry, under the conditions that the fluid densities are nearly equal and interfacial tension is relatively large, Benjamin [J. Fluid Mech. 245, 401 (1992)] derived a weakly nonlinear long-wave evolution equation which admits a new kind of elevation solitary wave that features decaying oscillatory tails. We present computations of solitary waves of this type and examine their stability to small perturbations on the basis of the full Euler equations using numerical methods. Computed solitary-wave profiles are found to be in good agreement with solutions of the Benjamin equation even for Weber and Froude numbers well outside the formal range of validity of weakly nonlinear long-wave analysis. Furthermore, our computations reveal that, unless the upper fluid is very light relative to the lower fluid, moderately steep interfacial solitary waves behave qualitatively as predicted by the Benjamin equation: elevation waves with a tall center crest, akin to those found by Benjamin, are stable and coexist with depression solitary waves (not reported by Benjamin) which are unstable. As expected, for low enough density ratio, these two solution branches exchange stabilities and computed profiles resemble those of free-surface gravity-capillary solitary waves on deep water.

The stability of steep gravity–capillary solitary waves in deep water is numerically investigated using the full nonlinear water-wave equations with surface tension. Out of the two solution branches that bifurcate at the minimum gravity–capillary phase speed, solitary waves of depression are found to be stable both in the small-amplitude limit when they are in the form of wavepackets and at finite steepness when they consist of a single trough, consistent with observations. The elevation-wave solution branch, on the other hand, is unstable close to the bifurcation point but becomes stable at finite steepness as a limit point is passed and the wave profile features two well-separated troughs. Motivated by the experiments of Longuet-Higgins & Zhang (1997), we also consider the forced problem of a localized pressure distribution applied to the free surface of a stream with speed below the minimum gravity–capillary phase speed. We find that the finite-amplitude forced solitary-wave solution branch computed by Vanden-Broeck & Dias (1992) is unstable but the branch corresponding to Rayleigh’s linearized solution is stable, in agreement also with a weakly nonlinear analysis based on a forced nonlinear Schrödinger equation. The significance of viscous effects is assessed using the approach proposed by Longuet-Higgins (1997): while for free elevation waves the instability predicted on the basis of potential-flow theory is relatively weak compared with viscous damping, the opposite turns out to be the case in the forced problem when the forcing is strong. In this régime, which is relevant to the experiments of Longuet-Higgins & Zhang (1997), the effects of instability can easily dominate viscous effects, and the results of the stability analysis are used to propose a theoretical explanation for the persistent unsteadiness of the forced wave profiles observed in the experiments.

Steady, finite-amplitude internal-wave disturbances, induced by nearly hydrostatic stratified flow over locally confined topography that is more elongated in the spanwise than the streamwise direction, are discussed. The nonlinear three-dimensional equations of motion are handled via a matched-asymptotics procedure: in an ‘inner’ region close to the topography, the flow is nonlinear but weakly three-dimensional, while far upstream and downstream the ‘outer’ flow is governed, to leading order, by the fully three-dimensional linear hydrostatic equations, subject to matching conditions from the inner flow. Based on this approach, non-resonant flow of general (stable) stratification over finite-amplitude topography in a channel of finite depth is analysed first. Three-dimensional effects are found to inhibit wave breaking in the nonlinear flow over the topography, and the downstream disturbance comprises multiple small-amplitude oblique wavetrains, forming supercritical wakes, akin to the supercritical free-surface wake induced by linear hydrostatic flow of a homogeneous fluid. Downstream wakes of a similar nature are also present when the flow is uniformly stratified and resonant (i.e. the flow speed is close to the long-wave speed of one of the modes in the channel), but, in this instance, they are induced by nonlinear interactions precipitated by three-dimensional effects in the inner flow and are significantly stronger than their linear counterparts. Finally, owing to this nonlinear-interaction mechanism, vertically unbounded uniformly stratified hydrostatic flow over finite-amplitude topography also features downstream wakes, in contrast to the corresponding linear disturbance that is entirely locally confined.

Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.

The phenomenon of shelf generation by long nonlinear internal waves in stratified flows is investigated. The problem of primary interest is the case of a uniformly stratified Boussinesq fluid of finite depth. In analysing the transient evolution of a finite-amplitude long-wave disturbance, the expansion procedure of Grimshaw & Yi (1991) breaks down far downstream, and it proves expedient to follow a matched-asymptotics procedure: the main disturbance is governed by the nonlinear theory of Grimshaw & Yi (1991) in the ‘inner’ region, while the ‘outer’ region comprises multiple small-amplitude fronts, or shelves, that propagate downstream and carry O(1) mass. This picture is consistent with numerical simulations of uniformly stratified flow past an obstacle (Lamb 1994). The case of weakly nonlinear long waves in a fluid layer with general stratification is also examined, where it is found that shelves of fourth order in wave amplitude are generated. Moreover, these shelves may extend both upstream and downstream in general, and could thus lead to an upstream influence of a type that has not been previously considered. In all cases, transience of the main nonlinear wave disturbance is a necessary condition for the formation of shelves.

Symme tric gravity–capillary solitary waves with decaying oscillatory tails are known to bifurcate from infinitesimal periodic waves at the minimum value of the phase speed where the group velocity is equal to the phase speed. In the small-amplitude limit, these solitary waves may be interpreted as envelope solitons with stationary crests and are described by the nonlinear Schrödinger (NLS) equation to leading order. In line with this interpretation, it would appear that one may also co nstruct asymmetric solitary waves by shifting the carrier oscillations relative to the envelope of a symmetric solitary wave. This possibility is examined here on the basis of the fifth-order Korteweg–de Vries (KdV) equation, a model for g ravity–capillary waves on water of finite depth when the Bond number is close to 1/3. Using techniques of exponential asymptotics beyond all orders of the NLS theory, it is shown that asymmetric solitary waves of the form suggested by the NLS theory in fact are not possible. On the other hand, an infinity of symmetric and asymmetric solitary-wave solution families comprising two or more NLS solitary wavepackets bifurcate at finite values of the amplitude parameter. The asymptotic results are consistent with numerical solutions of the fifth-order KdV equation. Moreover, the asymptotic theory suggests that such multi-packet gravity–capillary solitary waves also exist in the full water-wave problem near the minimum of t he phase speed.

There is evidence from previous analytical work based on a model equation and from numerical computations that gravity–capillary solitary waves in a liquid layer are nonlocal—they feature oscillatory tails of constant amplitude—when the Bond number τ is less than 1/3. Here, the full gravity–capillary wave problem is examined and these tails are calculated asymptotically in the weakly nonlinear regime. For given values of τ (0

The flow of a continuously stratified fluid over a smooth bottom bump in a channel of finite depth is considered. In the weakly nonlinear-weakly dispersive régime ε = a/h [Lt ] 1, μ = h/l [Lt ] 1 (where h is the channel depth and a, l are the peak amplitude and the width of the obstacle respectively), the parameter A = ε/μp (where p < 0 depends on the obstacle shape) controls the effect of nonlinearity on the steady lee wavetrain that forms downstream of the obstacle for subcritical flow speeds. For A = O(1), when nonlinear and dispersive effects are equally important, the interaction of the long-wave disturbance over the obstacle with the lee wave is fully nonlinear, and techniques of asymptotics ‘beyond all orders’ are used to determine the (exponentially small as μ → 0) lee-wave amplitude. Comparison with numerical results indicates that the asymptotic theory often remains reasonably accurate even for moderately small values of μ and ε, in which case the (formally exponentially small) lee-wave amplitude is greatly enhanced by nonlinearity and can be quite substantial. Moreover, these findings reveal that the range of validity of the classical linear lee-wave theory (A [Lt ] 1) is rather limited.

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg–de Vries equation to leading order, and locally confined solitary waves with a ‘sech’ profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n > 1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.

Previous experimental and numerical work indicates that an initially symmetric deep-water wave pulse of uniform frequency and moderately small steepness evolves in an asymmetric manner and eventually separates into distinct wave groups, owing to higher-order modulation effects, not accounted for by the nonlinear Schrödinger equation (NLS). Here perturbation methods are used to provide analytical confirmation of this group splitting on the basis of the more accurate envelope equation of Dysthe (1979). It is demonstrated that an initially symmetric multisoliton wave envelope, consisting of N bound NLS solitons, ultimately breaks up into N separate groups; a procedure is devised for determining the relative speed changes of the individual groups. The case of a bi-soliton (N = 2) is discussed in detail, and the analytical predictions are compared to numerical results.

Rapid distortion theory is applied to stratified homogeneous turbulence that is sheared in a rotating frame. Insight into the stabilizing and destabilizing effects of the combined stratification and frame rotation is gained by considering initial fields that are two-dimensional, with the axis of independence aligned with the flow direction. For these conditions, we derive solutions for the Fourier components of the flow variables, and for one-point statistics in physical space. The analytical results are in qualitative agreement with the exact numerical solution for initially isotropic homogeneous turbulence, and they could be a reference point for the development of turbulence models. © 2007 American Institute of Physics. 19

In the present study, we investigate, using inviscid rapid distortion theory, the evolution of sheared turbulence in a rotating frame as a function of the rotation rate (including stable, transitional, and unstable regimes), and examine the sensitivity of the results for various nonisotropic initial conditions. Analytical solutions are derived for the evolution of the stresses and the structure dimensionality tensor components for three one-dimensional and three two-dimensional initializations. From these solutions, we calculate the asymptotic states of the turbulence, which are compared to the exact numerical solution of the three-dimensional initially isotropic case. From the investigation it is shown that the qualitative characteristics of the isotropic solution in the unstable regime are represented quite accurately when the initial turbulence is dependent at least on the axis of the rotation of the frame. For the transitional and the stable regimes, though, the initial dependence of the turbulence on the axis of the mean flow is also crucial. © 2007 American Institute of Physics. 19

We apply inviscid rapid distortion theory to the limiting hyperbolic case of turbulent plain strain flow in a rotating frame and investigate the dependence of the evolution of the turbulent kinetic energy on the frame rotation rate. We derive an analytical two-dimensional solution that, unlike previous oversimplified pressureless analyses, allows for an accurate approximation of the three-dimensional initially isotropic problem. From the analytical solutions, we determine the correct stability criterion for the evolution of the turbulent kinetic energy in this flow. Also, we calculate the asymptotic states of the turbulence, in terms of the normalized Reynolds stresses and structure dimensionality tensor components, which coincide with the exact three-dimensional numerical results. © 2007 American Institute of Physics. 19

We apply inviscid rapid distortion theory to the generalized case of turbulent hyperbolic flow in a rotating frame and investigate the dependence of the evolution of the turbulent kinetic energy on the frame rotation rate. We derive an analytical two-dimensional solution which allows for an accurate approximation of the three-dimensional initially isotropic problem. From the analytical solutions we determine a new generalized stability criterion for the evolution of the turbulent kinetic energy in this class of flows. © 2007 American Institute of Physics. 19

In this study we apply rapid distortion theory to the case of nonstratified homogeneous turbulence that is sheared in a frame that counter-rotates at a rate that matches in magnitude the rotation associated with the mean shear. In the inviscid case, analytical solutions are worked out for the evolution of the components of the Reynolds stresses and the structure dimensionality tensor, and these are shown to equal each other. The results are compared to direct numerical simulations data with which they proved to be in good agreement, especially in terms of the Reynolds shear stress and of the dimensionless tensor components. Finally, the development of the structure of a passive scalar field with a constant mean gradient is investigated, and remarkable analogies are shown to exist between this case and the case of shear in a fixed frame. © 2006 American Institute of Physics. 18

A theoretical study is made of fully localized solitary waves, commonly referred to as ‘lumps’, on the interface of a two-layer fluid system in the case that the upper layer is bounded by a rigid lid and lies on top of an infinitely deep fluid. The analysis is based on an extension, that allows for weak transverse variations, of the equation derived by Benjamin (J. Fluid Mech. vol. 245, 1992, p. 401) for the evolution in one spatial dimension of weakly nonlinear long waves in this flow configuration, assuming that interfacial tension is large and the two fluid densities are nearly equal. The phase speed of the Benjamin equation features a minimum at a finite wavenumber where plane solitary waves are known to bifurcate from infinitesimal sinusoidal wavetrains. Using small-amplitude expansions, it is shown that this minimum is also the bifurcation point of lumps akin to the free-surface gravity–capillary lumps recently found on water of finite depth. Numerical continuation of the two symmetric lump-solution branches that bifurcate there reveals that the elevation-wave branch is directly connected to the familiar lump solutions of the Kadomtsev–Petviashvili equation, while the depression-wave branch apparently features a sequence of limit points associated with multi-modal lumps. Plane solitary waves of elevation, although stable in one dimension, are unstable to transverse perturbations, and there is evidence from unsteady numerical simulations that this instability results in the formation of elevation lumps.

Two-dimensional (plane) solitary waves on the surface of water are known to bifurcate from linear sinusoidal wavetrains at specific wavenumbers k = k 0 where the phase speed c(k) attains an extremum (dc/dk| 0 =0) and equals the group speed. In particular, such an extremum occurs in the long-wave limit k 0 =0, furnishing the familiar solitary waves of the Korteweg-de Vries (KdV) type in shallow water. In addition, when surface tension is included and the Bond number B = T/(ρgh 2 ) 1/3 in shallow water, gravity-capillary lumps, in the form of locally confined wavepackets, are found for B < 1/3 in water of finite or infinite depth; like their two-dimensional counterparts, they bifurcate at the minimum phase speed and are governed, to leading order, by an elliptic-elliptic Davey-Stewartson equation system in finite depth and an elliptic two-dimensional NLS equation in deep water. In either case, these lumps feature algebraically decaying tails owing to the induced mean flow.

The flow of a Boussinesq density-stratified fluid of large depth past the algebraic mountain (Witch of Agnesi) is studied in the hydrostatic limit using the asymptotic theory of Kantzios & Akylas (1993). The upstream conditions are those of constant velocity and Brunt-Vaisala frequency. On the further assumptions that the flow is steady and there is no permanent alteration of the upstream flow conditions (no upstream influence), Long's model (Long 1953) predicts a critical amplitude of the mountain (e = 0.85) above which local density inversions occur, leading to convective overturning. Linear stability analysis demonstrates that Long's steady flow is in fact unstable to infinitesimal modulations at topography amplitudes below this critical value, 0.65 ≤ e < 0.85. This instability grows at the expense of the mean flow and may be attributed to a discrete spectrum of modes that become trapped over the mountain in the streamwise direction. The transient problem is also solved numerically, mimicking impulsive startup conditions. In the absence of instability, Long's steady flow is reached. For topography amplitudes in the unstable range 0.65 ≤ e < 0.85, however, the flow fluctuates about Long's steady state over a long timescale ; there is no significant upstream influence and no evidence of transient wave breaking is found for e ≤ 0.75.

A theoretical model is presented for the propagation of long, weakly nonlinear water waves along a channel bounded by sloping sidewalls. The theoretical predictions are in quantitative agreement with experiments and confirm the presence of significant three-dimensional effects, not accounted for by previous theories.

In the vicinity of a caustic of a dispersive wave system, where the group velocity is stationary and hence dispersive effects are relatively weak, the nonlinear Schrodinger equation (NLS) breaks down, and the propagation of the envelope of a finiteamplitude wavepacket is governed by a modified nonlinear Schrodinger equation (MNLS). On the basis of the MNLS, a search for wave envelopes of permanent form is made near a caustic. It is shown that possible solitary wave envelopes satisfy a nonlinear eigenvalue problem. Numerical evidence is presented of symmetric, double-hump solitary-wave solutions. Also, a variety of periodic envelopes are computed. These findings are discussed in connection with previous analytical and numerical work.

The temporal and spatial evolution of large-scale modulations of weakly nonlinear edge waves on a uniformly sloping beach is studied using the full water-wave formulation for beach angles α = π/2N. Equations governing the evolution of envelopes of edge waves, excited by resonant interactions with incident wavetrains, are derived. It is deduced that a uniform train of free periodic edge waves is always unstable to large-scale variations, so that envelope solitons will develop; the resulting three-dimensional solitons are described in detail. In addition, it is shown that steady-state standing subharmonic edge waves, excited by incident wavetrains on a long, mildly sloping beach, can be unstable to large-scale modulations. The possible physical significance of these findings is discussed.

A study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth. The linearized equations of motion are first used to obtain the large-time asymptotic behaviour of the disturbance in the far field; the linear response consists of long waves with temporally growing amplitude, so that the linear approximation eventually breaks down owing to finite-amplitude effects. A nonlinear theory is developed which shows that the generated waves are actually of bounded amplitude, and are governed by a forced Korteweg-de Vries equation subject to appropriate asymptotic initial conditions. A numerical study of the forced Korteweg-de Vries equation reveals that a series of solitons are generated in front of the pressure distribution.

The three-dimensional wave pattern generated by a moving pressure distribution of finite extent acting on the surface of water of depth h is studied. It is shown that, when the pressure distribution travels at a speed near the linear-long-wave speed, the response is governed by a forced nonlinear Kadomtsev-Petviashvili (KP) equation, which describes a balance between linear dispersive, nonlinear and three-dimensional effects. It is deduced that, in a channel of finite width 2w, three-dimensional effects are negligible if w [Lt ] h2/a, a being a typical wave amplitude; in such a case the governing equation reduces to the forced Korteweg-de Vries equation derived in previous studies. For aw/h2 = O(1), however, three-dimensional effects are important; numerical calculations based on the KP equation indicate that a series of straight-crested solitons are radiated periodically ahead of the source and a three-dimensional wave pattern forms behind. The predicted dependencies on channel width of soliton amplitude and period of soliton formation compare favourably with the experimental results of Ertekin, Webster & Wehausen (1984). In a channel for which aw/h2 [Gt ] 1, three-dimensional, unsteady disturbances appear-ahead of the pressure distribution.

The excitation of surface waves on a viscous fluid by shear flows is studied. Turbulent and laminar air flows over oil of low and high viscosity are considered. It is found that the dominant wave-generation mechanism depends crucially on the shear-flow profile: for a turbulent flow, long surface waves are generated at low wind speeds due to the work done by the stress components in phase with the surface slope, while Kelvin-Helmholtz instability is responsible for the excitation of short waves at higher wind speeds. On the other hand, for a laminar shear flow, direct resonance between surface waves and Tollmien-Schlichting waves in the shear flow is the dominant wave-generation mechanism.

A numerical investigation of finite-amplitude, non-axisymmetric disturbances, in the form of travelling spiral waves, is made in pipe flow with superimposed solid-body rotation. Rotating pipe flow is found to be supercritically unstable both in the rapid and slow-rotation regimes. Earlier weakly nonlinear calculations, suggesting subcritical instability in the slow-rotation limit, are shown to be in error. Bifurcating neutral waves are calculated for various axial and azimuthal Reynolds numbers and wavenumbers. For fixed axial mean pressure gradient, the axial mean flow induced by these waves gives rise to a significant flux defect, in certain cases as large as 40-50% of the undisturbed mass flux; the possible relevance of this finding to the phenomenon of vortex breakdown is pointed out. In non-rotating pipe flow, no neutral disturbances in the assumed form of spiral waves are found for moderate Reynolds numbers; this indicates that previous conjectures, regarding a possible connection between nonlinear spiral waves in slowly rotating pipe flow and the asymptotic neutral states of Smith & Bodonyi (1982) in non-rotating pipe flow, are not valid.

A study is made of time-harmonic wave modes in an elastic layer in plane strain with traction-free surfaces. It is demonstrated that, at points of the Rayleigh-Lamb frequency spectrum where the group velocity vanishes, there exist nonseparable time-harmonic modes in which the amplitudes of the displacements vary linearly in the direction along the layer. These modes are used to explain the terrace-like structure of the free-vibration frequency spectrum of a circular disk.

A theoretical study is made of the stability of pipe flow with superimposed rigid rotation to finite-amplitude disturbances at high Reynolds number. The non-axisymmetric mode that requires the least amount of rotation for linear instability is considered. An amplitude expansion is developed close to the corresponding neutral stability curve; the appropriate Landau constant is calculated. It is demonstrated that the flow exhibits nonlinear subcritical instability, the nonlinear effects being particularly strong owing to the large magnitude of the Landau constant. These findings support the view that a small amount of extraneous rotation could play a significant role in the transition to turbulence of pipe flow.

The envelope equation of Dysthe (1979), which provides an extension of the nonlinear Schrodinger equation (NLS) to fourth order in wave steepness, is used to discuss higher-order modulation effects on the long-time evolution of solitary wave envelopes in deep water. The Dysthe equation admits solitary-wave solutions, similar to those of the NLS. Using perturbation methods, it is shown that an initial disturbance in the form of a solitary wave group of the NLS evolves to a solitary wave of the Dysthe equation having lower peak amplitude and moving with higher speed than the original wave; the increase in wave speed is caused by a downshift in wavefrequency. Asymptotic expressions are derived for this amplitude decrease and frequency downshift, which are consistent with numerical and experimental results.