Physics And Astronomy (34)
Journal of Fluid Mechanics (34)
Ryan Test (21)
test society (13)
Effect of background mean flow on PSI of internal wave beams
Boyu Fan, T. R. Akylas
Journal: Journal of Fluid Mechanics / Volume 869 / 25 June 2019
Published online by Cambridge University Press: 23 April 2019, R1
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.
On the interaction of an internal wavepacket with its induced mean flow and the role of streaming
Boyu Fan, T. Kataoka, T. R. Akylas
Journal: Journal of Fluid Mechanics / Volume 838 / 10 March 2018
Published online by Cambridge University Press: 18 January 2018, R1
Print publication: 10 March 2018
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.
Tilting at wave beams: a new perspective on the St. Andrew's Cross
T. Kataoka, S. J. Ghaemsaidi, N. Holzenberger, T. Peacock, T. R. Akylas
Journal: Journal of Fluid Mechanics / Volume 830 / 10 November 2017
Print publication: 10 November 2017
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.
On resonant triad interactions of acoustic–gravity waves
Usama Kadri, T. R. Akylas
Journal: Journal of Fluid Mechanics / Volume 788 / 10 February 2016
Published online by Cambridge University Press: 22 December 2015, R1
Print publication: 10 February 2016
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.
On three-dimensional internal gravity wave beams and induced large-scale mean flows
T. Kataoka, T. R. Akylas
Journal: Journal of Fluid Mechanics / Volume 769 / 25 April 2015
Print publication: 25 April 2015
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).
Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains
Hussain H. Karimi, T. R. Akylas
Journal: Journal of Fluid Mechanics / Volume 757 / 25 October 2014
Published online by Cambridge University Press: 17 September 2014, pp. 381-402
Print publication: 25 October 2014
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.
Stability of internal gravity wave beams to three-dimensional modulations
Journal: Journal of Fluid Mechanics / Volume 736 / 10 December 2013
Published online by Cambridge University Press: 01 November 2013, pp. 67-90
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.
Oblique collisions of internal wave beams and associated resonances
T. R. Akylas, H. H. Karimi
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 , a behaviour akin to that of forced waves at cut-off frequencies.
Resonantly forced gravity–capillary lumps on deep water. Part 2. Theoretical model
YEUNWOO CHO, JAMES D. DIORIO, T. R. AKYLAS, JAMES H. DUNCAN
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.
Resonantly forced gravity–capillary lumps on deep water. Part 1. Experiments
JAMES D. DIORIO, YEUNWOO CHO, JAMES H. DUNCAN, T. R. AKYLAS
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.
Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline
T. R. AKYLAS, R. H. J. GRIMSHAW, S. R. CLARKE, ALI TABAEI
Published online by Cambridge University Press: 23 November 2007, pp. 297-313
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.
On gravity–capillary lumps. Part 2. Two-dimensional Benjamin equation
BOGUK KIM, T. R. AKYLAS
Published online by Cambridge University Press: 12 June 2006, pp. 237-256
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.
On gravity–capillary 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 $(\left. \hbox{d}c/\hbox{d}k \right |_{0}\,{=}\,0)$ 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 $B\,{=}\,T/(\rho gh^2)\,{<}\,1/3$ ($T$ is the coefficient of surface tension, $\rho$ the fluid density, $g$ the gravitational acceleration and $h$ the water depth), $c(k)$ features a minimum at a finite wavenumber from which gravity–capillary solitary waves, in the form of wavepackets governed by the nonlinear Schrödinger (NLS) equation to leading order, bifurcate in water of finite or infinite depth. Here, it is pointed out that an entirely analogous scenario is valid for the bifurcation of three-dimensional solitary waves, commonly referred to as 'lumps', that are locally confined in all directions. Apart from the known lump solutions of the Kadomtsev–Petviashvili I equation for $B\,{>}\,1/3$ 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.
Nonlinear effects in reflecting and colliding internal wave beams
ALI TABAEI, T. R. AKYLAS, KEVIN G. LAMB
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 $n\omega\,{<}\,N_{0}$ 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.
Wave trapping and upstream influence in stratified flow of large depth
DILIP PRASAD, T. R. AKYLAS
Journal: Journal of Fluid Mechanics / Volume 491 / 25 September 2003
Print publication: 25 September 2003
A theoretical study is made of continuously stratified flow of large depth over topography when small periodic vertical fluctuations are present in the Brunt–Väisälä 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.
Nonlinear internal gravity wave beams
ALI TABAEI, T. R. AKYLAS
Journal: Journal of Fluid Mechanics / Volume 482 / 10 May 2003
Print publication: 10 May 2003
Based on linear inviscid theory, a two-dimensional source oscillating with frequency $\omega_{0}$ in a uniformly stratified (constant Brunt–Väisälä frequency $N_{0}$) Boussinesq fluid induces a steady-state wave pattern, also known as St Andrew's Cross, that features four straight wave beams stretching radially outwards from the source at angles $\pm\cos^{-1}(\omega_{0}/N_{0})$ relative to the vertical. Similar wave beams are generated by oscillatory stratified flow over topography and also appear in simulations of thunderstorm-generated gravity waves in the atmosphere. Uniform plane-wave beams of infinite extent are in fact exact solutions of the nonlinear inviscid equations of motion, and this property is used here to study the propagation of finite-amplitude wave beams taking into account weak viscous and refraction effects. Oblique beams ($\omega_{0}\,{<}\,N_{0}$) are considered first and an amplitude-evolution equation is derived assuming slow modulations along the beam direction. Remarkably, the leading-order nonlinear terms cancel out in this evolution equation and, as a result, the steady-state similarity solution of Thomas & Stevenson (1972) for linear viscous beams is also valid in the nonlinear régime. Moreover, for the same reason, nonlinear effects are found to be relatively unimportant for two-dimensional and axisymmetric beams that propagate nearly vertically ($\omega_{0}\,{\approx}\,N_{0}$) in a Boussinesq fluid. Owing to the fact that the group velocity vanishes when $\omega_{0}\,{=}\,N_{0}$, 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–Väisälä frequency.
Stability of steep gravity–capillary solitary waves in deep water
DAVID C. CALVO, T. R. AKYLAS
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.
Three-dimensional aspects of nonlinear stratified flow over topography near the hydrostatic limit
T. R. AKYLAS, KEVIN S. DAVIS
Published online by Cambridge University Press: 13 February 2001, pp. 81-105
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.
The effect of the induced mean flow on solitary waves in deep water
T. R. AKYLAS, F. DIAS, R. H. J. GRIMSHAW
Journal: Journal of Fluid Mechanics / Volume 355 / 25 January 1998
Print publication: 25 January 1998
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.
On the generation of shelves by long nonlinear waves in stratified flows
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.