The study of internal gravity waves provides many challenges: they move along interfaces as well as in fully three-dimensional space, at relatively fast temporal and small spatial scales, making them difficult to observe and resolve in weather and climate models. Solving the equations describing their evolution poses various mathematical challenges associated with singular boundary value problems and large amplitude dynamics. This book provides the first comprehensive treatment of the theory for small and large amplitude internal gravity waves. Over 120 schematics, numerical simulations and laboratory images illustrate the theory and mathematical techniques, and 130 exercises enable the reader to apply their understanding of the theory. This is an invaluable single resource for academic researchers and graduate students studying the motion of waves within the atmosphere and ocean, and also mathematicians, physicists and engineers interested in the properties of propagating, growing and breaking waves.

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We present measurements of the density and velocity fields produced when an oscillating circular cylinder excites internal gravity waves in a stratified fluid. These measurements are obtained using a novel, non-intrusive optical technique suitable for determining the density fluctuation field in temporally evolving flows which are nominally two-dimensional. Although using the same basic principles as conventional methods, the technique uses digital image processing in lieu of large and expensive parabolic mirrors, thus allowing more flexibility and providing high sensitivity: perturbations of the order of 1% of the ambient density gradient may be detected. From the density gradient field and its time derivative it is possible to construct the perturbation fields of density and horizontal and vertical velocity. Thus, in principle, momentum and energy fluxes can be determined.In this paper we examine the structure and amplitude of internal gravity waves generated by a cylinder oscillating vertically at different frequencies and amplitudes, paying particular attention to the role of viscosity in determining the evolution of the waves. In qualitative agreement with theory, it is found that wave motions characterized by a bimodal displacement distribution close to the source are attenuated by viscosity and eventually undergo a transition to a unimodal displacement distribution further from the source. Close quantitative agreement is found when comparing our results with the theoretical ones of Hurley & Keady (1997). This demonstrates that the new experimental technique is capable of making accurate measurements and also lends support to analytic theories. However, theory predicts that the wave beams are narrower than observed, and the amplitude is significantly under-predicted for low-frequency waves. The discrepancy occurs in part because the theory neglects the presence of the viscous boundary layers surrounding the cylinder, and because it does not take into account the effects of wave attenuation resulting from nonlinear wave–wave interactions between the upward and downward propagating waves near the source.

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

We investigate the dynamics of a gravity current that propagates along the interface of a two-layer fluid. The results of the well-studied symmetric case are reproduced in which the upper-and lower-layer depth of the ambient are equal and the density of the intrusion is the average density of the ambient. In addition, we present the first detailed examination of asymmetric circumstances in which the density of the intrusion differs from the mean density of the ambient and in which the upper-and lower-layer fluid depths are unequal. The general equations derived by J. Y. Holyer & H. E. Huppert (J. Fluid Mech. vol. 100, 1980, pp. 739-767,), which predict the speed and vertical extent of the gravity current head, are re-expressed in a simpler form that employs the Boussinesq approximation. Approximate analytic solutions are determined using perturbation theory. The predictions are compared with the results of laboratory experiments. We find excellent agreement if the density of the gravity current is the average of the upper-and lower-layer densities weighted by the respective depths of the two layers. However, exact theory significantly underpredicts the gravity current speeds if the current density differs from this weighted-mean average. The discrepancy is attributed to the generation of waves that lead and trail the gravity current head. Empirical support for this assertion is provided through an examination of the observed wave characteristics.

Laboratory experiments are performed to measure the amplitude of internal waves generated by an elliptical cylinder oscillating vertically with different frequencies and amplitudes in a uniformly stratified fluid. The experimental results are compared with the theoretical predictions of Hurley and Keady ͑1997͒. Though in qualitatively good agreement with experiments, the theory underestimates the amplitude of low-frequency waves and overestimates the amplitude of high-frequency waves. The measured beam width is underestimated by theory, which neglects the dynamics of viscous boundary layers surrounding the cylinder. When the cylinder oscillates at a frequency less than half the buoyancy frequency, experiments reveal that two sets of waves beams are generated. The secondary set of waves have double the frequency of the primary waves and are excited due to nonlinear processes.

[1] The evolution and breaking of internal solitary waves in a shallow upper layer as they approach a constant bottom slope is examined through laboratory experiments. The waves are launched in a two-layer fluid through the standard lock-release method. In most experiments, the wave amplitude is significantly larger than the depth of the shallow upper layer so that they are not well described by Korteweg-de Vries theory. The dynamics of the shoaling waves are characterized by the Iribarren number, Ir, which measures the ratio of the topographic slope to the square root of the characteristic wave slope. This is used to classify breaking regimes as collapsing, plunging, surging, and nonbreaking for increasing values of Ir. For breaking waves, the maximum interface descent, H i ? , is predicted to depend upon the topographic slope, s, and the incident wave's amplitude and width, A sw and L sw , respectively, asThis prediction is corroborated by our experiments. Likewise, we apply simple heuristics to estimate the speed of interface descent, and we characterize the speed and range of the consequent upslope flow of the lower layer after breaking has occurred.

Horizontally periodic, vertically localized internal wavepackets evolve nonlinearly due only to interactions between the waves and their wave-induced mean flow. The corresponding weakly nonlinear equation that describes the evolution of the amplitude envelope before the onset of parametric subharmonic instability is examined. The results are compared with fully nonlinear numerical simulations and are shown to lie in excellent agreement for over 15 buoyancy periods. Analysis of the equation shows that the evolution is modulationally unstable if the wave frequency exceeds that of waves with the fastest vertical group speed and if the amplitude is sufficiently large. Waves that move close to the fastest vertical group speed are unstable even if their relative amplitude is a tiny fraction of the inverse relative vertical extent of the wavepacket. At late times in the evolution of an unstable wavepacket third-order dispersion terms become non-negligible and act in conjunction with weakly nonlinear effects to retard the vertical advance of the wavepacket as a whole.