Bruce R. Sutherland

Visualization and measurement of internal waves by 'synthetic schlieren'. Part 1. Vertically oscillating cylinder

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.

Finite-amplitude internal wavepacket dispersion and breaking

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 ] c , where Θ c = 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 ] > Θ c decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] c 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 A CV = cot Θ (1 + cos 2 Θ)/2π, where A CV 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 A SA = sin 2Θ/(8π 2 ) 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°.

Internal wave excitation by a vertically oscillating elliptical cylinder

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 (1997). 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.

Internal wave excitation by a vertically oscillating sphere

The properties of waves generated by a vertically oscillating sphere in a uniformly stratified fluid are examined both theoretically and experimentally. Existing predictions for the wave amplitude and phase structure are modified to account fo rt he effects of viscous attenuation. As with waves generated by an oscillating cylinder, the main effect of attenuation is to broaden the two peaks of the amplitude envelope on either flank of the wave beam so that far from the sphere the wave beam exhibits a single peak with a maximum along the centreline. The transition distance from bimodal to unimodal wave beam structure is shown to occur closer to the source than the corresponding distance calculated for the oscillating circular cylinder. For laboratory experiments, a recently developed ‘synthetic schlieren’ method is adapted so that quantitative measurements may be made of an axisymmetric wave field. This non-intrusive technique allows us to evaluate the amplitude of the waves everywhere in space and time. Experiments are performed to examine the amplitude of waves generated by small and large spheres oscillating with a range of amplitudes and frequencies. The wave amplitude is found to scale linearly with the oscillation amplitude A for A/a as large as 0.27, where a is the radius of the sphere. Generally good agreement between theory and experiment is found for the small sphere experiments. However, the theory overpredicts both the amplitude and the bimodal-to-unimodal transition distance for waves generated by the large sphere.

Internal wave excitation from stratified flow over a thin barrier

We perform laboratory experiments in a recirculating shear flow tank of non-uniform salt-stratified water to examine the excitation of internal gravity waves (IGW) in the wake of a tall, thin vertical barrier. The purpose of this study is to characterize and quantify the coupling between coherent structures shed in the wake and internal waves that radiate from the mixing region into the deep, stationary fluid. In agreement with numerical simulations, large-amplitude internal waves are generated when the mixing region is weakly stratified and the deep fluid is sufficiently strongly stratified. If the mixing region is unstratified, weak but continuous internal wave excitation occurs. In all cases, the tilt of the phase lines of propagating waves lies within a narrow range. Assuming the waves are spanwise uniform, their amplitude in space and time is measured non-intrusively using a recently developed ‘synthetic schlieren’ technique. Using wavelet transforms to measure consistently the width and duration of the observed wavepackets, the Reynolds stress is measured and, in particular, we estimate that when large-amplitude internal wave excitation occurs, approximately 7% of the average momentum across the shear depth and over the extent of the wavepacket is lost due to transport away from the mixing region by the waves. We propose that internal waves may act back upon the mean flow modifying it so that the excitation of waves of that frequency is enhanced. A narrow frequency spectrum of large-amplitude waves is observed because the feedback is largest for waves with phase tilt in a range near 45°. Numerical simulations and analytic theories are presented to further quantify this theory.

Internal waves generated from a turbulent mixed region

Using two nonintrusive visualization methods, laboratory experiments are performed to examine the internal wave field underlying a turbulent region generated by a vertically oscillating grid. The first method uses dye-lines to mark the vertical motions of isopycnal layers and the second uses “synthetic schlieren” to visualize the entire wave field. In a range of experiments, the strength of the stratification is varied so that the buoyancy frequency N=0.33–1.40 s−1. In all cases, large tank-scale standing wave modes are established which last throughout the experiment. The amplitudes of the isopycnal lines, Aξ, follow a power law relation Aξ∼N−1.5. The synthetic schlieren technique allows us to visualize turbulent eddy-scale waves and to isolate the properties of the strongest downward propagating waves at the base of the turbulent layer. These waves have a surprisingly narrow range of frequencies and vertical wavenumbers. The angles of wave propagation from the vertical for the dominant waves lie in the ...

Weakly nonlinear internal gravity wavepackets

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.

Intrusive gravity currents in two-layer fluids

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.

Interfacial gravity currents. II. Wave excitation

We examine the response of a two- and three-layer salt-stratified fluid to the collapse of a mixed region intruding along the middle layer. For sufficiently deep middle layers, the intrusion (an interfacial gravity current) excites a double-humped solitary wave appearing in the interfacial layer in front of the intrusion head. When the solitary wave is generated the current stops propagating. Trailing the intrusion are large-amplitude trapped internal waves. We study the effect of middle-layer depth and density difference to determine the conditions under which a solitary wave is generated. We propose that this transition occurs because the intrusion resonantly couples with trapped internal waves for a sufficiently thick interface.

Internal wave tunnelling

We present the first laboratory evidence of internal gravity wave tunnelling through weakly stratified fluid patches and we derive analytic theories for energy transmission by the waves in two distinct circumstances. In one, the computed transmission coefficient is directly analogous to the textbook calculation for quantum tunnelling of a free electron incident upon a potential barrier. In the other, we consider the partial reflection and transmission of internal waves through a mixed region bounded by discontinuities in the density profile. The results reveal a linear resonance between vertically propagating internal waves and interfacial waves that exist on either flank of the mixing region. The resonance permits perfect transmission of internal waves that would otherwise strongly reflect from the weakly stratified region. We discuss a specific application of our results to deep convective storm-generated internal waves that tunnel through the mesosphere to the ionosphere.