Session HH: Internal Waves II

Weakly Nonlinear Non-Boussinesq Internal Gravity Wavepackets

Internal gravity waves induce a horizontal mean flow that interacts with the waves if they are of moderately large amplitude. Previous studies of finite-amplitude Boussinesq wavepackets have shown that this interaction results in modulational instability for waves with frequency $\omega>2^{-1/2}N$. Otherwise, the waves are modulationally stable, and their amplitude decreases faster than predicted by the theory of linear dispersion. In this work, a theoretical form for the wave-induced mean flow of non-Boussinesq internal gravity waves and the corresponding weakly nonlinear Schr\"{o}dinger equation will be presented. The solution of the Schr\"{o}dinger equation is numerically integrated for horizontally periodic, vertically localized wavepackets. We consider Gaussian wavepackets with small initial amplitude, propagating upwards through a background in which density decreases exponentially with height. As they propagate, the wave amplitude increases and weakly nonlinear effects develop. The Schr\"{o}dinger equation results, when compared with fully nonlinear numerical simulations, are found to capture the dominant characteristics of the wave evolution, in the linear and weakly nonlinear regimes. In particular, hydrostatic waves propagate well above the level at which linear theory predicts they should overturn. Implications for gravity wave drag parameterization in the atmosphere are discussed.   

Schlieren measurements of internal waves in non-Boussinesq fluids

Previous experiments that have examined the generation of internal gravity waves by a monochromatic source have been restricted to small amplitude forcing in Boussinesq stratified fluids. Here we present measurements of internal waves generated by a circular cylinder oscillating at large amplitude in a non-Boussinesq fluid. The ``synthetic schlieren'' optical measurement technique is extended to stratifications in which the fluid index of refraction may vary nonlinearly with density. The method is applied to examine disturbances in uniformly stratified ambients consisting either of NaCl or NaI solutions whose concentrations are near-saturation at the bottom of the tank. We report upon the first extensive measurements of the optical properties of NaI solutions as they depend upon concentration and density. Using these results in experiments, we find that large amplitude forcing generates oscillatory turbulence surrounding the cylinder, thereby increasing the effective cylinder size and decreasing the amplitude of the waves in comparison with linear theory predictions. We parameterize the influence of the turbulent boundary layer in terms of an effective cylinder radius and forcing amplitude. Upward propagating waves are observed to grow in amplitude due to non-Boussinesq effects, in agreement with expectations. Surprisingly, however, the waves are observed to break despite propagating over only a fraction of the density scale height.   

Direct numerical simulations of the large-amplitude internal waves

The instability of large amplitude internal waves is investigated by direct numerical simulations. Realistic values of parameters, such as the density contrast, fluid viscosity, and the ratio of wave amplitude to wave length for the internal waves in the ocean are used. From the simulation results the critical wave amplitude is determined. The dependence of the critical wave amplitude on the physical parameters will be reported. The numerical code is carefully validated, and its performance and convergence will also be reported. Diagnostics and analysis are conducted to compare with existing experimental findings and observations.   

Turbulent Shear and Internal Waves

A series of experiments is presented that model the generation of non-hydrostatic internal gravity waves in upper ocean by the forcing of wind driven turbulent eddies in the surface mixed layer. A turbulent shear layer is forced by a conveyor belt with affixed flat plates near the surface of a stratified fluid and downward propagating internal waves are generated. The turbulence in the shear layer is characterized using particle image velocimetry to measure the kinetic energy as well as length and time scales. The internal waves are measured using synthetic schlieren to determine the amplitudes, frequencies, momentum fluxes, and the energy of the generated waves. The fraction of energy that leaks from the mixed layer to the internal wave field is presented. Consistent with other studies, it is found that the frequencies of internal waves generated by turbulence are an approximate constant fraction of the buoyancy frequency. Implications to internal waves propagating into the deep ocean will be discussed.   

Internal Waves in Shear Flow

Internal waves propagating through a region of shear flow can exchange energy with the mean flow. This effect is most pronounced at a ``critical level,'' a depth where the wave horizontal phase velocity matches the local mean flow speed. Waves may be reflected or transmitted through the critical level, gaining or losing energy by exchange with the mean flowfield. The linear theory of internal waves suggests that an increase in wave energy, or amplification, may be possible if the Richardson number is less than 1/4 at the critical level. Several laboratory experiments on internal waves interacting with a critical level have been published, but none have demonstrated wave amplification. New experiments are underway to study internal waves interacting with a critical level to determine if/when wave amplification can occur. This presentation will summarize preliminary results from these experiments.   

Wind-forced evolution of long internal waves in a large lake

A wind forced, weakly nonlinear, weakly dispersive evolution model is derived for a continuously stratified, circular lake of slowly varying depth under the effect of earth rotation. The model was numerically integrated to investigate the evolution of long internal waves of vertical mode one for various sets of environmental parameters. It is demonstrated that Kelvin waves steepen as they propagates, and the steepened front subsequently generates a train of oscillatory waves. It is demonstrated that Poincare waves do not steepen, but their amplitudes are modulated in time, exhibiting a pseudo recurrence character. The model was applied to the wind forced problem, confirming that Kelvin and Poincare waves are the dominant response. The energy partition between Kelvin and Poincare wave modes is estimated. For large lakes, the dominant wave amplitude is found in the Kelvin wave mode, but the Poincare wave mode clearly dominates the total energy deposited by an applied wind stress.   

Internal wave tunnelling: Laboratory experiments

Heuristics based upon ray theory are often used to predict the propagation of internal waves in non-uniform media. In particular, they predict that waves reflect from weakly stratified regions where the local buoyancy frequency is less than the wave frequency. However, if the layer of weak stratification is sufficiently thin, waves can partially transmit through it in a process called tunnelling. In this work, the first laboratory evidence of internal gravity wave tunnelling through a weakly stratified region is analysed. Generated by a vertically oscillating circular cylinder, internal gravity waves partially reflect from and transmit through a weakly stratified region. Using a non-intrusive optical technique called ``synthetic schlieren,'' the wave amplitude and structure can be measured and the transmission coefficient is computed as the ratio of the transmitted and incident energy of the waves. Using an appropriate superposition of plane waves to reproduce the structure of the incident wave beam, the theoretical transmission coefficients are calculated using the experimentally measured background density profiles. We form a corresponding weighted sum of transmission coefficients and so predict the transmission of the beam. These results are compared with experimental measurements.   

Spatial characterization of vortical structures and internal waves in stratified turbulent wake using POD

Proper orthogonal decomposition (POD) is applied to 2-D slices of vorticity and horizontal divergence obtained from the 3-D DNS of the stratified turbulent wake of a towed sphere at \textit{Re=}5x10$^{3}$ and \textit{Fr}=4. Slices are sampled along the stream-depth (Oxz) and stream-span planes (Oxy) at 231 times during the interval $Nt\in [12,35]$. POD was chosen amongst the available statistical tools due to its advantage in characterization of simulated and experimentally measured velocity gradient fields, as previously demonstrated for turbulent boundary layers. In the Oxz planes, at the wake centerline, the higher most energetic modes reveal a structure similar of the structure of late-time stratified wakes. Off-set from centerline, the signature of internal waves in the form of forward-inclined coherent beams extending into the ambient becomes evident. The angle of inclination becomes progressively vertical with increasing POD mode. Lower POD modes on the Oyz planes show a layered structure in the wake core with coherent beams radiating out into the ambient over a broad range of angles. Further insight is provided through the relative energy spectra distribution of the vorticity eigenmodes. POD analysis has provided a statistical description of the geometrical features previously observed in instantaneous flow fields of stratified turbulent wakes.   

The Interaction between a Stably-Stratified Jet and an Internal Gravity Wave Field

Direct Numerical Simulation of an internal wave field penetrating a stratified jet is performed. The jet is linearly stratified such that the gradient Richardson number is greater than 0.25 everywhere. Internal waves, generated with specified amplitude and group velocity, propagate downward toward the jet situated below. A critical layer is observed in the upper-flank of the jet where the jet velocity is equal to the wave horizontal phase speed. In the vicinity of the layer, a significant amount of wave energy is reflected back to the region above the jet while a small fraction tunnels through to the lower-flank. The wave field in the region below the jet is strongly attenuated compared to the incident field although the propagation direction remains unchanged. In the region above the jet the incident and reflected waves interact resulting in an unsteady wave field. Furthermore, the interaction results in an induced turbulent dissipation rate which is an order of magnitude higher than that inside the jet.