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A stably stratified, two-layer fluid system with a free surface admits resonant triads, including ones involving one surface and two internal waves. Resonant triads that couple surface and internal gravity waves are thought to play an important role in transferring energy from wind-generated surface waves to internal waves. Ball used a geometric construction to show that the linearized dispersion relation for a two-layer liquid system with a free upper surface and no stream velocity admits resonant triads.
A laboratory study has been undertaken to measure the momentum transfer from surface to internal gravity waves in a nonlinear, resonant interaction. The interacting waves form triads for which σ1s - σ2s " σI = 0 and κ1s - κ2s " κI = 0; σj and κj being the frequency and wavenumber of the jth wave. In particular, the experiment is designed to model a generating mechanism for high frequency, oceanic internal waves. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. The growth to steady state of a resonant internal wave is observed while two deep water surface eigen modes are simultaneously forced by a paddle. Results are compared to theoretical predictions which assume, ab initio, all waves to be standing. Inclusion of viscous side wall dissipation and slight detuning permit predictions of steady state amplitudes and phases as well as initial growth rates. Good agreement is found between predieted and measured amplitudes and phases. The experiments also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves.
A laboratory study was undertaken to measure the momentum transfer from surface to internal gravity waves in a nonlinear, resonant interaction. The interacting waves form triads for which (Sigma sub is)- (Sigma sub 2S) plus or minus (Sigma subI) = 0 and (Kappa sub is)-(Kappa sub 2S) plus or minus (Kappa sub I) = 0; Sigma sub j and Kappa sub j being the frequency and wavenumber of the jth wave. In particular, the experiment is designed to model a generating mechanism for high frequency, oceanic internal waves. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. The growth to steady state of a resonant internal wave is observed while two deep water surface eigen modes are simultaneously forced by a paddle. Results are compared to theoretical predictions which assume, ab initio, all waves to be standing. Inclusion of viscous side wall dissipation and slight detuning permit predictions of steady state amplitudes and phases as well as initial growth rates. (Author).
The dynamics of the coupling of linear internal gravity waves and linear surface gravity waves on the ocean is studied using a Hamiltonian formalism and action-angle variables. The dynamic equations are solved both numerically and in some analytic approximations. The results compare favorably with the interaction experiments of Lewis, Lake and Ko the 'resonant triad' experiments of Joyce and some satellite observations of Apel et al. The growth time for internal waves generated by the resonant interaction of surface waves is calculated using the Garrett-Munk ocean model and the Phillips spectrum for surface waves. Energy exchange rates are deduced.
This monograph creates a systematic interpretation of the theoretical and the most actual experimental aspects of the internal wave dynamics in the ocean. Firstly, it draws attention to the important physical effects from an oceanographical point of view which are presented in mathematical descriptions. Secondly, the book serves as an introduction to the range of modern ideas and the methods in the study of wave processes in dispersive media. The book is meant for specialists in physics of the ocean, oceanography, geophysics, hydroacoustics.
Internal waves are propagating disturbances within stratified fluids, arising from a balance of gravity, buoyancy, and rotation. As well as being of fundamental scientific interest, they are ubiquitous in a variety of forms in the Earth's oceans, where they are responsible for driving vertical mixing. And it is the rule, rather than the exception, that internal waves propagate through a varying background density stratification. We begin by theoretically studying internal waves that are harmonically forced at a horizontal level above a semi-infinite, non-uniform density stratification. Starting with a two-layer model, we identify the existence of resonance peaks and diminution troughs in the wave transmission spectra, and provide physical insight through the application of ray theory. Thereafter, we proceed to consider smoothly varying stratifications, demonstrating that these resonance and diminution features persist beyond simple models. We conclude by considering the relevance of the results to geophysical settings. As an example, we demonstrate that an ocean stratification is inherently tuned to transmit internal wave energy to the deep ocean at specific combinations of wavelength and frequency. Subsequently, we perform a laboratory experimental study of an internal wave field generated by harmonic, spatially-periodic surface forcing of a strongly-stratified, thin upper layer sitting atop a weakly-stratified, deep lower layer. In linear regimes, the energy flux associated with relatively high frequency internal waves is prevented from entering the lower layer by virtue of evanescent decay. In the experiments, however, we find that the development of parametric subharmonic instability (PSI) in the upper layer transfers energy from the forced primary wave into a pair of subharmonic daughter waves, each capable of penetrating the weakly-stratified lower layer. We find that around 10% of the primary wave energy penetrates into the lower layer via this nonlinear wave-wave interaction for the regime we study. With an emphasis on assessing the role of interference in tuning wave transmission, we perform a series of laboratory experiments in order to measure resonance and diminution in the aforementioned non-uniform stratification. We find that the occurrence of destructive interference in the upper stratification layer naturally yields diminution of the transmitted wave. Conversely, constructive interference results in a notable amplification of the wave field over time scales on the order of the forcing period; the development of nonlinear wave-wave interactions due to wave amplification is observed over longer time scales. Good agreement is obtained between the experimental results and a weakly viscous, long wave model of our system within the linear regime. Given the ubiquity of layering in environmental stratifications, an interesting example being double-diffusive staircase structures in the Arctic water column, we furthermore present the results of a joint theoretical and laboratory experimental study investigating the impact of multiple layering on internal wave propagation. We first present results for a simplified model that demonstrates the nontrivial impact of multiple layering. Incident waves of particular length and time scales can experience constructive interference taking place within the alternating stratified and mixed layers, which in turn appreciably enhances wave transmission. Thereafter, utilizing a weakly viscous, linear model that can handle arbitrary vertical stratifications, we perform a comparison of theory with experiments finding excellent qualitative and quantitative agreement. We conclude by applying this model to a case study of a staircase stratification profile obtained from the Arctic Ocean, finding a rich landscape of transmission behavior.
This monograph creates a systematic interpretation of the theoretical and the most actual experimental aspects of the internal wave dynamics in the ocean. Firstly, it draws attention to the important physical effects from an oceanographical point of view which are presented in mathematical descriptions. Secondly, the book serves as an introduction to the range of modern ideas and the methods in the study of wave processes in dispersive media. The book is meant for specialists in physics of the ocean, oceanography, geophysics, hydroacoustics.