Interaction Of Internal Waves Research Articles

Internal solitary wave refraction at Dongsha Atoll, South China Sea

AbstractInternal solitary waves (ISWs) are regularly generated at the Luzon Strait, propagate westward into the South China Sea, and dissipate on the continental shelf after persisting for more than 4 days. The Dongsha Atoll stands in the middle of their propagation path and the incident ISW bifurcates as it interacts with, and passes the atoll. As the two arms meet on the western side of Dongsha, they produce complex patterns from nonlinear internal wave interactions. In this study, new satellite observations are presented that show unusual ISW refraction patterns at the Dongsha Atoll. In these satellite images the incident wave is located on one side of Dongsha, and the portion of the wave closest to the atoll continues to propagate around the atoll circularly, while the remaining portion of the arm continues its westward propagation. The process is adequately modeled using a function relating ISW phase speed to ocean depth.

Unstable modes of a sheared pycnocline above a stratified layer

Internal waves incident on a sheared ocean pycnocline are studied using analytic and numerical methods. Linear analysis of the unstable modes of a sheared ocean pycnocline is used to demonstrate interactions between internal waves and shear instabilities. A new analytic solution for an asymmetric shear layer over a stratified layer is presented, illustrating modes which couple to internal waves, in addition to the well-known Holmboe modes. The robustness of these solutions is demonstrated using numerical methods for realistic shear profiles. Fully nonlinear numerical simulations illustrate the growth of these modes and demonstrate the excitation of shear instabilities by incident internal waves. The results may have implications for internal wave interactions with the ocean pycnocline and the local generation of internal solitary waves.

Fluid circulation and seepage in lake sediment due to propagating and trapped internal waves

It has long been known that surface gravity waves induce significant seepage through the porous layer found at the lake bottom. Away from coastal regions, however, the pressure signature of surface waves at the lake bottom is weak. We consider fully nonlinear internal gravity waves, whose long wavelength and slow motion implies a sustained and strong pressure perturbation even in the deep regions of the lake. We argue that internal waves can induce significant seepage through the sediment layer, in regions where surface gravity waves have negligible impact. The pressure profile at the fluid–porous layer interface is computed from the “exact” Dubreil‐Jacotin‐Long theory, giving a reliable profile even for large waves. This profile is used in conjunction with Darcy's law to compute the seepage within the porous region. We find that the geometric distribution of seepage is strongly controlled by both the ratio of porous media thickness to the horizontal length of the pressure perturbation, and the bottom topography, when it is present. Based on work on the interaction of internal solitary waves with the bottom boundary layer, we develop a model to account for the changes in permeability due to wave‐induced instabilities in the bottom boundary layer and enhanced benthic turbulence. This turbulence acts to unplug the pores near the surface by lifting the detritus that clogs them. The resulting changes in permeability significantly enhance exchange between the free fluid and the porous medium on the downstream side of the wave.

NONLINEAR OBLIQUE INTERACTION OF LARGE AMPLITUDE INTERNAL SOLITARY WAVES

Solitary waves are typical nonlinear long waves in the ocean. The two-dimensional interaction of solitary waves has been shown to be essentially different from the one-dimensional case and can be related to generation of large amplitude waves (including 'freak waves'). Concerning surface-water waves, Miles (1977) theoretically analyzed interaction of three solitary waves, which is called "resonant interaction† because of the relation among parameters of each wave. Weakly-nonlinear numerical study (Funakoshi, 1980) and fully-nonlinear one (Tanaka, 1993) both clarified the formation of large amplitude wave due to the interaction ("stem† wave) at the wall and its dependency of incident angle. For the case of internal waves, analyses using weakly nonlinear model equation (ex. Tsuji and Oikawa, 2006) suggest also qualitatively similar result. Therefore, the aim of this study is to investigate the strongly nonlinear interaction of internal solitary waves; especially whether the resonant behavior is found or not. As a result, it is found that the amplified internal wave amplitude becomes about three times as much as the original amplitude. In contrast, a "stem" was not found to occur when the incident wave angle was more than the critical angle, which has been demonstrated in the previous studies.

Laboratory experiments on internal wave interactions with a pycnocline

Laboratory experiments have been performed to investigate the interaction of internal waves with a pycnocline. An oscillating cylinder generated internal wave beams, which were observed using the synthetic schlieren technique. Internal waves incident on the pycnocline layer excited higher-frequency modes. In the absence of shear, a discrete spectrum of harmonic modes was generated due to nonlinear effects. These harmonic modes might play a role in the formation of internal solitary waves which have been observed in ocean pycnoclines. With shear, a continuous spectrum of excited modes was found.

Tsunami‐induced phase and amplitude perturbations of subionospheric VLF signals

A network of VLF receivers, sited in the Far East, has been used to observe the response of the lower ionosphere to tsunamis resulting from the November 15, 2006 (Kuril region) and the March 11, 2011 (Tohoku region) earthquakes. Specific perturbations in the phase and amplitude of VLF signals have been found for both cases. A qualitative interpretation of the observed effects is suggested in terms of the interaction of internal gravity waves with lower ionosphere.

The surface signature of internal waves

AbstractOceans that are stratified by density into distinct layers support internal waves. An internal soliton gives rise to characteristic features on the surface, a signature of its presence, in the form of a ‘rip’ region, as reported in Osborne & Burch (Science, vol. 208, 1980, pp. 451–460), which results in a change in reflectance as seen in NASA photographs from the space shuttle. In the present paper, we give a new analysis of this signature of an internal soliton, and the ‘mill pond’ effect of an almost completely calm sea after its passage. Our analysis models the resonant interaction of nonlinear internal waves with the surface modes, where the surface signature is generated by a process analogous to radiative absorption. These theoretical results are illustrated with numerical simulations that take oceanic parameters into account.

Oblique wave‐wave interactions of nonlinear near‐surface internal waves in the Strait of Georgia

Although nonlinear interactions that occur when two large internal waves collide at oblique angles are often observed in the natural world, quantitative and theoretical aspects of these interactions are only poorly understood. The available analyses are generally theoretical or the result of limited numerical experimentation, with few (if any) quantitative field measurements. Here we describe four cases of internal wave interactions, two of which involve the fundamentally non‐steady generation of “Mach stems” at the site of the interaction, observed in the Strait of Georgia, Canada. Details of the interactions are quantified using time sequences of photogrammetrically rectified oblique images obtained from a circling aircraft, while simultaneous observations are made of water column properties from a surface vessel. The imaging technique allows us to estimate the speed and direction of each wave, as well as details of phase shifts induced by interactions. An existing small‐amplitude theory is applied but is found to overestimate the likelihood of Mach interactions at large amplitude.

Resonant and Near-Resonant Internal Wave Interactions

Abstract The spectral energy density of the internal waves in the open ocean is considered. The Garrett and Munk spectrum and the resonant kinetic equation are used as the main tools of the study. Evaluations of a resonant kinetic equation that suggest the slow time evolution of the Garrett and Munk spectrum is not in fact slow are reported. Instead, nonlinear transfers lead to evolution time scales that are smaller than one wave period at high vertical wavenumber. Such values of the transfer rates are inconsistent with the viewpoint expressed in papers by C. H. McComas and P. Müller, and by P. Müller et al., which regards the Garrett and Munk spectrum as an approximate stationary state of the resonant kinetic equation. It also puts the self-consistency of a resonant kinetic equation at a serious risk. The possible reasons for and resolutions of this paradox are explored. Inclusion of near-resonant interactions decreases the rate at which the spectrum evolves. Consequently, this inclusion shows a tendency of improving of self-consistency of the kinetic equation approach.

The Stability of Short Symmetric Internal Waves on Sloping Fronts: Beyond the Traditional Approximation

AbstractThe interaction of internal waves with geostrophic flows is found to be strongly dependent upon the background stratification. Under the traditional approximation of neglecting the horizontal component of the earth’s rotation vector, the well-known inertial and symmetric instabilities highlight the asymmetry between positive and negative vertical components of relative vorticity (horizontal shear) of the mean flow, the former being stable. This is a strong stratification limit but, if it becomes too low, the traditional approximation cannot be made and the Coriolis terms caused by the earth’s rotation vector must be kept in full. A new asymmetry then appears between positive and negative horizontal components of relative vorticity (vertical shear) of the mean flow, the latter becoming more unstable. Particularly conspicuous at low latitudes, this new asymmetry does not require vanishing stratification to occur as it operates readily for rotation/stratification ratios 2Ω/N as small as 0.25 (the stratification still dominates over rotation) for realistic vertical shears. Given that such ratios are easily found in ocean–atmosphere boundary layers or in the deep ocean, such ageostrophic instabilities may be important for the routes to dissipation of the energy of the large-scale motions. The energetics show that, depending on the orientation of the internal wave crests with respect to the mean isopycnal surfaces, the unstable motions can draw their energy either from the kinetic energy or from the available potential energy of the mean flow. The kinetic energy source is usually the leading contribution when the growth rates reach their maxima.

Investigation of High-Frequency Internal Wave Interactions with an Enveloped Inertia Wave

Using ray theory, we explore the effect an envelope function has on high-frequency, small-scale internal wave propagation through a low-frequency, large-scale inertia wave. Two principal interactions, internal waves propagating through an infinite inertia wavetrain and through an enveloped inertia wave, are investigated. For the first interaction, the total frequency of the high-frequency wave is conserved but is not for the latter. This deviance is measured and results of waves propagating in the same direction show the interaction with an inertia wave envelope results in a higher probability of reaching that Jones' critical level and a reduced probability of turning points, which is a better approximation of outcomes experienced by expected real atmospheric interactions. In addition, an increase in wave action density and wave steepness is observed, relative to an interaction with an infinite wavetrain, possibly leading to enhanced wave breaking.

The interaction of turbulent mixing region with local density perturbation in a pycnocline

Efficient numerical models were developed for the dynamics of local and turbulent formations in a pycnocline for which the averaged flow is characterized by appearance of internal waves of soliton type. Numerical analysis of the process of initiation, interaction, and subsequent propagation of internal waves generated by two local density perturbations in a pycnocline is carried out. Anisotropic turbulence degeneration in the turbulent mixing region in a pycnocline is numerically modeled. The problem of interaction of the turbulent mixing region with the local density perturbation in a pycnocline is numerically investigated in a wide range of local perturbation parameters.

Propagation speeds of strongly nonlinear near-surface internal waves in the Strait of Georgia

[1] A novel aerial observational method for studying internal features in the coastal ocean is developed and tested in a study of large nonlinear internal solitary-like waves. Photogrammetrically rectified oblique photo images from a circling aircraft are used to track a number of internal wave packets for periods of up to one hour in the Strait of Georgia, British Columbia, Canada. Combining these sequences with coincident water column data allows us to obtain a more complete view of the spatial structure of internal waves. Highly accurate measurements of wave propagation speeds and directions are possible. The applicability of various weakly nonlinear theories in modeling propagation of the observed large-amplitude waves is tested. The measured wave speeds enable us to differentiate between classic internal wave models. The linear, KdV (Korteweg-de Vries), and BO (Benjamin-Ono) models are applied with and without background shear. After background shear effects are included, it is found that a continuously stratified BO equation can predict propagation speeds within observational error, and that this is not true for other theories. The technique may be useful in future studies of oblique internal wave interactions.

Interaction and generation of long-crested internal solitary waves in the South China Sea

[1] Oblique nonlinear interactions based on the Kadomtsev-Petviashvili equation (also known as the two-dimensional Korteweg–de Vries equation) are extended to internal solitary waves (ISWs) to explain why the amplitude does not decrease owing to the geometric spreading of the cylindrical wavefronts in the South China Sea (SCS). This resonance theory is used to explain a satellite image exhibiting special features, and it is proposed that wave arcs of different amplitude resonate, providing a mechanism for reinforcing a wave by boosting the amplitude. The present theory suggests the amplitude of the ISW that propagates across the SCS basin depends on the interaction of ISWs originating from different sources; hence, studying the generation of an ISW from a single source location cannot predict the ISW correctly.

Decay of an internal tide due to random topography in the ocean

We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.

Decay of an internal tide due to random topography in the ocean

We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.

A model for nonlinear interactions of internal gravity waves with saturated regions

A model for interactions between non-hydrostatic gravity waves and deep convective narrow hot towers is presented. The starting point of the derivation are the conservation laws for mass, momentum and energy for compressible flows combined with a bulk micro-physic model. Using multiscale asymptotics, a set of leading order equations is extracted, valid for the specific scales of the investigated regime. These are a timescale of 100 s, a horizontal and vertical lengthscale of 10 km for the wave dynamics plus a second horizontal lengthscale of 1 km for the narrow hot towers. Because of the comparatively short horizontal scales, Coriolis effects are negligible in this regime. The leading order equations are then closed by applying conditional averages over the hot tower lengthscale, leading to a closed model for the wave-scale that retains the net effects of the smaller scale dynamics. By assuming a systematically small saturation deficit in the ansatz, the small vertical displacements arising in this regime suffice to induce leading order changes of the saturated area fraction. The latter is the essential parameter in the model arising from the micro-physics.

Large Amplitude Internal Solitary Waves due to Solitary Resonance

To clarify the resonance of fully-nonlinear internal solitary waves which is one of the reasons for the occurrence of large amplitude internal waves, fully-nonlinear and strongly-dispersive internal wave equation model was applied, which attempted to investigate interaction of internal solitary waves in a two-dimensional plane. The 3rd order theoretical solutions for internal waves in a two-layer system was used for the initial conditions and progress of internal solitary wave was confirmed. Seven different incident wave angles were given, in which 'stem' was confirmed to appear when incident wave angle is less than critical angle. As a result, it is found that the amplified internal wave amplitude becomes about three times as much as the original amplitude.

Mechanism of Energy Transfers to Smaller Scales Within the Rotational Internal Wave Field

We discuss the effect of the earth rotation on the two-triad inter- action and the oceanic energy distribution processes that occur between five coupled internal gravity waves. The system we study is a two-triad test wave system consisting of an initial wave of the tidal M2 frequency interacting with four recipient waves forming two resonant triads. It is shown that the general mechanism of an arbitrarily large number of internal wave interactions can be described by a three classes of interactions which we call the sum, middle and difference interaction classes. The four latitude singularities are distinguished for the particular case of five interacting waves and all three classes of resonant interactions are studied separately at those critical values. It is shown that the sum and difference interaction classes represent the latitude- inferior and latitude-predominant classes respectively. The phenomenon of coalescence of the middle and difference interaction classes is observed along latitude 48.25 ◦ N. It shown that at this value of latitude, the coalescence phe- nomenon provides the analogy between rotating and reflecting internal waves from slopes.

On the interaction of internal gravity waves with a magnetic field - II. Convective forcing

We present results from numerical simulations of the interaction of internal gravity waves (IGW) with magnetic fields in the radiative interior of the Sun. In this second paper, the waves are forced self-consistently by an overlying convection zone and a toroidal magnetic field is imposed in the stably stratified layer just underneath convection zone. Consistent with the results of previous analytic and simple numerical calculations, we find a strong wave-field interaction, in which waves are reflected in the field region. The wave-field interaction and wave reflection depend on the field strength as well as adopted values of the diffusivities. In some cases wave reflection leads to an increased mean flow in the field region. In addition to reproducing some of the features of our simpler models, we find additional complex behaviour in these more complete and realistic calculations.