scholarly journals An Energy Compartment Model for Propagation, Nonlinear Interaction, and Dissipation of Internal Gravity Waves

2014 ◽  
Vol 44 (8) ◽  
pp. 2093-2106 ◽  
Author(s):  
Carsten Eden ◽  
Dirk Olbers
Keyword(s):  
Gravity Waves ◽  
Bottom Topography ◽  
Compartment Model ◽  
Tidal Energy ◽  
Internal Gravity Waves ◽  
Internal Tides ◽  
Continental Margins ◽  
Wave Interactions ◽  
Dissipation Energy ◽  
Parametric Subharmonic Instability

Abstract The recently proposed Internal Wave Dissipation, Energy and Mixing (IDEMIX) model, describing the propagation and dissipation of internal gravity waves in the ocean, is extended. Compartments describing the energy contained in the internal tides and the near-inertial waves at low, vertical wavenumber are added to a compartment of the wave continuum at higher wavenumbers. Conservation equations for each compartment are derived based on integrated versions of the radiative transfer equation of weakly interacting waves. The compartments interact with each other by the scattering of tidal energy to the wave continuum by triad wave–wave interactions, which are strongly enhanced equatorward of 28° due to parametric subharmonic instability of the tide and by scattering to the continuum of both tidal and near-inertial wave energy over rough topography and at continental margins. Global numerical simulations of the resulting model using observed stratification, forcing functions, and bottom topography yield good agreement with available observations.

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Stratified Shear Flow ◽  
Weak Resonance ◽  
Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

Scattering of internal tides by irregular bathymetry of large extent

2014 ◽  
Vol 747 ◽  
pp. 481-505 ◽  
Author(s):  
Yile Li ◽  
Chiang C. Mei
Keyword(s):  
Gravity Waves ◽  
Method Of Characteristics ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Internal Gravity Waves ◽  
Internal Tides ◽  
Two Dimensional ◽  
One Dimensional ◽  
Stratified Ocean ◽  
Stationary Random Function

AbstractWe present an analytical theory of scattering of tide-generated internal gravity waves in a continuously stratified ocean with a randomly rough seabed. Based on a linearized approximation, the idealized case of constant mean sea depth and Brunt–Väisälä frequency is considered. The depth fluctuation is assumed to be a stationary random function of space, characterized by small amplitude and a correlation length comparable to the typical wavelength. For both one- and two-dimensional topographies the effects of scattering on the wave phase over long distances are derived explicitly by the method of multiple scales. For one-dimensional topography, numerical results are compared with Bühler & Holmes-Cerfon (J. Fluid Mech., vol. 678, 2011, pp. 271–293), computed by the method of characteristics. For two-dimensional topography, new results are presented for both statistically isotropic and anisotropic cases.

Scattering of internal tide energy to super-tidal frequencies in global HYCOM 

2021 ◽  
Author(s):  
Miguel Solano ◽  
Maarten Buijsman
Keyword(s):  
Internal Tide ◽  
Tidal Energy ◽  
Horizontal Resolution ◽  
Internal Tides ◽  
Numerical Dispersion ◽  
Global Ocean ◽  
Wave Interactions ◽  
Continental Shelves ◽  
Lee Wave ◽  
Ocean Models

<p>Energy decay in realistically forced global ocean models has been mostly studied in the diurnal and semi-diurnal tidal bands and it is unclear how much of the tidal energy in these bands is scattered to higher frequencies. Global ocean models and satellite altimetry have shown that low-mode internal tides can propagate thousands of kilometers from their generation sites before being dissipated in the ocean interior but their pathway to dissipation is obscured due to lee-wave breaking at generation, wave-wave interactions, topographic scattering, shearing instabilities and shoaling on continental shelves. Internal tides from some generation sites, such as the Amazon shelf and the Nicobar and Andaman island chain, have large amounts of energy resulting in a steepening of the internal waves into solitary wave trains due to non-hydrostatic dispersion. In HYCOM, a hydrostatic model, this process is partially simulated by numerical dispersion. However, it is yet unknown how the dissipation of internal tides is affected by the numerical dispersion in hydrostatic models. In this study we use the method of vertical modes and rotary spectra to quantify the scattering of internal tides to higher-frequencies and analyze the dissipation processes in global HYCOM simulations with 4-km horizontal resolution.</p>

On the damping of internal gravity waves in a continuously stratified ocean

1966 ◽  
Vol 25 (1) ◽  
pp. 121-142 ◽  
Author(s):  
Paul H. LeBlond
Keyword(s):  
Internal Waves ◽  
Gravity Waves ◽  
Internal Gravity Waves ◽  
Bottom Friction ◽  
Internal Tides ◽  
Attenuation Rate ◽  
Mixing Coefficients ◽  
Stratified Ocean ◽  
Shallow Seas ◽  
Strongly Damped

The problem studied here is that of the attenuation of internal waves through turbulent mixing in a weakly and exponentially stratified fluid. The equations are linearized and it is assumed that the action of turbulence can be parametrically represented by eddy mixing coefficients and that the influence of bottom friction is restricted to a thin bottom boundary layer. The simple case where there is no rotation and only one component to the stratification is first examined in detail, and the modifications caused by introducing rotation and a second component are subsequently investigated. Subject quantitatively to the choice made for the eddy coefficients, but qualitatively not strongly dependent on that choice, the following conclusions are drawn: (i) very short internal waves (length < 100 m) are strongly damped in basins of all depths; (ii) long internal waves or seiches in shallow seas (depth ≃ 100 m) will not last more than a few cycles as free oscillations; (iii) the attenuation rate for long internal tides is small enough that these should be observable very far from the coasts, but large enough to exclude the possibility of oceanic standing wave systems; (iv) for very long internal waves the damping is predominantly due to the effect of bottom friction, and the attenuation rate becomes almost independent of the actual form of the stratification present in the fluid.

scholarly journals Internal Tides in the Southwestern Japan/East Sea

2006 ◽  
Vol 36 (1) ◽  
pp. 22-34 ◽  
Author(s):  
Jae-Hun Park ◽  
D. Randolph Watts
Keyword(s):  
Bottom Topography ◽  
Energy Levels ◽  
Internal Tide ◽  
Tidal Energy ◽  
High Energy ◽  
Shelf Break ◽  
Internal Tides ◽  
East Sea ◽  
Thermocline Depth ◽  
Generation Region

Abstract This paper investigates the internal tidal energy distribution in the southwestern Japan/East Sea using vertical round-trip travel time (τ) data from 23 pressure-sensor-equipped inverted echo sounders (PIES). The τ records are analyzed by bandpass filtering to separate time-dependent variability of the semidiurnal and diurnal bands. The semidiurnal internal tides exhibit a horizontal beam pattern of high energy, propagating into the open basin. They originate from a restricted portion of the shelf break where the Korea Strait enters the Ulleung Basin. The generation appears to occur at ∼200-m water depth near 35.5°–35.7°N and 130°–131°E, where the slope of bottom topography matches that of the wave characteristics, coinciding with the location where the semidiurnal barotropic cross-slope tidal currents are strongest. Maximum vertical displacement of the thermocline interpreted as a long-wave first baroclinic mode from the measured τ is about 25 m near the generation region. Annual and monthly variations of the propagation patterns and generation energy levels are observed, and these are closely associated with changes in the mesoscale circulation and stratification. Eastward (westward) refraction is observed when a warm (cold) eddy crosses the path of internal tide propagation. Moreover, when the generation region is invaded by cold eddies that spoil the match between shelf break and thermocline depth, the internal tidal energy level decreases by a factor of about 2. A simple geometric optics model is proposed to explain the observed horizontal refraction of the beam of semidiurnal internal tides in which stratification and current shear play essential roles. In contrast, diurnal internal tides are observed to be trapped along the continental slope region around 36°N.

Theoretical and experimental studies of gravity wave interactions

1967 ◽  
Vol 299 (1456) ◽  
pp. 104-119 ◽  
Keyword(s):  
Gravity Waves ◽  
Experimental Studies ◽  
Internal Gravity Waves ◽  
Stokes Wave ◽  
Wave Interactions ◽  
Principal Characteristics ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Interaction Distance ◽  
Wave Modes

This paper is concerned with various aspects of the resonant interactions among waves. An experiment was suggested by Longuet-Higgins (1962) to detect this type of interaction among surface waves. This was subsequently performed by Longuet-Higgins & Smith (1966) and by McGoldrick, Phillips, Huang & Hodgson (1966). The results of the two sets of experiments are compared. Together they demonstrate very clearly the principal characteristics of the interaction; the maximum response at resonance and the linear growth with interaction distance, the decrease in band width with interaction distance and the shift of the resonance point that results from the amplitude dispersion. It is shown further that the instability of the Stokes wave, discovered and analysed by Benjamin & Feir, can be described in terms of these interactions and that it is not restricted to purely two dimensional motion. A Stokes wave is unstable to a disturbance containing a pair of wavenumbers defined by any point in the zone just inside the figure-of-eight loop shown in figure 12. Another example of resonant wave interactions is provided by short, internal gravity waves in a stratified fluid with constant Brunt-Väisälä frequency. The interactions among Fourier modes are considered, and it is shown that there arise both free and forced modes. In the latter, the dispersion relation for internal waves is not satisfied; there is no particular relation between wavenumber and frequency. The amplitudes of these are small compared with those of the internal wave modes provided the harmonic mean of the vorticity in the two interacting waves is small compared with the Brunt-Väisälä frequency. The motion then consists of interacting internal gravity waves, whose interaction sets are closed. On the other hand, if the forced components are comparable in magnitude with the wave modes, these interact strongly and indiscriminately; a ‘cascade’, characteristic of turbulence, develops.

scholarly journals Resolving the horizontal direction of internal tide generation

2019 ◽  
Vol 864 ◽  
pp. 381-407 ◽  
Author(s):  
Friederike Pollmann ◽  
Jonas Nycander ◽  
Carsten Eden ◽  
Dirk Olbers
Keyword(s):  
Internal Wave ◽  
Energy Flux ◽  
Gravity Waves ◽  
Large Scale ◽  
Bottom Topography ◽  
Internal Gravity Waves ◽  
Global Ocean ◽  
Wide Field ◽  
Practical Advantage ◽  
Analytical Approaches

The mixing induced by breaking internal gravity waves is an important contributor to the ocean’s energy budget, shaping, inter alia, nutrient supply, water mass transformation and the large-scale overturning circulation. Much of the energy input into the internal wave field is supplied by the conversion of barotropic tides at rough bottom topography, which hence needs to be described realistically in internal gravity wave models and mixing parametrisations based thereon. A new semi-analytical method to describe this internal wave forcing, calculating not only the total conversion but also the direction of this energy flux, is presented. It is based on linear theory for variable stratification and finite depth, that is, it computes the energy flux into the different vertical modes for two-dimensional, subcritical, small-amplitude topography and small tidal excursion. A practical advantage over earlier semi-analytical approaches is that the new one gives a positive definite conversion field. Sensitivity studies using both idealised and realistic topography allow the identification of suitable numerical parameter settings and corroborate the accuracy of the method. This motivates the application to the global ocean in order to better account for the geographical distribution of diapycnal mixing induced by low-mode internal gravity waves, which can propagate over large distances before breaking. The first results highlight the significant differences of energy flux magnitudes with direction, confirming the relevance of this more detailed approach for energetically consistent mixing parametrisations in ocean models. The method used here should be applicable to any physical system that is described by the standard wave equation with a very wide field of sources.

Interharmonics in internal gravity waves generated by tide-topography interaction

2008 ◽  
Vol 611 ◽  
pp. 61-95 ◽  
Author(s):  
ALEXANDER S. KOROBOV ◽  
KEVIN G. LAMB
Keyword(s):  
Internal Wave ◽  
Gravity Waves ◽  
Wave Spectrum ◽  
Tidal Flow ◽  
Internal Gravity Waves ◽  
Wave Band ◽  
Wave Interactions ◽  
Highly Nonlinear ◽  
Tidal Frequency ◽  
Forced Waves

The dynamics and spectrum of internal gravity waves generated in a linearly stratified fluid by tidal flow over a flat-topped ridge are investigated at five different latitudes using an inviscid two-dimensional numerical model. The resulting wave field includes progressive freely propagating waves which satisfy the dispersion relation, and forced waves which are trapped non-propagating oscillations with frequencies outside the internal wave band. The flow is largely stable with respect to shear instabilities, and, throughout the runs, there is a negligibly small amount of overturning which is confined to the highly nonlinear regions along the sloping topography and where tidal beams reflect from the boundaries. The wave spectrum exhibits a self-similar structure with prominent peaks at tidal harmonics and interharmonics, whose magnitudes decay exponentially with frequency. Two strong subharmonics are generated by an instability of tidal beams which is particularly strong for near-critical latitudes where the Coriolis frequency is half the tidal frequency. When both subharmonics are within the free internal wave range (as in cases 0°–20° N), they form a resonant triad with the tidal harmonic. When at least one of the two subharmonics is outside of the range (as in cases 30°–40° N) the observed instability is no longer a resonant triad interaction. We argue that the two subharmonics are generated by parametric subharmonic instability that can produce both progressive and forced waves. Other interharmonics are produced through wave–wave interactions and are not an artefact of Doppler shifting as assumed by previous authors. As the two subharmonics are, in general, not proper fractions of the tidal frequency, the wave–wave interactions are capable of transferring energy to a continuum of frequencies.

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