Oceanic Internal Gravity Waves Research Articles

Estimating the Location of Acoustic Sources with Oceanic Internal Gravity Waves via Frequency-Difference Source Localization

Frequency-difference source localization (FDSL) is a relatively new method for locating sources emitting acoustic or electromagnetic waves from recorded data. Estimates of location are produced by matching a nonlinear function of the received data with the same nonlinear function of modeled data. This so-called matched-field approach depends on the accuracy of the modeled field. FDSL exhibits less sensitivity to model errors in some cases of interest. Fluctuations of the speed of sound in the ocean are affected by a wide variety of phenomena and internal gravity waves is one where their spectrum is usually well known. There is no analytical theory for predicting how these fluctuations of sound speed affect the accuracy of FDSL, leaving a numerical evaluation as the alternative for the focus of this paper. Experimental application of FDSL was previously made for an acoustic source of 100 Hz bandwidth surrounding 250 Hz for a 129 km section in the Philippine Sea with data recorded on a long vertical array [Geroski and Dowling, Long-range frequency-difference source localization in the Philippine Sea, J. Acoust. Soc. Am. 146 (2019) 4727–4739.]. The role of internal waves is quantified with a realistic FDSL simulation using models based on the parabolic and ray approximations of the linear acoustic wave equation. The modeled vertical variation of FDSL error is similar to observed while the modeled horizontal error is too small. The complicated software model is automatic but is computationally intensive, with our version needing about 1.5 mo to compute on a PC with 16 cpu cores.

A Model of Energy and Spectral Shape for the Internal Gravity Wave Field in the Deep Sea: The Parametric IDEMIX Model

Abstract The spectral description of the energy of oceanic internal gravity waves is generally represented by the Garrett–Munk (GM) model, a function with a power-law decrease of spectral energy in wavenumber–frequency space. Besides the slopes of these power laws, the spectrum is expressed as a function of energy and a bandwidth parameter that fixes the range of vertical modes excited in the respective state. Whereas concepts have been developed and agreed upon of what processes feed the wave spectrum and what dissipates energy, there is no explanation of what shapes the spectral distribution, i.e., how the power laws come about and what sets the bandwidth. The present study develops a parametric spectral model of energy and bandwidth from the basic underlying energy balance in terms of forcing, propagation, refraction, spectral transfer, and dissipation. The model is an extension of the IDEMIX (Internal Wave Dissipation, Energy and Mixing) models where bandwidth was taken as a constant parameter. The current version of the model is restricted to single-column mode and the slopes of the spectral power laws are fixed. A coupled system of predictive equations for energy and bandwidth (for up- and downward propagating waves) results. The equations imply that bandwidth relates to energy by a power law with an exponent given by the dynamical parameters. It agrees favorably with energy, bandwidth, and slope data from previously published fits of the GM model to Argo float observations. Numerical solutions of the coupled energy–bandwidth model in stand-alone modus are presented.

Bidirectional Whitham type equations for internal waves with variable topography

In the context of oceanic internal gravity waves, one of the most widely-used theories on examining wave dynamics and interpreting observational data is the Korteweg–de Vries (KdV) equation. Nonetheless, the characters of unidirectional propagation and unbounded phase and group velocities restrict its application to some general cases (Benjamin et al., 1972). Thus, using the Dirichlet–Neumann operator with the rigid-lid approximation, we derive both bidirectional and unidirectional Whitham type equations in the Hamiltonian framework, which retain the full linear dispersion relation of the Euler equations. The effect of topography is also incorporated in modeling due to its practical relevance, although the invoked scaling plausibly excludes the accommodation of a significant bottom variation. There are no analytic solutions of internal solitary waves explicitly given in the newly proposed equations, even though these equations possess a concise form. Therefore, a modified Petviashvili iteration method is implemented to obtain the numerical solutions to circumvent this difficulty. Facilitated by these techniques, several numerical experiments are investigated and compared among different models: the KdV equation, the Whitham type equations, and the primitive equations. The discrepancies and similarities between the various models jointly indicate the advantage of full dispersion and bidirectional propagation and, thus, the effectiveness of the Whitham type equations.

Simulating turbulent mixing caused by local instability of internal gravity waves

Abstract

Global Characterization of the Ocean’s Internal Wave Spectrum

AbstractA key ingredient of energetically consistent ocean models is the parameterized link between small-scale turbulent mixing, an important energy source of large-scale ocean dynamics, and internal gravity wave energetics. Theory suggests that this link depends on the wave field’s spectral characteristics, but because of the paucity of suitable observations, its parameterization typically relies on a model spectrum [Garrett–Munk (GM)] with constant parameters. Building on the so-called “finestructure method,” internal gravity wave spectra are derived from vertical strain profiles obtained from Argo floats to provide a global estimate of the spatial and temporal variability of the GM model’s spectral parameters. For spectral slopes and wavenumber scales, the highest variability and the strongest deviation from the model’s canonical parameters are observed in the North Atlantic, the northwest Pacific, and the Southern Ocean. Internal wave energy levels in the upper ocean are well represented by the GM model value equatorward of approximately 50°, while they are up to two orders of magnitude lower poleward of this latitude. The use of variable spectral parameters in the energy level calculation hides the seasonal cycle in the northwest Pacific that was previously observed for constant parameters. The global estimates of how the GM model’s spectral parameters vary in space and time are hence expected to add relevant detail to various studies on oceanic internal gravity waves, deepening the understanding of their energetics and improving parameterizations of the mixing they induce.

On the wave turbulence theory for stratified flows in the ocean

After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions.

Generation of oceanic internal gravity waves by a cyclonic surface stress disturbance

Atmospheric cyclones with strong winds significantly impact ocean circulation, regional sea surface temperature, and deep water formation across the global oceans. Thus they are expected to play a key role in a variety of energy transport mechanisms. Even though wind-generated internal gravity waves are thought to contribute significantly to the energy balance of the deep ocean, their excitation mechanisms are only partly understood.The present study investigates the generation of internal gravity waves during a geostrophic adjustment process in a Boussinesq model with axisymmetric geometry. The atmospheric disturbance is set by an idealized pulse of cyclonic wind stress with a Rankine vortex structure. Strength, radius and duration of the forcing are varied. The effect upon wave generation of stratification with variable mixed-layer depth is also examined.Results indicate that internal gravity waves are generated after approximately one inertial period. The outward radial energy flux is dominated by waves having structure close to vertical mode-1 and with frequency close to the inertial frequency. Less energetic higher mode waves are observed to be generated close to the sea floor underneath the storm. The total radiated energy corresponds to approximately 0.02% of the wind input. Deeper mixed-layer conditions as well as weaker stratification reduce this fraction.The low energy transfer rates suggest that other processes that drive vertical motion like surface heat fluxes, turbulent motion, mixed region collapse and storm translation are essential for significant energy extraction by internal gravity waves to occur.

Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra

Abstract Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett–Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky–Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

Theory of the Eulerian tail in the spectra of atmospheric and oceanic internal gravity waves

Observed atmospheric and oceanic internal wave spectra, when analysed in an Eulerian frame of reference, exhibit a large-wavenumber ‘tail’. In one-dimensional vertical-wavenumber (k3) spectra, it is typically proportional to |k3|−3.In 1989, K. R. Allen and R. I. Joseph showed that a large-wavenumber tail was to be anticipated as a consequence of Eulerian nonlinearity, and they derived relations for the coefficients of both horizontal and vertical spectra of the form |k|−3. The coefficients were obtained only for the wave-induced vertical-displacement spectra, and only for an input spectrum having a certain ‘canonical’ frequency variation derived on other grounds.The present work builds on that of Allen & Joseph. It is more general in some respects, more limited in others. It provides a more transparent form of analysis, it treats a broad class of wave variables, and it does so for input (Lagrangian) spectra that can be chosen by the user, free from any constraint to canonical or other restricted forms. It provides relations whereby the full Eulerian spectrum may be determined numerically, once the input spectrum has been chosen, and it provides analytic forms applicable at large wavenumbers for horizontally isotropic spectra. The derived one-dimensional vertical-wavenumber spectra are discussed in relation to observations.Certain shortcomings in the development, both as given by Allen & Joseph and as found here, are identified and discussed.

Theory of the Eulerian tail in the spectra of atmospheric and oceanic internal gravity waves

Observed atmospheric and oceanic internal wave spectra, when analysed in an Eulerian frame of reference, exhibit a large-wavenumber tail'. In one-dimensional vertical-wavenumber (k 3 ) spectra, it is typically proportional to |k 3 | -3 . In 1989, K. R. Allen and R. I. Joseph showed that a large-wavenumber tail was to be anticipated as a consequence of Eulerian nonlinearity, and they derived relations for the coefficients of both horizontal and vertical spectra of the form |k| -3 . The coefficients were obtained only for the wave-induced vertical-displacement spectra, and only for an input spectrum having a certain canonical' frequency variation derived on other grounds. The present work builds on that of Allen & Joseph. It is more general in some respects, more limited in others. It provides a more transparent form of analysis, it treats a broad class of wave variables, and it does so for input (Lagrangian) spectra that can be chosen by the user, free from any constraint to canonical or other restricted forms. It provides relations whereby the full Eulerian spectrum may be determined numerically, once the input spectrum has been chosen, and it provides analytic forms applicable at large wavenumbers for horizontally isotropic spectra. The derived one-dimensional vertical-wavenumber spectra are discussed in relation to observations. Certain shortcomings in the development, both as given by Allen & Joseph and as found here, are identified and discussed.

Direct numerical simulation of the roll‐off range of internal wave shear spectra in the ocean

Oceanic internal gravity waves play an important role in the dynamics of the ocean, providing a link in the overall energy cascade from large forcing scales to small dissipation scales. Quantifying the cascade of energy available for mixing processes in the ocean interior, for example, is essential to accurate modeling of the oceanic general circulation. In the present study, based on an extensive numerical model that resolves most of the internal wave spectrum, the dynamics of the process by which internal waves dissipate their energy in the deep ocean are investigated. Putting all the calculated results together, we propose here a model for the dynamic balance of the internal wave spectrum wherein a downscale energy flux into the high vertical wavenumber near‐inertial portion of the spectrum is balanced by energy dissipation at critical layers formed by high vertical wavenumber near‐inertial flows.

Scattering of Oceanic Internal Gravity Waves off Random Bottom Topography

Abstract The scattering of oceanic internal gravity waves off random bottom topography is analyzed under the assumptions that (i) the height of the topography is smaller than the vertical wavelength and (ii) the slope of the topography is smaller than the wave slope. For each frequency, scattering redistributes the incoming energy flux in horizontal wavenumber space. The scattered wave field approaches an equilibrium state where the energy flux is equipartitioned in horizontal wavenumber space. For incoming red spectra, this implies a transfer from low to high wavenumbers. For typical internal wave and bottom spectra, about 6.8% of the incoming energy flux is redistributed. While this might be less than the flux redistribution caused by reflection off a critical slope, the scattering process transfers the energy flux to higher wavenumbers than the reflection process. Scattering might thus be equally or more efficient than reflection in causing high shears and mixing near the bottom.

Internal gravity waves and mixing

Oceanic internal gravity waves span a range between mesoscale eddies and smallscale turbulence, providing an important link in the overall energy cascade from the large scales of generation to the small scales of dissipation. Progress in the understanding of internal‐wave dynamics was discussed at the sixth ‘Aha Huliko'a Hawaiian Winter Workshop, held January 15–18, 1991, in Honolulu. The workshop brought together oceanographers and meteorologists, who came to the following conclusions on the topic. Considerable progress is being made in predicting the diapycnal mixing rates associated with the shear instability of small‐scale internal waves (Figure 1). Large‐scale internal waves are forced by the atmosphere and (possibly) other oceanic flows. These waves are the most energetic and can propagate over distances of O(1000 km).

Doppler sonar measurements from FLIP

A pulse-to-pulse incoherent Doppler sonar has been developed at the Marine Physical Laboratory for use in the observation of oceanic internal gravity waves. Mounted on the Research Platform FLIP, the sonar transmits at frequencies between 65 and 90 kHz at a peak power of −32 kW. During preliminary tests, measurements have been made out to a range of 1.6 km, depths of 1.2 km, with a range resolution of 25 m. The sensitivity of the sonar measurements to a given internal wave group depends on both the orientation of the sonar beam and the direction of propagation of the wave packet. Since the vertical angle of wave propagation is a function of frequency, a predictable pattern can occur in sonar derived estimates of the interal wave frequency spectrum. These can be used to infer a limited amount about the azimuthal directional properties of the wavefield. A more direct approach is to use multiple sonars, operating simultaneously but pointed in different directions. A four sonar system is currently being constructed for use on FLIP. This should permit accurate estimates of azimuthal directionality of waves up to kilometer scales.

Internal waves, finestructure, microstructure, and mixing in the ocean

Progress in measuring, interpreting, and understanding oceanic internal gravity waves and fine and microstructure is reviewed; we emphasize the quadrennium 1975–1978. The context is how these subjects contribute to oceanic mixing. The overlap between the areas is examined, as is the relevance of the subjects to other aspects of oceanography.Present trends and suggestions for future work are included, and we offer some speculation on possible progress during the next quadrennium, which may be substantial especially for finestructure understanding.

Internal waves in the ocean

Progress in measuring, describing, and understanding oceanic internal gravity waves during the quadrennium 1971–1974 is reviewed. Emphasis is given to trends in techniques and ideas that hold promise for the future. An extensive bibliography is provided.