Energy Spectra of the Ocean’s Internal Wave Field: Theory and Observations (original) (raw)
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Hamiltonian Formalism and the Garrett-Munk Spectrum of Internal Waves in the Ocean
Physical Review Letters, 2001
Wave turbulence formalism for long internal waves in a stratified fluid is developed, based on a natural Hamiltonian description. A kinetic equation appropriate for the description of spectral energy transfer is derived, and its self-similar stationary solution corresponding to a direct cascade of energy toward the short scales is found. This solution is very close to the high wavenumber limit of the Garrett-Munk spectrum of long internal waves in the ocean. In fact, a small modification of the Garrett-Munk formalism includes a spectrum consistent with the one predicted by wave turbulence.
Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra
Journal of Physical Oceanography, 2010
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 non-rotating 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. We identify a small domain 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 which potentially permit a balance between the infra-red and ultra-violet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky-Raevsky spectrum.
On a spectrum of nonlinear internal waves in the oceanic coastal zone
Nonlinear Processes in Geophysics, 2007
This paper studies the internal wave band of temperature fluctuation spectra in the coastal zone of Pacific ocean. It is observed that on the central Mexican Pacific Shelf in the high-frequency band of temperature spectra the spectral exponent tends to ∼ω −1 at the time of spring tide and ω −2 at the time of neap tide. On the western shelf of the Japan/East Sea, in the ≪ω≪N * range, where N * is the representative buoyancy frequency and is the inertial frequency, the rate tends to ∼ω −3. These features of spectra are simulated by the model spectrum of nonlinear internal waves in the shallow water. Interaction of high-frequency internal waves with an internal wave of semidiurnal frequency is considered. It is shown that as a result of the interaction the spectrum of high-frequency internal waves take the universal form and the spectral exponent tends to ∼ω −1 .
Interpretations and observations of ocean wave spectra
Ocean Dynamics, 2010
The paper starts with a discussion of the linear stochastic theory of ocean waves and its various nonlinear extensions. The directional spectrum, with its unique dispersion relation connecting frequency (ω) and wavenumber (k), is no longer valid for nonlinear waves, and examples of (k, ω)-spectra based on analytical expressions and computer simulations of nonlinear waves are presented. Simulations of the dynamic nonlinear evolution of unidirectional free waves using the nonlinear Schrödinger equation and its generalizations show that components above the spectral peak have larger phase and group velocities than anticipated by linear theory. Moreover, the spectrum does not maintain a thin well-defined dispersion surface, but rather develops into a continuous distribution in (k,ω)-space. The majority of existing measurement systems rely on linear theory for the interpretation of their data, and no measurement systems are currently able to measure the full spectrum in the open ocean with high accuracy. Nevertheless, there exist a few low-resolution systems where data may be interpreted within a minimal assumption of a non-restricted (k,ω)-spectrum.
TOWARD REGIONAL CHARACTERIZATIONS OF THE OCEANIC INTERNAL WAVEFIELD
Reviews of Geophysics, 2011
Many major oceanographic internal wave observational programs of the last 4 decades are reanalyzed in order to characterize variability of the deep ocean internal wavefield. The observations are discussed in the context of the universal spectral model proposed by Garrett and Munk. The Garrett and Munk model is a good description of wintertime conditions at Site-D on the continental rise north of the Gulf Stream. Elsewhere and at other times, significant deviations in terms of amplitude, separability of the 2-D vertical wavenumber -frequency spectrum, and departure from the model's functional form are noted.
Intense short-period internal waves in the ocean
Journal of Marine Research, 2005
Trains of quasi-periodic high-frequency internal waves (IWs) of large amplitude are common in the upper thermocline of the ocean. Sources for these waves may be different ones but it is not always possible to experimentally establish them for certain. We analyzed results of many IW experiments carried out in different representative regions of the World Ocean, including continental margins in the Mid-Atlantic Bight, in the northwestern Pacific at Kamchatka, the Seyshelles-Mascarene bottom rise, and some regions of the open ocean where the intense short-period IWs occur. Comparative analysis of the intense IWs observed in the Mid-Atlantic Bight and at Kamchatka revealed similarity and difference in the IW field in these regions differing by their bottom topography. Most of the observed trains in the Mid-Atlantic Bight propagate shoreward from the shelf break in the form of soliton packets or solibores and do not occur seaward from the shelf. The soliton trains in the northwestern Pacific at Kamchatka are common not only at the shelf edge but also in deep water where they propagate in various directions that seem to be related to the supercritical steepness and complicated form of the continental slope. Observation of generation and evolution of the IW trains at the Seyshelles-Mascarene bottom rise where huge internal solitons have been encountered has shown that the undular bore generated at the lee side of the bottom rise gradually evolves in a train of solitons with the trailing linear waves. Large solitons are generated also in deep water as a result of ray propagation of the internal tide emanated from the rise as happens in the Bay of Biscay. Certain consequences of the IW interaction with the background current leading to intensification of the high-frequency waves were observed in several regions of the open ocean. Revealed dependency of the intense wave propagation direction on the current direction, and closeness of the wave frequency to the frequency at which the waveguide steeply tapers may be regarded as clear evidences for the important role which currents play in the IW intensification.
Journal of Geophysical Research, 1993
Statistics of high-frequency (0.2-0.5 cph) fluctuations are derived from moored upper ocean measurements of currents and temperatures at four latitudes spanning the equator along 140øW. Some of the more unusual statistics include (1) nonunity ratios of kinetic energy to potential energy; (2) nonunity ratios of zonal to meridional kinetic energy; (3) nonzero current-temperature coherence amplitudes, with depth-dependent phases; and (4) high vertical coherence amplitudes, With approximately 180 ø phases, between current measurements spanning the thermocline. A simple model of •hear-modified internal waves is employed to gain insight into the causes of the latitudinal variability of the statistics. Much of this variability can be attributed to the vertical advection of significantly different mean vertical shears by a spectrum of internal waves. The statistics also suggest that the spectrum of high-frequency internal waves in the upper equatorial Pacific differs in important ways from canonical spectral models. The statistics are consistent with a model based on vertical modes which neglect advection by the mean flow, provided the energy in the first mode is much less than (about 0.3 funes) that in the spectrum described by and Munk (1981) and two to four times as much energy propagates eastward as westward. Some of the statistics are inconsistent with the simple internal wave model examined, possibly indicating contamination by mooring motion. 0.5 cph) with canonical models of linear internal waves in an ocean with no mean currents. As a first step toward rationalizing the spectral statistics, an interpretation in terms o f stable, shear-modified internal waves is attempted under the extreme, simplifying assumption that the waves are neithei' advected by the mean flow nor refracted in the vertical by the mean shear but only vertically advect the mean flow. Predictions from this model in various shear flows are compared with observations in order to identify the unique features of the data which cannot be explained by an isotropic spectrum of no-mean-flow modes (per Garrett and Munk [1972, 1975, 1979] and Munk [1981], hereinafter collectively referred to as GM) and to gain insight into which features can be explained by kinematic modifications of the wave field and which features require modification of the energy spectrum. The empirically based GM spectral model of the deepocean internal wave field depends on the following assUmptions: (1) the no-mean-flow linear internal wave dispersion relation is valid; (2) the internal Wave field is vertically symmetric and horizontally isotropic with respect to energy propagation; (3) the distinction between a mode and the sum of upward and downward propagating waves Of equal energy is not important; and (4) the internal wave energy spectrum is separable into a function of frequency times a function of wavenumber, where the function of frequency is indepen-18,089 18,090 BOYD ET AL.: HIGH-FREQUENCY INTERNAL WAVES IN THE EQUATORIAL PACIFIC dent of wavenumber and the function of wavenumber is independent of frequency except for a bandwidth scale factor. Statistics of the internal wave field derived from the GM model spectrum fit numerous observed statistics in the deep ocean quite well, while not matching others (e.g., vertical coherence of velocity components). Some of the model-data inconsistencies have been postulated to be due to fine structure and noise contamination [e.g., Miiller et al., 1978], while others appear to be related to the presence of sources and sinks of energy, especially near boundaries (for example, see the review by Olbers [1983]). Despite these known inadequacies of the GM spectrum, it nevertheless remains a well-known and useful benchmark against which observed internal wave characteristics can be compared in searching for unusual behavior. Several of the GM model assumptions are expected to be violated in the strong vertically sheared near-surface zonal mean flows at the equator along 140øW, including the following: (1) the linear dispersion relation will be modified by the vertical curvature of the mean flow and by Doppler shifting in the intrinsic frequency (Doppler shifting of downstream waves to intrinsic frequencies, i.e., in mean-flow coordinates, that are lower than fixed coordinate frequencies results in a failure of linearization near the critical layer depths where the intrinsic frequency approaches zero); (2) the mean zonal currents may impose directional asymmetry on the energy spectrum through critical layers involving east-west propagating waves; and (3) in the neighborhood of the ocean surface, a fixed phase relationship between upward and downward propagating components is expected. Adding to these contradictions of the GM assumptions, the fact that linear internal waves advect the mean flow readily leads to the expectation that many of the observed statistics of fluctuations at internal wave frequencies in the upper equatorial Pacific will differ significantly from predictions derived from the GM spectrum. Following brief descriptions of the data and mean-flow conditions at the equator (sections 2 and 3, respectively), we introduce our internal wave model in section 4. We present statistics derived from the observations in section 5 and the analysis of those statistics in section 6. This analysis proceeds on the hypotheses (1) that the recorded fluctuations of temperature and velocity are due to internal wave motions only and (2) that a single internal wave energy spectrum can explain observations from all of the moorings between 3øS and 1.5øN. Our model makes a simplifying approximation to the internal wave modes which retains the kinematic effects of internal waves in mean shear flows while ignoring the dynamical modifications to the wave vertical structure. These modes are combined using a GM-like separable spectrum to which modifications (suggested by comparison with the observed statistics) are made to the mode number and azimuthal dependences of the spectrum; the frequency dependence is not addressed here, since we average over a small part of the internal wave frequency band in this paper. The shear-modified modes that we have avoided here have been calculated by Boyd [1989], who shows that the low vertical modes are not strongly affected by the shallow, energetic equatorial mean flows. Boyd's [1989] spectral model using the shear-modified modes validates the basic conclusions reached with the simpler model presented here. Specifically, the empirical statistics suggest that the internal wave energy spectrum must have much less mode 1 energy than in the GM spectrum and that the spectrum must be east-west asymmetric. Section 7 contains a discussion of the comparison between model and data statistics, and section 8 summarizes the conclusions about the wave field based on those statistics. Some of the statistics do not appear to be in agreement with the simple wave model presented, possibly indicating contamination by mooring motion. 2. DATA The Tropic Heat experiment (see Eriksen [1985a] for an overview) included a period of intensive measurements from approximately November 1984 to June 1985, during which time four tautly moored surface floats supported current meters and temperature/pressure gauges at fixed depths in the upper 300 m of the equatorial ocean along 140øW at nominal latitudes of were deployed by R. Knox and D. Luther, and the mooring at the equator was deployed by D. Halpern under the auspices of both Tropic Heat and the National Oceanic and Atmospheric Administration's EP-OCS program. The 1.5øN, 1.5øS, and 3øS moorings were instrumented with vector-averaging current meters (VACMs) and vector-measuring current meters (VMCMs) that stored data every 8 or 15 min. The Draper Laboratory temperature/pressure recorders on these moorings sampled at 16-rain intervals. On the equatorial mooring, temperature recorders and VACMs recorded at 15-min intervals. The available measurements are listed in Table 1. For details of the data-editing procedures, see work by Halpern et al. [1988] and R. A. Knox et al. (manuscript in preparation, 1993, hereinafter referred to as Knox et al., 1993). The focus of this study is on the "high-frequency" fluctuations, where 0.2 cph -< w -< 0.5 cph. The lower limit was chosen in order to avoid potentially deterministic tidal energy, and the upper limit was chosen because the VACM response appears to be different from the VMCM response for •o > 0.5 cph at 1.5øN and 1.5øS. Furthermore, currents and temperatures observed in the 0.2-to 0.5-cph band at midlatitudes in the deep ocean have been shown [Miiller et al., 1978] to be dominated by a spectrum of internal waves that is more horizontally isotropic, vertically symmetric, and free from contaminating fine structure than any other frequency band where free internal waves exist. For the record durations shown in Table 1, the 0.2-to 0.5-cph band has substantial degrees of freedom, containing between 416 and 1579 Fourier transform harmonics; for most records the band contains more than 1000 harmonics. Conductivity-temperature-depth (CTD) profiles were obtained by D. Halpern near the equatorial mooring site when moorings were deployed and recovered. For the Tropic Heat period from October 1983 to October 1985, seven CTD profiles are available from the 0øN, 140øW site. 3. MEAN CONDITIONS The mean upper ocean currents in the central equatorial Pacific during non-El Nifio years are well known [e.g., Firing et al., 1981]. Figure la shows the average zonal velocity profiles for the Tropic Heat data listed in Table 1. The 3øS mooring is in a region of relatively low vertical shear (0.25 cm s-• m -x , maximum) typical of the southern hemisphere BOYD ET AL.' HIGH-FREQUENCY INTERNAL WAVES IN THE EQUATORIAL PACIFIC 18,091
Features of the nonlinear internal wave spectrum in the coastal zone
Geophysical Research Letters, 2005
This paper studies two examples of internal wave band temperature fluctuation spectra from moored instruments on the central Mexican Pacific Shelf and the western shelf of the Japan/East Sea. It is observed that for band f (w (N, where N is the buoyancy frequency and f is the inertial frequency, the spectral falloff rate with frequency w tends to w À3. These features of spectra are simulated by the model spectrum of nonlinear internal waves in the shallow sea. It is shown that nonlinear internal waves with frequencies f (w (N are governed by the modified simple wave equation. This equation underlie of the model spectrum of nonlinear internal waves on the shelf. The model spectrum shows w À3 falloff rate with frequency in contrast with model spectrum proposed with Garrett and Munk, which shows w À2 falloff rate.
Weak Turbulence in Ocean Waves
2002
Further develop weak turbulence theory, which is used to predict the evolution of spectral energy density in ocean surface waves and internal waves. Isolate, identify and quantify sources of possible discrepancies between numerical solutions of weak turbulence modeling and observations of ocean waves. Obtain theoretical predictions of forms of steady state spectral energy distributions for surface and internal ocean waves.
Toward regional characterizations of the oceanic internal wavefield. in preparation
2007
Many major oceanographic internal wave observa-tional programs of the last four decades are reanalyzed in order to characterize variability of the deep ocean internal wavefield. The observations are discussed in the context of the universal spectral model proposed by Garrett and Munk. The Garrett and Munk model is a good description of wintertime conditions at Site-D on the continental rise north of the Gulf Stream. Elsewhere and at other times, significant deviations in terms of amplitude, separability of the 2-D vertical wavenumber- frequency spectrum, and departure from the model’s functional form are reported. Specifically, the Garrett and Munk model overesti-mates annual average frequency domain spectral levels both at Site-D and in general. The bias at Site-D is associated with the Garrett and Munk model being a