Internal-inertial waves and crossfrontal circulation in the upper ocean (original) (raw)
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Observations of the Vertical Structure of Internal Waves
Journal of Geophysical Research, 1975
Velocity profiles in the open ocean reveal variations in the current structure throughout the water column. Superimposed on a smooth low-frequency shear profile are many layers of large time-variable shear. Repeated profiles at one location show that the time-dependent structure is dominated by rotary currents of diurnal-inertial period. Coherence calculated between profiles lagged in time indicates downward energy propagation. The kinetic energy of these internal waves varies with depth in a manner similar to that of the Brunt-Viisili frequency, but over a brief series of profiles there can be localized zones which are more energetic than might be expected. Based on velocity shear measured over 10-dbar intervals and a time mean Brunt-Viisili profile, Richardson numbers between 1/2 and 4 are observed over much of the water column. Simultaneous profiles are most similar at a separation of 100 m, gradually becoming more different for larger separations of up to 10 km. This paper briefly describes some aspects of internal waves as revealed by nearly continuous profiles of ocean velocity as a function of depth. Profiler methods are relatively new and only recently have been used extensively in the study of internal waves. It is more customary to observe water velocity at one location for an extended period of time than to obtain a velocity profile at one time over an extended depth interval. In many instances the lack of a long time series of observations at several depths may be more than compensated for by knowledge of the velocity structure throughout the whole water column. For example, in a wave field which is transient or spatially intermittent the complete profile allows near real time identification of zones of unusual or interesting activity. Also multiple or serial profiles are most suitable for deriving modal structure, vertical energy propagation, Richardson numbers, and the joint distribution of energy as a function of frequency and vertical wave number. Several instruments deployed to fall simultaneously permit the observation of spatial structure to the velocity field. Last, a mobile profile method is well suited to observing the structure within intense currents, such as the Gulf Stream, w,here moorings are diffi- cult to maintain. The velocity Profiles described here were collected as part of the Mid-Ocean Dynamics Experiment (Mode 1) in the spring of 1973. Mode 1 SOught to define the structure and evolution of a mesoscale eddy in the western North Atlantic. In order to describe the vertical structure of the low-freqUency eddy a con- siderable number of velocity profiles were obtained in a variety of locations. The data reported here are part of those collected and hence represent a preliminary presentation. The emphasis of the discussi øn is on the visual rather than the statistical aspects of internal waves. In this way the presentation seeks to increase awareness of the spatial and temporal variability of velocity profiles, to guide more statistical analysis of the data, and to stimulate further model formula- tion and theoretical Consideration of vertical structure. THE METHOD
Near-inertial waves in the ocean: beyond the
J Fluid Mech, 2005
The dynamics of linear internal waves in the ocean is analysed without adopting the 'traditional approximation', i.e. the horizontal component of the Earth's rotation is taken into account. It is shown that non-traditional effects profoundly change the dynamics of near-inertial waves in a vertically confined ocean. The partial differential equation describing linear internal-wave propagation can no longer be solved by separation of spatial variables; it was however pointed out earlier in the literature that a reduction to a Sturm-Liouville problem is still possible, a line that is pursued here. In its formal structure the Sturm-Liouville problem is the same as under the traditional approximation, but its eigenfunctions are no longer normal vertical modes of the full problem. The question is addressed of whether the solution found through this reduction is the general one: a set of eigenfunctions to the full problem is constructed, which depend in a non-separable way on the two spatial variables; these functions are orthogonal and form, under mild assumptions, a complete basis.
Deep Sea Research Part I: Oceanographic Research Papers, 2007
From ADCP measurements in the thermocline over the continental slope of the Bay of Biscay the vertical variation of the contribution of the inertio-gravity waveband to the kinetic energy and variance of the current shear was analysed. The semi-diurnal tides together with near inertial waves appeared to provide over 70% of the high-frequency kinetic energy (> 1 / 3 cpd). Over the vertical range of the ADCP bins, about 400 M, the phase of the M2 tide changed up to 155°, showing the importance of the contribution of internal waves to the observed tidal motion. Both semi-diurnal internal tidal waves and near-inertial waves were organized in wave beams with a limited vertical extent, probably about 50 to 60% of the vertical wavelength. The relatively large shear in the inertio-gravity wave band, supported an annual mean gradient Richardson number well below 1, and was probably capable of maintaining turbulent mixing for a large part of the time.
Near-inertial motions in the coastal ocean
1995
Internal-inertial waves are frequently observed in the upper ocean and below the thermocline. A three-dimensional general circulation model with turbulence-closure mixed layer is used to study the generation and propagation of near-inertial motion below the mixed layer. In particular, the problem of effect of a coastal wall on the wind induced inertial-internal wave field is re-examined using a fully nonlinear model. Responding to a wind pulse, a sharp wavefront propagates offshore. After the wavefront passage, strong near-inertial internal waves, marked by the tilting velocity isolines and the interface oscillations, are generated. The predicted near-inertial motion is consistent with the wave dispersion relation. Downward energy propagation occurs after the wavefront passage, and both kinetic and potential energy are strongly modified. After several inertial periods, the kinetic energy in the upper layer can be completely removed. The theoretical results, which are supported by observations, indicate that internal-inertial wave are important for mixing in the upper coastal ocean. 0924-7963/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO924-7963(94)00030-l
Near-inertial waves in the ocean: beyond the ‘traditional approximation’
Journal of Fluid Mechanics, 2005
The dynamics of linear internal waves in the ocean is analysed without adopting the 'traditional approximation', i.e. the horizontal component of the Earth's rotation is taken into account. It is shown that non-traditional effects profoundly change the dynamics of near-inertial waves in a vertically confined ocean. The partial differential equation describing linear internal-wave propagation can no longer be solved by separation of spatial variables; it was however pointed out earlier in the literature that a reduction to a Sturm-Liouville problem is still possible, a line that is pursued here. In its formal structure the Sturm-Liouville problem is the same as under the traditional approximation, but its eigenfunctions are no longer normal vertical modes of the full problem. The question is addressed of whether the solution found through this reduction is the general one: a set of eigenfunctions to the full problem is constructed, which depend in a non-separable way on the two spatial variables; these functions are orthogonal and form, under mild assumptions, a complete basis.
Internal waves across the Pacific
Geophys. Res. …, 2007
1] The long-range propagation of the semidiurnal internal tide northward from the Hawaiian ridge and its susceptibility to parametric subharmonic instability (PSI) at the ''critical latitude,'' l c = 28.8°N, were examined in spring 2006 with intensive shipboard and moored observations spanning 25 -37°N along a tidal beam. Velocity and shear at l c were dominated by intense vertically-standing, inertially-rotating bands of several hundred meters vertical wavelength. These occurred in bursts following spring tide, contrasting sharply with the downward-propagating, wind-generated features seen at other latitudes. These marginally-stable layers (which have inverse 16-meter Richardson number Ri 16 À1 = 0.7) are interpreted as the inertial waves resulting from PSI of the internal tide. Elevated near-inertial energy and parameterized diapycnal diffusivity, and reduced asymmetry in upgoing/ downgoing energy, were also observed at and equatorward of l c . Yet, simultaneous moored measurements of semidiurnal energy flux and 1-km-deep velocity sections measured from the ship indicate that the internal tide propagates at least to 37°N, with no detectable energy loss or phase discontinuity at l c . Our observations indicate that PSI occurs in the ocean with sufficient intensity to substantially
Internal gravity waves in the upper eastern equatorial Pacific: Observations and numerical solutions
Journal of Geophysical Research, 1997
On the basis of data froxn a towed thermistor chain collected near 140øW on the equator during April 1987, the zonal wavenumber and vertical structure of internal gravity waves were observed to vary significantly between wave events. Our hypothesis is that this variability is due to changes in the vertical structure of mean horizontal velocity and density. Assuming that the observed waves were the fastest growing modes for shear instability, we solve the Taylor-Goldstein equation, using different analytical basic states, including a zonal and meridional flow, to simulate the different conditions during 4 nights of intense internal wave activity. We find that while the observed waves are of finite amplitude, linear sheeu' instability is sufficient to explain the wavelength and vertical structure of vertical displacement for most of the waves. The fastest growing, unstable, mode-one solutions have e-folding growth times of less than 10 min. These solutions show wave phase speeds and vertical structures to be highly dependent upon the velocity structure of the uppermost 40 m. Near the base of the mixed layer at a flow inflection point the kinetic energy of the mean flow is extracted for wave growth. Wave vertical displacement is maximum near this inflection point. Zonal phase speeds range from-0.8 to-0.1 m/s. The propagation direction of waves with growth rates of 75% of the maximum growth rate can range from about 45 ø north to 45 • south of the zonal direction. The vertical wave-induced Reynolds stress divergence could explain a discrepancy in zonal momentum budgets of the upper 90 m of this region. Estimates of this stress divergence show that only about 100 days of intense internal wave activity is needed per year for these internal waves to explain estimated residuals of the mean zonal momentum budgets of this region at 50-to 100-m depth. eral mechanisms for this diurnal cycling have been considered, including diurnal cycles in solar heating, surface wind stress, and mean velocity shears, only internal waves can account for the cycle in turbulence extending well below the mixed layer. Indeed, the correspondence of the deeper mixing with the presence of internal waves subsequently was confirmed by two sets of independent observations. The first, a towed thermistor chain [
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
Resolving the horizontal direction of internal tide generation
Journal of Fluid Mechanics, 2019
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...
The Close Relationship between Internal Wave and Ocean Free Surface Wave
Journal of Marine Science and Engineering, 2021
A numerical model was used to simulate the propagation of internal waves (IW) along the surface layer. The results show that strong water exchange during IW propagation results in strong free surface flow and produces small but distinct free surface waves. We found a close relationship between the internal and ocean surface waves. Our intuitive reaction is that by training the relationship between the water surface wave height and the internal wave waveform, the internal wave waveform can be reversed from the water surface wave height value. This paper intends to validate our intuition. The artificial neural network (ANN) method was used to train the Fluent simulated results, and then the trained ANN model was used to predict the inner waves below by the free surface wave signal. In addition, two linear internal wave equations (I and II) were derived, one based on the Archimedes principle and the other based on the long wave and Boussinesq approximation. The prediction by equation (...