1. Introduction
The currents associated with the semidiurnal barotropic tides are nearly orthogonal to the topography of the Hawaiian Ridge, setting the stage for intense barotropic–baroclinic conversion in the central Pacific Ocean (Rudnick et al. 2003). The ridge is also a relatively isolated topographic feature, enabling studies of the generation, dissipation, and propagation of the internal tide with limited interference from competing, nonlocal phenomena.
Using a global tidal model constrained by Ocean Topography Experiment (TOPEX)/Poseidon (T/P) altimeter data, Egbert and Ray (2000, 2003) estimated that 18–25 GW of tidal energy is lost from the M2 barotropic tide along the ridge. Of the converted energy, some fraction is dissipated in boundary layers immediately adjacent to the topography, some propagates to middepths prior to wave instability and subsequent mixing, and some escapes the proximity of the conversion site, propagating into the pelagic ocean. Ray and Mitchum (1996, 1997) detected the sea surface height signature of the first-mode internal tide in T/P data. They found that the first-mode internal tide is somewhat spatially coherent and can propagate over great distances (>1000 km). They observed a slow decay in amplitude away from the Hawaiian Ridge that can be due to either (or a combination of) the internal tide losing energy or its phase becoming more variable in space and time (losing coherence). TOPEX/Poseidon-based estimates of the radiated baroclinic energy flux range from the Ray and Mitchum (1996) rough estimate of 15 GW to 5.4 GW in Kang et al. (2000) and 6 GW in Ray and Cartwright (2001). The uncertainty is still very large. Using a tomographic array located 1000 km north-northeast of Oahu, Dushaw et al. (1995) also found a large baroclinic energy flux coming from the ridge.
This paper focuses on the propagating baroclinic energy, attempting to quantify the energy flux within the generating region, where the spatial structure of the flux is complex, and 450 km offshore, along the path of the propagating waves. We discuss two series of in situ measurements of the internal wave field obtained from the Research Platform (R/P) Floating Instrument Platform (FLIP) as an aspect of the Hawaii Ocean Mixing Experiment (HOME).
In recent years, E. Kunze and colleagues (Kunze et al. 2002; Althaus et al. 2003) have pioneered the use of profiling instruments to estimate energy fluxes. With several profiles per day, they define a mean velocity and density field, against which the perturbation velocities and relative isopycnal displacement are computed. Perturbation pressure is calculated from integrating the hydrostatic equation. This technique has proven to be remarkably effective in regions in which the semidiurnal internal tide dominates. In more complex regions, where fluxes at several wave frequencies are anticipated, a more intensive time series approach is required.
Using the R/P FLIP, we have developed the capability to profile the oceanic velocity and density fields from the surface to approximately 800-m depths at 4-min intervals. The high sampling frequency and multiweek duration of the FLIP observations enable estimates of energy flux continuously in frequency with minimal statistical error. In addition to providing a first look at the frequency dependence of the flux at Hawaii, the FLIP measurements can provide valuable error bars for the flux estimates obtained by colleagues with more limited time sampling. A description of the FLIP observations is given in section 2, followed by a display of the internal wave energy flux spectrum (section 3). The vertical structure (section 4) and time variability (section 5) of the energy fluxes are discussed with a focus on the semidiurnal and diurnal frequency bands, providing insight to the generation and propagation of the internal tide. Our observations also suggest the existence of a nonlinear interaction that transfers energy from low-mode semidiurnal waves to higher-mode waves at one-half of their frequency.
2. Data
a. Sites
Data were obtained during two 6-week cruises of the R/P FLIP, one at an active internal-wave generation site (Nearfield) and the other 430 km offshore (Farfield), along the anticipated propagation path of the baroclinic tide (Fig. 1).
During the autumn 2002 Nearfield program, FLIP was trimoored in the Kauai Channel between Oahu and Kauai, at the southwest edge of the Kaena Ridge (21.68°N, 158.63°W). Water depth at this site is 1100 m. The Kaena Ridge is one of the most active regions of internal tide generation found in the HOME Survey program (Rudnick et al. 2003). Despite the strong tidal currents, the 3-point mooring maintained FLIP’s position within 500 m.
The current ellipses of the M2 and K1 barotropic tides obtained from the TPXO.5 regional tidal model (Egbert and Erofeeva 2002) are shown in Fig. 2. The current vectors at time = 0 (0000 UTC 1 January 2001) and the sense of rotation along the ellipse are indicated in black. Note that the barotropic currents are very polarized, especially the K1 tide. The barotropic currents of the M2 and K1 tides rotate clockwise at all locations in and around the Kauai Channel. The strongest currents are seen over the ridge, particularly near the FLIP mooring site. At 21.75°N, 158.5°W, the amplitude of the M2 currents is roughly 2.5 times that of the K1 currents.
The interference of other tidal components (primarily the S2 and O1) modulates the semidiurnal and diurnal currents in a fortnightly cycle; the current range in amplitude is shown in Fig. 2 as the shaded ellipses. On top of the ridge, the maximum currents during spring tide are about 3 times the maximum currents during neap tide.
The Farfield (autumn 2001) site is located 430 km southwest of Oahu, approximately along the expected line of internal tidal propagation from the Kauai Channel. The R/P FLIP was (single point) moored for 40 days at 18.39°N, 160.70°W, on the northern edge of the North Equatorial Current. The water depth at this site is about 5200 m. FLIP’s position varied by as much as 6 km, under the influence of the mean and inertial currents, as well as an intense mesoscale eddy that advected the platform northward toward the end of the cruise.
The along- and cross-ridge directions define a right-handed coordinate system aligned with the ridge (respectively −30° and +60° with respect to the east). Throughout this paper, x (and u for currents) refer to the along-ridge direction and y (and υ) refer to the cross-ridge direction.
b. Instruments
In both field programs, temperature, conductivity, and density profiles were obtained at 4-min intervals from the surface to 800 m. Two Sea-Bird Electronics, Inc., SBE11 CTDs were used, each profiling a 400-m vertical range at 3.6 m s−1 (Fig. 3). Sensors were sampled at 24 Hz.1 The Nearfield dataset consists of over 12 000 profiles, collected over 36 days. For the Farfield, close to 10 000 CTD profiles were obtained, spanning 29 days.
To reduce salinity spiking, the response of the conductivity sensors was matched to that of the temperature sensors using a technique described in Ferrari and Rudnick (2000). Physically reasonable functional forms for the responses of the sensors were obtained by fitting the phase of the conductivity and temperature cross-spectrum in a region of small salinity variation. Pressure, temperature, conductivity, and density were subsequently processed following Anderson (1993). The profiles obtained by the upper CTD overlapped those measured by the lower instrument by about 20 m at depths around 400 m, enabling the cross calibration of upper and lower profilers. After response corrections and calibration, the vertical resolution of density structure was slightly better than 2 m.
Cruise-averaged (respectively, 36- and 29-day averages) profiles of buoyancy frequency for the Nearfield and the Farfield are shown in Fig. 4. Both profiles peak at values around 10 cycles per hour (cph) at 100 m, dropping exponentially with depth to reach 2 cph at 800 m. Note that the mixed layer in the Nearfield is slightly shallower than in the Farfield. Seasonal temperature and salinity profiles from the World Ocean Atlas 2001 (Stephens et al. 2002; Boyer et al. 2002) have been used to complete the profiles from 800 m to the seafloor.
An eight-beam Doppler sonar (the Deep-8) was deployed at a depth of 400 m in both experiments. It measured velocities in the same range profiled by the CTDs (0–>800 m). This sonar had four beams oriented upward (170 kHz) and four beams facing down (140 kHz). Velocity profiles were recorded with 4-m depth resolution and 30-s temporal resolution. Repeat sequence codes (Pinkel and Smith 1992) were transmitted, with a bandwidth of ±8 kHz. Ocean velocities were estimated by combining the Doppler velocities into east, north, and up components, taking into account the tilt and rotation of the sonar as well as the slow drift velocity associated with the motion of FLIP. Baroclinic velocities were estimated by subtracting modeled barotropic tidal velocities (M2, S2, K1, O1, N2, K2, P1, and Q1) from the sonar/GPS estimate of absolute velocity using the TPXO.5.1 model (Egbert 1997; Egbert and Erofeeva 2002).
c. Depth–time maps
Representative data collected during each of the cruises are presented in Fig. 5. Depth–time maps of velocity and density (smoothed by 1 h in time and 6 m in depth) are presented for a 3-day period around the spring tide.2
The baroclinic velocities observed at both sites are large relative to the barotropic velocity that forces them. The Nearfield (Fig. 5a) data are dominated by velocities and isopycnal displacements of semidiurnal period and long (full ocean depth) vertical scales. Downward phase (upward energy) propagation with time is evident, consistent with local topographic generation. In contrast, a broader range of frequencies is present in the Farfield (Fig. 5b). Energetic velocity signals are seen at periods of 24 h and longer, as well as in the semidiurnal band (especially near the surface). Isopycnal displacement also displays strong semidiurnal and diurnal variance.
Depth–time maps of the semidiurnal root-mean-square (rms) displacement for both deployments are shown in Figs. 6a,b. Vertical displacements of over 100 m (40 m rms) are observed at depths of 600–800 m over periods of few days during spring tides in the Nearfield and become very small during neap tides. Similar variability is seen in the Farfield.
The semidiurnal and diurnal motions have strikingly different vertical structures. This is seen by bandpass filtering the velocity and displacement fields at these specific frequencies (total bandwidth of 1/80 cph). Maps of the rms isopycnal displacement in the diurnal (Figs. 6c,d) band show considerable variability in both depth and time, at both sites. The semidiurnal variability (Figs. 6a,b) reflects the dominance of very large scale vertical motions.
To provide a more quantitative analysis, Farfield modal fits to vertical displacement can be obtained using temperature data from a tomographic mooring located just 30 km from the FLIP site (B. Dushaw and P. Worcester 2004, personal communication). A fit to data obtained at depths of 150–1900 m indicates, in the semidiurnal band, that mode 1 accounts for over 60% of the variance, and the first three modes account for almost all of the variance. In the diurnal band, the first three modes account for less than one-half of the variance, with mode 1 accounting for less than 10%. A similar contrast between the semidiurnal and diurnal bands is observed in the Nearfield, although the presence of well-defined beams of tidal energy (section 4) suggests that a discussion in terms of vertical modes is not adequate.
3. Energy flux as a function of frequency
In the ideal case, perturbation pressure p′(z) could be estimated from the FLIP density profiles by integrating the hydrostatic equation from the sea surface to fixed depth z for each of the ∼104 density profiles and subsequently subtracting the temporal mean. However, in field measurements, pressure is used as a surrogate for depth and a different approach is required (appendix A; Kunze et al. 2002). It is necessary to estimate the perturbation to sea surface elevation induced by the baroclinic waves themselves. For the low modes, the elevation signal represents a significant fraction of the perturbation pressure. To estimate this signal, we fit theoretical modal forms to the data over the ∼750-m observation window and calculate the contribution to surface displacement from each mode (appendix A). Perhaps a surprise is that, even in the Farfield, where the observations span only one-half of the “WKB stretched” water column, the fits are robust. Surface elevation effects are negligible for all but the first three modes.
The magnitude of the depth-integrated energy flux density vector (F2x + F2y)1/2 is shown in Figs. 7a,b. Note that this is not the magnitude of the cross-spectrum but rather is the magnitude of the vector defined from the along- and cross-ridge components (the cospectrum of p and u, and that of p and υ). Individual components are also shown (Figs. 7c,d). Absolute values of the energy flux density in each direction are plotted, with the cross-ridge component indicated by red (toward 60°) or black (240°). In a similar way, the along-ridge energy flux density is plotted in blue (150°) or green (330°). The upper limit of integration is chosen to be 80 m in both cases to avoid the noisy velocities associated with the surface reflection of sidelobes of the sonar and to stay out of the mixed layer, where isopycnal displacements (and thus p′) are not well defined. In a similar way, deep isopycnals are sometimes displaced below the range of the measurements, making it difficult to estimate displacements. The deepest isopycnal staying at all times in the domain sampled by the CTDs conservatively dictates the lower limit. This limit is chosen to be 730 m in the Farfield. In the Nearfield, the semidiurnal flux reverses direction at middepth. Here, the energy flux density was integrated to 500 m to capture only the southward-propagating tidal energy and to avoid cancellation by the deeper northward-propagating beam. Vertical profiles of energy flux are presented in section 4.
Inertial ( f ), diurnal (D1), semidiurnal (D2), and 6-h period (D4) motions are evident in the spectra (Fig. 7). The semidiurnal band clearly dominates the energy flux density. Other frequencies are, however, present, and at significant levels. The energy fluxes contained in each band are listed in Tables 1 (Nearfield) and 2 (Farfield). appendix B describes the estimation of relative errors. Higher harmonics (D4) are virtually absent in the Nearfield but carry a significant energy in the Farfield (10% of semidiurnal flux). There is also a large energy flux in the diurnal band at both sites (respectively, 13% and 19% of semidiurnal flux).
From this set of observation, the energy flux in the Nearfield does not appear to be much larger than that at the Farfield. This is misleading, since the Nearfield site is on top of the ridge, inshore of a downward-propagating beam generated at the sharp slope break on the south flank of the ridge (Merrifield et al. 2001; Nash et al. 2006). The Nearfield observations thus capture only one of the two beams that propagate south-southwest toward the Farfield.
An underlying continuous spectrum of flux magnitude appears to be present in Fig. 7 in addition to the specific bands described above, in particular at higher frequencies. The magnitude of this “continuum flux” is in part due to the bias in the cospectral magnitude, which is necessarily ≥0 even when the true coherence between p and u is zero. The relatively short length of our deployments makes it difficult to determine whether there is a nonzero true coherence for frequencies above M2.
In Fig. 8, the energy flux density (Figs. 7c,d) of all frequencies lower than ω is integrated as a function of ω, showing the net contributions of each frequency. Energy flux integrated over both the shallow (80–500 m) and deeper (500–700 m) depth ranges are shown for the Nearfield, and all of the resolved depth range (80–730 m) has been used in the Farfield. The final values (at ωmax) correspond to the total values listed in Table 1 and Table 2.
The energy fluxes associated with the diurnal, semidiurnal, and D4 bands described above, and their relative amplitudes, are evident in the cumulative integrals shown in Fig. 8. The diurnal and semidiurnal frequency bands carry most of the energy. Some hint of inertial energy fluxes can be seen; the inertial band is well resolved in the Farfield but is not distinguishable from the diurnal band in the Nearfield. The first M2 superharmonic (D4) is only present in the Farfield, suggesting generation during the propagation of the principal M2 internal tide.3 In the Farfield, high frequencies appear to carry a small but constant energy flux to the east. This is also evident from the bias in the cospectrum magnitude (Fig. 7f). The small flux originates from a robust 〈pu〉 correlation of high-frequency, long-vertical-wavelength waves that is maintained for periods of 1–2 days at a time. In particular near the buoyancy frequency, it seems probable that high frequencies carry a real nonzero energy flux. The signal at D4 and higher frequencies seen in the FLIP measurement is the focus of a future paper.
In the context of HOME, the remainder of this paper focuses on the semidiurnal and diurnal frequency bands.
4. Vertical structure of the energy flux
a. Vertical profiles
The Nearfield cross- and along-ridge flux profiles are presented in Figs. 9 and 10. These are obtained by integrating the cospectra [Cx(ω, z), (Cy(ω, z)] over the diurnal and the semidiurnal frequency bands. The total integration bandwidth is 1/80 cph, centered around ½M2 (D1) and M2 (D2). The profiles give the averaged direction and magnitude of the energy flux over the duration of the cruises.
The semidiurnal energy flux (Fig. 9a) is directed toward the southwest in the upper 500 m and reverses for depths greater than 500 m. It is apparent that separate beams of tidal energy are generated on each flank of the ridge, crossing in the middle (Fig. 9a). Because FLIP was moored just south of the center, the observed energy flux is asymmetric. The depths at which the beams are observed are consistent with linear ray theory (dashed lines). In a similar way, a deep southward diurnal flux is observed (Fig. 9b), possibly corresponding to a ray originating from the north flank. The flux in this band has a more complex vertical structure than does the semidiurnal flux. Note that the flux in the diurnal band is a factor of 10 smaller than in the semidiurnal band.
In the along-ridge direction, the Kauai Channel bathymetry is relatively more complex than in the cross-ridge direction (Fig. 10). Moving westward from Oahu (Kaena Point), the bottom rapidly drops to 1000 m. Several topographic features between 600 and 1200 m populate the ridge for about 70 km. It then drops to nearly 3000 m in a channel separating the islands of Oahu and Kauai. The semidiurnal energy flux in the along-ridge direction is much smaller than in the cross-ridge direction. There are suggestions of beams generated near the bottom.
Diurnal internal tide characteristics cannot link the energy flux observed in the upper water column to any local topography, with the exception of the upper flanks of Oahu and Kauai. Although there is a strong component of flux toward Oahu near the surface, the principal direction of diurnal propagation is also cross ridge. Nonlinear generation of the diurnal motions in the mid- and upper water column over the ridge is a possibility.
In the Farfield, the fluxes are directed away from the ridge in both diurnal and semidiurnal bands. The cross-ridge semidiurnal flux profile (Fig. 11a) has lost its beamlike character and exhibits instead a nearly exponential decay with depth, consistent with the structure of a primarily mode-1 flux. The Farfield diurnal flux is also quasi exponential but shows more depth variability than the semidiurnal band. Note here the factor-of-5 difference in scale between the bands, not the factor of 10 used in the Nearfield.
b. Vertical wavenumber spectra
As shown in section 2c, the vertical structure of isopycnal displacement in the semidiurnal and diurnal bands is very different. The semidiurnal internal waves are mostly low mode, but higher modes have more importance in the diurnal band. The same is true for the energy fluxes, although 〈p′u〉 emphasizes the low wavenumbers. Indeed, since both p′ and u have red spectra (dominated by low wavenumbers), varying respectively as k−4z and k−2z for vertical wavenumbers less than 1/10 cpm in the Garrett–Munk model spectrum (Gregg and Kunze 1991), the cross-spectrum of p′ and u is also likely to be a red spectrum.
Vertical-wavenumber cross-spectra of p′ and u can be calculated from the FLIP data to obtain energy flux density as a function of vertical wavenumber (of mode number). The magnitude of the energy flux density vector [(C2x + C2y)1/2] in the Farfield as a function of vertical wavenumber is shown in Fig. 12. Note that this is not the vertical spectrum of the energy flux but rather is the energy flux associated with each vertical wavenumber (or mode). Prior to calculating the cross-spectra, the velocity and perturbation pressure field have been bandpassed around the ½M2 (D1) and M2 (D2) frequencies with a bandwidth of 1/80 cph. Perturbations p′ and u have been WKB normalized and stretched to compensate for the effects of nonuniform buoyancy frequency. The average of the buoyancy frequency over the range of the FLIP observations has been used (N0 = 4 cph), leading to equivalent bottom depth HWKB = 1410 m. For each frequency band, Cx(t, kz) and Cy(t, kz) are calculated and then averaged in time to form the mean cospectral magnitude estimates shown in Fig. 12 for the diurnal (gray) and semidiurnal (black) bands. Wavenumbers are expressed in terms of vertical modes (kz × 2HWKB). Spectra are initially estimated at 2–16 dof. Additional stability is then gained by averaging in time (there are 5–15 independent measurements for the semidiurnal band, increasing with mode number; see appendix B), leading to a larger ndof.
Using a least squares fit, modes can be fitted to the pressure and velocity fields derived from the FLIP data, and the energy flux associated with mode n can be directly estimated using 〈p′nun〉. The results of these calculations (three modes on p and υ) are shown in Fig. 12. The dots joined by thin lines represent the energy flux normalized by the width of the wavenumber band associated with a single mode [δkz = (2HWKB)−1]. The semidiurnal mode 1 dominates the spectra and carries almost an order of magnitude more energy than diurnal mode 1. At higher wavenumbers, the pressure and velocity have roughly the same spectrum levels in both bands, leading to the same energy flux density (although more energetic diurnal velocity at vertical wavenumber 100–250 m can be seen).
For reference, the mode dependence of the energy flux generated by a knife edge, calculated simply by matching an oscillating barotropic M2 flow and satisfying the boundary conditions at an infinitely narrow ridge (St. Laurent et al. 2003) is plotted in Fig. 12 as the thin gray line. During the Nearfield program, J. Nash (2004, personal communication) found good agreement between the M2 energy flux at the 3000-m isobath (about 20 km from the top of the ridge) associated with the first 20 modes and the knife-edge model of internal tide generation by St. Laurent et al. (2003). His observations were obtained from free-falling profilers, and the energy flux was estimated from a technique described in Kunze et al. (2002). The low modes seem to follow the slope predicted by St. Laurent et al. (2003), but modes 4 and higher (continuous line) fall more quickly than in this simple model. This is consistent with the conceptual model that higher modes do not propagate far, being dissipated or dispersed before reaching the Farfield.
The normalized cross-ridge cospectrum [Eq. (5) but as a function of wavenumber] shows little coherence at mode numbers above 10 (Fig. 12b). A similar quantity can be calculated for each mode, where 〈p′nυn〉 is compared with (〈pn′2〉〈υ2n〉)1/2, where 〈〉 is an average over time and depth. This quantity is shown in Fig. 12b as the solid circles. Perturbation pressure and velocity have a high coherence and are in phase for the first semidiurnal mode (0.8), but higher modes are uncorrelated. Diurnal p′ and υ show an “in-phase coherence” of about 0.5 for the first two modes, before dropping to near zero. The spectral level observed at high mode numbers is presumed to be due to the bias in the estimate of the cospectral magnitude.
c. Farfield full-ocean depth energy flux
Knowing the amplitude of each vertical mode, the energy flux can be obtained from the surface to the bottom. Modes have been fitted successively (i.e., mode 1 is fit first, then mode 2 on the remainder, and so on) to the semidiurnal and diurnal p′(t, z), u(t, z), and υ(t, z) fields measured from FLIP between 80 and 730 m (bandpass filtered, with a bandwidth of 1/80 cph). The cruise-averaged values of the diurnal and semidiurnal depth-integrated (over the entire water column, 0–5200 m) energy flux for modes 1, 2, and 3 are listed in Table 3. The total energy fluxes integrated over the entire water column are about twice the 80–730-m integral. Note that the total of the first three modes (modes 1–3) is a vector sum and therefore is not equal to the sum of the magnitudes of each mode.
In the diurnal band, modes 1–3 are on average directed away from the ridge, with modes 1 and 2 both being equally important (as seen in Fig. 12). In contrast, the total energy flux in the semidiurnal band is strongly dominated by mode 1, with the magnitude of mode 2 being about 20% that of mode 1. The direction of mode 2 is almost parallel to the ridge (directed southeast), and its time variability does not follow an offset fortnightly cycle. The results from the numerical model of Merrifield et al. (2001),4 which predicts a depth-integrated M2 energy flux of 1759 W m−1, are very close to the mode 1 flux (1739 W m−1) directly measured from FLIP. The Farfield measurements contribute to the validation of the model.
5. Temporal variability of the energy fluxes
It is possible to resolve slow temporal variations of the energy flux by computing the cross-spectrum over shorter time intervals. In a compromise between resolution and statistical precision, blocks of 4 days have been chosen. Flux spectral densities are integrated over frequency bands and in depth. The widths of the integration bands are chosen to be 1/40 cph, centered around ½M2 (D1) and M2 (D2) frequencies.
To quantify the forcing, the magnitude of the barotropic tidal cross-shelf current on top of the ridge (21.70°N, 158.50°W) is plotted in Fig. 13a for both the diurnal and semidiurnal bands. Barotropic currents are obtained from the TPXO.5 tidal model (Egbert 1997; Egbert and Erofeeva 2002). As expected, a big fortnightly cycle is evident in the semidiurnal forcing. The interference of the K1 and O1 tides also creates a 14-day cycle, shifted significantly in phase relative to its semidiurnal counterpart. Maximum currents in the diurnalbands are smaller by about a factor of 3 than their semidiurnal counterparts. Note that, in particular in the semidiurnal band, not all spring tides are the same.
For the Nearfield, the 80–500-m depth-integrated energy flux vectors are shown in Fig. 13, with the 500–700-m flux (the deep northward beam) given in Fig. 14. The origin of the arrows indicates the center of the 4-day window. A significant fraction of the northward-propagating semidiurnal flux might occur below our observation window and be missed in this calculation. The time variability of the semidiurnal flux follows the cycle of the forcing, with values reaching up to 3 kW m−1 at spring tide and vanishing at neap tide. The semidiurnal time-averaged flux agrees well with the modeled M2 flux (Merrifield et al. 2001). However, given the large variability in the flux magnitude, it is critical to account for the timing in the fortnightly cycle when interpreting flux estimates obtained over only a few days.
At times when the diurnal forcing is the largest (around yearday 285), large fluxes directed toward the southwest can be seen at depth (Fig. 14c). These are associated with the beam observed in Fig. 9 and possibly result from the direct generation of diurnal internal waves by the K1 barotropic tide. Big eastward fluxes correlated with the diurnal forcing function can also been seen in the upper water column. Topographic features on Kaena Ridge northwest of FLIP are too close and too deep to be the origin of this flux (Fig. 10), but it is possible that barotropic currents interacting with topography between 1000 and 1500 m near Kauai generate diurnal internal waves propagating eastward.
In addition to the signal linked with the diurnal forcing, large diurnal fluxes are also seen at times of big semidiurnal fluxes but small diurnal forcing. This is particularly true in the upper water column (Fig. 13) but can also be seen in Fig. 14 where southward diurnal flux can be distinguished when the semidiurnal forcing is large.
The Farfield depth-integrated energy flux vectors are presented in Fig. 15. The integration is from 80 to 730 m, and frequency bands of 1/40 cph centered on each tidal are used, as before.
The Farfield semidiurnal energy flux (Fig. 15b) is directed away from the ridge and shows a surprisingly strong variability. It follows the fortnightly cycle in the semidiurnal forcing, but it effectively disappears, and even reverses in some occasions. Note that maximum fluxes are seen roughly 2 days after maxima in the semidiurnal forcing at the ridge, corresponding to the time required by mode 1 to propagate from the ridge to the Farfield.
The diurnal flux (Fig. 15c) also shows a strong time variability, but it does not appear to follow the diurnal forcing at the ridge. Strong diurnal energy fluxes away from the ridge are seen when the semidiurnal energy flux is big. This observation suggests that the Farfield diurnal band is strongly linked to the semidiurnal baroclinic tide.
6. Summary
With continuous depth–time records of displacement and velocity, the baroclinic energy flux is calculated as a function of frequency and depth. By computing the cospectra of p′ and u, large internal wave energy fluxes are found to exist at tidally significant frequencies in both the Nearfield and the Farfield. In addition to providing a first look at the frequency dependence of the flux, the FLIP measurements provide valuable error bars for flux estimates obtained with more limited time sampling. Several tidal beams are evident in the Nearfield observations, whereas the energy fluxes reveal mode-like structures in the Farfield. In the Nearfield, the FLIP measurements capture the upward-propagating beams of internal wave energy generated on the flanks of the ridge but not those propagating downward and contributing to the large fluxes observed off ridge (Rudnick et al. 2003; Lee et al. 2006).
The averaged depth-integrated M2 energy flux (over the entire water column) in the Farfield is 1.7 ± 0.3 kW m−1, directed away from the ridge (mode 1 only). However, values measured in the course of the 6-week cruise include depth-integrated fluxes up to 4 kW m−1 and even small fluxes toward the ridge at neap tide. Energy fluxes in the diurnal frequency band represent about 15%–20% of the semidiurnal energy flux at both sites. Higher harmonics (D4) are virtually absent in the Nearfield but carry a significant energy flux in the Farfield (10% of semidiurnal flux).
We see excellent agreement between the direct observations and the predictions of the Princeton Ocean Model run by Merrifield et al. (2001). Observations are generally consistent with the Nash et al. (2006) single-day full-depth estimates near the ridge.
The semidiurnal energy flux at both sites follows the fortnightly cycle of the semidiurnal barotropic tide. However, in particular in the Farfield, the magnitude of the diurnal energy flux depends on the presence of semidiurnal internal waves rather than on the diurnal forcing at the ridge. The linkage between the semidiurnal and diurnal energy fluxes is suggestive of a nonlinear energy transfer, either at the ridge itself or along the propagation path. Parametric subharmonic instability (PSI) has been identified as a candidate nonlinear interaction that transfers energy from a wave of frequency ω to a wave at ½ω with a higher vertical wavenumber (e.g., M2 to ½M2). However, initial calculations of the time scale of this interaction yield values that are too slow (over 100 days) to be relevant here (Pomphrey et al. 1980; Olbers and Pomphrey 1981). In the classical application of PSI to the internal-wave problem, the modulation of the buoyancy frequency by the driving wave forces the subharmonic growth. The Farfield semidiurnal tide is primarily first mode and engenders only slight modulation of N 2. Near the ridge, order-1 variation in buoyancy frequency is seen immediately above the topography. In recent nonlinear numerical models (Hibiya et al. 2002; MacKinnon and Winters 2003, 2005), a fast energy transfer between semidiurnal and diurnal frequency bands (specifically M2 to ½M2) is being seen. However, the Farfield diurnal fluxes are borne primarily by the first three modes, and contemporary models indicate that the primary PSI response is at much smaller vertical scales. Additional analysis combining the different HOME datasets, coupled with further numerical study, is required to understand the mechanism of this energy transfer.
Acknowledgments
We thank Eric Slater, Mike Goldin, Mai Bui, Jerry Smith, Lloyd Green, and Tyler Hughen for their help in the design, construction, deployment, and operation of the sensors used during this experiment. We acknowledge Captain Golfinos and the crew of the R/P FLIP for their excellent support. We also thank the HOME principal investigators for numerous discussions, the National Science Foundation, and the Office of Naval Research.
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APPENDIX A
Estimation of Perturbation Pressure
APPENDIX B
Error in Cospectral Estimates
The final depth-averaged spectra are used in Eq. (B2) to compute the relative errors. Following this technique, the relative errors in the 80–500-m depth-integrated energy fluxes for the Nearfield (Table 1) are around 15% and the relative errors in the Farfield 80–730-m values are roughly 10% (Table 2).
Location of the Nearfield and Farfield sites (stars), near the Hawaiian Ridge. The thin black line indicates the 1000-m contour. Depth-integrated energy flux vectors from a numerical model (Merrifield and Holloway 2002) are also indicated.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Barotropic current ellipses in the Kauai Channel for (a) the M2 and (b) the K1 barotropic tides (gray line). The shading represents the range of the currents during the fortnightly cycle (due to the interference with the S2 and O1 tides, respectively). The black lines represent the current vector at time = 0. The sense of rotation is also indicated. Bathymetric contours are plotted every 1000 m, and the Nearfield site is indicated by the star.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Schematic diagram of CTDs and Doppler sonar aboard FLIP.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Mean buoyancy frequency profiles from observations and extrapolated using the World Ocean Database, for the Farfield (black line) and the Nearfield (gray line): (a) the complete water column for the Farfield and (b) the top 1100 m (domain of the Nearfield).
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Three-day depth–time maps of velocity and displacement for the (a) Nearfield and (b) Farfield. Colors indicate cross-ridge velocity, and black contours are the isopycnals, whose mean depths are separated by 50 m.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Depth–time maps of the rms displacements in the (top) semidiurnal and (bottom) diurnal bands for the (a), (c) Nearfield and (b), (d) Farfield. The phases of the moon are indicated on the upper axis.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Depth-integrated energy flux spectral density in the (left) Nearfield and (right) Farfield. (a), (b) The magnitude of the flux density, (c), (d) the along- and cross-ridge components, and (e), (f) the normalized cospectra of p′ and υ (see text) are plotted. Energy flux density is integrated from 80 to 500 m in the Nearfield and from 80 to 730 m in the Farfield. Frequencies corresponding to inertial ( f ), diurnal (D1), semidiurnal (D2), and 6-h periods (D4) are indicated. In (c) and (d), the directions of the energy flux density along the cross-ridge (black or red) and along-ridge (blue or green) axes are indicated. The cospectrum magnitudes are shown in (e) and (f), together with an estimate of the expected bias for zero coherence.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Cumulative integrals of energy flux densities (cospectra) shown in Figs. 7a,c, from 0 to 0 < ω < ωmax plotted vs ω for (a) the Nearfield [depth integrated from 80 to 500 m (solid lines) and 500 to 700 m (dashed lines)] and (b) the Farfield (from 80 to 730 m).
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Cross-ridge profiles of the energy flux in the (a) semidiurnal and (b) diurnal bands in the Nearfield. The topography of the ridge is shown in black. Rays showing the propagating paths of M2 and K1 waves generated at the critical slopes are also shown. Note the factor-of-10 change of scale between fluxes in (a) and (b).
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Along-ridge profiles of the energy flux in the (a) semidiurnal and (b) diurnal bands in the Kauai Channel. The topography of the ridge (Kauai on the left, Oahu on the right) is shown in black. Rays showing the propagating paths of M2 and K1 waves generated at the critical slopes are also shown. Note the factor-of-10 change of scale between fluxes in (a) and (b).
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Cross-ridge profiles of the energy flux in the (a) semidiurnal and (b) diurnal bands in the Farfield. The topography of the bottom is shown in black. The vertical structure in both bands is close to that of mode 1, although greater depth variability is seen in (b). Note the different scale factor between (a) and (b).
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
(a) Magnitude of the Farfield energy flux density (C2x + C2y)1/2 as a function of vertical mode number for the semidiurnal (black line) and diurnal (gray line) bands. Energy flux density [energy flux normalized by δkz = (2HWKB)−1] associated with the first three modes is also shown. The time required for an M2 internal wave to propagate from the ridge to the Farfield is shown by a dashed line (scale on right). Energy fluxes for the first 300 modes generated from a knife ridge (St. Laurent et al. 2003) are shown by thinner gray lines. (b) Normalized cross-ridge cospectra [Eq. (5)] and mode in-phase coherences in the same bands (see text). Expected bias for zero coherence is shown by the dashed line.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Nearfield depth-integrated (80–500 m) energy flux vectors as a function of time in the (b) semidiurnal and (c) diurnal bands. (a) The forcing (cross-ridge currents at the ridge) in the diurnal (gray line) and semidiurnal (black line) bands.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
As in Fig. 13, but for 500–700 m. The scale of (b) is a factor of 5 smaller than in Fig. 13b.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
As in Fig. 13, but for the Farfield (80–730 m). Vectors are plotted in an east/north coordinate frame. Note the different scales for each band.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Fig. A1. (a) Surface pressure for each mode, expressed as a surface displacement. (b) Offset in the surface displacement associated with each mode when the integration constant is not calculated over the entire water column. All modes are assumed to have a 10-m normalized amplitude.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Fig. A2. Representative perturbation pressure profiles for (a) the Nearfield and (b) the Farfield. Dashed lines are the results of the integral of the displacement [Eq. (A2)], and the thick black lines are the (adjusted) p′ profiles. The baroclinic pressure profiles associates with modes 1 and 2 are shown in gray, and the thin black lines are the sum of the modes.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Fig. B1. Number of independent observations (a) when averaging the displacement (black) and velocity (gray) frequency spectra over all depths (80–820 m) and (b) when averaging the displacement (black) and velocity (gray) vertical wavenumber spectra over 21 days. This is calculated for the Nearfield.
Citation: Journal of Physical Oceanography 36, 6; 10.1175/JPO2882.1
Depth-integrated (80–500 m) energy flux for the Nearfield. Positive fluxes are toward −30° (along ridge) and 60° (cross ridge), where the angles refer to directions with respect to east. Relative errors are about 15%.
Depth-integrated (80–730 m) energy flux for the Farfield. Positive fluxes are toward −30° (along ridge) and 60° (cross ridge). Relative errors are about 10%.
Magnitude of the depth-integrated (surface to bottom) energy flux in the Farfield. Relative errors are estimated to be less than 20% for the semidiurnal mode 1.
Microconductivity cells were also aboard each profiler, sampled at 96 Hz. Data from these sensors will be discussed separately.
Throughout this paper, times are expressed in terms of yeardays, defined to be zero at 0000 UTC 1 January.
Lamb (2004) has recently studied the generation of harmonics numerically, finding significant cross-frequency transfer at surface and seafloor reflections of the M2 tide. Williams (1985) found significant bicoherence between D4 and D2 in isopycnal displacement observations off California. He concluded that part of the D4 signal is associated with a forced wave.
The internal tide at the Farfield extracted from the numerical model is effectively only mode 1. The model is run for 4 days, with the forcing ramping up during the first day. The M2 baroclinic signals are obtained from fitting harmonics on the last day. Because mode 2 takes 3.25 days to propagate to the Farfield from the Ridge (in contrast to 1.75 days for mode 1), only mode 1 has time to propagate fully through the 430 km separating the Farfield from the ridge.
