Observations of Tidal Internal Wave Beams at Kauai Channel, Hawaii

S. T. Cole Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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D. L. Rudnick Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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B. A. Hodges Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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J. P. Martin Applied Physics Laboratory, University of Washington, Seattle, Washington

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Abstract

To observe the across-ridge structure of internal tides, density and velocity were measured using SeaSoar and a Doppler sonar over the upper 400–600 m of the ocean extending 152 km on each side of the Hawaiian Ridge at Kauai Channel. Eighteen sections were completed in about 18 days with sampling intentionally detuned from the lunar semidiurnal (M2) tide so that averaging over all sections was equivalent to phase averaging the M2 tide. Velocity and displacement variance and several covariances involving velocity and displacement showed one M2 internal wave beam on each side of the ridge and reflection of the beams off of the surface. Theoretical ray slopes aligned with the observed beams and originated from the sides of the ridge. Energy flux was in agreement with internal wave generation at the ridge. Inferred turbulent dissipation was elevated relative to open ocean values near tidal beams. Energy flux was larger than total dissipation almost everywhere across the ridge. Internal wave energy flux and dissipation at Kauai Channel were 1.5–2.5 times greater than at the average location along the Hawaiian Ridge. The upper 400–600 m was about 1/3 to 1/2 as energetic as the full-depth ocean. Tidal beams interact with each other over the entire length of the beams causing gradients along beams in almost all covariances, momentum flux divergences, and mean flows. At Kauai Channel, momentum flux divergences corresponded to mean flows of 1–4 cm s−1.

Corresponding author address: Sylvia Cole, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Drive, Mail Code 0213, La Jolla, CA 92093-0213. Email: stcole@ucsd.edu

Abstract

To observe the across-ridge structure of internal tides, density and velocity were measured using SeaSoar and a Doppler sonar over the upper 400–600 m of the ocean extending 152 km on each side of the Hawaiian Ridge at Kauai Channel. Eighteen sections were completed in about 18 days with sampling intentionally detuned from the lunar semidiurnal (M2) tide so that averaging over all sections was equivalent to phase averaging the M2 tide. Velocity and displacement variance and several covariances involving velocity and displacement showed one M2 internal wave beam on each side of the ridge and reflection of the beams off of the surface. Theoretical ray slopes aligned with the observed beams and originated from the sides of the ridge. Energy flux was in agreement with internal wave generation at the ridge. Inferred turbulent dissipation was elevated relative to open ocean values near tidal beams. Energy flux was larger than total dissipation almost everywhere across the ridge. Internal wave energy flux and dissipation at Kauai Channel were 1.5–2.5 times greater than at the average location along the Hawaiian Ridge. The upper 400–600 m was about 1/3 to 1/2 as energetic as the full-depth ocean. Tidal beams interact with each other over the entire length of the beams causing gradients along beams in almost all covariances, momentum flux divergences, and mean flows. At Kauai Channel, momentum flux divergences corresponded to mean flows of 1–4 cm s−1.

Corresponding author address: Sylvia Cole, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Drive, Mail Code 0213, La Jolla, CA 92093-0213. Email: stcole@ucsd.edu

1. Introduction

Vertical mixing in the deep ocean, which keeps the ocean stratified and helps to maintain global overturning circulation, is primarily accomplished by the dissipation of internal waves. Internal waves are forced by basin-scale winds and tides and dissipate energy to small-scale turbulence. Tidal and wind dissipation are estimated to be of roughly equal importance to maintaining open ocean stratification (Munk and Wunsch 1998; Wunsch and Ferrari 2004; Garrett and Kunze 2007). We focus on the tidal energy cascade, from basin-scale barotropic tides to internal waves to turbulence, because tides have discrete frequencies that are convenient to observe. The lunar semidiurnal (M2) tide is the dominant frequency of the barotropic tide over most of the ocean.

Internal waves of tidal frequency, or internal tides, are generated by the barotropic tide flowing over topography, propagate as internal wave beams, and dissipate over large distances. Flow of the barotropic tide over any topographic slope, such as continental shelves, ridges, and seamounts, generates internal tides. The amount of energy transferred to internal tides is largest for topography occupying a large fraction of the water column, topographic slope equal to or exceeding internal wave slope, and strong barotropic tides (Holloway and Merrifield 1999; Khatiwala 2003; Llewellyn-Smith and Young 2003; Munroe and Lamb 2005; Legg and Huijts 2006). Theoretical, numerical, and laboratory studies have shown that internal tides propagate as internal wave beams, which are spatially compact and coherent regions of energy with slopes that depend on frequency (Mowbray and Rarity 1967; Prinsenberg et al. 1974; Sutherland et al. 2000; Merrifield and Holloway 2002; Llewellyn-Smith and Young 2003; Khatiwala 2003; St Laurent et al. 2003; Legg and Huijts 2006). Internal tides are susceptible to dissipation and energy transfer to other frequencies, particularly at surface and bottom boundaries, the thermocline, or when intersecting other internal wave beams (Thorpe 1998; Sutherland 1999; Davies and Xing 2003; Lamb 2004; Tabaei et al. 2005; Gerkema et al. 2006a,b). Internal tide dissipation occurs over hundreds to thousands of kilometers (Dushaw et al. 1995; Ray and Mitchum 1996; Egbert and Ray 2001; Lozovatsky et al. 2003). Large-scale velocity and density fields vary over this distance, changing the direction of wave propagation and decreasing coherence along a beam (Müller 1976; Rainville and Pinkel 2006b).

Internal wave beams generated by tidal flow over topography have been observed. Full depth measurements with coarse horizontal resolution provide evidence of M2 tidal beams over 50–100-km horizontal distances (Pingree and New 1989; Pingree and New 1991; Althaus et al. 2003). Finely spaced horizontal measurements have captured 5–15-km segments of M2 tidal beams (Lueck and Mudge 1997; Petruncio et al. 1998; Lien and Gregg 2001; Kitade and Matsuyama 2002; Lam et al. 2004) and segments at Kauai Channel, Hawaii, of 50 km (Martin et al. 2006) and 20 km (Nash et al. 2006). The previous studies have collectively observed internal wave beams in several quantities including displacement and velocity magnitude, internal wave phase, turbulent kinetic energy density, energy flux, velocity shear, and turbulent dissipation.

We will show observations of internal wave beams along one section across the Hawaiian Ridge. The Hawaiian Ridge is an efficient generator of internal tides because of steep topography and barotropic tidal flows perpendicular to the ridge. The Hawaii Ocean Mixing Experiment (HOME) focused on how internal tide properties varied along and across the ridge. The locations with largest internal tide energy and dissipation were French Frigate Shoals, west of Nihoa Island, and Kauai Channel (Merrifield and Holloway 2002; Klymak et al. 2006; Lee et al. 2006; Martin et al. 2006). This paper uses observations at Kauai Channel, the most geographically convenient location with enhanced energy and dissipation. One section across the ridge was repeatedly observed with timing intentionally detuned from the M2 tide so that the M2 tide could be phase averaged. A representative picture was obtained of the across-ridge structure of internal wave propagation and dissipation over a substantial potion of a tidal beam.

The remainder of this paper is constructed as follows: Section 2 presents the observations. Section 3 discusses velocity variance, displacement variance, velocity shear, and turbulent dissipation. The relationship between the tidal beams and topography is determined. Section 4 focuses on covariances, including energy flux, momentum flux, and interactions between tidal beams. The relationships between internal wave energy density, energy flux, and dissipation are discussed. Conclusions are presented in section 5.

2. Observations and methods

From 2 to 19 October 2002, SeaSoar and a Doppler sonar were used to measure velocity and density along one section across the ridge. The 50-kHz Doppler sonar gave usable velocity data from 45 to 585 m. SeaSoar is a towed platform that cycled from 20 to 400 m in less than 12 minutes and had a horizontal resolution finer than 3.3 km. The M2 frequency was the dominant frequency at the Hawaiian Ridge, and the desired horizontal coverage was one wavelength of a mode-one M2 internal wave, 150 km, on each side of the ridge. With a towing speed for SeaSoar of 4 m s−1, less than 90 km could be traveled in half an M2 period and tidal frequencies were not resolved. A range of tidal phase was observed by completing 18 sections along a 304-km track (Fig. 1). The center of the cruise track was roughly the midpoint of the ridge. The sections were intentionally timed to detune from the M2 tide. The sampling pattern was successful, with observations within 152 km of the ridge crest approximately equally spaced in tidal phase (Fig. 2). Averaging over all sections (i.e., averaging in the vertical in Fig. 2) approximates averaging over the M2 phase.

SeaSoar and Doppler sonar observations were processed according to Martin et al. (2006). The 24-Hz raw SeaSoar density measurements were averaged into one-second records, and then SeaSoar and Doppler sonar data were averaged into 12-min profiles. Profiles had an 8.6-m constant vertical spacing. Horizontal spacing averaged over all profiles was 2.81 km with a standard deviation of 0.22 km.

An across-ridge (x positive toward northeast), along-ridge (y positive toward northwest), vertical (z positive upward) coordinate system is aligned with the direction of the cruise track, 34.2°T. The cruise track was chosen based on topography and models to coincide with the direction of maximum energy flux (Merrifield and Holloway 2002). A sample section during spring tide shows that across-ridge velocity was stronger than along-ridge velocity and both were largest within 100 km of the ridge (Fig. 3). Velocity was surface intensified but commonly greater than 10 cm s−1 at all depths. Large internal waves were observed throughout the section. The M2 internal wave ray paths calculated from linear theory are shown and discussed further in section 3.

We perform an average over all sections and into 8-km horizontal bins, maintaining the 8.6-m vertical spacing. The average is denoted by angle brackets (e.g., 〈υ〉, where υ is the along-ridge velocity). Density bins from 45 to 330 m and all velocity bins contained 40–55 observations within 152 km of the ridge. All bins with less than 25 observations were ignored. Perturbations from the average are denoted as primed variables, for example, υ′ = υ − 〈υ〉. Displacement, ζ′, is the deviation of each isopycnal from its mean depth. Barotropic fields were not removed from the perturbation fields in any way. Averaged quantities have time scales larger than 18 days. Perturbations have time scales shorter than 18 days and include semidiurnal, diurnal, and inertial motions. Mean buoyancy frequency, 〈N2〉, was largest at the base of the mixed layer, which was in the upper 100 m, and decreased below the mixed layer base (Fig. 4). Mean density, 〈ρ〉, had horizontal density gradients, which were presumably in geostrophic balance as they persisted for at least 18 days.

3. Variance

a. Velocity and displacement variance

Total velocity variance, 〈(u′)2 + (υ′)2〉, and displacement variance, 〈(ζ′)2〉, had some similarities in across-ridge structure (Fig. 5). Velocity and displacement variance were largest on either side of the ridge crest (x = 0 km) and approximately symmetric in pattern and magnitude about the ridge crest. Both had a minimum over the ridge crest that was widest near the surface. Below 400 m, velocity variance was elevated over the ridge crest. Velocity variance was surface intensified and displacement variance was large deeper in the water column because density surfaces are displaced by a larger amount where N2 is smaller. Many regions of elevated velocity and displacement variance coincided with each other. There were locations where either velocity or displacement variance was elevated, such as at ±125 km.

The M2 ray slopes coincided with regions of elevated velocity and displacement variance. The ray slope |μ|, a consequence of the internal wave dispersion relation, is
i1520-0485-39-2-421-e1
with constant Coriolis parameter, f, and 2π/ω = 12.42 h for the M2 frequency. A distinction will be made between the terms tidal beam and ray path or ray slope. Ray path or ray slope will refer to the curve calculated from (1). Tidal beam will refer to features of velocity or density fields that coincide with a ray path. Tidal beams have a width and amplitude while ray paths do not. The M2 ray slopes were placed in the horizontal to coincide best with total velocity variance (Fig. 5a). These ray paths are repeated in subsequent figures placed at the same location to facilitate comparisons between figures and with theoretical M2 ray slopes. The region of low variance directly over the ridge corresponded to the region between the ray paths (Figs. 5a,b). Regions with largest variance were consistent with M2 ray paths and are M2 tidal beams. Both velocity and displacement variance showed M2 tidal beams and their reflection off of the surface. Considering individual sections (e.g., Fig. 3 during spring tide), ray slopes agreed well with observed features. Locations where either velocity or displacement variance were elevated, such as ±125 km, were far from the M2 ray paths and did not correspond to tidal beams as variances were mainly vertically coherent. Only M2 beams were apparent, even though velocity and displacement variance included all frequencies less than 18 days. The M2 frequency could have been dominant for two reasons: 1) M2 motions had larger amplitudes than other motions, such as diurnal and near-inertial motions, and 2) the sampling pattern allowed for effective averaging over M2 phase but not over diurnal or near-inertial phase.

Ray paths were extended below 355 m to determine their relationship with the topography. Ray slopes below 355 m were calculated using the mean buoyancy frequency of HOME (Klymak et al. 2006). Ray paths intersect the sides of the ridge (Fig. 6), consistent with internal wave generation at the ridge (Baines 1982; Holloway and Merrifield 1999; Lamb 2004; Petrelis et al. 2006).

b. Velocity shear and turbulent dissipation

Mean-square vertical shear of horizontal velocity, 〈(uz)2〉 + 〈(υz)2〉, was elevated near the ridge. Velocity shear was largest in the upper ocean and decayed away from the ridge at all depths (Fig. 7a). Beams in velocity shear were harder to distinguish than in velocity variance (Fig. 5a). Mean-square shear from mean velocity was less than one-tenth of mean-square shear from velocity perturbations.

Velocity shear is used to parameterize the turbulent dissipation rate ε and the diapycnal eddy diffusivity Kρ. Along the Hawaiian Ridge, dissipation from microstructure measurements agreed better with parameterizations from Gregg (1989) (within a factor of 2 for Kρ) than with Gregg et al. (2003) both with and without shear-strain ratio and latitude factors (Martin and Rudnick 2007). From Gregg (1989),
i1520-0485-39-2-421-eq1
i1520-0485-39-2-421-e2
where N0 = 5.24 × 10−3 s−1 is a reference buoyancy frequency, SGM is the total variance in vertical shear for the Garrett–Munk (GM) spectrum at wavelengths less than 10 m (Cairns and Williams 1976; Gregg and Kunze 1991), and SOBS2 is the observed vertical shear (Fig. 7a) multiplied by a correction factor to account for averaging from the Doppler sonar (Martin and Rudnick 2007).

Dissipation and diffusivity were elevated relative to open ocean values (Figs. 7b,c). Maximum values below 95 m were 1.3 × 10−7 W kg−1 and 3.6 × 10−4 m2 s−1 compared with open ocean values of 10−10 W kg−1 and 10−5 m2 s−1. Both quantities decayed to open ocean values tens of kilometers from the tidal beams. Tidal beams were most apparent in Kρ (Fig. 7c) because of appropriate scaling by N2.

c. Variance ratios

Energy density and shear-strain ratios give information about the frequency content of internal waves. For a hydrostatic two-dimensional inviscid plane wave or internal wave, denoted by caret variables (e.g., υ̂), energy density and shear-strain ratios are
i1520-0485-39-2-421-e3
which is 1.34 for the M2 frequency, where (1/2)ρ0N2〉〈(ζ̂)2〉 is potential energy density, (1/2)ρ0〈(û)2 + (υ̂)2〉 is kinetic energy density, and 〈N2〉〈(ζ̂z)2〉 is vertical strain (Fofonoff 1969; Lien and Müller 1992); ρ0 = 1026 kg m−3 is a reference density used throughout this analysis. Energy density and shear-strain ratios have a value of 3 for a GM spectrum of internal waves (Munk 1981). Observations of internal tides at the Hawaiian Ridge (Lee et al. 2006; Martin et al. 2006) and other locations (Toole et al. 1997; Kunze et al. 2002) have corresponded to the M2 frequency for both ratios in (3).

The observed shear-strain ratio showed that the M2 frequency was dominant. A scatterplot of vertically integrated shear versus strain clustered around the M2 ratio from (3) (Fig. 8). There was little variation in the ratio, even though locations where beams had been observed (within 68 km of the ridge, Figs. 5 and 6) and where beams had not been observed (68–152 km from the ridge) were included, as well as locations north and south of the ridge. Integrated shear and strain was largest where beams were observed. The energy density ratio (not shown) also clustered around the M2 ratio, but had more scatter as it contained larger vertical scales. Larger vertical scales include variability such as geostrophic motions and barotropic tides in addition to internal waves. The M2 energy density and shear-strain ratios were consistent with our observations of M2 tidal beams and other HOME observations showing increased energy at the M2 frequency (Aucan et al. 2006; Rainville and Pinkel 2006a).

4. Covariances

Tidal beams were evident in several covariances (Figs. 9 and 10). Covariances involving displacement, 〈uζ′〉, 〈uζx〉, and 〈uxζ′〉, showed beams with equal magnitudes along all portions of the ray paths (Figs. 9a and 10a,b). Beams in 〈uυx〉 and 〈uxυ′〉 were thinner and more variable and had strongest magnitudes along the portions of ray paths closest to the ridge (Figs. 10c,d). Covariances 〈uζ′〉, 〈uυx〉, and 〈uxυ′〉 had different signs north and south of the ridge (Figs. 9a and 10c,d). Covariances 〈uζx〉 and 〈uxζ′〉 changed signs where the beams reflected off of the surface (Figs. 10a,b), while 〈υζ′〉 was smaller in magnitude than 〈uζ′〉 and did not have clear tidal beams (Fig. 9). Tidal beams in other covariances, such as 〈uζz〉 or 〈uυz〉, were evident but not as uniform as those in Figs. 9 and 10. These phase-averaged observations clearly show tidal beams and changes in sign at the ridge and upon reflection from the surface.

To interpret covariances, internal tides at Kauai Channel are modeled as linear, hydrostatic, two-dimensional, and inviscid. Nonlinear terms are neglected as the relevant nondimensional parameter was at most 0.01 (Martin et al. 2006). The hydrostatic approximation is valid as ω2 / N2 = O(10−4). Neglecting dissipation and along-ridge gradients, the basic equations of motion are
i1520-0485-39-2-421-e4a
i1520-0485-39-2-421-e4b
i1520-0485-39-2-421-e4c
i1520-0485-39-2-421-e4d
i1520-0485-39-2-421-e4e
with the bottom boundary condition
i1520-0485-39-2-421-e4f
where P* is the pressure perturbation, h(x) is the boundary, (ũ, ) are barotropic velocities, and asterisks distinguish theoretical fields, which are baroclinic, from observed perturbation fields, which include barotropic fields (Gill 1982). If motions with a single frequency are averaged over a complete cycle, covariances are independent of time: 〈υ*υ*〉t = (1/2)〈υ*υt*〉 = 0, for example. Motions consistent with (4) and appropriately averaged over a tidal cycle will have 〈u*υ*〉 = 0 [multiply (4b) by υ* and average] and 〈u*υx*〉 = − 〈ux*υ*〉. Observed 〈uυ′〉 was small in magnitude and did not have clear tidal beams (not shown), and only minor differences between 〈uυx〉 and − 〈uxυ′〉 were found (Figs. 10c,d), indicating that the dynamics of (4) are sensible and that the observations and averaging procedure were adequate.

The barotropic tide is included in the observed perturbation fields but showed no clear effect on covariances. For a barotropic tide of the form ũ = U0 cos(ωt), 〈ũũ〉 is a background field with no beam pattern. Covariances between baroclinic fields and ũ, 〈ũζx*〉, for example, have beam patterns and may differ in sign from covariances with u*. Magnitudes of u* and ũ are equal at the top of a ridge. As barotropic transport U0h(x) is constant because of mass conservation, ũ becomes smaller than u* as water depth increases, and covariances involving the barotropic tide would have magnitudes dependent on water depth. The observed covariances 〈uu′〉, 〈uζ′〉, 〈uζx〉, and 〈uυx〉 (Figs. 5a, 9a, 10a, 10c) and variance ratios (section 3c; Fig. 8) showed no obvious dependence on water depth, so the barotropic tide did not apparently affect covariances. Observed covariances are interpreted as resulting from the baroclinic tide.

Velocity components parallel and perpendicular to the direction of wave propagation are uncorrelated. Along the cruise track, 34.2°T, the correlation of u′ with υ′ was 0.1 ± 0.3 over 95–585 m. Rotated along 31.7°T, the correlation was 0.0 ± 0.3 over 95–585 m. North and south of the ridge, zero correlation was obtained for directions of 36.8°T and 27.0°T, respectively. The direction of internal wave propagation was within eight degrees of the cruise track.

a. Covariance 〈u′ζ′〉 and interactions between tidal beams

Tidal flow over steep topography generates upward and downward beams that can interact and covary with each other. Following Llewellyn-Smith and Young (2003), a streamfunction is introduced into (4), where ũ = U0 cos(ωt), (u*, w*) = ( −ψz,ψx),

and
i1520-0485-39-2-421-e5
For a knife-edge ridge at x = 0 with height h0,
i1520-0485-39-2-421-eq2
i1520-0485-39-2-421-e6
where X = πx/μH and Z = πz/H are nondimensional variables, n = 1,2,3… ∞ is a mode number, and H is total water depth (Llewellyn-Smith and Young 2003). The coefficients An are determined from boundary conditions. The spatial components of the streamfunction, ϕR and ϕI, each contain two terms that correspond to beams with upward and downward energy flux. Using horizontal energy flux as an example, covariances have the form 〈u*P*〉 = 〈u*P*〉 + 〈u*P*〉 + 〈u*P*〉 + 〈u*P*〉, where brackets denote phase averaging and subscript arrows refer to upward and downward beams. Covariances between components in the same vertical direction, for example 〈u*P*〉 and 〈u*P*〉, will be referred to as beams acting in isolation and are equivalent to a single beam extending infinitely in one direction. Covariances between upward and downward components, for example 〈u*P*〉 and 〈u*P*〉, will be referred to as beam interactions. Beams traveling in different vertical directions can result from different beams, or the same beam that has reflected off of the surface or bottom. Beam interactions can occur for gently sloping topography, which generates only upward beams, provided that beams reflect off of the surface. Interactions between beams are possible because there are oscillating fields within and far from each beam.

Energy flux is an illustrative example of beam interactions. A knife-edged ridge has four beams originating from the top of the ridge (x = 0 km, z = 3200 m), with one initially sloping upward and one downward on each side of the ridge (Fig. 11). Total energy flux (Fig. 11a) is largest along the tidal beams, positive north of the ridge and negative south of the ridge. Energy flux from isolated beams is constant in magnitude along all of the beams (Fig. 11b). Energy flux from beam interactions varies in sign and magnitude along all of the beams and abruptly changes sign where two beams intersect (Fig. 11c). The magnitudes of energy flux from beam interactions and isolated beams are similar. Total energy flux (Fig. 11a) has gradients along each beam, which are largest where beams intersect because of beam interactions.

Covariance 〈u*ζ*〉 is zero for isolated beams but not for interacting beams. From (4)(5), 〈u*ζ*〉 = U02ω−1[ −ϕxRϕzI + ϕzRϕxI] and is zero if (ϕxR,ϕzI) ∝ (ϕxR,ϕxI). For a single beam [e.g., only the upward beams in (6)], as in a plane wave, horizontal and vertical derivatives are proportional and displacement and across-ridge velocity are uncorrelated. Physically, internal wave oscillations parallel ray paths so that maximum across-ridge and vertical displacements coincide and across-ridge velocity is zero when vertical displacement is a maximum. With multiple beams, as in (6), horizontal and vertical derivatives are not proportional and 〈u*ζ*〉 ≠ 0. Covariance 〈u*ζ*〉 is largest along tidal beams and varies in magnitude and sign along tidal beams (Fig. 12) because the covariance is caused entirely by beam interactions.

The sign of 〈u*ζ*〉 in the upper ocean (Fig. 12) agrees with observed 〈uζ′〉 (Fig. 9a), suggesting that beam interactions are a reasonable explanation for the observed covariance. A knife-edge ridge was used because this simple topography leads to tractable solutions. A model with more realistic topography would have covariances with different magnitudes but similar signs. The processes of internal wave propagation are essentially the same, so a knife-edge ridge is an adequate model.

b. Interpretation of 〈u*ζx*〉 and 〈u*υx*〉

Covariances 〈u*ζx*〉, 〈ux*ζ*〉, 〈u*υx*〉, and 〈ux*υ*〉 show the direction of energy flux along the beams. For a knife-edge ridge, horizontal energy flux changes sign at the ridge and is positive in the positive x direction (Fig. 11a); vertical energy flux changes sign at surface and bottom reflections and is initially positive (negative) along upward- (downward) sloping beams (Fig. 13). Covariances 〈u*υx*〉 and −〈ux*υ*〉 (Figs. 14c,d) have the same signs as horizontal energy flux (Fig. 11a), and −〈u*ζx*〉 and 〈ux*ζ*〉 (Figs. 14a,b) have the same signs as vertical energy flux (Fig. 13). Covariances 〈u*ζx*〉, 〈ux*ζ*〉, 〈u*υx*〉, and 〈ux*υ*〉 have barely perceptible gradients along the beams, in contrast to 〈u*P*〉 and 〈w*P*〉 because beam interactions are an order of magnitude smaller than covariances from isolated beams. The direction, but not the magnitude, of energy flux can be obtained from 〈u*ζx*〉, 〈ux*ζ*〉, 〈u*υx*〉, and 〈ux*υ*〉. The observed covariances (Fig. 10) resemble the covariances for a knife-edged ridge in the upper ocean (Fig. 14) in having 1) the same sign and 2) only minor differences between −〈uζx〉 and 〈uxζ′〉 as well as 〈uυx〉 and −〈uxυ′〉. At Kauai Channel, horizontal energy flux was directed away from the ridge (Figs. 10c,d), and vertical energy flux was initially upward nearest the ridge and then downward after reflection off of the surface (Figs. 10a,b), consistent with internal wave generation at the ridge.

c. Across-ridge energy flux, 〈u′P′〉

Pressure perturbations were determined by fitting displacement to vertical modes (Gill 1982). Using vertical modes enforces the constraint that pressure perturbations vertically integrate to zero over the full ocean depth. Vertical modes were calculated from full depth buoyancy frequency using observed horizontally averaged buoyancy frequency above 355 m and the mean buoyancy frequency of HOME (Klymak et al. 2006) below 355 m. Velocity and displacement were fit to baroclinic modes by minimizing a cost function that included measures of misfit and magnitudes of the modal coefficients. The barotropic mode was not included in the fit. Magnitudes of the modal coefficients were minimized so that pressure perturbations did not become unrealistically large below 355 m. Ten modes were used in the fit as vertically integrated energy flux using 4–10 modes was essentially the same (averaged across the ridge, the rms variation over modes 4–10 was 10% of the total), and 10 modes showed the beam pattern better than 4 modes. Misfits between observed and fitted u′ and ζ′ averaged over all bins were 0.03 m s−1 and 3 m, compared with magnitudes of 0.08 m s−1 and 9 m. Misfits were largely baroclinic and mainly had vertical scales smaller than can be represented with 10 modes. Including across-ridge variations in depth or N2 did not change the results significantly.

Across-ridge energy flux was away from the ridge and largest along M2 ray paths (Fig. 15a), consistent with internal wave generation at the ridge. Energy flux was meaningful over 50–355 m, where observations of density perturbations were available, and over 0–50 m because properties were uniform with depth in the mixed layer. The largest energy fluxes on both sides of the ridge occurred near the surface reflections of the ray paths. Vertically integrated energy flux is considered over two depth ranges: 0–335 m, which is the largest range with meaningful energy flux, and 95–355 m, which is below the mixed layer base. Vertically integrated energy flux over 95–355 and 0–355 m had reasonable magnitudes and across-ridge structures (Fig. 15b). Maximum energy flux integrated over 0–355 m was 6.7 and −4.7 kW m−1 north and south of the ridge, respectively, compared to 3.5 and −2.3 kW m−1 over 95–355 m.

Errors in energy flux arose from several sources. A formal error in 〈uP′〉 was estimated by assuming random errors in u′ and ζ′ equal to the magnitude of the misfits between the observations and the fit to modes. For vertically uncorrelated errors in u′ and ζ′, the formal error was 0.25 kW m−1 over 95–355 m and 0.39 kW m−1 over 0–355 m. Some portion of barotropic energy flux might have been included in 〈uP′〉 because the barotropic mode is not orthogonal to the baroclinic modes in the upper ocean. For vertically correlated errors in u′ and ζ′, the formal error in energy flux was 1.14 kW m−1 over 95–355 m and 1.74 kW m−1 over 0–355 m. Greater than 100 km from the ridge, energy flux was typically 1–2 kW m−1 and might have resulted from nontidal mesoscale features.

d. Relationships between energy density, energy flux, and dissipation

Energy density and energy flux were correlated (Fig. 16a). For each vertical mode, energy flux equals energy density multiplied by group velocity (Klymak et al. 2006; Alford and Zhao 2007). Observed vertically integrated energy flux was regressed against vertically integrated energy density to find a horizontal group velocity of 0.92 m s−1 directed away from the ridge (Fig. 16a). There was no obvious difference in the magnitude of the group velocity north and south of the ridge. The theoretical group velocity of the third mode and the average group velocity of the first eight modes were both equal to 0.91 m s−1, agreeing well with the observed 0.92 m s−1.

Energy flux and dissipation varied across the ridge in different ways and are compared through the decay length scale. Tidal beams in dissipation (Fig. 7b) were less uniform than in energy flux (Fig. 15a) and harder to distinguish, particularly for the beams farthest from the ridge. The minimum over the ridge crest between the tidal beams was clearer in energy flux than in dissipation. Velocity shear and dissipation fields represent smaller vertical scales than energy flux fields, which may explain some of these differences. The decay length scale, |∫〈uP′〉dz|/∫ρ0〈ε〉dz, is the distance an internal wave must propagate for its total dissipation to equal initial energy flux. Energy flux was integrated over 0–355 m and dissipation over 95–355 m because the parameterization is not valid in the mixed layer. Averaged within 52 km of the ridge, the decay length scale was 380 km with a standard deviation of 340 km. The largest decay length scales north and south of the ridge were 2550 and 850 km, respectively, at the locations of largest energy flux. Even though the decay length scale varied across the ridge, it was similar to previous estimates at Kauai Channel of 400–1000 km for the M2 frequency (Lee et al. 2006).

Dissipation and energy density were correlated as previously observed (Klymak et al. 2006; Lee et al. 2006; Martin and Rudnick 2007). A correlation results because dissipation is proportional to energy flux through the decay length scale and energy flux for each mode is the product of energy density with group velocity. From full depth observations and models along the Hawaiian Ridge:
i1520-0485-39-2-421-eq3
where P = 1 ± 0.5 (Klymak et al. 2006). In our observations, energy density and energy flux (Figs. 5, 15a, and 16a) were correlated better than energy flux and dissipation (Figs. 7b and 15a). With integration over 95–355 m, P = 1.3 (Fig. 16b). This differed from the estimate at Kauai Channel of P = 0.7 ± 0.2 made during spring tide in 2000 (Martin and Rudnick 2007), but both were within the uncertainty of the Klymak et al. value.

Dissipation and energy flux in the upper ocean are extrapolated to the Hawaiian Ridge using a 2500-km ridge length, as in Martin and Rudnick (2007). Outward energy flux was elevated near where beams were observed with maximum values in the upper 355 m of 16.8 GW and 11.8 GW north and south of the ridge, respectively (Fig. 16c). Dissipation over 95–355-m depth was integrated from the ridge crest outward and so increased from zero to 1.6 GW and 1.5 GW at −152 and 152 km from the ridge, respectively. Energy flux was almost everywhere larger than total dissipation. Averaged within 52 km of the ridge crest, ∫ ∫ρ0〈ε〉dxdz/|∫〈uP′〉dz|, a measure of the radiation efficiency, was 0.16 with a standard deviation of 0.15. The radiation efficiency was similar to previous full depth M2 estimates of 0.10 at Kauai Channel and 0.08–0.25 along the Hawaiian Ridge (Klymak et al. 2006; Nash et al. 2006).

The observed dissipation and outward energy flux are compared to the 20 ± 6 GW of energy believed lost from the barotropic tide at the Hawaiian Ridge (Egbert and Ray 2001). At 52 km from the ridge crest, observed outgoing energy was 15.1 GW and integrated dissipation was 2.0 GW. Thus, the sum of outward internal wave energy and dissipation was 17 GW. Our vertically integrated estimates of energy density [3–6 kJ m−2 (Fig. 16a)], energy flux [1–5 kW m−1 (Fig. 15b)], and dissipation [2–10 mW m−2 (Fig. 16b)] were a factor of 2–3 smaller than previous full depth estimates at Kauai Channel (10–20 kJ m−2, 4–21 kW m−1, and 10–30 mW m−2, respectively) (Merrifield and Holloway 2002; Rudnick et al. 2003; Klymak et al. 2006; Lee et al. 2006; Nash et al. 2006; Rainville and Pinkel 2006a; Zaron and Egbert 2006). Applying this factor of 2–3 implies a loss of 34–51 GW, or 1.5–2.5 times greater than the Egbert and Ray (2001) value, consistent with Kauai Channel being a hot spot along the ridge.

e. Momentum flux and mean velocity

Mean fields of velocity and density had notable features near M2 ray paths. Mean along-ridge velocity changed sign and mean across-ridge velocity was a maximum near M2 ray paths (Fig. 17). Mean isopycnals were displaced toward the surface near M2 ray paths (Fig. 4). We investigate the possibility that these correspondences were caused by internal waves forcing mean flows through momentum flux divergences. If mean fields are forced only by divergences of internal wave covariances, the governing equations are
i1520-0485-39-2-421-e7a
i1520-0485-39-2-421-e7b
i1520-0485-39-2-421-e7c
i1520-0485-39-2-421-e7d
i1520-0485-39-2-421-e7e
Observed f−1uu′〉x was 1–4 cm s−1, f−1uυ′〉x was 1–2 cm s−1, and 〈uζ′〉x was 10−4 m s−1. If the energy in the internal wave field is steady, 〈w*ζ*〉 is zero from (4d) and mean vertical velocity is given by 〈u*ζ*〉x + 〈w*ζ*〉z = 〈u*ζ*〉x. Through mass conservation (7e), mean across-ridge velocity is 〈u*ζ*〉z. Observed 〈uζ′〉z was 2–3 cm s−1, which is in rough agreement with f−1uυ′〉x (7b). Compared with observed mean velocities along the tidal beams of 5–15 cm s−1 (Fig. 17), internal wave forcing of mean flows was not quite an order of magnitude smaller. Momentum fluxes are partly responsible for the relationships between mean fields and ray paths, but the relationships are not fully understood.

Mean flows forced by the momentum flux divergences in (7) require gradients along tidal beams, which are typically thought to be caused by dissipation (Lighthill 1978) but can result from beam interactions in the absence of dissipation. Beam interactions cause (ϕzR, ϕzI) and (ϕxR, ϕxI) to be disproportionate, which causes 〈u*ζ*〉 ≠ 0 (section 4a) forcing a mean vertical velocity (7d) and causes 〈u*u*〉x + 〈u*w*〉z and 〈u*υ*〉x + 〈υ*w*〉z to be nonzero [see (4)(6)] forcing mean horizontal velocities (7a, b). Interactions between tidal beams result in mean flows in all directions. For a knife-edge ridge with parameters as in Fig. 11, 〈u*u*〉x + 〈u*w*〉z and 〈u*υ*〉x + 〈υ*w*〉z corresponded to velocities of 1–5 cm s−1.

5. Conclusions

Notable features of this study include 1) repeated sections intentionally detuned from the M2 frequency, so that M2 phase was well sampled everywhere across the ridge, 2) observations of tidal beams in several covariances, including changes in sign at the ridge and upon reflection from the surface, 3) a representative picture of the across-ridge structure of internal wave energy density, energy flux, and inferred dissipation, which allowed many of the goals of HOME to be investigated from a single set of observations, and 4) investigations of interactions between tidal beams, including effects on covariances and momentum flux divergences.

The systematic timing of sections in this study differed from previous methods of spatially surveying internal tides. Several studies have taken repeat sections and found tidal constituents by fitting (Loder et al. 1992; Petruncio et al. 1998; Kitade and Matsuyama 2002; Dale et al. 2003; Lam et al. 2004). Some studies fit to short time series taken at a number of stations (Pingree and New 1989; Kunze et al. 2002; Lee et al. 2006; Nash et al. 2006). Other studies have avoided fitting and averaged over all sections with imperfect distributions of tidal phase (Lien and Gregg 2001; Martin et al. 2006). In this study, intentionally timing sections to detune from the M2 tide provided adequate averaging over the M2 phase, which allowed key features of the internal tide to be investigated.

Two semidiurnal internal wave beams were evident as regions of elevated variance and covariance that coincided among many quantities including velocity and displacement variance, energy density, energy flux, and covariances 〈uζ′〉, 〈uυx〉, 〈uxυ′〉, 〈uζx〉, and 〈uxζ′〉. Both beams were observed over 60-km horizontal distances, paralleled M2 ray slopes, and reflected off of the surface. Beams in velocity shear and dissipation were harder to distinguish, particularly for the downward beams. Differences between the beam to the north of the ridge and the beam to the south were smaller than variations within each beam. The M2 ray slopes originated from the sides of the ridge near the ridge peak in agreement with several studies (Baines 1982; Holloway and Merrifield 1999; Lamb 2004; Petrelis et al. 2006). Phase averaging of the observations was adequate to observe sign changes in covariances at the ridge and when beams reflected off of the surface. The barotropic tide was not removed from perturbation fields and did not appear to affect covariances. The directions of horizontal and vertical energy flux along the beams as determined from 〈uυx〉, 〈uxζ′〉, and 〈uP′〉 corresponded to internal wave generation at the ridge.

These observations addressed some of the goals of HOME. The across-ridge structure of internal tide energy density, energy flux, and dissipation was quantified over a 152-km distance on both sides of the ridge. Turbulent dissipation, as parameterized from velocity shear, was elevated relative to open ocean values along M2 ray paths. Averaged within 52 km of the ridge, the total energy dissipated to mixing was 16 ± 15% of outward energy flux. Compared with 20 ± 6 GW of energy lost from the barotropic tide at the Hawaiian Ridge (Egbert and Ray 2001), a 2500-km ridge of Kauai Channel topography would have total dissipation plus outward energy flux of 34–51 GW, consistent with Kauai Channel being a hot spot along the ridge. It is desirable to explicitly separate the M2, diurnal, and near-inertial frequencies, but in order to phase average observations obtained over a 300-km distance, all frequencies less than 18 days were included in the perturbation fields. The M2 frequency most likely was the dominant frequency, and only M2 tidal beams were observed.

There are oscillating fields within and far from tidal beams allowing different beams to interact and covary with each other. Observed 〈uζ′〉 was the key factor in focusing on this process because 〈u*ζ*〉 is zero for isolated beams and is caused entirely by interactions between beams. Beam interactions cause gradients along beams in almost all covariances, and therefore cause momentum flux divergences and mean flows in the absence of dissipation. The observed covariances suggest wave-forced mean flows of 1–4 cm s−1, about an order of magnitude smaller than observed mean flows. It was not clear whether internal wave forcing of mean flows at Kauai Channel was caused by beam interactions, dissipation, or beam interactions and dissipation. Further work on this topic could proceed with the addition of dissipation to the internal wave model used here.

Acknowledgments

This work was supported by the National Science Foundation under Grants OCE98-19521, OCE98-19530, and OCE04-52574.

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Fig. 1.
Fig. 1.

Bathymetry near Kauai Channel, Hawaii. The black line shows 304 km of the cruise track centered at 21.67°N, 158.58°W (black square). Velocity and density were measured over the upper 400–600 m along the cruise track 18 times from 2 to 19 October 2002.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 2.
Fig. 2.

The M2 tidal phase as a function of distance along the cruise track. The center of the cruise track is 0 km, roughly the midpoint of the ridge. Each point shows the tidal phase at one particular location for one particular section across the ridge. Phase is relative to the tide at the beginning of data collection and does not indicate high or low tide. Subsequent plots range from ±152 km (gray vertical lines), where there was complete sampling of tidal phase.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 3.
Fig. 3.

Sample section of potential density, ρ, across-ridge velocity, u, and along-ridge velocity, υv, taken during spring tide (7 Oct 2002). Black lines are M2 ray slopes as in Fig. 5.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 4.
Fig. 4.

Average buoyancy frequency (color) and average density surfaces (contours). Density contours are separated by 0.2 kg m−3. The average is over all 18 sections into bins with 8-km horizontal spacing and 8.6-m vertical spacing. Thick black lines are M2 ray slopes as in Fig. 5.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 5.
Fig. 5.

(a) Total velocity variance, 〈(u′)2 + (υ′)2〉, and (b) displacement variance, 〈(ζ′)2〉. Displacement is the deviation of each isopycnal from its mean depth. Black lines are M2 ray slopes with the location of the surface reflections chosen by eye to best coincide with regions of largest velocity variance.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 6.
Fig. 6.

Across-ridge bathymetry and M2 internal wave ray paths. Bathymetry is as measured along the cruise track with an echosounder. Horizontal placement of the ray paths (red) was based on velocity variance (Fig. 5a) and not the ridge profile. The mean buoyancy frequency of HOME was used to calculate ray slopes below 355 m. The upper 400 m where density observations were available is shaded. Ray paths reached the surface at +41 and −35 km. Gray lines are shown at these surface reflections “Ray at 0 m,” and where the reflected ray paths leave the region of available data, “Ray at 400 m.” The gray line at 0 km identifies the ridge crest.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 7.
Fig. 7.

Mean (a) velocity shear squared 〈(uz)2 + (υz)2〉, (b) turbulent dissipation rate 〈ε〉, and (c) diapycnal eddy diffusivity 〈Kρ〉: ε and Kρ were parameterized based on velocity shear. The mean buoyancy frequency of HOME was used below 355 m in the parameterization. Ray paths as in Fig. 5.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 8.
Fig. 8.

Shear-strain ratio. Each symbol represents the vertically integrated average value in an 8-km bin. Different symbols are used to distinguish between north (circles) and south (triangles) of the ridge crest as well as where tidal beams have been observed (black symbols are within 68 km of the ridge and gray symbols are greater than 68 km from the ridge). Theoretical values for the GM spectrum and inviscid M2 plane waves are shown. The reference density is ρ0 = 1026 kg m−3. Vertical integration is over 95–355 m.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 9.
Fig. 9.

Covariance of displacement with (a) across-ridge velocity 〈uζ′〉 and (b) along-ridge velocity 〈υζ′〉. Ray paths as in Fig. 5.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 10.
Fig. 10.

Covariances between u′, υ′, and ζ′ involving horizontal gradients. (a) −〈uζx〉, (b) 〈uxζ′〉, (c) f−1uυx〉, and (d) −f−1uxυ′〉. Ray paths as in Fig. 5.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 11.
Fig. 11.

Across-ridge energy flux for a knife-edge ridge. The solid black line at x = 0 km represents the ridge. (a) Across-ridge energy flux is the sum of (b) across-ridge energy flux from isolated beams and (c) across-ridge energy flux from beam interactions. In (c) 〈u*P*〉 = 〈u*P*〉 and vertically integrates to zero. Parameters used in Eqs. (4)(6) are n = 100, U0 = 0.1 m s−1, h0/H = 0.8, N2 = 1 × 10−5 rad2 s−2, and f = 5.4 × 10−5 s−1.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 12.
Fig. 12.

The 〈u*ζ*〉 from (4)(6) with parameters as in Fig. 11. The covariance is caused by interactions between tidal beams.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 13.
Fig. 13.

Vertical energy flux, 〈w*P*〉, from (4)(6) with parameters as in Fig. 11.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 14.
Fig. 14.

Covariances between u*, υ*, and ζ* involving horizontal gradients calculated from (4)(6). (a) −〈u*ζx*〉, (b) 〈ux*ζ*〉, (c) f−1u*υx*〉, and (d) −fux*υ*〉. Dashed gray lines mark the upper 10% of the water column. Parameters as in Fig. 11. Signs should be compared to Figs. 11a, 13, and to Fig. 10 in the upper ocean.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 15.
Fig. 15.

Energy flux: (a) 〈uP′〉 as fit to 10 vertical modes. Dashed lines show the average region where density was observed. Ray paths as in Fig. 5. (b) 〈uP′〉 vertically integrated over 95–355 and 0–355 m. Gray vertical lines represent changes in the sign of the ray slope in the upper 400 m as shown in Fig. 6.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 16.
Fig. 16.

Relationships between integrated energy density, energy flux, and dissipation. (a) Energy density vs energy flux. The best linear fit is shown (black). (b) Energy density vs turbulent dissipation. The black curve is the best-fit power law of 1.3. Vertical integration in (a) and (b) is over 95–355 m. Symbols in (a) and (b) as in Fig. 8. (c) Integrated energy flux and dissipation. Energy flux is integrated vertically over 0–355 m and horizontally along a 2500-km ridge (solid line). Dissipation is integrated vertically over 95–355 m, horizontally from 0 km across the ridge, and horizontally along a 2500-km ridge (dashed line). Gray vertical lines as in Fig. 6. ρ0 in (b) and (c) as in Fig. 8.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

Fig. 17.
Fig. 17.

Mean (a) across-ridge velocity 〈u〉 and (b) along-ridge velocity 〈υ〉. Ray paths as in Fig. 5.

Citation: Journal of Physical Oceanography 39, 2; 10.1175/2008JPO3937.1

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