## 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 (*M*_{2}) 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 *M*_{2} 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 *M*_{2} 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 *M*_{2} tide so that the *M*_{2} 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 *M*_{2} frequency was the dominant frequency at the Hawaiian Ridge, and the desired horizontal coverage was one wavelength of a mode-one *M*_{2} 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 *M*_{2} 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 *M*_{2} 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 *M*_{2} 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 *M*_{2} 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, 〈*N* ^{2}〉, 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 *N ^{2}* 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.

_{2}ray slopes coincided with regions of elevated velocity and displacement variance. The ray slope |

*μ*|, a consequence of the internal wave dispersion relation, is

*f*, and 2π/

*ω*= 12.42 h for the

*M*

_{2}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

*M*

_{2}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

*M*

_{2}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

*M*

_{2}ray paths and are

*M*

_{2}tidal beams. Both velocity and displacement variance showed

*M*

_{2}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

*M*

_{2}ray paths and did not correspond to tidal beams as variances were mainly vertically coherent. Only

*M*

_{2}beams were apparent, even though velocity and displacement variance included all frequencies less than 18 days. The

*M*

_{2}frequency could have been dominant for two reasons: 1)

*M*

_{2}motions had larger amplitudes than other motions, such as diurnal and near-inertial motions, and 2) the sampling pattern allowed for effective averaging over

*M*

_{2}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, 〈(*u _{z}*)

^{2}〉 + 〈(

*υ*)

_{z}^{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.

*K*. Along the Hawaiian Ridge, dissipation from microstructure measurements agreed better with parameterizations from Gregg (1989) (within a factor of 2 for

_{ρ}*K*) than with Gregg et al. (2003) both with and without shear-strain ratio and latitude factors (Martin and Rudnick 2007). From Gregg (1989),

_{ρ}*N*

_{0}= 5.24 × 10

^{−3}s

^{−1}is a reference buoyancy frequency,

*S*

_{GM}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

*S*

_{OBS}

^{2}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} m^{2} s^{−1} compared with open ocean values of 10^{−10} W kg^{−1} and 10^{−5} m^{2} s^{−1}. Both quantities decayed to open ocean values tens of kilometers from the tidal beams. Tidal beams were most apparent in *K _{ρ}* (Fig. 7c) because of appropriate scaling by

*N*.

^{2}### c. Variance ratios

*υ̂*), energy density and shear-strain ratios are

*M*

_{2}frequency, where (1/2)

*ρ*

_{0}〈

*N*

^{2}〉〈(

*ζ̂*)

^{2}〉 is potential energy density, (1/2)

*ρ*

_{0}〈(

*û*)

^{2}+ (

*υ̂*)

^{2}〉 is kinetic energy density, and 〈

*N*

^{2}〉〈(

*ζ̂*)

_{z}^{2}〉 is vertical strain (Fofonoff 1969; Lien and Müller 1992);

*ρ*

_{0}= 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

*M*

_{2}frequency for both ratios in (3).

The observed shear-strain ratio showed that the *M*_{2} frequency was dominant. A scatterplot of vertically integrated shear versus strain clustered around the *M*_{2} 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 *M*_{2} 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 *M*_{2} energy density and shear-strain ratios were consistent with our observations of *M*_{2} tidal beams and other HOME observations showing increased energy at the *M*_{2} 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, 〈*u*′*ζ*′〉, 〈*u*′*ζ*′_{x}〉, and 〈*u*′* _{x}ζ*′〉, showed beams with equal magnitudes along all portions of the ray paths (Figs. 9a and 10a,b). Beams in 〈

*u*′

*υ*′

_{x}〉 and 〈

*u*′

*′〉 were thinner and more variable and had strongest magnitudes along the portions of ray paths closest to the ridge (Figs. 10c,d). Covariances 〈*

_{x}υ*u*′

*ζ*′〉, 〈

*u*′

*υ*′

_{x}〉, and 〈

*u*′

*′〉 had different signs north and south of the ridge (Figs. 9a and 10c,d). Covariances 〈*

_{x}υ*u*′

*ζ*′

_{x}〉 and 〈

*u*′

*′〉 changed signs where the beams reflected off of the surface (Figs. 10a,b), while 〈*

_{x}ζ*υ*′

*ζ*′〉 was smaller in magnitude than 〈

*u*′

*ζ*′〉 and did not have clear tidal beams (Fig. 9). Tidal beams in other covariances, such as 〈

*u*′

*ζ*′

_{z}〉 or 〈

*u*′

*υ*′

_{z}〉, 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.

*ω*

^{2}/

*N*

^{2}=

*O*(10

^{−4}). Neglecting dissipation and along-ridge gradients, the basic equations of motion are

*P** is the pressure perturbation,

*h*(

*x*) is the boundary, (

*ũ*,

*w̃*) 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**

*υ**〉 = 0, for example. Motions consistent with (4) and appropriately averaged over a tidal cycle will have 〈

_{t}*u**

*υ**〉 = 0 [multiply (4b) by

*υ** and average] and 〈

*u**

*υ**〉 = − 〈

_{x}*u**

_{x}*υ**〉. Observed 〈

*u*′

*υ*′〉 was small in magnitude and did not have clear tidal beams (not shown), and only minor differences between 〈

*u*′

*υ*′

_{x}〉 and − 〈

*u*′

*′〉 were found (Figs. 10c,d), indicating that the dynamics of (4) are sensible and that the observations and averaging procedure were adequate.*

_{x}υThe barotropic tide is included in the observed perturbation fields but showed no clear effect on covariances. For a barotropic tide of the form *ũ* = *U*_{0} cos(*ωt*), 〈*ũũ*〉 is a background field with no beam pattern. Covariances between baroclinic fields and *ũ*, 〈*ũζ _{x}**〉, 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

*U*

_{0}

*h*(

*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 〈

*u*′

*u*′〉, 〈

*u*′

*ζ*′〉, 〈

*u*′

*ζ*′

_{x}〉, and 〈

*u*′

*υ*′

_{x}〉 (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* = *U*_{0} cos(*ωt*), (*u**, *w**) = ( −*ψ _{z}*,

*ψ*),

_{x}*x*= 0 with height

*h*

_{0},

*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

*A*are determined from boundary conditions. The spatial components of the streamfunction,

_{n}*ϕ*and

^{R}*ϕ*, 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 〈

^{I}*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**〉 = *U*_{0}^{2}*ω*^{−1}[ −*ϕ _{x}^{R}ϕ_{z}^{I}* +

*ϕ*] and is zero if (

_{z}^{R}ϕ_{x}^{I}*ϕ*,

_{x}^{R}*ϕ*) ∝ (

_{z}^{I}*ϕ*,

_{x}^{R}*ϕ*). 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 〈

_{x}^{I}*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 〈*u*′*ζ*′〉 (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*ζ_{x}*〉 and 〈u*υ_{x}*〉

Covariances 〈*u***ζ _{x}**〉, 〈

*u**

_{x}*ζ**〉, 〈

*u**

*υ**〉, and 〈

_{x}*u**

_{x}*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**

*υ**〉 and −〈

_{x}*u**

_{x}*υ**〉 (Figs. 14c,d) have the same signs as horizontal energy flux (Fig. 11a), and −〈

*u**

*ζ**〉 and 〈

_{x}*u**

_{x}*ζ**〉 (Figs. 14a,b) have the same signs as vertical energy flux (Fig. 13). Covariances 〈

*u**

*ζ**〉, 〈

_{x}*u**

_{x}*ζ**〉, 〈

*u**

*υ**〉, and 〈

_{x}*u**

_{x}*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**

*ζ**〉, 〈

_{x}*u**

_{x}*ζ**〉, 〈

*u**

*υ**〉, and 〈

_{x}*u**

_{x}*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 −〈

*u*′

*ζ*′

_{x}〉 and 〈

*u*′

*′〉 as well as 〈*

_{x}ζ*u*′

*υ*′

_{x}〉 and −〈

*u*′

*′〉. 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.*

_{x}υ### 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 *N ^{2}* did not change the results significantly.

Across-ridge energy flux was away from the ridge and largest along *M*_{2} 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 〈*u*′*P*′〉 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 〈*u*′*P*′〉 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, |∫〈*u*′*P*′〉*dz*|/∫*ρ*_{0}〈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 *M*_{2} frequency (Lee et al. 2006).

*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, ∫ ∫*ρ*_{0}〈ε〉*dxdz*/|∫〈*u*′*P*′〉*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 *M*_{2} 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

*M*

_{2}ray paths. Mean along-ridge velocity changed sign and mean across-ridge velocity was a maximum near

*M*

_{2}ray paths (Fig. 17). Mean isopycnals were displaced toward the surface near

*M*

_{2}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

*f*

^{−1}〈

*u*′

*u*′〉

_{x}was 1–4 cm s

^{−1},

*f*

^{−1}〈

*u*′

*υ*′〉

_{x}was 1–2 cm s

^{−1}, and 〈

*u*′

*ζ*′〉

_{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 〈

*u*′

*ζ*′〉

_{z}was 2–3 cm s

^{−1}, which is in rough agreement with

*f*

^{−1}〈

*u*′

*υ*′〉

_{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 (*ϕ _{z}^{R}*,

*ϕ*) and (

_{z}^{I}*ϕ*,

_{x}^{R}*ϕ*) to be disproportionate, which causes 〈

_{x}^{I}*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 *M*_{2} frequency, so that *M*_{2} 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 *M*_{2} tide provided adequate averaging over the *M*_{2} 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 〈*u*′*ζ*′〉, 〈*u*′*υ*′_{x}〉, 〈*u*′* _{x}υ*′〉, 〈

*u*′

*ζ*′

_{x}〉, and 〈

*u*′

*′〉. Both beams were observed over 60-km horizontal distances, paralleled*

_{x}ζ*M*

_{2}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

*M*

_{2}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 〈

*u*′

*υ*′

_{x}〉, 〈

*u*′

*′〉, and 〈*

_{x}ζ*u*′

*P*′〉 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 *M*_{2} 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 *M*_{2}, 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 *M*_{2} frequency most likely was the dominant frequency, and only *M*_{2} 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 〈*u*′*ζ*′〉 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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