## Abstract

Strong semidiurnal internal tides are observed near Mendocino Escarpment in full-depth profile time series of velocity, temperature, and salinity. Velocity and density profiles are combined to estimate the internal tide energy flux. Divergence of this flux demonstrates that its source is the barotropic tide interacting with the escarpment. A baroclinic energy flux of 7 kW m^{−1} radiates from the escarpment, corresponding to 3% of the 220 kW m^{−1} fluxing poleward in the surface tide. Energy and energy flux are concentrated in packets that emanate from the flanks of the ridge surmounting the escarpment and one site ∼90 km north of the escarpment. Coherent beamlike structure along semidiurnal ray paths remains identifiable until the first surface reflection. Beyond the first surface reflection north of the escarpment, the energy flux drops by 2 kW m^{−1} and beams are no longer discernible. Turbulence, as inferred from finescale parameterizations, is elevated by over two orders of magnitude relative to the open-ocean interior in localized 500-m-thick layers at the bottom over the ridge crest, near the surface at the station closest to the first surface reflection to the north, slightly north of the first bottom reflection to the north, and on the south flank of the escarpment. Despite its intensity, turbulent dissipation integrated over the ridge crest is only 1% of the energy flux in the internal tides. Thus, the bulk of surface tidal losses at the escarpment is radiating away as internal waves. High turbulent dissipation rates near the surface reflection suggest that loss of energy flux there may be turbulent. This turbulence may arise from (i) Wentzel–Kramers–Brillouin amplification of semidiurnal shear as the internal tide propagates into high near-surface stratification or (ii) superposition of incident and reflected waves enhancing nonlinear transfers to small scales and turbulence production. Localized mixing due to internal tide beams impinging on the base of the mixed layer may be an important unconsidered cause of nutrient and water-mass fluxes between the surface layer and the upper pycnocline.

## 1. Introduction

Recent observations have revealed the importance of deep-ocean tide–topography interactions for dissipation of the astronomical tide (Egbert and Ray 2001), generation of internal waves (Ray and Mitchum 1997), and possibly mixing in abyssal waters (Munk and Wunsch 1998). The bulk of the global tidal dissipation of 3700 GW (2500 GW in *M*_{2}) has long been ascribed to bottom friction in shallow shelves and seas (Jeffries 1920; Wunsch 1975; Egbert et al. 1994; Kantha et al. 1995). Internal tide generation was thought to play a miniscule role based, in part, on estimates of semidiurnal losses of only 17 GW from continental shelf breaks (Baines 1982). Although some shelf breaks, such as the Bay of Biscay (Pingree and New 1989, 1991) and near the Queen Charlotte Islands (Cummins and Oey 1997), are sites of intense internal tide generation, surface tides tend to propagate along rather than across shelves, inducing only weak across-shelf currents to interact with shelfbreak topography.

Stronger internal tide generation is expected for topographic obstacles lying across the path of barotropic tidal currents (Baines 1974; Sjöberg and Stigebrandt 1992; Morozov 1995) as found in the deep ocean. For example, numerical simulations by Holloway and Merrifield (1999) find that the semidiurnal internal-wave energy flux from an idealized ridge topography with a length-to-width ratio of 3:1 was an order of magnitude larger than the baroclinic energy flux from a symmetric obstacle of the same bathymetric slope.

Satellite altimetry has led to a renaissance in deep-ocean internal tide research by detecting strong surface (barotropic) tidal losses at deep-ocean ridges, island chains, trenches, and fractures zones (Egbert and Ray 2001) and energetic internal tides radiating from these same topographic features (Ray and Mitchum 1997). These energy fluxes are considerably larger than those from continental shelf breaks. In situ measurements have confirmed many of these internal tide sources (Morozov 1995; Dushaw et al. 1995; Feng et al. 1998; Cummins et al. 2001). [Note that, since barotropic tides *O*(1–10 cm s^{−1}) are much larger than *O*(0.1 cm s^{−1}) eddy and mean-flow bottom currents in general, tidal currents should generate larger internal waves than geostrophic flows (St. Laurent and Garrett 2002).] Egbert and Ray (2001) estimate that as much as one-third of the global tidal dissipation (600–800 GW) occurs in the deep ocean based on the residual for a global fit of satellite altimetry to the Laplace tidal equations. Morozov (1995) obtained *O*(1000 GW) surface tidal losses by applying a simplified version of Baines's (1982) internal tide generation theory to ridges and fracture zones in the deep ocean. Other models (Sjöberg and Stigebrandt 1992; Kantha and Tierney 1997; Gustafsson 2001) predict similar losses. These studies are leading to a reevaluation of surface tidal losses in the deep ocean. Here, we describe direct measurements of internal tide radiation and turbulent dissipation generated by tide–topography interactions across the Mendocino Escarpment.

## 2. Setting

To better understand tide–topography interactions in the deep ocean, vertical profiles of water properties were collected across and along the Mendocino Escarpment in the eastern North Pacific. The Mendocino Escarpment is a zonal transform fault 3000 km long in the deep water off of northern California (Fig. 1). It originated 11 million years ago and broached the surface roughly 5.5 million years ago (Fisk et al. 1993). South of the escarpment, the ocean floor is relatively flat at 4400 m. To the north, bathymetry is more variable at 1800–3000 m over the Gorda Ridge. The escarpment is surmounted by a 10-km-wide transverse ridge that is 1200 m deep at its shallowest point (Fig. 2). Thus, the escarpment presents a significant barrier to meridional barotropic tidal currents. Based on where the bottom slope is supercritical for semidiurnal internal waves, that is, steeper than the slope of semidiurnal ray paths, the escarpment extends westward 400 km from the continental slope off of north California (Fig. 1). Semidiurnal critical slopes are also found on the Gorda Ridge to the north.

Near the Mendocino Escarpment, the semidiurnal barotropic tide propagates poleward as a Kelvin wave with surface elevations of ±55–60 cm and depth-independent meridional velocities of ±5 cm s^{−1} (Munk et al. 1970; Schwiderski 1979; Noble et al. 1987; Egbert et al. 1994; Mofjeld et al. 1995; Ray and Mitchum 1997). Thus, the escarpment is well suited to efficiently generate internal tides.

## 3. Data

A total of 115 full-depth profiles of horizontal velocity (*u, **υ*), temperature *T,* salinity *S,* and pressure *P* was collected at 21 stations (Fig. 1) using the Absolute Velocity Profiler (AVP; Sanford et al. 1985; Stahr and Sanford 1999). Seventeen stations made up the main transect ranging from 50 km south to 125 km north of the escarpment across the 1200-m apex of the ridge (Fig. 2). Because of the along-ridge variability, generation and subsequent propagation of the internal tide may not be strictly two-dimensional. Four additional stations were occupied to the east and west (Fig. 1). Sampling spanned 28 days (two fortnightly cycles) during October 1997. Three stations on the main transect were reoccupied in order to investigate temporal variability in the internal tide, in particular, fortnightly (*M*_{2} + *S*_{2}) modulation of the semidiurnal forcing.

During each station occupation, 3–9 AVP profiles were obtained over a 12–15-h interval in order to isolate semidiurnal (∼12 h) fluctuations from the ambient internal wave field. At most stations, the AVP time series were extended with full-depth CTD casts and 1600-m expendable current profiler (XCP) profiles before and after the AVP measurements. Most occupations included 5–6 AVP profiles. Stations with less than four profiles (dots in Fig. 1) could not be used to isolate the semidiurnal tide and so are not included. Only down profiles are used here; up profiles give similar results.

Relative horizontal velocity is measured electromagnetically throughout the water column, with an estimated uncertainty of 1 cm s^{−1}. An acoustic Doppler measurement of absolute velocity, also with an uncertainty of 1 cm s^{−1}, is obtained in the bottom several hundred meters of each AVP profile. These velocity measurements are combined to obtain absolute velocity throughout the water column with uncertainties of 1.5 cm s^{−1}. Data are gridded to 10-m vertical resolution. Two microstructure shear probes were attached to the profiler, but these measurements are contaminated by the wake of the weight release off the ridge and so are not used. Instead, a finescale parameterization for turbulence is employed in section 4g.

## 4. Results

### a. Barotropic velocities

Barotropic velocities were computed as the depth-mean of the horizontal velocity profiles obtained with the AVP, (〈*u*〉_{z}, 〈*υ*〉_{z}). Observed AVP barotropic velocities fit to a sinusoid at the lunar semidiurnal frequency (*M*_{2}) are compared with the TPXO.3 model tidal predictions south of the escarpment at 39°N in Fig. 3; TPXO.3 is a more recent version of the global inverse model described by Egbert et al. (1994) in which satellite altimetry is assimilated into approximate Laplace tidal equations constrained by continuity to obtain a solution for each of eight major tidal constituent frequencies (*M*_{2}, *S*_{2}, *N*_{2}, *K*_{2}, *K*_{1}, *P*_{1}, *O*_{1}, *Q*_{1}). The predicted and AVP barotropic tidal velocities agree to within the 1 cm s^{−1} uncertainty of the AVP measurements except over steep topography (○) and 33 km north of the ridge (+). Over steep topography (○), (i) barotropic currents may depend on local topography not resolved by TPXO.3 and (ii) AVP Doppler velocities may be less accurate because of difficulties tracking the bottom. The station 33 km north of the escarpment (+) exhibited strong bottom-intensified fluctuations with no discernible relationship to the semidiurnal tide. These may be associated with bottom-trapped motions at unresolved smallscale bathymetry, but our measurements are inadequate to identify their dynamics. Barotropic tidal excursions are *O*(0.5 km) south and north of the escarpment, and *O*(1.5 km) above the ridge crest, and so any internal tides generated by the topographic features resolved in Fig. 1 (see also Fig. 7) should be linear.

### b. Baroclinic perturbation velocities and displacements

In order to remove barotropic and subtidal velocity signals, the depth-mean velocities (〈*u*〉_{z}, 〈*υ*〉_{z}) and baroclinic station time-means [〈*u*′(*z*)〉_{t}, 〈*υ*′(*z*)〉_{t}] were subtracted from each profile [*u*(*z, **t*), *υ*(*z, **t*)]. The baroclinic profile is thus *u*′(*z, **t*) = *u*(*z, **t*) − 〈*u*(*t*)〉_{z} and the baroclinic perturbation profile *ũ*(*z, **t*) = *u*(*z, **t*) − 〈*u*(*t*)〉_{z} − 〈*u*′(*z*)〉_{t}. Wentzel–Kramers–Brillouin (WKB)-scaled (*N*_{0} = 2.5 × 10^{−3} rad s^{−1}) rms baroclinic perturbation meridional and zonal velocities are typically 4–7 cm s^{−1}, with 10–15 cm s^{−1} found at some stations over the ridge crest and at one station 60 km north of the ridge. Rms meridional velocities are 1–2 cm s^{−1} larger than zonal velocities.

Following Desaubies and Gregg (1981), isopycnal displacement profiles *ξ̃*(*z, **t*) were constructed relative to station time-mean profiles with respect to potential density *σ*_{θ}[〈*z*(*σ*_{θ})〉_{t}]:

WKB-scaled (*N*_{0} = 2.5 × 10^{−3} rad s^{−1}) rms vertical displacements are 12–20 m off ridge. Higher values of 30–35 m are found above the ridge and at one station 60 km north of the ridge.

### c. WKB scaling

As a result of depth-varying stratification, horizontal velocity, energy density, and energy density flux tend to be intensified near the surface where stratification is strong, while isopycnal displacements are amplified in the deep water where stratification is weak. Surface temperatures were several degrees above normal because of the 1997 El Niño so that stratification increased exponentially all the way to the surface. To obtain a description of the vertical structure of the internal tide without the complicating influence of variable stratification, the data were WKB normalized. The appropriate scaling for horizontal velocity is

(Leaman and Sanford 1975) where the caret denotes the WKB-scaled value, *N*(*z*) is the time-mean measured buoyancy frequency, and *N*_{0} = 2.5 × 10^{−3} rad s^{−1} is a constant reference buoyancy frequency based on the depth-average buoyancy frequency 15 km north of the escarpment. The scaling for vertical displacement is

Energy density and energy density flux scale as

where

and the angle brackets 〈 · 〉_{ϕ} denote an average over wave phase (in this case, over the station occupation). The small *w*^{2} contribution to kinetic energy has been ignored here. The stretched-depth coordinate is

This scaling is equivalent to converting the ocean to one of constant stratification in which (i) there is no spatial variability due to changes in stratification, (ii) vertical standing modes are sinusoids, and (iii) internal-wave ray paths are straight lines. The WKB approximation degrades as the vertical length scale of the wave becomes comparable to or larger than the scale of the stratification; in particular, the mode-1 zero crossing can be offset in depth by as much as 10%.

### d. Semidiurnal fitting

Profiles taken 6 h apart are mirror images of each other (Fig. 4), as would be expected if the dominant signals had semidiurnal (12 h) periods. Mirror imaging is less evident in profiles taken 9 h (0.5 inertial period) apart. This motivates fitting the station time series to semidiurnal oscillations.

To extract the semidiurnal signal, baroclinic velocity [station time-mean plus perturbation, e.g., *u*′(*z, **t*) = *u*(*z, **t*) − 〈*u*(*z, **t*)〉_{z} = *ũ*(*z, **t*) + 〈*u*′(*z, **t*)〉_{t}], profiles were least squares fit to a time-mean plus semidiurnal sinusoids at each depth for example,

where *ω* is the frequency of the sinusoidal component (*M*_{2} in this case), and similarly for the meridional velocity *υ.* Isopycnal displacements *ξ̃*(*z*) were fit solely to the oscillatory component because *ξ̃* is defined such that its station time-mean is zero. Given the short duration of the station time series, the lunar *M*_{2} and solar *S*_{2} semidiurnal constituents cannot be separated. During the 28-day measurement period, there will be fortnightly modulation of the semidiurnal barotropic forcing and hence the baroclinic response.

Because data coverage at each station is limited to just over 12 hours, the semidiurnal fit represents energy in a range of frequencies about *M*_{2}. Monte Carlo simulations suggest that, for 5–6 profiles in a typical deep-ocean internal wave field (*N*_{0} = 2.5 × 10^{−3} rad s^{−1}), semidiurnal velocities should be resolvable to within rms amplitude uncertainties of ±1.5 cm s^{−1} and phase uncertainties less than ±30°. Isopycnal displacement amplitudes should be resolved to ±4.5 m with phase uncertainties less than ±10°. Amplitude uncertainties depend on the number of profiles collected while phase uncertainties are most sensitive to the strength of the semidiurnal signal. Uncertainties are relatively insensitive to the strength of near-inertial (18 h) fluctuations. The signal-to-noise ratio exceeds 3 for typical rms signals of 4–7 cm s^{−1} in horizontal velocity and 12–20 m in vertical displacement.

Semidiurnal fits accounted for 70%–90% of the rms vertical displacement *ξ,* 60%–80% of the rms meridional velocity *υ,* and 50%–70% of the rms zonal velocity *u.* The lower values are found over the ridge where higher-frequency harmonics were evident. The fits for vertical displacement and meridional velocity contained more than 4 times more variance than the residuals.

To further investigate the relative contribution from semidiurnal and inertial motions, stations with six or more profiles were fit to a sum of semidiurnal (*M*_{2}) and local inertial (*f*) frequency components. The inertial period at this location is approximately 18 h. Semidiurnal fit amplitude and phase structure are displayed in Fig. 5 for the station 33 km north of the escarpment. Corresponding inertial fit amplitude and phase structure is shown in Fig. 6. The semidiurnal part of the two-frequency fit is not significantly different from a single-frequency semidiurnal fit. Its phase tends to increase with depth in all three components, implying downward (upward) propagation of phase (energy). Downward phase (upward energy) propagation is seen at all stations except those 3.7 and 13 km south of the ridge where upward phase (downward energy) propagation is observed in the bottom part of the profile. These inferences about vertical energy propagation are consistent with energy-flux estimates described later (see Fig. 9). The inertial amplitudes are less energetic, particularly the isopycnal displacements *ξ* and meridional velocity *υ.* A weaker potential energy (*ξ*) contribution is expected for near-inertial waves. There is no preferred direction of vertical phase propagation in the inertial fits. Energy-flux contributions (see section 4e) from near-inertial fluctuations will be smaller than those from semidiurnal signals both because the near-inertial energy is weaker (Figs. 5 and 6) and the group velocity is slower (*C*_{g} ∝ *ω*^{2} − *f*^{2}).

### e. Energetics

The profiles are used to estimate the baroclinic perturbation horizontal kinetic energy density HKE = 〈*ũ*^{2} + *υ̃*^{2}〉_{ϕ}/2 and available potential energy density APE = *N*^{2}〈*ξ̃*^{2}〉_{ϕ}/2. Energy density profiles based on semidiurnal fits resemble (but are less energetic than) those derived from the raw baroclinic profiles. Baroclinic energy densities are WKB normalized (*N*_{0} = 2.5 × 10^{−3} rad s^{−1}) as described in section 4c.

The largest energy densities (0.01–0.03 m^{2} s^{−2}) are near the bottom over the ridge crest (Fig. 7) and, away from the escarpment, in energetic packets 3–10 times typical oceanic energies of 0.003 m^{2} s^{−2} (GM: Garrett and Munk 1979). Though at most stations, ratios of horizontal kinetic to available potential energies in the semidiurnal fits, that is, HKE/APE, lie near the semidiurnal value of 2.6 (±0.5), energy ratios are both high (5–15) and low (0.8) over the ridge crest and are high at the northmost station where there is little energy.

The packets appear to lie along trajectories parallel to semidiurnal ray paths of slope

(blue lines in Fig. 7) where *ω* is the intrinsic wave frequency, *f* is the Coriolis frequency, and *N* is the buoyancy frequency. North of the ridge, a beam parallel to ray paths is particularly evident between 15 and 60 km (Fig. 7) where a sequence of packets can be followed from 2000-sm (stretched meter) depth at 40°30′N (15 km north of the ridge) upward to the upper 500 sm at 40°55′N (60 km). This beam appears to originate from the supercritical upper flanks of the ridge. The particularly energetic packet in the upper 500 sm at 40°55′N may be a superposition of an incident upgoing beam with the reflected downgoing beam. A deeper but less well defined beam can be followed from 2500 sm at 40°35′N (25 km) upward to 1200 sm at 40°55′N (60 km). The slope of this sequence of packets is slightly gentler than semidiurnal ray paths. It may originate east or west of the main transect (Fig. 1) with zonal propagation carrying it across our observational transect north of the ridge. This packet may undergo reflection off the bottom and originate as a downgoing wave on the supercritical flanks of the ridge, but this cannot be established with our data. Neither beam has a clearly discernible expression north of 41°N (70 km) though weak packets around 1200 sm at 41°15′N (100 km) and 1800 sm at 41°22′N (110 km) might be post-surface-reflection expressions of the shallower beam. These beams appear to originate from the supercritical north flank of the ridge rather than critical slopes (blue lines) that are often cited as sources of internal tides.

South of the ridge, a packet between 1200–1800 sm at 40°15′N (−13 km) can be followed along a ray path trajectory upward to a surface intensification at 39°55′N (−50 km), which again may be a superposition of an upgoing incident wave and a downgoing reflected wave. Enhanced energy below 2500 sm at 40°15′N can be followed through a bottom reflection to a midwater packet between 2000 and 2800 sm at 39°55′N (55 km). These two beams appear to originate from the ridge crest and upper south flank. Energetic packets lying along ray paths are consistent with internal tide generation theory (e.g., Rattray 1960; Rattray et al. 1969; Prinsenberg and Rattray 1975; Baines 1982). A similar beam phenomenon has been observed in laboratory experiments (Mowbray and Rarity 1974; Baines and Fang 1985) as well as oceanic measurements (DeWitt et al. 1985; Pingree and New 1989; Lien and Gregg 2001).

For internal gravity waves, baroclinic energy density flux is the product of the baroclinic reduced pressure anomaly *p̃* and perturbation velocity **v** = (*ũ, **υ̃*, *w̃*)

Perturbation horizontal velocity (*ũ, **υ̃*) is measured directly. Vertical velocity profiles are inferred from the semidiurnal fits to the vertical displacement time series, *w̃*(*z*) = ∂*ξ̃*(*z*)/∂*t.* The baroclinic reduced pressure perturbation *p̃* is computed by integrating the hydrostatic balance

to obtain

where *p̃* represents the perturbation reduced pressure *P̃*/*ρ*_{0}, and *P̃* the perturbation pressure. The *p̃*(0) term represents the contribution from the internal wave's surface displacement that must be included to correctly represent the vertical structure (see also Kunze et al. 2002a). This last term follows from recognizing that the depth integral of the internal-wave baroclinic perturbation pressure vanishes, allowing one to infer the internal-wave surface displacement from baroclinic measurements. At the station 33 km north of the ridge (Figs. 5 and 6), the meridional energy flux 〈*υ̃**p̃*〉_{ϕ} is dominated by semidiurnal oscillations with a negligible contribution from the inertial component (Fig. 8).

A WKB-scaled (*N*_{0} = 2.5 × 10^{−3} rad s^{−1}) across-escarpment section reveals meridional–vertical energy fluxes (〈*υ̃**p̃*〉_{ϕ}, 〈*w̃p̃*〉_{ϕ}) radiating away from the escarpment both to the north and south (Fig. 9). Vertical energy fluxes are predominantly upward and are consistent with the direction of vertical energy propagation inferred from semidiurnal fit phase structure (e.g., Fig. 5). Away from boundaries, fluxes are strikingly parallel to semidiurnal ray paths (8). Flux arrows with gentler slopes may be due to (i) superpositions of up- and downgoing rays, particularly near surface and bottom boundaries (e.g., near the surface and bottom at 15 km north), or (ii) some zonal propagation. Steeper flux arrows are found where upgoing beams propagating in contrary meridional directions superimpose (e.g., upper 1000 sm between 30 and 40 km).

Over the ridge crest, energy fluxes are negligible despite energy densities 10–30 times typical open-ocean GM levels (Fig. 7) because velocity and pressure fluctuations are uncorrelated (see Fig. 13). The region over the ridge crest may be energized by internal tides generated on the flanks and radiating over the crest in opposing beams from either side so that a “standing wave” results with little net flux. Crossing beam structure has been seen in recent numerical simulations (M. A. Merrifield and P. E. Holloway 2002, personal communication).

Off ridge, 70%–80% of the perturbation velocity and pressure variances participate in the meridional energy flux away from the ridge; that is, the normalized cor-relation 〈*υ̃**p̃*〉_{ϕ}/〈*υ̃*^{2}〉_{ϕ}〈*p̃*^{2}〉_{ϕ} = 0.7–0.8. The four beams that could be identified in the energy section (Fig. 7) can also be discerned in the energy flux, though with more difficulty. South of the escarpment, well-defined packets of high energy flux coincide with those seen in energy (Fig. 7). Fluxes are consistent with a 1000–1800-sm upgoing packet at 40°15′N (−13 km) originating from the ridge crest (upper panel) and impinging the surface around 39°55′N (−50 km). The downgoing packet below 2800 sm at 40°15′N (−13 km) can be traced from the supercritical southern flank of the escarpment (upper panel, 40°20′N). It survives a bottom reflection to become the upward-propagating packet between 2200 and 2800 sm at 39°55′N.

Packets are less well defined north of the escarpment. A broad region of more or less northward and upward energy flux above 2000 sm between 40°30′ and 41°N seems to emanate from the water column above the ridge crest, its lower edge shoaling with distance from the escarpment. This is consistent with the internal tide generation theory of Baines (1982) in which barotropic flow over sloping topography induces isopycnal displacements, *ξ*_{BT} = *U*_{BT}*H*_{x}*z*/(*ωH*), which act as vertically distributed forcing throughout the water column. At greater depths, a weaker packet propagates up- and northward, most clearly visible between depths of 2000–2500 sm at 40°40′N (33 km) and 1000–1200 sm at 40°55′N (60 km). Farther north, fluxes are weakly northward in the upper 1000 sm and vanish completely at the northmost station (40°30′N, 125 km). Small middepth (∼1800 sm) upgoing packets traveling south at 40°55′N and north at 40°22′N may originate from the near-critical bottom slope at 41°10′N (87 km). The additional vertical flux from this north generation site may be responsible for steepening the energy-flux vectors in the upper water column 20–40 km north of the ridge. Normalizing station profiles by the barotropic tidal forcing to account for the fortnightly cycle did not qualitatively alter Figs. 7 and 9.

Zonal energy fluxes 〈*ũp̃*〉_{ϕ} (Fig. 10) are generally smaller than meridional energy fluxes (Fig. 9). While meridional energy fluxes tend to have uniform direction throughout the water column, zonal fluxes exhibit reversals. This suggests that internal tide generation may be distributed unevenly along the ridge. There are no strong zonal energy fluxes along the flanks of the ridge (Fig. 10), indicating that bottom-trapped topographic waves are weak or nonexistent at Mendocino Escarpment.

Depth-integrated baroclinic energy fluxes ∫ 〈**v***p̃*〉_{ϕ }*dz* are consistently directed away from the escarpment (Fig. 11). While the predominant orientation is meridional, there tends to be south-southeastward (north-northwestward) flux to the south (north) of the ridge. The southmost station in particular has a substantial eastward energy-flux component.

The depth-integrated meridional energy flux (Fig. 12) exhibits considerable meridional variability across the Mendocino Escarpment. Over the ridge crest, fluxes are near zero. Off ridge, in an intermediate region within 10–60 km of the escarpment, fluxes are 4 (−3) kW m^{−1} to the north (south). Energy appears to be radiating away from the ridge with little dissipation or generation in this range as the fluxes show little variability with latitude. In the farfield, more than 80 km north of the escarpment, fluxes drop to 2 kW m^{−1}, then near zero at the northmost station. The largest energy fluxes are in the 10–60-km range, representing a total outgoing flux (north plus south) of 7 kW m^{−1}.

The surface tide semidiurnal flux across the Mendocino Escarpment following Pugh (1987),

is 220 kW m^{−1} poleward using amplitudes and phases from TPXO.3 (Egbert et al. 1994), where *H* = 4200 m is the water depth, *ξ*_{0} = 0.5 m is the amplitude of the surface elevation associated with the barotropic tide, *υ*_{0} = 10 cm s^{−1} is the amplitude of the barotropic tidal velocity, *g*_{υ} = 165° is the phase of the tidal velocity, and *g*_{ξ} = 148° is the phase of the tidal elevation. Therefore, the efficiency for internal tide generation of 7 kW m^{−1} is about 3%, somewhat less than the 6% at the Aleutian Ridge (Cummins et al. 2001) and 10% at the Hawaiian Ridge (Ray and Mitchum 1997). The Aleutian and Hawaiian Ridges are likely more efficient at generating internal tides because they block more of the surface tidal flow.

### f. Spectra and cospectra

Vertical wavenumber spectra are computed from WKB-scaled profiles of vertical displacement *ξ̃*, reduced-pressure anomaly *p̃,* and the two horizontal velocity components (*ũ, **υ̃*) both by (i) least squares fitting to standing modes, and (ii) Fourier transforming the Hanning-tapered profiles. The vertical spacing in stretched depth coordinates is not uniform and so the stretched profiles were averaged into 50-sm depth bins over most of the water column for the Fourier transform. In the uppermost 400 sm, where there were not enough data to average, profiles were interpolated onto the 50-sm grid. The mode fits resolve the lowest 20 modes. The Fourier transforms resolve vertical wavenumbers corresponding to modes 2 and higher.

Based on the flux properties (Fig. 12), the study region is divided into four regions: (i) stations south of the escarpment, (ii) stations over the ridge, (iii) intermediate north stations 10–90 km from the ridge, and (iv) far north stations. Spectral estimates from these regions were averaged to obtain variance-preserving vertical wavenumber spectra for energy *k*_{z }*S*[E](*k*_{z}) (Fig. 13a) as well as meridional velocity–pressure coherence coh[*υ̃**p̃*](*k*_{z}) (Fig. 13b), phase *ϕ*[*υ̃**p̃*](*k*_{z}) (Fig. 13c), and variance-preserving cospectra *k*_{z}co*S*[*υ̃**p̃*](*k*_{z}) (Fig. 13d).

Over the ridge, variance-preserving energy spectra are more energetic than the canonical Garrett–Munk internal-wave model spectrum at intermediate wavenumbers *k*_{z} = (0.6–2) × 10^{−2} rad sm^{−1} (vertical wavelengths *λ*_{z} = 300–1000 sm; *N*_{0} = 2.5 × 10^{−3} rad s^{−1}). Elsewhere, the spectra are comparable to or lower than the GM model.

The coherence between reduced pressure and meridional velocity coh[*υ̃**p̃*](*k*_{z}) is high at low wavenumber *k*_{z} = (2–10) × 10^{−3} rad sm^{−1} (*λ*_{z} = 600–3000 sm) in the south and intermediate north regions (Fig. 13b). In the far north, significant coherence is found only at intermediate wavenumbers *k*_{z} = (5–10) × 10^{−3} rad sm^{−1} (*λ*_{z} = 600–1200 sm) where energy densities are low (Fig. 13a). Meridional velocity and pressure are 180° out of phase to the south and over the ridge and in phase to the north (Fig. 13c), consistent with meridional energy fluxes away from the ridge.

Variance-preserving vertical wavenumber cospectra between meridional velocity and reduced pressure *k*_{z}co*S*[*υ̃**p̃*](*k*_{z}) (Fig. 13d) reveal the partition of meridional energy flux by vertical wavenumber. Most of the energy-flux covariance is at low wavenumbers *k*_{z} < 10^{−2} rad sm^{−1} (*λ*_{z} > 600 sm), implying that modes higher than 10 contribute negligible energy flux. One would expect the energy-flux cospectra to be redder than the energy spectra by *k*^{−1}_{z}. To the south (north), modes 1 and 2 contain 40% (56%) and 27% (33%) of the total energy flux, respectively. The cospectra exhibit negative (positive), or southward (northward) energy fluxes, at all wavenumbers to the south (north), consistent with the spatial-domain fluxes (Fig. 9). Though energy density is high atop the ridge (Figs. 7 and 13a), the cospectrum there is only weakly south, consistent with Figs. 9 and 12. Cospectra are weakly north in the far north stations.

### g. Turbulence estimates

Microstructure measurements were made with two shear probes mounted at the bottom end of AVP. However, these were contaminated by the wake of drop weights hanging below the profiler. Therefore, turbulent kinetic energy dissipation rates ɛ here are estimated using a finescale parameterization (Sun and Kunze 1999; Gregg 1989; Polzin et al. 1995) based on internal wave–wave interaction theory (McComas and Müller 1981; Henyey et al. 1986)

where ɛ_{0} = 4.1 × 10^{−11} W kg^{−1}, *N*_{0} = 5.2 × 10^{−3} rad s^{−1} is the reference buoyancy frequency, 〈*V*^{2}_{z}〉 is the shear variance based on the shear spectral level at wavenumbers below the finescale break in spectral slope (at vertical wavelengths of about 10 m), 〈_{GM}*V*^{2}_{z}〉 is the open-ocean (GM: Garrett and Munk 1975; Cairns and Williams 1976; Gregg and Kunze 1991) shear spectral level calculated in the same way, *R*_{ω} = 〈*V*^{2}_{Z}〉/(*N*^{2}〈*ξ*^{2}_{z}〉) is the shear/strain ratio, and we use the upper bound

from Sun and Kunze (1999). This shear-and-strain parameterization gives similar estimates as that of Polzin et al. (1995). Here, shear and strain variances are calculated spectrally in half-overlapping 32-point (320 m long) profile segments starting from the bottom. Dissipation rates ɛ are calculated from non–WKB-scaled individual profiles.

An across-escarpment section of station-average turbulent kinetic energy dissipation rate 〈ɛ〉_{t} (Fig. 14) reveals localized 500-m-thick layers with dissipation rates in excess of 10^{−8} W kg^{−1} above the ridge crest [(2–5) × 10^{−8} W kg^{−1}] and near the surface at the station closest to the first surface reflection 60 km north of the ridge. Integrating the ridge crest dissipation rates with depth and across 3.5 km yields an upper-bound estimate of local turbulent dissipative losses of 0.1 kW m^{−1} associated with spring tide forcing, only 1% of the ∼7 kW m^{−1} of tidal energy radiating away as internal waves (Fig. 12). Thus, the bulk of the surface tide loss radiates away from the escarpment as internal waves and only 1% is lost locally at the escarpment to turbulence.

Turbulent eddy diffusivities are calculated following Osborn (1980) with a “mixing efficiency” *γ* = 0.2 consistent with high-Reynolds-number turbulence,

An across-escarpment section of turbulent eddy diffusivity *K* (Fig. 15) reveals 100–500-m-thick layers with eddy diffusivities exceeding 10^{−3} m^{2} s^{−1} near the bottom above the ridge crest [(30–50) × 10^{−4} m^{2} s^{−1}], on the south flank and poleward of the first bottom reflection to the north. Elsewhere, diffusivities are *O*(0.05 × 10^{−4} m^{2} s^{−1}), consistent with typical open-ocean values. Because stratification is high near the surface, the high dissipation rates near the first surface reflection to the north are not associated with high diffusivities *K* as seen by the *N*^{−2} dependence in (15).

### h. Stratification anomalies

An across-escarpment section of stratification anomalies, where anomalies are defined relative to the southmost station at 39°35′N (Fig. 16), reveals layers of weakened stratification (*δN*^{2}/*N*^{2} < −0.5) several hundred meters thick (sandwiched above and below by layers of elevated stratification) over the ridge crest and at stations within 10 km of the escarpment. The available potential energy anomaly associated with these stratification anomalies is 1–7 J kg^{−1}. If the observed turbulent mixing (Fig. 15) is responsible for the weakened stratification (4 × 10^{−8} W kg^{−1}), it would take 1–5 years to produce the anomaly, requiring extremely long residence times. Water-mass anomalies of warmer saltier water are evident at depths of 200–500 m above the ridge but not in the bottom few hundred meters and so cannot account for the stratification anomalies.

The stratification anomalies imply anticyclonic flow around the ridge's topographic high (Fig. 2). Scaling of the thermal-wind balance suggests geostrophic currents of *O*(10 cm s^{−1}). Similar time-mean zonal velocities are seen atop the ridge (not shown). While it is possible that water over the ridge has residence times of years in which to mix, the doming and anticyclonic flow is reminiscent of vortex caps found atop seamounts. In the absence of strong mean flows, Kunze and Toole (1997) demonstrated that the vortex cap atop Fieberling Guyot was maintained by tidal rectification, not Taylor–Proudman dynamics. Similar forcing may occur here.

## 5. Summary and discussion

Full-depth profiles across the Mendocino Escarpment (Figs. 1 and 2) measure barotropic tidal currents consistent with satellite altimetry (TPXO.3: Egbert et al. 1994) and strong baroclinic semidiurnal (12 h) signals (Figs. 4–5), which account for 60%–90% of the rms WKB-normalized (*N*_{0} = 2.5 × 10^{−3} rad s^{−1}) meridional velocities of 4–7 cm s^{−1} and isopycnal displacements of 12–20 m. Energy (Figs. 7, 9, 11–12) radiates away from the escarpment, confirming that the baroclinic tides are produced by surface tide currents interacting with the abrupt topography of Mendocino Escarpment. Vertically integrated fluxes from the escarpment are 7 kW m^{−1}. The northward energy flux drops by 2 kW m^{−1} near the first surface reflection, then to zero at the northmost station (Fig. 12). Elevated near-surface turbulent dissipation rates at the station closest to the surface reflection (Fig. 14) suggest losses to turbulent dissipation. Sampling did not extend far enough equatorward to evaluate fluxes beyond the first surface reflection south of the ridge.

Localized energetic packets appear to follow trajectories parallel to semidiurnal ray paths. These packets can be traced upward to the first surface reflection 50–70 km from the escarpment and, to the south, through bottom reflections. Localized packets are consistent with contributions from many modes as predicted by internal tide generation theory (e.g., Rattray et al. 1969; Prinsenberg and Rattray 1975) and found in past observations (Torgrimson and Hickey 1979; DeWitt et al. 1985; Pingree and New 1989; Lien and Gregg 2001) though the lowest modes dominate (Fig. 13).

Over the ridge crest, energies are high (Fig. 7) but energy fluxes are weak because of low coherence between meridional velocity *υ̃* and reduced pressure *p̃* (Figs. 9 and 12). Similarly, high energy but weak fluxes are found over Kaena Ridge between Oahu and Kauai (Kunze et al. 2000, 2002b). Numerical simulations suggest that beams produced on opposing flanks cross over the ridge produce zero net fluxes (M. A. Merrifield and P. E. Holloway 2002, personal communication).

While satellite altimetry (Ray and Mitchum 1997; Kantha and Tierney 1997) provides global estimates of baroclinic energy flux, these authors emphasize that these are underestimates because repeat paths occur only once every 9.9 days, requiring multiyear records to isolate the semidiurnal (*M*_{2}) contribution. Thus, satellite estimates include only the signal coherent with the surface tide, failing to detect the substantial internal tide known to exist in the Bay of Biscay (e.g., Pingree and New 1989, 1991), for example. Further, repeat tracks are separated by 150 km, more than the horizontal wavelength of even the lowest baroclinic mode, making it possible to miss a localized surface signal. Near the Mendocino Escarpment, our peak in situ energy flux of 4 kW m^{−1} is almost 10 times the 0.5 kW m^{−1} estimate from satellite altimetry (R. Ray 2002, personal communication). Last, altimetry only reliably detects mode one. While mode-one fluxes dominate north of the Mendocino Escarpment (Fig. 13d), over one-half of the energy flux to the south is in modes 2 and higher.

The outgoing baroclinic energy flux of 7 kW m^{−1} represents only 3% of the poleward semidiurnal surface tidal energy flux of 220 kW m^{−1}. Another possible sink for surface tidal energy is local turbulent dissipation (Munk and Wunsch 1998). Turbulent dissipation rates over the ridge crest are 100 times typical open-ocean interior values. Despite this, integrated vertically and across the ridge crest (Fig. 14), turbulent dissipations are only 0.1 kW m^{−1}. Thus, the bulk of the surface tide loss radiates away as low-mode internal waves with only ∼1% locally lost to turbulence. Similar partitions have been inferred at the Hawaiian Ridge (Klymak et al. 2002) and the sill in Knight Inlet (Klymak and Gregg 2002, manuscript submitted to *J. Phys. Oceanogr.*). The low internal wave modes that carry the bulk of the energy flux are unlikely to break to produce turbulence while the very high wavenumbers likely to be locally dissipated as turbulence contain very little energy flux.

Our results contrast with Munk and Wunsch's (1998) assumption that 70% of the surface tide losses are lost to abyssal turbulence. However, their assumption may still be globally valid if the bulk of the radiated internal tide field breaks near the bottom. For the Mendocino Escarpment, one-half of this appears to occur within 100 km based on the 2 kW m^{−1} decrease in energy flux 50–70 km north of the escarpment (Fig. 12) and dissipation rates in excess of 10^{−8} W kg^{−1} in the upper 500 m near the first surface reflection (Fig. 14). Thus, tide–topography interactions at the Mendocino Escarpment contribute little to abyssal mixing. This does not appear to be universal because mode-1 internal tides have been observed to propagate with little dissipation 1000 km through many surface and bottom reflections from the Hawaiian Ridge and Tuomuto Archipelago (Ray and Mitchum 1997) and the Aleutians (Cummins et al. 2001). St. Laurent and Garrett (2002) suggest that less than 10% of the low-mode energy flux will be scattered to high wavenumbers upon each bottom reflection off subcritical abyssal hills. Over a broad field of rough topography such as the Mid-Atlantic Ridge, Polzin (2002, manuscript submitted to *J. Phys. Oceanogr.*) argues that low-mode internal tides are lost to abyssal turbulence more efficiently through successive bottom scattering to high wavenumber. In the absence of such bottom roughness, for example, away from slow tectonic spreading centers, other possible sinks include breaking on continental slopes, which are often near-critical at the semidiurnal frequency (Cacchione et al. 2002; Nash et al. 2002, manuscript submitted to *J. Phys. Oceanogr.*), refraction by strong currents, and parametric subharmonic instability internal wave–wave interactions equatorward of 28° (Hirst 1991; Hibiya et al. 1998).

Our outgoing energy flux of 7 kW m^{−1} is somewhat larger than the 4.7 kW m^{−1} inferred by Morozov (1995) but includes an *S*_{2} contribution. The Mendocino flux can be compared with the 6–15 kW m^{−1} observed near the Hawaiian Ridge (Ray and Mitchum 1997; Merrifield et al. 2001; Kunze et al. 2000, 2002b), 5 kW m^{−1} into the mouth of the Monterey Submarine Canyon (Kunze et al. 2002a), 3 kW m^{−1} from the Aleutian Ridge (Cummins et al. 2001), and 1 kW m^{−1} onshore from the slope off of the Australian North West Shelf (Holloway et al. 2001). While not the largest, and below satellite altimetry surface tide dissipation noise levels (M. D. Egbert 2002; personal communication), internal tide radiation from the Mendocino Escarpment confirms that a substantial fraction of surface tide losses occur in the deep ocean through tide–topography interactions that generate internal waves. Numerous other small-scale topographic features in the ocean may contribute significantly to surface tide dissipation while individually being unresolved by satellite altimetry.

Assuming uniform semidiurnal internal tide generation along the 400-km length of the Mendocino Escarpment critical slopes, the total integrated energy flux radiating from the Mendocino Escarpment (north plus south) is ∼3 GW; this may be an overestimate because off-transect stations suggest smaller fluxes (Fig. 11). It is a small fraction of the 600–800 GW total believed to dissipate in the deep ocean (Egbert and Ray 2001) (though coincidentally the fraction is identical to the fraction of the ocean floor covered by our observations). For comparison, integrated fluxes are 15 GW from the Hawaiian Ridge (Ray and Mitchum 1997), 2 GW from Amukta Pass in the Aleutian Ridge (Cummins et al. 2001), 1 GW from Chatham Rise east of New Zealand (Chiswell 2000), and roughly 100 GW each from the Mid-Atlantic Ridge, Micronesia/Melanesia and the west. Indian Ocean (Egbert and Ray 2001).

The 2 kW m^{−1} drop in northward energy flux 70–90 km north of the escarpment (Fig. 12) appears to be the result of dissipative losses associated with elevated turbulence near the first surface reflection (Fig. 14). The case for this will be made below after ruling out (i) temporal aliasing due to fortnightly modulation in the forcing and (ii) lateral spreading with distance due to finite length of the ridge.

The lowest two baroclinic modes, which dominate the energy flux north of the ridge, have horizontal group velocities of *C*_{gH} = 3 and 1.5 m s^{−1}, allowing them to travel 200 and 100 km day^{−1}. Therefore, fortnightly modulation in the forcing of fluxes should be communicated rapidly across the entire transect as compared with a fortnight (Fig. 13). Scaling the baroclinic fluxes (Fig. 12) with the barotropic forcing did not remove the flux convergence between 70–90 km north of the escarpment, suggesting that fortnightly modulation of the semidiurnal signal (i) is not responsible for 2 kW m^{−1} drop in energy flux.

Idealized numerical simulations that exhibit little radial spreading also show a sharp drop in the outgoing energy flux roughly a mode-1 horizontal wavelength from a finite-length ridge (Holloway and Merrifield 1999). The more variable three-dimensional topography of the real ocean may produce greater spreading, and our 2D picture could be further complicated by zonally narrow beams moving on and off our sampling transect as well as offridge internal tide generation by topography north of the escarpment. While our spatial sampling is inadequate to address the issue of lateral spreading, the orientations of the depth-integrated energy-flux vectors (Fig. 11) are normal to the Mendocino Escarpment with little spreading as in the simulations, suggesting we can rule out (ii).

Average turbulent kinetic energy dissipation rates exceed 10^{−8} W kg^{−1} during spring tide in a 500-m-thick surface layer at the station 65 km north of the ridge that is closest to the first surface reflection (Fig. 14). This station coincides with the large drop in outgoing energy flux (Fig. 12). Such surface intensification is not observed at other stations. Integrated over 500 m vertically and 50 km meridionally, the inferred dissipation rates 〈ɛ〉_{t} yield total dissipations of 0.5 kW m^{−1}, as compared with the 2 kW m^{−1} needed to balance the drop in outgoing energy flux (Fig. 12). Given our coarse meridional sampling, it is easy to imagine that higher dissipation rates were present north or south of this station's position. This suggests that the 2 kW m^{−1} energy-flux convergence is lost to turbulence at the first surface reflection. Higher-resolution sampling of the surface reflection would be desired in future measurements to confirm this. Perhaps the particularly strong near-surface stratification due to the 1997 El Niño led to stronger turbulent losses than usual near the Mendocino Escarpment.

Elevated turbulent dissipation rates near the first surface reflection could arise from instability of the internal tide through WKB scaling as it propagates into high near-surface stratification. Horizontal velocity scales as *N*^{1/2} and vertical wavenumber *k*_{z} by *N* so that shear *υ*_{z} scales as *N*^{3/2} and Richardson number as *N*^{−1}. Assuming WKB-scaled vertical wavelengths of 100 sm, deep WKB-scaled energies would have to exceed 0.006 m^{2} s^{−2} for the near-surface buoyancy frequencies of 20 × 10^{−3} rad s^{−1} to give rise to unstable shears; such energy levels are observed in the energetic packets (Fig. 7). Instability of the internal tide might also arise from the superposition of incident up- and reflected downgoing wave packets. Rapid energy cascades from semidiurnal frequencies to turbulence through parametric subharmonic instability are confined to latitudes equatorward of 28° (Hirst 1991; Hibiya et al. 1998) and so are not expected at the Mendocino Escarpment.

Bottom reflections 10–20 km north and ∼30 km south of the ridge are not associated with large flux convergences (Fig. 9) or dissipation rates (Fig. 14). Weak abyssal stratification implies weaker horizontal flows, reducing nonlinearity since WKB scaling of momentum nonlinearity goes as (**v** · ∇)**v** ∝ *N*^{2}. Nonlinear effects will be more important near the surface where high stratification amplifies horizontal velocities and their gradients. On the other hand, weaker stratification at depth amplifies abyssal eddy diffusivities *K* ∼ *N*^{−2} (Fig. 15). In 100–500-m-thick bottom layers over the ridge crest, on the south flank and north of the first bottom reflection, eddy diffusivities are more than 100 times as large as the 0.05 × 10^{−4} m^{2} s^{−1} typically found in the ocean interior, exceeding 10 × 10^{−4} m^{2} s^{−1} (Fig. 15). Similarly elevated diffusivities have been reported over other ridges (Ledwell et al. 2000; Klymak et al. 2002) and seamounts (Nabatov and Ozmidov 1988; Kunze and Toole 1997; Lueck and Mudge 1997). Kunze and Toole (1997) demonstrated that these boundary mixing levels may be sufficient to elevate basin-average diffusivities above 10^{−4} m^{2} s^{−1} as proposed by Munk and Wunsch (1998), but only below 4000-m depth. It is possible that sufficiently elevated bottom boundary mixing has yet to be sampled at shallower depths but such eddy diffusivities would have to exceed 1000 × 10^{−4} m^{2} s^{−1} to impact basin averages above 4000 m in light of sampling to date. Bryden and Nurser (2003) suggest that hydraulically controlled mixing over deep sills easily dominates over interior and boundary mixing in the abyssal ocean. These processes tend to be confined to depths below 4000 m as well.

Further modeling and observational studies are needed to evaluate the partition between internal tide radiation and turbulence generation, as well as the ultimate fate of radiated internal tides. These are likely to depend on properties of the topography (supercritical vs subcritical, isolated vs homogeneous) and so will require multiple approaches. Considerable recent theoretical effort has revisited Bell's (1975) subcritical slope model (Llewellyn Smith and Young 2002; Khatiwala 2003; M. Li 2002, personal communication; St. Laurent and Garrett 2002; Polzin 2002), extending predictions to near-critical slopes for some specific topographies. However, Egbert and Ray's (2001) maps of surface tidal losses show hot spots often localized at isolated supercritical topography where the approaches of Prinsenberg and Rattray (1975) or Baines (1982) seem more appropriate. For complicated 2D supercritical topography, numerical modelling (e.g., Merrifield et al. 2001) may be necessary.

More detailed studies near surface and bottom reflections are needed to quantify turbulent losses. Localized strong turbulent mixing where internal tide beams approach the surface may be important sites for nutrient and water-mass exchange between the surface mixed layer and upper pycnocline. Ffield and Gordon (1996) reported evidence for fortnightly modulation of sea surface temperature in Indonesian seas. If the mixing was strongly localized as here, it would easily be missed by in situ microstructure measurements without consideration of internal tide dynamics. The Hawaiian Ocean Mixing Experiment will provide new insights into tide-generated internal waves and turbulence at isolated supercritical topography.

## Acknowledgments

The authors thank the captain and crew of the R/V *Wecoma* for all their help: engineers John Dunlap, Art Bartlett, and Bob Drever, as well as Eric Boget, Dicky Allison, and James Girton. Chris Fox and Andra Bobbitt are thanked for providing the NOAA Vents Program Seabeam bathymetry over the Mendocino Escarpment. Lou St. Laurent and an anonymous reviewer provided numerous helpful suggestions for improving the manuscript. This research was funded under NSF Grant OCE 95-29735. Alana Althaus was funded by a 1998 NDSEG Fellowship.

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## Footnotes

*Corresponding author address:* Eric Kunze, APL and School of Oceanography, University of Washington, 1013 NE 40th Street, Seattle, WA 98105-6698. Email: kunze@ocean.washington.edu