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  • View in gallery

    (left) Plan view of Kaena Ridge. Darkest colors correspond to 5000-m depth. FLIP was trimoored on the south rim (white X). A hill rises ~500 m above the ~1000-m ridge crest, 8 km to the north. (right) The average buoyancy profile during the Nearfield experiment.

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    (a) Cross section of Kaena Ridge showing instantaneous CR current during the flow reversal of the barotropic forcing. (b) A cross section showing the measured energy fluxes at the FLIP site and theoretical semidiurnal ray trajectories (Rainville and Pinkel 2006a). (c) A schematic of the site (on the right) with FLIP’s location, wave ray patterns, and particle trajectories (red) indicated. The blue arrows illustrate the barotropic forcing velocity. For an isolated wave beam, the vertical divergences of the measured stress d(u2u3)/dx3 is presumably balanced by the unmeasured gradient . Particle trajectories are linear within an isolated beam and elliptical in the near field of generation and reflection sites (following Sandstrom 1966).

  • View in gallery

    (a) A schematic of the FLIP sensing systems showing the sampling patterns of the Deep-8 sonar and the profiling CTD. In fact, two CTDs were profiled to cover the 800-m observation window. (b) The Deep-8 sonar being deployed. (c) The 4.5-m nominal depth resolution of the sonar was increasingly degraded by the spreading of the sonar beam with range.

  • View in gallery

    A 7-day record of zonal velocity from the Deep-8 sonar. The shading is proportional to vertical shear. The downward-propagating D2 crests, dominant over 300–700 m, are scarred by the aliased seafloor echo at ~550 m. The scar at ~400 m indicates the position of the sonar. A 35-day record was obtained in the HOME Nearfield experiment.

  • View in gallery

    Profiles of variance vs squared buoyancy frequency for (top) vertical displacement and vertical velocity and (bottom) LR and CR horizontal velocity. (left) Variance across the full internal wave band, 0.06–180 cpd, and (right) variance from the D2 tidal band, 1.5–2.4 cpd. In a WKB wave field with separable wavenumber and frequency dependence, displacement and vertical velocity variance are proportional to N−1 while horizontal velocity variance goes as N+1. Note that the abscissa differs in each plot. Displacement profiles are reduced by a factor of (top left) 10−6 and (top right) 10−7 relative to the vertical velocity profiles.

  • View in gallery

    (a) The CR spectrum of horizontal velocity and the (b) coherence, (c) cospectrum, and (d) quadrature spectrum of CR and vertical velocity 〈u2u3〉. Spectral estimates are averaged in frequency and in depth, from 100–200 m (at top) to 600–700 m (at bottom). Inertial (red); D1 (magenta); and D2, D4, D6, and D8 frequencies are indicated. In estimating statistical stability, it is assumed that the 100-m vertical average is equivalent to an average of two independent spectral estimates. In (a), upper and lower 95% confidence bounds are presented. In (b), the magenta lines give the expected value of coherence when the true coherence is nil.

  • View in gallery

    The spectrum of an Eulerian record of vertical velocity at 200 m (black) displays peaks in the D1, D2, and D4 bands and a pronounced cutoff at the buoyancy frequency. The D2 filtered vertical and CR velocity spectra (black and red) isolate the tidal peak but also include variance from the spectral line at D2 + f. The raw Reynolds product u2u3 (blue) has variance at low frequencies and in the D4 band. A low-pass filter removes the D4 variance and leaves a stress signal with a ~1-cpd bandwidth (green).

  • View in gallery

    The D2 bandpass filtered (top) vertical displacement and (middle) LR and (bottom) CR horizontal velocities. Measurements are presented in an Eulerian frame, with displacement taken as the time integral of the filtered vertical velocity. The subtle phase shifting of these long-wavelength motions is responsible for the depth variability of the stress.

  • View in gallery

    A 5-day record of the D2 Reynolds product u1u3 during the second spring tide of the experiment. When horizontal and vertical velocities are large and in quadrature, alternating red and blue bands are seen at frequency 2D2 (e.g., yearday 277 and depths 600–800 m). When the velocities are in phase or antiphase, the difference-frequency signal is the Reynolds stress, which is seen as the changing mean blue or red color of the image.

  • View in gallery

    The LR and CR (a),(b) Reynolds stress and (c),(d) mass flux for the 35-day Nearfield record. A 24.8-h running mean filter converts perturbation maps (Fig. 9) into these flux maps. The maps are shifted in time to account for the lag of the filter. The gray region at depths below 700 m starting at yearday 280 represents data lost because of CTD failure.

  • View in gallery

    Cruise-averaged summary of (a) D2 Reynolds stress, (b) lateral mass flux, (c) E–P flux, and (d) the azimuth of the (upward) fluxes. Azimuth is given in terms of ridge coordinates, such that 0° and 360° represent an upward LR flux toward Oahu. The green reference marks the offshore (negative) CR azimuth. Black vertical reference lines indicate Earth-frame azimuth. Data are smoothed in the vertical over 30 m here and in Figs. 12, 13, and 16.

  • View in gallery

    Stress and flux estimates for the first (blue; yeardays 260–268) and second (red; yeardays 277–284) spring tides of the cruise and the intervening neap tide (magenta; yeardays 269–276). Columns give (left) LR, (middle) CR fluxes, and (right) flux azimuth.

  • View in gallery

    Vertical profiles of (a) upward Reynolds stress, (b) lateral mass flux, and (c) E–P flux over the south rim of Kaena Ridge.

  • View in gallery

    (top) CR Reynolds stress (m2 s−2), (middle) CR mass flux and LR Reynolds stress (m2 s−2), and (bottom) energy flux (W m−2) for the free-wave linear WKB solution for barotropic D2 flow over a 2D ridge. The contribution from the first five vertical modes is presented. The color scale (positive values are red) is the same for (top) and (middle).

  • View in gallery

    As in Fig. 14, but with the first 20 vertical modes included.

  • View in gallery

    (top) Reynolds stress, (middle) mass flux, and (bottom) E–P flux for the linear 2D solutions for flow over Kaena Ridge, from the linear 2D WKB model of Echeverri and Peacock (2010), using 20 vertical modes (red). The nonlinear, 3D POM predictions are given in blue. Observations are presented in black. Columns give (left) LR fluxes, (middle) CR fluxes, and (right) flux azimuth.

  • View in gallery

    The ridge-crest topography with the D2 rays arriving at FLIP traced backward to their presumed site of generation. The black dot represents FLIP, to scale; the blue line below is the section of the water column sampled; and the light blue extension gives the water depth at the measurement site. The color of the dots at the point of origin of each ray corresponds to the magnitude of the Reynolds stress along the ray (m2 s−2).

  • View in gallery

    The correlation between horizontal and vertical velocities in the band cph.

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Semidiurnal Baroclinic Wave Momentum Fluxes at Kaena Ridge, Hawaii

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  • 1 Scripps Institution of Oceanography, La Jolla, California
  • 2 Applied Physics Laboratory, University of Washington, Seattle, Washington
  • 3 School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada
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Abstract

Kaena Ridge, Hawaii, is a site of energetic conversion of the semidiurnal barotropic tide. Diffuse baroclinic wave beams emanate from the critical-slope regions near the ridge crest, directed upward and southward from the north flank of the ridge and upward and northward from the south flank. Here, the momentum fluxes associated with generation at the ridge are estimated. Continuous vertical profiles of density and velocity from 80 to 800 m were obtained from the Research Platform Floating Instrument Platform (FLIP) over the southern edge of the ridge, as an aspect of the Hawaii Ocean Mixing Experiment. Data are used to estimate the Reynolds stress, Eulerian buoyancy flux, and the combined Eliassen–Palm flux in the semidiurnal band. An upward–southward stress maximum of ~0.5 × 10−4 m2 s−2 appears at depths of 300–500 m, generally consistent with beam-like behavior. A strong off-ridge buoyancy flux (~0.3 × 10−4 m2 s−3) combines with large along-ridge Reynolds stresses to form an Eliassen–Palm flux whose along-ridge and across-ridge magnitudes are comparable. The stress azimuth rotates clockwise with increasing altitude above the ridge crest. The principal upward–southward beam is found to be at depths 100–300 m shallower than are predicted by an analytic two-dimensional (2D) model and a 3D numerical simulation. This discrepancy is consistent with previous observations of the baroclinic energy flux. If these observed tidal momentum fluxes were to diverge in a 100-m-thick near-surface layer, the forcing would be comparable to a moderate wind stress. Pronounced lateral gradients of baroclinic tidal stresses can be expected offshore of Hawaiian topography.

Corresponding author address: Robert Pinkel, Scripps Institution of Oceanography, La Jolla, CA 92093-0213. E-mail: rpinkel@ucsd.edu

Abstract

Kaena Ridge, Hawaii, is a site of energetic conversion of the semidiurnal barotropic tide. Diffuse baroclinic wave beams emanate from the critical-slope regions near the ridge crest, directed upward and southward from the north flank of the ridge and upward and northward from the south flank. Here, the momentum fluxes associated with generation at the ridge are estimated. Continuous vertical profiles of density and velocity from 80 to 800 m were obtained from the Research Platform Floating Instrument Platform (FLIP) over the southern edge of the ridge, as an aspect of the Hawaii Ocean Mixing Experiment. Data are used to estimate the Reynolds stress, Eulerian buoyancy flux, and the combined Eliassen–Palm flux in the semidiurnal band. An upward–southward stress maximum of ~0.5 × 10−4 m2 s−2 appears at depths of 300–500 m, generally consistent with beam-like behavior. A strong off-ridge buoyancy flux (~0.3 × 10−4 m2 s−3) combines with large along-ridge Reynolds stresses to form an Eliassen–Palm flux whose along-ridge and across-ridge magnitudes are comparable. The stress azimuth rotates clockwise with increasing altitude above the ridge crest. The principal upward–southward beam is found to be at depths 100–300 m shallower than are predicted by an analytic two-dimensional (2D) model and a 3D numerical simulation. This discrepancy is consistent with previous observations of the baroclinic energy flux. If these observed tidal momentum fluxes were to diverge in a 100-m-thick near-surface layer, the forcing would be comparable to a moderate wind stress. Pronounced lateral gradients of baroclinic tidal stresses can be expected offshore of Hawaiian topography.

Corresponding author address: Robert Pinkel, Scripps Institution of Oceanography, La Jolla, CA 92093-0213. E-mail: rpinkel@ucsd.edu

1. Introduction

In the ocean, internal waves are ubiquitous (Garrett and Munk 1972, 1975; Munk 1981), and their ability to transport momentum is potentially significant. For example, if an upward-propagating plane wave with horizontal velocity u1 ~ 0.1 m s−1 and vertical velocity u3 = 0.001 m s−1 is completely dissipated as it encounters the sea surface, the stress on the surface is roughly 1 N m−2, comparable to that exerted by a 5–10 m s−1 wind.

For the atmosphere, it has long been appreciated (e.g., Lindzen 1973) that gravity wave momentum fluxes play a significant role in the global zonal momentum balance. The development of remote sensing techniques (e.g., Lhermitte 1968; Reid and Vincent 1987) led to confirmation of wave momentum transport in the middle atmosphere and to the identification of topography, boundary layer turbulence, convective activity, etc., as important wave sources.

However, there is no corresponding open-ocean evidence for wave momentum fluxes of significant magnitude. In the pioneering investigations of Ruddick and Joyce (1979), Brown and Owens (1981), and Plueddemann (1987), statistically significant wave Reynolds stresses were not observed.1 The quasi isotropy of the oceanic wave field in the horizontal is associated with near-zero correlations between horizontal and vertical velocities and correspondingly small momentum fluxes.

Recently, evidence of anisotropic, beam-like propagation has been detected associated with tidal-frequency motions. Zones of enhanced turbulent dissipation have been found emanating from topographic features (Lueck and Mudge 1997; Lien and Gregg 2001), outlining the form of semidiurnal (D2) tidal-frequency wave beams.2 The suggestion is that the tidal shear is sufficiently intense in these beams to trigger local instability.

Beam-like patterns in energy density and related signals are also seen in the Hawaii Ocean Mixing Experiment (HOME). The beams are detected in spatial transects oriented normal to Kaena Ridge, Hawaii (Fig. 1), (Martin et al. 2006; Cole et al. 2009), a site of intense D2 barotropic to baroclinic conversion. The horizontal D2 wave energy flux is also confined in distinct beams (Lee et al. 2006; Rainville and Pinkel 2006a). Can significant momentum fluxes be detected in this highly anisotropic situation?

Fig. 1.
Fig. 1.

(left) Plan view of Kaena Ridge. Darkest colors correspond to 5000-m depth. FLIP was trimoored on the south rim (white X). A hill rises ~500 m above the ~1000-m ridge crest, 8 km to the north. (right) The average buoyancy profile during the Nearfield experiment.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

In a rotating, horizontally homogeneous environment, the horizontal Eulerian wave pseudomomentum and the vertical pseudomomentum flux3 are given by
eq1
and
eq2
Time changes in the pseudomomentum at a fixed position are associated with a divergence of the flux in the vertical and a corresponding acceleration of the quasigeostrophic background flow. Here, b = −g(ρρ0)/ρ0 is the buoyancy perturbation associated with wave passage; {ui} are the components of wave velocity, with u3 upward; N2 = −(g/ρ0)0/dx3 is the squared buoyancy frequency; and f is the Coriolis frequency. The brackets 〈 〉 indicate an average over many wave periods. Here, Βi3 is referred to as the Eliassen–Palm (EP) flux [Eliassen and Palm 1961; Müller 1976; Bühler 2009, Eq. (8.30)]. For σf, the Eliassen–Palm flux becomes the classical wave Reynolds stress, Βi3 = 〈uHu3〉.

As an aspect of HOME, the Research Platform Floating Instrument Platform (FLIP) was moored at the south rim of Kaena Ridge, collecting a 35-day record of the density and velocity fields in the upper 800 m of the water column. The time continuity of the measurements facilitates the extraction of tidal-frequency signals from the energetic nontidal signals at the site, leading to stress estimates that are of greater precision than those obtained from transect measurements. In the D2 frequency band, the coherence between horizontal and vertical velocity exceeds 0.5, reflecting the extreme anisotropy and quasi-deterministic nature of the generation process. The combination of signal strength and high coherence renders the task of estimating wave momentum transport at Kaena Ridge tractable relative to the early open-ocean studies.

Unfortunately, depth–time measurements at a single station are not adequate to map the spatially intricate propagation patterns expected near an isolated source such as the ridge. For a horizontally homogeneous wave field, a vertical profile of the EP flux is adequate to quantify wave–mean flow interaction. However, for an isolated source and ray-like wave propagation, the vertical derivative of the EP flux dBi3/dx3 can be balanced by its horizontal derivative, dBii/dxi, (Fig. 2c), such that there is no acceleration of the background flow. In HOME, these horizontal derivatives go unmeasured. [Bühler and McIntyre (1999) and Polzin (2010) discuss the complete specification of the momentum balance associated with waves from a localized source.]

Fig. 2.
Fig. 2.

(a) Cross section of Kaena Ridge showing instantaneous CR current during the flow reversal of the barotropic forcing. (b) A cross section showing the measured energy fluxes at the FLIP site and theoretical semidiurnal ray trajectories (Rainville and Pinkel 2006a). (c) A schematic of the site (on the right) with FLIP’s location, wave ray patterns, and particle trajectories (red) indicated. The blue arrows illustrate the barotropic forcing velocity. For an isolated wave beam, the vertical divergences of the measured stress d(u2u3)/dx3 is presumably balanced by the unmeasured gradient . Particle trajectories are linear within an isolated beam and elliptical in the near field of generation and reflection sites (following Sandstrom 1966).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

Nonetheless, accurate depth–time measurements can significantly constrain subsequent conjecture. Here, we attempt to quantify principal terms in the Eliassen–Palm flux and to identify the motions that support the flux. Descriptions of the site, experiment, and notation below are followed by presentations of the raw semidiurnal signals, the averaged terms in the flux, and a comparison with several model results.

2. Observations

HOME (Pinkel et al. 2000; Pinkel and Rudnick 2006) was organized to study wave and turbulent processes at a suspected site of intense barotropic to baroclinic conversion. The 2002–03 HOME Nearfield program focused on Kaena Ridge, an 80-km ridge extending west-northwest from the Hawaiian island of Oahu (Fig. 1). The ridge crests at approximately 1000 m in a background ocean 5 km deep. Energetic baroclinic tides have been detected emanating from the ridge in satellite studies of sea surface topography (Ray and Mitchum 1996). Initial HOME findings have documented the conversion process (Nash et al. 2006); the resulting baroclinic energy fluxes (Lee et al. 2006; Rainville and Pinkel 2006a); the locally generated turbulence (Klymak et al. 2006; Levine and Boyd 2006; Aucan et al. 2006); and, together with the Internal Waves Across the Pacific experiment (IWAP; Alford et al. 2007; Zhao et al. 2010), the departing baroclinic tide (Rainville and Pinkel 2006b).

The flanks of the Kaena Ridge are supercritical to semidiurnal baroclinic tides. The site of most intense tidal conversion is the ridge “shoulder,” the fringe surrounding the crest with critical slope |∇h| = |kH|/|k3|. Here, h(x1, x2) gives the height of the topography, kH = (k1, k2, 0) is the horizontal wavenumber of the D2 baroclinic tide, and k3 is the vertical wavenumber. The cross-ridge (CR) extent of the critical regions, ~5–10 km, is small relative to the overall width of the ridge, ~20–30 km. Models of the generation process (Merrifield and Holloway 2002; Fig. 2a) indicate that upward and southward energy flux emanates from the north shoulder of the ridge, whereas an upward and northward flux is initiated at the southern shoulder. A similar pattern for momentum is anticipated.

In September–October 2002, the Research Platform FLIP was trimoored at the southern edge of the Kaena Ridge, in 1100-m water (Fig. 1). The ridge is the primary conduit for a myriad of undersea communications cables connecting Hawaii with the U.S. mainland. The south-rim site avoided the cables and provided an interesting vertical sampling of anticipated patterns. The watch circle of the moor was approximately 500 m, centered at 21°40.8′N, 158°37.7′W.

An eight-beam Doppler sonar, designed and fabricated at Scripps Institution of Oceanography (SIO), was used to monitor the velocity field at the ridge (Figs. 3a,b). The sonar was deployed at a depth of 400 m. The four upward-looking beams operated at 160 kHz, whereas the downward-looking beams transmitted at 140 kHz. All beams were oriented 30° off vertical. Transmitting at 250 watts per beam, velocity estimates were obtained over the depth range 50–900 m.4

Fig. 3.
Fig. 3.

(a) A schematic of the FLIP sensing systems showing the sampling patterns of the Deep-8 sonar and the profiling CTD. In fact, two CTDs were profiled to cover the 800-m observation window. (b) The Deep-8 sonar being deployed. (c) The 4.5-m nominal depth resolution of the sonar was increasingly degraded by the spreading of the sonar beam with range.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The beam widths of the transducers, measured from the 3-dB points of the beam pattern, are ±1.5° and ±1° for the horizontal and vertical spread of the beams. A four-bit transmit sequence (Pinkel and Smith 1992) with 6.6-kHz bandwidth was repeated 11 times, yielding a nominal depth resolution of 4.5 m. Subsequent averaging during processing results in an effective convolution of the data with a triangular weighting function of base 9.0 m and width at half height of 4.5 m. Data are binned at 2.25 m, such that the Nyquist wavenumber corresponds to the fundamental resolution limit.

Actual depth resolution is strongly affected by sonar beamwidth. As range from the sonar increases, the vertical width of the acoustic beam becomes greater than the along-beam extent of the transmitted pulse (Fig. 3c). At vertical distances greater than ±220 m from the instrument, finite-beamwidth degradation of vertical resolution exceeds the finite-pulse-duration lower bound.

The echo from the 1100-m seafloor is detected in the downward-looking beams at the aliased depth of ~540 m. Given the slope of the seafloor, each beam is affected at a slightly different depth. The net result is a 10–20-m scar in the velocity estimates. The degradation is imperceptible in the D2 Eulerian data to be discussed. Higher-frequency motions, particularly when viewed in an isopycnal-following frame, are contaminated over a broader range of density space.

The TPXO.5 modeled M2 and S2 barotropic tidal velocities, calculated for this site (Egbert and Erofeeva 2002), are subtracted from the observations prior to scientific processing. FLIP’s drift velocity within the watch circle of the trimooring, as determined by GPS, is also removed.

To sample the density field, a pair of Seabird-911 CTDs was profiled at 4-min intervals. Dropping at a rate of 3.6 m s−1, one CTD transited the depth range 20–420 m, whereas the other covered depths of 370–800 m. The response times of conductivity and temperature cells are matched using a Fourier approach (Sherman and Pinkel 1991), such that the density signal is resolved to 2-m vertical scale. Gaps in the time series are rare, caused primarily by collisions or entanglement of the CTDs and the sonar. The gaps are typically short relative to tidal periods and are not apparent in the tidally filtered data. A collision on yearday 280 damaged the lower CTD cable, such that the profiling depth is subsequently restricted to 710 m. Following editing, a set of 11 768 density profiles and synchronized 4-min-averaged velocity records are available for subsequent analysis.

3. Conventions

Observations are presented in a Cartesian frame with axes (x1, x2, x3) centered at FLIP and oriented (115°, 25°, upward) such that the positive x1 axis is parallel to the ridge and toward Oahu. Positive velocities (u1, u2, u3) refer to motion in the along-ridge (LR), CR, and upward directions. The terms “eastward,” “northward,” etc., will henceforth refer to this rotated coordinate frame, such that northward flow is in the direction 25° east of geographic north. Earth-based references will be subsequently termed “geographic,” as in “geographic north.”

4. Site characterization

At the south-rim measurement site, the D2 tide appears as a predominantly low-mode phenomenon, with roughly one-quarter vertical wavelength spanning the 1100-m water column (Fig. 4). Tidal crests display downward phase propagation in the middle water column, consistent with a ridge-crest generation site. In the upper 250 m, the phase pattern becomes irregular, possibly a result of the surface reflection of tidal signals generated at distant ridge-top locations.

Fig. 4.
Fig. 4.

A 7-day record of zonal velocity from the Deep-8 sonar. The shading is proportional to vertical shear. The downward-propagating D2 crests, dominant over 300–700 m, are scarred by the aliased seafloor echo at ~550 m. The scar at ~400 m indicates the position of the sonar. A 35-day record was obtained in the HOME Nearfield experiment.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The shading in Fig. 4 highlights regions of high-shear, principally due to near-inertial waves. The upward inertial phase propagation suggests a near-surface source for these motions or a nearby surface reflection. The slowly propagating inertial shear is vertically advected by the tide. It can be used as a tracer of tidal vertical velocity. In the top 400 m, shear layers are generally seen to rise (upward tidal velocity) during periods of westward (green) flow. Descending layers are found during eastward (red) phases of the tide. This is the signature of a net upward–westward D2 momentum flux. The patterns are more apparent when density rather than shear is used as the tracer of vertical motion.

Profiles of wave vertical displacement and vertical velocity variance (Figs. 5a,b) depart significantly from classical Wentzel–Kramers–Brillouin (WKB) scaling (Garrett and Munk 1975; Munk 1981; Gregg and Kunze 1991), a consequence of the proximity of the sea surface, the active generation at the seafloor, and the large vertical scale of the dominant waves relative to depth changes in N2. Below 400 m, variance levels grow rapidly with increasing depth. Maximum values for rms vertical velocity approach 0.02 m s−1 across the full internal wave band and 0.003 m s−1 for a band of width 0.9 cpd centered on the D2 tide. Broadband vertical displacement (Fig. 5a) grows irregularly with increasing depth, decreasing N2. Peak displacements of ~30-m rms are found at depths greater than 600 m. In the tidal band, displacement grows as N−4, peaking at 25-m rms below 600 m.

Fig. 5.
Fig. 5.

Profiles of variance vs squared buoyancy frequency for (top) vertical displacement and vertical velocity and (bottom) LR and CR horizontal velocity. (left) Variance across the full internal wave band, 0.06–180 cpd, and (right) variance from the D2 tidal band, 1.5–2.4 cpd. In a WKB wave field with separable wavenumber and frequency dependence, displacement and vertical velocity variance are proportional to N−1 while horizontal velocity variance goes as N+1. Note that the abscissa differs in each plot. Displacement profiles are reduced by a factor of (top left) 10−6 and (top right) 10−7 relative to the vertical velocity profiles.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

LR horizontal velocity variance (Figs. 5c,d) varies as N−1, in good agreement with the WKB scaling of linear waves. CR velocity exceeds LR by a factor of 2 (full band, Fig. 6c) or more (below 600 m). In the D2 band, a clear excess of variance is seen between 200 and 400 m. This velocity peak corresponds to the irregular structure in the D2 vertical displacement (Fig. 5b) and is thought to be associated with a tidal beam.

Fig. 6.
Fig. 6.

(a) The CR spectrum of horizontal velocity and the (b) coherence, (c) cospectrum, and (d) quadrature spectrum of CR and vertical velocity 〈u2u3〉. Spectral estimates are averaged in frequency and in depth, from 100–200 m (at top) to 600–700 m (at bottom). Inertial (red); D1 (magenta); and D2, D4, D6, and D8 frequencies are indicated. In estimating statistical stability, it is assumed that the 100-m vertical average is equivalent to an average of two independent spectral estimates. In (a), upper and lower 95% confidence bounds are presented. In (b), the magenta lines give the expected value of coherence when the true coherence is nil.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

5. Flux estimation

Radar and acoustic sounding instruments are adept at estimating off-diagonal elements of the Reynolds stress tensor. With
eq3
the ocean velocity field, and
eq4
specifying the look-directions of back-to-back, vertically slanting sonar beams (beams I and III), the signal sensed by each sonar beam at range r is
eq7
Here, r = rγj and measurement imprecision is given by .
An estimate of the Reynolds stress is obtained from the difference in back-to-back beam variances,
eq8
(Lhermitte 1968; Vincent and Reid 1983; Plueddemann 1987).

If the velocity and noise fields are homogeneous over the lateral distance spanned by the separated beams, the stress estimate is valid. By extension, the difference in slant-velocity power spectra, at like depths/ranges, gives the Reynolds stress as a function of wave frequency.

Estimating stress in stratified fluids is rendered difficult by the extreme anisotropy, , that is commonly encountered. Estimates 〈uHu3〉 are critically sensitive to the definition of “up” and to error in its determination (appendix A). For example, if a back-to-back sonar beam pair is rotated by δ in the plane of the beams, the stress estimate 〈u1u3〉 is altered by δ〈|uH|2〉. Similarly, if the beam pair is tilted out of a vertical plane by δ, out-of-plane horizontal flow will be interpreted as vertical velocity. The estimate of 〈u1u3〉 will be altered by δu1u2〉. Horizontal flows can contribute an apparent stress that is comparable to 〈uHu3〉 when δ|uH| is on the order of |u3|.

In HOME, estimates of u3 are obtained from tracking the vertical displacement of the density field. This reduces the sensitivity to errors in array orientation, at the expense of introducing noise associated with the lateral advection of spatially variable density structures. The use of density to estimate u3 has the added benefit of avoiding the considerable signal associated with the diel vertical migration of the organisms that contribute to the back-scattered acoustic signal. Although the primary migration signal is at D1 frequency, variance is spread to higher harmonics.

The buoyancy term in the EP flux −f/N2bu2, −bu1〉 is an extremely robust signal at Kaena Ridge. The challenge is to produce an Eulerian description of the flux that is not sensitive to so-called fine structure contamination. It is convenient to describe b in terms of a “semi Lagrangian” (s-L) wave vertical displacement η such that b(x3, t) = −N2η(x3, t) and f/N2buH〉 = −fηuH. Here, η(x3, t) is the vertical displacement of that isopycnal currently residing at depth x3, an exact index of the buoyancy perturbation at (x3, t). We calculate η by integrating the D2 bandpassed Eulerian vertical velocity (appendix B). This effectively recreates the density field initially used to calculate the vertical velocity. By bandpass filtering the Eulerian vertical velocity, which is not fine structure contaminated, rather than the density signal itself, which is, we avoid including the density signature of D1 motions that can produce a D2 first harmonic if the background density profile has curvature.

6. Cross-spectral analysis

In studies of turbulence and other broad-spectrum phenomena, cross-spectral analysis is used to quantify flux terms as a continuous function of encounter frequency ω. Specifically, the cospectrum of perturbation velocities ui and uj gives the Reynolds stress tensor as a function of frequency. The corresponding coherence provides a measure of the statistical precision of the stress estimates. In the present case, quasi-deterministic D2 baroclinic motion is the major contributor to the momentum flux. The coherence between the various components of D2 velocity is a useful measure of degree of repeatability in the tidal cycle.5

The coherence, cospectrum, and quadrature spectrum are presented in Fig. 6 for CR u3 flow. The coherence in the D2 band exceeds 0.8 in the 200–500-m depth region. Significant coherence is seen in the D4 band as well at middepths. The CR cospectrum clearly indicates the dominance of the D2 tide. Negative values of the cospectrum above 500 m correspond to the upward and southward stress seen below (in Figs. 10b, 11a). Weaker upward–northward stress is seen below 500 m. The CR D2 quadrature spectrum is uniformly positive. This corresponds to upward vertical velocity preceding southward horizontal velocity, a signature of mode-like southward propagation. The strong quadrature spectrum is an indication that the CR momentum flux is maintained by rather mild distortions of slowly varying profiles of u2 and u3 that are principally in quadrature.

In the LR direction (not shown), D2 coherence varies from 0.6 to 0.9. The along-crest cospectrum indicates weak upward–westward stress above 300 m and upward–eastward stress from 3 to 600 m. The LR D2 quadrature spectrum is negative at all depths. It is of comparable magnitude to the cospectrum but with opposite sign, consistent with a generally eastward energy flux. The D4 coherence peak is convincing only in the 400–600-m depth range.

The strong coherence between horizontal and vertical velocity in the D2 band in the HOME Nearfield contrasts sharply with the pioneering open-ocean stress-estimation efforts. The coherence results from the highly anisotropic generation process at the ridge and enables estimates of Reynolds stress and buoyancy flux at very high signal-to-noise ratios (Fig. 7). In addition to the fortnightly cycle, D2 stress estimates fluctuate not because the underlying generation process is stochastic but rather because the background mesoscale ocean is.

Fig. 7.
Fig. 7.

The spectrum of an Eulerian record of vertical velocity at 200 m (black) displays peaks in the D1, D2, and D4 bands and a pronounced cutoff at the buoyancy frequency. The D2 filtered vertical and CR velocity spectra (black and red) isolate the tidal peak but also include variance from the spectral line at D2 + f. The raw Reynolds product u2u3 (blue) has variance at low frequencies and in the D4 band. A low-pass filter removes the D4 variance and leaves a stress signal with a ~1-cpd bandwidth (green).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

7. The D2 signal

Given the high signal-to-noise ratio in the D2 band, it is attractive to bandpass filter the data and examine flux variability in the depth–time domain. In subsequent analyses, Eulerian-frame horizontal and vertical velocity data are bandpass filtered using a three-pole Butterworth filter, with 1.5 < < 2.4 cpd. The effect of this time filtering is summarized in Fig. 7.

Our focus is on the D2 tide and its associated momentum transport. Here, we examine the motions that support the E–P flux in the depth–time domain (Fig. 8). The principal observation is that the D2 horizontal velocity and vertical displacement signals are not confined in distinct vertical zones, consistent with ray-like behavior. Indeed, the vertical wavelengths are extremely long, with roughly a quarter wavelength filling the 1100-m water column above the ridge.

Fig. 8.
Fig. 8.

The D2 bandpass filtered (top) vertical displacement and (middle) LR and (bottom) CR horizontal velocities. Measurements are presented in an Eulerian frame, with displacement taken as the time integral of the filtered vertical velocity. The subtle phase shifting of these long-wavelength motions is responsible for the depth variability of the stress.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The displacement signal increases in amplitude with increasing depth, revealing the spring tides of yeardays 259–266 and 274–281. Displacements exceed ±50 m at the bottom of the observation window, still 300 m above the seafloor.6 Pronounced downward phase propagation is seen at depths below ~300 m, consistent with upward energy propagation. The lack of phase progression near the surface might be the signature of nearby surface reflection, as indicated schematically in Fig. 2c.

The D2 CR current also displays the downward phase propagation of a long-wavelength tide. Here, the phase pattern is undistorted near the sea surface but highly altered in the lower water column, as the ridge crest is approached. The LR currents show greater phase variability and less amplitude variability with depth.

The D2 LR Reynolds product u1u3 (Fig. 9) displays the various interacting factors that establish the mean stress. The records consist of a series of vertically coherent stripes at frequency 2D2, overlaying a difference-frequency signal that changes rather slowly in time. The vertical scale of the stripes is comparable to that of the D2 signals. Downward phase propagation is seen at middepths. The sense of phase propagation is less consistent near the surface and near the bottom (of the observational window). A time average of uHu3 over multiple tidal cycles removes the 2D2 variability, leaving the difference-frequency signal as the Reynolds stress. The depth variability of the stress can be quite rapid (e.g., Fig. 9, 300 m), in spite of the fact that the dominant motions are of large vertical scale. The apparently irregular deformation of the velocity fields in Figs. 4 and 8, due to the presence of small-scale waves, act in concert to create the sharp stress patterns. The time variability of this signal is limited by the bandwidth, 0.8 cpd, of the D2 bandpass filter.

Fig. 9.
Fig. 9.

A 5-day record of the D2 Reynolds product u1u3 during the second spring tide of the experiment. When horizontal and vertical velocities are large and in quadrature, alternating red and blue bands are seen at frequency 2D2 (e.g., yearday 277 and depths 600–800 m). When the velocities are in phase or antiphase, the difference-frequency signal is the Reynolds stress, which is seen as the changing mean blue or red color of the image.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The entire Nearfield record of D2 Reynolds stress and mass flux fηuH〉 is presented in Fig. 10. The existence of the fortnightly cycle is apparent in both the stress and mass flux fields. Upward and northward stresses are seen at depths below 600 m and upward and southward fluxes above, roughly consistent with Fig. 2. Flux magnitudes increase at all depths with the onset of spring tides, suggesting a rapid penetration time from the ridge into the interior. The flux patterns vary slightly from one spring tide to the next, possibly due to low-frequency current variability. LR stresses are of the same magnitude as the CR. Strong upward and eastward stresses are seen below 300 m during spring tides.

Fig. 10.
Fig. 10.

The LR and CR (a),(b) Reynolds stress and (c),(d) mass flux for the 35-day Nearfield record. A 24.8-h running mean filter converts perturbation maps (Fig. 9) into these flux maps. The maps are shifted in time to account for the lag of the filter. The gray region at depths below 700 m starting at yearday 280 represents data lost because of CTD failure.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The mass flux signals are remarkably regular, with magnitude roughly half that of the stress. The mass flux nearly vanishes during the neap tides, with some hint of an actual reversal in the LR flux.

Cruise-averaged stress and mass flux profiles (Fig. 11) summarize the observed patterns. The LR Reynolds stress is consistently eastward, at depths of 300–600 m. Stress magnitude is ~0.4 × 10−4 m2 s−2, decaying as the surface is approached and surface reflection becomes important. The CR stress appears consistent with the pattern of Fig. 2, with upward and northward flux at depth and upward and southward flux from 200 to 500 m. The decay of both flux components near the bottom of the observational window is perhaps fortuitous given that the seafloor lays 300 m below.

Fig. 11.
Fig. 11.

Cruise-averaged summary of (a) D2 Reynolds stress, (b) lateral mass flux, (c) E–P flux, and (d) the azimuth of the (upward) fluxes. Azimuth is given in terms of ridge coordinates, such that 0° and 360° represent an upward LR flux toward Oahu. The green reference marks the offshore (negative) CR azimuth. Black vertical reference lines indicate Earth-frame azimuth. Data are smoothed in the vertical over 30 m here and in Figs. 12, 13, and 16.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The azimuth of the upward stress is toward the north at great depth, rotating clockwise with increasing altitude through the upward–southward direction at ~300 m and then coming nearly full circle (Fig. 11d). The steadiness of this phase progression is a striking aspect of the signal. The rotation rate is ~0.015 m−1.

By contrast, the mass flux is oriented toward geographic north at all depths. When the EP flux is formed by combining the stress and mass flux, it is seen that rotational effects indeed reduce the magnitude of the “main” upward–southward beam emanating from Kaena Ridge. However, the deep, upward–northward beam and the eastward LR beam are in fact enhanced when rotational effects are included. This behavior is not consistent with the linear theory for wave generation at a 2D ridge, where the LR component of the EP flux should vanish.

To investigate the stability of this pattern, the temporal averages are repeated for the first and second spring tides and the intervening neap in Fig. 12. In spite of the pronounced variation in energy between the first spring tide and the following neap (Fig. 7), the Reynolds stress and mass flux are generally similar in magnitude and form. The second spring tide brings a significant increase in the CR stress (Fig. 12b) and mass flux (Fig. 12e). This CR mass flux, in turn, combines with the LR Reynolds stress (Fig. 12a) to yield an EP flux that is enhanced in both LR and CR directions. Maximum stress exceeds 10−4 m2 s−2 in the second spring tide.

Fig. 12.
Fig. 12.

Stress and flux estimates for the first (blue; yeardays 260–268) and second (red; yeardays 277–284) spring tides of the cruise and the intervening neap tide (magenta; yeardays 269–276). Columns give (left) LR, (middle) CR fluxes, and (right) flux azimuth.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The azimuth of the upward Reynolds stress (Fig. 12c), the lateral mass flux (Fig. 12f), and the resultant EP flux (Fig. 12i) are remarkably consistent throughout the experiment. The largest variations in azimuth are found when the stresses are near zero and azimuth becomes indeterminate.

The combined effect of magnitude and azimuth variability is seen in vector-profile plots (Fig. 13). The clockwise rotation of both the upward Reynolds stress and the EP flux with increasing height above the ridge is clearly seen, as is the generally northward flux of mass at all heights. The peak in the upward southward Reynolds stress between 300 and 400 m (Fig. 13a) is toward geographic south, slightly to the right of the CR offshore azimuth. The EP flux maximum (Fig. 13c) is shifted even farther to the right.

Fig. 13.
Fig. 13.

Vertical profiles of (a) upward Reynolds stress, (b) lateral mass flux, and (c) E–P flux over the south rim of Kaena Ridge.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

8. Model comparisons

It is of value to compare the rather complicated patterns in stress and mass flux with existing models of Kaena Ridge. Professor Tom Peacock, Massachusetts Institute of Technology (MIT), graciously provided a two-dimensional, linear, finite-topography WKB model (Echeverri and Peacock 2010), which is a valuable benchmark for wave field properties. Dr. Shaun Johnston has given us output of the Princeton Ocean Model (POM) appropriate for the FLIP site. The POM output adds realistic 3D topography and nonlinearity to the simulation.

Cross-ridge sections of CR stress, mass flux, and energy flux for the Echeverri and Peacock model are presented in Figs. 14 and 15 for solutions based on 5 and 20 vertical modes, propagating in a quiescent ocean with Hawaiian stratification. The figures represent only the free-wave component of the overall solution at the ridge, but they are sufficient to illustrate a number of consequences of the nonhomogeneous seafloor topography and the assumed reflective sea surface. Specifically, note that, although the energy flux is largely concentrated in the upper ocean (Figs. 14, 15, bottom), over a depth range established by the buoyancy profile, the Reynolds stresses (Figs. 14, 15, top) are maintained throughout the ocean depths. The spatial pattern of the tidal beam reflects both the buoyancy profile and the depth of the original generator, the ridge crest.

Fig. 14.
Fig. 14.

(top) CR Reynolds stress (m2 s−2), (middle) CR mass flux and LR Reynolds stress (m2 s−2), and (bottom) energy flux (W m−2) for the free-wave linear WKB solution for barotropic D2 flow over a 2D ridge. The contribution from the first five vertical modes is presented. The color scale (positive values are red) is the same for (top) and (middle).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

Fig. 15.
Fig. 15.

As in Fig. 14, but with the first 20 vertical modes included.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

The Eulerian mass flux fu2η〉 (Figs. 14, 15, middle) is always toward the ridge above the ridge crest and away below the crest. This mass flux is a property of the linear wave solutions and extends, in principle, indefinitely into the far field. For free waves, cross-ridge velocity and vertical displacement/buoyancy perturbation are in quadrature, and no mass flux should be found. The mass flux is clearly associated with sea surface and bottom reflections (Fig. 2c), where the relative phases of various wave constituents are altered. For linear waves, it is easy to demonstrate fu2η〉 = 〈u1u3〉, and the resulting along-ridge Eliassen–Palm flux vanishes (Fig. 16g).

Fig. 16.
Fig. 16.

(top) Reynolds stress, (middle) mass flux, and (bottom) E–P flux for the linear 2D solutions for flow over Kaena Ridge, from the linear 2D WKB model of Echeverri and Peacock (2010), using 20 vertical modes (red). The nonlinear, 3D POM predictions are given in blue. Observations are presented in black. Columns give (left) LR fluxes, (middle) CR fluxes, and (right) flux azimuth.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

Agreement between the two-dimensional model and the HOME observations, given in Fig. 16 for a 20-mode simulation, is unimpressive. Although the mass flux estimates agree in sign and magnitude, the LR observed Reynolds stress is upward and toward Oahu (Fig. 16a) from 400 to 700 m, whereas the modeled equivalent has reverse sign. The observed upward CR stress is negative in a main beam centered at 350 m, whereas the modeled beam is split with centers at 400 and 700 m. Although the structure of the modeled beam is sensitive to the number of modes used in the simulation, the center of the beam is always deeper than that of the observed upward–southward beam.

It is reasonable that these discrepancies are associated with 3D aspects of the ridge that the POM addresses numerically. POM solutions for horizontal and vertical velocity were averaged over three 1-km grid cells near the FLIP site (21°40.2′N and 158°38.4′W, 158°37.8′W, and 158°37.2′W). Flux products, averaged over an M2 period, are also plotted in Fig. 16. The POM Reynolds stress displays a different vertical structure than the 2D linear model, but they are similar in character. Mass flux estimates from both models are in reasonable agreement with observations. Surprisingly, although the POM is indeed 3D, it too disagrees with the sign of the observed LR Reynolds stress in the 400–600-m depth range (Fig. 16a). There is no sign of a “stress spiral” in either model output (Fig. 16c). The POM EP flux is constantly normal to the ridge at all depths.

9. Discussion

The steady clockwise rotation of the Reynolds stress vector with increasing elevation is one of the surprise findings of the experiment. In spite of the failure of the POM to replicate the spiral, we feel that the three dimensionality of the ridge is the key factor in its creation.

The distance between the FLIP site and the westward end of the ridge, ~40 km, is comparable with the horizontal wavelength of a mode-three wave and only slightly greater than the width of the ridge at the site. One can back trace the Reynolds stress vectors arriving at FLIP (Fig. 13a), identifying their point of origin on the ridge (Fig. 17). Not surprisingly, the deepest observed arrivals are generated on the south flank of the ridge and are associated with upward and northward propagation. The clockwise upward spiral generally follows the topography of the west end of the ridge. It is notable that at many of the sites of origin the rays are locally tangent to the topography. This is particularly true for the strongest sites on the north face of the ridge. The D2 rays can only reach the upper 150 m of FLIP’s observation window if launched from the shallow topography where the ridge shoals toward Oahu. This is indeed the azimuth of the observed (weak) arrival.

Fig. 17.
Fig. 17.

The ridge-crest topography with the D2 rays arriving at FLIP traced backward to their presumed site of generation. The black dot represents FLIP, to scale; the blue line below is the section of the water column sampled; and the light blue extension gives the water depth at the measurement site. The color of the dots at the point of origin of each ray corresponds to the magnitude of the Reynolds stress along the ray (m2 s−2).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

In examining the D2 horizontal energy fluxes at the FLIP site, Rainville and Pinkel (2006a) find an identical spiral (their Figs. 9a, 10a), with LR energy fluxes of magnitude comparable to the CR fluxes. These results suggest that baroclinic waves are generated along the azimuth of critical topographic gradients, rather than parallel to the barotropic forcing or normal to the large-scale orientation of the ridge.

10. Summary

We have described the semidiurnal Eliassen–Palm fluxes observed above Kaena Ridge, Hawaii, during the HOME Nearfield experiment. In contrast to the pioneering open-ocean efforts of Ruddick and Joyce (1979), Brown and Owens (1981), and Plueddemann (1987), the stress signals on the ridge are robust, exceeding 10−4 m2 s−2.7

To detect these signals, it proved advantageous to deploy an eight-beam up–down profiling sonar at middepth (400 m) rather than using a four-beam downward-looking sonar deployed near the surface (Pinkel 1981). With the middepth deployment, higher acoustic frequencies could be used, enabling relatively high depth resolution (4.5 m) over the 800-m observational window. It was also helpful to combine the sonar-derived horizontal velocity estimates with synchronous vertical velocity estimates from profiling CTDs. When compared with purely acoustic stress estimation, the hybrid approach is less sensitive to uncertainty in the direction of “up.” It is also free of the errors in sonar-derived estimates of vertical velocity that stem from the diel vertical migration of the zooplanktonic scatterers. The migration signal is nonsinusoidal with D1 fundamental frequency. However, the first harmonic of the migration signal can significantly degrade D2 stress estimates. The magnitude of this degradation varies with depth, location, and season.

Over Kaena Ridge, the principal D2 motions have large vertical scale. The 1100-m ridge depth corresponds to roughly a quarter wavelength. Beam-like patterns are detected in off-ridge energy (Lee et al. 2006; Rainville and Pinkel 2006a; Martin et al. 2006) and momentum fluxes. The beams are confined in regions several hundred meters thick: small compared to the dominant vertical wavelength. This is achieved through subtle phase distortions of the dominant fields caused by the smaller-scale D2 motions.

The finite depth extent of the beams reflects the horizontal scale of the active generating regions near the ridge crest, as well as the wavenumber bandwidth of the generated baroclinic waves. The fraction of the barotropic forcing borne by each baroclinic mode is determined by the topography and buoyancy fields (Llewellyn-Smith and Young 2002; St. Laurent et al. 2003). In a linear generation model, it is independent of the magnitude of the forcing. However, the propagation of small-scale motions with |cpH| < |Ubarotropic| is presumably impeded through interaction with the larger-scale flow field (Klymak et al. 2010). The threshold wavenumber is κ3 > αD2|Ubarotropic|−1h/H, where and h/H is the ratio of crest depth to total depth. For |Ubarotropic| ~ 0.05 m s−1 and N = 2 cph, α ~ 27 and the threshold corresponds to scales smaller than λ3 ~ 413 m. Presumably, tidal beams both weaken in strength and sharpen spatially as the fortnightly cycle wanes.

The broad tidal beams in HOME contrast markedly with the observations of Lueck and Mudge (1997) and Lien and Gregg 2001. They detected ray-like signatures in depth-range maps of turbulent dissipation. The elevated turbulence is presumed to result from large tidal shears associated with compact rays. Midwater regions of intense tidal shear are not seen at Kaena Ridge. We conjecture that, offshore of a conversion site, enhanced tidal shear is probably associated with weak barotropic forcing.

At Kaena Ridge, northward and upward fluxes are seen below ~600 m. They are presumably generated at the critical-slope region at the south edge of the ridge (Fig. 2a). Farther aloft, 300–500-m depths, upward–southward fluxes are seen, associated with the beam generated at the north critical slope. The azimuth of the upward flux rotates clockwise with increasing height above the ridge, at a constant rate of 0.015 radians m−1.

The Eulerian lateral mass flux (fηu2〉 ~ 0.5 × 10−4 m2 s−2) induced at the generation site is onshore, geographically northward, corresponding to a southward buoyancy flux on the order of 〈bu2〉 ~ 0.3 × 10−4 m2 s−3 and a mass transport velocity 〈ρu2〉/ρ ~ 0.3 × 10−5 m s−1. Linear free-wave solutions replicate this behavior above the crest of a 2D ridge. The model mass flux reverses at depths below the crest, indicating a recirculating Eulerian flow on either side of the ridge.

Surface and bottom reflections alter the phase between CR and vertical velocities. LR velocities, in quadrature with the CR flow, thus can be partially in phase with vertical velocity. This leads to a puzzling along-ridge Reynolds stress that, in a 2D linear model, extends indefinitely offshore. Although this stress cannot diverge in the along-ridge direction, it does diverge in depth and offshore. This stress is exactly balanced by the CR buoyancy flux, such that the LR component of the Eliassen–Palm flux vanishes in the 2D model. Measurements of Reynolds stress alone, in the absence of corresponding buoyancy flux estimates, have the potential to mislead, if motions at frequencies near f are being considered.

In the observations, robust LR Reynolds stresses are seen directed upward and toward Oahu. This is opposite to the sign of the 2D and 3D POM predictions. When observed rays are back traced to their point of origin on the ridge, it is seen that the greatest momentum fluxes are sourced from near-critical slopes, where the rays are tangent to the topography. The discrepancy with the POM regarding the directionality of the flux, and the discrepancy with ray theory concerning the depth of the principal beams deserves further study.

The E–P flux magnitude can be expressed in terms of a friction velocity m s−1. This corresponds to a of 0.3 m s−1, equivalent to the wind stress resulting from a 5–10 m s−1 wind. If the flux were to diverge at the surface reflection of the tide—for example, over a 100-m-thick mixed layer—the associated mean flow acceleration is of order 10−6 m s−2 sufficient to change the flow speed on the order of 0.1 m s−1 over an inertial period.

Even though the wave Reynolds stress is small compared to the surface stress from high winds, the spatial scale of variability is much smaller than in the open ocean. Thus, the curl of the stress is large comparable to open-ocean values. Chavanne et al. (2002) have noted that the wind field around the Hawaiian Islands is strongly perturbed by island topography, resulting in wind stress variations at the same scales as the tidal surface imprints, but with different spatial patterns. These two means of forcing subinertial flows are roughly comparable during moderate winds. Can the fortnightly modulation of the stress (divergence) be detected in subinertial flows?

Acknowledgments

The authors thank Eric Slater, Mike Goldin, Lloyd Green, Tony Aja, and Tyler Hughen for the design, construction, and operation of the instruments used in this experiment. Prof. Tom Peacock provided the Echeverri and Peacock (2010) model and Dr. Shaun Johnston graciously provided POM output for the FLIP site. Captain Tom Golfinos and the crew of the R/P FLIP ably deployed the trimooring on the edge of Kaena Ridge crest and operated the platform for the duration of the experiment. This work was supported by NSF as an aspect of the Hawaii Ocean Mixing Experiment.

APPENDIX A

Stress Estimation and Error

The estimation of stress in a stratified fluid is rendered difficult by the extreme anisotropy, , that is commonly encountered. The estimates are critically sensitive to the definition of “up” and to error in its determination.A1

Recapitulating from the text, if a back-to-back sonar beam pair is rotated by δ in the plane of the beams, the stress estimate 〈u1u3〉 is altered by δ〈|uH|2〉. If the beam pair is tilted out of a vertical plane by δ, out-of-plane horizontal flow will be interpreted as vertical velocity. The estimate of 〈u1u3〉 will be altered by δu1u2〉. Horizontal flows can contribute an apparent flux that is comparable to 〈uHu3〉 when δ|u2| is on the order of |u3|.

For sonar-derived oceanic stress estimates there is an added complication: the acoustic scatterers can swim. Many zooplanktonic species migrate vertically in response to changing light level. This migration signal significantly degraded our earlier attempts at vertical flux estimation. To sidestep both this issue and concern about the direction of “up,”A2 here we estimate the vertical velocity from the apparent vertical motion of the density field,
eq9
Here, η(ρ, x, t) gives the vertical excursion the density surface ρ as a function of lateral position and time and gives the fictitious velocity resulting from lateral advection. For the Kaena Ridge tide, ~ 0.002, corresponding to a 100-m-amplitude tide varying over a 50-km half wavelength. With |uH| more than an order of magnitude greater than |u3|, 10% errors in are possible. For a single dominant sinusoid, uH and are in quadrature for a vertical standing wave and are in phase for vertically propagating motions. The self-advection of the tide is associated with both uH-tide and /dx2 fluctuating at tidal frequencies. The product signal is at frequencies 2D2 and near zero. It is happily excluded from the analysis by the D2 filter.
This density-derived is combined with the “Janus” estimate of horizontal velocity,
eq10
For , the Janus estimate of is tolerant of small tilt errors.
The stress estimate combines CTD and sonar data,
eq11
For a single dominant waveform, the triple products of these highly correlated quantities vanish.A3 The remaining error bias is associated with uncertainty δ in the direction of up. The magnitude of this bias depends on vertical not horizontal velocity variance. Thus, the combined use of sonar and CTD information leads to a significantly more user-friendly stress estimateA4 than when using sonar alone, apart from the issue of swimming scatterers.

APPENDIX B

Estimation of Eulerian Vertical Velocity

Using the depth-continuous CTD data, it is possible to form Eulerian vertical velocity time series that do not suffer from so-called fine structure contamination. For the HOME Nearfield, a set of isopycnal depth–time records is first formed, using 372 reference densities that are, on average, separated by 2 m in depth. The s-L vertical velocity is taken as the time difference in the encounter depth of these reference densities over the 11 767 profile pairs in the edited Nearfield run. In spite of the fact that the underlying density profiles are irregular in depth, the s-L velocity profiles generally vary smoothly with increasing density. Occasional jumps are seen as isopycnals transition low gradient regions of the density field. The Eulerian record of vertical velocity is formed by interpolating each s-L profile to a set of reference depths. Being a difference of two profiles separated by 4 min, the vertical velocity record is time shifted relative to the original density and horizontal velocity grids. By averaging adjacent horizontal velocity estimates in time, the time synchrony of the uH and u3 signals is maintained.B1

APPENDIX C

Correlation of D2 Horizontal and Vertical Velocity

The baroclinic tides are here treated as a narrowband random process sampled in a low-noise background. However, the relationship between horizontal and vertical velocities is potentially deterministic, a consequence of the repetitive nature of the barotropic forcing and the absence of random distortion in the baroclinic response. The degree of determinism can be quantified by the correlation between horizontal (i = 1, 2) and vertical tidal velocities,
ec1
The correlations vary in sign with the stress itself (Fig. C1), but the peak magnitudes are very large relative to those found in stochastic situations. The cpd half bandwidth of the tidal filter corresponds to a response time days, admitting only ~20 independent tidal measurements over a spring or neap record. The corresponding expected correlation when the true correlation is zero is ~(0.05)1/2 = 0.224. The LR correlations are significant above 600 m. The normalization of the covariance removed the variations in magnitude associated with the fortnightly cycle. CR correlations (Fig. C1b) are significant between 200 and 450 m and are remarkably similar throughout the fortnightly cycle. The azimuth of the correlation again displays a characteristic clockwise rotation with altitude.
Fig. C1.
Fig. C1.

The correlation between horizontal and vertical velocities in the band cph.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0124.1

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1

Using a buoyancy-driven profiling instrument, Jacobs and Cox (1987), Duda and Cox (1989), and Duda and Jacobs (1998) have detected small but statistically significant stress associated with the propagation of short, high-frequency waves in a background inertial shear. van Haren et al. (1994) have detected wave stress in stratified water above the seafloor in the Strait of Juan del Fuca.

2

It is convenient to refer to the semidiurnal (D2) and diurnal (D1) tides generically and note their fortnightly cycles, rather than discuss individual tidal constituents, such as M2, S2, etc., which are imprecisely resolved in the present 35-day record.

3

The prefix “pseudo,” introduced to geoscience by McIntyre (1981) from classical mechanics, acknowledges the intellectual challenge of isolating the momentum of a wave from that of the medium through which it propagates.

4

The deep (>600 m) acoustic scattering cross section was much greater over Kaena Ridge than in the open ocean, enabling the good depth penetration of the sonar.

5

For temporally stationary processes, the cross-spectral approach is the accepted standard for flux estimation. However, when the background flow is periodic, time variability can cause high-frequency phenomena to “disappear” in cross-spectral flux descriptions. For example, small-scale topography on the ridge can be expected to trigger high-frequency lee waves. These might support a significant momentum flux, but direction of the flux will reverse with each reversal of the D2 current. Cross spectra that are estimated from records of duration sufficient to resolve the D2 momentum flux will blur the view of higher-frequency fluxes. Cross-spectral analysis does not distinguish between a large but reversing flux and a steady small flux.

6

On the upper ridge flanks, vertical displacements exceeding ±150 m are seen (Levine and Boyd 2006).

7

These signals would be easily detectable with Plueddemann’s 1983 technology.

A1

In laterally homogeneous situations, it is often practical to define local coordinates such that 〈u3〉 = 0. However, applying the criterion 〈u3〉 = 0 in the presence of lee waves yields a spatially varying definition of “up.”

A2

For a back-to-back sonar beam pair rotated by δ in the plane of the beams, errors in u1 and u2 are O(δ2) or smaller.

A3

For broadband, quasi-linear wave fields, the triple products will also vanish.

A4

This result depends on the fact that slope and velocity vary at the same frequency. A “frozen,” frontal slope, under lateral advection, leads to a fictitious vertical stress.

B1

Although immune to fine structure contamination, the advective error still contaminates the record.

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