The M2 internal tide in Monterey Submarine Canyon is simulated using a modified version of the Princeton Ocean Model. Most of the internal tide energy entering the canyon is generated to the south, on Sur Slope and at the head of Carmel Canyon. The internal tide is topographically steered around the large canyon meanders. Depth-integrated baroclinic energy fluxes are up canyon and largest near the canyon axis, up to 1.5 kW m−1 at the mouth of the upper canyon and increasing to over 4 kW m−1 around Monterey and San Gregorio Meanders. The up-canyon energy flux is bottom intensified, suggesting that topographic focusing occurs. Net along-canyon energy flux decreases almost monotonically from 9 MW at the canyon mouth to 1 MW at Gooseneck Meander, implying that high levels of internal tide dissipation occur. The depth-integrated energy flux across the 200-m isobath is order 10 W m−1 along the majority of the canyon rim but increases by over an order of magnitude near the canyon head, where internal tide energy escapes onto the shelf. Reducing the size of the model domain to exclude remote areas of high barotropic-to-baroclinic energy conversion decreases the depth-integrated energy flux in the upper canyon by 20%. However, quantifying the role of remote internal tide generation sites is complicated by a pressure perturbation feedback between baroclinic energy flux and barotropic-to-baroclinic energy conversion.
Submarine canyons are a common feature along continental shelves. They are estimated to cover approximately 20% of the shelf along the west coast of North America (Hickey 1995). Canyons can be efficient generators of internal tides (internal gravity waves with tidal frequencies) through scattering of the barotropic tides from sloping topography (Bell 1975; Baines 1982). They are also thought to trap internal waves from outside the canyon, through reflection from the sloping topography, and channel the energy toward the canyon head (Gordon and Marshall 1976; Hotchkiss and Wunsch 1982).
Unlike internal tide generation, which may be dominated by topographic blocking (Garrett and Kunze 2007), reflection of internal tide beams is entirely determined by the topographic slope. Reflection is often considered in only two dimensions; when internal waves approach a slope from deep water, the onshore–offshore direction of propagation after reflection is determined by the ratio of the topographic slope to the internal wave characteristic slope,
where H is the total depth, x is distance, ω is the angular frequency of the wave, f is the inertial frequency, and N is the buoyancy frequency. If α < 1 (subcritical), waves will continue to shoal after reflection. If α > 1 (supercritical), reflected waves will propagate back into deeper water. If α = 1 (critical), linear theory breaks down, leading to nonlinear effects and potentially wave breaking. In a three-dimensional ocean, it is more apt to visualize internal wave “sheets” rather than beams (Carter et al. 2006; Jachec 2007). The slope of an internal wave beam in a horizontal–vertical section will depend on the orientation of the section with respect to the internal wave sheet. If the section is in the same plane as the sheet, the slope of the beam will equal swave. If the section is perpendicular to the sheet, the beam will be horizontal.
Internal waves above the rim of the canyon are focused toward the canyon floor by supercritical reflection from the steep canyon walls (Gordon and Marshall 1976), whereas internal waves entering the canyon from offshore are focused toward the canyon head by subcritical reflection along the typically gentle sloping canyon floor (Hotchkiss and Wunsch 1982). In both cases, energy density increases upon reflection because the separation between adjacent internal wave characteristics narrows, concentrating the energy into a smaller area. Increases in internal wave energy density have been observed in Hydrographers Canyon (Wunsch and Webb 1979), Hudson Canyon (Hotchkiss and Wunsch 1982), and recently in Gaoping Canyon off Taiwan (Lee et al. 2009).
Monterey Submarine Canyon (MSC) is located within and offshore of Monterey Bay in central California. It is the largest submarine canyon along the west coast of the United States, extending over 100 km from the abyssal plain at the base of the continental slope to within 100 m of Moss Landing in the center of the bay. The bathymetry of the canyon is very complex, with several sharp meanders in the upper reaches and two smaller side canyons (Fig. 1). Following the canyon thalweg (the deepest part of the canyon axis) from Moss Landing, the first major meander is between 12 and 20 km, referred to as Gooseneck Meander (GM). Between 30 and 55 km, there are two large meanders, Monterey Meander (MM) and San Gregorio Meanders (SM), that form an S-shaped bend. Soquel Canyon merges with MSC from the north at around 32 km, just up canyon of the apex of Monterey Meander. Carmel Canyon merges with MSC at around 60 km, down canyon of San Gregorio Meander. Farther down canyon, the canyon intersects two regions of smooth shelf slope, Smooth Ridge to the north and Sur Slope to the south. The canyon floor is gently sloping (stopog = 0.03–0.06) and is subcritical to semidiurnal internal tides down canyon of San Gregorio Meander; up canyon of the meander, the floor is near critical. The canyon walls are steep (slopes up to 0.7) and supercritical along the whole length of the canyon.
The currents in MSC are dominated by a semidiurnal internal tide. Velocity amplitudes > 0.2 m s−1 have been observed in the canyon (Shepard et al. 1974; Rosenfeld et al. 1994; Petruncio et al. 1998) and are intensified near the bottom (Xu et al. 2002). Key (1999) observed strongly bottom-intensified currents near the head of the canyon that were associated with internal tidal bores. Semidiurnal vertical isopycnal displacements with 30- to 60-m amplitudes have also been observed near the canyon head (Broenkow and McKain 1972; Petruncio et al. 1998; Carter and Gregg 2002).
The M2 is the dominant tidal constituent in the canyon and the four largest constituents (M2, S2, K1, and O1) account for 90% of the total current variability (Xu and Noble 2009). Near-inertial oscillations are absent (Kunze et al. 2002), possibly because of the presence of the steep canyon walls. At all depths, M2 tidal ellipses are more rectilinear than predicted by linear theory and the semimajor axes are aligned in the along-canyon direction (Petruncio et al. 1998; Xu and Noble 2009), suggesting that the internal tide is topographically steered.
From a mass balance approach, Petruncio et al. (1998) estimated barotropic tidal velocities in Monterey Bay to be small, around 0.006 m s−1. They considered this barotropic flow to be negligible compared to the measured depth-varying velocities and so considered all velocities to be baroclinic. However, numerical model simulations of the tides in Monterey Bay (Jachec 2007; Rosenfeld et al. 2009; Wang et al. 2009; Carter 2010) have suggested barotropic M2 velocities may be significantly larger. Jachec (2007) showed barotropic velocities in the upper canyon reach 0.06 m s−1. Both Rosenfeld et al. (2009) and Wang et al. (2009) quote barotropic current magnitudes in the range 0.03–0.04 m s−1 for the Monterey Bay area. Most recently, Carter (2010) showed depth-averaged velocities in the bay can exceed 0.1 m s−1 and are spatially variable.
Kunze et al. (2002) measured depth-integrated internal tide energy fluxes of 5 kW m−1 around San Gregorio and Monterey Meanders, decreasing to order 1 kW m−1 around Gooseneck Meander. Other energy flux measurements have been made up canyon of Monterey Meander. Petruncio et al. (1998) estimated the semidiurnal internal tide energy flux to be 1.3 kW m−1 at a station 22 km along the thalweg, slightly down canyon of Gooseneck Meander, and 0.8 kW m−1 at 6 km, near the canyon head. Carter and Gregg (2002) made several measurements of internal tide energy flux, between 2 and 11 km along the thalweg,1 ranging from 0.3 to 1.9 kW m−1. These internal tide energy fluxes are larger than those typically found at continental shelf edges (order 0.1 W m−1; Sherwin 1988; Green et al. 2008) but less than half that observed at the mouth of Gaoping Canyon (Lee et al. 2009).
Previous measurements of internal tide energy fluxes have been predominantly in a net up-canyon direction, suggesting the majority of internal tide generation occurs offshore or in the lower reaches of the canyon. On the basis of internal tide characteristics, Petruncio et al. (1998) suggested Smooth Ridge and the steep ridge inside San Gregorio Meander as likely generation sites, but later studies found no evidence for internal tide generation at these locations (Kunze et al. 2002; Jachec et al. 2006; Carter 2010). Kunze et al. (2002) and Carter and Gregg (2002) identified along-canyon flux divergences and down-canyon energy fluxes near Gooseneck Meander, suggestive of local internal tide generation in the upper canyon.
Petruncio et al. (2002) used a three-dimensional numerical model to investigate the generation of semidiurnal internal tides in idealized canyons with dimensions similar to MSC but with varying floor slopes and canyon geometry. Strong shoreward propagating internal tides were generated in canyons with near-critical floor slopes along much of their length. Along-slope velocity and energy density were bottom intensified, consistent with internal tide characteristics emanating from the shelf break at the foot of the canyon. The largest internal tides were generated in a canyon that was near critical at its foot. However, little internal tide energy made it onto the shelf because the canyon floor was supercritical farther shoreward.
From baroclinic energy flux divergence in a three-dimensional, nonhydrostatic, unstructured grid model of the M2 tide in the Monterey Bay region, Jachec et al. (2006) identified Sur Slope and Sur Platform as key generation sites for the internal tide in MSC. Other generation sites occur in the canyon itself. However, energy flux divergence can only provide a lower bound on internal tide generation, because it does not account for local baroclinic energy dissipation (Carter et al. 2008).
High levels of mixing are observed at the head of MSC. Carter and Gregg (2002),2 estimated the average turbulent kinetic energy dissipation rate to be 1.9 × 10−7 W kg−1, two orders of magnitude higher than typical values for the interior of the open ocean (Gregg 1989; Ledwell et al. 1993; Toole et al. 1994), and in agreement previous estimates by Lueck and Osborn (1985). The enhanced mixing is assumed to be caused by dissipation of the internal tide; approximations of dissipation rate from the convergence of internal tide energy fluxes (Petruncio et al. 1998; Carter and Gregg 2002; Kunze et al. 2002) have shown reasonable agreement with microstructure measurements (Lueck and Osborn 1985; Carter and Gregg 2002). Jachec et al. (2006) calculated 8.3-MW net dissipation in the canyon from baroclinic M2 energy flux divergence.
This study focuses on generation of the internal tide on Sur Slope and its propagation through MSC using a high-resolution numerical model of the M2 tide in the Monterey Bay region. In section 2, the model setup is described. Generation of the internal tide on Sur Slope is assessed in section 3. Propagation of the internal tide through MSC and onto the shelf is investigated in sections 4 and 5. In section 6, possible remote internal tide generation sites are considered. A summary is given in section 7.
2. Numerical model setup
A modified version of the Princeton Ocean Model (POM; Blumberg and Mellor 1987) is used to simulate the M2 tide in the Monterey Bay region. POM is a three-dimensional, nonlinear, hydrostatic, free-surface, finite-difference, terrain-following (σ coordinate) primitive equation model. The Mellor and Yamada (1982) second-moment turbulence-closure scheme is used to calculate diffusivities and viscosities in the vertical, and the Smagorinsky (1963) scheme is used in the horizontal.3 The model setup and domain is the same as used by Carter (2010). The Flather condition (Flather 1976) is applied at the boundaries so that barotropic energy is transmitted out of the domain. Baroclinic energy is absorbed at the boundaries using the relaxation scheme described by Carter and Merrifield (2007).
The model domain extends from 35°31′13″N, 123°43′59″W to 37°9′50″N, 121°44′8″W with 250-m horizontal resolution; the bathymetry is derived from Monterey Bay Aquarium Research Institute multibeam data and is higher resolution than previous models of the region. A total of 51 evenly spaced sigma levels are used, giving a vertical resolution between 0.3 and 80 m.
The hydrostatic pressure assumption is valid when the horizontal scales of motion are much greater than the vertical scales (Mahadevan 2006). This is true for internal tides, except during wave breaking, when steepening to form highly nonlinear solitary waves, and during the formation of hydraulic jumps. The horizontal resolution of the model is too course to resolve these nonhydrostatic processes. For example, nonlinear solitary-like waves have been observed on the continental shelf in Monterey Bay (Carter et al. 2005), but the horizontal scale of the waves is less than half the horizontal resolution of the model. Jachec (2007) showed that nonhydrostatic pressure effects are generally small in the Monterey Bay region but may be important within MSC.
Initial conditions are no flow and horizontally uniform stratification; the initial temperature and salinity profiles are the average of 7 CTD casts at 36°36′30″N, 123°00′00″W taken over 12 h on 18–19 February 2009. The model is forced at the boundaries with M2 barotropic velocities. Elevations and normal velocities used to calculate the Flather boundary condition are taken from the TPXO6.2 inverse model (Egbert 1997; Egbert and Erofeeva 2002; http://volkov.oce.orst.edu/tides/).
Because the simulation is of a generic M2 tide, surface buoyancy and momentum fluxes are set to zero. Diffusivities are not applied to the temperature and salinity fields, so the stratification is not eroded by mixing in the absence of a restoring buoyancy flux. This setup, with a single tidal constituent and nonevolving stratification, allows a relatively short simulation time because model spinup is rapid. The simulation is run for 20 M2 tidal cycles (10.35 days); an M2 harmonic analysis is performed over the last 6 tidal cycles. Previous applications of the model, to the Hawaiian Islands (Carter et al. 2008) and Mid-Atlantic Ridge (Zilberman et al. 2009), showed near steady-state conditions were reached within 12 tidal cycles. In comparison, Rosenfeld et al. (2009) forced their model with 8 tidal constituents and allowed stratification to evolve with time. They found 22 days of model spinup was required, longer than the entire simulation time used here.
Sigma coordinate models, such as POM, suffer erroneous low-frequency velocities from the calculation of horizontal pressure gradients over steep topography (e.g., Haney 1991; Mellor et al. 1994). These erroneous velocities are effectively removed by harmonic analysis. In addition, the erroneous velocities are interleaved vertically, so they cancel out during depth integration.
Carter (2010) found the harmonic analysis output for this simulation to be in excellent agreement with surface elevation measurements from tide gauges (RMS error ∼1%) and reasonable agreement with velocity measurements from moored and shipboard ADCPs within the canyon (RMS error 30%–209%). In comparison, the RMS error of interannual variation observed at the Monterey Bay Aquarium Research Institute mooring in MSC was 46%–157%. No further model validation is provided here.
Depth-averaged velocities in Monterey Bay, both in the canyon and on the shelf, are significantly larger than barotropic velocities inferred from a diagnostic model run in which temperature and salinity are not advected. Carter (2010) explained this phenomenon as the result of a feedback in which sea surface elevation anomalies associated with the internal tide change the pressure gradient forces driving depth-averaged currents in the bay. Despite this, we define baroclinic velocity as total minus depth averaged to allow comparison with previous work.
In this study, the depth-dependent structure of the internal tide is of interest; therefore, baroclinic energy flux and barotropic-to-baroclinic energy conversion are calculated from the harmonic analysis output rather than using the depth-integrated energy equations derived by Carter et al. (2008). Only a ∼1% difference between quantities calculated from the depth-integrated energy equations and those calculated from the harmonic constants were seen around the Hawaiian Islands (Carter et al. 2008).
The internal (baroclinic) hydrostatic tidal energy flux is calculated as F = 〈u′p′〉, where u′ is the velocity perturbation, p′ is the pressure perturbation, and 〈〉 denotes an average over a tidal cycle (e.g., Kunze et al. 2002; Nash et al. 2005). The perturbations u′ and p′ are reconstructed from the harmonic constants. Kelly et al. (2010) noted that surface tide pressure contains a depth-dependent component (due to isopycnal heaving by movement of the free surface) as well as the depth-averaged component. Neglecting the depth-varying component introduces an error to the energy flux calculation, but this error is small in regions with large internal tides such as MSC. Barotropic-to-baroclinic energy conversion is calculated as
where u is depth-averaged velocity from the harmonic analysis output (Niwa and Hibiya 2001). Positive conversion indicates energy being transferred from the barotropic tide to the baroclinic tide. Negative conversion indicates work being done on the barotropic tide by the baroclinic tide (Zilberman et al. 2009).
3. Internal tide generation on Sur Slope
Interpretation of internal tide energy fluxes in the MSC region is complicated by the existence of multiple generation sites. Rainville et al. (2010) showed that internal tides from multiple sources constructively and destructively interfere, resulting in depth-integrated energy flux beams appearing and disappearing in the open ocean. However, it is possible to make realistic inferences about the internal tide generation by comparing depth-integrated baroclinic M2 energy fluxes (Fig. 2a) with the spatial distribution of barotropic-to-baroclinic M2 energy conversion (Fig. 2b).
Most of the internal tide energy entering the upper reaches of MSC (<60 km along the thalweg) originates to the south, on Sur Slope and at the head of Carmel Canyon. Depth-integrated baroclinic energy fluxes at the mouth of the upper canyon (defined here as between Smooth Ridge and Carmel Canyon) are directed northeast and the appearance of depth-integrated energy fluxes on Sur Slope and at the head of Carmel Canyon is consistent with areas of high (>0.1 W m−2) barotropic-to-baroclinic energy conversion. This is in agreement with the inferences of internal tide generation from baroclinic energy flux divergence in the Monterey Bay region by Jachec et al. (2006). Other, more remote, internal tide generation sites are considered in section 6.
Internal tide generation on Sur Slope can be roughly divided into four areas where barotropic-to-baroclinic energy conversion is high: the northern part of the slope near Monterey and Carmel Canyons, the flanks of Sur Platform, the lower slope west of Sur Ridge, and the southern part of the slope (including the northern rim of Sur Canyon). The internal tide generated on the northern part of the slope and along the northern flank of Sur Platform propagates north, into MSC. The internal tide generated on the southern part of the slope and along the southern flank of Sur Platform appears to mostly propagate southeast, along the shelf slope. Finally, the internal tide generated on the lower slope appears to at least partly propagate offshore.
The internal tide does not propagate shoreward along the whole length of MSC. Over the lower reaches of the canyon, to the northwest of the Sur Slope (about 36°35′N, 122°20′W), the depth-integrated energy flux is down canyon (Fig. 2a). The switch to up-canyon energy flux occurs approximately 80 km along the thalweg, between Smooth Ridge and Sur Slope, as baroclinic energy enters from the south. At the mouth of the upper canyon, depth-integrated energy fluxes increase up to 1.5 kW m−1. Across-canyon section A, located across the mouth, shows the up-canyon energy flux is focused near the bottom, over the right-hand slope when looking up canyon (Fig. 3). This core of baroclinic energy enters the upper canyon from the southwest, apparently originating where Carmel Canyon merges with MSC.
4. Internal tide propagation through MSC
In the upper reaches of MSC, the depth-integrated baroclinic M2 energy fluxes are almost entirely up canyon and largest near the canyon axis (Fig. 4a). Maximum energy fluxes (>4 kW m−1) occur around San Gregorio and Monterey Meanders and are in good agreement with previous observations of the internal tide in the canyon by Kunze et al. (2002; see Carter 2010). Petruncio et al. (1998) assumed the internal tide could not navigate San Gregorio Meander without significant dissipation or scattering. However, the modeled internal tide has no difficulty propagating around this meander or the other sharp meanders in the upper canyon. The spatial distribution of depth-integrated energy flux in MSC is in agreement with Jachec et al. (2006) and qualitatively similar to Wang et al. (2009), although the magnitude of the energy fluxes in the latter study are smaller than calculated here by roughly a factor of 2.
The narrow band of maximum energy follows a less meandering path through the canyon than the thalweg. This is particularly evident at Gooseneck Meander, where depth-integrated energy flux vectors are directed over the ridge inside the meander rather than around it. Similarly, at approximately 8 km along the thalweg, the majority of baroclinic energy enters Butterfly Bowl rather than continuing toward the canyon head. In contrast, around the larger San Gregorio and Monterey Meanders, depth-integrated energy flux vectors are orientated along isobaths. This suggests that although the internal tide is topographically steered around the large canyon meanders, it is not affected by smaller-scale bathymetric features.
Barotropic-to-baroclinic energy conversion in the upper canyon is positive and negative in roughly equal proportions (Fig. 4b). This implies local internal tide generation in areas of positive conversion is at least partially compensated for by work done on the barotropic tide by the baroclinic tide in adjacent areas of negative conversion. Negative barotropic-to-baroclinic energy conversion results from phase differences between remote and locally generated internal tides and so is evidence for multiple generation sites (Zilberman et al. 2009).
There are some trends in the spatial distribution of energy conversion; at the ridges inside the meanders, positive conversion tends to occur on the up-canyon flank, whereas negative conversion tends to occur on the down-canyon flank. Kunze et al. (2002) observed a decrease in depth-integrated energy flux at Monterey Meander and suggested local internal tide generation by scattering of the barotropic tide from canyon topography could account for the increase in energy flux farther up canyon. Although positive energy conversion on the up-canyon flank of the ridge inside Monterey Meander is consistent with this argument, there is no noticeable decrease in depth-integrated energy flux around the meander. At the apex of Monterey Meander, Kunze et al. (2002) only measured the energy flux at a single location, so the decrease is more likely a result of undersampling the band of maximum energy.
a. Tidal ellipses
Horizontal baroclinic M2 tidal ellipses along the canyon thalweg and the thalwegs orientation are shown in Fig. 5. The down-canyon direction of the thalweg is primarily toward the west but varies in its north–south orientation. However, at a few locations the thalweg turns back on itself so that the down-canyon direction is toward the southeast: for example, between San Gregorio and Monterey Meanders and at the apex of Gooseneck Meander (Figs. 4a, 5a).
Baroclinic tidal ellipses every 5 km along the thalweg are shown in Fig. 5b. Horizontal baroclinic velocity amplitudes and phases are vertically averaged in 100-m bins before the ellipse parameters are calculated. Down canyon of 65 km, the tidal ellipses display a range of eccentricities (semimajor axis–semiminor axis) and orientations. There is, however, a trend of increasing tidal amplitude toward the surface and bottom, consistent with dominance by low-mode internal waves. The ellipses with large amplitudes also tend to have low eccentricities with many close to ω/f = 1.6, the theoretical eccentricity for a propagating internal wave.
At 65 km the ellipses in the bottom 500 m of the water column have eccentricities <0.2 and are orientated northeast–southwest, the same direction as the thalweg. This agreement between semimajor axis and thalweg orientation suggests topographic steering of the internal tide also occurs down canyon of San Gregorio Meander. By 50 km, the near-bottom ellipses are almost rectilinear (i.e., the semiminor axis is negligible compared to the semimajor axis) and consistently orientated in the same direction as the thalweg. This increase in tidal ellipse eccentricity and alignment along the canyon axis has previously been observed by Petruncio et al. (1998) and Xu and Noble (2009).
As well as becoming increasingly rectilinear, the semimajor axes of the near-bottom tidal ellipses increase to >0.15 m s−1 up canyon of 65 km, indicating bottom intensification of the along-canyon tidal flow. Bottom-intensified along-canyon tidal currents have previously been observed at San Gregorio Meander by Xu et al. (2002) and near the canyon head by Key (1999). Up canyon of 30 km, the along-canyon tidal flow is intensified at all depths.
b. 3D structure
The three-dimensional structure of the internal tide is shown using six sections across the upper canyon (B–G, Fig. 3) and one section along the thalweg (Fig. 6). For the across-canyon sections, the along-canyon component of the baroclinic energy flux is shown. This is simply defined as the energy flux normal to the section. Distance across the canyon is referenced to the thalweg; positive distances are toward the shelf north of the canyon (i.e., on the left when looking up canyon). For the along-thalweg section, both the along-canyon and across-canyon components of the baroclinic energy flux are shown. These are defined as the energy flux tangential and normal to the local orientation of the thalweg. Distance along the canyon is referenced to the Moss Landing.
The along-thalweg section is noisier than the across-canyon sections because of the meandering nature of the thalweg when compared to the smoother path followed by the band of maximum energy. For example, the depth-integrated along-canyon energy flux along the thalweg (Fig. 6a) features several down-canyon energy fluxes in the upper 20 km of the canyon. These occur at locations where the thalweg turns back on itself (such as at the apex Gooseneck Meader), so the down-canyon direction varies by >90° depending on whether it is defined as the orientation of the thalweg or the orientation of the band of maximum energy. The down-canyon energy fluxes are therefore an artifact of the ambiguous definition of down canyon, rather than actual fluxes of energy toward the canyon mouth.
The depth-integrated along-canyon energy flux increases from near zero at 80 km along the thalweg to >4 kW m−1 between San Gregorio and Monterey Meanders (Fig. 6a). It then decreases to 2 kW m−1 between Monterey and Gooseneck Meanders and to near zero at 4 km. The minimum near the apex of Monterey Meander is not an actual decrease in up-canyon energy flux but is due to the band of maximum energy deviating from the path of the thalweg (see Fig. 4). Therefore, at this location along the canyon, the energy flux over the thalweg does not fully represent of the band of maximum energy.
The along-canyon internal tide energy flux is up canyon or near zero at all depths, except near the canyon head, where weak down-canyon energy fluxes occur (Fig. 6b). The up-canyon energy flux is bottom intensified between 55 and 25 km, suggesting topographic focusing occurs. Maximum energy fluxes at the seabed are >15 W m−2. Around the 24- and 12-km marks, there are weak down-canyon energy fluxes near the bottom. These are actual fluxes of energy toward the canyon mouth (rather than artifacts of the meandering thalweg) and are also apparent in across-canyon sections E and G (Fig. 3). The down-canyon energy fluxes may result from supercritical reflection of the up-canyon propagating internal tide from the steep canyon walls. Near-bottom, down-canyon energy fluxes were observed by Kunze et al. (2002) up canyon of Monterey Meander.
The energy flux across the canyon is spatially incoherent and mostly smaller than the along-canyon energy flux (Fig. 6c). The small-scale spatial structure is primarily due to the meandering nature of the thalweg relative to the band of maximum energy.
The bottom intensification of up-canyon energy flux is a result of correlation between large velocity and pressure perturbations. The real component [A cos(ϕ)] of along-canyon velocity perturbation is mostly positive in the upper half of the water column and negative in the lower half (Fig. 6d). Up canyon of 65 km, the real component of pressure perturbation is also positive in the upper half of the water column and negative in the lower half (Fig. 6e). Between approximately 30 and 10 km, the real component of the along-canyon velocity perturbation is up canyon near the bottom, explaining the near-bottom, down-canyon energy fluxes.
c. Kinetic and potential energy
Baroclinic horizontal kinetic energy (HKE) density is calculated as HKE = (1/4)ρ(uA2 + υA2), where uA and υA are perpendicular horizontal velocity amplitudes. Depth-integrated HKE is maximum (∼4 kJ m−2) around San Gregorio and Monterey Meanders (Fig. 7a) and closely matches the spatial distribution of depth-integrated baroclinic energy flux in the canyon (Fig. 2a). HKE is also elevated in Carmel Canyon. The spatial distribution of HKE is similar to that shown by Jachec (2007), but the absolute values calculated here are larger by roughly a factor of 2.
The amplitude of vertical displacement by the internal tide ξA is inferred from vertical velocity amplitude and has a maximum (∼250 m) around San Gregorio and Monterey Meanders (not shown). Farther toward the head of MSC, displacement amplitudes are smaller, order 50 m, and in general agreement with previous observations of the internal tide near the canyon head (Broenkow and McKain 1972; Petruncio et al. 1998; Carter and Gregg 2002). The vertical displacement maximum occurs in the lower third of the water column, consistent with bottom intensification of the energy flux. The associated available potential energy (APE) density is calculated from linear theory, APE = (1/4)ρN 2ξA2, a good approximation if stratification is slowly varying (Carter et al. 2008; Kang and Fringer 2010). Depth-integrated APE is intensified over a larger area of the canyon than HKE (Fig. 7b). Elevated APE occurs all the way to the head of MSC and in Carmel Canyon; maximum values are >5 kJ m−2.
In the upper reaches of MSC, HKE/APE is smaller than the theoretical value for an M2 internal tide, (ω2 + f 2)/(ω2 − f 2) = 2.2 (Gill 1982). This is explained by Petruncio et al. (1998) as an effect of the canyon topography constraining across-canyon motion; if the motion is considered irrotational, the theoretical HKE/APE value is one. However, in 81% of the upper canyon by area, the energy ratio less than one (i.e., APE > HKE). This is inconsistent with free hydrostatic internal waves. Kunze et al. (2002) argue that the excess APE is due to vertical isopycnal displacements induced by barotropic tidal flow over the sloping bottom. They show that removing the barotropic contribution increases the energy ratio to near the theoretical value.
Following Kunze et al. (2002), we calculate the barotropic contribution to vertical displacement ξbt as a linear least squares fit to ξ(z) with zero at the surface. Then, ξbt is subtracted from ξ before recalculating APE. This decreases depth-integrated APE throughout the upper canyon (not shown) and reduces the fractional area in which APE > HKE to 33%. The mean energy ratio in the upper canyon is increased from 0.7 to 2.0, close to the theoretical value.
An alternate explanation for the areas of excess APE in the canyon is the existence of partly standing internal waves. A standing wave (the superposition of two free waves, with the same frequency and amplitude, propagating in opposite directions) was observed in MSC by Petruncio et al. (1998) during October 1994, whereas in April of that year the internal tide was observed to propagate up canyon. They attributed the difference to changes in stratification. Martini et al. (2007) showed that for partially standing waves (where the two free waves have different amplitudes) HKE/APE oscillates between zero and infinity with half the wavelength of the incident waves. It is possible that partially standing waves occur in the canyon, from superposition of the up-canyon propagating internal tide and reflected waves from the supercritical canyon walls. Indeed, down-canyon energy fluxes are apparent near the bottom in across-canyon sections E and G. However, the meandering canyon topography complicates the diagnosis, and no coherent pattern in HKE/APE is observed in this study.
d. Canyon energy budget
To better assess the flux of internal tide energy around the meanders in the upper reaches, the canyon is divided into 17 separate regions. These regions are bounded by 16 across-canyon sections, 17 sections along the 200-m isobath (the approximate depth of the shelf break), 4 sections along the ridges inside San Gregorio and Monterey Meanders, and one section across the mouth of Soquel Canyon (Fig. 4). Net baroclinic energy flux between adjacent regions is the energy flux normal to the section between them, depth and horizontally integrated along the section. In each region, baroclinic energy flux divergence (net energy flux out of the region), net barotropic-to-baroclinic energy conversion, and baroclinic energy dissipation (defined as net energy conversion minus energy flux divergence) are calculated.
Hotchkiss and Wunsch (1982) note that, for the case of internal waves propagating along the axis of a canyon toward its head, energy density should increase, not only because of the depth change but because the walls tend to converge. In the absence of dissipation, the energy density increase should therefore be inversely proportional to the decrease in canyon cross-sectional area. This effect is accounted for in the area (depth and along section) integrals of along-canyon energy flux at the across-canyon sections. If there is no internal tide generation or dissipation in the canyon and no energy flux across the canyon rim, net energy flux will be constant along the canyon.
Net along-canyon energy flux at the mouth of the upper canyon (section A) is 9.0 MW. This accounts for all energy entering the upper canyon from offshore, including Sur Slope, Carmel Canyon, and Smooth Ridge. At the apex of San Gregorio Meander, the along-canyon energy flux decreases to 7.0 MW and continues to decrease almost monotonically to 1.1 MW at Gooseneck Meander (Fig. 8a). Up canyon of Gooseneck Meander, the along-canyon energy flux remains around 1 MW. The decrease in along-canyon energy flux between the mouth and Gooseneck Meanders implies that either high levels of internal tide dissipation occur in the canyon or a large fraction of the baroclinic energy escapes over the canyon rim onto the shelf. The second explanation is discounted because the depth-integrated energy flux across the 200-m isobath is small (order 10 W m−1 along the majority of the canyon rim) compared to the along-canyon energy flux (order 1 kW m−1).
Net energy flux across the 200-m isobath sections and ridge sections inside San Gregorio and Monterey Meanders is shown by the gray line and right axis in Fig. 8a. The energy flux across the 200-m isobath between each across-canyon section is typically <25 kW, compared with a 500-kW average decrease in along-canyon energy flux between the sections. However, at two locations the energy flux onto the shelf is substantial: Gooseneck Meander (18 km), where there is a 0.2 MW energy flux over the southern rim, and near the canyon head (7 km), where there is a 0.9 MW flux over the northern rim, mostly into Butterfly Bowl. The energy flux onto the shelf here, as well at other locations along the shelf break, is examined in section 5. The energy fluxes across the ridges inside the meanders are large compared to the typical energy flux across the 200-m isobath but small compared to the along-canyon energy flux. Inside Monterey Meander, the across-ridge energy flux is 0.4 MW, apparent as a maximum at 41 km and a minimum at 30 km. The across-ridge energy flux inside San Gregorio Meander is 0.5 MW, apparent as a minimum at 39 km.
As expected from the decrease in net along-canyon energy flux toward the head canyon head, baroclinic energy flux divergence is negative (i.e., convergent) along the majority of the upper canyon (Fig. 8b). The two exceptions are Monterey and Gooseneck Meanders, where there are small positive energy flux divergences. Net energy conversion alternates between positive and negative values along the canyon, implying that there are areas of local internal tide generation (positive values) as well as areas where work is done on the barotropic tide by the baroclinic tide (negative values). The largest positive values occur at Monterey Meander, whereas the largest negative values occur at Gooseneck Meander and between San Gregorio and Monterey Meanders.
There is not, however, any consistent evidence for significant dissipation and regeneration of the internal tide around Monterey Meander, as suggested by Kunze et al. (2002). The small increase in along-canyon energy flux on the up-canyon side of Monterey Meander is consistent with the band of positive energy conversion on the up-canyon flank of the ridge inside the meander, but it is small compared to the large decrease in energy flux between San Gregorio and Monterey Meanders. The decrease in depth-integrated energy flux at Monterey Meander observed by Kunze et al. (2002) is more likely a result of undersampling the band of maximum energy.
Total baroclinic energy dissipation in the upper canyon (a summation of all 17 regions between section A and the canyon head), inferred from only baroclinic energy flux divergence, is 7.6 MW. This is comparable to the 8.3-MW net dissipation in the canyon calculated by Jachec et al. (2006). However, the estimate of total baroclinic dissipation should also include any barotropic-to-baroclinic energy conversion that occurs in the upper canyon. Net energy conversion in the upper canyon is positive but only 50 kW. It therefore has little effect on total baroclinic energy dissipation. Dividing total barotropic energy dissipation by the area of the upper canyon (225 km2) yields a depth-integrated dissipation rate of 0.03 W m−2.
e. Rotational effects
Across the mouth of the upper canyon, the along-canyon baroclinic energy flux is asymmetrically distributed. The up-canyon energy flux is focused over the right-hand slope when looking up canyon (Fig. 3a). This asymmetry is also apparent but to a lesser extent farther toward the canyon head at across-canyon sections E–G. Petruncio et al. (2002) find a similar energy density asymmetry with idealized canyon topography and suggest it is due to rotational effects.
To test this hypothesis, the model is rerun without rotation. Interestingly, the across-canyon distribution of the along-canyon energy flux is not significantly altered in the nonrotating case (not shown). At section A, the up-canyon energy flux remains focused over the right-hand slope when looking up canyon. However, the amount of internal tide energy entering the upper canyon is reduced in the nonrotating case (Fig. 9). The net along-canyon energy flux at section A is only 6.9 MW, a decrease of 23% relative to the original rotating case. In the upper reaches of the canyon, depth-integrated energy fluxes are decreased by up to 2 kW m−1, with the largest reductions occurring around San Gregorio Meander. This suggests that, although rotation may aid the funneling of internal tide energy into canyons, within meandering canyons such as MSC topographic steering is the more important control on across-canyon energy distribution.
5. Internal tide propagation onto the shelf
Depth-integrated baroclinic M2 energy fluxes on the continental shelf are small compared to those in MSC, typically <10 W m−1. However, near the head of the canyon and along the northern edge of Sur Platform energy fluxes >250 W m−1 occur (Fig. 10a). These energy fluxes are mostly onshore or along the shelf break.
The large energy fluxes near the canyon head are the end of the band of maximum energy that follows the canyon axis. The majority of the baroclinic energy enters the Butterfly Bowl, but some continues toward the canyon head. Baroclinic energy also escapes over the southern rim at Gooseneck Meander. At these three locations, the internal tide is almost entirely dissipated by the 100-m isobath.
On either side of Smooth Ridge, there are smaller, ∼50 W m−1 energy fluxes, where baroclinic energy is funneled up Cabrillo Canyon and Horseshoe Scarp and onto the shelf. In contrast, the internal tide on the Sur Platform is most likely generated locally, on the flanks of the platform, where barotropic-to-baroclinic energy conversion is high.
The baroclinic energy flux across the shelf break is taken to be the energy flux normal to the 200-m isobath (Fig. 10b). Similar to the along-thalweg section, the energy flux across the 200-m isobath is noisy because of the winding nature of the isobath, especially at the head of MSC and Carmel Canyon. The depth-integrated energy flux across the 200-m isobath is order 10 W m−1 along the majority of the shelf break, but it increases by over an order of magnitude near the head of MSC, where internal tide energy escapes onto the shelf. Maximum energy flux onto the shelf, 500 W m−1, occurs in Butterfly Bowl (2 km along the shelf break). On the southern rim of MSC, there is a 200 W m−1 energy flux onto the shelf at Gooseneck Meander (−6 km) and a smaller energy flux off of the shelf between Gooseneck Meander and the canyon head. At the head of Carmel Canyon and along the northern edge of Sur Platform (between −80 and −35 km), energy fluxes are both onto and off of the shelf, including a 300 W m−1 energy flux into Carmel Canyon (−57 km) from Sur Platform to the south.
Levels of mixing from microstructure measurements on Smooth Ridge and at the head of MSC are described by Lien and Gregg (2001) and Carter and Gregg (2002). The microstructure profiles on the shelf from these studies are reanalyzed by Carter et al. (2005); the locations of these profiles are shown in Fig. 10a and are clustered in four groups. One group of profiles [those from Lien and Gregg (2001), denoted “fan” in Carter et al. (2005)] is at the top of the Smooth Ridge, between Cabrillo Canyon and Horseshoe Scarp. Two groups (“north” and “south”) are on the rim of MSC, up canyon of Gooseneck Meander. The final group (“test”) is on the shelf north of MSC, down canyon of Gooseneck Meander.
Carter et al. (2005) find no significant difference in mean turbulent kinetic energy dissipation rate from the fan, north, and south profiles (∼5 × 10−8 W kg−1 for each group). However, mean dissipation rate from the test profiles is 5 times lower. They suggest the difference may result from the test profiles being made during neap tide or the presence of coastal upwelled water on the shelf. The model results presented here suggest an alternative explanation. The fan, north, and south profiles are located close to areas where internal tide energy escapes onto the shelf and is rapidly dissipated, resulting in elevated turbulent kinetic energy dissipation rates. In contrast, the test profiles are in an area where the baroclinic energy flux is negligible. Even higher dissipation rates may be expected in Butterfly Bowl and on Sur Platform, where the energy fluxes gradients are largest.
6. Remote internal tide generation
By comparing depth-integrated baroclinic M2 energy fluxes with the spatial distribution of barotropic-to-baroclinic M2 energy conversion, we infer that most of the internal tide energy entering the upper reaches of MSC is generated on Sur Slope and at the head of Carmel Canyon. It is of interest, however, to quantitatively assess how much of the internal tide energy in MSC originates at more remote sites. To this end, the model is run for a smaller domain, from 35°59′9″N, 122°39′52″W to 37°9′50″N, 121°44′8″W, a subset of the original large model domain (Fig. 11). The small domain excludes three remote areas of high barotropic-to-baroclinic energy conversion that feature in the large domain (see Fig. 12a of Carter 2010); these are Davidson and Guide Seamounts and the corrugated shelf slope northwest of MSC (about 122°50′W, 37°0′N). However, it retains the areas of high conversion on Sur Slope and the flanks of Sur Platform. The same initial temperature and salinity profiles are used and the model is again forced at the boundaries with M2 barotropic velocities from TPXO6.2.
Depth-integrated baroclinic energy flux in the upper reaches of MSC from the small domain model run still features a band of maximum energy that follows a topographically steered path around San Gregorio and Monterey Meanders and dissipates toward the canyon head (Fig. 12). However, the magnitude of the energy flux is smaller throughout the canyon; energy fluxes around San Gregorio and Monterey Meanders are reduced by up to 1 kW m−1. The decrease relative to the energy flux from the large domain run is around 20% throughout the upper canyon. Net along-canyon energy flux at the mouth of the upper canyon (section A) is 7.3 MW in the small domain run, 1.7 MW less than the 9.0 MW in the large domain run (19% relative decrease). In Carmel Canyon, depth-integrated energy fluxes are reduced by ∼800 W m−1 (60%).
The energy flux decrease in the canyons is not a simple subtraction of internal tide energy generated outside the area common to both domains. The hydrostatic pressure perturbation due to internal waves (p′) affects barotropic-to-baroclinic energy conversion through Eq. (2). Therefore, any change to the internal wave field potentially feeds back on internal wave generation. Differences in energy conversion between the two domains may also result from adjustment of depth-averaged velocity u. In this case, however, reducing the size of the domain has a negligible effect on depth-averaged currents inside the common area; differences in energy conversion (Fig. 13) are almost entirely the result of changes to the internal wave field. Barotropic-to-baroclinic energy conversion is decreased on the northern part of Sur Slope, in Carmel canyon, and along the flanks of Sur Platform. These are areas where the internal tides in MSC and Carmel Canyon are assumed to be generated, which may explain the energy flux decrease in the canyons.
Net energy conversion in the common area is 56.7 MW for the large domain run and 49.6 MW for the small domain run (Table 1), a decrease of 7.1 MW (13%). This is the result of a 16.7-MW decrease in positive conversion, mostly compensated by a 9.5-MW decrease in negative conversion. Integrating over the area of the upper canyon only, net energy conversion actually increases from 50 kW to 2.2 MW, but this is the result of a 3.2-MW decrease in negative conversion (partially compensated by a 0.9-MW decrease in positive conversion) rather than an actual increase in internal tide generation.
Although the 7.1-MW energy conversion decrease in the common area is more than enough to explain the reduction in energy entering the upper canyon (1.7 MW), the direct influence of internal tides generated at remote sites cannot be ruled out. Unfortunately, the exact value of the energy flux into the common area is unknown for the large domain run, because the incoming and outgoing internal tides cannot be separated (〈u′ p′〉 only gives the net energy flux). An approximate value can be arrived at by assuming the outgoing energy flux is the same as for the small domain run.4 This yields an incoming energy flux of 4.5 MW. However, this assumption is unlikely to be strictly valid because of the pressure perturbation feedback between baroclinic energy flux and barotropic-to-baroclinic energy conversion.
A modified version of the Princeton Ocean Model is used to simulate the M2 internal tide in Monterey Submarine Canyon. The origin of internal tide energy in the canyon is inferred by comparing depth-integrated baroclinic M2 energy fluxes with the spatial distribution of barotropic-to-baroclinic M2 energy conversion. Most of the internal tide energy entering the upper reaches of the canyon is generated to the south, on Sur Slope and at the head of Carmel Canyon. Positive energy conversion on the up-canyon flanks of the ridges inside the meanders, implies some local internal tide generation occurs in the canyon, but this is at least partially compensated for by adjacent areas of negative conversion.
In the upper reaches of the canyon, the internal tide is topographically steered around the large meanders. Depth-integrated energy fluxes are almost entirely up canyon and largest near the canyon axis, up to 1.5 kW m−1 at the mouth of the upper canyon and increasing to over 4 kW m−1 around Monterey and San Gregorio Meanders. The narrow band of maximum energy follows a less meandering path through the canyon than the thalweg. Near the apex of Monterey Meander, the band of maximum energy deviates from the path of the thalweg. The decrease in depth-integrated energy flux observed by Kunze et al. (2002) at this meander is most likely a result of undersampling the band of maximum energy with a single energy flux measurement over the thalweg.
The up-canyon energy flux is bottom intensified between 55 and 25 km (along the thalweg), suggesting that topographic focusing occurs. The bottom intensification is a result of correlation between large velocity and pressure perturbations.
Baroclinic M2 tidal ellipses in the upper canyon typically have eccentricities larger than ω/f = 1.6, the theoretical eccentricity for a propagating internal wave, and are orientated in the same direction as the thalweg. The ratio of HKE to APE is smaller than the theoretical value for an M2 internal tide (2.2) and, in the majority of the upper canyon, less than one. The energy ratio can be brought closer to the theoretical value if the barotropic contribution to vertical displacement is first removed.
Net along-canyon energy flux decreases almost monotonically from 9 MW the mouth of the upper canyon to 1 MW at Gooseneck Meander, implying that high levels of internal tide dissipation occur. Net barotropic-to-baroclinic energy conversion in the upper canyon is positive but only 50 kW. Total baroclinic energy dissipation is 7.6 MW, comparable to the value calculated by Jachec et al. (2006).
Asymmetric distribution of the along-canyon energy flux across the mouth of the upper canyon is not an effect of rotation, because it is also apparent in a nonrotating model run. However, the net along-canyon energy flux at the mouth is decreased by 23% in the nonrotating case. This suggests rotation aids the funneling of internal tide energy into the canyon, but topographic steering is the more important control on across-canyon energy distribution.
The depth-integrated energy flux across the 200-m isobath is order 10 W m−1 along the majority of the canyon rim but increases by over an order of magnitude near the canyon head, where internal tide energy escapes onto the shelf. The internal tide is almost entirely dissipated by the 100-m isobath.
Reducing the size of the model domain to exclude remote areas of high barotropic-to-baroclinic energy conversion decreases the depth-integrated energy flux in the upper canyon by 20%. However, quantifying the role of remote internal tide generation sites is complicated by a pressure perturbation feedback between baroclinic energy flux and barotropic-to-baroclinic energy conversion. Care must therefore be taken when comparing internal wave fields in numerical models with different domain sizes, because any change to the energy flux can affect internal wave generation in other areas of the domain.
Helpful comments on the manuscript were provided by Mike Gregg, Matthew Alford, Ren-Chieh Lien, Danielle Wain, and three reviewers. This work was funded by the National Science Foundation Grant OCE0751226.
Corresponding author address: Rob A. Hall, Department of Oceanography, University of Hawaii at Manoa, Marine Science Building, 1000 Pope Road, Honolulu, HI 96822. Email: firstname.lastname@example.org
The distances stated here are longer than those reported by Carter and Gregg (2002) because higher resolution bathymetry data is used to define the thalweg. This results in a longer along-thalweg distance between any two points on the canyon axis because small-scale bends between the points are resolved.
Mean horizontal viscosity over the entire model domain is 0.016 m2 s−1.
The incoming energy flux at the boundaries of the small domain (i.e., the boundaries of the common area) is zero because the forcing is only barotropic. The net energy flux is therefore equal to the outgoing energy flux.