Energetics of M₂ Barotropic-to-Baroclinic Tidal Conversion at the Hawaiian Islands

a. Model setup

For this study we use the Princeton Ocean Model (POM), a three-dimensional, nonlinear, free-surface, finite-difference primitive equation model (Blumberg and Mellor 1987). For computational efficiency, the model calculates the fast-moving surface gravity waves separately from the internal structure, using a technique known as mode, or time, splitting. POM has been used for a number of previous studies of baroclinic tidal processes over idealized (e.g., Holloway and Merrifield 1999; Johnston and Merrifield 2003) and realistic (e.g., Cummins and Oey 1997; Merrifield et al. 2001; Niwa and Hibiya 2001; Merrifield and Holloway 2002; Johnston et al. 2003) topography.

POM uses the hydrostatic approximation, wherein the pressure is simply related to the weight of the water column. This is valid as long as the horizontal scales of motion are much greater than the vertical scales (Hodges et al. 2006; Mahadevan 2006). Internal tides typically meet this criterion, although exceptions include wave breaking and steepening into highly nonlinear (solitary) waves. Venayagamoorthy and Fringer (2005) found that in a bolus [a high (∼2:1) aspect-ratio, self-advecting vortex core], the nonhydrostatic pressure was 37% of the total pressure. With sufficient horizontal resolution, a hydrostatic model can identify the presence of such features but cannot accurately describe them (Holloway et al. 1999; Hodges et al. 2006); lacking the dispersive (nonhydrostatic) processes, hydrostatic models tend to overestimate the steepness of the features (Hodges et al. 2006). When comparing hydrostatic and nonhydrostatic simulations, Mahadevan (2006) found it was difficult to identify the effect of the nonhydrostatic term at a horizontal resolution of 1 km.

The Mellor and Yamada (1982) level-2.5, second-moment turbulence closure scheme (MY2.5) is used by POM to calculate the vertical eddy diffusivities. This k–l (turbulent kinetic energy–mixing length) submodel has been used extensively for a range of applications (over 1650 citations to date, according to the Web of Science database), including the internal tide studies listed above. It should be noted that this submodel was developed for application to atmospheric and oceanic boundary layers and does not explicitly include the wave–wave interaction dynamics expected to dominate the dissipation of internal tide energy. Warner et al. (2005) found that the MY2.5 underestimated mixing in steady barotropic and estuarine flows but overestimated mixing in a wind-driven mixed layer deepening study. We find that the combination of MY2.5 and Smagorinsky horizontal diffusivity gives reasonable agreement with microstructure observations (section 4c). Finally, a quadratic bottom friction is used in POM with a logarithmic layer formulation for the coefficient (Mellor 2004).

The simulation domain extends over 20°21.8′–23°0.3′N and 160°48.3′–155°22.5′W, that is, including the main Hawaiian Islands with the exception of the Big Island (Fig. 1). The model grid is derived from multibeam survey data and has a horizontal spacing of 0.01° (1111.9 m in latitude and 1023.5–1042.4 m in longitude), with 61 sigma levels spaced evenly in the vertical. The stratification is specified using time-averaged temperature and salinity profiles obtained over 10 yr at Station ALOHA, the Hawaii Ocean Time-series (HOT) site located 100 km north of Oahu (Fig. 1). This background stratification is horizontally uniform throughout the domain. Carter et al. (2006) and Klymak et al. (2006) found little variation outside the surface layer in stratification observed over the Hawaii Ridge. Surface buoyancy and momentum fluxes are set to zero, but the background stratification is preserved because neither horizontal nor vertical diffusivity is applied to temperature and salinity (i.e., these fields are simply advected). The model is forced at the lateral boundaries using the Flather condition (Flather 1976; Carter and Merrifield 2007), with M₂ tidal elevation and barotropic velocity from the Hawaii region TPXO6.2 inverse model (Egbert 1997; Egbert and Ray 2001; Egbert and Erofeeva 2002). Following Carter and Merrifield (2007), the baroclinic velocity fluctuations and isopycnal displacements are relaxed to zero over a 10-cell-wide region. They show that this modified relaxation scheme does not reflect energy even when a range of internal tide modes are present.

The simulations are run for 18 tidal cycles (9.3 days) from a quiescent, horizontally uniform state. Over the last six tidal cycles (3.1 days), singular value deposition (SVD) analysis is performed to obtain barotropic (depth averaged) and baroclinic (total minus depth averaged)¹ harmonic amplitudes and phases. Also, the barotropic and baroclinic energy equations [Eqs. (4) and (5) in section 3] are averaged over the same six tidal cycles.

b. Model validation

Using sea level data, Larson (1977) noted a 46° (1.5 h) M₂ phase shift across the island of Oahu, between Mokuoloe and Honolulu. Ray and Mitchum (1997) argued that most of this phase difference is likely due to the presence of the shallower topography along the Hawaiian Ridge, with additional contributions from the baroclinic tides. This is confirmed by comparing cotidal plots of our baroclinic (Fig. 2a) and the TPXO barotropic (Fig. 2b) simulations. Although most of the phase difference is in the barotropic field, the inclusion of baroclinic tides adds significant small-scale variability. The baroclinic tide has a noticeable effect on the amplitude around Oahu and Kauai. In particular, the amplitudes are reduced on the north shore of Oahu and increased on the western coast compared to the barotropic calculation.

A first check on the model skill is provided by comparing the modeled surface elevation amplitude and phase with coastal sea level observations (Table 1). National Oceanic and Atmospheric Administration’s (NOAA) Center for Operational Oceanographic Products and Services (CO-OPS) maintains long-term, quality-controlled records from gauges at Port Allen, Nawiliwili, Honolulu, Mokuoloe, and Kahului (location shown with stars in Fig. 2a; the data and harmonic constants are available online at http://tidesandcurrents.noaa.gov/). Following Cummins and Oey (1997), a quantitative comparison is given using an absolute RMS error (RMSE),

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where subscripts o and m denote observed and modeled amplitudes (A) and phases (G). The model and observations are in good agreement, with three sites having E < 9 mm. The Honolulu and Mokuoloe gauges, where amplitude differences are 10–13 mm, are in Honolulu Harbor and Kaneohe Bay, respectively, which are not fully resolved by the model. The largest RMSE at Kahului is due to a 5.9° (∼12 min) phase difference, which presumably is due to the narrow harbor entrance.

The baroclinic cotidal plot (Fig. 2a) shows that the surface elevation amplitudes and phases, particularly between Oahu and Kauai, are influenced by the surface bounce of the baroclinic tide. The yellow–red band is where the near-surface baroclinic displacement is in phase with the barotropic tide, resulting in increased elevation, and, conversely, the dark blue bands are out of phase. To evaluate how well we have simulated the surface elevation away from land, a comparison is made to the along-track M₂ fits of satellite altimetry data (Fig. 3). In addition to the three Ocean Topography Experiment (TOPEX)/Poseidon tracks (TP-003, TP-112, and TP-041) that cross the domain, we also consider data from the European Space Agency’s European Remote Sensing Satellite (ERS; track 396), which goes though the middle of the Kauai Channel. The positions of the tracks are shown as gray lines in Fig. 2a. Overall, there is very good agreement in both amplitude and phase between our simulation and the satellite altimetry. Often the model lies within one standard error of the satellite data (Fig. 3, gray shading). The average RMSE along each track, E = n⁻¹Σ_nE, ranges from 0.9 to 1.0 cm. The largest amplitude differences are on the order of 2 cm and occur at the first surface bounce along the ERS-396 track and around Kauai (TP-003).

This POM simulation, like the TPXO model from which the boundary conditions are derived, calculates the height of the ocean surface relative to the seabed. The satellite altimetry, however, gives the surface height relative to a reference geoid. The difference between these two measurements is the “load tide,” the deformation of the solid earth due to the weight of the water above it. In the above comparisons, the satellite observations have had the load tide included using the TPXO6.2 load model. The dashed lines in the amplitude panels show the original (without load tide) satellite elevations. Inclusion of the load tide reduces E for TP-041 by 54%; for TP-112 by 26% and for ERS-396 by 2%; but it increases E for TP-003 by 10%. This is the first time, to our knowledge, that a process tide model has been shown to be accurate enough to require the load tide correction when validating to satellite data.

An intensive field experiment, the HOME Nearfield, which was conducted at Kaena Ridge from 2001 to 2003, resulted in detailed observations of the currents and mixing patterns. As a check on the validity of the model, we compare model output to M₂ harmonic fits from two moorings. One mooring, A2, was located on the southern edge of Kaena Ridge (Fig. 1, northern star) with three ADCPs giving coverage of most of the water column (Boyd et al. 2005). The second mooring, C2, was located south of the ridge in ∼4000-m water depth (Fig. 1, southern star) with coverage only in the upper 700 m.

The magnitudes and vertical structure of the model amplitudes and phases agree well, for the most part, with the mooring observations (Fig. 4). The E range from 0.026 to 0.035 m s⁻¹. The largest amplitude differences are near the seafloor in A2 (Fig. 4a), where the model overpredicts both the u and υ currents. An across-ridge section (not shown) indicates that the model predicts the formation of a near-bed downward- propagating beam at A2. Such beams have been observed elsewhere on the Kaena Ridge (e.g., Nash et al. 2006; Aucan et al. 2006), but their formation is dependent on the criticality of the local slope (Balmforth et al. 2002; Garrett and Kunze 2007). It is possible that near A2 the 0.01° resolution topography is closer to critical than the actual topography.

At mooring C2, the model predicts a surface intensification in υ velocity, which is not seen in the observations (Fig. 4b). Important near-surface forcing processes such as the wind and mesoscale activity are not included in this simulation and could easily alter the surface velocities. High-frequency radar observations at Kaena Ridge, taken as part of HOME, show that the M₂ surface velocity pattern predicted by the model is usually masked or altered by mesoscale processes (Chavanne 2007).

Current ellipses at depths where the model levels are nearly collocated with the ADCP measurements show good agreement at subsurface depths (Fig. 4). Near the surface, flows at both moorings are more rectilinear than predicted by the model.

Our ability to verify the simulation with the suite of observations described above gives us a high level of confidence in the model’s ability to simulate the M₂ internal tide around the steep topography of the Hawaiian Islands. Including the baroclinic tide improves the prediction of the water level around the Hawaiian Islands. The RMSEs from the along-track satellite data (E ≈ 1.0 cm) are approximately one-third those for the barotropic M₂ TPXO model [Egbert and Erofeeva (2002) estimated the RMSE at 3.6 cm for ocean depths between 2500 and 4000 m, and Simmons et al. (2004) estimated it to be less than 3 cm over the entire domain].

c. Modeled internal tide structure

Although the focus of this work is on the energetics, it is constructive to briefly examine the structure of the internal tide generated at the Kaena Ridge. A transect across the ridge shows a complex vertical displacement and baroclinic current structure (Figs. 5a,c). Beamlike features emanate from the flanks of the ridge as well as from discontinuities deeper in the water column. This transect was chosen to pass through two stations occupied as part of the HOT experiment, including Station ALOHA where Chiswell (1994) observed internal tides from repeat hydrographic surveys. The structure is shown during maximum north-northeastward barotropic current (Fig. 5a) and the quadrature structure 3 h later during barotropic slack tide (Fig. 5c). Peak baroclinic currents (0.24 m s⁻¹) and displacements (92 m) are found along the tidal beams that originate on both sides of the ridge. During both plotted phases the displacement is upward on the south side and downward on the north side of the ridge. When the barotropic current slackens to near zero, 180° phase shifts of the baroclinic current occur in the beam, consistent with classic analytic descriptions of tidal beams (e.g., Rattray et al. 1969). The beams are more focused during maximum across-ridge barotropic flow than at slack flow.

The baroclinic energy flux (not shown) follows the same three main pathways seen in the vertical displacement: up and away from the ridge leading to surface bounces ∼40 km on either side of the crest; up and over the top of the ridge, with beams crossing over the crest resulting in weaker fluxes; and down and away with some contact with the bottom along the near and supercritical slope. The crossing beams over the crest of the ridge have been described as a quasi-standing wave by Nash et al. (2006) and Carter et al. (2006). The downward-propagating beams have been examined in the context of near-boundary mixing by Aucan et al. (2006) and Aucan and Merrifield (2008).

The horizontal variability in the baroclinic structure can be assessed by considering the total (barotropic plus baroclinic) surface currents (Figs. 5b,d). The banding of higher velocity that parallels the ridge corresponds to the interaction of the beam with the surface mixed layer. The first surface bounce has velocities of ∼0.2 m s⁻¹. Notice that near Station Kaena, this surface baroclinic current is comparable with the barotropic current over the ridge crest (Fig. 5a, blue arrows). Farther west, where the ridge crest is deeper, the baroclinic surface currents associated with the surface bounce exceed the barotropic currents over the ridge crest (Fig. 5b). Overall, the strongest surface currents (up to 0.55 m s⁻¹) are barotropic and are found over the shallow Penguin Bank, southwest of Molokai.

a. Energy balance

The terms of the barotropic and baroclinic energy equations [(4) and (5)] are averaged over the last 6 M₂ tidal cycles of an 18-tidal-cycle simulation. The area integrals presented in this section exclude the outer 12 cells along each boundary (20° 29.1′–22°53.1′N and 160°41.1′–155°29.7′W; gray line in Fig. 1). This exclusion is conservative, as there is no evidence that the effect of the relaxation layer extends beyond its 10-cell width (Carter and Merrifield 2007).

A total of 2.733 GW is lost from the barotropic tide within our domain (Table 2). The majority of this barotropic flux divergence is converted into baroclinic tides (2.286 GW), with the bottom friction (barotropic dissipation term) accounting for 0.163 GW. Of the energy converted from barotropic to baroclinic, 73% radiates out of the domain as baroclinic flux (∇ · Flux_bc = 1.701 GW). The majority of the remaining baroclinic energy is lost to dissipation within the domain (0.445 GW, 19%).

Although the simulation is forced only with M₂, nonlinear dynamics can transfer energy to higher harmonics (M₄, M₆, . . . ; e.g., Lamb 2004), to the subharmonic (1/2M₂, e.g., Carter and Gregg 2006), or into rectified tides. In (4) and (5), energy actively undergoing a nonlinear transformation would be in the advection term, whereas the tendency term includes the time rate of change of energy at all frequencies. The tendency and nonlinear advection terms are small in both the barotropic and baroclinic equations, suggesting little energy at, or being transferred to, other constituents. As the simulation started from a quiescent state, there was no “seed” energy at other frequencies to facilitate wave–wave interactions, and therefore, it is likely the nonlinear interactions are underestimated.

The computational mode splitting technique can produce an erroneous energy source (Simmons et al. 2004; Zaron and Egbert 2006b). Simmons et al. (2004) found that with a ratio of eight or less barotropic iterations to one baroclinic iteration, the energy error was <10%, which they considered to not have a qualitative impact on their analysis. Our barotropic error term is 10.8% of the flux divergence (−0.296 GW), which we attribute to the 50:1 mode split used in the present simulation. The baroclinic error is 5.3% of the conversion term, twice the difference between the two estimates of conversion (2.4%).

A 30 M₂ tidal-cycle simulation was performed to check the stability of the 18 tidal- cycle results. As in the 18-tidal-cycle simulation, the energy analysis was performed over the last 6 tidal cycles. The flux divergence and conversion values changed by ≲1% in both the barotropic and baroclinic balances. The absolute differences in dissipation were similar, 0.01–0.03 GW, which because of their small initial values results in differences of 3%–7%.

Sigma coordinate models such as POM generate erroneous currents through a pressure-gradient error (e.g., Mellor et al. 1994). In this simulation we observed these zero-frequency currents to be layered throughout the water column with the layers having opposite sign, such that they cancel in the vertical integral. Being zero frequency, they are excluded from the harmonic fits, so a further check on (4) and (5) is obtained by calculating the conversion from the harmonic fits using (6). This approach gives 2.368 GW of conversion, which is larger than our baroclinic value by only 1.2%. The mean and standard deviation of the RMS differences between the barotropic model output and the corresponding harmonic time series are (1.5, 0.8) mm, (2.0, 5.5) mm s⁻¹, and (1.9, 4.8) mm s⁻¹ for η, u, and , respectively. Only the barotropic field time series could be stored because of file size constraints. The largest departures from sinusoidal (RMS differences) occur near headlands on Oahu and Molokai (not shown). Therefore we conclude that non-M₂ currents, both physical (M₄, M₆, . . .) and erroneous, have little effect on the regional energy balances.

b. Structure of baroclinic energy fields

The ∼2.3 GW lost to internal tides in the barotropic energy balance becomes the source term in the baroclinic balance. The barotropic-to-baroclinic conversion occurs mainly on both sides of the Kaena Ridge, northwest of Oahu, and south of the 1000-m isobath surrounding Kauai and Niihau (Fig. 6a). The M₂ conversion is weak in the Kaiwi Channel (between Oahu and Molokai), although conversion occurs off Makapuu, the eastern tip of Oahu. Very little conversion occurs east of 157°W. Forty percent of the internal tide generation occurs between the 1000- and 2000-m isobaths, and 84% occurs over topography shallower than 3000 m (Fig. 6b).

From these generation sites, baroclinic energy radiates away from the ridge (Fig. 7). The largest fluxes are contained within two beams: one propagating northeast from the Kaena Ridge, and another southward beam formed from the interaction of the Niihau and Kaena generation sites. The broad dimensions of these beams are set by the length of the generation region; as such, similar features are seen in the coarser simulations of Merrifield and Holloway (2002) and Simmons et al. (2004). The global simulations of Simmons et al. (2004) show that in the absence of mesoscale disturbances these beams are coherent for large distances. The details and magnitude of the fluxes are, however, dependent on the slope and resolution of the topography.

Small-scale structures are observed emanating from localized generation regions. A second, southward beam occurs as a result of fluxes from the near-Oahu end of Kaena Ridge being steered by the topography south of Oahu. North of Molokai there is a ∼30-km wide beam parallel to the isobaths, and the generation off Makapuu Point leads to baroclinic energy fluxes that are steered around the south shore of Oahu. The smooth transitions between the flux regions are likely due to the linearity of the solutions (section 4a) as well as low numerical noise.

The rate of turbulent baroclinic energy loss within the model (Fig. 8) is a combination of the MY2.5 submodel (κ_m) and the Smagorinsky horizontal diffusivity (A_M). These quantities enter (5) through the D_x, D_y, and F_x, F_y terms, respectively. The total amount of vertical (submodel) mixing appears to be partly dependent on how noisy the simulation is, and in our simulations, the vertical mixing from the submodel is less than expected from observations. However, the combined horizontal and vertical baroclinic dissipation is consistent with the observed diapycnal mixing. From microstructure observations at a small seamount on the Kaena Ridge (21°43.8′N, 158°38.8′W), Carter et al. (2006) found an average turbulent dissipation rate of = 6.2 × 10⁻⁸ W kg⁻¹. If averaged over the water column ρ₀D, where D = 1000 m and ρ₀ = 1025 kg m⁻³, this would be ∼10⁻¹ W m⁻², similar to the modeled dissipation in that location. A more detailed comparison to microstructure observations is included in the next section.

c. Energy terms with distance from ridge

In this section we consider how the energy varies with distance from the ridge crest. As can be seen from Table 2, the baroclinic energy balance is predominately between three terms:

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We integrate these terms over regions bounded by the 1000-, 2000-, and 3000-m isobaths, as well as lines paralleling the 3000-m isobath at distances of 10, 20, 30, 40, 60, 80, 100, and 120 km (Fig. 9a). Each region is further divided into north and south by a line that passes through the main islands (Fig. 9a, heavy line).

The total southward flux peaks 10 km beyond the 3000-m isobath with a value of 0.95 GW (Fig. 9b). The northward flux is weaker (maximum of 0.84 GW) and peaks farther from the ridge, at a distance of 30 km from the 3000-m isobath. After peaking, the flux slowly decays with distance from the ridge. The flux 120 km from the 3000-m isobath is 3% (6%) lower than the maximum on the north (south) side of the ridge. However, it is not always possible to be within the model domain at 120 km from the 3000-m isobath (particularly south of the ridge); consequently, these flux decays are an overestimate.

The area-integrated conversion, baroclinic flux divergence, and dissipation are all larger on the south side of the ridge (Fig. 9c). Most of the conversion (and flux divergence) occurs within 10 km of the 3000-m isobath, in agreement with the analysis presented in Fig. 6b. In most regions, the conversion is slightly larger than the flux divergence, resulting in small dissipation. The exception is within the southern portion of the 1000-m isobath where the dissipation exceeds the flux divergence by 18.8 MW.

Thirty percent (132 out of 445 MW) of the total baroclinic dissipation in the model occurs within the 1000-m isobath south of the ridge crest. Of that 64.5 MW is dissipated between 158°12′ and 157°00′W, which encompasses Oahu minus Kaena Point through to the middle of Molokai. The majority of the relatively small amount of generation that occurs in this region is confined off Makapuu Point (Fig. 6a). The depth-integrated flux vectors (Fig. 10) show that the internal tide generated at Makapuu Point tends to either follow the coastline around into Mamala Bay (previously studied by Eich et al. 2004; Alford et al. 2006; Martini et al. 2007) or head south and be dissipated along the edge of Penguin Bank. In either case, very little of the internal tide energy generated here crosses the 1000-m isobath. Not only does this contrast sharply with the majority of the generation sites where most of the energy radiates significant distances but it means that 14% of the dissipation within the domain comes from a generation region with no radiative signature.

As part of the HOME Nearfield experiment, a total of 313 microstructure profiles were taken over topography less than 1000 m deep on the Kaena Ridge (Fig. 11). The data were collected with the loosely tethered deep Advanced Microstructure Profiler (AMP), which evaluates ε from centimeter-scale shear variance (Osborn and Crawford 1980; Gregg 1987; Wesson and Gregg 1994). These data allow a comparison between the modeled baroclinic dissipation and field observations. The AMP profiles were integrated over the water column from a 22-m depth (to avoid contamination from the ship) to ∼20 m above the bed (where profiling was stopped to avoid damaging the instrument on the rough volcanic seabed). The profiles were then binned according to bottom depth with a bin interval of 100 m; for example, all profiles taken over topography between 100 and 200 m deep were grouped together. The profiles within each bin were then bootstrap averaged and multiplied by the area associated with that depth range (with the eastern boundary being 158°12′W). The area-integrated dissipation was 28 MW with a 95% confidence interval of 17–42 MW.

The modeled baroclinic dissipation within the 1000-m isobath for the Kaena Ridge west of 158°12′W is 48.7 MW. This is above the upper limit of the 95% confidence interval on the area-weighted observations but within a factor of 2 of the observed mean (28 MW). A factor of 2 is often taken as the threshold for determining equivalence between ε measurements (Osborn 1980; Oakey 1982; Carter and Gregg 2002), so we consider this acceptable agreement between the modeled dissipation and the microstructure. It should be noted that the microstructure observations include dissipation of energy from all sources, not just M₂ tides. However, M₂ is the dominant tidal frequency in this region, both in terms of barotropic velocity (Carter et al. 2006) and barotropic-to-baroclinic conversion (Zaron and Egbert 2006a). The limited depth range used in the microstructure integration helps minimize the effect of surface and bottom boundary mixing on the comparison.