a. Model formulation

A two-layer, stratified, nearly horizontal flow is considered according to the definition sketch shown in Fig. 1. The governing equations for the two-layer flow are described under the spherical polar coordinate system. For a two-layered flow, the usual depth-averaged two-dimensional equations for shallow water waves can be alternatively represented in terms of layer-integrated velocities for each layer. The momentum and continuity equations of the upper and lower layers, without surface wind stress, are as follows.

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Lower layer:

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where χ is longitude, ϕ is latitude, Z_b is bottom topography (Fig. 1), h₀ (=H₀ + η₀) is total water depth of the lower layer, h₁ (=H₁ + η₁) is total water depth of the upper layer, Δz = 0.5(h₀ + h₁), U₁ (U₀) is mass flux of the upper (lower) layer in χ, V₁ (V₀) is mass flux of the upper (lower) layer in ϕ, R is radius of the earth, λ = ρ₁/ρ₀, K_b is the bottom friction coefficient, K_i is the interfacial shear stress coefficient, g is gravity acceleration, and F unresolved processes. Also H₀, η₀, H₁, and η₁ are the thickness of mean lower layer, the disturbance of lower layer, the thickness of mean upper layer, and the surface disturbance of upper layer, respectively, as shown in Fig. 1.

The full set of nonlinear equations is solved with an extension of the one-dimensional two-layer modeling scheme (Abbott 1979; Hodgins 1979; S. K. Kang 1986, unpublished manuscript) and a depth-averaged two-dimensional modeling method (Abbott et al. 1981; Kang et al. 1998) (see appendix A for additional details). Four component equations are solved in each directional sweep. That is the continuity and momentum equations for the upper and lower layers in the χ coordinate are solved simultaneously, with dynamical linking and without resorting to the separation of variables or modes that was used, for example, in Serpette and Maze (1989). This strategy of simultaneously computing each variable is more efficient than mode splitting techniques for obtaining the instantaneous surface response to interfacial disturbances.

The bottom friction term for the lower layer has been expressed in quadratic friction form, following the general applicability of the quadratic friction law for the semidiurnal tide modeling (e.g., Pingree and Griffiths 1987), while the interfacial stress is assumed to vary linearly with the velocity difference between the two layers. This was found to work reasonably well for the exchange flow tests done in previous works (e.g., Hodgins 1979; S. K. Kang 1986).

The momentum diffusion term was considered in terms of a subgrid-scale modeling technique since momentum diffusion effects induced by the subgrid term are often larger than dispersion effects. All motions induced by small-scale processes, not directly resolved by the model grid (subgrid scale), are parameterized in terms of horizontal diffusion processes. The terms F_χ and F_ϕ represent these unresolved processes and can be written as

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where (u_0,1, υ_0,1) are χ and ϕ velocity components for lower (subscript 0) and upper (subscript 1) layers. With constant C_s, A_M is defined as follows.

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where C_s is a constant, while Δχ and Δϕ denote the grid interval in the χ and ϕ directions. For the coefficient of subgrid-scale stress we used the Smagorinsky (1963) diffusivity concept. The technique of subgrid-scale stress modeling is described in detail by Abbott and Basco (1989) and the method was successfully employed in the large-scale tidal modeling of Kang et al. (1998).

b. Computation and experiments

The initial and boundary conditions required for the solution of Eqs. (1) to (6) are described as follows. At t = 0, the values of

η₀η₁U_0,1χ, ϕ, tV_0,1χ, ϕ, t

are specified at all points. The initial thickness of the upper layer was set to h₁ = 300, 400, 500, or 1000 m. The model boundary is open on all sides and the boundary values were specified from an update of the Schrama and Ray (1994) global barotropic M₂ tide. These open boundary elevations are prescribed for a tide according to the expression;

η₀tAωtG(8)

where A is the tidal amplitude and G the Greenwich phase lag of the M₂ tide. The boundary elevation was defined as a variation of the interfacial surface while keeping the thickness of upper layer constant as 300, 400, 500, or 1000 m. Since the propagation speed of a surface gravitation wave is about 100 times faster than the internal wave speed,

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the response of an internal wave to the barotropic tidal wave traveling over the submarine ridge can be investigated.

Grid resolution is ¹/₁₀° in latitude by ¹/₁₀° in longitude. Thus, resolving a 150-km wavelength requires about 14 grid cells. The model domain is 10°–40°N and 180°–160°W and the total number of grid cells is 60 000. Bathymetry was obtained from interpolation of the ETOPO5 dataset. Depth contours are shown in Fig. 2, with a contour interval of 200 m for depths less than 1000 m. The TOPEX/Poseidon tracks are also shown. Minimum water depth was set in the model to be 400 m, 500 m, 600 m, or 1000 m in respective experiments. Maximum depth is about 7100 m. The time step used is 87.0 s.

The bottom friction coefficient was chosen to be K_b = 0.0025. In contrast to the relatively well-known K_b, the magnitude of the interfacial stress coefficient (K_i) is not well known. Given the Hodgins (1977) reasonable results with exchange flow tests, we used K_i = 0.0015 in most of the cases (Table 1). A value of K_i = 0.0025 was also used for comparison. The Smagorinsky eddy viscosity model was extended to the present two-layer model and the empirical constant of Eq. (7c) was chosen as C_s = 0.3 for each layer. This yields different viscosity coefficients. A_M, for each layer. Here C_s = 0.3 is a little larger than the value (C_s ∼ 0.2) used by Deardorff (1971) in the three-dimensional study. This value was found to be proper for depth-averaged modeling (Kang et al. 1998).

Ray and Mitchum (1997) employed measurements from the HOTS experimental site (Tupas et al. 1995; Karl and Lukas 1996) to estimate density and layer thickness for a simple analytical two-layer model. We adopt their values as parameters for our experimental cases. Specifically, they estimated the upper- and lower-layer densities ρ_1,0 to be 1024.7 and 1027.7 kg m⁻³ and the upper-layer thickness H₁ to be 300 m. Although the permanent thermocline depth is known to vary with latitude (Pickard 1979), this layer thickness is in rough agreement with the commonly chosen value of H₁ = 400 m (e.g., Gill 1982, section 6.3). However, a modal decomposition yields approximately 1000 m for the depth of maximum vertical velocity of the first mode and near 300 m for the second mode (P. Cummins 1999, personal communication). In order to estimate model response for the case H₁ = 1000 m, an additional run (RUN5) was also carried out.

Considering the extent of the model region, sensitivity to the choice of H₁ was investigated using the first three experiments listed in Table 1. The case without advection and diffusion was examined for H₁ = 400 m. The magnitude of the interfacial stress coefficient K_i was set to 0.0015 for most of cases. The value K_i = 0.0025 was also tested (RUN6).

Continuous model runs were carried out over nine M₂ tidal cycles. A FFT analysis over the ninth cycle was used to calculate amplitudes and phase lags of the M₂ tide. Snapshots of the interfacial variation for the first, third, seventh and ninth tidal cycle were also stored (and shown in Figs. 5a–d) in order to investigate the propagation pattern of the internal waves. The model run was stopped at the end of cycle 9 to avoid the reflection of internal waves from boundaries.

a. Surface and interfacial disturbances

The computed M₂ surface elevation amplitudes are presented in Fig. 3, while the comparable Schrama and Ray (1994) values are shown in Fig. 4. (The computed surface elevation phases were not presented due to complicated features arising from a superposition of barotropic and baroclinic effects.) The barotropic M₂ tide propagates from northeast to southwest. An amphidromic point exists outside the model region, near 30°N, 140°W (Irish et al. 1971; Ray and Mitchum 1997). The general patterns in the two tidal charts agree well, implying that the barotropic surface elevation has been simulated reasonably well with the present two-layer model. The general agreement between large-scale variations in the computed and observed barotropic tidal amplitude is again clear from a comparison in Fig. 6, in which short-scale baroclinic waves are superposed on large-scale barotropic waves. Careful inspection shows that short-scale disturbances in the amplitude and phase lag are present in the two-layer results, as shown, for example, in the 10 cm co-amplitude line of Fig. 3. A detailed description of this disturbance is given in section 3b.

The interfacial disturbance for the first, third, seventh, and ninth tidal periods are presented in Figs. 5a, 5b, 5c, and 5d, respectively. The slowly curving lines in Figs. 5a–d denote TOPEX/Poseidon satellite tracks. After one tidal cycle (Fig. 5a), an interfacial disturbance amplitude of up to nearly 20 m is generated around the Hawaiian Ridge. Since a baroclinic signal that originated at the outer boundary could not have reached the ridge by this time, the driving force must be the barotropic tide at the Hawaiian Ridge (Fig. 5a). Figures 5a–d show how the internal waves, generated along the Hawaiian Ridge, propagate mainly toward the northeast and southwest. (In section 3d, this is also discussed in terms of the baroclinic energy flux.) The wavelength of the internal wave is about 150 km, virtually the same as the Ray and Mitchum (1996) value.

Obviously, the internal tide is generated from multiple sources along the ridge. It propagates in a horizontal beamlike pattern and has an interesting interference feature. This purely horizontal beamlike pattern differs from the customary characteristic paths of internal tide that propagate vertically, as well as horizontally, as beams. Due to this horizontal beamlike pattern, it is difficult to determine a precise e-folding decay distance of the internal tide from amplitude variations along the TOPEX/Poseidon tracks. Although Ray and Mitchum (1996) found about a 300-km decay distance in some tracks, Fig. 5d, along with Fig. 6a, suggest that the internal tide propagates well over 1000 km and has an approximate 1000-km decay scale. This compares well with the description that, for a reasonable range of values for the eddy viscosity, long internal tides (wavelength of about 200 km) will propagate over distances of 2000 km or more (LeBlond 1966). Using a large acoustical array located midway between Puerto Rico and Bermuda in the Atlantic, Dushaw and Worcester (1998) estimated the decay scale for the diurnal component to be about 2000 km.

Careful inspection of the propagation pattern along satellite tracks 87 and 125 (e.g., Fig. 5d) reveals an asymmetry between southwest and northeast propagation patterns. This is described in some detail in section 3d. It is also seen that the magnitude of the internal tidal amplitude is not uniform along all of the Hawaiian Ridge. This spatial nonuniformity could be caused by the different generation intensity from multiple sources, as well as by interference between the horizontal rays. The region that shows maximum power of internal tide generation is marked S in Figs. 2 and 11. The maximum amplitude in the internal tide appears between tracks 87 and 125.

b. Comparison of surface manifestation with TOPEX/Poseidon data

The propagation of the internal tide along the permanent thermocline creates a corresponding surface disturbance, as revealed by the Ray and Mitchum (1996) TOPEX/Poseidon analysis. Comparison between model (RUN2) and observed (TOPEX/Poseidon) M₂ amplitudes along four satellite tracks is shown in Fig. 6 [The analysis of TOPEX/Poseidon data is described in Cherniawsky et al. (1999, manuscript submitted to J. Atmos. Oceanic Technol.).]

As expected from the comparison between Figs. 3 and 4, the agreement in amplitudes at long wavelengths also indicates that the barotropic tide has been simulated reasonably well. The model results also show short wavelength oscillations in the amplitudes in Fig. 6. These oscillations compare well with the observed TOPEX/Poseidon values in both their magnitude and wavelengths. Specifically, a spectral analysis of the computed and observed oscillations along track 125 reveals the same peak at approximate wavelengths of 150 km. This is in close agreement with the wavelength observed by Ray and Mitchum (1996). The wavelength of the internal tide can also be analytically determined as (g′H₁)^1/2T ≈ 150 km, for h₁ ≪ h₀, where T is tidal period and reduced gravity g′ = (ρ₀ − ρ₁)g/ρ₀ is calculated using layer densities from RUN2.

The computed M₂ amplitude of the 150-km wave along track 125 is larger than in the observations by about 15%. Spectral analysis also indicates that the computed baroclinic oscillation of short wavelength leads the observed wave by about 25° along track 125. Such a pattern of phase lags is also visible in Fig. 6. Ray and Mitchum (1996) interpreted these oscillations as the surface manifestation of phase-locked internal tides propagating off various topographic features, primarily the Hawaiian Ridge. The interface variations shown in Fig. 5, as well as their phase progression (not shown here), suggest that the internal tide is generated along the Hawaiian Ridge and propagates toward the northeast and southwest.

Figures 7a–d present a comparison along track 125 between observations and results from model runs 1, 3, 5, and 4. Spectral analyses of computed results of runs 1, 3, 5, and 4 reveal peaks at wavelengths of 137, 175, 189, and 167 km, respectively. The ratios of computed to observed elevation amplitude at each peak wavelength are 0.91, 1.11, 1.22, and 1.20, respectively. Less agreement between RUN4 and RUN2 appears in spite of a general similarity between the two experiments. The peak at a slightly longer wavelength (RUN4) may be due to the different horizontal beam propagation characteristics, arising from neglecting internal tide advection effects by barotropic currents, or nonlinear interactions between internal tide beams. In general, our comparison between model results and altimeter observations indicates that the North Pacific Ocean is modeled best with an upper-layer thickness of H₁ = 400 m and with advection terms, as represented by RUN2 (see Fig. 6).

It is obvious that the internal wave will have both in-phase or out-of-phase contributions with the background barotropic tidal waves, resulting in fluctuating amplitudes of the barotropic (vertically averaged) tide as shown in Figs. 6 and 7. The magnitude of the surface manifestation of the internal tide is approximately Δρ/ρ₀ times η₀ (∼10⁻³) (LeBlond and Mysak 1978; Apel 1987). The propagation of internal waves with a dominant wavelength appears clearly in Figs. 6 and 7. This result also indicates that the surface manifestation, associated with internal tide propagation, can be reasonably well represented in terms of a two-layer system, even though this representation is fairly schematic.

c. Tidal current ellipses

Tidal current ellipses are presented in Fig. 8 for the two-layer-averaged (barotropic) velocity (u_bt, υ_bt)

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and in Fig. 9 for the upper-layer velocity (u₁, υ₁). The upper-layer current (u₁, υ₁) consists of the barotopic and baroclinic (u_1bc, υ_1bc) components,

u₁u_btu_1bcυ₁υ_btυ_1bc(10)

The barotropic tidal currents in Fig. 8 are almost uniform, with counterclockwise rotation and small speeds of 2–3 cm s⁻¹, which agree with previously reported results (Luyten and Stommel 1991). Larger currents appear around the Hawaiian submarine ridge, probably due to a current flowing over the ridge at nearly a right angle. (The straight lines in the ellipses denote the current vector position at the time when the tide-generating body passes through Greenwich.) In addition, rectilinear tidal ellipses appear at lower latitudes.

Tidal ellipses in the upper layer (Fig. 9) display irregular features, indicating the strong influence of a baroclinic component, with current speeds larger than in the barotropic ellipses. In many locations, the tidal current in the upper layer exceeds the barotropic current by a factor of 3–4. The maximum major-axis speed in Fig. 9 is about 10 cm s⁻¹. The nonuniformity of the tidal currents in Fig. 9 seems to reflect the ray path propagation of the internal tidal energy flux. This will be examined in section 3d.

Luyten and Stommel (1991) compared the observed M₂ tidal currents at 315 deep-sea locations with those computed by the numerical model of Schwiderski (1979). Such comparison is also made here and presented in Table 2. The current observed above Horizon Guyot by Noble et al. (1988) is also included in the comparison.

Our model barotropic currents show a somewhat better agreement with the Luyten and Stommel (1991) observations than do Schwiderski’s values. However, Dushaw et al. (1997), through comparison of the current data and TPX0.2 global tide model (Egbert et al. 1994), showed that considerable scatter is evident in the current-meter values for the North Pacific and North Atlantic. That is, Dushaw et al. (1997) suggest that the current-meter data presented by Luyten and Stommel suffer from considerable random noise, arising from the motion of the current-meter mooring (Hendry 1977), or contamination by the baroclinic component (Luyten and Stommel 1991). In contrast to other observations, the observed current speed above Horizon Guyot is noticeably stronger than the barotropic model result. However, the computed current speeds in the upper layer do compare well in magnitude, even though there is about 50° deviation in the inclination angle. Noble et al. (1988) rejected the likelihood that these large currents arise from internal tides propagating from afar, suggesting instead that they were generated locally. On the other hand, the Ray and Mitchum (1996) TOPEX/Poseidon observations show strong southward propagation throughout this region. The present study also shows that internal tides generated over the Hawaiian Ridge propagate over the Horizon Guyot and that the baroclinic component plays a dominant role in enhancing current speeds there.

d. Barotropic and baroclinic energy fluxes

Using analyzed harmonics of the surface and interfacial disturbances and of the barotropic and baroclinic currents, time-averaged barotropic and baroclinic energy fluxes for the two-layer system (see appendix B) have been computed for the ninth cycle model result. The corresponding energy flux vectors are presented in Figs. 10 and 11.

Figure 10 shows that the barotropic energy flux is from north-northeast to west-southwest in the direction of the M₂ tide. The barotropic energy flux shows relatively large variability with values of about 100 kW m⁻¹ in the northeast region, and nearly an order of magnitude less (10 kW m⁻¹) in the central region. The smaller magnitude in the central northwestern region is due to the smaller elevation amplitudes of 5–10 cm, as shown in the coamplitude charts (Figs. 3 and 4). The general features in Fig. 10 agree well with the barotropic tidal energy flux vectors shown by Ray and Mitchum (1997). It is worth noting that the barotropic energy fluxes appear to have some irregularity in their magnitude and direction. This spatial variability probably arises from baroclinic influences. It should not arise from diffraction effects since the modeled domain is located too far from Hawaiian Islands, as evident from the result of Larsen (1977). The irregular feature, especially in the amplitude, appears even farther to the southwest, down to 10°–15°N.

It is worth noting the diverging or asymmetric energy flux pattern in low latitudes, along tracks 87 and 125. A similar feature also appears in the model results, Figs. 5c and 5d. The depth contours south of the ocean ridge region S do not show any abrupt changes. Considering that the agreement between the model results of Fig. 6a and the observed TOPEX/Poseidon results is reasonable along the southern part of track lines 125 and 87, the diverging or asymmetric pattern of the baroclinic energy beam (Fig. 11), or the propagation of interfacial waves (Figs. 5c and 5d), must be real. However, a dynamical explanation for this divergence is not obvious.

The intensity of the generated baroclinic tide is noticeably larger around the crescent-shaped topography (inside the region S in Figs. 2 and 11), with a peak flux density of 12.9 kW m⁻¹. This could be due to a superposition, or interference, away from a line source. In order to estimate the spatial distribution of the generated baroclinic flux, relative to the barotropic flux density, the ratio of baroclinic to barotropic energy flux density was calculated from the model results and the contours of equal percentage shown in Fig. 12. Ratio values are larger near the Hawaiian Ridge and values are in the range of 10%–40% near the region of crescent-shaped bottom topography (Fig. 2) around the ridge between tracks 125 and 87. Two locations showing values greater than 50% seem to be partly due to the relatively small barotropic energy flux, as seen in Fig. 12. The intensified baroclinic energy flux between tracks 125 and 87 is thought to appear because the crescent-shaped bottom topography feature focuses the baroclinic energy flux. It is also interesting to note that the ratios of baroclinic to barotropic energy flux fluctuate from 10% to 20(30)% in the northeastward direction. The distance between the fluctuating peaks is nearly equal to one internal tide wavelength of 150 km, and the locations of the peak value is nearly at the crests of the internal tidal waves in Fig. 5d.

In order to quantify the energy balance, both the baroclinic and barotropic components of the energy flux in region S (Figs. 2 and 11) are presented in Table 3. The integrated energy flux values are computed across four different sections: northern section (section S1), eastern section (section S2), southern section (section S3), and western section (section S4). Table 3 shows that the integrated barotropic M₂ energy flux across each section is about 5–20 times larger than the integrated M₂ baroclinic energy flux. The integrated baroclinic energy flux in a northeastern direction (sum of S1 and S2) is 1.15 GW, while this integrated energy flux in a southwestern direction (sum of S3 and S4) is 1.27 GW. The total integrated baroclinic M₂ energy flux emanating from the region S is therefore 2.42 GW. It is clear that energy flux emanating from any other energy region is much smaller than that from the box region. The total integrated baroclinic energy flux from the rectangular region T is 3.77 GW. Average northward, or southward, flux density is about 1.35 kW m⁻¹ in the T region. The present model domain does not include the eastern Hawaiian Islands (reaching to 155°W). But, if we take 2.7 kW m⁻¹ (1.35 kW m⁻¹ to each side) for the whole 2000-km-long Hawaiian Ridge, we obtain a total baroclinic energy flux of 5.4 GW. Using both models and observations, Morozov (1995) estimated the M₂ energy flux values, transferred into internal tidal motion, for several World Ocean ridges and obtained 8 GW for the Hawaiian Ridge. Using simple parameters for a two-layered ocean. Ray and Mitchum (1997) estimated the mean integrated energy flux into phase-locked, first mode, baroclinic tide at the Hawaiian Ridge to be 15 GW. This translates to a mean flux of 4 kW m⁻¹ for each of the northward and southward directions along this 2000-km-long ridge.

The agreement between the present result of 3.77 GW at western Hawaiian Ridge (5.4 GW if the whole 2000-km-long Hawaiian Ridge is considered) and the estimates of Morozov (1995) and Ray and Mitchum (1997) is not unreasonable, considering the approximate parameter values in their estimates and possible errors in the present calculation. These results aslo indicate that there is a large uncertainty in the value of energy dissipation around the Hawaiian Ridge. [The Munk (1997) estimate of 0.2 TW for the global M₂ dissipation via internal tide conversion may require between 14 and 37 Hawaiian Ridges.]

The barotropic energy flux propagates from the north-northeast to south-southwest in model region S, resulting in negative energy flux values in Table 3. The loss of barotropic energy flux over this box region is calculated by summing the incoming integrated energy flux (sum of S1 and S2) and subtracting the outgoing integrated energy flux (sum of S3 and S4). This gives a loss of barotropic energy flux in the box region of 2.64 GW. The dissipation (d) in the turbulent bottom boundary layer (BBL) was calculated, following Taylor (1919) and Munk (1997):

dC_dρ〈U³_s〉⁻²(12)

where C_d = 0.0025, ρ is mean density and

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Averaging over one M₂ period and integrating over the region S yields d = 0.029 GW. This small amount of bottom dissipation in the ocean ridge region S seems to reflect the Munk (1997) remark that 99% of the ocean are accounts for less than 1% of the dissipation. Eventually, the loss of barotropic energy (2.64 GW) compares well with the sum (2.45 GW) of 0.029 GW of bottom boundary layer dissipation and 2.42 GW of integrated M₂ baroclinic energy flux leaving the region.

At this stage, it should be mentioned that numerical simulation of the baroclinic energy flux can be sensitive to errors in topography, the barotropic open boundary conditions, and also the choice of coefficients for bottom and interfacial dissipation. In addition, this two-layer model has limited application because the topographic features must be below the interfacial depth and shallow ridges must be smoothed out, both of which may cause errors in the intensity and radiated direction of baroclinic energy flux.