Journal of Geophysical Research: Oceans

Numerical study of K1 internal tides in the Kuril straits

Authors


Abstract

[1] Tide-induced strong diapycnal mixing in the Kuril straits is thought to be one of the essential processes controlling water mass formation in the North Pacific. In order to make a definite quantification of diapycnal diffusivity in the Kuril straits, we drive a three-dimensional numerical model and examine the generation, propagation, and dissipation features of internal waves which play an essential role in transferring energy from the predominant K1 barotropic tide to diapycnal mixing processes. It is shown that most of the internal wave energy subtracted from the K1 barotropic tide is dissipated within the Kuril straits such that the local dissipation efficiency becomes 0.8–1.0, about three times the value previously employed. This is because the K1 tidal frequency is subinertial in this area so that significant amount of K1 tidal energy is fed into coastal trapped waves (CTWs) which stay around each island without propagating away from the straits; CTWs induce strong velocity shear near the ocean bottom causing bottom-confined intense mixing with a vertical decay scale ∼200 m, less than half the value previously employed, although there remains some uncertainties resulting from the employed parameterizations of viscosity and diffusivity. The average diapycnal diffusivity in the Kuril straits becomes ∼25 × 10−4 m2 s−1, about three times the value previously estimated, although it is still an order of magnitude less than assumed for the Kuril straits in the existing ocean general circulation models.

1. Introduction

[2] The Kuril straits are characterized by strong diurnal tidal currents with the amplitude occasionally exceeding 1 m s−1 [Kowalik and Polyakov, 1998; Rabinovich and Thomson, 2001; Katsumata et al., 2001, 2004; Ohshima et al., 2002] as well as prominent topographic features separating the Okhotsk Sea from the North Pacific. The strong diurnal tidal currents interact with bottom topography to generate large-amplitude internal waves which occasionally break causing vigorous diapycnal mixing [Nakamura et al., 2000; Nakamura and Awaji, 2004]. Actually, water mass properties that can be reasonably explained as resulting from diapycnal mixing are observed in the Kuril straits [Ono et al., 2007].

[3] The induced diapycnal mixing is thought to play an important role also in water mass formation in the North Pacific. Over a wide area at the intermediate depth in the North Pacific subtropical gyre, there exists a water mass termed North Pacific Intermediate Water (NPIW), which is characterized by the salinity minimum centered at around 26.8 σθ [Sverdrup et al., 1942; Reid, 1965; Talley, 1993]. The NPIW has its origin in the Okhotsk Sea where intermediate water with the characteristics of low temperature, low salinity, and low potential vorticity is produced via diapycnal mixing in the Kuril straits, in addition to the vertical convection due to sea ice formation over the northwestern shelf [Talley, 1991; Yasuda et al., 2002; Itoh et al., 2003]. This water outflows from the Okhotsk Sea through the Kuril straits to the North Pacific and mixes with the Western Subarctic Gyre water, forming the Oyashio water [Yasuda et al., 2001]. Part of the Oyashio water then flows southward to the Kuroshio-Oyashio interfrontal zone and merges with the Kuroshio Extension to produce the NPIW [Talley, 1997; Yasuda, 1997]. Strong diapycnal mixing in the Kuril straits is also thought to promote the southward intrusion of the Oyashio at the intermediate depth [Tatebe and Yasuda, 2004].

[4] Nakamura et al. [2006] employed an ocean general circulation model (OGCM) to show that the realistic NPIW could be reproduced only when large diapycnal diffusivity ∼200 × 10−4 m2 s−1 was assumed to exist in the whole Kuril straits, although the validity of this assumption was not checked. Recently, a close relationship between diapycnal mixing in the Kuril straits and the bidecadal climate variability in the North Pacific was also suggested [Osafune and Yasuda, 2006; Yasuda et al., 2006; Hasumi et al., 2008].

[5] Tanaka et al. [2007] were the first to quantify diapycnal diffusivity in the Kuril straits in terms of the baroclinic energy conversion which was parameterized through bottom stress terms incorporated into the momentum equations in a barotropic tide model. They carried out a series of numerical experiments to show that the model-predicted tidal elevations in the Okhotsk Sea best fitted the TOPEX/POSEIDON (T/P) altimeter data when ∼20 GW of energy was assumed to be subtracted from the predominant K1 barotropic tide in the Kuril straits. For this amount of energy, they estimated the average diapycnal diffusivity in the Kuril straits as ∼8 × 10−4 m2 s−1. This value is more than one order of magnitude less than assumed for the Kuril straits in the existing OGCMs [e.g., Nakamura et al., 2006]. However, the above quantification by Tanaka et al. [2007] is indirect one where the energy subtracted from the barotropic tide was evaluated only by tuning the model-predicted tidal elevation field to the T/P altimeter data. Furthermore, in quantifying the diapycnal diffusivity for this amount of energy, some ambiguous assumptions were made for the local dissipation efficiency (the fraction of internal wave energy locally dissipated by turbulent processes) and the vertical decay scale of the bottom-intensified mixing.

[6] In the present study, to make a definite quantification of diapycnal diffusivity in the Kuril straits, we investigate the way by which significant amount of K1 tidal energy is lost to dissipation processes. Using a high-resolution three-dimensional primitive equation model with realistic tidal forcing and bathymetric features, we numerically reproduce internal waves which play a crucial role in subtracting energy from the barotropic tide. Although Nakamura et al. [2000] claimed that tidal mixing was caused by breaking of lee waves near the top of the sill, several serious errors are found in their theory (see Appendix A). We look for more reasonable explanation for the strong tidal mixing in the Kuril straits based on the results from the present numerical experiments. Special attention is directed to the local dissipation efficiency as well as the vertical decay scale of the bottom-intensified mixing.

2. Numerical Experiment

[7] In the present numerical experiment (Experiment 1), the model domain covers most of the straits between the Kuril Islands including the Bussol Strait and the Kruzenshterna Strait as well as a large bank in the northeastern region (Figure 1). Note that the model domain is rotated anticlockwise by 44°. The model topography is constructed from the General Bathymetric Chart of the Oceans (GEBCO) data set which has a resolution of (1/60)°.

Figure 1.

Model domain including most of the Kuril straits. Numeral on each contour represents the depth in meters. Modal decompositions of coastal trapped waves in section 3.4 are carried out along the thick solid line and the dashed line.

[8] The numerical calculation is carried out using the Massachusetts Institute of Technology General Circulation Model (MITgcm) [Marshall et al., 1997] which solves the three-dimensional, free surface Boussinesq equations with a finite-volume method in space and a second-order Adams-Bashforth scheme in time. The model can incorporate arbitrary bottom topography through the use of finite-volume formulation, thus is suitable for simulating tide-topography interactions [Legg and Adcroft, 2003]. In the present numerical experiment, we employ the hydrostatic approximation which is valid so long as the horizontal scales of motion are much larger than the vertical scales. The governing equations are then given by

equation image
equation image
equation image
equation image

where D/Dt = ∂/∂t + u(∂/∂x) + v(∂/∂y) + w(∂/∂z) is the material derivative with t the time, (x, y, z) the space coordinates, u = (u, v) the velocity components in the (x, y) directions, and w the velocity component in the z direction; equation imageH = (∂/∂x, ∂/∂y) is the horizontal gradient operator; f is the Coriolis frequency; g is the acceleration due to gravity; ρ is water density; ρc is the reference water density; P is pressure; AH and AV are background horizontal and vertical eddy viscosity coefficients, respectively; KH and KV are background horizontal and vertical eddy diffusivity coefficients, respectively; and equation image is the vertically upward unit vector.

[9] The horizontal grid spacing is about 1000 m both in the x and y directions, and the vertical grid spacing is 20 m from the ocean surface down to a depth of 1800 m and gradually increased up to 300 m at the maximum depth of 8640 m. We employ the β-plane approximation where the Coriolis frequency is evaluated as f = f0 + βY with f0 ≃ 0.99 × 10−4 s−1, β ≃ 1.55 × 10−11 s−1 m−1, and Y the meridional distance from the origin of the (x, y) coordinate (Figure 1). In the present calculation, constant horizontal and vertical eddy viscosity coefficients (AH = 10 m2 s−1, AV = 10−3 m2 s−1) and diffusivity coefficients (KH = 1 m2 s−1, KV = 10−4 m2 s−1) are assumed. At the lowest level, bottom friction is applied through a quadratic friction law with a constant drag coefficient Cd = 2.5 × 10−3.

[10] In the present study, we focus on the predominant K1 tidal constituent. The model is forced at the open boundaries where we prescribe the K1 barotropic tide as obtained from the regional barotropic tide model of Tanaka et al. [2007] while applying a forced gravity wave radiation condition [Niwa and Hibiya, 2001a, 2001b, 2004; Carter et al., 2008]. Following Carter and Merrifield [2007], the baroclinic velocity fluctuations and isopycnal displacements are relaxed to zero over a 20 km-wide edge along each open boundary in order to avoid artificial wave reflection. Initially, we assume a horizontally uniform background density stratification by taking an area-average of the annual mean density values obtained from the World Ocean Atlas 2001 data set [Conkright et al., 2002]. The model is driven for 10 tidal periods (10 days) from an initial state of rest.

3. Results and Discussions

3.1. Model Validation

[11] Cotidal chart for the K1 tidal constituent is shown in Figure 2, where solid and dashed lines correspond to coamplitude (cm) and cophase (degree) lines, respectively. Note that the amplitude and phase are obtained by harmonically analyzing the calculated data for the final one tidal period. Tidally averaged root-mean-square differences between the model-predicted and T/P-observed ocean surface elevations are no more than 5 cm with the average 2.1 cm, small enough compared to the typical tidal elevation amplitude in the Kuril straits ∼30 cm. Furthermore, the cross-sectional distributions of the cross-ridge tide residual velocity (Figure 3) and the cross-ridge K1 tidal velocity (Figure 11a) in the Bussol Strait (along the thick solid line in Figure 1) are both found to agree well with those obtained from the intensive field observations [see Katsumata et al., 2004, Figures 7 and 8]. The general agreement between the model-prediction and the observation indicates that the barotropic and baroclinic responses are both simulated well in the present numerical model.

Figure 2.

Calculated coamplitude lines (solid) and cophase lines (dashed) of surface elevations for the K1 tidal constituent. The contour intervals are 5 cm for amplitudes and 10° for phases, respectively. The phases are relative to the Greenwich.

Figure 3.

Cross-sectional distribution of the cross-ridge component of tide residual velocity in the Bussol Strait (along the thick solid line in Figure 1).

3.2. Energetics of the K1 Tide

[12] The volume-integrated energy equation can be written as

equation image

where P0(z) is the hydrostatic reference pressure when the fluid is at rest, p is the pressure perturbation including the free surface contribution, K.E. and P.E. are kinetic energy and potential energy given by

equation image
equation image

respectively, Dis. is the energy dissipation rates caused by eddy viscosity

equation image

and (−Dif.) is the energy gain associated with buoyancy flux caused by eddy diffusivity [Vallis, 2006]

equation image

[13] For each group of terms in equation (5) integrated within the model domain except the outer 50 km-wide edge along each open boundary, the time series throughout the numerical experiment and the time average over the final one tidal period are calculated, respectively (see Figure 4 and Table 1). Note that energy flux pu is divided into barotropic component equation image and baroclinic component pu′ where equation image and ( )′ denote the depth-average and its subtraction from the total, respectively [Niwa and Hibiya, 2001a, 2004; Merrifield and Holloway, 2002; Simmons et al., 2004; Carter et al., 2008]. Figure 4 shows that a quasi-steady state is achieved already after 4 tidal periods from the start of the calculation, although there still remains slight increase of potential energy resulting from diapycnal mixing.

Figure 4.

Time series of each group of terms in equation (5), namely, kinetic energy tendency and advection (orange), potential energy tendency and advection plus energy flux divergence associated with hydrostatic reference pressure (green), barotropic energy flux divergence (purple), baroclinic energy flux divergence (blue), and dissipation plus diffusion (red), all integrated within the model domain except the outer 50 km-wide edge along each open boundary.

Table 1. Each Group of Terms in Equation (5) Averaged Over the Final One Tidal Period and Integrated Within the Model Domain Except the Outer 50 km-Wide Edge Along Each Open Boundary
ParameterValue (GW)
Tendency and advection of kinetic energy0.5
Tendency and advection of potential energy plus divergence of P0u2.4
Divergence of barotropic energy flux−30.2
Divergence of baroclinic energy flux0.6
Energy dissipation due to bottom friction5.6
Energy dissipation due to viscosity17.3
Energy change due to diffusivity−0.4
Error−4.2

[14] We can see that the amount of energy subtracted from the barotropic tide in the Kuril straits reaches 30.2 GW, about 1.5 times the value 20.4 GW previously estimated by Tanaka et al. [2007]. This discrepancy is, however, tolerable, considering uncertainties of bottom topography as well as background stratification employed in a three-dimensional numerical model [Niwa and Hibiya, 2004; Carter et al., 2008]. We can also find that most of the energy subtracted from the barotropic tide is dissipated near the straits, leaving very limited baroclinic energy 0.6 GW radiating out from the analyzed domain. In the present study, we calculate the local dissipation efficiency q defined by

equation image

or

equation image

Taking the same spatial integral and time average as in Table 1, q is estimated as 0.8–1.0, about three times the value ∼0.3 previously assumed by Tanaka et al. [2007] who just followed the suggestion by St. Laurent et al. [2002]. This is because the K1 tidal frequency is subinertial in this region and hence most of the resulting internal wave energy cannot leave its source area. The average diapycnal diffusivity in the Kuril straits then becomes also about three times the value ∼8 × 10−4 m2 s−1 previously estimated by Tanaka et al. [2007].

[15] Figure 5 shows the horizontal distribution of the depth-integrated energy dissipation rates. We can see that a large amount of energy is dissipated within each of the straits as well as over the northeastern bank. The energy dissipation in the northeastern (southwestern) part of the Kuril straits including the Kruzenshterna Strait (the Bussol Strait) reaches 7.9 GW (6.6 GW) and that over the northeastern bank reaches 6.2 GW, accounting for about 35% (29%) and 27%, respectively, of the total energy dissipation within the analyzed domain. Figure 5 confirms that thus obtained horizontal distribution of the energy dissipation rates is nearly identical to that obtained by Tanaka et al. [2007]. In order to estimate the vertical decay scale ζ of the bottom-intensified mixing, we calculate the energy dissipation rates horizontally averaged at each height above the ocean bottom (Figure 6). We can see that ζ becomes ∼200 m, less than half the value ∼500 m previously assumed by Tanaka et al. [2007].

Figure 5.

Model-predicted horizontal distribution of the depth-integrated energy dissipation rates.

Figure 6.

Energy dissipation rates horizontally averaged at each height above the ocean bottom.

[16] We examine the robustness of thus obtained values of q and ζ in the Kuril straits by changing the values of viscosity and diffusivity coefficients in a set of numerical experiments (see Appendix B). It is shown that, in contrast to the value of ζ, the value of q is quite robust, which indicates that the overall energetics is fairly independent of the employed parameterizations of viscosity and diffusivity.

3.3. Freely Propagating Waves Away From the Ridge

[17] Figure 7 is the snapshot showing the baroclinic cross-ridge velocity (v′) along the line x = 370 km passing through the Bussol Strait. Beam-like structures emanating from the top of the ridge can be clearly recognized although they gradually vanish with a distance from the ridge.

Figure 7.

Model-predicted cross-sectional snapshot of the baroclinic velocity (v′) along x = 370 km after 10 tidal periods from the start of calculation.

[18] Figure 8a shows the frequency spectra of baroclinic motions (u′, v′, and (′)/(ρcN)) averaged within the cross-sections along y = 80 km (dotted line), 250 km (solid line), and 390 km (dashed line). These spectra are calculated using data sampled at every 2 hours throughout the numerical experiment (10 days). The spectral peaks can be clearly seen at the K1 tidal frequency and its higher harmonics. This is qualitatively consistent with Bell [1975], Khatiwala [2003], and Legg and Huijts [2006] who showed that internal waves are generated not only at the forcing frequency but also at all its higher harmonics. In the proximity of the ridge (y = 250 km), the spectral peak at the K1 tidal frequency dominates the other spectral peaks. Away from the ridge (y = 80 km and 390 km), in contrast, the spectral peak at the K1 tidal frequency is much reduced and the largest peak is found at twice the K1 tidal frequency, consistent with the fact that the K1 tidal frequency is subinertial in this area. The solid line in Figure 8b shows the ratio of the kinetic to available potential energy both integrated within the cross-section along y = 390 km (away from the ridge). The calculated results are in close agreement with the theoretical prediction for linear internal waves (dashed line, r = (ω2 + f2)/(ω2f2)). The slope γ of the beam-like structures in Figure 7 also agrees well with the theoretical prediction γ2 = (ω2f2)/N2 [Gill, 1982] where ω is assumed to be twice the K1 tidal frequency.

Figure 8.

(a) Frequency spectra of baroclinic motions averaged within the cross-sections along y = 80 km (dotted), 250 km (solid), and 390 km (dashed). (b) Ratio of the kinetic to available potential energy both integrated within the cross-section along y = 390 km (solid) and the corresponding theoretical prediction for linear internal waves (dashed) as a function of frequency. Note that frequency is normalized by the K1 tidal frequency.

[19] In order to clarify the vertical structure of the propagating internal waves, we carry out modal decompositions of the model-predicted energy flux across y = 80 km and 390 km, respectively. Table 2 shows the estimated contribution of each vertical mode to the total baroclinic energy flux. For both lines, more than 90% of the total baroclinic energy flux can be accounted for by the sum of modes 1–4. We can thus conclude that the internal waves propagating away from the ridge consist of low vertical modes (modes 1–4) with twice the K1 tidal frequency.

Table 2. Contribution of Each Vertical Mode to the Model-Predicted Baroclinic Energy Flux Across y = 80 km and 390 km
Modey = 80 kmy = 390 km
123.529.3
227.234.0
324.022.6
414.77.4
56.32.9
61.50.6
70.60.2
80.20.1
Total98.097.1

3.4. Coastal Trapped Waves in the Straits

[20] As is seen in Figure 8a, the largest spectral peak of baroclinic motions along y = 250 km (near the ridge) occurs at the K1 tidal frequency. Figure 9 shows the spatial distribution of the harmonically analyzed along-isobath K1 tidal velocity at the ocean surface at each phase during the final one tidal period, which suggests the existence of coastal trapped waves (CTWs) propagating clockwise around each island [Rabinovich and Thomson, 2001; Ohshima et al., 2005].

Figure 9.

Model-predicted distribution of the harmonically analyzed along-isobath K1 tidal velocity at the ocean surface after (a) 9, (b) 9.25, (c) 9.5, and (d) 9.75 tidal periods from the start of calculation. Note that the current in the red (blue) shaded area flows with the shallower water on its right (left).

[21] In order to examine the characteristic features of these waves, we introduce here a linearized CTW model for an inviscid, continuously stratified ocean with realistic continental shelf topography [Wang and Mooers, 1976; Huthnance, 1978; Brink, 1982b]. In terms of the pressure perturbation p = P(x, z)cos(lyωt), the governing equation for these waves is written as

equation image

where (x, y) are the horizontal coordinates taken in the cross- and along-shelf directions, respectively. The appropriate boundary conditions are

equation image
equation image
equation image
equation image

where (13) indicates the free surface condition, (14) and (15) prescribe no normal flow at the ocean bottom and at the coast, respectively, and (16) indicates the necessary condition for the waves to be trapped along the coast. The differential equation (12) together with boundary conditions (13)(16) form a two-dimensional eigenvalue problem for a fixed frequency ω. We solve this eigenvalue problem numerically on a vertically stretched sigma coordinate while searching for the resonant response of the system to an arbitrary forcing [Wang and Mooers, 1976].

[22] The obtained modal structures of CTWs tend to be confined near the ocean bottom which are nearly barotropic in shallow regions with baroclinicity increasing seaward [Rabinovich and Thomson, 2001; Ono et al., 2006, 2008]. Figure 10a shows the cross-shelf distribution of the amplitude and phase of the model-predicted along-shelf K1 tidal velocity on the Okhotsk side of the Simushir Island (along the dashed line in Figure 1) and Figure 10b shows the corresponding one reconstructed using modes 0–5. Note that mode 0 is barotropic Kelvin wave [Brink, 1986] with a horizontal scale ∼103 km, so that the associated along-shelf velocity is assumed to be uniform within this cross-section. The contribution of each mode to the kinetic energy associated with the model-predicted along-shelf K1 tidal velocity is shown in Table 3, together with the along-shelf wavelength (i.e., the eigenvalue) of each mode. Since the modal functions are not exactly orthogonal, the amplitude and phase of each mode are determined using a least squares method so as to best reproduce the model-predictions. We can see that the structure of along-shelf velocity can be reproduced well with significant contribution from mode 1 (see Figure 10c). This analysis, however, implicitly assumes that topographic irregularities vanish in the along-shelf direction, which is not the case in the Kuril straits. For example, calculated along-shelf wavelength of mode 1 is about 165 km, which is longer than the Okhotsk side coastline of the Simushir Island.

Figure 10.

Cross-shelf distribution of the amplitude (color) and phase (contours) of (a) the model-predicted along-shelf K1 tidal velocity on the Okhotsk side of the Simushir Island (along the dashed line in Figure 1) and (b) the corresponding one reconstructed using modes 0–5. The contour interval is 30°. (c) Cross-shelf distribution of the calculated along-shelf velocity of mode 1 CTW with K1 tidal frequency. The contour units are arbitrary.

Table 3. Contribution of Each Mode of CTW to the Kinetic Energy Associated With the Model-Predicted Along-Shelf K1 Tidal Velocity on the Okhotsk Side of the Simushir Island and the Along-Shelf Wavelength of Each Modea
ModeContribution (%)Wavelength (km)
  • a

    The along-shelf wavelength is also known as the eigenvalue.

017.3-
165.6165
212.460
30.334
41.727
50.218
Error8.6-

[23] The same analysis is carried out along the slightly curved line crossing the Bussol Strait (the thick solid line in Figure 1) which is almost identical to the line of field observations by Katsumata et al. [2004]. As is shown from the model-prediction in this case (Figure 11a), CTWs exist on either side of the strait where normal flow is assumed to vanish. Figure 11b and Table 4 show the reconstructed along-shelf K1 tidal velocity using modes −4–4 with plus (minus) numbers denoting the CTWs propagating out of the page (into the page) along the southwestern coast (the northeastern coast). Again, we can see that bottom-intensified velocity structure can be reproduced well with significant contributions from modes −1–1 (see Figures 11c and 11d).

Figure 11.

(a) As in Figure 10a but for the Bussol Strait (along the thick solid line in Figure 1) and (b) the corresponding cross-strait distribution reconstructed using modes −4–4. As in Figure 10c but for (c) mode −1 and (d) mode 1 in the Bussol Strait.

Table 4. As in Table 3 but for the Bussol Strait
ModeContribution (%)Wavelength (km)
−40.219
−30.129
−21.051
−111.4126
038.2-
138.7123
21.394
34.427
40.018
Error20.0-

[24] The crucial role of CTWs in dissipating tidal energy within the Kuril straits can be demonstrated by carrying out an additional numerical experiment (Experiment 2) where CTWs are removed away by assuming f = 0 (no rotation of the Earth). The volume-integrated energy budget obtained assuming f = 0 is shown in Table 5 where we can see that the amount of energy drawn from the barotropic tide is reduced by half (30.2 GW vs. 14.4 GW). Since the local dissipation efficiency q becomes also about half, namely, 0.4–0.5, the volume-integrated energy dissipation rate is reduced to about one fourth (22.9 GW vs. 5.9 GW).

Table 5. As in Table 1 but for the Numerical Experiment Assuming f = 0
ParameterValue (GW)
Tendency and advection of kinetic energy0.1
Tendency and advection of potential energy plus divergence of P0u1.1
Divergence of barotropic energy flux−14.4
Divergence of baroclinic energy flux6.5
Energy dissipation due to bottom friction1.4
Energy dissipation due to viscosity4.5
Energy change due to diffusivity−0.4
Error−1.2

[25] These results indicate the way by which significant amount of K1 tidal energy is lost to dissipation processes in the Kuril straits; first, K1 barotropic tidal energy is fed into CTWs which then propagate clockwise around each island while giving up their energy to mixing processes. Some physical mechanisms such as shear instability, bottom friction [Brink, 1982a], and topographic scattering [Wilkin and Chapman, 1990] might be responsible for the dissipation of CTWs, although the most dominant mechanism remains to be identified in the future.

4. Conclusion

[26] In the present study, to make a definite quantification of diapycnal diffusivity in the Kuril straits, we have investigated the way by which significant amount of K1 tidal energy is lost to dissipation processes. Using a high-resolution three-dimensional primitive equation model with realistic tidal forcing and bathymetric features, we have numerically reproduced internal waves which play a crucial role in subtracting energy from the barotropic tide (Experiment 1). Special attention has been directed to the local dissipation efficiency as well as the vertical decay scale of the bottom-intensified mixing.

[27] The amount of energy subtracted from the predominant barotropic K1 tide in the Kuril straits has been estimated as 30.2 GW, comparable to 20.4 GW previously estimated by Tanaka et al. [2007]. Furthermore, it has been shown that most of the energy subtracted from the barotropic tide is dissipated within each of the straits as well as over the northeastern bank, leaving very limited energy carried away from the straits by low vertical modes (modes 1–4) with twice the K1 tidal frequency. Although the horizontal distribution of the depth-integrated energy dissipation rates is nearly identical to that obtained by Tanaka et al. [2007], the local dissipation efficiency has been estimated as 0.8–1.0, about three times the value previously employed. It follows that the average diapycnal diffusivity in the Kuril straits becomes ∼25 × 10−4 m2 s−1, also about three times the value previously estimated, although it is still an order of magnitude less than the value ∼200 × 10−4 m2 s−1 assumed for the Kuril straits in the existing OGCMs [e.g., Nakamura et al., 2006].

[28] The large value of local dissipation efficiency can be reasonably explained in terms of the coastal trapped waves (CTWs). Actually, CTWs propagating clockwise around each island with K1 tidal frequency have been clearly recognized and identified as consisting of a superposition of low modes. The velocity structures associated with CTWs tend to be confined near the ocean bottom causing the bottom-intensified mixing with a vertical decay scale of ∼200 m, less than half the value previously employed, although there remains some uncertainties resulting from the employed parameterizations of viscosity and diffusivity. The crucial role of CTWs in dissipating tidal energy within the Kuril straits has been demonstrated by carrying out an additional numerical experiment (Experiment 2) assuming f = 0 (no rotation of the Earth) where we have found that the volume-integrated energy dissipation rate is reduced to about one fourth.

[29] Although the physical processes associated with strong diapycnal mixing in the Kuril straits have thus been clarified, there remain some problems to be investigated in the future. First of all, the mechanism for the generation, propagation, and dissipation of CTWs should be examined in more detail. For this purpose, numerical experiments using highly resolved nonhydrostatic model are definitely necessary. The role of the mean flow (tide residual flow) which is as strong as K1 tidal flow in some places (e.g., Figure 3) must also be examined. Furthermore, since also the O1 tidal constituent (the other major diurnal tidal constituent) is expected to induce strong diapycnal mixing through the same physical processes clarified in the present study, the intensity of actual diapycnal mixing in the Kuril straits might exhibit spring-neap modulation. Apart from these remaining problems, the present study is a useful first step in understanding the mechanism for the formation of NPIW which has been inadequately explained in terms of the exaggerated tidal mixing in the Kuril straits.

Appendix A:: Errors From Nakamura et al.'s [2000] Study

[30] Nakamura et al. [2000] claimed that strong tide-topography interaction generates internal waves with the intrinsic frequency ω = −kU(t0) ± σtide at each instant of time t = t0, where k is the horizontal wave number of the bottom topography, U(t) = U0 cos(σtidet) is the tidal velocity with σtide the tidal frequency. They called such waves “unsteady lee waves,” insisting that these waves then propagate in the tidal flow while overlapping one after another.

[31] The derivation of the above dispersion relation is, however, misleading. The Fourier transformed bottom boundary condition for the vertical velocity is ŵ = ikequation imageU(t), which is further rewritten as

equation image

using the coordinate moving with the background tidal flow [Bell, 1975; Hibiya, 1986]. Nakamura et al. [2000] approximated U(t) in the argument of the above exponential function as U(t0) = const. during a short time interval t0t so that

equation image

Although this approximation is already invalid except during a limited time interval centered at the tidal flow maximum [Mohri et al., 2010], it is their inconsistent mathematical manipulation that should be pointed out here as another error; if the tidal flow is regarded as constant for the short time interval t0t, the other U(t) in equation (A1) should also be replaced by U(t0) = const. Then, we can correct the dispersion relation to ω = −kU(t0) representing quasi-steady lee waves which should stay over the bottom topography while keeping quasi-steady balance with the background tidal flow throughout each tidal cycle [Hibiya, 1986]. We believe the above inconsistent mathematical manipulation leads to somewhat mysterious term “unsteady lee waves.”

Appendix B:: Sensitivity of the Energetics to Viscosity and Diffusivity Parameterizations

[32] In order to examine the sensitivity of the calculated results to the employed parameterizations of viscosity and diffusivity, a set of additional numerical experiments is carried out. In Experiments B1 and B2, all the viscosity and diffusivity coefficients are decreased and increased by a factor of 5 compared to Experiment 1 (see section 2), respectively. In Experiment B3, the bottom drag coefficient Cd is decreased to 1.0 × 10−3. In Experiment B4, biharmonic horizontal viscosity and diffusivity are employed with constant coefficients of A4 = 107 m4 s−1L2 × AH and K4 = 106 m4 s−1L2 × KH where L is the horizontal grid spacing in the numerical experiments, AH and KH are the horizontal viscosity and diffusivity coefficients in Experiment 1.

[33] The results are summarized in Table B1. We can find that the calculated results are not significantly affected by the employed parameterizations of viscosity and diffusivity except the vertical decay scale of the bottom-intensified mixing ζ which is shown to decrease as the values of viscosity and diffusivity increase. The baroclinic energy flux associated with the internal waves propagating out of the analyzed domain, in particular, is not significantly increased even when the viscosity is decreased by a factor of 5, which indicates that the value of the local dissipation efficiency q is quite robust. Although the physical process controlling the vertical decay scale of the bottom-intensified mixing remains uncertain, it does not significantly affect the estimate of the average diapycnal diffusivity in the Kuril straits.

Table B1. Summary of the Sensitivity Experiments in Appendix Ba
ExperimentCommentsDivergence of Barotropic Energy Flux (GW)Divergence of Baroclinic Energy Flux (GW)Local Dissipation EfficiencyVertical Decay Scale of Bottom-Intensified Mixing (m)
  • a

    Area-integrated barotropic and baroclinic energy flux divergence, the local dissipation efficiency q, and the vertical decay scale of the bottom-intensified mixing ζ. Local dissipation efficiency q is estimated using equation (11).

1 −30.20.61.0200
B1AH, AV, KH, KV = 0.2 × (Experiment 1)−30.00.71.0530
B2AH, AV, KH, KV = 5 × (Experiment 1)−31.10.31.0100
B3Cd = 1.0 × 10−3−30.20.61.0170
B4Biharmonic horizontal viscosity and diffusivity−30.20.71.0180

Acknowledgments

[34] This work was supported by JSPS Research Fellowship. Figures were produced by GFD-DENNOU Library.

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