1. Introduction
The Kuroshio south of Japan, which is a part of the subtropical western boundary current in the North Pacific, has a bimodal nature having two types of paths, the nonlarge meander (NLM) paths and the large meander (LM) paths (Fig. 1; also see Kawabe 1995). This study is motivated by the nature of the nearshore NLM (nNLM; in Fig. 1) path of the Kuroshio. It detaches from the coastline and the bottom slope downstream of Cape Shiono-misaki. A region with weak current is present inshore of the boundary current with a distinct front on the inshore flank of the boundary current. However, it is not dynamically clear what makes this inshore inactive region, which is distinct from the jet. Most studies on the bimodality of the Kuroshio path takes the nNLM path to be trapped by the coastline (e.g., Masuda et al. 1999; Akitomo 2008). Although some studies represents this feature with path equations (e.g., Robinson and Taft 1972; Kawabe 1990), they assume the presence of the distinct boundary a priori.
Typical paths of the Kuroshio. Thin lines are 500-m isobaths. Adopted from Kawabe (1995).
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
Although what makes the Kuroshio’s nNLM path is not studied dynamically, it is reproduced by recent realistic numerical models with high resolution (e.g., Hurlburt et al. 1996; Endoh and Hibiya 2001; Mitsudera et al. 2004; Masumoto et al. 2004). In this study, we aim to understand the dynamical essence using an idealized model.
Western boundary currents such as the Kuroshio and the Gulf Stream are separated from the coastal area and flow eastward in the open ocean. Although the dynamics of the separated boundary currents are distinct from that of the nNLM path of the Kuroshio, they are common in that the boundary currents leave the coast and flow as internal jets. Therefore, some aspects of the separated boundary currents are reviewed here. Around the separated boundary currents, barotropic recirculations are found. An anticyclonic recirculation is present south of the separated Gulf Stream (e.g., Worthington 1976; Richardson 1985), and a cyclonic recirculation is present north of it (e.g., Hogg et al. 1986; Hogg 1992). Part of the northern recirculation extends westward to the continental slope around 70°W to make a westward countercurrent beneath the Gulf Stream. As counterparts in the Kuroshio, an anticyclonic recirculation is present south of the Kuroshio Extension (e.g., Kawai 1972; Mizuno and White 1983) and a cyclonic one is present north of it (Qiu et al. 2008).
In a dynamical point of view, it is pointed out that a pair of cyclonic and anticyclonic recirculations produces the separation, which is shifted southward from the latitude expected by the wind-driven Sverdrup volume transport (e.g., Nakano et al. 2008). Many other factors are pointed out to control the separation as reviewed in detail by Chassignet and Marshall (2008). Haidvogel et al. (1992) studies the role of the coastal viscous layer and points out that the coastal boundary condition is critical and a no-slip condition leads to more realistic results. Kiss (2002) indicates that an early separation is caused by an excessive supply of the cyclonic vorticity at the viscous coastal boundary layer. Dengg (1993) shows that the coastal topography of a cape tends to make the boundary current to separate associated with a cyclonic recirculation produced by cyclonic vorticity emanated from the coast. On the other hand, Özgökmen et al. (1997) shows that the existence of a bottom slope suppresses the separation. However, the nNLM path of the Kuroshio detaches from the coast, crossing the bottom slope with a relatively quiescent region in the inshore flank of the current.
In this study, detachment of the western boundary currents from the coasts is discussed, paying attention to effects of bottom slopes. Salmon (1992) studies the boundary current on a bottom slope analytically using a two-layer model. He assumes uniform lower potential vorticity (PV), which leads to nearly uniform lower thickness and controls the upper thickness. Although the reasons for this assumption are not described clearly, the Gulf Stream is represented properly. Nishigaki (1995) studies the boundary current on a bottom slope using a barotropic inertial model. In his model, a velocity front divides the basin into two regions, an active region filled with the water of the subtropical gyre and an inshore stagnant region. The condition that the pressure is uniform in the stagnant region controls the boundary current path. This condition is equivalent to constant velocity on the boundary (precisely, on the inshore end of the active region) because of the conservation of the Bernoulli function.
In this study, we aim to understand the dynamics of the subtropical western boundary currents over bottom slopes detaching from coasts with fairly quiet regions inshore of them, as in the nNLM path of the Kuroshio. We perform a series of numerical experiments using an isopycnal model with idealized conditions. The model has three isopycnic layers to represent an upper boundary current with vertical shear and lower currents in the bottom layer. The coastline has a steplike shape to represent the southern coast of Japan, where the coastline is oriented from southwest to northeast. Through the experiments, we investigate the roles of the bottom slopes and sensitivity to the volume transport of the boundary currents.
2. The numerical model
The numerical model. (a) The coastal topography and the Sverdrup transport of case 1. Labels are in Sv (1 Sv ≡ 106 m3 s−1). Zonal width of the basin is 100°. (b) The bottom topography in the western part. The depth is 200 m at the coast and 4000 m in the open ocean. The contour interval is 500 m.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
We make parameter studies in various conditions as listed in Table 1. Case 1 is the control case. The role of the bottom slope is studied with cases 1 and 2. The sensitivity to the volume transport of the boundary current is studied with cases 1 and 3–6, where the amplitude of the Sverdrup transport is controlled by τ1 and τ2.
Cases of the numerical experiments. Forcing is presented in terms of the wind-driven Sverdrup transport.
3. Results and discussions
a. Detachment with a pool region
Time-mean patterns of the sea surface height (SSH) and the bottom-layer Montgomery potential (pressure perturbation) in the control case (case 1) are shown in Fig. 3. The subtropical western boundary current leaves the coast and crosses the bottom slope in a zonal flow. A region with weak cyclonic circulation is present inshore of the boundary current, as in the nNLM path of the Kuroshio. We call this region a pool region. In the bottom layer (Fig. 3b), a pair of cyclonic and anticyclonic recirculations is present. The northern cyclonic recirculation extends to the lower bottom slope to make a deep countercurrent beneath the boundary current. This corresponds to the observed current in the recirculation north of the Gulf Stream (Hogg 1992) or the deep countercurrent beneath the Kuroshio south of Japan (Fukasawa et al. 1987; Umatani et al. 2001) and in the East China Sea (Nakamura et al. 2008). We note that essential natures of the nNLM path of the Kuroshio and the deep countercurrent are represented with quite simple conditions.
The time-mean pattern of case 1. (a) The SSH. The contour interval is 0.1 m. (b) The bottom-layer Montgomery potential. The contour interval is 0.05 m in equivalent geopotential. The thick dashed lines show the offshore ends of the bottom slopes.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
Distributions of the PV, which is defined by qk ≡ (f + ζk)/hk in conventional expressions, are shown in Fig. 4. The PV is relatively uniform offshore of the boundary current in all layers. In the pool region, the maxima of the top- and middle-layer PV are present. In the bottom layer (Fig. 4c), water with low PV expands to the lower bottom slope, which indicates homogenization of the PV. This homogenization is caused by the baroclinic instability associated with the local state shown schematically in Fig. 5a. The instability theory (e.g., Pedlosky 1987) shows that a flow pattern is unstable only if (i) the background PV gradient is positive somewhere and negative somewhere else and (ii) the product of the background eastward velocity and the background northward PV gradient must be positive somewhere. The condition (i) is satisfied by the negative (southward) PV gradient in the bottom layer just offshore of the slope (Figs. 4c, 5a). The condition (ii) is satisfied by the surface boundary current (Figs. 4a, 5a).
The time-mean patterns of the PV in case 1. (a) The top layer. (b) The middle layer. (c) The bottom layer. The contour labels are in 10−8 m−1 s−1. The shade in (c) shows PV above 4.0 × 10−8 m−1 s−1.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
A schematic view of a cross-slope section in case (a) 1 and (b) 2. Bottom-layer PV gradients within arrows are negative because of positive thickness gradients.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
In the control case, the high top- and middle-layer PV comes from the viscous coastal boundary. This is consistent with precedent studies that point out the importance of the coastal supply of the vorticity on the separation of the boundary currents (Haidvogel et al. 1992; Dengg 1993; Kiss 2002). However, we focus on another mechanism that makes the boundary current leave the coastline.
A case with a flat bottom (case 2) is presented in order to consider the role of the bottom slope. The time-mean SSH pattern is shown in Fig. 6. The detachment of the boundary current is not clear. In the time series, the boundary current has meanders with active variability, which leads to a relatively wide and meandering time-mean current. Comparison with case 1 shows that the bottom slope changes the boundary current markedly.
As in Fig. 3a, but for case 2.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
Distributions of the PV in case 2 (Fig. 7) show northward PV gradients in the upper two layers and a southward one in the bottom layer around the boundary current. In terms of linear baroclinic instability, the conditions of instability are satisfied in most part of the boundary current (Figs. 7c, 5b). How the absence of a bottom slope leads to active variation is illustrated in Fig. 5. In the case with a flat bottom, the area with negative bottom-layer PV gradient is present beneath the boundary current, which satisfies necessary conditions of the instability. In the case with a bottom slope, however, the area with negative bottom-layer PV gradient is limited because of the southward bottom slope. We note that the negative bottom-layer PV gradient is already reduced by eddy activity in Figs. 4c and 7c.
As in Fig. 4, but for case 2.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
Stabilization of currents by bottom topography is consistent with studies on the bimodality of the Kuroshio (e.g., Sekine 1990; Masuda et al. 1999), which show that the Kuroshio is likely to take the NLM path when a bottom slope is present. The result of case 2 has the same nature with Sekine’s (1990) flat-bottom case. Some of the flat-bottom cases of Özgökmen et al. (1997) show boundary current separations with meandering jets. In case 2, the path of the meandering jet is more variable, which makes the detachment unclear.
The geostrophic contours of case 1 are shown in Fig. 8. The idea that information is propagated along the geostrophic contours and controls the flow pattern in chapter 3 of Pedlosky (1996) explains some natures of the bottom-layer circulations. In the region filled with the first baroclinic geostrophic contours (Fig. 8a) from the eastern end, the motion in the bottom layer should be absent (Pedlosky 1996, chapter 3). In the closed contours, the motion in the bottom layer is allowed because the information from the eastern end of the basin does not enter the region, and the recirculations are made. The second baroclinic geostrophic contours (Fig. 8b) are also closed in the recirculations. This means that the second baroclinic waves from the east end cannot enter this region as well.
(a) The first baroclinic, (b) the second baroclinic, and (c) barotropic geostrophic contours of case 1. The contour intervals are (a),(c) 5 × 10−6 s−1 and (b) 2 × 10−5 s−1.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
The fact that the cyclonic recirculation north of the jet in case 1 (Fig. 3b) expands to the lower bottom slope is explained by the geostrophic contours. The cyclonic recirculation (labeled C in Fig. 8a) makes a minimum of QCF. On the bottom slope, the topographic term [the last term on the right side of (3)] is positive (c3 is negative) and makes the geostrophic contours in the cyclonic recirculation closed on the lower bottom slope. The geostrophic contours in the anticyclonic recirculation (labeled A in Fig. 8a), where QCF makes a maximum, split on the slope instead of closing. Therefore, the topographic term makes the cyclonic recirculation enter the lower bottom slope, whereas it keeps the anticyclonic recirculation from entering. Because the topographic term is independent of the flow pattern, the discussion above explains the nature of the recirculations. One may think that the coupling terms in (1) mean that the detailed physical meaning of these geostrophic contours becomes questionable over the slope, where wave modes are modified in vertical structure that are combinations of the barotropic and baroclinic modes. Nonetheless, the geometry of the contours, together with the fact that linear, Northern Hemisphere coastal-trapped waves all propagate with the coast to their right (Huthnance 1978), must be qualitatively correct even over the slope; the cyclonic recirculation north of the jet can penetrate onto the sloping region, whereas the anticyclonic recirculation to the south cannot.
The mechanism of the detachment with a pool region is discussed here. A schematic view of the PV distribution in the control case is shown in Fig. 9. The upper two layers are categorized into three parts: first, PV homogenized regions (labeled Q); including offshore flank of the boundary current and offshore anticyclonic recirculation; second, PV front regions (labeled F) on the inshore flank of the boundary current where the PV gradient is large; and last, pool regions (labeled P) inshore of the boundary current where the water of the boundary current with low PV does not enter. The bottom layer is categorized into two parts: a PV homogenized region (labeled Q) in the open ocean and a fraction of the lower slope and a stagnant region (labeled S) where the velocity is very small.
A schematic view of the PV distribution in case 1. Red lines show the boundaries in the top and middle layers. A green line shows that in the bottom layer. Blue dashed lines show the boundary current. The black dashed line shows the end of the bottom slope. The regions are as follows: Q, PV homogenized regions; P, pool regions; F, frontal regions; S, stagnant regions; and A, a part of Q3 that is on the bottom slope. Numbers mean indices of the layers.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
We note that, in a steady flow pattern, the conservation of the PV can be satisfied in three ways: (i) the streamlines are parallel to PV contours, (ii) the PV is uniform, and (iii) the velocity is zero. In the control case, (i) is applied to regions F and P, (ii) is applied to region Q, and (iii) is applied to region S.
Whether the bottom topography controls the surface currents depends on the state in the bottom layer. It does so where the bottom PV is homogenized (in Q) because the thickness of the bottom layer needs to be nearly uniform, which means that the lower isopycnal is nearly parallel to the bottom topography. It does not so where the bottom layer is at rest (in S) because the lower isopycnal is free from the conservation of the PV. The boundary current crosses the bottom slope in this region feeling no bottom topography. In case 1, it is only area A in Fig. 9 that the western boundary current feels the bottom slope, where the PV is homogenized. The nature of this area is produced by the extension of the deep anticyclonic recirculation as has been discussed. This area makes the boundary current feel the bottom slope and blocks the westward intensification. In the other area, the boundary current is intensified westward to the coast.
The geostrophic contours give another dynamical implication. The first baroclinic geostrophic contours indicate that a baroclinic western boundary layer is present where they converge. In case 1 (Fig. 8a), the convergence is present along the coast in the northern and southern parts, where the directions of the coastlines are meridional. This means that the baroclinic waves can intrude into the bottom slope region and reach the boundary if the coastline is oriented meridionally. Therefore, the western boundary current flows attached to the meridional coast. In the middle part, where the coastline is oblique, the convergence of the baroclinic geostrophic contours is not present because of the eastward component of the current. The barotropic geostrophic contours, which are not affected by the barotropic streamfunction, converge along the offshore edge of the bottom slope (Fig. 8c). It is indicated that this convergence controls the position of the jet because westward intensification by the baroclinic waves is absent there.
The mechanism of how the bottom slope controls the surface boundary current is compared to previous studies. It is similar to Nishigaki’s (1995) barotropic inertial boundary layer model in the following feature. Both models have two distinct regions, one filled with water of the boundary current and the other with coastal water: that is, the stagnant region in Nishigaki (1995) and the pool region in this study. However, the mechanisms that control the boundary between the two regions are different. In Nishigaki (1995), the condition of constant pressure on the boundary makes the velocity of the boundary current constant through Bernoulli’s law. The bottom slope controls the position of the boundary, which gives the path, through the effect on continuity of the fluid. In this study, the bottom slope traps the boundary current only in the northern cyclonic recirculation where the region of homogenized PV extends to the lower bottom slope.
Salmon’s (1992) two-layer boundary layer model assumes constant bottom PV under the boundary current on the bottom slope. The present numerical solution reveals that the PV conservation in the bottom layer is satisfied by the absence of motion in most part of the bottom slope instead of the homogenized PV.
b. Sensitivity to the volume transport
Sensitivity to the volume transport of the boundary current is investigated here to study the conditions that make the pool regions. Time-mean SSH patterns for cases with various Sverdrup transports (Table 1) are shown in Fig. 10. In cases with smaller forcing, cases 3, 1, and 4, the inshore pool regions are present. In cases with larger forcing, cases 5 and 6, the inshore flank of the boundary currents extends to the coast. We will call the former cases “subcritical” and the latter cases “supercritical.”
The time-mean SSH patterns of cases with various Sverdrup transports for case (a) 3, (b) 4, (c) 5, and (d) 6.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
The PV patterns of a supercritical case (case 5) are shown in Fig. 11. The most significant difference from subcritical cases (Fig. 4) is the absence of the top-layer PV maximum over the slope. This feature is associated with the outcropping of the upper isopycnal inshore of the boundary current, which occur in the cases with larger forcing.
As in Fig. 4, but for case 5.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
The time series of cases 5 and 1 are shown in Figs. 12 and 13, respectively. In case 5, the boundary current usually crosses the bottom slope offshore but intermittently sweeps the coast (Figs. 12c–f). This breaks the pool region and makes the mean current touch the coast. In case 1 (Fig. 13), although the boundary current shows some variation, the pool region is always present.
The time series of the SSH pattern in case 5 from day 5850 to 5900 every 10 days.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
The time series of the SSH pattern in case 1 from day 5770 to 5820 every 10 days.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
It is indicated that the intermittent sweep of the coastal region in case 5 is produced by local instability involving the top and middle layers. Figure 14 shows the isopycnals and the PV profiles on cross-current sections at x = 7° for snapshots corresponding to Figs. 12a and 13a. In case 5 (Fig. 14b), the top-layer PV gradient is positive and that of the middle-layer is negative inshore of the boundary current axis, which satisfies necessary conditions of instability in the upper two layers. In the time series, this feature is accompanied by an inflow of middle-layer water with high PV in the inshore flank of the boundary current and followed by a sweep (Figs. 12c–f). The positive top-layer PV gradient is made by an outcrop of the upper isopycnal (Fig. 14a). These suggest that the sweep is induced by the outcrop through an instability associated with the upper two layers. This instability is essentially the same with a surface-intensified baroclinic mode in Ikeda (1983), who studied linear instability of a current along a bottom slope using a three-layer model. In case 1, however, the top-layer PV gradient is negative in the pool region (north of 33.2°N in Fig. 14d), where the necessary condition of instability is not satisfied locally.
The isopycnals and the PV profiles on the meridional section x = 7° (a),(b) on day 5850 in case 5 and (c),(d) on day 5770 in case 1. Arrows in (b) and (d) show the inshore flanks of the boundary currents.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
The idea that an instability process involving the top and middle layers breaks the pool region is also supported by our results with a two-layer model, which represents the upper current with the top layer alone. The flow patterns with inshore pool regions are seen in all two-layer cases including one with three times of Sverdrup transport and one with very small upper water volume to make an outcrop (figures not shown).
A schematic view of a cross-current section.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
The discussion above can be applied to the present numerical results. The time-mean isopycnals in cross-current sections at x = 8° are shown in Fig. 16 for four cases. In the subcritical cases (Figs. 16a,b), the pool regions, where the isopycnals have small inclinations, are present. In the supercritical cases (Figs. 16c,d), the isopycnals have significant inclinations even in the upper slope and the upper isopycnal outcrops at the surface. In the subcritical regime, the increase in the volume transport is mainly supported by the increase in the difference of the upper-layer thicknesses: that is, the first term of the right side of (8). This suggests that the increase of the difference of the middle-layer thickness is suppressed by instability involving the PV of the middle and bottom layers. In the supercritical regime, where the inshore upper-layer thickness becomes zero, the difference of the middle-layer thicknesses increases with increasing forcing. This increase of the thickness difference is likely to enhance the boundary current variability because it enhances the conditions of instability. We note that the contributions of the bottom velocity terms [the last term of the right side of (8)] are small in all cases.
The time-mean isopycnals in the meridional section at x = 8° in cases (a) 3, (b) 1, (c) 5, and (d) 6. The right side is a description on the sensitivity to the volume transport.
Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-076.1
The sensitivity to the volume transport is summarized in the right part of Fig. 16. In the subcritical regime, increasing the volume transport leads to an increase of the cross-current difference of the top-layer thicknesses, whereas that of the middle-layer thicknesses is suppressed probably by an instability. At the critical volume transport, the upper isopycnal outcrops. The cross-current gradient of the PV becomes monotonically positive, which leads to an instability associated with the upper two layers and consequent intermittent sweep of the inshore region. Further increase of the volume transport makes the boundary current more unstable by an increase of the cross-current difference of the middle-layer thicknesses.
4. Conclusions
The dynamics of subtropical western boundary currents over bottom slopes detaching from coasts with inshore pool regions, where the water of the subtropical gyre does not enter and the velocity is small, are investigated in order to understand the nearshore path of the Kuroshio. Numerical experiments under idealized conditions are performed.
A flow pattern with a pool region similar to the nNLM path of the Kuroshio is produced under simple conditions. A deep countercurrent is present on the lower bottom slope, which represents observed deep currents. This is part of a deep cyclonic recirculation north of the jet, which extends to the lower bottom slope (region A in Fig. 9) despite steep topography. This extension can be explained by the geostrophic contours. In this region, the upper boundary current feels the bottom slope and the westward intensification is blocked. This is because, in terms of propagating waves, both barotropic waves and baroclinic waves cannot intrude into the slope region in the northern flank of the current. A pool region forms there, where motions are rather quiescent. In the other region where the bottom-layer velocity is very small, the upper boundary current is free from the bottom slope and westward intensification occurs at the coast.
The sensitivity to the volume transport of the boundary current is investigated by case studies. The pool regions are broken in cases with large volume transports. The mechanism of this sensitivity is as illustrated in Fig. 16. A large volume transport makes the isopycnal outcrop at the surface because the baroclinic volume transport is linked to the difference of the upper-layer thickness. It is indicated that the outcrop leads to instability and an unsteady pool region.
The control case of this study represents the nNLM path of the Kuroshio, which has an eastward flow crossing the bottom slope and an inshore pool region, with a distinct boundary between these two regions. Although the bottom topography south of Japan is more complicated than that of the model, a feature in the bottom topography of the Izu Ridge (on the nNLM path in Fig. 1) may block westward intensification playing the role of region A in Fig. 9. Some of the discussions in this study could be applied to the offshore NLM (oNLM) path (Fig. 1). Further, the Cape Shiono-misaki could be a distinct topographic feature that can cause the flow detachment from the coast. Nevertheless, we think it important to investigate effects of bottom slopes on the flow detachment from the coast. The Gulf Stream shows variable behavior around the separation point, which seems closer to the supercritical cases in this study. Applications to observed boundary currents need to be studied further because our numerical model and the reality are not close enough to each other.
Acknowledgments
This study was supported by the Grant for Joint Research Program of the Institute of Low Temperature Science, Hokkaido University. The calculations were partly executed by the computers in the Research Institute for Information Technology, Kyushu University. Two reviewers’ thoughtful and constructive comments improved the manuscript.
APPENDIX
Derivations of the Equations of the Geostrophic Contours
The equation of the first baroclinic geostrophic contour (1) is derived by [(A1) + aF × (A2) − (1 + aF) × (A3)] if we omit ∇2ψ terms in Jacobians. Similarly, that of the second baroclinic geostrophic contours (2) is derived by [(A1) + aS × (A2) − (1 + aS) × (A3)]. That of the barotropic geostrophic contour (5) is derived by [H1 × (A1) + H2 × (A2) + H3 × (A3)]/HT. Identities J(ψA, ψA) = 0 and J(ψA, CψB) = −J(ψB, CψA) are useful in the manipulations. For the numerical model, where H1 = 400 m, H2 = 600 m, H3 = 3000 m, γ1 = 0.013 24 m s−2 and γ2 = 0.005 88 m s−2 are applied, the values of constants are λF = 0.7311 × 
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