The Tsuchiya jets (TJs) are narrow eastward currents located along thermal fronts at the poleward edges of thermostad water in the Pacific Ocean. In this study, an oceanic general circulation model (OGCM) is used to explore the dynamics of the northern TJ. Solutions are found in a rectangular basin, extending 100° zonally and from 40°S to 40°N. They are forced by three idealized forcings: several patches of idealized wind fields, including one that simulates the strong Ekman pumping region in the vicinity of the Costa Rica Dome (CRD); surface heating that warms the ocean in the tropics; and a prescribed interocean circulation (IOC) that enters the basin through the southern boundary and exits through the western boundary from 2° to 6°N (the model’s Indonesian passages).
Solutions forced by all the aforementioned processes and with minimal diffusion resemble the observed flow field in the tropical North Pacific. A narrow eastward current, the model’s northern TJ, flows across the basin along the northern edge of a thick equatorial thermostad. Part of the TJ water upwells at the CRD upwelling region and the rest returns westward in the lower part of the North Equatorial Current. The deeper part of the TJ is supplied by water that leaves the western boundary current somewhat north of the equator. Its shallower part originates from water that diverges from the deep portion of the Equatorial Undercurrent (EUC); as a result, the TJ transport increases to the east and the TJ warms as it flows across the basin. These and other properties suggest that the dynamics of the model’s TJ are those of an arrested front, which in a 2½-layer model are generated when characteristics of the flow converge strongly or intersect.
Eddy form stress, due to instability waves generated at the CRD region, extends the TJ circulation to deeper levels. When diffusivity is increased to commonly used values, the thermostad is less well defined and the TJ is weak. In a solution without the IOC, the TJ is shifted to higher temperatures with its water supplied by the subtropical cell. When horizontal viscosity is reduced, the TJ becomes narrower and is flanked by a westward current on its equatorward side.
The Pacific subsurface countercurrents (SCCs) are eastward jets located along thermal fronts at the poleward edges of thermostad water. They were first reported by Tsuchiya (1972, 1975, 1981), and are now commonly referred to as Tsuchiya jets (TJs). Furue et al. (2007, hereafter F07) investigated the dynamics of the southern TJ using an ocean general circulation model (OGCM). In this follow-on paper to F07, we extend our earlier study to consider the dynamics of the northern TJ.
Basic TJ properties are summarized by Johnson and Moore (1997) and Rowe et al. (2000). The jets are only about 1.5° wide, attain peak speeds of 30–40 cm s−1 with a combined transport of 14 Sv (1 Sv ≡ 106 m3 s−1), are associated with a jump in potential vorticity across their core, and are flanked by westward flows on their equatorward sides. As TJs flow eastward, their velocity cores rise and shift poleward from about 300 m and ±3° in the western ocean to 150 m and ±6° in the east, and their core densities decrease. The northern TJ is often seen connected to the lower part of the North Equatorial Countercurrent (NECC) in observed zonal velocity sections but is regarded as dynamically separate from the latter (Kessler and Taft 1987; Rowe et al. 2000).
Although the TJs themselves are equatorially confined, their cool temperatures indicate that their source waters lie outside the tropics. Tsuchiya (1981) argued that TJ water is formed by subduction northeast of New Zealand, whereas Toggweiler et al. (1991) suggested Subantarctic Mode Water as another possibility. In either case, subthermocline and upper-intermediate waters circulate westward and northward about the South Pacific subtropical gyre, move to the equator within the lower portion of the New Guinea Coastal Undercurrent, and some then turn eastward near the equator to supply much of the water for the TJs (Tsuchiya et al. 1989; Tsuchiya 1991). Bingham and Lukas (1995) and Johnson and Moore (1997) also note the contribution of fresher Northern Hemisphere water but only on the northern flank of the northern TJ.
The sink regions of the jets are less clear. Rowe et al. (2000) argued that part of the TJs in both hemispheres turns equatorward in the eastern ocean to join the Equatorial Intermediate Current (EIC), a westward equatorial current usually located beneath the Equatorial Undercurrent (EUC). However, horizontal circulation patterns inferred from hydrographic observations (Johnson et al. 2001) suggest that the northern TJ recirculates in the interior ocean and that at least part of the southern TJ upwells along the South American coast where sea surface temperature (SST) can be as low as 15°C (Lukas 1986; Toggweiler et al. 1991).
The property that the northern TJ is part of an interior recirculation suggests that it may be one branch of a “beta plume” (Spall 2000), driven by Ekman suction and upwelling associated with the intertropical convergence zone (ITCZ; McCreary et al. 2002, hereafter MLY). In support of this idea, Kessler (2002) used hydrographic and expendable bathythermograph (XBT) data to investigate the circulation in the vicinity of the Costa Rica Dome (CRD; 8°N, 90°W), a region where the northern TJ abruptly turns northward. His analyses indicated that about half (∼3 Sv) of the TJ transport upwells through the thermocline, driven by the upwelling favorable wind curl there, with the rest turning westward to join the lower part of the North Equatorial Current (NEC; see also Kessler 2006). The Atlantic also has SCCs similar to the TJs, which flow into upwelling regions in the Guinea and Angola Domes (Schott et al. 2004). Doi et al. (2007) recently noted that the strength and position of the Atlantic’s southern SCCs are correlated with the strength of the Angola Dome in a high-resolution OGCM, suggesting a close dynamical connection between the dome and the SCC.
Three-generation mechanisms have been proposed for TJs, two of them involving “local” (y–z) dynamics and the third being basin scale. Marin et al. (2000) developed a y–z, two-dimensional model in which an ageostrophic meridional circulation is established with poleward flow just below the pycnocline; this current advects equatorial angular momentum poleward, generating eastward currents, their model TJs, on either side of the equator (see also Hua et al. 2003 and Marin et al. 2003). Jochum and Malanotte-Rizzoli (2004) used an eddy-permitting OGCM to study the Atlantic SCCs. Using an analysis based on the transformed Euler mean (TEM) framework, they concluded that eddy fluxes of momentum and density drive the model’s SCCs. Johnson and Moore (1997) used a nonlinear, inviscid, 1½-layer model, representing the thermostad layer, to produce a TJ-like, free, inertial jet driven by an inflow near the equator at the western boundary (or, equivalently, an outflow at the eastern boundary). Finally, as will be discussed next, MLY and F07 have explored the idea of the TJs being arrested fronts. It is noteworthy that the arrested front in MLY and F07 is dynamically similar to the one discussed by Johnson and Moore (1997), but MLY’s and F07’s solutions have a complete TJ circulation in which sources and sinks develop internally.
MLY used a hierarchy of models, varying from 2½-layer to 4½-layer systems to explore TJ dynamics, forcing the models both by winds and by a prescribed Pacific interocean circulation (IOC), the latter representing the outflow of water in the Indonesian passages and a compensating inflow from the Antarctic Circumpolar Current. Their solutions indicated that the sinks of the southern and northern TJs are quite different. The southern TJ is driven by upwelling along the South American coast, and the northern TJ by upwelling in the ITCZ band, consistent with Kessler’s (2002) analysis. The TJs vanished in a solution without the IOC, suggesting that they are a branch of the Pacific IOC.
The TJs in the MLY solutions are geostrophic currents along arrested fronts generated when Rossby waves propagate away from the sink regions along characteristic curves, xc(s), yc(s), that can converge, or intersect, in the interior ocean (Dewar 1991, 1992). In a 2½-layer model, the curves are determined by integrations of
where cr ≡ (β/f 2)(h1h2/h)g′12 is the speed of a linear, nondispersive, second-baroclinic-mode Rossby wave, h1 and h2 are the thicknesses of the two active layers, h ≡ h1 + h2, g′12 ≡ g(ρ2 − ρ1)/ρo is a reduced-gravity coefficient, and β ≡ fy. The quantities
are the components of geostrophic velocity averaged over the two active layers, where curlτ is the streamfunction of the Sverdrup transport, xe is the location of the eastern boundary, and curlτ is the vertical component of wind curl. For the winds in the tropical Pacific, ug < cr and υg are directed equatorward so that characteristics bend equatorward toward the west. Because the characteristics are also geostrophic streamlines of the layer-2 flow, a narrow jet can be formed in regions where characteristic curves converge or intersect.
Extending MLY’s study, F07 investigated the dynamics of the southern TJ in non-eddy-resolving OGCMs. Most solutions were obtained in a rectangular basin, extending 100° zonally and from 40°S to 10°N. They were forced by idealized zonal and meridional winds similar to those of MLY, by a prescribed IOC, and by surface heating that warms the ocean in the tropics. Solutions driven by all three forcings and with minimal diffusion resemble the observed flow field in the tropical South Pacific; in particular, a narrow eastward current, the model’s southern TJ, flows across the basin along the southern edge of a thick equatorial thermostad and upwells along the eastern boundary. Its deeper part is supplied by water that leaves the western boundary current somewhat south of the equator. Its shallower part originates from water that diverges from the deep portion of the EUC; as a result, the TJ transport increases to the east and the TJ warms as it flows across the basin. A major part of the water that upwells at the eastern boundary is supplied by the TJ, with a minor contribution from the southern boundary region. On the basis of these and other properties, the authors concluded that the dynamics of the model’s TJ are essentially those of an arrested front (MLY).
The TJs have only recently been simulated in OGCMs driven by realistic forcing, likely as a result of improved model resolution and decreased mixing [Maltrud et al. 1998; Ishida et al. 1998, 2005; Metzger et al. 2003; Masumoto et al. 2004; Maltrud and McClean 2005; also see Kitamura and Suginohara (1987) for a discussion of an earlier OGCM solution with weak TJs]. In this regard, F07 reported two solutions to a global OGCM, with open and closed Indonesian passages. In the latter solution, the southern TJ still exists, although it is warmed by 1°C, indicating that much of its water is supplied by an overturning cell confined within the Pacific basin, rather than by the IOC (also see McCreary et al. 2007).
c. Present research
In this study, we extend MLY’s and F07’s studies to the northern TJ using an OGCM similar to that of F07. Specifically, solutions are obtained in a rectangular basin and on a non-eddy-resolving grid, allowing us to conduct many sensitivity experiments. They are forced by idealized winds, by a prescribed IOC with transport M, and by a surface heat flux that relaxes the model’s surface temperature to a prescribed distribution. To illustrate the influence of each forcing, we obtain a hierarchy of solutions in which the forcings are sequentially added. In addition, we explore the sensitivity of solutions to the forcings and to mixing strengths and parameterizations.
Key results are as follows: Our standard solution simulates a northern TJ sharing basic TJ properties with the observations. Similar to MLY, it is generated by off-equatorial upwelling in the northeastern ocean forced by Ekman suction, is part of a basin-wide recirculation, and its deeper flows are driven by downward eddy momentum flux due to instability waves. As for the southern TJ (F07), northern TJ pathways change with depth, bending more sharply equatorward to the west at shallower depths; as a result, water at shallower depths bends poleward from the bottom of the EUC to join the top of the TJ, leading to an eastward increase in the overall temperature of the TJ. Overall, these and other properties are consistent with the mechanisms proposed by MLY.
Our OGCM is the Center for Climate System Research’s (CCSR) Ocean Component Model (COCO), version 3.4, a level model developed at the University of Tokyo. Details of COCO can be found in Hasumi (2000, 2006). Salinity is held constant at 35 psu in our study, so that surfaces of constant temperature and density coincide. Model configurations are straightforward modifications of those in F07, so only model aspects most relevant for our present purposes are discussed here.
a. Basin, boundary conditions, and grid
The model domain used by F07 was a box 100° wide, extending from 40°S to 10°N, and with a flat bottom at 4000 m. We extend this domain to 40°N to allow for the northern TJ. Our focus is on the response between the equator and 30°N; the extra 10° on the northern side of this region was included to minimize boundary effects. Along lateral boundaries, no-slip conditions are imposed for momentum and no-flux conditions for temperature, except at the inflow and outflow ports (see below).
Most experiments are carried out on a grid with a coarse horizontal resolution of Δx = 2° and Δy = 1°, which allows 100-yr integrations to be executed quickly. One solution is obtained on a “fine” grid with Δx = 1° and a variable Δy that is 0.25° between 10°S and 10°N, 1° poleward of ±20°, and ramps linearly between the two regions. The vertical grid has 36 levels with a uniform resolution of 20 m in the top 400 m, gradually decreasing to 540 m near the bottom.
b. Viscosity, diffusion, and advection
The parameterizations of subgrid-scale mixing are Laplacian, and the standard set of parameter values is almost the same as that of F07; the sole exception is that the maximum slope for isopycnal diffusion (Cox 1987) is increased from 0.01 to 0.1 to eliminate spurious subthermocline currents near the equator. The coefficient of horizontal eddy viscosity is νh = 108 cm2 s−1 in the interior and is increased to 20 × 108 cm2 s−1 near the western boundary to resolve the Munk layer, and the isopycnal diffusivity (Redi 1982; Cox 1987) is κI = 107 cm2 s−1. Background horizontal diffusion and the Gent and McWilliams (1990) thickness diffusion are not used. Vertical viscosity and diffusivity are given by the Pacanowski and Philander (1981) parameterization, with a background viscosity of 1 cm2 s−1, and the background diffusivity is set to zero. The numerical schemes for temperature advection are quadratic upstream interpolation for convective kinematics with estimated streaming terms (QUICKEST; Leonard 1979) in the vertical and uniformly third-order polynomial interpolation algorithm (UTOPIA; Leonard et al. 1993, 1994) in the horizontal with the universal limiter for transport interpolation modeling of the advective transport equation (ULTIMATE) limiter (Leonard 1991). Numerical diffusion associated with these higher-order upwind schemes allows integration without explicit background diffusion.
The wind stress forcing is an idealized version of the observed Pacific winds. It consists of an idealized meridional wind patch, τy = (0, τy), confined in the Southern Hemisphere to represent the southerly winds near South America, and two idealized patches of zonal winds to represent the trades: an “interior” patch, τx = (τx, 0), that extends throughout the entire basin and an “eastern” patch, τe = (τe, 0), confined to the eastern part of the Northern Hemisphere to simulate the strong Ekman pumping region in the vicinity of the CRD. (The zonal and meridional profiles τe and the meridional profile of τx are illustrated in Fig. 2.)
Each wind stress has the separable form τoαXα(x)Yα(y). The meridional wind is the same as F07’s, except that it is tapered to zero at the equator to minimize its influence on the Northern Hemisphere. Specifically,
where G(η, Δη) ≡ ½[1 + cos(πη/Δη)],
and τoy = 1.0 dyn cm−2. The interior zonal wind is symmetric about the equator and x = 50°, with
and a standard value for τox of −0.5 dyn cm−2. The eastern zonal wind has
and τoe = −0.75 dyn cm−2. With this definition, |τe| is zero south of y1 = 7°N, rapidly increases to a maximum at y2 = 15°N, and gradually decays to zero at the northern boundary. The associated Ekman pumping velocity wek = curl[τe/(ρof )] is positive from y1 to y′2 ≈ 14°N with its peak value near 10°N. North of y′2, wek is negative, attaining its minimum value (about 30% of the positive peak) at 15°N, and then it weakens slowly farther to the north. Note that because the interior wind τx does not have a minimum corresponding to the ITCZ, the NECC is driven solely by τe.
As will be shown, the Ekman upwelling associated with τe drives a TJ on its equatorward side. Since the meridional width of the EUC generated by τx is larger in our model than in observations because of large horizontal viscosity, the model TJ tends to merge with the EUC much farther east than it does in reality. To avoid this situation, we shifted the upwelling region due to τe a few degrees northward. In this way, the model TJ is located a few degrees farther north and is separate from the EUC along most of the equator.
Model SST is relaxed toward a prescribed distribution, T*(y), in the upper 20 m with a time scale of 30 days, where
With this choice, the ocean is increasingly warmed toward the equator, a key aspect for generating a realistic subtropical cell [section 3a(4)]. Keeping T* constant for |y| > 30° helps to reduce zonal currents induced by the SST gradient in these buffer zones.
Unless stated otherwise, an IOC is imposed with a transport M = 7.5 Sv by specifying a middepth inflow of cool water (its temperature ranging from 5° to 15°C) across the southern boundary from the sea surface to about 650 m and a uniformly distributed outflow of warm water at the western boundary from 2° to 6°N above 80 m. The IOC mimics the water exchange between the Pacific and Indian Oceans. The implementation of the IOC is exactly the same as in F07 (see their Fig. 1), where the interested reader can find more details.
Each integration starts from a state of no motion with a horizontally uniform temperature that is 15°C from the sea surface to 140 m, decreases linearly to 5°C from 140 to 685 m and is 5°C below. The integration time for most solutions is 120 yr. For solution H1 (no-wind case), which evolves much more slowly because of the lack of dynamical upwelling, the integration time is about 1400 yr with acceleration for the temperature equation (Bryan 1984; Wang 2001). For solution S1 (M = 0), which cannot maintain its initial quasi-equilibrium response to winds because of the lack of cold-water source, the integration time is about 1800 yr without acceleration. Results shown for solutions H1 and S1 are averages during the last four years of each integration. As discussed later, solutions to the coarse-resolution models forced by τe develop instability waves with a period of 13–15 months, which are well resolved by a 15-day sampling. For each coarse-resolution run (except for the aforementioned solutions H1 and S1), therefore, the integration is continued from year 120 for 12 more years, during which time 15-day averages are saved to obtain sufficient wave statistics. Results shown are averages of this 12-yr data. The fine-resolution solution has two types of oscillation: one with a period of 38 days and the other with a period of about 6.5 months, both of which are smooth in a 5-day sampling; the integration is therefore continued from year 120 for six more years, during which time 5-day averages are saved, and the results shown are averages of this 6-yr data.
We first present a hierarchy of solutions that adds forcings in an orderly manner, the final and most realistic of which is our standard run (solution H5, section 3a). Then we discuss the forcing of subsurface currents by unstable waves, a prominent aspect of most solutions (section 3b). Finally, we report sensitivities of the standard run to the strength of the IOC, background vertical diffusivity, and horizontal viscosity (section 3c). Table 1 lists the solutions discussed and summarizes the differences among them.
To illustrate the structure of solutions, we plot y–z sections of zonal velocity and temperature (Figs. 1 and 7) and x–y maps of transport/width vectors (i.e., velocity vectors integrated either between two isotherms or from the bottom to an isotherm) overlying various scalar quantities (Figs. 2, 4, 6, 8, and 9). We plot only the Northern Hemisphere part of the domain because solutions in the Southern Hemisphere are very similar to the corresponding ones from F07, both qualitatively and quantitatively. Most of the vector maps are integrations from the bottom to the 14°C isotherm, and we refer to this integrated flow as the solutions’ “subsurface” velocity field. Recall that the temperature of the inflow water ranges from 5° to 15°C. We restrict the upper limit of the integration somewhat because 14°–15°C water is very close to the surface or even outcrops at some locations and, hence, is affected by the Ekman layer. (In F07, we restricted the lower limit of integration to the 6°C isotherm on the grounds that the 5°C isotherm cannot be well defined. In the present paper, we extend the integration to the bottom because eddies can induce weak currents in the layer below the 6°C isotherm.)
The shaded regions in Fig. 2 indicate where water leaves (blue) and enters (red) a layer through its upper surface (appendix A). Wherever vertical temperature gradients are large, these estimates are prone to numerical error; in particular, fluxes into the subsurface layer (red) are sometimes questionable in solutions with minimal diffusion. Most of these upwelling and integrated-velocity fields are computed by first integrating the velocity field vertically and then averaging in time (appendix A), so that they include both Eulerian and eddy-induced components.
We begin with a solution forced only by M and then sequentially add forcings by τe, τy, and τx. This ordering is useful for understanding first how the CRD upwelling (driven by τe) affects the IOC and then how the upwelling-driven currents are modified by the other forcings.
1) Solution forced only by M
Solution H1 is essentially the same as its counterpart in F07 (see their Figs. 2a and 3a and their section 3a.1). In response to M, cold water enters the basin, warm water exits from it, and hence the thermocline shallows throughout the basin. Since diffusion is minimal and there are no regions of wind-driven upwelling in the interior ocean, the shallowing must continue until most of the inflow can flow directly to the outflow port along the southern and western boundaries. As a result, the equilibrium-state stratification is very unrealistic. Water colder than 15°C is present everywhere in the tropics below the first level of the model (similar to the stratification for solution H2 south of 2°N in Fig. 1). Moreover, the temperature at the bottom of the outflow port is quite cool (∼10°C); this value is still warmer than the coldest inflow water (5°C), an indication that some of the inflow water must be diffusively warmed during its journey to the outflow port. (In addition to this direct inflow-to-outflow circulation, there is also a weak, broad, interior circulation, not important for our purposes, driven by T*. It consists of a surface, eastward flow overlying a compensating westward, geostrophic current associated with a slight poleward thickening of near-surface isotherms.)
Table 2 summarizes the volume budget of subsurface water in the near-equilibrium solution: Out of the 7.0 Sv of inflow into the subsurface layer (below 14°C), 5.2 Sv exit through the outflow port without ever upwelling, 0.1 Sv upwells near the equator, driving a weak, eastward subsurface current and a compensating surface westward current along the equator (see F07, their Fig. 3a), and almost all of the remaining 1.7 Sv upwell near the inflow or outflow ports.
2) Solution forced by M and τe
When the model is forced by τe as well as M (solution H2), the steady-state response south of 2°N is almost unchanged from solution H1, but it differs markedly farther north because of the wind-driven circulation there. The barotropic part of the response rapidly adjusts to equilibrium via the radiation of barotropic waves. The resulting steady-state, barotropic circulation consists of a direct inflow-to-outflow boundary current plus a wind-driven interior flow field in Sverdrup balance (except for smoothing due to horizontal viscosity). Its meridional component Vs = (τe cosϕ)ϕ/(a cosϕ βρo) ≈ −τye/(βρo) exists only east of x = 64° [Eq. (4a)], where it is directed northward from y1 = 7°N to y2 = 15°N and southward north of y2 [Eq. (4b)]; also see the Ye(y) profile shown in Fig. 2. Its zonal component , plotted at x = 64° in Fig. B1 (curve U; below), is eastward from y1 to ym ≡ 11°N and westward from ym to y2, with its maximum amplitudes at y1 and y2. North of y2, there is a much weaker, westward current associated with the weak, positive curvature of Ye(y) there. There is an eastward current along the northern boundary that supplies the water for the southward Sverdrup flow there.
The baroclinic part of the response also adjusts toward a state of Sverdrup balance in the interior ocean plus an inflow-to-outflow boundary current, but it does so more slowly via the radiation of baroclinic waves. The baroclinic contribution tends to trap the overall Sverdrup circulation to the surface. Indeed, without upwelling or diffusion, the resulting flow would be entirely confined to the surface mixed layer (appendix B). As discussed next, however, this idealized state is not attainable because of upwelling in the CRD region.
Near equilibrium, the baroclinic adjustments require that the thermocline shoals westward within the CRD region where wek > 0. The shoaling is large enough to bring the thermocline to the surface at some locations, and subsurface water is quickly warmed to T*(y) there by surface heating. This upwelling drives cyclonic, subsurface and anticyclonic, surface recirculations that extend from the upwelling region to the western boundary, a beta plume (Spall 2000), both flows acting to spread the Sverdrup circulation into the deeper ocean (appendix B).
If unchecked, upwelling in the CRD region would continue to fill the basin with warm water until the thermocline no longer surfaces and the CRD upwelling ceases. The baroclinic adjustment due to the basin inflow and outflow, however, steadily drains warm water from the surface layer, and initially this effect more than counterbalances the thickening of the surface layer due to the CRD upwelling. As a result, the thermocline continues to shallow, eventually allowing some of the subsurface water to exit directly through the outflow port without upwelling through the 14°C isotherm, as in solution H1. The shallowing continues until the sum of this direct outflow and the upwelling in the CRD region matches the inflow transport. Table 2 indicates that out of 6.9 Sv of subsurface inflow in solution H2, only 2.8 Sv exit directly through the outflow port with 3.5 Sv upwelling in the CRD upwelling region. (There is a net upwelling transport of 0.6 Sv elsewhere. It includes the −0.5 Sv of downwelling just north of the CRD region, listed as 𝒲NG in Table 3, and other minor upwellings elsewhere.)
(ii) Vertical structure
Figures 1a and 1b illustrate the aforementioned and other features, plotting meridional sections of the u and T fields averaged over years 121–132 at x = 50° and x = 80°, respectively. At x = 80°, the eastward and westward branches of the Sverdrup circulation driven by the large curvature of τe for y1 < y < y2 are apparent. Note that the currents extend somewhat outside this band because of viscosity, which smoothes the otherwise sharp edges of the flows (appendix B). In addition, their vertical structures differ significantly because of differences in stratification and eddy forcing. There is upwelling in the latitude band of y1 < y < y′2, where wek > 0, which is strong enough near 11°N to bring water cooler than 12°C to the surface (this latitude agrees with that of the theoretical maximum thermocline tilt; see Fig. B1 and appendix B). In contrast, north of y′2, where wek < 0, subsurface isotherms slope downward toward the north, forming a well-defined, near-surface thermocline. As a result, south of y′2 both branches extend at least to the depths of the isotherms that are affected by diapycnal upwelling (>11°C) because of the weak, upper-ocean stratification, whereas north of y′2 the westward branch is confined within and above the near-surface thermocline (appendix B). As discussed in section 3b, both branches extend to even greater depths south of y′2 as a result of eddy forcing.
There are two other notable currents in the x = 80° section: eastward currents at a depth of 350 m at 17.5°N and near the surface in the top-right corner of the plot (Fig. 1). The former is an eddy-driven flow associated with the westward flow just south of it (section 3b). The latter is a geostrophic current driven by the surface cooling, which causes near-surface isotherms to tilt upward toward the north; it extends to 28°N (not shown) where the near-surface stratification becomes too weak to maintain it.
At x = 50° (Fig. 1a), the currents are broader, deeper, and have greater transport than at 80°, due to horizontal mixing, eddy forcing (section 3b), and additional wind forcing, respectively. Another striking difference is the thicker thermocline in the latitude band from 4° to 15°N, a consequence of the damping of baroclinic Rossby waves away from the upwelling region by viscosity. Consistent with this westward thickening, the near-surface flow is weakly southward between the westward and eastward currents from x = 64° to the western boundary layer. Interestingly, the eastward current has a subsurface core. This property is caused by the requirement to match the thicker thermocline to the shallower thermocline to the south; as a result, near-surface isotherms necessarily slope upward from 10° to 3°N, generating westward, near-surface shear.
(iii) Horizontal structure
Figure 2a illustrates the horizontal structure of the subsurface flow field, showing the recirculation gyre and its connection with the upwelling region (blue shading). The eastward and westward branches of the recirculation are almost entirely zonal, consistent with the corresponding solution to the 2½-layer model. Since there is no wind west of the upwelling region, the characteristic curves are oriented due west there [Eq. (1)]. The westward broadening of the zonal currents due to horizontal viscosity is apparent.
There are two patches of downwelling: northwest and southwest of the upwelling region (red shading). In the northwest region, Ekman downwelling penetrates into the subsurface layer likely because of enhanced vertical diffusivity as a result of weaker stratification and strong vertical shear there (Fig. 1b). This downwelling drives a weak anticyclonic recirculation. In contrast, the small downwelling patch southwest of the upwelling region is likely spurious, arising from the difficulty in defining the 14°C isotherm in regions with large, vertical temperature gradients.
Table 3 summarizes the time-mean, subsurface transports in the CRD region (64° < x < 100°, y1 < y < y2). There is a total inflow of 9.7 Sv into the region (𝒬CRD), consisting of an eastward transport of 6.0 Sv across 64° between y1 and ym (𝒰1m) and a northward transport of 3.7 Sv across y1 (𝒱S). Of this inflow, 3.5 Sv upwell in the CRD upwelling region (𝒲CRD), 5.8 Sv return westward (−𝒰m2), and a small residual of 0.4 Sv flows northward across y2 (𝒱2). A similar budget exists for the northern anticyclonic recirculation associated with the downwelling patch (Fig. 2a), the northern limit of which is about y3 ≡ 20°N and the boundary between its westward and eastward branches is near yn ≡ 18°N at x = 64°. We refer to the region, 64° < x < 100°, y2 < y < y3, as the “northern gyre (NG) region.” There is a total inflow of 1.7 Sv into the region (𝒬NG), consisting of an eastward transport of 0.5 Sv across 64° (𝒰n3) and a southward transport of 1.2 Sv across y3 (−𝒱N), and an additional 0.5 Sv from downwelling in the NG region (−𝒲NG). The total westward transport across 64° of the CRD and NG gyres is 8.4 Sv (−𝒰mn), of which 6.2 Sv (𝒬CRD − 𝒲CRD) come from the CRD gyre and 2.2 Sv (𝒬NG − 𝒲NG) from the NG. Note that the subsurface recirculations are stronger than the upwelling/downwelling transports that drive them, a property of beta plumes (Spall 2000; MLY).
Finally, we note that there is a subtropical gyre in the subsurface layer north of y′2, surprising because one expects that baroclinic adjustments should be able to trap the Sverdrup flow above 14°C. The absence of surface trapping is likely caused by the reversal of baroclinic Rossby wave characteristics as a result of the eastward northern-boundary current [MLY; Eqs. (1); appendix B shows a solution to a 2½-layer model, including a subtropical gyre in layer 2]. In support of this idea, solution H5 also has a subsurface subtropical gyre (Fig. 2d), whereas a test solution (not shown) similar to solution H5 except with T* = 25°C throughout the Northern Hemisphere does not: The introduction of stronger stratification near the northern boundary increases the intrinsic Rossby wave speed, allowing characteristics to extend westward despite the eastward northern-boundary current. Just what forces this subsurface flow is not clear, because there is little entrainment or detrainment across the top of the layer (14°C isotherm) north of the CRD region. One possibility is that it is forced by eddy stresses near the western boundary. Another is that it is forced by horizontal, as opposed to isopycnal, viscous stress across the sloping isopycnal of 14°C along the western and northern boundaries [the H term in Eq. (A4b) or ℋ term in Eq. (6)].
3) Solution forced by M, τe, and τy
When τy is included (solution H3), the Southern Hemisphere circulation changes markedly, as detailed in F07. A large portion of the subsurface inflow water leaves the western boundary south of the equator and flows eastward across the basin to provide water for eastern-boundary coastal upwelling forced by τy. This eastward current, the model’s southern TJ, is confined near the equator for most of its pathway along the edge of a thick equatorial thermostad, owing to the equatorward bending of Rossby wave characteristics by the curl of τy [Eq. (1)]. At the eastern boundary, the upwelled water is quickly warmed by surface heating and flows back to the western boundary in the surface layer. The Southern Hemisphere flow field of solution H3 is very similar to the corresponding solution from F07, both qualitatively and quantitatively (see their Figs. 2c and 3c).
Because of the additional coastal upwelling (3.9 Sv, Table 2), most of the basin inflow (7.0 Sv) now upwells to the surface layer in the interior ocean. As a result, the thermocline does not need to shallow as much as it did in solutions H1 and H2. The overall deepening of the thermocline reduces the direct outflow and the upwelling in the CRD region to 0.0 and 1.6 Sv, respectively, compared with 2.8 and 3.5 Sv for solution H2. The remaining 1.5 Sv upwell mostly near the inflow and outflow ports, with some weak equatorial upwelling (0.3 Sv) associated with noise in the temperature field.
(ii) Vertical structure
Figures 1c and 1d illustrate the vertical structure of solution H3. The flow fields are very similar in both plots to those of solution H2 (Figs. 1a and 1b), differing primarily in the presence of circulations near and south of the equator associated with the southern TJ. Because of the additional Southern Hemisphere upwelling, the thermocline is deeper throughout the Northern Hemisphere, an indication of the basin-wide adjustments that have taken place in response to the Southern Hemisphere upwelling. The deepening is particularly apparent south of about 5°N, where the near-surface thermocline deepens to 60 m. One result of this change is that near-surface isotherms now slope downward to the south from 10° to 5°N, generating eastward shear there. As a result, the eastward branch of the Sverdrup circulation is surface trapped at both sections. Another consequence of the deeper thermocline is that the eastward branch of the subsurface recirculation is somewhat narrower in solution H3 than in solution H2 (Figs. 1d and 1b), a result of the narrower CRD upwelling in solution H3 (see below).
(iii) Horizontal structure
Figure 2b shows the horizontal structure of the subsurface flow field of solution H3. The most prominent difference from solution H2 is that the zonal and meridional extents of the CRD upwelling patch are considerably less and its strength is much weaker, a result of the deeper thermocline there. In addition, the upwelling patch is shifted somewhat northward, consistent with the corresponding solution to a 2½-layer model when the thermocline (layer-1 thickness) in the eastern ocean is increased (appendix B).
According to Table 3, the total inflow transport into the CRD region is 6.1 Sv (𝒬CRD), consisting of 4.3 and 1.8 Sv of eastward and northward transports; 1.6 Sv are lost to diapycnal upwelling (𝒲CRD) and 1.9 Sv (𝒬NG − 𝒲NG) are added in the NG region, resulting in a westward transport of 6.3 Sv (𝒰mn). The transport into the CRD region (6.1 Sv) is 63% of that of solution H2 (9.7 Sv), whereas the upwelling in the CRD region (1.6 Sv) is only 46% of that of solution H2. As discussed in section 3b, the subthermocline recirculation is strengthened by eddy forcing, and the contribution from eddy forcing is larger in solution H3 than in solution H2. Another reason for the decrease in 𝒬CRD may be the reduction in the meridional width of the upwelling patch, a property of a beta plume (Spall 2000).
4) Solutions with all forcings
Here we discuss solutions H4 and H5 (which include τx and all other forcings), differing only in the magnitude of τx (τox = −0.25 and −0.5 dyn cm−2, respectively).
The steady-state response to τx includes shallow overturning circulations in both hemispheres, the north and south subtropical cells (STCs; McCreary and Lu 1994; Liu et al. 1994; Lu et al. 1998; Rothstein et al. 1998; Huang and Liu 1999). In these cells, surface water subducts into the thermocline in the subtropics, flows to the equator within the thermocline, upwells in the eastern equatorial ocean, and flows poleward in the surface layer to close the circulation. The zonal wind also drives an EUC, and the STCs are the main source of water for the shallower part of the EUC above 15°C (the minimum of T*). Poleward of 20°, τx also drives anticyclonic subtropical gyres, intensifying the one in the Northern Hemisphere already driven by τe.
The tropical thermocline tilts upward toward the east in response to τx, shallowing the thermocline in the eastern ocean and deepening it in the west, and hence strengthening the eastern-boundary upwelling from 3.9 Sv in solution H3 to 4.9 and 5.6 Sv in solutions H4 and H5, respectively (Table 2). The upwelling in the rest of the basin (the “other” region in Table 2) is reduced in solution H4 primarily as a result of a decrease in the upwelling near the outflow port. This upwelling region is visible in Fig. 2b but not in Fig. 2c, a natural consequence of the deeper thermocline in the west in solution H4. There is patchiness of upwelling and downwelling regions along and near the equator in both solutions H4 and H5; it results from interpolation error in determining the depth of the 14°C isotherm, and there is no large net interfacial mass flux in an area integral over these features. Figure 2d has two “downwelling” (red) patches near the western boundary; their cause is not clear but they are likely due to numerical diffusion.
(ii) Vertical structure
Figures 1e–1h illustrate the vertical structures of solutions H4 and H5. Along the equator, both solutions have an eastward EUC, just beneath the directly wind-driven, westward South Equatorial Current (SEC), and as expected the EUC is stronger in solution H5 (19 Sv, defined as an integration of mean eastward velocity from 4°S to 4°N and from the sea surface to 270 m at x = 50°) than in solution H4 (11 Sv). Also note the downward tilt to the west of the equatorial thermocline in both cases (cf. left and right panels): at 50°, it is centered near a depth of 100 m (150 m) in solution H4 (solution H5), considerably deeper than in solution H3 (60 m).
At 50°, the upper part of the eastward branch of the subsurface recirculation shifts equatorward to connect to the EUC. In solution H5, the eastward current has a well-defined core separate from the EUC, a property that holds for all longitudes from 10° to 80°. This eastward current is the model’s northern TJ. Consistent with the observations, the TJ core depth and temperature rise to the east and the thermostad thickens, the latter a consequence of the zonal tilt of the thermocline due to τx. In contrast to the observations, however, there is not an obvious poleward shift of the TJ core toward the east, the core remaining at 7°–8°N across the basin; a poleward shift is more apparent in the depth-integrated flow (Fig. 2d) and in individual sublayers, as will be discussed shortly.
There is a striking westward current flanking the EUC in solution H5 at 50°, and a relative minimum of eastward flow there in solution H4. This current is important because it splits the northern TJ from the EUC and the NECC, allowing it to be a distinct flow. Its cause is not clear. One possibility is that it is due to the nonlinear downwelling of the SEC, a process illustrated by Philander and Pacanowski (1980), but the current in solution H5 seems too deep to be accounted for by that process. A second possibility is that it results from the necessity to match the thermocline across the NECC, a process that is also at work in solutions H2 and H3: The isopycnal slopes are stronger equatorward of the core of the eastward current in solutions H4 and H5 than in solution H3; hence, the resulting eastward shear is stronger, thereby generating subsurface westward flow. A third possibility (dynamically linked to the second) is that it results from the equatorward bending of Rossby wave characteristics away from the CRD region for higher-order baroclinic modes [Eq. (1)]; these waves have no net depth-integrated transport, and the shallower westward flow is therefore an intrinsic part of the deeper, eastward TJ (MLY). Finally, a fourth possibility is that the westward current is part of an isopycnal recirculation on the northern flank of the EUC as a result of the tendency of EUC water to conserve PV as it flows eastward in a stretching water column (Lu et al. 1998).
(iii) Horizontal structure
In solution H4, the eastward branch of the subsurface recirculation (the model northern TJ) is shifted equatorward compared with that of solution H3, and the lower part of the EUC is visible along the equator (cf. Figs. 2b and 2c). In solution H5, the eastward flow has two distinct branches: the model TJ and the lower part of the EUC (Fig. 2d). Most of the water that diverges from the EUC joins the TJ, the transport of which increases as it flows eastward because of influx from both the EUC and the westward branch of the recirculation. Figure 3 shows the depth of the 11°C isotherm in solution H5. There is a well-defined front delimiting the northern edge of the thermostad and the TJ (see also Figs. 1g, 1h, and 2d). The overall pattern of this depth field is remarkably similar to that of MLY’s corresponding solution to the 2½-layer model (bottom panel of their Fig. 8c).
The CRD upwelling region shifts eastward in solution H4 and farther eastward in solution H5 compared with that of solution H3 (cf. Figs. 2b–2d) as a result of the thermocline being shallower along the eastern boundary (appendix B). The total inflow into the CRD region (𝒬CRD), a measure of TJ strength, is similar among solutions H3, H4, and H5 (Table 3), consistent with the fact that their upwelling transports are similar (the eddy forcing is also similar; see section 3b). The northward inflow (𝒱S) is, however, larger, and the eastward inflow (𝒰1m) is smaller in solution H4 than in solution H3, a consequence of the northeastward bending of water from the EUC (Fig. 2c).
To illustrate further the three-dimensional structure of the flow, Fig. 4 plots horizontal velocity vectors from solution H5 integrated over four sublayers. The panels illustrate three different types of flow paths. In the 14°–12°C sublayer (type 1) water from the bottom of the EUC bends northward in the eastern ocean to feed the upper part of the TJ. In addition, some water from the westward branch of the subsurface recirculation retroflects southeastward between x = 50° and x = 70° to join the TJ. This interior, partial closure of the recirculation before reaching the western boundary could be caused by horizontal viscosity; it may also result from the reversal of Rossby wave characteristics by the strong eastward Sverdrup flow (see MLY’s analytical solution A8 in their Fig. 7). In the 12°–11° and 10.5°–10°C sublayer (type 2), water in the bottom part of the EUC diverges to join the TJ farther to the west. The 10°–9.5°C sublayer (type 3) is deeper than the bottom of the EUC, and the source of its water is the western boundary. The two branches of the eastward flow seen in Fig. 2d are a vertical superposition of these three types of flow.
Figure 4 also makes it clear why the TJ core shifts to a lighter density as it flows eastward: it is continuously supplied by increasingly shallower EUC water as it flows eastward. To quantify this supply, we compute the northward transport below the 14°C isotherm across 4°N, approximately the northern edge of the EUC (Fig. 1g). Most of this water enters the northern TJ in the vertically integrated field (Fig. 2d). There is a transport of 3 Sv between x = 20° and the eastern boundary, 2.5 Sv of which occur between x = 40° and 90°.
A poleward shift of the TJ pathways to the east in each layer is apparent in the maps. It is consistent with the bending and convergence of characteristics [Eq. (1)] in the analytical, 2½-layer model in the region where υg ≈ τx/(ρofh) < 0. In support of this interpretation, the pathways in solution H5 are similar to those of the characteristics in the 2½-layer model (Fig. 7 of MLY). Furthermore, the pathways bend more sharply in shallower layers, which are representative of higher vertical modes with slower Rossby wave speeds. Because Rossby wave characteristics converge toward the west (Fig. 7 of MLY), one might expect the TJ to also narrow. Meridional profiles of zonal velocity on isotherms (not shown), however, do not show this tendency. A possible explanation for this lack of narrowing is that the TJ has already narrowed to the point where its width is set by horizontal viscosity, rather than by inviscid dynamics.
Figure 4 also plots potential vorticity (PV) on the isotherm at the middle of each sublayer. In each layer, PV is roughly conserved along the TJ and increases sharply across it to the north, consistent with Johnson and Moore (1997) and MLY. In contrast to the observations (Gouriou and Toole 1993; Rowe et al. 2000), however, PV is not homogenized equatorward of the TJ. This deficiency may be because the model TJ is not flanked by westward flow on its equator side and is weaker, broader, and farther from the equator than observed; thus, the equatorward increases in relative vorticity and layer thickness are too weak to balance the decrease in f.
b. Forcing of subsurface currents
In the previous sections, we have shown that the subsurface circulations in the CRD and NG latitudinal bands are mainly driven by the CRD upwelling. We have also seen that those circulations extend much deeper than expected from diapycnal upwelling alone. In this section, we confirm the former conclusion quantitatively and show that the latter property results from eddy forcing.
First, we describe unstable waves that develop in all our solutions forced by τe [section 3b(1)]. They are the source of the eddy forcing mentioned earlier. Then we discuss our method of analysis, which relates diapycnal upwelling and eddy forcing to subsurface currents in a quantitative manner [section 3b(2)]. Finally, we present the results of the analysis for the depth-averaged subsurface circulations [section 3b(3)] and then for their deeper part [section 3b(4)].
1) Unstable waves
Unstable waves, with a period of about 14 months and a westward propagation speed of 4–5 cm s−1, are prominent in solutions H2–H5. They are generated in the region 60° < x < 80°, 12°N < y < 18°N, likely by baroclinic or barotropic instability due to the upwelling-induced doming of the thermocline or to horizontal shear of the NEC or to both. They extend from the surface to 600 m and decay rapidly outside the generation region (interestingly, a similar distribution of eddies occurs in a numerical solution to the 2½-layer model used by MLY when horizontal viscosity is increased to the value used in our OGCM).
Similar waves are observed near 10°–13°N (e.g., Périgaud 1990) and attributed either to the barotropic instability of the meridional shear between the NECC and NEC or to the baroclinic instability of the NEC (Philander 1976; Périgaud 1990; Farrar and Weller 2006). The observed waves, however, have periods of 30–100 days, much shorter than those in our solutions; the long periods in our OGCM are possibly due to high-horizontal viscosity. In support of this idea, the period of the waves in our low-viscosity solution [section 3c(3)] reduces to 6.5 months (section 2d).
2) Diagnostic solutions
To investigate the processes that drive equilibrium-state, subsurface circulations, we obtained steady-state solutions to the diagnostic model in appendix A [Eq. (A4)]. It is a simplified version of the OGCM equations [Eq. (A1)] integrated over a subsurface layer (extending from the ocean bottom to the depth of an isotherm) and forced by time-mean mass flux and interfacial stresses taken from one of the OGCM solutions. Essentially, it is an extension of the beta-plume model of Spall (2000) that includes additional forcing terms.
Specifically, the steady-state, depth-integrated response of the layer is given by the equilibrium solution to (A5). With horizontal mixing (νh ≠ 0), solutions to (A5) are obtained numerically. Without horizontal mixing, the steady-state response to (A5) can be found analytically and is given by (A6) and (A7),
According to the first expression, a subsurface northward flow V results from diapycnal upwelling from the subsurface layer (), and the curls of the mean and eddy form stresses (ℱ̃ and ℱ*) and the horizontal and vertical, viscous, interfacial stresses (ℋ and 𝒱), where the forms of the stresses themselves are given in (A4c). (Refer to the end of appendix A for a discussion of the relationship of ℱ̃ and ℱ* to other formulations of eddy-mean flow interactions.) The zonal transport U is obtained by integrating the continuity relation from the eastern boundary at x = xe, where U = 0. Since, as argued later, contributions from ℋ and 𝒱 are small and because we have not found a reliable method to compute ℋ from our OGCM solutions, we omit them from the following discussion.
3) Subsurface layer (T ≤ 14°C)
Figure 5a plots meridional sections along 64° (the western edge of the CRD upwelling region) of various U fields for the entire subsurface layer (i.e., integrated from the bottom to 14°C), showing U from solution H2 (thick solid line) and from versions of the diagnostic model driven by , both with (long-dashed line) and without (thin solid line) horizontal mixing. The large difference between the viscid and inviscid diagnostic solutions demonstrates the significant effect of horizontal viscosity in our coarse-resolution model. The good comparison between the dashed and thick solid curves indicates that the subsurface layer in solution H2 is driven mainly by . The diagnosis does not change much when the forcing ℱ̃ + ℱ* is included in the viscous diagnostic model (short-dashed curve), the remaining discrepancies being due to either ℋ and 𝒱 or to numerical errors in the computations of ℱ̃, ℱ*, and .
Figure 5b plots meridional sections of various U fields for solution H5. In contrast to solution H2, the effect of mean and eddy form stress is significant. The viscous diagnostic model driven by ℱ* + ℱ̃ (dashed–dotted curve) and that driven by (long-dashed curve) have similar contributions to the transport of the eastward flow between 5° and 12°N, and the former is dominant farther north. This strong form stress driving is likely part of the reason why the subsurface recirculation (𝒬CRD in Table 3) reduces by only 41% between solutions H2 and H5, whereas the diapycnal upwelling (WCRD) reduces significantly more (by 64%).
The mean form drag ℱ̃ has a similar structure to that of ℱ* within the CRD upwelling region but with opposite sign and weaker amplitude (not shown); in addition, it is positive in a region directly south of the CRD upwelling region, where both and ℱ* are weak. Forcing ℱ̃ results from nonlinear terms analogous to those that cause characteristics to bend [appendix A; Eq. (1)]. In a test solution to the viscous diagnostic model driven only by ℱ̃, there is an eastward current between the equator and 5°N from the western boundary until x = 50°; farther east, under the influence of ℱ̃, it shifts northward and flows into the CRD region between x = 70° and 85°, consistent with the equatorward shift of the model northern TJ in solution H5. The ℱ* + ℱ̃ flow in Fig. 5b is thus largely due to ℱ̃ south of 5°N.
4) Deeper layer (T ≤ 10°C)
It is noteworthy that there is any flow at all in the deeper layers (bottom two panels of Fig. 4), which are too deep to be directly forced by . Figure 6 plots U and ℱ* fields when z1 is the depth of the 10°C isotherm and the forcing is taken from solution H2. Consistent with (6), the positive (blue) patch of ℱ* extending from x = 65° to x = 85° drives a cyclonic circulation, and the negative (red) patch to its north drives an anticyclonic circulation. (The mean form stress term ℱ̃ is much smaller than ℱ* in and near the CRD and NG regions.) These circulations explain the flow pattern below 10°C at 80° in Fig. 1b—namely, the eastward current at y = 17°N and z = 350 m, the deeper part of the westward current at y = 14°N, and the deeper part of the eastward current farther south at y = 8°N. As noted in section 3a(2), these recirculations are deeper at x = 50° (Fig. 1a) than at x = 80° because eddy forcing continues to deepen them from x = 85° to x = 70°; their depths remain approximately constant to the west of the CRD upwelling region (not shown).
In support of the importance of ℱ*, Fig. 5c plots meridional sections of U along 64° for the subsurface layer from the bottom to 10°C. The good comparison between U from solution H2 (thick solid line) and the one from the viscid version of the diagnostic model forced only by ℱ* (long-dashed line) indicates that the deep subsurface layer in solution H2 is driven primarily by ℱ*.
c. Sensitivity to parameters
Here we report the sensitivity of solutions to the three most influential parameters in the present study, namely, the strength of the IOC, background vertical diffusivity, and horizontal viscosity. (Refer to F07 for an extensive set of test solutions to forcings and mixing parameterizations.)
1) Influence of M
In solutions H2–H5, subsurface currents are driven by upwelling (and to some extent by eddy forcing). These solutions can be in equilibrium only when subsurface water is replenished and surface water is removed by M. To investigate the influence of M, we obtain a test solution with M = 0 (solution S1). In this solution, since upwelling drains subsurface water, the 14°C isotherm quickly deepens and the subsurface layer becomes warmer. At year 120 (not shown), the structure of the currents is similar to that of solution H5, except that the TJ is shifted to higher temperature: the TJ core is much less clear, closer to the EUC, and is located at higher temperature (∼13°C) than in solution H5, and there is no significant flow in the TJ below the 10°C isotherm.
As time passes, the 14°C isotherm continues to deepen and upwelling across it continues to weaken, resulting in slower subsurface warming. (Eventually, the only remaining process of subsurface warming is numerical diffusion associated with advection by subsurface currents.) Fig. 7a shows a meridional section of zonal velocity and temperature averaged over years 1796–1800, when the horizontal-mean rate of subsurface warming is about 0.2°C (100 yr)−1 at a depth of 650 m. The “thermostad” layer between 16° and 14°C is extremely thick, and the TJs are mostly confined to this layer. We conclude that a TJ will exist when the model is fully adjusted to equilibrium. In that state, the TJs must be associated with overturning in the STCs. Since the coolest temperature of subducted water in the model is 15°C, the minimum temperature in the thermostad layer will have this value.
2) Sensitivity to κb
In our previous solutions, vertical diffusivity is minimized by setting the background mixing coefficient κb to zero. To explore the influence of κb, we obtained a solution with κb = 0.1 cm2 s−1 (solution S2), the smallest value commonly used in OGCMs. The resultant TJ does not extend as deep as in solution H5, and it smoothly attaches to the EUC (Fig. 7b). In addition, the thermocline is more diffuse, and the thermostad is less well defined than in solution H5. Out of the 6.9-Sv inflow (Table 2), the upwelling transport across the 14°C isotherm is 3.9 Sv in the eastern-boundary region of the Southern Hemisphere and 1.5 Sv in the equatorial region, in comparison to 5.6 and 0.4 Sv, respectively, in solution H5. The upwelling transport in the CRD region, and hence the subsurface recirculation, is somewhat weaker than in solution H5 (Table 3). Finally, the TJ extends only to 10°C in solution S2 rather than to 8°C in solution H5 because either eddy forcing is weaker in solution S2 (not shown), or diffusion in the interior ocean warms the inflow water so that the bottom of the TJ is not as cold (compare the deep temperature in Fig. 7b with that in Fig. 1g), or both.
3) Influence of νh
Numerous studies have shown that the speed and width of the EUC are sensitive to the parameterization of horizontal viscosity (e.g., Maes et al. 1997; Large et al. 2001; Pezzi and Richards 2003). An eddy viscosity of νh ∼ 107 cm2 s−1 is inferred from observations (Bryden and Brady 1989) and when used in noneddy-resolving models, it allows for an EUC that is comparable to observations in speed and width. Here we report a solution on our fine grid with νh = 107 cm2 s−1 (solution S3), a smaller value of horizontal viscosity than is possible in our coarse-grid solutions.
Figure 7c plots meridional sections of zonal velocity and temperature at x = 50° for solution S3. One obvious change is that the EUC is narrower and stronger than in solution H5 (Fig. 1g). In addition, there are narrow westward currents on the equatorward sides of the northern and southern TJs and an eastward current below the EUC, both absent in solution H5. Finally, the NECC is now distinctly separated from the EUC by a westward flow, the upper part of which appears to be a downward extension of the South Equatorial Current as a result of the meridional-vertical advection of zonal momentum (Philander and Pacanowski 1980); if so, it is not clear why the flow extends nearly to the bottom of the thermocline.
Figure 8 plots velocity vectors (U) for the subsurface layer of solution S3. Also plotted is total form stress curl divided by the Coriolis parameter, (ℱ* + ℱ̃)/f, rather than diapycnal upwelling (), because the former dominates in this solution, indicating that the subsurface recirculation gyres are mainly driven by eddies. See also Table 3, which indicates that the subsurface, cyclonic gyre in solution S3 is stronger than that in solution H5, despite being much smaller. Similar to solution H5, ℱ*/f has a dipole structure with a positive (blue) pole extending from x = 60° to x = 90° and from 11° to 15°N and negative pole (red) from x = 70° to x = 90° and from 15° to 19°N, driving cyclonic and anticyclonic recirculation gyres, respectively. Forcing ℱ̃/f is much weaker than ℱ*/f in these regions but it has large cyclonic curl (blue) to the south, consistent with the northward bending of the eastward current there. In contrast to solution H5, water from the EUC reverses to flow westward before joining the TJ from x = 50°–90°. Most TJ water appears to originate from the western boundary, but that is due to a vertical superposition of flows in different density layers, as will be discussed next.
Figure 9 plots velocity vectors and PV for the same temperature bands as in Fig. 4. The general patterns are similar to those from solution H5 (Fig. 4) in the upper three sublayers, with water from the lower part of the EUC bending northward to join the TJ increasingly to the east at shallower depths. In comparison to solution H5, however, recirculation gyres are more pronounced: There is a cyclonic recirculation, visible in the upper three panels, north of the TJ from 20°–80° and from 6°–12°N; its eastward branch forms a separate core from the TJ, as can be seen in Fig. 7c at 8°N, z = −250 m. Another recirculation, most clearly visible in the 10.5°–10°C layer, is formed when part of the TJ water turns southwestward near x = 80° and joins the TJ again between x = 30° and x = 60°; the southeastward-flowing branch of this recirculation is the main source of the westward current on the equatorward side of the TJ in the meridional section (Fig. 7c). This westward current is also partly fed by a third weaker recirculation with its eastward-flowing branch along the equator, which in the 10.5°–10°C layer extends from x = 50° to x = 80°; this branch is the eastward current below the EUC mentioned for the meridional section (Fig. 7c).
The cause of the two recirculations that feed the westward current south of the TJ is not clear. One possibility is some form of eddy forcing. There is a strong, regular oscillation with a period of about 38 days confined to |y| < 8° in the region. Its vertical structure is that of mode 1, with the nodal point of velocity at about 700 m, below which temperature is uniformly at 5°C, and its horizontal structure is similar to that of an l = 1 equatorial Rossby wave. The amplitude of its meridional velocity anomaly is about 40 cm s−1 at the surface from x = 60° to 80° where it is largest; it is about 10 cm s−1 at 400 m in the same longitudinal range. This oscillation is likely to be one of the tropical instability waves discussed by McCreary and Yu (1992). Whether these subsurface recirculations have any counterpart in reality is questionable; in particular, the eastward current below the EUC is not observed, and hence is likely to be a model artifact.
Figure 9 also plots PV on the isotherm at the middle of each sublayer. Tongues of PV associated with the TJ are better defined than for solution H5 (Fig. 4), a result of weaker horizontal viscosity in solution S3. In this regard, the PV front along its northern flank is much sharper in the lower three layers.
4. Summary and discussion
In this paper, we investigate the dynamics of the northern TJ by obtaining a suite of solutions to an idealized OGCM. Solutions are forced by a meridional wind τy confined to the eastern part of the Southern Hemisphere, by a zonal wind τe with strong curl north of the equator in the eastern ocean to simulate Ekman suction in the CRD, and by basin-wide easterlies τx to represent trades. They are also forced by a relaxation of SST to T*(y) and a prescribed IOC transport M.
In a solution forced by M alone (solution H1), almost all the inflow water flows along the western boundary directly to the outflow port. When τe is added (solution H2), Ekman upwelling in the CRD region drives a subsurface, cyclonic recirculation gyre, in which part of the inflow water diverges from the western boundary to flow eastward into the CRD upwelling patch. It can be viewed either as a beta plume or as the spreading of the Sverdrup circulation into the subsurface layer (appendix B). When τy is added, another part of the inflow water flows across the basin in the Southern Hemisphere to upwell at the eastern coast, forming the model’s southern TJ. When τx is added (solutions H4 and H5), both the southern TJ and the eastward branch of the aforementioned subsurface recirculation, the model’s northern TJ, become narrower and shift equatorward. At deeper depths, water joins the TJs from the western boundary, whereas shallower water bends from the bottom of the EUC to join the top of the TJs, and this process leads to an eastward increase in overall temperature of the TJs. These and other properties support the idea that the dynamics of the model’s TJs are those of an arrested front, which in a 2½-layer model is generated when characteristics of the flow converge or intersect (see section 1; MLY).
The interior stratification also becomes increasingly realistic in the solution hierarchy, allowing the effect of each forcing to be assessed. In solution H1, the tropical “thermocline” is confined to the first level of the OGCM with water of 10°C at 80 m (the depth of the outflow port). In solution H2, the stratification is improved in the Northern Hemisphere, having a subtropical gyre and, hence, deep thermocline north of the CRD latitudes. In solution H3, the equatorial thermocline deepens near and south of the equator. Finally, in solutions H4 and H5, the basin stratification is quite realistic, with a deep thermocline in the subtropical gyres in both hemispheres, an upward tilt of the equatorial thermocline to the east, and an equatorial thermostad flanked by the TJs.
A diagnostic model, essentially an extension of the Sverdrup calculation to include horizontal viscosity and forcing by interfacial stresses and diapycnal upwelling, is used to investigate the roles of these forcings in driving subsurface flow. Applied to the entire subsurface layer (below the 14°C isotherm) of solution H2, the diagnostic model confirms the interpretation of the subsurface, cyclonic recirculation as a beta plume driven by diapycnal upwelling in the CRD region. In contrast, it shows that eddy form stress in solution H5 is as strong as diapycnal upwelling in driving the subsurface, cyclonic recirculation. (In all solutions forced by τe, low-frequency instability waves with a period of about one year are generated in the CRD region.) In the lower part of the subsurface layer (below the 10°C isotherm), where direct diapycnal upwelling does not reach, eddy form stress drives a cyclonic recirculation in which the eastward-flowing branch is the lower part of the northern TJ, and it also drives an anticyclonic recirculation just north of the cyclonic one. Mean form stress acts to shift the TJ equatorward, having an analogous effect to the bending of Rossby wave characteristics.
As in F07, solutions are sensitive to various parameters. When M = 0 (solution S1), the TJ core shifts to the layer between 16° and 14°C, with its water being supplied by overturning in the subtropical cell. We infer that a realistic thermostad and TJs could be obtained in a closed ocean by extending the basin to include the source regions of cooler “13°C” water (F07; McCreary et al. 2007). When the vertical diffusivity is increased to a value commonly used in OGCMs (solution S2), the thermostad becomes diffuse and the TJ somewhat weaker, likely due, in part, to the erosion of thermostad water by diffusion as it flows along the equator (F07). Finally, when horizontal viscosity is reduced (solution S3), the TJ becomes narrower and is flanked by a westward current on its equatorward side. The latter current is fed mainly by a portion of the TJ water that turns southwestward in the far eastern ocean and to a lesser extent by a weaker recirculation with its eastward-flowing branch along the equator below the EUC.
In conclusion, we have demonstrated through F07 and this study that a coarse-resolution OGCM reproduces northern and southern TJs that are similar to observations in many respects and whose basic dynamics are essentially those of arrested fronts. Our solutions also indicate the importance of mixing parameterizations and of basin-scale or interocean overturning circulations in the simulation of a realistic equatorial thermostad and TJs. In this regard, the peak zonal velocity of the model TJ in our coarse-resolution OGCM (∼3 cm s−1; Fig. 1g) is one order of magnitude smaller, and its meridional width (∼3°) is significantly larger than in observations (∼30 cm s−1 and ∼1.5°, respectively; Rowe et al. 2000), although the transport that enters the CRD region is not much lower (5.7 Sv; Table 3). One reason for these discrepancies is undoubtedly the large horizontal viscosity in our OGCM. Another may be that our model lacks eddy and other nonlinear effects that can sharpen and strengthen the TJ through the generation of local recirculations, such as those described by Ishida et al. (2005) [such recirculations may also account for another type of TJ-like current that does not bend poleward toward the east (Ishida et al. 2005)]. We were not able to extend our hierarchy of solutions to a fully eddy-resolving regime because the thermostad degraded at higher horizontal resolutions; this failure is a consequence of larger diapycnal (explicit or numerical) diffusion due to enhanced eddy activity (Roberts and Marshall 1998). The interaction of other processes with the mechanism we have explored here is an interesting topic for future research.
This research was sponsored by NSF Grant OCE02-41871; it was also supported by the Japan Agency for Marine-Earth Science and Technology (JAMSTEC), NASA under Grant NNX07AG53G, and NOAA under Grant NA17RJ1230 through their sponsorship of research activities at the International Pacific Research Center. The oceanic general circulation model, COCO, used in this research was provided by Hiroyasu Hasumi at the Center for Climate System Research, University of Tokyo. Discussions with Hidenori Aiki, Chueh-hsin Chang, Akio Ishida, Markus Jochum, Greg Johnson, Billy Kessler, Dennis Moore, and Peter Rhines were helpful. Thanks are also extended to anonymous reviewers for their constructive comments. The authors wish to acknowledge use of the Ferret program for analysis and graphics in this paper. Ferret is a product of NOAA’s Pacific Marine Environmental Laboratory (information about it is available online at www.ferret.noaa.gov).
The diagnostic model is a simplified version of our OGCM without the momentum acceleration terms; that is,
where ∇ = i∂x + j∂y and weρz represents the effect of all diffusive processes. We wish to relate the depth-integrated response of a subsurface layer, which extends from the flat ocean bottom at z = zb to the ρ1 density surface at z = z1(x, y, t), to forcings at z1 and zb. (For simplicity, the discussion assumes Cartesian coordinates, but the solutions shown in the main text are obtained using spherical coordinates.)
Dividing (A1d) through by ρz and using the identity zα|ρ = −(ρα|z)/ρz for α = x, y, t yields an expression governing the evolution of z(x, y, ρ, t) in density coordinates. Applied to the ρ1 surface, it is
where q1 = q(x, y, ρ1, t) and q is any of the variables u, w, and we.
where . A time average of this relation is
where the overbar denotes the time average. A similar vertical integration of (A1a) gives
are form stress F, horizontal H and vertical V viscous stresses, and ub is the bottom velocity.
In the main text, we report steady-state solutions to (A4a) and (A4b) when and F are time-averaged forcing terms taken from an equilibrium solution to our OGCM,1 and H and V are ignored. To compute , we first obtain U by vertically integrating the horizontal velocity components of the OGCM from the bottom to the target isotherm for each time slice. We then take the time average of U, followed by the divergence of U, to obtain using (A4a). This procedure is also used for the U and plotted in Fig. 2.
There is no obvious definition of diapycnal velocity we1 for a z-coordinate OGCM. Diapycnal velocity could be computed from the projection of the three-dimensional velocity vector onto the unit normal of the local isopycnal. Such a definition, however, is not unique because there is no obvious and unique definition of isopycnal surfaces from the discrete density field and because the three velocity components are not defined at the same grid points in most OGCMs’ grid systems, including COCO’s. Moreover, in our experience diapycnal velocity computed that way tends to be noisy and does not satisfy property (A4a), without which volume budget below an isopycnal cannot be meaningfully discussed. For this reason, we use (A4a) as a definition of diapycnal velocity. In doing so, we use the discrete version of the OGCM’s continuity equation (A1c) so that our volume budget is perfectly closed without any discretization error.
With horizontal viscosity (νh ≠ 0), we obtain the solution by integrating the equations
to steady state, where c is an arbitrary speed, F is divided into two parts: F̃ = ∇/ρo and F* = /ρo, and q′ ≡ q − q. The terms Ut and (1/c2)Pt are artificial terms, which play no role in the final steady state, and are introduced only to provide a means for finding the steady-state response.
The computational domain extends 100° zonally and from the equator to 40°N, and the horizontal resolution is the same as that of the OGCM. When the forcing includes , a matching sink or source is added at the southwestern corner of computational domain so that the net volume input is zero. The value of the horizontal viscosity is the same as the interior value for the OGCM. For simplicity, solutions to (A5) are obtained on a beta plane (although the forcing fields are computed on spherical coordinates).
where 𝒬 = k · curl Q. Equation (A4a) then yields
where xe is the eastern boundary of the basin. This integration is carried out on a beta plane.
An alternative formalism to eddy form stress F* involves the bolus transport (Greatbatch 1998 and references therein), which is widely used to parameterize eddy transports in coarse-resolution OGCMs (Gent et al. 1995). It can be shown that ℱ* ≃ f div U* + βV* under the geostrophic approximation, where U* = (U*, V*) is the vertical integration of the horizontal bolus velocity from the bottom to the depth of an isopycnal (Greatbatch 1998; see also Rhines and Holland 1979). In support of this relation, a map of div U* for the layer extending from the bottom to the 10°C isotherm in solution H2 is indeed very similar to the ℱ*/f field shown in Fig. 6 (βV* is negligible). A map of U* itself shows the bolus transport to be confined largely to the region 60° ≲ x ≲ 80°, 10° ≲ y ≲ 18°. The meridional overturning circulation (MOC) of bolus transport zonally integrated along isopycnals (isotherms) is centered around 14°N and 12.5°C, with northward transport at deeper isopycnals and a peak value of about 2 Sv in solution H2. A similar MOC exists in solution H5. The sense of the subsurface circulation is consistent with the weak-forcing regime of Berloff (2005), where southward eddy PV flux or northward bolus transport drives a westward current [see also Rhines and Holland (1979) and Holland and Rhines (1980); see Greatbatch (1998) for the relationship between bolus transport and eddy PV flux].
Sverdrup Flow in a 2½-Layer System
In this appendix, we illustrate the effect of interior upwelling on the wind-driven circulation within a simple dynamical framework, obtaining a steady-state solution to a 2½-layer model that allows across-layer transfer. We shall see that the primary effect of the transfer (upwelling) is to extend the wind-driven Sverdrup flow into the deeper ocean, consistent with our OGCM solutions.
The steady-state equations of a 2½-layer model without momentum advection and horizontal mixing are
i = 1, 2 is a layer index, h ≡ h1 + h2, and ρo is the mean water density (see McCreary et al. 2002 for details). For convenience, we use the equatorial β-plane approximation and let f = βy.
The rate at which water upwells from layer 2 to layer 1 is given by
where θ is a step function and h1min is a prescribed minimum thickness. According to (B1d), water will entrain into layer 1 whenever the wind-driven upwelling attempts to make h1 < h1min. To allow for an analytic solution, we assume that δt → +0 so that the entrainment is instantaneous.
The ocean basin is the half-plane x < xe, with the boundary conditions that
We also assume that h1min < H1 so that, as demonstrated below in (B7), the upwelling region driven by τe occurs offshore from the eastern boundary (see MLY for a similar solution with h1min = H1). Forcing is by wind stress τe as specified in (4) with the additional property that Ye(y) = 0 for y > yN ≡ 40°N.
Equations (B1) can be solved for the relation
where wek ≡ curl(τe/ρof ) is the Ekman pumping velocity and H ≡ H1 + H2. An additional constraint in regions where we = 0 (h1 ≥ h1min) is
where ug, υg, and cr are defined after Eqs. (1). This equation is a statement that values of h propagate into the interior ocean from boundaries along characteristics. As will be discussed next, these equations allow h1 and h to be determined separately; the currents in each layer can then be determined from (B1a) and (B1c). It is convenient to find the solution in four regions: y ≤ y1 (region 1), y1 < y ≤ y′2 (region 2), y′2 < y ≤ yN (region 3), and yN < y (region 4), where y′2 is defined after (4b). Figure B1 indicates these regions as well as the subregions defined below.
Regions 1 and 4
a state of rest in both layers.
For x ≥ 64° ≡ xw (the western edge of the wind), region 2 is divided into two subregions in which we is either zero or nonzero. Since ug is greater than zero in the latitude band where is also potentially positive depending on parameters such as H1 and g′12. Here we proceed by assuming that ug − cr < 0 everywhere in region 2. Near the eastern boundary (region 2a), characteristics emanate from the eastern boundary so that h is determined by its value along the eastern boundary. The resulting solution is
a statement that the solution is a Sverdrup flow confined to layer 1. This subregion extends westward from the boundary to the curve x′e(y) at which h1 = h1min, defined by2
Note that (B7) cannot be satisfied near the southern and northern edges of region 2 where wek ≈ 0, so that region 2a extends to xw near the edges. It is also clear from (B7) that as H1 approaches h1min, the curve x = x′e(y) expands meridionally and shifts eastward to approach the northern, southern, and eastern edges of region 2, a property consistent with the shrinking of the upwelling region from solution H2 to solution H3 (Fig. 2).
West of x′e (region 2b), h1 remains fixed at h1min because wek is still upwelling favorable, and hence the solution is
Solution (B8) describes a Sverdrup circulation in which the geostrophic flow is spread uniformly throughout both layers (since h1 is constant, there is no geostrophic shear between the two layers). The effect of we, then, is to deepen the Sverdrup flow, which would otherwise be confined to layer 1.
West of xw (region 2c), υg = 0 because there is no wind. It follows that h1x = hx = 0, and hence that
where h1w and hw are the values of h1 and h from (B8) evaluated at x = xw. (When the basin-wide trade wind τx is added, characteristics are bent equatorward into region 1 by υg as a result of τx. In that case, the southern branch of the layer-2 recirculation is shifted equatorward to form a narrow TJ, as in Fig. 7 of MLY.)
In region 3, characteristics propagate southwestward for x > xw, where υg < 0, and westward farther to the west. A curve xn(y) to be defined later extends southwestward from the point (xe, yN), dividing region 3 into two dynamically distinct subregions. South of xn(y) (region 3a), characteristics emanate from the eastern boundary so that h is determined by its value along the eastern boundary. The resulting solution is (B6).
North of xn(y) (region 3b), characteristics propagate southwestward away from the line y = yN for x > xw and then extend westward to fill the region x ≤ xw. Along y = yN, there are discontinuities in h and h1, which generate a positive ug proportional to δ(y − yN) that supplies water for the southwestward flow in the wind-forced region (x > xw). Clearly, ug − cr > 0 along yN so that the characteristic there flows eastward. We assume that layer-2 PV is constant across the jet with the value fN/H′2, where fN ≡ f (yN) and H′2 = h2(x, yN) = constant. It follows from PV conservation along characteristics that
in region 3b. The resulting value for h is
and then h1 can be found from the quadratic equation
where A is the rhs of (B3).
Because region 3b is isolated from the others, the value of fN/H′2 must be some average of the initial PV, f (y)/H2, but its precise value depends on details of the spinup. Furthermore, if H′2 ≠ H2 characteristics in regions 3a and 3b either converge to form a shock (Dewar 1991, 1992), which then defines xn(y), or they diverge from (xe, yN), so that there is a shadow zone where the solution is not determined. To illustrate a possible solution (Fig. B1), we therefore make the simplest choice H′2 = H2, so that the characteristic curve that starts from the point (xe, yN) defines xn(y).
The resulting layer-2 circulation in region 3b is not zero (as it is in region 3a) but rather represents a subsurface part of the subtropical gyre. This circulation is consistent with the corresponding subsurface flow in our OGCM solutions. A difference is that the eastward current along yN is broadened by horizontal mixing; hence, it is not strong enough to reverse the direction of characteristics all the way to the eastern boundary, and curve xn(y) intersects yN somewhat to the west (Figs. 2 and 8).
One can view the above solution as consisting of two parts: a directly wind-driven Sverdrup circulation confined to layer 1, and a beta-plume part driven by we in both layers. For wind stress τe, the beta-plume part consists of anticyclonic (cyclonic) flow in layer 1 (layer 2), with no net transport. The layer-1 anticyclonic flow, however, is overwhelmed by the cyclonic Sverdrup flow, so the anticyclonic surface flow of the beta plume is never seen.
Figure B1 illustrates the solution when H1 = 80 m, H = 300 m, H′2 = H2 = H − H1 = 220 m, h1min = 20 m, and g′ij = gα(Ti − Tj), where T1 = 25°, T2 = 11°, T3 = 5°C, and α = 2.5 × 10−4 (°C)−1. These parameters ensure that ug < cr. The left panel plots the horizontal structure of h (cf. Fig. 2a). The dotted curves indicate characteristics emanating from the eastern boundary; they define regions of no layer-2 flow where h = H, so that all the Sverdrup flow is confined to layer 1 (regions 1, 2a, 3a, and 4). The solid curves indicate contours of h, showing the subsurface extension of the Sverdrup cyclonic gyre in regions 2b and 2c and the subsurface subtropical gyre in region 3b. The right panel shows meridional profiles of h, h1, u2, u1, and U ≡ h1u1 + h2u2 at x = xw. In region 2b, the Sverdrup flow is deep and u1 = u2, but layer 2 carries most of the transport because layer 1 is thin (h1 = h1min). The surface currents (u1) have sharp edges at y = y1 and y2, where the Sverdrup circulation (U) is discontinuous, and both surface and subsurface currents have sharp edges on x′e(y). Note that |u1| increases toward x′e(y) in region 2a, with all the Sverdrup flow confined to layer 1 and h1 approaching h1min. The corresponding currents are smooth in the OGCM solution (Fig. 5a) because of horizontal viscosity. In the northern half of the subtropical gyre, the layer-2 transport is larger than the Sverdrup transport (U) because the layer-1 flow is—curiously—eastward there.
Corresponding author address: Ryo Furue, IPRC, University of Hawaii at Manoa, POST Bldg., 4th Floor, 1680 East-West Rd., Honolulu, HI 96822. Email: firstname.lastname@example.org
Pressure at z = z1(x, y, t) contains a huge component that is a function of z1 alone, p1o, which has no effect on dynamics because it does not contribute to the curl of F or ∇p. To minimize discretization error caused by p1o, we use in the computation of F, where η is the sea surface height and “overbar h” denotes a horizontal average over the domain.
For (B7) to be valid, all points on the curve x′e(y) must be reachable by characteristics emanating from the eastern boundary. It follows from (B6) that u2 = 0, and hence that hug = −(h1g′12/f )h1y,hυg = (h1g′12/f )h1x from (B1a) and (B1c). These relations are a statement that ug = (ug, υg) is parallel to h1 contours, so that the curve x′e(y) is parallel to (ug, υg). Recalling that υg > 0 in region 2, we see that in the northern half of region 2a, where ug < 0, the direction of characteristics (ug − cr, υg) is rotated counterclockwise from (ug, υg), so that they impinge on x′e(y) from the east. With the restriction that ug − cr < 0, they also do so in the southern half, where ug > 0, (ug, υg) is northeastward and (ug − cr, υg) is northwestward.
* School of Ocean and Earth Science Technology Contribution Number 7636 and International Pacific Research Center Contribution Number 586.