1. Introduction
Poleward surface currents along the eastern boundaries of the oceans are prominent features in the winter hemisphere. In the North Pacific and Atlantic they appear in the form of the Davidson Current (Hickey 1979) and the Iberian Poleward Current (Frouin et al. 1990; Haynes and Barton 1990). In the south Indian Ocean, the Leeuwin Current flows poleward throughout the year but is considerably stronger in the austral winter (Thompson 1984; Cresswell and Golding 1980). Modeling studies have shown that the Leeuwin Current and Iberian Poleward Current are forced by a large-scale meridional buoyancy gradient arising from differential heating over the subtropics (Godfrey and Weaver 1991; Peliz et al. 2003). The large-scale buoyancy gradient drives an eastward zonal flow at the surface, which causes water to pile up on the eastern boundary. The resulting zonal pressure gradient drives a geostrophic poleward current.
The large-scale meridional buoyancy gradient, which forces the poleward eastern boundary currents (EBCs), has little zonal structure, implying that the scale of these currents should be determined by internal dynamics. One would expect Rossby waves and eddies to radiate potential vorticity (PV) anomalies away from the eastern boundary, leaving behind very broad and slow currents. However, these currents are typically tens of kilometers wide and have speeds on the order of tens of centimeters per second. This fundamental question—what sets the narrow width of the buoyancy-forced currents—has yet to be conclusively answered.
The factors responsible for trapping these currents near the boundary are uncertain. Previous studies have attributed the trapping to bottom slope (Weaver and Middleton 1989, 1990; Csanady 1985; Furue et al. 2013; Benthuysen et al. 2014; Schloesser 2014) or high rates of entrainment and diapycnal mixing (McCreary 1981; McCreary et al. 1986; Pedlosky 2003; Pedlosky and Spall 2005). McCreary et al. (1992) suggested that Rossby waves emanating from the eastern boundary can be trapped by the background eastward flow. Cessi and Wolfe (2013, hereinafter CW13) found that, in eddy-resolving models, trapped EBCs arose in the adiabatic limit even with no bottom topography. They also found that Rossby wave propagation was not essential to their trapping mechanism.
All the eastern boundary currents are known to shed eddies. The observational studies of Griffiths and Pearce (1985), Pingree and Le Cann (1992), and Fang and Morrow (2003) found anticyclonic eddies being shed by surface poleward currents. Pelland et al. (2013) observed deep anticyclonic eddies shed by the California Undercurrent. Various modeling studies have been able to reproduce these rich eddy fields next to eastern boundaries. For example, Peliz et al. (2003) showed that the Iberian Poleward Current shed eddy pairs of opposite vorticity owing to the interaction of the current with the coastal topography. Kurian et al. (2011) investigated eddies shed by the California Current system, finding a greater prevalence of anticyclonic eddies at the depth of the poleward undercurrent. Molemaker et al. (2015) suggested that the interaction of bottom slope with the California Undercurrent was responsible for shedding anticyclonic eddies. Given this prevalence of eddies near the eastern boundaries of the oceans, it is not surprising that eddy processes played a dominant role in the trapping mechanism of CW13. In the present study, we observe a similar abundance of anticyclonic eddies near the eastern boundary; however, since we have straight coast and flat bottom, these eddies are formed by the instability of the EBC itself.
Following CW13, we use the thickness-weighted average (TWA) framework to study the effects of eddies on the EBC. The TWA framework was first proposed by De Szoeke and Bennett (1993) and refined by Young (2012) and Maddison and Marshall (2013). As the name suggests, the thickness-weighted average of a quantity is obtained by averaging over time along isopycnal layers after weighting the quantity with the layer thicknesses. The continuity and buoyancy equations combine to form a thickness equation, which is free of explicit eddy correlation terms. The TWA momentum equations contain the eddy correlations in the form of divergences of Eliassen–Palm (EP) flux tensors. Thus, the TWA system bundles the eddy correlation terms into the momentum equations and simplifies the eddy-mean flow interaction problem. The readers are referred to Young (2012) for mathematical notations and derivations.
CW13 found that the dominant momentum and PV balances in the EBC were between the Reynolds stress divergence and form drag. They argued that the former acted like a downgradient diffusion of thickness in the zonal direction, while the latter acted like a linear drag on meridional momentum. We will show that this balance is not, in itself, sufficient to produce a swift, narrow boundary current without accounting for the cross-shore distribution of form drag.
In the present study, we simulate a buoyancy forced EBC in an isopycnal-coordinate, eddy-resolving model with a flat bottom and low diapycnal diffusivity. The budgets of momentum, buoyancy, and thickness are analyzed in the TWA framework to determine the role played by eddies in boundary trapping. We expand on the idea presented in CW13 by showing that the zonal variation of the form drag is essential for producing a narrow EBC. The rest of this paper is arranged as follows: The model setup is described in section 2. The circulation observed in the model is described in section 3. Eddy shedding and its effect on the eastern boundary are discussed in sections 4 and 5, respectively. Budgets of momentum and thickness are examined in section 6. Parameterizations for eddy momentum fluxes are explored in section 7, and the essential physics necessary for confining a narrow EBC are illustrated in section 8 using a simple semianalytical model. Finally, the results are summarized in section 9.
2. Model setup
We use the Modular Ocean Model, version 6 (MOM6; a free and open-source model available at https://github.com/NOAA-GFDL/MOM6). MOM6 uses arbitrary Lagrangian/Eulerian coordinates, which allow it to seamlessly bridge the traditional divide between z-coordinate and isopycnal-coordinate ocean models (Adcroft and Hallberg 2006). Isopycnal coordinates are used for this study. The model is integrated forward in time using a mode-splitting algorithm (Hallberg and Adcroft 2009) with a baroclinic time step of 300 s and a barotropic time step determined at each forward integration. The continuity equation is solved using a piecewise-parabolic method and pressure-gradient force is accurately computed by combining an analytic formulation with the finite-volume method (Adcroft et al. 2008). Diapycnal diffusion is handled in accordance with Hallberg (2000).
The horizontal model domain is a Northern Hemisphere sector running from −25° to 0° in the zonal direction and from 10° to 60°N in the meridional direction. The depth of the model H is uniform at 3000 m. The horizontal grid spacing is uniformly 1/25°, which gives a uniform meridional grid resolution of 4.3 km and a zonal grid resolution decreasing from 4.3 km at the southern boundary to 2.2 km at the northern boundary. The high horizontal resolution ensures that the model resolves mesoscale eddies and their momentum and buoyancy fluxes. We use 30 isopycnal layers, uniformly spaced in buoyancy.


Forcing and mean state of the model. (a) Relaxation target for surface buoyancy. (b) Relaxation target for southern boundary buoyancy. (c) Meridional section showing the mean height of the isopycnal layers averaged over 0.5° closest to the eastern boundary. (d) Zonal section showing the mean height of the isopycnal layers near the eastern boundary at 36°N, indicated by the blue line in (c).
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1





Dissipation is provided by both a Laplacian horizontal viscosity (coefficient
3. Description of the circulation
Figure 2 shows the meridional overturning circulation (MOC) for the residual flow, integrated along isopycnals. Isopycnals above the green line are in contact with the surface at least 1% of the time; following CW13, we refer to this region as the swash zone. Isopycnals in the swash zone are intermittently exposed to diabatic forcing at the surface. For the residual MOC, streamlines crossing isopycnals indicate diapycnal flow. Nearly all diapycnal flow in the upper cell is confined to either the southern relaxation zone, where dense water is made lighter, or the swash zone, where light water is made denser. Thus, the flow in this cell away from the surface is very nearly adiabatic as diapycnal diffusion plays a negligible role. In contrast, the diapycnal flow outside the southern relaxation zone in the weak counterclockwise bottom water cell is driven entirely by diapycnal diffusion.
Meridional overturning streamfunction for the residual flow (Sv; 1 Sv ≡ 106 m3 s−1) as a function of latitude and buoyancy (contours and shading; contours are drawn at −5, −2.5, −1, 2.5, 5, 10, 15, and 20 Sv). The sense of the circulation is clockwise around highs. Isopycnals above the green line lie in the swash zone.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1


The mean and residual ZOCs are shown in Figs. 3a and 3b, respectively. While both ZOCs give a similar overall sense of the circulation—a strong clockwise overturning circulation within the swash zone over a weak counterclockwise bottom cell—they differ dramatically near the eastern boundary. The mean ZOC has strong upwelling within a western boundary layer balanced primarily by strong downwelling in an eastern boundary layer, with weak downwelling throughout the interior. In particular, the upwelling along the western boundary is stronger than the downwelling on the eastern boundary, in agreement with the results of Pedlosky and Spall (2005). The eastern downwelling boundary layer is entirely absent in the residual ZOC, and transformation of dense water to lighter water along the western boundary is balanced entirely by a broadly distributed densifying transformation throughout the interior of the domain. Thus, while the western boundary is a site of vigorous upwelling and water mass transformation, the eastern boundary does not contribute significantly to water mass transformation. The difference between the two pictures is due to eddies, which induce an eastern boundary circulation that cancels the mean downwelling almost perfectly.
Zonal overturning streamfunction for the (a) mean and (b) residual flow (Sv) as a function of longitude and buoyancy (contours and shading; contours are drawn at −5, −2.5, −1, 2.5, 5, 10, 15, and 20 Sv). The sense of the circulation is clockwise around highs.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
Taken together, the ZOCs and the residual MOC (which does not differ qualitatively from the mean MOC) suggest a 3D residual circulation where water upwells diabatically in the southwestern corner of the domain and spreads nearly adiabatically along shoaling isopycnals. Once the water reaches the swash zone, it is converted into denser water that returns—again, nearly adiabatically—to the southern boundary. The difference between the two ZOCs indicates that the actual trajectories of water parcels are more complicated than the residual flow suggests, with significant vertical excursions along the eastern boundary that do not result in water mass transformation. This decoupling between the locations of downwelling and transformation has been noted previously by Spall and Pickart (2001).
The mean isopycnals close to the eastern boundary are shown in Fig. 1c. The background stratification is set in the southern part of the domain where the isopycnals are nearly flat. Isopycnals begin to shoal northward of 20°N to accommodate the surface forcing. The relaxation at the southern boundary mimics the stratification set at the Southern Ocean (Wolfe and Cessi 2010; Nikurashin and Vallis 2011), while relaxation at the surface represents the interaction with a zonally uniform atmosphere (Haney 1971).
The SST mostly follows the meridional pattern of the imposed surface relaxation, except that the eastern half of the domain is warmer than the surface relaxation target (Fig. 4a) owing to two factors: 1) meridional advection of warm water by the poleward EBC and 2) warm-core eddies shed from the eastern boundary (more on this in section 5). The surface flow follows the SSH contours (Fig. 4a), flowing northward in a baroclinic western boundary current that diverges from the coast near 28° and drifts east-by-northeastward across the basin. This flow downwells when it impinges on the eastern boundary and the resulting isopycnal deflection drives a poleward surface jet between 25° and 55°N. The flow at intermediate depths (~1000 m) is similar to the surface flow but with the circulation reversed. A southward eastern boundary undercurrent is supported by the compressed isopycnals found near 1000 m along the eastern boundary (Fig. 1d). Water from the undercurrent detrains from the EBC and forms the intermediate-depth westward flow, which feeds into a southward deep western boundary current. In the Eulerian sense, the circulation is closed by connecting the surface and 1000 m circulation by upwelling and downwelling at the western and eastern boundaries, respectively (cf. Fig. 3a).
(a) Instantaneous SST (shading; °C) with mean SSH contours (m) overlaid. Contours are drawn at −0.4, −0.3, −0.15, 0, 0.15, 0.3, and 0.4 m. (b) Instantaneous vertically averaged vorticity (s−1). (c) Vertically averaged EKE (m2 s−2). Contours are drawn at 0.01, 0.03, and 0.05 m2 s−2.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
The snapshots of SST and vertically averaged vorticity (Figs. 4a,b), as well as the vertically averaged eddy kinetic energy (EKE; Fig. 4c) attest to the prevalence of eddies in the flow: these highly energetic mesoscale features occupy the entire region of eastward surface flow as well as a wedge near the eastern boundary north of 40°. As will be discussed in section 4, many of the eddies form on the eastern boundary. A considerable fraction also arise from the western boundary current or are formed spontaneously in the band of westward surface flow, which is supercritical with respect to baroclinic instability.
Wind forcing is absent in this model and the thick bottom layer (cf. Fig. 1) insulates the upper-ocean flow from the bottom drag; the barotropic flow (not shown) is therefore weak and disorganized compared to the baroclinic flow. The exceptions are two inertial recirculation gyres near the separation point of the western boundary current. These gyres only extend 5° from the western boundary and do not significantly affect the interior and eastern boundary flow.
The slope of the outcropping isopycnals in Fig. 1c is smaller than that in previous studies using z-coordinate models (e.g., de Verdière 1988; Marotzke 1997; Sumata and Kubokawa 2001; Cessi and Wolfe 2013). This difference can be attributed to the difference in the way convection is handled in z-coordinate models versus isopycnal-coordinate models (Park and Bryan 2001). Height-coordinate models produce excessive downwelling near the eastern boundary at the expense of northward transport. In contrast, the isopycnal-coordinate model in the present study produces a current that flows northward along the upward-sloping isopycnals. In the Eulerian-mean vertical velocity, this appears as an upwelling in the upper 500 m of the water column (see section 5). Despite the difference in the near-surface isopycnal slopes in height-coordinate and isopycnal-coordinate models, the boundary currents produced by both classes of models are very similar.
4. Eddy shedding from the eastern boundary
Individual eddies are tracked using the Okubo–Weiss parameter (Okubo 1970; Weiss 1991) at the surface and assigned to tracks based on the criteria set forth in Kurian et al. (2011). Eddies shed from the eastern boundary region propagate predominantly westward (Fig. 5). Most propagate faster than the theoretical midlatitude long Rossby wave speed, which is consistent with global observations of mesoscale eddy propagation (Chelton et al. 2011). Between 30° and 40°N, the number of long-lived eddies seems to be smaller (Fig. 5). This is because these latitudes coincide with relatively strong onshore flow and eddies frequently interact with other eddies in this region. Since our tracking algorithm does not account for eddy mergers or bifurcations, very few eddies in this region appear to be long-lived. Figure 6a shows total number of tracks originating from east of −2.5°, while Fig. 6b shows the average duration of tracks originating east of −2.5°. The former shows that there is a mild preference for anticyclonic eddies to be shed from the eastern boundary, while the latter shows that anticyclonic eddies outlast cyclonic eddies especially north of 30°N. This preference is due to the nature of the EBC, which is warmer at the surface than water just offshore, so eddies originating from the EBC tend to have warm cores and are thus anticyclones. Thus, the eastern boundary current feeds warm water into these anticyclonic eddies, which then deposit it in the interior as they propagate offshore and dissipate, leading to the zonally asymmetric surface buoyancy distribution seen in Fig. 4a. This finding has been formalized in the next section by examining the terms of the buoyancy budget. We shall see that the overwhelming abundance of anticyclonic eddies has a significant impact on the residual circulation near the eastern boundary.
Eddy tracks lasting at least 2 months originating from east of −2.5°. Black squares indicate the location of origin of each track. Anticyclones (cyclones) are in red (blue).
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
(a) Total number of eddy tracks originating east of −2.5° from each latitude. (b) Average duration of eddy tracks originating east of −2.5° as a function of latitude. Anticyclones (cyclones) are in red (blue).
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
5. Effect of eddies on circulation and heat transport near the eastern boundary
The Eulerian picture of circulation near the eastern boundary is shown in Fig. 7. The eastward surface flow, driven by the baroclinic pressure gradient imposed at the surface, downwells and returns westward at around 1000 m. The stretching and compressing of isopycnal layers at the surface and 1000 m, respectively, support a vertically stacked structure of meridional velocity, with poleward flow at the surface and equatorward flow at 1000 m.
Zonal section of mean velocity (m s−1), meridionally averaged between 35° and 37°N near the eastern boundary. The black contours represent mean positions of isopycnals at intervals of 0.003 m s−2.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1


Zonal section of mean TWA velocity (m s−1), averaged between 36° and 37°N, near the eastern boundary. Given are the residual (a)–(c) zonal, (d)–(f) meridional, and (g)–(i) vertical velocities and their respective components. For the horizontal velocities, the columns give the (left) time mean velocities at constant buoyancy, (center) the eddy-induced velocities, and (right) the residual velocities. The residual vertical velocity in the right column is divided into epipycnal (in the center column) and diapycnal (in the left column) components. See (8)–(10). The black contours represent mean positions of isopycnals at intervals of 0.002 m s−2.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
Along the eastern boundary, the mean zonal velocity (Fig. 8a) is nearly identical to the Eulerian-mean zonal velocity. The zonal eddy-induced velocity (Fig. 8b) is opposite to the mean zonal velocity and sufficiently strong to reverse the residual velocity relative to the mean, so that the residual circulation is westward near the surface and eastward just below it. The westward residual velocity is the result of anticyclonic eddies physically transporting water warmer than the mean SST from the surface EBC toward the interior. The mean meridional velocity (Fig. 8d) shows the surface current stacked on top of a deeper undercurrent; the eddy-induced velocity (Fig. 8e) marginally opposes the meridional flow, and only very close to the surface. The residual meridional velocity (Fig. 8f) is thus nearly identical to the mean meridional velocity. The contribution of the epipycnal component (Fig. 8h) to the residual vertical velocity (Fig. 8i) is far greater than the diapycnal component (Fig. 8g). The residual vertical velocity shows that the EBC is associated with upwelling rather than downwelling seen in the Eulerian-mean velocity. This upwelling is predominantly a manifestation of zonal and meridional transport along sloping isopycnals.








Zonal section of the terms of the residual-mean buoyancy budget from (13), averaged between 35° and 37°N, near the eastern boundary (m s−3). Advection by (a) zonal mean and (b) eddy-induced velocities is shown, as well as that by (c) meridional mean and (d) eddy-induced velocities. (e) Advection by vertical velocity and (f) tendency by diabatic transformation are also shown. The black contours represent mean positions of isopycnals at intervals of 0.003 m s−2.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
The mean meridional velocity advects warm water northward along the eastern boundary (Fig. 9c), which is then transported offshore by the zonal eddy-induced velocity (Fig. 9b). These two factors contribute to the poleward tilt in SST along the eastern boundary seen in Fig. 4a. The primary balance in the surface current is between warming due to mean meridional advection and cooling due to vertical advection (Figs. 9c and 9e, respectively). This balance represents water flowing adiabatically northward and upward along sloping isopycnals. Additional contributions to the budget come from the zonal advection—which results in a net heating in the upper 300 m due to cancellation between the cooling mean flow and warming eddy flow (Figs. 9a and 9b, respectively)—and weak cooling by eddy meridional advection and surface diabatic forcing (Figs. 9d and 9f, respectively). The buoyancy budget in the undercurrent is the opposite of the surface current, except for the absence of eddy meridional advection and surface cooling. Since eddies do not play a significant role in meridional advection of buoyancy (Fig. 9d), coarse-resolution models are able to produce the tilt in SST as well (e.g., Schloesser et al. 2012). In the absence of eddies, the zonal transport of buoyancy is achieved through horizontal diffusion.
6. Momentum and thickness budgets near the eastern boundary
Here we inspect the budgets of momentum and thickness near the eastern boundary to unravel the dominant physical processes that drive the eastern boundary current and maintain its cross-shore structure.




Very close to the boundary, the gradient of Montgomery potential (Fig. 10b) is balanced by the horizontal friction (Fig. 10d). Elsewhere, the primary balance in the meridional momentum equation (15) is between the Coriolis acceleration (Fig. 10a), gradient of Montgomery potential, and the EP flux divergence (Fig. 10c). Both the Coriolis force and the gradient of the Montgomery potential tend to accelerate the meridional flow and are countered by EP flux divergence. The EP flux divergence in (19) is dominated by the zonal Reynolds stress (Fig. 11a) and the form drag (Fig. 11b). The Reynolds stress is directed away from (toward) the eastern boundary above (below) 500 m, while the form drag transports momentum from the surface to the deeper levels. The overall role of the eddy momentum fluxes is to transport momentum from the surface current to the undercurrent. Since the currents are oppositely directed, this results in a slowing of both. Note, however, that the vertical transfer of momentum is not direct; instead, the EP flux emanates westward from the surface current, then turns downward at about a Rossby deformation radius from the eastern boundary, and then feeds eastward into the undercurrent. The downward flux of momentum is caused by the interfacial form drag resulting from the eddies shed from the eastern boundary. In the interior, the form drag homogenizes the momentum in the vertical direction, thereby ensuring that the EBCs are confined near the boundary. The eddies are themselves roughly a Rossby radius in size, so they can only form when they are at least a Rossby radius away from the eastern boundary. In the absence of form drag, the Montgomery potential gradient would accelerate the seaward flank of the EBC, resulting in a broader but slower current. The influence of form drag becomes small close to the coast, and it is this distribution of the form drag that sets the width of the EBC, as we show in section 8.
Zonal section of the dominant components of meridional TWA momentum budget (m s−2) from (15), namely, (a) the Coriolis acceleration, (b) the gradient of Montgomery potential, (c) the divergence of EP flux, and (d) the horizontal friction, averaged between 35° and 37°N, near the eastern boundary. The green line is the bottom boundary of the swash zone. Arrows indicate the direction of the EP fluxes in the x–z plane. The black contours represent mean positions of isopycnals at intervals of 0.003 m s−2.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
Zonal section of dominant components of meridional EP flux divergence (m s−2) from (19), namely, (a) divergence of the zonal Reynolds stress and (b) form drag, averaged between 35° and 37°N, near the eastern boundary. The green line is the bottom boundary of the swash zone. Arrows indicate the direction of the fluxes from (16). The black contours represent mean positions of isopycnals at intervals of 0.003 m s−2.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1




Zonal sections of the terms of the thickness budget in (21), averaged between 35° and 37°N, near the eastern boundary. (a),(e) Advection due to geostrophic velocity, (b) lateral friction, (c),(d) components of the EP fluxes, and (f) the diapycnal component. The black contours represent mean positions of isopycnals at intervals of 0.002 m s−2.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
7. Parameterization of Eliassen–Palm flux


Applying this form drag parameterization to the present case requires a spatially varying eddy diffusivity
(a) Lines of regression of
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1













(a) Zonal variation of
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1





Zonal sections of (a) Reynolds stress divergence, (b)
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
We use these parameterizations in the next section to illustrate the physics responsible for determining the zonal structure of the EBC.
8. A model of the eastern boundary current
The essential physics of the eastern boundary currents can be understood using a two-layer analog of the TWA equations linearized about a resting thickness profile. The two layers are intended to represent the poleward surface current and the equatorward undercurrent.























The total solutions
Plan views of (a)
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
The zonal sections of currents and interface displacement are shown in Fig. 17. Increasing
Zonal profiles of (a)–(c)
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0040.1
In the presence of β, the amplitude of the meridional velocity increases and the peak moves slightly offshore (Figs. 17d–f), while the meridional decay scale of meridional velocity decreases (not shown). The latter is due to the dependence of the meridional decay scale α on
The physics of the EBC is straightforward. The form drag term acts like a horizontal diffusion of PV (e.g., Treguier et al. 1997; Greatbatch 1998; McDougall and McIntosh 2001). If the diffusivity varies in space, the PV gradient must vary inversely to the diffusivity to ensure continuity of the PV flux (if all other things are equal). Since the diffusivity is smaller near the boundary than offshore, the PV gradient must be larger near the coast. The PV gradient is proportional to the curvature and vertical shear of the meridional velocity, so the meridional velocity must itself be larger near the eastern boundary.
An alternate interpretation can be made in terms of vertical momentum mixing, since the form drag term represents a vertical transfer of momentum. Vertical momentum mixing is small near the boundary, which allows the northward and southward EBCs to remain distinct. Further offshore, vertical mixing of momentum is strong, resulting in a vertical homogenization of meridional momentum. Since the northward and southward EBCs have nearly equal and opposite momentum, the homogeneous state has very small meridional velocity.
9. Summary
We simulate an eastern boundary current in an idealized eddy-resolving model forced by large-scale buoyancy forcing. This current has a baroclinic structure, with poleward flow stacked above equatorward flow, and sheds predominantly anticyclonic eddies at the surface. While sloping bottom topography and coastal shape has been argued to play an important role in the formation of anticyclonic eddies in poleward eastern boundary currents (Pelland et al. 2013; Molemaker et al. 2015; Southwick et al. 2016), this study shows that anticyclones are also formed in the absence of these effects. The alongshore currents are sufficiently strong that eddy formation is spontaneous and the prevalence of anticyclones at the surface is due to their warm cores, resulting from entrainment of buoyant water transported northward in the boundary current.
These eddies have a profound effect on the residual circulation and budgets of momentum and buoyancy. The westward zonal eddy-induced velocity overcomes the eastward mean velocity, resulting in a residual circulation that has the opposite sense as the mean in the upper water column. This broadens the downwelling limb of the residual-mean zonal overturning circulation so that diapycnal downwelling is evenly distributed across the interior instead of being concentrated near the eastern boundary. In the momentum budget, the efficiency of form drag increases dramatically 50–100 km from the eastern boundary and homogenizes momentum vertically. The maximum form drag is displaced seaward of the maximum current velocity, so the momentum flux cycle in the EBC involves offshore transport near the surface by Reynolds stresses, downward transport by form drag, and finally onshore transport by Reynolds stresses at the level of the undercurrent. The overall effect of the EP flux divergence is to mix momentum vertically but with an offshore excursion.
The form drag acts like a vertical viscosity (or horizontal diffusion of PV) with a coefficient that increases offshore before saturating at a large value. The effect of form drag is small near the boundary, so momentum is not homogenized vertically and there are two distinct jets. Away from the boundary, the form drag strengthens and homogenizes the two jets, effectively erasing the boundary currents. Thus, the scale over which form drag increases offshore sets the scale of the EBC. The diffusivity associated with the form drag and its zonal variation serve as good diagnostic tools to understand the cross-shore structure of the eastern boundary currents. A more complete theory would predict the zonal variation of the diffusivity and is the subject of ongoing research.
Acknowledgments
This research was funded by the NSF (OCE-1559065). This work used the Extreme Science and Engineering Discovery Environment (XSEDE) Stampede cluster at the Texas Advanced Computing Center through the allocation TG-OCE160008. The authors are thankful to the helpful and responsive team at NOAA/GFDL involved in developing the Modular Ocean Model 6 (MOM6) as well as to the Python community for providing free and open-source tools like Jupyter, Numpy, SciPy, Matplotlib, and Xarray.
The authors are grateful to the two anonymous reviewers whose feedback directly led to improvements in this manuscript. Finally, the authors thank Michael Spall for several helpful suggestions and comments.
APPENDIX
Two-Layer Model for the Residual Flow








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