a. Equations

We consider a nonlinear model for the steady-state circulation in a uniformly stratified ocean, with bottom topography declining in the cross-shore direction from an elevation h^* at the western boundary. The model is adiabatic and inviscid and conserves quasigeostrophic potential vorticity. To most clearly expose the parameter sensitivity and the structure of the circulation forced by the interaction of the Ekman transport with a boundary, we consider the nondimensional form of the equations. The vertical coordinate is scaled by the ocean depth far from the western boundary, H^*, and the horizontal length is scaled by L^*, which could be chosen to be the deformation radius. All variables with an asterisk are dimensional, and henceforth those lacking an asterisk are nondimensional.

The flow is driven by a uniform northward wind stress on a beta plane, therefore, a nearly meridionally uniform zonal transport within the Ekman layer is drawn away from the western boundary. The wind stress has no curl so that there is no Ekman pumping in the interior, allowing us to focus on the interaction between the Ekman layer and western boundary. It is assumed that, away from the boundary, the eastward Ekman transport is balanced by a westward interior geostrophic flow of strength $U^{*}={\tau }^{*}/{\rho }_0{f}_0{H}^{*}$, where f₀ is the dimensional Coriolis parameter at the meridional center of the model domain, and ρ₀ is the mean density in the Boussinesq approximation. The previously discussed theories (e.g., Choboter et al. 2005; Pedlosky 1978, 2013) illustrate that the response of the boundary current to the Ekman source and sink is baroclinic. For three-dimensional flows, the alongshore current is surface intensified with a vertical scale less than the full depth of the ocean, while the lower layer is not directly affected by the source and sink (Pedlosky 2013). This formulation is similar to the two-dimensional inviscid models of Pedlosky (1978) and Choboter et al. (2005, 2011). The primary difference is the addition of a third, along-coast, dimension, which will be shown to fundamentally alter the circulation and source waters for the Ekman upwelling.

Given that we have scaled the velocity with the barotropic interior flow U^*, U = 1. However, we keep the variable U in the following derivation because it helps to interpret the governing dynamics. Since q is conserved following the flow, the potential vorticity is a function of the streamfunction. This relationship will continue to hold on all streamlines as they approach the western boundary.

b. Solutions

It is clear that the homogeneous component contains two independent solutions, that forced by interaction between the zonal flow and the bottom topography and the source/sink solution due to Ekman pumping/sucking at the corner. These solutions will be labeled ${\phi }_{\text{ho}}^t$ and ${\phi }_{\text{ho}}^s$, respectively.

Note that the source/sink solution has no barotropic component, which is not surprising because the vertical integral of the first two terms on the left side of (12) is equal to the integral of the right-hand side.

a. The boundary pressure ψ₀

A representative solution for the streamfunction is shown in Figs. 1a and 1b at two depths, one averaged between z = 0.8 and 1.0 and the other near the bottom (z = 0.1). For this choice of ψ₀ = 5, the flow is symmetric in the meridional direction about y = 5. Deep in the water column, the interior flow approaches the western boundary and is diverted northward and southward in a narrow boundary layer. Note that, because the flow is inviscid, there is no bottom boundary layer and the onshore flow is limited in its ability to reach the boundary and upwell into the Ekman layer. However, near the surface, the flow develops a narrow boundary current with increasing strength away from the ψ₀ streamline. The flow in the boundary current is directed toward the latitude ψ₀ from both the north and south, in the opposite direction to the deep flow. There is also a component of the flow directed toward the boundary. This provides some of the source waters that are drawn into the Ekman layer in the corner. The balance in the potential vorticity terms is between relative vorticity and stretching vorticity, variations in planetary vorticity are locally relatively unimportant.

View Full Size

The streamfunction (a) averaged between z = 0.8 and z = 1 and (b) z = 0.1 of the source/sink solution. In this calculation S = 1, b = 100, U = 1, and ψ₀ = 5.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-22-0024.1

• Download Figure
• Download figure as PowerPoint slide

The solution depends on the choice of ψ₀, the pressure on the boundary. A choice of ψ₀ = 0 would result in all of the deep onshore transport turning toward the north with a strong boundary current near the surface, as seen in Fig. 1b for y > 5. It can be most easily understood by recognizing that the streamfunction holds as a constant line along the reference latitude far from the ocean interior to the boundary. Therefore, to the north (south) of this latitude, the pressure at the boundary is smaller (larger) than the ocean interior, which supports a northward (southward) flow near the boundary. It can also be understood from the vorticity balance. Since we assume the ocean interior is frictionless, the vorticity balance of the deep circulation within the boundary layer is between the relative vorticity and planetary vorticity. Furthermore, the main component of the relative vorticity is attributable to the meridional velocity because in the boundary layer the zonal scale is much smaller than the meridional scale. Therefore, the deep impinging flow moves either northward or southward, which depends on the boundary condition that we choose at x = 0.

The value of ψ₀ is determined by processes outside the local region of wind forcing. The boundary pressure is propagated along the boundary by waves (Allen 1976; Yoon and Philander 1982). In the case of spatially limited wind stress, the boundary pressure at the upstream (in the wave phase speed sense) limit of the wind stress would determine ψ₀. In that case the flow in the boundary current would be from the south toward the latitude where the wind stress ceases, as in the two-layer solutions of Allen (1976). For the cases considered here, with spatially uniform wind stress in a closed basin, the boundary pressure is determined by a contour integral around the whole basin, which would presumably involve distant wind forcing and dissipation. One can imagine a similar downwelling boundary layer on the eastern boundary that exports water to the south in a narrow boundary current that closes the circulation with this western boundary current in a Fofonoff-like inertial gyre. Although the flow direction depends on the boundary constant, the boundary layer width and vertical scale, the primary quantities of interest here, do not depend on ψ₀. Therefore, in the further analysis, we diagnose the boundary current structure at y = 0 without loss of generality.

b. The total solution

The total solution for the velocity shows that a southward meridional flow arises as the boundary is approached at all depths, with a stronger, narrower boundary current in the upper layers compared to the lower layers (Fig. 2a). An adjacent northward velocity develops in the upper ocean on the offshore side of this narrow flow. The deep meridional velocities have very weak vertical shear and are trapped near the western boundary with scale $L_I=\sqrt{U/b}$. This is the inertial boundary layer, which is governed by a potential vorticity balance between relative vorticity and the planetary vorticity gradient. The ratio of this inertial boundary layer width to the baroclinic deformation radius is given by $L_I/L_d=\sqrt{U/Sb}$, which is the square root of the ratio of the interior velocity to the baroclinic Rossby wave speed. For the present parameters, this ratio is much less than one, while for strongly forced, weakly stratified shallow shelves it can be greater than one. This is a significant difference between the present theory and those for which the width of the coastal current scales with the baroclinic deformation radius (e.g., Allen 1980).

View Full Size

(a) Meridional velocity of the total solution, (b) the source/sink forcing solution, (c) the particular solution, and the (d) topographic forcing solution, at y = 0. In this calculation, S = 1, b = 100, U = 1, ψ₀ = 5, λ = 5, and h₀ = 0.5.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-22-0024.1

• Download Figure
• Download figure as PowerPoint slide

Pedlosky (2013) found similar results for the interaction of surface Ekman transport with an island, with some of the streamlines feeding the eastern upwelling directly from the interior and some from along the island perimeter. If the island radius in Pedlosky (2013) is much larger than the deformation radius, the streamlines around y = 0 are similar to our straight boundary solution.

The total solution contains three parts, the flat bottom source/sink forcing solution, the particular solution, and the topographic homogeneous solution. Because the problem is linear, we may consider each of these components separately.

c. Flat bottom contribution

The flat bottom source/sink solution is much stronger than the particular and homogeneous solutions forced by topography (Fig. 2b). For the barotropic part of the interior flow, the stretching vorticity is negligible because the height variations are small over a length scale less than the order of the deformation radius in the quasigeostrophic framework. The deep flow potential vorticity balance of the source/sink solution is mainly between the relative vorticity and planetary vorticity, which are the first and third terms in (10a). Therefore, the streamfunction anomaly owing to the source/sink forcing is intensified at the western boundary, which decays eastward with the inertial boundary layer width. This is demonstrated in Fig. 3a in which the horizontal scale of the deep flow was diagnosed from a series of solutions with different values for b, as summarized in Table 1. The horizontal scale was diagnosed as the location of the e-folding of the boundary streamfunction anomaly at z = 0.5. The diagnosed boundary current width scales with $\sqrt{U/b}$ in Fig. 3a (solid line). This demonstrates that vortex stretching is negligible in the deep ocean and the balance is between relative vorticity and beta. Not surprisingly, the horizontal scale of the deep flow is not sensitive to S (not shown).

View Full Size

Comparison between (a) the horizontal scale of the deep streamfunction at the western boundary, (b) the vertical scale of the source/sink solution at the surface, and (c) the vertical scale of the particular solution and that predicted by the scaling theory. The colored dots denote the scale diagnosed from the analytic solutions for a wide range of the parameters. The red, blue, and purple dots represent the variations of b, S, and λ.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-22-0024.1

• Download Figure
• Download figure as PowerPoint slide

Table 1

Parameters for the theory calculation used in Fig. 3. The sensitivity calculations are based on the standard calculation, where S = 1, U = 1, b = 100, and λ = 5. In each calculation, only one parameter was altered.

View Table

The source/sink solution shows strong baroclinicity, especially as the flow in the upper ocean enters into the inertial boundary layer. The streamfunction of the source/sink solution shows a sharper gradient in the upper ocean than in the deep ocean (Figs. 1a,b). The delta function forcing at the surface results an intense, narrow structure, particularly as the surface is approached (Fig. 2b). The baroclinic structure shows a local maximum southward flow lying below the strong northward boundary current. However, if one averages the transport in the vertical, the upper-ocean baroclinic flow becomes evident (Fig. 1a). Below this the southward flow becomes independent of depth. The horizontal width of this baroclinic meridional boundary current scales as 1/α, which can be demonstrated from (13). The parameter scaling (18) shows that, for deep ocean scaling, the typical value of the nondimensional number b is much larger than S and U. Therefore, in the series solution, b/U > n²π²/S for small n. Meanwhile, the remaining terms play a decreasing role in the summation of the series solution as n increases, especially for small z, owing to the cosine function. Therefore, the horizontal scale 1/α can be approximated as $\sqrt{U/b}$ in the lower layers, consistent with Fig. 3a. However, as z increases from 0 toward 1, the remaining terms are beginning to play a more important role even for large n in the summation of the cosine series, resulting in the variability on a scale $1/\alpha <\sqrt{U/b}$, which means a sharper pressure gradient, and stronger currents, in the upper layer than in the lower layer. Therefore, this strong baroclinic boundary current is located above the depth where b/U and n²π²/S are comparable. Since in the cosine function of (13a) z is inversely proportional to nπ, the vertical scale of this strong baroclinic current can be inferred from (13b) to be $\sqrt{U/Sb}$, so for Sb ≫ 1 the flow is baroclinic while for Sb ≪ 1 the flow is barotropic. Below $\sqrt{U/Sb}$, the boundary flow is not directly affected by the source/sink, and can be approximated as an inertial boundary layer. Therefore, the flow shallower than $\sqrt{U/Sb}$ is effectively decoupled from the deep ocean.

The stretching vorticity starts to play an increasing role near the surface in the vorticity balance. Given that the stretching does not contribute to the barotropic vorticity and the baroclinic solution is surface intensified (Fig. 2b) for the source/sink solution in a flat bottom ocean, the streamfunction anomaly due to the baroclinic stretching is strong near the surface and decays with depth. The balance between the planetary vorticity and stretching vorticity in (10a) shows that the key parameter of the baroclinic current forced by the source/sink solution is U/Sb, which is the ratio of planetary vorticity to stretching vorticity as well as the ratio of the inertial boundary layer width and the deformation radius, squared. If U/Sb ≪ 1, the vertical length scale has to be small in order for the stretching term to balance the relative vorticity term. In the other limit, U/Sb ≫ 1, the stratification is weak and the vertical length scale approaches the bottom depth. As the flow moves into the boundary layer, the relative vorticity starts playing an increasingly role in the potential vorticity balance. In the boundary layer $x<\sqrt{U/b}$, the relative vorticity in (10a) exceeds the planetary vorticity advection, which then requires a smaller vertical scale so that the stretching vorticity can adjust to conserve q.

The vertical scale was diagnosed as the depth of the e-folding of the surface streamfunction anomaly at $x=\sqrt{U/b}$ for various values of S and b (Table 1). The vertical scale of the series solutions is plotted as a function of $\sqrt{U/Sb}$ in Fig. 3b (solid line), which is also well predicted by the leading-order behavior, especially for small$\sqrt{U/Sb}$. Physically, the larger the wind strength means stronger the onshore flow, which turns to the meridional flow in the boundary layer. To balance the same strength of planetary vorticity variation, the same relative vorticity forced by stronger onshore flow results in a larger horizontal scale for strong boundary flow. The vertical scale can also be understood from the balance between the relative vorticity and stretching vorticity in (10a), where the horizontal scale of the relative vorticity is the inertial boundary layer thickness$\sqrt{U/b}$. The vertical scale is obtained by taking the ratio of the first term for the second term in (10a). Therefore, the vertical scale depends on the horizontal scale through the vorticity balance in (10a), which is also larger for stronger wind stress. The stratification plays an important role in the stretching vorticity variation (10a). For weak/strong stratification, the stretching vorticity is also weak/strong, which needs a large/small vertical scale to balance vorticity variations. As the stratification tends to zero, the stretching vorticity is not effective and the solution becomes barotropic. The dependence of the scales on b can be understood from recognizing that a stronger planetary vorticity gradient requires stronger relative vorticity and stretching vorticity, which means a narrower boundary layer thickness.

d. The topographic contribution

Topography enters the problem in two ways. First, it alters the flat bottom solution, that is, the stretching vorticity is involved in the barotropic vorticity balance over the topographic length scale 1/λ. As the interior flow moves across the sloping topography it introduces stretching of planetary vorticity, fw_z, that tends to increase the potential vorticity, this is balanced by southward advection of planetary vorticity (Fig. 2c). Therefore, the horizontal scale of the stretching vorticity induced by topography depends on the topographic scale, which is distinct from the boundary layer thickness. As the flow impinges on the western boundary, the vorticity balance in the western boundary layer over topography is not only between relative vorticity and planetary vorticity, but the stretching vorticity is also active. Second, it provides a forcing at z = 0 through the no-normal flow condition at the bottom. This supports a bottom-intensified baroclinic current that decays upward with an e-folding scale of 1/m (Fig. 3c).

The vertical structure of the particular solution depends on the topographic slope (11a) and (11b). For wide topographic slopes (λ² < b/U), m is real, and the particular solution is a bottom-intensified flow over the slope, as in Fig. 2c. In this regime the topography is wider than the inertial boundary layer width, relative vorticity of the particular solution is small, and the potential vorticity balance is primarily between vertical stretching and advection of planetary vorticity. This is the most relevant regime for the midlatitude ocean. The strength of the particular solution is far smaller than the source/sink solution and the vertical scale of the particular solution is always larger than the source/sink solution since $m<\sqrt{Sb/U}$ in this limit. Physically, the dependence of vertical scale on stratification, geostrophic flow strength, and β is the same as the vertical scale of source/sink solution but trapped in the bottom layer.

If λ² = b/U, the topography is exactly the width of the inertial boundary layer and m = 0 so the balance is between relative vorticity of the particular solution and β and the particular response is barotropic.

For topography narrower than this (λ² > b/U) m becomes imaginary, which results in wave-like solutions in the vertical (Fig. 4). In this limit the relative vorticity produced by the narrow topographically induced flow is larger than can be balanced by advection of planetary vorticity and so the stretching term is required to balance. The stretching contribution is of opposite sign in this limit compared to the wide topography case. Therefore, compared with wide topographic slopes, the particular solution of narrow topographic slopes is not bottom-intensified but produces a vertical wave-like flow within a narrower boundary layer. The wavenumber increases as the topographic slope decreases.

View Full Size

The (a),(b) perturbation streamfunction and (c),(d) meridional velocity of the particular solution at y = 0. Panels (a) and (c) are for wide topographic slopes λ² < b/U; (b) and (d) are for narrow topographic slopes λ² > b/U.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-22-0024.1

• Download Figure
• Download figure as PowerPoint slide

The homogeneous topographic forced solution is a maximum at the western boundary and decays eastward with horizontal scale 1/α (17a). Since the particular solution satisfies the vertical boundary condition and the topographic forcing solution is a supplemental solution that matches the lateral boundary condition, the topographic forced solution shares the same vertical scale with the particular solution.

The source/sink forcing solution has no barotropic component. However, for nonzero topography, the particular and homogeneous topographic solution do contain barotropic components (Figs. 5a,b). Both the particular and homogeneous topographic solutions are boundary intensified but with different scales. The particular solution decays eastward with a scale depending on the topographic extension from the western boundary, while the homogenous topographic forcing solution decays over the inertial boundary layer width (the same decay scale as the source/sink solution), which is independent of the topography (Fig. 5b). The barotropic velocity over the topography increases as the width (1/λ) decreases or the maximum height at the boundary h₀ increases. Although the streamfunction anomaly of the particular solution at the boundary depends only on h₀ (11), the decay of the streamfunction anomaly depends on the topographic scale (1/λ), resulting in distinctive boundary currents with different topographic scales (Figs. 5a,b). The boundary perturbation streamfunction of the homogenous topographic solution is sensitive to both h₀ and λ. The velocity of the homogeneous topographic solution is reversed but with the same magnitude as the particular solution at the boundary. Therefore, the solutions forced by the topography have no meridional velocity at the western boundary, which is the no-slip boundary condition (16).

View Full Size

(a). The barotropic perturbation streamfunction of the particular solution (dotted line) at y = 0, the homogenous topographic forced solution (dashed line), and the total solutions relevant to the topography (solid line). Different colors denote the different topographic parameters. (b) As in (a), but for the barotropic meridional velocity.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-22-0024.1

• Download Figure
• Download figure as PowerPoint slide

e. The mass budgets

These solutions provide a framework for understanding the mass budget and the origin of water that is drawn into the Ekman layer. The zonal flow approaching the western boundary is exactly that required to balance the offshore Ekman transport. However, the deep flow turns parallel to the boundary, it does not upwell into the Ekman layer. This is a major difference between these three-dimensional solutions and two-dimensional solutions (e.g., Lentz and Chapman 2004; Choboter et al. 2005, 2011). So, the logical question is, if this deep water is not entering the Ekman layer, where does that transport into the Ekman layer come from?

The upper-ocean streamfunction in Fig. 1a shows that the upwelling is provided by the meridional flow in the baroclinic boundary current, which feeds into a vanishingly thin boundary layer that ultimately feeds the upwelling into the Ekman layer. Additional physics that include mixing and viscosity, not considered here, would be required to explicitly represent the balances in this narrow boundary region. The depth that marks the transition between the deep recirculating flow and the source waters for the Ekman layer is the vertical length scale for the baroclinic flow, $L_I/L_d=\sqrt{U/Sb}$. Given that the perturbation solution has zero depth-integrated meridional transport, the meridional transport shallower than this depth is of equal magnitude and in the opposite direction to the deep flow and thus provides the transport required by the offshore Ekman flow. As the boundary is approached, the deep zonal flow approaches zero over a horizontal length scale $\sqrt{U/b}$. To keep the source/sink forced perturbation solution purely baroclinic, the upper-layer zonal flow increases toward the boundary over this same length scale. Therefore, the water that gets upwelling into the Ekman layer comes primarily from the upper ocean shallower than $\sqrt{U/Sb}$ and is advected into the upwelling region through a combination of a meridional boundary current of the inertial boundary layer width and the onshore flow in the interior. The direction of the meridional flow that supplies the source waters in the baroclinic boundary current depends on the choice of boundary constant ψ₀. This means that the depth of the source waters that feed Ekman upwelling is not an inherent length scale that depends only on the local environmental parameters but instead deepens as the wind forcing strengthens, as the stratification weakens, or as the Coriolis parameter increases.

This also provides a scaling for the magnitude of the meridional velocity in the boundary current since it has to provide the transport into the Ekman layer that is not provided from the interior onshore flow. If one takes the deep southward transport, which scales as $U\left(1-\sqrt{U/Sb}\right)L_y$, and requires that this be provided in a boundary current of horizontal scale $\sqrt{U/b}$ and vertical scale$\sqrt{U/Sb}$, then the magnitude of the velocity in the baroclinic boundary current is $V=b\sqrt{S}\left(1-\sqrt{U/Sb}\right)L_y$, where L_y is the nondimensional distance from where ψ = ψ₀. For typical values of b ≫ 1, L_y ≫ 1, S = O(1), and U = O(1), V ≫ 1 and the baroclinic boundary current is very strong compared to the interior flow. Note that the boundary current is stronger for stronger stratification, larger β, and stronger onshore flow but it is not a simple linear dependence because both the width and depth of the boundary current depend nonlinearly on these parameters.

f. The eastward interior flow

Far from the boundary, the relative vorticity ζ_bt is zero. At the boundary, again since the meridional scale is much larger than the zonal scale, the relative vorticity is ∂υ_bt/∂x. For northward wind stress the westward zonal interior flow impinges on the western boundary and deflects either northward or southward. For northward flow, the relative vorticity in the boundary layer is negative, which can balance the increase in the planetary vorticity and vice versa for southward flow.

However, if there were southward wind stress the interior flow would be toward the east, away from the western boundary. The potential vorticity still holds as a constant along streamlines (10a) but the sign of the last term on the left-hand side is now positive. This does not support an exponentially decaying boundary current, as was found for the westward flow. Moreover, for southward wind and eastward zonal interior flow, the relative vorticity in the northward flowing boundary layer will increase, as does the planetary vorticity, violating potential vorticity conservation. Therefore, the eastward flow lacks the physical mechanism to support an inertial boundary layer on the western boundary. However, as we demonstrate in the next section, a numerical model with eastward interior flow produces a boundary current very similar to that for a westward interior flow. We assert that the dissipation in the model is large enough to trap short Rossby waves near the western boundary and allow for set up of a baroclinic boundary current structure analogous to the westward interior flow cases even though the boundary current in the model is viscous, not inertial.

a. Model configuration

The numerical model used is the MITgcm primitive equation model. The model is configured using a z-level vertical coordinate and with a partial cell treatment of the bottom topography. The domain is 960 km × 960 km and 2000 m deep with a flat bottom and closed boundaries. The model has a uniform horizontal grid spacing of 2 km and 45 levels in the vertical with spacing ranging of 10 m over the upper 200 m, gradually increasing to 200 m at the bottom. The initial stratification is uniform and a spatially uniform, steady, northward wind stress is applied. The model is run for a period of 120 days with the analysis taken over the final 90 days of integration. Subgridscale mixing is represented by a horizontal Smagorinsky viscosity (Smagorinsky 1963) with nondimensional coefficient 2.5 and vertical viscosity and diffusion with coefficients 10⁻⁴ and 10⁻⁶, respectively. Calculations with a uniform horizontal Laplacian viscosity with coefficient 20 m² s⁻¹ are essentially the same as those presented here, and additional calculations have shown that the basic results are not sensitive to these parameters. The Coriolis parameter at the southern limit of the domain is 3 × 10⁻⁵ s⁻¹ with meridional variation β = 2 × 10⁻¹¹ m⁻¹ s⁻¹. The wind stress for the example case is τ = 0.005 N m⁻². This weak wind stress was chosen to provide a central case that would produce moderate strength currents so as to be consistent with the quasigeostrophic approximation in the theory. Stronger wind stresses, up to 0.05 N m⁻², are applied in the following section. The initial stratification N² = 2.25 × 10⁻⁶ s⁻². After this central case is discussed, a series of model runs are carried out in which β, N², and τ are all varied, and the results are compared with predictions from the theory in the previous section.

The inertial boundary layer width varies between 2 and 10 km for these model runs. The Smagorinsky viscosity parameterization produces viscosities of O(10–50) m² s⁻¹, which gives a viscous boundary layer width of O(10) km. This is as wide or wider than the inertial boundary layer, so that friction is important in all cases and the model western boundary layer is not purely inertial.

b. Central case

A vertical section of the mean meridional velocity and density, averaged in time and between latitudes y = 200 and 400 km, is shown in Fig. 6. Note that this is only the upper 1000 m and westernmost 100 km of the basin. The flow is dominated by a northward surface intensified current and a weaker southward current below. The northward flow is a maximum just off the western boundary while the southward flow is a maximum on the boundary. The interior flow toward the boundary is O(10⁻⁴) m s⁻¹, so the boundary current is approximately two orders of magnitude stronger, consistent with the theory. The horizontal scale of the boundary current is O(10) km, an order of magnitude less than the baroclinic deformation radius. The isopycnals are flat in the interior but they are deflected within a few kilometers of the western boundary. In the upper 50 m the isopycnals rise, providing anomalously dense water near the boundary and a horizontal density gradient to support the surface intensified jet. Over the deeper half of the countercurrent the isopycnals are deflected downward, as required to balance the local maximum in southward flow. It is clear that near the surface the quasigeostrophic assumption of spatially uniform stratification is not well satisfied, yet the basic baroclinic current structure predicted by the theory is found in the model.

View Full Size

Zonal section of the mean meridional velocity (colors; m s⁻¹) and density field (white contours; contour interval 0.1 kg m⁻³) adjacent to the western boundary, averaged between y = 200 and 400 km. The bold black line is the zero velocity contour. The solid and dashed red lines are two measures of the vertical scale for the baroclinic flow, as described in the text.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-22-0024.1

• Download Figure
• Download figure as PowerPoint slide

There are differences between the model and the theory. Notably, the theory predicts that the countercurrent projects all the way to the surface in an ever-narrowing region along the western boundary. Its absence in the numerical model is not surprising, however, because the Ekman upwelling is not confined to a delta function in the corner and the model has a finite grid spacing and lateral viscosity that will erode such a narrow flow. The weak deep southward flow expected from the theory is also not apparent, but we find that the deep flow is time dependent as a result of basin modes that are excited by the forcing. They are sufficiently strong to alias the deep flow depending on what time period is chosen for averaging (but the stronger baroclinic flow in the upper ocean is not strongly affected). Friction is sufficiently small that they decay over a longer time scale than the model integration time. Longer time integrations result in instabilities of the western boundary current and further mask the basic current structure. Moreover, it is worth noting that there is a limitation on the applicability of our theory due to the offshore advection of density caused by Ekman transport in the surface layer. Absent a balancing surface heat flux, this will spread dense water offshore and result in convective mixing near the surface (Spall and Schneider 2016). Therefore, we focus on the early time mean vertical structure of the upper-ocean baroclinic flow in the following diagnostics. We note, however, that the basic current structure sets up quickly, on the order of 10 days, so our 120-day integration period is sufficient to capture the baroclinic current development.

c. Vertical scale

For the central parameters used for the model run depicted in Fig. 6, D = 127 m. The exponential decay scale used to diagnose the vertical scale in the previous section was found to be very sensitive and inconsistent when applied to the model output, especially for cases with very weak stratification. Instead, we use two different methods to diagnose the vertical scale from the model fields, one based on transport and one based on perturbation density near the boundary. The transport-based diagnostic is the depth at which the meridional transport in the boundary current is a maximum. This effectively distinguishes the northward flowing upper boundary current from the deeper counter current. The density anomaly metric is the depth at which the density anomaly adjacent to the boundary has dropped to 50% of its maximum value at the surface. Reassuringly, these two measures give very similar results, 135 and 115 m for this case, that are also close to the theoretical prediction. They are indicated on Fig. 6 by the solid and dashed red lines and compare well with the vertical scale of the baroclinic flow.

This is a scaling for the parameter dependence of the vertical decay scale of the baroclinic flow, so the best way to test this prediction is through a series of model calculations in which the governing parameters are varied. The same model configuration was used as for the central case but the values of N², β, and τ₀ were varied in various combinations. The wind stress was varied between 0.001 25 and 0.05 N m⁻², the stratification was varied such that the baroclinic deformation radius ranged between 10 and 200 km, and β was varied between 0.5 × 10⁻¹¹ m⁻¹ s⁻¹ and 2 × 10⁻¹¹ m⁻¹ s⁻¹. These values were chosen to provide a wide range of the primary scaling parameter L_I/L_d, which varied between 0.016 and 2 over 15 different model runs. The vertical length scale diagnosed in the model is compared to the scaling prediction in Fig. 7. The blue squares are for the transport-based diagnostic and the blue diamonds are for the density anomaly diagnostic. In general, the parameter dependence predicted by the theory is supported by the model calculations. The vertical length scale varies between about 60 and 1000 m in the model with an approximately linear dependence on (L_I/L_d)^2/3. At very weak stratification (L_I/L_d ≫ 1, large vertical length scales) the scaling theory overpredicts the vertical scale, but the model diagnostics become more sensitive to the detailed method in this limit. The model supports the prediction that the vertical length scale of the flow depends on the strength of the wind stress, not just the local environmental parameters.

View Full Size

A comparison between the vertical length scale diagnosed from a series of numerical model calculations (H_s) and the vertical scale predicted by the theory (L_I/L_d), where $H^{*}=2000\hspace{0.17em}\text{m}$ is the bottom depth). The squares are derived from a transport-based diagnostic while the diamonds are based on a density anomaly metric (as described in the text). The blue symbols are for positive wind stress and the red symbols are for negative wind stress. The green symbols are for the central case discussed in section 4b. The black asterisks are for uniform horizontal Laplacian viscosity of 20 m² s⁻¹.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-22-0024.1

• Download Figure
• Download figure as PowerPoint slide

The theory provides solutions only for cases in which the interior flow is toward the western boundary. To test the applicability of the scaling theory to downwelling favorable winds, for which the interior flow is eastward, we carried out 9 additional calculations in which the wind stress was varied between −0.001 25 and −0.05 N m⁻² and the baroclinic deformation radius was varied between 10 and 200 km. This produced values of L_I/L_d between 0.016 and 1.0. These model runs produce a similar vertical length scale as the upwelling winds and are also in general agreement with the theory (Fig. 7, red symbols). We attribute the ability of the numerical model to represent boundary layer solutions even with eastward interior flow to the weak but finite viscous dissipation in the model, which is able to damp short Rossby waves before they can propagate energy into the interior (Pedlosky 1965). It is also possible that the short Rossby waves propagate sufficiently slowly that the baroclinic structure remains over the 120-day integration time.