## 1. Introduction

The presence of midocean ridges adds new elements to the character of the oceanic general circulation. In particular, the presence of gaps in the midocean ridges allows for the communication through the ridges of even deep flows between contiguous ocean subbasins. In many cases the islandlike character of the ridge in the interval between the gaps adds complicating dynamical factors to the standard circulation theory. For example, in their discussion of the deep flow in the Angola Basin of the South Atlantic, Warren and Speer (1991, hereafter WS) attempt to apply classical Stommel–Arons theory to the flow but the “island” formed by the intragap ridge segment adds an element of apparent indeterminacy to the calculation.

From a theoretical point of view, the presence of such islands has led to the formulation of theories to deal with the non-simply connected region formed by the isolated ridge segment and to resolve the indeterminacy alluded to above. The work of Godfrey (1989) and Pedlosky et al. (1997, hereafter PPSH) attempts to resolve the ambiguity introduced by the “hole” in the fluid produced by the island by an application of Kelvin's theorem to a contour coincident with the island's boundary. In PPSH the circulation of a barotropic fluid around such islands was studied theoretically and experimentally, and one of the chief consequences of the theory is the prediction of a recirculation region in the linear limit for the region of flow just east of the isolated ridge segment. The recirculation arises since Kelvin's theorem demands that in the steady state the integral of the tangential component of the frictional force must vanish when integrated around the island. Since for steady flows of planetary scale the major currents appear on the *eastern* sides of such islands, the condition reduces to the requirement that the tangent component of the frictional force has zero average when integrated over the meridional length of the segment between the two gaps on the eastern side. For motion governed by linear dynamics this normally implies that the integral of the velocity itself must vanish when integrated over the eastern stretch of the island or ridge segment. The recirculation results if the Sverdrup flow has a zonal velocity of two signs over the meridional extent of the segment; otherwise no recirculation arises, but a single stagnation point appears along the segment so that the flow is of opposite signs on different portions of the segment. These results were verified for the barotropic case by both laboratory and numerical experiments in PPSH. Indeed, this basic constraint forms an element of Godfrey's “Island Rule,” which also holds for the barotropic Sverdrup transport in the linear limit.

However, in their investigation of the flow on the eastern flank of the Mid-Atlantic Ridge, WS found flow in the upper layer of the abyssal region that was unidirectional (see their Fig. 4) in a depth interval for which the ridge was bounded on the north and south by gaps, and indeed they conclude their paper with the remark, referring to the Kelvin constraint, “The frictional argument seems ineluctable, but it also seems to fail. Why?” That question forms a major part of the motivation of the present study.

In the situation studied by WS the flow was baroclinic and the geometry of the ridge was such that only the upper layer, in this region, was islandlike, while the lower layer(s) had, at most, a single entry-port into the Angola Basin. In this paper we will consider the baroclinic flow of a two-layer region that has that geometrical property.

Figure 1 shows the geometry of the model. A rectangular ocean basin is essentially bisected by a barrier representing the ridge. The model is imagined as a model just for the abyssal circulation. In the upper layer of the model the ridge is pierced by two gaps of width *d* in the intervals *y*_{s} < *y* < *y*_{s} + *d* and *y*_{n} − *d* < *y* < *y*_{n}. An assumption of the analysis, which simplifies the calculation, is the realistic one that *d* ≪ *L* where *L* is a characteristic large scale of the basin and the forcing. The flow in the model will be driven by a specified upwelling at the upper boundary and by a specification of the cross-isopycnal mass flux at the interface between the two layers. The formulation of the model is described in detail in section 2. Section 3 deals with the general integral constraints that obtain. There are two. The integral of the tangential component of the momentum equation around the contour bounding the isolated segment of the ridge in the upper layer, a version of Kelvin's theorem, leads to a fundamental constraint on the nature of the flow. It, along with the condition of mass conservation in the lower layer (McWilliams 1977), are sufficient to determine the two important free parameters of the problem, which arise in quasigeostrophic theory. They are 1) the constant value of the upper-layer streamfunction on the island segment, Ψ_{I}, and 2) the constant value of the streamfunction on the outer boundary of the basin in the lower layer, Ψ_{2}. The value of the streamfunction on the outer boundary of the basin in the upper layer can be chosen without loss of generality to be zero. Thus, Ψ_{2} determines the uniform thickness perturbation due to the motion on the outer boundary, while Ψ_{I} represents the geostrophic mass flux though the gaps in the upper layer. It is especially important to note that the boundary values of the streamfunctions in each layer extend to the portions of the ridge in contact with the basin boundary. In section 4 the solution to the linear problem is described. The incompleteness of the gaps in the lower layer mixes the barotropic and baroclinic responses to the forcing, and this has important consequences for the structure of the resulting flow and the nature of the recirculation produced in the eastern sub-basin. Such a mode mixing is also seen in the problem of Rossby wave transmission past barriers of this type (Pedlosky 2000). Section 5 discusses the results of the calculations based on the theory and, in particular, focuses on the role of the baroclinicity in coupling the two layers, altering the structure of the eastern basin recirculation cell and the strong zonal jets produced in the western sub-basin at the latitudes of the upper-layer gaps. Section 6 concludes with an overall summary of the results with some remarks directed to possible observational evidence for the jets.

Although the primary motivation for this study is the aforementioned abyssal circulation near the midocean ridge, the model clearly also applies to the wind-driven circulation around large islands and emphasizes the important role that mixing in regions close to the boundaries may play in affecting the large-scale circulation in the vicinity of the island and its interaction with the large-scale circulation.

## 2. Model formulation

Although the basic model to be employed is the classical two-layer quasigeostrophic model, details of the momentum equation within that implied expansion must be carefully used in the calculation, so it is helpful to briefly review the fundamentals of the model.

*n*= 1 referring to the upper layer and

*n*= 2 for the lower layer. The momentum equations employed in this study are

In (2.1) all variables are dimensionless with the exception of the layer thickness *h*^{*}_{n}*βL,* the horizontal velocities with *U,* and the pressure with *ρ*_{o}*f*_{o}*UL,* where *ρ*_{o} is the mean density of the two layers and *f*_{o} is the value of the Coriolis parameter at *y* = 0 (the southern boundary of the basin). The parameters appearing in (2.1) are *b* = *βL*/*f*_{o}, supposed small for the beta-plane assumption, and Ro = *U*/*f*_{o}*L,* which measures the nonlinearity in the momentum equation. The friction terms are represented by simple drag laws. The second term on the right-hand side of the equation yields a Stommel-like drag law proportional to the layer velocity itself with a drag coefficient *r,* while the final term represents a frictional interaction between the two layers whose coefficient is *r*_{i}. Lateral mixing of momentum could be included with no fundamental change in the results to be discussed. The Coriolis parameter in (2.1) is *f* = 1 + *by* and both the *x* and *y* variables are scaled with *L,* the meridional scale of the basin.

*g*′ is the reduced gravity based on the density difference between the two layers and

*H*

_{n}is the layer thickness of each layer in the absence of motion. The scaled variable

*h*

_{n}describes the variation of the layer thickness due to the motion. The parameters

*F*

_{n}are assumed to be

*O*(1) for the purposes of the expansion.

*h*

_{n}

^{n}

*p*

_{2}

*p*

_{1}

*βLκ*in dimensional units. The functions

*W*∗ and

*w*

_{e}are, in addition, specified forcing terms representing imposed cross-isopycnal velocities at the interface,

*W*∗, and at the upper surface where the velocity is labeled

*w*

_{e}in analogy with Ekman pumping. These vertical velocities are scaled with

*W*

_{o }where the horizontal and vertical scales for the velocity are related by

These imposed velocities represent potential vorticity sources in addition to the thermal and frictional dissipation terms in Eqns. (2.1) and (2.4); *H* is the total mean depth *H*_{1} + *H*_{2}, which is taken to be constant in this study.

*b*leads to the following momentum, mass and vorticity balances, where superscripts denote the order in the

*b*expansion:

*O*(

*b*) are simply

The zero-order velocities are horizontally nondivergent, but the higher-order velocities possess a divergence and enter into the mass budget that must be satisfied for the basin as a whole. I will assume that, except at specified sources and sinks on the boundaries of the basin, both the zero- and first-order velocities have zero normal component at solid boundaries.

On the outer boundary of the basin the streamfunction must be constant in each layer. In the upper layer we may take the value of that constant to be zero without loss of generality. In the lower layer that constant, which must be determined, will be designated Ψ_{2}. On the ridge segment in the upper layer, which is not connected to the outer boundary, the streamfunction is again constant and a priori unknown, and is called Ψ_{I}.

## 3. Integral constraints

*O*(1) and

*O*(

*b*) are assumed to vanish on the ridge, we obtain

*C*

_{I}is the contour that girdles the ridge segment and the integration proceeds in a counterclockwise path around the segment. Note that the Coriolis and nonlinear advection terms identically vanish as long as the velocity has no normal component to the segment. In the steady state the left-hand side of (3.1) is identically zero and the circulation condition involves only the integral of the velocity and, in the presence of frictional coupling between the layers, the integral of the shear between the two layers. When that coupling is zero, (3.1), in the steady state, reduces to the condition that the average of the tangential component of velocity, when integrated around the segment, must be zero. If the major currents are on the eastern side of the island, this leads to the condition that the average meridional velocity on the eastern side of the segment must be zero and at least one stagnation point on the eastern side of the ridge must exist. When the frictional coupling is nonzero, this is no longer true. In fact, in the limit when

*r*

_{i}/

*r*≫ 1 the steady condition simply reduces to the statement that the average value of the velocities must be the same in each layer. Since the lower layer's portion of the ridge is attached to the boundary, there is no constraint similar to (3.1) for the lower layer and the lower-layer meridional velocity need not have a zero average, which seems to imply that, at least in the case

*r*

_{i}/

*r*≫1, the upper-layer velocity could be unidirectional along the ridge. What is not so obvious, and what will be seen below, is that this can also occur when

*r*

_{i}/

*r*= 0 with sufficient baroclinic coupling.

*O*(1) flow, but the mass balance depends on the integral of the

*O*(

*b*) mass conservation condition expressed by (2.9). When (2.9) is integrated over the area of the total basin the condition of mass conservation leads to the constraint

*C*

_{B}is the contour encircling the boundary of the lower layer in the basin. The number

*S*

_{2}is the net input of mass into the second layer through the lateral boundary by the

*O*(

*b*) motion field, that is,

**n**is the outward normal to the boundary. In the steady state, since no temporary storage is possible, the condition (3.2) expresses the condition that the net

*O*(

*b*) flow entering through the lateral boundaries must exit through the interface into the upper layer. Note that there can be no

*net*quasigeostrophic flow entering the lower layer due to the condition that

*ψ*

_{2}be single valued.

_{I}and Ψ

_{2}. Godfrey (1989) showed how, under the conditions that certain simplifications obtain, the constant Ψ

_{I}could be determined directly from the forcing fields. Without repeating essentially the same derivation found in his paper it is straightforward to show that, starting from the

*O*(

*b*) momentum and mass equations in the upper layer, one can derive

In (3.6) the contour *C*_{w} runs (see Fig. 2) from the eastern boundary along a latitude circle at the northern tip of the segment, south along the *western side* of the segment, to the eastern boundary along the latitude line coincident with the southern tip of the segment, and then northward along the eastern boundary to close on itself. The area *A*_{e }is the area to the *east* of the segment enclosed by the contour and **n** is the outward normal to the contour. When dissipation and nonlinearity can be ignored everywhere except on the eastern side of the segment (where strong “western” boundary currents are expected), all the terms in (3.6) can be ignored except the first terms on the right-hand side and a simple expression for the streamfunction on the ridge segment. In the linear limit the determination is equivalent to what is obtained from (3.1). Of course, it is only after the solution is in hand that one can be sure that such terms can be neglected, and we proceed to examine the solution in detail in the next section. We shall find that, when the baroclinicity is important, several of these conditions, in particular, ignoring the velocity on the western side of the ridge in either (3.1) or (3.6) are no longer valid.

## 4. Linear problem

*δ*

^{2}

_{I}

*ĥ*

_{1}=

*ĥ*

_{2}= ½. Then (4.2) becomes

For the steady-state problem the time derivative is, of course, zero but it is retained in (4.3) and (4.4) to emphasize the similar roles of time dependence and dissipation in the model.

These conditions hold also on the portion of the ridge connected to the outer boundary.

_{I}, while on the portions of the ridge connected to the basin boundary the streamfunction is zero. As in Pedlosky and Spall (1999), we assume that in the narrow interval between the ridge and its segment connected to the outer boundary the streamfunction varies linearly from zero to Ψ

_{I}. Pedlosky and Spall checked this assumption by direct numerical integration and found it valid when, as in the present case,

*d*≪ 1. Thus on the longitude of the ridge at

*x*=

*x*

_{t}, it follows that

This has important implications for the flow in the lower layer. A similar approximation implies that there is no *O*(1) quasigeostrophic flow through the single gap in the lower layer, since the streamfunction on all portions of the ridge in the lower layer, and, thus on each side of the gap, is equal to Ψ_{2}. Nevertheless, it will be possible to deduce the net *O*(*b*) flow through that gap as shown below. That condition and (4.6) can easily be written in terms of the barotropic and baroclinic components to establish the appropriate boundary condition on the ridge segment, a step which is deleted for the sake of brevity.

Before describing the method of solution in detail it is useful to review the character of the boundary layer structures that obtain from (4.3) and (4.4) in the limit of small dissipation. Boundary layer theory, per se, will not be used in the calculation, but its results are most easily interpreted with the asymptotic structure of the boundary layers kept in mind. For the linear problem defined by (4.3) and (4.4) it is straightforward to deduce the boundary layer balances. They differ for the barotropic and baroclinic modes and differ for boundary layers on the meridional and zonal boundaries.

For the barotropic mode of motion, the boundary layers on the meridional boundaries will have the Stommel length scaled *δ*_{s} for all values of the stratification. On zonal boundaries or internal boundary layers that coincide with a latitude circle the length scale is (*δ*_{s}*L*_{x})^{1/2}, where *L*_{x} is a characteristic dimensionless distance in the *x* direction. The governing equation for this layer is the diffusion equation in which -*x* plays the role of the time variable and the solution parabolically spreads in *y* with increasing distance westward regardless of the direction of the flow as discussed in detail in PPSH. This will determine the structure of the zonal jets, which are tied to the gaps in the ridge and form the means of communication between the adjacent sub-basins.

_{T}

*κF*

^{−1}

In dimensional units this scale is just the length a long, baroclinic Rossby wave can propagate westward before thermal damping dissipates it.

For large values of *δ*_{T}, that is, for small values of *κ* or *F,* the boundary layer structure for the baroclinic mode is as follows. On meridional boundaries there is a western boundary layer scale, which is again the Stommel scale *δ*_{s}. In the vicinity of the eastern boundary there will be a zone in which the decaying Rossby wave is manifest and this will introduce a scale *δ*_{T} to the baroclinic zone near the eastern boundary. This region may, indeed, be very broad and it is sometimes included in the interior baroclinic solution. However, as *κF* increases, the eastern scale diminishes. When *δ*_{T} becomes as small as the Stommel scale, the boundary layer balances change in (4.4) and the baroclinic layers on both the eastern and western boundaries become equal and have the scale (*δ*_{s}*δ*_{T})^{1/2}. In this limit the boundary layer scales are similar to linear coastal upwelling layers on an *f* plane; that is, *β* plays no role and there is symmetry between east and west. On zonal boundaries the scale for the baroclinic solution is (*δ*_{s}*L*_{x})^{1/2} for large *δ*_{T} and the governing equation is the same diffusion equation as in the barotropic case. However, when *δ*_{T} becomes smaller than the basin scale, that is, *δ*_{T }< *L,* the boundary layer equation becomes identical to the balance obtained for the meridional scale for large F, that is, (*δ*_{s}*δ*_{T})^{1/2}. Thus, when *κF* is large enough the baroclinic boundary layer on *all* boundaries has this scale and there is a fundamental symmetry between east and west that is lacking in the barotropic mode. In this limit the governing boundary layer equation occurs as a balance of the terms in the second bracket on the left-hand side of (4.4), that is, a balance between frictional and thermal dissipation. Figure 3 shows the morphology of the boundary layer scales as a function of the baroclinic scale *δ*_{T} and shows the splitting of the boundary layer scales as the critical values *δ*_{s} and *L* are obtained. The baroclinic boundary layers may not, in the large *F* limit, carry much transport but they are important contributors to the dissipation integral in the circulation condition (3.1). The fact of the east–west symmetry in the large *F* limit (when the deformation radius is small with respect to the basin scale) will be seen to imply that the dissipation on the western side of the ridge can no longer be ignored for *O*(1) *κ,* and this vitiates the validity of the island rule.

As remarked above, it is not necessary to use the approximations of boundary layer theory to solve the linear problem. Indeed, since we will be interested in the change in structure as a function of increasing stratification, it is more efficient to have a solution valid for a wide range of parameters.

Although we are interested in the response to steady forcing, it is just as easy to consider the solution as a response to a periodic forcing at frequency *ω*_{o} although we consider in this paper only the solutions for that forcing frequency equal to zero.

*y*= 0 or

*y*= 1. That means neither the barotropic nor, more significantly, the baroclinic streamfunction is zero at the end points of the

*y*interval. To improve the convergence of the solution it is useful to write the solution in the form

*x,*representing a broad, basinwide upwelling and a part that is localized, in the present case, to a narrow region in the eastern sub-basin. That is,

*z*is the position of a narrow upwelling.

_{I}on the ridge segment and Ψ

_{2}on the outer boundary of the lower layer. Those constants are determined by the application of (3.1), the circulation constraint, and (3.2), the mass conservation condition. Since

*c*

_{1}

_{I}

*c*

_{2}

_{2}

*X*

_{c}

*c*

_{1}and

*c*

_{2}, as well as the constant

*X*

_{c}, are given in appendix B.

The application of the mass conservation constraint in the linear limit also gives rise to a linear equation in the streamfunction constants. It is useful, however, to carry out the integrals indicated in (3.2) separately for the two adjacent sub-basins. Since the two sub-basins are joined by the single gap in the lower layer, through which an *O*(*b*) flux can take place, the constraint must be applied to the sum of the two basins. However, the individual budgets will allow us to calculate the flux at higher order from one sub-basin to the next.

For simplicity, I will assume that the net *O*(*b*) mass flux into the lower layer from the lateral boundary, the term denoted by *S*_{2} in (3.2) is exactly balanced by the integral of the *imposed* cross-isopycnal flux W∗. This eliminates a mass source forcing term to the circulation and requires that the auto-induced cross-isopycnal mass flux due to the thermal dissipation integrate to zero over the whole lower layer. It is easy to relax this condition, but it would add a somewhat arbitrary forcing function to the problem.

_{I}

_{2}

*Y*

_{c}

In (4.17) the flux through the gap is given in terms of the mass flux through the interface in the eastern basin (the first and third terms in the equation) minus the flux into the *eastern* basin through its lateral boundary. The difference is the *O*(*b*) mass flux from the western to eastern basin in the lower layer. Although we have applied the condition that the total mass flux through the outer boundary match the imposed cross-isopycnal upwelling, no such constraint is required for the eastern basin alone. So, depending on our specification of the local boundary mass fluxes, the flow through the gap is determined only up to that specification. I shall not discuss this in more detail since it leaves a fairly arbitrary specification of this flux possible, but it is important to note that, in general, this mass flux is different from zero even though the quasigeostrophic mass flux as determined by the streamfunction is zero in this model.

## 5. Results

*κF*and small

*r*

_{i}/

*r.*Figure 4 shows the streamline pattern for the case where the forcing consists of

*w*

_{e}

*πy*

*δ*

_{s}has been chosen as 0.01 and

*κF*= 10

^{−6}and

*r*

_{i}/

*r*= 0. This makes the interface effectively rigid and slippery, so the problem reduces to that of the single-layer model studied in PPSH. We see a circulation consistent with the barotropic model of that earlier study. The circulation east of the ridge possesses a zone of trapped fluid traced by the dashed contour, and the boundary layer flow has two stagnation points at approximately

*y*= 0.35 and 0.65, well short of the ends of the segment, forming the extremities of the recirculation region. The solution has been calculated directly from the complete series solution, and satisfaction of the circulation condition has been exact without assuming that only the boundary layer on the eastern side of the ridge dominates the dissipation in the solution. Nevertheless, in this case, that would be an apt description of the dynamics, again in agreement with the development described in PPSH and in agreement with the consequences of Godfrey's (1989) “Island Rule.” The circulation in the lower layer is essentially zero in this case and is not shown.

The situation becomes more interesting when the coupling between the layers is strong. Consider the case first when the interface is slippery, that is, when the interfacial friction coefficient, *r*_{I} = 0 but when the baroclinic coupling is strong. Figure 5 shows the circulation pattern for the case when *κF* = 300 and *δ*_{s} is 0.01 so that *δ*_{T} is 0.003, and we are in the range where the baroclinic boundary layers have east–west symmetry (the left-hand edge of the schematic in Fig. 3).

In Fig. 5a we see the overall pattern of circulation in the upper layer and it is immediately obvious that the recirculation region, again shown by the dashed contour, has grown such that the flow along the ridge segment is all in one direction (which direction depends on the sign of the forcing; in the present case the flow is southward). Yet, with the interfacial friction equal to zero the constraint on the flow around the ridge segment given by (3.1) is simply that the total circulation around the segment between the gaps must be zero. If it were true that the only significant velocity (and dissipation) occurred on the eastern side of the island, this would appear to be inconsistent with this result and is similar to the conundrum posed by WS as described in the introduction.

Figure 5b shows a more detailed picture of the flow in the vicinity of the segment in the upper layer. The major boundary layer, as far as transport is concerned, is certainly on the eastern side in agreement with our expectations and in concordance with the requirements for the validity of the island rule. However, careful examination shows a contour encircling the western side of the segment. There is relatively little transport involved but the meridional velocity is not negligible. The contributor to this velocity on the western side of the ridge is given by the baroclinic portion of the flow shown in Fig. 5c. There is essential east–west symmetry for the baroclinic component whose characteristic length scale is (*δ*_{s}*δ*_{T})^{1/2} ∼ 0.005. The contribution that it makes to the velocity (and hence the dissipation) on the two sides of the ridge can be seen clearly in Fig. 5d, where the meridional velocity as a function of *y* along the ridge is shown. The upper panel shows *υ* on the eastern side of the ridge as a solid line (note it is everywhere negative), while the dashed curve yields the meridional velocity on the western side of the ridge, also negative for all *y.* Their integrals over the ridge segment length balance as imposed by the constraint condition (4.15). The lower panel in the same figure shows the relative contributions of the baroclinic (dashed) and barotropic (solid) meridional velocities on the western side of the ridge and, as anticipated, it is the baroclinic contribution that produces the balancing western flow. To balance that flow a strong barotropic southward component is required on the eastern side of the ridge, which produces the flow pattern seen in Fig. 5a.

The flow in the lower layer is shown in Fig. 5e where contours of *ψ*_{2} − Ψ_{2} are shown in the vicinity of the ridge. It is the unidirectional flow along the ridge in the lower layer, which does not directly feel the circulation constraint, that is then communicated vertically to the upper layer. There is, in fact, no flow through the gap at lowest order [but see (4.17)] as can be verified by calculating the zonal velocity to the immediate west of the ridge (see below). In the figure only the streamline corresponding to *ψ*_{2} = Ψ_{2} threads through the gap.

Figure 5f shows the profile of the zonal velocity at a position, *x* = 0.25, to the west of the ridge at *x* = 0.387. Note the sharp peaks in the profiles at the latitude of the gaps in the ridge. To emphasize this the ridge and its gaps are depicted at an arbitrary location on the right of the figure. The jets are very nearly barotropic, and it is important to note that, although the flow is toward the ridge in the south, the width of the jets decreases eastward in each jet as can be seen in Fig. 5a. To verify that there is no flow through the gaps in the lower layer at the ridge, Fig. 5g shows the same zonal velocity profiles at a position just west of the ridge. We note that the zonal velocity in the lower layer, seen in the lower panel, is now nearly zero.

*y*with a peak forcing concentrated just east of the ridge. Figure 6a shows the circulation in the upper layer, which again is unidirectional along the ridge, a result that appears robust with this strong interlayer coupling regardless of the zonal structure of the forcing. Figure 6b shows the jets to the west of the ridge at the same position as in Fig. 5f. The structure is nearly indistinguishable, although there is a difference in amplitude. Again, the structure seems sculpted by the interaction with the ridge rather than reflecting the details of the forcing.

If the coupling between the two layers is due to friction rather than to thermal damping, a similar result, that is, the expansion of the zone of recirculation, can take place although the mechanism is different. Figure 7 shows the circulation pattern in the case where the thermal damping coefficient *κ* is very small (0.005) but the ratio *r*_{i}/*r* = 10. The forcing is as in (5.1). In Fig. 7a the circulation pattern of the upper layer is shown and, again, the flow along the ridge segment is unidirectional. Figure 7b shows the streamline pattern in the lower layer, and again we note that the flow is unidirectional along the peninsula connected to the northern boundary of the basin in the interval of the upper layer's ridge segment. Figure 7c shows the meridional velocity on each side of the ridge. It is clear that the circulation constraint is not achieved by balancing the integrals of *υ* on each side of the ridge. Instead, it is the coupling term between the layers that dominates in (3.1), allowing a unidirectional flow along the ridge.

*w*

_{e}is zero and

*W*

*πy*

*r*

_{I}= 0,

*κF*= 300,

*δ*

_{s}= 0.01, as in Fig. 5. The direction of the upper-layer circulation is reversed since the forcing now leads to vortex compression in the upper layer. Figure 8b shows the circulation in the lower layer. Because the forcing is baroclinic, the major baroclinic response tends to be limited to narrow boundary layers [of width (

*δ*

_{s}

*δ*

_{T})

^{1/2}] as seen in Fig. 8c. There is, however, a strong barotropic response induced by the interaction with the topography, and this barotropic component is shown in Fig. 8d. If that figure is compared with Fig. 8a, it is clear that in the western basin, and the region just to the east of the ridge segment, the circulation is barotropic and topographically generated. This is also true of the zonal jets shown in Fig. 8e, which are clearly essentially barotropic in spite of the purely baroclinic forcing. This part of the solution is due almost entirely to the first term in Eq. (A.1a), that is, the part generated by the island constant Ψ

_{I}as a consequence of the Kelvin integral constraint (3.1). In Fig. 8f the meridional velocities on each side of the ridge segment are again shown and, again with no interfacial friction, the constraint is satisfied by the baroclinic production of meridional flow in the upper layer in the thin boundary layer to the west of the ridge.

Thus, the barotropic circulation in this parameter range, *δ*_{T} ≤ *δ*_{S}, seems largely independent of the structure of the forcing. It is the interaction of the forcing and the topography, exciting a barotropic response which reaches far to the west of the ridge in the form of zonal jets, that is a strongly robust feature of the physics.

## 6. Summary and conclusions

In this paper we have examined the response of a quasigeostrophic, two-layer model of the steady circulation of a steady flow, driven by upwelling at the upper boundary and the interface of the two layers as a model for the circulation of the abyss in the presence of a midocean ridge. The ridge is represented by a meridional barrier pierced by two gaps. One gap extends through both layers, while the other gap is limited to the upper layer alone. The geometry of the ridge and its gaps mixes the barotropic and baroclinic responses to the forcing, and this has a fundamental effect on the resulting circulation.

The isolated segment of the model ridge in the upper layer adds an islandlike feature to the circulation, and for the quasigeostrophic model this is manifested by the need to satisfy a version of the Kelvin circulation theorem on a circuit girdling the island. When the baroclinic coupling between the two layers is strong enough, the recirculation region in the upper layer grows in size beyond what its extent would be for a barotropic fluid to such an extent that the flow along the eastern flank of the ridge becomes unidirectional. This single direction of flow, which had been observed by Warren and Speer (1991), can be traced back to the important role of the baroclinic response of the fluid to the upwelling forcing. The baroclinic response for small deformation radius and *O*(1) thermal damping (in the sense that a baroclinic Rossby wave would damp on the timescale of its period) is limited to narrow zones near all solid boundaries, both east and west of the ridge. Although the major currents, in terms of transport, are east of the ridge due to the dynamical asymmetry associated with the *β* effect, there is sufficient velocity in the narrow baroclinic boundary layer to the west of the ridge to balance the dissipation of the unidirectional barotropic velocity east of the ridge to satisfy the Kelvin constraint (3.1).

Very strong interfacial frictional coupling provides an alternative mechanism to allow the growth of the eastern zone of recirculation near the isolated ridge segment, producing unidirectional flow along the ridge. In this case there is no longer the constraint that the meridional average of *υ* be equal on each side of the ridge. Indeed, in the limit of very large frictional coupling the requirement of (3.1) is simply that the upper-layer meridional velocity have a *y* average equal to that of the lower layer in the latitude band of the ridge segment. The fact that the lower-layer ridge does not form an isolated segment easily allows unidirectional flow in the lower layer, and the frictional coupling projects that flow upward into the upper layer.

Of course, this model is highly simplified. Both the coupling mechanisms, the thermal damping and the interfacial frictional coupling, are poor representations of rather more complicated turbulent processes of mixing of momentum and density. The nature of the interfacial friction in the model, serving as it does to smooth out the vertical shear of the mean flow, might be thought of as a crude representation of the reduction in vertical shear due to baroclinic instability not resolved by the calculation. It is important to be able to think of the coupling process that way rather than the simple effect of Ekman layers at the interface between the two layers since in that case the ratio *r*_{i}/*r* is fixed at a value of 0.5 while a much stronger value of the order (10) is required to produce unidirectional flow along the ridge. If *r* is thought of as a measure of the in-layer dissipation of momentum due to lateral shear instabilities and *r*_{I} is a measure of baroclinic instability, the ratio may be much larger than one, at least locally.

It is hard to argue for a particular value of *κ,* and in this paper I have chosen a value in order that the major baroclinic response is limited to narrow boundary layers around the basin and on the ridge so that the baroclinic response is not much broader than a deformation radius. This, and the timescale for wave decay that it implies, is an arbitrary choice at this point, but a smaller value of *κ* can be offset by a larger value of *F* in the theory.

One of the more robust results of the theory is the prediction of essentially barotropic jets in the region west of the forcing. The barotropic nature of the jet response is largely independent of the structure of the forcing, that is, whether that forcing is barotropic or baroclinic or some combination. It is the interaction of the flow with the ridge and its gaps that excites the jets. That has been seen earlier in barotropic models (e.g., PPSH) but it appears to be a robust response in a baroclinic model as well. The width of the zonal flow depends strongly on the degree of friction in the model. For the simple frictional parameterization of this model the width of the zonal jets is (*δ*_{s}[*x*_{T} − *x*])^{1/2}, where *δ*_{s} is Stommel's boundary layer thickness and *x*_{T} − *x* is the distance west of the ridge placed at *x*_{T}. Other parameterizations of mixing, for example, lateral mixing, would give different rules for the spreading of the widths of the zonal flows as a function of distance to the west. Indeed, if the momentum mixing is strong enough, the widths of the regions of zonal flow may be so broad as to bring into question the adjectival “jet.” For example, the zonal flows in Fig. 5f already extend over almost half the width of the basin well west of the jet and squeeze down to the widths of the gaps only quite near the ridge, as seen in Fig. 5g. Hogg and Owens (1999) in their analysis of the deep circulation within the Brazil Basin point out, from float data, a strikingly stable zonal flow at about 22°S just at the latitude of the Rio de Janeiro fracture zone in the Mid-Atlantic Ridge. The data are too sparse to decide whether the current is indeed a flow through the gap. It is possible it may join, east of the ridge, the Namib Col current that Warren and Speer (1991) (see also Speer et al. 1995) have associated with a similar gap in the Walvis Ridge even farther eastward. At this point I can only argue that it is suggestive that the zonal flow lines up with the gap in a manner consistent with the theory presented above.

From a fluid dynamical point of view the major result is that baroclinicity, while of course not vitiating the validity of Kelvin's circulation theorem, makes its application more complex in the steady state. That the narrow baroclinic boundary currents introduce important contributions to the resulting dissipation integral due to the flow to the west of the ridge segment means that the assumptions required for the validity of Godfrey's Island Rule will no longer be satisfied. This emphasizes the importance of the original constraint, which is the circulation integral of the momentum equation around the ridge segment itself.

## Acknowledgments

This research was supported in part by National Science Foundation Grant OCE 9901654. The author is grateful to Breck Owens and Nelson Hogg for helpful guidance concerning the Brazil Basin experiment and to Bruce Warren for stubbornly insisting that the observed flow along the ridge in the Angola Basin required serious theoretical consideration.

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,*J. Phys. Oceanogr***17****,**2294–2317.McWilliams, J. C., 1977: A note on a consistent quasi-geostrophic model in a multiply connected domain.

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,*J. Phys. Oceanogr***29****,**2332–2349.Pedlosky, J., L. J. Pratt, M. A. Spall, and K. R. Helfrich, 1997: Circulation around islands and ridges.

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## APPENDIX A

### Fourier Coefficients for (4.12a,b)

The Fourier coefficients for the solution as a sine series given by (4.12a,b) can be shown to be

*x*

_{e}and

*x*

_{w}are the eastern and western coordinates of the meridional boundaries of the basin. The constant

*a*

_{n}is defined as

In (A.1a) the function Θ(*ξ*) is the Heavyside step function, which is unity for positive values of its argument and zero when its argument is negative.

## APPENDIX B

^{*}

Woods Hole Contribution Number 10371.