on the separation of a barotropic western boundary current

On the Separation of a Barotropic Western Boundary Current from a Cape

DAVID R. MUNDAY AND DAVID P. MARSHALL

Department of Meteorology, University of Reading, Reading, United Kingdom

(Manuscript received 4 August 2003, in final form 14 March 2005)

ABSTRACT

The problem of western boundary current separation is investigated using a barotropic vorticity model.Specifically, a boundary current flowing poleward along a boundary containing a cape is considered. Themeridional gradient of the Coriolis parameter (the � effect), the strength of dissipation, and the geometryof the cape are varied. It is found that 1) all instances of flow separation are coincident with the presenceof a flow deceleration, 2) an increase in the strength of the � effect is able to suppress flow separation, and3) increasing coastline curvature can overcome the suppressive � effect and induce separation. These resultsare supported by integrated vorticity budgets, which attribute the acceleration of the boundary current tothe � effect and changes in flow curvature. The transition to unsteady final model states is found to haveno effect upon the qualitative nature of these conclusions.

1. Introduction

The dynamics controlling the separation, or contin-ued attachment, of western boundary currents remainsa poorly understood aspect of the large-scale ocean cir-culation. The problem is not aided by the fact that nu-merous hypotheses have been put forward, but no con-clusive explanation has been reached (Haidvogel et al.1992). These hypotheses include vanishing wind stresscurl (Munk 1950), outcropping of the main thermocline(Parsons 1969; Veronis 1973), interactions with thenorthern recirculation gyre (Hogg and Stommel 1985;Ezer and Mellor 1992), “vorticity crisis” (Cessi et al.1990; Kiss 2002), interactions with the deep westernboundary current (Thompson and Schmitz 1989; Agraand Nof 1993; Spall 1996a,b; Tansley and Marshall2000), and coastline/shelf geometry (Ou and de Ruijter1986; Stern and Whitehead 1990; Spitz and Nof 1991;Dengg 1993; Özgökmen et al. 1997; Stern 1998; Tansleyand Marshall 2001). Excellent reviews of some of thesehypotheses are given by both Haidvogel et al. (1992)and Dengg et al. (1996).

A classical paradigm of flow separation is nonrotat-ing flow past a cylinder. Batchelor (1969) interprets the

process of separation as occurring when the advectionof vorticity to a point is greater than the diffusion awayfrom it. In addition, he states that it is “an empirical factthat a steady state of the boundary layer adjoining asolid boundary cannot remain attached with an appre-ciable fall in the velocity of the external stream.” This isthe “classical” theory that flow along an adverse pres-sure gradient cannot remain attached and be at steadystate because of the formation of a pressure singularitywithin the boundary layer (Goldstein 1948).

It should be noted that “flow deceleration” and “ad-verse pressure gradient” are often used interchange-ably, although a strict equivalence is only true in theinviscid limit. However, the fundamental ingredient forseparation is a finite deceleration, rather than an ad-verse pressure gradient (e.g., see Smith 1982). Hence, inthis paper we refer to flow decelerations rather thanadverse pressure gradients.

The nonrotating flow past a cylinder is controlled bya single parameter, the Reynolds number, given by

Re �UL

�, �1�

where U is the velocity scale, L is the length scale, and� is the fluid viscosity. As the Reynolds number is in-creased the flow loses its upstream/downstream sym-metry and separation generally occurs for Re � 10. Atthe values one might expect to find in the oceanicboundary currents the flow is well into a fully turbulent

Corresponding author address: Dr. David R. Munday, Depart-ment of Meteorology, Earley Gate, University of Reading, Read-ing, Berkshire RG6 6BB, United Kingdom.E-mail: [email protected]

1726 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 35

© 2005 American Meteorological Society

JPO2783

Unauthenticated | Downloaded 04/19/22 02:56 AM UTC

regime, with typical Reynolds numbers of 1010 basedupon a length scale of 100 km, a velocity of 0.1 m s�1,and a molecular viscosity of 10�6 m2 s�2. This is inmarked contrast to the Reynolds numbers obtained inmany numerical ocean models used for climate predic-tion, in which assuming the values as above, except aviscosity of 104 m2 s�2, gives a Reynolds number of 1.

It is our hypothesis that the presence of a finite flowdeceleration may explain the high Reynolds numberseparation of western boundary currents, such as theGulf Stream and Kuroshio. Indeed, modeling studies atrelatively high resolution have already found the sepa-ration of a western boundary current to be coincidentwith the presence of an adverse pressure gradient(Haidvogel et al. 1992; Jiang et al. 1995; Baines andHughes 1996; Tansley and Marshall 2000).1 If this hy-pothesis is accepted, then the determination of the con-ditions under which a boundary current will separate,or remain attached, reduces to attributing the presence,or otherwise, of a flow deceleration to some plausiblephysical mechanism. The focus here is upon only one ofthe possibilities reviewed by Dengg et al. (1996)—thata change in direction of the coastline can decelerate aboundary current and induce its flow separation.

Although some studies, such as Stern and Whitehead(1990), have previously emphasized the role of bound-ary curvature in flow separation for rotating systems,the inclusion of differential rotation can have a pro-found effect upon a system’s behavior (see, e.g., Boyer1970; Merkine 1980; Boyer and Davies 1982; Page andJohnson 1990). Indeed, Stern and Whitehead explicitlystate that rotation does not enter into their theory.Rather, it acts to suppress three-dimensional turbu-lence, in the experimental section of their study, to givea two-dimensional flow.

Bryan (1963) makes the earliest, to the authors’knowledge, attempt at using an abrupt change in coast-line orientation to induce separation of a barotropicwestern boundary current. Sadly this attempt fails, be-cause of the south-facing wall of Bryan’s “barrier” hav-ing a free-slip boundary condition, rather than a no-slipcondition, applied to it [as Dengg (1993) reports]. How-ever, both Dengg and Özgökmen et al. (1997) showthat, when coupled with sufficient inertia, an abruptchange in direction of the coastline can “fix” the sepa-ration point of both barotropic and baroclinic boundarycurrents. They also show that proximity to the line of

zero wind stress curl can have implications for the sepa-ration state in a wind-driven gyre.

By area integrating the governing equations, Mar-shall and Tansley (2001) demonstrate that the tendencyfor a current to accelerate or decelerate (and henceseparate) can be described by the balance of forcesacting upon the flow. Most importantly, by using natu-ral coordinates this balance of forces is made to includethe flow curvature. Their results suggest that the � ef-fect will always accelerate a western boundary current,and thus tend to keep it attached. In contrast, flowcurvature can either accelerate or decelerate. They sug-gest that for a boundary current to separate, the radiusof curvature should satisfy the following inequality:

R � � U

�*�1�2

, �2�

where R is the radius of curvature of the coastline and�* is the gradient of the Coriolis parameter in the di-rection of the flow. This inequality serves as the basis ofour investigation into western boundary current sepa-ration, by supplying two complementary hypotheses.

1) A sufficient increase in the strength of the north–south gradient of the Coriolis parameter, a stronger� parameter in the parlance of Tansley and Marshall(2001), can suppress flow separation.

2) Regardless of the strength of the � effect, the coast-line can always induce separation by having suffi-cient curvature.

It is these twin hypotheses that we aim to test throughthe use of a barotropic vorticity model of a westernboundary current by placing an obstacle on the westernboundary. Although such a model is somewhat ab-stract, the problem remains simple enough that it ispossible to span the parameter space (within the con-fines imposed by numerical considerations). We believethat our broad findings should have wider relevance tomore realistic boundary currents with vertical structureand arbitrary coastlines, but such complications are notconsidered here.

In section 2, the numerics and setup of the chosenmodel are discussed, as are some of the issues surround-ing the choices made. Sections 3 and 4 investigatemodel solutions when an obstacle is added to the west-ern boundary at a variety of model parameters. Onlylong-term time-average or steady-state solutions areconsidered. An area-integrated vorticity budget is in-troduced, and derived, in section 5. The budget is thenused to interpret the results of the previous two sec-tions. In section 6, we close with a summary of ourresults and conclusions.

1 At low Reynolds number, colliding western boundary currentsare found to separate before the adverse pressure gradient isreached (Cessi 1991). However, Kiss (2002) disputes the interpre-tation of Cessi and argues that the observed flow separation isoccurring in a region of adverse pressure gradient.

OCTOBER 2005 M U N D A Y A N D M A R S H A L L 1727


2. Model numerics and domain

a. Model equations and nondimensional parameters

A barotropic vorticity model is chosen to investigatethe separation of western boundary currents by coast-line curvature. This type of model eliminates the com-plication of bottom topography and vortex stretching,as well as any other baroclinic effects. Any influencethat direct wind forcing may have is removed by usinga north–south aligned channel driven by “pumping”fluid in the southern end and out at the northern end(see section 2b for details of the model domain). We donot seek to directly simulate the Gulf Stream, or otherspecific boundary currents, but rather to investigate themore general problem of curvature forcing in westernboundary current separation.

For this system, the Coriolis parameter is written as f� f0 � �0y, where f0 and �0 are constants and y is thenorth–south distance from the latitude at which f � f0.The dimensional form of the barotropic vorticity equa-tion is then given by

��

�t� u · �� 0� � ��2�, �3�

where u is the fluid velocity, is the vertical componentof the relative vorticity, and � is the fluid viscosity. Thechannel width is taken as the system’s characteristiclength scale L, and a velocity scale is defined basedupon the net northward transport such that UL ��,where �� is the difference in streamfunction betweenthe east and west boundaries. The velocity scale U rep-resents the average velocity through the channel suchthat UL gives the net northward transport. Equation(3) is then nondimensionalized using the following re-lations: (x, y) � (x, y)/L, u � uL/��, � L/U,and t � (�0L)t.

The result of the nondimensionalization is given in(4) as

��

�t� u · ��

1Re

�2�. �4�

Here Re is the Reynolds number, given by (1), and � isthe � parameter of Tansley and Marshall (2001), givenby

� ��0L2

U� �L

I�2

, �5�

where I � (U/�0)1/2 is the width of an inertial boundarycurrent, with a higher � implying a lower inertial scale(Charney 1955).

Tansley and Marshall (2001) describe the � param-eter as quantifying the degree to which meridional dis-

placement is confined and also draw relevance to oceangyre theory. Moreover, 1/� is equivalent to the Rossbynumber used in the simulation of the barotropic wind-driven ocean circulation (Bryan 1963; Veronis 1966;Blandford 1971; Böning 1986). In this interpretation, �is seen as a measure of nonlinearity, with a higher �implying that nonlinear advection is less important tothe system’s evolution.

Note that the scales used to define Re and � are notthe same as in the flow past cylinder calculations ofTansley and Marshall (2001), who define L as the cyl-inder diameter. This means that solutions obtained atthe same parameter values are not directly comparable.However, it is still reasonable to expect that increasingRe will increase the tendency of the system to separate,while increasing � will have the opposite effect.

In ocean gyre theory, boundary layer scales are usu-ally defined based upon inertia and different types offriction (typically bottom friction and lateral friction).The largest scale is then the one that controls the west-ern boundary current.

The model results show that, particularly at low Re,systems with a higher value of � tend to produce asteady state. Essentially, this is because I → 0 as � →�, and at some point the boundary current must be-come defined by the dissipative scale [ M � (�/�0)1/3].

For this reason, and also because of the selected in-flow condition [see section 2b and Eq. (8)], it is moreinstructive to divide (4) through by � and write Mu �1/�Re. This gives, with tildes dropped,

��

�t�

1

�u · �� Mu�2�. �6�

In this case, Mu is referred to as the “Munk number,”since

Mu � �M

L �3

, �7�

where M is the width of a frictional boundary layer.One can interpret Mu as a nondimensional viscosity,which is appropriate for the flows presented here.

Using constant values of Mu to compare experimentsperformed at different � is more instructive becausevarying � then only changes one of the boundary layerscales. Throughout, flow visualizations will be shownwith all three parameter values (�, Mu, and Re) quoted,although only two are independent.

b. Experimental design

Equation (6) is integrated through time over a rect-angular domain with a sponge region at either end inwhich the flow is relaxed back toward a prescribed in-



flow/outflow region (the domain is illustrated in theschematic of Fig. 1). Although the obstacle appears tobe in fairly close proximity to the sponges, experimentswith a much longer channel showed no qualitative dif-ference to the results presented in the subsequent sec-tions.

The chosen inflow/outflow profile has the functionalform given by Pedlosky (1996) for the boundary layerpart of the linear problem first solved by Munk (1950);that is,

�x� � �I�1 � e�x�2M�cos��3x

2M�

�1

�3sin��3x

2M�� , �8�

where �I � 1 is the streamfunction in the limit x/ M →�. By using this functional form for the inflow, thespinup time of the model and the magnitude of theinitial shock are both reduced since the inflow is closeto the actual form of the western boundary current.Experiments with other functional forms for the inflow

condition show that there is little real impact upon theresults. This is not to say that the form of the inflow/outflow has no effect on the final state of the system.Rather, it is a qualitative one and our conclusions re-main unaffected.

The general numerics of the model are the same asthose of Tansley and Marshall (2001); that is, the ad-vection term uses the energy- and enstrophy-conservingArakawa Jacobian and the time stepping uses a leap-frog scheme with a Robert–Aselin filter. The vorticity isinverted to give the streamfunction using a multigridinverter and both the viscous and relaxation terms arebackground differenced in time for stability.

Two important differences between the experimentspresented here, and those of Tansley and Marshall(2001), are the form of the inflow/outflow conditionand the orientation of the channel. Tansley and Mar-shall restrict themselves to considering flow in an east–west aligned channel with an eastward directed inflow.Furthermore, this inflow is always uniform across itswidth. Our predominantly northward flow is arguablyin better agreement with the circumstances experiencedby real-world western boundary currents. Similarly, theuse of an inflow condition that closely mimics the ex-pected structure of a western boundary current is animportant step in understanding their separation dy-namics.

To promote flow separation of the boundary currentan obstacle is positioned halfway along the unspongedportion of the western boundary (at 750 km in Fig. 1).The obstacle’s shape is given by the following nondi-mensional equation:

o�y� �Wds

2

ds2 � �y � y0�2 , �9�

where o(y) is the obstacle width, W is the obstaclewidth at its crest, ds is the meridional “decay scale” ofthe obstacle, and y0 is the position of the obstacle crest.This is the “Witch of Agnesi” or “turning curve” and itstwo parameters allow us to tune the curvature at thepeak with some degree of exactitude. It is also interest-ing to note that two obstacles with the same aspectratio, given by W/ds, may not have the same maximumcurvature. This is because the maximum curvature ofthe obstacle, which occurs at its crest, varies as W/d2

s .Furthermore, the aspect ratio of the obstacles will gen-erally be greater than or equal to 1. This is in contrastto the study of flow separation and waves in a verticallystratified system (a physically and mathematicallyanalogous one to our � plane), where obstacles willgenerally have an aspect ratio much less than unity (see,e.g., Smolarkiewicz and Rotunno 1989; Sha et al. 1998).

FIG. 1. Diagram of the model domain. The region in which theflow is relaxed toward the prescribed inflow/outflow profile isshaded gray; W and ds refer to the obstacle parameters of (9).



c. Boundary conditions

For simplicity, both east and west boundaries haveno normal flow and no-slip boundary conditions ap-plied to them. It is worth noting that in barotropic stud-ies of the wind-driven ocean circulation the boundaryconditions can have an important impact on the flow. Inparticular, the combination of no-slip boundaries andlateral friction can lead to a boundary current that issubject to shear instabilities (Bryan 1963; Blandford1971). The choice of boundary condition is also some-what dictated by our desire to promote separation; asDengg (1993) shows, a free-slip solution is unable toseparate because of the nature of its boundary layers.

In the case of an obstacle being present, the bound-ary is treated in a piecewise-constant manner, muchlike many general circulation models. Adcroft and Mar-shall (1998) raise issues regarding the formulation ofboundary conditions with such a boundary, but thesewould seem to be less important when no-slip solutionsare sought. In addition, Tansley and Marshall (2001)use the same numerics to derive accurate solutions ofthe flow past a cylinder in both differentially rotatingand nonrotating conditions. In light of this, the outlinedmethod of treating the boundaries would seem justi-fied.

3. Solutions at Mu � 1/9000 (moderatedissipation)

In this section we present model solutions for � pa-rameter values of 150 and 750 at a fixed Munk numberof 1/9000 (corresponding to moderate dissipation): thisgives an effective Reynolds number of 60 and 12, re-spectively. A range of obstacle shapes are used to dem-onstrate that sufficiently high curvature can overcomethe � effect, at a fixed �, and that an increase in � cansuppress the flow separation, for a given obstacle shape.Since the western boundary is a streamline of fixedvalue (� � 0), to ensure no normal flow the criterionused for flow separation is outcropping of this stream-line in the interior, that is, negative values of stream-function. To highlight this, all figures of the flow havethe obstacle and boundary as solid black areas, whileareas with � � 0 are shaded a light gray. The model gridspacing is fixed at L /128, giving a dimensional value of�x � �y � 7.8125 km.

a. � � 150

Solutions at � � 150 and Mu � 1/9000 are presentedin Fig. 2. Results for a variety of obstacle parameters,including a straight western boundary (W � 0, ds � �),are displayed and clearly show the effect that an in-

crease in coastline curvature has upon the separationstate of the western boundary current.

All of the panels in Fig. 2 are long-term averages offluctuating flows. These fluctuations arise from shearinstability of the western boundary current, which is anexpected part of the solution when lateral friction isemployed (Bryan 1963; Blandford 1971). The presenceof the instability eddies can act to produce a systemwith a fluctuating separation state, that is, one that pe-riodically reattaches (not shown). However, we chooseto focus on the time-average characteristics of the flowseparation.

The panel with a straight western boundary, Fig. 2a,clearly does not separate, as expected. The slight diver-gence in the streamlines, which becomes stronger to thenorth, is due to the shear instability of the boundarycurrent manifesting itself in the form of eddies.

Returning to Fig. 2, it is clear that for the relativelyblunt obstacle of Fig. 2d, the streamlines shift aroundthe obstacle smoothly and the flow remains attached.However, there does appear to be evidence of stream-line divergence in the obstacle’s lee. In particular, thearea encompassed by the � � 0.2 streamline and thewestern boundary is larger downstream of the obstaclecrest than upstream of it. This is indicative of flow de-celeration and this flexing of the streamlines is a com-mon precursor to separation in the classical problem ofnonrotating flow past a cylinder (i.e., � � 0, see imagesin Van Dyke 1982).

In Fig. 2e, the obstacle decay scale has decreased,relative to Fig. 2d, such that the obstacle’s curvature issufficiently high to overcome the suppressive influenceof the � effect. The flow cannot remain attached and soseparates, creating a region of negative streamfunctionin the obstacle’s lee. This region is not clearly defined.However, in Figs. 2b, 2c, and 2f, the “separationbubbles” have vastly increased in size, and are nowclearly and smoothly defined by the � � 0.0 streamline.

The “strength” of separation can be judged by in-specting the minimum value of streamfunction for themodel solutions, with “stronger” separation expectedto give more negative minimum values. The �min valuesshown in the bottom right-hand corner of each panelconfirm what we can tell by looking at the streamfunc-tion plots of Fig. 2: that an obstacle with W � 0.1 and ds

� 0.1 produces relatively weak separation, probablybest described as marginal. In addition, a wider ob-stacle, one with larger W, gives stronger separation thanone that is narrower, but has the same aspect ratio(Figs. 2b and 2f).

An interesting point is that the sharpest obstacle,which is in Fig. 2c, does not give the strongest separa-tion, which is in Fig. 2b. This is because the instanta-



neous flow is in a new regime where the obstacle excitesstrong waves in the eastern part of the channel. Thiscreates a flow that is more strongly eddy driven thanthe others in Fig. 2.

b. � � 750

We have seen that sufficient coastline curvature canovercome the � effect and lead to flow separation. Fur-thermore, this separation is a function of both obstacleparameters, that is, of curvature rather than simplywidth or decay scale. This section seeks to demonstratethat Marshall and Tansley (2001) are correct: that anincrease in � can suppress separation.

Model solutions for the same obstacles and Munknumber (Mu � 1/9000), as in Fig. 2 but for � � 750, arepresented in Fig. 3. All of the panels in Fig. 3 are steady,in contrast to the time-average picture that Fig. 2 pre-sents. This is due to the boundary current being stableto shear at the current set of parameters. The shearinstability is a nonlinear effect, and increasing the �parameter has effectively made our flows more linear.

Plus, the inertial boundary layer scale ( I) is now lessthan the frictional boundary layer scale ( M), implyingthat the current is frictionally dominated.

The effect that the increased � has upon the flows isin line with our expectations; most of the flows thatpreviously separated now remain attached. Indeed,only Fig. 3c actually separates, and this is marginally so.However, downstream of the crest of the other ob-stacles there is, again, flexing and divergence of thestreamlines (Figs. 3b,e,f). This is less extreme for the W� 0.1, ds � 0.2 obstacle (Fig. 3d), suggesting that it ismore comfortably in the parameter space of an at-tached boundary current.

The change in separation behavior can be viewed asoccurring because of the flow being more linear, whichrestricts the role of advection of vorticity and allowsdiffusion to carry it away from the separation point. Asecond view is that the increase in � leads to a muchlarger suppressive force and so prevents the separationfrom occurring. The third, and perhaps simpler view isthat the increase in � reduces the effective Reynolds

FIG. 2. Long-term time-average streamlines for model solutions with Mu � 1/9000, � � 150, and Re � 60, with the obstacleparameters as shown in the panel captions. Obstacles are solid black and areas with � � 0, indicating flow separation, are light gray.The contour interval is �� 0.2, starting at 0 on the western boundary, and the flow is predominantly northward. The dotted contoursare at � � 1.0, and mark the region of relatively weak flow in the eastern part of the channel.



number (Re � 1/�Mu) of the flow to only 12 from aneffective value of 60 in section 3a. From the classicalview of flow past a cylinder, one would expect the flowto be more likely to remain attached although, as yet,there is no way to estimate the value of the Reynoldsnumber that might be “critical” for a given � and set ofobstacle parameters. All three views are equally validand are effectively a restatement of the same physicalprocess.

4. Solutions at Mu � 1/45 000 (weaker dissipation)

The results of section 3 suggest that increased bound-ary curvature or a decreased � can result in flow sepa-ration. This can be seen as the deceleration due to flowcurvature overcoming the stabilizing tendency of the �effect, or due to the nonlinear advection terms playingless of a role at higher �. The change in separationbehavior can also be interpreted as being the result of alower effective Reynolds number due to increasing � ata fixed Mu. Viewing the behavior in this light suggeststhat decreasing Mu should increase the tendency to

separate since it will raise the effective Reynolds num-ber.

The current section will investigate the effect that adecreased Mu of 1/45 000 has upon the flow by present-ing solutions obtained for both � � 150 and � � 750. Aswith section 3, the model grid spacing is fixed at L /128,and flow separation is viewed as having occurred when�min � 0.

a. � � 150

In Fig. 4 the Munk number has been decreased by afactor of 5, relative to Fig. 2, although the same range ofobstacles is used. This causes an increase in the effec-tive Reynolds number from 60 to 300, so one wouldexpect to get separation at a much lower obstacle cur-vature.

Most striking of the results in Fig. 4 is that the bound-ary current for the case with no obstacle is wider than atthe higher Munk number (cf. Fig. 4a with Fig. 2). Thisis an unexpected result since M, and thus the inflow, ismuch narrower. In fact, inspecting the instantaneousflow shows that a transition to a primarily eddy-driven

FIG. 3. Steady-state streamlines for model solutions with Mu � 1/9000, � � 750, and Re � 12, with the obstacle parameters as shownin the panel captions. Obstacles are solid black and areas with � � 0, indicating flow separation, are light gray. The contour intervalis �� 0.2, starting at 0 on the western boundary, and the flow is predominantly northward. The dotted contours are at � � 1.0 andmark the region of relatively weak flow in the eastern part of the channel.



regime has occurred. The instantaneous flow is domi-nated by large eddies, which act to broaden the bound-ary current and also considerably increase its transport.

The eddy-driven nature of this flow regime leads to avery different flavor to the results with obstacles. Therecirculation on the eastern flank is greatly increased insize and magnitude, and the weak flow in the easternside of the domain is now quite strong. In particular,although the sharpest obstacle (W � 0.2, ds � 0.05)gives the strongest separation, the intermediately sharpobstacles do not follow the logical progression ofsharper obstacle leading to stronger separation. Indeed,Fig. 4f has a �min that is only 3% of the value foundat the higher Munk number of 1/9000 (see Fig. 2f).

Although the details of the obstacle seem to be lessimportant to the separation process, they still holdsome relevance. The decrease in Mu has also led to theflow with a W � 0.1, ds � 0.1 obstacle (Fig. 4e) nowseparating. However, the particular sets of experimentsdo not contain enough evidence to completely elucidatethe effect that altering Mu has upon the separation be-havior of the system. For a more comprehensive look at

variations in the Munk number, the interested reader isreferred to Munday (2004).

b. � � 750

Already it has been seen that separation can be sup-pressed by an increased � and that decreasing Mu hasan effect on the behavior of the boundary current. Forthe combination of high � and low Mu one might expectresults to be somewhere in between those of Fig. 3 andFig. 4: that is, that separation is enhanced with respectto Fig. 3 but suppressed with respect to Fig. 4.

Figure 5 shows the effect of reducing Mu from 1/9000to 1/45 000 when � � 750. As with section 4a, thiscauses a fivefold increase in the effective Reynoldsnumber, in this case from 12 to 60. Such an increasemight well be expected to result in separated flows.However, unlike Fig. 3 all of the panels of Fig. 5 arelong-term time averages of an unsteady flow.

The images contained in Fig. 5 show that the modelbehaves as anticipated; the boundary currents of Figs.5a, 5b, 5d, and 5e all remain firmly attached, but in-creasing obstacle curvature eventually overcomes the

FIG. 4. Long-term time-average streamlines for model solutions with Mu � 1/45 000, � � 150, and Re � 300, with the obstacleparameters as shown in the panel captions. Obstacles are solid black and areas with � � 0, indicating flow separation, are light gray.The contour interval is �� 0.2, starting at 0 on the western boundary, and the flow is predominantly northward. The dotted contoursare at � � 1.0 and mark the region of relatively weak flow in the eastern part of the channel.



suppressive effect of � on flow separation. This resultsin Figs. 5c and 5f being in a separated state. Streamlinedivergence, indicative of flow deceleration, can clearlybe seen in Figs. 5b and 5e and less strongly in 5d. It isinteresting to note that the obstacles in Figs. 5b and 5fhave the same aspect ratio but that it is the obstacleswith the lower W that gives a separated boundary cur-rent. Whereas, in section 3a, for obstacles with the sameaspect ratio, the strongest separation tends to be givenby the obstacle with the largest W. This difference couldbe due to two different obstacle shape factors, whichare difficult to distinguish between and also related toeach other.

First, the separation condition of Marshall and Tans-ley (2001) really concerns the gradient of the Coriolisparameter in the direction of the flow [see (2)]. As such,we might expect that an obstacle that causes the flowupstream of its crest to deviate farther from true northwill be more likely to give separation since �* will besmaller. Second, the curvature at the crest of the ob-stacle is not given by W/ds, the obstacle aspect ratio, butrather by W/d2

s . This means that an obstacle with

smaller ds is actually more sharply curved at the crest.As a result, it might be expected to give stronger sepa-ration. Given the extreme streamline divergence of Fig.5b, it is reasonable to expect this flow to be very closeto separation. Similarly, the small size of the separationbubble in Fig. 5f suggests that this flow is close to at-tachment. If this is true, then either of the shape factorscould be responsible for tipping Fig. 5f over the edgeinto a separated state.

5. Vorticity diagnostics

As discussed in section 1 for the case of nonrotatingproblems, it is accepted that a flow will separate in thepresence of an “adverse pressure gradient” or, equiva-lently, a flow deceleration. Our hypothesis is that theseparation of a western boundary current can also beexplained by such a flow deceleration. Indeed, severalauthors find adverse pressure gradients to be coincidentwith boundary current separation (Haidvogel et al.1992; Jiang et al. 1995; Baines and Hughes 1996; Tans-ley and Marshall 2000). Furthermore, we seek to attrib-

FIG. 5. Long-term time-average streamlines for model solutions with Mu � 1/45 000, � � 750, and Re � 60, with the obstacleparameters as shown in the panel captions. Obstacles are solid black and areas with � � 0, indicating flow separation, are light gray.The contour interval is �� 0.2, starting at 0 on the western boundary, and the flow is predominantly northward. The dotted contoursare at � � 1.0 and mark the region of relatively weak flow in the eastern part of the channel.



ute the presence of the flow deceleration to increasingflow curvature induced by the presence of an obstacleon the western boundary. In this section a set of vor-ticity diagnostics is derived following the method ofMarshall and Tansley (2001). These diagnostics arethen applied to the model results of sections 3 and 4.

a. Method

To attribute the separation of the western boundarycurrent to flow deceleration, a “separation formula”must first be derived. The formula is essentially an area-integrated vorticity budget, which is integrated over aspecific part of the flow. It uses the barotropic vorticityequation to partition the flow deceleration of theboundary current, and hence attribute the flow separa-tion into terms due to flow curvature, advection of plan-etary vorticity, and dissipation of relative vorticity. Thefirst step is to write the steady-state barotropic vorticityequation (6) into its nondimensional flux form:

1

�� · ��u� �

1

�� · � fu� � Mu�2�, �10�

where f � �y is the Coriolis parameter and the othersymbols are as previously defined.

Following Marshall and Tansley (2001), (10) is inte-grated over a region of the flow bounded by twostreamlines and two lines orthogonal to streamlines.The two regions used for the analysis of sections 5b and

5c are illustrated in Fig. 6. The selection of the positionsof the points A, B, C, and D are considered below.

Applying the above form of integration to (10) yields

1

��

L

� · ��u� dA �1

��

L

� · � fu� dA

� Mu��L

�2� dA , �11�

where L is the region defined by the boundary ABCD,as shown in Fig. 6.

In evaluating (11), so-called natural coordinates areused. As such, the position vector is given by (s, n),where s is the unit vector tangent to the flow and n isthe unit vector normal to the flow. By convention, npoints to the left of the flow and the velocity vectorbecomes u � (Vs, Vn), with Vn having zero magnitudebut pointing in the direction of n [see, e.g., Holton(1992) for details].

The next step is to use the definition of vorticity innatural coordinates to partition vorticity into parts dueto normal shear and flow curvature. In natural coordi-nates, relative vorticity is given by

� �Vs

R�

�Vs

�n, �12�

where R is the radius of curvature of the flow. Equation(12) is substituted into (11), with subsequent rearrange-ment giving

1

��

L

� · ��Vs

�nu� dA �

1

��

L

� · �Vs

Ru� dA �

1

��

L

� · � fu� dA � Mu��L

�2� dA . �13�

After making use of the divergence theorem, the inte-grated vorticity budget becomes

1

��Vs

2

2 �A

D

� �Vs2

2 �B

C�shear

�1

��Vs

2

Rdn�

BA

CD

curv

�1

��

L

�*Vs dA

planet

� Mu��L

�2� dA

dissip

, �14�

where �* � �f/�s is the gradient of the Coriolis param-eter in the alongstream direction and [�]D

A � �D � �A.The labels will be used to refer to individual terms inthe integrated budget throughout the rest of this sec-tion. It is worth noting that the above integrated budgetonly differs from that of Marshall and Tansley (2001)by the inclusion of a term due to friction and the lack ofa term due to vortex stretching.

In the limit of weak flow along the far-boundarystreamline (the path BC), we can, as per Marshall and

Tansley (2001), interpret the shear term as representingthe acceleration or deceleration when following theflow. The other three terms (curv, planet, and dissip)then represent the acceleration or deceleration result-ing from different physical processes (curvature forc-ing, the � effect, and lateral friction, respectively).

The assumption of weak flow along the path BC isimportant to the interpretation of the budget as repre-senting acceleration/deceleration of the boundary cur-rent. In the case of there being flow along the path BC,



then shear will more correctly represent the change inkinetic energy along AD relative to the change in ki-netic energy along BC. Thus, when applying the budgetto the barotropic vorticity equation model results, theappropriate choice of the streamfunction along whichBC is placed is crucial.

In all cases considered in the following subsections,the path AD is a segment of the � � 0.4 streamline.This is chosen because, in the inflow, the point of maxi-mum current speed lies close to this value of stream-function. Similarly, the path BC is a segment of the � �1.1 streamline, because this value of streamfunction liesin the region of weak flow on the eastern flank of theinflow.

Meridional positions must also be selected for thelocations of the points A and D. These are chosen morearbitrarily than the streamfunction values at which toplace the paths AD and BC. In order for region 1 toencompass the area of positive curvature forcing up-stream of the obstacle crest, the values y � 0.55 and y� 0.75 are selected for A and D, respectively. Similarly,in order for region 2 to include the area of negativecurvature forcing downstream of the crest, the values y� 0.75 and y � 0.95 are selected for A and D, respec-tively.

At the value chosen for the path BC, � � 1.1, thechange in kinetic energy is typically �10% of the

change in kinetic energy along the path AD. In addi-tion, the actual flow along the path BC is always con-siderably weaker than that along AD, implying that,even if the change in kinetic energy is significant, theintegrated budget still represents why the strongest partof the flow is accelerating or decelerating relative to theweakest part. We believe that this justifies our inter-pretation of the integrated budget.

The above refers to a system in steady state. To con-struct a vorticity budget for systems that reach an un-steady final state, all of the variables must first be writ-ten as the sum of an average and a time-varying part;that is, � � �(x, y) � ��(x, y, t), � (x, y) � �(x, y, t),etc. The starting point for our vorticity budget thenbecomes the flux form of the time-averaged barotropicvorticity equation:

1

�� · ��Vs

�nu� �

1

�� · �Vs

Ru� �

1

�� · � f u� � Mu�2�

�1

�� · ��u��, �15�

where (�u�) is the time-averaged eddy flux of vorticity.The integration of (15), over the area defined by Fig.

6, then produces a vorticity budget for a time-averagesystem:

1

��Vs

2

2 �A

D

� �Vs2

2 �B

C�shear

�1

�� Vs

2

Rdn�

BA

CD

curv

�1

��

L

�*Vs dA

planet

� Mu��L

�2� dA

dissip

�1

��

L

� · ��u�� dA

eddy

.

�16�

FIG. 6. The steady-state barotropic vorticity equation is integrated over the shaded regionsABCD. Region 1 is selected to cover the area of positive curvature forcing upstream of theobstacle crest. In contrast, region 2 is selected to cover the area of negative curvature forcingdownstream of the obstacle crest. The solid black area is the obstacle and the dashed contoursare streamlines of the flow; only the part of the model domain immediately surrounding theobstacle crest is shown. The regions are bounded by two segments parallel to streamlines (ADand BC) and two segments perpendicular to streamlines (AB and CD).



The budget given in (16) produces shear, curv, planet,and dissip terms that differ from their steady compatri-ots [see (14)] only in the detail of them being defined bylong-term averages of the quantities involved. How-ever, a fifth term is introduced, due to the eddy flux ofvorticity, and this makes a crucial difference. The re-gions over which the area integration is performed areselected in the same manner as for the steady-statecase. In addition, the comments regarding the speedand kinetic energy along the paths AD and BC stillapply.

Regardless of the final model state, every term in thevorticity balance is calculated independently. Budgetclosure is good, with the residual term being two tothree orders of magnitude smaller than the largestterm.

b. Steady state

This section presents area-integrated vorticity bud-gets obtained from the steady-state solutions of section3b. The nondimensional parameters for these solutionsare � � 750, Mu � 1/9000, and Re � 12. The vorticitybudget for these solutions only require the four termsof (14): shear, curv, planet, and dissip.

Referring back to Fig. 3a, we see that a steady-statesystem with no obstacle shows little or no streamlinedivergence/convergence. As such, its integrated budgetshould show little or no acceleration with the dominantbalance being between the advection of planetary vor-ticity (planet) and the dissipation of relative vorticity(dissip). Indeed, this is the case, as Fig. 7 shows. In fact,planet and dissip have the same magnitude, to two ormore significant figures, both within and between inte-gration regions. The situation becomes more interest-ing when we consider a system with an obstacle on thewestern boundary.

In Fig. 8, the budgets obtained for the three modelsolutions shown in Figs. 3d, 3e, and 3f (from top tobottom) are all shown. The first column is for region 1

(upstream of the obstacle’s crest), the second column isfor region 2 (downstream of the obstacle’s crest). It isimmediately noticeable that the budget no longer re-flects a strict balance between planet and dissip.Rather, shear shows significant acceleration/decel-eration, which mirrors the sign of curv, and both planetand dissip vary between regions.

In region 1 (Figs. 8a, 8c, and 8e), the magnitude ofdissip has increased relative to the flow without an ob-stacle. This increase is quite significant, amounting to14%–23%, depending on the details of the obstacle.These changes are quite reasonable given that theboundary current is probably faster and/or moresheared than the straight western boundary case.

In contrast to the changes in dissip, the variations inthe magnitude of planet are rather minor. At ds � 0.2,planet has increased by 2.6%, relative to the straightwestern boundary case. As the obstacle becomes nar-rower, in the along-stream direction, planet tends toreturn toward the straight boundary value and may de-crease below it for sharp enough obstacles. These varia-tions reflect changes in the length of the integrationregion (it is longer for a system with an obstacle) andthe boundary current being deflected from true north,which decreases �* in (14).

The most important effect of the obstacle is seen inthe steady increase of curv across the three obstacles,which gives rise to a concomitant increase in shear. Theincrease in curv represents both the pinching of thestreamlines over the crest, which creates a faster cur-rent, and also the higher curvature of the flow. How-ever, the increase in the magnitude of dissip partly miti-gates the curvature forcing, so shear is always less thancurv.

In region 2 (Figs. 8b, 8d, and 8f), we find that planethas increased relative to the flow with no obstacle,more substantially than in region 1 (by 5%, as com-pared with �3%). In addition, dissip changes in mag-nitude much less than in region 1. For the obstacle with

FIG. 7. Integrated vorticity budget for � � 750, Mu � 1/9000, Re � 12 with W � 0. The key refers to the term labels of (14).



W � 0.1 and ds � 0.2, dissip increases in magnitude.However, for the other two obstacles, dissip actuallydecreases in magnitude, representing a decrease in thefrictional dissipation of the flow.

As expected, curv is negative, and increases in mag-nitude as the obstacle becomes more sharply curved (ds

decreases). However, the deceleration due to curv inregion 2 is slightly less in magnitude than the accelera-tion due to curv in region 1, regardless of the details ofthe obstacle. This is linked to the divergence of thestreamlines, noted previously. This divergence is a re-action to deceleration caused by the curvature forcing,and it also allows the current to select a slightlystraighter path in the obstacle’s lee. Furthermore, forcases in which flow separation occurs, the observedstraightening of the boundary current is often sufficientfor curv to decrease below the value found for a blunterobstacle that is incapable of promoting separation atthat particular combination of � and Mu.

The small, but consistent, increase in planet, relativeto region 1, represents the selection of a more north-ward path for the boundary current. The increase inplanet, along with the reduced frictional dissipation, isable to partly mitigate the curvature forcing, althoughnot to the extent that the increased dissipation compen-sates for the accelerative effect of curv in region 1.

c. Unsteady state

This section presents area-integrated vorticity bud-gets obtained from the time-average solutions of sec-tion 3a. The flow parameter values for these solutionsare � � 150, Mu � 1/9000, and Re � 60. The vorticitybudgets for these solutions require five terms: shear,curv, planet, and dissip, as well as eddy, as given in (16).

Figure 9 presents the integrated budget for a straightwestern boundary. In contrast to Fig. 7, it does notdisplay a strict balance between planet and dissip.Rather, the flow is decelerating in both regions, which

FIG. 8. Integrated vorticity budget for � � 750, Mu � 1/9000, Re � 12 with W and ds as per the panel captions. The key refers tothe term labels of (14).



the budget confirms to be as a result of eddy activity. Inaddition, it is interesting to note that although planetshows little variation from the values of Fig. 7, the mag-nitude of dissip is significantly smaller. As a result, theeddies partly act to dissipate relative vorticity, in order

to balance planet, as well as to broaden and deceleratethe flow.

Figure 10 shows the budget results for Figs. 2d, 2e,and 2f (from top to bottom). As with the straightboundary case, the budget diverges from the zero order

FIG. 9. Integrated vorticity budget for � � 150, Mu � 1/9000, Re � 60, and W � 0. The key refers to the term labels of (16).

FIG. 10. Integrated vorticity budget for � � 150, Mu � 1/9000, Re � 60 with W and ds as per the panel captions. The key refers tothe term labels of (16).



balance of dissipation of relative vorticity compensatingfor advection of planetary vorticity. An immediatelyobvious difference to the budgets shown in 5b is that itis now possible for the magnitudes of shear and curv tobe greater than the magnitudes of planet and dissip.This is due to the change in value of the � parameter,rather than the transition to an unsteady final state, andreflects the nonlinear advection terms becoming moreimportant to the flow’s evolution.

In region 1, the addition of an extra term to the bud-get has little effect on the patterns of variation de-scribed in section 5b. Principally, planet and dissip bothshow deviations from the straight western boundaryvalues. These deviations are manifestations of the cur-rent being turned away from true north, and so on,which are described above. Similarly, the addition ofthe obstacle introduces curvature forcing to the system,creating positive values of curv and shear.

In addition, the magnitude of eddy, for all three ob-stacles, is significantly less than for the case of a straightwestern boundary. It would appear that the longercoastline, and the increase in frictional dissipation, maywell have aided in stabilizing this section of the bound-ary current.

In region 2, the variations of the individual budgetterms are much the same as in section 5b, except thateddy has a much larger impact than in region 1. Forexample, as the streamlines in the obstacle’s lee un-dergo divergence, the boundary current becomes morenorthward in nature. This causes planet to increase.Similarly, the changes in the currents structure causesdissip to become smaller in magnitude as ds does. Thevariations in curv clearly demonstrate the loss of forc-ing achieved when separation occurs; that is, when ds ischanged from 0.1 to 0.05 and the boundary currentseparates, curv decreases. This is just as described insection 5b.

The remaining terms in the budget, shear and eddy,both behave differently. Unlike in region 1, eddy nowbecomes an important part of the budget. Although itchanges relatively little when ds decreases from 0.2 to0.1, a further decrease of ds to 0.05, eddy more thandoubles in magnitude. Similarly, shear continues to in-crease, even when curv does not, and it appears that theactual deceleration of the boundary current is nowheavily linked to curv and eddy.

The large increase in eddy activity, which occurs witha fully separated boundary current, is a result of howvariable the path of the boundary current is in the in-stantaneous case. Essentially, with the attached, ormarginally separated, cases the instantaneous positionof the boundary current is little different from the time-average position. As such, the eddies tend to either

squash the boundary current against the boundary orcause a flexing of the streamlines. However, in the caseswhere the time-average boundary current is fully sepa-rated, the eddies can actually change the position of theinstantaneous boundary current quite considerably.This can result in it being fully attached or even moreseparated. As a result, the eddy term in the time-average budget becomes much larger.

The size of eddy, for the fully separated cases, couldlead to the suggestion that it is this term, rather thancurv, that leads to the separation. However, the straightwestern boundary cases also display eddy activity, butnever separates. Thus, we believe that the increase inthe magnitude of eddy is caused by the act of separationrather than the other way about.

d. Summary of results

This section is a brief summary of the results ob-tained from the area-integrated vorticity budgets. It isintended to draw attention to the most important con-clusions.

The area-integrated vorticity budgets show that, up-stream of the obstacle crest, changes in flow curvaturelead to an acceleration of the boundary current. Thisremains true regardless of whether the system reachessteady state or is a long-term time average. In moststeady-state cases, the increase in dissipation due to anarrower, more curved, and faster boundary currentresults in some mitigation of the curvature forcing. Incontrast, in a time-average state, the interaction of theintegral effect of eddies with the dissipation term can becomplex. This can lead to the acceleration of theboundary current being enhanced with respect to theapplied curvature forcing.

Upstream of the obstacle crest, curvature forcing ap-plies a deceleration to the boundary current. In the caseof a time-average state, the interaction between theintegral effect of eddies and the other terms in the bud-get can lead to a complex pattern of changes. In par-ticular, the role of dissipation can be completely sub-verted by the eddies because of the current havingwidely varying position over the course of the eddygrowth cycle. However, curvature forcing is alwaysnegative, and always present, when the western bound-ary is not straight.

In both steady-state and time-average states, theboundary current is able to react to the flow decelera-tion prior to separating. This leads to the magnitude ofthe deceleration, in this region, growing more slowlythan the acceleration upstream of the crest. Further-more, the act of separation removes some of the cur-vature forcing and leads to the deceleration of the cur-rent being substantially less than one might expect.



If one considers both integration regions together,then the smaller magnitude of curv in region 2 willresult in there appearing to be a net acceleration of theboundary current. Despite this, the curvature forcingcan still cause flow separation because separation is avery local process. It is not necessary to decelerate theboundary current along its entire length, merely in asmall region. The resulting separation point will alwaysbe close to the region of deceleration.

6. Conclusions

The general problem of western boundary currentseparation, and specifically the separation of the GulfStream from the eastern seaboard of the United States,has given birth to a number of theories as to the mecha-nism for such separation. Classically, separation occursat high Reynolds number when flow is along an adversepressure gradient, resulting in a finite flow deceleration.In both ocean gyre models with straight boundaries(Haidvogel et al. 1992; Jiang et al. 1995) and those withmore complex shelf geometry (Tansley and Marshall2000), adverse pressure gradients have been found tobe coincident with flow separation. This is also true inthe physical laboratory experiments of Baines andHughes (1996). It should also be noted that Cessi (1991)has found that, for the case of colliding western bound-ary currents, separation can occur before the adversepressure gradient is encountered. However, these re-sults were obtained at relatively low Reynolds numberand Kiss (2002) argues that the observed separation inthe results of Cessi is occurring in a region of adversepressure gradient.

In this paper we have investigated the separation of awestern boundary current via a flow deceleration, usinga barotropic vorticity model in an idealized configura-tion. This removes any influence that the wind stresspattern might have, as well as the complications thatbaroclinicity and three dimensions bring. The experi-ments presented represent a departure from those thatpreviously considered the influence of boundary curva-ture upon rotating flows (e.g., Stern and Whitehead1990), because of the explicit impact of differential ro-tation upon the process of separation. The use of anorth–south aligned channel and an inflow conditionduplicating the structure of a “mature” western bound-ary current is also crucial in furthering our understand-ing of the dynamics surrounding the separation of west-ern boundary currents.

Our results confirmed the twin hypotheses of Mar-shall and Tansley (2001):

1) an increase in the gradient in Coriolis parametercan suppress flow separation, and

2) sufficiently strong boundary curvature can over-come the inhibiting effect of the � parameter andinduce flow separation.

Furthermore, reducing the Munk number, that is, de-creasing dissipation, clearly has an impact upon the sta-bility of the solutions and the separation states attained.In the presented solutions, it is unclear as to the exactdependence. However, in the extended range of experi-ments used by Munday (2004), it is clear that, forsteady-state solutions, a decrease in Munk numbermakes flow separation more likely.

The use of integrated vorticity budgets illuminatesseveral aspects of the model solutions. First, they at-tribute stronger flow acceleration upstream of the crest,and stronger deceleration downstream of the crest, toincreasing obstacle curvature. Second, they show thatthe “stretching” of streamlines prior to separation, andthe selection of a straighter path subsequent to separa-tion, results in weaker curvature forcing downstream ofthe crest. Third, they show that curvature forcing can bepartly mitigated by changes in the relative magnitude ofother terms in the budget. In addition, relatively smallchanges in the magnitude of planet seem to be relatedto deviations in flow direction from the true north. Inthe case of long-term time-average solutions, a largeincrease in the magnitude of eddy accompanied sepa-ration. This is caused by the large changes in boundarycurrent position made possible when the solid bound-ary is some distance from its western flank.

Marshall and Tansley (2001) suggest that there maybe a critical value of coastline curvature, as in (2), be-yond which a given flow should separate. For the re-sults presented here, the curvature at the peak of theobstacle is generally less than that predicted by (2).However, the inequality does appear to hold, broadly,if we introduce a dimensionless constant, that is,

R � � U

�*�1�2

. �17�

Calculations based upon our results indicate that � liesbetween 0.05 and 0.1.

One attraction of the idea that separation may beinduced by changes in flow curvature is that it mayexplain the steadiness of the Gulf Stream’s separationpoint. For example, the separation point of the GulfStream varies in north–south position by only �50 km(Auer 1987; Gangopadhyay et al. 1992). In contrast, theseparation points of the Brazil Current and the Malvi-nas Current vary in along coast position by 930 and850 km, respectively (Olson et al. 1988). Certainly,our steady-state and long-term time-average resultsshow flow separation is always initiated close to the



obstacle’s crest. However, the fixing of a separationpoint in a time-varying sense is one that requires fur-ther research, probably with more complex models toallow the inclusion of topography, baroclinicity, andperhaps wind forcing. In particular, Marshall and Tans-ley (2001) were able to include the effects of verticalstratification in their integrated vorticity budget. Thissuggests that it has an important role to play in settingthe net acceleration/deceleration of a western bound-ary current.

The issue of experiments at higher Reynolds number,in order to better simulate the real environment, is alsoimportant. At present, most ocean models have an un-realistically high dissipation, which may affect the sepa-ration process and explain why it has been consistentlymisrepresented.

Acknowledgments. This work was supported byNERC. The comments of one of the anonymous re-viewers led to a significant improvement to the analysisand the paper as a whole.

REFERENCES

Adcroft, A., and D. Marshall, 1998: How slippery are piecewise-constant coastlines in numerical ocean models? Tellus, 50,2259–2282.

Agra, C., and D. Nof, 1993: Collision and separation of boundarycurrents. Deep-Sea Res., 40, 2259–2282.

Auer, S. J., 1987: Five-year climatological survey of the GulfStream and its associated rings. J. Geophys. Res., 92, 11 709–11 726.

Baines, P. G., and R. L. Hughes, 1996: Western boundary currentseparation: Inferences from a laboratory experiment. J. Phys.Oceanogr., 26, 2576–2588.

Batchelor, G. K., 1969: An Introduction to Fluid Dynamics. Cam-bridge University Press, 615 pp.

Blandford, R. R., 1971: Boundary conditions in homogeneousocean models. Deep-Sea Res., 18, 739–751.

Böning, C. W., 1986: On the influence of frictional parameteriza-tion in wind-driven ocean circulation models. Dyn. Atmos.Oceans, 10, 63–93.

Boyer, D. L., 1970: Flow past a right circular cylinder in a rotatingframe. J. Basic Eng. Trans. ASME, 92, 430–436.

——, and P. A. Davies, 1982: Flow past a circular cylinder on a�-plane. Philos. Trans. Roy. Soc. London, 306, 533–556.

Bryan, K., 1963: A numerical investigation of a nonlinear modelof a wind-driven ocean. J. Atmos. Sci., 20, 594–606.

Cessi, P., 1991: Laminar separation of colliding western boundarycurrents. J. Mar. Res., 49, 697–717.

——, R. V. Condie, and W. R. Young, 1990: Dissipative dynamicsof western boundary currents. J. Mar. Res., 49, 697–717.

Charney, J. G., 1955: The Gulf Stream as an inertial boundarylayer. Proc. Nat. Acad. USA, 41, 731–740.

Dengg, J., 1993: The problem of Gulf Stream separation: A baro-tropic approach. J. Phys. Oceanogr., 23, 2182–2200.

——, A. Beckmann, and R. Gerdes, 1996: The Gulf Stream sepa-ration problem. The Warmwatersphere of the North Atlantic,W. Krauss, Ed., Gebruder Borntager, 253–289.

Ezer, T., and G. L. Mellor, 1992: A numerical study of variabilityand the separation of the Gulf Stream induced by surfaceatmospheric forcing and lateral boundary flows. J. Phys.Oceanogr., 22, 660–682.

Gangopadhyay, A., P. Cornillon, and D. R. Watts, 1992: A test ofthe Parsons–Veronis hypothesis on the separation of the GulfStream. J. Phys. Oceanogr., 22, 1286–1301.

Goldstein, S., 1948: On laminar boundary layer flow near a pointof separation. Quart. J. Mech. Appl. Math., 1, 43–69.

Haidvogel, D. B., J. C. McWilliams, and P. R. Gent, 1992: Bound-ary current separation in a quasigeostrophic, eddy-resolvingocean circulation model. J. Phys. Oceanogr., 22, 882–902.

Hogg, N. G., and H. Stommel, 1985: On the relationship betweenthe deep circulation and the Gulf Stream. Deep-Sea Res., 32,1181–1193.

Holton, J. R., 1992: An Introduction to Dynamic Meteorology.Academic Press, 511 pp.

Jiang, S., F. F. Jin, and M. Ghil, 1995: Multiple equilibria, periodic,and aperiodic solutions in a wind-driven, double-gyre, shal-low-water model. J. Phys. Oceanogr., 25, 764–786.

Kiss, A. E., 2002: Potential vorticity “crises”, adverse pressuregradients, and western boundary current separation. J. Mar.Res., 60, 779–803.

Marshall, D. P., and C. E. Tansley, 2001: An implicit formula forboundary current separation. J. Phys. Oceanogr., 31, 1633–1638.

Merkine, L., 1980: Flow separation on a �-plane. J. Fluid Mech.,99, 299–309.

Munday, D. R., 2004: On the flow separation of western boundarycurrents. Ph.D. thesis, Department of Meteorology, Univer-sity of Reading, 236 pp.

Munk, W. H., 1950: On the wind-driven ocean circulation. J. Me-teor., 7, 79–93.

Olson, D. B., G. P. Podesta, E. H. Evans, and O. B. Brown, 1988:Temporal variations in the separation of the Brazil andMalvinas Currents. Deep-Sea Res., 35, 1971–1990.

Ou, H. W., and P. M. de Ruijter, 1986: Separation of an inertialboundary current from a curved coastline. J. Phys. Ocean-ogr., 16, 280–289.

Özgökmen, T. M., E. P. Chassignet, and A. M. Paiva, 1997: Im-pact of wind forcing, bottom topography, and inertia on mid-latitude jet separation in a quasigeostrophic model. J. Phys.Oceanogr., 27, 2460–2476.

Page, M. A., and E. R. Johnson, 1990: Flow past cylindrical ob-stacles on a beta-plane. J. Fluid Mech., 221, 349–382.

Parsons, A. T., 1969: A two-layer model of Gulf Stream separa-tion. Deep-Sea Res., 39, 511–528.

Pedlosky, J., 1996: Ocean Circulation Theory. Springer-Verlag,453 pp.

Sha, W., K. Nakabayashi, and H. Ueda, 1998: Numerical study onflow pass of a three-dimensional obstacle under a strongstratification condition. J. Appl. Meteor., 37, 1047–1054.

Smith, F. T., 1982: On the high Reynolds number theory of lami-nar flows. IMA J. Appl. Math., 28, 207–281.

Smolarkiewicz, P. K., and R. Rotunno, 1989: Low Froude numberflow past three-dimensional obstacles. Part I: Baroclinicallygenerated lee vortices. J. Atmos. Sci., 46, 1154–1164.

Spall, M. A., 1996a: Dynamics of the Gulf Stream/Deep WesternBoundary Current crossover. Part I: Entrainment and recir-culation. J. Phys. Oceanogr., 26, 2152–2168.

——, 1996b: Dynamics of the Gulf Stream/Deep Western Bound-



ary Current crossover. Part II: Low-frequency internal oscil-lations. J. Phys. Oceanogr., 26, 2169–2182.

Spitz, Y. H., and D. Nof, 1991: Separation of boundary currentsdue to bottom topography. Deep-Sea Res., 38, 1–20.

Stern, M. E., 1998: Separation of a density current from the bot-tom of a continental shelf. J. Phys. Oceanogr., 28, 2040–2049.

——, and J. A. Whitehead, 1990: Separation of a boundary jet ina rotating fluid. J. Fluid Mech., 217, 41–69.

Tansley, C. E., and D. P. Marshall, 2000: On the influence of thebottom topography and the Deep Western Boundary Cur-rent on Gulf Stream separation. J. Mar. Res., 58, 297–325.

——, and ——, 2001: Flow past a cylinder on a � plane, with

application to Gulf Stream separation and the Antarctic Cir-cumpolar Current. J. Phys. Oceanogr., 31, 3274–3283.

Thompson, J. D., and W. J. Schmitz, 1989: A limited-area modelof the Gulf Stream: Design, initial experiments, and model-data intercomparison. J. Phys. Oceanogr., 19, 791–814.

Van Dyke, M., 1982: An Album of Fluid Motion. Parabolic Press,176 pp.

Veronis, G., 1966: Wind-driven ocean circulation. Part II: Numeri-cal solutions of the non-linear problem. Deep-Sea Res., 13,31–55.

——, 1973: Model of world ocean circulation. Part I: Wind-driven,two-layer. J. Mar. Res., 31, 228–288.



on the separation of a barotropic western boundary current

Documents