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. The meridional gradient of the Coriolis parameter (the β effect), the strength of dissipation, and the geometry of the cape are varied. It is found that 1) all instances of flow separation are coincident with the presence of a flow deceleration, 2) an increase in the strength of the β effect is able to suppress flow separation, and 3) increasing coastline curvature can overcome the suppressive β effect and induce separation. These results are supported by integrated vorticity budgets, which attribute the acceleration of the boundary current to the β effect and changes in flow curvature. The transition to unsteady final model states is found to have no effect upon the qualitative nature of these conclusions.
The dynamics controlling the separation, or continued attachment, of western boundary currents remains a poorly understood aspect of the large-scale ocean circulation. The problem is not aided by the fact that numerous hypotheses have been put forward, but no conclusive explanation has been reached (Haidvogel et al. 1992). These hypotheses include vanishing wind stress curl (Munk 1950), outcropping of the main thermocline (Parsons 1969; Veronis 1973), interactions with the northern recirculation gyre (Hogg and Stommel 1985; Ezer and Mellor 1992), “vorticity crisis” (Cessi et al. 1990; Kiss 2002), interactions with the deep western boundary current (Thompson and Schmitz 1989; Agra and Nof 1993; Spall 1996a, b; Tansley and Marshall 2000), and coastline/shelf geometry (Ou and de Ruijter 1986; Stern and Whitehead 1990; Spitz and Nof 1991; Dengg 1993; Özgökmen et al. 1997; Stern 1998; Tansley and Marshall 2001). Excellent reviews of some of these hypotheses are given by both Haidvogel et al. (1992) and Dengg et al. (1996).
A classical paradigm of flow separation is nonrotating flow past a cylinder. Batchelor (1969) interprets the process of separation as occurring when the advection of vorticity to a point is greater than the diffusion away from it. In addition, he states that it is “an empirical fact that a steady state of the boundary layer adjoining a solid boundary cannot remain attached with an appreciable fall in the velocity of the external stream.” This is the “classical” theory that flow along an adverse pressure gradient cannot remain attached and be at steady state because of the formation of a pressure singularity within the boundary layer (Goldstein 1948).
It should be noted that “flow deceleration” and “adverse pressure gradient” are often used interchangeably, although a strict equivalence is only true in the inviscid limit. However, the fundamental ingredient for separation is a finite deceleration, rather than an adverse pressure gradient (e.g., see Smith 1982). Hence, in this paper we refer to flow decelerations rather than adverse pressure gradients.
The nonrotating flow past a cylinder is controlled by a single parameter, the Reynolds number, given by
where U is the velocity scale, L is the length scale, and ν is the fluid viscosity. As the Reynolds number is increased the flow loses its upstream/downstream symmetry and separation generally occurs for Re > 10. At the values one might expect to find in the oceanic boundary currents the flow is well into a fully turbulent regime, with typical Reynolds numbers of 1010 based upon 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 in marked contrast to the Reynolds numbers obtained in many numerical ocean models used for climate prediction, in which assuming the values as above, except a viscosity of 104 m2 s−2, gives a Reynolds number of 1.
It is our hypothesis that the presence of a finite flow deceleration may explain the high Reynolds number separation of western boundary currents, such as the Gulf Stream and Kuroshio. Indeed, modeling studies at relatively high resolution have already found the separation of a western boundary current to be coincident with the presence of an adverse pressure gradient (Haidvogel et al. 1992; Jiang et al. 1995; Baines and Hughes 1996; Tansley and Marshall 2000).1 If this hypothesis is accepted, then the determination of the conditions under which a boundary current will separate, or remain attached, reduces to attributing the presence, or otherwise, of a flow deceleration to some plausible physical mechanism. The focus here is upon only one of the possibilities reviewed by Dengg et al. (1996)—that a change in direction of the coastline can decelerate a boundary current and induce its flow separation.
Although some studies, such as Stern and Whitehead (1990), have previously emphasized the role of boundary curvature in flow separation for rotating systems, the inclusion of differential rotation can have a profound effect upon a system’s behavior (see, e.g., Boyer 1970; Merkine 1980; Boyer and Davies 1982; Page and Johnson 1990). Indeed, Stern and Whitehead explicitly state that rotation does not enter into their theory. Rather, it acts to suppress three-dimensional turbulence, in the experimental section of their study, to give a two-dimensional flow.
Bryan (1963) makes the earliest, to the authors’ knowledge, attempt at using an abrupt change in coastline orientation to induce separation of a barotropic western boundary current. Sadly this attempt fails, because of the south-facing wall of Bryan’s “barrier” having a free-slip boundary condition, rather than a no-slip condition, applied to it [as Dengg (1993) reports]. However, both Dengg and Özgökmen et al. (1997) show that, when coupled with sufficient inertia, an abrupt change in direction of the coastline can “fix” the separation point of both barotropic and baroclinic boundary currents. They also show that proximity to the line of zero wind stress curl can have implications for the separation state in a wind-driven gyre.
By area integrating the governing equations, Marshall and Tansley (2001) demonstrate that the tendency for a current to accelerate or decelerate (and hence separate) can be described by the balance of forces acting upon the flow. Most importantly, by using natural coordinates this balance of forces is made to include the flow curvature. Their results suggest that the β effect will always accelerate a western boundary current, and thus tend to keep it attached. In contrast, flow curvature can either accelerate or decelerate. They suggest that for a boundary current to separate, the radius of curvature should satisfy the following inequality:
where R is the radius of curvature of the coastline and β* is the gradient of the Coriolis parameter in the direction of the flow. This inequality serves as the basis of our investigation into western boundary current separation, by supplying two complementary hypotheses.
A sufficient increase in the strength of the north–south gradient of the Coriolis parameter, a stronger β parameter in the parlance of Tansley and Marshall (2001), can suppress flow separation.
Regardless of the strength of the β effect, the coastline can always induce separation by having sufficient curvature.
It is these twin hypotheses that we aim to test through the use of a barotropic vorticity model of a western boundary current by placing an obstacle on the western boundary. Although such a model is somewhat abstract, the problem remains simple enough that it is possible to span the parameter space (within the confines imposed by numerical considerations). We believe that our broad findings should have wider relevance to more realistic boundary currents with vertical structure and arbitrary coastlines, but such complications are not considered here.
In section 2, the numerics and setup of the chosen model are discussed, as are some of the issues surrounding the choices made. Sections 3 and 4 investigate model solutions when an obstacle is added to the western boundary at a variety of model parameters. Only long-term time-average or steady-state solutions are considered. An area-integrated vorticity budget is introduced, and derived, in section 5. The budget is then used to interpret the results of the previous two sections. In section 6, we close with a summary of our results and conclusions.
2. Model numerics and domain
a. Model equations and nondimensional parameters
A barotropic vorticity model is chosen to investigate the separation of western boundary currents by coastline curvature. This type of model eliminates the complication of bottom topography and vortex stretching, as well as any other baroclinic effects. Any influence that direct wind forcing may have is removed by using a 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 do not seek to directly simulate the Gulf Stream, or other specific boundary currents, but rather to investigate the more general problem of curvature forcing in western boundary current separation.
For this system, the Coriolis parameter is written as f = f0 + β0y, where f0 and β0 are constants and y is the north–south distance from the latitude at which f = f0. The dimensional form of the barotropic vorticity equation is then given by
where u is the fluid velocity, ζ is the vertical component of the relative vorticity, and ν is the fluid viscosity. The channel width is taken as the system’s characteristic length scale L, and a velocity scale is defined based upon the net northward transport such that UL ∼ ΔΨ, where ΔΨ is the difference in streamfunction between the east and west boundaries. The velocity scale U represents the average velocity through the channel such that UL gives the net northward transport. Equation (3) is then nondimensionalized using the following relations: (x̃, ỹ) = (x, y)/L, ũ = uL/ΔΨ, ζ̃ = ζL/U, and t̃ = (β0L)t.
The result of the nondimensionalization is given in (4) as
where δI = (U/β0)1/2 is the width of an inertial boundary current, with a higher β̂ implying a lower inertial scale (Charney 1955).
Tansley and Marshall (2001) describe the β parameter as quantifying the degree to which meridional displacement is confined and also draw relevance to ocean gyre theory. Moreover, 1/β̂ is equivalent to the Rossby number used in the simulation of the barotropic wind-driven ocean circulation (Bryan 1963; Veronis 1966; Blandford 1971; Böning 1986). In this interpretation, β̂ is seen as a measure of nonlinearity, with a higher β̂ implying that nonlinear advection is less important to the system’s evolution.
Note that the scales used to define Re and β̂ are not the same as in the flow past cylinder calculations of Tansley and Marshall (2001), who define L as the cylinder diameter. This means that solutions obtained at the same parameter values are not directly comparable. However, it is still reasonable to expect that increasing Re will increase the tendency of the system to separate, while increasing β̂ will have the opposite effect.
In ocean gyre theory, boundary layer scales are usually defined based upon inertia and different types of friction (typically bottom friction and lateral friction). The largest scale is then the one that controls the western boundary current.
The model results show that, particularly at low Re, systems with a higher value of β̂ tend to produce a steady state. Essentially, this is because δI → 0 as β̂ → ∞, and at some point the boundary current must become defined by the dissipative scale [δM = (ν/β0)1/3].
For this reason, and also because of the selected inflow condition [see section 2b and Eq. (8)], it is more instructive to divide (4) through by β̂ and write Mu = 1/β̂Re. This gives, with tildes dropped,
In this case, Mu is referred to as the “Munk number,” since
where δM 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 experiments performed at different β̂ is more instructive because varying β̂ then only changes one of the boundary layer scales. Throughout, flow visualizations will be shown with all three parameter values (β̂, Mu, and Re) quoted, although only two are independent.
b. Experimental design
Equation (6) is integrated through time over a rectangular domain with a sponge region at either end in which the flow is relaxed back toward a prescribed inflow/outflow region (the domain is illustrated in the schematic of Fig. 1). Although the obstacle appears to be in fairly close proximity to the sponges, experiments with a much longer channel showed no qualitative difference to the results presented in the subsequent sections.
where ΨI = 1 is the streamfunction in the limit x/δM → ∞. By using this functional form for the inflow, the spinup time of the model and the magnitude of the initial shock are both reduced since the inflow is close to 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 the results. 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 remain unaffected.
The general numerics of the model are the same as those of Tansley and Marshall (2001); that is, the advection term uses the energy- and enstrophy-conserving Arakawa Jacobian and the time stepping uses a leap-frog scheme with a Robert–Aselin filter. The vorticity is inverted to give the streamfunction using a multigrid inverter and both the viscous and relaxation terms are background differenced in time for stability.
Two important differences between the experiments presented here, and those of Tansley and Marshall (2001), are the form of the inflow/outflow condition and the orientation of the channel. Tansley and Marshall restrict themselves to considering flow in an east–west aligned channel with an eastward directed inflow. Furthermore, this inflow is always uniform across its width. Our predominantly northward flow is arguably in better agreement with the circumstances experienced by real-world western boundary currents. Similarly, the use of an inflow condition that closely mimics the expected structure of a western boundary current is an important step in understanding their separation dynamics.
To promote flow separation of the boundary current an obstacle is positioned halfway along the unsponged portion of the western boundary (at 750 km in Fig. 1). The obstacle’s shape is given by the following nondimensional equation:
where o(y) is the obstacle width, W is the obstacle width at its crest, ds is the meridional “decay scale” of the obstacle, and y0 is the position of the obstacle crest. This is the “Witch of Agnesi” or “turning curve” and its two parameters allow us to tune the curvature at the peak with some degree of exactitude. It is also interesting to note that two obstacles with the same aspect ratio, given by W/ds, may not have the same maximum curvature. This is because the maximum curvature of the obstacle, which occurs at its crest, varies as W/d2s. Furthermore, the aspect ratio of the obstacles will generally be greater than or equal to 1. This is in contrast to the study of flow separation and waves in a vertically stratified system (a physically and mathematically analogous one to our β plane), where obstacles will generally have an aspect ratio much less than unity (see, e.g., Smolarkiewicz and Rotunno 1989; Sha et al. 1998).
c. Boundary conditions
For simplicity, both east and west boundaries have no normal flow and no-slip boundary conditions applied to them. It is worth noting that in barotropic studies of the wind-driven ocean circulation the boundary conditions can have an important impact on the flow. In particular, the combination of no-slip boundaries and lateral friction can lead to a boundary current that is subject to shear instabilities (Bryan 1963; Blandford 1971). The choice of boundary condition is also somewhat dictated by our desire to promote separation; as Dengg (1993) shows, a free-slip solution is unable to separate because of the nature of its boundary layers.
In the case of an obstacle being present, the boundary is treated in a piecewise-constant manner, much like many general circulation models. Adcroft and Marshall (1998) raise issues regarding the formulation of boundary conditions with such a boundary, but these would seem to be less important when no-slip solutions are sought. In addition, Tansley and Marshall (2001) use the same numerics to derive accurate solutions of the flow past a cylinder in both differentially rotating and nonrotating conditions. In light of this, the outlined method of treating the boundaries would seem justified.
3. Solutions at Mu = 1/9000 (moderate dissipation)
In this section we present model solutions for β parameter values of 150 and 750 at a fixed Munk number of 1/9000 (corresponding to moderate dissipation): this gives an effective Reynolds number of 60 and 12, respectively. A range of obstacle shapes are used to demonstrate that sufficiently high curvature can overcome the β effect, at a fixed β̂, and that an increase in β̂ can suppress the flow separation, for a given obstacle shape. Since the western boundary is a streamline of fixed value (ψ = 0), to ensure no normal flow the criterion used for flow separation is outcropping of this streamline in the interior, that is, negative values of streamfunction. To highlight this, all figures of the flow have the obstacle and boundary as solid black areas, while areas with ψ < 0 are shaded a light gray. The model grid spacing is fixed at L/128, giving a dimensional value of Δx = Δy = 7.8125 km.
a. β̂ = 150
Solutions at β̂ = 150 and Mu = 1/9000 are presented in 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 increase in coastline curvature has upon the separation state of the western boundary current.
All of the panels in Fig. 2 are long-term averages of fluctuating flows. These fluctuations arise from shear instability of the western boundary current, which is an expected part of the solution when lateral friction is employed (Bryan 1963; Blandford 1971). The presence of the instability eddies can act to produce a system with a fluctuating separation state, that is, one that periodically reattaches (not shown). However, we choose to focus on the time-average characteristics of the flow separation.
The panel with a straight western boundary, Fig. 2a, clearly does not separate, as expected. The slight divergence in the streamlines, which becomes stronger to the north, is due to the shear instability of the boundary current manifesting itself in the form of eddies.
Returning to Fig. 2, it is clear that for the relatively blunt obstacle of Fig. 2d, the streamlines shift around the obstacle smoothly and the flow remains attached. However, there does appear to be evidence of streamline divergence in the obstacle’s lee. In particular, the area encompassed by the ψ = 0.2 streamline and the western boundary is larger downstream of the obstacle crest than upstream of it. This is indicative of flow deceleration and this flexing of the streamlines is a common precursor to separation in the classical problem of nonrotating flow past a cylinder (i.e., β̂ = 0, see images in Van Dyke 1982).
In Fig. 2e, the obstacle decay scale has decreased, relative to Fig. 2d, such that the obstacle’s curvature is sufficiently high to overcome the suppressive influence of the β effect. The flow cannot remain attached and so separates, creating a region of negative streamfunction in the obstacle’s lee. This region is not clearly defined. However, in Figs. 2b, 2c and 2f, the “separation bubbles” have vastly increased in size, and are now clearly and smoothly defined by the ψ = 0.0 streamline.
The “strength” of separation can be judged by inspecting the minimum value of streamfunction for the model solutions, with “stronger” separation expected to give more negative minimum values. The ψmin values shown in the bottom right-hand corner of each panel confirm what we can tell by looking at the streamfunction plots of Fig. 2: that an obstacle with W = 0.1 and ds = 0.1 produces relatively weak separation, probably best described as marginal. In addition, a wider obstacle, one with larger W, gives stronger separation than one 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 separation, which is in Fig. 2b. This is because the instantaneous flow is in a new regime where the obstacle excites strong waves in the eastern part of the channel. This creates a flow that is more strongly eddy driven than the others in Fig. 2.
b. β̂ = 750
We have seen that sufficient coastline curvature can overcome the β effect and lead to flow separation. Furthermore, this separation is a function of both obstacle parameters, that is, of curvature rather than simply width or decay scale. This section seeks to demonstrate that Marshall and Tansley (2001) are correct: that an increase in β̂ can suppress separation.
Model solutions for the same obstacles and Munk number (Mu = 1/9000), as in Fig. 2 but for β̂ = 750, are presented in Fig. 3. All of the panels in Fig. 3 are steady, in contrast to the time-average picture that Fig. 2 presents. This is due to the boundary current being stable to shear at the current set of parameters. The shear instability is a nonlinear effect, and increasing the β parameter has effectively made our flows more linear. Plus, the inertial boundary layer scale (δI) is now less than the frictional boundary layer scale (δM), implying that the current is frictionally dominated.
The effect that the increased β̂ has upon the flows is in line with our expectations; most of the flows that previously separated now remain attached. Indeed, only Fig. 3c actually separates, and this is marginally so. However, downstream of the crest of the other obstacles there is, again, flexing and divergence of the streamlines (Figs. 3b,e,f). This is less extreme for the W = 0.1, ds = 0.2 obstacle (Fig. 3d), suggesting that it is more comfortably in the parameter space of an attached boundary current.
The change in separation behavior can be viewed as occurring because of the flow being more linear, which restricts the role of advection of vorticity and allows diffusion to carry it away from the separation point. A second view is that the increase in β̂ leads to a much larger suppressive force and so prevents the separation from occurring. The third, and perhaps simpler view is that the increase in β̂ reduces the effective Reynolds number (Re = 1/β̂Mu) of the flow to only 12 from an effective value of 60 in section 3a. From the classical view of flow past a cylinder, one would expect the flow to be more likely to remain attached although, as yet, there is no way to estimate the value of the Reynolds number that might be “critical” for a given β̂ and set of obstacle parameters. All three views are equally valid and are effectively a restatement of the same physical process.
4. Solutions at Mu = 1/45 000 (weaker dissipation)
The results of section 3 suggest that increased boundary curvature or a decreased β̂ can result in flow separation. This can be seen as the deceleration due to flow curvature overcoming the stabilizing tendency of the β effect, or due to the nonlinear advection terms playing less of a role at higher β̂. The change in separation behavior can also be interpreted as being the result of a lower effective Reynolds number due to increasing β̂ at a fixed Mu. Viewing the behavior in this light suggests that decreasing Mu should increase the tendency to separate since it will raise the effective Reynolds number.
The current section will investigate the effect that a decreased Mu of 1/45 000 has upon the flow by presenting solutions obtained for both β̂ = 150 and β̂ = 750. As with section 3, the model grid spacing is fixed at L/128, and flow separation is viewed as having occurred when ψmin < 0.
a. β̂ = 150
In Fig. 4 the Munk number has been decreased by a factor of 5, relative to Fig. 2, although the same range of obstacles is used. This causes an increase in the effective Reynolds number from 60 to 300, so one would expect to get separation at a much lower obstacle curvature.
Most striking of the results in Fig. 4 is that the boundary current for the case with no obstacle is wider than at the higher Munk number (cf. Fig. 4a with Fig. 2). This is an unexpected result since δM, and thus the inflow, is much narrower. In fact, inspecting the instantaneous flow shows that a transition to a primarily eddy-driven regime has occurred. The instantaneous flow is dominated by large eddies, which act to broaden the boundary current and also considerably increase its transport.
The eddy-driven nature of this flow regime leads to a very different flavor to the results with obstacles. The recirculation on the eastern flank is greatly increased in size and magnitude, and the weak flow in the eastern side 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 sharp obstacles do not follow the logical progression of sharper obstacle leading to stronger separation. Indeed, Fig. 4f has a ψmin that is only ∼3% of the value found at the higher Munk number of 1/9000 (see Fig. 2f).
Although the details of the obstacle seem to be less important to the separation process, they still hold some relevance. The decrease in Mu has also led to the flow with a W = 0.1, ds = 0.1 obstacle (Fig. 4e) now separating. However, the particular sets of experiments do not contain enough evidence to completely elucidate the effect that altering Mu has upon the separation behavior of the system. For a more comprehensive look at variations in the Munk number, the interested reader is referred to Munday (2004).
b. β̂ = 750
Already it has been seen that separation can be suppressed by an increased β̂ and that decreasing Mu has an effect on the behavior of the boundary current. For the combination of high β̂ and low Mu one might expect results to be somewhere in between those of Fig. 3 and Fig. 4: that is, that separation is enhanced with respect to Fig. 3 but suppressed with respect to Fig. 4.
Figure 5 shows the effect of reducing Mu from 1/9000 to 1/45 000 when β̂ = 750. As with section 4a, this causes a fivefold increase in the effective Reynolds number, in this case from 12 to 60. Such an increase might well be expected to result in separated flows. However, unlike Fig. 3 all of the panels of Fig. 5 are long-term time averages of an unsteady flow.
The images contained in Fig. 5 show that the model behaves as anticipated; the boundary currents of Figs. 5a, 5b, 5d and 5e all remain firmly attached, but increasing obstacle curvature eventually overcomes the suppressive effect of β̂ on flow separation. This results in Figs. 5c and 5f being in a separated state. Streamline divergence, indicative of flow deceleration, can clearly be seen in Figs. 5b and 5e and less strongly in 5d. It is interesting to note that the obstacles in Figs. 5b and 5f have the same aspect ratio but that it is the obstacles with the lower W that gives a separated boundary current. Whereas, in section 3a, for obstacles with the same aspect ratio, the strongest separation tends to be given by the obstacle with the largest W. This difference could be due to two different obstacle shape factors, which are difficult to distinguish between and also related to each other.
First, the separation condition of Marshall and Tansley (2001) really concerns the gradient of the Coriolis parameter in the direction of the flow [see (2)]. As such, we might expect that an obstacle that causes the flow upstream of its crest to deviate farther from true north will be more likely to give separation since β* will be smaller. Second, the curvature at the crest of the obstacle is not given by W/ds, the obstacle aspect ratio, but rather by W/d2s. 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 separation. Given the extreme streamline divergence of Fig. 5b, it is reasonable to expect this flow to be very close to separation. Similarly, the small size of the separation bubble in Fig. 5f suggests that this flow is close to attachment. If this is true, then either of the shape factors could be responsible for tipping Fig. 5f over the edge into a separated state.
5. Vorticity diagnostics
As discussed in section 1 for the case of nonrotating problems, it is accepted that a flow will separate in the presence of an “adverse pressure gradient” or, equivalently, a flow deceleration. Our hypothesis is that the separation of a western boundary current can also be explained by such a flow deceleration. Indeed, several authors find adverse pressure gradients to be coincident with boundary current separation (Haidvogel et al. 1992; Jiang et al. 1995; Baines and Hughes 1996; Tansley and Marshall 2000). Furthermore, we seek to attribute the presence of the flow deceleration to increasing flow curvature induced by the presence of an obstacle on the western boundary. In this section a set of vorticity diagnostics is derived following the method of Marshall and Tansley (2001). These diagnostics are then applied to the model results of sections 3 and 4.
To attribute the separation of the western boundary current to flow deceleration, a “separation formula” must first be derived. The formula is essentially an area-integrated vorticity budget, which is integrated over a specific part of the flow. It uses the barotropic vorticity equation to partition the flow deceleration of the boundary current, and hence attribute the flow separation into terms due to flow curvature, advection of planetary vorticity, and dissipation of relative vorticity. The first step is to write the steady-state barotropic vorticity equation (6) into its nondimensional flux form:
where f = β̂y is the Coriolis parameter and the other symbols are as previously defined.
Following Marshall and Tansley (2001), (10) is integrated over a region of the flow bounded by two streamlines 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 positions of the points A, B, C, and D are considered below.
Applying the above form of integration to (10) yields
where L is the region defined by the boundary ABCD, as shown in Fig. 6.
In evaluating (11), so-called natural coordinates are used. As such, the position vector is given by (s, n), where s is the unit vector tangent to the flow and n is the unit vector normal to the flow. By convention, n points to the left of the flow and the velocity vector becomes u = (Vs, Vn), with Vn having zero magnitude but pointing in the direction of n [see, e.g., Holton (1992) for details].
The next step is to use the definition of vorticity in natural coordinates to partition vorticity into parts due to normal shear and flow curvature. In natural coordinates, relative vorticity is given by
After making use of the divergence theorem, the integrated vorticity budget becomes
where β* = ∂f/∂s is the gradient of the Coriolis parameter in the alongstream direction and [γ]DA = γD − γA. The labels will be used to refer to individual terms in the integrated budget throughout the rest of this section. It is worth noting that the above integrated budget only differs from that of Marshall and Tansley (2001) by the inclusion of a term due to friction and the lack of a term due to vortex stretching.
In the limit of weak flow along the far-boundary streamline (the path BC), we can, as per Marshall and Tansley (2001), interpret the shear term as representing the acceleration or deceleration when following the flow. The other three terms (curv, planet, and dissip) then represent the acceleration or deceleration resulting from different physical processes (curvature forcing, the β effect, and lateral friction, respectively).
The assumption of weak flow along the path BC is important to the interpretation of the budget as representing acceleration/deceleration of the boundary current. In the case of there being flow along the path BC, then shear will more correctly represent the change in kinetic energy along AD relative to the change in kinetic energy along BC. Thus, when applying the budget to the barotropic vorticity equation model results, the appropriate choice of the streamfunction along which BC is placed is crucial.
In all cases considered in the following subsections, the path AD is a segment of the ψ = 0.4 streamline. This is chosen because, in the inflow, the point of maximum current speed lies close to this value of streamfunction. Similarly, the path BC is a segment of the ψ = 1.1 streamline, because this value of streamfunction lies in the region of weak flow on the eastern flank of the inflow.
Meridional positions must also be selected for the locations of the points A and D. These are chosen more arbitrarily than the streamfunction values at which to place the paths AD and BC. In order for region 1 to encompass the area of positive curvature forcing upstream 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 negative curvature forcing downstream of the crest, the values y = 0.75 and y = 0.95 are selected for A and D, respectively.
At the value chosen for the path BC, ψ = 1.1, the change in kinetic energy is typically ≲10% of the change in kinetic energy along the path AD. In addition, the actual flow along the path BC is always considerably weaker than that along AD, implying that, even if the change in kinetic energy is significant, the integrated budget still represents why the strongest part of the flow is accelerating or decelerating relative to the weakest part. We believe that this justifies our interpretation of the integrated budget.
The above refers to a system in steady state. To construct a vorticity budget for systems that reach an unsteady final state, all of the variables must first be written as the sum of an average and a time-varying part; that is, ψ = ψ(x, y) + ψ′(x, y, t), ζ = ζ(x, y) + ζ′(x, y, t), etc. The starting point for our vorticity budget then becomes the flux form of the time-averaged barotropic vorticity equation:
where is the time-averaged eddy flux of vorticity.
The budget given in (16) produces shear, curv, planet, and dissip terms that differ from their steady compatriots [see (14)] only in the detail of them being defined by long-term averages of the quantities involved. However, a fifth term is introduced, due to the eddy flux of vorticity, and this makes a crucial difference. The regions over which the area integration is performed are selected in the same manner as for the steady-state case. In addition, the comments regarding the speed and kinetic energy along the paths AD and BC still apply.
Regardless of the final model state, every term in the vorticity balance is calculated independently. Budget closure is good, with the residual term being two to three orders of magnitude smaller than the largest term.
b. Steady state
This section presents area-integrated vorticity budgets obtained from the steady-state solutions of section 3b. The nondimensional parameters for these solutions are β̂ = 750, Mu = 1/9000, and Re = 12. The vorticity budget for these solutions only require the four terms of (14): shear, curv, planet, and dissip.
Referring back to Fig. 3a, we see that a steady-state system with no obstacle shows little or no streamline divergence/convergence. As such, its integrated budget should show little or no acceleration with the dominant balance being between the advection of planetary vorticity (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 or more significant figures, both within and between integration regions. The situation becomes more interesting when we consider a system with an obstacle on the western boundary.
In Fig. 8, the budgets obtained for the three model solutions shown in Figs. 3d, 3e and 3f (from top to bottom) are all shown. The first column is for region 1 (upstream of the obstacle’s crest), the second column is for region 2 (downstream of the obstacle’s crest). It is immediately noticeable that the budget no longer reflects a strict balance between planet and dissip. Rather, shear shows significant acceleration/deceleration, which mirrors the sign of curv, and both planet and dissip vary between regions.
In region 1 (Figs. 8a, 8c and 8e), the magnitude of dissip has increased relative to the flow without an obstacle. This increase is quite significant, amounting to ∼14%–23%, depending on the details of the obstacle. These changes are quite reasonable given that the boundary current is probably faster and/or more sheared than the straight western boundary case.
In contrast to the changes in dissip, the variations in the magnitude of planet are rather minor. At ds = 0.2, planet has increased by 2.6%, relative to the straight western boundary case. As the obstacle becomes narrower, in the along-stream direction, planet tends to return toward the straight boundary value and may decrease below it for sharp enough obstacles. These variations reflect changes in the length of the integration region (it is longer for a system with an obstacle) and the boundary current being deflected from true north, which decreases β* in (14).
The most important effect of the obstacle is seen in the steady increase of curv across the three obstacles, which gives rise to a concomitant increase in shear. The increase in curv represents both the pinching of the streamlines over the crest, which creates a faster current, and also the higher curvature of the flow. However, the increase in the magnitude of dissip partly mitigates the curvature forcing, so shear is always less than curv.
In region 2 (Figs. 8b, 8d and 8f), we find that planet has increased relative to the flow with no obstacle, more substantially than in region 1 (by ∼5%, as compared with <3%). In addition, dissip changes in magnitude much less than in region 1. For the obstacle with W = 0.1 and ds = 0.2, dissip increases in magnitude. However, for the other two obstacles, dissip actually decreases in magnitude, representing a decrease in the frictional dissipation of the flow.
As expected, curv is negative, and increases in magnitude as the obstacle becomes more sharply curved (ds decreases). However, the deceleration due to curv in region 2 is slightly less in magnitude than the acceleration due to curv in region 1, regardless of the details of the obstacle. This is linked to the divergence of the streamlines, noted previously. This divergence is a reaction to deceleration caused by the curvature forcing, and it also allows the current to select a slightly straighter path in the obstacle’s lee. Furthermore, for cases in which flow separation occurs, the observed straightening of the boundary current is often sufficient for curv to decrease below the value found for a blunter obstacle that is incapable of promoting separation at that particular combination of β̂ and Mu.
The small, but consistent, increase in planet, relative to region 1, represents the selection of a more northward path for the boundary current. The increase in planet, along with the reduced frictional dissipation, is able to partly mitigate the curvature forcing, although not to the extent that the increased dissipation compensates for the accelerative effect of curv in region 1.
c. Unsteady state
This section presents area-integrated vorticity budgets obtained from the time-average solutions of section 3a. The flow parameter values for these solutions are β̂ = 150, Mu = 1/9000, and Re = 60. The vorticity budgets 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 straight western boundary. In contrast to Fig. 7, it does not display a strict balance between planet and dissip. Rather, the flow is decelerating in both regions, which the budget confirms to be as a result of eddy activity. In addition, it is interesting to note that although planet shows little variation from the values of Fig. 7, the magnitude of dissip is significantly smaller. As a result, the eddies partly act to dissipate relative vorticity, in order to balance planet, as well as to broaden and decelerate the flow.
Figure 10 shows the budget results for Figs. 2d, 2e and 2f (from top to bottom). As with the straight boundary case, the budget diverges from the zero order balance of dissipation of relative vorticity compensating for advection of planetary vorticity. An immediately obvious difference to the budgets shown in 5b is that it is now possible for the magnitudes of shear and curv to be greater than the magnitudes of planet and dissip. This is due to the change in value of the β parameter, rather than the transition to an unsteady final state, and reflects the nonlinear advection terms becoming more important to the flow’s evolution.
In region 1, the addition of an extra term to the budget has little effect on the patterns of variation described in section 5b. Principally, planet and dissip both show deviations from the straight western boundary values. These deviations are manifestations of the current being turned away from true north, and so on, which are described above. Similarly, the addition of the obstacle introduces curvature forcing to the system, creating positive values of curv and shear.
In addition, the magnitude of eddy, for all three obstacles, is significantly less than for the case of a straight western boundary. It would appear that the longer coastline, and the increase in frictional dissipation, may well have aided in stabilizing this section of the boundary current.
In region 2, the variations of the individual budget terms are much the same as in section 5b, except that eddy has a much larger impact than in region 1. For example, as the streamlines in the obstacle’s lee undergo divergence, the boundary current becomes more northward in nature. This causes planet to increase. Similarly, the changes in the currents structure causes dissip to become smaller in magnitude as ds does. The variations in curv clearly demonstrate the loss of forcing achieved when separation occurs; that is, when ds is changed from 0.1 to 0.05 and the boundary current separates, curv decreases. This is just as described in section 5b.
The remaining terms in the budget, shear and eddy, both behave differently. Unlike in region 1, eddy now becomes an important part of the budget. Although it changes relatively little when ds decreases from 0.2 to 0.1, a further decrease of ds to 0.05, eddy more than doubles in magnitude. Similarly, shear continues to increase, even when curv does not, and it appears that the actual deceleration of the boundary current is now heavily linked to curv and eddy.
The large increase in eddy activity, which occurs with a fully separated boundary current, is a result of how variable the path of the boundary current is in the instantaneous case. Essentially, with the attached, or marginally separated, cases the instantaneous position of 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 or cause a flexing of the streamlines. However, in the cases where the time-average boundary current is fully separated, the eddies can actually change the position of the instantaneous boundary current quite considerably. This can result in it being fully attached or even more separated. As a result, the eddy term in the time-average budget becomes much larger.
The size of eddy, for the fully separated cases, could lead to the suggestion that it is this term, rather than curv, that leads to the separation. However, the straight western boundary cases also display eddy activity, but never separates. Thus, we believe that the increase in the magnitude of eddy is caused by the act of separation rather than the other way about.
d. Summary of results
This section is a brief summary of the results obtained from the area-integrated vorticity budgets. It is intended to draw attention to the most important conclusions.
The area-integrated vorticity budgets show that, upstream of the obstacle crest, changes in flow curvature lead to an acceleration of the boundary current. This remains true regardless of whether the system reaches steady state or is a long-term time average. In most steady-state cases, the increase in dissipation due to a narrower, more curved, and faster boundary current results in some mitigation of the curvature forcing. In contrast, in a time-average state, the interaction of the integral effect of eddies with the dissipation term can be complex. This can lead to the acceleration of the boundary current being enhanced with respect to the applied curvature forcing.
Upstream of the obstacle crest, curvature forcing applies a deceleration to the boundary current. In the case of a time-average state, the interaction between the integral effect of eddies and the other terms in the budget can lead to a complex pattern of changes. In particular, the role of dissipation can be completely subverted by the eddies because of the current having widely varying position over the course of the eddy growth cycle. However, curvature forcing is always negative, and always present, when the western boundary is not straight.
In both steady-state and time-average states, the boundary current is able to react to the flow deceleration prior to separating. This leads to the magnitude of the deceleration, in this region, growing more slowly than the acceleration upstream of the crest. Furthermore, the act of separation removes some of the curvature forcing and leads to the deceleration of the current being substantially less than one might expect.
If one considers both integration regions together, then the smaller magnitude of curv in region 2 will result in there appearing to be a net acceleration of the boundary current. Despite this, the curvature forcing can still cause flow separation because separation is a very local process. It is not necessary to decelerate the boundary current along its entire length, merely in a small region. The resulting separation point will always be close to the region of deceleration.
The general problem of western boundary current separation, and specifically the separation of the Gulf Stream from the eastern seaboard of the United States, has given birth to a number of theories as to the mechanism for such separation. Classically, separation occurs at high Reynolds number when flow is along an adverse pressure 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 with more complex shelf geometry (Tansley and Marshall 2000), adverse pressure gradients have been found to be coincident with flow separation. This is also true in the physical laboratory experiments of Baines and Hughes (1996). It should also be noted that Cessi (1991) has found that, for the case of colliding western boundary currents, separation can occur before the adverse pressure gradient is encountered. However, these results were obtained at relatively low Reynolds number and Kiss (2002) argues that the observed separation in the results of Cessi is occurring in a region of adverse pressure gradient.
In this paper we have investigated the separation of a western boundary current via a flow deceleration, using a barotropic vorticity model in an idealized configuration. This removes any influence that the wind stress pattern might have, as well as the complications that baroclinicity and three dimensions bring. The experiments presented represent a departure from those that previously considered the influence of boundary curvature upon rotating flows (e.g., Stern and Whitehead 1990), because of the explicit impact of differential rotation upon the process of separation. The use of a north–south aligned channel and an inflow condition duplicating the structure of a “mature” western boundary current is also crucial in furthering our understanding of the dynamics surrounding the separation of western boundary currents.
Our results confirmed the twin hypotheses of Marshall and Tansley (2001):
an increase in the gradient in Coriolis parameter can suppress flow separation, and
sufficiently strong boundary curvature can overcome the inhibiting effect of the β parameter and induce flow separation.
Furthermore, reducing the Munk number, that is, decreasing dissipation, clearly has an impact upon the stability of the solutions and the separation states attained. In the presented solutions, it is unclear as to the exact dependence. However, in the extended range of experiments used by Munday (2004), it is clear that, for steady-state solutions, a decrease in Munk number makes flow separation more likely.
The use of integrated vorticity budgets illuminates several aspects of the model solutions. First, they attribute stronger flow acceleration upstream of the crest, and stronger deceleration downstream of the crest, to increasing obstacle curvature. Second, they show that the “stretching” of streamlines prior to separation, and the selection of a straighter path subsequent to separation, results in weaker curvature forcing downstream of the crest. Third, they show that curvature forcing can be partly mitigated by changes in the relative magnitude of other terms in the budget. In addition, relatively small changes in the magnitude of planet seem to be related to deviations in flow direction from the true north. In the case of long-term time-average solutions, a large increase in the magnitude of eddy accompanied separation. This is caused by the large changes in boundary current position made possible when the solid boundary is some distance from its western flank.
Marshall and Tansley (2001) suggest that there may be a critical value of coastline curvature, as in (2), beyond which a given flow should separate. For the results presented here, the curvature at the peak of the obstacle is generally less than that predicted by (2). However, the inequality does appear to hold, broadly, if we introduce a dimensionless constant, that is,
Calculations based upon our results indicate that α lies between 0.05 and 0.1.
One attraction of the idea that separation may be induced by changes in flow curvature is that it may explain the steadiness of the Gulf Stream’s separation point. For example, the separation point of the Gulf Stream varies in north–south position by only ±50 km (Auer 1987; Gangopadhyay et al. 1992). In contrast, the separation points of the Brazil Current and the Malvinas Current vary in along coast position by ∼930 and ∼850 km, respectively (Olson et al. 1988). Certainly, our steady-state and long-term time-average results show flow separation is always initiated close to the obstacle’s crest. However, the fixing of a separation point in a time-varying sense is one that requires further research, probably with more complex models to allow the inclusion of topography, baroclinicity, and perhaps wind forcing. In particular, Marshall and Tansley (2001) were able to include the effects of vertical stratification in their integrated vorticity budget. This suggests that it has an important role to play in setting the net acceleration/deceleration of a western boundary current.
The issue of experiments at higher Reynolds number, in order to better simulate the real environment, is also important. At present, most ocean models have an unrealistically high dissipation, which may affect the separation process and explain why it has been consistently misrepresented.
This work was supported by NERC. The comments of one of the anonymous reviewers led to a significant improvement to the analysis and the paper as a whole.
Corresponding author address: Dr. David R. Munday, Department of Meteorology, Earley Gate, University of Reading, Reading, Berkshire RG6 6BB, United Kingdom. Email: firstname.lastname@example.org
At low Reynolds number, colliding western boundary currents are found to separate before the adverse pressure gradient is reached (Cessi 1991). However, Kiss (2002) disputes the interpretation of Cessi and argues that the observed flow separation is occurring in a region of adverse pressure gradient.