a. f plane

We first consider a constant Coriolis parameter (i.e., β̂ = 0). The solutions correspond to those obtained in a nonrotating frame (e.g., Batchelor 1967; Van Dyke 1982) since the absolute value of the Coriolis parameter does not enter into the barotropic vorticity equation (3). For each solution we show snapshots of the streamfunction (ψ, defined such that ũ = k × ∇ψ) and the vorticity (q = <;ht;gz> + β̂ỹ).

A series of three solutions at different Reynolds numbers are shown in Fig. 2. First, at Re = 1 (Fig. 2a), the flow is steady and nearly symmetric upstream and downstream of the cylinder. There is no separation, but a closer inspection reveals slightly enhanced deflection of the streamlines downstream of the cylinder, associated with weak advection of the boundary vorticity anomalies by the background flow. At Re = 25 (Fig. 2b), the boundary vorticity anomalies are advected farther downstream, leading to separation and an enclosed region in which there are two weak recirculating eddies. As the Reynolds number increases the area of the downstream separated region increases and reaches a maximum at Re ≈ 50 (not shown); this compares favorably with the equivalent solution obtained at Re ≈ 41 in the classical nonrotating limit within an infinite domain. At Re > 50, the flow is unsteady with counterrotating eddies being shed periodically downstream of the cylinder to form a von Kármán vortex street (Fig. 2c).

b. β plane

We now allow the Coriolis parameter to vary linearly in the meridional direction. In Fig. 3 we first consider a relatively low Reynolds number. For small values of β̂ (Fig. 3a) there is little change from the f-plane case (Fig. 2b). This is to be expected since the wavelength of the standing Rossby waves (2πδ_I) is larger than the width of the channel (4L). As β̂ increases, upstream Rossby waves become apparent (White 1971). Figure 3b shows the flow for a value of β̂ = 7.5. The wavelength of the standing Rossby waves in this case corresponds to approximately 2L. As the value of β̂ is increased further (Fig. 3c), the wavelength of the downstream Rossby waves decreases and a western boundary current appears on the downstream face of the cylinder. The flow now fails to separate. There is direct analogy (e.g., Page and Johnson 1990) between these solutions and Long's model for stratified flow over a hill (Long 1955).

In some numerical calculations with negligible Ekman pumping, Merkine (1980) showed that the presence of β shifts the region of adverse pressure gradient downstream, and thus inhibits flow separation. For a value of β̂ = 1, the adverse pressure gradient shifts by approximately 20°, while for β̂ = 4 there is a 45° shift. The tendency of the β effect to suppress adverse pressure gradients has also been demonstrated analytically by Marshall and Tansley (2001). The suppression of the adverse pressure gradient can been seen in the numerical solutions through the bunching of streamlines on the downstream face of the cylinder as β̂ increases, indicating acceleration rather than deceleration of the boundary layer flow.

At the same time, a stagnant, blocked region appears upstream for high values of β̂. A similar blocking effect was found in both linear analysis and laboratory experiments by Hide and Hocking (1979). Page and Johnson (1990) describe how this region is formed by propagation of Rossby waves away from the cylinder so that eventually steady Rossby waves with a negative group velocity exactly balance the oncoming flow in the blocked region. Foster (1985) shows that the only steady-state solution for this upstream region is one of no flow. This is consistent with ocean gyre theory in which stable boundary currents are able to form only on the western margins of basins (e.g., Pedlosky 1987).

In Fig. 4 we show the equivalent results for a larger Reynolds number, Re = 200. At the lowest value of β̂ = 0.75 the flow remains similar to the nonrotating case, with counterrotating vortices shed periodically downstream of the cylinder (Fig. 4a). At a moderate value of β̂ = 7.5, the downstream convergence of streamlines is more pronounced compared with the lower Reynolds number case. Two meandering jets form downstream of the cylinder, with a small separated region in between (Fig. 4b). Finally, in the case in which β̂ = 75, two intense separated jets again form downstream of the cylinder. Compared with the previous case, the jets are narrower and hence penetrate a shorter distance due to the enhanced dissipation (Fig. 4c). In both of the above cases the flow fields are steady, indicating that these jets are generated by separation of the boundary layer rather than through turbulent processes.

c. Higher Reynolds number

In our model there is an upper limit to the size of the Reynolds number we can achieve, set by the requirements of numerical stability. (The details of what sets this upper limit are complex, depending on the relative importance of advection and the β effect.) One solution to this problem is to introduce biharmonic dissipation, as is often done in high-resolution ocean general circulation models. The nondimensional barotropic vorticity equation (3) becomes

View Expanded

where

View Expanded

is an “effective Reynolds number” associated with the biharmonic dissipation and K is a biharmonic dissipation coefficient. At large scales R₄ is very large, whereas at the scale of the model grid R₄ is typically O(1), indicating that the biharmonic dissipation is efficient at dissipating grid-scale structures that are inadequately resolved by the model, while leaving larger-scale structures alone.

In Fig. 5a we show a solution in which biharmonic dissipation (R₄ = 3 × 10⁵) is added to the case shown in Fig. 4c. The downstream separated jets are still very clear, but the effect of introducing the biharmonic dissipation is to slightly reduce their zonal extent, as would be expected. In Fig. 5b we show a solution in which the Reynolds number is increased to 500. The penetration of the jets downstream of the cylinder is vastly increased compared with the previous case. Finally, in Fig. 5c the Reynolds number is increased further to 1000, resulting in jets that virtually extend across to the sponge. Also note the presence of secondary jets to the north and south of the cylinder. The structure of these jets is clearer in the mean streamfunction field (Fig. 6), which indicates a transition from four to six jets a distance 4L downstream of the cylinder.

While the central jets are initially formed through separation of the boundary layer, the mechanism responsible for their extension, and for the existence of the secondary jets, is similar to the organization of freely decaying turbulence described by Rhines (1975). At higher Reynolds number, the Rossby waves break downstream of the cylinder to form a geostrophic turbulent wake, evident from the filamentation in the vorticity fields in Figs. 5b and 5c. In geostrophic turbulence there is a direct cascade of vorticity to small spatial scales, but an inverse cascade of energy that is arrested in the meridional wavenumber at the Rhines scale. Consequently, energy accumulates at the Rhines scale, resulting in a series of zonal jets. The role of eddies in maintaining the jets is confirmed in Fig. 7, in which we show the first two terms on the right-hand side of the time-mean vorticity equation,

View Expanded

Here overbars and primes denote mean and transient components respectively, and tildes have been dropped in the interests of clarity. The eddy vorticity flux convergence (Fig. 7a) contains a complex structure in the wake of the cylinder, but there is a clear tendency for the eddies to add vorticity to the north of the jets (unshaded contours) and to remove vorticity south of the jets (shaded contours); this pattern is consistent with acceleration of the mean flow. In contrast the effect of the viscous term (Fig. 7b) is the opposite; that is, it decelerates the mean flow. The dominant balance is between these two terms and the advection of vorticity by the mean flow, with the biharmonic dissipation being of secondary importance.

In section 4 we will demonstrate that the generation of these inertial jets has important implications for understanding and modeling the separated Gulf Stream.

d. Larger β̂

With the addition of a small amount of biharmonic dissipation, we can also explore solutions with larger values of β̂, corresponding to a smaller Rhines scale. In Fig. 8, we show a series of solutions in which β̂ is increased to 750, resulting in further inhibition of separation downstream of the cylinder. For a Reynolds number of 25, the flow does not separate (Fig. 8a). For a higher Reynolds number of 200, vortices form at the “shoulders” of the cylinder and the flow momentarily separates but then reattaches (Fig. 8b). If we increase the Reynolds number further to 1000, the small-wavelength Rossby waves fill the channel (Fig. 8c). The finite model resolution prevents us from decreasing the dissipation to the point that wave breaking can occur in these solutions. If the model resolution were increased and the dissipation appropriately reduced, we anticipate that Rhines jets might be obtained downstream of the cylinder, as in Fig. 5, although the number of jets would be increased due to the larger β̂. It is unclear whether the formation of such jets would modify our conclusions concerning the inhibition of separation at large β̂; this issue requires further study.

e. Summary

To summarize, the key results of relevance to the ocean are the following.

• In order for an eastward current to separate downstream of an obstacle, it is necessary to have both a sufficiently large Reynolds number and a moderate β parameter. At higher Reynolds numbers, the Rossby waves break downstream of the cylinder, leading to the formation of separated, inertial jets.

• For larger values of β̂, a stagnant region forms upstream of the obstacle, flanked by two inertial jets, and separation is inhibited downstream.

a. Gulf Stream separation

A perennial problem with ocean general circulation models is the failure of model boundary currents to separate from coastlines. In particular, models have great difficulty in simulating the separation of the Gulf Stream from the North American coastline at Cape Hatteras [for a review, see Dengg et al. (1996)]. Nevertheless, the most recent simulations at 0.1° or higher lateral resolution show improved separation behavior (Paiva et al. 1999; Smith et al. 2000). To demonstrate the application of our results to Gulf Stream separation, we modify the model domain to include a quarter cylinder with straight boundaries upstream and downstream (Fig. 9), mimicking the geometry of the North American coastline at Cape Hatteras.

We first consider the problem directly analogous to the cylinder calculations in which a uniform flow is prescribed across the sponge layer. Here the Reynolds number and β parameter are defined using the relaxation velocity within the eastern sponge, and the diameter of the equivalent cylinder. These conventions allow direct comparison between the solutions and earlier results.

In Fig. 9a we show a solution with a moderate Reynolds number (Re = 100) and a high β parameter (β̂ = 750), similar parameters as in Fig. 8. There is a broad similarity between the two solutions, except that the Rossby waves are damped to a slightly greater extent in the present case, most likely due to the stabilizing influence of the solid coastline boundary. The flow completely fails to separate. In Fig. 9b we show the solution in which the β parameter is reduced by an order of magnitude (Re = 100, β̂ = 75). The boundary flow forms a broad jet, that initially separates but quickly diffuses into a broad flow and reattaches to the coastline. There is some similarity with the solution in Fig. 5a, but with reduced Rossby wave activity due to the slightly lower Reynolds number and the influence of the coastline. Finally, in Fig. 9c, we additionally increase the Reynolds number by an order of magnitude (Re = 1000, β̂ = 75). Now a tighter jet forms adjacent to the boundary and separates cleanly. The separated jet penetrates a considerable distance downstream, although not as far as in the corresponding cylinder solution (Fig. 5c). There is a weak recirculation gyre to the north of the separated jet, somewhat reminiscent of that found north of the separated Gulf Stream. The filamentation in the vorticity field again emphasizes the importance of eddies in maintaining the separated jet. An analysis of the eddy vorticity flux convergence confirms that the eddy term is O(1) in the time-mean vorticity balance (not shown).

Finally, we illustrate the application of these results to boundary current separation by relaxing to a uniform flow only within the northern half of the channel. The nondimensional parameters are again evaluated using the relaxation velocity within the eastern sponge.

First, in Fig. 10a, with a low Reynolds number and high β parameter (Re = 100, β̂ = 750), the flow fails to separate and a damped Rossby wave forms against the cape, in a similar manner to the modeled Gulf Stream in many low-resolution ocean models [e.g., the 0.28° resolution experiment shown in Smith et al. (2000, Fig. 4b)]. In Fig. 10b we show the equivalent experiment with a larger Reynolds number and lower β parameter (Re = 1000, β̂ = 75). Here the boundary current separates and meanders propagate downstream, in a similar manner to observations of the Gulf Stream and the highest-resolution numerical simulations [e.g., the 0.1° resolution experiment shown in Smith et al. (2000, Fig. 4a)]. Filamentation in the vorticity field again indicates that eddies are active in the vorticity budget of the separated boundary current (not shown).

In comparison for the Gulf Stream, we estimate the following parameters: β ∼ 2 × 10⁻¹¹ m⁻¹ s⁻¹, a radius of curvature in the coastline at Cape Hatteras of L/2 ∼ 2.5 × 10⁵ m and U ∼ 10⁻¹ m s⁻¹ (note that the velocity of the separated boundary current is an order of magnitude greater than the relaxation velocity used to evaluate β̂ in Fig. 10b). These parameters give β̂ ∼ 50, a value that is most consistent with the separated solution in Fig. 10.

The above experiments emphasize the importance of achieving both a moderate β parameter (β̂ ∼ 10–100) and a sufficiently high Reynolds number (Re > 200) for a boundary current to separate. The role of the geostrophic turbulent cascades in forming the downstream separated jets in these calculations also points to an important role for geostrophic eddy fluxes in maintaining the separated Gulf Stream.

b. Antarctic Circumpolar Current

For a higher value of β̂, our results are relevant to the Antarctic Circumpolar Current, and in particular its interaction with topographic obstacles such as the Kerguelen Plateau. In Fig. 11 we reproduce a portion of the pressure field at 120 m from the Fine Resolution Antarctic Model (FRAM) simulation (Webb et al. 1991). These pressure contours are streamlines for the geostrophic flow. Upstream of the plateau there is a clear bunching of pressure contours into an inertial jet, whereas downstream of the plateau the pressure contours remain attached to the obstacle and feed into a series of standing Rossby waves.

Using the FRAM data we can estimate a typical Reynolds number and a β parameter. For consistency with our cylinder calculations, we identify the length scale with the north–south extent of the Kerguelen Plateau giving L ∼ 1.5 × 10⁶ m, and using geostrophic balance we estimate a mean zonal velocity of U ∼ 0.1 m s⁻¹. At the latitude of the plateau we have β ∼ 1.5 × 10⁻¹¹ m⁻¹ s⁻¹ and the viscosity employed in the simulation is ν = 200 m² s⁻¹. These values give a Reynolds number of Re ∼ 750 and a β parameter of β̂ ∼ 300, corresponding most closely to Figs. 8b and 8c. There are indeed many features in common between the character of the flow in these calculations and the circulation in the FRAM simulation, including the inertial jet upstream of Kerguelen Plateau, the attached streamlines downstream of the plateau, and the downstream standing Rossby waves. In an idealized study of Southern Ocean dynamics, using a two-layer model and similar values of β̂ and Re, we obtain similar jets upstream of the seamount together with a upstream blocked region (Tansley and Marshall 2001). These circulations also bear some similarities to the simple models of Webb (1993) and Pedlosky et al. (1997).