The southwest Indian Ocean is distinguished by discontinuities in the wind-driven Sverdrup circulation. These connect the northern and southern tips of Madagascar with Africa and the southern tip of Africa with South America. In an analytical barotropic model with a flat bottom, the discontinuities produce intense westward jets. Those off the northern tip of Madagascar and the southern tip of Africa are always present, while the strength of that off southern Madagascar depends on the position of the zero curl line in the Indian Ocean (the jet is strong if the line intersects Madagascar but weak if the line is north of the island). All three jets are barotropically unstable by the Rayleigh–Kuo criterion. The authors studied the development of the instability using a primitive equation model, with a flat bottom and realistic coastlines. The model produced westward jets at the three sites and these became unstable after several weeks, generating 200–300-km scale eddies. The eddies generated west of Madagascar are in accord with observations and with previous numerical studies. The model’s Agulhas eddies are similar in size to the observed eddies, both the anticyclonic rings and the cyclones that form to the west of the tip of South Africa. However, the model’s Agulhas does not retroflect, most likely because of its lack of stratification and topography, and so cannot capture pinching-off events. It is noteworthy nevertheless that a retroflection is not required to produce eddies here.
The Agulhas Current separates from the African coast and retroflects, continuing eastward into the Indian Ocean. Roughly six times per year the current pinches off an energetic anticyclonic (counterclockwise) ring. These are typically 100–300 km wide and extend deep into the water column (Lutjeharms and Gordon 1987; Olson and Evans 1986; Gordon and Haxby 1990; Van Aken et al. 2003). The rings drift west-northwest into the South Atlantic Ocean, decaying as they go (Byrne et al. 1995; Grundlingh 1995). Cyclones also form in the region, typically to the west of the retroflection (Penven et al. 2001; Boebel et al. 2003; Lutjeharms et al. 2003; Matano and Beier 2003). The cyclones are somewhat smaller than the rings and subsequently drift west–southwest into the South Atlantic (Boebel et al. 2003).
The situation off southern Madagascar is similar, with the southward-flowing Southeast Madagascar Current leaving the coast (Stramma and Lutjeharms 1997; Schott and McCreary 2001) and generating large (order of 100–200 km) eddies. Satellite images suggest that both signs of vortex form here (Quartly and Srokosz 2002; De Ruijter et al. 2004; Quartly et al. 2006) and many of these eddies subsequently drift westward toward Africa. A significant fraction later merge with the Agulhas Current (Grundlingh 1995; Schouten et al. 2002a).
Eddies also form west of the northern tip of Madagascar. The Northeast Madagascar Current leaves the northern tip and flows westward to join the East African Coastal Current (Stramma and Lutjeharms 1997; Schott and McCreary 2001). Current fluctuations with a period of 50 days and scales of several hundred kilometers were identified in the region by Quadfasel and Swallow (1986). These fluctuations were probably related to the 200-km scale anticyclones that form here, some or all of which drift south into the Mozambique Channel (De Ruijter et al. 2002; Ridderinkhof and De Ruijter 2003; Schouten et al. 2002b; Quartly and Srokosz 2004). These “Mozambique eddies” account for a large portion of the highly variable transport in the Mozambique Channel (Ridderinkhof and De Ruijter 2003).
The southwest Indian Ocean is thus a region of substantial variability. A number of authors have modeled the currents here, both analytically and numerically. The focus has been mostly on the retroflection of the Agulhas Current. In his seminal study of the linear, wind-driven circulation in the southern Indian/Atlantic basins, De Ruijter (1982) showed that a linear Agulhas Current does not retroflect but, instead, proceeds westward from the tip of South Africa into the South Atlantic. He suggested that inertia is required for the Agulhas to join the eastward wind-driven flow farther south. Subsequent analytical and numerical studies, however, showed that topography, stratification, current volume, and coastal orientation may also be important for retroflection (Boudra and De Ruijter 1986; Ou and De Ruijter 1986; Boudra and Chassignet 1988; Matano 1996; Biastoch and Krauss 1999). It is accepted nevertheless that the retroflection is responsible for Agulhas ring formation.
Numerous questions remain, however, about the formation of the many other vortices observed in the region. For instance, the dynamical origin of the cyclones that form northwest of the retroflection is not understood, nor do we know precisely why vortices form south of Madagascar. Lutjeharms (1988) suggested that the Southeast Madagascar Current retroflects on leaving the island and thus can pinch off anticyclones, like the Agulhas. However, it remains controversial whether the current does, in fact, retroflect (De Ruijter et al. 2004; Quartly et al. 2006; Palastanga et al. 2006), and cyclones are also found here (Grundlingh 1995; De Ruijter et al. 2004).
The Northeast Madagascar Current on the other hand does not retroflect, flowing westward after separation and generating large anticyclones. Quadfasel and Swallow (1986) and Schott et al. (1988) suggested that the eddy formation here resulted from barotropic instability of the separated boundary current because the observed 50-day oscillations could not be linked to the winds. Biastoch and Krauss (1999) also concluded that barotropic instability is important here, based on an energetics analysis of a numerical simulation of the region.
Hereinafter we consider a possible origin for the eddies in the region. We point out that the southwest Indian Ocean and the South Atlantic exhibit several discontinuities in the wind-driven Sverdrup function. These discontinuities, which are a consequence of the basin geometry, join the northern and southern tips of Madagascar with Africa and the southern tip of Africa with South America. Using a barotropic analytical model, we find that the discontinuities produce intense westward jets that are barotropically unstable. Simulations using a barotropic primitive equation model confirm this, displaying 200–300-km scale vortices at all three sites.
2. Analytical model
We illustrate the idea by using an idealized model of the Indian–South Atlantic basins, extending from South American to Australia (e.g., Fig. 1). Our model closely resembles that of De Ruijter (1982) except that we include a surrogate Madagascar island and use Cartesian coordinates (the solution in spherical coordinates is similar). The flow obeys the linear shallow water equations on the β plane, is driven by a zonal wind stress that varies only in y, and is damped by linear bottom (Ekman) drag. The flow thus obeys the barotropic vorticity equation (e.g., Pedlosky 1987):
We have nondimensionalized the variables using the north–south basin length and an advective time scale and imposed a rigid lid and flat bottom. Here ψ is the velocity streamfunction, β is the scaled meridional derivative of the Coriolis parameter, τ = τxi is the wind stress, and δ is the scaled bottom drag. The boundary conditions are ψ = 0 at the lateral walls and on the African continent. The streamfunction on Madagascar is also constant but is not necessarily zero. We determine the constant from Godfrey’s (1989) “island rule,” which, assuming a wind stress that varies only in y, can be written
in nondimensional form. Here yS and yN are the southern and northern latitudes of Madagascar and xe and xm are the positions of the eastern (Australian) boundary and of Madagascar.
With weak bottom drag (δ ≪ 1), the solution method follows Stommel (1948) for a wind-driven basin. Assuming a steady flow and neglecting friction in (1) leaves the Sverdrup relation. We integrate this westward from the eastern boundaries until striking a western boundary; the boundary condition is then satisfied in a frictional layer of thickness δ. The currents that occur in these western boundary layers represent the model’s Agulhas, East Madagascar, and Brazil Currents. For forcing we use the wind stress curl shown at the right in Fig. 2, which mimics the actual distribution here (e.g., Trenberth et al. 1989). The curl is negative in the south, positive at midlatitudes, and weakly positive in the north.
The resulting Sverdrup streamfunction is contoured in Fig. 1. The large-scale flow broadly resembles the currents in the region [as also noted by Godfrey (1989) and Schott and McCreary (2001)]. There are two and one-half gyres, with northward flow into the southern Indian Ocean and southward flow in the northern Indian Ocean. The flow in the mid Indian is westward toward Madagascar (as indicated in Fig. 2), and this is the model’s South Equatorial Current. South of the Indian Ocean the flow is eastward, originating where the Falklands Current leaves the South American coast. This corresponds to the northern portion of the Antarctic Circumpolar Current (ACC), which is purely Sverdrupian owing to the closed boundary south of Australia.
The central point for the present discussion, however, is that there are discontinuities in the streamfunction between the northern and southern tips of Madagascar and Africa and between the southern tip of Africa and South America. The discontinuity west of Africa has long been known (Welander 1959; Evenson and Veronis 1975; Godfrey 1989). Those off Madagascar have not been discussed but are visible, for example, in Godfrey (1989, Figs. 4 and 5). The discontinuities are the result of the streamfunction being “reset” at the African continent and the island. For instance, the streamfunction increases in the Indian Ocean westward from the eastern boundary but is reset to zero at Africa; it increases again in the South Atlantic. But south of Africa the streamfunction is not reset and continues to increase. So, there is a north–south discontinuity in the South Atlantic at the latitude of the African tip. Similar comments apply to the northern and southern tips of Madagascar, although the streamfunction is reset to the island constant rather than zero.
In the analytical model the discontinuities are smoothed out by bottom friction, which permits meridional boundary layers of thickness δ1/2. For example, the streamfunction correction to the west of the African tip can be shown to be
where the positive (negative) solution applies where y is greater (less) than the latitude of the African tip, y = g. Similar solutions are found west of the northern and southern tips of Madagascar (while also taking into account the nonzero island streamfunction). The layers spread out, as in a diffusive layer,1 in the westward direction (Fig. 2). The greater the bottom friction, the more pronounced the westward spreading. Using a different type of friction, like momentum diffusion, alters the boundary layer width but not its character. The boundary layer from the African tip in Fig. 2 is similar to that of De Ruijter (1982), despite his having used momentum diffusion.2
Associated with the zonal discontinuities are intense jets that link the western boundary currents. The northward Sverdrup flow entering the Indian Ocean to the east of Africa is carried in the jet connecting South Africa to South America whence it continues southward. The intensity of the jets depends both on the strength and the position of the wind stress curl. In the case shown in Fig. 2, the model’s South Equatorial Current splits into northward and southward branches on the western side of Madagascar. The two jets from island tips are accordingly of similar strength. As such, the circulation around the island is weak and the island constant is near zero; indeed, one obtains a flow nearly identical to that in Fig. 2 if one sets the island constant to zero.
The situation changes though if the zero curl lines shift north or south, as in Fig. 3 where the zero curl line lies to the north of the island. Now there is a strong cyclonic circulation around the island, and this weakens the southern jet and strengthens the northern one. We recover the southern jet if we set the island streamfunction to zero, showing that the island circulation is important here. The actual South Equatorial Current spans a range of latitudes, so such a weakening of the southern current is conceivable. In addition, the flow from the southern tip is eastward in Fig. 3, producing an apparent retroflection of the Southeast Madagascar Current (e.g., Lutjeharms 1988).
Note too that the jet from South Africa in Fig. 3 is substantially weaker than that in Fig. 2. This is because the zero curl line in the Southern Ocean has shifted northward so that there is less inflow into the Indian Ocean. The eastward flow south of the continent has also shifted northward, toward Africa, causing a partial retroflection of the model Agulhas. However, the model Agulhas lacks a full retroflection, and this is its primary deficiency (as noted previously by De Ruijter 1982).
Because the separated boundary currents are mostly zonal, we can evaluate their stability using the Rayleigh–Kuo criterion. This requires that the meridional gradient of the mean potential vorticity, β − Uyy, change sign for instability (e.g., Pedlosky 1987). We use the model’s zonal velocities to calculate this directly and plot the results in Fig. 4. All three zonal currents are sites of rapid variation in β − Uyy, with the largest changes occurring just to the west of the island and peninsula tips. With stronger friction, the variations are more confined to the tip regions because the fanning out of the currents decreases the meridional shear farther to the west. This suggests that the tip regions will be the most unstable, so eddies should form soon after the currents have detached from the boundaries.
The Rayleigh–Kuo criterion is a necessary rather than a sufficient condition for instability; therefore, this does not guarantee eddy formation. The flow, moreover, is not purely zonal. A more detailed stability analysis is of course possible, but it would be complicated because of the westward spreading of the jets and moreover would depend on the choice of bottom friction. So we chose instead to examine numerical solutions. These will permit us to observe the instabilities develop to finite amplitude and will also allow realistic coastlines, something that influences the character of the separating flow (Ou and De Ruijter 1986; Matano 1996; Dijkstra and De Ruijter 2001).
3. Numerical simulations
The model is the Regional Oceanic Modeling System (ROMS; e.g., Shchepetkin and McWilliams 2005). We configured this in a wedge-shaped basin (Fig. 5), with walls at the equator, 60°S, 60°W, and 120°E and with realistic Africa and Madagascar coastlines. The domain closely resembles those used by Matano (1996) and Biastoch and Krauss (1999). The maximum resolution was 0.12°, somewhat finer than that of Biastoch and Krauss and sufficient to resolve the 100-km eddies.
We simulated only the barotropic mode, with a free surface and a flat bottom. The depth was set at 1000 m; this is obviously shallower than the actual ocean, but hastens the model spinup. Using a shallower ocean has two effects: First it alters the barotropic deformation radius, which in turn changes the speed of gravity waves generated during spinup. Second, the velocity in the shallow-water equations is inversely proportional to the fluid depth (Gill 1982), so decreasing the depth by a factor of 5 magnifies the effective wind stress by the same amount (of course the fluid transport is independent of the total depth).3 In any case, we conducted additional experiments with different depths and obtained essentially the same results as described hereafter.
For forcing, we used the annual-mean Trenberth wind stress (Trenberth et al. 1989). The bottom friction coefficient was 10−4 m s−1, which with a depth of 1000 m implies a spinup time O(100 days). We used both Laplacian and biharmonic friction, to prevent small-scale instabilities, with relatively small viscosities (10 m2 s−1 and 106 m4 s−1, respectively).
a. Linear calculation
We first ran the model without advection, for comparison with the analytical model. The sea surface height (SSH) from a representative run is shown in Fig. 5. The ocean spins up from rest and Rossby waves, radiating from the eastern boundaries, propagate westward through the domain. The wave activity gradually decreases, as the waves are damped by bottom drag and small-scale dissipation at the western boundaries, and the model settles into an approximate steady state. The flow then resembles that seen in the analytical model. The southernmost gyre is the strongest (because of the greater longitudinal extent of the basin and the strength of the winds) while the gyre in the northern Indian is fairly weak.
There are strong meridional gradients in SSH west of Africa and Madagascar, as expected. Although not indicated in the figure, the flow is westward both off the tips of Madagascar and South Africa. So, the linear run conforms well to the analytical model, suggesting that the model setup is adequate for testing the stability of the boundary currents.
b. Nonlinear calculation
The gyres spin up in a similar fashion in the nonlinear runs. The difference is that the three zonal jets become unstable, with the instabilities manifesting themselves first as meanders in the currents and then as distinct vortices. Both cyclonic and anticyclonic eddies form in all three currents near the tips of Madagascar and Africa (Figs. 6, 7).
The Agulhas region after the onset of instability is shown in Fig. 6. A large anticyclone has pinched off from the tip of South Africa and accompanying it is a smaller cyclone, which forms in the lee of the continent. Both types of vortex subsequently drift westward into the South Atlantic.4
The Madagascar region at the same time is shown in Fig. 7. There are zonal currents flowing westward from both the northern and southern tips, and both are unstable. In the north, there is an anticyclone to the west of the island tip, and comparably sized cyclones forming to the north of the jet. All of the vortices drift westward toward Africa. Upon reaching Africa, the anticyclones proceed southward into the Mozambique Channel. Anticyclones are visible off the African coast, although they are decaying.
To gauge the size of the emerging vortices, we used the wavelet transform to calculate spatial spectra along the axes of the three jets. For this we used the meridional velocity, which is a more sensitive indicator of vortex motion than, say, the height. We calculated the transform over a range of latitudes and longitudes in the core of the jet and then averaged the result to obtain a longitudinal spectrum.5 We did this at each time, and contour the result in Fig. 8. The contours for the north Madagascar jet are on the left, those for the south jet in the middle, and those for the Agulhas jet on the right. The contours are normalized so that they have comparable amplitudes (otherwise the spectra from the more energetic Agulhas would have the largest amplitudes).
In all cases, the instabilities appear after roughly 60 days. The emerging eddies are roughly 200 km in size in the two Madagascar jets and nearer to 300 km in the Agulhas jet. The scales remain roughly constant while the eddies intensify, at least over the period shown. There are also indications of energy at other scales, particularly larger. The energy at 300 km, which occurs near day 140 in the north Madagascar case, reflects a large anticyclone that has pinched off the island. Nevertheless, the instability proceeds similarly in the three jets, albeit with consistently larger scales in the Agulhas case.
There is nevertheless a difference in the way the vortices form here. The north Madagascar and Agulhas jets generate anticyclones soon after separation but the south Madagascar jet has a more meandering appearance. These differences stem from inertia and are thus affected by coastal orientations and boundary current transport. The largest transport is in the Agulhas Current, which carries the entire inflow to the Indian basin. Its energetic southwesterly separation from the African coast favors anticyclone formation. The northwestern coast of Madagascar has a more north–south orientation, causing the model’s northeast Madagascar Current to flow nearly northward as it detaches. It turns to the west and then south before proceeding westward and this also favors anticyclone formation. The southeastern coast is instead oriented west-southwest, which allows the model’s southeast Madagascar Current to join smoothly with the westward flow. So, the instability proceeds more gradually here. Note that, if the currents were made to follow the shelf break rather than the coast, the separation details could be altered further.
As mentioned above, the spinup period is marked by Rossby wave propagation through the domain (Fig. 5). Additional Rossby waves appear later on in the simulations. These are excited in part by the local variability from the zonal jets and also by perturbations that propagate along the boundaries. The waves perturb the zonal currents and cause the generation of even larger eddies by introducing larger-scale disturbances on the jets. Similar perturbations may occur in the ocean, however, because the Indian Ocean lacks southern and eastern walls and, because the bottom is not flat, its large-scale waves are likely to be quite different. It is for this reason that we chose to focus on the earlier development.
The numerical simulations thus support the conclusions from the analytical model: that the wind generates three zonal currents and these are regions of significant eddy generation. The principal shortcoming is that the model’s Agulhas Current exhibits at best a weak retroflection; there is eastward flow to the south, in the model’s ACC, but only a fraction of the detached Agulhas joins that flow. Perhaps as a result of this difference, the model forms Agulhas eddies too frequently (at a rate of roughly 15–20 per year).
4. Summary and discussion
We have considered the wind-driven, barotropic circulation in the southern Indian and Atlantic Oceans, neglecting topography. The Sverdrup streamfunction here has three zonally oriented discontinuities, emanating from the northern and southern tips of Madagascar and the southern tip of Africa. An analytical calculation suggests that these discontinuities generate westward jets that are barotropically unstable. We confirmed this using a primitive-equation model with a single layer and a flat bottom. The model produced 200–300-km scale eddies, of both signs, at all three locations.
The southwest Indian Ocean is stratified and has complex bottom topography, so it is not obvious that a barotropic, flat bottom model is applicable here. Nevertheless, there are similarities between the present results and observations. The Northwest Madagascar Current does separate from the northern tip of Madagascar and proceed westward toward Africa, and anticyclonic eddies form in the area and drift southward into the Mozambique Channel (e.g., De Ruijter et al. 2002; Ridderinkhof and De Ruijter 2003). As noted, several previous authors have suggested that barotropic instability is the likely cause of the eddy formation here, from observations (Quadfasel and Swallow 1986; Schott et al. 1988) and from analyses of model energetics (Biastoch and Krauss 1999). So, the present model is in accord with those findings.
The situation south of Madagascar is more controversial. Lutjeharms (1988) suggested the Southeast Madagascar Current retroflects south of the island and proceeds eastward. Others (e.g., Quartly et al. 2006) maintain that the current actually flows westward but that anticyclone formation yields a false impression of a retroflection. It is also possible that a portion of the current retroflects, as suggested by Schott and McCreary (2001). Nevertheless, the satellite-derived observations of De Ruijter et al. (2004) reveal a “dipole street” of vortices extending from south Madagascar to Africa, and this closely resembles the situation shown in Fig. 7. The numerical simulations of Biastoch and Krauss (1999) exhibited vortex formation along the Madagascar coast, before the current even separated. However, our analytical model suggested that the instability should be greatest near the island tip, so this is also possibly consistent. In addition, we found that the southern jet can vanish if the zero wind stress curl line shifts north of the island (Fig. 3); therefore, the current may not be present at all times. This may account for some of the disagreement with regard to the observations.
On the other hand, the retroflection of the actual Agulhas Current is undisputed. We might have expected a retroflection in our numerical model, given the results of Dijkstra and De Ruijter (2001), who suggest that inertia alone can produce a retroflection.6 We found, instead, that most of the flow proceeded westward. A similar result was obtained by Matano (1996), who showed that the southwest orientation of the South African coast favors a smoother connection to the zonal flow than does a meridional wall (see also Ou and De Ruijter 1986; Chassignet and Boudra 1988; Dijkstra and De Ruijter 2001). Matano obtained a significant retroflection only after he included realistic topography. We conducted additional (barotropic) simulations with topography but these, in fact, produced too severe topographic steering. So, for example, the Southeast Madagascar Current proceeded southward along the Madagascar plateau rather than turning westward. Evidently one also requires stratification to reduce the bottom velocities, and hence the topographic steering, for a proper retroflection.
In any case, we cannot realistically claim to simulate Agulhas ring formation with this flat-bottom model. Agulhas rings pinch off from the retroflection, and modeling studies (e.g., Boudra and Chassignet 1988; Biastoch and Krauss 1999; Matano and Beier 2003) suggest that baroclinic instability is important in this. This may, in turn, explain why the formation rate is slower than observed here if the baroclinic growth rate is slower than the barotropic one.
Nevertheless, there are two points worth emphasizing. First, the present model may help explain the formation of the secondary vortices, particularly the cyclonic vortices seen to the west of South Africa (Penven et al. 2001; Boebel et al. 2003; Lutjeharms et al. 2003; Matano and Beier 2003). Second, the model demonstrates that eddies will form south of Africa even without a retroflection (at odds with Pichevin et al. 1999). Eddies are therefore an inevitable consequence of the wind-driven circulation in this geometry.
Thanks are given to Will De Ruijter, Peter Jan van Leeuwen, and two anonymous reviewers for comments on the manuscript. The work was supported by grants from the Norwegian Research Council.
Corresponding author address: J. H. LaCasce, Institute for Geophysics, University of Oslo, P.O. Box 1022, Blindern, 0315 Oslo, Norway. Email: firstname.lastname@example.org
The boundary layer equation can be converted to a diffusion equation by converting the x derivative to a time derivative. The same type of layer is found in Gill’s (1968) model of the ACC and at the meridional boundaries in solutions of wind- or thermally forced flows in a basin (e.g., Pedlosky 1969).
Isolated anticyclonic vortices in the Southern Hemisphere drift northwest due to self-advection on the β plane, and cyclones similarly drift southwest (e.g., Sutyrin and Flierl 1994). Such motion is observed in the South Atlantic (Boebel et al. 2003). The vortices are prevented from crossing each other in the model because of the westward zonal flow.
We used the Morlet wave in the Matlab software package. The scale shown is one-half of the wavelength produced by the wavelet routine, which yields a reasonable estimate of the scale of the emerging vortices. The meridional velocity is preferable to the SSH because it is a better indicator of the growing instability (υ is small in the unperturbed jet).