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NOTES AND CORRESPONDENCE
On Sverdrup Discontinuities and Vortices in the Southwest Indian Ocean
J. H. LACASCE
Institute for Geophysics, University of Oslo, Oslo, Norway
P. E. ISACHSEN
Norwegian Institute for Water Research, Oslo, Norway
(Manuscript received 28 June 2006, in final form 15 March 2007)
ABSTRACT
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.
1. Introduction
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;
Corresponding author address: J. H. LaCasce, Institute for Geophysics, University of Oslo, P.O. Box 1022, Blindern, 0315 Oslo,
Norway.
E-mail: j.h.lacasce@geo.uio.no
DOI: 10.1175/2007JPO3652.1
© 2007 American Meteorological Society
JPO3154
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;
DECEMBER 2007
NOTES AND CORRESPONDENCE
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
2941
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 ␤ 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):
⭸
⳵
⭸ 2
ⵜ ␺ ⫹ ␤ ␺ ⫽ ⵱ ⫻ ␶ ⫺ ␦ⵜ2␺ ⫽ ⫺ ␶ x共y兲 ⫺ ␦ⵜ2␺.
⭸t
⳵x
⭸y
共1兲
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 ␺ is the
velocity streamfunction, ␤ is the scaled meridional derivative of the Coriolis parameter, ␶ ⫽ ␶xi is the wind
stress, and ␦ is the scaled bottom drag. The boundary
conditions are ␺ ⫽ 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
␺I ⫽
␶ 共yN兲 ⫺ ␶ 共yS兲
共xE ⫺ xM兲
yN ⫺ yS
共2兲
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 (␦ K 1), the solution method
follows Stommel (1948) for a wind-driven basin. As-
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 37
FIG. 1. The (left) Sverdrup streamfunction in the analytical model, derived from the (right)
wind stress curl. The contour spacing is 0.25, the solid lines indicate positive values (counterclockwise circulation), and the dashed lines indicate negative values (clockwise circulation).
The curl is assumed to be invariant in longitude.
suming 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 ␦. 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, how-
ever, 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 ␦1/2. For example, the
streamfunction correction to the west of the African tip
can be shown to be
冋
册
1
y⫺g
⭸␶ x共g兲
erfc ⫾
␺ˆ ⫽ ⫾ 共xE ⫺ xM兲
,
2
⭸y
2公␦共xM ⫺ x兲
共3兲
DECEMBER 2007
NOTES AND CORRESPONDENCE
2943
FIG. 2. The full solution to the steady version of (1), with ␦ ⫽ 5 ⫻ 10⫺4. Velocity vectors
have been superimposed. The western boundary layers are very narrow and difficult to see in
the figure.
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
1
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).
2
There are, in addition, narrow boundary layers of thickness ␦
at the tips of Africa and Madagascar (e.g., Gill 1968; De Ruijter
1982). These affect the flow very near the tips but do not alter the
transport in the zonal currents.
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
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VOLUME 37
FIG. 3. The solution to (1) in the case that the zero wind stress curl line in the Indian Ocean
lies to the north of Madagascar. Note that the westward jet from the southern tip of the island
has essentially vanished.
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, ␤ ⫺ 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 ␤ ⫺ 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
FIG. 4. The Rayleigh–Kuo quantity, ␤ ⫺ Uyy, for the solution in
Fig. 2. The solid contours are positive and the dashed negative.
Regions where ␤ ⫺ Uyy changes sign satisfy the necessary condition for barotropic instability.
DECEMBER 2007
NOTES AND CORRESPONDENCE
2945
FIG. 5. The ROMS model domain. The contours indicate the SSH from a linear run, at day 200. The
ripples in the South Atlantic are residual Rossby waves, propagating westward. The color scale is ⫾1 m.
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
3
In idealized geometries, such an increase can favor an Agulhas
retroflection, even with a flat bottom (e.g., Dijkstra and De
Ruijter 2001), but this effect is less pronounced with realistic
coastlines (Matano 1996; Dijkstra and De Ruijter 2001).
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
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FIG. 6. A close-up view of the Agulhas region from a nonlinear run, after the separated current goes
unstable. Note the formation of a large anticyclone and an accompanying, smaller cyclone. The color scale
is ⫾1 m.
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
4
Isolated anticyclonic vortices in the Southern Hemisphere
drift northwest due to self-advection on the ␤ 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.
DECEMBER 2007
NOTES AND CORRESPONDENCE
2947
FIG. 7. A close-up view of the Madagascar region during the same nonlinear run shown in Fig. 6. The
separated zonal currents off the north and south tips of the island are forming vortices. The color scale is
⫾0.5 m.
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,
5
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 (␷ is
small in the unperturbed jet).
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
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FIG. 8. Longitudinal spectra as a function of time, calculated from the wavelet routine in
Matlab from the meridional velocities in the core regions of the three jets: (left) north Madagascar jet, (middle) south Madagascar jet, and (right) Agulhas jet. The spectra have been
normalized individually.
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 largescale 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;
DECEMBER 2007
2949
NOTES AND CORRESPONDENCE
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.
6
The lateral damping in our simulations was sufficiently weak
for retroflection, according to the criterion of Dijkstra and De
Ruijter (2001). Our nondimensional Ekman number was E ⫽
AH(2⍀r 20)⫺1 ⫽ 1.7 ⫻ 10⫺9.
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.
Acknowledgments. 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.
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