The Dynamics of North Brazil Current Retroflection Eddies

Journal of Marine Research, 54, 3553,1996

The dynamics of North Brazil Current retroflection eddies

by Hong Ma1

ABSTRACT This paper applies analytical and numerical methods to study possible causesof the formation and translation of North Brazil Current retroflection eddies.It showsthat the dynamicresponse of the equatorial westernboundary of the Atlantic Oceanto either a remote forcing which is representedby an incoming Rossbywave packet that deepensthe equatorial thermocline, or to the wind forcing over the tropical Atlantic Ocean can result in northwestward translating eddies up the coast of Brazil. The numerically simulated eddies closely resemble the features of the observed North Brazil Current retroflection eddies. The formation mechanismof theseeddies has to do with the short equatorial Rossbywaves (which can be generated by the said kinds of forcing), nonlinearity, and the boundary effect. The translation mechanismof theseeddies is related to the nonlinear effect causedby the eddies’ interaction with the boundary and to the variation of the Coriolisparameter with latitude.

1. Introduction In recent years, it was discovered that anticyclonic eddies frequently are pinched off from the retroflection region of the North Brazil Current which is a low-latitude western boundary current (Johns et al., 1990; Didden and Schott, 1993; Richardson et al., 1994; Fratantoni et al., 1995). The size of these eddies is about 200-400 km in diameter, and they drift northwestward along the coast at about 10 cm/s. Observa- tions indicated that the retroflection eddies extended down to 500-1000 m and contributed significantly in transporting northward South Atlantic water as well as high-nutrient Amazon water (-3 Sv) into the Caribbean. The mechanisms of formation and alongshore translation for these retroflection eddies have yet to be identified. The dynamical response of the equatorial ocean to seasonal and interannual variations in wind stress can generate vigorous wave motions in the equatorial waveguide. Many important equatorial phenomena can be interpreted with the concept of equatorial waves. For example, Lighthill (1969) pointed out that the Somali Current was a result of a “wave packet” of current pattern reaching the western boundary of the equatorial Indian Ocean. The waves to which Lighthill referred were short equatorial Rossby waves. Reflections of low-frequency equato-

1. Departmentof AppliedScience, Bldg. 490D, Brookhaven National Laboratory, Upton, New York, 11973,U.S.A. 35 36 Journal of Marine Research 15491 rial waves at the ocean’s meridional boundaries also were investigated by Moore (1968) Cane and Gent (1984) and McCalpin (1987). Ma (1992,1993) showed numerically that, with a high Reynolds number, the short equatorial Rossby waves which were generated in the reflection process of a long equatorial Rossby wave packet can form energetic vortices at a straight north-south western boundary. Among studies on the interaction of a vortex with a wall, the behavior of a dipole vortex pair approaching a plane surface at right angles in a nonrotating system was theoretically studied by SatTrnan (1979). He found that the vortices must approach the wall monotonically in the absence of viscous effects. As an experimental study, Van Heijst and Flor (1989) showed that when viscosity is present, vortices tend to rebound from the wall. The rebounding effect of the vortices was numerically simulated by Orlandi (1990), which not only confirmed, but also helped to explain, the observed rebounding effect. This paper applies analytical and numerical methods to study the translation mechanism of the retroflection eddies at the Atlantic equatorial western boundary (the coast of Brazil), and the possible cause of the formation of these eddies by short equatorial Rossby waves which are part of the response of this boundary to either a remote forcing from the ocean interior or to direct wind forcing. Comparisons were made between field observations and the results of the present study.

2. Governing equations The present study uses the following nondimensionalized shallow water equations

a~ a~ av z+"yg+vy+Yu= (2.2)

; + $ [u(l + h)] + ; [v(l + h)] = 0 where R, is the Reynolds number, l/R, = AH(UL)-1; F, and FY are the zonal and meridional wind stress terms, respectively. The equatorial nondimensionalization scales are defined as

I&y) = E-“4a(x,y) (2.4) H,,h’ = h (2.5)

Tt = E”4(2LR)-1t (2.6) u(u, v) = (gHo>“2(u, v) (2.7) 19961 Ma: North Brazil Current retrojlection eddies 37

Figure 1. Illustration of the coordinates. where the Lamb number E = 4C12a2(gH,,)-1, a is the radius of the earth, 0 = 2~ day-l, and Ho can be either the mean depth of the ocean for a barotropic model, or the so-called “equivalent depth” for a reduced gravity model. If we choose the equivalent depth Ho = 40 cm, then L = 295 km and T = 1.71 days.

3. Theoretical analysis In this section, we analyze why an eddy can translate along the equatorial western boundary, and how some factors affect this translation. For mathematical simplicity, we shall only investigate an idealized situation with an isolated eddy (not an eddy pair) on the equatorial p plane, in the neighborhood of a generalized western boundary. For an eddy at the equatorial western boundary, which is formed in the process of the western boundary’s reflection of incoming long equatorial Rossby waves, results of the present study only apply to the behavior of the eddy after the reflection takes place, since the wave reflection process is not considered here. If we rotate the coordinates in such a way that the y’ axis coincides with the western boundary (Fig. l), then the expressions for the new coordinates, x’ and y ‘, are

x’ = xcosa + ysincu y ’ = -xsincu + ycosol. (3-l) All equations from hereon are expressed in the new coordinates. We shall omit the primes, ‘, for the sake of simplicity. If nonlinearity is weak, Eq. (2.3) shows that low-frequency, shortwaves are nearly nondivergent. Then, we can introduce a stream function, +, so that

*x = v, -$ = u. (3.2) 38 Journal of Marine Research [54,1

The impulsive force per unit mass (Lamb, 1932) can be derived as follows

(3.3)

Zy = - J1”, dy &V2+ dx. (3.4)

The center of a vortex system, (i, j), is determined (Lamb, 1932; Saffman, 1979) by

(3.5)

(3.6) where I? has the definition

I- = s-;dyrV2JIdn. (3.7)

In the absence of viscosity, we can derive the vorticity equation on the equatorial p plane as av2* at + - aJI 2 . V(V2* + xsincu + ycoscu) = 0. (3.8) i ay ' ax1 It is easy to show that the area-averaged relative vorticity is a constant, i.e., dI’ -= dt ’ (3.9 in deriving which we applied Eq. (3.8), and assumed

4JI(X,Y) = 0 (3~) E afi (3.10) where XI is the domain boundary. From Eqs. (3.3), (3.4) and (3.8) we obtain

(3.11) + za* V2* - ycoscu z a* + ysinol - a* dx ay I

(3.12) - $ V2* - xcoso 2 + xsincu.?jj dx. I 19961 Ma: North Brazil Current retroflection eddies 39

Following principles similar to those applied by Saffman (1979) and assuming that r/~ and its derivatives vanish at infinity, we derive the translation of the center of a vortex on the equatorial p plane

(3.13)

According to Eq. (3.13) on the equatorial p plane, excluding viscosity, the center of an eddy approaches the generalized western boundary monotonically (d.F/dt < 0). Unlike the case studied by Saffman (1979) of a dipole vortex pair in a nonrotating system, where the vortex’s translation toward the wall is entirely due to the presence of the vortex’s mirror image, the cause for an eddy’s shoreward translation in the present study is caused by the p-effect. The smaller the angle 01 (angle between the western boundary and the north-south direction), the faster the center of the vortex approaches the boundary. This conclusion applies to both cyclonic and anticyclonic eddies on either side of the equator, since the vorticity and the stream function are always opposite in sign. If the orientation of the coastline does not change as the center of an eddy becomes closer and closer to the western boundary, it can make the shape of the eddy more and more elongated. With viscosity, however, the center of an eddy cannot approach the western boundary indefinitely. This is due to the fact that the viscosity effect prevents the width of a western boundary current from becoming indefinitely narrow (Lighthill, 1969). Eq. (3.14) shows that there are two distinctive mechanisms which can cause the alongshore translation of a vortex: the nonlinear effect caused by the eddy’s interaction with the western boundary, and the l3 effect.The nonlinear effect tends to cause an anticyclonic eddy to translate toward a higher latitude, and a cyclonic eddy a lower latitude. The p effect, on the other hand, always tends to enhance a westward translation, and whether it causes a tendency for an eddy to move toward a higher latitude or a lower latitude depends on the orientation of the coastline, (Y. Therefore, the roles of these two mechanisms sometimes enhance each other, sometimes undermine each other. This explains why the eddies formed at the equatorial Atlantic western boundary translate predominantly northwestward to the northern side of the equator, but not southeastward to the southern side of the equator. In the case of a north-south western boundary (CX= 0), an eddy’s alongshore translation speed is due entirely to the nonlinear effect caused by the eddy’s interaction with the boundary, as in a nonrotating system (Saffman, 1979). Equatorial short Rossby waves, which are generated as part of the equatorial western boundary’s response to equatorial low-frequency longwaves, can form retroflection currents at the equatorial western boundary (Lighthill, 1969). If the 40 Journal of Marine Research [54,1

retroflexing currents form an anticyclonic circulation, the above analysis shows that this retroflexing system has a tendency to translate northwestward in the northern equatorial B plane, or southwestward in the southern equatorial p plane. Due to the conservation of absolute vorticity, the associated water column is expected to gain relative vorticity by being displaced to a higher latitude. Therefore, the original retroflexing “currents” can become more and more eddy-like as their center translates alongshore toward high latitudes. There is a case where the alongshore translation of eddies to higher latitudes is unlikely. If an incoming gravest (first meridional mode) long Rossby wave packet is related to the shoaling of the thermocline, the resulting retroflexing currents caused by the short equatorial Rossby waves tend to form a cyclonic circulation on each side of the equator at the western boundary. Using a similar analysis as above, we can show that the centers of these two circulation systems move toward each other (i.e., toward the equator), instead of translating toward higher latitudes. This behavior was observed by the author in an earlier numerical simulation (Ma, 1992). In a nonrotating system, these two approaching eddies then should translate in the offshore direction along the equator. However, in a rotating system, the tendency to westward translation due to the B effect, as shown in Eq. (3.13), is going to partially or entirely overcome the offshore translation caused by the nonlinear interaction between the approaching eddies. Eventually then, the energy of the short equatorial Rossby waves is going to be consumed locally at the equatorial western boundary. This is probably why all the observed eddies that were drifting along the Guianan coast toward the Caribbean were anticyclonic ones. As a verification of the theoretical results given above, we use Eq. (3.14) to calculate the translation speed for North Brazil Current retroflection eddies, based on the observed features of these eddies (Fratantoni et al., 1995). We use average values for the stream function and vorticity as

V2* = - F (anticyclonic)

where h = 7.5 cm, 7 = 700 km, r = 100 km and v,, = 0.6 m/s are the area averaged sea surface height anomaly of a North Brazil Current retroflection eddy, the characteristic distance between the center of the eddy and the equator, the characteristic radius of the eddy, and the eddy’s maximum (averaged over depth) swirling velocity, respectively. By substituting the above values into Eq. (3.14) which should be converted into its dimensional form, and assuming that the average velocity along the part of the western boundary next to the eddy to be &!? v,,, we obtained a northwestward 19961 Ma: North Brazil Current retroj?ection eddies 41

70"N

35"

35"s ,j"w $0" 25" iO"E Figure 2. The spectralelement mesh for the Atlantic Ocean. alongshore translating speed of 18 cm/s, of which 9 cm/s is due to the p effect (o = 30”) and 9 cm/ s is due to the nonlinear effect. This theoretically predicted alongshore translating speed is close to the observed one of 8-16 cm/s for the North Brazil Current retroflection eddies (Fratantoni et al, 1995). If everything remains the same, except that the anticyclonic eddy is located south of the equator instead of being north of the equator, the alongshore translation speed would then be zero, because the translation tendency caused by the nonlinear effect is virtually canceled by that caused by the l3 effect. This is consistent with the numerical simulation results in the following section.

4. Numerical simulations Having analyzed the mechanism of the translating eddies at the tropical western boundary, we carried out two numerical simulations of eddy formation and translation at the western boundary of the tropical Atlantic Ocean. The first simulation is an initial value problem without direct wind forcing, in which the energy input for the equatorial western boundary comes from the ocean interior through the mechanism of the equatorial Rossby waves (“remote forcing”). In the second simulation, the model ocean was driven by the climatological wind stress over the Atlantic Ocean. In 42 Journal of Marine Research

70”N

35”

35”s

(4 Figure 3. Contour plots of upper-layer thickness displacement. Contour interval = 0.009. Labels scaled by 1000. (a) t = 10 (17 days); (b) t = 20 (34 days); (c)t = 30 (51 days); (d) t = 40 (68 days); and (e) t = 90 (153 days). both simulations, the energy dissipation is through horizontal eddy viscosity,& = 2 * lo6 cm2 s-l. The numerical model used in the present study is the spectral element model for the shallow water equations (Ma, 1993). The high convergence rate and geometrical flexibility of this spectral element numerical model are especially suited for capturing the vigorous eddy activities at the western boundary due to the short Rossby waves, and also for handling the realistic geometry of the tropical Atlantic. Figure 2 shows the spectral element mesh for present simulations. With recent development of parallel algorithms for the spectral element method, this model achieved an impres- sive performance rate on massively parallel computers (Ma, 1995).

a. Simulation results with “remote forcing”. In this simulation, we turn off the wind forcing and start the simulation with a gravest equatorial Rossby wave packet which initially is located in the central equatorial Atlantic Ocean. The asymptotic solution of this equatorial Rossby wave packet was given by Boyd (1980). Since this wave 19961 Ma: North Brazil Current retroflection eddies 43

70"N

60" i5" iO"E

(b) Figure 3. (Continued) packet initially is far enough from the western boundary, it propagates westward at a speed about 0.77 m s-l as predicted by the theory (Boyd, 1980). The fundamental frequency associated with this wave packet is seasonal. The maximum thermocline displacement and its equivalent sea surface height anomaly related to the initial Rossby wave packet under the current scaling system are 17 m and 8.5 cm, respectively. It is possible for an equatorial Rossby wave of this magnitude to be generated by wind in the real ocean. This particular initial condition is not a must in order to serve the main purpose of this section, i.e., to show that equatorial Rossby wave energy generated in the ocean interior can be a cause for the formation of the North Brazil Current retroflection eddies. We may use an initial equatorial Rossby wave packet which has a different shape and spatial scale from those of the present one, as long as the wave packet’s reflection from the western boundary produces short equatorial Rossby waves which form anticyclonic circulations. The chosen initial condition, however, has the advantages of being mathematically clean and able to create a localized disturbance which is easily adaptable to the numerical boundary conditions. 44 Journal of Marine Research [54,1

70"N

60" i5" 1'O"E Cc) Figure 3. (Continued)

Upon reaching the coast of South America (Fig. 3a), the equatorial Rossby wave packet undergoes a reflection process during which the equatorial Kelvin wave and the short equatorial Rossby wave components are generated. While the relatively fast equatorial Kelvin wave transports the bulk of the energy away from the western boundary toward the east, the short equatorial Rossby waves form “the North Brazil Current” and its retroflection current (Fig. 3b). Under the nonlinear effect and the 6 effect, as analyzed earlier, the center of the anticyclonic circulation formed by the North Brazil Current and its retroflection current translates northwestward along the Brazilian coast. These effects eventually cause the tip of the retroflection currents to be pinched off and to form an anticyclonic eddy near 7N, 55W (Fig. 3~). The maximum swirl speed of this newly formed eddy is 72 cm/s. Because of the northwest orientation of the Brazilian coast, the energy flux at the western boundary that responds to the initial westward-propagating Rossby wave packet should stream northeastward (Longuet-Higgins, 1964).This fact leaves much less short equatorial Rossby wave energy in the southern equatorial western boundary region than in the northern part. Furthermore, as explained in Section 3, the p 19961 Ma: North Brazil Current retroflection eddies

70"N

35% i5" 1'O"E (4 Figure 3. (Continued) effect weakens and, sometimes, eliminates the southeastward translation of an anticyclonic eddy on the southern side of the equator (Figs. 3b-d). Besides the part of the energy carried by the short equatorial Rossby waves in the western boundary region, the rest of the energy, which was originally carried by the Rossby wave packet, was transported eastward by equatorial Kelvin waves. Part of the energy of the equatorial Kelvin waves would be transported poleward by coastal Kelvin waves once the equatorial Kelvin waves arrive at the eastern boundary (Figs. 3b-d). By day 153 (t = 90), the first (most westerly) anticyclonic eddy has translated northwestward about 1350 km, and reached the Caribbean (Fig. 3e). However, the eddy’s strength is much weaker than when it was first formed because of the effect of viscosity. In the numerical simulation, the average northwest translation velocity for the eddies is 10.2 cm/s. The observed translation velocity of the North Brazil Current retroflection eddies is between 9 cm/s (Richardson et al., 1994) and 15 cm/s (Didden and Schott, 1993). The maximum diameter of the eddies along the Brazil coast in the numerical simulation is about 400 km, which is gradually reduced to about 200 km as the eddies translate northwestward toward the Caribbean. 46 Journal of Marine Research [54,1

70"N

35"

35"s SO" i5" iO"E (e) Figure 3. (Continued)

The above simulation results are in good agreement with the observations re- ported by Richardson et al. (1994) and by Didden and Shott (1993). For the Kelvin wave part of the equatorial western boundary response, we found that the realistic boundary shape of the Atlantic Ocean had only minor impact on the eastward energy flux carried by the equatorial Kelvin wave, compared to the results of a rectangular basin. This is consistent with a more general conclusion made by Cane and Gent (1984) namely, that accounting for the slant of the coast makes very little difference to the Kelvin wave energy generated by the gravest Rossby mode. For comparison, we also carried out a linear simulation and a simulation with an incoming Rossby wave packet that is related to a shoaling of the thermocline. Neither of these two simulations yielded retroflection eddies that could translate northwestward along Brazilian coast (compare Figs. 4 and 5 with Fig. 3~). This finding agrees with the analysis in the previous section. b. Simulation results with wind forcing. In this section, the numerical model is forced by the climatology wind stress over the Atlantic Ocean. The Servain wind stress data Figure 4. Contour plot of upper-layer thickness displacement for the linear simulation. t = 30 (51 days). Contour interval = 0.009. Labels scaled by 1000.

70"N

35”

35"s 9'5"W $0" 2-5" 1b"E Figure 5. Contour plot of upper-layer thickness displacement for the simulation with an incoming Rossby wave packet that is related to a shoaling of the thermocline. t = 30 (51 days). Contour interval = 0.009. Labels scaled by 1000. 48 Journal of Marine Research [54,1

8 10 20 38 40 50 t

Figure 6. The spatially integrated kinetic energy (over the entire model ocean) as a function of time.

are used for this simulation. We set “sponge layers” north of 30N and south of 20s to absorb the energy “leaked” to high altitudes by coastal Kelvin waves so that they cannot come back to the western equatorial ocean, which they can never achieve in real oceans due to various dissipation processes. The climatological wind stress for December is “switched on” at time t = 0 when the ocean is at rest. From the evolution of the spatially integrated kinetic energy (Fig. 6) we can see that the ocean is spun up within a time period oft = 15 (25 days), which is about the time it takes for the equatorial Kelvin wave to travel across the Atlantic. In the equatorial western boundary region, the North Brazil Current is gradually built up with the energy deposited there by the equatorial Rossby waves. Figures 7a and 7b show that while this current is gaining strength with time, it also becomes narrower and narrower. It takes about 35 nondimensional time units (60 days) for the North Brazil Current to become concentrated in a width of 100 km. Similar results were also found by Cane and Sarachik (1977) and Cane (1979) in their studies of the response of a bounded equatorial ocean to wind stress. Figures 7a and 7b also show that the North Brazil Current has a counter-current on its eastern side, which developed under a similar fashion as the North Brazil Current but much weaker. The results of the numerical simulation indicate that the counter-current is fed by the retroflection of the North Brazil Current, which occurs near 7N. This countercurrent eventually turns offshore near 3N to feed the eastward flowing North Equatorial Countercurrent. These results about the North Brazil Current and its countercurrent are consistent with the observations made by Flagg et al. (1986). 52W 47 42 37 32 longitude (4

0.15

c3 0.10 :: 5 a z 0.05

g ‘C B.BE E

-a.05

-0.18 52W 47 42 37 32 longitude

Figure 7. Velocity profiles along 5N between 52W and 32W at different points of the time integration (in nondimensionalunits). (a) Zonal velocity. (b) Meridional velocity. 50 Journal of Marine Research [54,1

70"N

35"

35"s 95"W SO" iO"E (4 Figure 8. Contour plots of upper-layer thickness displacement with Servain’s wind stress data. Contour interval = 0.011. (a) t = 25 (42.5 days); (b) t = 50 (85 days); (c) t = 100 (170 days).

By day 42 (t = 25) of the numerical simulation, an anticyclonic eddy is formed at the retroflection region of the North Brazil Current (Fig. 8a). This anticyclonic eddy has an oval shape and the long axis of which (about 500 km) is aligned with the coastline. Like the eddies in the numerical simulation with remote forcing, the anticyclonic eddy generated by direct wind forcing also has a northwestward alongshore translation speed, which on average is about 15 cm/s. After this anticyclonic eddy propagates out of the retroflection region, a second anticyclonic eddy is formed there (Fig. 8b). The later formed eddy is much smaller than the first one, about 250 km in diameter, and not as elongated. The centers of these two anticyclonic eddies are initially about 600 km apart, and they gradually become closer (to 400 km apart) as the eddies propagating toward the Caribbean (Fig. 8~). This is due to a fact that the second eddy is gaining momentum and catching up with the first one. Unlike the case with remote forcing, where eddies do not have continuous energy supply and, therefore, become smaller and smaller as they were weakened by dissipation, 19961 Ma: North Brazil Current retroflection eddies

70”N

the anticyclonic eddies in the present simulation have kept their size relatively well, mainly because the wind is constantly pumping energy into the system. Spinning up the ocean from rest, as we did here, is not a realistic initial condition. However, this initial condition provides a smooth start for a simulation and it avoids those nasty gravity waves which tend to bang off everywhere if the initial condition does not perfectly match the numerical model. Fortunately, in the case of an equatorial ocean, the fast equatorial waves make the adjustment time of the ocean to wind stress relatively short. In fact, in the present numerical simulation, by the time the first NBC retroflection eddy starts to form, the equatorial Atlantic Ocean has already been spun up and the equatorial currents as well as the low-latitude western boundary currents have been in place. The second anticyclonic eddy appeared one month later than the first one, Then, they translated continuously northwestward along the coastline for about four months until they reached the Caribbean. Although some of the initial impact might still remain, the circulation pattern in the model tropical Atlantic Ocean is reasonably realistic during the course of these eddies’ formation and translation. 52 Journal of Marine Research [54, 1

70"N

35"s 95"W SO" i5" iO"E Cc) Figure 8. (Continued)

5. Summary The present study shows that the dynamic response of the Atlantic equatorial western boundary to either a remote forcing, which is represented by an incoming Rossby wave packet that deepens the equatorial thermocline, or to direct wind forcing can generate northwestward translating eddies up the coast of Brazil. The characteristics of the numerically simulated eddies closely resemble those of the observed North Brazil Current retroflection eddies. The formation mechanism of these eddies has to do with the short equatorial Rossby waves (which can be generated by the said kinds of forcing), nonlinearity, and the boundary effect. The eddy translation mechanism involves both the nonlinear effect caused by the eddy’s interaction with the western boundary and the variation of the Coriolis parameter with latitudes.

Acknowledgments. This work was supported by The U.S. Department of Energy under contract No. DE-AC02-76CH00016, and by the National Science Foundation through grant OCE-9312324. The author is grateful to Editor George Veronis for his comments which 19961 Ma: North Brazil Current retroftection eddies 53 improved the clarity of the manuscript, to Dr. Eli J. Katz and Dr. Amal Chakraborty of Lamont-Doherty Earth Observatory for their help in implementing the wind-stress data.

REFERENCES Boyd, J. P. 1980. Equatorial solitary waves. Part 1: Rossby solitons J. Phys. Oceanogr., IO, 1699-1717. Cane, M. A. 1979. The response of an equatorial ocean to simple wind stress patterns: II. Numerical results. J. Mar. Res., 37, 253-299. Cane, M. A. and P. R. Gent. 1984. Reflection of low-frequency equatorial waves at arbitrary western boundaries. J. Mar. Res., 42, 487-502. Cane, M. A. and E. S. Sarachik 1977. Forced baroclinic ocean motions: II. The linear equatorial bounded case. J. Mar. Res., 35, 39.5432. Didden, N. and F. Schott. 1993. Eddies in the North Brazil Current retroflection region observed by Geosat altimetry. J. Geophys. Res., 98, 20,121-20,131. Flagg, C. N., R. L. Gordon and S. McDowell. 1986. Hydrographic and current observations on the continental slope and shelf of the western equatorial Atlantic. J. Phys. Oceanogr., 16, 1412-1429. Fratantoni, D. M., W. E. Johnsand T. L. Townsend. 1995.Rings of North Brazil Current: Their structure and behavior inferred from observationsand a numerical simulation. J. Geophys.Res., 100, 10633-10654. Johns,W. E., T. N. Lee, R. J. Zantopp and R. H. Evans. 1990.The North Brazil Current retroflection: Seasonalstructure and eddy variability. J. Geophys. Res., 95, 22,103-22,120. Lamb, H. 1932.Hydrodynamics, 6th ed. CambridgeUniversity Press. Lighthill, M. J. 1969.Dynamic responseof the Indian Ocean to the onset of the southwest monsoon.Phil. Trans. Roy. Sot. London A, 265,45-92. Longuet-Higgins,M. S. 1964.Planetary waves on a rotating sphere. Proc. Roy. Sot. A, 279, 446-473. Ma, H. 1992.The equatorial basinresponse to a Rossbywave packet: The effectsof nonlinear mechanism.J. Mar. Res., 50, 567-608. - 1993.A spectral elementbasin model for the shallowwater equations.J. Compt. Phys., 109, 133-149. - 1995.Parallel computation with the spectral element method, in Proceedings,Parallel ComputationalFluid Dynamics‘95. (Pasadena,CA, 1995). McCalpin, J. D. 1987.A note on the reflection of low-frequency equatorial Rossbywaves from realisticwestern boundaries. J. Phys. Oceanogr., 17, 1944-1948. Moore, D. W. 1968.Planetary Gravity Waves in an Equatorial Ocean. Ph.D. thesis.Harvard University, Cambridge,Massachusetts. Moore, D. W. and S. G. H. Philander. 1977.Modeling of the tropical oceaniccirculation, in The Sea6, 319-362. Orlandi, P. 1990.Vortex dipole rebound from a wall. Phys.Fluids A, 2, 1429-1436. Richardson,P. L., G. E. Hufford, R. Limeburner and W. S. Brown. 1994.North Brazil Current retroflection eddies.J. Geophys.Res., 99, 5081-5093. Saffman,P. G. 1979.The approachof a vortex pair to a plane surfacein inviscid fluid. J. Fluid Mech., 92, 497-503. Van Heijst, G. J. F. and J. B. Flor. 1989. Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence. J. C. J. Nihoul and B. M. Jouart, ed., Elsevier, Amsterdam. 591pp.

Received:2 May, 1995;revised: 26 September, 1995.