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The Dynamics of North Brazil Current Retroflection Eddies

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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 dynamic responseof the equatorial westernboundary of the Atlantic Ocean to 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 northwest- ward 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 these eddiesis related to the nonlinear effect causedby the eddies’ interaction with the boundary and to the variation of the Coriolis parameter 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 Applied Science,Bldg. 490D, Brookhaven National Laboratory, Upton, NewYork, 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 za* + 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).
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