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99 UPWELLING in the GULF of GUINEA Results of a Mathematical 99 UPWELLING IN THE GULF OF GUINEA Results of a mathematical model 2 2 6 5 0 A. BAH Mécanique des Fluides géophysiques, Université de Liège, Liège (Belgium) ABSTRACT A numerical simulation of the oceanic response of an x-y-t two-layer model on the 3-plane to an increase of the wind stress is discussed in the case of the tro­ pical Atlantic Ocean. It is shown first that the method of mass transport is more suitable for the present study than the method of mean velocity, especially in the case of non-linearity. The results indicate that upwelling in the oceanic equatorial region is due to the eastward propagating equatorially trapped Kelvin wave, and that in the coastal region upwelling is due to the westward propagating reflected Rossby waves and to the poleward propagating Kelvin wave. The amplification due to non- linearity can be about 25 % in a month. The role of the non-rectilinear coast is clearly shown by the coastal upwelling which is more intense east than west of the three main capes of the Gulf of Guinea; furthermore, by day 90 after the wind's onset, the maximum of upwelling is located east of Cape Three Points, in good agree­ ment with observations. INTRODUCTION When they cross over the Gulf of Guinea, monsoonal winds take up humidity and subsequently discharge it over the African Continent in the form of precipitation (Fig. 1). The upwelling observed during the northern hemisphere summer along the coast of the Gulf of Guinea can reduce oceanic evaporation, and thereby affect the rainfall pattern in the SAHEL region. Years of intense upwelling could be very dry and years of reduced upwelling should result in periods of relatively high rainfall. Clearly, a better understanding of the upwelling regime might contribute significantly to improved land-use. Many investigators have attempted to explain the generation and evolution of up­ welling in the Gulf of Guinea. It was soon apparent that localJwinds could not provide an adequate forcing mechanism (Houghton, 1976). Philander (197$) concluded that the 100 Fig. 1. Sketch of the monsoon wind pattern over the Gulf of Guinea (after Dhonneur, 1974). upwelling is not due to local oceanic circulation, but rather is part of the large scale oceanic circulation system. This explanation seems reasonable since Adamec and O'Brien (1978) have already shown that variation of the trade winds regime in the western tropical Atlantic Ocean could excite an equatorially trapped Kelvin wave which would induce upwelling on its way eastwards (see also Moore, 1968; Moore and Philander, 1977; Moore et al., 1978). But their study does not explain the irregular intensification of upwelling locally along the northern coast of the Gulf of Guinea. Some previous workers have tried to demonstrate the effects of zonal and meridio­ nal coasts on equatorial waves (Philander, 1979; Weisberg et al., 1979). Recently, the study of Clarke (1979) dealt with local longshore variations in the wind stress and the resulting long trapped waves travelling along the northern coast of the Gulf of Guinea. Yoshida (1967) and Arthur (1965) have described enhanced upwelling near a cape. Their results suggest that the role played by the irregular geometry of the northern coast of the Gulf of Guinea should be examined. This is the object of the present paper. r 101 1. THE NUMERICAL MODEL 1.1. Model geometry In contrast with the model developed by Adamec and O'Brien (1978), our model deals with a non-rectilinear coastline. The numerical simulation concerns the oceanic res­ ponse of an x-y-t two-layer model on the 3-plane to an increase in the wind stress X and y increase eastwards and northwards, respectively. The bottom of the Ocean is assumed flat, and the depth of the upper layer is constant. Fig. 2 shows the geometry of the basin, where horizontal dimensions Lx and L are respectively 3000 and 5000 km . 1000 km Fig. 2. The model geometry with the irregular coastline. 1.2. Model formulation The quasi-hydrostatic and Boussinesq approximations are made. The layer densities Pi and p2 are constant. The effects of atmospheric pressure, tides and thermohaline mixing are neglected. If there is no pressure gradient in the second layer (i.e. Vp2 = 0), the linea­ rized equations for a two-layer viscous flow reduce to : 5u (1) at PH0 gyu - g ^ (2) at 102 &♦«<&* g" ° The equations governing the mass transport are, on the other hand, = ßyV - g'H + — + A V2U (4) 3t 3x p f - (5) f * s * § - » where A and A h are the horizontal eddy viscosities (assumed constant), Ho the undisturbed depth of the upper layer, h the perturbation of the upper layer thick­ ness, H the total (disturbed) depth and P 2 “ Pi g ' _ g ---------------- P 2 the reduced gravity. Definitions of the other terms are obvious. This model repre­ sents the simplest formulation of the first baroclinie response. In non-dimensional form, the system of equations (1-3) can be rewritten (Bah, 1979; Bah et al., 1979) : 9u 9h T 92u 92u — = yv - a! — + a2 — + a 3 — - + ai+ - . (7) 9t 9x p 3X2 9y 9v 9h t 92v 92v ^ = - yu - ^ + a2 - + a3 + a, ^ (8) |h + ai |¡L + |v = 0 {9) at 9x 9y where ai = ; a2 = ß-^g'-374 hÔ 7/4 ; a , = A 4 B1 g'■ H.' 3/4 ; a, = Aß* (g ' H0 )' 3/4 X Ly /g'Ho ßLx Lx and L y are the zonal and the meridional length scales respectively. Working with the system of equations (4-6) one gets from the equation of conti­ nuity 9h 9u , 9v 9u 9h , 9v 9h . im — + a] — + — + aih — + a lU — + h -5- + v — = 0 (10) 9t 9x 9y 9x 9x 9y 9y 103 Thus, working with equations (1-3) relative to the mean velocity results in ne­ glecting the last four terms of equation (10), which is equivalent to neglecting contributions at least of the same order of magnitude as the respective second terms of the right side of eqs (7) and (8). The values for , a 2 , a 3 and a^ for the case of a 50 m depth are respec­ tively 2.24 , 4.47 X 1CT 2.24 X 10 3 and 4.47 x 10"* (Bah, 1979). 1.3. Numerical scheme The numerical model uses the numerical scheme developed by Ronday (1976) u and V are computed according to the following spatial grid. I I I I I I I I 2 1+2 9-- -©- 0 - for I i i for I i i for I i I i 2 I i ' “ t □ - - - AY I i 2 1 - 2 0 ----- — © - i ! I 2 J 2 J 2 J + 2 ■ AX Fig. 3. Spatial grid used for the computations. We have for the temporal grid, in the unidimensional case for example u n + 1 = a u n +, b, h, n n +1 n +1 n h = c u + d h u , V and h are computed at the same moment, but in the computation of h , esti­ mated values of u and v are used. All the boundaries are walls, and the normal velocity is assumed equal to zero at the coast. All calculations are done with a constant grid resolution of 50 km in both x and y directions. The stability condition At At < 1 Ax Ay requires At to be < 18 248 s (about 5 h) . The chosen temporal step was At = 3 h 104 (1/8 day). Other constants are : H 0 = 50 m ; A = IO2 m2s -1 ; g' = 2 x IO-2 m s -1 ; 3 = 2 x io'11 nf 1s“ 1 . Since we are interested in studying the probable effect of the irregular coastline geometry on the upwelling generation, evolution and intensification, we have chosen to study two major aspects : - the first one with prominent coastline features such as Cape Palmas, Cape Three Points, Cape Formoso and Cape Lopez; - the second, for comparison, with a linear coastline for the west african coast. For both cases, two situations are investigated : 1.3.1. a standard linear situation with an increase of respectively 0.025 Nm and 0.0125 N m 2 in the westward wind stress over the western 1 500 km of the basin (fig. 4) and without any meridional wind; 0 1 2 3 4 5 Distance km x 10 3 0 - 1 5 - 20 - Fig. 4. Spatial variation of the zonal wind stress. 1.3.2. a second situation, non linear, similar to the first but including advec- tive effects, local depth in the stress term, non-linearities in the continuity equa­ tion and an increase of 0.0125 N m -2 of the westward wind stress. In each situation, integrations are performed at least over 90 days, starting from rest (u = v = 0 at t = 0), with a sudden increase of wind stress (fig. 5) remaining constant throughout the following period of integration. To allow an easy comparison of the results, we have used the same denomination fo boundaries as did Adamec and O'Brien (1978) : the eastern boundary extending 500 km north of the Equator and 1500 km south is the south-eastern (S-eastern) boundary; the eastern boundary extending 500 km north of the Equator to 1500 km is the north­ eastern (N-eastern) boundary; the northern boundary is then the northern boundary of the Gulf of Guinea, extending approximately from Cape Palmas to Cape Formoso, i.e. r 105 ■i f 0 0 10 20 30 40 50 60 70 80 90 Days Fig. 5. Temporal variation of the wind stress. from 3 000 to 5 000 km east of the western boundary; the north-northern (N-northern) boundary is the northern boundary of the basin extending 3 000 km from the western boundary (fig.
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