Let us focus, first, on the inner part of the western shelf by building a stream function in a vertical plane to take advantage of the bay shape aspect of the western coast between 27°S and 33°S (see Figure 1). The net volume transport across the edge of a fixed subdomain with neither internal source nor sink of mass is virtually zero like, for instance, along the blue line drawn in Figure 1. The integration along this line of the cross-shore and vertical transport components following the model alongshore coordinate, j, provides, for each time step, a two-dimensional transport field (UU, WW):
where i, j, and k are the model coordinates, U and W are the model cross-shore and vertical transports, respectively. At any time, this field is nondivergent as both fulcra of the integration are located on land; it can be expressed as a stream function ψ:
Let us choose the reference (ψ = 0) so that ψ > 0 in standard upwelling conditions (i.e., a counterclockwise circulation in a cross-shore/depth plane with east to the right hand side). For each time step, the maximum value of ψ (ψmax) characterizes the amplitude of the meridionally integrated overturning circulation above the shelf. The pattern of variability of ψmax compares well with the time evolution of the alongshore wind stress over this subdomain, with a linear correlation coefficient close to 0.9 (Figure 15). The magnitude of the correlation is somewhat expected as wind stress fluctuations act directly on the volume of the subsurface water needed to compensate for the offshore surface Ekman flux. However, this result is here obtained within the scope of a realistic numerical simulation and provides a genuine estimate of the amount of water really involved in the upwelling process, with an appropriate visualization of the overturning. Its mean intensity over 1999–2003 is 0.49 Sv (1 Sv = 106 m3/s); this is 20% smaller than the theoretical Ekman transport (0.62 Sv) diagnosed from the mean alongshore wind stress component (0.065 N/m2), the average Coriolis parameter (f = −7.3 × 10−5 s−1) and the length of the considered arc (700 km). This difference can be explained by the depth of water over the shelf which allows interferences between the surface and bottom ocean boundary layers. Our calculation exemplifies the shortcoming of an upwelling intensity that could be purely diagnosed from surface wind values. Interestingly, the overturning almost vanishes in December 1999 (<0.2 Sv for 2 consecutive weeks), letting the surface summer heat flux warm up freely the shelf waters (Figure 16a). Conversely, in December 2000, a surge in the wind stress causes a doubling of the overturning (1 Sv for a few days (Figure 16b)). This surge is responsible for a concurrent, colder than usual, signature in SST with an upwelling circulation that penetrates over the shelf all the way to the shoreline. In the southern Benguela, our result constitutes a model-based validation of an upwelling index often used by fishery oceanographers [Bakun, 1973]. Despite a slight difference in the magnitude, the alongshore wind stress appears as a reasonable qualitative substitute for the overturning circulation and subsequent input of nutrient rich subsurface waters on the shelf. It is likely that such a good match between wind forcing and overturning circulation is not always the case. In areas like, for example, the Canary Current system, where upwelling sometimes moves from the nearshore domain to the shelf break [Barton et al., 1977], or the Gulf of Guinea where remotely forced coastal trapped waves contribute to upwelling [Picaut, 1983], wind-based upwelling indices are likely unable to condense the full underlying complex dynamics. By providing an estimate of overturning amplitude model-based indices can be very helpful in these regions.