The retroflection paradox

Abstract

The classical question of what happens when a warm western boundary current, such as the North Brazil Current (NBC), retroflects is addressed analytically using a reduced-gravity nonlinear model. The traditional view is that the northwestward flowing current separates from the wall, turns to the right (looking offshore), and forms a zonal boundary current that flows eastward. Integration of the steady inviscid momentum equation along the boundary gives the longshore momentum flux (or flow force) and shows that such a scenario leads to a paradox. To resolve the paradox the separated current must constantly shed anticyclones, which propagate to the northwest due to β and an interaction with the boundary. This new eddy shedding mechanism, which is not related to the traditional instability of a zonal jet, may explain why the NBC must produce rings. A nonlinear analytical solution to the problem is constructed with the aid of a powerful theoretical approach based on the idea that nonlinear periodic flows can be integrated over a control volume. This method enables us to extract all the details of the resulting features without solving for the details of the incredibly complicated three-dimensional and time-dependent generation process. Due to the strong nonlinearity of the problem, the method is quite different from the familiar averaging technique that requires the existence of a "mean" current. To employ the above method, however, it was necessary to derive a new nonlinear formula for the β-induced migration of eddies adjacent to a zonal boundary that slopes in the N-S direction. It turns out that the general problem involves an eddy retroflection length scale Rd/ε1/6 (where Rd is the parent current Rossby radius and ε = βRd/f0) that is greater than that of most eddies (Rd). Calculations show that, for the retroflected NBC, which transports about 45 Sv, eddies are shed approximately once every 90 days.