On the Role of Dissipation in Inertial Western Boundary Currents

Abstract

The problem of matching the nonlinear, frictional flow in a simple western boundary layer to a specified interior flow is considered. Two problems are discussed, using streamfunction as a coordinate across the boundary layer. First, a unidirectional flow is considered. The dissipation is considered to be some positive quantity, and it is shown that for a simple form of this, many different amounts permit a smooth match to the interior. The magnitude of the dissipation can be determined absolutely at the dividing point between in- and outflow. The dissipation south of this point must be smaller and north of this point must be larger; a simple equation describes the relationship between dissipations north and south of the dividing point. Second, a bidirectional boundary layer is permitted. A specific form of dissipation (a linear drag) is applied, with a constant coefficient. It is shown that in this case it still remains possible to match to a specified interior flow, although inertial overshoot occurs both into the next gyre polewards as well as equatorwards into the inflow region, if the drag is small enough. Thus, taken together with published results on Laplacian dissipation, these simple models suggest that western boundary layers are passive and can match to a specified interior flow without modifying that flow in any way (although this may not be the case for very low friction).