The momentum imbalance paradox

Abstract

The classical question of what happens when anomalous water enters an ocean via a meridional northward channel is addressed analytically using a reduced-gravity nonlinear model. The channel corresponds to either a conduit connecting 2 otherwise separated basins (e.g., the Yucatan Channel) or a conduit carrying water from an independent source. The traditional view is that, due to the Coriolis force, such an anomalous northward flowing current turns to the right (looking offshore) and forms a zonal boundary current that flows eastward. In this scenario, a front (corresponding to a surfacing interface) separates the oceanic and the anomalous water. Integration of the steady inviscid momentum equation along the boundary gives the long-shore flow-force and shows that such a scenario leads to a paradox. Specifically, such a flow corresponds to an unbalanced flow-force and, therefore, cannot exist. To balance the integrated momentum and resolve the paradox the inflow constantly sheds anticyclones which propagate to the left due to β. Under such conditions, the momentum of the eddies moving to the left balances the momentum of the current flowing to the right. This new eddy shedding mechanism may explain why the Loop Current produces loops and why other inflows produce anticyclones. A nonlinear analytical solution to the problem is constructed with the aid of a new and powerful theoretical approach which is based on the idea that, after each eddy generation process, the system returns to its original state. This implies that nonlinear periodic flows can be integrated over a control volume in a similar manner to the integration used in steady flows. This novel method enables us to extract the details of the resulting features (i.e., their size, speed, periodicity and depth of the shedded rings) without solving for the details of the incredibly complicated three-dimensional and time-dependent generation process. It turns out that the problem involves a new eddy length scale R^d/ε^1/6 (whereRd is the parent current Rossby radius and ε = βRd/f0) which is somewhat greater than that of most eddies. Calculations were made for both zero and finite potential vorticity flows; they show that, for currents such as the Loop Current which transports about 20 Sv, eddies are shed approximately once every 300 days. Quantitative numerical experiments using the Bleck and Boudra model show that, indeed, an inflow along a straight coastline produces eddies next to the source.