The Momentum Imbalance Paradox Revisited

The classical problem of a point source situated along a southern boundary emptying buoyant water into a (β plane) ocean is revisited. Pichevin and Nof (PN) have shown that, in contrast to the view prevailing at the time, such an inviscid outflow does not simply turn to the right. Rather, it bifurcates into two branches: a steady branch that does turn to the right (eastward) and an unsteady branch that periodically sheds eddies to the left (westward). This partition is because a simple turn to the right of the entire outflow leaves the outflow’s long-shore momentum flux unbalanced, creating a paradox. In contrast, the branching allows the westward-drifting eddies (westward branch) to balance the momentum flux of the steady current (eastward branch). Although the analytical PN solution is useful and informative, it is cumbersome and difficult to apply to actual outflows. Here, a considerably simpler nonlinear analytical solution is presented. Using the idea that the eddies grow slowly relative to their rotation rate, it is shown that an intense (i.e., large Rossby number) and large relative vorticity outflow dumps most of its mass flux (Q) into the eddies (66%). (The remaining 33% goes into the eastward long-shore current.) By contrast, a weak outflow (i.e., an outflow with weak anticyclonic vorticity −αf, where α is analogous to the Rossby number and is much smaller than unity and f is the Coriolis parameter) dumps most of its water into the downstream current [(1–2α)Q]. Unexpectedly, this partition of mass turns out to be the same as the one taking place on an f plane. (Note that this is not at all the case for southward outflow nor is it the case for either eastward or westward outflow, where β alters the balance drastically.) Although the above partition of mass is independent of β, the size of the eddies generated by the above process is a function of β. It is given by [768g′Q/βπf 2α(2 − α)(1 + 2α)]1/5, where g′ is the reduced gravity. This gives a reasonable estimate for the Loop Current eddies’ size and generation frequency. Numerical simulations are in agreement with the above nonlinear solution, though the agreement is not necessarily any better than that of PN.