Retroflection from slanted coastlines-circumventing the "vorticity paradox"
Abstract. The balance of long-shore momentum flux requires that the solution of zonally retroflecting currents involve ring shedding on the western side. An important aspect of the ring dynamics is the ring intensity α (analogous to the Rossby number), which reaches its maximum value of unity when the upstream potential vorticity (PV) is zero. Friction leads to a slow-down and a decrease in α. The main difficulty is that the solution of the system of equations for conservation of mass and momentum of zonal currents leads to the conclusion that the ratio (Φ) of the mass flux going into the rings and the total incoming mass flux is approximately 4α/(1+2α). This yields the "vorticity paradox" – only relatively weak rings (α≤1/2) could satisfy the necessary condition Φ≤1. Physically, this means, for example, that the momentum-flux of zero PV currents upstream is so high that, no matter how many rings are produced and, no matter what size they are, they cannot compensate for it.
To avoid this paradox, we develop a nonlinear analytical model of retroflection from a slanted non-zonal coastline. We show that when the slant of coastline (γ) exceeds merely 150, Φ does not reach unity regardless of the value of α. Namely, the paradox disappears even for small slants. Our slowly varying nonlinear solution does not only let us circumvent the paradox. It also gives a detailed description of the rings growth rate and the mass flux going into the rings as a function of time. For example, in the case of zero PV and zero thickness of the upper layer along the coastline, the maximal values of Φ can be approximately expressed as, 1.012+0.32exp(−γ/3.41)−γ/225. Interestingly, for significant slants γ≥300), the rings reach a terminal size corresponding to a balance between the β-force and both the upstream and downstream momentum fluxes. This terminal size is unrelated to the ultimate detachment and westward drift due to β. Our analytical solutions are in satisfactory agreement with the results of a numerical model that we run.
