Spin-up and spin-down in rotating fluid exhibiting inertial oscillations and frontogenesis

Abstract

Time-dependent, rotating flow in a finite depth of fluid is considered. Unbalanced initial conditions initiate flow in a shallow Ekman layer and in the inviscid interior, which is characterized by a state of zero potential vorticity. To determine the interior flow response to motion forced by the Ekman layer suction velocity, wB, an expansion of the flow to first-order in E1/2, where E is the Ekman number, is carried out. Frontogenesis, which occurs in both the baroclinic and barotropic parts of the geostrophic flow, modulates the inertial oscillation that enters at zero order. A baroclinic front (infinite relative vorticity) can occur in a finite-time, equal to or less than one-half the period of an inertial oscillation, πf−1 These fast-time processes are described in detail by Blumen (2000). Spin-up to the quasi-steady Ekman boundary layer solution also occurs during one-half the period of an inertial oscillation. Thereafter, wB varies on a slow-time scale, E−1/2f−1. Yet, a barotropic front may form in a finite-time if the initial anticyclonic relative vorticity exceeds f, a condition that favors nonlinear steepening in opposition to boundary layer dissipation. This analysis contributes to a theoretical understanding of the interplay between spin-down and frontogenesis in rotating fluid. Some values of the Ekman number, typical of mid-latitude flows, are introduced to compare theoretical predictions to observed conditions. It is concluded that the Ekman layer corrections are most likely smaller in magnitude than the magnitude of errors in current atmospheric wind measurement systems, and therefore, not verifiable. Oceanic flows are also difficult to measure at the required accuracy, and other processes compete with Ekman layer dissipation to explain slow-time spin-down in the oceans.