The transport in the Ekman surface layer on the spherical Earth

Abstract

The modification of the transport in the Ekman layer on the f-plane due to the Coriolis parameter’s variation with latitude and the curvature of Earth’s surface is analyzed by considering the temporal changes in the angular momentum. The latter plays the role of a dynamical variable of the model, replacing the zonal velocity component, and drag is modeled by Rayleigh friction. The steady transport, which on an f-plane is perpendicular to the applied wind stress, is recovered on the Earth as a special solution where the meridional velocity is time-independent. For zonal wind stress, the trajectory on Earth is simply a great circle that passes through the poles while for meridional wind stress the special solution can have a time-independent nonzero meridional component so the trajectory does not have to be purely zonal. This asymmetry between zonal and meridional wind stresses on the Earth is due to the Coriolis parameter’s variation with latitude only—an effect that is completely neglected on the f-plane. For steady wind forcing, the dynamical system is three-dimensional and its fixed points are located at the latitudes of vanishing wind stress. In the drag-free case, when the curl of the wind stress does not vanish at the fixed points, these points are always unstable; namely there exists at least one repulsive direction in (the 3D) phase space. When drag is included, these steady states still prevail but become stable for realistic values of the wind forcing and drag. An additional steady state, located right on the equator, exists in this case and its zonal velocity attains a constant value determined by the balance between the applied stress and the drag force. Although drag is present, this steady state is unstable for negative wind stress (i.e. easterly winds) so any deviation from a purely westward, equatorial, trajectory will grow exponentially in time. Naturally, no similar instability of the steady states occurs on the f-plane. The curl of the zonal wind stress at the latitudes where the stress itself vanishes determines the trajectory of a water column originating there via the nonlinear interaction between the motion due to inertial oscillations and that due to the wind-forced changes of the angular momentum. Temporal or zonal dependence of the wind stress has a profound effect on the trajectories, especially near the unstable latitudes due to the increase in the dimensionality of the system that enables more complex trajectories. The present simple model can quantitatively reproduce the observed fast dispersal of nearby launched drifters with steady and smooth wind stress. It can also explain qualitatively the different spectra of clusters of drifters launched in two field experiments in the NE Pacific Ocean under similar winds and the highly variable angle between the wind and the observed trajectories of clusters of drifters.