Extension of Ekman (1905) wind-driven transport theory to the β plane

The seminal Ekman (1905) f-plane theory of wind-driven transport at the ocean surface is extended to the β plane by substituting the pseudo-angular momentum for the zonal velocity in the Lagrangian equation. When the β term is added, the equations become nonlinear, which greatly complicates the analysis. Though rotation relates the momentum equations in the zonal and the meridional directions, the transformation to pseudo-angular momentum greatly simplifies the longitudinal dynamics, which yields a clear description of the meridional dynamics in terms of a slow drift compounded by fast oscillations; this can then be applied to describe the motion in the zonal direction. Both analytical expressions and numerical calculations highlight the critical role of the Equator in determining the trajectories of water columns forced by eastward-directed (in the Northern Hemisphere) wind stress even when the water columns are initiated far from the Equator. Our results demonstrate that the averaged motion in the zonal direction depends on the amplitude of the meridional oscillations and is independent of the direction of the wind stress. The zonal drift is determined by a balance between the initial conditions and the magnitude of the wind stress, so it can be as large as the mean meridional motion; i.e., the averaged flow direction is not necessarily perpendicular to the wind direction.