Wind-driven transport on the rotating spherical Earth

The dynamics of a water column at the surface of the ocean on the rotating spherical Earth forced by zonal wind stress is analyzed by substituting the angular momentum for the zonal velocity as one of the system's dependent variables. This substitution results in a model of the column's trajectory as a quasiparticle in a time dependent potential well. Explicit solutions are derived for the temporal changes in the angular momentum and the associated minima of the potential well as well as for the oscillations about these minima. The analytic results are confirmed by numerical solutions of the fourth-order nonlinear system of ordinary differential equations. For the eastward directed wind stress, our results provide exact formulas for the time it takes a column to reach the equator, where the dynamics is trivially described by the non-rotating paradigm of a particle subject to a constant force. In mid-latitudes, the analysis underscores the pivotal role played by the latitude where the wind-stress changes sign. Columns originating north or south of this latitude either converge to it or diverge away from it depending on whether the latitudinal change of the wind stress at this latitude is positive or negative. The oscillatory motion about this latitude is linearly unstable, and the growth rate of the amplitude is proportional to the gradient of the wind stress at that latitude.