The paper examines the applicability of known linear wave theories to numerical simulations of two, zonally invariant, fundamental problems of Physical Oceanography: Geostrophic adjustment and Ekman Adjustment. By simulating the problems with a modified version of the Massachusetts Institute of Technology General Circulation Model (MITgcm) we show that neither of the known wave theories can explain the results of the simulations in large and small meridional domains and for long and short times.

The study combines archived surface drifter trajectories along the Equator with a novel extension of Ekman's wind-driven theory to the equatorial β-plane in order to estimate the depth of the equatorial Ekman layer and the speed of upwelling into it. The analysis provides a direct estimate of the depth of the equatorial Ekman layer based on observed drifter trajectories and does not involve the 3D continuity equation, which is only used for estimating the upwelling speed. 

The transition from an arbitrary initial sea surface height to a geostrophic balance in which the velocity is steady was solved last century for constant Coriolis frequency, f(y), where y is the latitude. This study extends the theory to the realistic case in which f(y) is linear with y. We find that the variation in f(y) translates the steady geostrophic state westward as low-frequency Rossby waves that are harmonic in narrow domains and trapped near the equatorward boundary in wide ones.