An Analysis of Wind-Driven Ocean Circulation with a Limited Number of Fourier Components

Abstract

The equations of motion for a square ocean basin of dimension L on the β-plane are solved approximately for the case where a wind-strew curl of the form sin (x/L) sin (y/L)[0≤x,y≤πL] is applied to the surface. The stream function is expanded in a double Fourier sine series and this representation is truncated after only four terms. The resulting set of equations contains the effects of non-linearity, time dependence, linear variation of the Coriolis parameter, friction, and wind-stress. Multiple solutions to the steady-state equations exist when the wind-stress is sufficiently strong. One of the solutions can be related to Sverdrup's (1947) solution for an ocean basin with one longitudinal boundary. A second solution is dominated by the non-linear interactions of the system. Integration of the non-linear transient equations are carried out for the case where the wind-stress starts at some initial time. In some cases the system goes through a series of oscillations of decreasing amplitude before it settles down to a steady state. For a certain range of the Rossby number (or alternatively of the strength of the wind stress) the ocean never settles down to a steady state but, after an initial transient phase, enters a periodic limit cycle. However, if initial conditions are taken sufficiently close to the known steady state solution, the system always settles down into the steady state after the initial transient phase.