A simple model for the three-dimensional, thermally and wind-driven ocean circulation

As a generalization to box models of the large-scale, thermally and wind-driven ocean circulation, nonlinear equations, describing the evolution of two vectors characterizing the state of the ocean, are derived for a rectangular ocean on an ∫-plane. These state vectors represent the basin-averaged density gradient and the overall angular momentum vector of the ocean. Neglecting rotation, the Howard-Malkus loop oscillation is retrieved, governed by the Lorenz equations. This has the equations employed in box models, in the restricted sense where no distinction is made between the restoring time scales of the temperature and salinity fields, as a special case. In another approximation, with rotation included, the equations are equivalent to a set E. N. Lorenz introduced to describe the “simplest possible atmospheric general circulation model” . Although the atmospheric circulation may be chaotic, parameter values in the ocean are such that the circulation is steady or, at most, exhibits a self-sustained oscillation. For a purely thermally forced ocean, this is always a unique state. Addition of a wind-induced horizontal circulation allows for multiple equilibria, despite the neglect of the salinity field.