SIMPLIFIED DYNAMIC EQUATIONS APPLIED TO THE ROTATING-BASIN EXPERIMENTS

Abstract

By means of truncated Fourier-Bessel series, a two-layer geostrophic “numerical-prediction” model with heating and friction is reduced to a set of eight ordinary nonlinear differential equations in eight dependent variables. These equations allow the presence of disturbances of a single wave number. They permit the occurrence of baroclinic but not barotropic instability. They possess appropriate energy invariants if heating and friction are temporarily suppressed. The simplified equations are applied to the flow of a liquid in a symmetrically heated rotating basin. Exact solutions are determined for the steady Hadley and Rossby regimes, and the criterion for the stability of the Hadley regime is obtained. For high rotation rates the criterion for the disappearance of an established Rossby regime differs from the criterion for the onset of a Rossby regime. The equations are modified to allow for the presence of several wave numbers simultaneously. Each wave number interacts with the zonal flow, but the interactions between wave numbers are omitted. The criteria for the transitions between wave numbers are then obtained. The solutions agree qualitatively with Fultz's experiments in that with slow rotation there is no Rossby regime, with more rapid rotation the Rossby regime occurs with intermediate heating contrasts, and within the Rossby regime a smaller heating contrast leads to a higher wave number. It is concluded that the simplified equations are suitable for the study of baroclinic flow, and that the changes of regime are fundamental properties of the forced flow of a rotating fluid. It is suggested that the transitions in the experiments and the transitions described by the equations are manifestations of baroclinic instability having similar physical explanations.