Amplitude vacillation of baroclinic waves in a rotating fluid system

Abstract

Abstract Baroclinic instabilities and waves are fundamental to atmospheric dynamics and many mathematical and laboratory models have been developed to aid their study. Many distinctive time-dependent forms of behavior have been identified, including periodic or quasi-periodic variations in shape, amplitude or wavenumber of baroclinic waves, usually called “vacillations”. Here the phenomenon of amplitude vacillation of baroclinic waves in a rotating fluid is examined in the context of a three-layer channel model. A multiple instability theory is presented where the vacillation arises from the interference of two baroclinically unstable waves, each having the same horizontal wavenumber, but differing vertical structure and phase speed. The weakly nonlinear dynamics of these waves are investigated, the analysis leading to a simple system of ordinary differential equations in the wave amplitudes, describing a codimension-two double Hopf bifurcation. It is shown that the two unstable waves can coexist at finite amplitude to produce the vacillation, usually when one of them is linearly stable. The kinematic features of the model vacillations are compared with experimental observations, and found to agree in certain respects. Additionally, properties are found—such as the height-dependence of the vacillation stage and the possibility of triply periodic flow—which have not yet been reported in the literature.