Bifurcation Properties of the Evolution of a Rossby Wave Packet in Barotropic Flows on the Earth's δ 6-Surface

Abstract

Based on previous theory, which was made by using the Rossby wave-packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance system has been further investigated. In the presence of both the basic current and topography, bifurcation properties of the evolution of the wave packet due to symmetric topography and asymmetric topography have been studied analytically as the topography parameter changes, using bifurcation diagrams and the WKB phase space, i.e., the local wavenumber phase span. The results show that both topography and the basic current play very important roles in the bifurcation properties of the dynamics of geophysical flows. The results also show that the topological structure of the evolution of a Rossby wave packet in various basic currents and topographies can vary accordingly. Both subcritical and supercritical bifurcations have been found analytically. Moreover, reverse supercritical and sub-critical bifurcations also have been found. The effect of a zonal basic current on the bifurcation is different from that of a meridional basic current. For example, on concave topography the bifurcation will be supercritical (subcritical) when the packet is located on the left-hand side (right-hand side) of a westerly jet. However, in a meridional basic current, the evolution on the concave topography exhibits a reverse supercritical (subcritical) bifurcation on the left- (right) hand side of the current. On symmetric topography there exist only two kinds of equilibria, i.e., the largest scale state and the two pure latitudinal-scale states or the pure longitudinal-scale states. On asymmetric topography mixed-scale equilibrium states exist. The existence of the mixed-scale state suggests that there is a different kind of evolution for a Rossby wave packet. The results suggest that the evolution of a Rossby wave packet on one side of a basic current could be different from that on the other side, which is expected in real geophysical flows. At the center of a basic current, the dynamical system of the Rossby wave packet is structurally unstable. The “homoclinic orbits” and the ”heteroclinic orbits” also have been discussed, which are separatrices of the wave-packet vacillations in the WKB phase space. The structural vacillations in the Rossby wave packet and their implications in geophysical flows associated with the bifurcation properties of the evolution of a Rossby wave packet are investigated. The results suggest that the transitions between two kinds of wave-packet structural vacillations only can occur on one side of the basic current as the topography parameter is varied.