Secondary Bifurcation of the Evolution of a Rossby Wave Packet in Barotropic Flows on the Earth'sδ-Surface

Huijun Yang Geophysical Fluid Dynamics Institute, The Florida State University, Tallahassee, Florida

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Abstract

Based on a simplified mathematical model (Yang), using the Rossby wave packet approximation and the WKB method, the bifurcation properties of the evolution of a single geostrophic synoptic disturbance system were further studied. This analytical investigation was in the presence of an asymmetric basic current and utilized general topography as the topography parameter changes, using bifurcation diagrams and the WKB phase space.

The results showed that both the basic current and topography play very important roles in the bifurcation properties of geophysical flows. The results suggest that with the same type of topography, topological structure on one side of an asymmetric basic current will be different from that on the other side. On one side of an asymmetric basic current there exists only the primary bifurcation with three equilibrium states. However, on the other side of the asymmetric basic current, there are two distinct bifurcations. As the topography parameter reaches the first critical value, the primary bifurcation with five equilibrium states occurs. The equilibrium states are the 1argest spatial-scale state, two pure longitudinal-scale status and two pure latitudinal-scale status. As the topography parameter is increased further to the second critical value, a secondary bifurcation with the mixed-scale equilibrium states occurs. The evolution of a Rossby wave packet could exhibit the supercritical and subcritical primary bifurcations and the reverse supercritical and subcritical primary bifurcations accordingly, as the topography parameter varies. The secondary bifurcation, however, is the transcritical bifurcation. The WKB trajectories in the phase space are discussed for the different topography parameters. Both homoclinic orbits and heteroclinic orbits exist, as the separatrices of the wave packet structural vacillations. It has been shown that for the stable mixed-scale domain of the WKB phase space. For the unstable mixed-scale states the trajectories of the evolution constitute a family hyperbolic curves in the WKB phase space.

The structural vacilliations in the Rossby wave packet and their implications in geophysical flows are investigated. The results suggest that on side of an asymmetric basic current, the transitions can occur among three kinds of wave structural vacillations, while on the other side of the asymmetric basic current, the transitions can occur among three kinds of wave packet structural vacillations, while on the other side of the asymmetric basic current the transitions can only occur between two kinds of wave packet structural vacillations, as the topography parameter is varied.

Abstract

Based on a simplified mathematical model (Yang), using the Rossby wave packet approximation and the WKB method, the bifurcation properties of the evolution of a single geostrophic synoptic disturbance system were further studied. This analytical investigation was in the presence of an asymmetric basic current and utilized general topography as the topography parameter changes, using bifurcation diagrams and the WKB phase space.

The results showed that both the basic current and topography play very important roles in the bifurcation properties of geophysical flows. The results suggest that with the same type of topography, topological structure on one side of an asymmetric basic current will be different from that on the other side. On one side of an asymmetric basic current there exists only the primary bifurcation with three equilibrium states. However, on the other side of the asymmetric basic current, there are two distinct bifurcations. As the topography parameter reaches the first critical value, the primary bifurcation with five equilibrium states occurs. The equilibrium states are the 1argest spatial-scale state, two pure longitudinal-scale status and two pure latitudinal-scale status. As the topography parameter is increased further to the second critical value, a secondary bifurcation with the mixed-scale equilibrium states occurs. The evolution of a Rossby wave packet could exhibit the supercritical and subcritical primary bifurcations and the reverse supercritical and subcritical primary bifurcations accordingly, as the topography parameter varies. The secondary bifurcation, however, is the transcritical bifurcation. The WKB trajectories in the phase space are discussed for the different topography parameters. Both homoclinic orbits and heteroclinic orbits exist, as the separatrices of the wave packet structural vacillations. It has been shown that for the stable mixed-scale domain of the WKB phase space. For the unstable mixed-scale states the trajectories of the evolution constitute a family hyperbolic curves in the WKB phase space.

The structural vacilliations in the Rossby wave packet and their implications in geophysical flows are investigated. The results suggest that on side of an asymmetric basic current, the transitions can occur among three kinds of wave structural vacillations, while on the other side of the asymmetric basic current, the transitions can occur among three kinds of wave packet structural vacillations, while on the other side of the asymmetric basic current the transitions can only occur between two kinds of wave packet structural vacillations, as the topography parameter is varied.

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