Global Behavior of the Evolution of a Rossby Wave Packet in Barotropic Flows on the Earth's δ-Surface

Abstract

A concept of the δ-approxiniation for the earth's surface has been introduced. Using the Rossby wave packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance system has been further studied on the δ-surface of the earth. The global behavior of the structural changes of the wave packet due to the zonal, the meridional and the asymmetric basic currents and the variety of the topography on the δ-surface of the earth have been discussed and compared with those an the earth's,β-plane by using the WKB phase plane, i.e., the wave packet's local wavenumber phase plane. The results show that the governing system on the earth's δ-surface may be dynamically different from that on the earth's β-plane. Moreover, the wave packet structural vacillation has been found on both the β-plane and the δ-surface. Wave packet structural vacillation is characterized by the time-periodic changes of the wave packet's structure. Both the tilt and the spatial scales of the packet will evolve periodic changes simultaneously. The wave packet structural vacillations are also characterized by the closed WKB trajectories on the WKB phase plane. The results show that in the presence of the asymmetric basic currents, the WKB trajectories on the WKB phase plane appear simply to be elliptical, e.g., in the case of a southwesterly jet, or hyperbolic, e.g., in the case of a southeasterly jet. The results suggest that it is possible for the packet structural vacillations to exist in the presence of some asymmetric basic currents, e.g., a southwesterly jet. The behaviors related to topography in various distributions have also been discussed. It has been demonstrated that the quadratic east-west oriented topography modifies only the δ-effect, and that with some topographies, e.g., convex topographies, the wave packet structural vacillation can also exist. In some cases, however, the behaviors of the evolution of a packet will be qualitatively different on the earth's δ-surface from those on the earth's β-plane. For example, in the meridional basic current or on the north-south oriented topographies. only on the δ-surface of the earth do there exist such wave packet structural vacillations. On the other hand, in some cases, the wave packet solutions have been obtained on both the β-plane and the δ-surface. The wave packet vacillation suggests a possible mechanism of vacillations observed in the atmosphere.