External Rossby Modes on a ?–Plane with Realistic Vertical Wind Shear

Abstract

Free oscillations of small amplitude and low frequency in an isothermal atmosphere at rest have the vertical structure of an edge wave, energy density decaying exponentially away from the ground. In this paper we treat these external Rossby modes on a middle latitude β-plane to see how their vertical structure and phase speed are changed by the presence of a winter zonal wind with realistic vertical structure. We find that for planetary-scale modes the basic state interior potential vorticity gradient and the temperature gradient at the lower boundary produce roughly equal but opposite changes in phase speed. Energy density decays less rapidly with altitude when the basic state temperature gradient is included in the lower boundary condition. As wavenumber is increased, the phase speed of the external Rossby mode is found to approach the speed of the zonal wind at the surface, and there is a limiting wavenumber beyond which the mode does not exist. At this limiting value of wavenumber, the mode becomes the low-wavenumber bound for the rapidly growing unstable modes first investigated by Charney, which have similar vertical structure. In addition to the, neutral external free mode, we find also neutral internal free modes that exist because of wave trapping in regions of relatively weak zonal winds. The relationship between the neutral free modes and unstable modes is explored numerically for the realistic zonal wind profile and for a zonal wind increasing linearly with altitude. In contrast to past studies of the stability of the linear profile, the present study indicates the possibility of stability for a band of wavelengths on the low-wavenumber side of a critical stability curve.