Abstract
The most unstable normal modes have been obtained for a global model of the atmosphere which extends from the ground to 70 km. The model is quasi-geostrophic and in spectral form, including wavenumbers 1–6. Two sets of calculations were performed. In the first set, the basic state is representative of Northern Hemisphere winter solstice conditions. The long waves growing on this basic state are deep modes which have maximum kinetic energy in the stratosphere, and clearly are analogous to the stratosphere modes found for one-dimensional wind profiles by Geisler and Garcia. In the present calculations these internal modes exist even in the presence of a stratospheric wind minimum—but they propagate vertically into the stratosphere only to the north of this minimum, at the latitudes of the polar jet. The shorter wave normal modes are confined to the troposphere, and more closely resemble external (Charney) modes. The energy for all the modes ultimately derives from the available potential energy of the basic state in the lower troposphere, but local energy conversion in the stratosphere can play a role in supporting the deep, long waves.
The basic state for the second set of calculations was the axisymmetric solution corresponding to radiative equilibrium. The vertical wind shear in this state is very large, and all the modes are basically external, tropospheric Charney modes. In this case the large shear of the mean state causes the boundary between tropospheric (external) and stratospheric (internal) modes to fall to wavenumbers less than 1. Hence the latter cannot exist on the sphere.
