Linear Dynamics of Transient Planetary Waves in the Presence of Damping

Abstract

The model presented here extends the Charney model by including Newtonian cooling, Ekman dissipation, and a linear vertical variation of the stratification parameter. By using an integral representation of the solution and a Frobenius series expansion, we have shown that the dispersion equation and the vertical structure of the strongly unstable modes can be well approximated by a second-order transcendental equation and a generalized Laguerre polynomial multiplied by an exponential function, respectively. The midlatitude planetary wave 2, 3 and 4 belong to the intermediate scale motion between the Charney and Burger regimes and may be viewed as the atmospheric counterpart of the most unstable Green mode. The wavelength (growth rate) ratio of the most unstable Green mode to mm unstable Charney mode is about 2.5 to 3 (1/3 to 2/5) for typical midlatitude winter condition. That mode possesses a constant phase which tilts westward with height in the troposphere, and features a baratropic structure in the stratosphere; that mode extends to several density heights before being trapped, and exhibits a major peak in the stratosphere. Its available potential energy is converted in the lower troposphere, as well as in the stratosphere, and its kinetic energy is generated in both the middle troposphere and the middle stratosphere, with significant destruction near the tropopause. The Newtonian cooling was found to reduce the growth rate over most of the wavelength band especially for the Burger-Green modes and for the strong instabilities. Nevertheless, in the immediate vicinity of the critical wavelength small amount of Newtonian cooling has a destabilizing effect. The vertical increase of the static stability reduces the wavelength of the most unstable modes and affects the growth rate and vertical structure of the Green modes.