Linear Stability Analysis of CISK-Induced Disturbances: Fourier Component Eigenvalue Analysis

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Abstract
The physical stability of perturbation motions in a conditionally unstable tropical atmosphere at rest is examined under CISK forcing. The governing linear multi-layer primitive equations are formulated into an eigenvalue problem for the unconditional heating case. This analysis yields propagation speeds, growth rates and vertical structure for the modes supported by the model. The specified parameters are the vertical distribution of diabatic heating, the base state temperature structure, boundary layer specific humidity, lateral mixing coefficients for momentum and temperature, surface friction coefficient, Coriolis parameter and perturbation wavelength. A variety of unstable quasi-balanced modes are found, some of which have small e-folding times (on the order of days) with a cyclone-scale preferred wavelength; for these modes a pronounced short-wave cutoff is essentially independent of lateral mixing. For a well-delineated vertical heating profile, and certain values of the physical parameters, the vertical structure of these modes resembles that of the formative stage of a tropical cyclone. Internal gravity modes are found which are unstable; the external gravity mode damps in all cases. These results hold for both a three-layer and a seven-layer model. With coarse vertical resolution, the results are strongly dependent on the vertical staggering of the dependent variables; increasing the vertical resolution of the model eliminates differences due to the choice of staggering, and introduces a variety of unstable larger (cyclone) scale disturbances in response to variations in the vertical heating distribution.

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