Baroclinic instability of the zonal wind

Abstract

On the assumptions of a perfect gas, adiabatic motion, gravitational potential proportional to altitude, small Rossby number, and large Richardson number, the equations governing obliquely traveling, small disturbances relative to a zonal wind are reduced to a singular Sturm‐Liouville problem in which the latitude appears only as an implicit parameter. The structure of this problem is examined with the aid of an arbitrarily weighted quadratic integral and by transformation to an integral equation. Generalizations of Rayleigh's flex point and Howard's semicircle and rate‐of‐growth theorems are deduced from the quadratic integral. Approximations for small and large values of the wave number are deduced from the integral equation. It is shown that disturbances of sufficiently short wavelength in typical flows are always unstable but that they have a rate of growth that vanishes directly with the wavelength. Finally, it is shown that disturbances of all wavelengths in typical flows are always unstable for sufficiently small wind speeds. These conclusions lead to the conjecture that small disturbances in typical flows may be unstable for almost all wavelengths and all wind speeds.