A Simple Approximate Result for the Maximum Growth Rate of Baroclinic Instabilities

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Abstract
The Charney problem for baroclinic instability involves the quasi-geostrophic instability of a zonal flow on a β plane where the zonal flow is characterized by a constant vertical shear. The atmosphere is non-Boussinesq and continuous. The solution of this problem involves confluent hypergeometric functions, and the mathematical difficulty of the problem, for the most part, has precluded extracting simple results of some generality. In this note, it is shown that there does exist a very simple, powerful approximate result for the growth rate of the most rapidly growing instability, viz., that this growth rate is linearly proportional to the surface meridional temperature gradient. The coefficient of proportionality is also easily determined. Moreover, the result extends to substantially more general profiles than those in the Charney problem. Abstract The Charney problem for baroclinic instability involves the quasi-geostrophic instability of a zonal flow on a β plane where the zonal flow is characterized by a constant vertical shear. The atmosphere is non-Boussinesq and continuous. The solution of this problem involves confluent hypergeometric functions, and the mathematical difficulty of the problem, for the most part, has precluded extracting simple results of some generality. In this note, it is shown that there does exist a very simple, powerful approximate result for the growth rate of the most rapidly growing instability, viz., that this growth rate is linearly proportional to the surface meridional temperature gradient. The coefficient of proportionality is also easily determined. Moreover, the result extends to substantially more general profiles than those in the Charney problem.

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