Studies of CISK

A. WIIN NIELSEN

Abstract

The conditional instability of the second kind (CISK) is investigated. A quasi-geostrophic model with a variable static stability in the vertical direction is used to solve the CISK problem exactly. Due to the nature of the solution it cannot be calculated with any accuracy for small wavelenghts. Using vertical structure functions no numerical problems appear except for very small wavelenghts. The approximate solutions are in excellent agreement with the exact solution, when both of them can be evaluated with good accuracy. The frequency may be obtained by summing a series without solving the complete problem. The quasi-geostrophic case is generalized to the case of the primitive equations with a few vertical components. The existence of the gravity-inertia modes has an influence on the stability of the quasi-geostrophic mode. The long waves become more stable, but for short waves the difference between the two modes become very small. and in the limit where the wavelength goes to zero both of them will approach the so-called 'free-ride' solution. Large deviations from the quasi-geostrophic solution, as measured for example by the ratio of divergence to vorticity, are found for the short waves. A final result is that in using the primitive equations it is important to express the frictional vertical velocity in terms of the streamfunction, but not in terms of the geopotential. The latter procedure will lead to spurious modes due to an inconsistent use of the quasi-geostrophic equations. These spurious models have an e-doubling time which is a small fraction of a day.

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