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Baroclinic and Barotropic Instability Spectra as Functions of N in N-Level Models

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  • 1 Department of Atmospheric Sciences, The University of Arizona, Tucson, AZ 85721
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Abstract

The linear stabilities of simple baroclinic and barotropic flows are investigated by finite differencing of the quasi-geostrophic perturbation equations and reduction to the standard algebraic eigenproblem. In the baroclinic case, four different arrangements of winds and static stability are studied. In the barotropic case, cosine and hyperbolic secant jets are assumed between latitude walls. In all baroclinic and barotropic cases, only the number of levels is varied. In the baroclinic case, 2≤N≤26. In the barotropic case, 3≤N≤20, with solutions assumed to be symmetric across the wind maximum.

Increase of the number of levels, N, by one or a few does not assure a growth-rate spectrum closer to that for a great many levels. As N is increased from the minimum permissible number, the maximum growth rate, the corresponding phase velocity, and the wavelength of maximum growth rate, all describe irregular damped oscillations.

The results in the baroclinic cases depend greatly on the wind and temperature distributions. The two-level model may miss the existence of instability altogether, and in other cases small N may yield very poor descriptions of phase and amplitude, especially in the stratosphere and lower troposphere. Increase of N extends instability to shorter wavelengths, and secondary maxima varying with N are found at short wavelengths. In the barotropic case, small values of N may yield double maxima of similar strength at long wavelengths. For larger N, instability extends to shorter wavelengths, and a secondary maximum may appear at the shortwave end of the spectrum. The barotropic instability spectrum is very sensitive to subtleties of the wind distribution and shows tame variations as truncation errors vary with changing N.

Abstract

The linear stabilities of simple baroclinic and barotropic flows are investigated by finite differencing of the quasi-geostrophic perturbation equations and reduction to the standard algebraic eigenproblem. In the baroclinic case, four different arrangements of winds and static stability are studied. In the barotropic case, cosine and hyperbolic secant jets are assumed between latitude walls. In all baroclinic and barotropic cases, only the number of levels is varied. In the baroclinic case, 2≤N≤26. In the barotropic case, 3≤N≤20, with solutions assumed to be symmetric across the wind maximum.

Increase of the number of levels, N, by one or a few does not assure a growth-rate spectrum closer to that for a great many levels. As N is increased from the minimum permissible number, the maximum growth rate, the corresponding phase velocity, and the wavelength of maximum growth rate, all describe irregular damped oscillations.

The results in the baroclinic cases depend greatly on the wind and temperature distributions. The two-level model may miss the existence of instability altogether, and in other cases small N may yield very poor descriptions of phase and amplitude, especially in the stratosphere and lower troposphere. Increase of N extends instability to shorter wavelengths, and secondary maxima varying with N are found at short wavelengths. In the barotropic case, small values of N may yield double maxima of similar strength at long wavelengths. For larger N, instability extends to shorter wavelengths, and a secondary maximum may appear at the shortwave end of the spectrum. The barotropic instability spectrum is very sensitive to subtleties of the wind distribution and shows tame variations as truncation errors vary with changing N.

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