On the Formulation of Linear Instability and Forced Problems for a Zonally Varying Basic State in the Complex Domain

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  • 1 Cooperative Institute for Climate Studies, Department of Meteorology, University of Maryland, College Park, College Park, Maryland
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Abstract

It is shown that when the basic state has zonal variation, the linearized operator of an atmospheric spectral model acts on the perturbation spectral coefficients of both the nonnegative zonal indexes and their conjugates. Mathematically, this implies that the complex linear system is no longer analytic (i.e., the Cauchy–Riemann condition is not satisfied). This paper presents a method that solves the steady response and eigenvalue problems in the complex domain. It is also suggested that for a given computer memory capacity, the linear forced problem of a zonally varying basic state could be solved by the new method at a resolution twice as high as the methodologically more straightforward real matrix method.

Abstract

It is shown that when the basic state has zonal variation, the linearized operator of an atmospheric spectral model acts on the perturbation spectral coefficients of both the nonnegative zonal indexes and their conjugates. Mathematically, this implies that the complex linear system is no longer analytic (i.e., the Cauchy–Riemann condition is not satisfied). This paper presents a method that solves the steady response and eigenvalue problems in the complex domain. It is also suggested that for a given computer memory capacity, the linear forced problem of a zonally varying basic state could be solved by the new method at a resolution twice as high as the methodologically more straightforward real matrix method.

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