THE SPECTRA OF BAROTROPIC QUASIGEOSTROPHIC MODEL AND THE EVOLUTION OF DISTURBANCES

Abstract

The spectra of linearized barotropic quasigeostrophic model is one of classical but not yet completelysolved problems in dynamic meteorology. This paper is devoted to a detailed investigation of this problem.It is obtained that in the case of zonal basic flow μ(y), there exist two types of spectra, the discrete andcontinuous. Each discrete spectrum (eigenvalue) gives an eigenfunction (normal mode); and the real num-bers in the interval [μ_(min), μ_(max)] are the continuous spectra; every point c ? [μ_(min), μ_(max)] gives a spectralfunction which is bounded and whose derivative is unbounded but integrable. Every disturbance can berepresented as a linear combination of two parts, ψ'(x, y, t) and ψ'(x, y, t), the former is expressed bythe discrete spectra, and the latter by the continuum. ψ'_d can be easily derived from the initial conditionby the use of the generalized "weighted orthogonality" of normal modes. It is also proved that ψ'_c as wellas its energy E'_c and mean y-direction scale l_c all approach zero as i→∞. A finite difference method for calculating the spectra and spectral functions is suggested. This numer-ical scheme keeps the same global characteristics and the criteria of instability as the original differentialoperator has, although the continuous spectra are distorted as computational discrete spectra due to the fi-nite numbers of grid points. However, the evolutionary characteristics of ψ'c can be correctly representedby this method for a long time from the initial. Of course, the computational spectra in the intervalμ_(min)≤c≤μ_(max) become denser and denser as the grid size becomes smaller and smaller, and cover the wholeinterval as the grid size approaches zero.