On the Convergence of Spectral Series–A Reexamination of the Theory of Wave Propagation in Distorted Background Flows

Abstract

Through a critical analysis of the convergence properties of spectral series, it is shown that Clark's method of solution leads to a divergent series; hence all his recent results on quasi-geostrophic wave propagation in distorted background flows are erroneous. A general condition for convergence is derived. The convergent solution (if it exists) to a general second-order recurrence formula is given, which is then applied to Clark's problem, yielding an exact closed form solution. The solution consists of an interacting trio of waves whose wavenumbers add up to zero. With results thus obtained, it is found that the propagation of wavenumber 2 disturbances is not affected by wavenumber 1 finite-amplitude distortions in the background flow, in disagreement with the result of Clark.