Abstract
The stability analysis of stratified parallel shear flows is fundamental to investigations of the onset of turbulence in atmospheric and oceanic datasets. The stability analysis is performed by considering the
behavior of small-amplitude waves, which is governed by the Taylor–Goldstein (TG) equation. The TG
equation is a singular second-order eigenvalue problem, whose solutions, for all but the simplest background
stratification and shear profiles, must be computed numerically. Accurate numerical solutions require
that particular care be taken in the vicinity of critical layers resulting from the singular nature of the
equation. Here a numerical method is presented for finding unstable modes of the TG equation, which
calculates eigenvalues by combining numerical solutions with analytical approximations across critical
layers. The accuracy of this method is assessed by comparison to the small number of stratification and shear
profiles for which analytical solutions exist. New stability results from perturbations to some of these profiles
are also obtained.
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