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A General Numerical Method for Analyzing the Linear Stability of Stratified Parallel Shear Flows

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  • 1 School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada
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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.

Corresponding author address: Tim Rees, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3065 STN CSC, Victoria BC V8W 3V6, Canada. E-mail: timothy.rees@gmail.com

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.

Corresponding author address: Tim Rees, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3065 STN CSC, Victoria BC V8W 3V6, Canada. E-mail: timothy.rees@gmail.com
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