Finite-Amplitude Stability of a Zonal Shear Flow

A. G. Burns Department of Mathematics, McGill University, Montreal, Quebc H3A 2K6 Canada

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S. A. Maslowe Department of Mathematics, McGill University, Montreal, Quebc H3A 2K6 Canada

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Abstract

We consider the stability of a zonal shear flow = tanhy in the framework of the β-plane approximation. The objective of this study is to determine the effects of nonlinearity on the stability of the flow by perturbing about the known linear neutral solution obtained by Howard and Drazin and, independently, by Lipps. The procedure employed is a weakly nonlinear approach which leads to the familiar amplitude evolution equation of Stuan-Watson theory. The “Landau constant,” i.e., the coefficient of the nonlinear term, turns out to be positive in contrast with the result obtained earlier by Schade for the same flow in the absence of rotation. Thus, nonlinearity is found to be destabilizing.

Despite the fact that the linear solution is regular, critical point singularities are encountered in all higher order terms of the perturbation streamfunction expansion. These have been treated by methods Analogous to the Hadamard finite-part concept. The inviscid results obtained in this way agree reasonably well with high Reynolds number solutions of the viscous equations obtained numerically.

Abstract

We consider the stability of a zonal shear flow = tanhy in the framework of the β-plane approximation. The objective of this study is to determine the effects of nonlinearity on the stability of the flow by perturbing about the known linear neutral solution obtained by Howard and Drazin and, independently, by Lipps. The procedure employed is a weakly nonlinear approach which leads to the familiar amplitude evolution equation of Stuan-Watson theory. The “Landau constant,” i.e., the coefficient of the nonlinear term, turns out to be positive in contrast with the result obtained earlier by Schade for the same flow in the absence of rotation. Thus, nonlinearity is found to be destabilizing.

Despite the fact that the linear solution is regular, critical point singularities are encountered in all higher order terms of the perturbation streamfunction expansion. These have been treated by methods Analogous to the Hadamard finite-part concept. The inviscid results obtained in this way agree reasonably well with high Reynolds number solutions of the viscous equations obtained numerically.

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