The Nonlinear Evolution of Disturbances to a Parabolic Jet

G. Brunet Laboratoire de Météorologie Dynamique du CNRS, Ecole Normale Supérieure, Paris, France

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P. H. Haynes Centre for Atmospheric Science, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

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Abstract

It has been shown that the linearized equations for disturbances to a parabolic jet on a β plane, with curvature Un0(y) such that the basic-state absolute vorticity gradient β − Un0(y) is zero, ultimately become inconsistent in the neighborhood of the jet axis and that nonlinear effects become important. Numerical solutions of the nonlinear long-time asymptotic form of the equations are presented. The numerical results show that the algebraic decay of the disturbances as t−1/2 predicted by the linear equations is inhibited by the nonlinear formation of coherent vortices new the jet axis. These lead to a disturbance amplitude that decays only through the action of weak numerical diffusion but is otherwise as t0.

The linear theory is extended to the case when the basic-state absolute vorticity gradient is nonzero but weak. When the gradient is weak and negative the decay is modified and is ultimately as t−3/2. When the gradient is weak and positive, on the other hand, a discrete eigenmode is excited and asymptotic decay is inhibited. In both cases linear theory may give a self-consistent description if the amplitude is small enough. Numerical simulation shows that for both signs of the gradient there is a range of amplitudes for which nonlinear effects become directly important. Coherent vortices may form and either inhibit the decay or disrupt the linear mode. The structure of the nonlinear analog of the linear eigenmode is analyzed and shown to have a propagation speed, relative to the jet axis speed, that is a decreasing function of amplitude, tending to zero as the amplitude approaches a finite limiting value.

Abstract

It has been shown that the linearized equations for disturbances to a parabolic jet on a β plane, with curvature Un0(y) such that the basic-state absolute vorticity gradient β − Un0(y) is zero, ultimately become inconsistent in the neighborhood of the jet axis and that nonlinear effects become important. Numerical solutions of the nonlinear long-time asymptotic form of the equations are presented. The numerical results show that the algebraic decay of the disturbances as t−1/2 predicted by the linear equations is inhibited by the nonlinear formation of coherent vortices new the jet axis. These lead to a disturbance amplitude that decays only through the action of weak numerical diffusion but is otherwise as t0.

The linear theory is extended to the case when the basic-state absolute vorticity gradient is nonzero but weak. When the gradient is weak and negative the decay is modified and is ultimately as t−3/2. When the gradient is weak and positive, on the other hand, a discrete eigenmode is excited and asymptotic decay is inhibited. In both cases linear theory may give a self-consistent description if the amplitude is small enough. Numerical simulation shows that for both signs of the gradient there is a range of amplitudes for which nonlinear effects become directly important. Coherent vortices may form and either inhibit the decay or disrupt the linear mode. The structure of the nonlinear analog of the linear eigenmode is analyzed and shown to have a propagation speed, relative to the jet axis speed, that is a decreasing function of amplitude, tending to zero as the amplitude approaches a finite limiting value.

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