Bifurcation and Catastrophe in a Simple, Forced, Dissipative Quasi-Geostrophic Flow

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  • 1 Department of Meteorology, The Pennsylvania State University, University Park 16802
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Abstract

The steady solutions and their stability properties are investigated for a low-order spectral model of a forced, dissipative, nonlinear, quasi-geostrophic flow. A zonal flow is modified by two smaller scale disturbances in the model.

If only the zonal component (or only the smallest scale component) is forced, then the stationary solution is unique, always locally stable, and globally stable for weak forcing.

There is also a unique locally stable stationary solution for weak forcing of only the middle component. But as this forcing exceeds a critical value, a supercritical bifurcation to new solutions appears.

The entire solution surface for forcing of the zonal and middle components can be displayed graphically and is a form of the well-known cusp catastrophe surface. For forcing of all three components, the morphogenesis set is more complex, containing regions in which there are one, three or five solutions.

Numerical integrations of the phase-sparce trajectories of the solutions reveal that for forcing of the zonal and middle components 1) domains of attraction of stable steady solutions contain neighborhoods near the unstable steady solution, 2) there is a region near the cusp in which initial points produce periodic solutions, and 3) initial points further away from the cusp yield trajectories that quickly approach stable steady solutions.

The conclusion is that any successful theory of atmospheric climate will have to contend with multiple solutions and changing domains of attraction as external parameters are varied.

Abstract

The steady solutions and their stability properties are investigated for a low-order spectral model of a forced, dissipative, nonlinear, quasi-geostrophic flow. A zonal flow is modified by two smaller scale disturbances in the model.

If only the zonal component (or only the smallest scale component) is forced, then the stationary solution is unique, always locally stable, and globally stable for weak forcing.

There is also a unique locally stable stationary solution for weak forcing of only the middle component. But as this forcing exceeds a critical value, a supercritical bifurcation to new solutions appears.

The entire solution surface for forcing of the zonal and middle components can be displayed graphically and is a form of the well-known cusp catastrophe surface. For forcing of all three components, the morphogenesis set is more complex, containing regions in which there are one, three or five solutions.

Numerical integrations of the phase-sparce trajectories of the solutions reveal that for forcing of the zonal and middle components 1) domains of attraction of stable steady solutions contain neighborhoods near the unstable steady solution, 2) there is a region near the cusp in which initial points produce periodic solutions, and 3) initial points further away from the cusp yield trajectories that quickly approach stable steady solutions.

The conclusion is that any successful theory of atmospheric climate will have to contend with multiple solutions and changing domains of attraction as external parameters are varied.

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