Barotropic Quasi-Geostrophic Flow Over Anisotropic Mountains

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  • 1 Department of Astrogeophysics, University of Colorado, Boulder 80309
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Abstract

This paper discusses the nature of quasi-geostrophic β-plane flow over an idealized set of ridges with height h=F(y/Ly) cosx/Lx. When the mountains are highly anisotropic, with scale factor ratio Lx/Ly≪1, the asymptotically exact forced solution is governed by a simple set of three nonlinear ordinary differential equations similar to those obtained by Charney and DeVore (1978). For fixed forcing, the region of parameter space where multiple, stable steady solutions exist is mapped out. A cusp catastrophe occurs in which a rapid zonal flow over the ridges drops to a very low value as a parameter like the driving Rossby number decreases slightly below a certain critical point; and the zonal flow then remains at this low value for a large range of Rossby number on either side of the bifurcation value. The existence of limit cycle solutions is discussed. Such periodic solutions are shown to exist for the f-plane case, and probably exist for the β-plane as well. However, numerical solutions indicate. that the limit cycles are unstable, with the steady solutions being favored. The stationary solutions are also shown to be stable with respect to barotropic isotropic perturbations.

Abstract

This paper discusses the nature of quasi-geostrophic β-plane flow over an idealized set of ridges with height h=F(y/Ly) cosx/Lx. When the mountains are highly anisotropic, with scale factor ratio Lx/Ly≪1, the asymptotically exact forced solution is governed by a simple set of three nonlinear ordinary differential equations similar to those obtained by Charney and DeVore (1978). For fixed forcing, the region of parameter space where multiple, stable steady solutions exist is mapped out. A cusp catastrophe occurs in which a rapid zonal flow over the ridges drops to a very low value as a parameter like the driving Rossby number decreases slightly below a certain critical point; and the zonal flow then remains at this low value for a large range of Rossby number on either side of the bifurcation value. The existence of limit cycle solutions is discussed. Such periodic solutions are shown to exist for the f-plane case, and probably exist for the β-plane as well. However, numerical solutions indicate. that the limit cycles are unstable, with the steady solutions being favored. The stationary solutions are also shown to be stable with respect to barotropic isotropic perturbations.

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