On the Spectral Integration of the Quasi-Geostrophic Equations for Doubly-Periodic and Channel Flow

Geoffrey K. Vallis Scripps Institution of Oceanography, La Jolla, CA 92093

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Abstract

The spectral integration of the quasi-geostrophic equations is reexamined for simple boundary conditions in Cartesian geometry. For doubly-periodic flow, it is shown that the mean shear must be constant in time or its evolution specified; grid transform methods are then appropriate. In a channel model, a suitable choice of spectral expansion allows the mean shear to be determined internally, i.e., from the model equations, and also allows the meridional gradient of potential vorticity at the boundaries to be specified. However, the use of conventional transform techniques will lead to aliasing and energy nonconservation, and use must be made either of interaction coefficients or a combination of appended zero grid transforms and analytic Fourier expansions.

Abstract

The spectral integration of the quasi-geostrophic equations is reexamined for simple boundary conditions in Cartesian geometry. For doubly-periodic flow, it is shown that the mean shear must be constant in time or its evolution specified; grid transform methods are then appropriate. In a channel model, a suitable choice of spectral expansion allows the mean shear to be determined internally, i.e., from the model equations, and also allows the meridional gradient of potential vorticity at the boundaries to be specified. However, the use of conventional transform techniques will lead to aliasing and energy nonconservation, and use must be made either of interaction coefficients or a combination of appended zero grid transforms and analytic Fourier expansions.

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