THE SPECTRAL FORM OF THE VORTICITY EQUATION

George W. Platzman The University of Chicago

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Abstract

The nonlinear aspects of the vorticity equation for two-dimensional planetary circulations of the earth's atmosphere may be studied by expansion of the solution in spherical surface harmonics. Some of the main mathematical problems are discussed here that arise in an examination of the “spectral” form of the vorticity equation which results from such an expansion. The truncation of the spectral equations is discussed, and a proof is given of the invariance of mean square velocity and vorticity for truncated spectra. Some comments are made on low-order systems, which are to be followed by a detailed investigation of three-component systems in the second part of this study.

Abstract

The nonlinear aspects of the vorticity equation for two-dimensional planetary circulations of the earth's atmosphere may be studied by expansion of the solution in spherical surface harmonics. Some of the main mathematical problems are discussed here that arise in an examination of the “spectral” form of the vorticity equation which results from such an expansion. The truncation of the spectral equations is discussed, and a proof is given of the invariance of mean square velocity and vorticity for truncated spectra. Some comments are made on low-order systems, which are to be followed by a detailed investigation of three-component systems in the second part of this study.

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