Integration with the Spectral Vorticity Equation

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  • 1 Colorado State University
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Abstract

The barotropic vorticity equation in spectral form is integrated for a time period exceeding 80 days in two cases of hypothetical initial conditions with a time step of three hours without appreciable truncation error at the end of the period. The computational stability and truncation properties of the spectral system are discussed, the stability criterion for the two cases is computed, and the truncation errors are to some extent explained.

The results of the integrations show systematic periods of very pronounced energy exchange among the long waves, with little energy filtering down to the short waves. An analytic solution of a low-order spectral system suggests that the periodic exchange may be characteristic of the differential equation rather than dependent entirely on the initial conditions. The high-frequency components are examined for equilibrium of energy exchange after extended integration. Our results suggest that such an equilibrium does not exist in the model we have formulated.

Abstract

The barotropic vorticity equation in spectral form is integrated for a time period exceeding 80 days in two cases of hypothetical initial conditions with a time step of three hours without appreciable truncation error at the end of the period. The computational stability and truncation properties of the spectral system are discussed, the stability criterion for the two cases is computed, and the truncation errors are to some extent explained.

The results of the integrations show systematic periods of very pronounced energy exchange among the long waves, with little energy filtering down to the short waves. An analytic solution of a low-order spectral system suggests that the periodic exchange may be characteristic of the differential equation rather than dependent entirely on the initial conditions. The high-frequency components are examined for equilibrium of energy exchange after extended integration. Our results suggest that such an equilibrium does not exist in the model we have formulated.

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