Numerical Integration of the Global Barotropic Primitive Equations with Hough Harmonic Expansions

Akira Kasahara National Center for Atmospheric Research, Boulder, Colo. 80307

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Abstract

A new spectral model is formulated using Hough harmonics as basis functions to solve numerically the nonlinear barotropic primitive equations (shallow water equations) over a sphere. Hough harmonics are eigensolutions of free oscillations (normal modes) for linearized shallow water equations over a sphere about a basic state of rest and a prescribed equivalent height. Hough harmonics are expressed by Θls exp(isλ) with zonal wavenumber s, longitude λ and meridional index l. Hough vector functions Θls consist of three components-zonal velocity U, meridional velocity V and geopotential height Z, all functions of latitude. There are three modes with distinct frequencies for s≥1: eastward and westward propagating gravity waves and westward propagating rotational waves of the Rossby/Haurwitz type.

The advantage of using Hough harmonies for a spectral barotropic global primitive equation model is that the prognostic variables are efficiently represented because Hough harmonics are normal modes of the prediction model. Initialization is no longer a separate procedure but is built into the forecasting scheme. The characteristics of wave motions are associated with the expansion functions, so that the filtering of unwanted wave components can be performed easily.

The nonlinear advection term in the spectral equation is calculated by the transform method–a combination of Fourier transform and Gaussian quadrature. The results of a test calculation with a balanced non-divergent initial state (a Haurwitz wave) compare favorably with those of an integration using a fourth-order finite-difference model.

Abstract

A new spectral model is formulated using Hough harmonics as basis functions to solve numerically the nonlinear barotropic primitive equations (shallow water equations) over a sphere. Hough harmonics are eigensolutions of free oscillations (normal modes) for linearized shallow water equations over a sphere about a basic state of rest and a prescribed equivalent height. Hough harmonics are expressed by Θls exp(isλ) with zonal wavenumber s, longitude λ and meridional index l. Hough vector functions Θls consist of three components-zonal velocity U, meridional velocity V and geopotential height Z, all functions of latitude. There are three modes with distinct frequencies for s≥1: eastward and westward propagating gravity waves and westward propagating rotational waves of the Rossby/Haurwitz type.

The advantage of using Hough harmonies for a spectral barotropic global primitive equation model is that the prognostic variables are efficiently represented because Hough harmonics are normal modes of the prediction model. Initialization is no longer a separate procedure but is built into the forecasting scheme. The characteristics of wave motions are associated with the expansion functions, so that the filtering of unwanted wave components can be performed easily.

The nonlinear advection term in the spectral equation is calculated by the transform method–a combination of Fourier transform and Gaussian quadrature. The results of a test calculation with a balanced non-divergent initial state (a Haurwitz wave) compare favorably with those of an integration using a fourth-order finite-difference model.

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