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

Abstract

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 pr...