The Vector Harmonic Analysis of Laplace’s Tidal Equations

Abstract

A modal analysis is presented of the linearized shallow-water equations on the sphere called Laplace’s tidal equations, using the spherical vector harmonics. The present approach is more compact and conceptually simpler than past analyses in which the order of the differential equation was raised. The modes, called Hough harmonics, are expressed as a series in the spherical Vector harmonics whose coefficients are then computed as the eigenvectors of an infinite, banded, symmetric, linear system of equations. The frequencies of the modes are determined as the eigenvalues of the banded system. New zonal rotational modes for zonal wavenumber $m = 0$ are obtained as the limit of Hough vector harmonics as m tends to zero. Although the zonal rotational modes are not unique, this new set of functions shares many properties with the nonzonal rotational modes, including orthogonality of characteristic functions. Some limiting cases of the Hough harmonics are also discussed, including the Haurwitz modes, which are determined as the limit of solutions to the untransformed shallow-water equations as the equivalent height tends to infinity. Finally, a description of the software for computing the Hough harmonics as well as the Haurwitz modes is presented. This software package is available from the National Center for Atmospheric Research and consists of four user-entry FORTRAN subroutines. Subroutine SIGMA computes the frequencies of the normal modes. Subroutine ABCOEF computes the coefficients in the expansion of the normal modes in terms of the spherical vector harmonics. Subroutine UVH tabulates the components of the meridional structure of the Hough vector functions and subroutine UVHDER tabulates certain derivatives of the components.