On the normal modes of Laplace’s tidal equations for zonal wavenumber zero

Abstract

Normal modes of Laplace's tidal equations, referred to as Hough harmonics, are complete for zonal wavenumber m > 0 so that the longitudinal and meridional velocity components and the geopotential can be represented as a series of Hough harmonics. However, Hough harmonics corresponding to m= 0 are incomplete in that the second kind normal modes have all zero frequencies. To fill the need for orthonormal basis functions, Kasahara and Shigehisa have constructed two different sets of the rotational modes of Laplace's tidal equations for m= 0, referred to as the K-modes and the S-modes, respectively. In this study, we compared the characteristic differences between the K- and S-modes in their energy ratio and structures. The zonal-mean components of atmospheric data from the FGGE IIIb reanalysis are projected onto the K- and S-modes separately, in addition to the gravity modes. We showed that the K-mode representation captures the majority of observed zonal energy with a few terms, whereas the S-mode representation requires many terms. The K-mode series converges faster than the S-mode series, especially for small vertical-scale components in the observed zonal fields. The differences between the energy spectra projected upon the K- and S-modes are discussed along with the consideration of the merits of each set as expansion functions for the zonal atmospheric motions.