Normal Modes of Ultralong Waves in the Atmosphere

Abstract

Spherical harmonics have been used to analyze global meteorological data, and because they are the solutions of a linearized nondivergent vorticity equation, it is appropriate to use them as orthogonal basis functions for analysis and prediction. However, for ultralong waves?geostrophic motions of the second type?horizontal divergence plays as essential a role as the vertical component of vorticity. Hence, it will be advantageous to use the solutions of linearized primitive equations over a sphere as basis functions. This will also permit identification of the characteristics of wave motions for the initialization of primitive equation models. Such solutions have been investigated in the past in conjunction with atmospheric tidal theories and the basic mathematical tools already available piecewise in the literature. This paper reviews the mathematical development behind the construction of the eigensolutions (referred to as normal modes) of linearized primitive equations over a sphere. The basic state has no motion and the temperature is a function of height only. The solutions of both the vertical and horizontal structure equations are discussed. The horizontal parts of such normal modes are called Hough harmonics Θsl exp (is?), where s is zonal wavenumber, ? longitude and l meridional mode index. Hough vector functions Θsl consist of three components?zonal velocity ?, meridional velocity V? and geopotential height ?, all of which are functions of latitude. There are three modes with distinct frequencies: eastward and westward propagating gravity waves, and westward propagating rotational waves of the Rossby/Haurwitz type. Hough harmonics are orthogonal and are conveniently used to decompose wind and mass fields simultaneously. Some examples are presented of global data decomposition in terms of Hough harmonics for studying ultralong waves in the atmosphere.