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... 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...