The Use of Vector Spherical Harmonics in Global Meteorology and Aeronomy

Abstract

The nonlinear equations of motion of the atmosphere on the rotating earth are considered. The velocity field is expanded in terms of vector spherical harmonies, and the scalar fields (density, temperature, gravitational potentials and heat sources) are expanded in (scalar) spherical harmonies. Through the use of identities involving products of the vector and scalar spherical harmonies, the equations of motion reduce to sets of coupled equations for the radial functions which are nonlinear. In effect, a separation of variables has been accomplished. One can apply a linearization technique or one can try to solve the nonlinear equations exactly in special cases. As an example we derive an exact solution, discussed elsewhere, where the nonlinear equations are solved for the particularly simple case in which the velocity field is independent of time and longitude, the temperature is a function of radius only, the flow is adiabatic, and gravitational terms higher than the quadripole moment are ignored. Furthermore, the velocity is required to be the simplest possible function of the latitude and have no radial component. The solution corresponds to the rotation of the atmosphere about the spheroidal earth. In the present paper the atmosphere is assumed a perfect gas, and viscosity, ion drag, etc., are ignored. However, these effects can be included.