FINITE-AMPLITUDE THREE-DIMENSIONAL HARMONIC WAVES ON THE SPHERICAL EARTH

Abstract

By making use of the quasi-nondivergent approximation, the potential vorticity equation is reduced to an equation in the stream function ?. Assuming that the motion is of permanent wave type, a first integral of this nonlinear vorticity equation is obtained, which itself is a linear three-dimensional partial differential equation in ?. This equation has been solved as a boundary-value problem by the method of separation of variables. It is found that the latitudinal-amplitude functions of these waves satisfy a spheroidal-wave equation while the vertical-amplitude functions are given by Bessel and Hankel functions of the argument lp/p0, where l is a parameter depending on both the static stability and the nodal number r. The eigenvalues µmr of these wave solutions are connected with the parameter l2 by a transcendental relation. We have expanded µmr into a power series of l2 and obtained the various coefficients, up to that of the fourth power of l2. The latitudinal- and vertical-amplitude functions for the wave numbers m = 0, 3, and 6 have also been obtained. Because of the additional degrees of freedom introduced by the vertical variation of ?, it is possible to obtain a combination of the partial-wave solutions which can approximate the observed motions in the atmosphere more closely than the solutions of the purely two-dimensional vorticity equation can. When the effect of friction is included, these harmonic waves will be damped in time.