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

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Abstract
By making use of the quasi-nondivergent approximation, the potential vorticity equation is reduced to an equation in the stream function ψ. 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 ψ. 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 functio... Abstract By making use of the quasi-nondivergent approximation, the potential vorticity equation is reduced to an equation in the stream function ψ. 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 ψ. 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 functio...

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