Asymptotic normal modes of the Earth-II. Eigenfunctions

Abstract

Summary We derive asymptotic expressions for the toroidal- and spheroidal-mode eigenfunctions of a SNREI earth model with two discontinuities. The ray-theoretical wavefunctions of high-frequency SH and P-SV waves are expressed in terms of scalar potentials for which WKBJ solutions are readily available. The normal-mode eigenfunctions are then derived from the wavefunctions using the correspondence between the toroidal modes and SH waves, and the spheroidal modes and P-SV waves. Both oscillatory and evanescent solutions are included to enhance the accuracy of the asymptotic eigenfunctions in the vicinity of the inner-core and core-mantle boundaries. The analytic expressions of the asymptotic eigenfunctions only involve simple sine, cosine and exponential functions which can be easily manipulated and evaluated, making the asymptotic eigenfunctions very useful for further normal-mode studies. Numerical comparison of the asymptotic and exact eigenfunctions shows that there is an excellent agreement for a crustless version of earth model 1066A. A mistake in our previous asymptotic eigenfrequency analysis, which ignored the geometrical spreading effect, has been corrected. The consistency between the asymptotic eigenfrequency and eigenfunction analyses is also addressed.