Toroidal Free Oscillations of the Laterally Heterogeneous Earth

Summary The splitting of the eigenfrequencies of the Earth due to lateral heterogeneities is studied by means of the theory of perturbation of the degenerate eigenfrequencies of the spherically symmetric average earth. The eigen‐functions n u m l (r) associated with a degenerate frequency lead asymptotically (for large l ) to surface waves propagating along great circle paths determined by l and m . The lateral heterogeneities couple these eigenfunctions and produce a complicated interference pattern on the surface of the Earth. The perturbation expansion leads, to first order, to a matrix eigenvalue problem for a Hermitian matrix. The elements of the matrix express the coupling between different eigenvectors of the degenerate problem. We expand the lateral heterogeneities in a series of spherical harmonics (δρ t s ( r ) Y t s (θ, φ), δμ t s ( r ) Y t s (θ, φ)). Every term in this expansion couples the eigenfunctions selectively. These selection rules can all be deduced from a simple vector diagram. If there is no strong coupling with overtones or spheroidal‐toroidal coupling, the odd terms in the spherical harmonics expansion of the heterogeneities do not affect the eigenfrequencies. For instance, there would be no splitting if the Earth had a continental hemisphere and an oceanic hemisphere. Also, terms such that s > 2 l do not affect the eigenfrequencies of modes of order l .