Asymptotic normal modes of a laterally heterogeneous Earth: 2. Further results

In the geometrical optics approximation the split eigenfrequencies δω of an isolated multiplet on a laterally heterogeneous earth are certain eigencontours of the great circular average of the local eigenfrequency perturbation δωlocal. The normal to a surface wave orbital plane, averaged over an orbit, precesses around the contours of , and the quantization condition that selects the eigencontours is where is the orbital angular momentum and is averaged over a precession cycle. All but one of the split eigenfrequencies in this approximation are degenerate doublets because of the even parity of , and the spacing between adjacent doublets along a given branch of versus is Δωadjacent ≈ 2π/Tprecess, where Tprecess is the precession period. There is a separate branch of the action versus associated with every local minimum or maximum of on the surface of the earth, and the resulting interleaving of doublets can lead to very irregular line spacing. The doublets are split even on a nonrotating earth by tunneling between surface waves propagating in opposite directions about otherwise identical ray paths. Ordinarily, the tunneling time Ttunnel is much longer than Tprecess so the spacing Δωtunnel = 2π/Ttunnel between the singlets in a quasi‐degenerate doublet is extremely small. In general, the line spacing in a split multiplet is most complicated near the separatrix or slow‐fast transition, since it is there that the precession period is the greatest and the tunneling time is the least.