Mode-wave equivalence and other asymptotic problems in tsunami theory

A study is conducted of the asymptotic behavior of the gravity modes of an incompressible spherical oceanic layer, surrounding a rigid Earth, as its radius goes to infinity. The flat-layered Earth dispersion relation c= gh for the phase velocity of the tsunami wave is derived, and the existence of only one branch of tsunami modes is proved, a result fundamental for the use of mode theory in marigram synthesis. Studied further are the influence of such parameters as finite incompressibility in the ocean, and finite rigidity of the ocean floor, on the dispersion of tsunami modes, both theoretically and numerically for a number of models. It is concluded that for all physically acceptable models of both the ocean and its floor, tsunami dispersion is not significantly affected by either. This includes in particular the case of a sedimentary layer, which is found to have no effect on tsunami propagation. The maximum (and extremely small) rigidity allowed in the fluid before the tsunami modes disappear is also derived.