Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis

We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale ε (weak nonlinearity), and on the scale, σ, of the depth variation. In our model, dispersion does not affect the evolution of the wave even over the large distances that tsunamis may travel. We allow a distribution of vorticity, in addition to variable depth. Our solution is not valid for depth=O(ε4/5); the equations here are expressed in terms of the single parameter ε2/5σ and matched to the solution in deep water. For a slow depth variation of the background state (consistent with our model), we prove that a constant-vorticity solution exists, from deep water to shoreline, and that regions of isolated vorticity can also exist, for appropriate bottom profiles. We describe how the wave properties are modified by the presence of vorticity. Some graphical examples of our various solutions are presented.