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

Lookup NU author(s): Dr Adrian Constantin, Professor Robin Johnson


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We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale ε{lunate} (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 (ε{lunate}4 / 5); the equations here are expressed in terms of the single parameter ε{lunate}2 / 5 σ 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. © 2007 The Japan Society of Fluid Mechanics and Elsevier B.V.

Publication metadata

Author(s): Constantin A, Johnson RS

Publication type: Article

Publication status: Published

Journal: Fluid Dynamics Research

Year: 2008

Volume: 40

Issue: 3

Pages: 175-211

Print publication date: 01/03/2008

ISSN (print): 0169-5983

ISSN (electronic): 1873-7005

Publisher: Institute of Physics Publishing Ltd.


DOI: 10.1016/j.fluiddyn.2007.06.004


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