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

Constantin, Adrian LU and Johnson, R S (2008) In Fluid Dynamics Research 40(3). p.175-211
Abstract
We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale e (weak nonlinearity), and on the scale, a, 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(epsilon(4/5)); the equations here are expressed in terms of the single parameter epsilon(2/5) sigma 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... (More)
We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale e (weak nonlinearity), and on the scale, a, 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(epsilon(4/5)); the equations here are expressed in terms of the single parameter epsilon(2/5) sigma 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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
multiple, scales, shoreline, tsunami, vorticity, long waves, variable depth
in
Fluid Dynamics Research
volume
40
issue
3
pages
175 - 211
publisher
Elsevier
external identifiers
  • wos:000253404800002
  • scopus:38649087078
ISSN
1873-7005
DOI
10.1016/j.fluiddyn.2007.06.004
language
English
LU publication?
yes
id
db678908-8dcb-4d38-85cf-17119d5dd953 (old id 1193828)
date added to LUP
2008-09-10 09:18:03
date last changed
2017-07-09 03:43:39
@article{db678908-8dcb-4d38-85cf-17119d5dd953,
  abstract     = {We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale e (weak nonlinearity), and on the scale, a, 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(epsilon(4/5)); the equations here are expressed in terms of the single parameter epsilon(2/5) sigma 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.},
  author       = {Constantin, Adrian and Johnson, R S},
  issn         = {1873-7005},
  keyword      = {multiple,scales,shoreline,tsunami,vorticity,long waves,variable depth},
  language     = {eng},
  number       = {3},
  pages        = {175--211},
  publisher    = {Elsevier},
  series       = {Fluid Dynamics Research},
  title        = {Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis},
  url          = {http://dx.doi.org/10.1016/j.fluiddyn.2007.06.004},
  volume       = {40},
  year         = {2008},
}