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Weakly dispersive nonlinear gravity waves

Published online by Cambridge University Press:  20 April 2006

John Miles  and
Rick Salmon
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92093
Rick Salmon
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92093
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Abstract

The equations for gravity waves on the free surface of a laterally unbounded inviscid fluid of uniform density and variable depth under the action of an external pressure are derived through Hamilton's principle on the assumption that the fluid moves in vertical columns. The resulting equations are equivalent to those of Green & Naghdi (1976). The conservation laws for energy, momentum and potential vorticity are inferred directly from symmetries of the Lagrangian. The potential vorticity vanishes in any flow that originates from rest; this leads to a canonical formulation in which the evolution equations are equivalent, for uniform depth, to Whitham's (1967) generalization of the Boussinesq equations, in which dispersion, but not nonlinearity, is assumed to be weak. The further approximation that nonlinearity and dispersion are comparably weak leads to a canonical form of Boussinesq's equations that conserves consistent approximations to energy, momentum (for a level bottom) and potential vorticity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 157 , August 1985 , pp. 519 - 531
Copyright
© 1985 Cambridge University Press

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