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Weakly nonlinear waves in a stratified fluid: a variational formulation

Published online by Cambridge University Press:  21 April 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

The Lagrangian L for gravity waves of small but finite amplitude in an N-layer stratified fluid is constructed as a function of the generalized coordinates qv(t) ≡ {qvn(t)} of the N + 1 interfaces, where the qvn are the Fourier coefficients of the expansion of the interfacial displacement ηv(x, t) in a complete, orthogonal set {ψn(x)}. The density is constant in each layer, by virtue of which a velocity potential exists for that layer (even though the full flow is rotational). The explicit expansion of L is constructed through fourth-order in qν and $\dot{\boldmath q}_{\nu}$ through an extension of the surface-wave formulation (Miles 1976), in which the pressure appears as the Lagrangian density (Luke 1967). Three-dimensional progressive and standing interfacial waves in a two-layer fluid are treated as general examples, and the two-dimensional results of Hunt (1961) and Thorpe (1968) are recovered as explicit examples. It is shown that the spatial resonance between surface and internal waves conjectured by Mahony & Smith (1972) is impossible for the two-layer Boussinesq model.

The joint limit N ↑ ∞ and layer thickness ↓ 0 yields the Lagrangian density L for a continuously stratified, Boussinesq fluid as a functional of qn([yscr ]) and $\dot{q}_n$([yscr ]), where [yscr ], the counterpart of the layer index, is a Lagrangian (rather than Eulerian) coordinate. The coefficient C in the nonlinear dispersion relation (ω/ω1)2 = 1 + Ck2A2 for progressive waves of frequency ω, wavenumber k and amplitude A, where ω1 = ω1(k) for infinitesimal waves, is determined for any density profile for which the (linear) vertical structure problem can be solved. Explicit results are given for a fluid of finite vertical extent in which the buoyancy frequency is constant and for a vertically unbounded fluid in which the buoyancy frequency varies like sech ([yscr ]/h) and C = C(kh).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 172 , November 1986 , pp. 499 - 512
Copyright
© 1991 Cambridge University Press

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