Weakly nonlinear waves in a stratified fluid: a variational formulation

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 q v (t) ≡ {q v n (t)} of the N + 1 interfaces, where the q v n are the Fourier coefficients of the expansion of the interfacial displacement η v ( x , t) in a complete, orthogonal set {ψ n ( 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 q ν and q̇ν 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 q n ([yscr ]) and q̇ₙ ([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).