Internal waves in a compressible two-layer model atmosphere: Hamiltonian description

We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential in each of the layers (v_{1, 2} = ▿ϕ_{1, 2}) and neglecting the acoustic degrees of freedom (div($$ \bar \rho $$(z)▿ϕ_{1, 2}) ≈ 0, where $$ \bar \rho $$(z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ₁ ‑ ψ₂. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫$$ \bar \rho $$(η)η_tψdxdy ‑ ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential vorticity. We investigate a system with a model dependence $$ \bar \rho $$(z) ∝ e^‑2αz with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion corrections and derive an approximate nonlinear equation of the form u_t + auu_x ‑ b[‑$$ \hat \partial _x^2 $$ + α²]^1/2u_x = 0 (Smith's equation) for the slow evolution of a traveling wave.