Journal of Experimental and Theoretical Physics

, Volume 111, Issue 5, pp 776–785 | Cite as

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

  • V. P. Ruban
Solids And Liquids


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 = ▿ϕ1, 2) and neglecting the acoustic degrees of freedom (div(\( \bar \rho \)(z)▿ϕ1, 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 t ψdxdy − 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]1/2 u x = 0 (Smith’s equation) for the slow evolution of a traveling wave.


Vorticity Internal Wave Potential Vorticity Equilibrium Density Hamiltonian Structure 
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© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia

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