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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
  • 45 Downloads

Abstract

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.

Keywords

Vorticity Internal Wave Potential Vorticity Equilibrium Density Hamiltonian Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

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

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