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

Details Internal waves in a compressible two-layer atmospheric model: The Hamiltonian description Internal waves in a compressible two-layer atmospheric model: The Hamiltonian description

2010-04-22

arxiv.org/abs/1004.3844v2

JETP 111(5), 776-785 (2010)

Physics

Atmospheric and Oceanic Physics

revtex4, 8 pages, submitted to JETP, information about Eq.(44) added

Scientific paper

10.1134/S1063776110110099

Slow flows of an ideal compressible fluid (gas) in the gravity field in the presence of two isentropic layers are considered, with a small difference of specific entropy between them. Assuming irrotational flows in each layer [that is ${\bf v}_{1,2}=\nabla\phi_{1,2}$], and neglecting acoustic degrees of freedom by means of the conditions ${div}(\bar\rho(z)\nabla\phi_{1,2})\approx0$, where $\bar\rho(z)$ is a mean equilibrium density, we derive equations of motion for the interface in terms of the boundary shape $z=\eta(x,y,t)$ and the difference of the two boundary values of the velocity potentials: $\psi(x,y,t)=\psi_1-\psi_2$. A Hamiltonian structure of the obtained equations is proved, which is determined by the Lagrangian of the form ${\cal L}=\int \bar\rho(\eta)\eta_t\psi \,dx dy -{\cal H}\{\eta,\psi\}$. The idealized system under consideration is the most simple theoretical model for studying internal waves in a sharply stratified atmosphere, where the decrease of equilibrium gas density with the altitude due to compressibility is essentially taken into account. For planar flows, a generalization is made to the case when in each layer there is a constant potential vorticity. Investigated in more details is the system with a model density profile $\bar\rho(z)\propto \exp(-2\alpha z)$, for which the Hamiltonian ${\cal H}\{\eta,\psi\}$ can be expressed explicitly. A long-wave regime is considered, and an approximate weakly nonlinear equation of the form $u_t+auu_x-b[-\hat\partial_x^2+\alpha^2]^{1/2}u_x=0$ (known as Smith's equation) is derived for evolution of a unidirectional wave.

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