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

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 ${\bf v}_{1,2}=\nablaϕ_{1,2}$], and neglecting acoustic degrees of freedom by means of the conditions ${div}(\barρ(z)\nablaϕ_{1,2})\approx0$, where $\barρ(z)$ is a mean equilibrium density, we derive equations of motion for the interface in terms of the boundary shape $z=η(x,y,t)$ and the difference of the two boundary values of the velocity potentials: $ψ(x,y,t)=ψ_1-ψ_2$. A Hamiltonian structure of the obtained equations is proved, which is determined by the Lagrangian of the form ${\cal L}=\int \barρ(η)η_tψ\,dx dy -{\cal H}\{η,ψ\}$. 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 $\barρ(z)\propto \exp(-2αz)$, for which the Hamiltonian ${\cal H}\{η,ψ\}$ can be expressed explicitly. A long-wave regime is considered, and an approximate weakly nonlinear equation of the form $u_t+auu_x-b[-\hat\partial_x^2+α^2]^{1/2}u_x=0$ (known as Smith's equation) is derived for evolution of a unidirectional wave.