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

TL;DR

This paper develops a Hamiltonian framework for internal waves in a compressible, two-layer atmosphere, deriving equations of motion and analyzing weakly nonlinear wave regimes with explicit Hamiltonian expressions.

Contribution

It introduces a Hamiltonian structure for internal waves in a compressible two-layer atmosphere, including explicit forms for specific density profiles and weakly nonlinear wave equations.

Findings

Hamiltonian structure for internal wave equations established

Explicit Hamiltonian derived for exponential density profile

Weakly nonlinear wave equation obtained and analyzed

Abstract

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.