Hamiltonian Description of Idealized Binary Geophysical Fluids

Abstract

A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity. The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (???·????)/?. Here, ? is the absolute vorticity, ? is the total density of the fluid, and ? is any thermodynamic variable. For example, ? can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature. Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies. A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive. The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.