A variational derivation of the thermodynamics of a moist atmosphere with rain process and its pseudoincompressible approximation

Abstract

ABSTRACTIrreversible processes play a major role in the description and prediction of atmospheric dynamics. In this paper, we present a variational derivation for moist atmospheric dynamics with rain process and subject to the irreversible processes of viscosity, heat conduction, diffusion, and phase transition. This derivation is based on a general variational formalism for nonequilibrium thermodynamics which extends Hamilton's principle to incorporate irreversible processes. It is valid for any state equation and thus also includes the treatment of the atmosphere of other planets. In this approach, the second law of thermodynamics is understood as a nonlinear constraint formulated with the help of new variables, called thermodynamic displacements, whose time derivative coincides with the thermodynamic force of the irreversible process. In order to cover the case of atmospheric dynamics, the original variational principle is extended in three directions in this paper: the inclusion of the rain process, the inclusion of phase changes, and the treatment of constraints. The variational formulation is written both in the Lagrangian and Eulerian descriptions and can be directly adapted to oceanic dynamics. The proposed variational formulation yields a general approach for the modelling of thermodynamically consistent models in atmospheric dynamics, thereby extending previous variational methods that were restricted to the reversible Hamiltonian case. We illustrate this point, by deriving a moist pseudoincompressible model with general equations of state and subject to the irreversible processes of viscosity, heat conduction, diffusion, and phase transition.