Improving Moist Thermodynamics in Weather and Climate Models

Abstract

Approximations in the moist thermodynamics of atmospheric models can often be in- consistent; different parts of numerical models may handle the thermodynamics in dif- ferent ways, or the approximations may disagree with the laws of thermodynamics. To address these problems all relevant thermodynamic quantities may be derived from a de- fined thermodynamic potential; approximations are instead made to the potential itself — this guarantees self-consistency, as well as flexibility. Previous work showed that this concept is viable for vapour and liquid water mixtures in a moist atmospheric system using the Gibbs potential. However, on extension to include the ice phase an ambiguity is en- countered at the triple-point. To resolve this ambiguity, here the internal energy potential is used instead. Maximisation/minimisation methods on the entropy/energy can be used to find the system equilibrium state of a simple moist system, but some difficulties are encountered where the solution suggests negative mass. Conveniently, Lagrange multipli- ers can be used to enforce the equality and inequality constraints, and properly solve for the equilibrium state. However, a further extension is necessary for realistic atmospheric systems where many important non-equilibrium processes take place; for example, freezing of super-cooled water, and evaporation into subsaturated air. To fully capture processes such as these, the previous method must be reformulated to involve finite rates of approach towards equilibrium. The principles of non-equilibrium thermodynamics are therefore used; beginning with a set of phenomenological equations a framework is developed to represent the non-equilibrium processes in a moist atmospheric parcel. To implement the proposed approach it is necessary to translate conventional atmospheric microphysics ex- pressions for fluxes of matter and entropy in and around a cloud droplet into the formalism of non-equilibrium thermodynamics. This procedure is first derived for some simple ide- alised cases, beginning with liquid droplet growth by vapour diffusion, and proceeding to more complex three-phase cases. A single parcel model is then explored as a test case before developing a 2D model. The single parcel model couples the thermodynamics to some rudimentary dynamics and solves both the equilibrium and non-equilibrium regimes for comparison. To represent a more realistic atmospheric system, the moist non-equilibrium framework is subsequently coupled to a 2D semi-implicit semi-Lagrangian dynamical core. For the equilibrium regime first of all, standard bubble test cases and simulations of ide- alised cloudy thermals are presented to demonstrate the viability of the approach. Once this is explored, the non-equilibrium processes are ‘turned on’ again. We then simulate some idealised cloudy thermals, comparing the equilibrium and non-equilibrium regimes — finding a robust decrease in the vertical velocity in the non-equilibrium regime, as ex- pected. Thus, this work demonstrates the feasibility of building a numerical model that includes a framework for consistently modelling the moist non-equilibrium thermody- namics of an atmospheric system and provides a step towards this type of more consistent atmospheric modelling.