A general formulation for conservation laws in moist isentropic coordinates

Isentropic analysis, i.e. the study of atmospheric flow on surfaces of constant entropy or potential temperature dates back to the early development of dynamical meteorology in the 1930s. As entropy evolves on time scales that are typically longer than that of atmospheric motions, it can be used as a quasi-lagragnian coordinate system to track the motions of air parcels. We will review here recent advances in analyzing the flow on moist isentropes - or surfaces of constant equivalent potential temperature - top study the global circulation, deep convection and hurricanes. The primary advantage of the moist isentropic analysis lies in that equivalent potential temperature is conserved in presence of phase transition, so that it can still be viewed as a slow varying state variable even when moist processes are taken into account.

These studies have also led to the emergence of a new framework to study the conservation laws in a complex turbulent flow. Indeed, an important conceptual difficulty arises from the fact that the equivalent potential temperature is rarely stratified. This makes the standard approach to coordinate transformation mathematically ill-posed. To address this issue, we introduce here a weak formulation of the coordinate transformation that is applicable to non-monotonic system. This method is then applied to derive the conservation laws for mass and momentum on moist isentropes.