Precipitating Quasigeostrophic Equations and Potential Vorticity Inversion with Phase Changes

Precipitating versions of the quasi-geostrophic (QG) equations are derived systematically, starting from the equations of a cloud-resolving model. The presence of phase changes of water from vapor to liquid and vice versa leads to important differences from the dry QG case. The precipitating QG (PQG) equations, in their simplest form, have two variables to describe the full system: a potential vorticity (PV) variable and a variable M including moisture effects. PV-and-M inversion allows the determination of all other variables, and it involves an elliptic partial differential equation (PDE) that is nonlinear due to phase changes between saturated and unsaturated regions. An example PV-and-M inversion is provided for an idealized cold-core cyclone with two vertical levels. A key point illustrated by this example is that the phase interface location is unknown a priori from PV and M, and it is discovered as part of the inversion process.