A generalization of Ertel's potential vorticity to a cloudy, precipitating atmosphere

Abstract

This paper discusses the generalization of Ertel's PV to a cloudy, precipitating atmosphere. The recommended generalization is P = ρ-1ζ ˙ ∇θρ, where ρ is the total density of moist air, ζ is the absolute vorticity, and θρ is the virtual potential temperature. Associated with this form are three important properties: (1) the solenoidal term is annihilated (i.e., ∇θρ˙ (∇ρ × ∇p) = 0, where p is the total pressure, the sum of the partial pressures of dry air and water vapor); (2) the limiting form for a dry atmosphere is the classical Ertel PV; (3) P is invertible, i.e., it carries all the necessary dynamical information about the balanced wind and mass fields. Two other possible generalizations are discussed,ρ-1ζ ˙∇θe and ρ-1ζ ˙∇θ*e, where θe is the equivalent potential temperature and θ*e is the saturation equivalent potential temperature. The former is rejected because properties (1) and (3) are lost, while the latter is rejected because property (2) is lost.