Potential vorticity in a moist atmosphere

The potential vorticity principle for a nonhydrostatic, moist, precipitating atmosphere is derived. The appropriate generalization of the well-known (dry) Ertel potential vorticity is found to be $P=\rho^{-1}(2{\bf \Omega}+\nabla\times{\bf u})\cdot\nabla\theta_\rho$, where $\rho$ is the total density, consisting of the sum of the densities of dry air, airborne moisture (vapor and cloud condensate) and precipitation, where ${\bf u}$ is the density-weighted-mean velocity of dry air, airborne moisture, and precipitation, and where $\theta_\rho$ is the virtual potential temperature, defined in such a way as to annihilate the solenoidal term. The $\theta_\rho$-surfaces are impermeable to $\rho P$. For balanced flows, there exists an invertibility principle which determines the total pressure (the sum of the partial pressures of dry air and water vapor) from the spatial distribution of $P$. In the special case of an absolutely dry atmosphere, $P$ reduces to the usual (dry) Ertel potential vorticity.