Potential Vorticity in a Moist Atmosphere

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Abstract
The potential vorticity principle for a nonhydrostatic, moist, precipitating atmosphere is derived. An appropriate generalization of the well-known (dry) Ertel potential vorticity is found to be P = ρ−1(2Ω + ∇ × u) · ∇θρ, where ρ is the total density, consisting of the sum of the densities of dry air, airborne moisture (vapor and cloud condensate), and precipitation; u is the velocity of the dry air and airborne moisture; and θρ = Tρ is the virtual potential temperature, with Tρ = p/(ρRa) the virtual temperature, p the total pressure (the sum of the partial pressures of dry air and water vapor), p0 the constant reference pressure, Ra the gas constant for dry air, and cPa the specific heat at constant pressure for dry air. Since θρ is a function of total density and total pressure only, its use as the thermodynamic variable in P leads to the annihilation of the solenoidal term, that is, ∇θρ · (∇ρ × ∇p) = 0. In the special case of an absolutely dry atmosphere, P reduces to the usual (dry) Ertel potential vorticity. For balanced flows, there exists an invertibility principle that determines the balanced mass and wind fields from the spatial distribution of P. It is the existence of this invertibility principle that makes P such a fundamentally important dynamical variable. In other words, P (in conjunction with the boundary conditions associated with the invertibility principle) carries all the essential dynamical information about the slowly evolving balanced part of the flow.

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