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• Author or Editor: Scott A. Hausman
• Journal of the Atmospheric Sciences
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# Potential Vorticity Structure of Simulated Hurricanes

Scott A. Hausman
,
Katsuyuki V. Ooyama
, and
Wayne H. Schubert

## Abstract

To better understand the processes involved in tropical cyclone development, the authors simulate an axisymmetric tropical-cyclone-like vortex using a two-dimensional model based on nonhydrostatic dynamics, equilibrium thermodynamics, and bulk microphysics. The potential vorticity principle for this nonhydrostatic, moist, precipitating atmosphere is derived. The appropriate generalization of the dry potential vorticity is found to be P = ρ −1 {(−∂υ/∂z) (∂θρ /∂r) + [ f + ∂()/rr] (∂θρ /∂z)}, where ρ is the total density, υ is the azimuthal component of velocity, and θρ is the virtual potential temperature. It is shown that P carries all the essential dynamical information about the balanced wind and mass fields. In the fully developed, quasi-steady-state cyclone, the P field and the θ̇ρ field become locked together, with each field having an outward sloping region of peak values on the inside edge of the eyewall cloud. In this remarkable structure, the P field consists of a narrow, leaning tower in which the value of P can reach several hundred potential vorticity (PV) units.

Sensitivity experiments reveal that the simulated cyclones are sensitive to the effects of ice, primarily through the reduced fall velocity of precipitation above the freezing level rather than through the latent heat of fusion, and to the effects of vertical entropy transport by precipitation.

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# Potential Vorticity in a Moist Atmosphere

Wayne H. Schubert
,
Scott A. Hausman
,
Matthew Garcia
,
Katsuyuki V. Ooyama
, and
Hung-Chi Kuo

## 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) · ∇θ ρ , 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 θ ρ = T ρ (p 0/p) R a /c Pa is the virtual potential temperature, with T ρ = p/(ρR a ) the virtual temperature, p the total pressure (the sum of the partial pressures of dry air and water vapor), p 0 the constant reference pressure, R a the gas constant for dry air, and c Pa the specific heat at constant pressure for dry air. Since θ ρ 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, ∇θ ρ  · (∇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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