3148 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 Potential Vorticity in a Moist Atmosphere WAYNE H. SCHUBERT,SCOTT A. HAUSMAN, AND MATTHEW GARCIA Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado KATSUYUKI V. O OYAMA Hurricane Research Division, Atlantic Oceanographic and Meteorological Laboratory, Miami, Florida HUNG-CHI KUO Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan (Manuscript received 11 October 2000, in ®nal form 27 April 2001) ABSTRACT The potential vorticity principle for a nonhydrostatic, moist, precipitating atmosphere is derived. An appropriate 21 generalization of the well-known (dry) Ertel potential vorticity is found to be P 5 r (2V1= 3 u)´=ur, where r is the total density, consisting of the sum of the densities of dry air, airborne moisture (vapor and cloud condensate), RaPa/c and precipitation; u is the velocity of the dry air and airborne moisture; and ur 5 Tr(p0/p) is the virtual potential temperature, with Tr 5 p/(rRa) 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 speci®c heat at constant pressure for dry air. Since ur 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, =ur ´(=r 3 =p) 5 0. In the special case of an absolutely dry atmosphere, P reduces to the usual (dry) Ertel potential vorticity. For balanced ¯ows, there exists an invertibility principle that determines the balanced mass and wind ®elds 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 ¯ow. 1. Introduction to the parameterization of the microphysics of the pre- cipitation process. At present, the dynamical basis for global numerical The purpose of the present paper is to extend the weather prediction and climate models is the set of qua- potential vorticity conservation principle to nonhydro- si-static primitive equations. Since the nonhydrostatic static models with Ooyama's (1990, 2001) form of moist motions typical of dry and moist atmospheric convec- dynamics and thermodynamics. We begin by reviewing tion have such small horizontal scales, and since they the exact, nonhydrostatic primitive equations for a moist cannot be accurately simulated with the quasi-static atmosphere in section 2. In section 3 we derive the primitive equations, the collective effects of these mo- generalized potential vorticity principle. Equations (20) tions must be parameterized in quasi-static primitive and (21) are our main results, the latter form being useful equation models. However, in the not-too-distant future, for physical interpretation. In section 4 we discuss one it will be possible to construct and run global numerical of the many possible invertibility principles (depending weather prediction and climate models based on the on the particular balance conditions) associated with the exact primitive equations, with much more accurate generalized potential vorticity. Section 5 contains a der- treatments of the moist thermodynamics, and with ivation of the ``equivalent potential vorticity'' and a cloud-resolving spatial discretization. In such nonhy- discussion of why this form is unacceptable from the drostatic models, cumulus cloud ®elds will be explicitly standpoint of possessing an invertibility principle. Con- simulated, eliminating the need for cumulus parame- clusions are given in section 6. terization. This pushes the frontier of empiricism back 2. Nonhydrostatic primitive equations for a moist Corresponding author address: Wayne H. Schubert, Department atmosphere of Atmospheric Science, Colorado State University, Fort Collins, CO 80523. Consider atmospheric matter to consist of dry air, air- E-mail: [email protected] borne moisture (vapor and cloud condensate), and pre- q 2001 American Meteorological Society Unauthenticated | Downloaded 09/25/21 05:25 AM UTC 1NOVEMBER 2001 SCHUBERT ET AL. 3149 1 cipitation. Let ra denote the mass density of dry air, rm total pressure p. In the determination of U, sr, Qr, Qs, 5 ry 1 rc the mass density of airborne moisture (con- and p, several other diagnostic variables are introduced. sisting of the sum of the mass densities of water vapor The additional diagnostic variables introduced in (6)± ry and airborne condensed water rc), and rr the mass (13) are the mass density of dry air ra, the thermody- density of precipitating water substance. The total mass namically possible temperatures T1 and T 2, the actual density r is given by r 5 ra 1 ry 1 rc 1 rr 5 ra 1 temperature T, the partial pressure of dry air pa, and the rm 1 rr. The ¯ux forms of the prognostic equations for partial pressure of water vapor py , the formulas for r, rm, and rr are given in (1)±(3), in which u denotes which are given below. To summarize, the diagnostic the velocity (relative to the rotating earth) of dry air and relations are airborne moisture, and u 1 U denotes the velocity (rel- r 5 r 2 r 2 r , (6) ative to the rotating earth) of the precipitating water sub- amr stance, so that U is the velocity of the precipitating water S2(ram, r 1 r r, T2) 5 s, (7) substance relative to the dry air and airborne moisture. The term Qr, on the right-hand sides of (2) and (3), is srr5 r C(T2), (8) the rate of conversion from airborne moisture to precip- S ( , , T ) , (9) itation; this term can be positive (e.g., the collection of 1 ramr 1 5 s 2 sr cloud droplets by rain) or negative (e.g., the evaporation T 5 max(T12, T ), (10) of precipitation falling through unsaturated air). p 5 r RT, (11) The total entropy density is s 5 sa 1 sm 1 sr, aaa consisting of the sum of the entropy densities of dry r 5 r , r 5 0, p 5 r RT, air, airborne moisture, and precipitation. Since the en- y mc yyy tropy ¯ux is given by s u 1 s u 1 s (u 1 U) 5 su if T 5 T12. T (absence of condensate), a m r (12) 1 srU, we can write the ¯ux form of the entropy con- ryy5 r*(T), rcm5 r 2 ryy, p 5 E(T), servation principle as (4), where Qs denotes diabatic processes such as radiation. if T 5 T21. T (saturated vapor), Next, we can write the equation of motion as (5), where p 5 p 1 p . (13) z 5 2V 1 = 3 u is the absolute vorticity; F the potential a y for the sum of the Newtonian gravitational force and the Note that, although there are four types of matter (with 21 centrifugal force; r =p the pressure gradient force, with densities ra, ry , rc, rr), there are only three prognostic p 5 pa 1 py the sum of the partial pressures of dry air mass continuity equations [(1)±(3)]. The separation of pa and water vapor py ; and F the frictional force per unit the predicted total airborne moisture density rm into the mass. The derivation of (5) is given in Ooyama (2001) vapor density ry and the cloud condensate density rc is and is reviewed in appendix B. accomplished diagnostically in the two alternatives of Consolidating our results so far, the prognostic equa- (12). In the absence of condensate, all the predicted tions for the total mass density r, the mass density of airborne moisture rm is in vapor form so that ry 5 rm airborne moisture rm, the mass density of precipitation and rc 5 0, while if the vapor is saturated, ry 5 r*y (T) content rr, the entropy density s, and the three-dimen- and rc 5 rm 2 ry . The formulas for the entropy density sional velocity vector u are functions S1 and S 2, from which the temperatures T1 and T are diagnosed, are given in appendix A. ]r 2 1 = ´(ru 1 rrU) 5 0, (1) In the context of numerical modeling, the procedure ]t for advancing from one time level to the next consists ]rm of computing new values of the prognostic variables r, 1 = ´(rmru) 52Q , (2) ]t rm, rr, s, and u from (1)±(5). The diagnostic variables required for the prognostic stage are determined by se- ]r r 1 = ´[r (u 1 U)] 5 Q , (3) quential evaluation of (6)±(13), namely, the diagnosis ]t rr of the dry air density ra from (6), the thermodynamically ]s possible (wet bulb) temperature T 2 from (7), the entropy 1 = ´(su 1 srU) 5 Qs, (4) density of precipitation from (8), the thermodynamically ]t possible temperature T1 from (9), the choice of the actual ]u 11 temperature from (10), the dry air partial pressure pa 1 z 3 u 1 = u ´ u 1F 1 =p 5 F. (5) from (11), the water vapor mass density r , the airborne ]t 122 r y condensate mass density r , and the water vapor partial The diagnostic variables appearing explicitly in (1)±(5) c pressure py from the appropriate condition in (12), and are the terminal fall velocity U, the entropy density of the total pressure from (13). Since they are not essential precipitation sr, the source terms Qr and Qs, and the to our discussion here, we have omitted the parameter- ization formulas for the terminal fall velocity U and the 1 The notation used here follows Ooyama (2001).