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 Ϫ1 ١, where´(ϫ u generalization of the well-known (dry) Ertel potential vorticity is found to be P ϭ (2⍀ϩ١ 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 ϭ T(p0/p) is the virtual potential
temperature, with T ϭ p/(Ra) 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 is a function of total density and total pressure only, its use as the
١p) ϭ 0. In the ١ ϫ)´ ١ ,thermodynamic variable in P leads to the annihilation of the solenoidal term, that is 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-
᭧ 2001 American Meteorological Society
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1 cipitation. Let a denote the mass density of dry air, m total pressure p. In the determination of U, r, Qr, Q, ϭ ϩ c 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)±
and airborne condensed water c), and r the mass (13) are the mass density of dry air a, the thermody- density of precipitating water substance. The total mass namically possible temperatures T1 and T 2, the actual density is given by ϭ a ϩ ϩ c ϩ r ϭ a ϩ temperature T, the partial pressure of dry air pa, and the m ϩ r. The ¯ux forms of the prognostic equations for partial pressure of water vapor p , the formulas for , m, and r 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 ϩ U denotes the velocity (rel- ϭ Ϫ Ϫ , (6) ative to the rotating earth) of the precipitating water sub- amr stance, so that U is the velocity of the precipitating water S2(am, ϩ r, T2) ϭ , (7) substance relative to the dry air and airborne moisture.
The term Qr, on the right-hand sides of (2) and (3), is rrϭ 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 am 1 ϭ Ϫ r cloud droplets by rain) or negative (e.g., the evaporation T ϭ max(T12, T ), (10) of precipitation falling through unsaturated air). p ϭ RT, (11) The total entropy density is ϭ a ϩ m ϩ r, aaa consisting of the sum of the entropy densities of dry ϭ , ϭ 0, p ϭ RT, air, airborne moisture, and precipitation. Since the en- mc tropy ¯ux is given by u ϩ u ϩ (u ϩ U) ϭ u if T ϭ T12Ͼ T (absence of condensate), a m r (12) ϩ rU, we can write the ¯ux form of the entropy con- ϭ *(T), cmϭ Ϫ , p ϭ E(T), servation principle as (4), where Q denotes diabatic processes such as radiation. if T ϭ T21Ͼ T (saturated vapor), Next, we can write the equation of motion as (5), where p ϭ p ϩ p . (13) ϫ u is the absolute vorticity; ⌽ the potential a ١ ϭ 2⍀ ϩ for the sum of the Newtonian gravitational force and the Note that, although there are four types of matter (with Ϫ1 ١p the pressure gradient force, with densities a, , c, r), there are only three prognostic centrifugal force; p ϭ pa ϩ p the sum of the partial pressures of dry air mass continuity equations [(1)±(3)]. The separation of pa and water vapor p ; and F the frictional force per unit the predicted total airborne moisture density m into the mass. The derivation of (5) is given in Ooyama (2001) vapor density and the cloud condensate density c 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 , the mass density of airborne moisture m is in vapor form so that ϭ m airborne moisture m, the mass density of precipitation and c ϭ 0, while if the vapor is saturated, ϭ * (T) content r, the entropy density , and the three-dimen- and c ϭ m Ϫ . 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. 2ץ u ϩ rU) ϭ 0, (1) In the context of numerical modeling, the procedure)´ ١ ϩ t for advancing from one time level to the next consistsץ
,m of computing new values of the prognostic variables ץ (mru) ϭϪQ , (2)´ ١ ϩ t m, r, , and u from (1)±(5). The diagnostic variablesץ required for the prognostic stage are determined by se- ץ (u ϩ U)] ϭ Q , (3) quential evaluation of (6)±(13), namely, the diagnosis]´ ١ r ϩ t rrץ of the dry air density a from (6), the thermodynamically possible (wet bulb) temperature T 2 from (7), the entropyץ u ϩ rU) ϭ Q, (4) density of precipitation from (8), the thermodynamically)´ ١ ϩ tץ possible temperature T1 from (9), the choice of the actual u 11 temperature from (10), the dry air partial pressure paץ ١p ϭ F. (5) from (11), the water vapor mass density , the airborne u ´ u ϩ⌽ ϩ ١ ϩ ϫ u ϩ t 2 ץ condensate mass density , and the water vapor partial The diagnostic variables appearing explicitly in (1)±(5) c pressure p 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 r, the source terms Qr and Q, 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). A list of symbols source terms Qr and Q. See Ooyama (2001) for further is given at the end of the paper. discussion.
Unauthenticated | Downloaded 09/25/21 05:25 AM UTC 3150 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 uץ To obtain a better feel for the somewhat unfamiliar D 1 ١ ϫϩ ϫ u ´ ١ ١ ϩ ´ system (1)±(13), it is interesting to note the limiting t ץforms of these equations for the case of a perfectly dry Dt [] atmosphere (i.e., ϭ ϭ ϭ 0). In that case the c r 1 ( U). (16)´ ١ ١ ´ ١˙ ϩ ´ rU term in (1) vanishes, Eqs. (2) and (3) are dropped, ϭ r the rU term in (4) vanishes, and the precipitation con- tribution to F (see appendix B) in (5) vanishes. In ad- We now combine (16) with the momentum equation (5). t ϩ ϫ u, noting thatץ/uץ dition, the diagnostic equations (7), (8), (10), and (12) Substituting from (5) for ,١B) ϭ 0 for any scalar functions A and B ١A ϫ)´١ ;are dropped; (6) and (13) reduce to ϭ a and p ϭ pa (9) reduces to cValn(T/T 0) Ϫ Raln(/a0) ϭ /, from (16) reduces to which T is diagnosed; and (11) reduces to p ϭ RaT, from which p is computed. D 11 ١´(ϫ F ١) ١p) ϩ ١ ϫ)´ ١ ١ ϭ ´ At present it is not feasible to numerically integrate Dt 2 moist, nonhydrostatic, ``full physics'' models over the whole globe with 1±2-km resolution. However, it is pos- 1 (rU). (17)´ ١ ١ ´ ١˙ ϩ ´ sible to perform such 1±2-km-resolution integrations ϩ over a single hurricane, for example. Such integrations advance the art of hurricane modeling to a new level For two-dimensional ¯ows with line symmetry or cir- ١p all lie in ١, and ,١ that involves much less physical parameterization. In cular symmetry, the vectors ,١ ١p is perpendicular to ١ ϫ order that such full physics models can be interpreted the same plane so that ١p) vanishes no ١ ϫ)´ ١ in terms of well-established principles of geophysical and the solenoidal term ¯uid dynamics, we now derive the potential vorticity matter how is chosen. However, for general three- principle associated with the system (1)±(13). As we dimensional ¯ows, how do we choose the scalar func- ١)´ ١ shall see, the only equations in the set (1)±(13) that are tion in such a way that the solenoidal term ١p) vanishes? To accomplish this, we follow Ooyama needed for the derivation of the potential vorticity prin- ϫ
ciple are (1) and (5). (1990) and ®rst de®ne the virtual temperature T by
ppa ϩ p T ϭϭ , (18) 3. The potential vorticity (PV) equation Raamra( ϩ ϩ )R a ϫ b) ϭ b so that T is the temperature that dry air would have if)´ ١ Consider the fundamental identities a ϩ a ´ its pressure and density were equal to those of the given ´ Aa) ϭ A١)´ ١ ϫ b) and ١)´ ϫ a) Ϫ a ١)´ ١A, for the arbitrary vector ®elds a and b, and the sample of moist air. Note that the virtual temperature u/ T is equal to the actual temperature T when p ϭ 0 andץarbitrary scalar ®eld A. Choosing a ϭٌ and b ϭ t)] ϭ m ϭ r ϭ 0. Next let us de®ne the virtual potentialץ/uץ) ١ ϫ]´ ١ t in the ®rst identity, we obtainץ temperature by ١ ١ ϭ 0 and ϭ 2⍀ ϩ ϫ ١ t), sinceץ/ץ)´ Ϫ١ t and a ϭ in the second ץ/ץϫ u. Choosing A ϭ identity, we obtain ´[ ( / t)] ´ ( / t), since ppp00 ( ϭ T ϭ , (19 ץ ץ ١ ϭ ץ ץ ١ ´ 0. Taking the difference of these two results ϭ p Rpa ١ we obtain where p 0 is a constant reference pressure and ϭ Ra/ c . Since can be written as a function of p and Pa ץ uץץ )p١. Sinceץ/ץ) p)١p ϩץ/ץ) ١ ϭ ١ ϫϪ ϭ 0. (14) only, we have ´ ١ ١) ϩ ´ ) ١p) ϭ 0, this ١ ϫ)´ ١ ١p) ϭ 0 and ١ ϫ)´ ١p tץ tץt ץ implies that ´( p) 0, so the choice2 ϭ ϭ ١ ١ ϫ ١ is the ١ ´ t ϩ uץ/ץDe®ning ˙ ϵ D/Dt, where D/Dt ϭ annihilates the ®rst term on the right-hand side of material derivative, and using the triple vector product ١ ϫ ( ϫ u), we can write (17). Thus, (17) reduces to ١) ϩ ´ ١) ϭ u( ´ (u (14) as DP 1 ( U)], (20a)´ ١˙ ϩ P١ ´ ١ ϩ ´(ϫ F ١)] ϭ Dt r uץץ ١ ϫϩ ϫ u ١) ϩ ´ u( ´ ١ ١) ϩ ´ ) t whereץt []ץ 1 (١ . (20b ´ P ϭ (١˙ . (15 ´ ϭ
Using the continuity equation (1), in the form D/Dt ϩ 2 Other choices of involving functions of [e.g., the ``virtual
rU) ϭ 0, we can put (15) in advective entropy'' ϭ cPa ln(/T0)] could also be used. For consistency with)´١ u ϩ´١ ,u to obtain the most widely used de®nition of potential vorticity in the dry case´١ form and eliminate we shall con®ne our attention to the choice ϭ .
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The generalized potential vorticity principle (20) is tangential wind and mass ®elds as continuously chang- our main result. It should be noted that (20) is a gen- ing from one hydrostatic and gradient balanced state to eralization of the well-known (dry) Ertel (1942) poten- another. If we have a method of predicting P(r, z, t), tial vorticity principle in three respects: (i) the total the invertibility problem is to diagnostically determine, density consists of the sum of the densities of dry air, at each time, the tangential wind (r, z, t), the total
airborne moisture, and precipitation; (ii) is the chosen density (r, z, t), the total pressure p(r, z, t), the virtual scalar ®eld; and (iii) precipitation effects are included temperature T(r, z, t), and the virtual potential tem- in F and in the last term of (20a). In a completely dry perature (r, z, t) from atmosphere, the total density reduces to the dry air pץ density, the virtual potential temperature reduces to f ϩ ϭ , (22) rץ the ordinary dry potential temperature, and precipitation r effects disappear, so that (20) then reduces to the or- dinary (dry) Ertel PV principle. pץ It is worth noting that (20a) can be written in a form Ϫg ϭ , (23) zץ ,that is more physically revealing for thermally forced frictionally controlled, balanced atmospheric motions. ١ | as the unit vector normal to p|/ ١ De®ning j ϭ T ϭ , (24) the surface and k ϭ /| | as the unit vector pointing Ra along the absolute vorticity vector becomes (20a)
rU) p0)´ ١ ١˙ ´ϫ Fk ١ ´ DP j ϭ P ϩϩ . (21) ϭ T , (25) ١ p ´ Dt j ´ k ץ (r)ץ ץ ץ This form emphasizes the exponential nature of the time 1 behavior of P for material parcels moving through a Ϫϩf ϩϭP. (26) zץ rץ[]rrץ zץ·region with approximately constant rate processes as- Ά
sociated with F, ˙ , and rU. For example, in the intense This constitutes a system of ®ve equations for the ®ve convective region of a hurricane, k tends to point up- unknowns (r, z, t), (r, z, t), p(r, z, t), T(r, z, t), and ward and radially outward. Since ˙ tends to be a max- (r, z, t) with given P(r, z, t). Note that the solution imum at midtropospheric levels, air parcels ¯owing in- of the invertibility problem gives us the total density ward at low levels and spiraling upward in the convec- (r, z, t), the total pressure p(r, z, t), and the virtual tive eyewall experience a material increase in P due to temperature T(r, z, t). The determination of the actual ١ ) term. This material increase of ´ ١˙ )/(k ´ the P(k temperature T, the partition of p between pa and p , and P can be especially rapid in lower tropospheric regions the partition of between a, m, and r is not possible ١) from knowledge of P only. In other words, solution of ´ ١˙ )/(k ´ near the eyewall, where both P and (k ١) term reverses the invertibility problem gives only the parts of the mass ´ ١˙ )/(k ´ are large. Although the (k sign in the upper troposphere, large values of P are often ®eld that are of direct dynamical signi®cance. found there because the large lower tropospheric values In principle it is possible to use (22)±(25) to eliminate are carried upward into the upper troposphere. , , and from (26) and thereby obtain a single partial It is also worth noting that the effects of precipitation differential equation relating the total pressure p(r, z, t) contained in the last term of (20a) can be distributed to the known PV distribution P(r, z, t). However, in (r, over the other three terms to obtain an equation that is z, t) coordinates, the resulting partial differential equa- formally similar to the one usually given for the dry tion is somewhat complicated. In the following two par- case. This alternative form is discussed in appendix C. agraphs we consider the transformation of the inverti- bility principle from the physical height coordinate to 4. Invertibility principle a total pressure type coordinate and to the virtual po- tential temperature coordinate. Both of these transfor- It is natural to ask if, under certain balance conditions, mations result in simpler forms of the invertibility prin- there exists an invertibility principle for P. The impor- ciple. tance of the existence of an invertibility principle is hard First consider the transformation to (r, zÃ, t) coordi- to overemphasize. It is the existence of such a principle nates, where zà ϭ zÃa[1 Ϫ (p/p 0) ] is the pseudoheight, that makes P such a dynamically interesting quantity. with zÃa ϭ cPaT 0/g. With this vertical coordinate, the /ץTo see that such a principle exists, consider an f-plane gradient and hydrostatic equations are ( f ϩ /r) ϭ
-zÃ, where ϭ gz is the geopoץ/ץr and (g/T 0) ϭץ case in which a large-scale axisymmetric ¯ow has a slowly evolving P(r, z, t) ®eld. The slow evolution of tential and where the radial derivatives are now under- the P ®eld is due to radial and vertical advection of P, stood to be at ®xed zÃ. In a similar fashion, the trans-
and the ˙ , F, and rU terms on the right-hand side of formation of (26), along with the use of the gradient (20a). Since the evolution is slow, we can consider the and hydrostatic relations, yields (27a), which can be
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regarded as a second-order nonlinear partial differential (r, , t) and ⌸(r, , t) can be calculated using the equation relating the geopotential to the PV. The gradient and hydrostatic relations. boundary conditions for (27a) are that goes to a spec- Of course, the zÃ-coordinate invertibility relation (27), → -/ the -coordinate invertibility relation (28), and the origץ i®ed far-®eld geopotential˜ (zÃ)asr ϱ and that -zà is given in terms of a known boundary at the top inal relations (22)±(26) are simply different mathematץ and bottom boundaries. Thus, the complete invertibility ical forms of the same physical principle. For the special principle in zà coordinates is case when all independent and dependent variables are interpreted in terms of their dry limits, the invertibility Ϫ1/2 relation (27) is equivalent to the one solved by Hoskinsץ 2 4 ץ 22ץ ץץ T 0 f 2 ϩ r3 Ϫ f 2 ϩ r et al. (1985) and Thorpe (1985, 1986), while (28) isץ zà ·rץrץzà ץ rץrץgà Ά[]r32 equivalent to the one solved by Schubert and Alworth (1987) and MoÈller and Smith (1994). In their calcula- ϭ P, (27a) tions these authors used a further transformation from → ˜ (zÃ)asr → ϱ, (27b) the physical radius r to the potential radius R, which is de®ned by ½ fR2 ϭ ½ fr2 ϩ r. Regardless of this ad- ditional transformation, we can interpret the results of g these previous ``dry'' invertibility studies in terms ofץ ϭ at lower and upper boundaries, (27c) zà T0 our moist model, since the dry and moist invertibilityץ problems are isomorphic under the interchanges ↔ (1Ϫ)/ whereà (zÃ) ϭ 0(1 Ϫ zÃ/zÃa) is the pseudodensity (a ↔ , T T, etc. known function of zÃ). The solution of the nonlinear, In passing we note that the invertibility principle (27), second-order partial differential equation (27a), with the expressed in the zà coordinate, or its equivalent, (28), ex- appropriate boundary conditions (27b) and (27c), yields pressed in the coordinate, represents only one member the geopotential (r, zÃ, t), from which (r, zÃ, t) and (r, of a family of invertibility principles. Other members of zÃ, t) can be calculated using the gradient and hydrostatic the family are generated by replacing the gradient wind relations. equation with different horizontal balance relations, for Now, assuming that is a monotonic function of example, the geostrophic equation, the nonlinear balance the physical height z, consider the transformation to equation, or the asymmetric balance equation (Shapiro and (r, , t) coordinates. With this vertical coordinate, the Montgomery 1993). In any event, the existence of such a gradient and hydrostatic equations are ( f ϩ /r) ϭ family of invertibility principles indicates that the potential , where M ϭ cPaT ϩ gz is theץ/Mץr and ⌸ϭץ/Mץ vorticity (20b), obtained through the choice ϭ , is the Montgomery potential (based on virtual temperature), form that maintains a direct connection to the balanced ⌸ϭcPa(p/p 0 ) is the Exner function (based on total dynamics and is hence the appropriate moist generalization pressure), and where the radial derivatives are now of the well-known (dry) Ertel PV. In the next section we understood to be at ®xed . In a similar fashion, the discuss a commonly suggested choice for that turns out transformation of (26), along with the use of the gra- to be unacceptable from the standpoint of possessing an dient and hydrostatic relations, yields (28a), which can invertibility principle. be regarded as a second-order nonlinear partial differ- ential equation relating the Montgomery potential M to the potential vorticity P. The boundary conditions 5. An alternative approach to the choice of for (28a) are that M goes to a speci®ed far-®eld Mont- Our approach in deriving (20) has been to choose Ä → ١p) term ١ ϫ)´ ١ is in such a way as to annihilate theץ/Mץ gomery potential M( )asr ϱ and that given in terms of a known boundary ⌸ at the top and on the right-hand side of (17). An alternative approach bottom boundaries. Thus, the complete invertibility attempts to choose in such a way as to annihilate the ١˙ term on the right-hand side of (17). One way to ´ principle in coordinates is accomplish this, at least in an approximate sense, is to 2M Ϫ1ץ M Ϫ1/2ץ M 4ץץ Ϫg⌫ f 2 ϩ rf3 2 ϩϭP, note that (4) can be written in the form ץ rץrrץ rץr32 [] Ds 1 ( U)], (29)´ ١ 28a) ϭ [Q Ϫ) Dt r → Ä → a M M()asr ϱ, (28b) where s ϭ /a is the ``dry-air-speci®c'' entropy of -M moist air. Now consider a physical situation that warץ ϭ⌸ at lower and upper boundaries, (28c) rants the neglect of the right-hand side of (29). Forץ example, such a situation might occur when radiation where ⌫(p) ϭ d⌸/dp ϭ ⌸/p. The solution of the non- and precipitation effects play a secondary role in the linear, second-order partial differential equation (28a), entropy budget. Then, de®ning the equivalent potential
with appropriate boundary conditions (28b) and (28c), temperature by e ϭ T 0 exp(s/cPa), (29) reduces to ˙ e
yields the Montgomery potential M(r, , t), from which ϵ De/Dt ഠ 0 and the choice ϭ e in (17) leads to
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D 1 of latitude); this means that the associated angular .١e momentum and energy principles are exact ´ Dt 2) The connection between dynamics and thermody- 11 namics is through the gradient of pressure, which (١ , (30´(ϫ F ١) ١p) ϩ ١ ϫ)´ ١ ഠ 3 eeincludes the partial pressures of dry air and water vapor. which is sometimes referred to as the equivalent po- 3) The ®rst law of thermodynamics is expressed in tential vorticity principle. The equivalent potential terms of , the entropy density of moist air; all the Ϫ1 -١ e has been used as a diagnostic in usual approximations associated with moist ther ´ vorticity several studies, for example, in the analysis of model modynamics are thereby avoided. simulations of the rotation and propagation of super- 4) There is no cumulus parameterization; the frontier cell thunderstorms (Rotunno and Klemp 1985) and of empiricism is pushed back to the microphysical model simulations of extratropical cyclones with em- parameterization of the precipitation process through bedded latent heat release (Cao and Cho 1995; Pers- U and Qr. son 1995). A variant, using Hauf and HoÈller's (1987) 5) The model is modular in the sense that ice can be entropy temperature for the choice of , has been used included by specifying E(T) to be synthesized from by Rivas Soriano and GarcõÂa DõÂez (1997) to study the the saturation formulas over water and ice (Ooyama effects of ice. 1990). The choice ϭ e is problematic in three respects: (i) in a precipitating atmosphere with radiative forcing, Even with a nonhydrostatic moist model of this gen- Ϫ1 ١˙ e is not exactly eliminated; (ii) be- erality, it is possible to derive a PV principle that is a ´ the term cause e does not depend on and p only, but also on straightforward generalization of the well-known (dry) and , the term ´( p) is not eliminated -Ertel PV principle. Using (20b) it is possible to con ١ ١ ϫ ١e m r from (30); and (iii) when attempting to set up an in- Ϫ1 struct PV maps as diagnostics of the nonhydrostatic -١e, analogous to the in ´ vertibility principle for Ϫ1 moist model. Such a PV diagnostic is a useful aid in ١, one ®nds ´ vertibility principle (22)±(26) for Ϫ1 understanding the relationship between nonhydrostatic ١e ®eld is not suf®cient ´ that knowledge of the and that additional information on the moisture ®eld is moist convection and the large-scale balanced ¯ow. required for determination of the balanced wind and Hausman (2001) has produced such PV cross sections Ϫ1 for a nonhydrostatic hurricane model based on the axi- ١e does not carry ´ mass ®elds. In other words, all the essential dynamical information on the balanced symmetric, f plane versions of (1)±(13). The PV struc- Ϫ1 ture associated with the intense hurricane stage of the ١ does. For balanced ¯ow and slow ´ ow like ¯ manifold dynamics, the relevant part of the vector is numerical simulation consists of an annular ring of very high PV (more than 200 PV units) extending perpendicular to the surfaces, not to the e surfaces. Ϫ1 through the whole troposphere between 10- and 15-km ,١e is not invertible ´ We conclude that, because (20) is more useful than (30) for the study of moist, radius. In a fully three-dimensional model, such a PV precipitating, balanced ¯ows. structure would be subject to combined barotropic± baroclinic instability, the barotropic aspects of which were studied by Schubert et al. (1999). Hausman also 6. Conclusions compared cross sections of P, computed from (26), The physical model that serves as the basis for the with cross sections of dry Ertel PV. The differences derivation of the generalized potential vorticity principle are quite small, indicating that dry Ertel PV and its (20) is the nonhydrostatic moist model [(1)±(13)]. In this invertibility principle can give an accurate description model, pressure is not used as one of the prognostic var- of the balanced aspects of hurricane dynamics if the iables, since it is not a conservative property and its use frictional and moist-diabatic source/sink terms for the as a prognostic variable would lead to an approximate dry PV are accurately parameterized. treatment of moist thermodynamics. Rather, the prognostic In closing we note that there is an impermeability prin- variable for the thermodynamic state is , the entropy of ciple associated with the ¯ux form of (20), that is, the moist air per unit volume, with temperature and total pres- surfaces are impermeable to P, even though they are sure (the sum of the partial pressures of dry air and water permeable to mass. This is discussed in appendix D. vapor) determined diagnostically. There are several unique aspects of this model that are worth noting. Acknowledgments. The authors would like to thank 1) The model dynamics are exact in the sense that there Michael Montgomery, Richard Johnson, Paul Ciesiel- is no hydrostatic approximation and no traditional ski, and three anonymous reviewers for many helpful approximation (i.e., selectively replacing the actual comments. This work was supported by NSF Grants radius by the constant radius to mean sea level and ATM-9729970 and ATM-0087072 and by NOAA Grant neglecting Coriolis terms proportional to the cosine NA67RJ0152 (amendment 17).
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APPENDIX A
List of Symbols Mass densities, temperatures, pressures, velocities
a Mass density of dry air
Mass density of water vapor c Mass density (as aerosol) of airborne condensate (droplets or ice crystals) r Mass density (as aerosol) of precipitating water substance (liquid or ice)
m ϭ ϩ c Mass density of airborne moisture (vapor plus airborne condensate) am ϭ a ϩ m Mass density of dry air and airborne moisture ϭ a ϩ m ϩ r Total mass density (dry air plus airborne moisture plus precipitation) T1 Temperature, when condensation does not occur or is not allowed T 2 Temperature if saturated, or wet-bulb temperature if unsaturated T ϭ max(T1, T 2) Temperature
T ϭ p/(Ra) Virtual temperature ϭ T(p 0/p) Virtual potential temperature e ϭ T 0 exp(s/cPa) Equivalent potential temperature
pa Partial pressure of dry air
p Partial pressure of water vapor p ϭ pa ϩ p Total pressure of moist air u Velocity of dry air and airborne moisture (relative to earth)
ur Velocity of precipitation (relative to earth) U ϭ ur Ϫ u Velocity of precipitation (relative to dry air and airborne moisture) u ϭ [(a ϩ m)u ϩ rur]/ Density-weighted-mean velocity Speci®c entropies (J kgϪ1 KϪ1) and entropy densities (J mϪ3 KϪ1)
sa Speci®c entropy of dry air, de®ned by s ϭ cVa ln(T/T 0) Ϫ Ra ln(a/a0) (1) (1) sm Speci®c entropy of airborne moisture in state 1, de®ned by sm ϭ cV ln(T/T 0) Ϫ R ln(m/)*m0 ϩ⌳0 (2) (2) sm Speci®c entropy of airborne moisture in state 2, de®ned by sm ϭ C(T) ϩ D(T)/m sr Speci®c entropy of condensed water (cloud or precipitation) s ϭ /a Dry-air-speci®c entropy of moist air a ϭ asa Entropy density of dry air (1) m ϭ msm Entropy density of airborne water substance for state 1 (2) m ϭ msm Entropy density of airborne water substance for state 2 r ϭ rsr Entropy density of precipitating water substance ϭ a ϩ m ϩ r Total entropy density (1) S1(a, m, T) Entropy density function for state 1, de®ned by S1(a, m, T) ϭ asa ϩ msm (2) S 2(a, m, T) Entropy density function for state 2, de®ned by S 2(a, m, T) ϭ asa ϩ msm Constants ⍀ Angular rotation rate of the Earth
Ra Gas constant of dry air
R Gas constant of water vapor cVa Speci®c heat of dry air at constant volume
cV Speci®c heat of water vapor at constant volume cPa ϭ cVa ϩ Ra Speci®c heat of dry air at constant pressure
cP ϭ cV ϩ R Speci®c heat of water vapor at constant pressure ϭ Ra/cPa p 0 Reference pressure, 100 kPa T 0 Reference temperature, 273.15 K 0 ϭ p 0/(RaT 0) Reference density for dry air E 0 ϭ E(T 0) Saturation vapor pressure at T 0
*0 ϭ * (T 0) Mass density of saturated vapor at T 0 ⌳ 0 ϭ⌳(T 0) Gain of entropy by evaporating a unit mass of water at T 0 zÃa ϭ cPaT 0/g Pseudoheight at which p ϭ 0
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De®ned functions of temperature
⌳(T) ϭ R T(d lnE(T)/dT) Gain of entropy by evaporating a unit mass of water at T C(T) Entropy of a unit mass of condensate at T as measured from the reference state T 0 D(T) ϭ dE(T)/dT Gain of entropy per unit volume by evaporating a suf®cient amount of water, *(T), to saturate the volume at T E(T) Saturation vapor pressure, which may be synthesized from the saturation vapor pres- sures over water and ice
* (T) ϭ E(T)/(R T) Mass density of saturated vapor Others
Fam Frictional force acting on am Fr Vertical drag force acting on r F Total frictional force per unit mass (including precipitation)
F ϭ F Ϫ (r/)U ϫ Frictional force appearing in (22) ϫ u Absolute vorticity vector ١ ϭ 2⍀ ϩ
١ | Unit vector normal to surface|/١ j ϭ k ϭ /| | Unit vector pointing along absolute vorticity vector Derivative following the dry air, water vapor, and airborne condensate ١ ´ t ϩ uץ/ץD/Dt ϭ Derivative following the density-weighted-mean velocity u ١ ´ t ϩ uץ/ץD/Dt ϭ (r) Derivative following the precipitation ١ ´ t ϩ urץ/ץD /Dt ϭ
˙ ϭ D/Dt Diabatic source term in (20a) ˙ ϭ D/Dt Diabatic source term in (22) Qr Conversion rate of m to r
Q Entropy source term (e.g., radiation) ⌽ Potential for sum of Newtonian gravitational force and centrifugal force ϭ gz Geopotential
M ϭ cPaT ϩ gz Montgomery potential (based on virtual temperature) ⌸ϭcPa(p/p 0) Exner function (based on total pressure) ⌫(p) ϭ d⌸/dp ϭ ⌸/p Derivative of the Exner function with respect to p zà ϭ zÃa[1 Ϫ (p/p 0) ] Pseudo-height (1Ϫ)/ Ã(zÃ) ϭ 0(1 Ϫ zÃ/zÃa) Pseudo-density (a known function of zÃ)
APPENDIX B of (B1) and (B2). The exact sum of (B1) and (B2), when converted to advective form, is Exact and Approximate Momentum Equations Du D(r)U ϩϩr 2⍀ ϫ U ϩ 2⍀ ϫ u To derive the single predictive equation for u, ®rst Dt Dt consider separately the momentum equations for am ϭ and , which are 1 (١p ϭ F, (B3 ١⌽ϩ a ϩ m r ϩ ( u)ץ ١⌽ϩ١p uu) ϩ 2⍀ ϫ u ϩ )´ ١ am ϩ t am am am whereץ 1 (u], (B4(١ ´ ϭ amF amϪ rF rϪ Q ru, (B1) F ϭ [ F Ϫ (U am am r ( rru)ץ (r) t ϩץ/ץand D /Dt ϭ ١ ´ t ϩ uץ/ץrrruu) ϩ r2⍀ ϫ u rϩ r١⌽ and where D/Dt ϭ)´ ١ ϩ (t (rץ Neglect of the (r/)(D U/Dt ϩ 2⍀ ϫ U) term .١ ´ ur ϭ rrF ϩ Q rru , (B2) in (B3) results in where F is the frictional force acting on and F Du 1 (١p ϭ F, (B5 ١⌽ϩ am am r ϩ 2⍀ ϫ u ϩ is the vertical drag force acting on r. Note that there Dt is no pressure gradient force in (B2) since the fractional volume of precipitation is assumed to be negligibly which is the approximate momentum equation used by small. As discussed by Ooyama (2001), there is a serious Ooyama (2001) in his numerical simulations.B1 We shall contradiction in using both (B1) and (B2), since pre- use this same approximate momentum equation in our
diction of both u and ur is equivalent to prediction of U, which we are assuming is a diagnostic variable. Ooy- B1 In Ooyama's numerical simulations, Fam ϭ 0 and ⍀ is anti- ama has suggested using an approximation of the sum parallel with U so that 2⍀ ϫ U ϭ 0.
Unauthenticated | Downloaded 09/25/21 05:25 AM UTC 3156 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 derivation of the PV principle. For the PV derivation, obtain an equation that is formally similar to the one it is most convenient to put this approximate momentum usually given for the dry case. To accomplish this we u ϭ ®rst de®ne the density-weighted-mean velocity as u ϭ(١ ´ equation in a rotational form. Thus, using (u u ´ u), we can convert the advective u ϩ (r/)U and the derivative following this velocity)١½ ϫ u) ϫ u ϩ ١) We then rewrite (20a) as .١ ´ t ϩ uץ/ץform (B5) to the rotational form asD /Dt ϭ u DP 1ץ (١ ϩ ˙ ϩ UP F ϫ)´ ١ ϫ u) ϫ u ϭ ١ ϩ (2⍀ ϩ t Dt rץ 1 11 ´(F ˙ ) ١ϩ ϫ ١ ١p ϭ F. (B6) ϭ u ´ u ϩ⌽ ϩ ١ ϩ 2 In passing we note that the particular approximation 11 ˙١ ´ ١ϩ ´(ϫ F ١) ϭ (B5) is not required for the derivation of the PV principle. In fact, the unapproximated sum of (B1) and (B2), written ˙١ ´ϫ Fk ١ ´ j as a prognostic equation for the density-weighted-mean ϭ P ϩ , (C1) ١ ´ velocity u ϭ u ϩ (r/)U, can be used as the starting j ´ k point for the derivation of the ``exact'' potential vorticity where F ϭ F Ϫ ( /)U ϫ , ˙ ϭ D /Dt, and in principle. The resulting exact PV principle (Hausman r going from the ®rst line to the second line we have used 2001) takes the form of (C1), but with slight modi®ca- ١ ) ϭ ´ the triple vector product UP ϭ ( /)U( ϫ r r ١ ϫ u replaces ϭ 2⍀ ϩ ١ tions (e.g., ϭ 2⍀ ϩ ١ ϫ [( /)U ϫ ]. The forms ١ ] ϩ ´ [( /)U u in the de®nition of P). However, it is important to note r r given on the ®rst two lines of (C1) are convenient be- that the difference between the density-weighted-mean cause they express the total source term for DP/Dt as velocityu and the velocity u tends to be quite small. In the divergence of a single vector ®eld, while the forms fact, the two are identical in nonprecipitating regions given on the last two lines are convenient because of (where ϭ 0), and their horizontal component is iden- r their formal similarity to the conventional forms for the tical in precipitating regions. Even in heavily precipitat- dry case. Note that the last line of (C1) is an alternative ing regions with rainfall rates of 36 mm hϪ1, the mag- Ϫ1 form to (21). nitude of (r/)U is only 0.01 m s , so that the vertical component of the density-weighted-mean velocity is ap- APPENDIX D proximately 0.01 m sϪ1 smaller than the vertical com- ponent of the dry air velocity. In addition, although it Impermeability Principle has a certain theoretical appeal, the use of the density- weighted-mean velocity presents practical dif®culties It is natural to ask if there is an impermeability principle both in observational analysis and in numerical modeling. associated with (20), that is to ask if the surfaces are
Observationally, r and the vertical component of u are impermeable to P even though they are permeable to both dif®cult to measure, so it may be impossible in mass. Indeed there does exist such an impermeability prin- practice to make a meaningful distinction betweenu and ciple, and its existence does not depend on the momentum u. In model results, r and the vertical component of u equation or on the choice ϭ . To see this, let us return are both highly dependent on model resolution. The use to (14) and recall the discussion given by Haynes and ofu as a prognostic variable is also problematic in nu- McIntyre (1987, 1990). Surfaces of constant move merical schemes that require a priori speci®cation of the through space, and an observer moving with the velocity t ϩ uץ/ץlower boundary condition. For example, with a ¯at lower u crosses these surfaces, since ˙ ϵ D/Dt ϭ ١ ± 0. We can regard the surfaces as being carried ´ boundary the vertical component of the dry air velocity u is zero but the vertical component ofu is not zero, through space by a hypothetical velocity ®eld uÃ, which ١ ϭ 0. Thus, an observer moving ´ t ϩ uÃץ/ץ and in fact changing, in precipitating regions where con- satis®es densed water substance is leaving the atmospheric model with the velocity ®eld u will cross surfaces while an domain. Thus, in both observational analysis and model observer moving with the hypothetical velocity ®eld uà will output diagnostics, compromising assumptions on the remain on the same surface. The ¯ux in (14) can thus ١). Since the triple ´ t) ϩ (uÃץ/uץ) ١ ϫ exact theory'' seem inevitable. For this reason, we have be written as`` ١ ١) ϩ ´ ١) ϭ uÃ( ´ chosen to introduce the plausible and consistent approx- vector product rule yields (uà imation (B5) at the start of the PV derivation. ϫ ( ϫ uÃ), we can write (14) as ץ (١ ´ APPENDIX C ( tץ uץ (An Alternative Form of the PV Principle (20a ,١ ϫϩ ϫ uà ϭ 0 ١) ϩ ´ uà ( ´ ١ ϩ tץIf the effects of precipitation contained in the last term [] of (20a) are distributed over the other three terms, we (D1)
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.t use and signi®cance of isentropic potential vorticity maps. Quartץ/uץ) ١ ϫ which has a form analogous to (15). The .١, that J. Roy. Meteor. Soc., 111, 877±946 ϩ ϫ uÃ) part of the ¯ux is perpendicular to MoÈller, J. D., and R. K. Smith, 1994: The development of potential ,.١) part of the ¯ux vorticity in a hurricane-like vortex. Quart. J. Roy. Meteor. Soc ´ is, along the surface. The uÃ( is not necessarily along the surface, but the associated 120, 1255±1265. velocity is uÃ, and an observer moving with velocity uà Ooyama, K. V., 1990: A thermodynamic foundation for modeling the remains on the same surface. Thus, it is impossible moist atmosphere. J. Atmos. Sci., 47, 2580±2593. ١ across a surface. In summary, the ÐÐ, 2001: A dynamic and thermodynamic foundation for modeling ´ to ¯ux the moist atmosphere with parameterized microphysics. J. At- .١, regardless of how the mos. Sci., 58, 2073±2102 ´ surface is impermeable to arbitrary scalar is ultimately de®ned. In particular, Persson, O., 1995: Simulations of the potential vorticity structure and when we make the choice ϭ , we can say that the budget of FRONTS 87 IOP8. Quart. J. Roy. Meteor. Soc., 121, surfaces are impermeable to P as de®ned by (20b). 1041±1081. Rivas Soriano, L. J., and E. L. GarcõÂa DõÂez, 1997: Effect of ice on the generation of a generalized potential vorticity. J. Atmos. Sci., REFERENCES 54, 1385±1387. Rotunno, R., and J. B. Klemp, 1985: On the rotation and propagation Cao, Z., and H.-R. Cho, 1995: Generation of moist potential vorticity of simulated supercell thunderstorms. J. Atmos. Sci., 42, 271± in extratropical cyclones. J. Atmos. Sci., 52, 3263±3281. 292. Ertel, H., 1942: Ein neuer hydrodynamischer Erhaltungssatz. Natur- Schubert, W. H., and B. T. Alworth, 1987: Evolution of potential wissenschaften, 30, 543±544. vorticity in tropical cyclones. Quart. J. Roy. Meteor. Soc., 113, Hauf, T., and H. HoÈller, 1987: Entropy and potential temperature. J. 147±162. Atmos. Sci., 44, 2887±2901. ÐÐ, M. T. Montgomery, R. K. Taft, T. A. Guinn, S. R. Fulton, J. Hausman, S. A., 2001: Formulation and sensitivity analysis of a non- P. Kossin, and J. P. Edwards, 1999: Polygonal eyewalls, asym- hydrostatic, axisymmetric tropical cyclone model. Ph.D. disser- metric eye contraction, and potential vorticity mixing in hurri- tation, Department of Atmospheric Science, Colorado State Uni- canes. J. Atmos. Sci., 56, 1197±1223. versity, 209 pp. Shapiro, L. J., and M. T. Montgomery, 1993: A three-dimensional Haynes, P.H., and M. E. McIntyre, 1987: On the evolution of vorticity balance theory for rapidly rotating vortices. J. Atmos. Sci., 50, and potential vorticity in the presence of diabatic heating and 3322±3335. frictional or other forces. J. Atmos. Sci., 44, 828±841. Thorpe, A. J., 1985: Diagnosis of balanced vortex structure using ÐÐ, and ÐÐ, 1990: On the conservation and impermeability the- potential vorticity. J. Atmos. Sci., 42, 397±406. orems for potential vorticity. J. Atmos. Sci., 47, 2021±2031. ÐÐ, 1986: Synoptic scale disturbances with circular symmetry. Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the Mon. Wea. Rev., 114, 1384±1389.
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