Moist Component Potential Vorticity

166 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 Moist Component Potential Vorticity R. MCTAGGART-COWAN, J.R.GYAKUM, AND M. K. YAU Department of Atmospheric an Oceanic Sciences, McGill University, Montreal, Quebec, Canaa (Manuscript receive 17 July 001, in final form 4 June 00) ABSTRACT The role that atmospheric water, in both its liqui an vapor phases, plays in cyclogenesis is ifficult to etermine because of the complex interactions between ynamic an thermoynamic forcings. From a potential vorticity (PV) perspective, it is possible to ecompose the atmospheric state into a set of superpose PV anomalies. The moification of these anomalies allows for sensitivity testing using numerical moels. Although this approach allows for the etermination of cyclogenetic contributions from iniviual PV features, its application has not accounte for the ynamically consistent moification of the moisture fiel. This paper evelops a PV-base variable that escribes the effects of water vapor, clou, an rainwater on balance ynamics. A special-case analytic form of this moist component PV is evelope an interprete using an iealie moel of the atmosphere. The application of the moist component methoology evelope here provies the basis for future work, which inclues sensitivity tests esigne to separate the impacts of ynamics an thermoynamics on cyclogenesis. 1. Introuction A metho for quantitatively assessing the relative importance of ynamics an thermoynamics to cyclogenesis woul be a valuable asset to operational an research meteorologists alike. This paper presents the evelopment of a potential-vorticity-base (PV-base) variable that escribes the balance ynamics associate with the atmospheric moisture fiel. An iealie moel atmosphere is use as a testbe to simplify the interpretation of the variable. We suggest how this moist component methoology may be use in sensitivity stuies to evaluate quantitatively the impacts of ynamics an thermoynamics on cyclogenesis. The utility of PV as a iagnostic variable for atmospheric motion has been recognie since its inception by Rossby (1939). Two properties make the PV variable of particular interest. The first is the Lagrangian conservation principle, which states that the PV of a parcel of air oes not change in frictionless, aiabatic flow. The secon important property of PV is invertibility, which remains conceptually vali even when frictional an iabatic effects are important. The invertibility principle states that given a PV fiel an a balance conition, the state of the atmosphere can be uniquely iagnose (for further escription of PV attributes, the reaer is referre to Hoskins et al. 1985). It is this secon Corresponing author aress: R. McTaggart-Cowan, Dept. of Atmospheric an Oceanic Sciences, McGill University, Burnsie Hall, 805 Sherbrooke Street, Montreal, PQ H3A K6, Canaa. E-mail: rmctc@ephyr.meteo.mcgill.ca quality of PV that will prove most valuable to the current research. In particular, a piecewise casting of the invertibility principle using Ertel PV (Ertel 194) an the nonlinear balance equation (Charney 1955) evelope by Davis an Emanuel (1991, hereafter DE91) will act as a framework for the propose moist component evelopments. A typical application of the piecewise PV proceure as propose by DE91 involves the ecomposition of the flow into a set of recogniable PV anomalies efine relative to a time mean. A subsequent inversion allows for the iagnosis of the balance flow associate with each feature (Davis 199a,b; Davis et al. 1996). Some authors, incluing Huo et al. (1998) an McTaggart- Cowan et al. (001), have performe sensitivity experiments base on the moification of these PV anomalies in moel initial conitions. Such moifications have the avantage of being consistent since ynamical balance is enforce by the inversion process. The effects of subsaturate moisture on the atmospheric balance may or may not be inclue in this process, epening on the form of hyrostatic balance employe in the moel or analysis cycle use to obtain the initial conitions. The research presente here focuses on the explicit calculation of the impact of water vapor, clou, an rainwater on balance ynamics by eveloping a set of iealie moist component perturbation equations that fit into the piecewise PV framework. Moification of the moist component member of the set of PV anomalies will allow for the ynamically consistent moification of the atmospheric moisture fiel. Such changes in moisture coul be use to test the sensitivity of cyclogenesis to 003 American Meteorological Society

1JANUARY 003 M C TAGGART-COWAN ET AL. 167 the water vapor fiel of a moel s initial state, an coul be use to a an extra egree of freeom to ensemble forecasts. As well, they coul provie the basis for an assessment of water vapor analysis uncertainty an its impact on operational forecasting. The following section presents a erivation of the general form of the moist component PV (PV mc ). A escription of the analytic moel s backgroun state an humiity structure is given in section 3a. The remainer of section 3 provies a set of analytic escriptions for the moist component variables. Physical interpretation of PV mc is presente in section 4. Section 5 escribes the result of a piecewise inversion of PV mc. The stuy conclues with a brief summary an a iscussion of the potential for the application of PV mc to the real atmosphere.. Moist component PV A general efinition of PV mc can be given as the portion of the full PV fiel that is irectly attributable to the presence of water in the atmosphere. To formalie this statement mathematically, we begin with the efinition of PV for a moist atmosphere, as given by Schubert et al. (001): 1 P. (1) Here, is the absolute vorticity an is the virtual potential temperature given by T (p o /p), where T (p a p )/[( a m r )R a ] is the virtual temperature an R a /c p a ; p a an p are the ry air an water vapor partial pressures, respectively (p p a p ); a is the ry air ensity; m is the summe water vapor an clou water ensity; r is the rainwater ensity ( a m r ); R a is the gas constant for ry air; an cp a is the specific heat at constant pressure for ry air. As note by Schubert et al. (001), the PV escribe by (1) retains the properties of conservation an invertibility for an atmosphere that contains moisture. Assuming hyrostatic balance yiels, P g( f ) p, () where p is the three imensional relative vorticity. Equation () is similar to that of Hoskins et al. (1985), except that the virtual potential temperature replaces the potential temperature. This is the isobaric form of PV for a moist atmosphere an p is the threeimensional graient operator in isobaric coorinates. In this erivation, kˆ (the vertical component of the ) is compute iagnostically from the mass fiel through a balance equation an therefore varies with a changing moisture fiel. Expaning () into components, [ ] u P g ( f ), (3) p p y p x an converting to an Exner function [ c (p/p o ) ] p a vertical coorinate, [ ] g u P ( f ). (4) p y x We note that /, /y u, an /x, where is the geopotential calculate hypsometrically using T, an is the corresponing balance streamfunction. Substitution of these expressions into (4) yiels [ g P ( f ) p x x, (5) y y] the PV equation for a moist atmosphere cast in terms of the geopotential an balance streamfunction. For the case of a completely ry atmosphere [( m, r ) 0], we start again with the efinition of PV, 1 P, (6) which is ientical to (1) except that,, an have reuce to T(p o /p a ), a, an for the ry case. Employment of the same approximations that yiele () results in the isobaric form of the ry PV P g( f ), p (7) where is the three-imensional ry balance vorticity vector calculate from the ry mass fiel. The erivation of the ry form of the PV equation continues exactly as outline above for the moist case except that all of the moisture-epenent variables in (1) to (5) are replace by their ry counterparts. Thus, [ g P ( f ) p x x (8) y y] is the PV equation for a ry atmosphere, where the subscripts inicate that the variables are compute for the ry state. Again, is relate to through an appropriate balance equation. As note by Schubert et al. (001), this is in fact the form of PV that is generally, but incorrectly, use in calculations. The ifference between this ry PV an the full PV given by (5) forms the basis of PV mc since it represents the effects on the PV fiel of the inclusion of water vapor. The moist component PV is itself simply efine as the ifference between the full (moist) an ry PV fiels of a balance atmosphere, Pmc P P. (9) This expression coul be employe irectly given a

168 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 knowlege of the ry an moist atmospheric states. In fact, the inversion of PV mc calculate from (9) woul yiel the moist component streamfunction an geopotential fiels. However, a iagnostic expression for PV mc is esirable for two reasons. First, the signal-to-noise ratio of (9) is very small, an secon, very little physical insight is gaine from this simple expression. Accoringly, we begin by expaning the full geopotential an balance streamfunction fiels of (5) in a Maclaurin series aroun the ry ( a ) state an introuce the total mixing ratio (w) as the inepenent moisture variable w w (10) 1 (w0) w w w0 w0 1 (w0) w w w0 w0 w w (11) Here, w w m w r, where w m m / a is the summe water vapor an clou water mixing ratio, an w r r / a is the rainwater mixing ratio. We note that (w0) an (w0) an retain only the eroth- an firstorer terms to obtain, mc, an (1) mc, (13) where mc (/w) w0 w is the moist component geopotential, an mc (/w) w0 w is the corresponing balance moist component streamfunction. Substitution of the expressions in (13) into (5) an subtraction of (8) yiels an expression for PV mc, where [ g mc mc mc p mc mc L(, mc) P ( f ) ] L(, mc) L( mc, mc ), (14) A B A B L(A, B), x x y y as efine by (9). Not suprisingly, the form of this equation is similar to that of the piecewise ecomposition of DE91. In this case however, instea of expaning the perturbations aroun a backgroun mean, the anomalies are efine relative to the instantaneous ry state. Also, the nonlinear terms in (14) are explicitly retaine rather than hien in the nonconstant coefficients of the linear ifferential operator. Equation (14) represents a efinition of the portion of the PV fiel irectly attributable to the presence of water in the atmosphere. We will now test the applicability of this expression using an iealie atmosphere an balance conition to ease interpretation of the results. 3. PV mc in an iealie moel atmosphere The complexity of (14) makes the interpretation of PV mc a aunting task. It is therefore instructive to consier the istribution of PV mc uner a set of simplifying conitions that make (14) tractable. We begin in section 3a with a escription of the iealie moel to be employe in eveloping an analytic expression for PV mc. Section 3b outlines the steps require for the calculation of PV mc, an calculations for each item are presente in sections 3c through 3h. a. Backgroun state an moisture structure For the purposes of efining a simple framework uner which to evelop an interpret the moist component variables, a simplifie atmospheric structure is assume. The ry atmosphere is efine to be isothermal (ry bulb temperature constant everywhere) an motionless. Dry balance implies that the vertical structure of the moel atmosphere is escribe by a hypsometric relation of the form p g ln Z, (15) p RT where all of the symbols have their stanar meanings an subscript refers to surface values. The moistbalance atmosphere is one in which the ry bulb temperature in (15) is replace by an approximate form of the virtual temperature T T(1 0.6w ), (16) an the ry subscripts are remove. For the remainer of the paper, we consier only subsaturate water vapor by reefining w w, the mixing ratio, in units of kilograms of water vapor per kilogram ry air. Thus, (16) becomes T T(1 0.6w). This simplifie form of the virtual temperature is use because of its linear epenence on the moisture fiel that makes an analytic escription of PV mc possible. In calculations for the real atmosphere, a generalie form of the virtual temperature that inclues the effects of clou an rainwater shoul be use (Schubert et al. 001). Since the mixing ratio will be allowe to vary with height, T cannot be remove from the integral as in (15). Consequently, the hypsometric equation for a moist-balance atmosphere takes the form p g Z ln. (17) p R T The Coriolis parameter ( f) is assume to be a constant of 10 4 s 1 everywhere an results in a nonero

1JANUARY 003 M C TAGGART-COWAN ET AL. 169 FIG.. Schematic representation of hyrostatic ajustment resulting from a point heat source elevate above the surface. e-foling height of the profile. The ry vertical coorinate ( ) is use since it is inepenent of the mixing ratio. The moisture variation in the horiontal is axisymmetric an therefore best escribe in cylinrical coorinates. In the following equations, r an represent the raial an aimuthal irections, respectively. Since the moisture fiel is aimuthally invariant, the full three-imensional structure of the mixing ratio is given by ((r /H w(r, ) we x)( /H )), (19) where H x is the horiontal length scale of the moisture bubble. The structure of the mixing ratio fiel is shown in Fig. 1. b. Introuction to PV mc in the moel atmosphere A brief outline of the metho use to obtain the moist component perturbation variables is containe in this section. The steps escribe here will be presente in FIG. 1. (a) Cross section of the water vapor mixing ratio with g kg 1 contour intervals. (b) Profiles of the mixing ratio taken at the center (right curve), H x (mile curve), an H x (left curve). ry-balance absolute vorticity, in spite of the motionless constraint in the ry case. The ifferences in the heights that serve as the upper bouns of the integration in (15) an (17) ( versus ) will form the basis of the moist component perturbations. The height ifferences follow irectly from the fact that the ensity of water vapor is less than that of air. Thus, the inclusion of water vapor in a hyrostatically balance column will result in increase thicknesses. The profile of moisture in the moel atmosphere is escribe by an exponential function of the form /H w( ) we, (18) where w is the mixing ratio at the surface, an H is the FIG. 3. Relative errors of approximations for mc from (5) (light curves) an W from (7) (heavy curves). Profiles are taken at the origin (soli lines) an at H x (ashe lines) as inicate.

170 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 FIG. 4. Cross sections of (a) the moisture integral, (b) the geopotential perturbation, (c) the relative vorticity perturbation, an () PV mc. (a) Contours plotte every 0.0 m K 1, (b) every 5 m s, (c) every 1 10 7 s 1, an () every 0.01 PVU (1 PVU 10 6 m Kkg 1 s 1 ). greater etail throughout the remainer of the section. Everywhere here, moist component perturbations (enote by a subscript mc ) are efine relative to the ry state as escribe by (15). The first step consists of iagnosing the height change of a pressure surface by ifferencing (15) an (17). As a lower bounary conition, we assume that the ry surface pressure is the same as the moist surface pressure, p ( 0) p ( 0), (0) since analyses of surface pressure are reasonably accurate. This is equivalent to assuming that for each unit mass of water vapor present in the moist column, an equivalent mass of ry air is absent. Another reasonable, though more complicate, choice of bounary conition involves the use of the height of the humiity maximum in the column ( w max ). In this case, we set p ( w ) p( w ), (1) max max so that surface pressure perturbations are restricte to be nonpositive. The physical meaning of (1) is best escribe by use of a simple thought exercise (Fig. ). A motionless point heat source locate between iscrete pressure levels p an p will lea to a hyrostatically increase separation istance (p p). Since the heating is confine by the extent of the source (the effects of iffusion are ignore in this exercise) the tempera-

1JANUARY 003 M C TAGGART-COWAN ET AL. 171 FIG. 5. Decomposition of the PV mc equation (39). (a) Term 1 represents ynamical inuction; (b) term, stability reuction; (c) term 3, the nonlinear moist component perturbation; an () term 4, the nonlinear metric. All cross sections are plotte in PVU with 0.01-, 0.01-, 0.0001-, an 0.000 0-PVU contour intervals for each of (a) (), respectively. tures in the layers below p an above p are unchange. Thus, the thicknesses p an p are constant an the height perturbations experience by the p an p surfaces are translate hyrostatically to all levels below an above p, respectively. This choice of bounary conition is characterie by null geopotential perturbations at w max, as evince by (1), an contains the intuitive avantage that the surface pressure nee not be fixe. In the simplifie atmosphere escribe here, an inee in general for the real atmosphere, the maximum water vapor mixing ratio occurs at the lowest level, so that (1) reuces exactly to (0). Even for cases in which, is slightly elevate above the lowest level, w max (1) likely remains approximately true an surface pressure perturbations will be small. With the lowest height (surface pressure) fixe, any nonero specific humiity will result in higher virtual temperatures in the column an thus larger thicknesses in the moist atmosphere. These ifferences in the geopotential heights of each pressure level in the moist atmosphere can be relate to ifferences in the streamfunction through a balance equation. In the interests of obtaining an analytic result, geostrophic balance is assume. A higher-orer balance equation (e.g., Charney 1955) might be more applicable for the real atmosphere, especially given the finescale structure of typical mois-

17 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 FIG. 6. Schematic representation of simple moisture profiles an their resulting PV mc. All panels are for a local maximum in the mixing ratio in the horiontal. For a local minimum, the signs of the vorticity-ominate anomalies are reverse. ture features. The Laplacian of this moist component streamfunction yiels the moist component relative vorticity, which appears in two of the terms in the PV mc equation (14). The changes to the static stability of the column are obtaine by consiering the ifference between the vertical graients of ry () an virtual ( ) potential temperatures. c. Moist component height perturbations In orer to track the height change of a pressure level, p an p in (15) an (17) are set equal, an the perturbation mc is obtaine. Differencing (15) an (17) an employing the conition escribe by (0) yiels Z Z, () T T 0 0 which may be rearrange to prouce where T Z 0.6W, (3) w T W Z (4) 0 is the moisture integral. In the simplifie moel atmosphere consiere here, the integral in (4) can be expresse analytically. Solving for the height perturbation yiels 0.6WT/H 0.6WT H ln[1 0.6w(e mc 1)]. (5) To a goo approximation, T may be consiere constant for small height changes ( mc ), so virtual temperature can be brought outsie the integral in (3) to yiel an approximation for mc, mc 0.6WT. (6) For our simplifie atmosphere, it is foun that (6) preicts height perturbations whose values are within 1% of those given by (5) (Fig. 3). We will therefore use (6) to iagnose moist component height perturbations in an effort to keep the subsequent analytic solutions as simple as possible without an appreciable loss of accuracy.. Moisture integral Using the iealie exponential profile of humiity given by (19), it is possible to integrate (4) using partial fractions. The complete form of the integral yiels where H 1 0.6wA W ln, (7) 0.6T 1 0.6w r /H A e x, an e /H. A highly simplifie result is obtaine if we assume that T T in the enominator of (4). This approximation results in HwA 1 W 1, (8) T which is accurate to within 1% for tropical moisture profiles (Fig. 3). A cross section of W through the center of the moisture maximum is shown in Fig. 4a. The free parameters H an H x are set to 3000 m an 3000 km,

1JANUARY 003 M C TAGGART-COWAN ET AL. 173 respectively, for all of the calculations in this stuy. Equation (8) will be use to escribe the moisture integral for the remainer of the paper. e. Moist component geopotential an streamfunction From (6), the expression for the moist component geopotential ( mc g mc ) is simply mc(r, ) 0.6gT (r, )W(r, ). (9) Using the approximate form of the moisture integral (8), the moist component geopotential can be expresse as 1 0.6gH w A(1 0.6w) 1, (30) mc an is isplaye in Fig. 4b. Assuming that the perturbations are geostrophically balance, the moist component streamfunction is trivially erive as 0.6gHw 1 mc A(1 0.6w) 1. (31) f Equations (30) an (31) escribe the ifference in the geopotential an streamfunction fiels between a ry balance atmosphere (15) an a moist balance atmosphere (17). As evience by Fig. 4b, the largest perturbations in the geopotential an streamfunction fiels occur at upper levels irectly above the moisture maximum. Although this result may initially seem somewhat counterintuitive since the humiity maximum occurs at the surface, it is a necessary consequence of the lower bounary conition for the moisture integral given by (0). The height of the column base is fixe an the perturbations are sign-efinite an positive; therefore, the moisture integral (an thus the moist component geopotential an streamfunction profiles) are monotonic increasing functions of height. This leas irectly to the result shown in Fig. 4b. f. Moist component relative vorticity The qualitative ifferences in the vorticity structures of the ry an moist hyrostatically balance states is evient from the thickness analysis escribe in the outline of this section. Since the lowest-level heights an surface pressures are not allowe to change, the consieration of moisture results in increase heights everywhere above the groun. Near the origin, where the specific humiity is largest, this effect will be most pronounce. We thus en up with a ome of higher heights an attenant anticyclonic flow at upper levels in the vicinity of the humiity maximum. This leas to a preominantly negative moist component relative vorticity, which can be expresse as mc mc 1 r r r r (in the axisymmetric moel employe here) [ ] x x.4gw H 1 1 f Hx r r A (1 1.w) 1 1. w. (3) H H For the iealie moisture istribution escribe by (19), the value of mc is generally negative (Fig. 4c) for r H x an positive for r H x. This is because the curvature component of the vorticity (consiere in a natural coorinate system) ominates insie H x an the shear component ominates at larger raii. For a local minimum in the moisture fiel, the moist component relative vorticity woul be positive near the center an negative beyon H x. g. Moist component static stability Absolute vorticity plays an important role in the calculation of PV; however, so oes the static stability / p. For consistency with (14), we will employ an alternative form containing the geopotential in Exner function coorinates: p. (33) p Using the simplifying qualities of the current moel, mc 0.6w an, (34) expansion of the static stability into ry an moist component parts results in a moist component static stability term 1 mc 0.6w, (35) gh which is negative for all reasonable H. The form of (35) epens strongly on the vertical graient of w; in fact, if w were to increase exponentially with height, the sign of the (ominant) first brackete term in (35) woul reverse. h. Moist component PV Moist component PV is erive in Cartesian coorinates as (14). The cylinrical coorinate form of the iagnostic PV mc equation is

174 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 where [ g mc mc mc p mc mc L(, mc) P ( f ) L(, ) L(, )], (36) mc mc mc A B 1 A B L(A, B), an r r r 1 1 r. r r r r Employing the simplifying assumptions of our moel (isothermal, motionless ry state, f plane), (36) reuces to g mc mc mc mc mc mc mc P f L(, ) p. (37) term 1 term term 3 term 4 The analytic form of the nonlinear metric term (term 4) in (37) is easily erive as 1 1.w r L( mc, mc ), (38) f H x from our knowlege of the three imensional istribution of mc an the balance equation. Thus, the full analytic form of the PV mc equation for the simplifie moel atmosphere is given by [ ] g.4gw H 1 r r 1 mc p f H H H x x x gh P 1 A (1 1.w) 1 1. w 0.6 fw term 1: ynamical inuction [ ] term : stability reuction 1.44gww H 1 r r 1 1 1.wr 1 A (1 1.w) 1 1. w f Hx Hx Hx gh f Hx. term 3: nonlinear moist component perturbation term 4: nonlinear metric (39) An investigation of each of the terms of (39) will be presente in section 4. Raius height plots of the iniviual terms appear in Fig. 5 an a cross section of PV mc is shown in Fig. 4. Calculations are performe only up to 300 hpa, since the static stability ( / ) at that level excees 0.6 K s 3 m, a typical value for the tropopause. Above this level lies the iealie moel s stratosphere although the tropopause in the moel is not efine by a sharp graient in stability ue to the isothermal state in which we o not consier the effects of water vapor ue to its negligible quantities at such high altitues. Like (9), (39) escribes the PV mc fiel: the PV that is present in the atmosphere solely as a result of the presence of subsaturate water vapor. It is conceptually the same as an iniviual component of the ecompose PV fiel of the type escribe by DE91. The moist component PV compute from (39) is therefore an ieal caniate for a piecewise PV inversion. Such an inversion woul yiel the balance perturbation geopotential an streamfunction fiels associate with subsaturate moisture. In this way, the ynamical influence of iniviual humiity structures, generally consiere to be relatively passive until saturation, can be irectly evaluate. This link between PV an atmospheric water is crucial for the type of sensitivity stuies propose by McTaggart-Cowan et al. (001), which involve moifications to the moisture fiel. 4. Interpretation of PV mc The utility of PV mc stems from its ability to escribe the balance portion of the ynamics irectly associate with the atmospheric moisture fiel. Although the analytic moel presente here eals only with subsaturate moisture, (9) is a general equation that can inclue the

1JANUARY 003 M C TAGGART-COWAN ET AL. 175 FIG. 8. Comparison of the original (ashe) an the retrieve (soli) mixing ratio following inversion. Contours are plotte every gkg 1. FIG. 7. (a) Perturbation temperature, an (b) height an win fiels following the inversion of the PV mc anomaly shown in Fig. 4. Temperatures are contoure at 1-K intervals, an heights at 1-am intervals. Positive win values are entering the page, an negative values are exiting it (contours every 0. m s 1 ). effects of both clou an rainwater. Local maxima in the moisture fiel generally correspon to negative PV mc, whereas local minima lea to positive PV mc. The vertical graient of humiity also plays an important role in etermining the sign of PV mc. An investigation of the iniviual terms in (39) yiels insight into how these anomalies are create. Term 1 of (39), the ynamical inuction term, escribes the role that subsaturate moisture plays in moifying balance atmospheric ynamics. The generation of a height ome centere above the humiity maximum results in balance anticyclonic flow aroun the origin. As note in section 3f, the competing contributions of curvature an shear vorticity result in a sign switch at r H x for the ynamical inuction term. In the analytic moel, the influence of curvature is greater for r T H x an results in the expecte anticyclonic vorticity near the center. For r t H x, this term becomes positive; however, its values are so small beyon H x as to be negligible (Fig. 5a). The stability reuction term (term ) in (39) escribes the effects of the moisture-moifie static stability on PV. Any monotonically ecreasing function of w with will have the effect of reucing the atmosphere s static stability as virtual temperatures below rise more than those aloft. This will necessarily ecrease the magnitue of PV everywhere, but most notably where the vertical graient of w is the largest. In the moel, w ecreases exponentially with height, so this term is a maximum near the surface, as shown in Fig. 5b. In the moel troposphere, this is the ominant term of the PV mc equation. Terms 3 (the nonlinear moist component perturbation ) an 4 (the nonlinear metric ) are two orers of magnitue smaller than the leaing terms (1 an ) as woul be expecte of perturbation correlations. As shown in Fig. 5c, the nonlinear moist component perturbation is largest in the region where both the ynamical inuction an the stability reuction terms are of significant magnitue. The nonlinear metric (Fig. 5) is ientically ero at the origin in the analytic moel, an reaches a maximum near the surface insie H x. As note by DE91, these nonlinear terms must be retaine in orer to ensure the aitive nature of the component perturbations uner the piecewise PV framework. Schematics of the leaing PV mc terms for several moisture profiles are shown in Fig. 6. All of the conceptual moels presente here are vali for a local moisture maximum in the horiontal. For a local minimum in humiity, the sign of the PV mc anomalies labele vorticity in Fig. 6 (term 1 in the escription above) shoul be reverse. Figure 6a summaries the case pre-

176 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 sente in this paper. Humiity ecreases monotonically with height, leaing to estabiliation primarily at lower levels, an a negative PV mc anomaly. The moisture integral is of moerate magnitue, an leas to a negative PV mc anomaly at higher levels. A constant mixing ratio with height, as shown in Fig. 6b, results in slightly increase static stabilities at all levels an a weak positive PV mc anomaly. At upper levels, the large moisture integral will lea to a strong negative anomaly, the effects of which will outweigh those of the stability reuction term (term ). In another unusual case in which the mixing ratio increases monotonically with height (Fig. 6c), the increase static stability at mile an upper levels will ten to prouce a positive anomaly. However, the continually increasing moisture integral leas to a negative PV mc anomaly. The sign of the resulting PV mc is case epenent. Figure 6 shows the PV mc structure associate with a milevel moisture maximum. In this case, static stabilities below the maximum are increase, while those above it are ecrease, leaing to an opposite-signe couplet in the PV mc fiel centere on the humiity peak. At upper levels, the moerately large moisture integral prouces moist component geopotentials that result in an upper-level negative PV mc anomaly. Although the list of possible states presente here is by no means exhaustive, the schematics of Fig. 6 will hopefully serve as a starting point from which to aress more complicate situations. 5. Inversion of PV mc As escribe in section 3h, PV mc represents a component perturbation on the backgroun (ry) state (9). It is thus possible to retrieve the balance win an mass fiels resulting from the presence of subsaturate water vapor using a piecewise PV inversion scheme such as that escribe by DE91. Such an inversion of the PV mc fiel (Fig. 4) yiels the balance win, temperature, an height anomalies shown in Fig. 7. The temperature perturbations arising from the PV mc anomalies shoul be exactly those given by the virtual temperature, allowing for the retrieval of the mixing ratio through rearrangement of (16): T w, (40) 0.6T where T is the inverte temperature perturbation an T is the ry bulb temperature of the (ry) backgroun state. The results of this retrieval are shown in Fig. 8. The small ifferences between the original an retrieve mixing ratio values near the surface reflect the errors associate with the geostrophic balance assumption applie uring the erivation of PV mc for the purposes of obtaining analytic solutions for this iealie case. The calculation of PV mc for more realistic atmospheres will involve the use of higher-orer balance equations, an may procee in one of two ways. The simplest approach employs (9), an consists of calculating from (15) an from a balance equation. Having obtaine these values, the calculation of P from (8) is simple, as is the solution of (9) for PV mc (P mc ). Alternatively, PV mc can be calculate irectly from (14) using (3) to compute mc, an a balance equation to etermine mc. The geopotential an streamfunction values for the ry atmosphere are again calculate from (15) an a balance equation, respectively. Both of these techniques result in three-imensional PV mc fiels which can be inverte using a piecewise PV inversion. For the case presente here (Fig. 8) the piecewise inversion of PV mc calculate irectly from (14) [an hence (39) for our simple atmosphere] serves to quantify the role that subsaturate moisture plays in ajusting the ynamics of the atmosphere. A local maximum (minimum) in the moisture integral results in anticyclonic (cyclonic) flow an increase (ecrease) heights. 6. Summary an iscussion A PV-base variable (PV mc ) escribing the effects of water vapor, clou, an rainwater on balance atmospheric ynamics was erive using the moist PV efinition of Schubert et al. (001) an the piecewise ecomposition framework of DE91. The complexity of the full equation (14) for PV mc makes interpretation ifficult for a realistic atmosphere. To simplify the problem, we focuse on escribing the ynamical influence of subsaturate water vapor using an iealie atmospheric moel. Using two forms of the hypsometric equation, (15) an (17), which assume ry an moist hyrostatic balance, respectively, perturbation equations for the geopotential, streamfunction, relative vorticity, static stability, an PV were evelope. When viewe as a component anomaly, the PV mc fiel escribe by (36) is an excellent caniate for PV inversion. The resulting mass an win fiels escribe the irect effect of the presence of atmospheric water on the ynamics of the atmosphere. It is also possible to retrieve the original specific humiity fiel through (40). A potential application of this moist component methoology is to the problem of sensitivity testing as outline by McTaggart-Cowan et al. (001). The sensitivity of the extratropical transition an reintensification of Hurricane Earl (1998) to PV anomalies in the initial conitions was investigate using PV removal an inversion. However, as note in that paper, it is very ifficult to assess the impact of the moisture fiel on the rapi reevelopment of the storm. The research presente here suggests that a link between the PV fiel an the moisture fiel exists through the PV mc variable. For example, the removal of the moisture, an balance mass an win fiels associate with temporally anomalous PV mc coul be couple with the removal of the ry PV anomaly associate with ex Hurricane Earl to completely remove the remnant hurricane from the initial conitions of a simulation. Such a sensitivity case stuy will be the topic of a future paper. The results

1JANUARY 003 M C TAGGART-COWAN ET AL. 177 from such research may eluciate the role of the nearsurface tropical features in rapi cyclogenesis. In such a way, we hope that PV mc will allow for a quantitative assessment of the relative importance of ynamics an thermoynamics in eveloping systems. Acknowlegments. We thank the members of the Mesoscale Research Group (McGill University) for their guiance an support throughout the course of this research. As well, we thank the three anonymous reviewers for their help in preparing this paper for publication. This work has been supporte by Natural Sciences an Engineering Research Council grants an by subventions from the Meteorological Service of Canaa. REFERENCES Charney, J., 1955: The use of the primitive equations of motion in numerical preiction. Tellus, 7, 6. Davis, C., 199a: A potential-vorticity iagnosis of the importance of initial structure an conensational heating in observe extratropical cyclogenesis. Mon. Wea. Rev., 10, 409 48., 199b: Piecewise potential vorticity inversion. J. Atmos. Sci., 49, 1397 1411., an K. Emanuel, 1991: Potential vorticity iagnosis of cyclogenesis. Mon. Wea. Rev., 119, 199 1953., E. Grell, an M. Shapiro, 1996: The balance ynamical nature of a rapily intensifying oceanic cyclone. Mon. Wea. Rev., 14, 3 6. Ertel, H., 194: Ein Neuer hyroynamischer Wirbelsat. Met. Z., 59, 71 81. Hoskins, B., M. McIntyre, an A. Robertson, 1985: On the use an significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877 946. Huo, Z., D. Zhang, an J. Gyakum, 1998: An application of potential vorticity inversion to improving the numerical preiction of the March 1993 superstorm. Mon. Wea. Rev., 16, 44 436. McTaggart-Cowan, R., J. R. Gyakum, an M. K. Yau, 001: Sensitivity testing of extratropical transitions using potential vorticity inversions to moify initial conitions: Hurricane Earl case stuy. Mon. Wea. Rev., 19, 1617 1636. Rossby, C. G., 1939: Relation between variations in the intensity of the onal circulation of the atmosphere an the isplacements of the semi-permanent centers of action. J. Mar. Res.,, 38 55. Schubert, W. H., S. A. Hausman, M. Garcia, K. V. Ooyama, an H. Kuo, 001: Potential vorticity in a moist atmosphere. J. Atmos. Sci., 58, 3148 3157.