Observations of Particle Size and Phase in Tropical ...

422 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2004 American Meteorological Society

Observations of Particle Size and Phase in Tropical Cyclones: Implications forMesoscale Modeling of Microphysical Processes

GREG M. MCFARQUHAR

Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois

ROBERT A. BLACK

NOAA/AOML/HRD, Miami, Florida

(Manuscript received 27 August 2002, in final form 27 August 2003)

ABSTRACT

Mesoscale model simulations of tropical cyclones are sensitive to representations of microphysical processes,such as fall velocities of frozen hydrometeors. The majority of microphysical parameterizations are based onobservations obtained in clouds not associated with tropical cyclones, and hence their suitability for use insimulations of tropical cyclones is not known. Here, representations of mass-weighted fall speed Vm for snowand graupel are examined to show that parameters describing the exponential size distributions and fall speedsof individual hydrometeors [through use of relations such as V (D) 5 aDb] are identically important for deter-mining Vm. The a and b coefficients are determined by the composition and shape of snow and graupel particles;past modeling studies have not adequately considered the possible spread of a and b values. Step variations inthese coefficients, associated with different fall velocity regimes, however, do not have a large impact on Vm

for observed size distributions in tropical cyclones and the values of a and b used here, provided that coefficientsare chosen in accordance with the sizes where the majority of mass occurs. New parameterizations for Vm aredeveloped such that there are no inconsistencies between the diameters used to define the mass, number con-centration, and fall speeds of individual hydrometeors. Effects due to previous inconsistencies in defined diameterson mass conversion rates between different hydrometeor classes (e.g., snow, graupel, cloud ice) are shown tobe significant.

In situ microphysical data obtained in Hurricane Norbert (1984) and Hurricane Emily (1987) with two-dimensional cloud and precipitation probes are examined to determine typical size distributions of snow andgraupel particles near the melting layer. Although well represented by exponential functions, there are substantialdifferences in how the intercept and slope of these distributions vary with mass content when compared toobservations obtained in other locations; most notably, the intercepts of the size distributions associated withtropical cyclones increase with mass content, whereas some observations outside tropical cyclones show adecrease. Differences in the characteristics of the size distributions in updraft and downdraft regions, whencompared to stratiform regions, exist, especially for graupel. A new representation for size distributions associatedwith tropical cyclones is derived and has significant impacts on the calculation of Vm.

1. Introduction

Model representations of the lifetime of clouds andof the production and evolution of precipitation are sen-sitive to the rate at which populations of solid phasehydrometeors fall. Using a single column model, Petchet al. (1997) showed that relatively small variations inice crystal velocities could produce large differences incloud mass and radiative properties. For the EuropeanCentre for Medium-Range Weather Forecasts(ECMWF) weather model, Klein and Jakob (1999)found that gravitational settling was the most important

Corresponding author address: Prof. Greg M. McFarquhar, Dept.of Atmospheric Sciences, University of Illinios at Urbana–Cham-paign, 105 S. Gregory Street, Urbana, IL 61801.E-mail: [email protected]

process controlling the abundance of ice in the highclouds of midlatitude cyclones, underscoring the needfor careful evaluation of parameterizations of solidphase microphysics. By comparing different cirrus mod-els for an idealized case, Starr et al. (2000) also notedthat the ice water fallout process has a dominant effecton the vertical distribution of ice water and on the in-tensity of circulation within cirrus clouds.

Microphysical processes also feed back on the dy-namics of tropical cyclones. Willoughby et al. (1984)showed that hurricane simulations with parameterizedice microphysics had a very different structure and evo-lution compared to those with liquid water microphys-ics. They, together with Lord et al. (1984) and Lord andLord (1988), showed that the extent and intensity ofmesoscale downdrafts associated with latent heat release

15 FEBRUARY 2004 423M c F A R Q U H A R A N D B L A C K

are determined by the horizontal advection of hydro-meteors from the convection, together with the fallspeeds of snow and graupel and the conversion ratesbetween hydrometeor species. These downdrafts con-tribute to the formation of multiple convective rings,which in turn modifies storm development (Willoughbyet al. 1984).

This study aims to acquire a better understanding ofhow the size, density, and shape of solid hydrometeorsaffects the fallout of snow and graupel in tropical cy-clones, and to develop better representations of suchprocesses for numerical models. This study is timely fora number of reasons. First, the majority of current mi-crophysical schemes are based on microphysical obser-vations obtained in midlatitudes (e.g., Gunn and Mar-shall 1958; Sekhon and Srivastava 1970), and it is notknown whether these observations are representative oftropical cyclones. McCumber et al. (1991) showed thatmodel simulations are sensitive not only to the choiceof parameterization scheme but also to details of as-sumed particle size distributions and hydrometeor con-version terms. Second, the basis for most bulk param-eterization schemes (e.g., Lin et al. 1983; Rutledge andHobbs 1983, 1984) was developed approximately 20 yrago when coarser resolution models were in use. Moredetails of microphysical processes (such as fallout orconversions between hydrometeor categories) and theirrelationships to the convection producing them can nowbe resolved. Finally, more observations of tropical cy-clones are now available (e.g., Black and Hallett 1986,1999) making such a study possible.

In many bulk microphysical schemes (Lin et al. 1983;Rutledge and Hobbs 1983, 1984; Dudhia 1989; Rotstayn1997; Reisner et al. 1998), ice-phase hydrometeors aresorted into three different categories: snow, graupel, andcloud ice. A mass-weighted terminal velocity Vm is cal-culated separately for each category. The rates of con-versions between liquid and solid hydrometeor cate-gories heavily depend on this estimate of Vm, which istypically predicted using relationships from Locatelliand Hobbs (1974) that describe the fall velocities ofindividual snow and graupel particles. For example, theReisner et al. (1998) microphysical scheme used in thefifth generation Pennsylvania State University–NationalCenter for Atmospheric Research (PSU–NCAR) Me-soscale Model (MM5) assumes that the fall speed ofsnow particles can be adequately described by relation-ships developed using observations of the fall speedsof unrimed radiating assemblages of plates, side planes,bullets, and columns. Such relationships do not repre-sent all situations. In tropical cyclones, most snow par-ticles exhibit some degree of riming because ice par-ticles begin as rimed particles in the convective regions,and then grow somewhat by diffusion in the stratiformregions; hence the relationships for unrimed radiativeassemblages may not be applicable.

Microphysical data collected during penetrations ofthe National Oceanic and Atmospheric Administration

(NOAA) P-3 aircraft into hurricanes are used here todetermine the relative importance of hydrometeor size,shape, and density for determining Vm. Snow and grau-pel are emphasized in this study because their fall speedsare more appreciable than that of ice and because theirrepresentation significantly impacts hurricane simula-tions. The effect of an inconsistency in the diameterused to define the size distributions and fall speeds ofindividual particles, first noted by Potter (1991), is ex-amined as well as the importance of a discontinuity inthe relationships used to define the fall speeds of in-dividual particles (Khvorostyanov and Curry 2002). Fi-nally, a new parameterization for Vm is produced usingobservations that describe tropical cyclones. Some pre-liminary studies of this nature have already been con-ducted. For example, Braun and Tao (2000) modifiedthe Goddard microphysical scheme (Tao et al. 1989; Taoand Simpson 1993) in their simulations of HurricaneBob to more accurately represent conditions observedin tropical cyclones. Note that although the slopes andintercepts calculated here to describe the size distribu-tions apply only to tropical cyclones, the general ap-proach of ensuring consistency in diameter definitionsis applicable to microphysical parameterizations used insimulations of any meteorological phenomena.

The remainder of the paper is organized as follows.In section 2 the relative importance of size, shape, anddiameter definition for the calculation of Vm is exam-ined; in section 3 the in situ measurements that werecollected in tropical cyclones are described; and in sec-tion 4 these observations are used to develop a newrepresentation for size distributions, which includes adependence on convective activity, that can be used inbulk microphysical schemes and that can be used tocalculate Vm. The significance of these results is dis-cussed in section 5.

2. Importance of size and shape for calculatingmass-weighted fall speed

a. Approach

In the aforementioned papers describing bulk cloudmicrophysical schemes used in mesoscale models, ex-pressions for mass-weighted fall speeds of hydrometeorpopulations are developed and are given by

`

V(D)m(D)N(D) dDE0

V 5 , (1)m `

m(D)N(D) dDE0

where V(D) represents the terminal velocity of an in-dividual particle, m(D) represents hydrometeor mass,and N(D) is the number density. Assumptions aboutV(D), m(D), and N(D) must be made in order to de-termine Vm. A detailed examination of the validity ofsome commonly used assumptions is made in this sec-


tion. First, a traditional approach of calculating Vm isreviewed.

In most parameterizations, fall speed–diameter rela-tions proposed by Locatelli and Hobbs (1974) are used,and are represented by

0.4P0bV(D) 5 aD , (2)1 2P

where a and b are habit, density, or size-dependent fitcoefficients that differ for snow and graupel; D is typ-ically the maximum particle dimension Dmax of the hy-drometeor; P is the atmospheric pressure; and P0 is aconstant given by 106 N m22. It should be noted thatthe parameterized dependence on pressure is an ad hocapproximation to the real change in velocities with pres-sure. Because analytic integration of Eq. (1) is desiredfor bulk parameterization schemes, it is conventionallyassumed that a and b are constant. The suitability ofthis assumption is discussed later. Typically, m(D) isrepresented by

p3m(D) 5 rD , (3)

6

where r represents the bulk density, defined as particlemass divided by particle volume, of snow (rs) or graupel(rg); and D is the volume equivalent diameter Dvol ofthe hydrometeor. The size distribution N(D) is repre-sented by an exponential function

N(D) 5 N exp(2lD),0 (4)

where N0 and l are the intercept and slope for the fit,which again differ for snow (N0s and ls) and graupel(N0g and lg). In Eq. (4), D can be Dvol , Dmax, or meltedequivalent diameter Dmelt, depending on the source ofobservations. Ignoring differences in diameter defini-tions between equations, and assuming that a and b areknown and independence of hydrometeor size, an an-alytic expression for Vm may be written as (e.g., Rut-ledge and Hobbs 1983)

0.4aG(4 1 b) P0V 5 . (5)m b 1 26l P

This analytic expression can be incorporated into mod-els without explicit integration, allowing for minimalcomputational time.

Potter (1991) was the first to point out the inconsis-tency between diameter definitions used in the devel-opment of such bulk parameterizations. This inconsis-tency makes the use of Eq. (5) inappropriate. An im-proved relation for Vm is developed here by properlycombining expressions for exponential hydrometeorsize distributions in terms of Dmelt (e.g., Gunn and Mar-shall 1958; Sekhon and Srivastava 1970) with publishedrelationships for V(D) (e.g., Locatelli and Hobbs 1974;Mitchell 1996; Heymsfield and Iaquinta 2000; Khvo-rostyanov and Curry 2002) that give velocity in terms

of Dmax, or on occasion, the diameter of a sphericalparticle with the same projected area Darea. Similar com-binations of size distributions have been used in the pastdevelopment of mesoscale parameterization schemes(e.g., Lin et al. 1983; Reisner et al. 1998), but differ-ences in diameter definitions have not been commonlyconsidered. The major problem with this inconsistencyis not that an exponential distribution in terms of Dmelt

is not strictly an exponential distribution in terms ofDmax, but rather than the parameters of the best-fit ex-ponential distribution will vary substantially for func-tions in terms of Dmelt and Dmax. The a and b coefficientsdescribing hydrometeor fall velocities also vary signif-icantly depending on whether the parameterization is interms of Dmelt or Dmax. Other studies have derived ex-ponential size distributions in terms of Dmax for snowand ice (e.g., Braham 1990; Houze et al. 1979; Herzeghand Hobbs 1985; Ryan 2000) and for graupel (Houzeet al. 1979), which permit more of a straightforwardapplication of Eq. (1), provided that the effective den-sity, mass divided by volume of sphere with diameterof maximum particle dimension, is used in Eq. (3) ratherthan the bulk density.

To develop an internally consistent representation ofVm, mass–diameter relationships of the form

bm(D) 5 aD (6)

are used, where a and b are coefficients that depend onsize, shape, and density of the snow and graupel. Usinga wide range of observations of particle mass, and a, bcoefficients derived by various authors, Mitchell (1996)catalogs a and b for a variety of different particles types.Combining these relationships with expressions for par-ticle fall speeds in terms of maximum dimension, a newvelocity relationship for individual particles is derived asV(Dmelt) 5 a9 (P0/P)0.4, whereb9Dmelt

b /bpr 3bwa 5 a and b9 5 . (7)1 26a b

Further, the mass of the melted equivalent particles usedto characterize the exponential distributions is expressedin terms of the density of water rw as m(Dmelt) 5prw /6. Substituting V(Dmelt) and m(Dmelt) in Eq. (1),3Dmelt

and assuming exponential distributions in terms of Dmelt ,gives

b /bpr 3bwa G 4 11 2 1 2 0.46a b P0V 5 , (8)m 3b /b 1 26l Pmelt

where lmelt is l defined in terms of Dmelt. Equation (8)is still an analytic expression for Vm and is based ontypical velocity coefficients in terms of Dmax, and canbe easily applied in numerical models. For cases wherethe size distributions of observed particles are param-eterized in terms of Dmax, with slope lmax, an alternateexpression for Vm can be derived as


FIG. 1. Terminal velocity Vm is a function of qs using (a, b) co-efficients from Mitchell (1996) describing unrimed assemblages ofcolumns, plates, side planes, and aggregates. Dashed line representsVm calculated using Eq. (8) (i.e., no inconsistency in diameter defi-nitions); solid line, using Eq. (5) (i.e., that used by majority of mi-crophysical parameterization schemes).

0.4aG(1 1 b 1 b) P0V 5 . (9)m b 1 2l G(1 1 b) Pmax

Brown and Swann (1997) considered the role of varyinga, b, a, and b coefficients in their single moment mi-crophysical scheme based on Dmax, but did not specif-ically present a generalized form of Eq. (9) in theirpaper. Caution must be taken to use either Eq. (8) orEq. (9) depending on the form of the particle obser-vations.

b. Effects due to diameter discrepancies

Figure 1 compares Vm calculated for distributions ofsnow particles using Eqs. (8) and (5), the latter repre-senting older parameterizations. A pressure equal to P0

is used in the calculations. For typical one-moment pa-rameterization schemes, N0 is assumed constant mean-ing that lmelt must vary with the mixing ratio. Integratingan expression for mixing ratio gives

1/46r Nw 0l 5 . (10)melt 1 2r qa

For Vm calculated using Eq. (5), l is defined differentlyfollowing relationships used in traditional bulk param-eterization schemes [e.g., Eq. (3b) of Rutledge andHobbs 1983] that do not account for the different di-ameter definitions discussed here. For Fig. 1, snow par-ticles are exponentially distributed with a constant N0

of 2.0 3 107 m24, and snow mass mixing ratio qs variesover ranges typically observed in tropical cyclones.

Figure 1 shows that failing to correct for the diameterdiscrepancy, and hence the values of both l and V(D),can cause a significant underestimate of Vm for snow;the underestimate is not as large for graupel because

graupel more closely approximate shapes and the b co-efficients are closer to 3. The trend noted in Fig. 1 isopposite that noted by Potter (1991), who found thatcorrecting for diameter discrepancies gave a decrease inVm. The difference is related to the manner in whichPotter (1991) applied the correction, using mass–velocityrelationships derived by Locatelli and Hobbs (1974) toobtain the corrected Vm. A combination of previouslypublished m–D, m–V, and m–D relationships do not yieldconsistent results suggesting that there are observationaluncertainties associated with the derivation of the a andb coefficients. The curves in Fig. 1 are not extendedbelow qs of 0.1 g kg21 because at such low mixing ratios,many smaller particles would be prevalent and the as-sumed constant a and b values would not apply to theseparticles. A further examination on the size dependenceof a and b is discussed below; Fig. 1 merely shows theimportance of a consistent diameter definition.

Despite the apparent importance of the inconsisten-cies noted by Potter (1991), only very few remote sens-ing studies (e.g., Braun and Houze 1995) and parame-terization schemes (Ferrier 1994) have implementedsuch corrections. These corrections have most likely notbeen incorporated into models because there are otherinconsistencies in conversion rates between hydrome-teor categories that also need to be addressed for theparameterizations to be consistent. For example, whencalculating the conversion of cloud water to snow byriming (PSACI), a term is required that involves thegeometric sweep out of a volume of cloud by a snow-flake. This term should be expressed by

` dM(D )s2l Ds sPSACI 5 N e dD ,E 0s sdt0

dM(D ) ps 25 r D V(D )q E , (11)a sa s i SIdt 4

where the s subscript represents snow, ESI is the snow/cloud ice collection efficiency, and Dsa is the area-equiv-alent diameter of snow. The use of Dsa allows moreaccurate representations of geometric sweep out ofcloud ice, which has mass mixing ratio qi. Mitchell(1996) defines relationships between projected area AP

and Dmax of the formsA 5 gD ,P max (12)

where g and s are coefficients that vary according tothe size, shape, and density of snow (g s or ss) or graupel(g g or sg). Equation (12) allows calculation of Darea bysubstituting AP 5 p /4. Hence, PSACI can be ex-2Darea

pressed as(s 1b )/bs s sr pwPSACI 5 a r g N q Es a s 0s i SI1 26as

3s 1 b )s sG 1 10.4[ ]bs P03 , (13)

113(s 1b )/b 1 2s s sl Ps


FIG. 2. Conversion rates between different hydrometeor categories(snow, graupel, cloud ice, cloud water, and rain). Solid lines representrates after corrections made to account for inconsistencies in diameterdefinitions used by schemes, and to account for calculation of geo-metric sweep out in terms of area-equivalent diameter; dashed linesare rates using Reisner et al.’s (1998) scheme without these correc-tions. Reisner et al. (1998) and appendix A give equations used, textdescribes cloud conditions assumed.

FIG. 3. Scatterplot of (a, b) coefficients used to describe fall speedof snow and graupel particles in terms of V 5 a . Different sym-bDmelt

bols correspond to observations of graupel and snow made by Lo-catelli and Hobbs (LH) and Mitchell (M96). The 3 and big squareindicate values for snow and graupel, respectively, currently used bythe Reisner et al. (1998) scheme of MM5 model.

which differs from Eq. (A43) of Reisner et al. (1998).Similar expressions for processes that convert mass be-tween other hydrometeor categories are also derived andare summarized in appendix A; all terms and symbolsare defined as in Reisner et al. (1998) and for simplicityit has been assumed that the pressure is equal to P0.Although more complex in form than the original equa-tions, listed in appendix B all equations are easily in-tegrable and require no extra computational expense.Therefore, there seems to be no reason not to use thesemore accurate equations. Figure 2 compares the mag-nitude of the conversions between the hydrometeor cat-egories as a function of qs or graupel mixing ratio qg

calculated by both applying and not applying the cor-rections noted above. The calculations are based upontemperatures similar to those observed in tropical cy-clones near the melting layer, namely, for a temperatureof 258C; pressure of 700 m; and assuming rainwaterqR and cloud mixing ratios qc of 1.0 and 0.1 g kg21,respectively. Differences between terms of at least afactor of 2, and sometimes more, frequently exist. Amore detailed modeling or observational study is neededto calculate average differences using simulated valuesof mixing ratios and densities, rather than using thesimple assumptions used to construct Fig. 2. The curvesare extended to low values of qs and qg, where as-sumptions of constant a and b values may be question-able. Nevertheless, Fig. 2 does show that these correc-tion terms should be considered for more accurate cal-culation of conversions between hydrometeor catego-ries.

c. Effect of uncertainties in (a, b) coefficientsAnother uncertainty associated with the model rep-

resentation of Vm is the choice of (a, b) coefficients

describing the fall speeds of snow and graupel particles.Typically, single values of (a, b) are used in simulations,but in reality, (a, b) vary according to the shapes anddensities of snow and graupel and their size (Mitchell1996). Based on Locatelli and Hobbs (1974) and Mitch-ell (1996), Fig. 3 shows the range of (a, b) coefficientsthat might be used to describe the fall of snow andgraupel. The plotted coefficients are not the values fromthe original papers, but rather are the (a9, b9) valuesgiven by Eq. (7) to characterize V in terms of Dmelt. Thevalues used in the Reisner et al. (1998) microphysicalscheme for snow and graupel, as incorporated in MM5,are shown for comparison. For the same shape, thereare different (a, b) values from Mitchell (1996) de-pending on particle diameter; values of (a, b) corre-sponding to the largest snowflakes are plotted in Fig. 3because analysis of Sekhon and Srivastava’s (1970) pa-rameterized snowflake size distributions shows that themajority of the mass would be contained in these sizes.Note that coefficients that are used to describe the fallvelocities of monohabit pristine crystals, with typicalsizes smaller than 100 mm, are not included in Fig. 3because such crystals would be represented as ice intypical numerical models (e.g., the minimum radius ofa snow crystal is 75 mm in MM5), and their fall speedsare negligible. This exclusion is important becausemass–diameter and velocity–diameter relationships mayprovide unrealistic results for sizes smaller than 100mm, and corrections are needed (e.g., Brown and Fran-cis 1995). Analysis of both the tropical cyclone datacollected here and Sekhon and Srivastava’s (1970) pa-rameterized distributions confirm that little mass is in-cluded in sizes below 100 mm for snow and graupel,


FIG. 4. Terminal velocity Vm is a function of N0 for (top) graupeland (bottom) snow distributions. Different line types correspond todifferent (a, b) coefficients, obtained from Locatelli and Hobbs (1974)but converted into expressions in terms of Dmelt (V 5 ). NotebaDmelt

(ag, bg) values given by (332, 0.45) for solid, (1078, 0.72) for dotted,(381, 0.36) for dashed, (1443, 0.84) for dashed–dotted, and (617,0.54) for dash–dot–dot. Note (as, bs) values given by (231, 0.24) and(332, 0.45) for solid, (336, 0.42) and (182, 0.24) for dotted, (359,0.33) and (307, 0.45) for dashed, (327, 0.48) and (307, 0.45) fordashed–dotted, and (215, 0.24) for dash–dot–dot. Thick lines rep-resent calculations for qs of 0.5 g kg21 and qg of 5 g kg21. Thin linesrepresent qs of 0.02 g kg21 and qg of 0.5 g kg21.

hence such corrections are not as important for thisstudy.

Substantial spread in (a, b) values is noted in Fig. 3because of the wide variety of crystal habits or mixturesof habits that make up the snow category and becauseof variations in the shape and densities of graupel. Forexample, from Locatelli and Hobbs (1974), the (a, b)values for lump graupel vary from (0.45, 332 cm0.55

s21) to (0.72, 1078 cm0.28 s21) to (0.36, 382 cm0.64 s21)for bulk densities between 0.05 to 1, 0.1 to 0.2, and 0.2to 0.45 g cm23. Density measurements are not availablefor many of the different snow crystal data, but it islikely that density variation also causes some of thisvariation in (a, b). This variation in (a, b) coefficientsmay have important ramifications for the simulation oftropical cyclones. Black (1990) noted substantial gra-dients in the densities of snow and graupel particles withlarger densities closer to the hurricane eyewall. Hence,it may be possible that varying (a, b) coefficients shouldbe used in the single simulation of a tropical cyclone.Khvorostyanov and Curry (2002) also developed an an-alytic function that describes the variation in (a, b) withparticle size; variation of mean size might therefore alsoaffects the calculation of Vm.

Figure 4 shows that variations in (a, b) have importantramifications for the calculation of Vm. The mass-weighted terminal velocity is plotted as a function ofN0, with different line types representing varying (a, b)coefficients, with some line types repeated for snow.Thin and thick lines correspond to different qg (0.5 and5 g kg21) and qs (0.02 and 0.5 g kg21), with the thickerlines representing higher mixing ratios. The slope of theexponential distribution in terms of Dmelt is determinedfollowing Eq. (10) to ensure consistency. It can be seenthat the choice of (a, b) is just as important as the choiceof N0 for determining Vm; for example, for N0g of 1.03 108 cm24, Vm for low-density lump graupel is only137 cm s21, whereas Vm for conical graupel is 287 cm s21.

Because of variations in the relationship between theReynolds and Best number depending on fall regime,Mitchell (1996) parameterized (a, b) to vary with par-ticle size. Khvorostyanov and Curry (2002) developedan analytic expression for the variation of the (a, b)coefficients that avoided the step functions or discon-tinuities in the Mitchell (1996) scheme, but their ex-pressions still do not allow for an analytic expressionfor Vm. Hence, an examination was made using Mitch-ell’s (1996) study of the size variation of a and b co-efficients to determine errors induced by assuming con-stant values of a and b. It is implicitly assumed that theMitchell (1996) stepwise relations adequately charac-terize the size variation of the fall speed coefficients.

Performing an analytic integration over the observedsnow and graupel size distributions collected in tropicalcyclones, to be discussed in section 3, allows a deter-mination of whether these variations of (a, b) with par-ticle size are important. Figure 5 compares Vm calculatedusing nonvarying values of (a, b) against Vm calculated

using a Simpson’s rule integration that accounts for thevariation of (a, b) coefficients following Mitchell(1996). Note that the (a, b) chosen for the comparisongive the closest match to the analytically calculated Vm.The different points represent calculations for varyingN0 values and for varying (a, b) using the values tab-ulated by Mitchell (1996). There are only minor dif-ferences between the two calculations, with average dif-ferences of 3.5% 6 2.2% for snow and 1% 6 2% forgraupel. Hence, given other uncertainties in the calcu-lation of Vm, it is not necessary to account for the var-iation of (a, b) coefficients with particle size for theseobserved size distributions provided that (a, b) coeffi-cients, based on those values where the majority of hy-drometeor mass is located, are used. This occurs becausethe majority of the mass in the snow and graupel sizedistributions occurs for the specific Reynolds/Best num-ber regime given by Mitchell (1996). The choice of (a,b) coefficients for the calculation of Vm may depend onthe size distribution or hydrometeor category being ex-


FIG. 5. Terminal velocity Vm is calculated using (a, b) coefficientsthat vary with particle size following Mitchell (1996) against Vm

calculated assuming a constant (a, b). The constant (a, b) are chosenas those values in Mitchell’s (1996) step functions that give Vm closestto that calculated considering the variation of (a, b). The calculatedvalues correspond to qg between 0.5 and 5 g kg21, and to qs between0.02 and 1 g kg21. All (a, b) coefficients listed in Mitchell’s (1996)Table 1 are used in calculations, after conversion using Eq. (7).

FIG. 6. Variation of (a, b) coefficients, in terms of Dmelt, obtainedfrom Locatelli and Hobbs (1974; solid lines), and Mitchell (1996;dashed lines), as function of Dmax. General trend for decrease of aand b with increasing particle size is seen.

amined; for example, Sekhon and Srivastava’s (1970)size distribution has more contributions from large snowthan do the observations presented here from tropicalcyclones, and therefore the use of different (a, b) co-efficients might be appropriate.

At first glance, it might appear that Fig. 5 shows that(a, b) coefficients are unaffected by variation in the sizesof snow or graupel. However, this is not the case. Forexample, the Mitchell (1996) coefficients for the largerparticle sizes and Reynolds number are always associ-ated with smaller (a, b) values than those coefficientsfor smaller particle sizes. Figure 6 shows that, despiteconsiderable scatter, such a general trend also existswhen comparing the (a, b) coefficients for differentshapes and sizes of snow using both the Mitchell (1996)and Locatelli and Hobbs (1974) coefficients. Matrosovand Heymsfield (2000) also demonstrated this and pa-rameterized these relations, and this is also seen fromKhvorostyanov and Curry’s (2002) parameterization.Examination of (a, b) values also shows that aggregatecrystals typically have smaller (a, b) coefficients thando monohabit pristine ice crystals for the habits de-scribed by Locatelli and Hobbs (1974) and Mitchell(1996). Because aggregates are larger than those pristinecrystals, this shows further evidence of the size-depen-dent trends of the (a, b) coefficients. Hence, if differentparts of clouds have differing size distributions or par-ticle sizes, the (a, b) coefficients might be expected to

differ, and a proper choice of (a, b) coefficients shouldbe made to reflect this.

Examination of a similar figure for graupel did notclearly exhibit such trends. Locatelli and Hobbs (1974)list three separate relationships for lump graupel de-pending on density and separate relationships for conicaland hexagonal graupel, and there is no clear dependenceon size; density variations dominate the relationships.This, plus the scatter in Fig. 6, shows that variations in(a, b) with size alone is not sufficient to understand thebehavior; variations due to different densities and shapesmust also be accounted for in parameterization devel-opment for numerical models.

Hence, to develop a new parameterization of Vm, thevariation of (a, b) with mean particle size, with particleshape, with density, and with other known effects mustbe accounted for. In reality, all effects combined meanthat a single (a, b) pair is not adequate for use in anumerical model. Extra information from in situ obser-vations, remote sensing, or other sources might be re-quired to deduce more representative (a, b) coefficients;further, probability distributions may be needed to char-acterize ranges of (a, b) coefficients because informationon the frequency of occurrence of different snow crystalshapes is not well known, and the percentages of dif-ferent shapes varies depending on cloud formationmechanism, geographic location, temperature, and otherfactors (Heymsfield and McFarquhar 2002). It is alsoimportant to know particle size distributions, namely,the intercept and slope parameter used to characterizethe exponential distribution. In the next section, in situobservations obtained in tropical cyclones are used tohelp develop a Vm parameterization that can be imple-mented in numerical models.

3. In situ observations in tropical cyclonesBlack (1990), Black and Hallett (1985, 1999), and

Black et al. (1994) describe the platforms and instru-


FIG. 7. Mass concentration of graupel as function of Dmelt for av-erage size distribution computed from all graupel size distributionswith graupel mass contents between 0.02 and 0.03 g m23, and between0.1 and 0.6 g m23.

ments used to collect in situ observations of particleshapes and sizes in tropical cyclones. During the 20 yrprior to 1992, two-dimensional cloud (2DC) and pre-cipitation (2DP) probes were installed on the NOAA P-3 aircraft, which routinely flew approximately 80-km-long radial legs through hurricanes at or just above themelting layer. In Fig. 1 of Black et al. (1990) an exampleis shown of a typical flight track made through Hurri-cane Emily. Because of inherent difficulties in makingmicrophysical observations in such an environment,data are not available from all flights. In this study,microphysical data collected in two hurricanes [Norbert(1984), Emily (1987)] are used. These cases are selectedfor analysis as they represent some of the highest qualitymicrophysical data collected in the past 20 yr and arereadily available.

Observations made in Hurricane Norbert (1984) andHurricane Emily (1987) also represent two differenttypes of conditions that may be encountered in hurri-canes. Most of the data collected on 24 September 1984in eastern Pacific Hurricane Norbert were obtained instratiform areas outside the eyewall (Black 1990) andthere was little strong convection anywhere in the storm.On the other hand, unusually strong updrafts and down-drafts were documented in the eyewall of HurricaneEmily on 22 September 1987 during its rapidly deep-ening phase (Black et al. 1994). Measurements weresampled at temperatures between 2118 and 18C in Hur-ricane Norbert, and between 278 and 88C in HurricaneEmily. During the flights, the pilots did not maneuverto either encounter or avoid heavy reflectivity regions,and hence the observations should be representative ofconditions present in these particular hurricanes. Fol-lowing Jorgensen et al. (1985), an updraft core is definedas a region where vertical motions were in excess of 1m s21 for at least 4 s, corresponding to a horizontaldistance of approximately 0.5 km. A downdraft core issimilarly defined as regions with vertical velocities lessthan 21 m s21 for 4 consecutive seconds. Stratiformareas are those areas outside these convective updraftand downdraft regions. This is slightly different thanthe definition of Houze (1993), who defined stratiformregions as locations with air motions less than the meanterminal velocity of hydrometeors. The definition dis-crepancy does not affect subsequent analysis.

The microphysical data were collected using ParticleMeasuring Systems (PMS) 2DC and 2DP probes. Blackand Hallett (1986) and Black (1990) describe the analysisprocedures followed. Separate size distributions are gen-erated for graupel, columns, liquid water, and cloud ice,hereafter snow, following an automated technique. The2DC resolution for the storms considered is 50 mm,meaning that it could measure particles as large as 1600mm, and the 2DP resolution is 0.2 mm, meaning that itcould measure particles as large as 6.4 mm. Any pre-sumed ice particle whose outline is close to circular, asdefined using measures of the image perimeter and areaperimeter ratio, is called graupel; actual raindrops are

elliptical because of airflow distortions. Graupel is onlyidentified from the 2DC data because the poor resolutionof the 2DP caused a fairly high false positive rate; ex-amination of mass size distributions showed that graupelparticles larger than 1 mm were very rare.

The sample volumes of the 2DC and 2DP are ap-proximately 10 L s21 and 56 L s21, respectively. The‘‘center-in’’ technique, which accepts particles whoselargest dimension is not on the edge of the array so longas there is nothing else wrong with them, is used toaccept particles. Heymsfield and Parrish (1979) describethis technique and the required adjustments to samplevolume. Because hydrometeor concentrations typicallydecrease with particle size, there might be a concernthat statistically significant numbers of large hydro-meteors would not have been measured, preventingcomputations of mass concentration and reflectivity fac-tor dominated by large particles. Figure 7 shows averagedistributions of mass concentration for graupel with re-spect to Dmelt for all size distributions with graupel masscontents between 0.02 and 0.03 g m23 and between 0.1and 0.6 g m23. There does not appear to be significantnumbers of graupel particles missing from the obser-vations as seen from the size distribution trends in theranges that the probes cover, even for the highest graupelmass contents observed, suggesting that the probe dataare adequate. In addition, flights by the NOAA P-3 inhurricanes over a 10-yr period that used a 2D gray probewith a maximum size of 9.6 mm never encounteredparticles larger than the 6.4-mm maximum size of the2DP probe used here; the sample volume of the 2D grayprobe used for these observations is approximately 50%larger than that of the 2DP probe used for the obser-vations presented here. The turbulent flow inside a hur-ricane prevents the formation of giant fluffy snowflakes,and the updrafts are not strong enough to create hail-stones. A similar plot of reflectivity factor Z showedthat, for mass contents between 0.02 and 0.03 g m23 Zpeaked at 450 mm, and for mass contents between 0.1and 0.6 g m23, Z peaked around 600 mm. This suggests


FIG. 8. Number concentration as function of diameter and as func-tion of time since the start of penetration for a constant altitude flightleg made through Hurricane Emily on 22 Sep 1987 from 2146:21 to2153:10 UTC. Size distributions obtained from composite of 2DCand 2DP data, and include contributions from all frozen and liquidhydrometeors. Solid line superimposed corresponds to vertical ve-locity deduced from aircraft measurements. Ambient temperature var-ied from 24.38 to 5.48C, with an average temperature of 22.08C.

FIG. 9. (top) Graupel (dashed line) and ice (dotted line) watercontent as function of time for time period plotted in Fig. 8. (bottom)Temperature (solid line) also shown.

that the suite of probes are adequate for detecting con-tributions to mass and reflectivity, though a few largeparticles contributing to Z may be missed.

Black (1990) describes a technique for estimating theeffective particle densities from regression equations link-ing radar reflectivity and ice water content from airborneradar and particle image data. Because the probes appearedto measure both mass and reflectivity adequately, this ap-proach should provide reasonable results. Typical densitiesusing this approach ranged from about 0.01 to 0.55 g cm23.The calculated Z values will then match the measured Zvalues because of the choice of effective density. This typeof analysis is available for Hurricane Emily, but not forHurricane Norbert. Hence, estimates of particle mass areprobably within a factor of 2 (Black 1990) for HurricaneEmily, but only within a factor of 3 for Hurricane Norbert.Because snow typically dominates particle masses, and toensure consistency between the two storms, graupel mas-ses are estimated using mass–diameter coefficients de-scribed by Mitchell (1996).

Liquid water content is available for some of the Em-ily cases as measured by a Johnson-Williams (JW) cloudliquid water meter. To obtain estimates of vertical ve-locity, the dynamic pressure was filtered with a 5-s one-sided filter in the forward direction, then backward in

time with the same filter. These data were averaged toget a filtered vertical airspeed, the difference betweenthis and the vertical aircraft speed computed from re-navigated inertial navigation equipment (INE) data, be-ing the vertical wind speed.

Figure 8 shows hydrometeor size distributions ob-tained during one radial leg through Hurricane Emilyplotted as a function of time, with the estimated verticalvelocity superimposed. These measurements were madenear the freezing level. Despite an expectation of highernumbers of large and small hydrometeors in the con-vective zones, if anything, there is a trend to smallernumber concentrations in these zones.

Large variations in the size distributions are seen onvery short time scales in Fig. 8. Similar variations areseen in the other radial penetrations that were plottedin a similar manner (figures not shown). Black and Hal-lett (1999) also reported that strong updrafts containingcloud water and millimeter raindrops, but no graupel,could occur within a few hundred meters of downdraftscontaining millimeter-diameter graupel, raindrops, andcloud water. Figure 9 shows the time variation of graupeland ice water contents as a function of time for the sametime periods depicted in Fig. 8, together with the tem-perature. The plotted ice water field includes contri-


FIG. 10. As in Fig. 8, except for time period between 2015:58 and2021:35 UTC on 22 Sep 1987.

FIG. 11. Images of snow and graupel particles obtained by 2DCduring time period depicted in Fig. 10. Each image bar correspondsto a height of 1600 mm. Substantial variations in types of particlesobserved during the penetration through the hurricanes are noted.

butions from both snow and graupel. Figure 9 showsthat immediately adjacent to the convection, graupelcontributes to a much higher percentage of the ice con-tent than for locations farther into the stratiform region.As Fig. 9 is plotted on a logarithmic scale, the per-centage variation is quite large, suggesting that it isunlikely that any difference in density would affect theconclusion. This may partly explain why larger sizes ofhydrometeors are seen farther away from the convectionin the stratiform region in Fig. 8, namely, because thesnow particles tend to have larger sizes than the moredense graupel particles. Near the convection, the trans-port of warmer air from lower levels is clearly seen;although there are no larger particles seen, Black andHallett (1999) showed that there was substantial liquidwater in such regions as detected by the JW probe.

Figure 10 shows another example of particle size dis-tributions measured during a penetration of HurricaneEmily, with the vertical velocity field superimposed.Here, the trends are somewhat different than those notedin Fig. 8, with greater numbers of larger hydrometeorsnoted near the updrafts. The major reason for these dif-ferences is the temperatures at which the penetrationswere made in the convection; for time periods plottedin Fig. 10, the temperatures are just below freezing.Separation of the hydrometeors into graupel and snow(figure not shown) again showed a greater contributionof graupel near the convective regions and substantialhorizontal fluctuations during the time period of themeasurements. Because the flight legs were flown at

constant altitudes, this bias in the convective measure-ments at warmer temperatures cannot be avoided, andhence must be accounted for in subsequent analysis.

Figure 11 shows examples of hydrometeors imagedby the 2DC for the constant altitude leg depicted in Fig.10; considerable horizontal variability in the images isnoted within the leg. For measurements in stratiformregions (e.g., 2016:13 to 2017:48 UTC), small ice crys-tals, columns, and medium-sized graupel are observed.For updraft regions (e.g., 2018:28, 2019:10, and 2019:19 UTC), small- to medium-sized solid hydrometeorsare seen; and for downdraft regions (e.g., 2018:45, 2019:10, and a weak downdraft at 2020:36 UTC), medium


FIG. 12. Frequency distribution of graupel and snow mass content,derived from 2DC and 2DP data, sorted according to whether inupdraft or downdraft core, defined following Jorgensen et al. (1985).Each count represents 6- to 10-s-averaged data. Median mass contentplotted in upper-left corner of each box.

FIG. 13. Number concentration of graupel plotted as function ofDmelt for average distribution from all observations with Mg between0.08 and 0.1 g m23. The solid line represents the best exponential fitto data obtained by minimizing x2 between observations and fit dis-tribution.

to large graupel, aggregates, and later raindrops are pres-ent. When examining all images, a lot of variability inthe types of particles detected, even on short spatialscales, was detected and no simple relation could ex-plain the types of particles observed.

A single parameterization describing the microphys-ical properties of a tropical cyclone might not be suitablefor both the convective and stratiform regions given thedifferences in microphysical properties between regions.These differences are especially important now giventhat mesoscale models are being run with resolutions assmall as 1 km, a scale comparable to that at whichmicrophysical observations are obtained. Bulk param-eterizations describing averages over much longer flightlegs are no longer appropriate. Because vertical velocityis a strong indicator of convective activity and is a prog-nostic variable in mesoscale models, it is hence logicalto parameterize the microphysical properties dependingupon whether processes are occurring in updrafts, down-drafts, or stratiform regions.

Figure 12 shows histograms of the relative frequencyof occurrence of snow and graupel in the updraft, down-draft, and stratiform regions for all temperatures in Hur-ricanes Norbert and Emily. Graupel occurs in muchhigher masses in the downdraft regions, whereas snowis more common in the stratiform regions that weresampled. The relative absence of both snow and graupelin the updraft regions can be attributed to the warmertemperatures at which these observations were made;as shown by Black and Hallett (1999), liquid waterwould be more prevalent in these regions. Graupelwould be expected to occur in higher concentrations inupdrafts at colder temperatures where there would havebeen more time for graupel to develop. In the followingsection, these tropical cyclone observations are used todevelop a new parameterization of mass-weighted fallspeed that depends upon vertical velocity.

4. Representation of Vm for tropical cyclones

In order to understand how microphysical propertiesaffect the calculation of Vm, it is important to know howthe size distributions of snow and graupel vary in bothconvective and stratiform regions. In this section, it isassumed that the fall speed–diameter relationship canbe described by a single set of (a, b) coefficients thatdo not vary with hydrometeor size; the uncertaintiesassociated with this assumption have already been dis-cussed in section 2 and are comparable in magnitude touncertainties associated with the size distribution. Inaddition, the (a, b) coefficients were derived from ob-servations not associated with tropical cyclones; how-ever, there are no such coefficients available from hur-ricane environments. The calculations performed heremust be interpreted in the context of these uncertainties.

In order to generate fits describing the size distri-butions, average size distributions describing snow andgraupel were first generated by stratifying the data intodifferent mass contents and then determining an averagesize distribution in terms of Dmelt for each mass contentrange. Exponential fits to these average size distribu-tions in terms of Dmelt were then performed. A bootstraptechnique (Efron and Tibshirani 1993), a modified Mon-te Carlo technique whose use is appropriate when notmuch is known about the nature or causes of the mea-surement errors, was used to determine uncertainties onthe fit parameters following the technique of Mc-Farquhar et al. (2002). Figure 13 shows an example ofa fit for a graupel size distribution in a stratiform regionfor graupel mass contents Mg between 0.08 g m23 and0.1 g m23. For each mass content range, a series of suchfits were generated using average size determined fromrandomly drawing, with replacement, samples from thepopulation of individual size distributions having Mg inthe appropriate range; the standard deviation of fits ob-


FIG. 14. Cumulative mass fraction for graupel as function of Dmelt,with different line types representing size distributions measured indowndrafts, updrafts, and stratiform regions of tropical cyclones. Cal-culations based on average size distributions measured for Mg be-tween 0.08 and 0.1 g m23.

FIG. 15. As in Fig. 14, except cumulative mass contents for snowsize distributions, with Ms between 0.1 and 0.2 g m23.

FIG. 16. Number of data points N0g is a function of Mg. Differentsymbols and thick line types correspond to size distributions measuredin updrafts, downdrafts, and stratiform regions. The thin solid linesrepresent the best fit to observations presented by Brown and Swann(1997).

tained for these average distributions defines the un-certainties in fit parameters. The approach of derivingfit parameters for average size distributions in differentmass content ranges is similar to that used by Marshalland Palmer (1948) for generating fits to raindrop sizedistributions, but differs from some recent studies thatfit exponential functions to individual size distributions,averaged over periods of 6 to 10 s (e.g., Brown andSwann 1997; Heymsfield et al. 2002). For the data pre-sented here, the averaging approach is preferable be-cause exponential functions did not always provide ad-equate fits to the individual size distributions, possiblybecause a statistically significant number of particleswere not detected and possibly because another func-tional form would have provided a better fit to the data.These outliers, representing deviations from exponentialbehavior, would have dominated the resulting analysis.

Figure 14 further illustrates the different microphysicalcharacteristics of the stratiform and convective regimes,justifying the need for parameterizations depending onvertical velocity. The cumulative mass fraction for graupelis plotted as a function of Dmelt, with different lines rep-resenting the fractions for updrafts, downdrafts, and strat-iform regions. Figure 15 performs similar analysis for cu-mulative snow mass fractions. Errors in Dmelt are likelyabout 21/3 or 31/3, as determined by the estimated uncer-tanties in particle mass. For graupel, the updraft and down-draft regions possess significantly more large particles thanthe stratiform regions. Examination of similar figures forother Mg showed that this was especially true for high Mg.Note that even though the updraft observations were typ-ically made at warmer temperatures, there is no such tem-perature bias in the downdraft observations, which, if any-thing, were made at cooler temperatures. For snow, theupdraft and downdraft regions seem to possess slightlymore large particles. This trend is not as clear as for the

graupel mass fractions, but is most noticeable for largesnow mass contents Ms.

This behavior is further illustrated when the interceptof the exponential distribution for graupel N0g is plottedas a function of Mg (Fig. 16). Uncertainty estimates forthe N0g data points are calculated from the bootstrap ap-proach, and Mg is accurate within a factor of 2 or 3, asdiscussed above. Since the size distributions were param-eterized in terms of Dmelt, specifying both Mg and N0g issufficient to determine the slope of the exponential dis-tribution lg. A few trends in Fig. 16 are worth noting.First, N0g for both the updraft and downdraft regions arelower than those for the stratiform regions, and this trendis statistically significant within the uncertainty estimates.


FIG. 17. As in Fig. 16, except for N0s as function of Ms. The thicksolid line represents the best fit to average size distributions fromconvective and stratiform regions. The thin solid line gives param-eterization currently used in MM5 based on Sekhon and Srivastava’s(1970) observations; the dashed line corrects this parameterizationfor inconsistencies in diameter definitions.

The uncertainties of N0g for the updraft and downdraftregions are larger mainly because there are fewer datapoints in the sample. Note that N0g is lower for updraftsand downdrafts because, given mass, the greater numbersof large crystals in these regions, corresponds to a smallerlg, which in turn forces smaller N0g to force mass con-servation. The solid line offers a best fit to N0g as a functionof Mg for the stratiform data, as given by

9 8 (0.6460.04)N 5 (2.6 3 10 6 5.0 3 10 )M ,0g g (14)

where standard deviations of fit coefficients are calcu-lated considering the uncertainties in the N0g values. Thedashed line provides a fit to the downdraft data, ex-pressed by

8 8 (0.5560.15)N 5 (9.0 3 10 6 5.3 3 10 )M ,0g g (15)

and the dotted line provides a fit to the updraft data,given by

8 8 (0.4160.10)N 5 (4.6 3 10 6 1.6 3 10 )M .0g g (16)

Comparison is made with analysis of graupel size dis-tributions recorded by Brown and Swann (1997) inweakly convective systems over southern England.There are substantial differences between the curves,especially at high Mg, and the differences are significantconsidering the uncertainties in the fits. There is an in-crease of N0g with Mg in the new parameterization,whereas there is a decrease in N0g with Mg in the Brownand Swann (1997) observations. The decrease of N0g

with Mg roughly corresponds to a situation with a fixednumber of graupel particles; as graupel particles growand Mg increases, N0g must decrease in order to conservenumber. The hurricane observations roughly correspondto a situation where there is not a fixed number of hy-drometeors, and the total number of particles is typicallylarger when mass contents are greater. However, the totalnumber increases at a slower rate than Mg; if the rateof increase were the same, the exponent of 0.64 in Eq.(14) would be 1. This indicates that the average particlesizes are increasing with Mg.

Figure 17 describes the variation of N 0s as a functionof Ms . There is not as clear a trend in how snow dis-tributions vary according to updraft velocity as thereis for graupel; for high and low Ms , the updraft anddowndraft regions possess more large particles (small-er N 0s), whereas the opposite might be true in an av-erage sense for intermediate Ms . A best-fit line of N 0s

to Ms is given by8 7 (0.7410.08)N 5 1.9 3 10 6 2.8 3 10 M .0s s (17)

Equation (17) is calculated from the average size dis-tributions from both convective and stratiform regions;separate curves are not generated because differencesbetween regions are marginal and not as significant sta-tistically. Because snow masses are estimated using re-lations in terms of radar reflectivity that vary betweensamples, the bootstrap technique cannot be easily usedto generate uncertainty estimates; instead, these esti-

mates come from variations in size distributions be-tween stratiform and convective regions.

When comparing against existing parameterizationsof N0s used in mesoscale models, there is a significantdifference compared to the uncertainty estimates, justas noted in Fig. 16 for graupel. The parameterizationshown is based on the observations of Sekhon and Sri-vastava (1970), and different versions that both includeand do not include corrections due to difference in di-ameter definitions are plotted. The differences can againbe attributed to different physical processes occurringin tropical cyclones than those that occur in other stormtypes, and parameterizations of mesoscale processesshould reflect these differences. The Sekhon and Sri-vastava (1970) observations again correspond to a sit-uation where total number is roughly conserved, andhence N0s must decrease as the particles grow by ag-gregation and diffusion increasing Ms. For a snowfallrate of 5 mm h21, 8% of mass and 40% of reflectivityfrom Sekhon and Srivastava’s (1970) observed distri-butions would not be detected by the 2D probes as itis found in particles larger than 6.4 mm. Because N0,and hence l, are higher for the size distributions intropical cyclones given the same mass content, the miss-ing mass and reflectivity would be significantly less forthe observations reported here.

Figure 18 shows the analysis in a slightly differentframework. Here, N0s is plotted as a function of ls, wherethe dots correspond to individual size distributions andthe solid circles correspond to the average size distri-butions that were computed for different Ms. Solid linescorrespond to different mass contents since a parame-


FIG. 18. Intercept N0s as a function of ls, where dots correspondto fit values obtained from individual 6- or 10-s-averaged size dis-tributions. Solid circles represent fits to average size distributionsgenerated for an array of varying mass contents. Solid lines representN0s/ls relationships for given mass content (g m23).

FIG. 19. Velocity Vm for snow determined using average size dis-tributions calculated for different mass content ranges vs Vm for snowcalculated using Reisner et al. (1998) parameterization based on Sek-hon and Srivastava’s (1970) observations. Different symbols repre-sent Vm calculated using in situ observations in updraft, downdraft,and stratiform regions of tropical cyclone. Asterisks compare Vm cal-culated from in situ stratiform observations vs Vm calculated usingthe Reisner et al. (1998) parameterization with corrections for in-consistent diameter definitions.

terization in terms of Dmelt ensures that there are only twoindependent parameters. An increase of N0s with increas-ing Ms is seen by comparing the positions of the soliddots with the positions of the solid lines. Note, that adirect examination of the relation between N0s and ls forthe individual data points would not allow this relation-ship to be seen as clearly. Analysis of N0g as a functionof lg showed similar trends (figure not shown).

Figure 19 compares Vm for snow calculated using theobserved tropical cyclone size distributions against bothexisting parameterization schemes based on Sekhon andSrivastava’s (1970) observations and against those cal-culated using a parameterization that properly accounts fordiscrepancies in diameter definitions and that uses sizedistributions measured in tropical cyclones. The (a, b)coefficients used for the calculations are from Mitchell’s(1996) study for aggregates with Dmax between 800 and4500 mm, or equivalently with Dmelt between 300 and 1000mm. Figure 15 suggests that only 15% of the mass is out-side of this size range. Potential variations in (a, b) co-efficients would also impact Vm as discussed in section 2.

Substantial differences are noted between the old andnew parameterizations. There is a much wider variationin Vm with mass for the midlatitude parameterizationsin use by current mesoscale models than for Vm cal-culated using the tropical cyclone size distributions.This occurs because of a large decrease in total numberwith mass as noted by the Sekhon and Srivastava (1970)observations that does not exist in the tropical cyclonedata. Such differences can have substantial ramificationsfor the conversion rates between various microphysicalspecies. An uncertainty estimate is not included in theplotted values of Vm because it is difficult to ascribe asingle value to this uncertainty. In reality, there are many

sources of uncertainty and to accurately account forthem all, it is probably necessary to use an ensembleor Monte Carlo modeling approach, where random (a,b) coefficients for fall speeds and distribution slopes aredrawn from a population of possible values; the pop-ulation drawn from may depend on the particular con-ditions being simulated.

Figure 20 does a similar comparison for Vm for grau-pel using the tropical cyclone size distribution param-eterizations and using a parameterization deduced usingBrown and Swann’s (1997) observations of graupel. Forhigher mass contents, Vm for graupel using the new pa-rameterization are smaller, whereas for smaller masscontents, Vm for graupel are larger. These differencesmay have important ramifications for mesoscale modelsof tropical cyclones, all of which suffer from a problemof overprediction of graupel at high levels right now.

Finally, an analysis is performed to determine howmass density ri changes according to updraft velocityand mass content. For all times when ri was calculatedfollowing Black (1990), the data were stratified accord-ing to maximum ice water content measured by the 2DCor 2DP probe, and according to updraft velocity. Massdensity ri varies from 0.13 6 0.15, to 0.087 6 0.057,to 0.14 6 0.15 for downdrafts, stratiform regions, andupdrafts, respectively, showing the prevalence of higher-density particles in updrafts and downdrafts. Since (a,b) coefficients depend on particle density, it suggests(a, b) values corresponding to faster falling particlesshould be used for simulations of such regions. Further,


FIG. 20. As in Fig. 19, except comparison of Vm using tropicalcyclone observations of graupel vs Vm derived using Brown andSwann’s (1997) observations of graupel. Different symbols representVm calculated using in situ observations in updraft, downdraft, andstratiform regions of tropical cyclone.

ri varies from 0.056 6 0.047, to 0.10 6 0.09, to 0.146 0.11 for ice contents less than 0.01 g m23, between0.01 and 0.2 g m23, and greater than 0.2 g m23, re-spectively. Since N0 increases at a slower rate than masscontent (Figs. 16 and 17), this shows that larger ri cor-respond to size distributions with larger mean diameters.Because of trends noted in Fig. 6, this again suggeststhat the use of different (a, b) coefficients, correspond-ing to faster falling particles, might be appropriate forthese regions. By examining possible (a, b) values inFig. 3, it may be possible to derive a relationship to usefor simulation of a tropical cyclone, provided that in-formation about hydrometeor density is known a priori.Hence, both the characteristics of the snow and graupelparticles, as well as their size distributions, are importantfor accurate calculations of Vm.

5. Summary

The foundation of microphysical parameterizationsused in current mesoscale models has been examinedto show that information about the sizes, shapes, den-sities, and phases of frozen hydrometeors are neededfor accurate prediction of mass-weighted terminal ve-locity Vm. Although in situ observations do not havesufficient detail to allow determination of shape effectson Vm, they can be used to provide information aboutthe size distributions of hydrometeors in tropical cy-clones. Since most current microphysical parameteri-zation schemes were not designed for simulations oftropical cyclones, a new bulk microphysical parame-terization is designed here to better represent tropicalcyclone conditions. Analysis of data collected in Hur-ricane Norbert (1984) and in Hurricane Emily (1987)

has shown that existing microphysical parameterizationsare not appropriate for representing the fallout of snowand graupel in tropical cyclones. Although the specificparameterization developed here applies to tropical cy-clones, the general approach of ensuring that conversionterms between hydrometeor categories are calculatedusing a consistent definition of diameter can be appliedto any atmospheric phenomena. The principal conclu-sions of this study can be summarized as follows.

1) The particle diameter and particle density that areused to characterize the mass–diameter and mass–velocity relationships must be defined consistentlywith those that are used to characterize the size dis-tributions.

2) To calculate conversion terms between hydrometeorcategories, a diameter based on equivalent projectedarea should be used to define the geometric sweepout of cloud water or cloud ice. Incorporation of thisterm into the microphysical conversions used by Linet al. (1983) and Rutledge and Hobbs (1983, 1984)can significantly affect the transfer of mass betweenhydrometeor categories.

3) Differences in (a, b) coefficients used to describevelocity–diameter relations of snow and graupel rep-resent an equally important uncertainty in the cal-culation of Vm as those uncertainties represented bycharacterizing the size distributions. Past modelingsimulations have used fixed (a, b) values, but thesecoefficients should vary with the density and meansize of snow and graupel particles.

4) In situ observations made in tropical cyclones showthat the composition (i.e., graupel or snow), num-bers, and sizes of particles can vary substantiallydepending on whether observations are collected inconvective or stratiform regions.

5) The intercept and slope of exponential distributionscharacterizing the tropical cyclone observations havebeen parameterized as a function of snow and graupelmass content and vary substantially from similar pa-rameterizations based on observations made outsidetropical cyclones. In particular, the intercept of theexponential distribution was found to increase withsnow or graupel mass, whereas it decreases with snowand graupel mass for current parameterizations.

6) Estimates of Vm obtained with the tropical cycloneobservations differ significantly from those obtainedusing parameterizations that are currently incorpo-rated in mesoscale models.

More sophisticated two-moment microphysicalschemes have recently been developed for use in nu-merical models. In such schemes, both the number con-centration and mass concentration of different classesof hydrometeors, such as snow and graupel, are pre-dicted (e.g., Ferrier 1994; Reisner et al. 1998). The slopeand intercept of the exponential distributions are thendetermined from the prognostic number and mass, al-leviating the need for diagnostic relations for the inter-


cepts, as derived in this study. However, the observa-tions acquired in the tropical cyclones and the resultinganalyses are still crucial for two reasons. First, evalu-ation of the new schemes is sorely needed to evaluatethe prognostic number especially against observations.Second, because of the computational costs in advectingmore variables and parameterizing additional sourcesand sink terms, this two-moment option requires muchmore computer time and is not always used, and a sim-pler alternative is sometimes required. In both one- andtwo-moment schemes, a simple expression for mass-weighted terminal velocity can be obtained by using theconsistent parameterization developed in this paper.

It should also be emphasized that this study was basedon observations acquired in only two tropical cyclonesbecause these data were available and because they rep-resent some of the highest quality microphysical dataobtained by NOAA in the past 20 yr. When a largerdatabase in situ observations becomes available, the ap-plicability of these findings, and especially the temper-ature dependence of these findings, should be examined.Ongoing work is concentrating on the implementationof these parameterizations in the MM5 model and inthe simulation of Hurricane Erin (2001) observed duringthe National Aeronautics and Space Administration(NASA) Convection and Moisture Experiment (CA-MEX) program. A subsequent paper will describe thesensitivity of tropical cyclones to the implementation ofthese new parameterizations.

Acknowledgments. This research was sponsored bythe National Aeronautics and Space AdministrationFourth Convection and Moisture Experiment Grant CA-MEX-0000-0024. These findings do not necessarily re-flect the views of the funding agency. Mike Timlin andHenian Zhang assisted in the preparation of the man-uscript. Greg McFarquhar is grateful for discussionswith Scott Braun that helped to focus the parameteri-zation development aspects of the study. Bob Rauberand Mohan Ramamurthy reviewed a preliminary ver-sion of the paper. The comments of three anonymousreviewers are acknowledged.

APPENDIX A

Corrected Mass Transfer Rates

Equations that describe the transfer of mass betweenhydrometeor categories (cloud water, cloud ice, rain,snow, and graupel) that have been corrected for incon-sistencies in diameter definitions noted in the text. Sym-bols are defined following the studies of Rutledge andHobbs (1983, 1984) and Reisner et al. (1998):

0.25pr Nw 0xl 5 (A1)x 1 2r qa x

b /bx xa G(4 1 3b /b ) prx x x wU 5 (A2)x 3b /b 1 2x x6l 6ax x

(s 1b )/bs s sprwPSACI 5 a r g N q Es a s 0s i si1 26as

G[1 1 3(s 1 b )/b ]s s s3 (A3)[113(s 1b )/b ]s s sls

(s 1b )/bs s sprwPSACW 5 a r g N q Es a s 0s c sc1 26as

G[1 1 3(s 1 b )/b ]s s s3 (A4)[113(s 1b )/b ]s s sls

22pN0sPSMLT 5 K (T 2 T )Z*BT1 (A5)a 0 sLf

4(S 2 1)Ni 0sPSDEP 5 Z*BT1 (A6)sA0 1 B0

4(S 2 1)N0sPMLTEV 5 Z*BT1 (A7)sA9 1 B9

(s 1b )/bg g gprwPGACI 5 a r g N q Eg a g 0g i gi1 26ag

G[1 1 3(s 1 b )/b ]g g g3 (A8)

[113(s 1b )/b ]g g glg

(s 1b )/bg g gprwPGACW 5 a r g N q Eg a g 0g c gc1 26ag

G[1 1 3(s 1 b )/b ]g g g3 (A9)

[113(s 1b )/b ]g g glg

2PGACR 5 p r E N N |U 2 U |w gr 0g 0r g r

2Z G[1 1 3s /(2b )]5 g g g3 1

6 5 [113s /(2b )]5 g gl l l lr g r g

2Z G[1 1 3s /(2b )]g g g1 (A10)

4 [113s /(2b )] 6g g4l lr g

2p(S 2 1)Ni 0gPGDEP 5 Z*BT2 (A11)gA0 1 B0

2p(S 2 1)N0gPMLTGE 5 Z*BT2 (A12)gA9 1 B92PRACS 5 p r E N N |U 2 U |w sr 0s 0r r s

5 2 0.53 1 1 (A13)

6 5 2 4 31 2l l l l l ls r s r s r

2PSACR 5 p r E N N |U 2 U |w sr 0s 0r r s

5 2 0.53 1 1 (A14)

6 5 2 4 31 2l l l l l lr s r s r s

0.5 s /(2b )s s4g prs wZ 5 (A15)s 1 2 1 2p 6as

0.5 s /(2b )g g4g prg wZ 5 (A16)g 1 2 1 2p 6ag


0.5 (b 11)/(2b )s sG[1 1 3s /(2b )] r a pr G[1 1 3(s 1 b 1 1)/(2b )]s s a s w s s sBT1 5 0.65 1 0.44 (A17)[113s /(2b )] [113(s 1b 11)/(2b )]5 1 2 1 2 6s s s s sl m 6a ls s s

0.5 (b 11)/(2b )g gG[1 1 3s /(2b )] r a G[1 1 3(s 1 b 1 1)/(2b )]prg g a g g g gwBT2 5 0.78 1 0.31 . (A18)[113s /(2b )] [113(s 1b 11)/(2b )]5 1 2 1 2 6g g g g gl m 6a lg g g

APPENDIX B

Original Mass Transfer Rates

As in appendix A, except equations have not beencorrected for inconsistencies between diameter defini-tions:

0.25pr Nx 0xl 5 (B1)x 1 2r qa x

a G(4 1 b )x xU 5 (B2)x bx6lx

pr a N G(b 1 3)a s 0s sPSACI 5 q E (B3)i si (b 13)s4 ls

pr a N G(b 1 3)a s 0s sPSACW 5 q E (B4)c sc (b 13)s4 ls

22pN0sPSMLT 5 K (T 2 T )a 0Lf

0.50.65 r a G[(b 1 5)/2]a s s3 1 0.442 (b 15)/25 1 2 6sl m ls s

(B5)

4(S 2 1)Ni 0sPSDEP 5A0 1 B0

0.50.65 r a G[(b 1 5)/2]a s s3 1 0.442 (b 15)/25 1 2 6sl m ls s

(B6)

4(S 2 1)N0sPMLTEV 5A9 1 B9

0.50.65 r a G[(b 1 5)/2]a s s3 1 0.442 (b 15)/25 1 2 6sl m ls s

(B7)

pr a N G(b 1 3)a g 0g gPGACI 5 q E (B8)i gi (b 13)g4 lg

pr a N G(b 1 3)a g 0g gPGACW 5 q E (B9)c gc (b 13)g4 lg

2PGACR 5 p r E N N |U 2 U |w gr 0g 0r g r

5 2 0.53 1 1 (B10)

6 5 2 4 31 2l l l l l lr g r g r g

2p(S 2 1)Ni 0gPGDEP 5A0 1 B0

0.5r a G[(b 1 5)/2]0.78 a g g3 1 0.31

2 (b 15)/25 1 2 6gl m lg g

(B11)

2p(S 2 1)N0gPMLTGE 5A9 1 B9

0.5r a G[b 1 5)/2]0.78 a g g3 1 0.31

2 (b 15)/25 1 2 6gl m lg g

(B12)2PRACS 5 p r E N N |U 2 U |s sr 0s 0r r s

5 2 0.53 1 1 (B13)

6 5 2 4 31 2l l l l l ls r s r s r

2PSACR 5 p r E N N |U 2 U |w sr 0s 0r r s

5 2 0.53 1 1 . (B14)

6 5 2 4 31 2l l l l l lr s r s r s

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