A Microphysical Bulk Formulation Based on Scaling

A Microphysical Bulk Formulation Based on Scaling Normalization of the Particle SizeDistribution. Part I: Description

WANDA SZYRMER

Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

STÉPHANE LAROCHE

Meteorological Research Branch, Meteorological Service of Canada, Dorval, and Department of Atmospheric and Oceanic Sciences,McGill University, Montreal, Quebec, Canada

ISZTAR ZAWADZKI

Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

(Manuscript received 3 December 2004, in final form 10 June 2005)

ABSTRACT

The authors address the problem of optimization of the microphysical information extracted from asimulation system composed of high-resolution numerical models and multiparameter radar data or otheravailable measurements. As a tool in the exploration of this question, a bulk microphysical scheme basedon the general approach of scaling normalization of particle size distribution (PSD) is proposed. Thisapproach does not rely on a particular functional form imposed on the PSD and naturally leads to power-law relationships between the PSD moments providing an accurate and compact PSD representation. Totake into account the possible evolution of the shape/curvature of the distribution, ignored within standardone- and two-moment microphysical schemes, a new three-moment scheme based on the two-momentscaling normalization is proposed. The methodology of the moment retrieval included in the three-momentscheme can also be useful as a retrieval algorithm combining different remote sensing observations. Thedeveloped bulk microphysical scheme presents a unified formulation for microphysical parameterizationusing one, two, or three independent moments, suitable in the context of data assimilation. The effective-ness of the scheme with different combinations of independent moments is evaluated by comparison witha very high resolution spectral model within a 1D framework on representative microphysical processes:rain sedimentation and evaporation.

1. Introduction

Accurate representation of moisture and microphysi-cal processes in numerical weather prediction (NWP)models is a very challenging problem and it is essentialfor good quantitative precipitation forecast (QPF). An-other important requirement for better QPF is accuratespecification of initial conditions for humidity, cloud,and precipitation through data assimilation. The micro-physical calculations describe the evolution of the par-ticle size spectra for different type of hydrometeors. A

good representation of the particle size distribution(PSD) is therefore central to any development of a mi-crophysics scheme. The PSDs also play an importantrole in radar meteorology since they link radar observ-ables such as reflectivity to hydrological quantities suchrainfall rate. Since the role of data assimilation is to usethe information extracted from observations to provideinitial conditions to NWP models, it is very important torepresent the PSD in a consistent manner for the as-similation of reflectivity or precipitation amount.

Two distinct approaches have been used to representthe evolution of PSD. In the first, the distributions ofdifferent water categories are discretized into small bins(e.g., Hall 1980; Kogan 1991). The growth of particlesthen is calculated by solving the quasi-stochastic growthequation. However, the microphysical processes cover

Corresponding author address: Wanda Szyrmer, Dept. of At-mospheric and Oceanic Sciences, McGill University, Montreal,QC H3A 2K6, Canada.E-mail: [email protected]

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a broad range of sizes, from 0.1 �m (nuclei) to severalcentimeters (hailstones), which means that a large, costprohibitive, number of bins must be used. In the secondtype of bulk microphysics scheme, all hydrometeors aredivided into categories and for each a specified continu-ous PSD is assumed (Flatau et al. 1989). The calcula-tions are done globally by integrating over the pre-scribed functional forms of the PSDs. These prognosticequations describe the evolution of the distributionfunctions allowing the modification of only one (in one-moment scheme), two (in two-moment scheme), orthree (in three-moment scheme) parameters of thesefunctional forms (Lin et al. 1983; Walko et al. 1995;Ziegler 1985; Ferrier 1994; Clark 1974). Normally, theintegral quantities are predicted rather than the distri-bution parameters themselves. Alternatively, micro-physical processes can be expressed empirically by theinterrelations between the moments of the PSD, with-out an explicit PSD form (e.g., Zawadzki et al. 1993;Tremblay and Glazer 2000).

The rationale of the bulk approach is that the pre-scribed functional form of the PSD remains validthroughout its different evolutionary stages and theevolution of a few bulk variables is sufficient to de-scribe the time dependence of the adjustable param-eters of the PSD function. The minimum number offree parameters of the assumed PSD should be suffi-cient to track the evolution of PSDs well enough toachieve an acceptable accuracy of the quantities of in-terest. In reality, each process contributes to the shap-ing of the PSD, and their combination may be complexand lead to a large variability of the actual PSD. Ananalytical investigation of the variability of the PSDparameters resulting from the active processes cannotbe done in any complex case. The greater the numbersof free parameters in the analytical PSD, the greater theflexibility of the function to describe actual PSDs. Aminimum of three parameters is essential: intensity (to-tal number concentration), characteristic/scaling size(representative particle size), and width (shape) of thedistribution. Thus, in the absence of any predeterminedinterrelation between these three parameters, a distri-bution of at least three degrees of freedom is requiredto represent the PSD variability.

From the observational point of view, a large numberof papers have been published on the subject of thePSD of rain, named drop size distribution (DSD), sincethe time of the seminal work of Marshall and Palmer(1948). Considerable efforts have been made to param-eterize the DSD using functional forms that dependsonly on a few free parameters. Over the past years, theconcept of scaling normalization has been found veryconvenient to formulate compact representation of the

DSD and to study its variability (Lee et al. 2004). It hasbeen suggested that scaling normalization, using tworeference moments (for obtained functional shape), isable to represent most of the natural variability of theobserved DSD. In this paper we therefore develop amicrophysics scheme based on this concept. This can beseen as merging the recent observational progress onthe DSD with modeling requirements such as develop-ing simplified, though realistic, microphysical schemes.This approach is also convenient for data assimilationpurposes, as shown in a companion paper (Laroche etal. 2005, hereafter LSZ).

Here, we will show that, based on power-law rela-tionships between moments of the PSDs within the nor-malization scaling approach, it is possible to develop ageneral and flexible microphysics formulation in whichthe number and order of predictive moments can bechosen, depending on the microphysical processes andquantities of interest and the observations available.We will only focus on the sedimentation and evapora-tion of rain as test processes for our study. We willevaluate the effectiveness of the parameterization bystudying individual physical processes in a 1D contextrather than within a full-blown 3D model simulation.

In section 2, we develop the relationships betweendifferent bulk variables describing a PSD in the scalingnormalization framework. The accuracy of the one-,two-, and three-moment schemes is studied. In section3, we introduce the basic formulation of microphysicalbulk scheme. Section 4 presents idealized one-dimensional simulations of precipitating rain in a cloud-free layer using the various schemes. Summary andconcluding remarks are presented in section 5.

2. Representation of the PSD

a. Variables related to the PSD

Various specific properties of individual particles,such as mass, fall speed, geometrical cross-sectionalarea, etc., are related to integral properties of the PSD.For a PSD expressed by n(D), if y(D) is the kernelfunction that describes a specific property of an indi-vidual particle as a function of particle diameter D, thecorresponding integral variable is defined as

Y � �0

�

y�D�n�D� dD. �2.1�

For y(D) expressed as a power of D,

y�D� � ayDby, �2.2�

or via a sum of such terms, the bulk quantities of dis-tribution that represent physical quantities, like water

DECEMBER 2005 S Z Y R M E R E T A L . 4207


content or radar reflectivity factor, are directly relatedto the moments defined by

Mp � �0

�

Dpn�D� dD. �2.3�

The number concentration Ntot (i.e., the total numberper unit volume of air) is the zeroth moment, M0. Thetotal volume of the whole population is expressed as(�/6)M3 (with D: diameter of equal-volume sphere).For particles with m(D) � amDbm the water content isequal to amMbm

. The sixth moment gives the radar re-flectivity factor for drops that are small relative to radarwavelength (in the Rayleigh approximation).

Characteristic sizes such as mean-volume diameter orthe effective diameter, used for radiative transfer mod-els, are defined in terms of two moments as

Di,j � �Mj �Mi�1��j�i�. �2.4�

b. Normalization

The PSD normalization, such as the one of Lee et al.(2004), is based on the assumption that the distributionsare self-similar and depend only on the scaling/reference bulk variables. In this case, the relative num-ber concentration of particles for a given size distribu-tion is a function only of particle size normalized byscaling/reference variables. This approach naturallygives rise to the occurrence of power-law relationshipsbetween different integral quantities of the PSD. A ref-erence variable represents the scaling quantity relatingall integral quantities to each other. For a given self-preserving distribution, the actual PSD can be calcu-lated from the reference variables. This concept hasalready been exploited in the field of aerosol for severaldecades (Friedlander and Wang 1966).

A normalization procedure applied to the PSDs ofrain has been proposed in the past (Sekhon and Srivas-tava 1971; Willis 1984). More recently, the normaliza-tion approach based on the concept of scaling has beenintroduced by Sempere-Torres et al. (1994), and in aunified manner by Lee et al. (2004). If the ith momentMi of the distribution is used as the reference (scaling)variable in one-moment normalization, all variables de-rived from the distribution can be expressed in terms ofMi. The PSD function is given by n(D, Mi) � M�i

i g(x),with x � DM� i

i and �i (i 1)i � 1. Here g(x) isindependent of the value of Mi and is called the generic(intrinsic) distribution function. The resulting expressionrelating any pth moment to the reference moment is

Mp � Cp�i �Mi

1�p�i��i, �2.5�

where the constant, C {i}p , is the moment of order p of

g(x): C {i}p � �

0 xpg(x) dx (Sempere-Torres et al. 1994;Lee et al. 2004).

This normalization approach, using one integralquantity, implies the interdependence of all parametersof the distribution. This one-moment (1-M) normaliza-tion is effective only if one universal form of the g(x)applies to all PSDs and the power laws between allmoments are well defined. However, observationstaken in various conditions show that there is quite anappreciable scatter around a mean shape, as shown bySempere-Torres et al. (1994). To capture some of thevariability of the general distribution function, the scal-ing methodology applied to the 1-M, has been recentlyextended to two moments by Testud et al. (2001), andin a more general form by Lee et al. (2004). In the latterwork the general function, g(x), was renormalized byintroducing an additional reference moment Mj. Thisimplies multiple power laws between different integralsof PSD. In the two-moment (2-M) normalization, thegeneral form of the PSD is written as n(D, Mi, Mj) �M(j1)/(j�i)

i M�(i1)/(j�i)j h(x), where Mi and Mj are two

reference moments, x � D(Mi/Mj)1/(j�i), and h(x) is the

generic PSD function. It follows that any moment oforder p can be obtained from

Mp � Cp�i,j �Mi

�j�p��j�i�Mj�p�i��j�i�. �2.6�

As before, C {i,j}p is the moment of order p of h(x) given

by C {i,j}p � �

0 xph(x) dx.The PSD normalized distribution is introduced with

no a priori assumption on specific functional forms andleads naturally to ubiquitous power-law relationshipsbetween moments, and consequently to the analyticalmore general description of the PSD in terms of mo-ments. Thus, it represents a very advantageous methodfor the modeling of bulk properties of the PSD in a bulkmicrophysics scheme for any hydrometeor category. Inthe modeling framework, the reference PSD moments aretaken as the predictive moments, while all other momentscan be calculated using the normalization relationships.

Microphysics modeling is usually based on the pre-diction of the lower moments of the PSD. Thus, accu-rate information about these lower moments is needed.On the other hand, observed larger moments (such asreflectivity) are not very sensitive to the lower mo-ments. When combining modeling with observations,information about both the lower and the higher mo-ments is required to accurately describe the moments ofinterest. Thus, a three-moment (3-M) microphysicsscheme using jointly the low and high reference mo-ments constitutes a more precise description as com-pared to the 2-M scheme.



To illustrate the above concepts Fig. 1a shows a set ofmeasured DSDs. The reader interested in the actualmeasurements used in this figure is referred to Lee andZawadzki (2005). Some of the presented DSDs in thisfigure were highlighted to show: a bimodal DSD (red),a very narrow DSD (green), exponential DSD (ma-genta), a broad gammalike DSD (blue), and a superex-ponential DSD (orange). In fact, these 15 DSDs con-tain all the shapes seen in five years of DSD observa-tions. Figure 1b shows the moment distributions(moments as a function of moment order) for the sameDSDs. As expected, all the fine details are gone in theseintegral quantities. For example, bimodalities cannot beseen unless the two modes are widely separated (to the

point of being better characterized as two types of hy-drometeors). Three parameters are required to de-scribe the variability of the curves in Fig. 1b: two mo-ments to account for the pivoting of the curves, and athird to account for the differences in curvature.

Figure 1c shows the two-moment normalization ofthe observed DSDs, where pivoting is accounted for.The curves represent the distribution of the moments ofh(x), that is, C {2,6}

p . The uncertainty due to the curvatureis given by the spread of the curves around the average.

c. Construction of a three-moment scheme

There are different ways to build a bulk parameter-ization 3-M scheme. Some previous studies (Khairout-

FIG. 1. DSD observations: (a) measured DSDs; (b) distribution of moments (in mmp m�3) of the measuredDSDs; (c) prefactors Cp calculated for the normalized moments, M2 and M6; and (d) prefactors Cp calculated forthe normalized moments, M2, M4, and M6, obtained using (2.9) with xp values found from the optimization in (2.8).


Fig 1 live 4/C


dinov and Kogan 2000) introduced an a priori choice ofa functional form of the PSD (a gamma function) withthree unknown parameters, including one shape pa-rameter. Three moments are sufficient to solve for thethree parameters; however, there are disadvantages inthis procedure. First, once the form of the function ischosen, the accuracy is fixed. This may be too restric-tive. Some forms of the PSDs may not be well repre-sented if the evolution has to be described by only oneshape parameter. Secondly, problems arise from thefact that the shape parameter may be very sensitive touncertainties in the reference moments, and thus im-pact the other calculated moments.

To reduce to a minimum the dependence on theshape of the prescribed function, we generalize (2.6) toa power-law relationship between any retrieved mo-ment of order p and three reference moments of orderi, j, and k. For convenience, we express this as a geo-metric weighted mean of (2.6) for the sets, (i, j) and( j, k) of reference moments as follows:

Mp � �Cp�i,j �Mi

�j�p��j�i�Mj�p�i��j�i��1�xp

� �Cp�j,k�Mj

�k�p��k�j�Mk�p�j��k�j��xp. �2.7�

Now, the terms depending on the shape of the PSD aregrouped into

Cp�i,j,k� � �Cp

�i,j ��1�xp�Cp�j,k��xp, �2.8�

leading to

Mp � Cp�i,j,k�Mi

�1�xp��j�p��j�i�Mj�1�xp��p�i��j�i�xp�k�p��k�j�

� Mkxp�p�j��k�j�. �2.9�

The value of xp and C {i,j,k}p are determined by minimiz-

ing the dependence of (2.8) on the shape parameter(s)of a general form of PSDs. The procedure for the op-timization is described in section 2d.

Figure 1d shows the result for our example of theobserved DSDs. Here, the C {2,4,6}

p s are obtained using(2.9) with xp values found by minimizing the differencebetween lhs and rhs in (2.8). For p � 1, C {2,4,6}

p are veryclose to 1 except for the seventh moment. By introduc-ing the third reference moment the spread (uncer-tainty) has been reduced by a factor of 2.

The power-law relationships between the moment oforder p and the set of the reference moments within 1-,2-, and 3-M schemes, as given in (2.5), (2.6), and (2.9),respectively, can be described in a unified form

Mp � Cp� �

m∈�

Mmemp

�

, �2.10�

where � � {i, j, . . .} is the set of orders of referencemoments. Its cardinality corresponds to the number ofreference moments.

d. Generalized gamma function as generaldistribution form

Hitherto, we have not specified any particular formof the generic functions g(x) or h(x). However, to de-termine the coefficients Cp, we must have informationon the shape of PSDs. One possible approach is to useobservations of the naturally occurring forms from dis-drometer and aircraft observations to adjust the param-eters by a minimization procedure used to constructFig. 1d. Alternatively, we can take a sufficiently gen-eral, analytical function that gives us all the flexibilityneeded to explore the limitations of our approach. Wefollow the later approach by using the generalizedgamma (GG) function to represent the generic func-tions because it shows enough flexibility of shape torepresent a Gaussian-type cloud droplet distribution,exponential PSD in stratiform rain, Gaussian-type PSDresulting from drop sorting in updrafts, etc. Moreover,the GG distribution follows the scaling properties (Leeet al. 2004) and its moments are easily integrable. Fi-nally, the family of GG distributions includes, as specialcases, forms that have been used in other studies: stan-dard gamma, Weibull, Khrgian–Mazin, exponential.

The general formulation of the GG function adoptedhere is f(x) � A(�x)��1 exp��(�x)�], where � and �are two shape parameters and � is a scaling parameter.The normalization factor, A, is determined from theadequate self-consistency requirement. When this GGfunction is used with x � D to describe the PSD interms of diameters, that is, n(D), the term A is obtainedfrom �

0 n(D) dD � Ntot, where Ntot is total concentra-tion. It leads to the following form of PSD:

n�D� � Ntot

�

��D��1 exp��D��, �2.11�

where �(�) is the complete gamma function and � isdirectly related to measurable physical variables. Thegeneral form for the pth moment of (2.11) is

Mp � Ntot��p��

p

��. �2.12�

Considering the GG as the general form of the func-tions g(x) and h(x), the term A is obtained from self-consistency constraints �

0 xig(x) dx � 1 and �0 xih(x) dx

� �0 xjh(x) dx � 1 respectively for 1-M(i) and 2-M(i, j)

schemes, leading to the following expressions:



for 1-M(i) scheme

Cp�i��, �� p��

�� i�� ii�p with

�i � const, �2.13a�

for 2-M(i, j) scheme

Cp�i,j ��, �� p��

�� i�� j��

�� i��i�p��j�i�

.

�2.13b�

In 1-M and 2-M schemes, both GG shape parametershave to be prescribed. In the 3-M scheme, the value ofC {i,j,k}

p (�, �) is given from (2.8) as

Cp�i,j,k��, �� Cp

�i,j ��, ��1�xp · �Cp�j,k��, ��xp. �2.13c�

The constant C {i,j,k}p is determined together with xp by

searching a constant cp and xp corresponding to a mini-mum of

�

��

|cp � Cp�i,j,k��, ��|. �2.14�

After minimizing (2.14), the constant value of C {i,j,k}p

corresponds to cp. The range of the variability of � and� is chosen to cover all shapes of calculated PSD. Thevalues of these parameters � ∈ [0.1, 6.] and � ∈ [1, 2]taken in the computations when minimizing in (2.14)are representative of the variability of PSD of precipi-tating particles.

e. Uncertainties in the retrieved moments

The dependence on the shape of the PSD in (2.10) isexpressed in terms of C�

p . Taking them as constantsimplies that changes of the distribution shape have anegligible impact on the values of the retrieved mo-ments. Here, we examine the errors of the retrievedmoments caused by using fixed values of C�

p in thethree schemes. These uncertainties can be evaluatedfrom (2.10) as �Mp/Mp, where the numerator is thedifference between the moment of the GG and the mo-ment Mp derived with C�

p constant. We limit our analy-sis to errors induced by the variability of the shapeparameters. From (2.10) we obtain

Ep�, �� Mp

Mp�

Cp��, �� Cp

�

Cp��, ��

. �2.15�

The coefficients C�p (�, �) are given by (2.13a), (2.13b),

and (2.13c), respectively for the 1-, 2-, and 3-Mschemes. For the 1-M and 2-M schemes, the values ofC�

p are calculated from (2.13a) and (2.13b) for � � � �1 as it is common for precipitation particles, assumingthe inverse exponential form. For the 3-M scheme, for

each pth moment we assign to C�p the constant value

found by minimizing (2.14).To compare the errors due to the shape variability

from the three schemes, we consider the following setsof predictive moments: {3} and {6} for 1-M [1-M(3) and1-M(6)], {0, 3} and {3, 6} for 2-M [2-M(0, 3) and 2-M(3,6)], and {0, 3, 6} for 3-M [3-M(0, 3, 6)]; C {0,3,6}

p and xp inthe 3-M(0, 3, 6) scheme are given in the appendix inTable A1. Here n-M(i, . . .) denotes the n-moment mi-crophysical scheme with (i, . . .) as the reference mo-ment orders.

We select four different forms of GG shapes de-scribed by four couples of � and � covering observedforms of the PSD of rain, as shown in Fig. 2. For thethree forms different from exponential inverse, the er-rors Ep(�, �) are presented in Fig. 3 as functions of theorder of the retrieved moment. With the 3-M scheme,the error for the three PSDs in all calculated moments,is smaller than 5%, except for the lower moments in thecase of superexponential form. The results of the 2-Mscheme calculations are, in general, satisfied for the mo-ments between the two known moments. However, theerrors increase rapidly for the extrapolated momentswith the distance between the calculated moment andthe closest reference moment (much in the same way asin Fig. 1). Thus, the use of the 1-M scheme with a wellchosen reference moment and related normalizationconstants may be a better choice for some cases.

We now evaluate more generally the accuracy of our3-M scheme only. Figure 4 shows the calculated errorsfor each calculated integer moment as a function of �for � � 1 and � � 2. Calculations are done using thesame three reference moments with the constants fromTable A1. Figure 4 shows that uncertainties of the

FIG. 2. Four forms of the generalized gamma (GG) functionthat represent different naturally occurring PSD.



3-M(0, 3, 6) scheme, for all calculated moments, aresmaller than 5% except for the superexponential form.

The methodology developed for the 3-M scheme, ifapplied to any three measured moments, allows theestimation of all other moments. Thus, the method maybe used as a retrieval algorithm combining differentremote sensing observations as in LSZ. In Fig. 5, we usethe set of independent moments that may be obtainedfrom observational data: second, fourth, and sixth, alsoused in Fig. 1d. The values of C {i,j,j}

p and xp, obtainedfrom (2.14), are given in Table A.1. The errors wouldrepresent the incertitude in the retrieved moments us-ing our 3-M(2, 4, 6) scheme. One can retrieve with agood accuracy moments of order p � 2 without explic-itly specifying the shape parameters, except for the su-perexponential distribution.

3. The microphysical scheme

The tendency of each reference mth moment is cal-culated from a separate predictive equation obtainedby multiplying the conservation equation of the par-

ticle-size number density n(D) � n(D, x, y, z, t) by Dm

and integrating over the entire PSD

dMm

dt� �

0

�

Dm�n�D�

�tdD. �3.1�

Then, the rate of change of Mm due to any microphysi-cal process PRC is related to the evolution of n(D) asfollows:

dMm

dt �PRC� �

0

�

Dm�n�D�

�t �PRC

dD. �3.2�

FIG. 5. As in Fig. 4 except with the 3-M(2, 4, 6) scheme.

FIG. 3. Analytical error as defined in (2.15) in moment estima-tion for the three forms, shown in Fig. 2, as a function of thecalculated moments for different combinations of the referencemoments. The couple (�, �) for each form is specified for eachplot.

FIG. 4. Analytical error as defined in (2.15) for moment esti-mation with the 3-M(0, 3, 6) scheme in function of the GG shapeparameter � for two values of the second shape parameter: (a)� � 1 and (b) � � 2. Curves represent different calculated integralmoments.



The expressions on the rhs of (3.2) are developed foreach process in terms of the PSD moments, integer ornot, and without explicit dependence on the PSD func-tional parameters. The general form is

dMm

dt �PRC� fPRC�Mm, Mp�. �3.3�

The nonreference moments Mp on the rhs are ex-pressed in terms of any set of one, two, or three refer-ence moments Mm using (2.10).

a. Sedimentation

Here, the sedimentation deals only with the relativefall speed between the precipitating hydrometeors andair. Assuming that the PSD varies linearly between twoadjacent levels z and z �z, the tendency of n(D) dueto sedimentation can be expressed as follows:

�n�D�

�t �SED

�1

z�u�D�n�D, z�fz�zz,z. �3.4�

Here fz accounts for the changes of fall speed with airdensity (changing with altitude). The integration of(3.4) with respect to D multiplied by Dm, gives the rateof change of the mth moment due to sedimentation,

dMm

dt �SED�

1z

�MmUm�zz,z, �3.5�

where Um represents the mean fall velocity weightedby Mm,

Um �

� Dmu�D�n�D� dD

Mmfz. �3.6�

The above derivation is similar to Tzivion et al. (1989).Several relationships between raindrop fall speed

and diameter is proposed in the literature: (i) the powerlaw with fixed coefficients (Spilhaus 1948, and others)or with varying coefficients (Khvorostyanov and Curry2002); (ii) the relationship containing the exponentialfunction (e.g., Best 1950); (iii) the polynomial form(e.g., Fowler et al. 1996). The fixed coefficients in (i) areappropriate only for limited interval of particle sizes. Ingeneral, for PSD that is not too narrow, particles ofvery different sizes dominate the sedimentation of dif-ferent moments: for example, the fall speed of the larg-est particles determines the sedimentation of high mo-ments, such as the sixth moment, while the much morenumerous small particles determine the sedimentationof the low moments, such as the zeroth moment. Themain contribution of the particles of various sizes is alsodifferent in different precipitation event. Thus, thepower-law relationship between the individual drop

size and their terminal velocity with fixed exponent,although computationally convenient, may introduceimportant errors. We use the polynomial form (iii) thatmay better cover the velocity variability over the wholeregime of particle sizes. The power-law formulationswith coefficients varying with D and the formulas (ii)agree in the large interval of sizes, but they are notconsistent with our normalization approach based onthe prescribed power-law relations between the mo-ments.

The third-order polynomial expression u(D) � �3k�0

au(k)Dk can be very accurate over the whole spectrum;introducing it into (3.6) it gives

Um � ��k�0

3

au�k��Mmk �Mm��fz. �3.7�

For drizzle/rain drops (between 80- and 6000-�m diam-eter size), the values of the constants au(k) are obtainedby a least squares fit of the Gunn–Kinzer measure-ments over the whole size interval as given in the ap-pendix.

b. Diffusional growth

The tendency of Mm due to vapor diffusion (VDF)has the general form

dMm

dt �VDF� ��

0

�

Dm�

�D �n�D�dD

dt�VDF� dD. �3.8�

This expression can be integrated by parts with limitsset to 0 and � with the assumption that the integrandvanishes at the two limits

dMm

dt �VDF� m�

0

�

Dm�1dD

dt�VDFn�D� dD. �3.9�

For (3.9) used to describe the tendency of moments viaa VDF process of the PSD of drops, the equation forgrowth by diffusion of an individual drop is given by

dD

dt �VDF� �wGwD�1fven, �3.10�

where �w � qv � qsw with qv and qsw as water vapor andsaturation water vapor mixing ratios, Gw is a thermo-dynamics function of temperature and pressure, andfven describes the ventilation effect that may be ne-glected for smaller particles (fven � 1).

The ventilation coefficient, fven, expressed in terms ofthe Reynolds number Re and Schmidt number Sc, fol-lowing Beard and Pruppacher (1971) leads to the ap-proximate expression

fven � aven bven,thD�1bu��2, �3.11�



obtained using u(D) � auDbu; bven,th is expressed interms of au, Sc, air density, and viscosity.

Introducing (3.10) and (3.11) into (3.9) and taking Gw

and �w constant within one time step, the tendency ofthe moment mth of the drop size distribution due to thevapor diffusion growth is

dMm

dt �VDF� m�wGw�avenMm�2 bven,thMm�bu � 3��2�.

�3.12�

For nonprecipitating hydrometeors with fven � 1, aven �1, and bven,th � 0 must be taken.

If the vapor diffusion process described above is anevaporation/sublimation process during which thesmallest particles completely transfer to vapor cat-egory, an additional term has to be added to the rhs of(3.9) describing the removal of these particles. Belowwe give a description of calculations done for completeraindrop evaporation. A very similar approach can bealso applied for complete sublimation of crystal par-ticles.

c. Rate of change raindrops concentration duringevaporation

Assuming that for complete evaporation of dropswithin �t, the effects of ventilation may be ignored and(3.10) gives

�D*

0

D dD � �t

tt

�wGw dt. �3.13�

From this, the diameter of the largest raindrop that justevaporated completely after �t, D*, can be easily cal-culated for given thermodynamic conditions. The rateof change of the zeroth moment due to the completeevaporation VDFr may be evaluated from

dM0

dt �VDFr� �

1t �0

D*

n�D� dD

�M0

t �M0�D*, ��

M0� 1�, �3.14�

where M0(D*, �) describes the incomplete momenttaken between D* and �. Using the GG distribution(2.11), the ratio M0(D*, �)/M0 � �(�, X*)/�(�) is aregularized incomplete GG function expressed in termsof X* � (�D*)� and �. This function is replaced by anapproximate function of X* and �, found by fitting

��, X*�� 1 � ��1 �X*

�

X�1 exp��X� dX

� 1 � Ce1�Ce2X*�Ce3, �3.15�

where X � (�D)�. The constants Ce# are given in theappendix where also the accuracy of the fitting is shownin the appendix in Fig. A1 as a function of � for somevalues of X*. Equation (3.14) combined with (3.15)gives

dM0

dt �VDFr�

M0

tCe1�Ce2X*�Ce3. �3.16�

To express � and X* in terms of the distribution mo-ments, we fit the actual distribution in the small dropregime by a standard gamma distribution assuming � �1. Then, using (2.12), we have � � (3 � 2M2M4/M2

3)/(M2M4/M2

3 � 1) and � � (� 2)M2/M3. Here, we willassume that the elimination of the completely evapo-rated drops enters as a loss term only in the tendency ofthe zeroth moment; the changes of the moments oflarger orders, due to small value of D*, are consideredhere as negligible.

4. Experiments

In this section, we assess the accuracy of the results ofthe calculations done using one-, two-, and three-reference moments. This analysis is a first step in theinvestigation of the optimal microphysical calculationsfor data assimilation. The 1D model simulations de-scribe the evolution of rain in the subsaturated layer.Figure 6 shows an example of evaporating rain formingthe trails as viewed on the time–height display of thevertically pointing X-band radar. Since the intent hereis to isolate, if possible, the bulk parameterization forindividual processes in combination with different ob-servable or calculated fields, first, the sedimentationprocess involving gravitational sorting within the trailsis considered in idealized conditions. Then, a simulationof rain evaporation in steady-state conditions is pre-sented.

a. Brief description of the 1D model calculations

The simulations are performed in an idealized one-dimensional framework using a simple time-dependentcolumn model. The vertical grid spacing and time stepare 25 m and 2.5 s, respectively. The model lid, wherethe precipitation is released, is located at 2 km abovethe ground. In the simulations presented here, no airmotion is assumed. The feedback of the microphysicson the dynamics and thermodynamics is also not in-cluded. The reader may find more details about themodel discretization in LSZ.

In our formulation, an explicit expression for g(x)and h(x) are not needed; only the moments of thesefunctions are required. However, for a better compari-son to the majority of parameterizations with imposed



distribution shape in the 1-M and 2-M schemes, we willassume that the PSD follows an inverse exponentialdistribution. Using this functional form, the shape de-pendent coefficients C�

p are calculated from (2.13a) and(2.13b) assuming � � � � 1. The two constants i and�i in the 1-M scheme are determined by imposing afixed intercept parameter N0 of the assumed exponen-tial distribution: i � (i 1)�1 and �i � [�(1 i)N0]i.In the 1-M scheme, the tested reference moments areM3 and M6. The 1-M(3) scheme, with � � � � 1 and N0

� 107 m�4, gives the same interrelations between themoments as in the Kessler (1969) scheme. Two pairs ofmoments are tested in the 2-M scheme: 2-M(0, 3) and2-M(3, 6). The first set represents the moments com-monly described in literature within two-moment mi-crophysical schemes. The predictive moments in the3-M scheme tested here are (0, 3, 6).

The accuracy of the bulk calculations is evaluated bycomparison to very high resolution spectral (bin) cal-culations. In the spectral model, the input drop popu-lation at the nth step is described by M0(ztop, n�t) ��nbins

i�1 n(Di, ztop, n�t)�D, where the height of the modeltop level is ztop. At any kth level of the model at theheight zk, the distribution is described by n(Di, zk, n�t).All initial moments at z � ztop are the same as in thebulk calculations. The spectral calculations are done for3000 bins and with the spatial and temporal resolution25 times greater than in the bulk model, that is, �z� ��z/25 � 1 m and �t� � �t/25 � 0.1 s.

b. Sedimentation process

To examine the situation where differential fall speedis the only process in changing raindrop spectra, wesimulate one-dimensional rainshafts in conditions ofsaturation with respect to water. The temporal evolu-

tion of the prescribed radar reflectivity near the lid issinusoidal in dBZ, simulating the passage of precipita-tion cells. The reflectivity varies between 0 and 50 dBZ.All reference moments in different schemes that haveto be initialized are calculated from (2.5) with the aid of(2.13a) for i � 6 and with fixed N0 � 0.8 � 107 m�4

from Marshall–Palmer distribution. Calculated by thebin-resolving model, vertical profiles of the reflectivityfactor Z � 10 log(constM6), liquid water content LWC� (�/6)�liqM3, and the number concentration Ntot �M0, are presented as a function of time in Fig. 7. Theprofiles of reflectivity and mean-volume diameter [D03

� (M3/M0)1/3], representing the higher and lower dis-tribution moments respectively, calculated by differentschemes at t � 7 and 14 min, are compared in Fig. 8.The calculations of the tendency of each reference/prognostic moment in different 1-, 2-, and 3-M schemesare done according to (3.5): first, the mean fall velocityweighted by the moment is calculated using (3.7), andthen the rate of its change due to sedimentation is cal-culated from the difference of the moment fluxes at twoadjacent levels, as in (3.5).

As shown in Fig. 8, the results of 3-M(0, 3, 6) are ingeneral similar to the spectral calculations. The twoschemes with M6 as a reference moment, 1-M(6) and2-M(3, 6), calculate the profile of Z with a very goodaccuracy as well. This indicates that the exponentialform of the DSD, introduced at the top of the model, ismaintained for the very large drops because of thesmall spread in the fall speeds of these drops (the slightdependence fall speed diameter). The D03 values aresystematically underestimated, except for 3-M(0, 3, 6).This is due to the lower moments being diagnosed fromthe larger moments, assuming the inverse exponentialform. With gravitational sorting, the actual distribu-

FIG. 6. Time–height section of reflectivity as viewed by the vertically pointing X-band radar during 20 min on 19 Aug 2002.


Fig 6 live 4/C


tions are more narrow and no longer exponential in thesmaller drops regime. In 1-M(3) and 2-M(0, 3), the re-flectivity is generally overestimated; the falling precipi-tation delay seen in the reflectivity profiles is due to thefall speed difference between the M6 and M3, the laterbeing slower.

From these results, we can conclude that, if the sedi-mentation is the dominating process, the larger ordermoments (like M6) can be rather well simulated withinthe 1-M scheme since their dependence on the lower

moments is weak and the slope of the distribution forthe larger sizes does not change significantly. In gen-eral, the 1-M and 2-M schemes cannot effectively simu-late the large and low moments at the same time. In thesimulated conditions the 3-M(0, 3, 6) scheme providesthe best results.

c. Steady-state rain evaporation

Here we present the results of simulations of rain-drop evaporation in a prescribed steady-state environ-

FIG. 7. Vertical profiles, calculated by the bin-resolving model, as a function of time, forthe three quantities: reflectivity factor, Z; liquid water content, LWC; and number concen-tration, Ntot.


Fig 7 live 4/C


ment with a pseudoadiabatic lapse rate of temperature.The vapor distribution is specified by prescribing a con-stant relative humidity. Conditions approaching a well-mixed subcloud layer with a constant dry-adiabaticlapse rate of temperature, and with a constant watervapor mixing ratio, were tested giving similar results(not shown).

We present here the results obtained assuming acloud-free layer with a constant relative humidity of50% and a near 25°C pseudoadiabatic temperature pro-file. The DSD imposed at the top level is assumed Mar-

shall–Palmer with reflectivity of 40 dBZ. The other ref-erence moments for each scheme are calculated fromthe reflectivity in the same manner as in the previoussection for the sedimentation experiment. The rate ofchange of any reference/prognostic moment in all theschemes is calculated from (3.12). The complete evapo-ration of drops, described by (3.16), is calculated onlyfor the zeroth moment. The calculated steady-state ver-tical profiles of reflectivity, precipitation rate, R � (�/6)M3U3, rate of temperature change due to evaporation(�dM3/dt |VDF), and mean-volume diameter, D03, are

FIG. 8. Vertical profiles of reflectivity and mean-volume diameter, calculated by different schemes with the sameinitialization as in Fig. 7. The presented profiles are at t � 7 and 14 min (indicated by two dashed lines in Fig. 7).



shown in Fig. 9. The 2-M(0, 3) calculations do not in-clude the removal of drops completely evaporated.

As shown in Fig. 9, the reduction in Z caused byevaporation is here relatively small. The larger thedrops, the smaller the size reduction by evaporation.Thus the DSD tail at the largest drop size remains al-most unchanged and keeps its initially imposed inverseexponential form, as shown in Li and Srivastava (2001).This explains why the 1-M(6) and 2-M(3, 6) schemes,using the larger-order moments as prognostic moments,are very accurate for predicting reflectivity. When Z isretrieved from the lower-order moments as in the1-M(3) or in the 2-M(0, 3), assuming exponential shape,the errors are very large over the entire profile. For theprecipitation rate and evaporative cooling, the accuracyof the 2-M(3, 6) scheme is very good; the error isgreater for the simulated D03; however, the calculatedD03 at the ground is larger than at the cloud base, aspredicted by the bin-resolving model. Nevertheless, in2-M(3, 6) and in other schemes, with M0 conservedduring evaporation (dM0/dt|VDF � 0) or diagnosed, sys-tematically greater evaporative cooling and smaller D03

are obtained. These errors may be very important. Thereduction of M0, due to complete evaporation in the2-M(0, 3) scheme (not shown), does not give satisfac-tory results since the removal is very sensitive to theshape of the DSD; even the 3-M(0, 3, 6) scheme cannotdescribe accurately this process as shown in D03 profile.

Usually, the only available information is the radarreflectivity, thus M6 of the DSD is the only known mo-ment. Figure 10 compares the bin-resolving model re-sults, for the same initial reflectivity and exponentialDSDs, for three values of the intercept parameter, N0:0.4 � 107, 0.8 � 107, 1.6 � 107 m�4. As shown, thedifferences are in the calculated initial values of quan-tities other than Z. In the layer just below the cloud, wecan see that the uncertainties in the precipitation rateand in the heating rate are very important and arelarger than the errors shown in Fig. 9. Lower in thesubsaturated layer, in Fig. 10, the profiles of precipita-tion rate and latent heat of evaporation converge for allN0. Conversely, the uncertainty of the results with re-duced number of the reference moments increases to-ward the ground, as shown in Fig. 9.

FIG. 9. The vertical profiles at steady state of radar reflectivity, precipitation rate, evaporative cooling, and the mean-volume diameter calculated by different variants of the bulk microphysical scheme. The solution obtained from thespectral calculations is also shown. The DSD, introduced at the top level, is exponential inverse with the interceptparameter N0 � 0.8 � 107 m�4.



The sensitivity of the lower moment calculations tothe initial distribution parameters is very large. Theuncertainties in N0, dependent on the middle and smallsizes regions of the DSD, would lead to very large er-rors in the profiles shown in Fig. 9. The differences arein the calculated initial values of R and D03 and in theirsubsequent evolution since the evaporative effects aremore important when the concentration of smallerdrops is larger (i.e., larger N0). Thus, the importance ofreducing the incertitude on the initial spectra for thecalculations for the subcloud region is evident. The dif-ference in N0, for the input exponential distributionsshown in Fig. 10, is equivalent to different distributionslopes, which may be provided via an additional infor-mation, as the reflectivity-weighted fall speed, U6, orvia data assimilation as shown in LSZ. Assuming thatthe terminal velocity of raindrops is described by apower law with the exponent on D of 0.67, it can easilybe shown that, for a given reflectivity, U6 � N�1/10

0 .In the experiments presented here, the 2-M(3, 6)

scheme provides relatively accurate profiles of all quan-tities and constitutes an interesting alternative for mod-eling a subcloud evaporating rain. For example, if theavailable observations are limited to the larger mo-ments, the extra calculations of the zeroth moment

might not be appropriate due to their uncertainties.The optimal choice may be the scheme 2-M(3, 6). How-ever, the situation may be different for weak rain rates(not shown) where the effects of evaporation becomerapidly important for the middle part of the spectrum.Moreover, the contribution of the self-collection andbreakup processes, neglected here, has to be taken intoaccount in the regions of high rates.

5. Summary and final remarks

The bulk quantities of hydrometeor PSD, providedby microphysical calculations, are important for the in-terpretation of remote sensing observations, and withinnumerical modeling; both aspects are inherent in dataassimilation. With Doppler radar, polarization and mul-tiwavelength techniques, and other remote sensingmethods, the number of potential microphysical vari-ables that may be inferred and used to maximize agree-ment between model solution and observations has be-come much larger. This is even more pertinent giventhe possibility to resolve fine scales in high-resolutionmodels. As a tool in the exploration of the question ofthe optimal microphysical calculation in the context ofdata assimilation into numerical models, we have de-

FIG. 10. As in Fig. 9 except the presented results are provided by the bin-resolving model for three different values ofthe intercept parameter at the model top level.



veloped a versatile bulk microphysical scheme havingthe following main features:

(i) The predictive variables of the scheme are thePSD moments; the evolution of the PSDs is de-scribed in moment space.

(ii) The number of reference moments may vary fromone to three (extension to a greater number is alsopossible).

(iii) The orders of reference moments may be chosenaccording to the availability of observations, dom-inant microphysical processes, quantities of inter-est, etc.

(iv) The conservation equation for each reference mo-ment is expressed in terms of other PSD mo-ments; each moment is given by a power-law re-lationship with the independent moments.

(v) The power-law relationships are obtained in thegeneral framework of the scaling normalizationused to represent the PSD of hydrometeor popu-lations.

(vi) The relationships between the moments havebeen derived using the generalized gamma func-tion and tested on a large database of observeddrop size distribution; alternatively, the relation-ships between moments derived from data alonecan also be used.

(vii) The scheme is suitable for variational data assimi-lation; the associated tangent-linear and adjointcodes have been constructed and tested in LSZ.

Simulations including other microphysical processes forrain and more hydrometeor categories are the subject ofongoing work and will be the object of a separate paper.

Acknowledgments. The authors wish to thank EduartHaruni for the calculations of the moments for the ob-served DSDs, and Edwin Campos for providing thepolynomial relation fall speed diameter for rain. We arevery grateful to Jennifer Lilly and Dr. Catherine Hey-raud for reading and correcting this manuscript and toProf. Frederic Fabry and Dr. GyuWon Lee for theirhelpful advice and comments during the course of thiswork. The authors would like to acknowledge theanonymous reviewers for their comments and sugges-tions. This research was supported by the CanadianFoundation for Climate and Atmospheric Sciences.

APPENDIX

Constant Coefficients Used in theMicrophysical Schemes

Constants in the polynomial expression for raindropfall speed used in (3.7) from E. Campos (2004, personalcommunication):

au�0� � �0.192 804,

au�1� � 4.962 56 � 103,

au�2� � �0.904 414 � 106,

au�3� � 0.056 581 7 � 109.

Constants in the approximate expression for a regular-ized incomplete GG function given in (3.15):

Ce1 � �1.35,Ce2 � 0.61,Ce3 � 0.96.

TABLE A1. Coefficients used in the 3-M schemes.

Order, p

0 1 2 3 4 5 6 7

3-M(0, 3, 6)Coefficient, xp — 0.365 0.580 — 0.835 0.925 — 1.06Prefactor, Cp — 1.11 1.07 — 0.96 0.96 — 1.10

3-M(2, 4, 6)Coefficient, xp �1.02 �0.450 — 0.335 — 0.820 — 1.16Prefactor, Cp 0.73 0.92 — 1.01 — 0.99 — 1.04

3-M(2, 3, 6)Coefficient, xp �1.03 �0.455 — — 0.605 0.820 — 1.16Prefactor, Cp 0.78 0.94 — — 0.99 0.98 — 1.05

FIG. A1. The plain line represents [�(�, X*)/�(�)] � 1, where�(�, X*)/�(�) is the regularized incomplete gamma function as afunction of shape parameter, �. The dashed lines are the approxi-mate expressions obtained using the fitting given in (3.15). Thepresented curves correspond to the values of X*, increasing fromleft to right: 0.0005, 0.005, 0.05, 0.1, 0.25. For � � 0.4, the value at� � 0.4 is assumed.



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A Microphysical Bulk Formulation Based on Scaling