Comparison of cloud microphysics parameterizations for simulation of mesoscale clouds and precipitation

Atmospheric Environment Vol. 26A~No. 15, pp. 2699 2712, 1992.

0004-6981/92 $5.00 + 0.00 © 1992 Pergamon Press Ltd

Printed in Great Britain.

COMPARISON OF CLOUD MICROPHYSICS PARAMETERIZATIONS FOR SIMULATION OF MESOSCALE CLOUDS AND PRECIPITATION IN YOUNG LEE Environmental Research Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. (First received 1 February 1991 and in final form 21 September 1991)

Cloud physics parameterizations are comprehensively reviewed in order to examine their theoretical bases and to evaluate their applicability to mesoscale modeling. The constant distribution functions of hydrometeors assumed in previous work are found to be unrealistic. To circumvent this problem, new parameterizations are developed for mixing ratios of water vapor, cloud water, rain water, cloud ice, graupel and snowflakes. Equations are derived from theoretical considerations and multiple regression applied to results of simulations with a cloud model that incorporates sophisticated treatments of microphysicsand dynamics.In a Lagrangian air parcel model under variable atmospheric conditions, the new parameterizations give consistent results and are computationally more efficient than previously available parameterizations. Abstract

Key word index: Cloud microphysics, parameterizations, cloud water, rain water, cloud ice, graupel,

snowflakes. 1. INTRODUCTION The microphysics of clouds and precipitation, in conjunction with macroscale storm dynamics, are of fundamental importance in determining the distributions of temperature, moisture, aerosols and chemical species. The evolution of water substances plays a basic role in determining how trace substances are taken up by cloud water, chemically transformed and accumulated in precipitation. Computationally efficient parameterizations are highly desirable for use in regional models whose grid cells characteristically range from 80 to 150 km. A flow diagram of the microphysical processes associated with development of clouds and liquid or solid precipitation is shown in Fig. 1. The total moisture mixing ratio accounts for water vapor, cloud droplets, raindrops, ice crystals, graupel and snowflakes. Strong reciprocal relationships exist between the various microphysical interactions. For example, cloud droplets formed by condensation of water vapor can coagulate to become raindrops. Cloud droplets and raindrops may freeze if they are transported into a subfreezing region of the cloud. Ice particles thus produced coagulate or grow through riming with liquid drops to produce snowflakes or graupel. Thus, macroscale parameters such as precipitation rates (rain or snow), storm efficiency and scavenging efficiency depend strongly on microphysical interactions. Parameterization of cloud microphysical processes, initiated by Kessler (1969), has since been explored by many investigators, including Ogura and Takahashi (1971), Berry and Reinhardt (1974b), Yau and Austin (1979), Tripoli and Cotton (1980), Lin et al. (t983), Nickerson et al. (1986), Ziegler (1988) and Lee (1989). In general, the parameterizations have been derived

...-.I

WATER

I\ooo..

CLOUD ~ . _ _ . . ~ ' ~ b r . , , , u o , . ~ . . ~ . _

WATER

"accreti .....

le sc ~ ) n c e ' ~ " ~ _

I

RAIN

~

1 c "=

I~.,io. ~

I I "'°0-/-/¢-I SNOW II "///.I

" x ~ ' v J' ' l /

.

.

.

.

.

I

.

/

Fig. 1. Schematic diagram of microphysical interactions among hydrometeors. by assuming that hydrometeors are distributed according to a simple density function and that bulk terminal velocities of rain water can be expressed as functions of their respective mixing ratios. However, formulas based on these assumptions may lead to an improper vertical distribution of hydrometeors and unrealistic dynamic fields within the life cycle of a simulated cloud system. Numerical studies of Lee et al. (1980) showed that variations in the chemical and physical properties of aerosols associated with particular air mass types can produce clouds with quite different drop-size distributions. The knowledge of these distributions is important for the initiation of

2699

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I.Y. LEE

rain by collision and coalescence, since cloud properties can vary significantly depending on air mass types in which they form. The objectives here are (1) to review cloud physics parameterizations comprehensively in order to examine their theoretical bases and evaluate their applicability to regional models and (2) to obtain more efficient, improved parameterizations by analysis of results of a detailed cloud model that incorporates treatments of microphysical processes and cloud dynamics too detailed for routine use in regional models. An overview of our approach is presented in Fig. 2. Although a comparison of the behavior of different parameterizations with observations is desirable, the microphysical parameterizations are calculated here through model simulations because appropriate experimental data are very limited. Multiple regressions are used to derive the numerical values of coefficients and exponents in the new parameterization formulas.

2. TECHNICALAPPROACH Our approach consists of three steps: (1) evaluation of initial-phase interaction schemes based on formulas

currently available, (2) derivation of formulas from data generated by detailed numerical simulations of clouds in a variety of atmospheric conditions and (3) comparison studies and sensitivity analyses. We have evaluated the cloud microphysics parameterizations currently available (Asai, 1965; Byers, 1965; Thorpe and Mason, 1966; Asai and Kasahara, 1967; Berry, 1968; Kessler, 1969; Mason, 1971; Koenig, 1971; Ogura and Takahashi, 1971; Cotton, 1972; Wisner et al., 1972; Berry and Reinhardt, 1974b; Orville and Kopp, 1977; Stephens, 1979; Hsie et al., 1980; Hales and Easter, 1982; Scott, 1982; Lin et al., 1983; Rutledge and Hobbs, 1983) for their conformation to the theoretical bases of cloud and precipitation physics and their applicability to mesoscale models. We found that the parameterization formulas must be applied with caution because the spectral evolution of both liquid and solid particles may depart significantly from an assumed spectrum such as the Marshall-Palmer distribution (Silverman and Glass, 1973; Lee, 1990). The literature can be summarized as follows: (1) Rates have been derived under the assumption that the particle size distribution functions for various types of hydrometeors are given;

Review and compile existing parameterization equations; analyze shcmcomings

Execute detailed storm model repeatedly for a wide range of conditions; record associated rates of individual cloud physics processes

Formulate advanced parameterization equations

Optimi~ ti~ free parameters of the advanced parameterizations

to maxin'D.eagreement with rates determined by detailed model

Compare predictions of old and new parameterization equations to measta~ performance improvement Fig. 2. Overview of the cloud microphysicai parameterizations.

Cloud microphysics parameterizations (2) Rates for condensation or evaporation of cloud water have been computed successfully for a quasi-saturated state; (3) Rates for autoconversion of cloud water have been expressed as functions of cloud-water content and spectral properties that are not readily available in regional models; (4) Rates for accretion of cloud water, cloud ice, snow or graupel have been expressed as functions of liquid or solid mixing ratios and collection efficiency, which varies from one formula to another. Accretion rates have been based on the concept of geometric sweep-out and on the assumption that the particles exhibit Marshall-Palmer distributions. Uncertainties exist because the collection efficiencies between various types of hydrometeors are much smaller than unity and because the specified size distributions may be significantly in error for some stages of cloud development; (5) Rates for evaporation of rain water have been expressed as functions of various dynamic and thermodynamic variables; (6) Rates for deposition or sublimation have been derived by integrating the growth rate, which should be applied only to a specified ice-crystal form, over all size ranges for an assumed icecrystal spectrum; (7) Rates for autoconversion, either from cloud ice to snow or from snow to graupel, have been formulated in a manner similar to that for autoconversion of cloud water to rain water; (8) Both freezing and melting rates have been expressed in simple forms in terms of macroscale parameters. Uncertainties arise from the specification of the spectral evolution and from inadequate knowledge of the behavior of ice crystals; (9) Rates for three-phase processes are formulated in terms of cloud-water mixing ratio, temperature and many hypothetical spectral properties of ice crystals. Our aim is to simulate microphysical processes in detail by using a cloud model that computes the spectral evolution of various types of hydrometeors while maintaining the main features of cumulus dynamics (Lee, 1990). For this purpose, we found a onedimensional, time-dependent cumulus model to be feasible. The description of adiabatic processes is similar to the one proposed by Asai and Kasahara (1967) except for the net buoyancy term. For computations of net buoyancy, we added the effect of drag by liquid or solid particles. As the flow diagram shows (Fig. 1), the total moisture mixing ratio accounts for water vapor, cloud droplets, raindrops, ice crystals, graupel and snowflakes. The material densities for cloud droplets and raindrops, ice crystals and graupel, and snowflakes are assumed to be 1.0, 0.9, and 0.5 g c m - 3, respectively. Size categories for hydrometeors and corresponding fall velocities computed at the 1000-rob level

2701

are presentea in Table 1. Cloud droplet and raindrop distributions are described by 15 radius ranges (bins) ranging from 1.0 to 32.0 pm and by 25 larger bins up to 8.2 ram, respectively. In practice, raindrops grow no larger than about 3.0 mm because of drop breakup. Ice crystals and graupel are described by 21 bins ranging from 1.0 to 101.6/zm and by 19 larger bins up to 8.2 mm, respectively. Graupel can grow without being broken up. Lastly, the snowflakes are categorized by 40 bins ranging from 1.0 to 8.2 ram. Berry and Reinhardt (1974a) studied the spectral evolution resulting from the stochastic collection process and found a 50-#m radius to be an appropriate dividing point between cloud droplets and raindrops. Our subdivisions for both liquid and solid particles are determined by numerical analysis with a Lagrangian cloud model, which computes the spectral evolution by considering the microphysical processes of coagulation among different ice and graupel particles and snowflakes. Our analyses show that the minimum volume falls between two modes, one controlled by cloud droplets and another by raindrops, and lies between radii of 25 and 40 #m. In contrast, the subdivision between cloud ice and graupel lies between radii of 80 and 128 #m. The governing equation for the diffusional growth rate of a drop is similar to that of Fitzgerald (1974), except for the treatment of the solute effect (Lee, 1990). The hygroscopicity of aerosols is determined from a formula proposed by Hanel (1976) under the assumption that the chemical composition of the deposit is uniform. The growth rate of an ice particle by deposition of water vapor is computed with the assumption that the capacitance C = r for ice crystals and graupel particles and C = 2 r / n for snowflakes. The time evolution of the particle-number density as a result of condensation-evaporation and of deposition-sublimation is governed by the spray equation (Arnason and Greenfield, 1972). The time evolution of the hydrometer size distribution by collection is complicated by the material-density differences among hydrometeors. The calculations of volumetric changes during collectional processes involving particles with different densities require careful analysis, because recomposition must proceed while the total mass is conserved. The collision efficiency E is determined from a formula proposed by Neiburger et al. (1974) that gives a close fit over most of the range. The terminal velocities for liquid and solid particles are computed as proposed by Beard (1974). The terminal velocity increases with altitude as well as with particle size. The terminal velocities for cloud ice, graupel and snowflakes are smaller than those for drops with identical size because of the smaller material density for solid particles and morphological differences. The terminal velocities of cloud droplets can be neglected, and those of raindrops can be limited to about 900 cm s- 1 because drops whose radii are greater than about 3 mm break up. The terminal velocities for cloud ice

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I.Y. LEE Table 1. Size categories and fall velocities (at 1000 mb) for hydrometeors Fall velocity (cm s- 1) Bin number

Radius (mm)

Drop

Ice/graupel (ice) 0.01 0.01 0.02 0.03 0.05 0.08 0.12 0.19 0.30 0.48 0.75 1.19 1.88 2.97 4.64

2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.0 1.3 1.6 2.0 2.5 3.2 4.0 5.0 6.4 8.0 10.1 12.7 16.0 20.2 25.4

16 17 18 19 20 21

32.0 40.3 50.8 64.0 80.6 101.6

(cloud droplets) 0.01 0.02 0.03 0.05 0.08 0.13 0.20 0.32 0.50 0.80 1.25 1.98 3.12 4.91 7.67 (raindrops) 11.83 17.91 26.53 38.31 53.89 73.92

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

128.0 161.3 203.2 256.0 322.5 406.4 512.0 645.1 812.8 1024.0 1290.2 1625.5 2048.0 2580.3 3251.0 4096.0 5160.6 6502.0 8192.0

99.12 130.43 169.14 217.01 276.01 347.60 430.75 514.87 603.00 698.26 795.23 881.74 943.20 971.52 974.55 981.52 997.22 997.22 997.22

1

and graupel range from 0~01 to 46 cm s - t and 61 to 645 cm s - 1, respectively. The fall velocities of snowflakes are smaller than those of other types of particles by a factor of two to three. Freezing of cloud droplets and raindrops depends on air temperature and drop size. The increment of frozen droplets is determined with the approach proposed by Danielson et al. (1972). The freezing rate increases with the decrease of air temperature, and larger drops freeze faster than smaller ones at the same temperature. Cloud ice, graupel and snowflakes melt when they fall through the freezing level. The rate of melting is controlled by a time constant determined by the balance of two rates, the rate of release of the latent heat of melting and the rate at which heat is transferred through the water layer. Melting occurs progressively as the particle size decreases. The model

7.17 10.89 16.19 23.47 33.15 45.64 (graupel) 61.40 81.00 105.25 135.23 172.22 217.27 270.09 325.56 382.17 443.56 507.13 565.63 609.46 632.07 636.18 638.03 644.91 644.91 644.91

Snow 0.00

0.00 0.01 0.01 0.01 0.02 0.03 0.05 0.08 0.13 0.21 0.33 0.52 0.82 1.28 2.01 3.10 4.69 6.94 10.03 14.10 19.35 25.94 34.14 44.27 56.80 72.24 90.98 114.19 136.97 160.92 186.91 213.96 239.07 258.19 268.37 270.48 270.29 270.29 270.29

assumes that a drop is produced at each time step as an ice particle melts, either by complete melting or by shedding. Adiabatic processes of dynamic variables and particle concentrations are computed first to produce intermediate values. Then values for water-vapor mixing ratio; saturation water-vapor mixing ratios over water and over ice; supersaturation ratios over water and over ice; air temperature; air density; air pressure; vertical velocity; horizontal velocity; and mixing ratios of cloud water, rain water, cloud ice, snow and graupel are stored for use in optimization calculations. Next, nonadiabatic effects are computed for condensation-evaporation, deposition-sublimation, melting, freezing and spectral evolution by coagulation among different types of hydrometeors. Condensation or deposition produces additional liquid water

Cloud microphysics parameterizations or solid water, reduces the water-vapor mixing ratio and warms the cloud environment. Therefore, the values for temperature, water-vapor mixing ratio and supersaturation ratio are adjusted accordingly to reflect the nonadiabatic effects. Coagulational processes control the spectral evolution of drops only until the air parcel ascends to the freezing level. Then drops start to freeze and produce ice particles, and coagulation among different types of particles proceeds. Large ice particles or graupels are produced because of additional freezing and riming, and snowflakes are produced mainly by ice-ice coagulation. By about -15°C, most of the drops will disappear because of freezing or riming on ice or snow. The computations of microphysics processes in the cloud model are outlined in the Appendix. The cloud model computes spectral evolution of cloud droplets, raindrops, cloud ice, graupel and snowflakes due to condensation, evaporation, deposition, sublimation, break-up, freezing, melting and coagulation, At each time step, 58 parameters are stored for use in optimization calculations. Fifteen bulk parameters [watervapor mixing ratio (q); saturation mixing ratios over water (qsw) and over ice (qsi); supersaturation ratios with respective to water (sw) and ice (sl); air temperature (T); air density (Pa); air pressure (P); vertical velocity (w); horizontal velocity (u); and mixing ratios for cloud water (Qw); rain water (Qf), cloud ice (Qi); graupel (Qg) and snow (Qs)] are stored after adiabatic processes only, and 43 macroscale rates as described in Equations (A1)-(A3) are stored after the microphysics computations. The number of data stored for each parameter is determined by the system; the only limitations are to exclude data smaller than the threshold values to avoid computational underflow. The limiting factors associated with the new parameterizations are variable for different macroscale rates obtained through optimization methods and thus will be discussed in detail along with the new regression formulas. The initial soundings for thermal and moisture structures are presented in Table 2. Two simulations were made to understand warm cloud processes (cases A and B) by using slightly different conditions of moisture content. Case B is designed to reduce the evaporation of raindrops as they fall through the subcloud layer, where the temperature decreases dry adiabatically and the relative humidity increases, reaching about 95% at the cloud base. A dry inversion layer at 500 m b inhibits extensive cloud development in these two cases. Cloud formation and dissipation processes occur mostly in the region between the subcloud and inversion layers, where the temperature and relative humidity decrease initially with lapse rates of about 5 - 6 ° C k m -1 and about 5% km -1, respectively. The initial soundings for cases C and D (Table 3) are designed to simulate warm and cold cloud interactions and cold cloud development, respectively. The freezing levels are located at 740 and 1000 mb for cases C and D, respectively. For case C,

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Table 2. Initial soundings chosen for warm cloud simulations with different moisture contents Air pressure (mb)

Air temperature (°C)

Relative humidity (%)

Case 10O0 900 850 700 600 500 400

29.0 20.0 16.2 5.5 -0.5 - 5.0 0.0

80.0 95.0 92.5 90.0 80.0 75.0 60.0

Case 100O 900 850 700 600 50O 400

29.0 20.0 16.2 5.5 -0.5 - 5.0 - 8.0

90.0 95.0 92.5 90.0 85.0 80.0 60.0

Table3. Initial soundings chosen for warm and cold cloud interactions (case C) and cold cloud developments (case D) Air pressure (mb)

Air temperature (°C)

Relative humidity (%)

Case C 10O0 900 800 700 600 500 400 300 250 200 1130

20.0 11.0 5.0 - 2.0 -- 11.0 - 22.5 - 36.5 - 53.0 - 60.0 - 60.0 -60.0

67.0 97.0 90.0 90.0 80.0 70.0 60.0 40.0 20.0 10.0 10.0

Case D 10O0 900 800 700 600 500 400 300 250 200 10O

0.0 - 8.5 - 16.0 - 23.5 -33.0 - 45.5 - 60.0 -65.0 -67.0 - 67.0 -67.0

67.0 97.O 93.0 90.0 85.0 80.0 50.0 30.0 10.0 10.0 10.0

the freezing level of 740 mb is in the lower portion of the convective layer, while in case D the freezing level is at the surface. In both cases, the temperature decreases dry adiabatically in the subcloud layer. Dry inversion layers are assumed at 250 and 400 mb for cases C and D, respectively, in order to inhibit extensive cloud development beyond about - 60°C. Cloud formation and dissipation processes occur between

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I.Y. LEE

the subcloud and inversion layers, where the temperature and relative humidity decrease initially with lapse rates of about 6 - 8 ° C k m -~ and about 5% km-1, respectively. Parameterized formulas are obtained through multiple regression for application in macroscale cloud simulations using the data provided by the four simulations with initial soundings presented in Tables 2 and 3. The first-order rates are expressed as functions of bulk parameters of thermodynamic and moisture variables readily available in regional-scale models. A total of 19,000 data sets is produced from four simulations. Each data set consists of 58 parameters, 15 bulk variables and 43 macroscale rates. However, the number of data sets used for regression parameterization varies from one rate to another. For most of the macroscale rates, the number of data used for multiple regression is greater than several hundred. For certain rates, on the other hand, appropriate data are very limited. For example, the freezing of raindrops proceeds too fast within a narrow spatial domain to allow enough data to accumulate. Some types of hydrometeors rarely coexist (e.g. cloud ice and raindrops), producing insufficient data for optimization of the production rate (e.g. of graupel by riming between cloud ice and raindrops).

(3)

(4)

(5)

(6) (7)

3. RESULTSAND DISCUSSION Computations were carried out to compare the rates produced by our regression parameterizations with those produced by the detailed model simulations and also to compare the behavior of different formulas from the literature with a Lagrangian air parcel model. The cloud microphysical process rates computed by using our parameterized formulas are compared with those obtained from the detailed simulations in Table 4. The rates for the loss of Qw and Qr by freezing and that for the gain of Qr by melting are not presented, because Q W f r = - Q l f r , QRfr=-QGfr and QRml = -(QGml +QSm0. The reduction rate for Qi by melting and the production rate for Qi by riming between snow and rain water are very small. In general, the rates computed from the regression formulas compare closely with those from the detailed treatments. The correlation coefficients range from about 0.81 for the rate of evaporation of cloud water to almost 1.0 for the rates of condensation of cloud water, accretion of cloud water, sublimation of graupel, freezing of cloud water and melting of graupel. The new parameterizations for a vapor-liquidsolid system can be summarized as follows: (1) Forty-three rates of cloud microphysical processes have been obtained through optimization methods by using data from numerical simulations; (2) Increases in the cloud-water mixing ratio are

(8)

mainly due to condensation of water vapor, while decreases are due to evaporation and to autoconversion and accretion in warm regions of the cloud. Decreases in the cloud-water mixing ratio can be quite rapid in subfreezing regions because of three-phase processes, freezing and riming, until Qw reaches about 10 -s. Thereafter, Qw can be reduced only by evaporation; The rain-water mixing ratio increases in warm regions of the cloud, mainly because of autoconversion and accretion of cloud water and melting of snow and graupel; it decreases rapidly in subfreezing regions of the cloud because of freezing and riming; Cloud ice, produced mainly by freezing of cloud water, increases through deposition of water vapor and decreases because of production of either snow through aggregation or graupel through riming. The melting of cloud ice can be ignored until a cloud reaches the dissipation stage; Graupel particles are mainly produced by riming among liquid and solid particles and by freezing of raindrops; Snowflakes are produced mainly by aggregation and are removed by melting and riming; The following microphysical processes were found to be insignificant: gain of graupel by autoconversion of cloud ice (QGau), gain of graupel by deposition of water vapor (QGvg), gain of cloud ice by transformation of graupel to cloud ice during sublimation of graupel (QGgi), melting of cloud ice (QlmO, gain of cloud ice by riming between rain water and snow (Qlrsr) and bulk terminal velocities for cloud water (Vqw) and cloud ice ( Vqi); The rates for graupel production through riming processes are large, but they are constrained by the availability of liquid particles.

We have examined the behavior of the parameterization formulas proposed by different investigators with our regression parameterizations in a Lagrangian air parcel cloud model. For warm clouds, the cloud droplets are produced initially by condensation of water vapor when the air is supersaturated with respect to water. As the cloud droplets grow large by condensation and coagulation, some of the large droplets become small raindrops. Thereafter, the evolution of raindrop distribution is mostly controlled by accretional processes involved with cloud droplets and raindrops. The evolution of liquid and solid particle distributions in cold clouds involves more complex processes. The cloud droplets and raindrops are produced by condensation and by autoconversion and accretion, respectively. As the air parcel rises through the freezing level, ice crystals and graupel particles are produced by freezing of cloud droplets and raindrops, respectively. The formation of ice crystals by deposition of water vapor on ice-forming nuclei is negligible because the concentration of iceforming nuclei is much smaller than that of cloud

Cloud microphysics parameterizations

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Table 4. Comparison of the rates produced by regression parameterizations and by detailed model simulations Rate Qw,~ QR.~ QWw~ QR,,, QR~., QRa. QRac

Ql~i QS~ QG~g Qli, QS~ QGgv QG~i Qlfr

QGf~ QGml

QS,,~ QG~. QS~i QS,~ Qs,, QS~** Ql~iw Ql~ir QG,iw QG~ QG~gw QG~g~ QG~ QG~r

v.,

Regression formula

No.*

Correl. coeff.

1 2 3 4

e-6"~vQ°'°Sq°w4° [(q/qs~)--l] °'a9

5

e -48"8 Qr°8° qs-~8"72 [l -(q/q,w)]

e 25SQr°'36qswl't8 [(q/qsw)-l] °'3";' e-2 "2 Q°'79 q°w19[1-(q/qsw)] °'66

e 17'lQr°'38q~-w121[1-(q/q~w)]l"22 TM

0.99 0.83 0.81 0.92 0.82 0.92 0.95 0.99 0.95 0.86

6a 6b 7 8

e l~~Q3Z3q~-~l°[(q/qsw)-l]-°69 e 14SQ2°SqZ.26'*[1-(q/q,~)]-°°3 e-55'*r~°'95Q°'94Q~°6 e °'51Q°"*°q°SS[(q/q~O-1]°'s3

9 10

e-21°Q~39q°si'36[(q/qsi)-l]°27 e~.5,2QO.63 qslio.9[(q/qsi)- 1] 1.52

11 12 13

e-l34Q°'SSq°'V3[l-(q/q~i)] T M e-283Q°SSq°i32[1-(q/qsi)]l°6 e '°°Q~°Sq°;'6[1-(q/q.,i)]1°3

14

e-16SQ°lO q ~ 3 7 [ 1 - ( q / q J ] °'89

17 18 20 21

e -4~~ Q°'lV(Qw+Q~)°st(273.16- T) ~51 e°'*~Qg°°ZQ~'sg(273.16- T) 1'22 e-6°4Q°'95Q°'°5(T-273.16) °'99 e 3°°Q~lSQ°2°(T-273.16)°'8°

0.98 0.98 0.99 0.86 0.99 0.87 0.99 0.96

24

e86.1 Qio.41 QO.12 qsliS.O

0.92

25 26

e6 2°(lOOOp,)la'aQ2"78Q°'18

27

28 29

eS'52(lOOOpa)V'19Qls'26Q~'3a e8 oo (1000p,)18.2 Qg2.a,Q~O.4O e12.8 Q2.31Qil.oo

0.96 0.87 0.92 0.97 0.92

30

e2.46 Q~ .o5 Qil.o3

0.98

31 33 34 35 36

e-8'~SQ°67 Q°'S* Q°'°7 elZ"*Q3w69Q°V4Q°18 e12°Q{°S Q°6S Q°'21 e145 (1000 p~)236 O°"*2Q~'°2 e23°(lOOOp~)-°9'* Q°95 Q1'°° elY'SQ°'87 QT M O0.03 eZt2 Q°'75Q°V° Q°'6°

0.96 0.96 0.96 0.87 0.94 0.97 0.95 0.95 0.95 0.90 0.99 0.90

37

38 39 40 41 42 43

e61.S(lOOOpa66.4Qls-23Q~ 0"3

e°'98 {1000 pa)- 2"66 O O'16 e837 (1000 pa)l'49 Q 0"2~" e3"2a (1000 pa)l'66 Q °'13 e6"49(1000 pa)- °"8 Q °'°°°3 e9.,,5 (1000 p,)2.2l Q0.49

0.88

Description Gain of Q~ by condensation Gain of Q,- by condensation Loss of Qw by evaporation Loss of Q, by evaporation Qr to Qw by evaporation Gain of Q, by autoconversion Gain of Qr by autoconversion Gain of Qr by accretion Gain of Q~ by deposition Gain of Q~ by deposition Gain of Qg by deposition Loss of Qi by sublimation Loss of Q~ by sublimation Loss of Q, by sublimation Q, to Qi by sublimation Gain of Qi by freezing Gain of Q, by freezing Loss of Qg by melting Loss of Q~ by melting Gain of Q8 by autoconversion Gain of Q~ by Q~-Q~ coagulation Gain of Q~ by Q~-Q~ coagulation Gain of Q, by Qs-Q, coagulation Gain of Q~ by Q,-Qg coagulation Gain of Qi by Qi-Qw riming Gain of Qi by Qi-QF riming Gain of Qi by Qs-Qw rimmg Gain of QB by Q i - Qg riming Gain of Q, by Q i - Qr riming Gain of Qg by Q,-Qw riming Gain of Q, by Q,-Q~ riming Gain of Q, by Qs-Qw riming Gain of Qg by Qs - QF riming Terminal velocity of Q~ Terminal velocity of Qr Terminal velocity of Qi Terminal velocity of Qg Terminal velocity of Qs

*Equation number in detailed treatment. Equations (6a) and (6b) are applicable when sw>0 and sw<0, respectively. The units are in gg ~for q, q~w,q~, Qw, Qw, Qr, Qi, Qg and Q~, in K for T, in gcm -3 for Pa, in gg- 1s- ~ for all dQ/dt rates, and in cm s- ~ for terminal velocities.

condensation nuclei. The depositional growth of ice crystals is augmented significantly by three-phase processes when ice crystals coexist with liquid drops. At this stage of cloud development, the drops evaporate rapidly to provide a strong vapor flux toward crystals. Snowflakes are mainly formed through aggregational processes involving ice crystals and graupel particles. The growth of graupel particles is augmented significantly by riming processes between solid particles and liquid drops. As the solid particles fall through the melting level, they melt to become raindrops.

Parameterizations are compared for a warm cloud in Figs 3-5. An air parcel saturated at 283 K and located initially at the 1000-mb level is assumed to rise with a vertical velocity of 1 m s- 1. The lapse rates are set to 9.0 and 6.5 K k m - 1 in calculating changes in cloud-water mixing ratio due to condensation and due to autoconversion and accretion, respectively. The saturation adjustment scheme of Asai (1965) and our parameterized formula for the condensation rate of cloud water (Table 4) are compared through evolution of the cloud-water mixing ratio (Fig. 3), initially set to 10 -6 . The formulas give similar results that

2706

I.Y. LEE 1.5-

..)

• = Lee (this article) = Asai (1965)

/ ~

1.0 ~ I

• ==Lee (this article) D ,= Kessler (1969) o Scott (1982) z~==Lin et el. (1983)

t~N.'~

0.8 -I

~~'~ ~'~"~

A

1.0-

0.6-

0.40.5-

o

O

0.2

0.0 0.0

5.0

10.0

0.0

15.0

I

0.0

i

5.0

t (rain)

~

0.8 v

0.6"t3

o (.9

0.4

• = Lee (this article) [] = Kessler (1969) o = Berry and Reinhardt (1974b)

0.2

[

0.0

I

5.0

15.0

t (min)

Fig. 3. Evolution of cloud-water mixing ratio due to condensation.

1.0 "

10.0

10.0

15.0

t (min)

Fig. 5. Evolution of cloud-water mixing ratio due to accretion.

the mass of the average cloud drop (Qo) considered. If Qo increases, the autoconversion rate increases rapidly. For example, if the average cloud drop size increases from 13 to 15 #m, the autoconversion rate will increase by a factor of two. At present, the Qo value is set to 10-a g, corresponding to a drop of about 13-#m radius. For accretion rates (Fig. 5), we compare formulas by Kessler (1969), Scott (1982) and Lin et al. (1983) with our formula. Our results agree closely with those of Scott during the first 9 min of cloud development and then become somewhat greater than Scott's, while Kessler and L i n e t al. predict a more rapid depletion of cloud water. Most of the cloud water can be transformed into rain water within 20 min in all cases. Parameterized formulas associated with cold cloud processes are presented in Figs 6-8. Figure 6 shows the evolution of Qi due to deposition. In this case, the

Fig. 4. Evolution of cloud-water mixing ratio due to autoconversion. 3.0• -~ Lee (this article)

closely follow the moist adiabatic line. However, our formula is more efficient, because Asai's formula requires saturation adjustment through iteration within the macroscale time step. Figures 4 and 5 present the changes in the cloud-water mixing ratio due to autoconversion and accretion, respectively. For autoconversion rates (Fig. 4), formulas by Kessler (1969) and Berry and Reinhardt (1974b) are compared with our formula (Table 4). In these cases, the cloudwater mixing ratios are initially set to 10 -3 . The Kessler formula reduces the water content by 3 5 o in 15 min, while the formula of Berry and Reinhardt reduces the water content by 10%. Our formula has an intermediate result, a 21% reduction after 15 min of cloud development. The formula of Berry and Reinhardt needs spectral properties that are not available in regional models. Their formula is quite sensitive to

2.5

----Ogura and Takahashi (1971), C=2.0 o = Ogura and Takahashi (1971), C=1.0

2.0 1.5

8

1.0 0.5 0.0

i

o.o

1

lO.O

2oo

30.0

t (min)

Fig. 6. Evolution of cloud-ice mixing ratio due to deposition.

Cloud microphysics parameterizations air parcel saturated with respect to ice at 268 K and located initially at the 850-mb level is assumed to rise adiabatically with a vertical velocity of 1 m s- 1. Our formula produces more cloud ice than that of Ogura and Takahashi (1971). The rapid growth in the cloudice mixing ratio occurs because our formula for the depositional growth rate of cloud ice is derived by optimization and uses a data set produced by accounting for the effects of three-phase processes. The growth of ice particles is accelerated by the increase of water vapor from evaporation of drops. The production rate by three-phase processes is not formulated separately as proposed by Lin et al. (1983). Snowflakes are produced by aggregation of ice crystals. Lin et al. (1983) applied the concept of autoconversion of cloud water to similarly derive the conversion rate for snow production from cloud ice. On the other hand, we assumed that snow is produced solely by aggregation among ice crystals. Figure 7 presents the evolution of graupel mixing ratios due to freezing of raindrops. In this case, the air parcel saturated at 273 K and located initially at the 850-mb level ascends in ambient air with a lapse rate of 8.0 K km-1. The rain-water mixing ratio is set initially to 2.0 x 10-3. The formulas of Ogura and Takahashi (1971), Wisner et al. (1972) and Hales and Easter (1982) predict fast freezing of rain water, while in our formula rain water freezes slowly during the first 2 min of cloud development and rapidly thereafter. In all cases, the rain water freezes completely within 10 min, by which time the air has cooled to about 263 K. Figure 8 shows the evolution of rain water due to the melting of snow. In this case, the air parcel saturated at 273 K and located initially at the 850-mb level is assumed to descend at a vertical velocity of 1 m s - 1. The mixing ratios for rain water and snow are initially set to 10 -4 and 1.9 x 10 -3, respectively. Formulas of Ogura and Takahasi (1971), Wisner et al. (1972) and Hales and Easter (1982) are compared. Our -

3.0

2.5

A 13)

2.0-

1.5

.c_ 1.0

0.5-

0.0

=

=

=

=

=

=

=

t

10.0 (rain)

15.0

20.0

Fig. 8. Evolution of rain-water mixing ratio due to melting of snow. results agree closely with those of Hales and Easter. Formulas of Ogura and Takahashi predict higher values, while those of Wisner et al. predict somewhat lower values. Most of the snow is transformed into rain water within 17 min. Finally, the combined effects of condensation, autoconversion and accretion on cloud water and rain water, of deposition and aggregation on cloud ice and snow; of freezing on rain water and graupel; and of accretion on rain water, graupel and snow are examined. In this case, both liquid and solid particles coexist in the cloud. The air parcel saturated at 280 K and located initially at the 850-mb level is assumed to rise adiabatically with a vertical velocity of 1 m s- 1. All the mixing ratios were initially set to 10-6, and computations were carried out for 60 min of real time. The results are presented in Fig. 9 for the changes in cloud water, rain water, cloud ice, graupel and snow

3.0 2.5J 2.0 o .B .i.-, a~

=

50

0.0

• = Lee (this article) [] = Ogura and Takahashi (1971) z~ ----Wisner et al. (1972) o = Hales and Easter (1982)

2.0-

• = Lee (this article) r7 = Ogura and Takahashi (1971) o = W i s n e r et al. (1972) A = Hales and Easter (1982)

A

3.0

2.5-

2707

• = Cloud w a t e r • = Rainwater o = Cloud ice = Graupel [] = S n o w

1.5

c-

1.5-

1.o

(3. -1

1.O-

ii

0.5-

0.O

i

0.0

5.0

i

I

10.O

15.0

0.0

10.0

200

30.0

4o.o

5o~o

5o.o

t (min) 20.0

t (rain) Fig. 7. Evolution of graupel mixing ratio due to freezing.

Fig. 9. Combined contribution of the parameterized formulas to the evolution of cloud water, rain water, cloud ice, graupel and snow mixing ratios.

2708

I.Y. LEE

mixing ratios. The cloud-water mixing ratio increases rapidly, closely following the moist adiabat, to a maximum value of 1.1 x 10 -3 after 12 min of cloud development, and then decreases rapidly to 0.18x 10 -3 during the next 12 rain. The production of rain water starts after 8 min of cloud development, increases rapidly during the next 13 min to about 1.53 x 10 -3, and then decreases rapidly to zero within 9 min. The quick reduction of rain water is triggered by freezing when the air parcel passes the freezing level after 20 rain of cloud development. Thereafter, the production of cloud ice by deposition and of graupel by riming proceeds rapidly. The graupel mixing ratio reaches 1.91 × 10 3 11 min after the air parcel is lifted above the freezing level and stays constant thereafter as the rain water is completely transformed into solid particles either by freezing or riming. The production of snow proceeds quite slowly by deposition alone but proceeds progressively faster by deposition plus aggregation when snow and cloud ice mixing ratios are greater than 0.28 × 10 -3 and 1.25 x 10 -3, respectively. The cloud-ice mixing ratio reaches a maximum value of 1.28 × 1 0 - 3 after 41 min of cloud development and decreases slowly thereafter. The results of our comparison studies can be summarized as follows: (1) The rates computed from our parameterized formulas compare closely with those from model simulations with detailed treatments of cloud physics including spectral evolution among liquid and solid particles. Because the evaluation uses a dependent data set, however, the expected agreement is undoubtedly overestimated; (2) The parameterizations of condensation of cloud water, accretion of cloud water, sublimation of graupel, freezing of cloud water and melting of graupel correlate closely with the rates obtained from the detailed cloud model; (3) The evolution of various mixing ratios in a Lagrangian cloud model shows that our parameterizations agree closely with other formulas in the literature; (4) The formulas for condensation of cloud water predict similar results that follow the moist adiabat. However, our formula is more efficient in terms of computational economy; (5) Some discrepancies are found in rain-water evolution when different formulas for the autoconversion rates are applied; (6) The rates for accretion of cloud water compare closely when the linear collection efficiencies in other formulas are smaller than one; (7) Some discrepancies are found in cloud-ice evolution by deposition; (8) Snowflakes are produced by deposition and by aggregation among ice particles. Our formula predicts that snow production occurs rapidly initially and then slows as the cloud ice mixing ratio reaches a threshold value of 1.26 x 10-3;

(9) The graupel mixing ratio increases rapidly because of freezing of rain water. Our formula predicts results similar to those of Ogura and Takahashi (1971), Wisner et al. (1972) and Hales and Easter (1982); (10) The graupel mixing ratio increases rapidly by riming when graupel or snow coexists with rain water. Our formulas compare closely with those of L i n e t al. (1983); (ll) Rain-water evolution by snow melting shows some discrepancies among the formulas used. Our formula predicts intermediate results.

4. C O N C L U S I O N S

The currently available parameterization rates for cloud microphysical interactions have generally been derived under the assumption that the size distribution functions for various hydrometeors are given. Uncertainties exist in assigning values for aerodynamic properties such as the bulk collection efficiency. Furthermore, the physical processes for various types of ice crystals, which are crucial in the life cycle of convective storms, are not sufficiently understood for accurate, multiphase cloud microphysics parameterizations. Such parameterizations must therefore be applied with caution because in reality the spectral evolution of various types of liquid and solid particles varies significantly during the stages of cloud development. New parameterizations made by multiple regression upon solution fields obtained from simulations of a cloud model incorporating sophisticated cloud microphysics generally reproduce fairly well the results of the sophisticated cloud model, and they perform better in terms of computational economy for potential use in mesoscale models. The evolution of various mixing ratios in a Lagrangian cloud model shows that our new parameterizations also agree closely with currently available formulas when certain aerodynamic and spectral properties in other formulas are modified. In the future, the new parameterizations will be applied to a fine-grid mesoscale model for derivation of cloud cover in terms of bulk atmospheric parameters readily available in mesoand large-scale models. The final product of this research will be an advanced set of cloud physics interactions that are applicable over variable temporal and spatial scales and under variable atmospheric conditions.

Acknowledgements--I wish to thank Drs J. D. Shannon and

M. L. Wesely of Argonne National Laboratory, Dr J. M. Hales of Pacific Northwest Laboratory and Dr H. D. Orville of South Dakota School of Mines and Technology for valuable discussions and their consistent interest in the problem of parameterizing microphysical processes in clouds. This work was supported by the U. S. Department of Energy,

Cloud microphysics parameterizations Assistant Secretary for Energy Research, Office of Health and Environmental Research, under contract no. W-31-109ENG-38.

REFERENCES

Arnason G. and Greenfield R. S. (1972) Micro- and macrostructures of numerically simulated convective clouds. J. atmos. Sci. 29, 342-367. Asai T. (1965) A numerical study of the air mass transformation over the Japan Sea in winter. J. Met. Soc. Jap. 43, 1-15. Asai T. and Kasahara, A. (1967) A theoretical study of the compensating downward motions associated with cumulus clouds. J. atmos. Sci. 24, 487-496. Beard K. V. (1974) Terminal velocity and deformation of raindrops aloft. In Preprint Volume, Conf. Cloud Physics, Tucson, AZ., Oct. pp. 116-119. American Meteorological Society. Berry E. X. (1968) Modification of the warm rain process. In Preprints First National Conf. Weather Modification, Albany, NY, April, pp. 81-88. American Meteorological Society. Berry E. X. and Reinhardt R. (1974a) An analysis of cloud drop growth by collection: Part I. Double distributions. J. atmos. Sci. 31, 1814-1824. Berry E. X. and Reinhardt R. (1974b) An analysis of cloud drop growth by collection: Part IV. A new parameterization. J. atmos. Sci. 31, 2127-2135. Brown Jr P. S. (1987) Parameterization of drop-spectrum evolution due to coalescence and breakup. J. atmos. Sei. 44, 242-249. Byers H. R. (1965) Elements of Cloud Physics. University of Chicago Press, Chicago. Cotton W. R. (1972) Numerical simulation of precipitation development in supercooled cumuli--Part II. Mon. Weath. Rev. 100, 764-784. Danielson E. F., Bleck R. and Morris D. A. (1972) Hail growth by stochastic collection in a cumulus model. J. atmos. Sci. 29, 135-155. Fitzgerald J. W. (1974) Effect of aerosol composition on cloud droplet size distribution. J. atmos. Sei. 31, 1358-1366. Hales J. M. and Easter R. C. (1982) Mechanistic evaluation of precipitation scavenging data using a one-dimensional reactive storm model. Pacific Northwest Laboratory report RP-2022-1, Richland, WA. Hanel G. (1976) The properties of atmospheric aerosol particles as a function of relative humidity at thermodynamic equilibrium with the surrounding moist air. Adv. Geophys. 19, 73-188. Hsie E. Y., Farley R. D. and Orville H. D. (1980) Numerical simulation of ice-phase convective cloud seeding. J, appl. Met. 19, 950-977. Kessler E. (1969) On the distribution and continuity of water substance in atmospheric circulation. American Meteorological Society Meteorological Monograph, Volume 10, Number 32. Koenig L. R. (1971) Numerical modeling of ice deposition. J. atmos. Sci. 28, 228-237. Lee I. Y, (1989) Evaluation of cloud microphysics parameterizations for mesoscale simulations. Atmos. Res. 24, 209-220. Lee I. Y. (1990) Parameterizations of microphysical processes in clouds. Argonne National Laboratory, report ANL/ER-2, Argonne, IL. Lee I. Y., Hanel G. and Pruppacher H. R. (1980) A numerical determination of the evolution of cloud drop spectra due to condensation on natural aerosol particles. J. atmos. Sei. 37, 1839-1853. Lin Y. L., Farley R. D. and Orville H. D. (1983) Bulk

2709

parameterization of the snow field in a cloud model. J. Clim. appl. Met. 22, 1065-1092. Low T. B. and List R. (1982) Collision, coalescence and breakup of raindrops. Part II. Parameterization of fragment size distribution. J. atmos. Sci. 39, 1607-1618. Mason B. J. (1971) The Physics of Clouds. Oxford University Press, London. Neiburger M., Lee I. Y., Lobl E. and Rodriguez L. Jr (1974) Computed collision efficiencies and experimental collection efficiencies of cloud drops. In Preprint Volume, Conf. Cloud Physics, Tucson, AZ, Oct., pp. 73-78. American Meteorological Society. Nickerson E. C., Richard E., Rosset R. and Smith D. R. (1986) The numerical simulation of clouds, rain, and airflow over the Vosges and Black Forest mountains: a mesofl model with parameterized microphysics. Mon. Weath. Rev. 114, 398-414. Ogura Y. and Takahashi T. (1971) Numerical simulation of the life cycle of a thunderstorm cell. Mon. Weath. Rev. 99, 895-911. Orville H. D. and Kopp F. J. (1977) Numerical simulation of the history of a hailstorm. J. atmos. Sei. 34, 1596-1618. Rutledge S. A. and Hobbs P. V. (1983) The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones: VIII. A model for the "seeder-feeder" process in warm frontal rainbands. J. atmos. Sci. 40, 1185-1196. Scott B. C. (1982) Prediction of in-cloud conversion rates of SO2 to SO,, based upon a simple kinetic and kinematic storm model. Atmospheric Environment 16, 1735-1752. Silverman B. A. and Glass M. (1973) A numerical simulation of warm cumulus clouds: Part I. Parameterized vs nonparameterized microphysics. J. atmos. Sci. 30, 1620-1637. Srivastava R. C. (1978) Parameterization of raindrop size distribution. J. atmos. Sci. 35, 108-117. Stephens M. A. (1979) A simple ice phase parameterization. Colorado State University Atmospheric Science Paper Number 319, Fort Collins. CO. Thorpe A. D. and Mason B. J. (1966) The evaporation of ice spheres and ice crystals. Br. J. appl. Phys. 17, 541-551. Tripoli G. J. and Cotton W. R. (1980) A numerical investigation of several factors contributing to the observed variable intensity of deep convection over south Florida. J. appl. Met. 19, 1037-1063. Wisner C., Orville H. D. and Myers C. (1972) A numerical model of hail-bearing cloud. J. atmos. Sci. 29, 1160-1180. Yau M. K. and Austin P. M. (1979) A model for hydrometeor growth and evolution of rain drop size spectra in cumulus cells. J. atmos. Sci. 36, 655-668. Ziegler C. L. (1988) Retrieval of thermal and microphysical variables in observed convective storms. Part II. Sensitivity of cloud processes to variation of the microphysical parameterization. J. atmos. Sci. 45, 1072-1090. APPENDIX

COMPUTATIONS OF MICROPHYSICS RATES When the supersaturation ratio with respect to water is greater than zero, the cloud droplets and raindrops grow by water-vapor diffusion, while evaporation occurs if the air is unsaturated. The rates for production of cloud water (Q wvw) and rain water (QRvr) by condensation and the rate for reduction of cloud water (QWwv) by evaporation can be expressed as

Q ,W-paAt

nl (ri),+a,--

vinf i=1

+ v, na1( v,+ l --(4n/3)j)]+ a, ~ \ Vl+l--Vl / ( (4n/3) (r'])~+~' ~ ], s, >~O, vx /A

(A1)

I. Y. LEE

2710 I.

1 ,=,+I

The rate of rainwater production by accretion (QR.,) may be expressed as

L

nf(rp):+A,-

c i=,+

ntui 1

(

+C

vI+I-vJ

[ vt$

QfL.=~k~

WWrX+~,-v,

,+,nf

1

>

Kff,nj].

i

(A7)

j=l+l

The spontaneous and collisional break-up of large rain-

642) drops to form fragments has been investigated by Srivastava i ~in+.
(A3)

where ni. ri and r, denote hydrometeor number, radius and volume in ith bin. respectively; L and I are the total numbers of size bins and of cloud droplet bins, respectively; P, and pw are the densities of water and air, respectively; s, is the supersaturation ratio over water; and superscript d denotes cloud droplets or raindrops. When raindrops evaporate, some of the drops in size bin I + 1 (the smallest raindrops) are transferred to the size bin I (the largest cloud droplets). Therefore, the rate of raindrop evaporation may be divided into two rates, rain water to water vapor (QRrv) and rain water to cloud water (QR,), as L QR,, = 5 (4n/3) ; &rll)?+,, - 1 UinP

a

-

(4ni3) (I!+,

QR,, =k

P.At

i=,+*

i=,+2

+P,+,n?+,

VI+1 v,nf+,

(4n/3) (r?+ I I?+,,- 4

)Z+At ,s,
>

(A4)

(‘48)

uI+ , - (4n/3) (r?+ I ):+m

“1+1-VI QL=s

Vn:3)(4+ 1IT+,, , S&O. ( ’ 1’1 In Equation (Al), the first term denotes the cloud water integrated over the size range rl to r,- 1 at t+At, the third term the contribution of r, at r + At to cloud water, and the second term the total cloud water at t. In Equation (A2), the first and second terms denote the rain water integrated through the raindrop size range at t+ At and the total rain water at t, respectively. The third term represents the contribution of cloud water to rain water due to the growth of the largest cloud drop r,. Equation (A3) describes the change in cloud water produced by droplet evaporation. Equations (A4) and (A5) represent the transfer of rain water to vapor and to cloud water, respectively. Therefore, the first term of Equation (A4) represents the rain water integrated over the size range r,+2 to L at r+Af, excluding the contribution of r, + , The second term in Equation (A4) denotes the total rain water at time t. The evaporation of raindrops with size r,+ 1transfers water to the cloud environment in the form of vapor as described in the third term of Equation (A4) and to the cloud water as described in Equation (AS). The rate of rain-water production by autoconversion (QR,,) represents the total mass transfer rate from the largest cloud droplet to the smallest raindrop by condensation and coagulation. This rate may be written as

(

+X-

(1978), Low and List (1982) and Brown (1987) on the basis of numerical and experimental analyses. Brown (1987) showed that the number of fragments per break-up is independent of the sizes of the colliding drops and that each collision produces approximately seven smaller drops at stationary state. At present, the raindrops greater than 2.58 mm in radius are assumed to break into three smaller drops. One raindrop in size bin i, where i735, will be distributed into smaller size bins by producing one drop in size bin i- 1 and two drops in size bin i-2. The rate of rain-water loss by break-up is zero, because the smallest drops produced by break-up will be about 1.6 mm in radius, which is far larger than the largest cloud droplet size of 25.4~pm radius. The rates for production of cloud ice (Qr,,), snowflakes (QS,,) and graupels (QG,,) by deposition of water vapor and the rates for reduction of cloud ice (Qr,,) and snowflakes (QS.,) by sublimation can be expressed as

til + 1n! (4n/3) (rD?+A,- or (47r/3)(rf)?+dl )I* 01+1 vr+1-“I

(A6)

where K denotes the collection kernel. In Equation (A6), the first term represents the rate of gain of thesmallest raindrops in size bin I + 1 by means of coagulation of cloud droplets in size bin I’with other smaller cloud droplets. The second term represents the transfer rate of water by condensation from size bin I to I+ 1.

B

QG..=%

B [

(4n/3) i n%?):+ari=1 (4n/3)

+uJ+lnf

i

(A9)

n:(r%+&,-

i=.r+1 WNr%+A,-~~J uJ+I -L’J ! (AlO)

Qs.=%( a

(4n/3) i

I=1

nf(r:):+A,- f

,=I

v,nf),s,
(All) (Al2)

where J denotes the total number of cloud ice bins; p. and pn are densities of snowflakes and cloud ice or graupel, respectively; and superscripts g and s indicate graupel and snow, respectively. In Equation (A8) the first term represents the cloud ice integrated over the size range rl to rJ_l at t + At, the second term the total cloud ice at t, and the third term the net contribution to rJ due to the depositional growth of the largest cloud ice. In Equation (A9), the first and second terms represent the total snowflakes at t + At, respectively. In Equation (AlO), the first and second terms denote the graupel mixing ratios at t + At and t, respectively, and the third term represents the contribution due to the transfer of cloud ice to graupel when the largest cloud ice grows by deposition to become the smallest graupel particles. In Equations (Al 1) and (A12), the first and second terms represent the cloud ice and snow mixing ratios at time t + At and t, respectively. When graupel particles sublimate, some of the particles in size bin J + 1, the smallest graupel particles, are transferred to the largest cloud ice in size bin J. Therefore, the rate of graupel sublimation is divided into two rates, graupel to

Cloud microphysics parameterizations

paAtL

, , 3 n,(r,),+a,Z v,n~

(4~/3)

j=J+2

vI+z--Y~Y~

p~ c p.mti=l

vapor (QG,~) and graupel to cloud ice (QG,I):

QG,, =

2711

L

\v,+~-v, v,

-F y'-v~ Y~)]+ ~ yTnT[C,+D,

i=J+l

UI+ 1 --U I UI+ 1

\

vj+l-v'1

/

x

( (4~/3)(r'+ ')~+'~'11,%<0

/A

v'1+1

-----

.

\UI+I--VIU l

(A13)

(A23)

UI+ 1 --L~I UI+ 1

Equations (A19)-(A23) are valid when 0 < x S < l

with

(A14)

xB=6rw(T--273.16) At/[ogLm(r~)2], 0 < x S < 1 with x ' = 6 x , (T-273.16) At/[O, Lm(r~)2] and T>273.16 K. In Equations (A22) and (A23), y~=p,vxS/pw a n d / = p , vx'/p,~. In Equation (A22), A g = l if y~
In Equation (A13), the first term represents the graupel mixing ratio obtained after integrating over the size range r'1+ 2 to L at t + At, excluding the contribution of particles in size bin J + 1. The second term is the total graupel mixing ratio at time t. The sublimation of graupel with size rs+x transfers water to the cloud environment as vapor as described by the third term of Equation (A13) and to cloud ice as described by Equation (AI4). Freezing of cloud drops and raindrops produces solid particles. The rates of loss of cloud water (QWf,) and rain water (QRfr) are given by

y~ < vt and Bs = 1 if vt < y~ < vl + 1. In -Equation (A23), on the other hand, Cg = 1 if y ~> vl + 1, DR= 1 if vl < yi' < vt + 1, C, = 1 if y~ > v~+ 1 and D, = 1 if vl < Y7< vl + 1. Otherwise, the values for A, B, C, D are set to zero. Equations (A19)-(A23) are valid when T>273.16 K and are constrained by At because some of the solid particles melt completely within At. In Equations (A22) and (A23), A, B, C and D work as delta functions to properly distribute the equivalent liquid volume from melting into cloud water and rain water. The rate of graupel production by autoconversion (QG,,) is defined by the mass transfer rate by deposition and coagulation from the largest cloud ice to the smallest graupel particle. It may be written as

QG,, = P,p'At v'1n'+l ( v'~+~-(4n/3)(r' +, - v'1+' )~+~t/ ( (4~/3)(r~+l)~+atI, v'1

si<0.

/

Q W ~ , = - P-'~. exp[ - 0 ' 6 8 ( r - 2 6 6 " 1 6 ) ] k

~=x

p,At

v,ne~,

(AI5)

~.L

L

QRr,=-P~

PaAt

exp[-0.68(T-266.16)]

v,n~.

~

J-1

(A16)

i=l + 1

The rates for production of cloud ice (Qltr) and graupel

i=1 \

'

UJ+I--1)JVJ+I/

{ Vs+fV,'~ +Vs+1~j.snsnjl -V s + 2-- v ' 1 + 1 /lkl - -v'1+i I / \ •-sd

d//vs*2-(vs+Uvs)'~

.

(QGr~) by freezing are expressed as At.

,/~.

Vj+I--V J

/\

.i-1~ (v'1+,_fv'1~fvi_vivi)], +f vine\ vi+l_vl Vi+l--ViVi+l i=1

1A24) (A17)

QGf,= p~. e x p [ - 0 . 6 8 ( T - 2 6 6 . 1 6 ) ] paAt

UI+I - - U i

Here the first term represents the rate of gain of the smallest graupel in size bin d + 1 by coagulation between the largest cloud ice in size bin J and drops
Ui+1 --Ui~i+l/..J

Melting of cloud ice, graupel and snowflakes produces cloud water and rain water. The rates of loss of cloud ice (QlmO,graupel (QGml) and snow (QSm3 and the rates of gain of cloud water (Q Wm~) and rain water (QRm~)by melting may be written as J

QI~,=_ o, ~ v~n~x~

Pg

QScl i

QS,~

(A21)

W,,,,-_ ~.p. f ~,=, yrnr , .[-L A, + n,~,['vt+,--y'y' ,.,-------S ,-',+,,:.

Pa

-

-

+

-

AE(A) 26:I5-D

,

(A22)

,, , - i ~ K~jn~h ,

(A26)

j=i

i=1

Ki..injh"g

-I

,

(A27)

j=J+l

t,,n~ ~. K~,Sn~ ,

(A2S)

j='1+ 1

where h=ps/Pv The rates of production of cloud ice by riming of cloud ice with cloud water (QI.w), cloud ice with rain water (QI..), snow with cloud water (QI.~) and snow with rain water (Qlr.) may be written as

Pak~=1

i=1

v.,+~--v~v, v~+l--vtvz+t./])

v i n ,i L

Pa j = ' 1 + 1

Ql~iw=ps UI+I --UI DI+I

1 i=1

QS~g=--P' I vin~ ~

(A20)

i=J+l

am,- -p.A~t i~'-- v,n,x,,

j=i

QS,,,= p-g

L

QG=,=- p~At P__L ~ vin~x~,

P'

= - -

Pa

Ki,jnj,

vlng

Pa i = 1

(awl

p.At i= 1

x

/J

U/+ 1

Qlr,=p~At exp [-O.68( T - 266.16)] [ fv'1n~( ~ ' 1 )

v,n~ K[djn~f +v'1

K'1.in'1n,

j

×(v'1+v'1~v")(~)]},

(A29,

2712

I.Y. LEE

x "~' (K~?.,n~,nf fv, O+fv,~+v,+ '

QI.,= p= '-' v,n~ y r~§n~/ +v~ i

i=l+l

j=I+l

r,<,,n..,; J'` '~ ' I,,vx+,-t,-~-~) \ V , l _ l

J-I

V

,=,~,L

j

(A30)

x

'

VJ+I--UJ DJ+I J

Z" ( K i t

'°'

" - l , l n /' + 1111

,~=]- 1

v]+l--v]

v]+l

/J (A34)

gl.w ='°, { £ •Oa

(v,n,h-' j ~.= l K~n d~'~ I'J J J )

Ps t - 1

QGnw= -

i=1 \

(

~

Pa i f J+l

( hv''+v.s -~+fv''~/_])' ] l

~--I vln~h \

~" j=l+l

(A35)

K i lid j n j fd

,

Vini ~

Ki,jnjf

+Vj+ln]+l

jffil+l

(A36)

(A31) P, Li=]+2

i

,

~.

v, nf

QG,.,,,= p"

Q[rsr=~

)

j=l

Q G , v = ~ , =~+ i

i=l

vin~ ~ K~n~f

K~Jn<]i

x y.

/

/=l

(KSd+~.md hVz+l+fv'-vjhvj+'+fv'~l,

i=l

v]+I--V]

VJ+I

f ..J

(A37)

,,,,+:+;9

sd d +vj+~n.~+~ s QG~v= vfn~ K~.jn~ P, Li=J+I\ ./=,I+1 ['_. x

(A32)

1L

dhV]+l-kfvi-v.lhv]+l+fvl~]

lr'-z+l,ini

-

i=/+i\

-

l _

VI+I--vJ

v]+l

}J" (A38)

The production rates of graupel particles by riming of cloud ice with cloud water (QG,~w),cloud ice with rain water (QG,ir), graupel with cloud water (QG,.,,), graupel with rain water (QGr=r),snow with cloud water (QG.,,,),and snow with rain water (QGr,,) may be expressed as

Lastly, the bulk terminal velocities for cloud water (Vq.), rain water (Vq,), cloud ice (Va,), graupel (Vqs) and snow (Vq.) are determined from

Vq.,= QGriw =p~ vjn~ Z IK~.~n~ Pa i= 1 k

×(

io,

(A33)

j=J+l

V.i

(v,n~V~)

=

(A39)

(A40) i=

i=l

__E .,n~ Z K~3n~:

gdj r lgj n dj f'+ vj + l + v j K j,

@,n'~ vf) iffil+ l

W",+I¢,-1

i 1

(v,nf),

li=l

v..=

,,,+,-,,/t ~-;;:T+,) J'

eo.,=

(vinfV~) t=1

±F

(vine),

(A41)

i

v,, =

(v,n~ v~) i=J+l

(vine).

(A42)

/i=J+l

/

v..=

(v~n~VD

(v,n~).

(A43)