NOAA Technical Report ERL 381-APCL 40 Simulation of Cold Cloud Precipitation in a Three-Dimensional Mesoscale Model Charles F. Chappell David R. Smith October 1976 U. S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration Environmental Research Laboratories NOAA Technical Report ERL 381-APCL 40 'MCNT Of C Simulation of Cold Cloud Precipitation in a Three-Dimensional Mesoscale Model Charles F. Chappell David R. Smith Atmospheric Physics and Chemistry Laboratory Boulder, Colorado October 1976 U. S. DEPARTMENT OF COMMERCE Elliot Richardson, Secretary National Oceanic and Atmospheric Administration g- Robert M. White, Administrator Environmental Research Laboratories o ■~ Wilmot Hess, Director o a o vi '^6-^ CONTENTS Page Abstract 1 1. INTRODUCTION 1 2. MODEL CHARACTERISTICS 2 3. MODEL EQUATIONS 3 3.1 . NUCLEATION 3 3.2 ICE GROWTH AND SUBLIMATION 4 3.2.1. Unrimed Crystals 4 3.2.2. Partially Rimed Crystals 5 3.2.3. Graupel 6 3.3. SEDIMENTATION AND FLUX DIVERGENCE 7 3.3.1. Unrimed Crystals 7 3.3.2. Partially Rimed Crystals 7 3.3.3. Graupel 8 3.4. ELIMINATION OF ICE PARTICLES BY SUBLIMATION 8 3.4.1. Unrimed Crystals 8 3.4.2. Partially Rimed Crystals 9 3.4.3. Graupel 9 3.5. CONVERSION FUNCTIONS 9 3.5.1. Transfer of Unrimed to Partially Rimed Crystals 9 3.5.2. Transfer of Partially Rimed Crystals to Graupel 10 3.6. PREDICTION EQUATIONS 11 3.6.1. Unrimed Crystals 11 3.6.2. Partially Rimed Crystals 11 3.6.3. Graupel 12 4. DYNAMICAL MODEL AND INITIAL CONDITIONS 12 4.1. CHARACTERISTICS OF THE DYNAMICAL MODEL 12 4.2. TOPOGRAPHY 13 4.3. INITIAL CONDITIONS 14 5. NUMERICAL RESULTS 14 6. SUMMARY AND DISCUSSION 23 7. ACKNOWLEDGMENTS 24 8. REFERENCES 24 Appendix: SYMBOLS 25 SIMULATION OF COLD CLOUD PRECIPITATION IN A THREE-DIMENSIONAL MESOSCALE MODEL Charles F. Chappell David R. Smith A cold cloud microphysical model is developed and merged with a 15-level primitive equation mesoscale dynamical model. The microphysical model partitions ice particles into three categories: unrimed, partially rimed, and graupel. Each is considered to have a Marshall-Palmer distribution, and is assigned its own set of physical characteristics. Criteria for converting from one ice particle category to another are based on a comparison of accretional to depositional growth rates. Six prognostic equations predict the concentra- tion and mixing ratio for each ice particle category. Processes in the model include deposi- tion, accretion, condensation, evaporation, sublimation, nucleation, and sedimentation. Microphysical processes are coupled back into the dynamical prediction equations so that the effect of ice processes on the air flow can be investigated. The mesoscale model is used to investigate numerically the development and distribution of snowfall over a mountain massif during a 4500-second integration. It appears to simulate realistically the nucleation and growth of ice particles and their transport and sedimentation to the mountain surface. Snowfall distribution, including the leeward extent of precipitation, also appears reason- able and consistent with that observed along the continental divide of Colorado for similar wind regimes. When further developed the model will be applicable to studies relating to prediction and modification of snowfall over mountainous terrain. 1. INTRODUCTION Models link theory and observations. During model development current knowledge is eval- uated and organized into a coherent entity. This task includes identification of the relevant physics, differentiation of the importance of various physical processes to the specific prob- lem and modeling goal, and formulation of the vital processes into a coherent set of physical concepts defined by mathematical expressions. The accomplishment of this task then allows one to compare theory with observations, and eventually extends those observations both spatially and temporally through prediction. Application of numerical models to weather modification problems provides a physical rationale for modification attempts by 1) per- mitting numerical testing of modification hypotheses prior to field experimentation, 2) better defining the observations required to evaluate hypotheses, and 3) more clearly desig- nating measurements and the instrumentation needed to acquire relevant observations. Numerical models also have application in designing operational programs to produce maximum benefits. Forecasts of precipitation from cold oro- graphic clouds depend upon satisfactory predictions of airflow over mountainous ter- rain, and realistic treatment of the interactions of microphysical processes at various levels within the cloud. Current models that predict mountain snowfall are limited by their one- dimensional or two-dimensional nature, assumptions of cloud homogeneity, or inade- quate treatment of settling particles. Examples of these are the models of Jiusto (1971), Cotton (1972), Hobbs et al. (1973), Chappell and Johnson (1974), and Plooster and Fukuta (1975). Young (1974) developed a microphysical model that extended treatment of cloud pro- cesses to more than one level through use of a continuous bin technique. Since Young mod- eiea only microphysical processes ^prediction aquations for air flow were not included), he as able to treat :hem in considerable detail. Many important processes related to snow- fall over mountainous terrain can be simulated three-dimensional models. Blocking, deflecting, and tunneling of air flow ever in are examples . These pro- loud formation and the spatial oud condensate ai recipi- to ic ie ©cesses, :■ T mesoscale - ncai -■~ a cped and combined with :: die tysical model. This paper nbes mainv the cold cloud microphysical .uscusses its application to the pre- diction of snowaxi over complex mountainous terrain, a\. detailed desenption of the dynamical model is given by Nickerson and Magazmer 1976). 2. MODEL CHARACTERISTICS In the development of the microphysical model presented here, the desire for more detail has been tempered by the usual limitations imposed by computer storage and economics. T he overall approach has placed equal empha- sis on the prediction of dynamical and micro- physical processes, which has made it necessary to parameterize certain aspects of the precipi- tation process. However, this approach has also permitted the coupling of the micro- physical thermodynamic forcing to the govern- ng dynamical equations so that the dynamic -esponse of the air flow to condensation and ice nowth is quantitativelv defined. In developing this microphysical model, the approach was to treat ice growth and sublima- tion in greater detail than condensation and evaporation processes. Microphysical processes included in the model are shown in the flow dia- gram of Figure 1. Liquid cloud water is formed in the model if the saturation vapor mixing ratio is less than the predicted total mixing ratio of water vapor plus liquid cloud water, since supersaturation with respect to water is not allowed. Condensation is the only source of liquid cloud water since melting is not consid- ered. Evaporation of liquid cloud water occurs when the total mixing ratio of water vapor plus liquid cloud water is predicted to be less than figure 1. Schematic diagram showing microphysi- cal processes included in the model. Processes are: accretional growth (A), condensation (C), deposi- iional growth (D), evaporation (E), sublimation (S), nucleation (N), conversion of partially rimed crystals to graupel (P-G), and conversion of un- rimed crystals to partially rimed crystals (U-P). the saturation vapor mixing ratio. The mois- ture source for ice growth can be both vapor and liquid cloud water. Precipitation occurs only as snow and the amount is equal to the ice exiting the liquid water cloud plus any growth, or minus any sublimation, that occurs below water saturation during transit to the surface. Ice 'particles are nucleated whenever effective ice nuclei and liquid cloud water are present simultaneously. The rate of nucleation depends on a temperature activation spectrum for ice nuclei and the temperature and cooling rate predicted by the mesoscale dynamical model. At the present time, no attempt is made to include effects of supersaturation on ice nu- cleation or ice crystal multiplication. Perhaps these features can be added in the future when they are better understood and can be ex- pressed in terms of variables predicted by the model. The total ice particle population is partitioned into three categories; unrimed crystals, partially rimed crystals and graupel. Conversion from one category to another is dependent on the ratio of crystal growth by accretion to growth by deposition. Prediction equations for the mix- ing ratio of ice within the three particle cate- gories are formulated within an Eulerian framework. Three additional prediction equa- tions for the corresponding particle concen- trations close the system of equations. Particles within the three categories are constrained by integrable size distribution functions that depend upon ice mixing ratio and crystal con- centration. Truncated Marshall-Palmer distri- butions are employed for the three categories. These distribution functions permit the growth equations for individual crystals to be converted into growth equations for individual ice cate- gories. Observations of graupel and planar crystals in Colorado mountain snowfall (Vardi- man and Hartzell, 1973) suggest that particles below cloud base have size distributions that might be approximated by an exponential rela- tion (Figs. 2 and 3). 3. MODEL EQUATIONS Prediction equations for the concentration and mixing ratio of ice in the form of unrimed, partially rimed, and graupel particles can be formulated by considering nucleation, ice growth, sublimation, sedimentation, flux di- vergence processes, and the rates at which crystals convert from one ice category to an- other. Expressions for these processes are for- mulated below for an x, y z coordinate system using the meter-kilogram-second system of units. A list of symbols is included in the Appendix. 3.1. NUCLEATION The temperature activation spectra of natural and artificial ice nuclei are often approximated by exponential functions for temperatures above that corresponding to homogeneous nucleation (<235K), or dT aN.exp [a(273.16-T)]. (1) Chappell (1970) reported values of 0.2575 m -'' and 0.435 K _l for N, and a, respectively, for an average activation spectrum observed during non-seeded days of the Climax Experiment. These values are used in the integration that is reported on in a later section. The observed flat- tening of activation spectra at extremely cold temperatures is approximated by assigning a value of 235K to the temperature in (1) for all temperatures below 235K. Unrimed crystals are nucleated when super- cooled cloud water (^10~ 5 kgm kgm ') and cool- ing exist simultaneously. From (1) the particle nucleation rate is then C„ = dN„ dT dT dt ' (2) and the rate at which ice is introduced into a unit mass of air is P„ = m„ dN„ dT p dT dt ' (3) where m„ is the average mass of a new crystal just after nucleation. Air density and the cool- ing rate are output from the mesoscale dynam- ical model. • y Planar Crystals 10 20 0.30 40 Crystal Diameter (-cm) 50 60 Figure 2. Distribution of planar crystals and graupel observed on 21 March 1973 at Wolf Creek Pass, Colorado (1452 MST-1558 MST). Planar Crystals 0.10 20 30 40 0.50 0.60 Crystal Diameter (~cm) Figure 3. Distribution of planar crystals and graupel observed on 10 February 1973 at Wolf Creek Pass, Colorado (1748 MST-2027 MST). 3.2 ICE GROWTH AND SUBLIMATION 3.2.1 Unrimed Crystals The growth rate of an unrimed planar crystal by deposition is dm,, 4GS,F„D„, ' dt ),, where G ={LJR,KT i + R,Tle,Dy\ L = 2.83658(10 6 ), K =0.0243 + 0.00008 (T- 273.16), (4) D =2.26(T/273.16)'- 81 (l/p), R, =461.6, 21.87456 (T- 273.16) e, = 610.78 exp (T-7.66) and S, is supersaturation with respect to a plane ice surface. The ventilation factor in the depositional growth equation written in terms of Reynolds Number is F U = 1+0.22(R,) 1 ' 2 . (5) The Reynolds Number for a planar crystal may, in turn, be expressed as a function of its dimensions, settling speed, and kinematic viscosity, or R ( , = 1.82r,^ :, c u " :, V u / l '. (6) A mean value of 2(10~"') m- seer 1 for the kine- matic viscosity is selected as representative of cold orographic clouds. Observations of Auer and Veal (1970) show that the ratio of crystal radius to crystal thickness for planar crystals varies from about 5 for r„ = 100 microns to 30 for r„ = 2500 microns. We therefore assign to (6) an average value of 20. Finally, settling speeds for unrimed planar crystals are taken from Brown (1970) to be V M = 1.6834 D!J (7) After substituting the above quantities into (6), the ventilation factor is F„ = l +36.96 Dg- 61 and (4) becomes, dm,, dt 4GS,D„ + 147.84GS,D,V ,;| (8) (9) The growth of planar crystals by accretion is d ^A = EApq ril V„. (10) at I,, The crystal cross-sectional area is given by A= 0.64952 Df u and the crystal settling speed is given by (7). The mesoscale dynamical model predicts the mixing ratio of liquid cloud water ((],„). The collection efficiency E of crystals for supercooled droplets is made a function of liquid cloud water content. This function accounts for the "collectability" of the liquid water present, and ranges from near zero for small droplets and low liquid water contents to a large percentage for large droplets and high liquid water contents (Fig. 4). The function is £ = 0.047 + 2274 (pq ru ) - 1564000 {pq (,, = l + 83.35D,°- 65 [17) Substituting (17) into (15), the depositional growth rate is dm, dt 5GS,D„ + 416.75GS,D,V K5 (18) The growth rate of partially rimed planar crystals by accretion of supercooled droplets is dm, IF EApq ru V pi (19) where £ is defined by (11). The same procedure used to develop (12) is followed, and upon substitution of (16) into (19) the growth rate bv accretion becomes dm,, 7T 4AUpq r ,cEDjr 10) Again the quantity of importance is the growth of ice over the entire partially rimed crys- tal distribution. The production of total ice by growth processes in this truncated Marshall- Palmer distribution is pJu dt (21) Substituting (18) and (20) into (21), the pro- duction of partially rimed ice is P m , = (rr„/p)(5GSi [ "pV-VWD,, J ii + 416.75GS, [ D) ) ^e- K v"vdD„ + 4A14pq r uE I Df; :i e- x p"pdD,X (22) 3.2.3 Graupel The growth rate of a graupel particle by deposition is ^A =6.2GD g F 9 S,. (23) The settling velocity of graupel is taken from Nakaya and Terada (1934) to be \/„ = 76.18 D!J B:i:! - (24) The ventilation factor for graupel is derived similarly to (8) except that, for this case, graupel is assumed to be spherical. The ventilation fac- tor becomes with this assumption F„ = 1 + 459. 75 DV-. (25) Substituting (25) into (23), the depositional growth rate is dm,, dt ),, = 6.2GD y S ( + 2850.4GD,) H -S,. (26) The growth rate of graupel by accretion is where £ is given by (11). (27) If we follow the same procedure used to develop (12), and substitute from (24) for the settling speed, (27) becomes ^ - 59. 83pq cu - EDf™. (28) dt !< The quantity of importance is again the pro- duction of ice over the entire graupel distri- bution. If a truncated Marshall-Palmer distribu- tion is assumed for graupel, the growth of graupel ice is 1 i, dm u I'd i P„„= n u e-W> g -^dD u (29) Substituting (26) and (28) into (29), the total production of graupel ice is P OT = (Vp)(6.2GS f D„e- K n"odD„ + 2850.4GS, f D^-e-WudD,, J ii + 59.83pq rl cE I D; ' i:!:t i'-VWD„). (30) 3.3. SEDIMENTATION AND FLUX DIVERGENCE 3.3.1 Unrimed Crystals The production of unrimed ice due to flux divergence is then '() ft r) P """~d^^" U ^~d^^" V ^~f Z ^" W ^ ' (38) The vertical flux of unrimed crystals relative to the updraft is f iw = P°Lv» e-*«'WD„. (3i) 3.3.2 Partially Rimed Crystals j (i Substituting for V„ from (7), the vertical flux of unrimed crystals due to sedimentation is F IIP = | 1.6834h„D<>- 17 ?-*«"« dD„. (32) The production of ice particles due to sedi- mentation is obtained next by differentiating the flux of particles with height, or C„, = — I 1.6834»„D',!- 17 t < A «"«(fD„. (33) '>- J (i Crystals are carried along with the vector wind, which is predicted by the mesoscale dy- namical model. The production of unrimed particles due to flux divergence is then C„„ = -^-(N„u)~(N„v)—^-(N u iv). (34) nX ay aZ The vertical flux of ice in the form of unrimed crystals relative to the updraft is f "' F ui = n„m„V„c K """dD„. J n (35) Substituting from (7) again for V„, and using the empirical relationship of Nakaya and Terada (1934) for unrimed crystals (w„ = 0.0038 DH) to convert from crystal mass to size, the vertical flux of ice due to sedimentation is F„,= 0.0064h„D5- 2,7 i' K »""dD lt . (36) The production of ice in the form of unrimed crystals due to sedimentation is obtained by differentiating the flux of ice with height, or in terms of unit mass P IIS = — I (0.0064/p)H„Dj;-"- l7 c K »"»dD„ (37) dz The vertical flux of partially rimed crystals relative to the updraft is F„„ = n^e-WpdD,, (39) Substituting for V ; , from (16), the vertical flux of particles due to sedimentation is F PII = 6.796 ^D^'t'-VWD,,. (40) The production of partially rimed ice particles by sedimentation is obtained next by differen- tiating the flux of particles with height, or -, r in C,„ = — 6.796 n,,DVe VWD„. (41) "2 J» Production of partially rimed crystals due to flux divergence is then C llll ^-£(N ll u)-£(N ll v)-~(N ll zc). (42) o\ cty dz The vertical flux of ice in the form of partially rimed crystals relative to the updraft is f '" F„i= n„m„V„e VWD,,. J n (43) Using the empirical relationship of Nakaya and Terada (1934) for partially rimed crystals (/;/,, = 0.027D/,) to convert from crystal mass to size and substituting again from (16) for l 7 ; ,, the vertical flux of ice due to sedimentation is r in F,„= 0. 1835 /;,,D/, ; c VVrfD,,. (44) The production of partially rimed ice due to sedimentation is obtained by differentiating the flux of ice with height, or in terms of unit mass : | (0.l835tp)n l ,Df l «e-W>pdD lt . (45) The production of partially rimed ice due to flux divergence is then *\ *\ *\ ~ ^ (gdD„. (51] The empirical relationship of Nakaya and Terada (1934) for graupel, m,, = 65D,',\ is used to convert from graupel mass to size. Substituting this relationship into (51), the vertical flux of graupel ice due to sedimentation is F„,= P 4951.7 h,,D;}- k,:, i?-VWD,,. (52) The production of graupel ice due to sedi- mentation is obtained by differentiating the flux of ice with height, or in terms of unit mass P„s = — (4951.7/p)//, / D;;' ;il r '">"■> dD„. (53) rlZ 3.4 ELIMINATION OF ICE PARTICLES BY SUBLIMATION 3.4.1 Unrimed Crystals The sublimation rate for small unrimed crystals, for which ventilation effects are negli- gible, is from (9) dm,, dt AGSiD,,. (55) If the Nakaya and Terada (1934) expression that relates unrimed crystal mass and size is differentiated, we obtain dm M = 0.0076D H dD„. (56) Substituting (56) into (55) and integrating over a time step of the model, It. as crystal diameter reduces from D„. v to zero, we obtain 'M ,(4GS,70.0076)df. (57) Performing the integration and simplifying, we obtain D„, = -526.31 GS, At, (58) where D u » is the diameter of a crystal that will just sublimate away during a model time step. Clearly, crystals of a smaller diameter will also sublimate entirely. If we integrate the truncated Marshall-Palmer distribution with respect to crystal diameter from zero to D,„, we obtain an expression for the number of unrimed crystals that disappear through sublimation during a model time step, or N,„: n„ i.'^""" dD„. (59) 3.4.2 Partially Rimed Crystals The sublimation rate for small partially rimed crystals, for which ventilation effects are negligible, is taken from (18) to be dm dt t = 5GS i D, l . (60) If the relationship of Nakaya and Terada (1934) that converts partially rimed crystal mass to size is differentiated, we obtain dm,, = 0. 054 D,,t1D„. (6i; Substituting (61) into (60), and integrating over a time step of the model as crystal diameter reduces from D px to zero, we obtain dD„ (5GS//0.054) dt. (62) Performing the integration and simplifying, we obtain D ; , s = -92.59GS,Ar, (63) where D p „ is the diameter of a partially rimed crystal that just sublimates entirely during a model time step. If we integrate the truncated Marshall-Palmer distribution with respect to crystal diameter from zero to D jlXl we obtain an expression for the number of partially rimed crystals eliminated by sublimation during a model time step, or lh,x N/«= n„e K i'"i>dD„. (64) 3.4.3 Graupel The sublimation rate for small graupel par- ticles is from (26) dm dt l = 6.2GD u Si. (65) We differentiate the empirical relationship of Nakaya and Terada (1934) that converts graupel mass to size to obtain dm,. \95D;,dD„. (66) Substituting (66) into (65) and integrating over a time step of the model as graupel diam- eter reduces from D (/ , to zero, we obtain D.dD,, (6.2GS,-/ 195) dt. At (67) Performing the integration and simplifying, we obtain D„ x = (-.0636GS,-Af)' (68) where D,,.„ is the diameter of graupel that will just sublimate during a model time step. The truncated Marshall-Palmer distribution is integrated with respect to diameter from zero to D,, s to obtain an expression for the number of graupel particles that are eliminated by sublima- tion during a model time step. Thus N„s n,, e V'w dD,,. (69) 3.5 CONVERSION FUNCTIONS 3.5.1 Transfer of Unrimed to Partially Rimed Crystals A conversion from unrimed crystals to partially rimed crystals is assumed when the accretional growth rate of a crystal attains a spec- ified factor, f ]f of the depositional growth rate. From (9) and (12), this conversion condition can be expressed as M4GS,D, + 147.84GS,D, IK1 ) = 1.0934p< ?( ,, £D;-- 17 . (70) If we let y - D" Kl , then to good approximation (70) can be written as a quadratic equation in y. This equation can then be solved to obtain an expression for the crystal diameter that satisfies the conversion criteria. The general form of the solution is D, 147.84GS,/i 2.1868pq,.„.E + \ 21856.7 (C5,/,)- + 17.5CS,pq,.„- Ef, 2.1868pi/,.„ E (71) Solutions for (71) are shown in Figure 5 for fi =1, which is the factor presently used in the microphysical model. After defining the crystal diameter that satis- fies the conversion criteria, those crystals in the distribution with diameters larger than D r are transferred to the partially rimed category. The number of such crystals is defined by integrat- ing the truncated Marshall-Palmer distribution with respect to crystal diameter over sizes greater than D, , or N lw = r n u e-WudD u , (72) J/v and N M/ , = if D,. >0.01. The mass of ice to be transferred to the par- tially rimed category (or the mass of ice in crystals with diameters greater than D,- and less than 0.01 meters) is c 3o5.2 Transfer of Partially Rimed Crystals to Gran A conversion from partially rimed crystals to graupel is assumed when the accretional growth rate of a crystal reaches a specified factor, f 2/ of the depositional growth rate. Using (18) and (20), this conversion criterion can be expressed fa (5GSi D„ + 416.75GS,-D,V 85 ) .i ±AUpq rw EUtt (74) If we let y — D$" 65 , then (74) can be solved as a quadratic equation in y. An expression can then be obtained for the crystal diameter that satisfies the conversion criterion, or D n 416.75G5,/-. 8.828pq rw E VT73680 (GSif 2 )- + 8S.28GSipq r ,r Ef-> 8.828pq r „ E (75) Solutions for (75) are shown in Figure 6 for f-> = 7, which is the factor currently used in the microphysical model. 10 ■ <=> 10' 10 : 0.25 50 30 20 10 Temperature (°C) Figure 5. Crystal diameter for conversion of un- rimed crystals to partially rimed crystals as a func- tion of environmental temperature and cloud liq- uid water content (gm m '). 10 ,-2 0.75 — c 1 00 O - 10 1 50 o 2 00 — 0> *N> 75 ^~\^\ ^ 1.00 — ^\\ 1 50 i ^ 2 00 -30 -20 -10 Temperature ( C) : igure 6. Crystal diameter for conversion of par- tially rimed crystals to graupel as a function of environmental temperature and cloud liquid water content (gm m '). !() The number of crystals to be transferred each model time step is defined by integrating the truncated Marshall-Palmer distribution with respect to crystal diameter from D,i to 0.01, or Ji),i (76) The mass of ice to be transferred to graupel is then q lltl = (0.027/p) n„ Dj, e r VV dD„. (77) h),i 3.6 PREDICTION EQUATIONS 3.6.1 Unrimed Crystals A prediction equation for the mixing ratio of ice in the form of unrimed crystals is formulated by combining the production terms (3), (14), (37), and (38), or + {n„lp) dq u _ m„ dNji dT dt ~ p dT dt f "' \GS,\ D u e- x ^dD u J i) (78) + 147.84GS, D,', fil e-W* dD u The prediction equation for the concentration of unrimed crystals is obtained by combining the production terms (2), (33), and (34), or dN u dN„dT d f?' „. _.,.. . . ,_ -r- = ^^ — + — 1.6834n ll D°~ 17 e- x »">'dD u dt dl at dz J (i -^-(H„u)-^-(N ll v)-±(N ll w). (81) ax ay dz The concentration of unrimed ice crystals at time n is obtained from N?* = N?r 2 +2At dN u \" dt (82) The value IV,}'* is then adjusted by subtracting the number of unrimed crystals that sublimated or that were transferred to the partially rimed category at time n. The adjusted value is then N» = N»*-N;;.,-Nv (83) 3.6.2 Partially Rimed Crystals A prediction equation for the mixing ratio of ice in the form of partially rimed crystals is formulated by combining the production terms (22), (45) and (46), or §3lL. dt (Vp) 5GSi Dpe-WpdD,. + 1.0934pi/,. H E Drr n e- x ""udD„ + — ! (0.0064/ p)n„Di- 2ll e- k u'>udD u dz J„ ri ri ri - ^ (v) - ^ (q»v) - ^ ('?»«')• The mixing ratio of unrimed ice at time n is obtained from the centered finite difference equation w q',r 2 +2\t dq„ dt (79) The value q',\* is then adjusted by subtracting q" t ,, the unrimed mixing ratio to be transferred to the partially rimed category at time n, so that q'i = > = N;j* + N',', p - Njj, - N',',,,. (89) 3.6.3 Graupel A prediction equation for the graupel mixing ratio is formulated by combining the production terms (30), (53) and (54), or 6.2GS, D„t'-VWD„ »=(%/p) clt + 2850.4GS, DJ K - t'~ V'(y + y () ) 2 ] } + fl.i exp [a , (x - x ( ,)'-' + «.-, y'-], where a = 1.6xl0 :l 3.7836x10- a- = - 0.532 xlO", «:, = 1.4382 x 10\ a A = -0.5204 x 10~ K , rt, = - 1.0284 xlO" 1 ', x = 1.3 x10 s , y„ = 2.823 x 10 ! , and all distances are expressed in meters. The shape of the mountain is identical (except for a vertical scale exaggeration) to that reported on by Lin et al. (1974). The mountain massif rises from a plain that is 1500 meters above sea level, and reaches 3640 meters at the top of the twin peaks. L3 tuw ^ 1 1 ' 1 1 "•^.^N, V \ A. 500 \\\ > i E 305K \\\ \\\ \vV E V\ Q 3 to t/1 V Q. 700 A\ A\ \\ 800 ;/ i i/ 50 ■4( -30 -20 Temperature ( C) -10 Figure 8. Sounding of temperature (solid line) and dew point temperature (dashed line) used for in- itializing mesoscale model. Moist layer is at 90% relative humidity. 4.3 INITIAL CONDITIONS The temperature and moisture sounding used to initialize the model is depicted in Figure 8. Note that the sounding is stable with temperature lapse rates less than the moist adiabatic lapse rate. The very stable layer next to the surface is frequently observed in the Rocky Mountain area. The atmosphere is moist (90% relative humidity) up to about 480 mb, and the temperature at this level is approximately -25C. Some lifting is therefore required to satu- rate the airmass and cause the development of clouds. A uniform westerly wind of 5 m sec" 1 is used. The dynamical and microphysical prediction equations were started simultaneously and the model integrated for 4500 seconds. Output from these prediction equations at certain points during the integration are presented next. 5. NUMERICAL RESULTS In Figures 9 through 13 results are displayed for a west-east cross-section (along the wind) through the south peak, and also for a south- north cross-section (normal to the wind) located 10 km, or one grid point, upwind of the twin peaks. (Locations of these cross-sections are depicted in Fig. 7.) The vertical motion profile for a section along the wind at 600 seconds is shown in Figure 9. Vertical motion exceeds 10 cm sec" 1 upwind of the crest and -10 cm sec ' downwind of the peak. The upwind area of positive vertical motion slopes slightly downward toward the crest. Using the definition that cloud exists when the liquid water mixing ratio (q cu ) equals or exceeds 0.01 gm kgm" 1 , note in Figure 10 that three blobs of cloud water have formed after 600 seconds. These are located upwind of the peak, one at low level and two near the top of the moist layer. The locations of these initial cloud parcels are important, since ice nucle- ation occurs in the model when supercooled clouds undergo further cooling. Figure 11 depicts the vertical motion field at 1200 seconds into the integration. Although there are minor changes in the field since 600 sec- onds, the basic features remain relatively un- changed. The liquid water cloud at 1200 seconds into the run is shown in Figure 12. The lower cloud just above the upwind slope has con- tinued to develop and now has a q cw ^ 0.10 gm kgirr 1 . One of the initial high-level cloud par- cels persists, while the other has disappeared. It is evident from the total ice mixing ratio (^,) at 1200 seconds (Fig. 13) that the latter parcel was converted to ice through glaciation. Note that q,> 0.05 gm kgm"' in this area. Mixing ratios of cloud water and total ice at 1200 seconds normal to the wind flow are shown in Figures 14 and 15. Note that q ru . > 0.20 gm kgm" 1 in the saddle area between the peaks, while q t > 0.05 gm kgm "' at the 6 km level just to the saddle side of the twin peaks. At 2400 seconds into the run, Figures 16 and 17 indicate that cloud water has continued to grow in the lower part of the orographic cloud, while substantial ice growth has glaciated the 14 Z1KMI 90 110 W-E(KM) 130 150 jgure 9. Vertical motion (cm sec ') along the wind at 600 seconds into the integration. 2 (KM) 90 110 W-E(KM) 130 150 Figure 10. Mixing ratio of cloud water (gm kgm -1 ) along the wind at 600 seconds into the integration. upper portion of the cloud. The mixing ratio of cloud water has increased to above 0.30 gm kgm -1 in the lowest levels just upwind of the crest; the mixing ratio of total ice exceeds 0.20 gm kgm -1 in middle sections of the cloud just up- wind of the peaks. Note that the 0.01 gm kgm -1 contour for q t reaches the mountain top and snow is now falling on the peaks. From Figures 18, 19, and 20 it is evident that this first snowfall is mostly in the form of graupel. Note that the 0.01 gm kgm -1 contours for unrimed ice (q u ) and partially rimed ice (q p ) are still suspended above the peaks, while the graupel mixing ratio {cfg) has reached 0.05 gm kgm" 1 on the upwind surface. Also, note how unrimed ice is being carried over the crest by the westerly wind flow, and how the two original ice nucleating cells remain distinct entities while growing with time. Z(KM) 90 110 w-E:km; 130 150 Figure 11. Vertical motion (cm sec ') along the wind at 1200 seconds into the integration 15 I? Z«M) 6 - - I 1 I r - ^-,01 O 7 .OK -r10 Vw 1 I 1 50 '0 SO 110 W-ECKM) 130 150 Figure 12. Mixing ratio of cloud water (gm kgm"') along the wind at 1200 seconds into the integra- tion. 15 ^EiKM) Figure 13. Mixing ratio of total ice (gm kgm ') along the wind at 1200 seconds into the integration. ZfKM) , 01 ^~- . 1 \ - 1 1 1 - 50 110 IfiO N-StKM) 200 Figure 14. Mixing ratio of cloud water (gm kgm ) normal to the wind at 1200 seconds into the inte- gration. Cross-sections normal to the wind at 2400 seconds (Figs. 21-25) show that cloud water has increased to over 0.40 gm kgm ' in the saddle area between the peaks. Total ice exceeds 0.30 gm kgm _1 at about the 5-km level and just to the saddle side of the peaks. Nearly all ice above 3km is in the form of unrimed crystals. Partially rimed crystals and graupel make up most of the ice over the peaks below 4.5km. At 3750 seconds into the run some interesting changes are. apparent. Note in Figure 26 that cloud water has been depleted near the summit, and the 0.30 gm kgm ' contour displaced fur- ther down the upwind slope. The reason for this is evident in Figures 27-30, which show that substantial quantities of cloud water have been converted to ice near the summit. Total ice in excess of 0.30 gm kgm ' is now present near the peaks, and is composed mainly of par- tially rimed crystals and graupel. Unrimed crystals have now reached the mountain sum- mit, but many continue to be carried over the crest to the leeward side. Cross-sections normal to the wind at 3750 seconds (Figs. 31-35) show that cloud water has decreased to 0.30 gm kgm ' in the saddle area. Cloud water has also been considerably depleted on the upwind flanks of the high peaks. Conversely, total ice has increased to near 0.40 gm kgm -1 at the 4. 5-km to 5-km level above the peaks, and also near the surface on the saddle side of the peaks. The maximum of total ice at high levels is largely due to unrimed crystals, whereas partially rimed crystals con- tribute most to the lower maximum. Graupel has decreased over the peaks as cloud water has been depleted, but continues to increase over the saddle area. At the end of the nin, or 4500 seconds, cloud water in excess of 0.30 gm kgm" 1 has been dis- placed even further down the upwind slope (Fig. 36). The two original maxima in total ice, Z.KMI 110 1C0 ht-S.m; roo Figure 15. Mixing ratio of total ice (gm kgm ') normal to the wind at 1200 seconds into the inte- gration. .' > n 150 Figure 16. Mixing ratio of cloud water (gm kgm ') along the wind at 2400 seconds into the integra- tion. 16 Z(KM 90 110 W-E(KM) 1^0 Figure 17. Mixing ratio of total ice (gm kgm ' ) along the wind at 2400 seconds into the integration. Z(KM) 90 110 W-E(KM) 150 Figure 18. Mixing ratio of unrimed crystals (gm kgm ') along the wind at 2400 seconds into the integration. although continuing distinct, now show signs in Figure 37 of merging. Total ice in excess of 0.30 gm kgm" 1 is found at the 4-km level just upwind of the peaks. Figures 38, 39, and 40 show that unrimed crystals are now mainly falling on the peaks, and also are extending well over the leeward side. Partially rimed crys- tals and graupel are confined to the upwind slope, and their respective maxima have devel- oped further upwind. Cross-sections normal to the wind at 4500 seconds (Figs. 41^15) show cloud water has decreased in the saddle to about 0.20 gm kgm" 1 . Maxima of total ice near 0.40 gm kgm" 1 are lo- cated just above the peaks and to the saddle side of the summits. The upper portion of the oro- graphic cloud is glaciated and composed solely of unrimed crystals. More unrimed ice is now set- tling on the higher peaks, whereas riming and graupel development is most pronounced in the saddle area. Figures 46 and 47 show cloud water and total ice for a cross-section along the wind and through the middle of the saddle area. Note that the upwind edge of the 0.01 gm kgm" 1 contours for cloud water and total ice are dis- placed 30 km to 40 km farther downwind than in a similar cross-section through the south peak. This reflects a different slope of the ter- rain along this cross-section. Total snowfall during the 4500-second period is depicted in Figure 48. Note that the snow- fall maxima are located upwind and to the sad- dle side of the peaks. Snowfall extends farther upwind along the higher flanks of the peaks than upwind of the lower saddle area. Measur- able snowfall has also occurred to a point about 15 km leeward of the peaks. Snowfall maxima are slightly in excess of 0.20 cm of water equiva- lent, or about one inch of snow. ZIKM 90 110 W-E(KM) Figure 19. Mixing ratio of partially rimed crystals (gm kgm ') along the wind at 2400 seconds into the integration. Z(KM 150 Figure 20. Mixing ratio of graupel (gm kgm ') along the wind at 2400 seconds into the integration. 17 ZIKH ZlKM MO 140 N-S(KM) 200 110 140 N-S(KM) 170 200 Figure 21. Mixing ratio of cloud water (gm kgm -1 ) normal to the wind at 2400 seconds into the inte- gration. Figure 22. Mixing ratio of total ice (gm kgm -1 ) normal to the wind at 2400 seconds into the inte- gration. Z:km] Z:km 110 140 n-s;km; 110 140 N-S(KM) 200 Figure 23. Mixing ratio of unrimed crystals (gm kgm ' ) normal to the wind at 2400 seconds into the integration. Figure 24. Mixing ratio of partially rimed crystals (gm kgm ') normal to the wind at 2400 seconds into the integration. 2 YVi, ZlKM) 110 140 N-SlKM) 200 90 110 W-ElKMJ 150 Figure 25. Mixing ratio of graupel (gm kgm ') normal to the wind at 2400 seconds into the inte- gration. Figure 26. Mixing ratio of cloud water (gm kgm" 1 ) along the wind at 3750 seconds into the integra- tion. 18 Z(KM 90 110 W-E(KM) 150 Figure 27. Mixing ratio of total ice (gm kgm ') along the wind at 3750 seconds into the integration. Z(KM) - — .01 , i i l J ■ 50 70 90 110 W-E(KM) 130 I 50 Figure 28. Mixing ratio of unrimed crystals (gm kgm ') along the wind at 3750 seconds into the integration. Z(KM) 6 - 3 - 50 r-^~~T1J>==2JK^V\ 1 1 1 1 70 90 110 W-E(KM) 130 150 Figure 29. Mixing ratio of partially rimed crystals (gm kgm ~' ) along the wind at 3750 seconds into the integration. ZIKMJ 90 110 W-E(KM) 1 SO 150 Figure 30. Mixing ratio of graupel (gm kgm ' ) along the wind at 3750 seconds into the integration. Z:km; 110 iqo Figure 31. Mixing ratio of cloud water (gm kgm"') normal to the wind at 3750 seconds into the inte- gration. ZIKM) Figure 32. Mixing ratio of total ice (gm kgm ') normal to the wind at 3750 seconds into the inte- gration. Dotted area denotes^, > 0.40 gm kgm 1 . 19 2 (KM 110 140 N-S(KM) 170 200 Figure 33. Mixing ratio of unrimed crystals (gm kgm" 1 ) normal to the wind at 3750 seconds into the integration. z:km) 110 140 N-S(KM) 200 Figure 34. Mixing ratio of partially rimed crystals Tgm kgm" 1 ) normal to the wind at 3750 seconds into the integration. Dotted area denotes^ > 0.20 gm kgm" 1 . Z: 0.20 gm kgm 1 . ZKM) 90 110 W-E(KM) '50 Figure 40. Mixing ratio of graupel (gm kgm ) along the wind at 4500 seconds into the integration. Dot- ted area denotes q p > 0.20 gm kgm"'. Z xm: 50 - " / — -T^ .01 \ _r- .10 -\^~, \ 1 /~— -<~ .20 ^\f- V ] 1 110 140 N-S(KM) 170 200 Figure 41. Mixing ratio of cloud water (gm kgm 1 ) normal to the wind at 4500 seconds into the inte- gration. ZKM) 110 140 N-S(KM) 200 Figure 42. Mixing ratio of total ice (gm kgm" 1 ) normal to the wind at 4500 seconds into the inte- gration. Z KM) 110 140 N-S(KM) Figure 43. Mixing ratio of unrimed crystals (gm kgm -1 ) normal to the wind at 4500 seconds into the integration. Z KM) 110 140 N-S(KM) 200 Figure 44. Mixing ratio of partially rimed crystals (gm kgm" 1 ) normal to the wind at 4500 seconds into the integration. Dotted area denotes^ > 0.20 gm kgm" 1 . 21 j 110 140 n-s/km) 200 Figure 45. Mixing ratio of graupel (gm kgm - ') normal to the wind at 4500 seconds into the integration. Z(KM 90 110 W-EIKM) 150 Figure 46. Mixing ratio of cloud water (gm kgm ') along the wind and through the saddle at 4500 seconds into the integration. 7 • M 200 I-, Figure 47. Mixing ratio of total ice (gm kgm ' ) along the wind and through the saddle at 4500 seconds into the integration. 90 110 W-E!KM) 150 Figure 48. Snow depth (cm FLO equivalent) at 4500 seconds into the integration. Stars denote loca- tions of mountain peaks. 22 6. SUMMARY AND DISCUSSION A three-dimensional mesoscale model with cold cloud microphysics has been developed. The microphysical model partitions ice particles into three categories, each with its own set of physical characteristics. Growth by deposition and accretion of supercooled droplets is integrated over the particle populations in each category and summed to define the total ice growth. Besides deposition and accretion, other processes included in the model are condensation, evaporation, sublimation, sedimen- tation, and nucleation. Criteria for converting from one ice particle category to another are based on a comparison of accretional to depositional growth rates. The model has been used to investigate the development and distribution of snowfall over a mountain massif. It appears to simulate realistically the nucleation and growth of ice particles, and their transport and sedimentation to the mountain surface. The leeward extent of snowfall (about 15 km) appears consistent with that frequently observed along the continental divide of Colorado for similar wind regimes. Concentrations of ice particles were typically 3 to 10 liter -1 during the run, which seems reasonable for the particular input sounding and ice nucleus activation spectrum used in the run. The maximum snow depth of approximately 0.2 cm H 2 equivalent for the 4500-second run also appears reasonable. This converts to approximately one inch of snowfall for the period. Much development work remains. We plan to investigate the sensitivity of the model to the upper limit of integration specified for the Marshall-Palmer distributions. Further work is also needed to define the minimum concentration of particles necessary to constitute such a distribution. This is a problem when nucleation of ice crystals proceeds very slowly or when particle concentrations are sublimating away. In preparation for verifying the model, we plan to introduce real terrain into the model and obtain stable flow solutions from the dynamical equations. Then, a first attempt will be made to compare the model with an actual precipitation episode over the Colorado Rockies. Observations of precipitation rates, crystal sizes, and riming should help "tune" the model by furnishing the data required to define collection efficiencies and the conversion criteria. The presence of the mountain barrier leads to an adjustment in the initially uniform horizontal wind field, and the development of a vertical motion field within the model domain. This evolutionary process has important consequences when microphysical processes are included. The evolving vertical motion field can initiate condensation and start a sequence of microphysical events. Characteristics of this chain of events are therefore sensitive to the initial timing and location of condensation. It follows that future validation studies of precipitation using real data will require a careful examination of the initialization procedure. Several areas of uncertainty remain with respect to the prediction and modification of snowfall over mountainous terrain. One is the effect of blocking on the air flow, and the transport of seeding materials toward and over the mountain barrier. Second, there is a need to better differentiate the net areal modification of snowfall from possible illusory results that arise from the redistribution of snowfall over the mountain barrier. Third, there are possible effects of natural seeding as ice crystals fall out from higher cloud layers, survive a trip through a cloudless layer, and resume growth as they enter a lower cloud system. Also, it is possible that ice crystals exiting an orographic barrier may survive to seed cloud layers further downwind. The mesoscale model presented here has the capability to investigate and explore these uncertainties. Furthermore, since microphysical processes are coupled back into the dynamical prediction equations, this model can define dynamic effects associated with various seeding treatments. Finally various seeding strategies can be studied within the model to develop a design for bringing about beneficial results, including targeting of snowfall over mountainous regions. 13 7. ACKNOWLEDGMENTS We thank E. L. Magaziner, E. C. Nickerson, and J. M. Fritsch for many helpful discussions during the model development. This work has been partially supported by the Division of Atmo- spheric Water Resources Management, Bureau of Reclamation, Dept. of the Interior, Contract 14-06-D 7676, for which we are grateful. 8. REFERENCES Arakawa, A., and Y. Mintz, 1974: The UCLA" general circulation model. Notes distributed at the workshop, 25 March-4 April. Dept. of Meteorology, UCLA. Auer, A. H., Jr. and D. L. Veal, 1970: The dimen- sion of ice crystals in natural clouds. /. Atmos. So., 27, 919-926. Brown, S. R., 1970: Terminal velocities of ice crystals. Atmos. Sci. Paper 170, Dept. of Atmos. Sci., Colorado State Univ., Ft. Col- lins, 52 pp. Chappell, C. F, 1970: Modification of cold oro- graphic clouds. Ph.D. Thesis, Colorado State University, 196 pp. Chappell, C. F and F L. Johnson, 1974: Poten- tial for snow augmentation in cold oro- graphic clouds. /. Appl. Meteorol., 13, 374-382. Cotton, W. R., 1972: Numerical simulation of precipitation development in supercooled cumuli. Part II. Mon. Weather Rev., 100, 764-784. Deardorff, J. W. , 1972: Parameterization of the planetary boundary layer for use in general circulation models. Man. Weather Rev. 100, 93-106. Hobbs, P. V, R. C. Easter, and A. B. Fraser, 1973: A theoretical study of the flow of air and the fallout of solid precipitation over mountainous terrain. Part II. Microphysics. /. Atmos. Sci., 30, 813-823. Jiusto, J. E., 1967: Nucleation factors in the devel- opment of clouds. Ph.D. Dissertation, Pennsylvania State Univ., 124 pp. Jiusto, J. E., 1971: Crystal development and gla- ciation of supercooled cloud. /. Rech. Atmos., 5, 2, 69-85. Lin, J. T, H. T. Liu, and Y. H. Pao, 1974: Labora- tory simulation of plume dispersion in stably stratified flows over complex terrain, Flow Research Report No. 29, Flow Research, Inc., Kent, Washington. Nakaya, U. and T Terada, 1934: Simultaneous observations of the mass, falling velocity, and form of individual snow crystals. /. Fac. Sci., Hokkaido Univ., Ser. 2, 1, 191-201. Nickerson, E. C. and E. L. Magaziner, 1976. The numerical simulation of orographically induced non-precipitating clouds. NOAA Tech. Report, ERL 377-APCL 39. Plooster, M. N. and N. Fukuta, 1975: A numer- ical model of precipitation from seeded and unseeded cold orographic clouds. /. Appl. Meteorol., 14, 859-867. Vardiman, L. and C. L. Hartzell, 1973: An inves- tigation of precipitating ice crystals from natural and seeded winter orographic clouds. Report SR-359-33 by Western Sci- entific Services, Inc. to the Bureau of Rec- lamation under Contract No. 14 - 06 -D- 6644, 122 pp. Young, K. C, 1974: A numerical simulation of wintertime orographic precipitation. Part I. Description of model microphysics and numerical techniques. /. Atmos. Sci., 31, 1735-1748. 24 Appendix SYMBOLS a Parameter in the temperature activation spectrum for ice nuclei A Cross-sectional area of crystal c„ Thickness of unrimed planar crystal C„ Production of crystals by nucleation C Hft Production of graupel particles by flux divergence C gs Production of graupel particles by sedimen- tation Cud Production of partially rimed crystals by flux divergence C p . s Production of partially rimed crystals by sedimentation C ud Production of unrimed crystals by flux divergence C us Production of unrimed crystals by sedi- mentation D Diffusivity of water vapor in air D r Crystal conversion diameter (unrimed to partially rimed) D lt Crystal conversion diameter (partially rimed crystal) D„ Diameter of graupel particle D,, Diameter of partially rimed crystal D„ Diameter of unrimed crystal D„ x Diameter of maximum graupel particle that will sublimate in a model time step D IJS Diameter of maximum partially rimed crys- tal that will sublimate in a model time step D us Diameter of maximum unrimed crystal that will sublimate in a model time step f; Saturation vapor pressure over a plane surface E Collection efficiency of crystals for super- cooled cloud droplets /i Ratio of accretional growth to depositional growth for unrimed crystals fi Ratio of accretional growth to depositional growth for partially rimed crystals F„ Ventilation factor for graupel particles F p Ventilation factor for partially rimed crystals F„ Ventilation factor for unrimed crystals F ai Vertical flux of graupel ice relative to the updraft F m , Vertical flux of graupel particles relative to the updraft F pi Vertical flux of partially rimed ice relative to the updraft F,„, Vertical flux of partially rimed crystals rela- tive to the updraft F„, Vertical flux of unrimed ice relative to the updraft F UI , Vertical flux of unrimed crystals relative to the updraft G Thermodynamic function K Thermal conductivity of air L s Latent heat of sublimation trig Mass of a graupel particle m„ Average mass of a new crystal after nuclea- tion m„ Mass of a partially rimed crystal m„ Mass of an unrimed crystal n g Number of small graupel particles in the distribution n„ Number of small partially rimed crystals in the distribution n„ Number of small unrimed crystals in the distribution N e Parameter in the temperature activation spectrum for ice nuclei N n Concentration of ice nuclei effective at a given supercooling N y Total concentration of particles in the grau- pel distribution N,, Total concentration of particles in the par- tially rimed crystal distribution 25 \ Total concentration of particles in the unnmed crystal distribution N^ Concentration of graupel particles with diameters less than D y , JV P „ Concentration of partially rimed crystals converting to graupel particles N ps Concentration of partially rimed particles with diameters less than D,,, N UI , Concentration of unrimed crystals con- verting to partially rimed crystals N lis . Concentration of unrimed particles with diameters less than D„ s p Pressure P„ Nucleation rate of unrimed ice P,„i Production of graupel ice by flux diver- gence P gg Production of graupel ice by growth pro- cesses P g „ Production of partially rimed ice by growth processes P gs Production of graupel ice by sedimentation P y „ Production of unrimed ice by growth processes P pd Production of partially rimed ice by flux divergence P ps Production of partially rimed ice by sedi- mentation P ud Production of unrimed ice by flux diver- gence Pus Production of unrimed ice by sedimen- tation q g Mixing ratio of graupel ice in the distri- bution 17, Mixing ratio of total ice q„ Mixing ratio of partially rimed ice in the distribution q„ Mixing ratio of unrimed ice in the dis- tribution q ru . Mixing ratio of cloud water c\ po Mixing ratio of partially rimed ice convert- ing to graupel ice q up Mixing ratio of unrimed ice converting to partially rimed ice r u Radius of unrimed planar crystal R,. Reynolds Number R, Specific gas constant for water vapor S, Supersaturation with respect to a plane ice surface t Time T Temperature u West component of the horizontal wind velocity v South component of the horizontal wind velocity V g Fall speed of graupel particle V,, Fall speed of partially rimed crystal V„ Fall speed of unrimed crystal w Vertical velocity x,y Horizontal coordinates z Altitude and vertical coordinate Af Model time step \ g Parameter in graupel particle distribution \ p Parameter in partially rimed crystal distri- bution K„ Parameter in unrimed crystal distribution p Air density v Kinematic viscosity 26 f L ABO HAT O RIES The mission of the Environmental Research Laboratories (ERL) is to conduct an integrated program of fundamental research, related technology development, and services to improve understanding and prediction of the geophysical environment comprising the oceans and inland waters, the lower and upper atmosphere, the space environment, and the Earth. 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