Numerical Simulations of the Physical Process for Hailstone Growth

VOL.19 NO. 1ACTA METEOROLOGICA SINICA2005Numerical Simulations of the Physical Process forHailstone Growth*FANG Wen1,3 (房文), ZHENG Guoguang2 (郑国光), and HU Zhjin3 (胡志晋)'Nanjing Univereity of Inforrmation Science and Technology, Nanjing 2100442 State Meteorological Administration of China, Beijing 1000813 Chinese Academy of Meterological Sciences, Bejjing 100081(Received March 30, 2004; revised August 31, 2004)ABSTRACTa determinant role in growth rates and different structures. However, many numerical model researchersmade extrapolation of the key heat transfer cofficient of the thermal balance expression from measurementsof evaporating water droplets obtained under small Renolds numbers (Re≤200) introduced by Ranz andMarshall, leading to great diference from reality. This paper is devoted to the parameterization of measuredheat transfer cofficients under Renolds numbers related to actual hail scales proposed by Zheng, which arethen applied, to Hu-He 1D and 3D models for hail growth respectively, indicating that the melting rateof a hailstone is 12% 50% bigger, the evaporation rate is 10%-20% higher and the dry-wet growth rate is10%-40% larger from the present simulations than from the prototype models.Key words: hail, paramneterization, numerical simulation, heat transfer1. Introductiontransfers are a lot more vigorous from oblate than fromspherical particles. Based on accurate measurementsTheoretical and experimental studies on the phys-of the surface temperatures of ice particles cooled onical processes of hail growth (Schumann, 1938; Lud-1.1 x 104≤Re≤5.2 x 104 using a thermal imag-lan, 1958; List, 1963) showed that its growth rate anding system, Zheng (1994) developed a numerical modelstructural characteristics depend on the heat and massfor defining a heat transfer coefficient denoted as Nutransfers; its dynamic characteristics determine hail-standing for Nusselt number and experiments indicatestone's movement and stay in clouds and damage donethat the obtained Nu is approximately 30% biggerto ground bodies, actually controlling the growth in-compared to the one coming from RM expression, 40%side clouds. As we know, the heat transfers affectslarger from oblate than from spherical particles withdirectly hailstone's wet growth, melting and evapo- the diameter equal to the major axis of the formerration. In their expression of heat balance equation,and even twice as large from coarse particles as fromhowever, a lot of researchers dealing with cloud-physicsspheroids of the same diamneter.models have adopted the heat transfer coefficient mea-In past studies, meteorological scientists employedsured from evaporating water drops at small Renoldsonly the measurement of evaporating water drops atnumbers (Re≤200) produced by Ranz and MarshallRe≤200 (vide ante) as the heat transfer cofficient(RM) and extended it to1x10≤Re≤1x 107, thus that is a key component of the thermal balance equa-leading to great difference from the conditions of ac-tion for hail growth, with the RM cofficient showntual hail particles (List, 1989; Zheng, 1994). Macklininvestigated by experiment the heat and vapor trans-Nu= 2.0+ 0.53 x Re-/2,(1)fer cofficients from melting particles, discovering thatcont中国煤化工the obtained transfers are considerably stronger com-pared to the equivalents given by RM and that theYHCNMHG.57,(2)"Supported by the National Natural Science Foundation of China under Grant No. 49775255.04ACTA METEOROLOGICA SINICAVOL.19Nu= 0.313 x Re0.599,(3)For convenience of later discussion Nu of Eqs.(1)Nu= 0.114 x Re0.74,(4to (4) are denoted as Nu1 through Nus, in order andother physical quantities calculated using Nu1 - Nuswhere Nu of Eqs.(2)-(4) measured at 1 x 104will be given by corresponding subscripts 1 throughRe≤5.2 x 104 proposed by Zheng who used a verti- 4. Nu1 to Nua, respectively, fom Eqs(1) to (4) atcally pesusenrtrollable wind tunnel for hail growth.1x103 ≤Re≤5.2x 104 are given in Table 1, in whichEquation (2) is applicable to smooth sphere, i.e., theirwe see that Nu2 is by 2%-30% larger than Nu, aboutaspect ratioa = 1; Eq.(3) holds for smooth oblate 14%, on average, higher; Nus exceeds Nu1 by 4%-particles witha= 0.67 and Eq.(4) is true for rough 69%, averaging roughly 35% larger; Nua(B = =2% par-oblate particles (a = 0.67) with surface having coarse ticle) is 83% greater compared to Nu, but at largerness β =2%.Re, Nus is by 150% larger than Nu1.Table 1. Comparison of Nu nunbers of hiltones with various characteristic at1x10°≤Re≤5.2x 10*R1x 1031.5x 1032.8x 1033.4 x 1034.4x 1035.2x 1036.5x 103Nu118.88722.68230.25733.13737.42140.50745.052u216.92421.32530.43633.99839.38143.41549.189Nu19.65125.05236.41040.90147.73152.75560.299Nua18.91925.53940.53246.79556.632 .64.08475.589Re7.6x 1039.4x 1031x 1042.2x 1043x 1043.7x 105.2x 10448.55353.77355.40081.20594.491104.717123.771Ne53.77560.70162.88098.559117.618132.551160.92966.21975.21178.051125.166150.720170.895209.538Nu484.86199.316103.97186.336234.409273.762352.166Equations (1)-(4) are parameterized and put, sep- interest in this study.arately, into the 1D and 3D time-dependent cumulusIn the derivation of the model expressions of Hu-models developed by Hu and He (1988; 1989)， with He for wet-dry growth, melting and evaporation ofthe simulations for comparison.hailstones, the integral portion is approximated by anempirical expression. If, for example, the hail subli-2. Parameterization schememation expression were treated with that way,The 1D cloud model was employed to deriveSvh = πrkap(Qv - Qgo)NupooNoD'9exp( -入h D)dDthe specific water contents of in-cloud vepor, cloudJDgdroplets, the combination of graupels, ice crystals, rain= 2πkap(Qv - Qs0)0.29√pAvh/μwater and hail particles, and the conversion rate ofJ.NpDl-9exp(-AnD)dDspecific number concentration from 26 primary micro-physical processes, each of which is denoted by a cap-≈2πkap(Qv - Qso)0.29v pAvh/ μNnH:9ital letter for the process and two subscripted letters,:[(XnD.)+9 + r(2.9)(0.9XnD. + 1),the first for the phase of consumption and the second for production or action, which are also used to then there would result in roughly 10% error. Instead,indicate the change rate of specific mass during mi- we, by means of the results from integral by parts andcrophysics. Of the physical quantities, the Nu-related numerical integral, re derive its precise parameteriza-hail sublimation (Svh), melting (Mhr) and the critical tion fo中国煤化工; in error on the ordervalue of hail dry-wet growth (Chw) are of particular of 2%YHCNMHGNO.1FANG Wen, ZHENG Guoguang and HU Zhjin952.1 Sublimation (Svh) of hailstones2.1.1 For the wet growth of hailstones (k=1)Son1 = 2πkap(Qv - Q.o)0.26√pAvh/ TuNnAzl:9[(\nD.)-1.9+1.827exp( - -2.375)*(\nD.1)0.611 + 1.827],(5)Syn2 = 2πkap(Qv - Q0)0.17(Avh/ 1.57Nma)2:08C[AnD.)2.03 + 2.03()nD.)203+2.06exp( - 3.363)(>hD.)0.595 + 2.06],(6)Srh3 = 2πkap(Qr - Qo)0.156(pAvn/ 1).59 MNA-2.08 [(\nD.)208 + 2.08(\nD.)1.8+2.16exp(-2.375)(\hD.).1 + 2.16]，(7)Svb4 = 2mkap(Qv - Q.o)0.057(pA.n/ 1)4N7:752.[()D.)2.3.3 + 233(\nD.).3+1.049(2nD.).744 + 277].(8)2.1.2 For the dry growth of hailstones (kk=0)Sunl = {0.()- IykepQ(L - 1)(C++ ()}.[1+ LokdDQn( - 1)-1,(9)ktT\ RTktT RTSuna = {Eq.(6)- ykapQe(系- 1)(Can+ C)}.[1+ LokaDQ(系_ 1)-,(10)ktT( RTktT\RT.Sun3 ={Eq.(7)- Lygap9e(二n - 1)(Ca+(Cq)}.[1+ LskdDQm(o - 1)-1,(11)ktT\ RThtT\ RTSvns = {Ea.(8)- LykaDCo(o - 1)(Ca+ Cn)}.[1+ Lo.kaOQe(二- 1)-1.(12)kT RTktT RI2.2 Hail's melting (Mnr)Mhr1 = | Nohexp(-λnD)[e(T - To) + Lvkap(Qv - Q2o]0.26√AvhP/ 7μD0.9dD+=(Cc + Ch)(T - To)0.52LjVAvhP/ Wk(T - To) + Lykap(Qv - Qo)]Nh):19(\bD,)1.9 + 1.827exp(-2.375).(\nD.).811 + 1.827]+ E + (Ccn + Crn)T - T),(13)LMnra =0.33r (Anp/).57TIt(T - To)+ ILvkap(Qv - Q0)]NAl9[(AD.)2.03 + 2.03()nD.).08+2.06(AnD.).595 + 2.06]+ + (Ccn + Cr)(T -To),(14)fIMhr3 =033r(Aonp/u)0. 59e(T - To) + LwJkap(Q, - Q)AE2O8C[(\nD.)2.08 + 2.08()nD.)2.08Lf+2.16exp(-2.375)(\nD.)0.611 + 2.16] + + (Can + Crn)(T - To),Mhr4 =0.114π(AvhP/ (10i74Iit(T - Tn) + Lvkap(Qv .中国煤化工2.33(\hD,)-33MHCNMHG+2.77exp(- -0.979)(\)nD.).74 + 2.77]+兴+ (Ccn +urn(1-10).(16)Lp96ACTA METEOROLOGICA SINICAVOL.192.3 Critical value of wet-dry growth (Chw)Cuw1 ={oex()D)2D19.6/AvhpP[kt(T - T)+ Lokap(Q。- Qo)]dD .-CihC:(T - To){/Lp +Cw(T -T)]≈{.53/AvpPlx(T - To) + Lokap(Q。- Qo)NA\191(\D.)1.9 + 1.827exp(- 2375):(\hD.).11 + 1.827]+ CinC(T - To))}/1u+Cu(T- ToI，(17)Chw2 ={ 033r( AvVP)-51[e(T - To) + Lvkap(Qv - Q0)1NAaEz.03[(AnD.)2.03 + 2.03(\nD.)1.03u+2.06()nD.).595 + 2.06] + CinC:(T - T)}/[u;+Cu(T -T0小,(18)Crws = {0.313r( A.yJ.s0[e(T - To) + Lokap(Qv - Qw)\x2.0.(\n.2.00 + 2.08()nD.)2.08+2.16exp(-2.375)(\nD)0.811 + 2.16] + CinC(T - To)}/u;(I,+Cw(T- T),(19)Chw4 =114141().7.th(T - To) + LofapQ。- Qm1-a1a*(D.1-33 + 23(3xD.)1.3H+7px(-079)9)D.).7444 +.7]+ CunC(T -I)}/iu+Co(T-T).(20)3. Calculationsinto the model, the hail particles start wet growth atminute 21 as well except for lowered height (at 6.0 km3.1 Calculations from the 1D cloud modellevel), with -11.050°C observed; at minute 54 the wetAs a case study we computed soundings made at growth layer extends from the cloud base to the 6.0 kmDezhou of Shandong Province on 1 July 1989, with altitude. At this time the critial value is 2.76x10-3gthe results shown below.kg-1 s-1 for the wet-dry growth, with the critical tem-3.1.1 Critical value of the wet-dry growth of hai-perature reaching - 12.023°C. As Nu3 is put into thestones (Chw)model the wet growth occurs in model minute 24 at aTrends of the growth development denoted as 5.8 km height with the temperature of -9.9349C, fol-Chw1 to Crw4 are similar from Eqs.(17)-(20) but Chw2 lowed by the growth layer gradually thickened, reach-to Chw4 are bigger compared to Chw1, indicating ing its maximum layer extending to the height fromthat Chw2, Chw3 and Chw4 are ~10%， ~30% and cloud base at model minute 51, with dry-wet growth~40% larger in magnitude than ChwI, respectively critical value of 2.35x 10-3 gkg-1 s-1. Finally, we(see Fig.1). With Nu1 inserted into the model, hail substitute Nu4 into the 1D model hail wet growth hap-particles begin wet growth from minute 21 at the pens at minute 24 at 5.8 km level where temperature6.2 km level where T = -12.411°C, fllwed by the is -9.934°C, and the growth layer extends from cloudgrowth layer thickened progressively and finally ex- base thereto in minute 51. At this time the dry-wettending from the cloud base up to the 6.2 km height at growth critical value is 2.61x 10-3gkg-1 s-1 and theminute 54. At this time the critical value is 2.89x10- -3 critic中国煤化工put into the model,gkg-1 s- 1 for hail dry-wet growth, with the criticalMYHCNMH Gtemperature of 12.029°C. When Nu2 is substituted the wo.。..... T.g.nning at minute 24,NO.1 .FANG Wen, ZHENG Guoguang and HU Zhjin97results in different depth of growth layers. The Nu2-minute 27 compared to the Nu1 case, both arrivingrelated depth is comparable to that of the Nu1 caseat the same depth subsequent to minute 78. In theexcept an extra 200 m thickness calculated at minutes Nu4 experiment the moist growth layer is reduced by21 and 24 in the former case. With Nu3 used the400-600 m from model minute 27 in contrast to Nu1wet growth thickness diminishes by 200-400 m fromrun, reaching the same calculations after minute 81.100010000 -; (a).(8000; 600060004000置40020004.02.00.0Critical values of wet- dry growth (10~'g kg'.s")Critical values of wet- -dry growth (10-g kg'.s")Fig.1. The height-dependent distribution critical values of wet-dry growth in units of 10-3gkg-1 s-, atminutes 51 (a) and 54 (b).10000(a)b)8000-s 6000号4002000 -o十.03.Critical values of wet- -dry growth (10~g kg".g")Critical values of wet- dry growth (10-'g k'.s)Fig.2. Hail's sublimation and evaporation rates in units of 10-3 gkg-1 s-1 at minutes 60 (a) and 63 (b).3.1.2 Hail's sublimation (Svh)minute 63. On the average, Svh2 , Svh3 and Svh4 are bySvh1 to Svh4 from Eqs.(5) to (8), respectively, ~10%, 130% and 200% higher than Svh1, respectively,all reach the order of magnitude of 10- 6 at theindicating that the aspect ratio and surface coarseness6400 m level at minute 33, which increases there- of hail particles have great impacts on sublimation.fter. Svh1 (Svh2) arrives at its maximum of 1.93 x 3.1.3 Hail melting (Mur)10-4(2.20x 10-4) g kg-1 s-1 at 5400 (5600 m) atEquations (13)-(16) related Mhr1 to Mnr4 eachminute 63 (see Fig.2a). Svh3(Svha) has its maximum give中国煤化工)-0 gkg-1 s-1 priorof3.67x 10-4(3.83x10-4)gkg-1s-1 at 5600 (5400) to rYHmelting value reachesm at minute 60 (Fig.2b), both reducing bit by bit after its nCNM,HG 10-3,8.6x 10-8 and98ACTA METEOROLOGICA SINICAVOL.191.06x 10-3gkg-1s-1 for Mhrl to Mhr4, in order, as 2.8433, 3.3773, 3.5741 and 3.1423 g kg-1 at the heightshown in Fig.3b, with subsequent decline and termina-of 5600, 5200, 5200 and 5000 m at minutes 66, 69, 69tion at minute 93 (refer to Fig.3). On the average, the and 69, respectively.melting in Mhr2, Mnr3 and Mhr4 is 12%-15%, ~30% 3.1.5 Rainfall and hailfall with Nu1 to Nu4 intro-and ~50%, respectively, higher compared to that induced into the 1D modelMhr1.The Nu1 to Nug-relative rainfall and hailfall are3.1.4 Scale (Xh) and content of hail particles (Qn)17.358 and 2.518, 18.002 and 2.410, 18.815 and 2.124,Nu2, Nu3 and Nua- related hail scale (Xn) and and 19.471 and 1.694 mm in depth, in order. Thiscontent of particles (Qh) are larger than those asso- means that owing to hail's aspect ratio and coarsenessciated with Nu1. Qh1 to Qn4 have the maximum of its melting is augmented, leading to increase in rainfall0001000010000 7 (@)1 (b)800060006003艳葛4000蓬4000鲁400020002Melting rate (10°g kg+.s-) .Melting rate (10~°g kg'.s")Melting rate (10-'g kg'●s)Fig.3. Hail's melting rate at minutes 75 (), 81 (b) and 83 (c), in order. Unite: 10-3 gkg~°s-'.and decrease in hailfall.,It is apparent from Fig.6 that graupel collectingcloud water for growth (Crg) acts a8 the most signif-3.2 Calculation by the 3D cloud modelicant mechanism. Ice crystals are located at a higherHailfall occurred on 10 June 1996 in such sub-level with respect to cloud water, so that the higherurbs of Beijing as Haidian, Xuanwu, Shijingshan andcentral concentration for graupel production is at aMiyun Districts, with measured hailstone having itslower part of its profle. The graupel-growth by col-major (minor) axis, 8 (4) cm long for larger size andlecting ice crystals (Cig) is increased as quantity of the1.5 (0.6) cm in length for smnall particles. Echoes in-crystals grows. But the coalescence of rain dropletschuding those from 11 km level, with 6.5 km-level cen-with ice crystals for the growth is losing effect becausetral intensity of 70 dBz. Based on Zheng's heat trans-of gradually reduced rising velocity and no existencefer cofficients (Nu2 to Nu4) introduced into the 3Dof super-cooled rain drops. Growth rate of graupel viamodel, calculations show that hailfall is diminished,sublimation from vapor (Svg) changes with the amountparticle's size increased, to some degree, and rainfallof graupel, and sublimation makes insignifcant contri-strengthed compared to those from the original model.bution thereto throughout the process, and acts as aNu1-associated rainfall and haifall are 8.21 x 108 anddominant mechanics for the increase of graupel in the4.86 x 108 kg, respectively in contrast to rainfall oflater stage of cloud development. The automatic8.23x 108,8.44x 108 and 8.66x 108 kg and hailfallconversion of ice crystals into graupel (Aig) is a chiefof 4.84x 108,4.54x 108 and 4.23 x 108 kg in relationprocess of its production (see Fig.6) but neverthelessto the introduced Nu2, Nug and Nu4, in order (seeAigPl中国煤化工xquent stage of grau-pel in;ibution made by theFigs.4 and 5).:fYHCNMHGNO.1FANG Wen, ZHENG Guoguang and HU Zhijin)912.00 -12.00 710.00 -8.00里8.006.00国6.00-←Nu4.00昌4.00琶2.0020 30 4050607080 901002030405060708090100Time (min)Time (nin)Fig.4. Temporally-varying total weight ofFig.5. As in Fig.4 but for the total rainfall.hailstones (105 kg) from the 3D model intowhich are introduced Nu1 to Nu4.2.507 (回)(0).00→A十M.1.50豆1.501.000.500.000 " 1020304050608090100.0020.0040.00 60.0080.00100.00Fig.6. The time -varying specific mass (i unitsof 10 kg 1 s~1) of graupel production through the 4mechanisms in (田) and conversion of ice crystals into graupel (Aig) and melting hailstone for graupel (Mng)in (b).conversion of cloud droplets into graupel to be given and hail growth at the expense of cloud droplets (Cch)in the figure. However, the coalescence of rain drops make up a larger proportion and hail growth on rainand ice crystals (Ci) generates a certain amount of droplets (Crth), ice srystals (Crb) and sublimation fromgraupel. Only a small quantity of water from meltingvapor (Svh) in combination are smaller by 1-2 ordershail particles that thus reduce their scale is changed of magnitnde compared to the first two factors put to-into graupel, which is, however, noticeably larger in getheramount compared to graupel from ice crystals.中国煤化工droplet growth byFigure 7 portrays the evolution of mechanisms for collecMYHC N M H Gutomatic conversionhail genesis, of which graupel-to-hail conversion (Agh) of cloud (water) into rain water (Acr) represent the100ACTA METEOROLOGICA SINICAVOL.190.600.01 7.|(a;(6)。0.40 .9 0.01 -唇0.20-0.00 20.00 40.00 60.00 80.00 100.000.00 20.0040.00 60.00 80.00 100.00Time (min)Time (mmin)Fig.7. Hail growth by means of graupel to hailstone conversion (Agh) and at the expense of cloud droplets(Ccn) in (a) and by collecting rain drops (Crb), ice crystals (Cin) and the sublimation from vapor (Svb) in(b).most important mechanisms for the augmentation ofinitial stage of cloud development. In addition, gr81u-rain water in the early stage of cloud development.pel growth via collecting super- cooled raindrops (Crg)The conversion of could into rain water is responsibleconsumes a certain amount of rain water and evapo-for incipient rain droplets, which grow fast upon C0a-ration of rain drops (Svr) serves as a main mechanicslescence with cloud water. Graupel begins to melt atfor rain water consumption throughout cloud develop-minute 22 when entering the warm section of cloud andment, with large quantities of rain water evaporatedthe melting serves as the most prominent mechanismin the later stage, sufficiently big to account for nearlyfor producing rain water and also as the extremelyhalf water from melting graupel, indicating that rainsignificant source for subsequent persistent rainfall.droplets from graupel melting in the warm portion ofThe coalescence of super coolded rain droplets withcloud are evaporated in big quantities, failing to reachice crystals (Cri) to produce graupel is the predom-ground as rainfall.inant mechanism for rain water consumption in the2.00b)1.60-1.602 1.20置1.20售0.80-0.80三0.40-20.40-M0.0020.00 40.0060.0080.00 100.0020.0040.00rime (min)Fig.8. Time dependent 4 mechanisms for rain wate中国煤化工xion ().TYHCNMHGNO.1FANG Wen, ZHENG Guoguang and HU ZhijinAs shown in Fig.9, each of the Mhr1 to Mhr4 has Renolds numbers (Re ≤200) used in past studiesa maximum of 2.217x10 -3 2.222x10- , 2.659x 10-3would result in bigger difference frorn reality.and 3.038x10-3 gkg-1 s-1, in order. Figure 10 gives(2) Nu2 to Nuz of Eqs.(2) to (3) developed bythe maximum of Svh1 to Srhs, which is, respectively, Zheng (1994) are put, separately, into the 1D and 3D2.413x 103, 2.403x 103,3.925x 103 and 5.165x 103.time dependent cumulus models, leading to increased0.004(decreased) rainfall (ailall). Results related to Nu2to Nus change with the aspect ratio and coarseness ofhail particles, with heat transfer stronger for elliptical0.003and coarse particles than for spherical and smoothones, showing that the melting, evaporation rates andcritical value of hail dry-wet growth are bigger in the0.002former than in the latter case. on the average, therates of hail melting and evaporation (critical value of0.001its dry-wet growth) are 12%-15% and 10%-200%, inorder, (10%- 40%)} higher with Nu2 to N u4 introducedinto the models than with Nu1 employed in the nu-0.000merical study.20800Time (min)REFERENCESFig.9. Time -varying hailstone melting in rlation to inserted Nu1 to Nu4.Hu Zhijin and He Guanfang, 1988: Numerical simulation0.006 -of microphysical processes in cumulonimbu8. ParI: Microphysical model. Acta Meteor. Sinica, 2(4):471-489.Hu Zhijin and He Guanfang, 1989: Numerical simulation0.004-of microphysical processes in cumulonimbus. PartgII: Case studies of shower, hailstorm and torrentialrain. Acta Meteor. Sinica, 3(2), 185 199.List, R., 1963: General heat and mas exchange of spher-ical hailstones. J. Atmos. Sci, 20, 189-197.List, R., 1989: Analysis of sensitivities and error prop-agation in heat and mass transfer of spheroidal20.0 40.060.080.0100.0hailstones using sphread sheet. J. Appl. Meteor.,28, 118-1127.Fig.10. As in Fig.9 but for hailstone evaporation.Ludlam, F. H, 1958: The hail problem. Nubila, 1, 12-96.Schumann, T. E. W., 1938: The theory of hailstone for-4. Concluding remarksmation. Quart. J. Roy. Meteor. Soc, 64, 3-21.(1) The heat transfer cofficient for hailstone dZheng Guoguang, 1994: An experimental investigationvelopment is an innegligible parameter of its growthof convective heat transfer of rotating and gyrat-rate and structure. The Nusselt number from the mea-ing hailstone models. Ph. D. dissertation, Dept ofsurement of water droplet evaporation under smallPhysics, University of Toronto, Canada, 121 pp.中国煤化工MYHCNMHG