Numerical Simulations of the Physical Process for Hailstone Growth

Transcription

1 NO.1 FANG Wen, ZHENG Guoguang and HU Zijin 93 Numerical Simulations of te Pysical Process for Hailstone Growt FANG Wen 1,3 ( ), ZHENG Guoguang 2 ( ), and HU Zijin 3 ( ) 1 Nanjing University of Information Science and Tecnology, Nanjing State Meteorological Administration of Cina, Beijing Cinese Academy of Meteorological Sciences, Beijing (Received Marc 30, 2004; revised August 31, 2004) ABSTRACT Teoretical and experimental studies sow tat during ail growt te eat and mass transfers play a determinant role in growt rates and different structures. However, many numerical model researcers made extrapolation of te key eat transfer coefficient of te termal balance expression from measurements of evaporating water droplets obtained under small Renolds numbers (Re 200) introduced by Ranz and Marsall, leading to great difference from reality. Tis paper is devoted to te parameterization of measured eat transfer coefficients under Renolds numbers related to actual ail scales proposed by Zeng, wic are ten applied, to Hu-He 1D and 3D models for ail growt respectively, indicating tat te melting rate of a ailstone is 12%-50% bigger, te evaporation rate is 10%-200% iger and te dry-wet growt rate is 10%-40% larger from te present simulations tan from te prototype models. Key words: ail, parameterization, numerical simulation, eat transfer 1. Introduction Teoretical and experimental studies on te pysical processes of ail growt (Scumann, 1938; Ludlan, 1958; List, 1963) sowed tat its growt rate and structural caracteristics depend on te eat and mass transfers; its dynamic caracteristics determine ailstone s movement and stay in clouds and damage done to ground bodies, actually controlling te growt inside clouds. As we know, te eat transfers affects directly ailstone s wet growt, melting and evaporation. In teir expression of eat balance equation, owever, a lot of researcers dealing wit cloud-pysics models ave adopted te eat transfer coefficient measured from evaporating water drops at small Renolds numbers (Re 200) produced by Ranz and Marsall (RM) and extended it to Re , tus leading to great difference from te conditions of actual ail particles (List, 1989; Zeng, 1994). Macklin investigated by experiment te eat and vapor transfer coefficients from melting particles, discovering tat te obtained transfers are considerably stronger compared to te equivalents given by RM and tat te transfers are a lot more vigorous from oblate tan from sperical particles. Based on accurate measurements of te surface temperatures of ice particles cooled on Re using a termal imaging system, Zeng (1994) developed a numerical model for defining a eat transfer coefficient denoted as Nu standing for Nusselt number and experiments indicate tat te obtained N u is approximately 30% bigger compared to te one coming from RM expression, 40% larger from oblate tan from sperical particles wit te diameter equal to te major axis of te former and even twice as large from coarse particles as from speroids of te same diameter. In past studies, meteorological scientists employed only te measurement of evaporating water drops at Re 200 (vide ante) as te eat transfer coefficient tat is a key component of te termal balance equation for ail growt, wit te RM coefficient sown as Nu = Re 1/2, (1) contrasted wit Nu = 0.33 Re 0.57, (2) Supported by te National Natural Science Foundation of Cina under Grant No

2 94 ACTA METEOROLOGICA SINICA VOL.19 Nu = Re 0.599, (3) Nu = Re 0.74, (4) were Nu of Eqs.(2)-(4) measured at Re proposed by Zeng wo used a vertically pressure-controllable wind tunnel for ail growt. Equation (2) is applicable to smoot spere, i.e., teir aspect ratio α = 1; Eq.(3) olds for smoot oblate particles wit α = 0.67 and Eq.(4) is true for roug oblate particles (α = 0.67) wit surface aving coarseness β =2%. For convenience of later discussion N u of Eqs.(1) to (4) are denoted as Nu 1 troug Nu 4, in order and oter pysical quantities calculated using Nu 1 Nu 4 will be given by corresponding subscripts 1 troug 4. Nu 1 to Nu 4, respectively, from Eqs.(1) to (4) at Re are given in Table 1, in wic we see tat Nu 2 is by 2%-30% larger tan Nu 1, about 14%, on average, iger; Nu 3 exceeds Nu 1 by 4%- 69%, averaging rougly 35% larger; Nu 4 (β =2% particles) is 83% greater compared to Nu 1, but at larger Re, Nu 4 is by 150% larger tan Nu 1. Table 1. Comparison of Nu numbers of ailstones wit various caracteristic at Re Re Nu Nu Nu Nu Re Nu Nu Nu Nu Equations (1)-(4) are parameterized and put, separately, into te 1D and 3D time-dependent cumulus models developed by Hu and He (1988; 1989), wit te simulations for comparison. 2. Parameterization sceme Te 1D cloud model was employed to derive te specific water contents of in-cloud vapor, cloud droplets, te combination of graupels, ice crystals, rain water and ail particles, and te conversion rate of specific number concentration from 26 primary micropysical processes, eac of wic is denoted by a capital letter for te process and two subscripted letters, te first for te pase of consumption and te second for production or action, wic are also used to indicate te cange rate of specific mass during micropysics. Of te pysical quantities, te N u-related ail sublimation (S v ), melting (M r ) and te critical value of ail dry-wet growt (C w ) are of particular interest in tis study. In te derivation of te model expressions of Hu- He for wet-dry growt, melting and evaporation of ailstones, te integral portion is approximated by an empirical expression. If, for example, te ail sublimation expression were treated wit tat way, S v = πk d ρ(q v Q s0 )Nu D 8 = 2πk d ρ(q v Q s0 )0.29 ρa v µ D N D D 1.9 exp( λ D)dD N 0 D 1.9 exp( λ D)dD 2πk d ρ(q v Q s0 )0.29 ρa v µn λ 1.9 [(λ D ) Γ(2.9)(0.9λ D + 1)], ten tere would result in rougly 10% error. Instead, we, by means of te results from integral by parts and numerical integral, re-derive its precise parameterization formula tat give calculations in error on te order of 2%.

3 NO.1 FANG Wen, ZHENG Guoguang and HU Zijin Sublimation (S v ) of ailstones For te wet growt of ailstones (kk=1) S v1 = 2πk d ρ(q v Q s0 )0.26 ρa v µn λ 1.9 [ (λ D ) exp( 2.375) (λ D 1 ) ], (5) S v2 = 2πk d ρ(q v Q s0 )0.17(ρA v µ) 0.57 N λ 2.03 [ (λ D ) (λ D ) exp( 3.363)(λ D ) ], (6) S v3 = 2πk d ρ(q v Q s0 )0.156(ρA v µ) N λ 2.08 [ (λ D ) (λ D ) exp( 2.375)(λ D ) ], (7) S v4 = 2πk d ρ(q v Q s0 )0.057(ρA v µ) 0.74 N λ 2.33 [ (λ D ) (λ D ) (λ D ) ]. (8) For te dry growt of ailstones (kk=0) S v1 = {Eq.(5) k d ρq si S v2 = {Eq.(6) k d ρq si S v3 = {Eq.(7) k d ρq si S v4 = {Eq.(8) k d ρq si 2.2 Hail s melting (M r ) M r1 = RT 1)(C c + C r )} [1 + L sk d ρq si RT 1)(C c + C r )} [1 + L sk d ρq si RT 1)(C c + C r )} [1 + L sk d ρq si RT 1)(C c + C r )} [1 + L sk d ρq si N 0 exp( λ D) 2πD [k t (T T 0 ) + L v k d ρ(q v Q s0 )]0.26 A v ρ µd 0.9 dd + C w (C c + C r )(T T 0 ) RT 1)] 1, (9) RT 1)] 1, (10) RT 1)] 1, (11) RT 1)] 1. (12) = 0.52 Av ρ µ[k t (T T 0 ) + L v k d ρ(q v Q s0 )]N λ 1.9 [(λ D ) exp( 2.375) (λ D ) ] + C w + (C c + C r )(T T 0 ), (13) M r2 = 0.33π (A v ρ µ) 0.57 [k t (T T 0 ) + L v k d ρ(q v Q s0 )]N λ 1.9 [(λ D ) (λ D ) (λ D ) ] + C w + (C c + C r )(T T 0 ), (14) M r3 = 0.33π (A v ρ µ) [k t (T T 0 ) + L v k d ρ(q v Q s0 )]N λ 2.08 [(λ D ) (λ D ) exp( 2.375)(λ D ) ] + C w + (C c + C r )(T T 0 ), (15) M r4 = 0.114π (A v ρ µ) 0.74 [k t (T T 0 ) + L v k d ρ(q v Q s0 )]N λ 2.33 [(λ D ) (λ D ) exp( 0.979)(λ D ) ] + C w + (C c + C r )(T T 0 ). (16)

4 96 ACTA METEOROLOGICA SINICA VOL Critical value of wet-dry growt (C w ) { C w1 = N 0 exp( λ D)2πD C w2 = C w3 = C w4 = } C i C i (T T 0 ) /[ + C w (T T 0 )] A v ρ µ [k t(t T 0 ) + L v k d ρ(q v Q s0 )]dd { A v ρ 0.53π µ [k t(t T 0 ) + L v k d ρ(q v Q s0 )]N λ 1.9 [(λ D ) exp( 2.375) }/ (λ D ) ] + C i C i (T T 0 ) [ + C w (T T 0 )], (17) { 0.33π( A vρ µ )0.57 [k t (T T 0 ) + L v k d ρ(q v Q s0 )]N λ 2.03 [(λ D ) (λ D ) 1.03 }/ +2.06(λ D ) ] + C i C i (T T 0 ) [ + C w (T T 0 )], (18) { 0.313π( A vρ µ )0.599 [k t (T T 0 ) + L v k d ρ(q v Q s0 )]N λ 2.08 [(λ D ) (λ D ) 1.08 }/ +2.16exp( 2.375)(λ D ) ] + C i C i (T T 0 ) [ + C w (T T 0 )], (19) { 0.114π( A vρ µ )0.74 [k t (T T 0 ) + L v k d ρ(q v Q s0 )]N λ 2.33 [(λ D ) (λ D ) 1.33 }/ +2.77exp( 0.979)(λ D ) ] + C i C i (T T 0 ) [ + C w (T T 0 )]. (20) 3. Calculations 3.1 Calculations from te 1D cloud model As a case study we computed soundings made at Dezou of Sandong Province on 1 July 1989, wit te results sown below Critical value of te wet-dry growt of ailstones (C w ) Trends of te growt development denoted as C w1 to C w4 are similar from Eqs.(17)-(20) but C w2 to C w4 are bigger compared to C w1, indicating tat C w2, C w3 and C w4 are 10%, 30% and 40% larger in magnitude tan C w1, respectively (see Fig.1). Wit Nu 1 inserted into te model, ail particles begin wet growt from minute 21 at te 6.2 km level were T = C, followed by te growt layer tickened progressively and finally extending from te cloud base up to te 6.2 km eigt at minute 54. At tis time te critical value is g kg 1 s 1 for ail dry-wet growt, wit te critical temperature of C. Wen Nu 2 is substituted into te model, te ail particles start wet growt at minute 21 as well except for lowered eigt (at 6.0 km level), wit C observed; at minute 54 te wet growt layer extends from te cloud base to te 6.0 km altitude. At tis time te critical value is g kg 1 s 1 for te wet-dry growt, wit te critical temperature reacing C. As Nu 3 is put into te model te wet growt occurs in model minute 24 at a 5.8 km eigt wit te temperature of C, followed by te growt layer gradually tickened, reacing its maximum layer extending to te eigt from cloud base at model minute 51, wit dry-wet growt critical value of g kg 1 s 1. Finally, we substitute Nu 4 into te 1D model ail wet growt appens at minute 24 at 5.8 km level were temperature is C, and te growt layer extends from cloud base tereto in minute 51. At tis time te dry-wet growt critical value is g kg 1 s 1 and te critical temperature is C. As we see, wit Nu 1 to Nu 4 put into te model, te wet growt, altoug beginning at minute 24,

5 NO.1 FANG Wen, ZHENG Guoguang and HU Zijin 97 results in different dept of growt layers. Te Nu 2 - related dept is comparable to tat of te Nu 1 case except an extra 200 m tickness calculated at minutes 21 and 24 in te former case. Wit Nu 3 used te wet growt tickness diminises by m from minute 27 compared to te Nu 1 case, bot arriving at te same dept subsequent to minute 78. In te Nu 4 experiment te moist growt layer is reduced by m from model minute 27 in contrast to Nu 1 run, reacing te same calculations after minute 81. Fig.1. Te eigt-dependent distribution critical values of wet-dry growt in units of 10 3 g kg 1 s 1, at minutes 51 (a) and 54 (b). Fig.2. Hail s sublimation and evaporation rates in units of 10 3 g kg 1 s 1 at minutes 60 (a) and 63 (b) Hail s sublimation (S v ) S v1 to S v4 from Eqs.(5) to (8), respectively, all reac te order of magnitude of 10 6 at te 6400 m level at minute 33, wic increases tereafter. S v1 (S v2 ) arrives at its maximum of ( ) g kg 1 s 1 at 5400 (5600 m) at minute 63 (see Fig.2a). S v3 (S v4 ) as its maximum of ( ) g kg 1 s 1 at 5600 (5400) m at minute 60 (Fig.2b), bot reducing bit by bit after minute 63. On te average, S v2, S v3 and S v4 are by 10%, 130% and 200% iger tan S v1, respectively, indicating tat te aspect ratio and surface coarseness of ail particles ave great impacts on sublimation Hail melting (M r ) Equations (13)-(16) related M r1 to M r4 eac give values smaller tan g kg 1 s 1 prior to minute 45, after wic te melting value reaces its maximum of , , and

6 98 ACTA METEOROLOGICA SINICA VOL g kg 1 s 1 for M r1 to M r4, in order, as sown in Fig.3b, wit subsequent decline and termination at minute 93 (refer to Fig.3). On te average, te melting in M r2, M r3 and M r4 is 12%-15%, 30% and 50%, respectively, iger compared to tat in M r Scale (X ) and content of ail particles (Q ) Nu 2, Nu 3 and Nu 4 - related ail scale (X ) and content of particles (Q ) are larger tan tose associated wit Nu 1. Q 1 to Q 4 ave te maximum of , , and g kg 1 at te eigt of 5600, 5200, 5200 and 5000 m at minutes 66, 69, 69 and 69, respectively Rainfall and ailfall wit Nu 1 to Nu 4 introduced into te 1D model Te Nu 1 to Nu 4 -relative rainfall and ailfall are and 2.518, and 2.410, and 2.124, and and mm in dept, in order. Tis means tat owing to ail s aspect ratio and coarseness its melting is augmented, leading to increase in rainfall Fig.3. Hail s melting rate at minutes 75 (a), 81 (b) and 83 (c), in order. Units: 10 3 g kg 1 s 1. and decrease in ailfall. 3.2 Calculation by te 3D cloud model Hailfall occurred on 10 June 1996 in suc suburbs of Beijing as Haidian, Xuanwu, Sijingsan and Miyun Districts, wit measured ailstone aving its major (minor) axis, 8 (4) cm long for larger size and 1.5 (0.6) cm in lengt for small particles. Ecoes including tose from 11 km level, wit 6.5-km-level central intensity of 70 dbz. Based on Zeng s eat transfer coefficients (Nu 2 to Nu 4 ) introduced into te 3D model, calculations sow tat ailfall is diminised, particle s size increased, to some degree, and rainfall strengted compared to tose from te original model. Nu 1 -associated rainfall and ailfall are and kg, respectively in contrast to rainfall of , and kg and ailfall of , and kg in relation to te introduced Nu 2, Nu 3 and Nu 4, in order (see Figs.4 and 5). It is apparent from Fig.6 tat graupel collecting cloud water for growt (C rg ) acts as te most significant mecanism. Ice crystals are located at a iger level wit respect to cloud water, so tat te iger central concentration for graupel production is at a lower part of its profile. Te graupel-growt by collecting ice crystals (C ig ) is increased as quantity of te crystals grows. But te coalescence of rain droplets wit ice crystals for te growt is losing effect because of gradually reduced rising velocity and no-existence of super-cooled rain drops. Growt rate of graupel via sublimation from vapor (S vg ) canges wit te amount of graupel, and sublimation makes insignificant contribution tereto trougout te process, and acts as a dominant mecanics for te increase of graupel in te later stage of cloud development. Te automatic conversion of ice crystals into graupel (A ig ) is a cief process of its production (see Fig.6) but neverteless A ig plays a small role in te subsequent stage of graupel increase, wit too small contribution made by te

7 NO.1 FANG Wen, ZHENG Guoguang and HU Zijin 99 Fig.4. Temporally-varying total weigt of ailstones (10 6 kg) from te 3D model into wic are introduced Nu 1 to Nu 4. Fig.5. As in Fig.4 but for te total rainfall. Fig.6. Te time-varying specific mass (in units of 10 6 kg 1 s 1 ) of graupel production troug te 4 mecanisms in (a) and conversion of ice crystals into graupel (A ig) and melting ailstone for graupel (M g ) in (b). conversion of cloud droplets into graupel to be given in te figure. However, te coalescence of rain drops and ice crystals (C ri ) generates a certain amount of graupel. Only a small quantity of water from melting ail particles tat tus reduce teir scale is canged into graupel, wic is, owever, noticeably larger in amount compared to graupel from ice crystals. Figure 7 portrays te evolution of mecanisms for ail genesis, of wic graupel-to-ail conversion (A g ) and ail growt at te expense of cloud droplets (C c ) make up a larger proportion and ail growt on rain droplets (C r ), ice srystals (C r ) and sublimation from vapor (S v ) in combination are smaller by 1-2 orders of magnitude compared to te first two factors put togeter. From Fig.8 we see tat rain droplet growt by collecting cloud water (C cr ) and automatic conversion of cloud (water) into rain water (A cr ) represent te

8 100 ACTA METEOROLOGICA SINICA VOL.19 Fig.7. Hail growt by means of graupel to ailstone conversion (A g ) and at te expense of cloud droplets (C c ) in (a) and by collecting rain drops (C r ), ice crystals (C i ) and te sublimation from vapor (S v ) in (b). most important mecanisms for te augmentation of rain water in te early stage of cloud development. Te conversion of could into rain water is responsible for incipient rain droplets, wic grow fast upon coalescence wit cloud water. Graupel begins to melt at minute 22 wen entering te warm section of cloud and te melting serves as te most prominent mecanism for producing rain water and also as te extremely significant source for subsequent persistent rainfall. Te coalescence of super-coolded rain droplets wit ice crystals (C ri ) to produce graupel is te predominant mecanism for rain water consumption in te initial stage of cloud development. In addition, graupel growt via collecting super-cooled raindrops (C rg ) consumes a certain amount of rain water and evaporation of rain drops (S vr ) serves as a main mecanics for rain water consumption trougout cloud development, wit large quantities of rain water evaporated in te later stage, sufficiently big to account for nearly alf water from melting graupel, indicating tat rain droplets from graupel melting in te warm portion of cloud are evaporated in big quantities, failing to reac ground as rainfall. Fig.8. Time-dependent 4 mecanisms for rain water production (a) and consumption (b).

9 NO.1 FANG Wen, ZHENG Guoguang and HU Zijin 101 As sown in Fig.9, eac of te M r1 to M r4 as a maximum of , , and g kg 1 s 1, in order. Figure 10 gives te maximum of S v1 to S v4, wic is, respectively, , , and Renolds numbers (Re 200) used in past studies would result in bigger difference from reality. (2) Nu 2 to Nu 3 of Eqs.(2) to (3) developed by Zeng (1994) are put, separately, into te 1D and 3D time-dependent cumulus models, leading to increased (decreased) rainfall (ailfall). Results related to Nu 2 to Nu 4 cange wit te aspect ratio and coarseness of ail particles, wit eat transfer stronger for elliptical and coarse particles tan for sperical and smoot ones, sowing tat te melting, evaporation rates and critical value of ail dry-wet growt are bigger in te former tan in te latter case. on te average, te rates of ail melting and evaporation (critical value of its dry-wet growt) are 12%-15% and 10%-200%, in order, (10%-40%) iger wit Nu 2 to Nu 4 introduced into te models tan wit Nu 1 employed in te numerical study. Fig.9. Time -varying ailstone melting in relation to inserted Nu 1 to Nu 4. REFERENCES Hu Zijin and He Guanfang, 1988: Numerical simulation of micropysical processes in cumulonimbus. Part I: Micropysical model. Acta Meteor. Sinica, 2(4), Hu Zijin and He Guanfang, 1989: Numerical simulation of micropysical processes in cumulonimbus. Part II: Case studies of sower, ailstorm and torrential rain. Acta Meteor. Sinica, 3(2), List, R., 1963: General eat and mass excange of sperical ailstones. J. Atmos. Sci., 20, Fig.10. As in Fig.9 but for ailstone evaporation. 4. Concluding remarks (1) Te eat transfer coefficient for ailstone de velopment is an innegligible parameter of its growt rate and structure. Te Nusselt number from te measurement of water droplet evaporation under small List, R., 1989: Analysis of sensitivities and error propagation in eat and mass transfer of speroidal ailstones using spread seet. J. Appl. Meteor., 28, Ludlam, F. H., 1958: Te ail problem. Nubila, 1, Scumann, T. E. W., 1938: Te teory of ailstone formation. Quart. J. Roy. Meteor. Soc., 64, Zeng Guoguang, 1994: An experimental investigation of convective eat transfer of rotating and gyrating ailstone models. P. D. dissertation, Dept of Pysics, University of Toronto, Canada, 121 pp.