COLLISION RATE ENHANCEMENT IN TURBULENT CLOUDS OF DIFFERENT TYPES. M. Pinsky and A. Khain* The Hebrew University of Jerusalem, Israel

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1 P.4 COLLISION RATE ENHANCEMENT IN TURBULENT CLOUDS OF DIFFERENT TYPES M. Pnsky and A. Khan* The Hebrew Unversty of Jerusalem, Israel. INTRODUCTION It s wdely accepted now that turbulence enhances the rate of partcle collsons see overvews by Pnsky et al 000; Vallancourt and Yau 000; Shaw 003). There exst three major mechansms that turbulence affects the collson rate: a) ncrease n the relatve partcle velocty or ncrease n the swept volume); ths effect s also known as turbulent transport effect; b) formaton of concentraton nhomogenety partcle clusterng), and c) turbulence effect on the hydrodynamc drop nteracton HDI) that leads to an ncrease n the collson effcency. Publshed reports dedcated to turbulence effects on HDI are qute scarce. Small amount of such studes s surprsng because gravtynduced values of the collson effcences are small ) Pruppacher and Klett 997) keepng a large volume for turbulence to ncrease the collson effcency and the collson rate. Almeda 979) and Kozol and Leghton 996) analyzed the effects of turbulent vortces of scales smaller than HDI, whch normally fall well nto the vscous range. The effect of these * Correspondng author address: Prof. Alexander Khan, The Hebrew Unversty of Jerusalem, Dept. Atmos. Scences, Israel, khan@vms. huj.ac.l low - energy vortces on the collson effcency was shown to be neglgbly small Pruppacher and Klett, 997). Further estmatons made by Pnsky et al 999a) and Pnsky and Khan 004), who took nto account the effects of turbulent vortces wthn the nertal and transton ranges, ndcated that turbulence can sgnfcantly by hundred percent) ncrease collson effcences and kernels between cloud droplets. Drect numercal smulatons DNS) performed recently by Frankln et al 004) and Wang et al 005a) also ndcate a pronounced ncrease n the collson effcency n a turbulent flow generated by DNS models. Note that DNS are conducted wth the Taylor mcroscale Reynolds numbers Re ranged from 70 to 00. These values are much smaller than those typcal of atmospherc 4 turbulence about 5) 0 ). Pnsky and Khan 004) calculated the collson effcences and kernels n a turbulent flow wth hgh Re typcal of atmospherc clouds. In that study, however, only the Lagrangan acceleratons were taken nto account, whle the effects of turbulent shears were dsregarded. Pnsky et al 006a) analyzed the effects of both turbulent acceleratons and shears on droplet collsons n the absence of HDI. To perform the calculatons, a statstcal representaton of a turbulent flow was proposed based on the results of the scale

2 analyss of characterstcs of turbulence and droplet moton. The statstcal propertes of turbulent flow were represented by a set of noncorrelated samples of turbulent shears and the Lagrangan acceleratons. Each sample can be assgned to a certan pont of a turbulent flow. Each such pont can be surrounded by a small "elementary" volume wth lnear length scales of the Kolmogorov length scale, wthn whch the Lagrangan acceleraton and turbulent shears were regarded as unform n space and nvarable n tme. Usng the statstcal model Pnsky et al 004), long seres of turbulent shears and acceleratons were generated reproducng the probablty dstrbuton functons PDFs) of these quanttes at hgh Reynolds numbers as they had been obtaned n recent laboratory and theoretcal studes. The swept volumes of droplets were calculated for each sample of the acceleraton-shear par, and the PDF of swept volumes was calculated for the parameters typcal of cloud turbulence. It was found that the magntude of the mean swept volume ncreases both wth the Reynolds number and the dsspaton rate. At the same tme such ncrease for cloud droplets dd not exceed ~60 % even under turbulent condtons typcal of strong cumulus clouds Re 3 ε = 0. m s ). = 0 4, Pnsky et al 006b) extended the approaches developed n Pnsky et al 006a) for the purpose of calculatng turbulence effects on HDI for cloud droplets wth rad below 0 µ m. In ths presentaton we brefly descrbe the method of calculaton of collson effcency and collson kernels developed by Pnsky et al 006b) and present tables of enhancement factors for collson kernels as compared to the gravtaton collson kernels. Collson effcences and collson kernels between cloud droplets are calculated under turbulent condtons typcal of actual clouds at 000 mb and 500 mb pressure levels. Effects of turbulence on the droplet sze dstrbuton DSD) s llustrated by solvng the stochastc collson equaton for dfferent dsspaton rates and Re.. PHYSICAL MODEL The essental parts and general concepts of the study are descrbed by Pnsky et al 006a). For the sake of convenence, we brefly repeat here the man ponts and defntons ntroduced theren, and concentrate on the calculaton of collson effcences. Droplet moton n a turbulent flow at small scales s determned by gravty, the turbulent Largangan acceleratons A and the tensor of turbulent shears S, j =,, 3 ). Statstcal j propertes of a turbulent flow are represented here by a set of non-correlated samples of turbulent shears and Lagrangan acceleratons. Each sample can be assgned to a certan pont of the turbulent flow. Snce the spatal and temporal scales of turbulent shears and acceleraton exceed the correspondng Kolmogorov scales several fold, each such pont can be surrounded by a small "elementary" volume wth the characterstc lengths and tme scales equal to the correspondng Kolmogorov scales, n whch the Lagrangan acceleraton and turbulent shears are consdered to be unform n space and nvarable n tme.

3 The quanttes characterzng collsons were calculated for each elementary volume. As t was shown by Pnsky et al 006a), droplet velocty relatve to the envronment flow relaxes n the majorty of cases to ts adapted quasstatonary) value V j ad wthn the tme perods shorter than the temporary scales of turbulent shears and acceleratons. It means that wthn each elementary volume we can use adapted relatve droplet veloctes when dealng wth droplet collsons. The equaton system for the moton of solated droplets can be wrtten n ths case as: ad V j δj + Sj = A + Vt S3 dx dt τ j j t 3 + ad ) = V = S x + V δ V, ) ad where V = V W Vtδ 3, and V, W and V t are droplet velocty, ar velocty and the termnal fall velocty nduced by gravty, respectvely; τ s the drop relaxaton tme that s the measure of droplet nerta. For cloud droplets τ = / 9) ρ / ρ ) a / ν ), where ρ w and ρa w a are the water and ar denstes, respectvely, and a s the droplet radus. Eq. ) s vald f τ s less than a certan crtcal value dependng on the shear tensor. Cases when ths condton was not satsfed for droplets wth rad below 0 µ m were qute rare and rejected from the analyss. Takng nto account that these cases are very rare, neglectng those does not affect the mean values of swept volumes and collson effcences. As regards the PDF of collson effcences, these rare cases contrbute to the far tal only. The equaton for the relatve droplet velocty between two non-nteractng droplets ~ = V V wthn an elementary turbulent V volume can be wrtten as see Pnsky at al 006a): ~ dx ~ ~ V = = Sj x j + V 3) dt where ~ = s the poston vector between x x x the centers of two droplets, and ~ ad ad V = V V + Vt Vt ) δ 3. One can see from 3) that the relatve velocty between nonnteractng droplets does not depend on the locaton of the droplet par wthn the elementary volume, but rather on the vector connectng ther centers only. The growth rate of a droplet wth radus a, caused by collsons wth droplets of radus a, s determned by the flux Φ of the a -radus droplets onto the a -radus droplet. Ths flux s assumed to cross the sphercal surface of a + ) radus, and hence can be expressed as: Φ = a + a Ω + ~ N V ~ x d Ω + a, 4) where N s the concentraton of the a - radus droplets and Ω+ s the fracton of the sphercal ~ surface, where V ~ < 0. Fg. llustrates ths x defnton of the drop flux. In the fgure, the a ) + a radus sphere represents the target;

4 the curves wth arrows represent relatve droplet trajectores. Fgure. Illustraton of the drop flux defnton. The a + ) - radus sphere represents the a Droplet flux Relatve droplet trajectores Ω + a + +a a a target; the curves wth arrows represent relatve droplet trajectores. Surface Ω + represents the area where relatve trajectores are drected nsde the target. Takng nto account that the nerta-nduced fluctuatons of droplet concentraton for cloud droplets are qute small especally f the dfferental sedmentaton velocty s taken nto account, see abstract 4.3), the concentraton N can be consdered constant wthn elementary volumes. Consequently, nstead of the droplet flux, t appears more convenent to deal wth relatve velocty fluxes whch can be Φ defned as F =. Usng 3) and 4), one can N express the swept volume normalzed to ts value n the pure gravty case as: S v t Ω+ S ~ x ~ ~ x + V ~ j j x ) = d 3 π a + a ) V V t Ω + 5) As t follows from eqs. ) and 5), to calculate droplet flux n the absence of HDI one must know the Lagrangan acceleraton and the turbulent shear n each elementary volume. Pnsky et al 006a) calculated long seres of acceleraton-shear pars or samples of elementary volumes) usng a statstcal model by Pnsky at al 004). Ths model reproduces the probablty dstrbuton functons PDF) of acceleratons and the shears at hgh Re as they were obtaned n the recent laboratory studes Beln et al 997, La Porta et al 00) and theoretcal studes Hll 00). The same seres of acceleratons and turbulent shears are used to calculate collson effcences. 3. DEFINITION AND CALCULATION OF COLLISION EFFICIENCY 3. Defnton of the collson effcency The collson effcency E between two populatons of a and a - rad droplets s generally defned as the rato of the droplet fluxes: E a, a ) = Φ Φ, 6a) where Φ HDI HDI s the flux of a rad droplets onto a -radus droplet when colldng droplets experence HDI, and Φ n the absence of HDI. s the droplet flux As the droplet

5 concentraton s assumed to be unform throughout the elementary volume, the collson effcency can be also defned as the rato of relatve velocty fluxes: FHDI E a, a ) = 6b) F Thus, evaluaton of the collson effcency requres calculaton of these fluxes under turbulent condtons typcal of atmospherc clouds. The procedure of calculaton of droplet fluxes s descrbed below. a. The technque of calculaton of a flux n the absence of HDI. To calculate ths velocty flux see Pnsky et al 006a for more detal), the a ) + a - radus target surface s dvded nto hexagonal pxels of equal square d Ω+ usng the cosahedron-based method Tegmark 996). The number of pxels has been chosen so as to be large enough to provde a hgh accuracy of calculatons. The hgh resoluton s especally mportant when calculatng small collson effcences, snce the fluxes are determned wth the accuracy equal to the magntude of the flux crossng a sngle pxel. In the absence of HDI, the velocty flux df k correspondng to the k-th pxel s calculated as: ~ S ~ ~ ~ j x0 j x0 + V x0 ) dω+ dfk = 7) a + a The flux F was calculated by ntegratng 7) over the surface Ω +. b. Calculaton of the droplet flux n the presence of HDI. To perform ths task, the followng procedure was used. We start the descrpton wth the smplest case, namely when droplet trajectores n the presence of HDI do not devate sgnfcantly from the non-dsturbed trajectores see Fgure ). Usng Eq. 3) we calculated the non-dsturbed relatve trajectores of droplets, whch cross the sphercal surface ~ x a a = + the dashed crcle n Fg. ) 0 durng droplet approach n the absence of HDI. For ths purpose, Eq. 3) was ntegrated back n tme usng the ntal postons located on the target surface as t was dscussed above. Such trajectores were calculated begnnng from the centre of each pxel. The backward trajectores were calculated for the same tme perods, by the end of whch the droplets were separated by the dstance of about 0 rad of the larger droplet n the droplet par. Ths dstance s large enough to provde accurate calculaton of mutual droplet tracks Pruppacher and Klett, 997). The fnal ponts of the backward trajectores form the surface of the second order referred to n Fg. as A. As a result, there s a one-toone pxel correspondence between the ponts of surface A and the ponts on the target surface ~ x a 0 = a +. Trajectores connectng the correspondng pxels e.g. A-A, B-B ) can be regarded as a stream tube that transports the flux df k 7). To determne the velocty flux n the presence of HDI, we calculated the absolute trajectores of droplets of radus a and a forward n tme, as shown n Fgure. These trajectores start at the fnal ponts of the backward trajectores,.e., from the centers of

6 the pxels located on the surface A. Two examples of such trajectores that start from ponts A and B are schematcally shown n Fg.. Due to HDI, only a certan fracton of the trajectores leads to collsons for example, the droplet startng at pont A colldes, whle the droplet that starts at pont B does not). Takng nto account that each of these trajectores s responsble for ts own velocty flux df, we calculated the total velocty flux F HDI n the presence of HDI by summng up the elementary fluxes of the trajectores whch lead to droplet collsons. The rato of the fluxes wth and wthout HDI determnes the value of the collson effcency. k dvded nto pxels, thus the k-th pxel on the surface corresponds to the velocty flux df k. The absolute trajectores of a -radus droplets start from the centers of all the pxels stuated on the A surface. In the presence of HDI, only a fracton of these trajectores leads to droplet collson. Fgure 3 llustrates the defnton of the collson effcency. The black sphere n the fgure represents the target of a + ) radus. The surface flux F a A represents the source of the total. The fracton of ths surface denoted as A HDI forms the flux F HDI of droplets that collde wth the target, when HDI s taken nto account. A Surface A B B A Relatve backward trajectores no HDI) Droplet of radus a Droplets startng from the red surface collde the target wth hydrodynamc nteracton taken nto account no collson collson Droplet of radus a Absolute droplet trajectores n the presence of HDI Droplets startng wth the green surface collde the target wthout hydrodynamc nteracton target Fgure. Scheme llustratng the procedure of calculaton of droplet velocty fluxes n case HDI s taken nto account. Notatons: the a ) + a radus sphercal surface s the target. see text for more detal) The fnal ponts of the backward relatve trajectores e.g. AA, BB ) form A surface. The A surface s Fgure 3. An example of the droplet collson geometry for 0 µ m and 0 µ m - rad droplet par for one of the samples of the turbulent flow.

7 Notatons: the blue sphere s the a + ) a radus target. The A surface formed by the fnal ponts of the backward-n-tme trajectores s shown n the upper part of the fgure, marked green. Droplets that start movng from ths surface collde wth the target n the absence of hydrodynamc droplet nteracton. Droplets startng from the area A HDI, marked red, collde wth the target n the presence of hydrodynamc droplet nteracton. The fgure llustrates the smplest case when the A HDI area s fully located nsde A area. Fg. 3 llustrates the case when the surface A HDI s fully located wthn the surface A. In some cases, however, HDI affects the tracks of approachng droplets so sgnfcantly that the area A HDI s not fully located wthn the surface A. To take these cases nto account, a more complcated algorthm was used related to utlzaton of an ancllary sphercal surface of a larger radus that surrounds the a + ) -radus target. a It s noteworthy that n the pure gravty case the defnton 6) of collson effcency as the rato of velocty fluxes reduces to the commonly used defnton of collson effcency as the rato of collson areas and areas of geometrcal cross-sectons Pruppacher and Klett 997). As t follows from eq.3), n calm ar the relatve velocty between droplets at the nfnty s equal to the dfference between ther termnal fall veloctes. Thus, n expresson 7) for droplet fluxes, ths constant relatve velocty can be factored out of the ntegral, and fluxes of relatve velocty or droplets) thus become proportonal to the area of the cross-sectons of the flux tubes the areas of pxel projectons on the horzontal plane). As a result, n the pure gravty case the rato F F HDI s transformed nto the rato of the collson and geometrcal cross-secton areas. 3. The modfcaton of the superposton method In order to perform collson effcency calculatons, the problem of hydrodynamc nteracton between two spheres movng n the arflow s consdered. We use the superposton method Pruppacher and Klett 997), accordng to whch each sphere s assumed to move n a flow feld nduced by ts counterpart movng alone. The method was frst proposed by Langmur 948), and was later successfully used by many nvestgators Shafrr and Gal-Chen 97; Ln and Lee 975; Schlamp et al 976) for the calculaton of collson effcences between droplets wthn a wde range of szes. The superposton method was used by Pnsky et al 00) for calculaton of detaled tables of gravty-nduced collson effcences for droplets wthn the µ m to 300 µ m -rad range. The calculated values of collson effcences allowed us to obtan an accurate reproducton of drop-collector growth measured n the vertcal wnd tunnel at the Unversty of Manz Vohl et al 999). Wang et al 005b) have recently shown that the center-pont formulaton of the superposton method provdes a good agreement wth the exact solutons avalable. Pnsky et al 999a, 00, the present study) use

8 the center-pont formulaton of the superposton method for calculaton of collson effcency n a turbulent flow. Accordng to the superposton method, the perturbed velocty feld nduced by each drop n calm ar s calculated as for the case of a sngle solated droplet. Each of the nteractng drops experences the velocty feld nduced by ts counterpart droplet. The expresson for the nduced velocty feld can be found n Pruppacher and Klett 997) and n Pnsky et al 999). In the present study we apply a modfed superposton method allowng smplfcaton of the governng equatons. The dea of the modfcaton s the followng: each droplet veloctyv relatve to the surroundng ar s represented as the sum of two components: a non-dsturbed velocty V ) wthout the mpact of the counterpart droplet, and a dsturbed veloctyv, nduced by the counterpart droplet. As t was shown by Pnsky et al 006a), rapdly adapts to ad V V ), whch depends on the Largangan acceleraton, turbulent shear and the termnal fall velocty of the droplet. Wthn each elementary volume, ad V s constant wth a hgh accuracy. Ths allows us to smplfy the correspondng moton equaton by utlzng the condton V ) = ad V see eqs. and 8b). In the course of HDI the velocty component V can experence sgnfcant changes caused by the rapdly varyng velocty feld nduced by the counterpart droplet. Hence, the tme dervatve n the equaton for the component V should be taken nto account see eq. 8a). As a result, the system of equatons governng the moton of any nteractng droplets can be reduced to the followng form: dv ) dt * = V j δ j + Sj + u x ) τ τ V δ j + Sj = A + Vt S3 V τ 8a) 8b) dx ) = = Sj x j + Vt δ 3 + V + V 8c) dt To descrbe the mutual moton and collson of two nteractng droplets, two coupled equaton systems smlar to 8a-8c) should be solved. Snce V tends to zero at large separatons ~0 rad of the larger droplet), the ntal droplet velocty s equal to ad V = Sjx j + V + Vtδ 3 9) In the absence of HDI, = 0 V and eqs. 8) concde wth eqs. -3 n Pnsky et al 006a) for the moton of non-nteractng droplets. Note also that n calm ar A 0; S = 0) eqs. 8) are = j reduced to the classcal system of equatons for the superposton method n the pure gravty case. The advantage of the approach proposed s that equaton 8a) for V contans the vscous term the frst rght-hand one), whch allows the utlzaton of a smple and computatonally effcent tme-explct mddle pont numercal scheme. The followng verfcaton of the accuracy of calculatons was performed:

9 Senstvty of trajectory ntegraton to value of tme step. The tme step t of ntegraton of eqs 8a-c) was chosen t 0.τ s, where τ s s the characterstc relaxaton tme of the smallest droplet n the droplet par. Calculatons showed that further decrease of the tme step dd not nfluence the results. Senstvty to the number of pxels used for calculaton of droplet fluxes. The accuracy s determned by the resoluton that can be evaluated as 4/n, where n s the number of pxels coverng the target surface. For spheres of a + ) radus, n=5 was used, a so that the error n the calculatons of droplet velocty fluxes dd not exceed 0. %. As was mentoned above, the number of pxels was ncreased proportonally to the surface square of the ancllary sphere to preserve hgh resoluton. Senstvty wth respect to the numercal scheme used for the calculaton of droplet trajectores was analyzed by comparng the results the values of collson effcency) obtaned usng a one-step tme-explct numercal scheme appled n the calculatons, wth the results obtaned usng the tme consumng Runge-Kutta scheme of the 5-thorder wth the automatc choce of tme steps. Smulatons showed that the smple scheme provded the same values of collson effcency. Senstvty to the ntal separaton dstance between droplets. Supplemental experments ndcate that the utlzaton of the ntal separaton dstance of ~ largest droplet rad, nstead of the separaton dstance of ~0 rad of the largest droplet, results n a few percent change n collson effcences. These results ndcate that that the HDI zone falls well nto elementary volumes. Therefore, calculatons were performed wth the ntal separaton dstance equal to ~0 rad of the largest droplet. Senstvty wth respect to the sze of the ancllary sphere. The analyss of a great number of smulatons showed that utlzaton of the ancllary sphere wth the maxmum radus of a + ) was sutable for actually any 3 a combnatons of acceleraton-shear and droplet rad. Further ncrease of the ancllary sphere radus dd not affect collson effcency. Verfcaton of the modfed superposton method has been performed n several boundary cases. In the absence of HDI, the method reproduces the values of the swept volume calculated wth accuracy better than 0.% see Pnsky et al 006a). For the pure gravty case, the dfference between the values of collson effcency obtaned usng eqs. 8a-c) and those obtaned usng the classcal method Pnsky et al 00) s about 3% and does not exceed 5%. An mportant evdence of the valdty of the calculatons s that Ea, a )= Ea, a ), whch means that the algorthm s nvarant wth respect to the choce of droplets ndces a and a. Senstvty to the volume of statstcs. Senstvty experments wth dfferent volumes of statstcal nformaton related to turbulent shear flows ndcate that the ncrease of statstcal data set n addton to the set used n the present study for the calculaton of average values and 5 0 for the calculaton of PDF) has a neglgble effect on the results. Relatve errors are wthn the range 0.4-3%.

10 We beleve, therefore, that the overall error n the calculaton of the collson effcency and the collson kernel does not exceed a few percent. 4. RESULTS 4. The magntudes of the collson effcences and collson kernels Collson effcency Collson effcences were calculated for the turbulent condtons typcal of clouds. Probablty dstrbuton functons of collson effcency and collson kernels were performed n Pnsky et al 006b). We present here the dependences of averaged values of the collson effcences and kernels on drop sze, dsspaton rate and Re. Fgure 4 shows the collson effcences between 5 µ -rad rght) µ m -left) and 0 m collectors wth smaller droplets under dfferent dsspaton rates and Re. The pure gravty values of the collson effcences are shown as well. One can see that strong turbulence ncreases sgnfcantly the collson effcences between cloud droplets. Especally sgnfcant ncrease n the collson effcency takes place for droplets of close sze. Collector 5µm ε = *0-3 m s -3, Re = 5*0 3 ε = *0 - m s -3, Re = *0 4 ε = *0 - m s -3, Re = *0 4 Gravtatonal Fgure 4. Collson effcences between 5- µ m frst) and 0- µ m rad second) collectors wth smaller droplets under dfferent dsspaton rates and Re after Pnsky et al 006b). Results obtaned by 3 Wang et al 005) forε = 00cm s are marked by crosses on the second panel. Comparson of the values of collson effcences obtaned by Pnsky and Khan 006b) calculated under the condtons relatvely to those used by Wang et al 005a) = 00cm s 3 ε, drop collector radus 0 m µ ) ndcates a reasonably good agreement n the results. Fgure 5 shows the dependence of the averaged normalzed collson kernel for the 0 µ m- and 0 µ m- rad droplet par on the dsspaton rate ε under dfferent Re Droplet radus, µm

11 Normalzed collson kernel Re = *0 Re = 0 3 Re = Re = Dsspaton rate, m s -3 accountng for the effect of Re s as mportant as accountng for the effect of the dsspaton rateε. Fgure 6 shows collson kernel enhancement factor for three typcal cases: stratform clouds 3 3 ε = 0.00 m s Re = 5 0, ) upper panel), 3 4 cumulus clouds ε = 0.0 m s, Re = 0 ) mddle) and cumulonmbus 3 4 ε = 0. m s, Re = 0 ) lower panel). stratform cumulus cumulonmbus clouds clouds clouds Fgure 5. Dependence of the averaged normalzed collson kernel for the 0 µ m- and 0 µ m - rad droplet par on the dsspaton rate ε under dfferent 006b). Re after Pnsky et al Whle the factor of the swept volume ncrease was found to be.6 at very strong turbulence ntensty Pnsky et al 006a), the collson kernel ncreases by the factor as large as 4.8. Thus, the effect of turbulence on the HDI appears to be the man mechansm by means of whch turbulence ncreases the rate of cloud droplets collsons. Note that the 0 µ m -0 µ m - rad droplet par ndcates the mnmum turbulent enhancement factor. The ncrease n the collson kernel of droplet pars contanng droplets of close sze or droplets smaller than ~3 µ m n rad s much more pronounced. One can see n Fg. 6 that the collson kernel ncreases both wth the ncrease n ε and Re. Thus, the

12 Fgure 6. Collson kernel enhancement factor as compared to the gravty value) for three typcal cases: stratform clouds ε = 0.00 m s 3 3 Re = 5 0, ) upper panel), 3 cumulus clouds ε = 0.0 m s, Re = 0 4 ) mddle) and cumulonmbus 3 ε = 0. m s, Re 4 = 0 ) lower panel). The collson kernel enhancement factors are presented n tables -3 for correspondng levels of turbulence ntensty. Table. Stratocumulus clouds ε = 0.00 m s 3 3 Re = 5 0, ) µm µm 3 µm 4 µm 5 µm 6 µm 7 µm 8 µm 9 µm 0 µm µm µm 3 µm 4 µm 5 µm 6 µm 7 µm 8 µm 9 µm 0 µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm

13 3 Table. Cumulus clouds ε = 0.0 m s, Re = 0 4 ) µm µm 3 µm 4 µm 5 µm 6 µm 7 µm 8 µm 9 µm 0 µm µm µm 3 µm 4 µm 5 µm 6 µm 7 µm 8 µm 9 µm 0 µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm Table 3. Cumulonmbus ε = 0. m s, Re = 0 4 ) µm µm 3 µm 4 µm 5 µm 6 µm 7 µm 8 µm 9 µm 0 µm µm µm 3 µm 4 µm 5 µm 6 µm 7 µm 8 µm 9 µm 0 µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm µm

14 One can see that collson enhancement factor s qute sgnfcant, especally for drops of close szes as well as for droplets of very dfferent szes. Collson enhancement factors were calculated for 500 mb level as well. The enhancement factor at 500 mb level s larger than that for 000 mb level by factors.5- that agrees wth the results obtaned by Pnsky et al 00). The factor ncreases because the ncrease n the relatve sedmentaton velocty caused by the decrease of ar densty wth heght. Note that enhancement factors presented n Fgure 6 and tables -3 were calculated wth no effect of droplet clusterng. 4. Parameterzaton of droplet clusterng effect To parameterze effects of droplet clusterng n a turbulent flow we used an emprcal dependence of clusterng ntensty on the St number, presented by Pnsky and Khan 003). The dependence has been obtaned as a result of statstcal analyss of a long seres of drop arrval tmes measured n stu n ~60 cumulus clouds. The parameterzaton formula s as follows: N N where / concentraton, 0.37 = f St) = St, 0) N s the cloud averaged droplet / N s the r.m.s. of the concentraton fluctuatons. Under assumpton of strong spatal correlaton of concentraton fluctuatons of cloud droplets of dfferent sze, we ntroduced a correcton factor of the collson kernel between droplets characterzed by the Stokes numbers St and St as follows: G St, St ) = N N N = St St ) N 0.37 = + f St ) f St ) = ) Functon G St, St ) depends on both droplet szes and on the ntensty of turbulence on the dsspaton rate). 4.3 Examples of droplet spectrum evoluton To llustrate effects of turbulence on DSD development and randrop formaton we ntegrated the stochastc collson equaton ) df dt 0 = m / 0 f m) f m m, m) K m m, m) dm ) f m) f m) K m, m) dm for the perod of 30 mn. In ) f s the DSD functon. Two droplet spectra typcal of ntermedate droplet concentraton 3 3 N=500 cm, CWC=.8 gm ) and more 3 contnental clouds N=800 cm, CWC=.3 3 gm ) were chosen. The ntal DSD are centered at rad of ~ 9 µ. Calculatons were performed both for pure gravty case, as well as for turbulent condtons. The calculatons under turbulent condtons were performed both wth and wthout effects of droplet clusterng taken nto account. In smulatons wthout the m

15 clusterng effect f m ) f ) was m calculated as f m ) f ) n accordance m wth the common practce n cloud modelng. To take the clusterng effect nto account the expresson f m ) f ) has been multpled m by factor G St, St ) determned n ). In all turbulent smulatons collson kernel K m, m ) has been multpled by the enhancement factors presented n Tables -3. Note that turbulence effects were taken nto account only for cloud droplets wth rad below µ m. No changes for drops of larger sze were made. Thus, the result possbly underestmates the turbulent effect. At the same tme, t allows us to evaluate effects of turbulence on formaton of large cloud droplets and small randrops whch trgger precptaton formaton. Results of calculatons are presented n Fgure 7, showng DSD obtaned toward t=30 mn. n smulatons mentoned above.

16 Fgure 7. DSDs obtaned by solvng the stochastc collson equaton durng t=30 mn. DSD obtaned toward t=30 mn n case of pure gravty case are marked by green. Three values of the dsspaton rate were used: ε = 0.0 m s,0.0 m s,0.05 m s and 0. m s. DSD obtaned when collson kernel enhancement s determned only by HDI and transport effect Tables -3) are marked by green. DSDs obtaned when the enhancement factor by clusterng effect s taken nto account s marked by red. One can see that at hgh droplet concentraton and low turbulent ntensty the effect of turbulence s not sgnfcant at least durng the tme perod of 30 mn). Turbulence of larger ntensty sgnfcantly accelerates collsons, ndcatng the formaton of randrops, whle no randrops were formed n pure gravty case. 5. CONCLUSIONS The collson effcences and collson kernels were calculated for condtons typcal of real cloud of dfferent type for - mcron- rad droplets. Detaled tables of collson kernel enhancement are presented. A parameterzaton of the effect of droplet concentraton fluctuaton s proposed. The method s based on statstcal analyss of n stu data n 60 clouds. Effect of turbulence s llustrated by calculaton of droplet sze dstrbuton evoluton usng a stochastc collson equaton. A sgnfcant acceleraton of ran formaton n a turbulent flow was demonstrated for droplet spectra typcal of ntermedate and contnental clouds. It s shown that ncrease n the collson rate between cloud droplets may serve as a trggerng mechansm for randrop formaton, at least for not extremely martme clouds. The tables of the collson kernel enhancement factors are avalable upon request. Acknowledgements The study was supported by the Israel Scence Foundaton grant 73/03) and the European Project ANTISTORM. The authors are grateful to Prof. N. Kleeorn for valuable dscussons and to Professor M. Tegmark who provded us wth codes for sphere pxelzaton. REFERENCES Almeda, F. C., 979: The collsonal problem of cloud droplets movng n a turbulent envronment-part II: Turbulent collson effcences. J. Atmos. Sc., 36, Beln F, Maurer J, Tabelng P. and Wllame H. 997: Velocty gradent dstrbutons n fully developed turbulence: An expermental study, Phys. Fluds, 9, Frankln, C.N., P.A. Vallancourt, M.K. Yau and P. Bartello, 004: Cloud droplet collson rates n evolvng turbulent flows, 4-th Internatonal conference on Clouds and Precptaton, Bologna, Italy, 9-3 July 004,

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