S. Ghosh, P. R. Jonas

Transcription

1 Ann. Geophyicae 16: 628±637 (1998) Ó EGS ± Springer-Verlag 1998 On the application of the claic Keler and Berry chee in Large Eddy Siulation odel with a particular ephai on cloud autoconverion, the onet tie of precipitation and droplet evaporation S. Ghoh, P. R. Jona Departent of Phyic, UMIST, Mancheter, M60 1QD, UK Fax: Received: 30 June 1997 / Revied: 5 Deceber 1997 / Accepted: 8 Deceber 1997 Abtract. Many Large Eddy Siulation (LES) odel ue the claic Keler paraeteriation either a it i or in a odi ed for to odel the proce of cloud water autoconverion into precipitation. The Keler chee, being linear, i particularly ueful and i coputationally traightforward to ipleent. However, a ajor liitation with thi chee lie in it inability to predict di erent autoconverion rate for aritie and continental cloud. In contrat, the Berry forulation overcoe thi di culty, although it i cubic. Due to their di erent for, it i di cult to atch the two olution to each other. In thi paper we ingle out the procee of cloud converion and accretion operating in a deep odel cloud and neglect the advection ter for iplicity. Thi facilitate exact analytical integration and we are able to derive new expreion for the tie of onet of precipitation uing both the Keler and Berry forulation. We then dicu the condition when the two chee are equivalent. Finally, we alo critically exaine the proce of droplet evaporation within the fraework of the claic Keler chee. We iprove the exiting paraeteriation with an accurate etiation of the di uional a tranport of water vapour. We then deontrate the overall robutne of our calculation by coparing our reult with the experiental obervation of Beard and Pruppacher, and nd excellent agreeent. Key word. Atopheric copoition and tructure á Cloud phyic and cheitry á Pollution á Meteorology and atopheric dynaic á Precipitation 1 Introduction Repreentation of icrophyical procee that control the partition of water between cloud and precipitation Correpondence to: S. Ghoh are crucial to the realitic prediction of cloud odel including the large eddy iulation (LES) odel. Keler (1969) pioneered the introduction of the e ect of varying updraught into cloud odel. In hi approach, all water i rt condened a cloud water, with all drop ize (roughly 5±30 l) and negligible terinal velocity. Then a proce called autoconverion begin. Thi involve the foration of precipitation particle either by the aggregation of everal cloud particle or by the action of giant alt nuclei, or iilar procee. Water cloud can perit for a long tie without precipitating and variou eaureent how that cloud water content 1 g 3 are uually aociated with precipitation (Maon, 1957; Singleton and Sith, 1960). Two di erent autoconverion chee are widely ued (Sipon and Wiggert, 1969): dm ˆ k 1 a g 3 1 ; 1 dt and dm dt ˆ 2 g 3 1 ; :0366N b D b where Eq. (1) i due to Keler (1969) and Eq. (2) to Berry (1968). M and are the precipitation water content and the cloud water content, repectively, k 1 the autoconverion rate 1, a the autoconverion threhold g 3,N b and D b are the droplet nuber denity no.c 3 and droplet relative diperion at the cloud bae. We hall rt dicu the iplication of the Keler forula. 2 The Keler autoconverion chee Keler aued that the rate of autoconverion increae with the cloud water content but i zero for oe value below the threhold a, below which cloud converion doe not occur. The paraeter k 1 i the

2 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee 629 reciprocal of the 1=e ``converion tie'' of the cloud water. In odel which ue the Keler chee, the converion of cloud water to precipitation doe not tart until the cloud content produced in aturated updraught ha exceeded a. Thereafter, the cloud water repond to it accretion by relatively large precipitation particle in addition to procee of vertical advection and condenation. The real atophere' analogue to the autoconverion threhold can be height dependent. There abound coniderable arbitrarine with regard to the choice of appropriate value for k 1 and a, although the recoended value for k 1 by Keler i 10 3, actual value of a can only be acertained fro experient which are often di cult to conduct. In the abence of uch upporting experiental data variou odeller have ued variou cobination of value for k 1 and a. In oe of our recent LES odelling tudie (Jona and Ghoh, 1997) we odelled a tratocuulu cloud widely tudied during the EUCREX (European Cloud and Radiation Experient) iion 204 of 18 April We found that with an adopted value of for k 1 when a i unrealitically et at 1:0 gkg 1, there wa no evidence of precipitation and the liquid water path (LWP) howed a low increae to the end of the iulation. It hould be noted that the icrophyical paraeteriation are applied at each point and thi iplie that the average cloud water content ay be le that the autoconverion threhold. Only when a wa lowered to 0:3 gkg 1 wa there a gradual decreae in the LWP, indicating the initiation of precipitation in the iulation. In Table 1 we how the axiu layer averaged rain a for the odel tratocuulu cloud after 4 h of iulation. Fro Table 1 we nd that the rain a increae with lower value of a and the iulation correpondingly howed that the cloud liquid water content increaed with increaing value of a. The paraeter k 1 i the rate of autoconverion to precipitation of cloud content in exce of the threhold a. When k 1 i large, the approach to teady tate i hatened once > a; when k 1 i all, the cloud content continue to increae longer in the updraught after precipitation ha tarted (epecially if the accretion proce i alo weak). Decreaing the autoconverion coe cient k 1 a ect the precipitation developent in uch the ae way a doe increaing the converion threhold a. Thi i clear fro Keler (1969, Fig and 12.8) where the cloud and precipitation pro le correponding to k 1 ˆ 10 3 and 10 5 in Fig of that Table 1. Raina with varying autoconverion threhold in a iulated tratocuulu cloud fro Jona and Ghoh (1997) a gkg )1 raina g kg ) ) )3 paper are virtually identical with a ˆ 0:5 and 2.0 in Fig Fro the preceding paragraph it i clear that Keler' treatent of the autoconverion proce, although very ueful and aenable to incorporation into cloud odel traightforwardly, i rather intuitive. Berry (1968) on the other hand ha treated the proce ore rigorouly, and thi we dicu in the next ection. 3 The Berry autoconverion chee Berry' equation [Eq. (2)] i developed theoretically fro a odel of initial cloud growth by condenation and coalecence of cloud-ized particle with each other. The early droplet pectru near the cloud bae ha a nuber concentration of N b drop c 3 and a relative diperion D b deterined by the condenation nucleu pectru. The relative diperion i de ned a D b ˆ ra a ; 3 where r a i the tandard deviation of the droplet radii and a i the ean. Equation (2) correpond to a choice of the 200-l boundary between cloud and precipitation drop. Thi i realitic for the following reaon: 1. A drop of 200 l diaeter ha a terinal velocity of 1±2 1 and i thu beginning to fall at a peed coparable to cuulu updraught. It can urvive a fall of everal hundred etre in ubaturated air without coplete evaporation. 2. Mot 10-c radar begin to how an echo of a cloud when nuerou drop of about thi ize are preent (Sipon and Wiggert, 1969). Jona and Maon (1974) tudied the evolution of droplet pectra by the cobined e ect of condenation and coalecence in cuulu cloud. Their coputed pectra cloely reeble eaured pectra and reproduce the biodal tructure oberved by Warner (1969). Their tudy reveal that the onet of precipitation in continental cloud containing a high cloud condenation nucleu concentration depend on the value ued for the collection e ciency. Due to enhanced turbulence the growth rate of larger drop would increae igni cantly. The work of Jona and Maon (1974) further validate the reult of Berry (1968). Berry (1968) nd that the initial droplet ditribution generated jut above the cloud bae by condenation in a war cuulu cloud et the tage for future droplet growth by collection. If thee droplet are large or the pread of the ditribution broad, the rate at which precipitation droplet will be fored i fat. In a cloud which ha a liited lifetie governed by the eocale circulation thi initial rate of droplet growth ay ake the di erence between rain and no rain. Twenty-eight typical, condenation-produced pectra were ued to cover a range of droplet nuber and relative diperion. The ubequent growth were found to t a pattern of two ditinct region ± initial and nal, with the boundary between the two regie at a radiu

3 630 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee of 40 l ± an approxiate radiu of thoe droplet which contain ot of the liquid water (Berry, 1968). In the initial growth region, the elaped tie `T ' required for the drop to attain a radiu of 40 l, excluding the e ect of collection, ha been calculated by Berry (1968). The ux of cloud water paing r ˆ 40 l averaged over any parcel at variou tage of developent i jut the cloud water content of the parcel L c divided by the tie `T ', which i the autoconverion rate given in Eq. (2). Upon exaining Eq. (1) and (2) it becoe clear that Berry' (1968) forula, although epirical, account for the iportant icrophyical paraeter like the droplet nuber denity and the relative diperion, wherea the Keler chee doe not. Other than by coparion of the prediction of a cloud odel with eld experient there can be no direct way to link the value of k 1 and a in the Keler forula in Eq. (1) with the icrophyical paraeter of the cloud. The Keler forula i linear in while the Berry forula i cubic; a a reult the Keler chee i ore traightforward to ue in LES odel. Since both Eq. (1) and (2) repreent the ae phyical proce, one way to link the value of k 1 and/or a in Eq. (1) with icrophyical paraeter would be to ue Eq. (2) to optiie the choice of k 1 and a for a particular iulation. In thi way one would be able indirectly to incorporate the e ect of the nuber denity and the relative diperion without altering the baic tructure of the Keler chee ued within the fraework of an LES odel. An iportant feature of Berry' equation (2) i that a di erent autoconverion rate i predicted for aritie and for continental cloud. Typically for aritie cloud N b 50 c 3, D b 0:366 and for a continental cloud N b 2000 c 3, D b 0:146 (Sipon and Wiggert, 1969). The tratocuulu cloud tudied extenively during the EUCREX iion [and alo by u in our LES odelling (Joan and Ghoh, 1997)] ha the nuber denity and the relative diperion in between the aritie and the continental extree. For thi cloud N b 200 c 3, D b 0:24 (Pawlowka and Brenguier, 1996), uggeting that although the nuber concentration i not a high a that of a continental cloud, it had at oe tage of it developent paed over pollution ource. Thee feature are illutrated in Fig. 1, where we have hown the rate of change of cloud water content (d/dt) a a function of for both the Keler and the Berry forulation. In the unodi ed Keler forulation [hown a Keler (original)] k 1 ˆ and thi i ore repreentative of a aritie cloud. Thi i clearly evident when we copare thi graph with the graph correponding to the Berry (aritie) cae. Figure 1 clearly illutrate that the autoconverion rate of aritie cloud are vatly di erent fro thoe of continental cloud. We alo how the rate of change of the cloud water aount for a cloud with icrophyical propertie in between thee liit [Berry (interediate)]. We have ued the Berry chee directly a well a the optiied Keler chee with k 1 ˆ 5: , where it now becoe clear that with thi procedure one can achieve coparable reult fro both chee. One of the iportant phyical paraeter that could poibly link the two paraeteriation chee i the tie of onet of precipitation. Thi i dicued in the following ection, where we alo dicu the procedure for optiiing the Keler chee in conjunction with the Berry chee. 4 On the onet of precipitation of a unifor cloud We conider cloud converion and accretion operating in a deep odel cloud of water content and neglect the advection ter for the ake of iplicity. Change in cloud and precipitation water content then arie a a reult of autoconverion and collection of cloud drop by precipitation. 4.1 The Keler forulation Fig. 1. Rate of change of cloud water content calculated fro the claic Keler (1969) and Berry (1968) chee The equation decribing the aount of cloud water can be expreed a dm ˆ d dt dt ˆk 1 a k2 0 M 7=8 ; 4 where k 1 > 0 when > a and k2 0 > 0. The contant k2 0 ˆ k 2EN 1=8 0 where E i the collection e ciency and k 2 ˆ 6: (Keler, 1969). Since there i no vertical otion we alo require that M ˆ 0 ; 0 t 1 ; 5 where 0 i the aount of cloud water when t ˆ 0 and ince M 7=8 ' M ; d dt ˆ k0 2 2 k 1 k k 1 a ; 6 7

4 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee 631 or, Z d t ˆ 0 k2 0 2 k 1 k2 0 0 k 1 a : 8 Integration of thi equation will give u a handle on the tie of onet of precipitation. A convenient eaure would be to evaluate the tie over which reduce fro 0 to 0 =e (thi can be eaily obtained fro plot of a a function of t). In order to integrate Eq. (8) we et k2 0 ˆ C ; k 1 k2 0 0 ˆ2B ; k 1 aˆa ; and we have 1 t ˆ p tan 1 c B 0 p 9 AC B 2 AC B 2 when AC > B 2 ; 10 or " p # 1 t ˆ p 2 C B B ln 2 0 AC p B 2 AC C B 11 B 2 AC when B 2 > AC : 12 Fro the two new olution given by Eq. (9) and (11) it i clear that the tie of onet of precipitation will vary according to the condition ipoed by Eq. (10) and (12) which are function of the autoconverion coe cient k 1, the autoconverion threhold a, the initial cloud water aount 0 and the accretion coe cient k2 0. In Fig. 2a we exaine the e ect of the initial cloud water content with xed value of the autoconverion coe cient k 1 ˆ 5: and threhold a ˆ 0:3 g 3. We nd that with lower initial cloud water content the onet tie are longer. Note that the kink in the curve how the region when ˆ a, below which point only the proce of accretion i operative. In Fig. 2b we x 0 at1g 3 and a at 0:3 g 3 and exaine the e ect of varying k 1. For low value of k 1, the precipitation onet tie are large a i expected. In Fig. 2c we exaine the e ect of varying the autoconverion threhold and nd that the onet tie are higher for larger threhold value. 4.2 The Berry forulation In the Berry forulation (Berry, 1968) the equation equivalent to Eq. (4) i dm dt ˆ d dt ˆ :0366N k2 0 M 7=8 : 13 b D b Fig. 2a±c. Tie of onet of precipitation calculated fro the analytical olution of Eq. (9) and (11) uing the Keler (1969) chee. a E ect of varying initial cloud water content with xed autoconverion coe cient and threhold. b E ect of varying autoconverion coe cient with xed initial cloud water content and autoconverion threhold. c E ect of varying autoconverion threhold with xed initial cloud water content and autoconverion coe cient

5 632 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee Typically, N b 200 c 3, 0:5 g 3, and D b 0:24 and for thi cae 0:0366N b D b 5:0, and therefore approxiating Eq. (13) we can write dm ˆ d dt dt 'k 1 3 k2 0 M ; 14 where k 1 ˆ 60 0:0366N b =D b. Hence, Z d t ˆ 0k 1 3 k2 02 k2 0 0 Further by etting k2 0 0 ˆ A 0 ; k2 0 ˆ 2B0 ; k 1 ˆ C 0 we obtain for the cae when B 02 > A 0 C 0 t ˆ 1 A 0 ln Š 0 1 2A 0ln A0 2B 0 C 0 2 B 0 p 2A 0 B 0 2 A 0 C " 0 p ln C0 B 0 # B 0 2 A 0 C 0 0 p C 0 B 0 : B 0 2 A 0 C 0 : A an illutrative exaple, a we have een before, for the EUCREX tratocuulu cloud, N b 200 c 3, D b ˆ 0:24; thi yield k 1 ˆ 5: Setting N 0 ˆ 10 7, E ˆ 1:0, we obtain k 2 ˆ 5: For an initial cloud water content of 1 g 3, we nd that A 0 C 0 < B 02 and therefore indeed Eq. (16) i valid. In contrat for typical aritie cloud N b 10 c 3, D b 0:4, yielding k 1 ˆ 1: For thi ituation A 0 C 0 > B 02 and we obtain a di erent olution: t ˆ 1 A 0 ln Š 0 1 2A 0ln A0 2B 0 C B 0 p tan 1 C 0 B 0 0 p : 17 2A 0 B 0 2 A 0 C 0 A 0 C 0 B 0 2 Equation (16) and (17) are alo two new olution and predict di erent onet tie for icrophyical paraeter appropriate for continental and aritie cloud and alo provide ueful inight on the relative iportance of the procee of autoconverion and accretion. In Fig. 3a we how cloud water content a a function of tie for a typical continental cloud, a aritie cloud, and a cloud with icrophyical propertie in between thee two extree uch a the EUCREX tratocuulu. The curve are obtained fro the olution in Eq. (16) and (17) and how that the onet tie of precipitation for a continental cloud i 1200 a copared to 600 for a aritie cloud. In Fig. 3b we how the e ect of initial cloud water content with N b 200 c 3 and D b 0:24. Larger initial cloud water content yield aller onet tie. Thi trend i alo oberved fro the Keler forulation (ee Fig. 2a). One of the objective of thi analyi i to etablih an equivalence between the claic Keler paraeteriation and the Berry (1968) paraeteriation. Although the Berry (1968) paraeteriation i able to di erentiate between aritie and continental cloud, it i not a widely ued in any LES odel. Mot bulk-paraeteried cloud odel continue to utilie oe odi ed verion of the Keler chee [for exaple Rutledge and Hobb (1983); the widely ued U.K. Meteorological O ce LES odel (Derbyhire et al., 1994)]. With the preent analyi we are now in a poition judiciouly to pecify the autoconverion coe cient and the threhold in any cloud odel uing the Keler chee rather than ue arbitrary value. We ugget that when cloud icrophyical obervation are available, the oberved value hould be ued a far a poible for the purpoe of odel contrainent. Then we recoend the ue of the Berry (1968) forulation, Eq. (2), to calculate the autoconverion rate. Auing that Eq. (1) and (2) are identical rate, one can now Fig. 3a, b. Tie of onet of precipitation calculated fro the analytical olution of Eq. (16) and (17) uing the Berry (1968) chee. a Cloud water content for di erent cloud type. b E ect of varying initial cloud water content with N b 200 c 3, D b 0:24

6 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee 633 calculate k 1 with a precribed value of a, or vice vera. In thi way we need to pecify one rather than two adjutable paraeter. For exaple if we aue that a ˆ 0:33 kg 3 then we obtain a value of k 1 ˆ 5: a the correct input to the Keler chee. Thi enable one alo to acertain the condition under which the two chee can yield nearly identical cloud water content. We how thi in Fig. 4, where we obtain nearly the ae cloud water content a a function of tie. Uing thi approach we nd igni cant iproveent in the predictive capacitie of our LES odelling of the EUCREX tratocuulu cloud. In thi context it i worth entioning that Weintein (1970) howed fro a enitivity analyi that the autoconverion threhold a i a crucial paraeter. Rutledge and Hobb (1983) deduced a value of kg kg 1 0: kg 3 fro airborne obervation of cloud and rainwater content in war tratifor cloud. Fro the procedure jut dicued with the above value of a the calculated value of k 1 to be ued in a cloud odel involving the Keler chee turn out to be 1: , very cloe to the value of 1: ued by Rutledge and Hobb (1983). Thi exaple deontrate the overall iplicity of the outlined procedure. In the next ection we exaine the proce of raindrop evaporation within the fraework of the Keler chee and include an accurate etiation of the di uion coe cient of water vapour ued in the chee. 5 Iproved paraeteriation for the evaporation of raindrop with an accurate etiation of di uional a tranport of water vapour Fig. 4. Equivalence of the optiied Keler (1969) chee and the Berry (1968) chee for N b 200 c 3, D b 0:24 In order to account for the rate of change of a of an evaporating raindrop one nd that variou paraeteried expreion are frequently ued in nuerical odel, particularly in LES odel. Many of thee odel incorporate the calculation of Keler (1969). A careful evaluation of thi chee reveal that oe of the paraeteriation involved with the quanti cation of the droplet evaporation ter are directly dependent on the di uion coe cient of water vapour. Modeller ue di uion data reported under laboratory condition which are often di erent fro atopheric condition. Secondly, any odel, and in particular oe of the LES odel ignore the teperature di erence between the evaporating droplet and the environental air. In thi paper we re-evaluate the evaporation ter involved in the Keler (1969) chee by calculating the exact value of the di uion coe cient of water vapour (accounting alo for the nite dipole oent of water vapour) and taking account of the teperature di erence between the drop and the abient air. We copare our reult with the obervation of Kinzer and Gunn (1951), with Keler (1969), and alo with the ore recent and de nitive obervation of Beard and Pruppacher (1971). Signi cant iproveent are obtained by uing our approach and our reult agree very well with the obervation of Beard and Pruppacher (1971). 5.1 On the evaporation of a rain droplet For an evaporating drop, if one aue that over a liited range the aturation vapour denity at the urface of the drop i a linear function of the abolute teperature, then q a q b ' q b 1 U U T b T a dq dt ; 18 where U i the fractional relative huidity and dq=dt i the ean lope of the aturation vapour denityteperature curve deterined at the teperature of the drop and the environent, q a and q b are the aturation denity of water vapour at the teperature of the drop, and the denity of water vapour in the environental air (Kinzer and Gunn, 1951). Alo, ince U ˆ qb ; 19 q a q a q b ' q a 1 U T b T a dq dt : 20 In Eq. (18) and (20), T b T a i the teperature di erence between the evaporating drop and the environent. Fro the experiental obervation of Kinzer and Gunn (1951), it wa found that the teperature of the freely falling drop are very cloe to the wet-bulb teperature. Equation (20) can be ued to etiate the vapour denity di erence fro which the evaporation rate i deterined only when T b T a dq dt < q a 1 U : 21

7 634 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee Fro the eaureent of the variation of aturation denity of water vapour (Maon, 1957; Sithonian Table, 1958) it i clear that dq=dt increae rapidly with increaing teperature. Uing repreentative value of the relevant quantitie we nd that Eq. (21) i true for teperature colder than the freezing point of water. In thi tudy we are ainly concerned with layer cloud without ice. Fro a record of the teperature eaureent of a tratocuulu cloud tudied during the EUCREX eaureent (Pawlowka and Brenguier, 1996), it i found that the teperature within the cloud ranged between 0:5 C and 0 C. Fro a uary of obervation on non-freezing layer cloud in id-latitude, it i known that drizzle ay often fall fro cloud which contain no ice crytal and i therefore produced by coalecence of droplet. For upercooled cloud in which no ice crytal were detected, about 30% were colder than 12 C, only 12% colder than 16 C, and only 5% colder than: 20 C. Continuou precipitation can be aintained only if the cloud water i replenihed by a peritent vertical otion fed, perhap over a large region, by low-level convergence in the wind eld (Maon, 1957). Fro the preceding paragraph we nd that for a vat ajority of layer cloud over the id-latitude, we ut be concerned with teperature ranging fro 0 Cto about 10 C, a teperature range afely applicable to ue Eq. (20). Soe of the LES code currently in ue, including the U.K. Meteorological O ce LES code (Derbyhire et al., 1994) ue the Keler (1969) chee for paraeteriing the evaporation rate of precipitation, and thi i baed on the eaureent of Kinzer and Gunn (1951). Thee author, however, report eaureent covering a teperature range of 0±40 C. Hence, uing the Kinzer and Gunn (1951) eaureent to derive epirical paraeteriation for the raindrop evaporation calculation at low teperature (particularly ub-zero teperature) i likely to yield inaccurate reult. In fact, Keler (1969) point out that hi etiation of D v q b q a (where D v i the di uivity of water vapour in air) could involve an error bar a large a 40%. It i poible that thi error can be greatly reduced if the ter D q b q a i etiated a accurately a poible. We hall rt etiate the agnitude of the teperature di erence between the drop and the abient air, for the condition typical to tratocuulu cloud oberved over the id-latitude. For the EUCREX obervation of 18 April 1994, the relative huidity wa 70%. Auing T b T a ' T b T w, where T w i the wetbulb teperature, and neglecting the weak dependence of L (latent heat of condenation) on the teperature (Roger and Yau, 1989), T b T w ˆ L C p W W ; 22 where W and W are the water vapour ixing ratio and the aturation ixing ratio, repectively. We can alo write, T b T w ˆ L W 1 U : C p 23 L Typically C p 2: K, and for U ˆ 0:7 and W ˆ kg kg 1 between 800±900 b and at teperature between 0±4 C, we nd that T b T w 3 C. Thi teperature di erence i often not accounted for in any of the LES code. Neglecting the teperature di erence, we have the following approxiate relationhip: q a q b ' q a 1 U : 24 However, a we have een earlier, in cold and huid condition, a teperature di erence of 3 C can contribute igni cantly to the aturation vapour denity di erence at the urface of the drop, and it i ore appropriate not to aue that the drop i at the ae teperature a the environent. In Table 2 we how the aturated vapour denity di erence at varying teperature including the contribution of the teperature di erence uing Eq. (20) and copare thi with the cae when the teperature di erence i neglected [uing the approxiate Eq. (24)]. In thi table we how reult correponding to a relative huidity of 70%. The lat colun of thi table how the percentage error that would be encountered if the teperature di erence i not accounted for. Fro Table 2 we nd that at a relative huidity of 70% and within a teperature range of 10 Cto5 C, a typical teperature range encountered in any cloud, it i very iportant to account for the teperature di erence between the evaporating droplet and the environent. The rate of change of a of a freely falling water drop in g 1 i given by dm i dt ˆ 4pa 1 F D v q a q b Š ; 25 where a i the drop radiu, i the equivalent thickne of the tranition hell outide the drop, F, a dienionle factor, and D v the di uion coe cient. Kinzer and Gunn' tabulation of the rt ter how that thi ter i nearly independent of the abient teperature and huidity and can be tted to an accuracy of about 20% (Keler, 1969) by: 4pa 1 F ˆ 2: D 8=5 ; 26 Table 2 Saturated water vapour denity di erence a a function of teperature T, C T b T a, C q a q b [Eq. (20)] q a q b [Eq. (24)] %error ) )

8 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee 635 where D ˆ 2a i the drop diaeter in etre. According to Keler (1969), the econd ter i related linearly to the aturation de cit a a D v q a q b ˆ10 5 a g 1 1 Š : 27 Since the rt ter in quare bracket in Eq. (25) i independent of the teperature and the huidity, the paraeteriation in Eq. (26) i adequately repreented. However, the econd ter in quare bracket in Eq. (25) doe trongly depend upon the teperature and the relative huidity. Hence it i very iportant to paraeterie thi ter a accurately a poible and preferably in a anner that can allow one to incorporate the teperature dependence. Uing Eq. (24) we can write D v q a q b 'D v q a 1 U : 28 When one copare Eq. (27) and (28), it i found that in order for Eq. (27) to be dienionally correct, the right-hand ide of Eq. (27) hould involve a factor that hould have the unit of a di uion coe cient L 2 T 1. Indeed, the factor 10 5 on the right-hand ide hould have unit of D v, and thi i poible when D v ˆ 0:1 c 2 1. Or, in other word, the Keler (1969) chee a decribed in Eq. (27) iplicitly aue that the di uion coe cient of water vapour in air i equal to 0:1 c 2 1. However, the oberved value of the di uion coe cient of water vapour in air are uch higher. For exaple, at 0 C D v ˆ 0:2 c 2 1 (Sithonian Table, 1958). Auing a value of 0:1 c 2 1 would therefore at the very leat underpredict the evaporation rate by a factor of 2. However, the ot accurate procedure i to ue Eq. (20) and one now obtain: dq D v q a q b 'D v q a 1 U T b T a : dt 29 Thu, an iediate and obviou iproveent to the Keler (1969) icrophyical chee would be to ue Eq. (29) intead of Eq. (27). Experiental value of di uion coe cient are uually reported for laboratory condition only, and odeller are often contrained by the availability of di uion coe cient data relevant to the real atophere, particularly a a function of altitude. Hence, it would be very ueful if one could calculate the di uion coe cient theoretically at any deired level in the atophere a a function of both teperature and preure intead of adopting arbitrarily precribed value. In the next ection we how the ue of a 6±12 Lennard-Jone odel to obtain the binary di uivity of water vapour in air. Although the ethod i decribed in Ghoh (1993) and Ghoh et al. (1995) we outline the general procedure for the ake of copletene. In thi tudy however, we have extended the original routine decribed in Ghoh (1993) and Ghoh et al. (1995) to account for the dipole oent of water, ince H 2 Oia trongly polar olecule with a dipole oent of 1.8 Debye. 5.2 The Lennard-Jone ethod The theory decribing di uion in binary ga ixture at low to oderate preure ha been well developed and the reult are credited to Chapan and Enkog. Sipli ed verion can be found in tandard textbook of phyical cheitry (e.g. Maitland et al., 1982). Baed on the kinetic theory, the di uion coe cient ay be expreed a (Reid et al., 1987): 3=2 0:00266T D AB ˆ PM 1=2 AB r2 AB X ; 30 D where, M A, M B ˆ olecular weight of A and B, M AB ˆ 2 1=M A 1=M B Š 1, T ˆ teperature, K, P ˆ preure, bar, r AB ˆ characteritic length, A Ê, X D ˆ diffuion colliion integral, dienionle, D AB ˆ diffuion coe cient, c 2 1. The key to the ue of Eq. (30) i the election of an interolecular force law and the evaluation of X D. Reid and Sherwood (1958) ephaie that of all the available ethod for correlating the interolecular energy w between two olecule to the ditance of eparation r, the ot accurate ethod i that given by Lennard-Jone: r 12 r 6 w ˆ 4 ; 31 r r with and r a the characteritic Lennard-Jone energy and length, repectively. To ue Eq. (30) oe rule ut be choen to obtain r AB fro r A and r B. Alo, it can be hown that X D i a function only of kt = AB (k being Boltzann' contant), where again oe rule ut be elected to relate AB to A and B. The following iple rule are uually eployed: AB ˆ A B 1=2 ; 32 r AB ˆ r A r B =2 : 33 X D i tabulated a a function of kt = for the 6±12 Lennard-Jone potential and variou analytical approxiation are alo available. We ued the relation of Neufeld et al. (1972): X D ˆ A T B C exp DT E exp FT G exp HT ; 34 where, T ˆ kt = AB, A ˆ 1:06036, B ˆ 0:15610, C ˆ 0:19300, D ˆ 0:47635, E ˆ 1:03587, F ˆ 1:52996, G ˆ 1:76474 and H ˆ 3: Application of the Chapan-Enkog theory to the vicoity of pure gae ha led to the deterination of any value of and r (Reid et al., 1987; Bzowki et al., 1990). Polar gae. If one or both coponent of a ga ixture are polar, a odi ed Lennard-Jone relation, uch a

9 636 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee the Stockayer potential, i often ued. A di erent colliion integral relation becoe neceary and Lennard-Jone r and value are not u cient (Reid et al., 1987). A odi ed colliion integral X D i now given a (Reid et al., 1987) X D ˆ X D [Eq.(34)] 0:19d2 AB T ; 35 where T ˆ kt AB and d ˆ 1:94103 l 2 p V b T b, l p ˆ dipole oent, debye, V b ˆ liquid olar volue at the noral boiling point, c 3 ol 1, T b ˆ noral boiling point (1 at), K k ˆ 1:18 1 1:3d2 T b ; 36 r ˆ 1:585V 1=3 b 1 1:3d 2 ; 37 Ê d AB ˆ d A d B 1=2 ; 38 AB k ˆ A B 1=2 k k ; 39 r AB ˆ r A r B 1=2 : 40 For H 2 O, V b ˆ 18:7 c 3 ol 1, T b ˆ 375:15 K; and uing the foregoing relation we nd that d H2 O ˆ 0:9, k H 2 O ˆ 903:97 K and r H 2 O ˆ 2:43 A. 5.3 Re-evaluation of the droplet evaporation ter Uing the preceding forulation we are able to calculate a typical di uivity pro le of water vapour. In order to do thi we needed teperature and preure pro le. We ued the U.K. Meteorological O ce LES odel contrained with EUCREX teperature obervation. Thi i hown in Fig. 5, fro which we clearly nd that the di uion coe cient of water vapour between 900± 1200 (a typical region where the tratocuulu cloud are oberved) i 0:2 c 2 1. In the ae gure we have alo hown the teperature pro le. Having obtained the di uivity pro le for water vapour we hall now copare our etiation of the ter D v q a q b uing Eq. (29) intead of Eq. (27) with thoe of Kinzer and Gunn (1951) and Beard and Pruppacher (1971). Thee author report their experiental value of their etiation of D v q a q b at a preure of one atophere for varying relative huiditie a well a teperature. We have ued their reult for a teperature of 0 C correponding to the condition relevant to the tratocuulu cloud iulated by the LES code. Thi i hown in Fig. 6, where the reult of the Keler (1969) chee are alo hown. We oberve excellent agreeent between the obervation of Beard and Pruppacher (1971) and the preent approach. We nd that wherea in the Keler paraeteriation one Fig. 5. Calculated di uivity pro le of H 2 O uing the Lennard-Jone odel and the teperature pro le fro the LES odel oberve an overall underprediction, in our analyi we have uch better agreeent at all huidity level. Unfortunately, there are no experiental data available for a coparion alo at lower teperature (e.g. at 10 C). The analyi in thi ection clearly point to the iportance of aiing toward accurate a tranport etiate involved in cloud icrophyical procee, and in particular thi tudy deontrate the uefulne of the Lennard-Jone odel for an accurate etiation of the di uion coe cient ± a coe cient that one frequently encounter in a-tranfer calculation. 6 Concluion In thi paper we have carefully exained the proce of cloud autoconverion a decribed in the claic Keler Fig. 6. D v q a q p a a function of relative huidity

10 S. Ghoh, P. R. Jona: On the application of the claic Keler and Berry chee 637 (1969) and Berry (1968) chee. We dicu in detail the applicability of thee chee in LES odel. We are alo able to derive new analytical expreion for the tie of onet of precipitation uing both the Keler and Berry forulation. We then dicu the condition when the two chee are equivalent. Our tudy deontrate that by rt calculating the autoconverion coe cient a a function of droplet nuber denity and droplet diperion uing the Berry (1968) chee, one can then calculate the autoconverion threhold rather than arbitrarily pecify two adjutable paraeter in the Keler forula. In thi way one can till ue the iple and linear Keler forula in LES odel and alo indirectly incorporate the e ect of di erent cloud type via the Berry (1968) forulation a decribed. Finally, we alo critically exaine the proce of droplet evaporation within the fraework of the claic Keler chee. We iprove the exiting paraeteriation with an accurate etiation of the di uional a tranport of water vapour. We then deontrate the overall robutne of our calculation by coparing our reult with the experiental obervation of Beard and Pruppacher (1971). Thi coparion how an excellent agreeent. Acknowledgeent. J. P. Duvel thank T. Trautann and another referee for their help in evaluating thi paper. Reference Beard, K. V., and H. R. Pruppacher, A wind tunnel invetigation of the rate of evaporation of all water drop falling at terinal velocity in air, J. Ato. Sci., 28, 1455±1464, Berry, E. X., Modi cation of the war rain proce, Proc. Firt Natl. Conf. Weather odi cation, Ed. Aerican Meteorological Society, State Univerity of New York, Albany, pp. 81±88, Bzowki, J., J. Ketin, E. A. Maon, and F. J. Uribe, Equilibriu and tranport propertie of ga ixture at low denity: eleven polyatoic gae and ve noble gae, J. Phy. Che. Ref. Data, 19, 1179±1232, Derbyhire, S. H., A. R. Brown, and A. P. Lock, The eteorological o ce large eddy iulation odel. Met O (APR) Turbulence and Di uion Note, No., 213, Bracknell Ghoh, S., On the di uivity of trace gae under tratopheric condition, J. Ato. Che., 17, 391±397, Ghoh, S., D. J. Lary, and J. A. Pyle, Etiation of heterogeneou reaction rate for tratopheric trace gae with a particular reference to the di uional uptake of HC1, ClONO 2, HNO 3, and N 2 O 5 by polar tratopheric cloud, Ann. Geophyicae, 13, 406±412, Jona, P. R., and S. Ghoh, Siulation of a tratocuulu topped planetary boundary layer with a particular ephai on the ode of energy tranfer between updraught and downdraught. To appear in a EUCREX pecial iue, Touloue, France, Jona, P. R., and B. J. Maon, The evolution of droplet pectra by condenation and coalecence in cuulu cloud. Q. J. R. Meteorol. Soc., 100, 286±295. Keler, E., On the ditribution and continuity of water ubtance on atopheric circulation. Meteorol. Monogr., 10 (32), 84, Kinzer, G. D., and R. Gunn, The evaporation, teperature and theral relaxation tie of freely falling water drop. J. Meteorol., 8, 71±83, Maitland, G. C., M. Rigby, E. B. Sith, and W. A. Wakeha, Interolecular force ± their origin and deterination, Oxford Science Publication, Oxford, Maon, B. J., The phyic of cloud, Clarenden Pre, Oxford, Neufeld, P. J., A. R. Janen, and R. A. Aziz, Epirical equation to calculate 16 of the tranport colliion integral X l; for the Lennard-Jone (12-6) potential, J. Che. Phy., 57, 1100±1102, Pawlowka, H., and J. L. Brenguier, A tudy of the icrophyical tructure of tratocuulu cloud. Proc. 12th Int. Conf. Cloud and precipitation, Zurich, Ed. P. R. Jone, Publihed by Page Bro., Norwich, U.K., Vol. I, pp. 123±126, Reid, R. C., and T. K. Sherwood, The propertie of gae and liquid, McGraw-Hill, New York, Reid, R. C., J. M. Praunitz, and B. E. Poling, The propertie of gae and liquid, McGraw-Hill, New York, Roger, R. R., and M. K. Yau, A hort coure in cloud phyic, Pergaon Pre, Oxford, Rutledge, S. A., and P. V. Hobb, The eocale and icrocale tructure and organiation of cloud and precipitation in idlatitude cyclone. VIII: a odel for the ``eeder-feeder'' proce in war-frontal rainband, J. Ato. Sci., 40, 1185± 1206, Sipon, J., and V. Wiggert, Model of precipitating cuulu tower, Mon. Weather Rev., 97, 471±489, Singleton, F., and D. J. Sith, Soe obervation of drop-ize ditribution in low-layer cloud. Q. J. R. Meteorol. Soc., 86, 454±467, Sithonian Meteorological Table, The Sithonian Intitution, Wahington, D.C., Warner, J., The icrotructure of cuulu cloud. Pt. 2. the e ect of droplet ize ditribution of the cloud nucleu pectru and updraft velocity, J. Ato. Sci., 26, 1272±1282, Weintein, A. I., A nuerical odel of cuulu dynaic and icrophyic, J. Ato. Sci., 27, 246±255, 1970.