Timescale analysis of aerosol sensitivity during homogeneous freezing and implications for upper tropospheric water vapor budgets

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1 Timescle nlysis of erosol sensitivity during homogeneous freezing nd implictions for upper tropospheric wter vpor budgets Jennifer E. Ky 1 nd Robert Wood 2 1 Ntionl Center for Atmospheric Reserch, Boulder, CO, jenky@ucr.edu, ph: Deprtment of Atmospheric Sciences, University of Wshington, Settle, WA Abstrct: Using timescles for the genertion nd depletion of wter vpor, we predict erosol sensitivity in clouds formed by homogeneous freezing. Our timescle nlysis explins why erosol sensitivity increses drmticlly with ice deposition coefficients (α i ) << 0.1, nd lso why erosol sensitivity increses s verticl velocity increses, temperture decreses, N decreses, nd erosol size decreses. We combine existing in-situ observtions with dibtic prcel modeling to constrin α i ~ 0.1 for smll ice crystls forming t high ice supersturtions. Two importnt implictions for understnding nd modeling upper tropospheric wter vpor budgets emerge from our results: 1) erosol sensitivity cn be pprecible t low tempertures nd slow updrfts found in the upper tropicl troposphere, 2) reconciling our results with recent lbortory mesurements supports theory tht α i increses with ice supersturtion nd/or decreses with ice crystl size. 1. Introduction The sensitivity of clouds to erosol properties is n importnt re of climte reserch. Twomey (1974) described the first indirect effect of incresing erosol concentrtions (N [m -3 ]) on clouds: For fixed wter content, the drop number concentrtion nd brightness of wrm clouds increses. In contrst, modeling studies hve shown tht the number of ice crystls (N i [m -3 ]) resulting from homogeneous freezing is reltively insensitive to upper tropospheric N (e.g., Jensen nd Toon (1994), DeMott et l. (1997), Kärcher nd Lohmnn (2002, 2002b), Kärcher nd Ström (2003)). In other words, these studies imply wek erosol sensitivity, or tht incresing N hs negligible effect on N i or η << 1 where η is n erosol sensitivity prmeter defined s: d( ln N i ) η (1) d ln( N )

2 Observtions show positive but wek correltion between N i nd N during cold cloud formtion (Siefert et l., 2004). If η is lrger thn these studies suggest, n increse in nthropogenic N will increse cold cloud N i. For fixed wter content, incresing N i will increse cold cloud lbedos nd lter rditive fluxes. In ddition, incresing N i will increse the drwdown of supersturtion in the upper tropicl troposphere nd therefore lter the wter vpor budget of the strtosphere. Given the influence of cold cloud microphysicl properties on rditive fluxes nd wter vpor budgets, it is importnt to understnd the tmospheric conditions under which cold clouds re sensitive to chnges in N. In this pper, we investigte the physicl fctors tht determine η in cold clouds formed by homogeneous freezing. Although erosols tht serve s heterogeneous ice nuclei cn lter cold cloud properties, we do not consider the impct of heterogeneous freezing on cold cloud erosol indirect effects. Heterogeneous freezing is not well constrined by observtions or theory. In ddition, observtions of lrge ice supersturtions (e.g., Ovrlez et l. (2002)) nd low ice nuclei concentrtions (e.g., DeMott et l., (2003)) suggest tht homogeneous nucletion is n importnt ice formtion mechnism in the upper troposphere. In Section 2, we use n dibtic prcel model with binned ice microphysics (Ky et l., 2006) to demonstrte tht simple timescle rtio explins the dependence of η upon thermodynmic fctors including verticl velocity (w [m s -1 ]) nd co-vrying temperture (T [ C]) nd pressure (P [mb]), nd microphysicl fctors including N, hydrted erosol rdius (r [m]), nd the ice deposition coefficient or mss ccommodtion coefficient (α i ). In Section 3, we discuss the implictions of our results for the tmosphere. In Section 4, we summrize our results nd provide suggestions for future work. Our work builds on the nlyticl results of Kärcher nd Lohmnn (2002,b) (herefter KL). KL found tht N i is primrily controlled thermodynmiclly rther thn microphysiclly, with strong dependence upon w. KL compred ice crystl growth timescles to the timescle of the freezing event, nd found two homogeneous freezing regimes: fst-growth nd slow-growth regime. Yet, KL did not consider two importnt microphysicl fctors: 1) KL did not ddress the current uncertinty in α i, i.e., the frction of impinging wter vpor molecules tht re incorported into n ice crystl lttice (Pruppcher nd Klett, 1997). KL ssumed α i =0.5 in their nlysis, but lbortory mesurements of α i vry from to 1 (e.g., Hynes et l. (1992), Mgee et l. (2006)). 2) KL did not explicitly tret the depletion of N by freezing, crucil fctor when N i re limited by N. 2. Wht determines η? For simplicity, we consider cold cloud formtion during dibtic scent t constnt w. The supersturtion with respect to ice (S i ) increses during scent nd, t the beginning of freezing

3 (time t = 0), the homogeneous freezing rte (J hom [m -3 s -1 ]), n exponentilly incresing function of S i reches threshold vlue (J o [m -3 s -1 ]). Freezing stops t lter time (t event [s]) when vpor deposition on newly formed ice crystls cuses S i to decrese nd J hom to decrese below J o. Given this physicl picture of cloud formtion, the N i generted in homogeneous freezing event cn be pproximted s: N i = t event 0 J 4 3 ( S i( t)) π r ( t) N ( t dt (2) 3 hom ) With this physicl model of cold cloud formtion, the prtitioning of wter between the ice nd vpor phse depends on competition between lifting (cooling), which increses S i, nd ice crystl growth, which depletes S i.. Using timescle nottion, the dependence of S i on the competition between lifting nd growth cn be expressed s: 78 ds i dt = S i τ lift S i τ growth (3) where τ lift is the timescle for increse of S i vi cooling through scent, nd τ growth is timescle for growth of freshly- nucleted ice crystls by vpor deposition. Becuse ice crystl growth results from vpor deposition, τ growth is lso timescle for the drwdown of S i. We hypothesize tht η cn be predicted by the timescle rtio, R. τ growth R (3b) τ lift In other words, η is entirely determined by the competition between the rtes of lifting (cooling) nd ice crystl growth. When R >> 1 ice crystl growth is reltively slow when compred with the cooling tht increses S i. Consequently, lrge S i vlues occur for long periods, nd lmost ll of the vilble erosol cn freeze (η pproches 1). When R << 1, ice crystl growth is reltively fst when compred to cooling. As result, lrge S i re quickly reduced, nd only smll number of the vilble erosol cn freeze (η pproches 0). Note tht R>>1 is roughly equivlent to KL s fst growth regime while R << 1 is roughly equivlent to KL s slow growth regime. To evlute if R cn quntittively predict the erosol sensitivity prmeter η, nlyticl expressions for τ lift nd τ growth re required. In n nlyticl nlysis of rising dibtic prcel bsed on Eq. (2) nd KL, the time constnts τ lift nd τ growth rise nturlly nd re defined s follows:

4 τ = [ Q w (4) lift with Q ] Γ L s ( Si + 1) 5 = T RvT 2 where Q 1 is thermodynmic constnt, Γ = K m -1 is the dry dibtic lpse rte (pproprite for the low tempertures being considered), L s = J kg -1 is the ltent het of sublimtion, nd R v = 461 J K -1 kg -1 is the gs constnt for wter vpor. 2 / 3 * 1 τ growth = [( KN ) ( Si D )] (5) with K = 2 ( ρ ρ ) nd D * v st i Dv = r Dv + r + λ r α i i v v 2πM w R T idel Dv = r λ + r + λ α r i where K is constnt, ρ st-i is the sturtion vpor density with respect to ice (kg m -3 ), ρ i = 900 kg m -3 is the density of ice, * D v is the modified vpor diffusivity (m 2 s -1 ) (Eq in Pruppcher nd Klett (1997)) which includes impednces to growth due to vpor diffusivity nd surfce processes but neglects the reltively smll therml impednce to growth, D v is the vpor diffusivity (m 2 s -1 ) (Eq in Pruppcher nd Klett (1997)), λ is the moleculr men free pth (m) (Eq in Jcobson (1999)), M w = kg mole -1 is the moleculr weight of wter, nd R idel = J K -1 mole -1 is the idel gs constnt. By clculting τ lift nd τ growth in number of model experiments (dibtic prcel model with binned microphysics, configurtion described in Tble 1), we evlute if η cn be predicted by R lone. In ll cses, we clculted R using the prcel model output t the timestep before freezing begins, herein defined s when the ice prticle production rte (dn i /dt) exceeds 1 m -3 s -1. To introduce our prcel model experiments, we first show n experiment in which we only vry α i (Figure 1). Decresing α i increses R by incresing τ growth without ffecting τ lift. Indeed, the increse in S i nd J hom resulting from long τ growth is why smll α i led to lrge N i (Gierens et l., 2003). The α i lifting experiment revels the drmtic effect of R on the sensitivity of N i to N nd on the

5 drwdown of S i. When efficient growth is ssumed (R<<1, α i > 0.1, blue curves Figure 1), J hom nd S i re quickly reduced by ice crystl growth, nd N i is not sensitive to N. In contrst, with inefficient growth (α i <<0.1, R>>1, red curves Figure 1), J hom nd S i rech lrge vlues nd the sensitivity of N i to N increses. It is interesting to note tht only the lowest α i (α i =0.001) result in persistent high S i (S i > 40%) fter the freezing event ends. Surprisingly, S i is depleted fster when α i =0.01 thn when α i = 1. This counter-intuitive result is explined s follows: When α i decreses, individul prticles grow inefficiently llowing both the pek S i nd J hom to increse. When the pek J hom increses, N i nd the totl surfce re drmticlly increse. Thus, even though individul prticles re growing inefficiently, the increse in totl ice surfce re llows the S i drwdown to be fster when α i =0.01 thn when α i = 1. Although the α i lifting experiment reveled tht there re complex reltionships between η nd tmospheric vribles, our prcel model runs suggest tht the erosol sensitivity prmeter η cn be predicted by R lone (Figure 2). When multiple prcel model experiments re plotted on one R vs. η grph (Figure 2, pnel A), they collpse onto single curve. In other words, one cn predict chnges in η by evluting the effect of chnging externl tmospheric conditions on R (Eq. 3). Using sensitivity test pproch, we explored the influence of plusible vritions in thermodynmic (w, T, P) nd microphysicl (α i, r ) vribles on η nd R (Figure 2b-d). Through R, we cn understnd the physicl bsis for the influence of these prmeters on η. Aerosol sensitivity increses with w through τ lift. With α i > 0.1, η is smll; however when α i < 0.1, τ growth increses nd η increses drmticlly. Aerosol sensitivity increses when T << -70 C becuse low T increses τ growth. Finlly, vritions in r hve only limited influence on η. As r decreses, η slightly increses becuse for fixed N, reducing r increses τ growth. 3. Implictions for the tmosphere When α i <0.1, plusible vritions in α i cn drmticlly chnge η, nd lter the sensitivity of η to vritions in other microphysicl nd thermodynmic vribles. The influence of α i on τ growth is more importnt t low α i becuse r ( r λ ) + λ α ir + λ αir * D v does not depend directly on α i, but on + 1 (Eq. 5). When λ α i r << 1, the precise vlue of α i is unimportnt becuse diffusive impediments to growth re more importnt thn surfce impediments to growth. Reviews of lbortory mesurements t cold cloud tempertures suggest tht α i for ice crystls could be s low s nd s high s 1 (Hynes et l., 1992); recent lbortory mesurements found α i =0.006

6 (Mgee et l., 2006). Given the sensitivity of η to α i when α i <0.1, discrepncies between α i mesurements must be resolved. Fortuntely, existing observtions cn be used to constrin α i for smll ice crystls forming t high S i in the tmosphere. In generl, observed N i ( cm -3 (e.g., Mce et l. 2001, Kärcher nd Ström, 2003) rrely pproch observed N ( cm -3 (e.g., Rogers et l, 1998, Minikin et l. 2003). The INCA field cmpign (Kärcher nd Ström, 2003) provides unique opportunity to constrin α i. Using INCA mesurements, we require α i 0.1 to simultneously mtch the men N i, N, T, nd w in lifting prcel model experiments (Tble 2). With α i = (Mgee et l., 2006), modeled N i re orders of mgnitude lrger thn INCA-observed N i. If present, shttering of ice crystls by ircrft probes (e.g., Field et l, 2003, 2006) would reduce observed N i nd increse the vlue of α i required to mtch INCAobserved vlues with prcel modeling experiments. Uncertinty in the observed w lso hs n importnt effect on the constrined α i. If lrge rnge of w re considered (3 cm/s < w < 50 cm/s), much lrger rnge of α i (0.01 < α i < 1) re consistent with the men observed N i. In summry, tmospheric observtions suggest tht η rrely pproches 1 nd α i re ~ 0.1 for smll ice crystls forming t high S i. Therefore, lbortory observtions of α i = (Mgee et l., 2006), which were mde t S i < 20%, my only be pproprite for lrge ice crystls or t low S i. There is theoreticl bsis for the ltter possibility (Nelson nd Bker (1996), Wood et l. (2001)), but further mesurements re required to constrin the behvior of α i s function of S i nd ice crystl size. Assuming α i = 0.1 for smll ice crystls forming t lrge S i, cold clouds in the tmosphere primrily form in regime where η << 1 (Figure 3). In other words, the N i resulting from homogeneous freezing is generlly thermodynmiclly-limited, not erosol-limited. This outcome grees with KL, Hoyle et l. (2005), Kärcher nd Ström, (2003), nd Ky et l. (2006), who ll found tht tmospheric N i re primrily controlled by w. With α i = 0.1, our modeling results do suggest there re conditions under which N i does depend on N. First, η increses t very lrge w (pproximtely w > 100 cm s -1 when T=-50 C nd α i =0.1). Second, there is significnt increse in η t tempertures such s those where cirrus clouds form in the tropicl upper troposphere. Finlly, η increses s N decreses, which cn be importnt t high w or low T.

7 Summry nd Discussion In this study, we used nlyticl nlysis, prcel modeling, nd observtions to understnd the sensitivity of ice crystl concentrtion produced by homogeneous freezing to chnges in erosols (η - Eq. 1). Our primry findings were: The dependence of η on lrge number of microphysicl nd thermodynmic vribles cn be explined nd predicted using single timescle rtio, R (Eq. 3) In modeling sensitivity experiments, η increses drmticlly when α i < 0.1, but lso when w increses, T decreses. N decreses, or r decreses. Using existing tmospheric observtions nd simple modeling, we suggest α i 0.1 for smll ice crystls forming t high S i. As consequence η is smll under most tmospheric conditions, but my increse t lrge w (w > 100 cm s -1 ) or t low T (T< -70 C). In order to reconcile lbortory nd tmospheric observtions, we suggest tht α i likely decreses with ice crystl size or increses with S i. We suggest tht future studies investigte the implictions of this study for wter vpor budgets t low T. Hoyle et l. (2005) proposed tht co-vrying T nd P decreses hve competing effects on ice crystl growth rtes, nd s result, the N i generted by homogeneous freezing does not depend on T. In contrst, our results suggest the blncing of T nd P effects on τ growth is not universl nd tht τ growth, η, nd N i do increse t low T. Simultneous observtions of N nd N i t low T could be used to estimte tmospheric α i nd to evlute the dependence of η on T presented in this study, but it will be criticl to mke ccurte estimtes of w in ddition to microphysicl prmeters in order to determine these constrints. Finlly, future work should constrin vritions in α i s function of S i nd ice crystl size. Constrining vritions in α i will be especilly importnt when evluting if α i cn explin tmospheric observtions of lrge persistent S i in the upper troposphere (Peter et l., 2006). Acknowledgments: JEK ws supported by NSF-ATM nd by the Office of Biologicl nd Environmentl Reserch of the U.S. Deprtment of Energy under contrct DE-AC06-76RL01830 to the Pcific Northwest Ntionl Lbortory s prt of the Atmospheric Rdition Mesurement Progrm. The uthors both thnk Dr. Mrci Bker for her encourgement nd her contributions to this work. References DeMott, P. J., D. C. Rogers, nd S. M. Kreidenweis (1997), The susceptibility of ice

8 formtion in upper tropospheric clouds to insoluble erosol components, J. Geophys. Res., 102(D16), 19,575-19,584. DeMott, P. J., D. J. Cziczo, A. J. Prenni, D. M. Murphy, S. M. Kreidenweis, D. S. Thomson, R. Borys, nd D. C. Rogers (2003), Mesurements of the concentrtion nd composition of nuclei for cirrus formtion, Proceedings of the Ntionl Acdemy of Sciences, 100:25, Field, P. R., R. Wood, E. Hirst, R. Greenwy, P. Kye, P. R. A. Brown, nd J. A. J. Smith, 2003: Ice prticle interrrivl times mesured with fst FSSP. J. Atmos. Ocenic Technol., 20, , Field, P. R., A. J. Heymsfield, nd A. Bnsemer (2006), Shttering nd prticle interrrivl times mesured by opticl rry probes in ice clouds, J. Atmos. Ocenic Technol., 23(2), 1,357 1,371. Gierens, K. M., M. Monier, nd J. Gyet (2003), The deposition coefficient nd its role for cirrus clouds, J. Geophys. Res., 108 (D2), 4069, doi: /2001jd Hynes, D. R., N. J. Tro, nd S. M. George (1992), Condenstion nd evportion of H20 on ice surfces, J. Phys. Chem., 96, 8,502-8,509. Hoyle, C. R., B. P. Luo, nd T. Peter (2005), The origin of high ice crystl number densities in cirrus clouds, J. Atmos. Sci., 62(7), 2,568-2,579. Jcobson, M. Z. (1999), Fundmentls of Atmospheric Modeling, 656 pp., Cmbridge University Press, New York. Jensen, E. J. nd O. B. Toon (1994), Ice freezing in the upper troposphere: sensitivity to erosol number density, temperture nd cooling rte, Geophys. Res. Lett., 21 (18), 2,019-2,022. Ky, J. E., Bker, M., nd D. Hegg (2006), Microphysicl nd dynmicl controls on cirrus cloud opticl depth distributions, J. Geophys. Res., 111, D24205, doi: /2005jd Kärcher, B., nd U. Lohmnn (2002), A prmeteriztion of cirrus cloud formtion: Homogeneous freezing of supercooled erosols, J. Geophys. Res, 107(D2), 4010, doi: /2001jd Kärcher, B., nd U. Lohmnn (2002b), A prmeteriztion of cirrus cloud formtion: Homogeneous freezing including effects of erosols size, J. Geophys. Res, 107(D23), 4698, doi: /2001jd Kärcher, B. nd J. Ström (2003), The roles of dynmicl vribility nd erosols in cirrus cloud formtion, Atmos. Chem. Phys., 3, Koop, T., B. Luo, A. Tsls, nd T. Peter (2000), Wter ctivity s the determinnt for homogeneous ice nucletion in queous solutions, Nture, 406, Mce, G. G., E. E. Clothiux, nd T. P. Ackermn (2001), The composite chrcteristics of

9 cirrus clouds: bulk properties reveled by one yer of continuous cloud rdr dt, J. Clim., 14, Mgee, N., A. M. Moyle, nd D. Lmb (2006), Experimentl determintion of the deposition coefficient of smll cirrus-like ice crystls ner 50 C, Geophys. Res. Lett., 33, L17813, doi: /2006gl Nelson, J. T., nd M. B. Bker, 1996: New theoreticl frmework for studies of vpor growth nd sublimtion of smll ice crystls in the tmosphere. J. Geophys. Res., 101, Minikin, A., A. Petzold, J. Strom, R. Krejci, M. Seifert, P. vn Velthoven, H. Schlyer, nd U. Schumnn (2003), Aircrft observtions of the upper tropospherice fine prticle erosol in the Northern nd Southern Hemispheres t mid ltitudes, Geophys. Res. Lett., 30(10), 1503, doi: /2002gl Ovrlez, J., J-F Gyet, K. Gierens, J. Strom, H. Ovrlez, F. Auriol, R. Busen nd U. Schumnn (2002), Wter vpour mesurements inside cirrus clouds in Northern nd Southern hemispheres during INCA, Geophys. Res. Lett., 29(16), 1813, /2001GL Peter, T., C. Mrcolli, P. Spichtinger, T. Corti, M. B. Bker, nd T. Koop (2006), When dry ir is too humid, Science, 314, 1,399-1,402. Pruppcher, H. R. nd J. D. Klett (1997), Microphysics of Clouds nd Precipittion, 2 nd ed., 954 pp., Kluwer Acdemic Publishers, Boston. Rogers, D. C., P. J. DeMott, S. M. Kredenweis, nd Y. Chen (1998), Mesurements of ice nucleting erosols during SUCCESS, Geophys. Res. Lett., 25:9, Seifert, M., J. Ström, R. Krejci, A. Minikin, A. Petzold, J.-F. Gyet, H. Schlger, H. Ziereis, U. Schumnn, nd J. Ovrlez (2004). Aerosol-cirrus interctions: A number bsed phenomenon t ll?, Atmos. Chem. Phys., 4, Twomey, S. A. (1974), Pollution nd the plnetry lbedo, Atmos. Env., 8, Wood, S. E., Bker, M.B., nd D. Clhoun, (2001). New model for the vpor growth of hexgonl ice crystls in the tmosphere, J. Geophys. Res, 106,

10 Figure Cptions Figure 1. Time series of prcel model lifting experiments (Tble 1) with w = 10 cm s -1, T 0 = -50 C, P 0 = 250 mb, N =100 cm -3, nd r =0.2 µm. Figure 2. Aerosol sensitivity vs. R for ll prcel model runs (pnel A) nd for sensitivity test with rnge of α i (pnel B), T 0 (pnel C) nd r -dry (pnel D). For pnels B-D: 1) circles w=2 cm s -1, dimonds w=10 cm s -1, nd squres=100 cm s -1. 2) Aerosol sensitivity ws clculted using the chnge in N i from model runs with N =100 cm -3 nd N =500 cm -3 (see Eq. 1). 3) Unless otherwise indicted, vlues re bse vlues: T 0 = -50 C, P 0 = 250 mb, nd r =0.2 µm. Figure 3. Mximum N i contoured s function of verticl velocity (w) nd erosol number concentrtion (N ) from the prcel model lifting experiments with α i =0.1, T 0 = -50 C, P 0 = 250 mb, nd r =0.2 µm. Colors indicte the erosol sensitivity prmeter η (Eq. 1) nd rnge from the thermodynmiclly limited nucletion regime (η =0) to the erosol-limited nucletion regime (η =1). The green line shows where η =0.5 line for T=-80 C nd P 0 =100 mb. The INCA field cmpign observtions re indicted in the trnsprent white circle (10-90% percentile vlues tken from Kärcher nd Ström (2003) nd Minikin et l. (2003)).

11 Tbles: Tble 1. Prcel model description nd configurtion used for this study. In ll model runs, α i does not depend on ice crystl size or on ice supersturtion. Prcel model description Model configurtion description nd vlidtion in Ky et l. prcel lifted t constnt verticl (2006) velocity (w) binned ice microphysics (300 bins) homogeneous nucletion (Koop et l., erosol ctivtion using Köhler curve, 2000) only monodisperse sulfuric cid erosol ice crystl fllout included with with n individul dry weight of prcel depth=100 m kg nd vrible number concentrtion sturtion vpor pressures e s nd es ice from Murphy nd Koop (2005) Tble 2. INCA observtions (Kärcher nd Ström, 2003) nd prcel model predicted N i. Prcel model N i re found from constnt lifting experiments (Tble 1) with INCA-observed w, T, nd N, nd P 0 = 250 mb. α i vlues of nd 0.13 were required to mtch the men Scotlnd nd Chile observtions respectively. INCA Observtions Prcel Model N i (Kärcher nd Ström, 2003) (# cm -3 ) w T (cm s -1 ) (ºC) N N i α i = α i = α i = α i = (# cm -3 ) (# cm -3 ) Scotlnd Chile

12 N i (# cm -3 ) J hom (m -3 sec -1 ) C. A Time (minutes) d N i \ dt (# cm -3 sec -1 ) Ice Supersturtion - S i (%) B D Time (minutes) R=15, α=0.001 R=1.5, α=0.01 R=0.15, α=0.1 R=0.017, α=1

13 Aerosol Sensitivity (dlnn i /dlnn ) A. model runs in pnels B-D ll model runs in this pper C. T 0 =-80 C, P 0 =100 mb T 0 =-60 C, P 0 =170 mb T 0 =BASE (-50 C, P 0 =250 mb) T 0 =-40 C, P 0 =340 mb 0.0 Aerosol Sensitivity (dlnn i /dlnn ) B. α i = α i = 0.01 α i = BASE (0.1) α i =1 D. R -dry = 0.1 μm R -dry = BASE (0.2 μm) R -dry = 0.5 μm τ growth /τ lift (Eq. 3) τ growth /τ lift (Eq. 3)

14 500 erosol limited freezing Verticl velocity - w (cm s -1 ) η =0.5 (T=-80 C) thermodynmiclly limited freezing INCA Observtions Aerosol number concentrtion - N (# cm -3 ) Aerosol sensitivity - η (T=-50 C)

Terminal Velocity and Raindrop Growth

Terminal Velocity and Raindrop Growth Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,