parameterization of cirrus cloud formation in large-scale...

Parameterization of cirrus cloud formation in large-scale models:

Homogeneous nucleation

Donifan Barahona1 and Athanasios Nenes1,2

Received 5 September 2007; revised 22 January 2008; accepted 21 February 2008; published 14 June 2008.

[1] This work presents a new physically based parameterization of cirrus cloudformation for use in large-scale models which is robust, computationally efficient, andlinks chemical effects (e.g., water activity and water vapor deposition effects) with iceformation via homogenous freezing. The parameterization formulation is based onascending parcel theory and provides expressions for the ice crystal size distribution andthe crystal number concentration, explicitly considering the effects of aerosol sizeand number, updraft velocity, and deposition coefficient. The parameterization isevaluated against a detailed numerical cirrus cloud parcel model (developed during thisstudy), the equations of which are solved using a novel Lagrangian particle-trackingscheme. Over a broad range of cirrus-forming conditions, the parameterization reproducesthe results of the parcel model within a factor of 2 and with an average relative error of�15%. If numerical model simulations are used to constrain the parameterization, errorfurther decreases to 1 ± 28%.

Citation: Barahona, D., and A. Nenes (2008), Parameterization of cirrus cloud formation in large-scale models: Homogeneous

nucleation, J. Geophys. Res., 113, D11211, doi:10.1029/2007JD009355.

1. Introduction

[2] The effect of aerosols on clouds and climate is one ofthe major uncertainties in anthropogenic climate changeassessment and prediction [Intergovernmental Panel onClimate Change (IPCC), 2007]. Cirrus is one of the mostpoorly understood cloud types, yet they can strongly impactclimate. Cirrus are thought to have a net warming effectbecause of their low emission temperatures and smallthickness [Liou, 1986]. They also play a role in regulatingthe ocean temperature [Ramanathan and Collins, 1991]and maintaining the water vapor budget of the uppertroposphere and lower stratosphere [Hartmann et al.,2001]. Concerns have been raised on the effect of aircraftemissions [Penner et al., 1999; Minnis, 2004; Stuber et al.,2006; IPCC, 2007] and long-range transport of pollution[Fridlind et al., 2004] changing the properties of uppertropospheric clouds, that is, cirrus and anvils, placing thistype of clouds in the potentially warming components ofthe climate system.[3] Cirrus clouds form by the homogenous freezing of

liquid droplets, by heterogeneous nucleation of ice onice nuclei, and the subsequent growth of ice crystals[Pruppacher and Klett, 1997]. This process is influenced bythe physicochemical properties of the aerosol particles (i.e.,size distribution, composition, water solubility, surface

tension, and shape), as well as by the thermodynamicalstate (i.e., relative humidity, pressure, and temperature) oftheir surroundings. Dynamic variability (i.e., fluctuationsin updraft velocity) also impact the formation of cirrusclouds, potentially enhancing the concentration of smallcrystals [Lin et al., 1998; Karcher and Strom, 2003; Hoyleet al., 2005].[4] The potential competition between homogeneous and

heterogeneous freezing mechanisms has an important impacton cirrus properties. For instance, by enhancing ice forma-tion at low relative humidity, heterogeneous effects maysuppress homogeneous freezing and decrease the ice crystalconcentration of the newly formed cloud [DeMott et al.,1994; Karcher and Lohmann, 2002a; Gierens, 2003; Haaget al., 2003b]. It has been suggested that, in the absence ofIN, homogeneous nucleation is not probable for pollutedareas [Chen et al., 2000; Haag et al., 2003b; Abbatt et al.,2006], at low updraft velocities (less than 10 cm s�1)[DeMott et al., 1997, 1998; Karcher and Lohmann, 2003],and at temperatures higher than �38�C [Pruppacher andKlett, 1997; DeMott et al., 2003]. Conversely, homogenousfreezing is thought to be the prime mechanism of cirrusformation in unpolluted areas, high altitudes, and lowtemperatures [Heymsfield and Sabin, 1989; Jensen et al.,1994; Lin et al., 2002; Haag et al., 2003b; Cantrell andHeymsfield, 2005; Khvorostyanov et al., 2006].[5] A major challenge in the description of cirrus forma-

tion is the calculation of the nucleation rate coefficient, J,that is, the rate of generation of ice germs per unit ofvolume. Historically this has been addressed with applica-tion of classical nucleation theory [DeMott et al., 1997;Pruppacher and Klett, 1997; Tabazadeh et al., 1997], orusing empirical correlations [i.e., Koop et al., 2000]. The

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, D11211, doi:10.1029/2007JD009355, 2008ClickHere

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1School of Chemical and Biomolecular Engineering, Georgia Instituteof Technology, Atlanta, Georgia, USA.

2School of Earth and Atmospheric Sciences, Georgia Institute ofTechnology, Atlanta, Georgia, USA.

Copyright 2008 by the American Geophysical Union.0148-0227/08/2007JD009355$09.00

D11211 1 of 15

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former requires the accurate knowledge of thermodynamicproperties, such as surface and interfacial tensions, densi-ties, and activation energies [Cantrell and Heymsfield,2005]. With appropriate extensions [i.e., DeMott et al.,1994, 1997; Khvorostyanov and Sassen, 1998; Chen et al.,2000; Lin et al., 2002], theory included in cirrus formationsimulations shows agreement with experimental measure-ments and field campaigns [i.e., Chen et al., 2000;Archuleta et al., 2005; Khvorostyanov et al., 2006]. Still,the physical properties of aqueous solutions and ice at lowtemperatures are subject to large uncertainty. Until now,the most reliable methods to calculate J are based onlaboratory measurements [Lin et al., 2002]. Koop et al.[2000] used experimental data to develop a parameteriza-tion showing J as a function of water activity andtemperature (rather than on the nature of the solute),which has been supported by independent measurementsof composition and nucleation rate during field campaignsand cloud chamber experiments [i.e., Haag et al., 2003a;Mohler et al., 2003].[6] The formation of cirrus clouds is modeled by

solving the mass and energy balances in an ascending(cooling) cloud parcel [e.g., Pruppacher and Klett, 1997].Although models solve the same equations (described insection 2.1), assumptions about aerosol size and compo-sition, J calculation, deposition coefficient, and numericalintegration procedure strongly impact simulations. Thiswas illustrated during the phase I of the Cirrus Parcel ModelComparison Project [Lin et al., 2002]; for identical initialconditions, seven state-of-the-art models showed variationsin the calculation of ice crystal concentration, Nc, (for purehomogeneous freezing cases) up to a factor of 25, whichtranslates to a factor of two difference in the infraredabsorption coefficient. Monier et al. [2006] showed that3 orders of magnitude difference in the value of J, whichis typical among models at temperatures above �45�C,will account only for about a factor of two variation in Nc

calculation. The remaining variability in Nc results fromthe numerical scheme used in the integration, the calcula-tion of the water activity inside the liquid droplets at themoment of freezing, and the value of the water vapordeposition coefficient.[7] Introducing ice formation microphysics in large-

scale simulations requires a physically based link be-tween the ice crystal size distribution, the precursoraerosol, and the parcel thermodynamic state. Empiricalcorrelations derived from observations are available [i.e.,Koenig, 1972]; their applicability, however, for the broadrange of cirrus formation presented in a GCM simulationis questionable. Numerical simulations have been used toproduce prognostic parameterizations for cirrus formation[Sassen and Benson, 2000; Liu and Penner, 2005], whichrelate Nc to updraft velocity and temperature (the Liu andPenner [2005] parameterization also accounts for thedependency of Nc on the precursor aerosol concentration,and was recently incorporated into the NCAR Commu-nity Atmospheric Model (CAM3) [Liu et al., 2007]).Although based on theory, these parameterizations are con-strained to the values of parameters (i.e., deposition coeffi-cient, aerosol composition and characteristics) used duringthe parcel model simulations (the uncertainty of which is

quite large). Karcher and Lohmann [2002b, 2002a] intro-duced a physically based parameterization solving analyti-cally the parcel model equations. In their approach, a‘‘freezing timescale’’ is used (related to the cooling rate ofthe parcel) to obtain an approximated crystal size distributionat the peak saturation ratio through a function describing thetemporal shape of the freezing pulse. This function, alongwith the freezing timescale, should be prescribed (the freez-ing pulse shape and freezing timescale may also change withthe composition and size of the aerosol particles). Ananalytical fit of the freezing timescale based on Koop et al.[2000] data was provided by Ren and Mackenzie [2005]. Theparameterization of Karcher and Lohmann [2002a, 2002b]has been applied in GCM simulations [Lohmann andKarcher, 2002] and extended to include heterogeneousnucleation and multiple particle types [Karcher et al.,2006]. All parameterizations developed to date providelimited information on the ice crystal size distribution, whichis required for computing the radiative properties of cirrusclouds [Liou, 1986].[8] This study presents a new physically based parame-

terization for ice formation from homogeneous freezing.The parameterization unravels much of the stochastic natureof the cirrus formation process by linking crystal size withthe freezing probability, and explicitly considers the effectsthe deposition coefficient and aerosol size and number, onNc. With this approach, the requirement of prescribedparameters is relaxed and the size distribution, peak satura-tion ratio, and ice crystal concentration can be computed.The parameterization is then evaluated against a detailednumerical parcel model (also developed here), which solvesthe model equations using a novel Lagrangian particle-tracking scheme.

2. Numerical Cirrus Parcel Model

[9] Homogenous freezing of liquid aerosol droplets is astochastic process resulting from spontaneous fluctuations oftemperature and density within the supercooled liquid phase[Pruppacher and Klett, 1997]. Therefore, only the fraction offrozen particles at some time can be computed (rather than theexact moment of freezing). At anytime during the freezingprocess, particles of all sizes have a finite probability offreezing; this implies that droplets of the same size andcomposition will freeze at different times, so freezing of aperfectly monodisperse droplet population will result in apolydisperse crystal population. This conceptual model canbe extended to a polydisperse droplet population; eachaerosol precursor ‘‘class’’ will form an ice crystal distributionwith its own composition and characteristics, which if super-imposed, will represent the overall ice distribution. In thefollowing sections, the formulation of a detailed numericalmodel, taking into account these considerations, is presented.The equations of the model share similar characteristics withthose proposed by many authors [Pruppacher and Klett,1997; Lin et al., 2002, and references therein] as the ascend-ing parcel framework is used for their development.

2.1. Formulation of Equations

[10] The equations that describe the evolution of icesaturation ratio, Si (defined as the ratio of water vaporpressure to equilibrium vapor pressure over ice), and

D11211 BARAHONA AND NENES: PARAMETERIZATION OF CIRRUS CLOUD

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temperature, T, in an adiabatic parcel, with no initial liquidwater present, are [Pruppacher and Klett, 1997]

dSi

dt¼ � Map

Mwpoi

dwi

dt� 1þ Sið Þ DHsMw

RT2

dT

dt� gMa

RTV

� �; ð1Þ

dT

dt¼ � gV

cp�DHs

cp

dwi

dt; ð2Þ

whereDHs is the latent heat of sublimation of water, g is theacceleration of gravity, cp is the heat capacity of air, pi

o is theice saturation vapor pressure at T [Murphy and Koop,2005], p is the ambient pressure, V is the updraft velocity,Mw and Ma are the molar masses of water and air,respectively, and R is the universal gas constant. Forsimplicity, radiative cooling effects have been neglected inequation (2), although in principle they can be readilyincluded. By definition, the ice mixing ratio in the parcel,wi, is given by

wi ¼rira

p6

ZDo;max

Do;min

ZDc;max

Dc;min

D3cnc Dc;Doð ÞdDcdDo; ð3Þ

where ri and ra are the ice and air densities, respectively. Dc

is the volume-equivalent diameter of an ice particle(assuming spherical shape), Do is the wet diameter of the

freezing liquid aerosol, nc(Dc, Do) =dNc Doð ÞdDc

is the ice

crystal number distribution function, Nc(Do) is the numberdensity of ice crystals in the parcel formed at Do; Do,min, andDo,max are the limits of the droplet size distribution, andDc,min and Dc,max are the limits of the ice crystal sizedistribution. Taking the time derivative of equation (3) weobtain

dwi

dt¼ ri

ra

p2

ZDo;max

Do;min

ZDc;max

Dc;min

D2c

dDc

dtnc Dc;Doð ÞdDcdDo; ð4Þ

where the term Dc3@nc Dc;Doð Þ

@twas neglected as instanta-

neous nucleation does not substantially deplete water vaporfrom the cloudy parcel. The growth term in equation (4) isgiven by [Pruppacher and Klett, 1997]

dDc

dt¼

Si � Si;eq� �G1Dc þ G2

; ð5Þ

with

G1 ¼riRT

4poi DvMw

þDHsri4kaT

DHsMw

RT� 1

� �

G2 ¼riRT2poi Mw

ffiffiffiffiffiffiffiffiffiffiffiffi2pMw

RT

r1

ad

;

ð6Þ

where ka is the thermal conductivity of air, Dv is the watervapor diffusion coefficient from the gas to ice phase, Si,eq isthe equilibrium ice saturation ratio, and ad is the water vapordeposition coefficient on ice.

[11] The crystal size distribution, nc(Dc, Do) is calculatedby solving the condensation equation [Seinfeld and Pandis,1998]

@nc Dc;Doð Þ@t

¼ � @

@Dc

nc Dc;Doð Þ dDc

dt

� �ð7Þ

subject to the boundary and initial conditions (neglectingany change of volume upon freezing),

@nc Dc; tð Þ@t

Dc¼Do

¼ no Do; tð Þ @Pf Do; tð Þ@t

� y Do; tð Þ;

nc Dc;Do; 0ð Þ ¼ 0; ð8Þ

where no(Do, t) is the liquid aerosol size distribution function,y(Do, t) is the nucleation function (which describes the num-ber concentration of droplets frozen per unit of time), andPf (Do, t) is the cumulative probability of freezing, given by[Pruppacher and Klett, 1997]

Pf Do; tð Þ ¼ 1� exp �p6

Z t

0

D3oJ tð Þdt

� �ð9Þ

and

@Pf Do; tð Þ@t

¼ p6D3

oJ tð Þ exp �p6

Z t

0

D3oJ tð Þdt

� �; ð10Þ

where J is the homogeneous nucleation rate coefficient, beingthe number of ice germs formed per unit volume of liquid perunit of time [Pruppacher and Klett, 1997].[12] Equation (7) is a simplified version of the continuous

general dynamic equation applied to the ice crystal popula-tion [Gelbard and Seinfeld, 1980; Seinfeld and Pandis,1998], where the nucleation term has been set as a boundarycondition to facilitate its solution. This can be done sincethe size of the ice particles equals the size of the precursoraerosol only at the moment of freezing.[13] The evolution of the liquid droplets size distribu-

tion, no(Do, t), is calculated using an equation similar toequation (7),

@no Do; tð Þ@t

¼ � @

@Do

no Doð Þ dDo

dt

� �� y Do; tð Þ: ð11Þ

The first term of the right-hand side of equation (11)represents the growth of aerosol liquid particles bycondensation of water vapor, and the second term theremoval of liquid particles by freezing. Boundary and initialconditions for equation (11) are simply the initial aerosolsize distribution and the condition of no particles at zerodiameter.

2.2. Numerical Solution of Parcel Model Equations

[14] Equations (1)–(11) are solved numerically using aLagrangian particle-tracking scheme; this uses a particle-tracking grid for the ice crystal population (the growth ofgroups of ice crystals is followed after freezing) coupled toa moving grid scheme (the liquid aerosol population isdivided into bins the size of which is changing with time),


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for the liquid aerosol population (Figure 1). For each timestep, the number of frozen aerosol particles is calculatedusing equation (9) and placed in a node of the particle-tracking grid, after which their growth is subsequentlyfollowed. This group of ice crystals represents a particularsolution of equation (7) for which all particles freeze at thesame time, and have the same size and composition. Since aparticular solution of equation (7) can be obtained for eachtime step and droplet size, the general solution of equation(7) is obtained from the superposition of all generated icecrystal populations during the freezing process; wi can thenbe calculated and equations (1)–(4) readily solved. Todescribe the evolution of no(Do, t), a moving grid isemployed, where frozen particles are removed from eachliquid drop size bin in each time step.[15] The particle tracking scheme allows ice particles to

grow in each node of the particle-tracking grid (no sortingafter each time step is needed); thus, the effect of numericaldiffusion on the simulation is minimized without losinginformation about ice crystal compostion. The discretizationof equation (7) transforms the partial differential equationinto a system of ordinary differential equations, each ofwhich represents the growth of a monodisperse ice crystalpopulation. Thus, simple integration schemes can be usedwithout compromising solution accuracy, although a largenumber of points may be required (the total number ofnodes in the particle-tracking grid is the number of timesteps times the number of nodes of the liquid aerosolmoving grid). However, the particle-tracking grid size canbe substantially reduced by grouping the newly frozenparticles in a fewer number of sizes [i.e., Khvorostyanovand Curry, 2005],

@nc Dc;D0o; t

� �@t

Dc¼D0

o

¼ 1

Dupper � Dlower

Z Dupper

Dlower

no Do; tð Þ @Pf Do; tð Þ@t

dDo; ð12Þ

where Do0 is the assumed size of the frozen particles. If all

aerosol particles freeze at the same size, the integral inequation (12) is evaluated over the entire size spectrum ofthe liquid aerosol population. A further reduction in the size

of the particle-tracking grid is achieved by considering thatthe freezing process occurs after a water vapor saturationthreshold is reached [Sassen and Benson, 2000; Karcherand Lohmann, 2002b]. The initial time step size in themodel is set to 2 V�1 s, and reduced to 0.05 V�1 s (with V inm s�1) when ice nucleation starts (J > 104 m�3 s�1) andgrowth of ice particles needs to be accounted for.

2.3. Baseline Simulations

[16] The formulation of the parcel model was tested usingthe baseline protocols of Lin et al. [2002]. Pure ice bulkproperties were used to calculate the growth terms(equations (5) and (6)). Do was assumed to be the equilib-rium size at Si, given by Kohler theory [Pruppacher andKlett, 1997], and solved iteratively using reported solutionbulk density and surface tension data [Tabazadeh andJensen, 1997; Myhre et al., 1998]. This assumption maybias the results of the parcel model simulations at low T andhigh V [Lin et al., 2002]. Alternatively, the aerosol size canbe calculated using explicit growth kinetics, although thewater vapor uptake coefficient from the vapor to the liquidphase is uncertain [Lin et al., 2002] (recent measurementsindicate a value between 0.4 and 0.7 [Gershenzon et al.,2004]). Although any model can be employed for J, theKoop et al. [2000] parameterization was used owing to itssimplicity and its widely accepted accuracy for a broad rangeof atmospheric conditions [i.e., Abbatt et al., 2006]. The dryaerosol population was assumed to be pure H2SO4, lognor-mally distributed with geometric mean diameter, Dg,dry =40 nm, geometric dispersion, sg,dry = 2.3, and total numberconcentration, No = 200 cm�3. The runs were performedusing 20 size bins for the liquid aerosol; the newly frozenparticles were grouped into 4 size classes, producing a gridbetween 1500 and 2000 nodes; numerical results showed thatlittle accuracy was gained by using a finer grid (not shown).Runs of the parcel model using a regular PC (2.2 GHzprocessor speed and 1 GB of RAM), usually took between5 and 12 min.[17] Figure 2 shows results of the performed simulations

for the protocols of Lin et al. [2002] and ad = 1. The valueof ad is still uncertain and may impact Nc [Lin et al., 2002].Simulations using ad = 0.1 (not shown) produced Nc (cm

�3)of 0.20, 2.87, 24.06, for the cases Ch004, Ch020 and Ch100

Figure 1. Lagrangian particle-tracking scheme for a hypothetical population of two liquid dropletsduring three successive time steps. Circles represent supercooled liquid droplets, and hexagons representice crystals. Arrows indicate growth, aging, and onset of freezing. Transition of light to darker shadingindicates temporal evolution.


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D11211

(using the nomenclature of Lin et al. [2002]) respectively,and 0.043, 0.535, and 5.98 for the cases Wh004, Wh020and Wh100, respectively. Results from the INCA campaignsummarized by Gayet et al. [2004] indicate Nc � 1.71 cm�3

for T between �43 and �53�C, and Nc � 0.78 cm�3 for Tbetween �53 and �63�C (V was mainly below 1 m s�1, atconditions favorable for homogeneous freezing) [Haag etal., 2003b]. These values are consistent with a low value forad (around 0.1) which is supported by independent studies[Gierens et al., 2003; Hoyle et al., 2005; Khvorostyanov etal., 2006; Monier et al., 2006]. However, direct comparisonof the parcel model with experimental results is a rathersimplistic view of the cirrus formation process, and over-looks other important effects (i.e., variation in aerosolcharacteristics, V and T fluctuations [Karcher and Strom,2003; Karcher and Koop, 2005]). Theoretical calculationsand direct experimental observations have reported ad

values from 0.03 to 1 at temperatures ranging from 20 to263 K [i.e., Haynes et al., 1992; Wood et al., 2001]. Owingto these considerations, ad is explicitly introduced in theparameterization.

3. Parameterization of Ice Nucleation andGrowth

3.1. Parameterization of nc(Dc, Do)

[18] The ultimate goal of this study is to develop anapproximate analytical solution of equations (5)–(12) topredict number and size of ice crystals as a function of cloudformation conditions. For this, a link should be establishedbetween ice particle size and their probability of freezing atthe time of nucleation, so that nc(Dc, Do) can be defined ateach instant during the freezing pulse. nc(Dc, Do) is deter-mined for a given Si profile by tracing back the growth of agroup of ice crystals particles of size Dc down to Do

(Figure 3). In the following derivation we assume that mostof the crystals are nucleated before maximum saturationratio, Si,max, is reached (the implications of this assumption

are discussed in section 4). We start by writing a solution ofequation (7) in the form

nc Dc;Doð Þ ¼ �no Doð Þ@Pf S0o

� �@Dc

; ð13Þ

where So0 is a value of Si < Si,max at which the ice crystals

were formed and Pf (So0) represents the current fraction of

crystals of size Dc, that come from liquid aerosol particles ofsize Do. no(Do) is the average no(Do) during the freezinginterval, and is taken to be constant since Nc is usually muchless than No [i.e., Lin et al., 2002] and freezing occurs over avery narrow Si range [Karcher and Lohmann, 2002b]. Sincein a monotonically increasing Si, field Pf(So

0) decreases withincreasing Dc (as explained below), a negative sign isintroduced in equation (13).

Figure 3. Sketch of the parameterization concept. Circlerepresents the supercooled liquid droplet, and hexagonrepresents the ice crystal.

Figure 2. Si, Nc, and ice water content (IWC) profiles obtained from the numerical solution of theparcel model. Cases considered are taken from Lin et al. [2002]. Solid lines indicate To = 233 K, anddashed lines indicate To = 213 K. The top line is for V = 0.04 m s�1, the middle line is for 0.2 m s�1, andthe bottom line is for 1 m s�1.


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[19] Calculation of So0 is key for solving equation (13);

this is done by combining equations (1) and (2),

dSi

dt¼ aVSi � b

dwi

dt; ð14Þ

where a =gDHsMw

cpRT 2� gMa

RT, b =

Map

Mwpoi

DH2s Mw

cpRT2. Before

the nucleated ice crystals substantially impact saturation

(known as ‘‘free growth,’’ Figure 3),dwi

dt 0, and the

integration of equation (14) from So0 to Si,max gives

t � to 1

aV1� S0o

Si;max

� �; ð15Þ

where the approximation ln(x) x � 1 has been used.Equation (15) is similar to the ‘‘upper bound’’ expressionderived by Twomey [1959] for liquid water clouds. Numericalsimulations (section 4) support that ‘‘free growth’’ holds upto Si values very close Si,max; for Si ! Si,max however,equation (15) may underestimate t � to and its effect isdiscussed in section 4.[20] By definition, t � to should equal the time for growth

of the ice particles from Do to Dc (Figure 3), which is foundby integration of equation (5),

t � to ¼1

Si � 1ð ÞG1

2D2

c � D2o

� �þ G2 Dc � Doð Þ

� �; ð16Þ

where Si has been assumed constant. This is supported byparcel model simulations that suggest that nucleation occursin a very narrow Si range (i.e., Figure 5). The calculation ofPf (So

0) using equation (9) requires the knowledge of J as anexplicit function of Si; this can be further simplified giventhat nucleation occurs on a very narrow interval ofsaturation so that J(Si) can be approximated with J(Si,max)[i.e., Khvorostyanov and Curry, 2004],

lnJ Sið Þ

J Si;max

� � k Tð Þ Si � Si;max

� �: ð17Þ

k(T) is obtained by fitting the Koop et al. [2000] data forJ between 108 and 1022 m�3 s�1,

k Tð Þ ¼ 0:0240T2 � 8:035T þ 934:0; ð18Þ

with T in K. k(T) can also be derived for any aerosoltype and composition using classical nucleation theory[Khvorostyanov and Sassen, 1998; Khvorostyanov andCurry, 2004]. Using equations (14) and (17), equation (9)is solved under the assumption of free growth to give

Pf Do; S0o

� � 1

aVSi;max

Z S0o

So

voJ Sð ÞdS

¼J Si;max

� �vo

aVk Tð ÞSi;max

exp �k Tð Þ Si;max � S0o� � �

; ð19Þ

where vo =pD3

o

6, and the approximation 1 �

exp �Z t

0

voJ tð Þdt� �

Z t

0

voJ(t)dt has been used.

[21] The lower integration limit, So, in equation (19)represents the beginning of the freezing pulse, assumedto be where Pf (Do, So) < 10�6 Pf (Do, Si,max), thatis, exp[�k(T)(Si,max � So

0)] exp[�k(T) (Si,max � So)]. Theintegration is not very sensitive to the latter assumption, asmost of the ice crystals form for Si close to Si,max [i.e.,Karcher and Lohmann, 2002a]. Combining equations (15)and (16) to find Si,max � So

0, and replacing into equation (19)gives

Pf Do;Dcð Þ ¼J Si;max

� �vo

aVk Tð ÞSi;max

exp �mG Dc;Doð Þ½ ; ð20Þ

where m =aVk Tð ÞSi;max

Si;max � 1� � , and G(Dc, Do) =

G1

2(Dc

2 � Do2) +

G2(Dc � Do). The ice number distribution function at Si,max

is obtained after computing@Pf Do;Dcð Þ

@Dc

from equation (20)

and substituting into equation (13),

nc Dc;Doð ÞjSi;max

¼ no Doð ÞJ Si;max

� �vo

Si;max � 1� � G1Dc þ G2ð Þ exp �mG Dc;Doð Þ½ : ð21Þ

Since ice particles attain large sizes after freezing, thespectrum of Dc values spans over several orders ofmagnitude [i.e., DeMott et al., 1994; Monier et al., 2006].Typically variation in Do is much smaller and is furtherreduced because Pf is significant only for a fraction of theliquid droplets (generally those with larger sizes and lowsolute concentrations [Pruppacher and Klett, 1997]). Withthis, the value of exp[�mG(Dc, Do)] will be dominated byvariation in Dc, so that exp[�mG(Dc, Do)] can be approxi-mated with exp[�mG(Dc, Do)]. Obtaining the crystal sizedistribution then is done by integration of equation (21) overthe contribution from each droplet size class,

nc Dcð ÞjSi;max¼

J Si;max

� �Si;max � 1� � G1Dc þ G2ð Þ exp �mG Dc;Do

� � ��Z

no Doð ÞvodDo; ð22Þ

which for a lognormal distribution of no(Do) gives

nc Dcð ÞjSi;max

¼Novoe

9=2 ln2 sg J Si;max


� � �ð23aÞ

or

nc lnDcð ÞjSi;max¼ dNc

d lnDc

Si;max

¼Novoe

9=2 ln2 sgDcJ Si;max


� � �;

ð23bÞ

where vo =p6Do3 and sg is the geometric dispersion of the

droplet size distribution. Equation (23) is the final


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expression used for the ice crystal size distribution at Si,max.Equations (21)–(23) demonstrate the probabilistic characterof ice nucleation: at any time particles of all sizes have finitefreezing probabilities; that is, the population of ice crystalsof a given size Dc results from the freezing of droplets withdifferent sizes at different times. As a result, the freezing ofa monodisperse aerosol size population produces a poly-disperse ice crystal population. Since Si increases mono-tonically with time before reaching Si,max, the number ofcrystals generally increases as Dc decreases. For a givendroplet size, Pf (Do, Si) will increase with time so that thenumber of newly formed crystals will increase. Thesecrystals in turn will have less time to grow before Si reachesSi,max; in other words, the most recently formed crystals willhave the largest probability of freezing, Pf (Dc = Do, Si,max).The maximum in the distribution may be shifted toDc >Do ifthe timescale for the growth of the newly formed particles islarger than the timescale of change of probability (section 4).

3.2. Calculation of Nc at Si,max

[22] The deposition rate of water vapor upon ice crystalscan be approximated by substituting equation (23) intoequation (4),

dwi

dt¼ rira

p2Novoe

9=2 ln2 sg J Si;max

� � ZDc;smax

Do

D2c

� exp � m2G

D2c � D

2

o

� �� dDc; ð24Þ

where Dc;smaxis the equivalent diameter of the largest ice

crystal at Si,max (calculated in section 3.3) and G is calculatedusing the approach of Fountoukis and Nenes [2005],

G ¼

ZDc;smax

Do

Dc

G1Dc þ G2

dDc

Dc;smax� Do

¼ �

G1

1þ G2

G1

lnG2 þ G1Dc;smax

G2 þ G1Do

� �Dc;smax

� Do

2664

3775;ð25Þ

where g is a correction factor defined in section 3.3.Integration of equation (24) gives

dwi

dt¼ rira

p2Novoe

9=2 ln2 sg J Si;max

� � em

2GD

2

o

4m2G

� �3=2

�ffiffiffip

perf

ffiffiffiffiffiffim2G

rDc

� �� 2

ffiffiffiffiffiffim2G

rDce

� m

2GD2

c

� �Dc;smax

Do

: ð26aÞ

An order of magnitude estimation of parameters based onparcel simulations (not shown) suggest that Do � 10�7 m,

Dc;smax� 10�5 m and

m2G

� 1010 m�2. Therefore, the

bracketed term in equation (26a) tends to approachffiffiffip

pin

most conditions, that is,

dwi

dt¼ ri

raNovoe

9=2 ln2 sg J Si;max

� � em

2GD

2

o

2mpG

� �3=2ð26bÞ

after substitution of equation (26b) into equation (14),

atdSi

dt= 0,

aVSi;max ¼ brira

Novoe9=2 ln2 sg J Si;max

� � em

2GD2

o

2mpG

� �3=2; ð27Þ

the fraction of frozen particles, fc, at Si,max is found atthe maximum Pf , given by equation (20)

fc ¼1

No

Zno Doð ÞPf Dc ¼ Do;Do

� �dDo ¼

voe9=2 ln2 sg J Si;max

� �aVk Tð ÞSi;max

;

ð28Þ

combining equations (27) and (28) we obtain,

fc ¼rari

k Tð Þ½ 1=2

bNo

2aVSi;max

pG Si;max � 1� �

" #3=2

exp � aVk Tð ÞSi;max

2G Si;max � 1� �D2

o

" #:

ð29Þ

The exponential term in equation (29) approaches unity,which is a result of the assumption made in equation(13) that freezing depletes a negligible amount ofaerosol (i.e., no(Do) is constant during the nucleationprocess). Since J(Si,max) has been eliminated fromequation (29), fc is not strongly influenced by smallvariations in Si,max. Thus, Si,max can be taken as thesaturation freezing threshold obtained by solving theparameterization of Koop et al. [2000] for J(Si,max) =1016 s�1 m�3, which represents an average of J(Si, max) overa wide set of simulations (section 4), and is close to thenucleation rate of pure water at �38�C [Pruppacher andKlett, 1997; Sassen and Benson, 2000]. The total numberof crystals would be given by Nc = No fc; however, sucha result is not limited by No. Instead, lifting theassumption made in equation (19), fc can be associatedwith the solution of the integral in equation (9); that is,fc

Z t

0

voJ(t)dt, so

Nc ¼ Noe�fc 1� e�fc

� �; ð30Þ

where Noe�fc represents the number of remaining unfrozen

droplets.

3.3. Calculation of Dc,smax

[23] Dc;smaxis required for calculating the ice growth

rate (equation (26b)). Two methods are used to calculateit. The first one is based on theoretical arguments(therefore g = 1 in equation (25)), and assumes Dc;smax

as the diameter of the ice crystal at whichPf Dc;Doð Þ

Pf Dc ¼ Do;Doð Þ =

10�6 (i.e., the size above which the number of crystals isbelow 10�6 Nc). With this, equation (20) can be solved forDc;smax

,

D2c;smax

þ 2G2

G1

Dc;smax�2 ln 10�6ð Þ Si;max � 1

� �aVk Tð ÞSi;maxG1

¼ 0; ð31Þ


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where we have assumed Dc;smax Do (supported by

numerical simulations). Since G2 / 1

ad

(equation (6)), the

value of Dc;smaxwould increase as ad decreases. For low

values of ad ice crystals grow slowly, and noncontinuumeffects limit the condensation rate; when ice become largeenough, gas-to-particle mass transfer is in the continuumregime and the crystals grow quickly [Pruppacher andKlett, 1997]. When growth of the newly frozen ice crystalsis delayed, water vapor water tends to preferentiallycondense on preexisting ice crystals. Slow uptake effectsbecame important for ad lower than 0.1, that is, when G2

becomes comparable to G1 [Lin et al., 2002; Gierens et al.,2003]. Alternatively, Dc;smax

can be computed using anempirical fit to numerical simulations obtained with theparcel model,

Dc;smax¼ min

(1:6397� 10�14T � 3:1769� 10�12� �

� V�0:05 NoD3g;dry

� ��0:373; 10�4

); ð32Þ

where V is in m s�1, T is in K,No in cm�3 andDg,dry in meters.

For this case, g = 1.33 in equation (25). Equation (32) wasgenerated over a broad set of T, p, V, No, Dg,dry , and ad

(Table 1, section 4). The T and V dependencies inequation (32) are introduced to adjust the effective growth ofthe particles correcting for the ‘‘free growth’’ assumption(section 3.1). Variability in the formulation of equation (30)from aerosol property changes is not accounted for. Sinceno(Do) is determined by the dry aerosol size distribution, alarger Dg,dry will enhance Pf, as the total volume of theliquid aerosol particles is increased. This would produce alonger freezing pulse and increase Nc. The same effect canbe achieved by reducing the effective growth of theparticles, and explains why Dc,Smax scales with the averagevolume of the dry aerosol population, that is, �(NoDg,dry

3 )1/3

(equation (32)).

3.4. Implementation of the Parameterization

[24] Application of the parameterization is presented inFigure 4. Inputs are the conditions of cloud formation T, P,V, ad and the aerosol size distribution (i.e., No and Dg,dry);the outputs are Nc and ice crystal size distribution. To applythe parameterization, first G is calculated (equation (25)),using Dc;smax

computed from either equation (31) or (32), thelatter being preferred. Si,max is obtained by solving J(Si,max) =1016 s�1 m�3 [Koop et al., 2000], for which reported fits canbe employed [i.e., Ren and Mackenzie, 2005]. Nc is calcu-lated from equation (30) using fc calculated from equation

(29). After fc is calculated, J(Si,max) can be corrected using

J(Si,max) =fcaVk Tð ÞSi;max

voe9=2 ln2 sg

(equation (28)); nc(Dc, Do) is then

obtained from equations (21)–(23).

4. Evaluation of the Parameterization

[25] The parameterization was evaluated against the de-tailed numerical solution of the parcel model over a widerange of T, V, Dg,dry, No, and ad (Table 1), for a total of 1200runs that reflect the uncertainty in ad [Lin et al., 2002] andthe range of cirrus cloud formation conditions expected in aGCM simulation. The parcel model was run using initial Si =1.0; initial p was estimated from hydrostatic equilibrium at Tusing a dry adiabatic lapse rate. The dry aerosol wasassumed to follow a lognormal size distribution and becomposed of pure H2SO4. ‘‘Aged,’’ ‘‘unperturbed,’’ and‘‘perturbed’’ aerosol is represented by setting the geometricdispersion of the dry aerosol distribution, sg,dry to 1.7, 2.3,and 2.9, respectively [Seinfeld and Pandis, 1998]. Theaerosol was assumed to be deliquesced and in equilibriumwith Si in all simulations. Calculated Nc ranged from 0.001to 100 cm�3 and Si,max varied between 1.4 and 1.6, whichagrees with the reported values for cirrus formation byhomogeneous freezing [i.e., Heymsfield and Sabin, 1989;Lin et al., 2002].[26] Two main assumptions were introduced in the devel-

opment of the parameterization: (1) Nc is calculated at Si,max

rather than at the end of the freezing pulse, that is, Nc NcjSmax

, and, (2) the newly formed crystals have a negligibleimpact on Si before Si,max is reached (‘‘free growth’’). Figure 5shows how these assumptions may impact the results of theparameterization for high and low V and T (1–0.04 m s�1,233–203 K). By using free growth regime to estimate theevolution of Si (Figure 5, dotted black line), Si,max is over-estimated by�0.5% at low V (Figures 5c and 5d) and�2% athigh V (Figures 5a and 5b). At high T (Figures 5b and 5d) thisoverestimation does not impact parameterization perfor-mance, as Si,max is low and small overestimations thereofdo not significantly influence J(Si,max). However, as Tdecreases and V increases, Si,max reaches higher valuesduring the parcel ascent; J (which is a very nonlinearfunction of Si,max), and Pf become very sensitive to smallchanges in Si,max. As a result, an overestimation in Nc maybe expected. Conversely, Figures 5c and 5d show that at lowV, NcjSmax

underestimates the actual Nc by nearly a factor of2; since few ice crystals (low Si,max) are formed, it takes alonger time to deplete the available water vapor, resulting inlonger freezing pulses. At high V (Figures 5a and 5b),NcjSmax

is close to Nc, since Si,max is reached rapidly and alarger number of crystals is formed; this effect will be moreimportant at low T as higher Si,max values are reached.

4.1. Calculation of nc(Dc)

[27] Figure 6 presents the parameterized nc(Dc) (equation(23b)) for two representative cirrus cases, along withnc Dcð ÞjSi;max obtained from parcel model simulations. Al-though the effect of the droplet size distribution parametersis explicitly considered in equation (23), it cancels out withits effect on fc (which is a result of assuming constantno(Do)), and a single crystal size distribution is obtained forthe three values of sg,dry tested (although small variations in

Table 1. Conditions Used in Parameterization Evaluation

Property Values

T, K 200–235V, m s�1 0.02–5ad 0.05–1.0sg,dry 1.7, 2.3, 2.9No, cm

�3 10–5000Dg,dry, nm 20–160


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Figure 4. Parameterization algorithm.

Figure 5. Si (solid lines, left axes) and Nc (dash-dotted lines, right axes) evolution calculated by theparcel model. T and V conditions shown are (a) 233 K and 1 m s�1, (b) 203 K and 1 m s�1, (c) 233 K and0.04 m s�1, and (d) 203 K and 0.04 m s�1. Dashed lines represent the time at which Si,max is reached, anddotted lines represent Si evolution under ‘‘free growth’’. No = 200 cm�3, and Dg,dry = 20 nm.


9 of 15

D11211

nc Dcð ÞjSi;maxmay occur owing to differences in T at Si,max).

The latter is not a critical issue, as Nc variations with respectto sg,dry are generally small (i.e., Figure 6). Do wasapproximated with the equilibrium size of Dg,dry at Si,max.nc(Dc) was calculated using the outline in section 3.4(Figure 4), from which J(Si,max) is corrected using equations

(30) and (32) (therefore enforcing Nc ZDc;max

Dc¼Do

nc(Dc)dDc).

Generally, the parameterized nc(Dc) reproduces well thenumerical results at Si,max; however, the size of the icecrystals is underestimated (which is a result of the assumptionof ‘‘free growth’’ which underestimates t � to, equation (15)

andNcjSmaxis overestimated.’’ The adjusted J(Si,max) will then

be slightly below the obtained in numerical simulations tosatisfy Nc = NcjSmax

. Thus, the correction in J(Si,max) producesa reduction in the peak of the ice crystal size distribution. The

influence of these factors on the resulting effective radius ofthe cirrus cloud, and its radiative properties, will require thetime integration of nc(Dc), and the comparison with numer-ical simulations at different stages after cloud formation.Although such a task may be readily achieved, it is out of thefocus of this manuscript and will be undertaken in a futurestudy.

4.2. Calculation of Nc

[28] Figure 7 shows the comparison of Nc predicted bythe parcel model and the parameterization (equation (30))using the theoretical calculation of Dc;smax

(Figure 7, right;equation (31)) and using the empirical correlation for Dc;smax

(Figure 7, left; equation (32)). The effects of assuming ‘‘freegrowth’’ and the approximation of Nc as NcjSmax

are expectedto cancel out at moderate T and V. However, at low V andhigh T (hence low Nc), the parameterization tends tounderestimate Nc with respect to the parcel model results;

Figure 7. Ice crystal number concentration calculated by the parcel model and the parameterization.Dc;smax

was calculated using either (left) equation (32) or (right) equation (31). Gray scale represents thevalue of ad used in the calculations; dashed lines represent ±50% error.

Figure 6. Crystal size distribution calculated by the parcel model and the parameterization at Si,max:(left) T = 213 K, V = 1.0 m s�1; (right) T = 233 K, V = 0.2 m s�1. sg,dry is the geometric dispersion of theaerosol size distribution.


10 of 15

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the opposite tendency occurs at very high V (>2 m s�1) andlow T (hence, high Nc). The behavior at these T, V extremesis not typical of cirrus formation [i.e., Heymsfield andSabin, 1989; Gayet et al., 2004], hence not expected to bea source of significant bias in GCM simulations. Overall,using the theoretically calculated Dc;smax

gives a parameter-ized Nc that agrees within a factor of 2 of the parcel modelsimulations, with a mean relative error about 29% (for allruns in Table 1). When using the empirically calculatedDc;smax

(equation (32)), the parameterized Nc is much closerto the numerical parcel model (average relative error 1% ±28%), as equation (32) allows more flexibility in reproduc-ing the parcel model results, and accounts for the additionalvariability due to the effect of aerosol size and number.

4.3. Comparison Against Other Parameterizations

[29] The new parameterization is evaluated against severalpublished schemes for different combinations of V, No,Dg,dry, T, and ad. The parameterizations used in this sectionare those of Liu and Penner [2005] (LP2005), Sassen andBenson [2000] (SB2000), Karcher and Lohmann [2002a](KL2002), and Fountoukis and Nenes [2005]. SB2000 isbased on an empirical fit to numerical simulations relatingNc to T and V. A similar approach is used in LP2005 wherean additional dependency on No is included. In both cases, Jis calculated through classical nucleation theory (the latterusing the effective temperature method [i.e., DeMott et al.,1994]). KL2002 is physically based and employs thefreezing timescale and the threshold supersaturation as inputparameters. It resolves explicitly the dependency of Nc on T,V, ad, and Do, and uses No as upper limit for Nc. Althoughthe freezing of polydisperse aerosol is discussed in KL2002,not explicit solution is presented; their monodisperse solu-tion is therefore used for comparison. The freezing time-scale and supersaturation threshold are calculated using theanalytical fits to Koop et al. [2000] data provided by Renand Mackenzie [2005]. Do in this case was taken as inequilibrium with the volume-weighted geometric mean

diameter of the dry size distribution. By using this definitionof Do, the best agreement between the parcel model simu-lations and the results of the KL2002 parameterization wasobtained. The three parameterizations were compared to thesolution of equation (30) using theoretically calculatedDc,max, equation (31) (although termed ‘‘theoretical’’, k(T)was derived from an empirical fit to J), and using theempirically adjusted Dc,max equation (32), (termed ‘‘adjust-ed’’). Do was calculated as the equilibrium size of Dg,dry atSi,max. All parameterizations are evaluated using T obtainedat Si,max from the parcel model simulations. ad was set to0.1 and 1.0 to test both diffusionally and nondiffusionallylimited cases (see section 3.3).4.3.1. Dependency on V[30] Figure 8 presents Nc predicted from all parameter-

izations at To = 213 (Figure 8, left; T between 208.6 and209.4 K) and 233 K (Figure 8, right; T between 228.8 and229.2 K), and ad = 0.1 (black line) and 1.0 (gray line). AtTo = 233 K, all parameterizations agree fairly well whenad = 0.1 and V < 1 m s�1. At higher V, KL2002 andLP2005 predict a larger Nc, whereas SB2000 predicts alower Nc with respect to the parcel model results andbecomes significant when V > 3 m s�1. At theseconditions the adjusted parameterization follows the par-cel model very well, whereas the theoretical parameter-ization slightly underpredicts Nc at low V. Runs madeusing ad = 1 (gray line) showed a good agreementbetween the parcel model and KL2002, the adjustedand theoretical parameterizations. This is not surprising,as equation (29) bears the same dependency on V and pi

o,reported by KL2002 in their ‘‘fast growth’’ solution (i.e.,Nc / V3/2), and further emphasized by more recent work[Gierens, 2003; Ren and Mackenzie, 2005]. At high fc (i.e.,low T, low ad, and high V) the exponential term inequation (30) dampens the effect of V (also becauseDc;smax

scales with V) and Nc scales almost linearly withV. Results for ad = 1 lie below those of LP2005 and

Figure 8. Ice crystal number concentration as a function of the updraft velocity, calculated using theparcel model (circles), KL2002 (dashed line), LP2005 (dash-dotted line), SB2000 (dotted line), and thetheoretical (asterisks) and adjusted (solid line) parameterizations. Results are shown for (right) T = 233 K,p = 34,000 and (left) T = 213 K, p = 17,000 Pa, and ad = 0.1 (black) and ad = 1.0 (gray). No = 200 cm�3,and Dg,dry = 40 nm.


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SB2000, who used lower ad values for their numericalsimulations (LP2005 used ad = 0.1 and SB2000 used ad =0.36 [Lin et al., 2002]). At To = 213 K and ad = 0.1(Figure 8, left, black line), the parameterizations agreeonly for V below 0.3 m s1 whereas for large V theydiverge, with KL2002 giving the largest Nc over the wholeV interval. At very high V (>3 m s�1) the adjustedparameterization underpredicts Nc with respect to theparcel model results, which is a result of the exponentialterm introduced in equation (30). As with To = 233 K,KL2002, the adjusted and theoretical parameterizationsagree well with the numerical results when ad = 1,and To = 213 K (Figure 8, left, gray line).4.3.2. Dependency on No

[31] Figure 9 presents Nc as a function of No for V =0.2 m s�1 (Figure 9, left) and V = 1.0 m s�1 (Figure 9,right) and To = 213 (black line) and 233 K (gray line);Dg,dry was set to 40 nm and ad to 0.1. In all cases ofFigure 9, LP2005 and the adjusted parameterization showthe best agreement with the parcel model results. Still, atTo = 213 K, V = 1.0 m s�1, and No below 20 cm�3, theadjusted parameterization underpredicts with respect tothe numerical results and LP2005 overpredicts. In bothcases the difference with the parcel model results isabout ±50%, which is not critical as these very low No

are atypical of cirrus forming conditions [i.e., Pruppacherand Klett, 1997]. In all cases of Figure 9, KL2002predicts larger Nc than the parcel model; however, thedifference becomes much smaller at large No. SB2000predicts Nc close to the average of the parcel modelresults at To = 213 K; when To = 233 K, SB2000 isclose to the parcel model results at large No. Thetheoretical parameterization tends to agree better withthe parcel model results at high No. As proposed byLP2005, the parcel model results can be reasonably wellexpressed in the form Nc = aNo

b where a and b arefunctions of T, V, Dg,dry and ad. The dependency of Nc

on No generally increases when T and ad decrease andwhen V increases. For the cases of Figure 9, b lies

between 0.19 (To = 233 K, V = 0.2 m s�1) and 0.61 (To =213 K, V = 1 m s�1). This is consistent with experimentaland numerical studies that report a factor of 2 to 4 increasein Nc for a tenfold increase in No [i.e., Heymsfield andSabin, 1989; Jensen and Toon, 1994; Seifert et al., 2004].4.3.3. Dependency on Dg,dry

[32] Figure 10 presents Nc as a function of Dg,dry for To =213 K (Figure 10, left) and To = 233 K (Figure 10, right),and ad = 0.1 (black line) and ad = 1 (gray line); for thesesimulations V = 1.0 m s�1 and No = 200 cm�3. To applyKL2002, Dg,dry was converted into the volume-weightedmean diameter in equilibrium with Si,max. In all conditionsof Figure 10, parcel model results suggest Nc scales almostlinearly with Dg,dry, which is also found by the combinationof equations (29) and (32). This was also observed inLP2005 and is a result of the increased Pf due to the largervolume of the liquid aerosol particles in equilibrium withthe aerosol dry distribution (see section 3.3 and equation (9)).An inverse tendency predicted by KL2002 (Nc / Do

�1), andalso by the theoretical parameterization (although with amuch weaker dependence). In the latter, the effect ofincreased Pf is not accounted for owing to the assumptionof an infinite aerosol source (section 3.1, equation (13)).Although LP2005 does not take into account this depen-dency, it predicts Nc in agreement with parcel model resultsat Dg,dry = 80 nm, and ad = 0.1. SB2000 predictions agreewith the parcel model results at Dg,dry = 40 nm and To = 213,and at Dg,dry = 120 nm and To = 233, when ad = 0.1. Thedependence of NcwithDg,dry is much stronger in the left thanin the right panel of Figure 10, which suggests once more thatsize effects would be more important at low T and high V. Inall cases of Figure 10, the adjusted parameterization closelyreproduces the parcel model results. While the comparisonof the different parameterizations was carried out over acomprehensive set of conditions, common values of Dg,dry

often ranges between 40 and 100 nm (and between 100 and500 cm�3 for No) [e.g., Heymsfield and Miloshevich, 1995;Gayet et al., 2004]. Figures 8–10 show that the effect of Tand V variations on Nc is much stronger than that of Dg,dry

Figure 9. Ice crystal number concentration as a function of the aerosol number concentration. Symbolsas in Figure 8. Results are shown for (left) V = 0.2 m s�1 and (right) 1 m s�1, and T = 233 K, p = 38,812Pa (gray) and T = 213 K, p = 27,350 Pa (black); ad = 0.1, and Dg,dry = 40 nm.


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and No. The relative importance of each parameter remainsto be assessed in global model studies.

5. Summary and Conclusions

[33] To address the need for improved ice cloud physicsin large-scale models, we have developed a physicallybased parameterization for cirrus cloud formation, whichis robust, computationally efficient, and links chemicaleffects (e.g., water activity and uptake effects) with iceformation via homogenous freezing. This was accomplishedby tracing back the growth of ice crystals to their point offreezing, relating their size to freezing probability. Usingthis approach, an expression for the ice crystal size distri-bution is derived, the integration of which yields the numberconcentration.[34] The parameterization is evaluated against the pre-

dictions of a detailed numerical parcel model developedduring this study. The parcel model equations were inte-grated using a Lagrangian particle-tracking scheme; theevolution of the ice crystal size distribution is describedby the superposition of growing of monodisperse crystalpopulations generated by the freezing of single classes (ofsame size and composition) of supercooled droplets.[35] Two versions of the parameterization are developed

and evaluated; one based solely on theoretical arguments,and one with adjustments in the ice crystal growth rate usingnumerical parcel simulations. When compared against thepredictions of the numerical parcel model over a broad setof conditions, the theoretically based parameterization forNc robustly reproduced the results of the parcel modelwithin a factor of two and with an average relative errorof about 29%. When numerical simulations are used toadjust the ice crystal growth rate, the relative error wasreduced to 1 ± 28%, which is remarkable given thesimplicity of the final expression obtained for Nc, the broadset of conditions tested, and the complexity of the originalparcel equations.

[36] The new parameterization presented in this workoffers an analytical and physically based relationship be-tween the parameters affecting cirrus ice formation viahomogeneous freezing. The prediction skill of the parame-terization is robust across a wide range of parameters (e.g.,ad, aerosol characteristics). As shown in Figure 7, theaccuracy with which the parameterization reproduces theparcel model results is independent of these parameters. Inthis regard, only the KL2002 parameterization shares thischaracteristic; the presented parameterization, however, (1)explicitly links the variables that control the freezingtimescale of the particles and (2) successfully reproducesthe effect of the aerosol number on Nc.[37] The results given are applicable for cirrus formation

on predominantly homogenous freezing conditions. Fre-quently, heterogeneous freezing and competition betweenmultiple particle types can significantly impact cloud for-mation. Both the numerical model and the parameterizationcan be readily extended to include these processes, and willbe the focus of future work.

Notation

a, b parameters defined in equation (14).ad water vapor deposition coefficient on ice.cp heat capacity of air.

DHs heat of sublimation of water.Dc;smax

diameter of the largest ice particle at Si,max.Dc volume sphere-equivalent diameter of an ice

particle.Dg,dry geometric mean diameter of the dry aerosol

size distribution.Do diameter of a liquid aerosol particle.Dv bulk water vapor diffusion coefficient.fc fraction of frozen particles at Si,max.g acceleration of gravity.

G(Dc, Do) growth function defined in equation (20).G effective growth parameter defined in

equation (25).

Figure 10. Ice crystal number concentration as a function of the aerosol mean dry diameter. Symbols asin Figure 8. Results are shown for (right) T = 233 K, p = 34,000 Pa and (left) T = 213 K, p = 17,000 Pa,and ad = 0.1 (black) and ad = 1.0 (gray). No = 200 cm�3, and V = 1 m s�1.


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G1, G2 parameters defined in equation (6).J nucleation rate coefficient.

k(T) constant defined in equation (18).ka thermal conductivity of air.m parameter defined in equation (20).

Mw, Ma molar masses of water and air, respectively.nc(Dc, Do) ice crystal size distribution function origi-

nated at Do.nc(Dc) ice crystal size distribution function.no(Do) liquid aerosol size distribution function.no(Do) average liquid aerosol size distribution func-

tion during freezing.NcjSmax

ice crystal number concentration at Si,max.Nc ice crystal number concentration.No aerosol number concentration.P ambient pressure.pio ice saturation vapor pressure.

Pf, Pf (Do, t) cumulative probability of freezing.ri, ra ice and air densities, respectively.

R universal gas constant.sg geometric dispersion of the liquid aerosol

size distribution.sg,dry geometric dispersion of the dry aerosol size

distribution.Si water vapor saturation ratio with respect to

ice.Si,eq equilibrium ice saturation ratio.

Si,max maximum ice saturation ratio.So ice saturation ratio at the beginning of

freezing.So

0 ice saturation ratio at which an ice crystalwas formed.

T temperature.t time.to time of freezing.V updraft velocity.vo volume of a liquid droplet of size Do.wi ice mass mixing ratio.

y(Do, t) ice nucleation function.

[38] Acknowledgments. This study was supported by NASA MAP,NASA EOS-IDS-CACTUS, and a NASA New Investigator Award. Wewould also like to thank B. Kahn, V. Khvorostyanov, and two anonymousreviewers for comments that substantially improved the manuscript.

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��D. Barahona, School of Chemical and Biomolecular Engineering,

Georgia Institute of Technology, Atlanta, GA 30332-0340, USA.A. Nenes, School of Earth and Atmospheric Sciences, Georgia Institute

of Technology, 331 Ferst Drive, Atlanta, GA 30332-0340, USA. ([email protected])


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