parameterizing the competition between homogeneous and

Atmos. Chem. Phys., 9, 369–381, 2009www.atmos-chem-phys.net/9/369/2009/© Author(s) 2009. This work is distributed underthe Creative Commons Attribution 3.0 License.

AtmosphericChemistry

and Physics

Parameterizing the competition between homogeneous andheterogeneous freezing in cirrus cloud formation –monodisperse ice nuclei

D. Barahona1 and A. Nenes1,2

1School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, USA2School of Earth and Atmospheric Sciences, Georgia Institute of Technology, USA

Received: 30 June 2008 – Published in Atmos. Chem. Phys. Discuss.: 18 August 2008Revised: 8 December 2008 – Accepted: 8 December 2008 – Published: 16 January 2009

Abstract. We present a parameterization of cirrus cloud for-mation that computes the ice crystal number and size dis-tribution under the presence of homogeneous and heteroge-neous freezing. The parameterization is very simple to ap-ply and is derived from the analytical solution of the cloudparcel equations, assuming that the ice nuclei population ismonodisperse and chemically homogeneous. In addition tothe ice distribution, an analytical expression is provided forthe limiting ice nuclei number concentration that suppressesice formation from homogeneous freezing. The parameteri-zation is evaluated against a detailed numerical parcel model,and reproduces numerical simulations over a wide range ofconditions with an average error of 6±33%. The parameter-ization also compares favorably against other formulationsthat require some form of numerical integration.

1 Introduction

Cirrus clouds are key components of climate and can havea major impact on its radiative balance (Liou, 1986; Hart-mann et al., 2001; Lohmann et al., 2004). They can be af-fected by anthropogenic activities such as aircraft emissionsin the upper troposphere (Seinfeld, 1998; Penner et al., 1999;Minnis, 2004; Stuber et al., 2006; Karcher et al., 2007), andfrom long range transport of pollution and dust (Sassen etal., 2003; Fridlind et al., 2004). When studying ice-cloud-climate interactions, the key cloud properties that need tobe computed are ice crystal concentration distribution, and

Correspondence to:A. Nenes([email protected])

ice water content (e.g. Lin et al., 2002); in large-scale mod-els such parameters are either prescribed or computed witha parameterization. Due to the limited understanding of thephysics behind ice nucleation, and the diversity and complex-ity of interactions between atmospheric particles during cir-rus formation and evolution, the anthropogenic effect on cir-rus clouds remains a major source of uncertainty in climatechange predictions (e.g. Baker and Peter, 2008).

Cirrus can form by either homogeneous or heterogeneousfreezing. Homogeneous freezing occurs spontaneously fromtemperature and density fluctuations within a supercooledliquid phase (Pruppacher and Klett, 1997); whether it hap-pens however largely depends on the temperature, dropletvolume, and water activity within the liquid phase (Chen etal., 2000; Koop et al., 2000; Cziczo and Abbatt, 2001; Linet al., 2002). The rate of formation of ice germs by homo-geneous nucleation has been extensively studied and can becomputed using a number of relationships based on experi-ments and theory (e.g. DeMott et al., 1994; Khvorostyanovand Sassen, 1998; Koop et al., 2000; Tabazadeh et al.,2000). Most cirrus clouds form at conditions favorable forhomogeneous freezing, which is believed to be the pri-mary mechanism of cirrus formation in pristine environments(e.g. Heymsfield and Sabin, 1989; Cantrell and Heymsfield,2005).

Heterogeneous freezing encompasses a group of nucle-ation “modes” (deposition, contact, immersion, and conden-sation) in which ice formation is mediated by insoluble solids(Pruppacher and Klett, 1997) or surfactant layers (Zobristet al., 2007) that lower the energy barrier for the forma-tion of an ice germ. Macroscopically, this can be expressedas a decrease in the freezing saturation threshold (or an in-crease in the freezing temperature) with respect the value

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370 D. Barahona and A. Nenes: Homogeneous and Heterogeneous Freezing

for homogeneous freezing. Aerosol particles that contributesuch surfaces and undergo heterogeneous freezing are termedice nuclei (IN) (Pruppacher and Klett, 1997); the nature of INis very diverse and not well understood. They can be com-posed of mineral dust (DeMott et al., 2003b; Sassen et al.,2003), sulfates (Abbatt et al., 2006), crustal material (Zu-beri et al., 2002; Hung et al., 2003), carbonaceous material(Beaver et al., 2006; Cozic et al., 2006; Zobrist et al., 2006),metallic particles (Al, Fe, Ti, Cr, Zn, and Ca) (Chen et al.,1998; DeMott et al., 2003a), and biological particles (e.g.Mohler et al., 2007). Sources of IN have been associated withlong range transport of dust (DeMott et al., 2003b; Sassen etal., 2003), pollution (Fridlind et al., 2004), and direct aircraftemissions (Penner et al., 1999; Minnis, 2004). Heteroge-neous freezing saturation thresholds,Sh, depend primarily onthe IN size, composition, and molecular structure, as well ason particle history (Cziczo and Abbatt, 2001; Mohler et al.,2005; Abbatt et al., 2006). Experimental studies have shownSh ranging from 1.1 to 1.5 (e.g. Heymsfield et al., 1998; De-Mott et al., 2003a; Haag et al., 2003; Archuleta et al., 2005;Field et al., 2006; Prenni et al., 2007).

Field campaign data generally show that ice crystals col-lected at conditions favorable for homogeneous freezing con-tain a moderate fraction of insoluble material (e.g. DeMott etal., 2003a; Prenni et al., 2007), which suggests the poten-tial for interaction between homogeneous and heterogeneousfreezing during cirrus formation. Therefore, a comprehen-sive understanding of ice formation requires a combined the-ory of homogeneous and heterogeneous freezing (e.g. Vali,1994; Karcher and Lohmann, 2003; Khvorostyanov andCurry, 2004; Zobrist et al., 2008). Modeling studies how-ever suggest (Karcher and Lohmann, 2003; Khvorostyanovand Curry, 2005; Liu and Penner, 2005; Monier et al., 2006)that at atmospherically relevant conditions, a single mech-anism (i.e. homogeneous or heterogeneous) dominates thefreezing process. This is largely because IN tend to freezefirst, depleting water vapor, and inhibiting homogenous nu-cleation before it occurs (DeMott et al., 1997); only if INconcentration is low enough can homogeneous freezing oc-cur. Thus, IN and supercooled droplets may be consideredseparate populations that interact through the gas phase, butundergo heterogeneous or homogeneous freezing, respec-tively, at different stages during cloud formation (DeMott etal., 1997). This distinction may become less clear for hetero-geneous freezing in the immersion mode for low concentra-tion of immersed solid, or, if the solid is not an efficient icenuclei (i.e.Sh is high, close to the value for homogeneousfreezing) (Khvorostyanov and Curry, 2004; Marcolli et al.,2007); we will assume for the rest of this study that we arefar from such conditions, given that their importance for theatmosphere is not clear.

The presence of IN may drastically reduce the crystalconcentration compared to levels expected for homogeneousfreezing (DeMott et al., 1994, 1998), creating a potentialanthropogenic indirect effect of IN on climate (DeMott et

al., 1997; Karcher and Lohmann, 2003). Therefore, param-eterizations of cirrus formation for use in climate modelsshould explicitly resolve the interaction between homoge-neous and heterogeneous freezing. Using a simplified ap-proach to describe the growth of ice particles and box modelsimulations, Gierens (2003) obtained a semi-analytical ex-pression to estimate the limiting IN number concentrationthat would prevent homogeneous nucleation. Later, Ren andMackenzi (2005) derived analytical constraints for the effectof monodisperse IN on homogeneous freezing; these analyt-ical constraints are not generally applicable to all cloud for-mation conditions. Liu and Penner (2005) used parcel modelsimulations for combined homogeneous and heterogeneousfreezing cases to obtain numerical fits relatingNc to T , V ,andNIN . Although based on simulations, such parameteri-zations are limited to the choice of IN type, concentration,freezing thresholds and assumptions on crystal growth (i.e.deposition coefficient) made during their derivation. The firstcomprehensive physically-based parameterization of cirrusformation was presented by Karcher et al. (2006) in whichhomogeneous and heterogeneous nucleation are considered.In this scheme, the competition between different freezingmechanisms and particle types is resolved by using numeri-cal integration to calculate the size and number of ice crystalsat different stages of the parcel ascent. Although accurate,this method may be computationally expensive and does notanalytically unravel the dependence ofNc on key IN proper-ties, likeSh andNIN .

This study presents an analytical parameterization of cir-rus formation that addresses the issues identified above andexplicitly considers the competition between homogeneousand heterogeneous nucleation. We develop a novel methodto account for the growth of the heterogeneously frozen icecrystals and incorporate their effect within the physically-based homogeneous nucleation framework of Barahona andNenes (2008, hereinafter BN08). The new parameterizationexplicitly resolves the dependency ofNc on dynamical con-ditions of cloud formation (i.e. temperature, pressure, andupdraft velocity), aerosol number, size, the characteristics ofIN and their effect on ice saturation ratio. To clearly presentthe approach used to unravel the interactions between IN andsupercooled droplets, we assume IN are monodisperse andchemically uniform. The extension of the parameterizationto polydisperse IN (in size and chemical composition) willbe presented in a companion study.

2 Parameterization development

The parameterization is based on the ascending parcel con-ceptual framework. Contained within the cooling parcel is apopulation of supercooled droplets externally mixed with INthat compete for water vapor when their characteristic freez-ing threshold is met. Heterogeneous and homogeneous freez-ing occur sequentially as the parcel water vapor saturation

Atmos. Chem. Phys., 9, 369–381, 2009 www.atmos-chem-phys.net/9/369/2009/

D. Barahona and A. Nenes: Homogeneous and Heterogeneous Freezing 371

ratio over ice,Si , increases during the cloud formation pro-cess. Homogeneous freezing is restricted to liquid dropletsand described using the formulation of BN08. The IN popu-lation can be characterized by its number concentration,NIN ,and freezing threshold,Sh. Upon freezing, ice grows on theIN, depleting and redistributing the available water vapor be-tween the homogeneously and heterogeneously frozen crys-tals (i.e. IN-frozen crystals) (e.g. Lin et al., 2002; Ren andMackenzie, 2005; Karcher et al., 2006). Water vapor com-petition also affects the size distribution of homogeneouslyfrozen crystals. To account for this competition effect, thechange in homogeneous freezing probability due to the pres-ence of IN-frozen crystals (i.e. the amount of water vaporcondensed upon IN), and the size of the IN-frozen crystalsat the homogenous freezing threshold, should be calculated.Both effects can be directly related to the impact of IN-frozencrystals on the rate of change ofSi (Barahona and Nenes,2008). A method to represent such interactions is presentedbelow.

According to the homogeneous nucleation framework ofBN08, the fraction of frozen droplets for a pure homoge-neous pulse is given by

fc =ρa

ρi

[k(T )] 1/2

βNo

[2αV Smax

π 0(Smax − 1)

]3/2

(1)

where symbols are defined in Appendix A and B. Equa-tion (1) was obtained under the assumption that homoge-neously nucleated ice crystals do not substantially impactSi ,up to the point of maximum ice saturation ratio (in reality,Si is affected, and is accounted for by adjusting the effectivegrowth parameter0 and settingSmax equal to the homoge-nous nucleation threshold).

TheαV Smax term in Eq. (1) represents the rate of changeof Si at Smax. The presence of IN perturbsSmax, but Eq. (1)can still be applied ifαV Smax is replaced with the (perturbed)rate of change ofSi , Smax,IN (calculated in Sect. 2),

fc,IN =ρa

ρi

[k(T )] 1/2

βNo

[2Smax,IN

π 0(Smax − 1)

]3/2

(2)

wherefc,IN is the fraction of droplets frozen from homo-geneous freezing in presence of IN (Eq. 2 can be formallyderived using the procedure presented in BN08). The useof Eq. (2) implies thatSmax,IN is almost constant duringthe homogeneous freezing pulse, even when IN are present.This is because, by the time homogeneous freezing occurs,the IN have reached large sizes (and grow very slowly);since the condensation rate is proportional to the availablesurface area (which can be considered constant over thetimescale ofSmax, which is around 10−3V s, with V inm s−1), Smax,IN≈const.

Dividing Eqs. (1) and (2) by parts we obtain,

fc,IN

fc

=

[Smax,IN

αVSmax

]3/2

(3)

as all other terms in Eqs. (1) and (2) depend mainly onT

(Appendix A). Equation (3) expresses the impact of IN onthe fraction of frozen droplets, which results from the redis-tribution of water vapor to IN-frozen crystals, and, the in-crease in the average size of the homogeneously frozen crys-tals due to a longer freezing pulse, which for the pure homo-geneous case (in s) is approximately25

V(with V in m s−1)

(Barahona and Nenes, 2008). Both of these effects tend todecreasefc,IN relative tofc. When no IN are present, thefreezing pulse is purely homogeneous andfc,IN=fc. If fc,INis known, the number of crystals in the cirrus cloud,Nc, canbe calculated as

Nc = Noe−fc,IN (1 − e−fc,IN ) + NIN (4)

whereNo is the liquid droplet number concentration in theparcel prior to freezing. The first term in Eq. (4) is the contri-bution from homogeneously frozen droplets (Barahona andNenes, 2008), while the second term is from heterogeneousfreezing.

2.1 Calculation of the size of the heterogeneously frozencrystals atSmax

Smax,IN is required to findfc,IN , and can be determined bysolving a system of coupled ODEs that express the growth ofheterogeneously frozen ice crystals and their impact on theparcel ice saturation ratio,

dSi

dt= Si = αV Si − β

dwIN

dt(5)

dwIN

dt=

ρi

ρa

π

2NIND2

c,INdDc,IN

dt(6)

dDc,IN

dt=

(Si − 1)

01Dc,IN + 02(7)

where symbols are defined in Appendix B. Equation (5) ex-presses the rate of change of parcel ice saturation ratio, whichincreases from cooling (first term of right hand side), and de-creases from deposition on the growing IN (second term ofright hand side). Equation (6) represents the rate of water va-por deposition on IN-frozen crystals. Equation (7) expressesthe rate of change of crystal size due to water vapor conden-sation.

Equation (5), evaluated atSmax providesSmax,IN . If theinitial conditions of the parcel are known, Eqs. (5) to (7) canbe numerically solved, as closed solution of Eqs. (5) to (7)

is challenging. IfNIN andSh are such thatdSi

dt

∣∣∣Smax

=0, then

Eqs. (5) to (7) become very stiff asSi→Smax, so even nu-merical integration of the system becomes difficult.

Although a general closed solution to Eqs. (5) to (7) is notpossible, an approximation can be derived. Based on numer-ical simulations of Eqs. (5) to (7), and examining the impactof IN on Si ,

dSi

dt, andSmax,IN , the behavior outlined in Fig. 1

www.atmos-chem-phys.net/9/369/2009/ Atmos. Chem. Phys., 9, 369–381, 2009


Fig. 1. Schematic relation betweenx, y, Si , and dSidt

.

(top panel) is observed: (1) IfNIN=0 (blue line), thendSi

dtvs.

Si is linear andSmax,IN=αV Smax (Eq. 5). (2) IfNIN is low(green curve), IN-frozen crystals affectdSi

dtonly after they

grow substantially large; the resultingSmax,IN is slightly be-low its value atSh, Sh=αV Sh, (shown asSmax,IN1). (3) Asimilar behavior is obtained for higherNIN (purple curve),however IN impactdSi

dtmore quickly, and a lowerSmax,IN

is reached (pointSmax,IN2). (4) If NIN is large enough, thenthe deposition rate on the ice crystals equals the rate of cool-

ing from expansion, so thatdSi

dt

∣∣∣Smax

=0, i.e.Smax,IN=0 (red

curve) and corresponds to the inhibition of homogeneous nu-cleation due to IN freezing (Gierens, 2003; Ren and Macken-zie, 2005).

The behavior outlined above can be described in terms ofthe following dimensionless variables,

x =Si − Sh

1Sh

(8)

where1Sh=Smax−Sh, and

y =Sh −

dSi

dt

Sh

(9)

where,Sh=αV Sh, is the rate of change ofSi at the instantwhen IN freeze.x represents a normalization ofSi by thedifference between homogeneous and heterogeneous freez-ing thresholds, andy represents a normalizeddSi

dtwith re-

spect to its value atSh. The relation betweenx, y, Si , and

dSi

dtis presented in Fig. 1. IfNIN=0 (blue line),x vs. y is

linear and the value ofy at x=1 (termed “ymax”) is equal to−

1Sh

Sh(Eq. 9). IncreasingNIN is reflected by increase inymax

(ymax 1andymax 2, for green and purple curves, respectively),in these cases the change in the sign ofy (Fig. 1, bottompanel) is related to the onset of significant depletion of wa-ter vapor, which impactsdSi

dt. Smax,IN=0 is represented as

ymax=1 (red curves); at this pointSi passes by a maximum atSmax so that there are at least two values ofy for each valueof x, one in the increasing branch ofSi and the second asSi decreases (dashed red curves). Thus, atx=y=1 the func-tional dependency betweenx andy ceases to be univocal andlimy→1

dydx

=∞, or, limy→1

dxdy

=0. At this limit, IN impactdSi

dtvery

quickly after freezing, andy changes sign at very lowx.Based on the above analysis an approximate mathematical

relation can be obtained betweenx andy. Sincex is less

than 1, together withdxdy

∣∣∣y=1

=0, a Taylor series expansion

aroundy=1 gives,

d2x

dy2

∣∣∣∣∣y=1

= −K + O(y, y2, ...) (10)

whereK is a positive constant, andO(y, y2, ...) are higherorder terms that are neglected (the consequences of this as-sumption are discussed in Sect. 3). Since to zeroth order, thecurvature around the pointy=1 is constant (equal to−K), x

is related toy as,x=−Ky2+ay+b (beinga andb integra-

tion constants). Sincex=0 wheny=0, and,dxdy

=0, for y=1,thenb=0,K=

a2 , and

Ax = y(y − 2) (11)

whereA=−1a

is an arbitrary constant. As we desirex andy

to be normalized, we selectA such thatx=1 wheny=ymax,(Fig. 1), orA=ymax(ymax−2). Substitution into Eq. (11) andrearranging gives

y = 1 −√

1 + Ax (12)

Equation (12) represents an approximation ofdSi

dtvs.Si writ-

ten in normalized form, and expresses the dependency be-tween dSi

dtandSi considering cooling from expansion, and

condensation of water vapor on ice crystals.To find Smax,IN , the rate of water vapor condensation on

IN-frozen crystals atSmax,dwINdt

∣∣∣Smax

, must be calculated

(Eq. 5); however, due to the coupling between Eqs. (6) and(7), this requires the calculation of the size of the IN-frozencrystals atSmax, Dmax,IN . Equation (12) combined withEq. (7) can be used for the latter, as follows. First, rearrang-ing Eq. (7) and integrating fromSh to Smax we obtain

01

2D2

max,IN+02Dmax,IN−0(Do,IN)=

t (Smax)∫t (Sh)

(Si−1)dt (13)



where0(Do,IN)=012 D2

o,IN + 02Do,IN , andDo,IN is the ini-tial size of the IN at the point of freezing. In most cases,0(Do,IN) in Eq. (13) can be neglected, since generallyDmax,IN�Do,IN . From Eqs. (8) and (9), and using Eq. (12)

Si = x1Sh + Sh (14a)

and

dSi

dt= 1Sh

dx

dt= Sh(1 − y) = Sh

√1 + Ax (14b)

substituting Eq. (14a) and (14b) into Eq. (13), gives

t (Smax)∫t (Sh)

(Si − 1) dt =1S2

h

Sh

1∫0

x√

1 + Axdx

+1Sh(Sh − 1)

Sh

1∫0

1√

1 + Axdx (15a)

integrating Eq. (15a), after substitutingA = ymax(ymax − 2)

and rearranging, gives

1S∗

h

Sh

=

t (Smax)∫t (Sh)

(Si − 1) dt =1

Sh

21Sh

(2 − ymax)[1Sh(3 − ymax)

3(2 − ymax)+ (Sh − 1)

](15b)

substituting Eq. (15b) into Eq. (13), and solving forDmax,INgives,

Dmax,IN=−02

01+

√(02

01

)2

+2

01

[1S∗

h

αV Sh

+0(Do,IN)

](16)

where we have usedSh=αV Sh. Equation (16) uniquely de-fines Dmax,IN , and represents an approximated solution ofthe ODE system Eqs. (5) to (7), at the point of maximumice saturation ratio in the parcel;ymax can be computed aftercombining Eqs. (6) and (7) and using Eq. (14b),

dSi

dt

∣∣∣∣Smax

= αV Sh(1 − ymax)

= αV Smax − βρi

ρa

π

2NIND2

max,IN(Smax − 1)

01Dmax,IN + 02(17)

2.2 Limiting NIN

Besides defining a maximum inSi , the conditiondSi

dt

∣∣∣Smax

=0, (red curves, Fig. 1) also defines the maximum

condensation rate that can be achieved right before homoge-neous freezing is prevented; any increase indwIN

dtbeyond this

value implies pure heterogeneous freezing (Gierens, 2003).Figure 1 shows that this condition is uniquely determined by

ymax=1; the limiting size of the ice crystals,Dlim , at the max-imum water vapor deposition rate can then be computed fromEq. (16) for1S∗

h calculated forymax=1,

Dlim=−02

01+

√(02

01

)2

+2

01αV Sh

(4

31S2

h+21Sh(Sh−1)

)(18)

where0(Do,IN) has been neglected becauseDlim�Do,IN .Equation (18) can be used to compute the limiting number

concentration of IN that will prevent homogeneous nucle-

ation,Nlim , as follows. SincedSi

dt

∣∣∣Smax

=0, Eqs. (5) and (6)

can be combined to give

βρi

ρa

π

2NlimD2

limdDlim

dt= αV Smax (19)

after substitution of Eqs. (18) and (7) into Eq. (19),Nlim canbe obtained as

Nlim =αV

β

ρa

ρi

2

π

(Smax

Smax − 1

)(01Dlim + 02

D2lim

)(20)

2.3 Calculation offc,IN

The set of Eqs. (15) to (17) represents a closed systemthat can be used to findSmax,IN (hencefc,IN). Althoughthe dependence ofDmax,IN on ymax suggests an iterativeprocedure for calculation ofSmax,IN , a faster alternative(developed below) can be used. Assuming thatDmax,IN islarge enough so non-continuum effects can be neglected (i.e.01Dmax,IN�02), Eq. (6), after substituting Eq. (7) and set-ting Si=Smax, becomes

dwIN

dt

∣∣∣∣Smax

=ρi

ρa

π

2

(Smax − 1)

01NINDmax,IN (21)

rearranging Eq. (20) , and assuming01Dlim�02, gives

Smax − 1

01=

(β

αV Smax

π

2

ρi

ρa

NlimDlim

)−1

(22)

after substituting Eq. (22) into Eq. (21) we obtain

dwIN

dt

∣∣∣∣Smax

=αV Smax

β

(NIN

Nlim

Dmax,IN

Dlim

)(23)

which after substituting into Eq. (7), atSmax, gives

Smax,IN

αV Smax= 1 −

(NIN

Nlim

Dmax,IN

Dlim

)(24)

Equation (24) can be used to determineSmax,IN , if a rela-tion is found betweenDmax,IN andDlim . Considering thatat NIN=Nlim (henceDmax,IN=Dlim) the surface area of thecrystal population is the largest (which follows from Eq. (16)since for allNIN<Nlim , ymax<1) then the total surface areaof the crystal population atNIN can be related to its value atNlim by,



Fig. 2. Parameterization algorithm.

NIND2max,IN+(Nlim−NIN)(Dlim−Dmax,IN)2

=NlimD2lim (25)

which after expanding the quadratic terms and rearranging,gives,

NIN

Nlim=

D2max,IN − 2DlimDmax,IN

D2lim − 2DlimDmax,IN

(26)

typically, by the timeSi reachesSmax, ice crystals havegrown to sizes comparable toDlim , i.e. Dmax,IN∼Dlim , andEq. (26) can be approximated as,

NIN

Nlim≈

D2max,IN

D2lim

(27)

Equation (27), when introduced into Eq. (24) gives,

Smax,IN

αV Smax= 1 −

(NIN

Nlim

(NIN

Nlim

)1/2)

(28)

which can be solved forSmax,IN as,

Smax,IN = αV Smax

[1 −

(NIN

Nlim

)3/2]

(29)

Smax,IN calculated from Eq. (29) can then be used into Eq. (3)to find fc,IN . The restriction thatSmax,IN>0 for all caseswhen homogeneous freezing is not prevented (e.g. Fig. 1)implies thatfc,IN=0, if NIN≥Nlim .

2.4 Applying the parameterization

The application of the parameterization is outlined in Fig. 2.First, Dlim is computed from Eq. (18) and used in Eq. (20)to calculateNlim ; if NIN≥Nlim thenfc,IN=0. If NIN<Nlim ,Smax,IN is then obtained from Eq. (29), which after substi-tution into Eq. (3) (using Eq. 1 to obtainfc), givesfc,IN .Equation (5) is used then to calculateNc from combined het-erogeneous and homogeneous freezing.

3 Results and discussion

The parameterization was evaluated against the numericalparcel model developed in BN08, modified to include the ef-fect of IN (by allowing the deposition of water vapor ontothe IN whenSi exceedsSh). The performance of the param-eterization for pure homogeneous freezing conditions wasdiscussed in BN08, so the evaluation here is focused on thecompetition between homogeneous and heterogeneous freez-ing. The conditions used for evaluation covered a wide rangeof T , V , αd , Sh andNIN (Table 1) for a total of 800 simu-lations; initialp was estimated from hydrostatic equilibriumatTo using a dry adiabatic lapse rate. For simplicity,αd wasassumed to be the same for the homogeneously frozen andIN-frozen ice crystal populations (although this assumptioncan easily be relaxed).



Fig. 3. Nc vs.NIN for all simulation conditions of Table 1. Lines represent parameterization results and symbols correspond to parcel modelsimulations. Symbols are colored bySh, being 1.1 (blue), 1.2 (yellow), 1.3 (red), and 1.4 (black).V is equal to 0.04, 0.2, 0.5, 1, and 2 m s−1,from bottom to top lines in each panel, respectively.

3.1 Evaluation against parcel model

Figure 3 presentsNc versusNIN using the parameterization(lines) and the parcel model (symbols), for all the simula-tions of Table 1. At lowNIN (∼ less than 10−4 cm−3), theimpact of IN onNc is minimal andNc is close to the ob-tained under pure homogeneous freezing. AsNIN increasesto about 0.1Nlim , IN considerably impactNc, so thatNc

reduces proportionally toNIN up to the prevention of ho-mogeneous nucleation atNIN=Nlim , at which a minimumNc=Nlim is reached. WhenNIN is greater thanNlim , freezingis only heterogeneous andNc=NIN . Figure 3 shows that athighV and lowT , large concentrations of IN (Nlim∼1 cm−3)

are required to impact homogeneous freezing, which is con-sistent with existing studies (e.g. DeMott et al., 1997); giventhat highNIN are rare in the upper troposphere (e.g. Gayetet al., 2004; Prenni et al., 2007), our results are consistentwith current understanding that homogeneous nucleation isthe primary freezing mechanism at these conditions.

Overall, the parameterization reproduces very well theparcel model results (Fig. 3), and clearly captures the effectof NIN on Nc. When calculating the error between parcelmodel and parameterization, care should be taken to exclude

Table 1. Cloud formation conditions used in evaluation.

Property Values

To (K) 215, 230V (m s−1) 0.04–2αd 0.1, 1.0σg,dry 2.3No (cm−3) 200Dg,dry (nm) 40Sh 1.1–1.4NIN/Nlim 0.01–3

points where homogeneous nucleation is prevented, as boththe parameterization and the parcel model predictNc=NIN(i.e. fc,IN=0 Eq. 29). Including these points in the analysiswould bias the error estimation towards low values. Omit-ting points whereNc=NIN (i.e. where homogeneous freezingdoes not occur), the average relative error for the calculationof Nc using the parameterization with respect to parcel modelresults for all the runs of Table 1, is about 6±33%. The



Fig. 4. Nlim vs. Sh (solid line) andDlim vs. Sh (dashed line) forV =0.2 m s−1, αd=0.1.

small value of the error indicates no significant systematicbiases in the parameterization over the broad range of con-ditions considered. Still, some overprediction is found atNc

at V =2 m s−1, and a slight underprediction atV =0.04 m s−1

andT =230 K. The former bias is not critical as such highV

are typically not encountered during cirrus formation. Thelatter can be attributed to the inherent error of the homoge-neous framework of BN08 (1±28%), as a much better agree-ment is obtained iffc (homogeneous) is replaced with thenumerical solution.

3.2 Understanding sources of error

Discrepancy between the parameterization and the parcelmodel originate from (i) assumptions made in obtainingEq. (29), which determines the shape of the curvesNc vs.NIN , and, (ii) from errors in the calculation ofNlim , Eq. (20),which determines the minimumNc. The relative contribu-tion of each source of error can be assessed by calculat-ing the average difference inNc between the parcel modeland the parameterization, normalized by the correspondingNc (from parcel model results) obtained from pure homoge-neous freezing (i.e.Nc for NIN→0, Fig. 3); which is about1±10% for all simulations of Table 1. This suggests thatmost of the discrepancy between parameterization and parcelmodel results occurs at lowNc (NIN close toNlim), therefore,most of the error in the parameterization should result fromthe calculation ofNlim . This is supported by the simulationsin Fig. 3, which show that the shape of the parameterizedNc

vs. NIN follows well the parcel model results, and discrep-ancy between the parameterization and parcel model mainlyoriginates from slight “shifting” inNlim .

Potential sources of error in the calculation ofNlim can beidentified by analyzing Eq. (20). First,Smax has been as-sumed to remain constant during the homogeneous pulse.Although this is a good approximation for computingNc

from homogeneous nucleation,1Sh, which is dependenton Smax (Eq. 18), may be in error and bias the calcula-tion of Dlim (henceNlim). This is shown in Fig. 4, whichpresentsDlim (dotted lines) andNlim (solid lines) as a func-tion ofSh, for T =215 K (Smax=1.51, blue lines) andT =230 K(Smax=1.46, black lines). At lowSh (1Sh>0.1), Dlim de-creases almost linearly withSh. For 1Sh<0.1, the de-pendency ofDlim on Sh changes so thatDlim→0 whenSh→Smax. As a result, for1Sh>0.1, Nlim is almost linearin Sh (Eq. 22) and insensitive to small errors inSmax. AsSh→Smax, Nlim increases steeply and becomes very sensitiveto errors inSmax. This is strongly evident forT =230 K whenSh>1.4 (Fig. 4). At these conditions,1Sh<0.06, the param-eterization becomes very sensitive to the value ofSmax (Fig. 3for Sh=1.4 andT =230 K, i.e. black lines and circles), andtends to overestimateNlim because the parameterizedSmaxis belowSmax reached in the parcel model simulations. Thiseffect is more pronounced at highV (Fig. 3) asSmax tendto reach high values at such conditions. The overestimationin Nlim at Sh=1.4, however, is less pronounced atT =215 K(Fig. 3 upper panels); for these conditions1Sh>0.1, so theparameterization is insensitive to the error inSmax. It is alsonoticeable that at lowV (<0.2 m s−1) the parameterizedSmaxdoes not introduce significant errors inNlim .

Another potential source of error in the parameteriza-tion lies in neglecting the higher order terms in Eq. (10),O(y, y2, ...). Because of this, Eq. (12), will not predict achange in the sign ofy (e.g. green line, Fig. 1), and cannotdescribe the time “lag” before appreciable condensation ratesare achieved upon the IN and the impact of ice crystals ondSi

dtbecomes significant. Since Eq. (12) predicts an instan-

taneous effect of IN ondSi

dt(e.g. red line Fig. 1), neglecting

O(y, y2, ...) become less important when the impact ofNINon dSi

dtis the highest, i.e.NIN=Nlim . Moreover, the error

introduced by neglectingO(y, y2, ...) is in general not crit-ical to calculatefc,IN , since at lowNIN the impact of IN onhomogeneous nucleation is small.

Some bias may also be introduced whenSh is low. Atthese conditions, recently frozen crystals are exposed to lowSh therefore grow slowly, so even atNIN=Nlim there is a sub-stantial time lag before they can impact ice saturation ratio.Since Eq. (12) does not account for this,Dlim is slightly over-predicted andNlim underpredicted (e.g. Fig. 3). This “growthlag” effect is even stronger at lowT and lowαd since con-densation (growth) rates are further decreased. This is seenin the slight underprediction of the parameterization shownin Fig. 3 atSh=1.1 andT =215 K (upper panels, blue linesand circles).

3.3 Evaluation against other parameterizations

We first compare the new parameterization against the for-mulation of Karcher et al. (2006, K06). K06 was devel-oped for a polydisperse aerosol distribution, but it can be



compared against the new parameterization provided thatSmax>Sh (for such conditions, the polydisperse populationwith a single freezing threshold can be well approximatedby a chemically uniform and monodisperse IN with freez-ing thresholdSh). Simulations are carried out forTo=210 K,230 K, p=22000 Pa, andαd=0.5; NIN was set to 1, 5, and30 l−1, and,Sh to 1.3. Results for K06 were obtained fromKarcher et al. (2006). Figure 5 shows the comparison be-tween the two parameterizations; both formulations agreeremarkably well. Slight discrepancies between the param-eterization and K06 are found atTo=210 K andNIN=1 and5 l−1, which is explained by the differences in the homoge-neous scheme of BN08 and K06, as analyzed in Barahonaand Nenes (2008). For very lowV (<0.03 m s−1), Smax<Sh,and, comparison against K06 is not possible. Such conditionsrequire a polydisperse formulation, and will be addressed ina companion study.

Finally, we compared Eq. (20) to the expressions providedfor Nlim by Gierens (2003, G03) and Liu and Penner (2005,LP05). LP05 (Fig. 6, green circles) is based on detailed par-cel model simulations and resolves the dependency ofNlimon T andV . G03 (Fig. 6, stars) uses a semi-analytical ap-proach, where box model simulations are employed to fit thetime scale of growth of the ice crystals; it resolves explic-itly the dependency ofNlim on T , V , Sh andp. Figure 6presents the results of the three expressions as functions ofT

(left panel,V =0.1 m s−1) andV (right panel,T =215 K), andSh=1.3 (blue) and 1.4 (black). Equation (20) is presentedin Fig. 6 for αd=0.1 (dashed lines) and 1 (solid lines). Ata givenV (Fig. 6, left panel),Nlim decreases almost expo-nentially with increasingT . The difference betweenNlimpredicted by G03 and Eq. (20) is the largest atT =200 K, butis minimal atT =230 K. The convergence at highT is likelyfrom the large crystal size attained, which implies that themass transfer coefficient is consistent with the approxima-tion taken in G03 of neglecting kinetic effects. LP05 predictsNlim about one order of magnitude lower than Eq. (20), po-tentially because they included both deposition and immer-sion freezing in their simulations; deposition IN would act asa second population (with differentSh andNIN) that wouldcompete for water vapor with the existing crystals loweringNlim . Equation (20) cannot currently represent such a case,but can be expanded to include it and will be the subjectof a future study. At fixedT , G03 and Eq. (20) predict aNlim∼V 3/2 dependency (Fig. 6, right panel). LP05 shows ahigher order dependency, likely due to the effect of deposi-tion IN depleting water vapor.

4 Summary and conclusions

We developed an analytical parameterization for cirrus for-mation that explicitly considers the competition between ho-mogeneous and heterogeneous nucleation. Using functionalanalysis of the crystal growth and ice saturation ratio evolu-

Fig. 5. Comparison of new parameterization (lines) against the for-mulation of Karcher et al. (2006) (symbols). Simulations shown arefor To=210 K, 230 K,p=22 000 Pa, and,αd=0.5. NIN was set to 1,5, and 30 l−1 , and,Sh to 1.3.

tion equations for a cloud parcel, an expression relatingdSi

dtto Si was found, and used to calculate the limiting size ofthe ice crystals,Dlim , and limiting concentration of IN thatwould prevent homogeneous nucleation,Nlim . These two pa-rameters were employed to obtain a simple, yet accurate, ex-pression to account for the reduction indSi

dtdue to the pres-

ence of previously frozen crystals which, when incorporatedwithin the physically-based homogeneous nucleation frame-work of Barahona and Nenes (2008), allows the calculationof the number of crystals produced when heterogeneous andhomogeneous freezing mechanisms compete for water vaporduring cirrus cloud formation.

Previous work, using numerical parcel simulations, de-termine “heterogeneously-dominated” and “homogeneously-dominated” regimes and correlate them with cloud forma-tion conditions (V , T , and P ). However, it was demon-strated that at conditions where homogeneous freezing typi-cally dominates (i.e., highV and lowT ), ice crystal forma-tion can be affected by IN, provided thatNIN is high enough.Thus,V andT cannot uniquely define heterogeneously andhomogeneously-dominated regimes, and, parameterizationsusing such approaches are inherently limited. It was shownthat the contribution of each freezing mechanism to crystalnumber concentration, depends only onNIN

Nlim. By provid-

ing an expression forNlim , the need for defining “dominant”regimes of ice formation is eliminated.

When evaluated over a wide range of conditions, the newparameterization reproduced the results of the detailed nu-merical solution of the parcel model equations, with anaverage error of 6±33%, which is remarkable low giventhe complexity and nonlinearity of Eqs. (5) to (7). Thenew parameterization also compares favorably against other



Fig. 5. Comparison of new parameterization (lines) against the formulation of Karcher et al. (2006) (symbols).

Simulations shown are forTo = 210 K, 230 K,p = 22000 Pa, and,αd = 0.5. NIN was set to 1, 5, and 30 l−1 ,

and,Sh to 1.3.

Fig. 6. Nlim vs.T at V =0.1 m s−1 (left panel) andNlim vs.V at T=215 K (right panel), calculated using the

parameterizations of Gierens (2003 , G03), Liu and Penner (2005, LP05), and Eq. (20).

26

Fig. 6. Nlim vs. T at V =0.1 m s−1 (left panel) andNlim vs. V at T =215 K (right panel), calculated using the parameterizations ofGierens (2003 , G03), Liu and Penner (2005, LP05), and Eq. (20).

formulations (which require numerical integration). The ap-proach presented here overcomes common difficulties in in-cluding the competition between homogeneous and hetero-geneous modes in large scale models. In this study we as-sumed a single type, monodisperse, chemically uniform, INpopulation. Multiple IN populations and freezing modes thatreflect cirrus formation for atmospheric relevant aerosol isthe focus of a future study. Even though much work still re-mains to be done on the prediction of IN concentrations andfreezing thresholds, this work offers a computationally ef-ficient and rigorous approach to include such knowledge inatmospheric and cloud models, once available.

Appendix A

A summary of parameters developed for the BN08 parame-terization is provided here. The effective growth parameterin Eq. (1) is defined as

0=

Dc,smax∫Do

Dc

01Dc+02dDc

Dc,smax−Do

=γ

01

1+02

01

ln(

02+01Dc,smax02+01Do

)Dc,smax−Do

(A1)

whereγ =1.33, andDc,smax is the maximum size reached bythe homogeneously nucleated ice crystals, given by

Dc,smax = min{

(1.6397×10−14T − 3.1769×10−12)

V −0.05(NoD3g, dry)

−0.373; 10−4

}(A2)

with V is in m s−1, T is in K, No in cm−3, andDg,dry in m.The parameter, k(T ) in Eq. (1) is defined as

k(T )=ln[J (Si )/J (Smax)]

Si−Smax. For the purposes of this study

we fit k(T ) to the data of Koop et al. (2000) forJ between108 and 1022 m−3 s−1:

k(T ) = 0.0240T 2− 8.035T + 934.0 (A3)

with T in K.

Smax is obtained by solvingJ (Smax)=1016 s−1 m−3 (Koopet al., 2000), using the analytical fit of Ren and Mackenzie(2005).

Appendix B

List of symbols

A ymax(ymax−2), defined in Eq. (11)

αg1HsMw

cpRT 2 −gMa

RT

βMap

Mwpoi

αd Water vapor to ice deposition coefficient

cp Specific heat capacity of air

1Hs Specific heat of sublimation of water

Dc,smax Equivalent diameter of the largesthomogeneously frozen ice particle atSmax

Dc Volume sphere-equivalent diameter of anice particle

Dc,IN Volume sphere-equivalent diameter of anIN-frozen crystal

Dg,dry Geometric mean diameter of the dryaerosol size distribution

Dlim Limiting size of the IN-frozen crystals

Dmax,IN Equivalent diameter of the IN-frozen icecrystals atSmax

Do Average diameter of the supercooled liquiddroplet population



Do,IN Initial size of the IN at the point of freezing

Dv Water vapor diffusion coefficient

1Sh Smax − Sh

1S∗

h Growth integral defined in Eq. (15)

fc, fc,IN Fraction of frozen particles atSmax withand without IN present, respectively.

g Acceleration of gravity

0 Effective growth parameter defined inEq. (A1)

0(Do,IN)012 D2

o,IN+02Do,IN

01ρiRT

4poi DvMw

+1Hsρi

4kaT

(1HsMw

RT−1)

02ρiRT

2poi Mw

√2πMw

RT1αd

K Curvature constant defined in Eq. (10)

k(T ) Homogeneous freezing parameter definedin Eq. (A3)

ka Thermal conductivity of air

Mw, Ma Molar masses of water and air, respectively

Nc Ice crystal number concentration

NIN Number concentration of the INpopulation

Nlim Limiting NIN that would preventhomogeneous nucleation

No Number concentration of the supercooledliquid droplet population

O(y, y2, ...) High order terms in Eq. (10)

p Ambient pressure

poi Ice saturation vapor pressure

ρi , ρa Ice and air densities, respectively

R Universal gas constant

Si Water vapor saturation ratio with respectto ice

Smax Maximum ice saturation ratio;homogeneous freezing threshold

Sh Freezing threshold of the IN population

Smax,IN Rate of change ofSi atSmax with INpresent

Sh Rate of change ofSi atSh

T Temperature

To Initial temperature

t Time

V Updraft velocity

wIN Ice mass mixing ratio of theheterogeneously frozen crystals

x, y Dimensionless variables defined inEqs. (8) and (9)

ymax Value ofy atx=1

Acknowledgements.This study was supported by NASA MAP anda NASA New Investigator Award.

Edited by: T. Koop

References

Abbatt, J. P. D., Benz, S., Cziczo, D. J., Kanji, Z., and Mohler, O.:Solid ammonium sulfate as ice nuclei: a pathway for cirrus cloudformation, Science, 313, 1770–1773, 2006.

Archuleta, C. M., DeMott, P. J., and Kreidenweis, S. M.: Ice nu-cleation by surrogates for atmospheric mineral dust and mineraldust/sulfate particles at cirrus temperatures, Atmos. Chem. Phys.,5, 2617–2634, 2005,http://www.atmos-chem-phys.net/5/2617/2005/.

Baker, M. B. and Peter, T.: Small-scale cloud processes and climate,Nature, 451, 299–300, 2008.

Barahona, D. and Nenes, A.: Parameterization of cirrus formationin large scale models: Homogenous nucleation, J. Geophys. Res.,113, D11211, doi:11210.11029/12007JD009355, 2008.

Beaver, M. R., Elrod, M. J., Garland, R. M., and Tolbert, M. A.: Icenucleation in sulfuric acid/organic aerosols: Implications for cir-rus cloud formation, Atmos. Chem. Phys., 6, 3231–3242, 2006,http://www.atmos-chem-phys.net/6/3231/2006/.

Cantrell, W. and Heymsfield, A. J.: Production of ice in tropo-spheric clouds, B. Am. Meteorol. Soc., 86, 795–807, 2005.

Chen, Y., Kreidenweis, S. M., McInnes, L. M., Rogers, D. C., andDemott, P. J.: Single particle analyses of ice nucleating aerosolsin the upper troposphere and lower stratosphere, Geophys. Res.Lett., 25, 1391–1394, 1998.

Chen, Y., DeMott, P. J., Kreidenweis, S. M., Rogers, D. C., andSherman, D. E.: Ice formation by sulfate and sulfuric acidaerosol particles under upper-tropospheric conditions, J. Atmos.Sci., 57, 3752–3766, 2000.

Cozic, J., Verheggen, B., Mertes, S., Connolly, P., Bower, K., Pet-zold, A., Baltensperger, U., and Weingartner, E.: Scavengingof black carbon in mixed phase clouds at the high alpine siteJungfraujoch, Atmos. Chem. Phys. Discuss., 6, 11 877–11 912,2006,http://www.atmos-chem-phys-discuss.net/6/11877/2006/.

Cziczo, D. J. and Abbatt, J. P. D.: Ice nucleation in NH4HSO4,NH4NO3, and H2SO4 aqueous particles: Implications for cirrusformation, Geophys. Res. Lett., 28, 963–966, 2001.

DeMott, P. J., Meyers, M. P., and Cotton, R. W.: Parameteriza-tion and impact of ice initiation processes relevant to numericalmodel simulations of cirrus clouds, J. Atmos. Sci., 51, 77–90,1994.


http://www.atmos-chem-phys.net/5/2617/2005/


http://www.atmos-chem-phys-discuss.net/6/11877/2006/


DeMott, P. J., Rogers, D. C., and Kreidenweis, S. M.: The suscep-tibility of ice formation in upper tropospheric clouds to insolu-ble aerosol components, J. Geophys. Res., 102, 19 575–19 584,1997.

DeMott, P. J., Rogers, D. C., Kreidenweis, S. M., Chen, Y., Twohy,C. H., Baumgardner, D. G., Heymsfield, A. J., and Chan, K. R.:The role of heterogeneous freezing nucleation in upper tropo-spheric clouds: Inferences from SUCCESS, Geophys. Res. Lett.,25, 1387–1390, 1998.

DeMott, P. J., Cziczo, D. J., Prenni, A. J., Murphy, D. M., Krei-denweis, S. M., Thompson, D. S., Borys, R., and Rogers, D. C.:Measurements of the concentration and composition of nucleifor cirrus formation, Proc. Natl. Acad. Sci. USA, 100, 14 655–14 660, 2003a.

DeMott, P. J., Sassen, K., Poellot, M. R., Baumgardner, D. G.,Rogers, D. C., Brooks, S. D., Prenni, A. J., and Kreidenweis,S. M.: African dust aerosols as atmospheric ice nuclei, Geophys.Res. Lett., 30, 1732, doi:1710.1029/2003GL017410, 2003b.

Field, P. R., Mohler, O., Connolly, P., Kramer, M., Cotton, R.,Heymsfield, A. J., Saathoff, H., and Schnaiter, M.: Some icenucleation characteristics of Asian and Saharan desert dust, At-mos. Chem. Phys., 6, 2991–3006, 2006,http://www.atmos-chem-phys.net/6/2991/2006/.

Fridlind, A. M., Ackerman, A. S., Jensen, E. J., Heymsfield, A.J., Poellot, M. R., Stevens, D. E., Wang, D., Miloshevich, L.M., Baumgardner, D. G., Lawson, P., Willson, J., Flagan, R. C.,Seinfeld, J. H., Jonsson, H. H., VanReken, T. M., Varutbangkul,V., and Rissman, T. A.: Evidence for the predominance of mid-tropospheric aerosols as subtropical anvil cloud nuclei, Science,304, 718–722, 2004.

Gayet, J. F., Ovarlez, J., Shcherbakov, V., Strom, J., Schumann,U., Minikin, A., Auriol, F., Petzold, A., and Monier, M.: Cirruscloud microphysical and optical properties at southern and north-ern midlatitudes during the INCA experiment, J. Geophys. Res.,109, D20206, doi:20210.21029/22004JD004803, 2004.

Gierens, K.: On the transition between heterogeneous and homoge-neous freezing, Atmos. Chem. Phys., 3, 437–446, 2003,http://www.atmos-chem-phys.net/3/437/2003/.

Haag, W., Karcher, B., Strom, J., Minikin, A., Lohmann, U., Ovar-lez, J., and Stohl, A.: Freezing thresholds and cirrus formationmechanisms inferred from in situ measurements of relative hu-midity, Atmos. Chem. Phys., 3, 1791–1806, 2003,http://www.atmos-chem-phys.net/3/1791/2003/.

Hartmann, D. L., Holton, J. R., and Fu, Q.: The heat balance of thetropical tropopause, cirrus, and stratospheric dehydration, Geo-phys. Res. Lett., 28, 1969–1972, 2001.

Heymsfield, A. J. and Sabin, R. M.: Cirrus crystal nucleation byhomogenous freezing of solution droplets., J. Atmos. Sci., 46,2252–2264, 1989.

Heymsfield, A. J., Miloshevich, L. M., Twohy, C. H., Sachse, G.,and Oltmans, S.: Upper-tropospheric relative humidity observa-tions and implications for cirrus ice nucleation, Geophys. Res.Lett., 25, 1343–1346, 1998.

Hung, H., Malinowski, A., and Martin, S. T.: Kinetics of hetero-geneous ice nucleation on the surfaces of mineral dust cores in-serted into aqueous ammonium sulfate particles, J. Phys. Chem.A, 107, 1296–1306, 2003.

Karcher, B. and Lohmann, U.: A parameterization of cirrus cloudformation: Heterogeneous freezing, J. Geophys. Res., 108, 4402,

doi:10.1029/2002JD003220, 2003.Karcher, B., Hendricks, J., and Lohmann, U.: Physically

based parameterization of cirrus cloud formation for use inglobal atmospheric models, J. Geophys. Res., 111, D01205,doi:01210.01029/02005JD006219, 2006.

Karcher, B., Mohler, O., DeMott, P. J., Pechtl, S., and Yu, F.: In-sights into the role of soot aerosols in cirrus cloud formation,Atmos. Chem. Phys., 7, 4203–4227, 2007,http://www.atmos-chem-phys.net/7/4203/2007/.

Khvorostyanov, V. I. and Sassen, K.: Toward the theory of homo-geneous nucleation and its parameterization for cloud models,Geophys. Res. Lett., 25, 3155–3158, 1998.

Khvorostyanov, V. I. and Curry, J. A.: The theory of ice nucleationby heterogeneous freezing of deliquescent mixed CCN. Part I:critical radius, energy and nucleation rate, J. Atmos. Sci., 61,2676–2691, 2004.

Khvorostyanov, V. I. and Curry, J. A.: The theory of ice nucleationby heterogeneous freezing of deliquescent mixed CCN. Part II:parcel model simulations, J. Atmos. Sci., 62, 261–285, 2005.

Koop, T., Luo, B., Tslas, A., and Peter, T.: Water activity as the de-terminant for homogeneous ice nucleation in aqueous solutions,Nature, 406, 611–614, 2000.

Lin, R., Starr, D., DeMott, P. J., Cotton, R., Sassen, K., Jensen,E. J., Karcher, B., and Liu, X.: Cirrus parcel model compari-son project. Phase 1: The critical components to simulate cirrusinitiation explicitly, J. Atmos. Sci., 59, 2305–2328, 2002.

Liou, K.: Influence of cirrus clouds on weather and climate pro-cesses: a global perspective, Mon. Weather Rev., 114, 1167–1199, 1986.

Liu, X. and Penner, J. E.: Ice nucleation parameterization for globalmodels, Meteorol. Z., 14, 499–514, 2005.

Lohmann, U., Karcher, B., and Hendricks, J.: Sensitivity studies ofcirrus clouds formed by heterogeneous freezing in the ECHAMGCM, J. Geophys. Res., 109, doi:10.1029/2003JD004443, 2004

Marcolli, C., Gedamke, S., Peter, T., and Zobrist, B.: Efficiencyof immersion mode ice nucleation on surrogates of mineral dust,Atmos. Chem. Phys., 7, 5081–5091, 2007,http://www.atmos-chem-phys.net/7/5081/2007/.

Minnis, P.: Contrails, cirrus trend, and climate, J. Climate, 17,1671–1685, 2004.

Mohler, O., Buttner, S., Linke, C., Schnaiter, M., Saathoff, H., Stet-zer, O., Wagner, R., Kramer, M., Mangold, A., Ebert, V., andSchurath, U.: Effect of sulfuric acid coating on heterogeneousice nucleation by soot aerosol particles, J. Geophys. Res., 110,D11210, doi:11210.11029/12004JD005169, 2005.

Mohler, O., DeMott, P. J., Vali, G., and Levin, Z.: Microbiology andatmospheric processes: the role of biological particles in cloudphysics Biogeosciences, 4, 1059–1071, 2007,http://www.biogeosciences.net/4/1059/2007/.

Monier, M., Wobrock, W., Gayet, J. F., and Flossman, A.: Develop-ment of a detailed microphysics cirrus model tracking aerosolparticles histories for interpretation of the recent INCA cam-paign, J. Atmos. Sci., 63, 504–525, 2006.

Penner, J. E., Lister, D. H., Griggs, D. J., Dokken, D. J., and McFar-land, M.: Aviation and the global atmosphere – A special reportof IPCC working groups I and III. Intergovernmental Panel onClimate Change, Cambridge University Press, 365 pp., 1999.

Prenni, A. J., DeMott, P. J., Twohy, C. H., Poellot, M. R., Krei-denweis, S. M., Rogers, D. C., Brooks, S. D., Richardson, M.







http://www.biogeosciences.net/4/1059/2007/


S., and Heymsfield, A. J.: Examinations of ice formation pro-cesses in Florida cumuli using ice nuclei measurements of anvilice crystal particle residues, J. Geophys. Res., 112, D10121,doi:10110.11029/12006JD007549, 2007.

Pruppacher, H. R. and Klett, J. D.: Microphysics of clouds and pre-cipitation 2nd ed., Kluwer Academic Publishers, Boston, MA,1997.

Ren, C. and Mackenzie, A. R.: Cirrus parameterization and the roleof ice nuclei, Q. J. Roy. Meteorol. Soc., 131, 1585–1605, 2005.

Sassen, K., DeMott, P. J., Prospero, J. M., and Poellot, M.R.: Saharan dust storms and indirect aerosol effects onclouds: CRYSTAL-FACE results, Geophys. Res. Lett., 30, 1633,doi:1610.1029/2003GL017371, 2003.

Seinfeld, J. H.: Clouds, contrails and climate, Nature, 391, 837–838, 1998.

Stuber, N., Forster, P., Radel, G., and Shine, K.: The importanceof the diurnal and annual cycle of air traffic for contrail radiativeforcing, Nature, 441, 864–867, 2006.

Tabazadeh, A., Martin, S. T., and Lin, J.: The effect of particlesize and nitric acid uptake on the homogeneous freezing of aque-ous sulfuric acid particles, Geophys. Res. Lett., 27, 1111–1114,2000.

Vali, G.: Freezing rate due to heterogeneous nucleation, J. Atmos.Sci., 51, 1843–1856, 1994.

Zobrist, B., Marcolli, C., Koop, T., Luo, B., Murphy, D. M.,Lohmann, U., Zardini, A., Krieger, U. K., Corti, T., Cziczo, D.J., Fueglistaler, S., Hudson, P. K., and Peter, T.: Oxalic acid as aheterogeneous ice nucleus in the upper troposphere and its indi-rect aerosol effect, Atmos. Chem. Phys., 6, 3115–3129, 2006,http://www.atmos-chem-phys.net/6/3115/2006/.

Zobrist, B., Koop, T., Luo, B., Marcolli, C., and Peter, T.: Hetero-geneous ice nucleation rate coefficient of water droplets coatedby a nonadecanol monolayer, J. Phys. Chem. C, 111, 2149–2155,2007.

Zobrist, B., Marcolli, C., Peter, T., and Koop, T.: Heterogeneousice nucleation in aqueous solutions: the role of water activity, J.Phys. Chem. A., 112, 3965–3975, 2008.

Zuberi, B., Bertram, A. K., Cassa, C. A., Molina, L. T., and Molina,M. J.: Heterogeneous nucleation of ice in (NH4)2SO4-H2O par-ticles with mineral dust immersions, Geophys. Res. Lett., 29,1504, doi:1510.1029/2001GL014289, 2002.



parameterizing the competition between homogeneous and

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