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An Empirical Parameterization of Heterogeneous Ice Nucleation for Multiple Chemical Species of Aerosol

September 30, 2008

By Phillips, Vaughan T J DeMott, Paul J; Andronache, Constantin

ABSTRACT A novel, flexible framework is proposed for parameterizing the heterogeneous nucleation of ice within clouds. It has empirically derived dependencies on the chemistry and surface area of multiple species of ice nucleus (IN) aerosols. Effects from variability in mean size, spectral width, and mass loading of aerosols are represented via their influences on surface area. The parameterization is intended for application in largescale atmospheric and cloud models that can predict 1) the supersaturation of water vapor, which requires a representation of vertical velocity on the cloud scale, and 2) concentrations of a variety of insoluble aerosol species.

Observational data constraining the parameterization are principally from coincident field studies of IN activity and insoluble aerosol in the troposphere. The continuous flow diffusion chamber (CFDC) was deployed. Aerosol species are grouped by the parameterization into three basic types: dust and metallic compounds, inorganic black carbon, and insoluble organic aerosols.

Further field observations inform the partitioning of measured IN concentrations among these basic groups of aerosol. The scarcity of heterogeneous nucleation, observed at humidities well below water saturation for warm subzero temperatures, is represented. Conventional and inside-out contact nucleation by IN is treated with a constant shift of their freezing temperatures.

The empirical parameterization is described and compared with available field and laboratory observations and other schemes. Alternative schemes differ by up to five orders of magnitude in their freezing fractions (-30[degrees]C). New knowledge from future observational advances may be easily assimilated into the scheme’s framework. The essence of this versatile framework is the use of data concerning atmospheric IN sampled directly from the troposphere.

(ProQuest: … denotes formulae omitted.)

1. Introduction

Clouds govern the transfers of radiation that drive the circulation of the earth’s atmosphere. How cloud radiative properties might change in response to changing concentrations and composition of aerosol in the future is a great source of uncertainty in the prediction of global climate change. Recently, there has been much interest in predicting both the number and mass of cloud particles (e.g., Lohmann and Feichter 1997; Lohmann et al. 1999; Ming et al. 2006) in general circulation models (GCMs). The goal of such a “doublemoment” approach is to predict the size of cloud particles, and hence the radiative and microphysical properties of clouds, as a function of the aerosol in the environment.

Heterogeneous nucleation of ice, by which crystals are formed on the insoluble components of aerosol particles, is one of the processes to be represented when predicting the crystal number with models. Current empirical formulas for heterogeneous nucleation (e.g., Meyers et al. 1992) rely on data from instruments that measure ice activation by exposure of aerosol to controlled conditions of temperature and humidity representative of supercooled clouds. The continuous flow diffusion chamber (CFDC) is one such device. Some direct comparisons by Rogers and DeMott (1995,2002) between number concentrations of ice nuclei (IN) measured by the CFDC and of ice crystals formed in orographie wave clouds at similar temperatures demonstrate agreement to within a factor of 2 (see also Prenni et al. 2007a). Those particular wave clouds were outside the region where the Hallett-Mossop (H-M) process (Hallett and Mossop 1974) of ice multiplication is active (-3[degrees] to -8[degrees]C) and below levels where homogeneous freezing can occur.

Although the residence time of aerosol exposure to specified conditions inside the CFDC is short (7-15 s, depending on the project), there is observational evidence that activation of most IN occurs very soon after their ambient conditions for freezing are reached. First, nucleation of ice by mineral dust (e.g., by deposition) was seen by Mohler et al. (2006, p. 1554) to stop in less than about 10 s just after the supersaturation with respect to ice, s^sub i^, reached its peak value at the cessation of cooling. Similarly, Vali (1994) observed that the freezing rate of drops was reduced by 97% in less than 30 s just after a cessation of supercooling. Ice formation ensues sharply within a very short distance of cloud edge during smooth flow into it (Cooper and Vali 1981). second, the probability of freezing per second for a given IN particle has been observed to increase from zero to an extremely high value over an interval of about 1 K during supercooling near its characteristic freezing temperature, which is invariant over many freezing cycles (Shaw et al. 2005). That is why the error in the freezing fraction from an assumption of perfectly instantaneous freezing is seen to be equivalent to a temperature error of about 1[degrees]C (Vali 1994, p. 1852; see also Vali 1969, 1971). Third, laboratory observations (e.g., DeMott 1990; Mohler et al. 2006) show that frozen fractions of IN populations are independent of the cooling rate. This is all consistent with the assumption of almost instantaneous freezing when IN reach their characteristic freezing temperatures (the “singular hypothesis”; see Rau 1944; Levine 1950; Langham and Mason 1958; Vali 1971), which may depend on s,- and have a probability distribution among a given population of IN.

Vali and Stansbury (1966) and Vali (1994) observed a stochastic aspect superimposed on nucleation temperatures defined by the singular nature of nucleation sites on the surface of IN. However, the first-order description of the nucleation that they observed is provided by the singular hypothesis according to their analysis (e.g., Vali 1994, p. 1852). Marcolli et al. (2007) have confirmed this picture with laboratory studies of immersion freezing. Marcolli et al. inferred a distribution of rare but efficient active sites over the IN surface where ice embryos can form, as invoked by the singular hypothesis (see section 2).

Evidence supporting the singular hypothesis is now sufficient for it to no longer be viewed as a mere hypothesis, as far as immersion freezing is concerned. It may also be valid for other modes such as deposition nucleation (e.g., Mohler et al. 2006), if the characteristic freezing temperatures are a strong function of s,. In fact, S1 has been seen to be a good single descriptor of deposition nucleation (Huffman 1973) in observations of natural IN between – 12[degrees] and -2O0C (at much colder temperatures there may be an additional, weaker temperature dependence). Consequently, measured IN concentrations may be expressed in terms of s, (e.g., Meyers et al. 1992), an approach followed here.

A disadvantage of formulas based on CFDC data hitherto has been that they include no explicit dependencies on the multiple chemical species of insoluble aerosol, because of the lack of simultaneous aerosol data from other probes. Chen et al. (1998) have shown that ice-nucleating aerosol species at least include categories that can be described as carbonaceous, metallic, and dust aerosols. Within a given category, IN activity has been seen to increase with the aerosol loading (Georgii and Kleinjung 1967; Berezinskiy et al. 1986).

There is a need for parameterizations of heterogeneous ice nucleation to be developed that reflect the diversity of IN chemistry and that are empirical, based on observations of IN sampled from the earth’s atmosphere. Almost all other parameterizations proposed in the past are based either on laboratory observations (Lohmann and Diehl 2006; Diehl and Wurzler 2004; Diehl et al. 2006) or on a form of classical theory that does not assume the singular hypothesis (Karcher and Lohmann 2003; Khvorostyanov and Curry 2004; Liu and Penner 2005; Pruppacher and Klett 1997,341-344).

In the present paper, a versatile framework for parameterizing atmospheric ice nucleation is formulated that 1) accounts for contributions from different chemical species of aerosol, as constrained by in situ measurements of IN activity (e.g., by the CFDC) and composition, and 2) represents the relative scarcity of nucleation seen at humidities well below water saturation at temperatures warmer than -4O0C (“warm subzero temperatures”). An advantage of this approach is that it is empirical, being based mostly on observations of natural IN sampled from the background free troposphere.

In the next section there is a discussion of the empirical basis of the parameterization, including the assumed dependency on aerosol surface area. This allows the scheme to be applied to a wide range of aerosol scenarios globally, despite being based on field measurements at a single location. In subsequent sections, there is a detailed description of the scheme’s framework for known modes of nucleation, followed by a comparison with independent observations and alternative existing schemes. Advice about how to implement the scheme is given in another section. Its merits and limitations, and our vision for its future development, are discussed in the concluding two sections.

2. Observational basis for key assumptions of the parameterization of heterogeneous ice nucleation a. Classification of IN by chemical composition

Chen et al. (1998) observed that the set of icenucleating aerosol species includes carbonaceous, metallic, and dust aerosols. A common representation of insoluble aerosols in GCMs is in terms of dust, organic carbon, and inorganic black carbon. Consequently, we partition the components of IN identified by Chen et al. for compatibility with current GCM design, in terms of the following basic groups: 1) dust/metallic aerosols (DM); 2) inorganic black carbon (BC); and 3) insoluble organic particles (O). The last group includes IN from bacteria (Vali et al. 1976; Lindow et al. 1978), leaf litters (Schnell and Vali 1972,1976), pollen (Diehl et al. 2001; Diehl and Wurzler 2004), and perhaps oxalic acid dihydrate (OAD; Zobrist et al. 2006).

An area of uncertainty concerns the choice of a single aerosol species to represent the insoluble organic group of IN. Oxalic acid occurs in aerosols (e.g., Narukawa et al. 2003; Murphy et al. 1998) and has a solid hydrate, OAD, that can nucleate ice (Zobrist et al. 2006). But whether solid OAD occurs significantly in the atmosphere is a moot point.

Biogenic aerosols may constitute about 10% of all submicron aerosols in the troposphere (Jaenicke 2005). A few species of bacteria can nucleate ice (e.g., Vali et al. 1976; Morris et al. 2004) and originate from plants (Lindemann et al. 1982; Lindemann and Upper 1985). Pollen (exceeding 10 [mu]m) and leaf litter, for example, can nucleate ice, as noted above, but pollen is too large to stay in the atmosphere for long. If pollen or leaf litter break up, their fragments may be numerous at submicron sizes but have a nucleating ability that few studies have examined. That of ice- nucleation active (INA) strains of bacteria has been quantified in several studies (e.g., Vali et al. 1976; Lindow et al. 1978; Hirano et al. 1985). For these reasons, the nucleating ability of the O group of IN is constrained by observations of INA bacteria, as well as by analysis of crystal residual material (e.g., Targino et al. 2006).

b. Relation of numbers of active IN to total surface area of a given aerosol species

A fundamental assumption of the proposed scheme is that the number of active IN of a particular species of insoluble aerosol is approximately proportional to the total surface area of its aerosol particles (see section 3a). There is a strong theoretical basis for such a dependence on surface area (e.g., Pruppacher and Klett 1997). Heterogeneous nucleation is an interface phenomenon involving the formation, on the surface of the IN material, of critical ice embryos on the nanometer scale at specific sites. Such active sites have a certain probability of occurrence per unit area of the surface of a given IN material. The sites are determined by (e.g., crystallographic) features of the surface, and each one nucleates ice close to a unique, characteristic temperature (e.g., Vali 1994; Shaw et al. 2005). Such active sites have a high compatibility with the lattice structure of the ice embryo, but they are rare. Larger IN particles have more and better active sites than smaller ones, which is why they are more likely to have a highernucleating efficiency. The natural singular character of IN emerges from the probability distribution of nucleation efficiencies, and hence of characteristic freezing temperatures, among IN particles (e.g., Marcolli et al. 2007). This distribution of nucleation efficiencies arises from that of the “contact angle” (i.e., suitability of an individual site to trigger nucleation) among active sites on IN particles.

At least two strands of observational evidence support this assumed dependence on surface area for soot and dust, respectively. First, the number of crystals nucleated by immersion freezing of acetylene soot has been observed to be proportional to its total surface area (DeMott 1990). Second, observations of atmospheric IN sampled from the free troposphere by the CFDC described by Rogers et al. (2001a) are analyzed here for two contrasting cases: 1) the First Ice Nuclei Spectroscopy Study (INSPECT-I) during 1-19 November 2001 at Mt. Weraer [106.73[degrees]W, 40.45[degrees]N; 3.22-km altitude above mean sea level (MSL)] in Colorado (DeMott et al. 2003a), with aerosol surface area derived from measurements with various probes and aerosol loadings measured by the Interagency Monitoring Program for Visual Environment filter data (IMPROVE; http://vista.cira.colostate.edu/improve) at Mt. Zirkel nearby; and 2) the second Ice Nuclei Spectroscopy Study (INSPECT-2) during April- May 2004 also at Mt. Werner (Richardson et al. 2007), where similar aerosol measurements were made. Surface area of dust (larger than 500 nm) was inferred from aerosol size distribution data from the TSI Aerodynamic Particle Sizer (APS) instrument (INSPECT-1), Differential Mobility Analyzer (DMA), and an optical particle counter (INSPECT-2; Richardson et al. 2007), combined with measurements of dust loading either near Mt. Werner on a daily basis (INSPECT-2) or by IMPROVE nearby at Mt. Zirkel (INSPECT-1; DeMott et al. 2003a). Spherical equivalent sizes and p^sub DM^ = 2.3 g cm^sup – 3^ were assumed.

Figure 1 shows daily averages of the ratio (zeta) of the number of active IN to total surface area of dust particles inside the CFDC (smaller than about 1 [mu]m in equivalent spherical diameter) as a function of the supersaturation with respect to ice, s^sub i^, imposed inside the CFDC. Despite the average dust loadings of both campaigns (INSPECT-I and -2) differing by an order of magnitude (about 0.2 and 1.3 mug m~3) and despite limited fluctuations in dust size, approximately the same relation between zeta and s^sub i^ is seen throughout both cases. This normalized IN number increases logarithmically with s^sub i^, resembling the trend seen by Meyers et al. (1992). Mineral dust was the most prevalent IN measured (by CFDC) during INSPECT (e.g., section 3a; Fig. 2). The constancy of the relation between zeta and s^sub i^ seen throughout both cases is consistent with the above assumption of proportionality between IN activity and total surface area of the dust, at any given value of s^sub i^.

Also shown in Fig. 1 are two brief episodes (10 min each) of data from dusty days (28-29 July 2002) of the Cirrus Regional Study of Tropical Anvils and CUTUS Layers-Florida-Area Cirrus Experiment (CRYSTALFACE) (DeMott et al. 2003b). The upper size limit for surface area relevant to the CFDC measurements was 1.5 [mu]m in CRYSTAL-FACE, instead of 1.0 [mu]m in the INSPECT studies, because of the choice of the inlet impactor used for all instruments. The surface area of dust (larger than 500 nm) in CRYSTAL-FACE is derived from aircraft measurements from the Cloud and Aerosol Spectrometer (CAS; Droplet Measurement Technologies, Boulder, Colorado). Although all samples displayed in Fig. 1 are significant with regard to total numbers of IN acquired, the sampling of the troposphere is much less extensive in CRYSTAL-FACE. The Saharan atmospheric dust (80 [mu]g m^sup -3^) seen in CRYSTAL-FACE was advected to Florida within a dry layer (all particles larger than 500 nm were assumed to be dust). The value of zeta is slightly higher (CRYSTAL-FACE) by almost an order of magnitude than for the background troposphere (INSPECT). We speculate that dust in the layer is likely to be freshly emitted and younger than dust in the background free troposphere. Moreover, our analysis of laboratory observations by Archuleta et al. (2005) of unprocessed Asian dust (0.1 and 0.2 [mu]m in diameter) sampled from the earth’s crust reveals values of zeta consistently higher by almost two orders of magnitude than for INSPECT over a wide range of s^sub i^ (Fig. 1). These observations are at least consistent with a hypothesis of microphysical and/or chemical atmospheric processing of dust acting to lower the nucleating ability during long-range transport after emission from the earth’s surface (section T).

In summary, concentrations of active IN of a given species are proportional to its aerosol surface area (in the large mode, for atmospheric IN). However, the constant of proportionality differs with aerosol age and other environmental factors (section 7).

c. Scarcity of heterogeneous ice nucleation caused by subsaturated conditions at warm subzero temperatures

An area of uncertainty concerns the degree to which heterogeneous nucleation occurs in the troposphere at low humidities well below water saturation at warm subzero temperatures (i.e., warmer than about -40[degrees]C). Heymsfield and Miloshevich (1995) observed that ice formed only at relative humidities exceeding 90% in the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment, phase II (FIRE-11), for temperatures between – 35[degrees] and -40[degrees]C. Soot particles produced virtually no nucleation when held in constant conditions subsaturated with respect to water at temperatures warmer than -30[degrees]C (Dymarska et al. 2006). DeMott et al. (1999) observed that nucleation of ice by soot at -30[degrees]C occurred only at water saturation (see also Mohler et al. 2005a). When humidities well below water saturation were imposed in the Aerosol Interaction and Dynamics in the Atmosphere (AIDA) chamber, dust was seen to act as IN (e.g., by deposition) at temperatures colder (but not warmer) than – 15[degrees]C with “active fractions” (fraction of the number lost by ice nucleation) that were higher than 0.1% (the threshold for their detection) and lower than for the immersion mode (3%-8% at about – 20[degrees]C) at the same temperature (O. Mohler 2006, personal communication). Field et al. (2006) observed no heterogeneous nucleation of desert dust at detectable active fractions higher than 0.5% for temperatures warmer than about -40[degrees]C at humidities well below water saturation at AIDA. Consequently, the active fraction for dust studied at AIDA is on the order of 0.1% at – 20[degrees]C at humidities well below water saturation. 3. Description of empirical parameterization of heterogeneous ice nucleation for dust, black carbon, and insoluble organic particles

For compatibility with current GCM design, icenucleating aerosol species are partitioned into three groups: dust/metallic, black carbon, and insoluble organic aerosols (see section 2). The proposed parameterization of heterogeneous ice nucleation for these species is described here. Variables and constants are defined in appendix A.

a. Nucleation of ice by condensation and immersion freezing and by deposition


A reference activity spectrum of the average number concentration of IN, n^sub IN,1,*^. (number of active IN per kilogram of air), is constructed from field observations with the CFDC (Rogers et al. 2001a) for a backgroundtroposphere scenario (denoted by the subscript *) at the Storm Peak Laboratory on Mt. Werner (see section 2) for 1-19 November 2001 (INSPECT-1; DeMott et al. 2003a). The reference spectrum describes the IN activity at water saturation for this scenario, as inferred from the CFDC observations. The CFDC was modified for the field experiment to operate at temperatures colder than -60[degrees]C. The scenario is so called because the INSPECT-1 data are assumed to be representative of the background state of the free troposphere. It includes a diverse mixture of IN species.

During the background-troposphere scenario, loadings of aerosol particles with equivalent spherical (dry) diameters smaller than 2.5 [mu]m were measured by IMPROVE at Mt. Zirkel near the site at Mt. Werner (DeMott et al. 2003a). Measurements with the CFDC were made at temperatures between about -40[degrees] and -60[degrees]C, at humidities below those for the onset of any homogeneous aerosol freezing, and at aerosol (dry) diameters less than 1 [mu]m (denoted by the subscript 1; the aerodynamic size of the CFDC inlet impactor). They are characterized by this expression, which is extrapolated to -35[degrees]C and to temperatures (T) colder than – 70[degrees]C (the deposition mode depends mainly on S1-; see section 1):

n^sup C^^sub IN,1,*^ = c^sub 1^ {exp[12.96(S^sub i^ - 1.1)]}^sup 0.3^/p^sub c^ for T = – 35[degrees]C and 1

S^sub i^(T, Q^sub v^) is the saturation ratio of water vapor with respect to ice, where Q^sub v^ is the vapor mixing ratio and S^sup hom^^sub i^ is the value of S^sub i^ at the onset of homogeneous aerosol freezing. Also, c^sub 1^ = 1000 m^sup -3^, and p^sub c^ = 0.76 kg m^sup -3^ is the air density inside the CFDC (for an operating pressure of 500 mb). This empirical formula includes nucleation by deposition and by condensation and immersion freezing, resembling that presented by DeMott et al. (2004). These CFDC measurements were done below water saturation and only involved ice nucleation by interstitial IN. At high humidities approaching water saturation, the full contribution from condensation and immersion freezing is difficult to measure because of the onset of homogeneous freezing. Equation (1) is extrapolated to the value of S1 at water saturation, S^sup w^^sub i^(T). A factor of gamma [almost equal to] 2 then yields the reference activity spectrum, describing heterogeneous nucleation in the background-troposphere scenario at water saturation:

n^sub IN,i,*^(T, S^sub i^) = n^sup C^^sub IN,i,*^(T, S^sub i^)gamma for T = 35[degrees]C and 1

Justification for our choice of gamma is that the IN concentration measured by the CFDC between -30[degrees] and – 35[degrees]C in INSPECT-1 (DeMott et al. 2003a) doubles from about 3 L^sup -1^ [[approximate] n^sup C^^sub IN,i,*^(T, S?)pc] to 6 L^sup – 1^ as the relative humidity with respect to water increases from 97% to 100%. Such jumps of IN activity are explicable in terms of an intensification of condensation and immersion freezing when IN- containing aerosols swell with uptake of water as water saturation is approached, diluting the associated dissolved solute and raising the heterogeneous freezing temperature. This is consistent with our estimate (appendix B) that the dissolved solute’s depression of the heterogeneous freezing temperature must change from about 4 K (97% relative humidity) to less than about 0.1 K (water saturation) at – 30[degrees]C, boosting IN activity by about a factor of 2. Jumps of IN activity are excluded from Eq. (1) because they coincide with onset conditions for homogeneous aerosol freezing. Both forms of freezing, with and without IN, are suppressed by the humidity- sensitive solute concentration.

The reference spectrum above is then extrapolated to temperatures warmer than -25[degrees] by rescaling the formula from Meyers et al. (1992) with a normalization factor psi:

n^sub IN,i,*^(T, S^sub i^) = psici exp[12.96(5, - 1) - 0.639] for T >/= -25[degrees]C and 1 = S^sub i^ S^sup w^^sub i^. (3)

Here, psi is selected to be 0.058707y/pc m^sup 3^ kg^sup -1^, so as to match it with nINa. in Eq. (2) at -3O0C (with extrapolation of both equations) and water saturation (hi the case of condensation- and immersion-freezing modes). The factor psi is much less than unity, probably because of a difference in aerosol mass loading due to height and other factors (e.g., time of year, geographic location) between the two datasets from Meyers et al. (1992) (continental boundary layer, rich with IN) and DeMott et al. (2003a) (free troposphere).

For T between -35[degrees] and -250C, nIN4, is interpolated between values and n^sub IN,i^ (appendix A) obtained from extrapolation to T of Eqs. (2) and (3), respectively:

… (4)

… (5)

… (6)

… (7)

Such an interpolation is necessary to acquire a smooth transition between temperature ranges of validity of Eqs. (2) and (3). For Eqs. (2)-(7), the input value of S^sub i^ is artificially prevented from exceeding water saturation.


CFDC measurements in the background-troposphere scenario are assumed to have contributions from the basic groups [DM, BC, and O (roman upper case "O," short for "organic")] of insoluble aerosol (n^sub IN,1,*^ = n^sub IN,DM,*^, + n^sub IN,BC,*^ + n^sub IN,0,*^). In general, for any scenario of aerosol loading, the number mixing ratio of active IN (all aerosol sizes) also has contributions from the same groups:

… (8)

where X = DM, BC, O for dust/metallic, black carbon, and organic aerosols, respectively. As justified in section 2, the number concentration of active IN from group X within the size interval d logD^sub x^ increases with aerosol surface area:

… (9)

… (10)

The term Omega^sub x^ is the total surface area of all aerosols with dry diameters larger than 0.1 [mu]m, per unit mass of air (the surface area mixing ratio) in group X (including interstitial IN and IN immersed in cloud liquid), and dOmega^sub x^/dn^sub x^ [asymptotically =] psiD^sup 2^^sub x^. Here, mu^sub x^ is the average of the number of activated ice embryos per insoluble aerosol particle of size Dx. This integer number is assumed to be statistically (Poisson) distributed. The number mixing ratio of aerosols in group X is n^sub x^. This minimum value of IN diameter (0.1 [mu]m) has been introduced because central (aerosol) residual particles of snow crystals have been observed to be usually larger than that size (e.g., Pruppacher and Klett 1997; Chen et al. 1998; Prenni et al. 2007a; see also Marcolli et al. 2007). Here, Omega^sub X,1,*^, is the component of Omega^sub X^ due to aerosols with diameters between 0.1 and 1 [mu]m in the background-troposphere scenario. Such aerosols caused the observed ice nucleation inside the CFDC. Note that at low freezing fractions (e.g., at warm subzero temperatures) mu^sub X^ [much less than] 1 and n^sub IN,X^ [asymptotically =] (S^sub i^, T) xi (T)(alpha^sub X^n^sub IN,1,*^Omega^sub X,1,*^) x Omega^sub X^. Hence, Eqs. (9)-(10) express the fundamental assumption justified in section 2 that the number concentration of active IN from aerosol particles (larger than 0.1 i[eth]a) in group X is approximately proportional to their surface area.

Values of Omega^sub X^ or dn^sub X^/d logD^sub X^ may be inferred from the predicted mass (and/or number) mixing ratio Q^sub X^ (and/ or n^sub X^) of environmental aerosols in group X, for an assumed form of the aerosol size distribution. Until the cloud-free environment is reached, Omega^sub X^ and Q^sub X^ (and/or nx) are not depleted by ice nucleation.

The term Hx(S^sub i^, T) in Eq. (10) is an empirically determined fraction (0 = H^sub X^ = 1) representing the scarcity of heterogeneous nucleation of ice seen in substantially subsaturated conditions (see section 2). At water saturation, H^sub X^ = 1. In the backgroundtroposphere scenario, alpha^sub X^ is the fractional contribution from aerosol group X to the IN concentration (when H^sub X^ = 1) inferred from CFDC data. The factor xi takes into account the observation that drops containing IN are typically seen not to freeze at temperatures warmer than about -2[degrees]C (e.g., Fig. 9). At temperatures colder than -5[degrees]C and warmer than – 2[degrees]C, xi is assigned values of unity and zero, respectively, with a cubic interpolation (delta; appendix A) in between. Measurements with the CFDC cannot be made at such warm temperatures, as the instrument relies on a gradient of temperature being established between its walls to create supersaturation.

Laboratory observations suggest that when S^sub i^ and T ([degrees]C) are less than critical thresholds (S^sup X^^sub i,0^ and T^sup X^^sub 0^), rates of nucleation of ice are negligible or very low. Consequently, H^sub X^ for X = DM and BC is defined as

… (11)

… (12)

Here, f^sub c^ represents the contribution to H^sub X^ from modes of nucleation (e.g., deposition) in subsaturated conditions giving rise to the CFDC data fitted by Eq. (1). This data was measured at humidities below the onset of homogeneous aerosol freezing. At such humidities H^sub X^ ~ 1/gamma, because the contributions from group X to the IN concentration observed (alpha^sub X^n^sup C^^sub IN,1,*^) and predicted [Eqs. (2) and (9)-(10)] for the backgroundtroposphere scenario in the same aerosol size range (approximately n^sub IN,X,*^ OmegaX,1,*/Omega^sub X,*^) must then be equal. The extra term in Eq. (11) involving S^sub w^, which is the saturation ratio with respect to water, represents enhancement of immersion and condensation freezing at higher humidities approaching water saturation (the jump noted above). Uptake of liquid by soluble coatings on IN particles then reduces the concentration of dissolved solute, making freezing more likely. Here, delta^sup b^^sub a^ provides an interpolation (appendix A) over intervals DeltaT and DeltaS^sub i^ of temperature and saturation ratio with respect to ice, while h^sup x^ is the small fraction to which H^sub X^ is reduced by warming over DeltaT. Figure 3 displays H^sub X^ for dust and black carbon.

The number of crystals generated in a time step Deltat is given by

Deltan^sub i^ = Sigma^sub x^ max(n^sub IN,X^ -n^sub X,a^,(13)

where n^sub X,a^ is the number mixing ratio of IN from group X that has already been activated. When incrementing n^sub i^, by Deltan^sub i^, then n^sub X,a^ is also incremented by Deltan^sub X,a^ = max(n^sub IN,X^ – n^sub X,a^,0) for each group X. In the cloud-free environment, n^sub X,a^ is set to zero in the manner of Cohard and Pinty (2000), while Q^sub X^ (and/or n^sub X^) is reduced to account for previous losses from ice nucleation. This implicitly assumes that insoluble matter from the three basic groups of IN is not mixed together (“internally”) in the same aerosol particle, which seems realistic for dust (e.g., Clarke et al. 2004). Nonetheless, fragments of biogenic material may stick to dust in the troposphere (e.g., Schnell and Vali 1976; Griffin et al. 2001). Insoluble organic and inorganic carbon often tends to be mixed internally; such internal mixing may optionally be treated by modifying Eq. (13) (e.g., increment n^sub X,a^ for each insoluble component of an internal mixture whenever one component nucleates ice).

In summary, Eqs. (1)-(13) predict the change in crystal number Deltan^sub i^ which is the output from the empirical parameterization. The inputs are S^sub i^ and T, while Omega^sub X^ (e.g., inferred from Q^sub X^, n^sub X^), as well as n^sub X,a^ are all both inputs and outputs. The essence of the scheme’s versatile framework lies in Eqs. (8)-(10), which may easily be adapted for extra groups of IN and empirical refinements to parameters (e.g., H^sub X^).

In principle, it would be possible to resolve explicitly the nucleation of ice by IN immersed in cloud liquid (condensation- and immersion-freezing modes) created by the cloud condensation nuclei (CCN) activation of their solute, if “in-cloud” scavenging of IN, involving the removal of their cloud liquid by precipitation, is included (see section 5). The number of immersed (or interstitial), active IN, n^sub 1N,X,imm^ (or n^sub IN,A,int^), could be derived from Eqs. (9)-(10) by replacing ohm^sub X^ with ohm^sub X,imm^, (or ohm^sub X,int^) and applying it to n^sub IN,1,*^ a shift, DeltaT^sub X,imm^, of the freezing temperature of IN between interstitial and immersed states at water saturation (or applying none). However, DeltaT^sub X,imm^ is so small (


Following Chen et al. (1998), DeMott et al. (2003a), and Heintzenberg et al. (1996), and in view of the observed composition of IN from heterogeneous crystals collected in six field campaigns shown in Fig. 2, we assign alpha^sub DM^ = 2/3, requiring that alpha^sub BC^ + alpha^sub O^ = 1/3, the fraction for all carbonaceous aerosol. The relative assignments of values for alpha^sub BC^ and alpha^sub O^ are not evident from such studies, as few data were collected to distinguish the nature of the carbonaceous aerosols. Although the fractions alpha^sub X^ in Eq. (10) might be expected to depend on temperature and humidity, in the absence of unequivocal observational data they are assumed to have constant values. Future advances in knowledge may change this (section 7).

For group O, INA strains of bacteria have been selected as partially representative (see section 2). Cells belonging to INA strains may form only a small fraction, f^sub INA^ ~ 1% (Lindemann et al. 1982), of all bacteria in the troposphere. In view of the wide range of available estimates (~10-1000 L^sup -1^; appendix C), the concentration of all bacterial cells in the backgroundtroposphere scenario is assigned as N^sub bac^ = 100 L^sup -1^. The freezing fraction of cells belonging to INA strains is assumed to be about 10^sup -4^ at -5[degrees]C (Gross et al. 1983; Hirano et al. 1985) and about g^sub fr,30^ = 0.1 at -30[degrees]C by extrapolation with observed temperature dependencies (Vali et al. 1976; Gross et al. 1983). Impurities in water used for diluting cultures contributed to a lack of observational data at colder temperatures.

The estimated number of crystals nucleated by bacteria in the background-troposphere scenario at water saturation and – 30[degrees]C is g^sub fr,30^/f^sub INA^N^sub bac^ ~ 0.1 L^sup -1^. In view of CFDC observations in such conditions [Eq. (3)], INA bacteria may contribute about 0.03 to alpha^sub O^. Yet there is significant uncertainty in N^sub bac^ (appendix C), f^sub INA^, and alpha^sub O^. Also, there is uncertainty about how representative Pseudomonas syringae (Ps) is for the nucleation by bacteria in the troposphere. Other species with INA strains (e.g., Erwinia herbicola) have been seen to be sometimes more significant in ice nucleation than Ps (e.g., Lindow et al. 1978; Kaneda 1986).

In nature, there may be many contributions to alpha^sub O^ from nucleation of ice by various types of biogenic and/ or nonbiogenic insoluble organic aerosol. Of the residual aerosol particles from heterogeneous ice crystals sampled from wave clouds, 14% was found to be composed of organic carbon and not sulfate (Targino et al. 2006). Other analyses (e.g., DeMott et al. 2003a; Cziczo et al. 2004; Richardson et al. 2007) have revealed that about 13%-20% of residual particles are dominated by either sulfate or organic carbonaceous matter. This organic/sulfate residual material might not have been responsible for nucleation of the ice sampled. Also, sulfate can crystallize and then nucleate ice (e.g., Abbatt et al. 2006). In view of the paucity of observational data, a parsimonious assumption is that up to about half of residual particles classified as partially organic may have acted as insoluble organic IN. That would suggest alpha^sub O^ is between 0.03 and 0.1, approximately. An intermediate value of alpha^sub O^ = 0.06 is assumed here, so that alpha^sub BC^ = 1/3 – 0.06 from the constraint noted above. This value of a0 is much larger than the fraction (0.005) of oxalate- containing heterogeneous crystals observed in CRYSTAL-FACE (Cziczo et al. 2004).

Origins of parameters for the factor H^sub x^ and most other parameters in Eqs. (10)-(12) are summarized in Table 1 (see also appendix A). Soot in the free troposphere is assumed to be coated with so much soluble material that its onset of nucleation occurs by condensation and immersion freezing. Experimental data in subsaturated conditions are scarce (e.g., Levin et al. 1980; Diehl et al. 2001), so it is assumed that H^sub O^ = H^sub BC^ always. Values of ohm^sub x,1,*^. (Table 2) are partially constrained by IMPROVE measurements of the mass loading of aerosols smaller than 2.5 [mu]m in (dry) diameter (Q^sub X,2.5,*^; appendixes A and D) made during observing periods of the CFDC in the backgroundtroposphere scenario. For the carbanaceous aerosol, recent tropospheric observations of average sizes and spectral widths of aerosol size distributions and bulk densities (Table 3) are invoked to infer ohm^sub X,1,*^ from Q^sub X,2.5,*^

b. Conventional and inside-out contact freezing

The fundamental assumption here is that each IN particle can nucleate ice either by contact freezing or by immersion or condensation freezing, depending on local conditions of temperature, humidity, and availability of supercooled liquid. Shaw et al. (2005) observed that a given IN particle of about 0.1 mm in size has a freezing temperature for the surface mode (contact nucleation) that is DeltaT^sub CIN^ [asymptotically =] 4.5[degrees]C higher than for the bulk-water mode (immersion and condensation freezing), irrespective of the following factors: 1) whether the contact-IN (CIN) approaches the air-water interface from inside or outside the liquid; 2) the chemical composition of IN particles examined, which were glassy and/or crystalline; and 3) their size (larger than 0.1 mm; R. Shaw 2006, personal communication). Thus, it is assumed that the same temperature difference applies to atmospheric IN. Yet there is Significant experimental uncertainty associated with this assumption (e.g., DeMott 1995).

The number mixing ratio of potentially active, interstitial CIN n^sub X,cn^ is then simply related to that of immersion- and condensation-freezing IN [Eqs. (2) and (3)]:

… (14)

Here, ohm^sub X,int^ is the component of ohm^sub X^ for interstitial IN. The fundamental behavior of water molecules near the air-water interface is known strongly to favor freezing there (e.g., Vrbka and Jungwirth 2006). This is consistent with the above observation that the exact nature of the IN is not the first-order dependence of DeltaT^sub CIN^.

For an interstitial CIN to nucleate an ice crystal, it must collide with a supercooled cloud droplet. Sources for the rate of CIN-droplet collisions arise from the forces of Brownian diffusion (Br), thermophoresis (Th), and diffusiophoresis (Di) acting on the CIN aerosol. Electrophoretic forces may also be included, if deemed significant. The increment of crystal number mixing ratio n^sub i^ due to contact nucleation in time step Deltat is given by …. (15)

All three source terms are described by Young (1974), Cotton et al. (1986), Pruppacher and Klett (1997, 724728), and Ovtchinnikov and Kogan (2000). The thermophoretic source, which usually prevails, is a function of the in-cloud supersaturation. These sources are a function of max (n^sub X,cn^ – n^sub X,a^ 0). Contact nucleation is performed separately for each jth size bin of cloud droplets.

Thermal conductivities of CIN are K^sub aero,X^ = 0.25,4.2, and 0.2 W m^sup -1^ K^sup -1^ for X = DM, BC, and O (Seinfeld and Pandis 1998; Ovtchinnikov and Kogan 2000). Their mean radii (R^sub aero,X^) may De prescribed (appendix A) or predicted.

The exotic mechanism of inside-out contact nucleation (Durant and Shaw 2005; Shaw et al. 2005) may also be represented, if in-cloud scavenging of insoluble aerosol is treated (see section 5). An extra source then arises:

…. (16)

Here, n^sub X,a,imm^ and ohm^sub X,imm^ are components of n^sub X,a^ and ohm^sub X^, respectively, for IN immersed in cloud liquid. Then n^sub X,a,imm^ [asymptotically =] n^sub X,a^ohm^sup X,imm^/ S^sub X^, where ohm^sub X^ = ohm^sub X,int^ + ohm^sub X,imm^. All CIN particles immersed inside evaporating cloud droplets are assumed to be brought into contact, eventually, with their vanishing liquid surfaces. Here, |Deltan^sub w^| is the number of supercooled cloud droplets depleted by total evaporation (e.g., Phillips et al. 2007) in Deltat. Also, n^sub w^ is their number at the start of the time step. Finally, there is the option of incrementing n^sub X,a^ by the number of CIN lost by freezing during Deltat.

4. Results from empirical parameterization

a. Comparison with observational data for dust, black carbon, and insoluble organic IN

Available laboratory data are compared here with the empirical parameterization for each of the three basic groups of insoluble aerosol. All laboratory data concern specific types of artificial aerosol, either manufactured or sampled from beneath the surface of the earth’s crust. By contrast, the scheme is mostly based on a very different set of observations (from CFDC and crystal residual analysis) of the diverse mixture of natural atmospheric IN. Even within the same group of IN, vast differences in nucleating ability are seen for different aerosol samples (section 7). Consequently, exact agreement in the comparison is not to be expected, and discrepancies do not necessarily imply inaccuracy of the scheme.

Only a limited subset of this laboratory data has informed selection of the scheme’s parameter values during design (see section 3). In the following figures, the few data points already used to determine such parameters are plotted differently (open symbols) from laboratory data, which are independent (plotted with filled symbols or other line styles). Temperature and humidity were varied as prescribed inputs to the empirical parameterization. For predictions of condensation and immersion freezing, the prescribed humidity corresponded to exact water saturation at all temperatures [S^sub i^ = S^sup w^^sub i^ (T)]. For nucleation (e.g., by deposition) in subsaturated conditions, the vapor mixing ratio was prescribed as Q^sub v^ = Q^sub s,i^ + (Q^sub s,w^ – Q^sub s,i^)(2/ 3) for S^sub w^ less than S^sub w,0^. Formulas for vapor pressures at water and ice saturation are from Murphy and Koop (2005).

The frozen (or freezing) fraction for group X is defined here as the ratio between the numbers of its crystals nucleated and of its insoluble aerosols (per kilogram of air). It is proportional to the average surface area per insoluble aerosol particle [ohm^sub X^/ n^sub X^; see Eq. (10)]. Constant values from tropospheric observations are assumed for the average size and spectral width of (lognormal) aerosol size distributions (Table 3). Thus, the average surface area per particle here is independent of the group’s aerosol concentration. Tropospheric values are used here because the scheme and many of the observations are intended to portray atmospheric nucleation. Because the predicted freezing fraction is independent of aerosol concentration, the comparison is robust for a wide range of aerosol loadings.

Uncertainty in the predicted freezing fraction arises from that of the scheme’s parameters (e.g., alpha^sub X^) and from natural variability of average surface area per particle (estimated from Table 3). In view of nonlinear dependencies of the predicted freezing fraction on some of the uncertain parameters, a statistical model has been used to estimate its relative error. The model has seven uncertain input parameters for X = DM [lnD^sub g,DM^ and sigma^sub DM^ for both modes; gamma, ln(n^sup C^^sub IN,1,*^/ ohm^sub DM,1,*^), and lnalpha^sub DM^], and eight for X = BC, O (lnD^sub g,X^ and sigma^sub X^ twice; rho^sub X^, gamma, lnQ^sub X,2.5,*^, and lnalpha^sub X^ all assumed to be independent. For each uncertain input parameter, 10^sup 6^ synthetic values were generated by drawing random perturbations from an independent normal distribution and adding them to the observed mean value. The variance of the predicted freezing fraction was estimated from the variance of the sample of 106 frozen fractions evaluated from the synthetic sets of values of input parameters. For dust, n^sup C^^sub IN,1,*^/ohm^sub DM,1,*^ is one such parameter with a relative error (-55% to 120%) at any given value of zeta being assumed to arise from sampling uncertainty due to the natural variability of zeta in the local troposphere during INSPECT-1. This relative error corresponds to a 95% confidence interval for the average value of zeta there and is derived from the standard deviation of the sample of daily values of ln[zeta(s^sub i^)] ln[[left angle bracket]zeta[right angle bracket](s^sub i^)], where TY denotes the line of best fit plotted on Fig. 1.Af distribution of their sample mean is assumed. Finally, error lines for the prediction are plotted in the following figures.


Field et al. (2006) have measured the frozen fractions of dust sampled from the earth’s crust in Asian and Saharan desert regions (Mohler et al. 2006) in the AIDA aerosol chamber between – 20[degrees] and -60[degrees]C. Figure 4 shows that the frozen fraction predicted by the parameterization has the same order of magnitude as most of the observed fractions at AIDA for immersion and condensation freezing (S^sub i^ > S^sup w^^sub i^ – 0.15). The observed fractions are consistently higher (by about one standard deviation) than the scheme’s prediction, possibly because the artificial dust studied in the AIDA chamber was sampled from the earth’s crust and not the atmosphere (see section 7). The low- average diameter of their artificial dust (about 400 nm) has prevented this difference from being even larger.

Figure 5 shows a comparison with corresponding AIDA observations for nucleation in subsaturated conditions (S^sub i^ = S^sup w^^sub i^ – 0.15; Field et al. 2006). The scheme’s prediction has a similar order of magnitude compared to the AIDA data, which again is significantly higher (by about two standard deviations). The prediction is slightly higher than for Asian aeolian dust sampled from the earth’s crust (Archuleta et al. 2005), because of the latter’s very low size (200 nm). By design, in warm subsaturated conditions (S^sub i^

Figure 6 shows the frozen fraction of dust as a function of increasing humidity at a constant temperature (-50[degrees], then again at -60[degrees]C). The prediction is compared with laboratory observations of Asian dust sampled from the earth’s crust (Archuleta et al. 2005). Their observed saturation ratio, and that of giant dust (larger than 10 [mu]m) immersed in sulfate solution (Zuberi et al. 2002), differ by about one and three standard deviations, respectively, from the prediction, which is intermediate between both, partly because of differences in size compared to atmospheric dust. Moreover, the predicted freezing onset occurs at a saturation ratio that is only about 0.15 lower than that observed in situ for cirrus formation in the interhemispheric differences in cirrus properties from anthropogenic emissions project (INCA, the United Kingdom; Haag et al. 2003). Dust is assumed to be the ice nucleant responsible for their in situ observations. Agreement with cirrus observations from FIRE-II (Heymsfield and Miloshevich 1995) is by design.

Finally, the predicted active fraction agrees with AIDA observations at -50[degrees]C of dust sampled from the earth’s crust (Mohler et al. 2006), also shown in Fig. 6, though it is slightly lower (by about one standard deviation mostly). At -60[degrees]C, the prediction becomes similar to, though slightly lower than, the freezing fraction seen by Archuleta et al. (2005) when the latter is adjusted to correspond to a more realistic dust size (e.g., 400 nm, as studied by Mohler et al., would correspond to a freezing fraction of 0.04 at S^sub i^ = 1.36), assuming proportionality to surface area per particle (see section 2). The freezing fraction seen by Mohler et al. is much higher-by about one to two orders of magnitude- than either the prediction or the data from Archuleta et al. at most observed humidities at that temperature. There would still be a discrepancy between both observational datasets even after the same type of adjustment of the data at -60[degrees]C from Archuleta et al. (2005) to a more realistic size. More experiments are needed to clarify reasons for this discrepancy between the datasets of Archuleta et al. and Mohler et al. In summary, differences between the prediction and the various sources of observational data are less than or comparable to differences between these sources of data.


Figure 7 shows a comparison of the parameterization with various laboratory observations for condensation and immersion freezing of artificial soot, coated with sulfate. The prediction from the empirical parameterization agrees with recent observations by DeMott et al. (1999) and Mohler et al. (2005a), though it is slightly lower than the former, which has the more realistic size. Median diameters were about 240 and 90 nm, respectively. The freezing fraction for acetylene soot (DeMott 1990) is significantly higher (by about one standard deviation) than the other observations and the prediction.

Figure 8 shows the relative humidity for the freezing onset of black carbon predicted by the parameterization at a freezing fraction of 0.1% and observed in two laboratory studies (DeMott et al. 1999; Mohler et al. 2005a). Of course, agreement with one of the two datasets is by design (H^sub x^). This is not the case for the other (Mohler et al. 2005a), which agrees with the prediction.


Figure 9 shows the predicted frozen fraction for all insoluble organic particles, compared with various laboratory studies for immersion and condensation freezing by bacteria. The predicted frozen fraction is within the range of values inferred from laboratory data not used to constrain the scheme. Their spread is great, reflecting a lognormal distribution of freezing fractions seen among individual INA strains in nature (Hirano et al. 1985). In summary, available experimental data on the freezing fraction of bacteria are consistent with the prediction at warm subzero temperatures.

b. Comparison with existing schemes

The empirical parameterization was compared with four other schemes that predict heterogeneous freezing, by Lohmann and Diehl (2006, referred to as L-D), Liu and Penner (2005, L-P), Khvorostyanov and Curry (2004, K-C) and Meyers et al. (1992, MDC). The MDC scheme has no dependence on aerosol chemistry or amounts, and here it is applied only at temperatures warmer than – 30[degrees]C (e.g., Phillips et al. 2005).

For all schemes, water saturation was artificially maintained at 500 mb during cooling to -70[degrees]C. All contact and nonheterogeneous nucleation of ice were prohibited. Note that the test is idealized in the sense that vapor growth of crystals does not affect the humidity. Nevertheless, it mimics typical conditions in a natural mixed-phase cloud containing supercooled cloud liquid, which can only persist if the humidity is close to water saturation.

Aerosol concentrations were prescribed with typical values observed by Clarke et al. (2007) in pollution and biomass-burning plumes during Intercontinental Chemical Transport Experiment/ International Consortium for Atmospheric Research on Transport and Transformation (INTEX/ICARTT) in 2004 (Q^sub BC,2.5^ = 3 x 10^sup – 10^; Q^sub O,2.5^ = 3 x 10^sup -9^ kg kg^sup -1^; Q^sub DM,2.5^ = 2.9 x 10^sup -10^ kg kg^sup -1^ was arbitrarily prescribed). Size distribution parameters specified in Table 3 were assumed, to infer Omega^sub X^/n^sub X^ and freezing fractions.

Figure 10 shows the predicted number concentration of crystals nucleated by various IN species. At temperatures colder than – 30[degrees]C, the empirical parameterization as well as the MDC and L-P schemes all predict similar orders of magnitude for the total crystal concentration (about 10-100 L^sup -1^). All five schemes differ by up to almost 5 orders of magnitude in their predicted concentrations (and freezing fractions) at temperatures of about – 30[degrees]C. The K-C and L-D schemes predict extremely high concentrations, with absolutely all insoluble aerosols in each represented group being predicted to freeze at all temperatures colder than about -20[degrees]C. (For the L-D scheme, the freezing fraction for all IN is less than unity and the peak IN concentration is less than for the K-C scheme, because no freezing by insoluble organic IN is represented by it). The K-C scheme neglects the probability distribution of contact angles among active sites on the IN surface, which in reality determines that of the nucleation efficiencies among IN, so the singular character of IN is not represented naturally (see sections 1 and 2). The L-D scheme is based on data from artificial drops that may each have contained multiple IN (see section 6) and neglects a dependence of nucleating ability on IN size, in contrast to the empirical parameterization.

The empirical parameterization predicts a concentration of heterogeneous crystals of 29 L^sup -1^ at -30[degrees]C, with 40% from insoluble organic (X = O) IN, 15% from dust (X = DM), and 45% from soot (X = BC). This composition of IN differs from that seen in the background troposphere (Fig. 2) because an aerosol mixture for the above plumes has been assumed. The L-P scheme also predicts that most crystals originate from soot.

Also shown in Fig. 10 is the freezing fraction from all schemes. Comparison with all observations (Figs. 4, 7) of condensation and immersion freezing by dust (Archuleta et al. 2005; Field et al. 2006) and soot (DeMott 1990; DeMott et al. 1999; Mohler et al. 2005a) generally shows better agreement for the empirical parameterization than for alternative schemes at most subzero temperatures. Its prediction is mostly intermediate, compared to other schemes. The empirical parameterization reproduces the qualitative trend of a smooth increase in freezing fraction during extensive supercooling, as seen in laboratory observations at water saturation. Moreover, it represents insoluble organic carbonaceous IN.

5. Implementation of empirical parameterization in a large-scale or cloud model

In the troposphere, insoluble IN material is mixed with various soluble compounds inside the same aerosol particle (e.g., Chen et al. 1998; Clarke et al. 2004). Consequently, in-cloud scavenging of a typical IN particle in an updraft may occur when its soluble material activates a cloud droplet, which is then removed by precipitation.

Optionally, such scavenging of IN can be represented by assuming that an IN particle’s soluble material constitutes a certain fraction of its total mass (e.g., appendix B). If the supersaturation reaches a critical value, then the soluble material activates as a cloud droplet and the IN material is assumed to become immersed in it. The critical supersaturation emerges from kappa-Kohler theory (Fetters and Kreidenweis 2007, Eqs. (7) and (10) therein; see also Snider et al. 2003). Each group X of IN is then partitioned into prognostic components that are interstitial and immersed in cloud liquid. Extra scalars constraining the IN size distributions are the number mixing ratio of interstitial IN lost by becoming immersed in cloud liquid without necessarily freezing (n^sub x,a,liq^), and the actual number mixing ratio of immersed IN (n^sub x,imm^). A temporary grid of size bins for IN is applied. The immersed and interstitial IN size distributions have a cutoff size determined by n^sub X,a,liq^ and Q^sub X^. Inside clouds, Q^sub X^ is hypothetical insofar as it is unaltered there, either by immersion of IN in cloud droplets or by ice nucleation. The fraction of clouddroplet number accreted onto precipitation equals the fraction removed of n^sub X,imm^ and n^sub X,a,imm^, while n^sub X,a^ is also reduced.

Inclusion of in-cloud scavenging of IN means that heterogeneous freezing of rain may be represented as follows: accretion of cloud liquid by rain provides the source of mass of IN contained inside rain, Q^sub X,rain^, which falls with the rain mass. A temporary grid of size bins discretizes the raindrop size distribution. For each bin with mass mixing ratio dQ^sub r^ of rain, the number of the rain’s IN particles that activate in Deltat given by

… (17)

For that size bin, dOmega^sub X,rain^ = Omega^sub X,rain^dQ^sub r^/Q^sub r^ is the surface area mixing ratio of IN contained inside its raindrops and v^sub t^ is their fall speed, while w is the vertical air velocity. Optionally, Omega^sub X,rain^ [asymptotically =] Omega^sub X,imm^Q^sub r^/Q^sub w^. The integer number of active IN in any drop in the bin is assumed to follow a Poisson distribution and the fraction frozen of the bin’s drops is then [1 - ^sub exp^(-d[mu])] (e.g., Phillips et al. 2001), where d[mu] is the average number of IN activated per drop in Deltat;.

A challenge when implementing the parameterization in a model is that the supersaturation with respect to ice is an input and must be predicted. A method for predicting the supersaturation was given by Phillips et al. (2007). Ascent is the source of supersaturation with respect to water in liquid or mixed phase clouds [e.g., Howell 1949; Mordy 1959; Rogers and Yau 1989, Eq. (7.29), p. 110, therein] and with respect to ice in ice clouds. Consequently, a representation of the vertical velocity on the cloud scale is required in general for prediction of the supersaturation. Several ways of representing the subgrid-scale variability of vertical velocity have been proposed (e.g., Donner 1993; Chuang and Penner 1995; Lohmann et al. 1999) for models that cannot resolve convective elements. For global models, evaluation of the supersaturation with which to represent nucleation remains one of the most difficult problems because of the multiscale nature of mesoscale dynamics.

Finally, if the model in which the scheme is applied cannot represent accurately the average sizes of insoluble aerosol, then it is best to prescribe values of parameters of the aerosol size distribution according to Table 3. Naturally, heterogeneous ice nucleation is one of many possible avenues (e.g., Hallett and Mossop 1974; Heymsfield et al. 2005) for crystal formation in the atmosphere (e.g., Phillips et al. 2007). They must all be included in the large-scale or cloud model. 6. Discussion of the scheme’s accuracy

In construction of the empirical parameterization, coincident field measurements of aerosol particles (from IMPROVE), their size distributions, and IN (from INSPECT-1; CFDC data between – 40[degrees] and -62[degrees]C, from ice saturation to the onset of homogeneous freezing) were combined. They have been extrapolated to the wider range of conditions found throughout the troposphere (temperatures of 0[degrees] to -70[degrees]C or colder, humidities from ice to water saturation, and all conditions of aerosol loading, chemistry, and size) as follows: first, a reference activity spectrum has been fitted to the CFDC data and extrapolated to warmer temperatures using other CFDC datasets (Meyers et al. 1992). The extended set of all CFDC data used here ranges from -7[degrees] to – 62[degrees]C. Second, extra observations of nucleation in subsaturated conditions (H^sub x^) have been combined with the spectrum. Third, the observed composition of residual IN from heterogeneous crystals from the troposphere has determined contributions (a^sub X^) from aerosol groups. Finally, other field observations (section 2) justify extrapolation of the spectrum into any scenario of aerosol species.

There is some uncertainty in these steps of extrapolation. For instance, when determining H^sub x^, observations of artificial aerosol have necessarily been used. Data about the role of soluble coatings on IN are scarce, which limits the accuracy of H^sub x^. Dependencies on temperature of empirical parameters (e.g., alpha^sub X^, gamma) are uncertain. Although such omissions may introduce limited biases, they are not overly serious for three reasons. First, for all three basic groups of IN, differences between the scheme’s prediction and independent observational data not used to construct it are either comparable with or less than differences among the sources of this data. The parameterization has better agreement with these observations than with alternative schemes. second, each step is justified with observations of IN. Lastly, a flexible framework [Eqs. (8)-(10)] expresses species’ IN activity in terms of observable quantities. Any biases may be reduced as observations improve (see section 7).

Limited inconsistencies between the scheme and laboratory observations may be partly due to 1) crystallographic, chemical, and size differences of the insoluble IN material between the artificial laboratory samples and the real atmosphere; 2) differences in composition and amount of soluble material mixed inside individual IN particles between laboratory samples and the atmosphere; 3) the limited sampling of the troposphere for data constraining the scheme; and 4) any limitations of the residual analysis and CFDC.

The CFDC has at least three potential limitations, though they do not cause very great inaccuracy here. First, the CFDC can measure neither contact nucleation nor all of the immersion and condensation freezing at temperatures colder than -40[degrees]C at humidities approaching water saturation. The latter is not such a problem because homogeneous aerosol freezing is prolific in such conditions, producing many more crystals than heterogeneous nucleation typically. second, the IN activity of aerosols larger than 1-1.5 [mu]m was not measured in datasets used here. Also, the optical detection technique presently used, requiring growth of crystals beyond about 1 [mu]m, may limit the ability to detect nucleated ice at very cold temperatures. Both detection issues may soon be resolved.

Last, the CFDC cannot resolve the separate contributions to the total activity from IN that are interstitial and immersed in cloud liquid at a given (positive) supersaturation (s^sub w^) with respect to water imposed within it. However, this is not a great limitation because whether an IN is interstitial or immersed does not affect its heterogeneous freezing temperature very greatly for estimated soluble coatings on IN at water saturation (appendix B). Essentially, uptake of liquid water by an interstitial IN dilutes the dissolved solute, minimizing its depression of the freezing temperature. This is why the sensitivity to positive s^sub w^ above water saturation is often seen to be quite limited (e.g., Al-Naimi and Saunders 1985; Rosinski and Morgan 1988; Meyers et al. 1992; Rogers et al. 2001b). Measured IN concentrations only changed by a factor of 2 when s^sub w^ increased from zero to 5% in Arctic flights (Rogers et al. 2001b); this must have immersed the IN inside the CFDC (see appendix B; Rogers and Yau 1989, p. 89). Hence, the ice-nucleating behavior of many interstitial and immersed IN seems similar at water saturation. Yet sometimes a higher sensitivity to S14, inside the CFDC is seen (DeMott et al. 1998; Rogers et al. 1998), for example, due to poor soluble coatings.

The CFDC does not measure the volume of cloud droplets that immerse IN but does not need to, so this is not a limitation. The volume dependence of freezing fractions of drops seen in early laboratory studies (e.g., Bigg 1953) arose because drops artificially sampled from bulk water contain multiple IN with a spectrum of freezing temperatures (Langham and Mason 1958; Vali 1971). The larger the artificial drop, the more numerous were its IN and the warmer was the maximum freezing temperature among its multiple IN, which is the drop’s freezing temperature. Consequently, such volume dependence is not expected for most cloud droplets in nature, which each form by condensation onto only one aerosol particle.

At low subsaturated humidities, the parameterization represen