JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D3, 4026, 10.1029/2001JD000483, 2002
A parameterization of aerosol activation 3. Sectional representation Hayder Abdul-Razzak Mechanical Engineering Department, Texas A&M University-Kingsville, Kingsville, Texas, USA
Steven J. Ghan Pacific Northwest National Laboratory, Richland, Washington, USA Received 3 February 2001; revised 16 July 2001; accepted 20 July 2001; published 8 February 2002.
 A parameterization of the activation of a lognormal size distribution of aerosols to form cloud droplets is extended to a sectional representation of the aerosol size distribution. For each section, number concentration and chemical composition are uniform functions of particle radius. The parameterization is applied by calculating an effective critical supersaturation of all sections from which the maximum supersaturation of the air parcel is calculated using the previously derived parameterization. The Ko¨hler theory is used to relate the aerosol size distribution and composition to the number activated for each section as a function of maximum supersaturation. For most cases, parametric results are within 10% of those obtained by detailed numerical computations for both idealized and measured aerosol size distributions. The parameterization thus provides an accurate method of treating the activation process for models that use a sectional representation of the aerosol size distribution. INDEX TERMS: 0305 Atmospheric Composition and Structure: Aerosols and particles (0345, 4801); 0320 Atmospheric Composition and Structure: Cloud physics and chemistry; 0345 Atmospheric Composition and Structure: Pollution –urban and regional (0305); 0322 Atmospheric Composition and Structure: Constituent sources and sinks; KEYWORDS: aerosol, activation, cloud, parameterization, sectional
1. Introduction  Aerosol activation is the process by which aerosol particles grow by water condensation to form cloud droplets. It is called activation because it involves a threshold wet particle size, above which particles will continue to grow spontaneously unless the supersaturation with respect to water decreases fast enough to deactivate the particles [Nenes et al., 2001].  The aerosol activation process is important for several reasons. First, it determines the formation of cloud droplets and hence affects droplet number concentration, droplet effective radius, cloud albedo, and (via its influence on the coalescence of droplets to form precipitation) cloud liquid water content and cloud lifetime; accurate parameterizations of the process are therefore needed for estimates of the so-called indirect forcing by anthropogenic aerosols through their role as cloud condensation nuclei. Second, by separating particles into those activated and those not activated, the aerosol activation process determines which particles can increase their solute mass through aqueous chemistry in cloud droplets and also which particles are removed from the atmosphere through precipitation of the cloud droplets; accurate parameterizations of the process are therefore needed for models of cloud processing of aerosols and nucleation scavenging of aerosols.  Representing the effects of activation on aerosol particles, cloud droplet number concentration, cloud liquid water content, and cloud albedo is one of the most challenging problems facing the atmospheric sciences community today. The problem is difficult enough for a single cloud or cloud system, but is especially difficult because of the need to treat it for the whole Earth so that the potential global climate impacts can be estimated. Any treatment of these effects in a model of the global climate system must be computationally affordable, yet must address the essential Copyright 2002 by the American Geophysical Union. 0148-0227/02/2001JD000483
physics of the problem. It is therefore not enough to apply all we know about the physics to a climate model; the knowledge must be condensed enough to permit multidecade global climate simulations.  The effects of aerosols on clouds can be separated into two processes, namely, the effects of the aerosol particles on cloud droplet number, and the effects of droplet number on cloud liquid water content and cloud albedo. This paper is concerned only with the first effect. In particular, we focus on the activation of aerosols to form cloud droplets (droplet number concentration is also influenced by the coalescence process and by evaporation, which require separate treatments).  Our knowledge about the aerosol activation process is summarized in text [Pruppacher and Klett, 1997; Seinfeld and Pandis, 1998] and in numerical models having detailed treatments of aerosol activation [Jensen and Charlson, 1984; Flossmann et al., 1985; Ahr et al., 1989; Liu and Wang, 1996; Abdul-Razzak et al., 1998]. However, the treatment in these numerical models is too computationally demanding to be applied to global climate models. Condensed parameterizations of the aerosol activation process are therefore required.  All previous parameterizations have relied upon simplified representations of the aerosol size distribution. The earliest parameterizations [Squires, 1958; Twomey, 1959; Squires and Twomey, 1960] implicitly assumed the aerosol size distribution follows a power law [Jiusto and Lala, 1981; Herbert, 1986]. More recent parameterizations have assumed a single mode [Ghan et al., 1993; Abdul-Razzak et al., 1998] or multiple mode [Ghan et al., 1995; Abdul-Razzak and Ghan, 2000] lognormal size distribution.  Although parameterizations based on such simplified aerosol size distributions accurately predict droplet nucleation when the aerosol size distribution can be accurately approximated by the simplified representation, they cannot be expected to perform well when the aerosol size distribution is very different from the idealized form. In addition, although existing parameterizations
ABDUL-RAZZAK AND GHAN: SECTIONAL AEROSOL ACTIVATION
can predict the impact of nucleation on the total number and mass concentrations of aerosols, they cannot predict changes in details of the size distribution because the size distribution is constrained by the simplified representation. In particular, unless many narrow lognormal modes are used, the parameterizations fail to predict the sharp transition between particles large enough and those too small to be activated (Figure 1), and the influence of the transition on subsequent aerosol activation. Resolving the transition is important within clouds, which contain both activated and interstitial particles.  Fortunately, the aerosol dynamics modeling community has developed a much more general representation of the aerosol size distribution that has the potential to resolve such sharp transitions. The so-called sectional representation of the aerosol predicts aerosol number concentration for each of an arbitrary number (typically 10 – 100) of size sections. Treatments of many aerosol processes (nucleation, coagulation, condensation, impaction scavenging, gravitational settling, and dry deposition) have been developed for sectional models [Zhang et al., 1999, and references therein]. However, no parameterization of aerosol activation has yet been developed for a sectional model of the aerosol size distribution. This limits the application of sectional models to cloud-free conditions (which obviously precludes their application to estimates of effects of aerosols on clouds or vice versa), or requires a computationally intensive explicit treatment of the activation process [Russell and Seinfeld, 1998] (which precludes their application to multiyear global simulations). What is clearly needed is an aerosol activation parameterization that is applicable to a sectional representation of the aerosol size distribution.  In this paper we extend our previously developed aerosol activation parameterization for a lognormal representation of the aerosol size distribution [Abdul-Razzak et al., 1998] (hereafter denoted part 1) to a sectional representation. Given the updraft velocity and the size, number concentration, and composition of each size section, the parameterization predicts the maximum supersaturation of the rising air parcel and the number and mass activated for each size section.  In the following section, the procedure by which we can apply the parameterization of part 1 to sectional representation is described. The evaluation of this procedure is presented in section 3. Concluding remarks and summary of results are outlined in section 4.
2. Parameterization  The parameterization of the single mode version (part 1) relates the maximum supersaturation Smax of the rising air parcel to four dimensionless parameters as follows: Sm
Smax ¼ f ðs Þ
3=2 V h
þ g ðsÞ
3=4 1=2 ;
where f ðsÞ 0:5 exp 2:5 ln2 s
g ðsÞ 1 þ 0:25 lns
are functions of the standard deviation s of the lognormal size distribution of the aerosol particles, 2 A 3=2 Sm pﬃﬃﬃ B 3am
Figure 1. The number fraction of aerosol particles activated for each section from a numerical simulation with 100 bins (diamond) and from the sectional parameterization for 10 bins (square). The updraft velocity is 1 m s1, the aerosol size distribution is lognormal with a number concentration of 1000 cm3, a geometric mean radius of 0.05 mm and a standard deviation of 2, the temperature is 15C, and the pressure is 90,000 Pa.
is the critical supersaturation of a particle with a dry radius equal to the geometric mean dry radius am of the size distribution, and 2A aV 1=2 z 3 G
ðaV =GÞ3=2 ; 2prw gN
where A accounts for surface tension effects in the Ko¨hler equilibrium equation (see part 1), B is the hygroscopicity, V is the updraft velocity, G accounts for diffusion of heat and moisture to the particle, rw is the density of water, N is the total aerosol number concentration, and g and a are coefficients in the supersaturation balance equation. The hygroscopicity, defined
vfMw ra Ma rw
depends on particle composition through its dependence on the number of ions n, the osmotic coefficient f, the aerosol molecular weight Ma, and the aerosol material density ra. Note that f (s) has been modified by Abdul-Razzak and Ghan .  For a sectional representation of the size distribution one might consider simply applying the Abdul-Razzak and Ghan  multimodal parameterization to a sectional representation. In the multimodal parameterization each mode is assumed to be a lognormal distribution. To adapt it to a sectional representation, one would have to choose an appropriate geometric standard deviation for each mode. However, such a treatment is inconsistent with a sectional representation, which typically assumes the number distribution is uniform rather than gaussian in log radius within each section, with no overlap of sections. We have found that a more accurate parameterization for the activation of a sectional distribution of aerosols is to set the standard deviation in equation (1) to a value of one and replace the critical super-
ABDUL-RAZZAK AND GHAN: SECTIONAL AEROSOL ACTIVATION
Mi mi ;
where the number and mass fraction activated for each section, ni and mi, are given by ni ¼ mi ¼ 0 ni ¼
Figure 2. The number fraction of aerosol particles activated as a function of geometric mean radius for an updraft velocity of 0.5 m s1, a number concentration of 1000 cm3, and a standard deviation of 2 at a temperature of 15C and a pressure of 90,000 Pa. The solid line represents single mode parameterization described in part 1; S10 (triangle) and S30 (square) are sectional results for 10 and 30 sections, respectively; N100 (cross) is numerical result with 100 bins. saturation Sm by an effective critical supersaturation that is calculated from I P
. 2= ; I P Ni Si 3
where I is the number of sections, Si is the critical supersaturation at the middle of section i, and Ni is the number of particles in each section. We cannot offer a rigorous physical basis for (8), but note that (8) provides a crude estimate of Sm2/3 via (4); other equally plausible formulations for Se2/3 were tested but did not perform as well as (8). Note that because according to (4) the critical supersaturation depends on the hygroscopicity and hence the particle composition, each section in (8) can have a different composition.  Using the effective critical supersaturation and assuming s = 1, the maximum supersaturation of the air parcel can be calculated as follows: Se Smax ¼ : 3=2 2 3=4 1=2 Se V 0:5 h þ hþ3z
Note that assuming s = 1 implies the sections are delta functions. This is consistent with the treatment of activation in numerical models, which consider the growth of only the central size in each section. However, for a coarse representation of the aerosol size distribution (with, for example, 10 sections) it is necessary to treat the size distribution within each section as uniform so that the number activated is a continuous function of updraft velocity. This treatment is described next.  Once the maximum supersaturation of the air parcel is determined, the activation of each section is determined by comparing Smax with Sil and Siu (the lower and upper critical supersaturation bounds of the section) and by comparing the dry particle radius aimax corresponding to Smax with the radii ail and aiu corresponding to Sil and Siu. The dry particle radii are related to supersaturation by equation (4). By integrating the uniform distribution of number within each section, the concentrations of the total number and mass activated are then N¼
I X i¼1
a3i max a3il a3iu a3il
ni ¼ mi ¼ 1
if Sil Smax Siu
if Sil Smax Siu
if Siu < Smax :
Analogous functions ni and mi could also be derived if the volume rather than number distribution is assumed to be uniform within each section. Note that aimax need not be the same for every section if the hygroscopicity is different for each section. The number and mass activated for such a treatment are continuous functions of Smax and hence updraft velocity.
Se 3 ¼
logðSmax =Sil Þ logðSiu =Sil Þ
if Smax < Sil
 To evaluate the activation parameterization for a sectional representation of aerosol particles, the aerosol activation is simulated for a variety of aerosol size distributions and a wide range of updraft velocities using the numerical model described in part 1. The simulated number fraction activated (defined as the fraction of particles with radius larger than their critical radius for activation at the time of maximum supersaturation) is compared with the estimates obtained from the parameterization outlined in section 2. All simulations are initialized with a temperature 288 K, a pressure of 90,000 Pa, and a supersaturation of zero, with particles initially assumed to be in equilibrium with a relative humidity of 100%. The gas kinetic parameters are the same as those used in part 1, namely, 2.16 105 cm for thermal jump length, 1.096 105 cm for vapor jump length, 1.0 for condensation coefficient, and 0.96 for thermal accommodation coefficient.  Two different types of aerosol size distributions are considered. In the first the aerosols are composed of ammonium sulfate with a lognormal size distribution. The geometric mean radius am ranges from 0.001 to 0.1 mm, standard deviation s ranges from 1.12 to 3.0, number concentration N ranges from
Figure 3. As in Figure 2, but as a function of number concentration for a geometric mean radius of 0.05 mm.
ABDUL-RAZZAK AND GHAN: SECTIONAL AEROSOL ACTIVATION
Figure 4. As in Figure 2, but as a function of updraft velocity for a geometric mean radius of 0.05 mm.
10 to 10,000 cm3, and updraft velocity V ranges from 0.02 to 10 m s1. The baseline conditions are am = 0.05 mm, s = 2, N = 1000 cm3, and V = 0.5 m s1. For each numerical simulation the aerosol population is discretized into 100 sections with radius increasing by a constant factor. For each sectional parameterization estimate the aerosol population is discretized into 10 or 30 sections with radius increasing by a constant factor.  Figure 1 illustrates the performance of the parameterization for a particular updraft velocity and aerosol size distribution. It shows that the sectional parameterization accurately distinguishes between size bins with particles too small to be activated and those with particles large enough to be activated. By treating fractional activation for each section, the parameterization also accurately treats the transition between particles too small to be activated and those large enough to be activated.  Figures 2, 3, 4, and 5 show the number fraction activated as a function of geometric mean radius, aerosol number, updraft velocity, and standard deviation, respectively, as simulated using 100 sections, as parameterized using 10 and 30 sections, and as parameterized according to the lognormal parameterization described in part 1. The number fraction activated according to the sectional parameterization is in close agreement (within 10%) with the numerical simulation for the full range of conditions. The close agreement for the fraction activated means that the parameterization of the maximum supersaturation is in close agreement with the numerical simulation. For a size distribution with uniform composition, the close agreement for the fraction activated also means that the transition between activated and unactivated sections is treated accurately, to within the resolution
Figure 5. As in Figure 2, but as a function of standard deviation for a geometric mean radius of 0.05 mm.
Figure 6. Distribution of aerosol number concentration with size for a free troposphere aerosol [Raes et al., 1997], for a marine aerosol [O’Dowd et al., 1997], for a polluted marine aerosol [Collins et al., 2000], and for an urban aerosol [Hughes et al., 1998].
of the size distribution by the sections. The figures also show that the number of sections has the same effect on the accuracy of numerical and parametric results, and that the sectional and numerical results agree as well as the lognormal and numerical results.  The second type of aerosol size distribution is measured in the field. Figures 6 and 7 show the size distributions of number and hygroscopicity, respectively, of four different measured aerosols. The first is a free tropospheric aerosol (FT-II) measured by Raes et al. . The distribution is dominated by a single mode that is approximated quite well by a lognormal distribution with a number concentration of 480 cm3, a number mode radius of 0.03 mm, and a geometric standard deviation of 2.1. For simplicity, the chemical composition of the free tropospheric aerosol is assumed to be pure ammonium bisulfate. The sectional resolution is taken to be the same as that of the measurements, namely, 50 bins ranging in radius from 0.003 to 0.25 mm. The fraction activated is plotted as a function of updraft velocity in Figure 8. The parameterization estimates the fraction activated to within 5% at all updraft velocities.  The second measured aerosol size distribution is a marine aerosol [O’Dowd et al., 1997, Figure 1] typical of the northeast
Figure 7. Distribution of aerosol hygroscopicity with size for the same aerosols as shown in Figure 6.
ABDUL-RAZZAK AND GHAN: SECTIONAL AEROSOL ACTIVATION
Figure 8. The sectional parameterization and numerical simulation of the fraction of aerosol particles activated as functions of updraft velocity for the free troposphere aerosol.
Figure 10. The sectional parameterization and numerical simulation of the fraction of aerosol particles activated as functions of updraft velocity for the polluted marine aerosol.
Atlantic for average wind speeds (8 m s1) and a relative humidity of around 80%. The measurements provide the number concentrations for 61 bins ranging in radius from 0.005 to 25 mm. The size distribution is characterized by the presence of multiple modes (ultrafine, nucleation, accumulation, and sea salt). The chemical composition of the aerosol as a function of size is taken from Katoshevski et al. . The hygroscopicity ranges from that of ammonium sulfate (0.7) for ultrafine particles to that of sodium chloride (1.3) for coarse particles. The total number concentration is 903 cm3. Figure 9 shows the simulated and parameterized fraction activated as functions of updraft velocity. Both the numerical simulations and the sectional parameterization predict low activation fractions for this aerosol because most of the aerosol number is at radii less than 0.01 mm while most of the aerosol surface area is at radii greater than 1 mm. The numerical and parameterized estimates of activation agree to within 10% for updrafts stronger than 0.5 m s1, but the parameterized activation is twice as large as the numerical simulation for updrafts weaker than 0.1 m s1.  The third measured size distribution is a polluted marine aerosol corresponding to conditions where aged continental aerosol from Europe is mixed with marine aerosol, producing a bimodal distribution with a pronounced accumulation mode [Collins et al., 2000]. The total number concentration is 1457 cm3, and the geometric mean radius is 0.068 mm. The chemical composition is assumed to be ammonium sulfate. The measurements cover 100
bins with radii from 0.002 to 1 mm. Figure 10 compares the simulated and parameterized fraction activated as functions of updraft velocity. The numerical model’s results are similar to those for the lognormal size distribution illustrated in Figure 4. The parameterization agrees with the numerical simulation to within 5% for updrafts stronger than 0.5 m s1, but underestimates the fraction activated by as much as a factor of 3 for updrafts weaker than 0.1 m s1.  The fourth measured aerosol is an urban aerosol [Hughes et al., 1998]. It is dominated by fine and ultrafine particles, with a low soluble fraction producing a hygroscopicity ranging from 0.05 for ultrafine particles to 0.1 for coarse particles and 0.25 for accumulation mode particles. The total number concentration is 3707 cm3, but judging from the shape of the measured size distribution the measurements probably fail to sample many particles with radius smaller than 0.035 micron; however, the unsampled particles are too small to be activated for all but the strongest updrafts, so we do not expect the limited size range of the measurements to significantly influence the estimate of the number or mass of activated particles. Figure 11 compares the simulated and parameterized fraction activated as functions of updraft velocity. The parameterized number agrees with the numerical simulation to within 10% for all updraft velocities. Although the hygroscopicity decreases abruptly from 0.2 to 0.1 for particles with radius between 0.48 mm and 0.54 mm, the fraction activated is a monotonic function of particle size for all
Figure 9. The sectional parameterization and numerical simulation of the fraction of aerosol particles activated as functions of updraft velocity for the marine aerosol.
Figure 11. The sectional parameterization and numerical simulation of the fraction of aerosol particles activated as functions of updraft velocity for the urban aerosol.
ABDUL-RAZZAK AND GHAN: SECTIONAL AEROSOL ACTIVATION
updraft velocities because for all updraft velocities all particles in that size range are activated.
4. Conclusions  In this study we have extended the aerosol activation parameterization of part 1 to the case of a sectional representation of aerosol particles. The number activated according to the sectional parameterization agrees with numerical simulations to within 10% for idealized lognormal size distributions under a wide variety of conditions. Agreement for measured size distributions with sizedependent composition is almost as good, except when the updraft velocity is weaker than 0.1 m s1. This success has been achieved by simply calculating an effective critical supersaturation that is compatible with our physical understanding of the relationship between the fraction activated and four dimensionless parameters. This affirms that the relationship accurately represents the aerosol activation process.  The aerosol activation parameterization described in this paper should be suitable for any sectional representation of the aerosol size distribution. Because it is expressed in terms of the critical supersaturation of each bin, it can be applied to an arbitrary aerosol composition in each section. Thus the parameterization can be applied to multiple size distributions with different compositions (e.g., an external mixture of sea salt, sulfate, organic carbon, and dust particles). For sections that are composed of an internal mixture of different materials, Abdul-Razzak and Ghan  describe how the mean hygroscopicity and critical supersaturation of the section can be calculated. By combining the treatment for an external mixture of aerosol components described here with the treatment for an internal mixture, the activation of virtually any distribution (externally and internally mixed) of aerosol composition with size can be predicted by the parameterization.  Acknowledgments. Richard Easter’s comments led to improvements in the manuscript. This study was primarily supported by the NASA Earth Science Enterprise under contract NAS5-98072 and grants NAG58797 and NAG5-9531. Support was also provided by the U.S. Department of Energy Atmospheric Radiation Measurement Program, which is part of the DOE Biological and Environmental Research Program. The Pacific Northwest National Laboratory is operated for the DOE by Battelle Memorial Institute under contract DE-AC06-76RLO 1830.
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H. Abdul-Razzak, Mechanical Engineering Department, Texas A&M University-Kingsville, Kingsville, TX 78363, USA. S. J. Ghan, Pacific Northwest National Laboratory, Richland, WA 99352, USA. ([email protected]