1. Introduction
In the study of atmospheric aerosols, a major discovery was that with time, tropospheric aerosols might lose their “birthmarks” and acquire a size distribution independent of the physical properties of the medium (e.g., temperature, viscosity, and density) and time (Pruppacher and Klett 1997; Clark and Whitby 1967). This phenomenon puzzled scientists in the first half of the last century before Friedlander established the self-preserving size distribution (SPSD) theory (Friedlander 1960) to research turbulence; here, the SPSD was attributed to aerosol coagulation, resulting in a shift from small aerosol condensation nuclei to the Aitken range in the upper end of the spectrum.
In the SPSD theory (Swift and Friedlander 1964; Friedlander and Wang 1966), a similarity transformation method (STM) was introduced to obtain asymptotic solutions to the kinetic collection equation (KCE). The STM assumes that particle size distribution (PSD) is a function of only particle volume normalized by the average particle volume. Substituting Eq. (28) into Eq. (1) with different coagulation kernels for the continuum and free-molecular regimes yields ordinary integrodifferential equations for ψ, with η the independent variable after some complex processing; however, analytical solutions to the ordinary integrodifferential equations can only be found for the upper and lower ends of the distribution by making suitable approximation (Friedlander 2000), or numerical results can be obtained using numerical approaches such as the finite-difference method, the detailed numerical process can be found in Friedlander and Wang (1966).
Lee (1983) further developed the SPSD theory based on the concept of the method of moments (MOM) by presuming that size distribution during the coagulation process can be represented by a time-dependent lognormal function (Lee 1983; Lee et al. 1984; Park and Lee 2001; Wang et al. 2019). This method is called “log MOM” in the present article. The SPSD function ψ(η) presented by log MOM was obtained by combining Eq. (28) and the lognormal distribution function n(υ, t), in which the parameters with respect to time will approach their limit values when time tends to infinity; thus, the analytical expression of the parameters should be obtained first. The SPSD theory via log MOM not only avoids the difficulty of solving the ordinary integrodifferential equations in the STM but directly yields the analytical expressions of SPSD. Furthermore, the analytical solutions to the KCE provided by log MOM are more accurate than those through the STM by Friedlander (Lee 1983).
Since these findings, numerous studies have developed and tested the SPSD theory (Dekkers and Friedlander 2002; Hidy 1965; Junge 1969; Lai et al. 1972; Liu and Whitby 1968; Nathans et al. 1970; Frenklach 1985). Typically, the SPSD theory has played a major role in explaining particle coagulation in gases, liquids, and recently gelation (Goudeli et al. 2015; Park et al. 2000; Söderlund et al. 1998; Zhao et al. 2015). This theory has also been used to explain particle evolution after nuclear air bursts and aerosol distribution in the urban atmosphere; size distribution calculated based on the SPSD theory was consistent with that obtained through experimental procedures (Clark and Whitby 1967; Nathans et al. 1970). Moreover, based on the SPSD theory, analytical solutions to the KCE can be obtained, which is a logical procedure that yields a relatively accurate solution and has strict functional forms in explicit expressions, from which any corresponding value associated with distribution function (such as kth moment, mean, variance, kurtosis, skewness) can be calculated. Because of their generality, analytical solutions can intuitively reflect the relationships among various aerosol mechanisms and can be used for both qualitative and quantitative analyses (Turco and Yu 1998, 1997; Kelesidis and Goudeli 2021; Hussein et al. 2020, 2004; Ensor 2011).
The lognormal distribution was introduced to the description of dispersed particles by Kolmogorov (1941) and was introduced to the field of atmospheric science by Foitzik (1965) for describing optical aerosol properties (Kolmogorov 1941; Foitzik 1965). With its extended methods, the lognormal distribution has been the most widely used function for characterizing atmospheric aerosol size distribution (Cai et al. 2016; Riemer et al. 2009; Wang et al. 2012; Wang et al. 2019). Recently, a three-parameter inverse Gaussian distribution (IGD) was demonstrated to be a more suitable function than the lognormal distribution for characterizing atmospheric aerosol size distribution, and a new MOM, namely, the inverse Gaussian distributed MOM (IGDMOM), was established for solving the KCE involving Brownian coagulation, turbulent shear coagulation, and condensation (Shen et al. 2020b). IGD dates back to 1915 when Schrödinger (1915) and Smoluchowski (1915) presented independent derivations of the density of the first passage time distribution of Brownian motion with positive drift (Schrödinger 1915; Smoluchowski 1915). The modern statistical community was acquainted with this law by the pioneering work of Tweedie (1957). Seshadri (1999) provided a general discussion of the early history; later, IGD was first used to describe the motion of pollen particles in water as well as lifetime distribution by (Chhikara and Folks 1977), who elaborates the advantages in the use of the inverse Gaussian over the lognormal. To the best of our knowledge, adopting the generalized IGD as an aerosol/cloud PSD model was first proposed by Alexandrov and Lacis (2000) to characterize aerosol radiative properties. We later observed that the IGDMOM can characterize the skewness and kurtosis of size distribution more accurately than the log MOM (Shen et al. 2020b).
This study derived explicit analytical solutions with the realistic collision kernel for aerosol particle coagulation in the continuum and free-molecular regimes. In the following sections, the analytical solutions of the parameters in the IGD with respect to time are presented by rigorous mathematics derivation; the asymptotic expressions of the zeroth moment, second moment, and shape factor are then obtained as time tends to infinity; and finally, the analytical expressions of the SPSD through the IGDMOM and the function of time required to reach the SPSD are both determined theoretically. Compared with the other methods, including recognizable STM and log MOM, the SPSD obtained by the IGDMOM has the advantages of logical and rigorous derivation process, concise analysis expression, and competitive precision. Noted here only coagulation is considered in the present study because other dynamics, such as condensation, might make the SPSD nonexistent, which is shown in appendix G. Notably, all nomenclature, Greek letters, and abbreviations appearing in this article are presented in appendix H.
2. Theory
a. Numerical solution to KCE using the IGDMOM
The method for obtaining b_{0} and b_{2} in Eq. (11), as well as their expressions, are presented in appendix A.
b. Analytical solutions to the KCE
The study of the analytical solution to the KCE is not new; Smoluchowski (1917) was the first to provide an analytical expression for the early coagulation stage of an aerosol consisting of monodisperse aerosols, and his work was subsequently developed by Friedlander (1960), Lee et al. (Lee et al. 1990; Lee 1983), and Yu et al. (2016, 2015a). The detailed mathematical expressions for these analytical solutions are presented in appendix B.
Other works on the analytical solutions to the KCE include the study of cloud droplet coalescence in which Scott (1968) solved the KCE analytically using three types of unrealistic collection kernels. In the study of the steady-state, source-enhanced atmospheric aerosol, Klett obtained the analytical solution to the KCE by assuming the collection kernel K = K_{0}u^{α}υ^{α}, where K_{0} and α are constants (Klett 1975). Moreover, the analytical solutions to the KCE were executed considering plume expansion, condensation, and nucleation by Turco et al. (Turco and Yu 1997, 1998, 1999; Yu and Turco 1998) and puff/plume coagulation effects in the presence of atmospheric dispersion by Anand and Mayya (2011); the collision formulas in these studies are simplified to be constant or reduced forms to include maximum dynamics.
1) Brownian coagulation in the continuum regime
Notably, s = 1/(Ω − 1). It also should be mentioned that a(s) in Eqs. (15)–(18) can be assigned as the initial value a(s_{0}) for simplified calculation, which has been discussed in appendix D.
2) Brownian coagulation in the free-molecular regime
It must be mentioned that a_{k}(s) and b_{k}(s) (k = 0, 2) in Eqs. (20)–(24) can be assigned as the initial values a_{k}(s_{0}) and b_{k}(s_{0}) (k = 0, 2) for simplified calculation, which has been discussed in appendix E.
Notably, benefiting from the unique characteristics of the kth moment of the IGDMOM, the analytical solution can be obtained directly and simply from the ordinary differential equations of the first three moments, which is completely different from that of the other methods, namely, the STM by Friedlander, log MOM by Lee et al., and TEMOM by Yu et al. (Chen et al. 2014; Lee 1983; Lee et al. 1984, 1997; Yu et al. 2015a,b).
3) SPSD formulas
Equations (32) and (33) are the SPSD expressions proposed in this study for Brownian coagulation in the continuum and free-molecular regimes, which are verified by comparing with other general expressions, as detailed in section 4.
4) Self-preserving time τ_{∞}
It is critical to know how long it takes to reach SPSD, that is, the self-preserving time (SPT)—as aerosols are polydisperse. The SPT strongly depends on the initial aerosol size distribution (Lee 1983; Vemury et al. 1994; Frenklach 1985). For the first time, Hidy and Lilly gave the SPT for an initially monodisperse aerosol undergoing Brownian coagulation in the continuum regime (Hidy and Lilly 1965; Hidy 1965). Lee et al. derived expressions of the SPT in both continuum and free-molecular regimes as a function of the initial geometric standard deviation of the PSD (Lee et al. 1990). They defined the SPT as the time required for the geometric standard deviation of the distribution to approach its SPSD value within ±10%. In this article, following Lee et al.’s definition of SPT, the expression of SPT with respect to Ω_{0} is given.
From Eqs. (35) and (37), the SPT can be determined based on only the initial shape factor; that is, Ω_{0}, whereas Lee et al. reported that the SPT is determined based on only the initial geometric standard deviation σ_{g0} of SPD (Lee et al. 1990). The relation between Ω_{0} and σ_{g0} is presented in Eq. (47).
3. Numerical implementation
To verify the obtained analytical solutions to the KCE as well as SPSD expressions, we compared the results with highly recognizable schemes (Friedlander 1960, 1961; Lee 1983; Smoluchowski 1917; Yu et al. 2015a). Furthermore, the results were compared with a prestigious method, namely, the sectional method (SM; Landgrebe and Pratsinis 1990), which does not assume a particular size distribution during coagulation and has been extensively used to validate other methods as a reference (Lee 1983; Lee et al. 1984, 1990; Otto et al. 1999; Yu and Lin 2009a,b, 2017, 2018; Wang et al. 2019). In this paper, the SM is also used as a reference to validate the method proposed in this study. To ensure high numerical accuracy of this SM method, the section number is 800 and the section spacing factor is 1.045 (Shen et al. 2020a).
Here, to discover the ability of the analytical and asymptotic schemes proposed in this study to characterize the evolution of atmospheric aerosol and warm cloud droplets under high concentration conditions (Otto et al. 1999), the following initial parameters for the comparison calculations were chosen:
N_{0} = 1 × 10^{16} m^{−3},
ρ_{p}= 1000.0 kg m^{−3},
T = 300 K,
μ = 1.83 × 10^{−5} kg s^{−1} m^{−1},
d_{p0} = 1.0 nm (Brownian coagulation in the free-molecular regime),
d_{p0} = 0.5 μm (Brownian coagulation in the continuum regime),
${\sigma}_{g0}=1.2\text{\hspace{1em}}\left[\text{i}.\text{e}.,\hspace{0.17em}{\Omega}_{0}={e}^{9{\mathrm{ln}}^{2}\left(1.2\right)}=1.349\right]$ .
The definition for M_{k0} presented in Eq. (38) is derived from the work of Lee (1983), who assumed the atmospheric aerosol size distribution to be a lognormal size distribution, which was also used by Pratsinis (1988) and our previous study (Shen et al. 2020b). For convenient analysis, all the quantities presented in section 4, including total particle number concentration and the second moment, are presented in dimensionless form.
4. Results and discussion
By applying the IGD to the description of atmospheric aerosol size distribution, we noted that Friedlander’s SPSD theory of coagulating atmospheric aerosols can be efficiently presented, which is reflected in the following four analytical solutions to the SCE, asymptotic analysis, SPT, and dimensionless self-preserving size distribution expression.
a. Analytical solutions to the SCE
Figure 1 illustrates the variance of the dimensionless total particle number concentration M_{0}/M_{00} (Fig. 1a), the dimensionless second moment M_{2}/M_{20} (Fig. 1b), and the shape factor Ω (Fig. 1c), with time as well as the relative errors of such quantities to the referenced SM (Figs. 1d–f) for Brownian coagulation in the continuum regime. In the figure, the analytical solutions to the KCE proposed by Smoluchowski (1917), Friedlander and Wang (1966), Lee (1983), and Yu et al. (2015a) are implemented and are compared with the solutions presented in Eqs. (15) and (18a). In the analytical solutions of Smoluchowski (1917) and Friedlander and Wang (1966), no information regarding M_{2}/M_{20} is provided; thus, no curves are available for their solutions in Figs. 1b, 1c, 1e, and 1f. As proposed in the literatures (Landgrebe and Pratsinis 1990; Yu and Lin 2009a), the initial distribution of all MOMs here is chosen as lognormal distribution.
The analytical solution of Friedlander and Wang (1966) produces the largest errors for M_{0}/M_{00}, followed by that of Smoluchowski (1917). The IGDMOM, log MOM, and TEMOM overlap, and it is difficult to distinguish them, even from the relative errors presented in Fig. 1d; the relative errors of these three solutions to the SM are less than those of solutions by Smoluchowski (1917) and Friedlander and Wang (1966). This is because the solution by Smoluchowski accurately applies only to monodisperse aerosols, whereas the solution by Friedlander and Wang applies only to aerosols that have attained the SPSD distribution. For both M_{2}/M_{20} and Ω, Figs. 1b, 1c, 1e, and 1f illustrate that the present solution (IGDMOM) gives nearly the same results as that produced by the log MOM and TEMOM. Thus, the analytical solution proposed in this article has the same accuracy for characterizing the evolution of aerosol quantities, including M_{0}/M_{00}, M_{2}/M_{20}, and Ω as the log MOM and TEMOM, although they have different mathematical expressions.
Similarly, the analytical solutions to the KCE for Brownian coagulation in the free-molecular regime are displayed in Fig. 2 for the three typical quantities, namely, M_{0}/M_{00}, M_{2}/M_{20}, and Ω, through the implementation of the present solution and solutions by Smoluchowski (1917), Lai et al. (1972), Lee et al. (1984), and Yu et al. (2015a). The present solutions, presented in Eqs. (20) and (24a), are used for the calculation. For the dimensionless total particle number concentration, M_{0}/M_{00}, the solution by Smoluchowski generates the largest errors relative to the SM. The present solution generates curves that overlap with the TEMOM and log MOM, whose relative errors are nearly the same as the solution by Lai et al. (1972), although the solution by Lai et al. (1972) theoretically only applies to aerosols that have attained the SPSD due to similarity transformation. The present solution is nearly the same as that by the TEMOM and log MOM for M_{2}/M_{20}, whereas for the shape factor Ω presented in Figs. 2c and 2f, the present solution produces significantly fewer errors than the log MOM.
Variances of dimensionless total particle number concentration M_{0}/M_{00}, dimensionless second moment M_{2}/M_{20}, and the shape factor Ω with time, as well as the relative errors of such quantities to the referenced SM for Brownian coagulation in the free-molecular regime. The scaled time is
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Variances of dimensionless total particle number concentration M_{0}/M_{00}, dimensionless second moment M_{2}/M_{20}, and the shape factor Ω with time, as well as the relative errors of such quantities to the referenced SM for Brownian coagulation in the free-molecular regime. The scaled time is
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Variances of dimensionless total particle number concentration M_{0}/M_{00}, dimensionless second moment M_{2}/M_{20}, and the shape factor Ω with time, as well as the relative errors of such quantities to the referenced SM for Brownian coagulation in the free-molecular regime. The scaled time is
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
In summary, through the examination of three typical quantities, namely, M_{0}/M_{00}, M_{2}/M_{20}, and Ω, the analytical solutions proposed in this study have higher accuracy than solutions by Smoluchowski (1917) and other researchers (Friedlander and Wang 1966; Lai et al. 1972). The present solutions provide similar or higher accuracy than the log MOM for M_{0}/M_{00}, M_{2}/M_{20}, and Ω, although all these solutions have different mathematical expressions and are obtained from different size distributions.
b. Asymptotic analysis for SPSD
1) Brownian coagulation in the continuum regime
Figure 3a illustrates the evolution of the dimensionless zeroth moment predicted using different methods. All the results are in excellent agreement, especially those given by the present study and by Lee (1983) because the value of α given by these two methods is almost the same.
Evolution of the dimensionless (a) zeroth moment and (b) second moment for the SPSD in the continuum regime. The scaled time is τ = K_{CO}M_{00}t.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Evolution of the dimensionless (a) zeroth moment and (b) second moment for the SPSD in the continuum regime. The scaled time is τ = K_{CO}M_{00}t.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Evolution of the dimensionless (a) zeroth moment and (b) second moment for the SPSD in the continuum regime. The scaled time is τ = K_{CO}M_{00}t.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
2) Brownian coagulation in the free-molecular regime
Evolution of the dimensionless (a) zeroth moment and (b) second moment for the SPSD in the free-molecular regime. The scaled time is
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Evolution of the dimensionless (a) zeroth moment and (b) second moment for the SPSD in the free-molecular regime. The scaled time is
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Evolution of the dimensionless (a) zeroth moment and (b) second moment for the SPSD in the free-molecular regime. The scaled time is
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Figure 4b illustrates the evolution of the dimensionless second moment predicted using different methods. The present IGDMOM, TEMOM, and log MOM produce overlapped curves, which cannot be distinguished from each other.
In conclusion, we observed that for dimensionless moments, namely, M_{0}/M_{00} and M_{2}/M_{20}, unified expressions exist in all existing methods. There are only differences for specific parameters among all the methods. The analytical expressions derived using the IGDMOM provide great precision.
c. SPT
According to the SPSD theory by Friedlander (1960), a coagulating aerosol in an atmospheric environment should reach the same steady-state size distribution with time regardless of the aerosol’s initial size distribution. When this steady state is reached, the shape of aerosol size distribution dealt by similarity transformation treatment does not change with time (Friedlander 1960, 1961; Friedlander and Wang 1966; Wang and Friedlander 1967). Clark and Whitby (1967) used the theory to successfully explain the general shape of the observed size distribution of atmospheric aerosols. Although in the atmospheric environment, the aerosol size distributions in any position may be different, the size distribution can be locally SPSD at all times. Thus, for predicting the property of atmospheric aerosol at a given time, determining the time required to attain the SPSD; that is, SPT is essential.
Time to attain the self-preserving distribution in the (a) continuum regime and (b) free-molecular regime as a function of the initial geometric standard deviation for various methods.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Time to attain the self-preserving distribution in the (a) continuum regime and (b) free-molecular regime as a function of the initial geometric standard deviation for various methods.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Time to attain the self-preserving distribution in the (a) continuum regime and (b) free-molecular regime as a function of the initial geometric standard deviation for various methods.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Although the MOMs, including the IGDMOM, TEMOM, and log MOM, generate nearly overlapping results for all σ_{g0}, substantial differences exist between the MOMs and the SMs (υ^{2}- and υ-based models). The obvious difference between the log MOM and SMs was addressed by Vemury et al. (1994). Figure 5b illustrates the SPT in the free-molecular regime as a function of the initial geometric standard deviation σ_{g0}. Notably, at the SPSD state near σ_{g0} ≈ 1.345, the present model overlaps with the log MOM, whereas away from σ_{g0} ≈ 1.345, the present study approaches the TEMOM. Similar to the discussion in Fig. 5a, all three MOMs deviate substantially from the SMs (υ^{2}- and υ-based models). In Table 1, the SPTs are presented for the various methods when Ω_{0} = 1. For comparison, the υ^{2}-based model, υ-based model, log MOM, and TEMOM are selected as references. The present IGDMOM yields the same SPT for Brownian coagulation in the continuum regime as log MOM, and the value lies between that obtained using the log MOM and TEMOM for Brownian coagulation in the free-molecular regime.
SPT when Ω_{0} = 1. For υ^{2}-based and υ-based models f_{s} = 1.08.
Figure 5 illustrates that the further the value of σ_{g0} is from the value in the SPSD at time τ = 0, the longer it takes for the aerosol to reach the SPSD. This finding is quantitatively in agreement with Hidy (1965) in that the self-preserving spectrum is independent of n(υ, 0) although the time required to reach the SPSD depends on the shape of the initial distribution.
d. SPSD
The SPSD theory has become a critical tool for exploring atmospheric particle coagulation mechanisms (Kreidenweis et al. 2019; Pruppacher and Klett 1997). Friedlander and Wang (1966) studied the theoretical coagulation equation in the continuum regime by using the STM, and Lai et al. (1972) used the same similarity solution to study the coagulation equation in the free-molecular regime. On the basis of the assumption that the PSD is lognormal distribution, through using the transformation formula in STM [i.e., Eq. (28)], Lee (1983) and Lee et al. (1984) derived new analytical SPSD expressions for atmospheric aerosols underlying Brownian coagulation in both continuum and free-molecular regimes. Such solutions by Friedlander and Lee have become the main tools for determining atmospheric aerosol asymptotic distribution after long periods.
Therefore, comparing the SPSD obtained from the IGDMOM with solutions obtained from the STM by Friedlander and log MOM by Lee is essential, and such a comparison is presented in Fig. 6. In Fig. 6a, two direct numerical solutions to the KCE regarding Brownian coagulation in the continuum regime are displayed as a reference, namely, the finite-difference method used by Friedlander and Wang (1966) and the SM used by Vemury and Pratsinis (1994). These two numerical methods nearly overlap each other, although different numerical techniques are used. Both the log MOM and IGDMOM provide a continuous spectrum over the whole size distribution, whereas the STM by Friedlander and Wang provides only reasonable spectra in the lower and higher ends of size distributions. Friedlander and Wang (1966) claimed that the STM agrees well only with the numerical solutions for η < 0.1 and η > 0.1, which are also presented in Fig. 6a. Comparing the IGDMOM with the log MOM and STM, the IGDMOM is much closer to the log MOM over the whole size distribution, although these two methods seem different from the referenced numerical solutions.
Self-preserving size distribution (SPSD) for different methods in the (a) continuum regime and (b) free-molecular regime.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Self-preserving size distribution (SPSD) for different methods in the (a) continuum regime and (b) free-molecular regime.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Self-preserving size distribution (SPSD) for different methods in the (a) continuum regime and (b) free-molecular regime.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Figure 6b compares the present IGDMOM, log MOM, and STM for the SPSD regarding Brownian coagulation in the free-molecular regime. For reference, we adopted the SM used by Vemury and Pratsinis (1994) and the Adams extrapolation method used by Lai et al. (1972). SPSD expressions in the log MOM and the STM are proposed by Lee et al. (1984) and Lai et al. (1972). The STM in Fig. 6b gives only the fitted values to the numerical solutions in the upper and lower ends of the size distribution, but the fitting is not as good as the STM in Fig. 6a. Figure 6b presents that the satisfying agreement between the STM and numerical solutions is only for η < 0.01 and η > 5.0. Similar to the presentation in Fig. 6a, the IGDMOM is much closer to the log MOM than to other solutions.
As is illustrated in Figs. 6a and 6b, the STM can only provide the spectra in the upper and lower ends of the size distribution, although these two little ends fit the referenced solution given by the SM well. The present IGDMOM straightly gives the analytical and explicit expression for the whole size distribution. The difference between the IGDMOM and the referenced solution is due to the assumption of the time dependent distribution.
5. Conclusions
A scheme to realize the SPSD theory is presented in this study. The scheme is executed by first introducing IGDMOM into the description of the atmospheric aerosol size distribution and then by analytically solving the KCE in terms of realistic Brownian coagulation kernels in the continuum and free-molecular regimes. The KCE is solved using the concept of the MOM in which the governing equations for the first three moments are needed. We demonstrate the ability of the new scheme for realizing the SPSD theory by presenting new reasonable mathematical expressions for the SPSD and SPT.
The present scheme is mainly compared with other several recognizable schemes, including the SM, STM, log MOM, and TEMOM. Regarding analytical solutions to the KCE, the SPT, and SPSD expression, our new scheme is verified to realize the SPSD theory with nearly the same accuracy as or higher accuracy than the log MOM, and shows obvious higher accuracy than the schemes by Smoluchowski (1917) and Friedlander and Wang (1966). The scheme presented in this study demonstrates the SPSD theory more reasonably than the other schemes, such as the STM and log MOM, since the derivations for expressions of the SPSD and SPT are only based on rigorous theoretical analysis.
The SPSD expressions from the present study are straightforward and explicit functions for the whole spectra. Previously, analytical solutions could only be obtained from extreme ranges. To obtain a full range of solutions through STM (Friedlander and Wang 1966), the integrodifferential equations were handled using numerical methods, with several key coefficients obtained from other numerical solutions (Friedlander and Wang 1966; Lai et al. 1972).
The present scheme provides additional proof for the SPSD theory and has important implications for developing tropospheric atmospheric aerosol dynamics models involving the evolution of size distribution.
Acknowledgments
The authors thank the National Natural Science Foundation of China (11872353, 91852102) and the Zhejiang Provincial Natural Science Foundation of China (LR16A020002) for their support. Additionally, we are grateful to the anonymous reviewers for their suggestions.
APPENDIX A
Method for Obtaining b_{0} and b_{2} Presented in Eq. (11)
Here, the solution for b_{i} differs from that by Lee et al. in that the log MOM has lower accuracy than the IGDMOM (Lee et al. 1984; Shen et al. 2020b).
APPENDIX B
Analytical Solutions to the KCE by Various Scientists
APPENDIX C
Derivation of Eq. (14) for Different Methods
a. IGDMOM
b. Log MOM
c. TEMOM
APPENDIX D
Derivation of Eq. (15)
The variance of a(s) is very small in the interval (0, +∞), and it has a high convergence rate to 1. When s = 0.5, the value of a(0.5) is equal to 1.1388. Some values of a(s) with s are presented in Table D1, and the curve displaying the variance of a(s) with s is illustrated in Fig. D1.
Values of a for s.
Three parameter functions in the analytical solutions.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Three parameter functions in the analytical solutions.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Three parameter functions in the analytical solutions.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
The parameter a(s) can be regarded as a constant when solving Eq. (D2) because it does not diverge but always converges quickly to a certain value during coagulation, and the difference between the maximum and the minimum is negligible. To facilitate the calculation, we choose the initial value a(s_{0}). Next, Eq. (15) can be obtained from Eq. (D2).
APPENDIX E
Derivation of Eq. (20)
The properties of a_{0}(s) imply that the variance with s is very small in the interval (0, +∞), and it has a high convergence rate to 2 [e.g., a_{0}(0.5) = 1.9502], and notably, a_{2}(s) also has a high convergence rate to 2 [e.g., a_{2}(0.5) = 2.3714]. Some values of a_{0}(s) and a_{2}(s) with s are presented in Table E1, and the curves displaying the variance of them with s are illustrated in Fig. D1.
Values of a_{0}, a_{2}, b_{0}, and b_{2} for s.
APPENDIX F
SPT of υ^{2}-Based Model and υ-Based Model
For the υ^{2}-based model, σ_{gsc} = 1.445, A_{c} = 13.1, B_{c} = 180.3, C_{c} = 7.3, D_{c} = −2.44, and E_{c} = 2.63, whereas for the υ-based model, σ_{gsc} = 1.446, A_{c} = 17.3, B_{c} = 182.6, C_{c} = 4.4, D_{c} = −2.46, and E_{c} = 2.55.
APPENDIX G
The Case of SPSD in the Condensation
The description of condensation can be referenced in Shen et al. (2020b). Here, it will be discussed in more detail, including its numerical and analytical solution.
Variances of (a) the first moment and (b) the shape factor with time in the condensation.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Variances of (a) the first moment and (b) the shape factor with time in the condensation.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
Variances of (a) the first moment and (b) the shape factor with time in the condensation.
Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0273.1
APPENDIX H
Nomenclature, Greek Letters, and Abbreviations
a. Nomenclature
υ | Particle volume |
N | Particle number concentration density |
M_{k} | kth moment of particle size distribution |
N(t) | Total particle number concentration at time t |
N_{0} | Initial total particle number concentration |
C | Cunningham correction factor |
k_{B} | Boltzmann constant, J·K |
Kn | Particle Knudsen number |
t | Time, s |
T | Temperature, K |
r | Aerosol particle radius |
d_{p0} | Initial mean diameter of aerosol |
b. Greek letters
ν | Kinematic viscosity, m^{2} s^{−1} |
β | Particle collision kernel |
μ | Gas viscosity kg m^{−1} s^{−1} |
λ | Mean free path of the gas, m |
ζ | Volume fraction of aerosol material |
ρ | Mass density of the particles |
Ω | Shape factor of size distribution |
σ_{g} | Geometric mean deviation of size distribution |
ϕ | Mean of IGD |
φ | Shape parameter of IGD |
ψ | Size distribution density function in SPSD |
η | Particle volume in the SPSD |
τ | Scaled time |
c. Abbreviations
KCE | Kinetic coagulation equation |
IGD | Inverse Gaussian distribution |
MOM | Method of moments |
SPSD | Self-preserving size distribution |
STM | Similarity transformation method |
TEMOM | Taylor series expansion method of moments |
IGDMOM | Inverse Gaussian method of moments |
Log MOM | Lognormal method of moments |
SM | Sectional method |
ODE | Ordinary differential equation |
SPT | Self-preserving time |
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