Modal Bin Hybrid Model: A surface area consistent, triple‐moment sectional method for use in process‐oriented modeling of atmospheric aerosols

Abstract

[1] A triple‐moment sectional (TMS) aerosol dynamics model, Modal Bin Hybrid Model (MBHM), has been developed. In addition to number and mass (volume), surface area is predicted (and preserved), which is important for aerosol processes and properties such as gas‐to‐particle mass transfer, heterogeneous reaction, and light extinction cross section. The performance of MBHM was evaluated against double‐moment sectional (DMS) models with coarse (BIN4) to very fine (BIN256) size resolutions for simulating evolution of particles under simultaneously occurring nucleation, condensation, and coagulation processes (BINx resolution uses x sections to cover the 1 nm to 1 µm size range). Because MBHM gives a physically consistent form of the intrasectional distributions, errors and biases of MBHM at BIN4‐8 resolution were almost equivalent to those of DMS at BIN16–32 resolution for various important variables such as the moments M_k (k: 0, 2, 3), dM_k/dt, and the number and volume of particles larger than a certain diameter. Another important feature of MBHM is that only a single bin is adequate to simulate full aerosol dynamics for particles whose size distribution can be approximated by a single lognormal mode. This flexibility is useful for process‐oriented (multicategory and/or mixing state) modeling: Primary aerosols whose size parameters would not differ substantially in time and space can be expressed by a single or a small number of modes, whereas secondary aerosols whose size changes drastically from 1 to several hundred nanometers can be expressed by a number of modes. Added dimensions can be applied to MBHM to represent mixing state or photochemical age for aerosol mixing state studies.

1 Introduction

[2] A variety of atmospheric aerosol properties are important for the environment. Aerosols alter the energy balance of the Earth substantially [Forster et al., 2007] by serving as cloud condensation nuclei (CCN) and ice nuclei (IN) and by scattering and absorbing light. Aerosols are important carriers of acids and nutrients, which affect ecosystems by several depositional pathways, such as precipitation, fog, and dry deposition [Burns et al., 2011]. Aerosols that carry hazardous materials such as metals, persistent organic pollutants (POPs), and radioactive nuclides aggravate animal health through inhalation and exposure [Cohen et al., 2004; Kaneyasu et al., 2012].

[3] However, because of the wide variation in their properties such as size, bulk and surface compositions, mixing state, phase state, and shape, there are still large uncertainties in estimates of their impacts [Forster et al., 2007; Carmichael et al., 2008; Carmichael et al., 2010; Gusev et al., 2010]. In addition, some processes are still poorly understood, such as gas uptake where uptake coefficients vary by several orders of magnitude [Shiraiwa et al., 2011; Abbatt et al., 2012] and new particle formation where nucleation rates vary by more than 10 orders of magnitude [Zhang et al., 2012].

[4] An atmospheric aerosol model is a useful tool to connect physicochemical theories and laboratory and field experiments and obtain a comprehensive understanding of behaviors and impacts of aerosols in nature. There are two common approaches to solve dynamics of atmospheric aerosols: the modal method and the sectional (or bin) method. The modal method represents the aerosol as multiple, distinct populations of aerosols referred as modes [Whitby et al., 1991; Whitby and McMurry, 1997]. The aerosol within each mode is represented by a prescribed functional form, usually a lognormal distribution. The sectional (bin) method is based on dividing the particle size domain into discrete sections (or bins), then simulating the mass of each aerosol chemical component (and sometimes number) in each section [Gelbard et al., 1980]. The modal methods are basically more computationally efficient than the sectional methods, because the modal methods usually categorize aerosols into a few modes that are frequently observed in the atmosphere (e.g., Aitken mode (diameter ~10 nm), the accumulation mode (~100 nm), and the coarse mode (~1 µm)). Sectional methods generally require about 10 or more sections (bins) to represent the multimodal nature of atmospheric aerosols. On the other hand, the modal methods are less accurate because they cannot simulate the deviations from the prescribed distribution function.

[5] As recent laboratory and field experiments have revealed more about the complexity of atmospheric aerosols (e.g., composition, mixing state, morphology) and the importance of the complexity, more complexity has been added to the aerosol modules. Aerosol modeling approaches now range from the one‐dimensional size‐resolving modal or sectional approach (conventional internal mixture assumption in regional chemical transport models), the category approach (conventional external mixture assumption in global climate models), and the category approach with a few mixing state categories [Wilson et al., 2001; Vignati et al., 2004; Stier et al., 2005; Wang et al., 2009; Aquila et al., 2011; Kajino and Kondo, 2011; Liu et al., 2012; Kajino et al., 2012a, 2012b] to the computationally expensive two‐dimensional (size and mixing state) mixing state resolving approach [Russell and Seinfeld, 1998; Jacobson, 2001; Oshima et al., 2009; Matsui et al., 2013], the multiple category with mixing state resolving approach [Jacobson, 2001; Bauer et al., 2008; Bergman et al., 2012], and ultimately to the particle‐resolving approach [Riemer et al., 2009; Zaveri et al., 2010].

[6] Efficiency is critically important for process modules in three‐dimensional atmospheric models, because of the need to represent other processes as well: Photochemistry often requires more than 100 chemical species, weather forecasts require finer spatial grid resolutions (< ~km), and climate simulation needs a global domains and longer time integrations (> ~ decades). To permit simulations with such complexity, much effort has been exerted to obtain high efficiency in solving costly aerosol processes such as multicomponent thermodynamic equilibrium [Nenes et al., 1998, Jacobson, 1999; Zaveri et al., 2005] and fast evolution in the nucleation mode [Kerminen and Kulmala, 2002].

[7] To obtain high efficiency as well as accuracy for use in process‐oriented modeling of atmospheric aerosols, Kajino [2012] proposed the new Modal Bin Hybrid Model (MBHM) approach, so named because it is based on sorting a number of lognormal modes by size into a fixed bin structure. Modal methods are efficient but less accurate, whereas the sectional methods are accurate but less efficient. MBHM combined the two methods to obtain high levels of efficiency and accuracy at the same time. Kajino [2012] concluded that the result of MBHM for coarser size resolution (about four to eight bins between 1nm and 1 µm) was consistent with that for finer resolution (64 bins between 1 nm and 1 µm).

[8] The purpose of the study is to characterize and compare the performance of MBHM to that of existing double‐moment methods that are used in current three‐dimensional chemical transport models. We use box model simulations for growth under simultaneously occurring nucleation, condensation, and coagulation processes with various size resolutions. The model description is given in section 2.1 and the merit of MBHM versus other methods is described in section 2.2. The simulation setup and comparisons of MBHM with DMS results are shown and discussed in section 3. Major findings are summarized in section 4. To focus on the evaluation of MBHM against DMS in the paper, other topics are discussed in the appendices. Appendix A presents detailed derivations of methods for calculating condensation and coagulation rate integrals. Appendix B presents comparisons between the harmonic mean and the more accurate methods for condensation and coagulation rates in MBHM.

2 Modal Bin Hybrid Model

[9] Abbreviations and definitions of terms used in the paper are listed in Table 1.

Table 1. Definition of Acronyms and Abbreviations

Term	Definition
BINx	Discretization into x of bins (or sections) over the 3 orders of magnitude in size Δ(logD = 3/x). BIN4, 8, 16, 32, 64, 128, and 256 are performed.
CCNx	Number concentration of particles larger than a critical diameter x (nm) (x: 50, 100)
CCVx	M3 (∝ volume) concentration of particles larger than a critical diameter x (nm)
DMM	Double‐Moment Modal method
DMS	Double‐Moment Sectional method
F(Dg)	Fuchs [1964] approach for coagulation and condensation with a correction factor as a function of Dg.
FFS	Fuchs [1964] approach for coagulation and Fuchs and Sutugin [1971] approach for condensation
FFS(Q)	FFS approach solved using the quadrature method
GMA	Gradual merging approach (renaming or reallocation)
HM	Harmonic mean approach for coagulation and condensation
LDM	Linear discrete method [Simmel and Wurzler, 2006]
LNSD	Lognormal size distribution
MADMS	Modal Aerosol Dynamics Model for Multiple Modes and Fractal Shapes
MBHM	Modal Bin Hybrid Model
MCA	Moving‐center approach [Jacobson, 1997]
MDE	Moment dynamics equation (equations A1a, A1b, and A10)
MaxNE*	Maximum normalized error (MaxNE) with a sign of its bias (equation 16b)
MNB	Mean normalized bias (equation 16a)
NMaxE*	Normalized maximum error (NMaxE) with a sign of its bias (equation 16d)
NMB	Normalized mean bias (equation 16c)
NME	Normalized mean error (equation 16c)
PLA	Piecewise Lognormal Approximation [von Salzen, 2006]
TMS	Triple‐Moment Sectional method (=MBHM)

2.1 Model Description

[10] MBHM is a triple‐moment sectional (TMS) aerosol dynamics model because there are three moments (number, volume, and surface area) in each section. It is based on sorting by size a number of modes of the Modal Aerosol Dynamics Model for Multiple Modes and Fractal Shapes (MADMS) [Kajino, 2011] into a fixed bin structure. MADMS is an extension of the Modal Aerosol Dynamics (MAD) model [Whitby and McMurry, 1997], in which the Brownian coagulation process of multimodal aerosols is extended to cover the full aerosol size range and arbitrary fractal dimensions.

[11] The particle size coordinate, D (diameter), is divided into a set of size sections (bins) with constant logarithmic spacing. Let D_lw,i and D_up,i be the lower and upper boundaries of section i. Then D_up,i = D_lw,i + 1 = xD_lw,i, where x is the constant spacing parameter (if D_lw,1 is 1 nm and x = 2, then D_up,1 = D_lw,2 = 2 nm, D_up,2 = D_lw,3 = 4 nm, …). For each section, the aerosol number, surface area, and chemical component mass concentrations are predicted. The volume concentration, which is also needed, is calculated from the component mass concentrations and particle density.

[12] As basis functions, we use lognormal size distribution (LNSD) functions defined as

(1)

where n_i is the intrasectional number distribution of bin i and N_i, D_g,i, and σ_g,i are the three LNSD parameters, number concentration, number equivalent geometric mean diameter, and geometric standard deviation of bin i, respectively. There are three reasons why we use LNSD: (1) Observed aerosol size distributions are often well approximated by a sum of several LNSD modes, (2) LNSD is mathematically convenient, and (3) there are three parameters in LNSD, which can be obtained from the three moments predicted by MBHM. Note that unlike other sectional methods, the intrasectional distribution n_i is allowed to extend beyond the bin boundaries D_lw,i and D_up,i. The tails of the basis functions thus overlap.

[13] The kth moment of bin i, M_k,i, is defined and integrated by applying the Gaussian integral as

(2)

M_0,i is identical to number concentration N_i and M_2,i and M_3,i are proportional to surface area and volume, respectively. Using equation 2, the three LNSD parameters are given by

(3)

where k₁ = 2 and k₂ = 3.

[14] Time evolution of moments due to emission, nucleation, condensation, coagulation, and deposition processes is described by the Moment Dynamics Equations (MDEs):

(4)

where the time derivatives within the integral have different functional forms for different process. Operator splitting is applied to the MDEs, so that for each model time step, the MDEs are integrated for one process (or subset of processes) at a time. For each process integration step, the beginning moment values (M_0,i, M_2,i, and M_3,i) give the beginning LNSD parameter values. These are used to evaluate the moment time derivatives, which are then used in the calculation of updated moment values.

[15] The MDE integration for coagulation is somewhat different because MDEs for M_0,i, M_3,i, and M_6,i are integrated. This is done to allow analytical evaluation of the coagulation rate integrals (equations A1a and A1b), as proposed by Whitby et al. [1991]. The beginning value of M_6,i is calculated from the beginning LNSD parameters. The coagulation MDEs are integrated to give updated moment values. Updated LNSD parameters are then calculated from the updated M_0,i, M_3,i, and M_6,i, and these are used to calculate the updated M_2,i, using equation 3 with k₁ = 3 and k₂ = 6 and equation 2.

[16] MBHM uses a fixed bin structure, so particles must be transferred to adjacent bins if they grow (or shrink) beyond the bin boundaries. The moving‐center approach (MCA) [Jacobson, 1997] is used for this transfer. When D_g,i becomes larger or smaller than the upper or lower boundaries (D_up,i or D_lw,i), all the number, surface area, and mass within the bin are transferred to the neighboring bin i + 1 or i − 1, respectively. Consequently, all the moments in bin i become 0, unless there is a transfer into bin i from another bin. MCA is applied after each condensation and coagulation operator.

[17] The MDEs are described briefly in section 2.1.1 and in detail in Appendix A. Treatment of the width of each LNSD is given in section 2.1.2.

2.1.1 Integration of Moment Dynamics Equations

[18] Full descriptions of the Moment Dynamics Equations (MDEs) [e.g., Whitby and McMurry, 1997, Equation 5] can be found in the previous literature [Whitby et al., 1991; Whitby and McMurry, 1997; Binkowski and Shankar, 1995], while MADMS is completely described by our contributions [Kajino, 2011; Kajino and Kondo, 2011; Kajino et al., 2012a]. Modal models with LNSD before MADMS used Otto's approximation [Otto et al., 1997] for Brownian coagulation in the free‐molecular regime, which restricted the model to intramodal coagulation. Kajino [2011] extended Otto's approximation for intermodal coagulation (and for arbitrary fractal dimensions of aggregates, too). Thus, strictly speaking, the idea of MBHM became feasible only after MADMS.

[19] The moment time derivatives that are described by the MDEs involve integrals over the LNSD for each bin and double integrals for coagulation. For some processes, analytical forms of the integrals do not exist, so numerical approximations are required. These approximations involve trade‐offs between accuracy and efficiency. Different approaches for evaluating the coagulation and condensation MDE integrals are discussed in Appendix B.

[20] In this study, we conduct simple box model simulations of growth from 1 to several hundred nanometer size under simultaneously occurring nucleation, condensation, and coagulation processes. Emission, deposition, and dilution processes are not considered. Operator splitting is used for time integration of the MDEs and is applied for each nucleation, condensation, and coagulation operator. Time integration for each operator is done over the host time step Δt_host.

[21] The explicit method (forward in time) is used for the time integration of the MDEs for each operator/process, and time substepping is also used under some conditions for stability and accuracy of the simulation. For condensation, the time substep Δt is never allowed to exceed 0.1 × τ_COND,i over all of the bins, where τ_COND,i = 1/k_w,i and k_w,i is the gas‐to‐particle mass‐transfer coefficient over bin i (s⁻¹). For coagulation, the time substep is never allowed to exceed 0.5 × τ_COAG,i over all of the bins. In the simulations presented in this paper, Δt_host is set to 1 s because we found that a 1 s time step was sufficient to yield an accurate nucleation rate under the conditions in our simulations. However, solving every aerosol dynamical processes with Δt_host = 1 s for 3‐D regional to global simulations is still far from feasible for current computational resources. Therefore, further optimization of the operator and time splitting is necessary for 3‐D simulation cases.

[22] For each time substep, the three moments at the beginning of the substep (time t), (k = 0, 2, and 3 or 0, 3, and 6) are known, and the three LNSD parameters at t, , , and are derived using equation 3. The right‐hand side of equation 4 is evaluated using a combination of moments or the LNSD at t, and are then calculated using the forward in time approach. Details of ways of calculating coagulation and condensation rate integrals with various approximations in the transition regime (harmonic mean, Fuchs [Fuchs, 1964] and Fuchs‐Sutugin [Fuchs and Sutugin, 1971] approaches) for the MBHM and DMS methods are presented in Appendix A.

2.1.2 Treatment of the Width of Each LNSD

[23] For most aerosol dynamical processes, the width variable σ_g,i is determined by (and is consistent with) the evolving moments of each bin. However, the σ_g,i must be determined when particles are introduced into the model (e.g., emitted), and σ_g,i may need to be adjusted sometimes during the simulation.

[24] Particles are introduced during a simulation by emissions, nucleation, and inflow at the spatial boundaries and also at the start of a simulation through the initial conditions. The size distribution of the particles being introduced by each of these processes will be specified through parameterization or assumptions, and this specified size distribution must be approximated by a set of LNSDs. If the specified size distribution is actually one or several LNSDs, then these LNSDs can be used directly by MBHM. For nucleation, the new particles introduced during a time step can be assumed monodisperse, which is LNSD with σ_g,i = 1. Size distributions for emissions are often specified as LNSDs [e.g., Dentener et al., 2006; Zender et al., 2003]. Note that in this case, the LNSD widths may be considerably broader than the bin widths.

[25] If the specified size distribution of introduced particles is not one or more LNSD, then the number, surface area, and volume/mass within each size bin (between D_lw,i and D_up,i) can be calculated, and these values in conjunction with equation 3 will give the LNSD parameters for particles being introduced into each bin. With this approach, the widths of the LNSDs will be similar to the bin widths. For example, if the specified size distribution is uniform, the LNSD widths can be shown to be . This approach can also be used when the specified size distribution of introduced particles is LNSD, but is considerably wider than the bins. Sensitivity studies of the different approaches for calculating LNSDs of introduced particles are described in section 3.4.

[26] Due to numerical errors during the simulation, ln σ_g,i could become negative depending on the combinations of the three moments. In this case, σ_g,i is adjusted to unity, preserving M_0,i and M_3,i but changing M_2,i and D_g,i.

[27] For larger values of σ_g,i, the MBHM simulation will not be disrupted but may become less accurate. As presented in Appendix A, we used three methods to calculate the coagulation and condensation rate integrals over LNSD: (1) the harmonic mean approximation, (2) the method in which deviation of the Fuchs/Fuchs‐Sutugin corrections from the harmonic mean is represented by the value at D_g (referred to as F(D_g), equations A6 and A15), and (3) the highly accurate (but costly) Gauss‐Hermite quadrature method which serves as a reference (referred to as FFS(Q), equations A7a, A7b, and A16). For the F(D_g) method, the deviation term should be larger when σ_g,i is large because the correction factors with D_g may not be representative near the edges of a broad LNSD.

[28] When Fuchs and Fuchs‐Sutugin corrections are made in the F(D_g) method, a certain representative diameter should be determined. However, those correction factors for a mode with large σ_g will vary significantly depending on which moment of D_gk is considered to be representative. The geometric mean diameter for the kth moment (D_gk) is always larger than D_g (=D_g0: number equivalent geometric mean diameter) when k > 0, as it is given by

(5)

[29] When D_g0 = 100 nm with a small σ_g of 1.3, the volume equivalent (k = 3) mean diameter D_g3 = 123 nm, whereas with a large value of σ_g = 2.5, D_g3 = 1.2 µm, more than 1 order of magnitude larger than D_g0. Using the mean diameter for the surface area (D_g2) might work well for the correction factor of condensational growth in the free‐molecular regime, but such a treatment does not work well for all correction factors. If σ_g is small enough, D_g3 /D_g0 is close to unity and thus D_g0 should be sufficiently representative for any of the correction factors. In this sense, accuracy can be ensured by setting an upper limit σ_g,max for σ_g.

[30] There are two ways to adjust σ_g,i: (1) When σ_g,i exceeds σ_g,max, it can be adjusted to σ_g,max, while preserving M_0,i and M_3,i, which results in a discrepancy in D_g,i and M_2,i; and (2) a mode with σ_g,i that exceeds σ_g,max can be divided into smaller and narrower modes. For the LNSD, arbitrary moments for diameters between D_lw,i and D_up,i can be expressed by using the error function (erf) as

(6a)

[31] Using the approximation [Ghan et al., 2011], equation 6a becomes an algebraic function that can reduce the computational cost considerably:

(6b)

where and , respectively. When σ_g,i of bin i exceeds σ_g,max, the mode can be discretized into three narrower child distributions with , , and , which are redistributed to the bins i − 1, i, and i + 1, respectively. However, even though the mode redistribution with equations 6a and 6b conserves all the moments after the redistribution of a mode, the shapes of their size distributions definitely change from their previous shape, which leads to different condensational and coagulation growth afterward.

[32] Therefore, accuracy can be ensured by setting an upper limit σ_g,max for σ_g, where σ_g,max should not be too small so that σ_g,i is adjusted too frequently. With σ_g of 1.3, 99.2%, 98.2%, and 96.8% of number, surface area, and volume (mass) are with the 0.5D_g0 to 2D_g0 size range (factor of 2), whereas only 55.1%, 13.6%, and 2.3% of them for σ_g of 2.5. In the current study, we set σ_g,max as 1.8, so that D_g3/D_g0 and D_g2/D_g0 are 2.8 and 2.0, and 76.3%, 49.2%, and 27.8% of number, surface area, and volume (mass) are with the 0.5D_g0 to 2D_g0 size range, and 98.0%, 82.5%, and 59.6% of them are with the 0.2D_g0 to 5D_g0 size range (factor of 5), respectively. It is noted here that in the simulation conditions presented in this paper, σ_g exceeded 1.8 and this artificial mode separation, equations 6a and 6b, was applied only for the bin resolution of BIN4 (four bins between 1 nm and 1 µm, with Δ(log D) = 0.75); equations 6a and 6b were never applied for the finer bin resolutions.

2.2 Merits of MBHM

[33] There are four merits of this approach:

1. [34]

   MBHM is a triple moment sectional aerosol dynamics model, and thus, surface area is conserved in addition to number and volume (mass) during transport, mixing, and transfer of particles between bins.

2. [35]

   MBHM gives a physically consistent form of intrasectional distributions.

3. [36]

   It is computationally efficient with low numerical diffusion.

4. [37]

   Only a single bin is sufficient to simulate full aerosol dynamical processes for particles whose size distribution can be approximated by a single lognormal mode. This discretized and/or single‐mode alternative is well suited for process‐oriented (multicategory and/or mixing state) modeling to efficiently simulate a variety of complex atmospheric aerosol properties and processes.

[38] These merits are described in detail in sections 2.2.1–2.2.4-sections 2.2.1–2.2.4.

2.2.1 Consistency in Aerosol Surface Area Modeling

[39] DMS methods are widely used for aerosol dynamics calculations in global‐ and regional‐scale chemical transport models, which explicitly treat aerosol chemical, thermodynamical, and microphysical processes [e.g., Jacobson, 1997; Zhang et al., 2004; Zaveri et al., 2008]. While aerosol number and mass concentrations are preserved in DMS, surface area is not always preserved: When two groups of aerosols with different diameters (D_i1 and D_i2) meet in the same bin i after a dynamical process (condensation and coagulation) or transport (advection or diffusion) in a 3‐D simulation, the moments of merged aerosols (D_i) in the bin of DMS will be

(7)

[40] In DMS, M_k,i is defined as the integrated moment strictly within the bin boundaries

(8)

[41] When the intrasectional distribution is assumed monodisperse, equation 8 is

(9)

where D_m,i is the mean diameter of particles in the bin, defined as (M_3,i/M_0,i)^1/3 in a so‐called moving‐center approach (MCA) [Jacobson, 1997]. Because the second moment (surface area) is not predicted in DMS, it is then diagnosed by the following equation:

(10)

[42] The preserved form of the second moment in DMS‐MCA in the event of equation 7 should be

(11)

which is different from that actually derived with DMS‐MCA (equation 10 plus equation 7) as

(12)

[43] The difference between equations 11 and 12 is a numerical error, generated whenever two different sizes of aerosols meet in the same bin, and might be accumulated for the whole time integration. The error may not be large with finer size resolutions, but it will increase as the resolution becomes coarser. In contrast, the Modal Bin Hybrid Model preserves the second moment and thus the surface area as

(13)

[44] There are similar discrepancies in surface area prediction with the double‐moment modal method (DMM), which is widely used in global climate models [Stier et al., 2005; Wang et al., 2009; Aquila et al., 2011; Liu et al., 2012]. In DMM, the double moment is obtained with prescribed geometric standard deviation σ_g, specific to each category of modes, and constant throughout the simulation. σ_g is typically set from 1.6 to 2.0. When σ_g is changed after any process operators (i.e., transport or aerosol dynamics processes), σ_g is adjusted to the constant value. Also, mode merging (renaming or reallocation) is done according to shrinking and swelling of a mode: Fractions of M₀ and M₃ transfer from one mode to another (e.g., nucleation mode to Aitken mode to accumulation mode) [Binkowski and Shankar, 1995; Wilson et al., 2001; Easter et al., 2004]. The mode merging using equations 6a and 6b is called a gradual merging approach (GMA) and is implemented in MBHM with an alternative of MCA, too.

[45] Figure 1 illustrates how the method with constant σ_g together with GMA while preserving M₀ and M₃ (double moment) produces discrepancies in other parameters such as D_g and M₂. If a mode with N = 10³ cm⁻³, D_g = 100 nm, σ_g = 1.6 (black solid line in Figure 1a) is divided at a certain diameter by equations 6a and 6b, a couple of modes (grey solid lines) were produced with D_g, smaller and larger than the initial diameter (100 nm) as shown in Figure 1b. The divided mode with smaller D_g is narrower (smaller σ_g) than the other one but both values of σ_g are smaller than the original diameter (Figure 1c). The sum of the divided modes (black dashed lines) is different than that of the initial mode, although the three moments are preserved (Figure 1a). This is the reason why GMA is not used as a standard method for MBHM.

Figure 1

(a) Number size distributions of a mode (N = 10³ cm⁻³, D_g = 100 nm, σ_g = 1.6) (black solid line), a couple of modes (grey solid line), and sum of the modes (black dashed line), divided at a diameter of 100 nm using the error function equations 6a and 6b (grey dashed line). (b) D_g of the couple of divided modes, divided at various diameters (10, 20, 50, 100, 200, 500, and 1000 nm), with the initial modes of N = 10³ cm⁻³, D_g = 100 nm, and various σ_g in x axis. The red dots indicate that the division was made at the central diameter, 100 nm. (c) Same as Figure 1b but for σ_g of the couple of the divided modes. (d) D_g after adjusting σ_g to a constant value (1.6), while preserving M₀ and M₃, from the initial modes with D_g = 100 nm and various σ_g in x axis (equation 14).

[46] With DMM, σ_g of a couple of modes, which are always smaller than σ_g of the mother mode (Figure 1c), is adjusted to the constant value after the mode merging (Figure 1d). With the lognormal distribution function, D_g and M₂ are written by

(14)

(15)

using equation 2. After adjusting σ_g of the divided modes larger to the constant value, D_g and M₂ of both modes become smaller artificially. (We have evaluated double‐moment versions of MBHM that use constant σ_g with either GMA or MCA. The double moment with GMA version did not perform well due to the artifact. The double‐moment method with MCA performed significantly better, although not as well as the triple‐moment MBHM. These results are not included in this study, which focuses on the triple‐moment MBHM. One can also refer to Li et al. [2008], which shows that in a 3‐D Eulerian framework, a triple‐moment modal method produces results in better agreement with the sectional method than a DMM.)

[47] This inconsistency in M₂ in DMS and DMM has been poorly addressed so far, even though several important aerosol processes depend on aerosol surface area: gas‐to‐particle mass transfer, heterogeneous reaction, and light extinction cross section. It also indirectly affects the number concentrations of aerosols and cloud condensation nuclei (CCN), because particle nucleation from gases with low vapor pressures (sulfuric acid and secondary organic gases) competes with their condensational loss rates.

2.2.2 Physically Consistent Form of Intrasectional Distributions

[48] A more accurate representation of the size distribution of particles within each bin (the intrasectional distributions, n_i(ln D)) allows more accurate calculation of aerosol dynamical processes (e.g., condensation, coagulation, removal) and a more accurate simulation. Although it may not be a big problem for very fine grid resolutions, it causes substantial errors at coarser resolution. There are several ways to determine the intrasectional distribution in DMS methods, such as the moving‐center approach which is monodisperse [Jacobson, 1997], the Zhang et al. [2002] approach which is log linear in size, linear discrete method (LDM) which is linear in volume [Simmel and Wurzler, 2006], and piecewise lognormal approximation (PLA) [von Salzen, 2006]. For example, PLA was proven to be as accurate as the results of a single‐moment bin approach with three times as many sections. However, even though the intrasectional distributions are determined by number and volume, the whole shape of the size distribution as a summation of intrasectional distributions is not necessarily a natural, contiguous, or physically consistent form. (The noncontiguous red lines at D_p of 1 µm in Figure 3 of von Salzen [2006] is a good example.)

[49] In contrast, the MBHM intrasectional distribution is a complete lognormal size distribution (equations 1 and 2), which is physically determined by the influence of aerosol dynamical processes on its three moments. Certainly, there is inconsistency between the lognormal size distribution and the true distribution in nature, but there is no nonphysical assumption made to determine the intrasectional distribution of MBHM. This is probably the main reason why MBHM performs well at coarser size resolutions.

[50] As discussed before in section 2.1.2, adjustment of σ_g,i should not be applied too frequently because it causes changes in shapes of LNSD that lead to different condensational and coagulation growth afterward. The adjustment of σ_g,i is better avoided also because the adjusted LNSD is not a physically consistent form.

2.2.3 Computational Efficiency and Low Numerical Diffusion

[51] MBHM uses the fixed bin boundary structure and the moving‐center approach (MCA) [Jacobson, 1997], which is accurate and produces less numerical diffusion than other methods such as stationary bin structure (fixed volume in each bin), semi‐Lagrangian, or Lagrangian techniques in condensational and coagulation growth simulations [Jacobson, 1997; Zhang et al., 2004]. In MBHM, when the number equivalent geometric mean diameter D_g,i of bin i becomes larger or smaller than the boundary diameters D_up,i or D_lw,i, all three moments in the bin are transferred to the next bins i + 1 or i − 1. MBHM with MCA is computationally efficient in the sense that it is not necessary to calculate the portion (or flux) of moments exiting and entering the grid cells. Comprehensive comparisons of computational efficiency between MBHM and DMS are presented and discussed in section 3.5.

[52] MBHM does require one additional transported tracer, M_2,i, for each bin. In many 3‐D CTMs that use DMS, the transported variables for each section include number and 10 or more chemical component masses (e.g., SO₄²⁻, NO₃⁻, Cl⁻, NH₄⁺, Na⁺, dust components, black carbon, primary organic aerosol, several lumped secondary organic aerosol, unidentified components, etc. Note that M_3,i is normally not transported but is diagnosed from mass concentrations of the chemical components and their densities.). By adding M_2,i, for each section, the number of tracers for MBHM is only about 10% (or less) greater than for DMS at the same size resolution. However, the greater accuracy of MBHM at coarser size resolution, compared to DMS, may allow use of fewer sections with MBHM, resulting in considerable computational savings in a 3‐D CTM.

2.2.4 Applications for Process‐Oriented Modeling

[53] MBHM is made by sorting by size a number of modes of MADMS. A single bin of MBHM is a single mode of MADMS. Therefore, using the integrodifferential moment dynamics equations, full dynamical processes such as emission, transport, nucleation, condensation, coagulation, and deposition can be solved for a single bin, in the same manner as MADMS (or MAD) to solve the processes for a single lognormal mode. Other size distribution functions can be alternatively used for the intrasectional distribution of the modal‐bin hybrid approach, e.g., a quadruple‐moment approach with modified gamma distribution for aerosol dynamics [Williams, 1986] or a triple‐moment closure with three‐parameter gamma developed for cloud microphysics [Milbrandt and Yau, 2005].

[54] This discretized and/or single‐mode alternative is well suited for process‐oriented (multicategory and/or mixing state) modeling to efficiently simulate a variety of complex aerosol properties and processes. Sizes of primary aerosols such as soot (~100 nm), sea salt (~1 µm), dust (~1 µm), and pollen (~10 µm) should not vary widely in time and space, so each type of primary aerosol can be represented by a single mode or a combination of a few modes. On the other hand, because secondary aerosols vary significantly from the nucleation mode (1 nm) to the submicron particles (~0.1–1 µm) in time and space, by 2 to 3 and 6 to 9 orders of magnitude in size and volume, respectively, those should be discretized into an adequate number of modes, depending on the accuracy required. An additional dimension can be added to MBHM to represent mixing state, in order to simulate changes in hygroscopicity and light absorption of aged black carbon more accurately, as in Oshima et al. [2009], Zaveri et al. [2010], and Matsui et al. [2013], or a dimension can be added to represent photochemical age for tracking changes in particle viscosity, hygroscopicity, and thus uptake coefficients of organic aerosols, as proposed by Abbatt et al. [2012]. Processes such as nucleation, condensation, and coagulation are consistently calculated over all sectional and categorical modes.

3 Simulation and Discussions

3.1 Simulation Settings and Definitions of Terms

[55] To characterize and compare the performance of the MBHM and DMS methods, we have conducted box model simulations for several ideal cases of growth under simultaneously occurring nucleation, condensation, and coagulation processes with various size resolutions. Performance is evaluated in terms of prediction accuracy in concentrations of total number and surface area as well as number and volume of particles larger than selected diameters which are used as proxies for CCN number concentrations and wet deposition amount, respectively.

[56] We use a simple power law model for the nucleation rate of sulfuric acid gas with parameters based on measurements in diverse atmospheric locations [Kuang et al., 2008], described as J₁ = K ⋅ [H₂SO₄]^P, where J₁ is the formation rate of 1 nm particles (cm⁻³ s⁻¹) and [H₂SO₄] is the number concentration of sulfuric acid molecules (cm⁻³). We chose 10^−12.4 for the prefactor K of the kinetic model nucleation rate (P = 2), an intermediate value among the observations. The new particles are assumed to be spheres, uniformly composed of sulfuric acid with a diameter of 1 nm. Throughout the 24 h simulation period, the production rate of sulfuric acid gas is set at 5 × 10⁻⁶ µg m⁻³ s⁻¹. This is a typical daytime value for a moderately polluted air mass, obtained from a three‐dimensional chemical transport simulation of East Asia. Nucleation is only allowed during the first 6 h, after which it is turned off. A second condensing gas, representing organics but assumed nonvolatile for simplicity, is also treated. Its production rate is set as 8 × 10⁻⁵ µg m⁻³ s⁻¹ throughout the simulation. The particle density is assumed constant as 1.83 g cm⁻³. Air pressure and temperature are set at 1000 hPa and 298.15 K, respectively.

[57] There are no specific targets of real atmospheric events for this simulation. We chose the combinations of values to focus on the evolution processes of aerosols from new particles to submicron within 24 h by simultaneously occurring nucleation, condensation, and coagulation. There are three sensitivity studies with different initial conditions for preexisting submicron particles which are LNSD with D_g and σ_g of 80 nm and 1.6. We set the three conditions to see the differences in the amount of nucleation and evolution of the new particles. No preexisting particles results in strong nucleation, a moderate preexisting particle concentration (10³ cm⁻³) gives weaker nucleation, and a high preexisting particle concentration (10⁴ cm⁻³) results in no apparent nucleation. A fourth sensitivity study uses 10³ cm⁻³ preexisting particles with a uniform number distribution between 25.38 and 206.67 nm diameter.

[58] As mentioned in section 2.1.1, the model time step Δt is set to 1 s throughout all the simulations, because this gave sufficiently accurate results for nucleation events with various combinations of nucleation rate parameters, sulfuric acid gas production rates, and preexisting particle concentrations.

[59] Before examining the simulation results, we define some terminology and performance metrics that are used in the next sections (these are also listed in Table 1). BINx indicates size resolution, where x is the number of bins over 3 orders of magnitude in size. For example, BIN64 indicates that there are 64 bins between 1 nm and 1 µm, and thus, Δ(logD) = 3/x = 0.046875. The bin central diameters range from 1 nm to ~10 µm, with two of the bins exactly at 1 nm and 1 µm, and with bin boundaries at logarithmic half intervals between each pair of central diameters. In this study, BIN4, 8, 16, 32, 64, 128, and 256 simulations are performed for each preexisting particle condition and each method.

[60] CCNx is the number concentration of particles with diameters larger than x nm. CCN100 and CCN50 are used as proxies for cloud condensation nuclei (CCN) concentrations at supersaturations of about 0.21 and 0.60 (assuming mean hygroscopicity of 0.3). CCVx is a M₃ (∝ volume) concentration for particles with diameters that are larger than x nm, and it is used as a proxy for the volume (or mass) of aerosol subject to nucleation scavenging at a particular supersaturation. Equations 6a and 6b can be used to obtain CCNx (with k = 0) and CCVx (with k = 3) for the MBHM approach. For DMS approaches, the intrasectional distribution must be specified. For DMS‐MCA, the intrasectional distribution is monodisperse. However, using this to calculate CCNx and CCVx results in substantial errors with coarser grid resolutions. Therefore, when calculating CCNx and CCVx for all of the DMS approaches, we assume that the intrasectional number distribution is a linear function of D³, consistent with LDM [Simmel and Wurzler, 2006].

[61] We used the following metrics to estimate simulation accuracy as listed in Table 1:

(16a)

(16b)

and

(16c)

(16d)

where χ^eval and χ^ref are predicted variables to from a simulation being evaluated and from a reference simulation, respectively. We used MNB (equation 16a) and MaxNE* (equation 16b) to evaluate the deviations of the harmonic mean approximation from the more accurate Fuchs and Fuchs‐Sutugin interpolations of the condensation and condensation terms ψ and β, respectively. Therefore, the bars in equations 16a and 16b are averages for diameter ranges between 1 nm and 10 µm. In contrast, we used NMB, NME (equation 16c), and NMaxE* (equation 16d) to evaluate model performance, for example, to compare M_k, dM_k/dt, CCNx, and CCVx from a simulation with BINx to those from a simulation with the highest resolution BIN256, or to compare results from a simulation with the harmonic mean approach to those from a simulation with the Fuchs and Fuchs‐Sutugin approaches with the same resolution. Therefore, the bars in equations 16c and 16d are averages over time.

[62] As listed in Table 2, six types of numerical methods are tested from Methods I to VI. In section 3.2, MBHM (Method I) is tested against the two DMS methods (Methods II and III) to see the differences between triple‐ and double‐moment sectional methods. Methods I–III all use harmonic mean interpolation for coagulation and condensation rates. Method II (DMS‐MCA‐HM) employs the moving‐center approach [Jacobson, 1997; Zhang et al., 2004] for condensational growth calculation, whereas Method III (DMS‐LDM‐HM) uses the linear discrete method (LDM) [Simmel and Wurzler, 2006; Zaveri et al., 2008] for condensational growth. MCA is employed for coagulation by the both Methods II and III. MCA uses fixed boundaries and M₀ and M₃ are fully transferred to neighboring bins when the diameter D = (M₃/M₀)^1/3 exceeds the boundaries, whereas LDM assumes that the intrasectional size distribution is linear in particle volume. The DMS‐MCA methods are documented in detail in Jacobson [1997] and Zhang et al. [2004] and the DMS‐LDM in Simmel and Wurzler [2006].

Table 2. List of Numerical Methods for Size Shift, Coagulation, and Condensation Theories for the Transition Regime, Equations to Describe the Theories (Coagulation and Condensation), and Sections in Which the Methods Are Compared and Evaluateda^a LDM was used for condensational growth, whereas moving‐center approach was used for coagulation.

Number Name		Size Shift	Coagulation	Condensation	Equations	Section
I	MBHM‐HM	Moving Center	Harmonic mean	Harmonic mean	(A3), (A11)	3.2, Appendix B
II	DMS‐MCA‐HM	Moving Center	Harmonic mean	Harmonic mean	(A4), (A9)	3.2
III	DMS‐LDM‐HM	Linear Discretea^a LDM was used for condensational growth, whereas moving‐center approach was used for coagulation.	Harmonic mean	Harmonic mean	(A4), (A9)	3.2, Appendix B
IV	DMS‐LDM‐FFS	Linear Discretea^a LDM was used for condensational growth, whereas moving‐center approach was used for coagulation.	Fuchs	Fuchs‐Sutugin	(A5), (A12a)	3.3, 3.4, Appendix B
V	MBHM‐F(Dg)	Moving Center	Fuchs	Fuchs	(A6), (A15)	3.3, 3.4, Appendix B
VI	MBHM‐FFS(Q)b^b Fuchs and Fuchs‐Sutugin methods are applied over LNSD using the quadrature method with the quadrature order n = 20.	Moving Center	Fuchs	Fuchs‐Sutugin	(A7a), (A7b), (A16)	Appendix B

• ^a LDM was used for condensational growth, whereas moving‐center approach was used for coagulation.
• ^b Fuchs and Fuchs‐Sutugin methods are applied over LNSD using the quadrature method with the quadrature order n = 20.

[63] Comparisons between MBHM/DMS using the harmonic mean approach (Methods I and III) and MBHM/DMS using the more accurate Fuchs and Fuchs‐Sutugin approaches (Methods IV–VI) are presented in Appendix B. There are two methods for MBHM with the more accurate approaches, MBHM‐F(D_g) (Method V) and MBHM‐FFS(Q) (Method VI). MBHM‐F(D_g) uses an approximated correction factor derived as a function of D_g in order to obtain the analytical solutions of the coagulation and condensation MDEs for LNSD, whereas MBHM‐FFS(Q) uses the quadrature method with quadrature order n = 20 to obtain the highly accurate integrals of MDEs for LNSD using the Fuchs and Fuchs‐Sutugin approaches. We found that the MBHM‐F(D_g) method was in close agreement with the most accurate MBHM method (FFS(Q), see discussion in Appendix B), and we thus used the F(D_g) approach for the simulation results presented in sections 3.3 and 3.4. Details of the equations and derivations of all the methods are presented in Appendix A.

[64] In section 3.3, comprehensive comparisons between Methods IV (DMS‐LDM‐FFS) and VI (MBHM‐F(D_g)) (the best options for MBHM and DMS in the study, respectively) are made for various size resolutions (BINx) and variables (M_k, dM_x/dt, CCNx, and CCVx). In section 3.4, tests of Method V (MBHM‐F(Dg)) with non‐lognormal initial size distributions are presented. In section 3.5, the computational efficiencies of the MBHM and DMS methods are discussed.

3.2 Triple‐ Versus Double‐Moment Sectional Methods

[65] Figure 2 compares the temporal evolution of number size distributions simulated with initial number concentrations of 0, 10³, and 10⁴ cm⁻³ of preexisting particles by MBHM‐HM (Method I) and DMS‐LDM‐HM (Method III) for the finest size resolution (BIN256). The MBHM and DMS simulations agree well. The left column of Figure 3 shows the gas‐to‐particle conversion rates of sulfate (dM₃/dt) due to nucleation, condensation to particles in the nucleation mode (NUC; defined here as particles smaller than D = 10 nm), condensation to particles in the Aitken and accumulation modes (AAA; defined as particles larger than D = 10 nm).

Figure 2

Time series of simulated number size distributions of (a, c, e) Method I (MBHM) and (b, d, f) Method III (DMS‐LDM) with the initial number concentrations of 0, 10³, and 10⁴ cm⁻³ of preexisting particles (D_g = 80 nm and σ_g = 1.6) for the highest size resolution (BIN256).

Figure 3

Time series of (a, c, e) gas‐to‐particle conversion rates of M₃ due to nucleation (red), condensation to particles in the nucleation mode (NUC, defined here as D < 10 nm) (blue), and condensation to particles in the Aitken and accumulation mode (AAA, defined here as D > 10 nm) (green) and (b, d, f) conversion rates of M₃ of NUC to AAA due to condensation (red) and coagulation (blue) to AAA particles for DMS‐LDM‐HM‐BIN64 simulations with preexisting particles concentrations of 0, 10³, and 10⁴ cm⁻³.

[66] In the absence of preexisting particles (Figures 2a and 2b), within 2 h, new particles form and grow by condensation to the nucleation mode (D < 10 nm) (red and blue in Figure 3a and red in Figure 3b). From 2 to 6 h, some of the particles grow continuously to Aitken mode (D ~ 10nm) by condensation, but more coagulate to the larger sizes (D ~ 10–50 nm) (blue in Figure 3b). After ceasing new particle formation at 6 h, particles grow by condensation and coagulate with AAA to a volume equivalent diameter of about 100 nm at 24 h. In the presence of preexisting particles of 10³ cm⁻³ (Figures 2c and 2d), the condensational loss of H₂SO₄ vapor is slow enough within 2–4 h (green in Figure 3c) so that the banana curve (equals the nucleation and condensational growth to the accumulation mode) is still evident and the same features are simulated as the case without preexisting particles. With a higher number of preexisting particles of 10⁴ cm⁻³ (Figures 2e and 2f), the banana curve is not evident because the condensation rate is much faster on submicron particles than on new particles (green in Figure 3e) and new particles coagulate almost immediately with preexisting particles (blue in Figure 3f).

[67] Figure 4 compares temporal evolutions of number size distributions simulated without preexisting particles by MBHM (Method I) and DMS‐LDM (Method III) for various size resolutions (BIN64, 16, 8, and 4). It should be noted here that the actual LNSD intrasectional distributions (denoted as full) are drawn for MBHM, whereas uniform (denoted as flat) rather than the method's actual intrasectional distributions are drawn for DMS. The MBHM and DMS‐LDM simulations agree well at the finest resolution (BIN64) and the patterns are visually similar at BIN16 resolution. The patterns between MBHM and DMS‐LDM become different at the coarser BIN8–4 resolutions. One important feature is the reddish shaded area for dN/dlogD > 500 × 10³ cm⁻³. The area is larger for MBHM‐BIN16 than for DMS‐LDM with the same resolution. This reflects less numerical diffusion with the MCA approach in MBHM. Another remarkable feature is that the evolution of the size distribution in MBHM after nucleation ends is consistent for BIN4 to 256.

Figure 4

Time series of simulated number size distributions of Method I (MBHM‐HM) and Method III (DMS‐LDM‐HM) with BIN64, BIN16, BIN8, and BIN4 no preexisting particles. Note that full intrasectional size distributions are used for plotting MBHM (denoted as full) but not for DMS (flat intrasectional distributions are used, denoted as flat).

[68] Figure 5 compares the temporal evolutions of number size distributions for BIN64 and 8 with MBHM (full), MBHM plotted using flat distribution (flat), DMS‐MCA, and DMS‐LDM. Patchy features for MBHM (flat) and DMS‐MCA are due to the moving‐center approach: The full moments are shifted to neighboring sections when the mean diameter exceeds the bin boundaries. For the simulations with BIN64, the size distribution at 24 h approaches LNSD (Figures 5a and 5g). However, with BIN8, the size distribution at 24 h is represented by one or two bins (Figures 5d, 5f, and 5h). Still, with the single bin of MBHM, this LNSD feature can be reproduced (Figure 5b).

Figure 5

Time series of simulated number size distributions of BIN64 and BIN8 with (a, b) MBHM‐HM (full), (c, d) MBHM‐HM (flat), (e, f) DMS‐MCA‐HM (flat), and (g, h) DMS‐LDM‐HM (flat). Results are for the no preexisting particles case.

[69] To see the structure and differences at coarser resolutions more clearly, Figures 6 and 7 illustrate snapshots of size distributions of number and the third moment, respectively, at 2, 6, 12, and 24 h for simulations with MBHM (full and flat) (Method I), DMS‐MCA (Method II), and DMS‐LDM (Method III) at resolutions BIN64, 16, and 4. The D_g,i and σ_g,i are depicted by the dots in the MBHM (flat) panels (Figures 6b, 6f, 6j, 7b, 7f, and 7j).For new particles, σ_g,i =1. Condensation growth never caused changes in σ_g,i when σ_g,i = 1, but condensational growth and nucleation make σ_g,i larger. Coagulation makes σ_g,i approach about 1.3. As time passes, D_g,i and σ_g,i become larger. A jagged structure (i.e., patchy features) is observed in MBHM and DMS‐MCA in Figures 4 and 5. This jagged structure is because MCA produces zero concentrations in some size bins. However, the jagged structure never causes numerical instability or significant numerical errors in total number and volume concentrations with either DMS‐MCA or MBHM.

Figure 6

Snapshots of simulated number size distribution of (a, e, i) MBHM‐HM (full), (b, f, j) MBHM‐HM (flat), (c, g, k) DMS‐MCA‐HM (flat), and (d, h, l) DMS‐LDM‐HM (flat) with BIN64, BIN16, and BIN4 at 2 (grey), 6 (red), 12 (green), and 24 h (blue) after the initial time. In the MBHM (flat) panels, D_g,i and σ_g,i are depicted with dots: The vertical position of each dot is σ_g,i (x axis) and the horizontal position is D_g,i (y axis). Results are for the no preexisting particles case.

Figure 7

Same as Figure 6 but for size distributions of M₃.

[70] The remarkable feature of MBHM is that even at the coarsest BIN4 resolution, the shapes of the second peaks (diameter larger than about 10 nm) at 2 and 6 h are consistent with those for all BINx of MBHM (full) and BIN64 of DMS‐LDM. The evolution of the size distribution in MBHM after nucleation ends is consistent from BIN4 to 256 (Figure 4). The distributions of the third moment simulated at 6, 12, and 24 h with MBHM (Figure 7) are also very similar for all BINx and finer resolutions of DMS‐MCA and DMS‐LDM. Condensation makes σ_g,i smaller. There is only one section (the third section between about 10 and 100 nm), filled at 24 h with BIN4 (Figures 6j and 7j). Within this section, the increase in D_g,i and decrease in σ_g,i by condensational growth are well reproduced. This feature is not reproduced by either DMS method at BIN4 resolution.

[71] MBHM simulates size‐integrated measures of the aerosol very well for all resolutions while DMS‐MCA and DMS‐LDM do not. Figure 8 shows the time series of total number M₀, total second moment M₂, CCN50, and CCV50 for MBHM, DMS‐MCA, and DMS‐LDM with the size resolutions BIN64, 32, 16, 8, and 4. With MBHM, there are no significant discrepancies between BINx for all variables shown in the figure, as well as M₃, CCN100, and CCV100, which are not shown. In contrast, with DMS‐MCA and DMS‐LDM, there are noticeable discrepancies in M₂ (about 10% overestimation for BIN4 against BIN64) because M₂ is not preserved in DMS. Also, there are significant discrepancies in CCN50 and CCV50 due to the linear intrasectional distribution assumption. The MBHM with coarser resolution gives more accurate intersectional distributions. The discrepancies in CCN50 and CCV50 are significantly larger with DMS‐MCA than those with DMS‐LDM because of the jagged structure produced by MCA. CCN50 and CCV50 are inaccurate even with BIN64 of DMS‐MCA. The better reproduction of CCN50 and CCV50 by DMS‐LDM than DMS‐MCA with coarser resolutions shows the intrasectional assumption of LDM produces better results for this type of metric. However, the lognormal intrasectional distribution of MBHM gives more accurate results than the two DMS methods.

Figure 8

Time series of M₀, M₂, CCN50, and CCV50 with various resolution of sizes, BIN4 (pink), BIN8 (blue), BIN16 (green), BIN32 (red), and BIN64 (black), for MBHM‐HM, DMS‐MCA‐HM, and DMS‐LDM‐HM. Results are for the no preexisting particles case.

[72] Figures 9-11 are identical to Figures 5, 7, and 8 but for the case with a moderate preexisting particle concentration (N = 10³ cm⁻³, D_g = 80 nm, and σ_g = 1.6). In Figure 10, the shapes of all the double peaks at 2, 6, 12, and 24 hr for BIN4 to BIN64 of MBHM (full) are consistent with those for BIN64 of DMS‐LDM. In Figure 11, the discrepancies of M₂ for DMS, which are still only about 10%, are somewhat larger than those without preexisting particles (Figures 8e and 8f).

Figure 9

Same as Figure 5 but with N = 10³ cm⁻³, D_g = 80 nm, and σ_g = 1.6 of preexisting particles.

Figure 10

Same as Figure 7 but with N = 10³ cm⁻³, D_g = 80 nm, and σ_g = 1.6 of preexisting particles.

Figure 11

Same as Figure 8 but with N = 10³ cm⁻³, D_g = 80 nm, and σ_g = 1.6 of preexisting particles.

[73] The reason why M₂ of DMS is always overestimated can be explained as follows. The overestimate of M₂ is there from the beginning, when the preexisting particle distribution is discretized into sections with the two moments M₀ and M₃. Figure 12 shows the ratio between M₂ after discretization by DMS (M₂^dis) and the true value (M₂^true), with various σ_g (1.0–2.5) and BINx (4–30). The true value is obtained by equation 3. The discretized value is obtained by the following procedure: equation 6a is used to obtain M_0,i and M_3,i in each section, then M_2,i of each section is derived by equation 11, and then . The discrepancy between M₂^true and M₂^dis becomes larger with larger σ_g and coarser resolution. Atmospheric aerosols are usually observed with σ_g smaller than about 2.0 and numerical models employ size resolutions finer than BIN4, so the discrepancy would not be very large. For example, with σ_g = 2.0, the overestimate of M₂ of a discretized mode is 16.0%, 5.6%, 1.5%, and 0.4% for BIN4, 8, 16, and 32, respectively.

Figure 12

The ratios of the second moments after discretization by DMS M₂^dis to the true second moments M₂^true with various σ_g (1.0–2.5) and size resolutions (BIN4–30).

3.3 Comprehensive Discussion of Errors With Different Methods and Size Resolutions

[74] In this section we compare simulation accuracy between the two methods DMS‐LDM‐FFS (Method IV) and MBHM‐F(D_g) (Method V) at various resolutions (BINx) with respect to the highest resolution BIN256, using the NME metrics for M_k and dM_k/dt of each process (Figure 13) and the NMaxE* metrics for M₂, CCNx, and dM_k/dt of each process (Figure 14). In this section and in Figures 13 and 14, MBHM‐F(D_g)‐BINx and DMS‐LDM‐FFS‐BINx are abbreviated to MBHM‐F(x) and DMS‐LDM‐FFS(x), respectively.

Figure 13

NMEs of BINx to BIN256 for (a) M₀, (b) M₂, (c) M₃, (d) dM₀/dt|_NPF,Cond, (e) dM₂/dt|_NPF,Cond, (f) dM₃/dt|_NPF,Cond, (g) dM₀/dt|_Coag., and (h) dM₂/dt|_Coag. of DMS‐LDM‐FFS(x: 4, 8, 16, 32, 64, 128) to DMS‐LDM‐FFS(256) (grey circles) and MBHM‐F(x: 4, 8, 16, 32, 64, 128) to MBHM‐F(256) (blue circles), and MBHM‐F(x: 4, 8, 16, 32, 64, 128, 256) to DMS‐LDM‐FFS(256) (red circles), without preexisting particles; dM_k/dt|_NPF,Cond and dM_k/dt|_Coag. indicate time derivative terms of moments due to each process, “new particle formation and condensation” and “coagulation”, respectively. Dashed black and red lines indicate the lines with slopes of −1 and −2 on a log‐log plane (y = 100 x⁻¹ and y = 100 x⁻²), thus showing the slopes of first and second orders of accuracy convergence, respectively.

Figure 14

NMaxE* of BINx to BIN256 for (a–c) M₂, (d–f) CCNx (x = 50 for N^t=0 = 0 and 10³ cm⁻³; x = 100 for N^t=0 = 10⁴ cm⁻³), (g–i) dM₃/dt|_NPF,Cond, and (j–l) dM₀/dt|_NPF,Cond with preexisting particles of N^t=0 = 0, 10³, and 10⁴ cm⁻³. Grey, blue, and red circles denote errors of DMS‐LDM‐F(x: 4, 8, 16, 32, 64, 128) to DMS‐LDM‐F(256), MBHM‐F(x: 4, 8, 16, 32, 64, 128) to MBHM‐F(256), and MBHM‐F(x: 4, 8, 16, 32, 64, 128, 256) to DMS‐LDM‐F(256), respectively. Note that in Figures 14a, 14c, 14g, and 14j, the red circles hide most of the blue circles.

[75] Figure 13 shows the accuracy convergence of the two methods (i.e., NMEs for various variables and their time derivatives versus resolution), for the case without preexisting particles. The slopes of both MBHM (blue) and DMS‐LDM (grey) are between −1 and 2 at finer resolutions, indicating first to second orders of accuracy convergence for the two methods. For DMS‐LDM, every slope becomes steeper from coarser to finer resolution and shows almost a second order of accuracy convergence between BIN64 and 128. This feature is not observed for MBHM. Therefore, accuracy convergence of MBHM‐F against DMS‐LDM‐FFS(256) (red) was shown here also. On the other hand, NMEs for MBHM (blue) are usually 1 to 2 orders of magnitude smaller than DMS‐LDM (grey), which indicates higher consistency of MBHM with coarser resolutions. Even MBHM against DMS‐LDM‐FFS(256) (red) showed about 1 order of magnitude better accuracy than DMS‐LDM (grey) at coarser resolution than BIN16–32. For all the variables and both methods, there are 3 orders of magnitude difference in NMEs between BIN4 and 128. Figure 13 only shows results for the case without preexisting particles, but similar features are found in the simulations with 10³ and 10⁴ cm⁻³ preexisting particles.

[76] Figure 14 is similar to Figure 13 but shows NMaxE* values for M₂, CCNx, dM₃/dt|_NPF &Cond., and dM₀/dt|_NPF &Cond. for all three preexisting particle cases. NMaxE* can indicate not only the maximum errors but their biases too. Note that the value of the error itself is not important in Figure 14, because the numerator of NMaxEast; is the maximum difference at a single time, while the denominator is a time average over the entire simulation. For example, the time average of M₀ is more weighted on the very low values for 6–24 h (Figures 8a–8c and 11a–11c) and thus, NMaxE* shows large negative values (<−100%) for dM₀/dt|_NPF &Cond., although NMEs are smaller than about 10% (Figure 11d). Thus, the relative magnitudes of the NMaxE* for the two methods at various resolutions are more important in Figure 14.

[77] Due to the surface area consistency in MBHM, errors in M₂ of MBHM‐F(x) are much smaller than those of DMS‐LDM at the same resolution and those of MBHM‐F(4) are even as small as DMS‐LDM‐FFS(32) in Figures 14a–14c, together with Figure 13b. Although the errors for M₀ and M₃ are less significant (Figures 13a and 13c), errors in CCNx of MBHM‐F(4–8) are as small as DMS‐LDM‐FFS(16–32) due to the more natural intrasectional distribution given by MBHM (Figures 14d–14f). In terms of the time derivatives shown in Figures 14g–14l, the maximum errors become larger for DMS‐LDM‐FFS(x) at coarser resolutions (x < 16). Due to the overestimation of M₂ for DMS for coarser resolutions, condensational growth rates of M₃ (proportional to M₂) were overestimated (Figures 14g–14i), and then the apparent nucleation rates dM₀/dt|_NPF &Cond. (= dM₀/dt|_NPF as condensation does not change number), competing with the condensation rates, were underestimated. Generally, the maximum errors in dM_k/dt of MBHM‐F(4–8) are as small as those of DMS‐LDM‐FFS(16–32) in Figures 14g–14l and 13d–13h.

3.4 Non‐LNSD Treatment in MBHM

[78] MBHM with coarser resolutions has been demonstrated to be accurate for the case with single LNSD preexisting particles. However, as discussed in section 2.1.2, the accuracy of MBHM is not assured for the cases when the size distributions of particles introduced into the model domain by initial and boundary conditions and emissions are not LNSD. LNSDs are often assumed for primary particle emissions [e.g., Dentener et al., 2006; Zender et al., 2003], but emitted size distributions for sea salt and mineral dust may not be lognormal [e.g., Martensson et al., 2003; Kok, 2011]. Even when emissions are treated as LNSD, if two or more of the emission LNSDs fall within a single MBHM section and have different D_g, the combination (i.e., sum) of these emissions is not a single LNSD. Also, an aerosol distribution that is initially LNSD will gradually evolve to non‐LNSD through growth and removal processes.

[79] In this section, in order to test the applicability of MBHM for arbitrary size distribution of initial and boundary conditions, MBHM is evaluated for two cases: one with preexisting particles having a LNSD that is discretized to multiple MBHM bins and a second with preexisting particles having square wave distribution with the same initial number and volume.

[80] The first case is the same as in Figures 9-11 (10³ cm⁻³ preexisting particles with D_g^t=0 = 80 nm and σ_g^t=0 = 1.6), but the initial distribution is discretized over multiple sections rather than being placed in a single LNSD section. The two moments and are discretized into sections using equations 6a and 6b as is done for DMS. There are three ways to determine the initial second moment , which is needed for MBHM: (1) calculate using equations 6a and 6b, (2) set , and (3) set . With all methods, the whole shape of the discretized distribution is not identical to the initial shape, but is preserved with method 1. Using method 2 with (or equivalently, method 3 with A = 0), the discretized initial size distribution is identical to DMS with MCA and is very discrete at coarser grid resolutions. Method 3 with A = 1 or 2, where σ_g,i is (logarithmically) proportional to the grid resolution, provides a more continuous distribution. Using one of these methods, an arbitrary size distribution can be discretized to a MBHM representation. We compare these methods with the reference simulations in which n^t=0(lnD) is placed in a single MBHM section rather than being discretized into multiple sections.

[81] Figure 15 illustrates time series of number size distributions of MBHM‐F(D_g) with high (BIN64) and low (BIN8) grid resolutions for M₂^t=0 derived using equations 6a and 6b (Figures 15a and 15b) and using (Figures 15c and 15d), and using with A = 1 or 2 (Figures 15e–15h), respectively. As noted in section 2.1.2, A = 1 is nearly equivalent to using equations 6a and 6b when the initial distribution is uniform (or nearly so) across a bin. There are ranges of produced by equations 6a and 6b as embedded in Figures 15a and 15b, but of a bin with maximum is close to A = 1. Figures 9a and 9b are the cases for BIN64 and 8 in which the preexisting particle distribution is not discretized, respectively. For BIN64 with = 1 (i.e., Figure 15c), the size distribution of preexisting particles are discrete during 0–12 h but the final size distribution for the all cases at 24 h is consistent with the reference case (Figure 9a). At the coarser BIN8 resolution, the size distribution of preexisting particles is discrete until 24 h for A = 0 (Figure 15d) or is merged with the secondary generated particles for A = 2 (Figure 15h) because was too large. NMEs (not shown) of various assumptions against the case without discretization of the preexisting particles indicate that derived using equations 6a and 6b performed best for most of the resolutions and model variables with NMEs mostly smaller than 1%. The monodisperse distribution assumption ( =1 or A = 0) performed well at finer resolutions (BIN32–64), with NMEs of 0.015–0.46%, compared to 0.011–0.31% for the equations 6a and 6b approach. At coarser resolutions (BIN4–16), where the equations 6a and 6b approach NMEs were 0.011–0.47%, the assumption with A = 1 also performed well (NMEs of 0.014–0.93%) and the monodisperse distribution assumption had the largest NMEs (0.018–1.68%).

Figure 15

Time series of simulated number size distributions of MBHM‐F(D_g) with high (BIN64) and low (BIN8) grid resolutions with the LNSD preexisting particles (N = 10³ cm⁻³, D_g = 80 nm, σ_g = 1.6). Discretization was made for (a) the three moments (M₀, M₂, and M₃) using equations 6a and 6b, (b) for the two moments (M₀ and M₃) of the initial size distribution with the prescribed standard deviations, (c, d) =1, (e, f) , and (g, h) . Ranges and values of are embedded in the bottom right of each panel.

[82] Figure 16 illustrates simulation accuracy of MBHM‐F(x) with discretized LNSD preexisting particles using equations 6a and 6b (denoted as MBHMdis‐F(x)) with respect to the finest resolution BIN256 of MBHMdis‐F(256) (blue circle) and DMS‐LDM‐FFS(256) (red circle). The simulation accuracy of MBHM‐F(x) but without discretization of initial LNSD (denoted as MBHMnodis‐F(x)) with respect to the finest resolution BIN256 of MBHMnodis‐F(256) (blue triangle) and DMS‐LDM‐FFS(256) (red circle) is also presented. The simulation accuracy of MBHMnodis‐F(x) to MBHMdis‐F(256) (green circle) and DMS‐LDM‐FFS(x) against DMS‐LDM‐FFS(256) (grey circle) is shown also for reference. Some features in Figure 16 are similar to those observed in Figure 13. For example, MBHMdis‐F(4–8) are as accurate as DMS‐LDM‐FFS(16–32). MBHMdis is generally more accurate than MBHMnodis at finer resolutions, because with MBHMnodis, the preexisting particles are forced to evolve as LNSD, which is only approximate. Therefore, errors of MBHMdis‐F(x) against DMS‐LDM‐FFS(256) (red circle) are lower than those of MBHMnodis‐F(x) (red triangle) for most of the variables for finer resolutions and MBHMdis‐F(x) (blue circle) generally showed better accuracy convergence than MBHMnodis‐F(x) (blue triangle). In contrast, errors of MBHMnodis‐F(x) (red triangle) become lower for coarser resolutions (BIN4–8), because the shapes of discretized initial distribution using equations 6a and 6b for coarser resolutions differ more from the true initial distribution.

Figure 16

Same as Figure 13 but with LNSD preexisting particles (N = 10³ cm⁻³, D_g = 80 nm, σ_g = 1.6) comparing MBHMdis‐F (with discretized LNSD preexisting particles using equations 6a and 6b), MBHMnodis‐F (without discretization of initial LNSD), and DMS‐LDM‐FFS. NMEs are illustrated for DMS‐LDM‐FFS(x: 4, 8, 16, 32, 64, 128) to DMS‐LDM‐FFS(256) (grey circles), MBHMdis‐F(x: 4, 8, 16, 32, 64, 128) to MBHMdis‐F(256) (blue circles), MBHMdis‐F(x: 4, 8, 16, 32, 64, 128, 256) to DMS‐LDM‐FFS(256) (red circles), MBHMnodis‐F(x: 4, 8, 16, 32, 64, 128) to MBHMnodis‐F(256) (blue triangles), MBHMdis‐F(x: 4, 8, 16, 32, 64, 128, 256) to DMS‐LDM‐FFS(256) (red triangles), and MBHMnodis‐F(x: 4, 8, 16, 32, 64, 128, 256) to MBHMdis‐F (256) (green circles). The order of the symbols are grey circle, green circle, blue circle, blue triangle, red circle, and red triangle from back to front.

[83] Figure 17 shows results from a simulation in which the preexisting particle size distribution is a square wave: n(logD) = 1100.6 cm⁻³ for 25.38 nm < D < 205.67 nm and n(logD) = 0 at other sizes. This initial distribution has the same M₀, M₂, and M₃ of as the LNSD in Figures 9 and 15. The initial size distribution of MBHM‐F(D_g) is wavy (left column) and becomes smooth during the simulation with higher resolutions (Figures 17a and 17c). Figure 18 shows simulation accuracy of MBHM‐F(x) for this square wave initial size distribution case. Features similar to the cases in Figures 13 and 16 are observed. The differences of MBHM‐F at coarser resolutions from DMS‐LDM‐FFS(256) are smaller than those of DMS‐LDM‐FFS at coarser resolutions, and the magnitude of the errors and differences is similar to the case with the initial LNSD in Figure 13. The results of these tests demonstrate that MBHM performs well in cases where the preexisting particle distribution is not LNSD, or is not treated as s single LNSD.

Figure 17

Time series of simulated number size distributions of (a, c, e) MBHM‐F(D_g) (full) and (b, d, f) DMS‐LDM‐FSS (flat) with BIN256, BIN64, and BIN8 grid resolutions with the square wave size distribution preexisting particles (N = 10³ cm⁻³, n(logD) = 1100.6 cm⁻³ for 25.38 nm < D < 205.67 nm).

Figure 18

Same as Figures 13 and 16 but with square wave size distribution preexisting particles (N = 10³ cm⁻³, n(logD) = 1100.6 cm⁻³ for 25.38 nm < D < 205.67 nm) for DMS‐LDM‐FFS(x: 4, 8, 16, 32, 64, 128) to DMS‐LDM‐FFS(256) (grey circles), MBHM‐F(x: 4, 8, 16, 32, 64, 128) to MBHM‐F(256) (blue circles), and MBHM‐F(x: 4, 8, 16, 32, 64, 128, 256) to DMS‐LDM‐FFS(256) (red circles).

3.5 Comparisons of Computational Efficiency Among MBHM, DMS‐MCA, and DMS‐LDM

[84] Figure 19 compares the central processing unit (CPU) time of the simulations by (I) MBHM‐HM, (II) DMS‐MCA‐HM, and (III) DMS‐LDM‐HM with BIN4–256 resolution for the no preexisting aerosol case (Figures 2a and 2b) and the square wave size distribution preexisting aerosol case (Figure 17). The CPU times of the three methods at coarser resolutions (BIN4–16) are within a factor of 2, but the difference between MBHM and the DMS methods becomes larger as finer resolutions, reaching a factor of 6 at BIN256. This large difference is due to coagulation. The CPU time for the condensation process is proportional to x at BINx resolution, whereas that for coagulation is proportional to x². Coagulation accounts for 60–70% of the total CPU time for BIN4 but 85–99% for BIN256. The significant difference of the CPU times between MBHM and the DMS methods is due to the difference in the number of empty bins, for which calculations can be skipped. The number of empty bins is significantly larger for MBHM than for the DMS methods (see Figures 5c, 5e, and 5g) due to a difference in a coagulation algorithm rule. For coagulation of bin i with bin j particles (where j ≥ i), the resulting particles (and moments) are put in bin j for MBHM (equation A1b). The bin j particles may then be moved to bin j + 1 if the bin's mean diameter exceeds the bin's upper bound, but this happens infrequently during coagulation. For DMS, the resulting particles are put in bin k, where k ≥ j ≥ i and (). Consequently, after a coagulation step, the number of nonempty bins usually remains unchanged for MBHM, whereas it often increases for DMS, especially at finer resolutions. The CPU time for a coagulation calculation involving a single pair of bins is somewhat higher for MBHM than for DMS, because of the additional second moment, but this difference is outweighed by the nonempty bin differences. Therefore, MBHM is computationally more efficient than DMS for this box model setup, and it may also be more efficient with Lagrangian models. If the MBHM coagulation rule was used with DMS, the DMS and MBHM CPU times might be closer, although this was not tested. The DMS‐LDM CPU times are slightly greater than the DMS‐MCA times, because the LDM condensation approach produces more nonempty bins than the MCA approach and is also somewhat slower.

Figure 19

CPU time of MBHM‐HM, DMS‐MCA‐HM, and DMS‐LDM‐HM versus resolution for (a) the no preexisting particles case and (b) the square wave size distribution preexisting particles (N^t=0 = 10³ cm⁻³) case.

[85] It is uncertain if these box model results can be extrapolated to 3‐D Eulerian models. Treating additional processes, particularly emissions, may reduce the number of empty bins. Transport (advection and mixing) of moments from the neighboring grid cells could also reduce the number of empty bins. The additional tracers for MBHM (the second moments) will increase the computational cost relative to DMS, possibly about 10% (see section 2.2.3). However, the computational cost of MBHM‐associated calculation (particularly coagulation) may not be large in comparison to the costs of tracer transport and detailed aerosol thermodynamics [e.g., Zaveri et al., 2008]. Finally, the real computational savings of MBHM may result from its higher accuracy allowing one to use coarser size resolution, with fewer bins and transport tracers.

4 Conclusions

[86] A triple‐moment sectional (TMS) model for aerosol dynamics, Modal Bin Hybrid Model (MBHM), has been developed. It is made by sorting a number of lognormal modes by size in the fixed bin structure. The moving‐center approach is applied for interbin transfer associated with the particle growth. In the double‐moment sectional (DMS) or double‐moment modal (DMM) methods, number and mass (volume) are preserved, while surface area is not always consistent. Because MBHM is TMS, consistency is assured for particle surface area too, which is important for gas‐to‐particle mass transfer, heterogeneous reaction, and light extinction cross section. This flexibility is useful for process‐oriented (multicategory and/or mixing state) modeling: primary aerosols whose size parameters would not differ substantially in time and space can be expressed by a single or a small numbers of modes, whereas secondary aerosols whose size changes from 1 nm to submicron meters could be discretized into sections. Processes such as nucleation, condensation, and coagulation are consistently calculated over all sectional and categorical modes.

[87] We evaluated the performance of MBHM against DMS with various size resolutions for simulating the evolution of particles from nucleation mode to accumulation mode under simultaneously occurring nucleation, condensation, and coagulation processes. Because MBHM gives a more physically consistent form of the intrasectional distributions than that provided by existing DMS methods and preserves surface area, the average and maximum errors (or biases) of MBHM at coarse resolutions (BIN4–8) are almost equivalent to those of DMS at intermediate resolutions (BIN16–32) for variables such as the moments M_k (k: 0, 2, 3), dM_k/dt, CCNx, and CCVx (definitions of BINx, CCNx, and CCVx are listed in Table 1).

[88] Usually, three‐dimensional Eulerian chemical weather or climate models with DMS or DMM predict about 10 transported variables (number and mass of several chemical components) for each section or mode. By predicting surface area for each section or mode, the increase in the number of tracers is only about 10%, while the number of sections could be reduced by about a factor of 4 (BIN16–32 to BIN4–8). This is a substantial saving of computational resources.

[89] Taking the current knowledge of complexity of aerosol microscale properties and their macroscale impacts into account, 3‐D models will have to predict new particle formation and CCN activity in a multiple category and mixing state resolving framework. MBHM can be extended by adding mixing state dimensions to accurately simulate changes in hygroscopicity and light absorption efficiency of aged black carbon particles. A dimension for photochemical age can also be added to track changes in uptake coefficients of trace gases by aerosols. The cost‐effective MBHM approach can make it more feasible to introduce new dimensions of complexity in either regional or global scale chemical weather or climate models.

[90] The evaluation of MBHM presented here has certain generality, but future work is necessary as only a limited number of test cases were presented. The light extinction coefficient of spherical particles at visible wavelengths increases dramatically with increasing particle diameter from Rayleigh to Mie regimes. Due to the natural intrasectional distribution and the surface area consistency in MBHM, it is reasonable to presume that MBHM will give better results with coarser sectional resolutions for the light extinction cross section, and this should be tested in a future study. Aerosol dynamics of coarse mode particles, which are dominated by emissions and dry and wet removal, were not evaluated here. The applicability of our constant D_g assumption to the Fuchs approach and/or the quadrature method in the whole size ranges as well as the Kelvin effect with multicomponent chemistry should be evaluated. Testing in a 3‐D Eulerian framework is clearly needed, as transport (advection and mixing) always increases errors in M₂ in DMS and in mode shape in MBHM by mode merging. Errors in M₂ are smaller as the size resolution increases in DMS, while merging of modes with different σ_g occurs in MBHM at any resolution. We believe that MBHM gives better results than DMS at coarser resolution, but DMS could be better for finer resolutions, and their accuracy and efficiency should be assessed in a 3‐D Eulerian framework. The accuracy, efficiency, and optimization of MBHM with a multicategory and mixing state resolving approach are topics of our ongoing research. Applying triple‐moment sectional methods in cloud models with size‐resolved microphysics parameterizations, where double‐moment methods are often used, could give similar benefits in accuracy and efficiency.

Acknowledgments

[115] M. Kajino was supported by the Japan Society for Promotion of Science (JSPS) Postdoctoral Fellowships for Research Abroad. This research was partly supported by the Ministry of Education, Science, Sports, and Culture (MEXT) Grant‐in‐Aid for Scientific Research on Innovative Areas 24110002, (B) 24340115, and (B) 23310018; and the Environment Research and Technology Development Fund (project A‐1101) of the Ministry of the Environment of Japan (MOE). S. J. Ghan and R. C. Easter were funded by the U.S. Department of Energy Atmospheric Systems Research program. Pacific Northwest National Laboratory is operated by Battelle Memorial Institute for the U.S. Department of Energy under contract DE‐AC05‐76RL01830. The authors thank Naga Oshima of Meteorological Research Institute, Japan, and Rahul A. Zaveri of Pacific Northwest National Laboratory, USA, for useful discussions.

Appendix A: Integrations of Coagulation and Condensation Equations in the Transition Regime for MBHM and DMS

[91] Here we describe the methods used for calculating coagulation and condensation rates for the moment dynamics equations. Since sizes of atmospheric aerosols range from nucleation mode (D ~ 1 nm) to coarse mode (D ~ 10μm), processes that influence the aerosol operate across a wide range of Knudsen number, Kn (ratio of mean free path of gas molecule to particle radius; defined specifically later as the definitions are different for condensation and coagulation equations). Thus, the condensational and coagulation growth equations for the free‐molecular regime (Kn > 10) to continuum regime (Kn < 0.1) are required. Because the growth rates in the two regimes are expressed in different forms and both forms overestimate the growth rate significantly in each other's regimes, an interpolation theory is needed in the transition range (Kn ~ 1) between the two regimes.

[92] Among several interpolations proposed, the harmonic mean approach has been widely applied to modal aerosol modeling because it allows the integral form of the integrodifferential MDEs to be reduced to the simple forms of equations A4 and A11 [Pratsinis, 1988; Binkowski and Shankar, 1995; Park et al., 1999; Park and Lee, 2000; Kajino et al., 2012a]. In order to use the more accurate interpolations such as Fuchs and Fuchs‐Sutugin approaches in the modal methods, highly accurate but expensive quadrature method [Giorgi, 1986], referred to collectively as FFS(Q), and the efficient harmonic mean method with lookup table correction factors [Whitby et al., 1991] were developed. Here we proposed another efficient method, the Fuchs correction factors derived as a function of D_g referred to as F(D_g) in the study. We also use a high‐order (20‐point) Gauss‐Hermite quadrature method with MBHM to evaluate errors caused by the F(D_g) and harmonic mean approximations. This quadrature method is highly accurate but computationally very costly.

A.1 Coagulation Equation

A.1.1 Coagulation Equation With Harmonic Mean Approach

[93] The MDEs for intrasectional (bin i) and intersectional (bins i and j) coagulation are written respectively as

(A1a)

and

(A1b)

[94] The first term on the right‐hand in equation A1a is the production of a merged large particle with a diameter of , by the collision of two smaller particles D_p1 and D_p2. The second term is the loss of moments of D_p1 due to coagulation with D_p2. As defined in equation A1b, i‐j coagulation (for i < j) goes to bin j even though the resulting particle could be larger than bin j. When i = j in equation A1b, the sum of the two equations yields equation A1a, and β is the coagulation kernel (m³/s). While the moments M₀, M₂, and M₃ are used for other processes to conserve number, surface area, and volume (mass), for coagulation, M₀, M₃, and M₆ are selected in order to remove the bracket in equations A1a and A1b as = for the convenience of integrating the MDE. Brownian coagulation kernels for spheres in the free‐molecular regime β_fm and the near‐continuum regime β_nc (m³/s) are given by

(A2a)

(A2b)

where k_B is the Boltzmann constant, ρ_p is the particle density, μ is the viscosity of air, and G is the Cunningham slip correction factor.

[95] For the MBHM method which uses LNSD, equations A1a and A1b have analytical solutions for β_fm with the introduction of Otto's approximation [Otto et al., 1997]. There are also analytical solutions for β_nc if a simpler form of G is selected, G = 1 + 1.296Kn. Here in the coagulation equation, Kn = 2λ/D, where λ is the mean free path of air molecules. The harmonic mean of dM_k/dt is then

(A3)

which is calculated for each i‐j pair of sections. Each term of equation A3 for intrasectional and intersectional coagulation is provided by equations 3–10 of Kajino et al. [2012a]. Equation A3 is used for the Method I (MBHM‐HM).

[96] For DMS methods, we assume monodisperse intrasectional distributions for the coagulation integral calculations, which reduces the integrals in equations A1a and A1b into simple function evaluations. This allows use of the harmonic mean kernel,

(A4)

in equations A1a and A1b, and this approach is used with Methods II (DMS‐MCA‐HM) and III (DMS‐LDM‐HM).

A.1.2 Coagulation Equation With Fuchs Approach

[97] On the other hand, the sectional approach allows one to use more complex and accurate interpolation for the transition regime, such as Fuchs [1964],

(A5)

where and δ_p1 is defined as where λ_p1 is the particle mean free path [Jacobson, 2005]. The thermal motion of a particle is assumed in the spherical region with a radius of D_p1 + δ_p1. At D_p1 ≪ λ_p1, x becomes small and β approaches β_fm. The Cunningham correction G was 1 + Kn(1.249 + 0.42exp(−0.87/Kn)) in the sectional method. The Fuchs kernel equation A5 is used with Method IV (DMS‐LDM‐FFS). Park et al. [1999] compared the harmonic mean method with the Fuchs coagulation and concluded that the overall error is always less than 20% (see also Table B2). Whether this discrepancy is unacceptably high or negligible in 3‐D chemical weather and climate modeling depends on which processes or parameters you focus on and how large the errors you can permit.

[98] With MBHM, there is no analytical solution to the coagulation integrals in equations A1a and A1b when the more accurate Fuchs coagulation kernel is used because of the term x(D_p1,D_p2) in equation A5 and the term exp(−0.87/Kn) in G. To obtain the analytical solution, D in the terms is substituted by the mean diameter D_g, thus making those terms constants as x(D_g1, D_g2) and exp(−0.87/Kn(D_g)) in the integration of equations A1a and A1b. Similarly as the harmonic mean approach equation A3, we obtained

(A6)

with x(D_g1, D_g2) and G = 1 + Kn(D)(1.249 + 0.42exp(−0.87/Kn(D_g))). This approach is referred to as F(D_g) and used for Method V (MBHM‐F(D_g)).

[99] Some DMM in global models used this kind of method as proposed in Wilson et al. [2001] and Vignati et al. [2004]. However, in their approach, because every diameter in equations A2a and A2b is replaced by the number equivalent geometric mean diameter, the discrepancy in coagulation rates from that with equation A5 increases as σ_g is larger. In contrast, in MBHM‐F(D_g), as σ_g increases, only the deviation term between the Fuchs and the harmonic mean is enhanced.

[100] Using the Gauss‐Hermite quadrature method, equations A1a and A1b with the Fuchs kernel (equation A5) are accurately integrated and equations A1a and A1b are turned to

(A7a)

where and

(A7b)

where X_i(x) = and X_j(x) = , respectively, and n, x, and w are the quadrature order, the abscissas, and the weight, respectively. The method is referred to as the coagulation part of FFS(Q) and used for Method VI (MBHM‐FFS(Q)).

[101] There are other forces than Brownian motion contributing on coagulation of atmospheric particles, such as convective Brownian diffusion enhancement, gravitational collection, and turbulence effects [Jacobson, 2005]. In the current paper, only Brownian coagulation is considered with the coalescence efficiency as unity, because we are focusing on aerosol dynamics from 1 nm to several micrometers. Integrated forms of MDEs for gravitational collection, turbulent shear, and turbulent inertia coagulation processes are presented in Kajino and Kondo [2011].

A.2 Condensation Equation

A.2.1 Condensation Equation With Harmonic Mean Approach

[102] In a similar manner, the harmonic mean method was used in the modal aerosol modeling for the following condensation (evaporation) equation for mass:

(A8)

where M is the total aerosol mass concentration, m_w,p and m_w,g are molecular weights of condensing (evaporating) materials in particulate and gas phases, respectively, and c_∞ and c_s are the gas‐phase mass concentrations in the bulk phase and near‐aerosol surface at equilibrium, respectively. ψ(D) is the size‐dependent term to determine the gas‐to‐particle mass transfer coefficient, given by

(A9)

where in the free‐molecular regime and ψ_co(D) = 2πD_vD in the continuum regime, α is a mass accommodation coefficient (assumed constant here as 0.1), is a mean velocity of the gas molecules, and D_v is the molecular diffusivity. The condensation MDE is given by

(A10)

where we have used , where v_p is volume of a particle. Using the harmonic mean approach equation A9 and neglecting the Kelvin effect for the equilibrium concentration on aerosol spherical surface, equation A10 is turned to

(A11)

where |A,B|_harm denotes the harmonic mean of A and B and, thus, AB/(A + B). Equation A10 is used for Method I (MBHM‐HM), while integration of equation A8 using DMS with equation A9 is used for Methods II (DMS‐MCA‐HM) and III (DMS‐LDM‐HM).

A.2.2 Condensation Equation With Fuchs and Fuchs‐Sutugin Approach

[103] In the sectional approach, more accurate correction factors for the transition regime can be used:

(A12a)

(A12b)

where Kn_g is the Knudsen number, which is different from Kn used in the coagulation equation. Kn_g is the ratio of the mean free path of the condensable gas λ_g to aerosol radius, whereas Kn is that of air. There are several theoretical definitions to determine λ_g and f ′ [Qu and Davis, 2001] so those should be consistent. In this paper, we used Fuchs and Sutugin's [1971] theory, where and with equation A12a as defined in Zaveri et al. [2008]. This method is used for Method IV (DMS‐LDM‐FFS).

[104] Park and Lee [2000] conducted condensational growth calculations with the harmonic mean method and with the Fuchs‐Sutugin's correction and found that the maximum error was 13.4% (see also Table B1).

Table B1. Deviations of the Harmonic Mean to Fuchs‐Sutugin Approximation of the Size‐Dependent Term of Condensation Rate, ψ(D) (m³ s⁻¹) for D of 1 nm to 10 µm for Various Conditionsa^a Central values of sections of BIN64 with which calculations were performed.

Temperature (K)	Pressure (hPa)	MNB (%)	MaxNE* (%)	D With MaxNE*a^a Central values of sections of BIN64 with which calculations were performed. (nm)	NME (%)
Reference Condition
298.15	1000	−1.26	−2.88	1000	1.73
Conditions With Ranges
223.15–314.15	10–1025	−0.15 to −1.30	−1.17 to −2.88	649–9646	0.87–2.77
Condition With Minimum MNB
223.15	1025	−1.30	−2.88	649	1.43

• ^a Central values of sections of BIN64 with which calculations were performed.

[105] To use the Fuchs‐Sutugin type of correction factor in the condensation MDE (equation A10) in MBHM, it can be calculated with D_g, making f(Kn_g(D), α) in equation A12a to be a constant, so

(A13)

[106] It is also noted here that some of the previous modal methods with harmonic mean approach neglected the Kelvin effect for the equilibrium concentration c_s(D) on aerosol spherical surface, which is important for nanoparticles [Zhang et al., 2012]. It can be also considered with D_g in MBHM by replacing c_s(D) in equation A10 to c_s(D_g). With this and equation A13, equation A10 is turned to

(A14)

[107] However, this approach produced larger discrepancy from equation A8 with equation A12a due probably to the nonuniform effect with σ_g. As we discussed with the coagulation rate, the nonuniform effect is supposed to affect only the deviation term between the Fuchs and the harmonic mean. However, here the condensation rate equation A14 could differ from the harmonic mean because the discrepancy in f becomes large for large σ_g. Therefore, we used Fuchs and Sutugin [1971] for DMS but for MBHM, we instead use Fuchs [1964] following Wilson et al. [2001] and Vignati et al. [2004] and propose another form of the MDE as

(A15)

where . This approach is referred to as F(D_g) and used for Method V (MBHM‐F(D_g)).

[108] Using the quadrature method, equation A8 can be reformulated accurately and turned to

(A16)

where and ψ is as defined in equation A12a. The method is referred to as the condensation part of FFS(Q) and used for Method VI (MBHM‐FFS(Q)).

[109] We used the following form of Gauss‐Hermite quadrature method in equations A7a, A7b, and A16 as

(A17)

and obtained the accurate results with the high quadrature order n = 20 in this study. To obtain the efficiency as well as accuracy with less numbers of the quadrature points (e.g., n = 2‐5), equations A7a, A7b, and A16 should be replaced by the following form of the quadrature:

(A18)

where and ln D_gk (= ln D_g + k ln ²σ_g) is the mean diameter for kth moment as defined in equation 5.

Appendix B: Harmonic Mean Versus Fuchs and Fuchs‐Sutugin Approaches Condensation and Coagulation Integrals in DMS and MBHM

[110] The different modal and sectional methods have “fundamental errors” associated with their basic numerical algorithms and additional errors associated with how accurately the condensation and coagulation rates are calculated. In this section, differences between DMS and MBHM and differences between the harmonic mean (HM) and the more accurate Fuchs [Fuchs, 1964] and/or Fuchs‐Sutugin [Fuchs and Sutugin, 1971] approaches (F/FS) are evaluated.

[111] Table B1 shows the MNBs and MaxNE* values of the harmonic mean ψ_hm (equation A9) to ψ (equation A12a) with Fuchs‐Sutugin approximation of the size‐dependent term of condensation rate for D of 1 nm to 10 µm at the reference condition (298.15 K, 1000 hPa), typical tropospheric and lower stratospheric condition ranges (223.15–314.15 K, 10–1025 hPa), and the condition with the largest discrepancy. Also, it shows D and Kn with MaxNE*. The NMEs are also shown. The MaxNE* and MNB increase at lower temperature and higher pressure, with the largest discrepancies of −2.88% and −1.30%, respectively, at 223.15 K and 1025 hPa, as shown in Figure B1a. The errors at the reference conditions (−2.88% and −1.26%) are almost the same as the largest errors.

Figure B1

(a) Size‐dependent condensation term ψ and (b–d) coagulation term β in the free‐molecular regime (FM, thin dashed black line), continuum or near‐continuum (CO or NC, solid black line), harmonic mean approach (HM, thick dashed black line), Fuchs‐Sutugin or Fuchs approximation (FS or F, solid pink line), and the difference of Fs or F minus HM (thick dashed pink line) under the conditions with largest discrepancies (minimum in MNB) between HM and F/FS, summarized in Tables B1 and B2 (223.15 K and 1025 hPa for condensation, 273.15 K and 50 hPa for Brownian coagulation). Figures B1b–B1d show dependency of β on D₁ with constant D₂ of 10 nm, 100 nm, and 1 µm, respectively.

Table B2. Deviations of the Harmonic Mean to Fuchs Approximation of the Coagulation Kernel, β(D₁, D₂) (m³ s⁻¹) (D₁ < D₂) for Ds of 1 nm to 10 µm for Various Conditionsa^a Central values of sections of BIN64 with which calculations were performed.

Temperature (K)	Pressure (hPa)	MNBa^a Central values of sections of BIN64 with which calculations were performed. (%)	MaxNE* [%]	D1 With MaxNE*a^a Central values of sections of BIN64 with which calculations were performed. (nm)	D2 With MaxNE*a^a Central values of sections of BIN64 with which calculations were performed. (nm)	NME (%)
Reference Condition
298.15	1000	−6.64	−19.67	1	246	4.34
Conditions With Ranges
223.15–314.15	10–1025	−6.43 to −8.53	−19.67 to −19.70	1–6.98	178–9646	3.57–18.23
Conditions With Minimum MNB
273.15	50	−8.53	−19.69	1	4532	17.35

• ^a Central values of sections of BIN64 with which calculations were performed.

[112] Table B2 is same as Table B1 but for coagulation kernels, showing the deviation of the harmonic mean approach β_hm (equation A3) to the Fuchs method β_f (equation A5) for D of 1 nm to 10 µm. The harmonic mean errors are greater than for condensation. The MaxNE* and MNBs are larger at higher temperature and lower pressure, showing largest discrepancies as −19.69% and −8.53% at 273.15 K and 50 hPa (Figures B1b–B1d), respectively. However, the variation of the MaxNE* and MNBs with temperature and pressure is relatively small.

[113] Table B3 summarizes NMBs and NMaxE* (with bracket) between the methods. In order to evaluate the applicability of the Fuchs and Fuchs‐Sutugin approaches to MBHM in the typical tropospheric condition ranges, comparisons between harmonic mean and Fuchs‐Sutugin approach and between MBHM and DMS‐LDM were made with the finest resolution (BIN256) and coarse resolution (BIN8) at the reference conditions. There are no apparent differences seen in the figures of time series of number size distributions (not shown). It should be noted that the errors (or biases) in Tables B1 and B2 and those here in Table B3 are not directly comparable, because of the difference in averaging (average over diameter for Tables B1 and B2, while average in time for Table B3). One can simply say from Rows #1 and #2 that the underestimate of condensation and coagulation rates with about −1.26% and −6.64% of MNBs, respectively, resulted in the increase of dM₀/dt|_Coag by 0.27% and 0.25% of NMBs (slower coagulation as the term is negative) and decrease of dM₃/dt|_Cond by −0.017% and −0.016% of NMBs (slower condensation as the term is positive) in the DMS and MBHM methods, respectively. Consequently, NMBs for M₀, M₂, and M₃ are 0.13 (0.12%), −0.0028 (−0.0035%), and −0.028 (−0.027%), in DMS (MBHM), respectively.

Table B3. NMBs and NMaxE (in Parentheses) (%) of Various Evaluation‐Reference Run Pairs at 298.15 K and 1000 hPa Without Preexisting Particlesa^a BINx is abbreviated to (x) in the second column.

Row #	Evaluation Run Reference Run	M0	M2	M3	dM3/dt|Cond.b^b Time derivative of third moment due to new particle formation and condensation processes.	dM0/dt|Coag.c^c Time derivative of zeroth moment due to coagulation process.
1	DMS‐LDM‐HM(256) DMS‐LDM‐FFS(256)	0.13 (0.62)	−0.0028 (−0.021)	−0.028 (−0.035)	−0.017 (−0.085)	0.27 (1.6)
2	MBHM‐HM(256) MBHM‐FFS(Q)d^d Quadrature order n is 20. (256)	0.12 (0.60)	−0.0035 (−0.019)	−0.027 (−0.032)	−0.016 (−0.082)	0.25 (1.5)
3	MBHM‐F(Dg)(256) MBHM‐FFS(Q)d^d Quadrature order n is 20. (256)	−0.080 (−0.38)	0.00031 (−0.015)	0.018 (0.022)	0.011 (0.052)	−0.17 (−0.99)
4	MBHM‐HM(256) MBHM‐F(Dg)(256)	0.20 (0.98)	−0.0038 (−0.032)	−0.044 (−0.055)	−0.026 (−0.13)	0.42 (2.5)
5	MBHM‐F(Dg)(16) MBHM‐FFS(Q)d^d Quadrature order n is 20. (256)	−0.44 (−3.2)	0.55 (0.70)	0.053 (0.063)	0.029 (0.31)	−0.59 (−5.6)
6	DMS‐LDM‐FFS(16) DMS‐LDM‐FFS(256)	−1.84 (−18.5)	3.75 (4.85)	0.34 (0.41)	0.17 (3.49)	−4.4 (−37.2)
7	DMS‐LDM‐HM(8) DMS‐LDM‐FFS(8)	0.11 (0.55)	−0.013 (−0.025)	−0.025 (−0.030)	−0.015 (−0.083)	0.26 (1.5)
8	MBHM‐HM(8) MBHM‐FFS(Q)d^d Quadrature order n is 20. (8)	0.08 (0.56)	−0.0065 (−0.019)	−0.025 (−0.031)	−0.015 (−0.76)	0.24 (1.4)
9	MBHM‐F(Dg)(8) DMS‐ FFS(Q)d^d Quadrature order n is 20. (8)	−0.064 (−0.38)	0.0012 (−0.013)	0.017 (0.022)	0.011 (0.051)	−0.17 (−1.0)

• ^a BINx is abbreviated to (x) in the second column.
• ^b Time derivative of third moment due to new particle formation and condensation processes.
• ^c Time derivative of zeroth moment due to coagulation process.
• ^d Quadrature order n is 20.

[114] Because all the differences between DMS‐HM and DMS‐FFS and those between MBHM‐HM and MBHM‐FFS(Q) are consistent and comparable (< 30%) with each other, the lower discrepancies between MBHM‐F(D_g) and the almost exact solution MBHM‐FFS(Q) (Row #3) indicate the successful implementations of F(D_g) approaches equations A6 and A15 to MBHM. All the differences between MBHM‐HM and MBHM‐F(D_g) (Row #4) are consistent and comparable (approximately a factor of 2) with Rows #1 and #2, indicating that the successful implementations of F(D_g), too. The differences between HM and F/FS (Rows #1, #2, and #4) are comparable or approximately 1 order of magnitude smaller than the differences between the resolution (BIN16 versus BIN256; see Rows #5 and #6) in their size ranges (1–100 nm). Tables B1 and B2 show that the discrepancies are largest for ~1 µm for condensation and between nanoparticles and supermicron particles for coagulation. Therefore, we conclude here that HM can provide results as accurate as F/FS methods in aerosol dynamical processes from the nucleation mode to accumulation mode ranges in typical tropospheric and lower stratospheric conditions at BIN16, the typical resolutions for 3‐D CTMs. The deviations between HM and FFS with DMS‐LDM and MBHM for the coarser resolutions (BIN8) (Rows #7 and #8) and the deviations between F(D_g) and FFS(Q) (Row #9) are consistent and in the same order of those with the finest resolutions (Rows #1, #2, and #3, respectively).