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Copyright © 2000, Charles S. Zender
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Contents

List of Tables

1 Introduction

This document describes mathematical and computational considerations pertaining to size distributions. The application of statistical theory to define meaningful and measurable parameters for defining generic size distributions is presented in §2. The remaining sections apply these definitions to the size distributions most commonly used to describe clouds and aerosol size distributions in the meteorological literature. Currently, only the lognormal distribution is presented.

1.1 Modal vs. Sectional Represenatation

mdlsxn Lu and Bowman (2004) designed and optimal non-linear least squares-based procedure for converting from sectional to modal representations.

1.2 Nomenclature

nomenclature There is a bewildering variety of nomenclature associated with size distributions, probability density functions, and statistics thereof. The nomenclature in this article generally follows the standard references, (see, e.g., Hansen and Travis, 1974; Patterson and Gillette, 1977; Press et al., 1988; Flatau et al., 1989; Seinfeld and Pandis, 1997), at least where those references are in agreement. Quantities whose nomenclature is often confusing, unclear, or simply not standardized are discussed in the text.

1.3 Distribution Function

This section follows the carefully presented discussion of Flatau et al. (1989). The size distribution function nn(r) is defined such that nn(r) dr is the total concentration (number per unit volume of air, or # m-3) of particles with sizes in the domain [r,r + dr]. The total number concentration of particles N 0 is obtained by integrating nn(r) over all sizes

(1)

The size distribution function is also called the spectral density function. The dimensions of nn(r) and N0 are # m-3 m-1 and # m-3, respectively. Note that n n(r) is not normalized (unless N0 happens to equal 1.0).

Often N0 is not an observable quantity. A variety of functional forms, some of which are overloaded for clarity, describe the number concentrations actually measured by instruments. Typically an instrument has a lower detection limit rmin and an upper detection limit rmax of particle sizes which it can measure.

(5)

These functions are often used to define nn(r) via

(6)

Note that the concentration nomenclature in (6) is N not N(r). Using N(r) would indicate that the concentration has not been completely integrated over all sizes. By definition, the total concentration N0 is integrated over all sizes, as defined by (1). A concentration denoted N(r) makes no sense without an associated size bin width Δr, or truncation convention, as in (2)–(4). We try to use N and N0 for normalized (N = 1) and non-normalized (N01, i.e., absolute concentrations). However this convention is not absolute and (1) defines both N and N0.

1.4 Probability Density Function

Describing size distributions is easier when they are normalized into probability density functions, or PDFs. In this context, a PDF is a size distribution function normalized to unity over the domain of interest, i.e., p(r) = Cnnn(r) where the normalization constant Cn is defined such that

(7)

In the following sections we usually work with PDFs because this normalization property is very convenient mathematically. Comparing (7) and (1), it is clear that the normalization constant Cn which transforms a size distribution function (1) into a PDF p(r) is N0^-1

(8)

1.4.1 Choice of Independent Variable

The merits of using radius r, diameter D, or some other dimension L, as the independent variable of a size distribution depend on the application. In radiative transfer applications, r prevails in the literature probably because it is favored in electromagnetic and Mie theory. There is, however, a growing recognition of the importance of aspherical particles in planetary atmospheres. Defining an equivalent radius or equivalent diameter for these complex shapes is not straighforward (consider, e.g., a bullet rosette ice crystal). Important differences exist among the competing definitions, such as equivalent area spherical radius, equivalent volume spherical radius, (e.g., Ebert and Curry, 1992; McFarquhar and Heymsfield, 1997).

A direct property of aspherical particles which can often be measured, is its maximum dimension, i.e., the greatest distance between any two surface points of the particle. This maximum dimension, usually called L, has proven to be useful for characterizing size distributions of aspherical particles. For a sphere, L is also the diameter. Analyses of mineral dust sediments in ice core deposits or sediment traps, for example, are usually presented in terms of L. The surface area and volume of ice crystals have been computed in terms of power laws of L (e.g., Heymsfield and Platt, 1984; Takano and Liou, 1995). Since models usually lack information regarding the shape of particles (exceptions include Zender and Kiehl, 1994; Chen and Lamb, 1994), most modelers assume spherical particles, especially for aerosols. Thus, the advantages of using the diameter D as the independent variable in size distribution studies include: D is the dimension often reported in measurements; D is more analogous than r to L.

The remainder of this manuscript assumes spherical particles where r and D are equally useful independent variables. Unless explicitly noted, our convention will be to use D as the independent variable. Thus, it is useful to understand the rules governing conversion of PDFs from D to r and the reverse.

Consider two distinct analytic representations of the same underlying size distribution. The first, nD n (D), expresses the differential number concentration per unit diameter. The second, nr n(r), expresses the differential number concentration per unit radius. Both nD n (D) and nr n(r) share the same dimensions, # m-3 m-1.

2 Statistics of Size Distributions

2.1 Generic

Consider an arbitrary function g(x) which applies over the domain of the size distribution p(x). For now the exact definition of g is irrelevant, but imagine that g(x) describes the variation of some physically meaningful quantity (e.g., area) with size. The mean value of g is the integral of g over the domain of the size distribution, weighted at each point by the concentration of particles

(13)

Once p(x) is known, it is always possible to compute g for any desired quantity g. Typical quantities represented by g(x) are size, g(x) = x; area, g(x) = A(x) ∝ x²; and volume g(x) = V (x) ∝ x³. More complicated statistics represented by g(x) include variance, g(x) = (x -)². The remainder of this section considers some of these examples in more detail.

2.2 Mean Size

The number mean size of a size distribution p(x) is defined as

(14)

Synonyms for number mean size include mean size, average size, arithmetic mean size, and number-weighted mean size (Hansen and Travis, 1974). Flatau et al. (1989) define n ≡, a convention we adopt in the following.

2.3 Variance

The variance σ2 x of a size distribution p(x) is defined in accord with the statistical variance of a continuous mathematical distribution.

(15)

The variance measures the mean squared-deviation of the distribution from its mean value. The units of σ2 x are [m2]. Because σ2 x is a complicated function for standard aerosol and cloud size distributions, many prefer to work with an alternate definition of variance, called the effective variance.

The effective variance σ2 x,eff of a size distribution p(x) is the variance about the effective size of the distribution, normalized by xeff (e.g., Hansen and Travis, 1974)

(16)

Because of the xeff^-2 normalization, σ2 x,eff is non-dimensional in contrast to typical variances, e.g., (15). In the terminology of Hansen and Travis (1974), σ2 x,eff = v.

2.4 Standard Deviation

The standard deviation σx of a size distribution p(x) is the square root of the variance (15),

(17)

σx has units of [m]. For standard aerosol and cloud size distributions, σx is an ugly expression. Therefore many authors prefer to work with alternate definitions of standard deviation. Unfortunately, nomenclature for these alternate definitions is not standardized.

3 Cloud and Aerosol Size Distributions

3.1 Gamma Distribution

Statistics of the gamma distribution are presented in http://asd-www.larc.nasa.gov/~yhu/paper/thesisall/node8.html. Currently, the mie program implements gamma distributions in a limited sense.

3.2 Lognormal Distribution

The lognormal distribution is perhaps the most commonly used analytic expression in aerosol studies. Table 1 summarizes the standard lognormal distribution parameters. Note that ≡ ln σg.

The statistics in Table 1 are easy to misunderstand because of the plethora of subtly different definitions. A common mistake is to assume that patterns which seems to apply to one distribution, e.g., the number distribution nn(D), apply to distributions of all other moments. For example, the number distribution nn(D) is the only distribution for which the moment mean size (i.e., number mean size n) equals the moment-weighted size (i.e., number-weighted size Dn). Also, the number mean size n differs from the number median size n by a factor of e ^2∕2. But this factor is not constant and depends on the moment of the distribution. For instance, s differs from s by e ², while s differs from s by e^{3 2∕2}. Thus converting from mean diameter to median diameter is not the same for number as for mass distributions.

						Table 1: Lognormal Distribution Relations123
Symbol	Value	Units	Description	Defining Relationship
N0	N0	# m-3	Total number concentration	N0 = ∫ 0∞n n(D) dD
D0	N0n exp( 2)	m m-3	Total diameter	D0 = ∫ 0∞Dn n(D) dD
A0	N0n2 exp( 2∕2)	m2 m-3	Total cross-sectional area	A0 = ∫ 0∞D2n n(D) dD
S0	πN0n2 exp(2 2)	m2 m-3	Total surface area	S0 = ∫ 0∞πD2n n(D) dD
V0	N0n3 exp(9 2∕2)	m3 m-3	Total volume	V0 = ∫ 0∞D3n n(D) dD
M0	N0ρn3 exp(9 2∕2)	kg m-3	Total mass	M0 = ∫ 0∞ρD3n n(D) dD
	n exp( 2∕2)	m #-1	Mean diameter	N0 = N0n = D0
	n2 exp(2 2)	m2 #-1	Mean cross-sectional area	N0 = N0s2 = A0
	πn2 exp(2 2)	m2 #-1	Mean surface area	N0 = N0πs2 = S 0
	n3 exp(9 2∕2)	m3 #-1	Mean volume	N0 = N0v3 = V0
	ρn3 exp(9 2∕2)	kg #-1	Mean mass	N0 = N0ρv3 = M0
N0	M0n-3 exp(-9 2∕2)	# m-3	Number concentration	N0 = ∫ 0∞n n(D) dD
n	1∕3 exp(-3 2∕2)	m	Median diameter	∫ 0n nn(D) dD =
Deff		m	Effective diameter	Deff = ∫ 0∞DD2n n(D) dD
S		m2 kg-1	Specific surface area	S = S0∕M0
n	n exp(- 2∕2)	m	Median diameter, Scaling diameter, Number median diameter. Half of particles are larger than, and half smaller than, n	∫ 0n nn(D) dD =
n, , Dn	n exp( 2∕2)	m	Mean diameter, Average diameter, Number-weighted mean diameter	n = ∫ 0∞Dn n(D) dD
s	n exp( 2)	m	Surface mean diameter	N0πs2 = N 0 = S0
v	n exp(3 2∕2)	m	Volume mean diameter, Mass mean diameter	N0v3 = N 0 = V0
s	n exp(2 2)	m	Surface median diameter	∫ 0s πD2n n(D) dD =
Ds, Deff	n exp(5 2∕2)	m	Area-weighted mean diameter, effective diameter	Ds = ∫ 0∞DD2n n(D) dD
v	n exp(3 2)	m	Volume median diameter Mass median diameter	∫ 0v D3n n(D) dD =
Dv	n exp(7 2∕2)	m	Mass-weighted mean diameter, Volume-weighted mean diameter	Dv = ∫ 0∞DD3n n(D) dD

Table 2 lists applies the relations in Table 1 to specific size distributions typical of tropospheric aerosols.

---

Table 2: Lognormal Size Distribution Statistics n v σg M Ref.a μm μm Patterson and Gillette (1977) b 0.08169 0.27 1.88 1           0.8674 5.6 2.2 1           28.65 57.6 1.62 1           Shettle (1984)c 0.003291 0.0111 1.89 2.6 × 10-4 2           0.5972 2.524 2.0d 0.781 2, 4           7.575 42.1 2.13 0.219 2           Balkanski et al. (1996)e 0.1600 0.832 2.1 0.036 3           1.401 4.82 1.90 0.957 3           9.989 19.38 1.60 0.007 3           Alfaro et al. (1998) f 0.6445 1.5 1.7 (0.22, 0.15) 5           3.454 6.7 1.6 (0.69, 0.76) 5           8.671 14.2 1.5 (0.09, 0.09) 5           Dubovik et al. (2002a), Bahrain (1998–2000) g h 0.1768 0.30 ± 0.08 0.42 ± 0.04 6           1.664 5.08 ± 0.08 0.61 ± 0.02 6           Dubovik et al. (2002a), Solar Village Saudi Arabia (1998–2000) i 0.1485 0.24 ± 0.10 0.40 ± 0.05 6           1.576 4.64 ± 0.06 0.60 ± 0.03 6           Dubovik et al. (2002a), Cape Verde (1993–2000) j 0.1134 0.24 ± 0.06 0.49 + 0.10τ ± 0.04 6           1.199 3.80 ± 0.06 0.63 - 0.10τ ± 0.03 6           Arimoto et al. (2006) k 0.0 1.1 0.0 7           0.0 5.5 0.0 7           0.0 14 0.0 7

---

Perry et al. (1997) and Perry and Cahill (1999) describe measurements and transport of dust across the Atlantic and Pacific, respectively. Reid et al. (2003) summarize historical measurements of dust size distributions, and analyze the influence of measurement technique on the derived size distribution. They show the derived size distribution is strongly sensitive to the measurment technique. During PRIDE, measured v varied from 2.5–9 μm depending on the instrument employed. Maring et al. (2003) show that the change in mineral dust size distribution across the sub-tropical Atlantic is consistent with a slight updraft of ~ 0.33 cm s-1 during transport. Ginoux (2003) and Colarco et al. (2003) show that the effects of asphericity on particle settling velocity play an important role in maintaining the large particle tail of the size distribution during long range transport.

Table 3 applies the relations in Table 1 to specific size distributions typical of tropospheric aerosols.

---

Table 3: Analytic Lognormal Size Distribution Statistics a b n Dn s Ds v Dv σg n rn s rs v rv σg μm μm μm μm μm μm 0.1 0.1272 0.2614 0.3323 0.4227 0.5373 2.0               0.1861 0.2366 0.4864 0.6184 0.7863 1.0 2.0               0.2366 0.3008 0.6184 0.7863 1.0 1.271 2.0               0.3009 0.3825 0.7864 1.0 1.271 1.616 2.0               0.3825 0.4864 1.0 1.271 1.616 2.055 2.0               0.5915 0.7521 1.546 1.966 2.5 3.179 2.0               0.8281 1.053 2.165 2.753 3.5 4.450 2.0               0.7864 1.0 2.056 2.614 3.323 4.225 2.0               1.0 1.272 2.614 3.323 4.227 5.373 2.0               1.183 1.504 3.092 3.932 5.0 6.356 2.0               2.366 3.008 6.184 7.863 10.0 12.71 2.0

---

Values in Table 3 are valid for radius and diameter distributions. Table 1 shows that all moments of the size distribution depend linearly on n (or n). Therefore all rows in Table 3 scale linearly (for a constant geometric standard deviation). For example, values in the row with n = 1.0 μm are ten times the corresponding values for the row n = 0.1 μm. Hence it suffices for Table 3 to show a decade range in n.

3.2.1 Distribution Function

The lognormal distribution function is

(18)

One of the most confusing aspects of size distributions in the meteorological literature is in the usage of σg, which is frequently called the geometric standard deviation. Some researchers (e.g., Flatau et al., 1989) denote by σg what most denote by ln σg. Thus the form of the lognormal distribution function sometimes appears

(19)

In practice, (18) is used more widely than (19) but the definition of σg in the latter may be more satisfactory from a mathematical point of view (Flatau et al., 1989) (and it subsumes the “ln”, which reduces typing). We adopt (18) in the following, and sometimes simplify formulae by using a convenient definition of ≡ ln σg. One is occasionally given a “standard deviation” or “geometric standard deviation” parameter without clear specification whether it represents σg (or ln σg, or exp σg, or σx) in (17), (18), or (19). A useful rule of thumb is that σg in (18) and eσg in (19) are usually near 2.0 for realistic aerosol populations. Since we adopted (18), physically realistic values of σg presented in this manuscript will be near 2.0.

Seinfeld and Pandis (1997) p. 423 describe the physical meaning of the geometric standard deviation σg. Define the particle size

(20)

The cumulative concentration smaller than Dσg, simplifies from (32) to

Direct substitution of D = 2r into (18) yields

3.2.2 Related Forms

Many important applications make available size distribution information in a form similar to, but hard to recognize as, the analytic lognormal PDF (18). The Aerosol Robotic Network, AERONET, for example, retrieves size distributions from solar almucantar radiances⁴ (Dubovik and King, 2000; Dubovik et al., 2000, 2002b). AERONET labels the retrieved size distribution dV (r)∕d ln r and reports the values in [μm3 μm-2] units. The correspondence between the AERONET retrievals and dN∕d ln r (18) in [# m-3 m-1] units is not exactly clear. Unfortunately, Table 1 does not help much here. Let us now show how to bridge the gap between theory and measurement.

First, total distributions contain N0 particles per unit volume and thus N0 applies as a multiplicative factor to (18)

(23)

Note that (23) is not normalized (cf. Section 3.3.2). Applying (6) to (23) yields

(24)

Multiplying each side of (24) by D and substituting d ln D = D^-1 dD leads to

(25)

The derivative in (25) is with respect to the logarithm of the diameter. The change in the independent variable of differentiation defines a new distribution which could be written nn(ln D) to distinguish it from the normal linear distribution nn(D) (6). However, the nomenclature nn(ln D) could be misinterpreted. We follow Seinfeld and Pandis (1997) and denote logarithmically-defined distributions with a superscript e for the distribution to re-inforce the use of ln D as the independent variable

(26)

The SI units of nn(D) (6) and ne n(ln D) (26) are [# m-3 m-1] and [# m-3], respectively.

Remote sensing application often sense columnar distributions rather than volumetric distributions. The columnar number distribution nc n(D), for example, is simply the vertical integral of the particle number distribution nn(D),

nc n(D)	≡		= ∫ z=0z=∞n n(D,z) dz		= same	(27a)
nc x(D)	≡		= ∫ z=0z=∞n x(D,z) dz		= ∫ z=0z=∞D2n n(D) dz	(27b)
nc s(D)	≡		= ∫ z=0z=∞n s(D,z) dz		= ∫ z=0z=∞πD2n n(D) dz	(27c)
nc v(D)	≡		= ∫ z=0z=∞n v(D,z) dz		= ∫ z=0z=∞D3n n(D) dz	(27d)
nc m(D)	≡		= ∫ z=0z=∞n m(D,z) dz		= ∫ z=0z=∞ρD3n n(D) dz	(27e)

Combining (27) with (25) leads to

ne,c n (ln D)	≡		= exp					(28a)
ne,c x (ln D)	≡		= exp					(28b)
ne,c s (ln D)	≡		= exp					(28c)
ne,c v (ln D)	≡		= exp					(28d)
ne,c m (ln D)	≡		= exp					(28e)

Measurements (or retrievals such as AERONET) are usually reported in historical units that can be counted rather than pure SI. The SI units for nv(D) = dV (D)∕dD are [m3 m-3 m-1], i.e., particle volume per unit air volume per unit particle diameter. These units condense to [m3 m-2], or, multiplying by 10⁶, [μm3 μm-2]. These condensed units may be confused with particle volume per unit particle surface area (V (D)∕S(D)), or with columnar particle volume per unit horizontal surface (e.g., ground or ocean) area (∫ V (z) dz). AERONET most definitely does not report any of these three quantities dV∕dr, V (D)∕S(D), or ∫ V (z) dz. AERONET reports ne,c v (ln D) the vertically integrated logarithmic volume distribution (28d), the logarithmic derivative of the columnar volume V c 0 .

3.2.3 Variance

According to (15), the variance σ2 D of the lognormal distribution (18) is

(29)

3.2.4 Common mistakes

Non-standard terminology leads to much confusion in the literature. For example, Dubovik et al. (2002a) provide precise analytic definitions of their supposedly lognormal size distribution parameters. However, their terminology is inconsistent with their definitions. Distributions computed according to their definitions are not lognormal distributions. Dubovik et al. (2002a) Equation A1 (their p. 606) defines the mean logarithmic radius v of the volume distribution which they confusingly name the volume median radius v. Dubovik et al. (2002a) Equation A2 (their p. 606) defines the standard deviation of the logarithm of the volume distribution. This differs from the geometric standard deviation σg of a lognormal distribution. The correct parameters of a lognormal distribution (18) are n and σg (or ≡ ln σg) For a lognormal volume path distribution ne,c v (ln D) (28d) the appropriate parameters are v and σg (or ≡ ln σg), not v and (29). Dubovik et al. (2002a) Equation 1 (their p. 593) is the correct form for ne,c v (ln D) (28d), but the incorrect parameter definitions will not yield a lognormal distribution.

3.2.5 Bounded Distribution

The statistical properties of a bounded lognormal distribution are expressed in terms of the error function (§5.2). The cumulative concentration bounded by Dmax is given by applying (2) to (18)

(30)

We make the change of variable z = (ln D - ln n)∕ ln σg

(33)

We are also interested in the bounded distributions of higher moments, e.g., the mass of particles lying between Dmin and Dmax. The cross-sectional area, surface area, volume, and mass distributions of spherical particles are related to their number distribution by

nx(D)	= D2n n(D)	(34a)
ns(D)	= πD2n n(D)	(34b)
nv(D)	= D3n n(D)	(34c)
nm(D)	= ρD3n n(D)	(34d)

(35)

3.2.6 Statistics of Bounded Distributions

All of the relationships given in Table 1 may be re-expressed in terms of truncated lognormal distributions, but doing so is tedious, and requires new terminology. Instead we derive the expression for a typical size distribution statistic, and allow the reader to generalize. We generalize (13) to consider

(36)

Note the domain of integration, D ∈ (Dmin,Dmax), reflects the fact that we are considering a bounded distribution. The superscript ^* indicates that the average statistic refers to a truncated distribution and reminds us that g^*g. Defining a closed form expression for p^*(D) requires some consideration. This truncated distribution has N* 0 defined by (33), and is completely specified on D ∈ (0,∞) by

(37)

The difficulty is that the three parameters of the lognormal distribution, n, σg, and N0 are defined in terms of an untruncated distribution. Using (33) we can write

(38)

If we think of p^* order to be properly normalized to unity, note that (fxm) Thus when we speak of truncated distributions it is important to keep in mind that the parameters n, σg, and N0 refer to the untruncated distribution.

The properties of the truncated distribution will be expressed in terms of * n, σ* g, and N* 0 , respectively.

Consider the mean size, D. In terms of (13) we have g(D) = D so that

(39)

3.2.7 Overlapping Distributions

Consider the problem of distributing I independent and possibly overlapping distributions of particles into J independent and possibly overlapping distributions of particles. To reify the problem we call the I bins the source bins (these bins represent the parent size distributions in mineral dust source areas) and the J bins as sink bins (which represent sizes transported in the atmosphere). Typically we know the total mass M0 or number N0 of source particles to distribute into the sink bins and we know the fraction of the total mass to distribute which resides in each source distribution, Mi. The problem is to determine matrices of overlap factors Ni,j and Mi,j which determine what number and mass fraction, respectively, of each source bin i is blown into each sink bin j.

The mass and number fractions contained by the source distributions are normalized such that

(40)

In the case of dust emissions, Mi and Ni may vary with spatial location.

The overlap factors Ni,j and Mi,j are defined by the relations

Using (33) and (40) we find

A mass distribution has the same form as a lognormal number distribution but has a different median diameter. Thus the overlap matrix elements apply equally to mass and number distributions depending on the median diameter used in the following formulae. For the case where both source and sink distributions are complete lognormal distributions,

3.2.8 Median Diameter

Substituting D = n into (32) we obtain

(45)

Thus the validity of n as the median diameter is now proven (5). The lognormal distribution is the only distribution known (to us) which is most naturally expressed in terms of its median diameter.

3.2.9 Multimodal Distributions

Realistic particle size distributions may be expressed as an appropriately weighted sum of individual modes.

(46)

where ni n(D) is the number distribution of the ith component mode⁵ . Such particle size distributions are called multimodal istributions because they contain one maximum for each component distribution. Generalizing (1), the total number concentration becomes

The median diameter of a multimodal distribution is obtained by following the logic of (30)–(33). The number of particles smaller than a given size is

3.3 Higher Moments

It is often useful to compute higher moments of the number distribution. Each factor of the independent variable weighting the number distribution function nn(D) in the integrand of (14) counts as a moment. The kth moment of nn(D) is

(51)

The statistical properties of higher moments of the lognormal size distribution may be obtained by direct integration of (51).

Applying the formula (53) to the first five moments of the lognormal distribution function we obtain

(54)

Table 1 includes these relations appear in slightly different forms.

The first few moments of the number distribution are related to measurable properties of the size distribution. In particular, F(k = 0) is the number concentration. Other quantities of merit are ratios of consecutive moments. For example, the volume-weighted diameter Dv is computed by weighted each diameter by the volume of particles at that diameter and then normalizing by the total volume of all particles.

The surface-weighted diameter Ds is defined analogously to Dv. Ds is more often known by its other name, the effective diameter (twice the effective radius). The term “effective” refers to the light extinction properties of the distribution. Light impinging on a particle distribution is, in the limit of geometric optics, extinguished in proportion to the cross-sectional area of the particles. Hence the effective diameter (or radius) characterizes the extinction properties of the distribution. Following (55), the effective diameter is

Moment-weighted diameters, such as the volume-weighted diameter Dv (55), characterize disperse distributions. A disperse mass distribution nm(D) behaves most like a monodisperse distribution with all mass residing at D = Dv. Due to approximations, physical operators may be constrained to act on a single, representative diameter rather than an entire distribution. The “least-wrong” diameter to pick is the moment-weighted diameter most relevant to the process being modeled. For example, Dv best represents the gravitational sedimentation of a distribution of particles. On the other hand, Ds (56) best represents the scattering cross-section of a distribution of particles.

3.3.1 Aspherical Particles

The useful relation (??) is a property of the lognormal distribution itself, rather than the particle shape. A lognormal distribution of aspherical particles also obeys (??). Important measurable properties of most convex aspherical habits may be represented by a constant times the k^th moment F(k) of the distribution. For example, the surface area Sh [m2] and volume V h [m3] of hexagonal prisms are given by (??)–(??). To be consistent with the diameter-centric expressions in Table 1, we introduce D_h, the hexagonal prism diameter. Adopting the convention that D_h ≡ 2a, the full-width of the basal face, we obtain

The functional forms for Sh and Vh consist of constants multiplying the diameter’s second and third moments, respectively. The surface area (πD²) and volume (πD³∕6) of spheres have the same form. Therefore the higher moments of aspherical particle distributions must be the same as spherical particle distributions modulo the leading constant expressions. Inserting Sh and Vh into (57), (58), and (??) leads to analytic expressions for the total surface area S0,h [m2 m-3] and volume V 0,h [m3 m-3] of a lognormal distribution of hexagonal prisms:

The total concentration N0,V∕S of equivalent V/S-spheres corresponding to a known distribution of hexagonal prisms must be computed numerically unless the size dependence of the aspect ratio Γ(D) takes an analytic form. In the simplest case, one can imagine or assume distributions of hexagonal prisms with constant aspect ratio, i.e, ΓΓ(D). In this idealized case, the ratio NV∕S∕Nh (??) is constant throughout the distribution. Then the analytic number concentration of equivalent V/S-spheres is simply NV∕S∕Nh times the analytic number concentration of hexagonal prisms which is presumably known directly from the lognormal size distribution parameters (cf. Table 1).

3.3.2 Normalization

We show that (18) is normalized by considering

(61)

where Cn is the normalization constant determined by (7). First we change variables to y = ln(D∕n)

4 Implementation in NCAR models

The discussion thus far has centered on the theoretical considerations of size distributions. In practice, these ideas must be implemented in computer codes which model, e.g., Mie scattering parameters or thermodynamic growth of aerosol populations. This section describes how these ideas have been implemented in the NCAR-Dust and Mie models.

4.1 NCAR-Dust Model

The NCAR-Dust model uses as input a time invariant dataset of surface soil size distribution. The two such datasets currently used are from Webb et al. (1993) and from IBIS (Foley, 1998). The Webb et al. dataset provides global information for three soil texture types: sand, clay and silt. At each gridpoint, the mass flux of dust is partitioned into mass contributions from each of these soil types. To accomplish this, the partitioning scheme assumes a size distribution for the source soil of the deflated particles.

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Soil Texture n σg Description Sand Sand         Silt Silt         Clay Clay         Soil Texture n σg Description Sand Sand         Silt Silt         Clay Clay         Table 4: Source size distribution associated with surface soil texture data of Webb et al. (1993) and of Foley (1998).

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Table 4 lists the lognormal distribution parameters associated with the surface soil texture data of Webb et al. (1993) and of Foley (1998). The dust model is a size resolving aerosol model. Thus, overlap factors are computed to determine the fraction of each parent size type which is mobilized into each atmospheric dust size bin during a deflation event.

4.2 Mie Scattering Model

This section documents the Mie scattering code mie. mie is box model intended to provide exact simulations of microphysical processes for the purpose of parameterization into larger scale models. mie provides instantaneous and equilibrium decriptions of many processes ranging from surface flux exchange, dust production, reflection of polarized radiation, and, as its name suggests, the interaction of particles and radiation. Thus the inputs to mie are the instantaneous state (boundary and initial conditions) of the environment. Given these, the program solves for the associated rates of change and unknown variables.

There is no time-stepping loop primarily because mie generates an extraordinary amount of information about the instantaneous state. Time-stepping this environment in a box-model-like format would be prohibitive if all quantities were allowed to evolve.

4.2.1 Input switches

The flexibility and power of mie can only be exercised by actively using the hundreds of input switches which control its behavior. This section describes how some of these switches are commonly used to control fundamental properties of the microphysical environment. A complete reference table for these switches, there default values, and dimensional units, is presented in Appendix 5.3.

The heart of mie is an aerosol size distribution. Most users will wish to initialize this size distribution to a particular type of aerosol, and to a particular shape. This is accomplished with the cmp_aer and psd_typ keywords. The linearity, range, and resolution of the grid on which the analytic size distribution is discretized are controlled by the sz_grd, sz_mnm, sz_mxm, sz_nbr switches, respectively. Compute size distribution characteristics of a lognormal distribution

4.2.2 Moments of Size Distribution

Determine the analytic (or resolved) moments of an arbitrary size distribution.

1. Generate the size distribution. (It may have more than one moment)
2. Select the statistics of interest

4.2.3 Generating Properties for Multi-Bin

On occasion, a seriouly masochistic scientist will decide to create the ultimate hybrid bin-spectral aerosol method by discretizing the size distribution into a finite number of bins each with an independently configurable analytic sub-bin distribution. Generating properties for all the bins in such a scheme requires enormous amounts of bookkeeping, or, if a computer is available, a relatively simple Perl batch script named psd.pl.

The psd.pl batch script calls mie repeatedly in a loop over particle bin. As input, psd.pl accepts concise array representations of each property of a bin. For example, --sz˘nbr={200,25,25,25} specifies that bin 1 is discretized into 200 sub-bins, and the remaining three bins are each discretized into only 25 sub-bins.

5 Appendix

5.1 Properties of Gaussians

The area under a Gaussian distribution may be expressed analytically when the domain is (-∞, +∞). This result may be obtained (IIRC) by transforming to polar coordinates in the complex plane x = r(cos θ + i sin θ).

(65)

This is a special case of a more general result

(66)

This result may be obtained by completing the square under the integrand, making the change of variable y = x + β∕2α, and applying (65). Substituting α = 1∕2 and β = 0 into (66) yields (65).

5.2 Error Function

The error function erf(x) may be defined as the partial integral of a Gaussian curve

(67)

Using (65) and the symmetry of a Gaussian curve, it is simple to show that the error function is bounded by the limits erf(0) = 0 and erf(∞) = 1. Thus erf(z) is the cumulative probability function for a normally distributed variable z (???). Most compilers implement erf(x) as an intrinsic function. Thus erf(x) is used to compute areas bounded by finite lognormal distributions (§3.2.5).

5.3 Command Line Switches for mie Code

Table 5 summarizes all of the command line arguments available to control the behavior of the mie program. This is a summary only—it is impractical to think that written documentation could every convey the exact meaning of all the switches⁶ . The most frequently used switches are described in Section 4.2.1. The only way to learn the full meaning of the more obscure switches is to read the source code itself.