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Copyright © 2000, Charles S. Zender
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. The license is available online at http://www.gnu.ai.mit.edu/copyleft/fdl.html.

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.

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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