Particle Size Distributions: Theory and Application to ...dust.ess.uci.edu/facts/psd/psd.pdf · 4 2 STATISTICS OF SIZE DISTRIBUTIONS and volume of ice crystals have been computed

Online: http://dust.ess.uci.edu/facts Updated: Mon 10th Jun, 2013, 11:45

Particle Size Distributions:Theory and Application to Aerosols, Clouds, and Soils

by Charlie ZenderUniversity of California, Irvine

Department of Earth System Science [email protected] of California Voice: (949) 891-2429Irvine, CA 92697-3100 Fax: (949) 824-3256

Copyright c© 1998–2013, Charles S. ZenderPermission is granted to copy, distribute and/or modify this document under the terms of the GNUFree Documentation License, Version 1.3 or any later version published by the Free SoftwareFoundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Thelicense is available online at http://www.gnu.org/copyleft/fdl.html.

Contents

Contents i

List of Tables 1

1 Introduction 11.1 Modal vs. Sectional Represenatation . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4.1 Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Statistics of Size Distributions 42.1 Generic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Mean Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Cloud and Aerosol Size Distributions 53.1 Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3.1 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3.2 Lognormal Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.3 Related Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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3.3.4 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.5 Non-standard terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.6 Bounded Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.7 Statistics of Bounded Distributions . . . . . . . . . . . . . . . . . . . . . 173.3.8 Overlapping Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.9 Median Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.10 Mode Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.11 Multimodal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Higher Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.1 Aspherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Implementation in NCAR models 244.1 NCAR-Dust Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Mie Scattering Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Input switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Moments of Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . 264.2.3 Generating Properties for Multi-Bin Distributions . . . . . . . . . . . . . . 27

5 Appendix 275.1 Properties of Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Command Line Switches for mie Code . . . . . . . . . . . . . . . . . . . . . . . 28

Bibliography 58

Index 61

List of Tables

1 Lognormal Distribution Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Measured Lognormal Dust Size Distributions . . . . . . . . . . . . . . . . . . . . 113 Analytic Lognormal Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Source Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Command Line Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 SWNB Output Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 CLM Output Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1 Introduction

This document describes mathematical and computational considerations pertaining to size distri-butions. The application of statistical theory to define meaningful and measurable parameters for

2 1 INTRODUCTION

defining generic size distributions is presented in §2. The remaining sections apply these defini-tions to the size distributions most commonly used to describe clouds and aerosol size distributionsin the meteorological literature. Currently, only the lognormal distribution is presented.

1.1 Modal vs. Sectional Represenatationmdlsxn Lu and Bowman (2004) designed and optimal non-linear least squares-based procedure forconverting from sectional to modal representations.

1.2 Nomenclaturenomenclature There is a bewildering variety of nomenclature associated with size distributions,probability density functions, and statistics thereof. The nomenclature in this article generally fol-lows 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 referencesare in agreement. Quantities whose nomenclature is often confusing, unclear, or simply not stan-dardized are discussed in the text.

1.3 Distribution FunctionThis section follows the carefully presented discussion of Flatau et al. (1989). The size distributionfunction nn(r) is defined such that nn(r) dr is the total concentration (number per unit volume ofair, or # m−3) of particles with sizes in the domain [r, r + dr]. The total number concentration ofparticles N0 is obtained by integrating nn(r) over all sizes

N0 =

∫ ∞0

nn(r) dr (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 nn(r) is only normalized if N0 = 1.0 (cf.Section 3.4.2).

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

N(r < rmax) =

∫ rmax

0

nn(r) dr (2)

N(r > rmax) =

∫ ∞rmax

nn(r) dr (3)

N(rmin, rmax) = N(rmin < r < rmax) =

∫ rmax

rmin

nn(r) dr (4)

Equations (2)–(4) define the cumulative concentration, lower bound concentration, and truncatedconcentration, respectively. The cumulative concentration is used to define the median radius rn.

1.4 Probability Density Function 3

Half the particles are larger and half smaller than rn

N(r < rn) = N(r > rn) =N0

2(5)

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

nn(r) =dN

dr(6)

Note that the concentration nomenclature in (6) is N not N(r). Using N(r) would indicate thatthe concentration has not been completely integrated over all sizes. By definition, the total concen-tration N0 is integrated over all sizes, as defined by (1). A concentration denoted N(r) makes nosense without an associated size bin width ∆r, or truncation convention, as in (2)–(4). We try touse N and N0 for normalized (N = 1) and non-normalized (N0 6= 1, i.e., absolute concentrations).However this convention is not absolute and (1) defines both N and N0.

1.4 Probability Density FunctionDescribing 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 domainof interest, i.e., p(r) = Cnnn(r) where the normalization constant Cn is defined such that∫ ∞

0

p(r) dr = 1 (7)

In the following sections we usually work with PDFs because this normalization property is veryconvenient mathematically. Comparing (7) and (1), it is clear that the normalization constant Cnwhich transforms a size distribution function (1) into a PDF p(r) is N−1

0

p(r) =1

N0

nn(r) (8)

1.4.1 Choice of Independent Variable

The merits of using radius r, diameter D, or some other dimension L, as the independent variableof a size distribution depend on the application. In radiative transfer applications, r prevails in theliterature 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. Defin-ing 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 defi-nitions, such as equivalent area spherical radius, equivalent volume spherical radius, (e.g., Ebertand Curry, 1992; McFarquhar and Heymsfield, 1997).

A direct property of aspherical particles which can often be measured is its maximum dimen-sion, i.e., the greatest distance between any two surface points of the particle. This maximumdimension, usually called L, has proven to be useful for characterizing size distributions of as-pherical particles. For a sphere, L is also the diameter. Analyses of mineral dust sediments in icecore deposits or sediment traps, for example, are usually presented in terms of L. The surface area

4 2 STATISTICS OF SIZE DISTRIBUTIONS

and volume of ice crystals have been computed in terms of power laws of L (e.g., Heymsfield andPlatt, 1984; Takano and Liou, 1995). Since models usually lack information regarding the shape ofparticles (early exceptions include Zender and Kiehl, 1994; Chen and Lamb, 1994), most modelersassume spherical particles, especially for aerosols. Thus, the advantages of using the diameter Das the independent variable in size distribution studies include: D is the dimension often reportedin measurements; D is more analogous than r to L.

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

Consider two distinct analytic representations of the same underlying size distribution. Thefirst, nDn (D), expresses the differential number concentration per unit diameter. The second, nrn(r),expresses the differential number concentration per unit radius. Both nDn (D) and nrn(r) share thesame dimensions, # m−3 m−1.

D = 2r (9)dD = 2 dr (10)

nDn (D) dD = nrn(r) dr (11)

nDn (D) =1

2nrn(r) (12)

2 Statistics of Size Distributions

2.1 GenericConsider 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 somephysically meaningful quantity (e.g., area) with size. The mean value of g is the integral of g overthe domain of the size distribution, weighted at each point by the concentration of particles

g =

∫ ∞0

g(x) p(x) dx (13)

Once p(x) is known, it is always possible to compute g for any desired quantity g. Typical quan-tities 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 − x)2.The remainder of this section considers some of these examples in more detail.

2.2 Mean SizeThe number mean size x of a size distribution p(x) is defined as

x =

∫ ∞0

p(x)x dx (14)

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

2.3 Variance 5

2.3 VarianceThe variance σ2

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

σ2x =

∫ ∞0

p(x)(x− x)2 dx (15)

The variance measures the mean squared-deviation of the distribution from its mean value. Theunits of σ2

x are [m2]. Because σ2x is a complicated function for standard aerosol and cloud size

distributions, many prefer to work with an alternate definition of variance, called the effectivevariance.

The effective variance σ2x,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)

σ2x,eff =

1

x2eff

∫ ∞0

p(x)(x− xeff)2 x2 dx (16)

Because of the x−2eff 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 DeviationThe standard deviation σx of a size distribution p(x) is the square root of the variance (15),

σx =√σ2x (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. Unfortu-nately, nomenclature for these alternate definitions is not standardized.

3 Cloud and Aerosol Size Distributions3.1 Gamma Distribution

Statistics of the gamma distribution are presented in http://asd-www.larc.nasa.gov/

˜yhu/paper/thesisall/node8.html. Currently, the aerosol property program mie im-plements gamma distributions in a limited sense.

3.2 Normal Distribution

The normal distribution is the most common statistical distribution. The normal distribution n(x)is expressed in terms of its mean x (14) and standard deviation σx (17)

n(x) =1√

2πσxexp

[−1

2

(x− xσx

)2]

(18)

http://asd-www.larc.nasa.gov/~yhu/paper/thesisall/node8.html

http://asd-www.larc.nasa.gov/~yhu/paper/thesisall/node8.html

6 3 CLOUD AND AEROSOL SIZE DISTRIBUTIONS

With our standard nomenclature for number distribution nn and particles diameter D, (18) appearsas

nn(D) ≡ dN

dD=

1√2πσD

exp

[−1

2

(D − Dn

σD

)2]

(19)

The cumulative normal distribution is called the error function and is discussed in Section (5.1).Integration of the error function shows that 68.3% of the values of (19) are in Dn ± σD, 95.4% arein Dn ± 2σD, and 99.7% are in Dn ± 3σD.

3.3 Lognormal Distribution

The lognormal distribution is perhaps the most commonly used analytic expression in aerosolstudies.

3.3.1 Distribution Function

In a lognormal distribution, the logarithm of abscissa is normally distributed (Section 3.2). Substi-tuting x = lnD into (18) yields

nn(lnD) ≡ dN

d lnD=

1√2π lnσg

exp

−1

2

(lnD − ln Dn

lnσg

)2 (20)

where σg and Dn are parameters whose physical significance is to be defined. In particular, there isno closed-form algebraic relationship between σD (19) and σg (20). The former is a true standarddeviation and the properties of the latter are as yet unknown.

Substituting d lnD = D−1 dD in (20) leads to the most commonly used form the lognormaldistribution function

nn(D) ≡ dN

dD=

1√2πD lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (21)

One of the most confusing aspects of size distributions in the meteorological literature is in theusage of σg, the geometric standard deviation. Some researchers (e.g., Flatau et al., 1989) prefera different formulation (21) which is equivalent to

nn(D) =1√

2π σgDexp

−1

2

(ln(D/Dn)

σg

)2 (22)

whereσg ≡ lnσg (23)

In practice, (21) is used more widely than (22) and we adopt (21) in the following.

3.3 Lognormal Distribution 7

The definition of σg in (22) may be more satisfactory from a mathematical point of view (Flatauet al., 1989), and it subsumes an “ln” which reduces typing. This is seen by transforming x in (18)with

x =1

σg

ln

(D

Dn

)D = Dn exp(σgx)

dD = σgDn exp(σgx) dx

dx = (σgDn)−1 dD

This maps x ∈ (−∞,+∞) into D ∈ [0,∞).One is occasionally given a “standard deviation” or “geometric standard deviation” parameter

without clear specification whether it represents σg (or lnσg, or expσg, or σx) in (17), (21), or (22).As a true standard deviation, σx has dimensions of x, whereas both σg and σg are dimensionlessso units cannot disambiguate them. A useful rule of thumb is that σg in (21) and eσg in (22)are usually between 1.5–2.5 for realistic aerosol populations. Since we adopted (21), physicallyrealistic values are σg ∈ (1.5, 2.5).

Seinfeld and Pandis (1997) p. 423 describe the physical meaning of the geometric standarddeviation σg. Define the special particle sizes

D+σg≡ Dnσg (24a)

D−σg≡ Dn/σg (24b)

The cumulative concentration smaller than D+σg

, simplifies from (38) to

N(D < D+σg

) =N0

2+N0

2erf

(1√2

)= 0.841344746069N0 (25)

Numerical integration must be used to obtain the final result, 0.841N0, as erf() has no closed-formsolution here. Using (25) to invert (24), we may define σg as the ratio of the diameter D+

σg(larger

than 84.1% of all particles) to the median diameter Dn. Monodisperse populations have σg ≡ 1.Similarly the cumulative concentration smaller than D−σg

, simplifies from (38) to

N(D < D−σg) =

N0

2+N0

2erf

(− 1√

2

)= 0.158655253931N0 (26)

where we have used the numerical result in (26) with the error function’s anti-symmetric property,erf(−x) = −erf(x). Subtracting (26) from (25) shows that 68.3% of all particles in a lognormaldistribution lie in D ∈ [D−σg

, D+σg

].By raising σg to any power x in (24), it is straightforward to verify that the number of particles

within D ∈ [Dnσ−xg , Dnσ

xg ] is

N(Dnσ−xg < D < Dnσ

xg ) = N0 erf(x/

√2) (27)

Application of (27) for small integer x shows that 68.3% of all particles lie within Dn/σg < D <Dnσg, that 95.4% of all particles1 lie within Dnσ

−2g < D < Dnσ

2g , and that 99.7% of all particles

1Seinfeld and Pandis (1997) p. 423 has a typo on this point. That page erroneously states that the bounds bracketing95% of a lognormal distribution are Dn/(2σg) < D < 2Dnσg.


lie within Dnσ−3g < D < Dnσ

3g . These bounds are identical to the fraction of particles enclosed

within one, two, and three standard deviations of the mean of a normal distribution (Section 3.2).

3.3.2 Lognormal Relations

Table 1 summarizes the standard lognormal distribution parameters. Note that σg ≡ lnσg. Thestatistics 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., thenumber distribution nn(D), apply to distributions of all other moments. For example, the numberdistribution nn(D) is the only distribution for which the moment mean size (i.e., number meansize Dn) equals the moment-weighted size (i.e., number-weighted size Dn). Also, the numbermean size Dn differs from the number median size Dn by a factor exp(σg

2/2). But this factoris not constant and depends on the moment of the distribution. For instance, Ds differs from Ds

by exp(σg2), while Ds differs from Ds by exp(3σg

2/2). Thus converting from mean diameter tomedian diameter is not the same for number as for mass distributions.


Tabl

e1:

Log

norm

alD

istr

ibut

ion

Rel

atio

ns23

4

Sym

bol

Val

ueU

nits

Des

crip

tion

Defi

ning

Rel

atio

n

N0

N0

#m−

3To

taln

umbe

rcon

cent

ratio

nN

0=

∫ ∞ 0

nn(D

)dD

D0

N0D

nex

p(σ

g2)

mm−

3To

tald

iam

eter

D0

=

∫ ∞ 0

Dn

n(D

)dD

A0

π 4N

0D

2 nex

p(σ

g2/2

)m

2m−

3To

talc

ross

-sec

tiona

lare

aA

0=

∫ ∞ 0

π 4D

2n

n(D

)dD

S0

πN

0D

2 nex

p(2σ

g2)

m2

m−

3To

tals

urfa

cear

eaS

0=

∫ ∞ 0

πD

2n

n(D

)dD

V0

π 6N

0D

3 nex

p(9σ

g2/2

)m

3m−

3To

talv

olum

eV

0=

∫ ∞ 0

π 6D

3n

n(D

)dD

M0

π 6N

0ρD

3 nex

p(9σ

g2/2

)kg

m−

3To

talm

ass

M0

=

∫ ∞ 0

π 6ρD

3n

n(D

)dD

DD

nex

p(σ

g2/2

)m

#−1

Mea

ndi

amet

erN

0D

=N

0D

n=D

0

Aπ 4D

2 nex

p(2σ

g2)

m2

#−1

Mea

ncr

oss-

sect

iona

lare

aN

0A

=N

0π 4D

2 s=A

0

SπD

2 nex

p(2σ

g2)

m2

#−1

Mea

nsu

rfac

ear

eaN

0S

=N

0πD

2 s=S

0

Vπ 6D

3 nex

p(9σ

g2/2

)m

3#−

1M

ean

volu

me

N0V

=N

0π 6D

3 v=V

0

Mπ 6ρD

3 nex

p(9σ

g2/2

)kg

#−1

Mea

nm

ass

N0M

=N

0π 6ρD

3 v=M

0

N0

6 πρM

0D−

3n

exp(−

9σg2/2

)#

m−

3N

umbe

rcon

cent

ratio

nN

0=

∫ ∞ 0

nn(D

)dD

Dn

( 6M

0

πN

0ρ

) 1/3ex

p(−

3σg2/2

)m

Med

ian

diam

eter

∫ D n 0

nn(D

)dD

=N

0 2

Deff

6M

0

ρS

0m

Eff

ectiv

edi

amet

erD

eff=

1 A0

∫ ∞ 0

Dπ 4D

2n

n(D

)dD

S6

ρD

eff

m2

kg−

1Sp

ecifi

csu

rfac

ear

eaS

=S

0/M

0


Tabl

e1:

(con

tinue

d)

Sym

bol

Val

ueU

nits

Des

crip

tion

Defi

ning

Rel

atio

n

Dn

Dn

exp(−σ

g2/2

)m

Med

ian

diam

eter

,Sca

ling

diam

eter

,Num

berm

edia

ndi

amet

er.H

alfo

fpar

ticle

sar

ela

rger

than

,and

half

smal

lert

han,

Dn

∫ D n 0

nn(D

)dD

=N

0 2

Dn,D

,D

n

Dn

exp(σ

g2/2

)m

Mea

ndi

amet

er,A

vera

gedi

amet

er,

Num

ber-

wei

ghte

dm

ean

diam

eter

Dn

=1 N

0

∫ ∞ 0

Dn

n(D

)dD

Ds

Dn

exp(σ

g2)

mSu

rfac

em

ean

diam

eter

N0πD

2 s=N

0S

=S

0

Dv

Dn

exp(3σ

g2/2

)m

Volu

me

mea

ndi

amet

er,M

ass

mea

ndi

amet

erN

0π 6D

3 v=N

0V

=V

0

Ds

Dn

exp(2σ

g2)

mSu

rfac

em

edia

ndi

amet

er∫ D s 0

πD

2n

n(D

)dD

=S

0 2

Ds,

Deff

Dn

exp(5σ

g2/2

)m

Are

a-w

eigh

ted

mea

ndi

amet

er,

effe

ctiv

edi

amet

erD

s=

1 A0

∫ ∞ 0

Dπ 4D

2n

n(D

)dD

Dv

Dn

exp(3σ

g2)

mVo

lum

em

edia

ndi

amet

erM

ass

med

ian

diam

eter

∫ D v 0

π 6D

3n

n(D

)dD

=V

0 2

Dv

Dn

exp(7σ

g2/2

)m

Mas

s-w

eigh

ted

mea

ndi

amet

er,

Volu

me-

wei

ghte

dm

ean

diam

eter

Dv

=1 V0

∫ ∞ 0

Dπ 6D

3n

n(D

)dD


For brevity Table 1 presents the lognormal relations in terms of diamterD. Change the relationsto befunctions of radius r is straightforward. For example, direct substitution of D = 2r into (21)yields

nn(D) =1√

2π 2r lnσg

exp

[−1

2

(ln(2r/2rn)

lnσg

)2]

=1

2

1√2π r lnσg

exp

[−1

2

(ln(r/rn)

lnσg

)2]

=1

2nrn(r) (28)

in agreement with (12).Table 2 lists applies the relations in Table 1 to specific size distributions typical of tropospheric

aerosols.

Table 2: Measured Lognormal Dust Size Distributions5

Dn Dv σg M Ref.µm µm

Patterson and Gillette (1977) 6

0.08169 0.27 1.88 ??0.8674 5.6 2.2 ??28.65 57.6 1.62 ??

Shettle (1984)7

0.003291 0.0111 1.89 2.6× 10−4 ??0.5972 2.524 2.08 0.781 ??, ??7.575 42.1 2.13 0.219 ??

Balkanski et al. (1996)9

0.1600 0.832 2.1 0.036 ??1.401 4.82 1.90 0.957 ??9.989 19.38 1.60 0.007 ??

6Detailed fits to dust sampled over Colorado and Texas in Patterson and Gillette (1977), p. 2080 Table 1. Originalvalues have been converted from radius to diameter. M was not given. Patterson and Gillette (1977) showed soilaerosol could be represented with three modes which they dubbed, in order of increasing size, modes C, A, and B.Mode A is the mineral dust transport mode, seen in source regions and downwind. Mode B is seen in the source soilitself, and in the atmosphere during dust events. Mode C is seen most everywhere, but does not usually correlate withlocal dust amount. Mode C is usually a global, aged, background, anthropogenic aerosol, typically rich in sulfateand black carbon. Sometimes, however, Mode C has a mineral dust component. Modes C and B are averages fromPatterson and Gillette (1977) Table 1 p. 2080. Mode B is based on the summary recommendation that rs = 1.5 andσg = 2.2.

7Background Desert Model from Shettle (1984), p. 75 Table 1.9Balkanski et al. (1996), p. 73 Table 2. These are the “background” modes of D’Almeida (1987).


Table 2: (continued)

Dn Dv σg M Ref.µm µm

Alfaro et al. (1998) 10

0.6445 1.5 1.7 (0.22, 0.15) ??3.454 6.7 1.6 (0.69, 0.76) ??8.671 14.2 1.5 (0.09, 0.09) ??

Dubovik et al. (2002a), Bahrain (1998–2000) 1112

0.1768 0.30± 0.08 0.42± 0.04 ??1.664 5.08± 0.08 0.61± 0.02 ??

Dubovik et al. (2002a), Solar Village Saudi Arabia (1998–2000) 13

0.1485 0.24± 0.10 0.40± 0.05 ??1.576 4.64± 0.06 0.60± 0.03 ??

Dubovik et al. (2002a), Cape Verde (1993–2000) 14

0.1134 0.24± 0.06 0.49 + 0.10τ ± 0.04 ??1.199 3.80± 0.06 0.63− 0.10τ ± 0.03 ??

Maring et al. (2003) 15

3.6± 0.3 none ??4.1 none ??

Arimoto et al. (2006) 16

0.0 1.1 0.0 ??0.0 5.5 0.0 ??0.0 14 0.0 ??

Mokhtari et al. (2012) 17

0.0 0.2 1.75 0.0008 ??0.0 1.67 1.76 0.0092 ??0.0 11.6 1.70 0.99 ??

10Mass fractions are reported as (a,b) for measurements and model, respectively, of Spanish soil sample at u∗ =66 cm s−1

11All Dubovik et al. (2002a) measurements follow certain conventions. Standard deviation of measurements follows± sign. Reported σg is not the geometric standard deviation. Their σg is defined as the standard deviation of thelogarithm of the size distribution (Dubovik et al., 2002a, p. 606, Equation A2).

12Bahrain is an island in the Red Sea.13The Solar Village AERONET station is located in empty land a few kilometers west of Riyadh’s international

airport.14Values of Dn were computed using σg based on τ = 0.1.15Measurements during PRIDE, July 2000, from Izana and Puerto Rico. Only Dv reported as measurements did

not fit lognormal distributions.16Original manuscript does not contain σg.17AMMA size distribution used in DEAD coupled with SURFEX.


Table 3: Analytic Lognormal Size Distribution Statistics ab

Dn Dn, Dn Ds Dv Ds Ds Dv Dv σg

rn rn, rn rs rv rs rs rv rv

µm µm µm µm µm µm µm µm0.1 0.1272 0.1619 0.2056 0.2614 0.3323 0.4227 0.5373 2.0

0.1861 0.2366 0.3009 0.3825 0.4864 0.6185 0.7864 1.0 2.0

0.2366 0.3008 0.3825 0.4864 0.6185 0.7864 1.0 1.272 2.0

0.3009 0.3825 0.4864 0.6185 0.7864 1.0 1.272 1.617 2.0

0.3825 0.4864 0.6185 0.7864 1.0 1.272 1.617 2.056 2.0

0.4864 0.6185 0.7864 1.0 1.272 1.617 2.056 2.614 2.0

0.5915 0.7521 0.9563 1.216 1.546 1.966 2.5 3.179 2.0

0.6185 0.7864 1.0 1.272 1.617 2.056 2.614 3.324 2.0

0.7864 1.0 1.272 1.617 2.056 2.614 3.324 4.225 2.0

0.8281 1.053 1.339 1.702 2.165 2.753 3.5 4.450 2.0

1.0 1.272 1.617 2.056 2.614 3.324 4.227 5.373 2.0

1.183 1.504 1.913 2.432 3.092 3.932 5.0 6.356 2.0

2.366 3.008 3.825 4.864 6.184 7.864 10.0 12.72 2.0

aShown are statistics for each moment equalling 1 µm, and for Dv = 0.1, 2.5, 3.5, 5.0, 10.0 µm.b Dn, Ds, and Dv are number, surface, and volume-mean diameters, respectively. Dn, Ds, and Dv are number,

surface, and volume median diameters, respectively. Dn, Ds, and Dv are number, surface, and volume-weighteddiameters, respectively.

Perry et al. (1997) and Perry and Cahill (1999) describe measurements and transport of dustacross the Atlantic and Pacific, respectively. Reid et al. (2003) summarize historical measurementsof dust size distributions, and analyze the influence of measurement technique on the derived sizedistribution. They show the derived size distribution is strongly sensitive to the measurment tech-nique. During PRIDE, measured Dv 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-tropicalAtlantic is consistent with a slight updraft of ∼ 0.33 cm s−1 during transport. Ginoux (2003) andColarco et al. (2003) show that the effects of asphericity on particle settling velocity play an im-portant 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 troposphericaerosols. Values in Table 3 are valid for radius and diameter distributions. Table 1 shows thatall moments of the size distribution depend linearly on Dn (or rn). Therefore all rows in Table 3scale linearly (for a constant geometric standard deviation). For example, values in the row withDn = 1.0 µm are ten times the corresponding values for the row Dn = 0.1 µm. Hence it sufficesfor Table 3 to show a decade range in Dn.


3.3.3 Related Forms

Many important applications make available size distribution information in a form similar to, buthard to recognize as, the analytic lognormal PDF (21). The Aerosol Robotic Network, AERONET ,for example, retrieves size distributions from solar almucantar radiances18 (Dubovik and King,2000; Dubovik et al., 2000, 2002b). AERONET labels the retrieved size distribution dV (r)/d ln rand reports the values in [µm3 µm−2] units. The correspondence between the AERONET retrievalsand dN/d ln r (21) in [# m−3 m−1] units is not exactly clear. Unfortunately, Table 1 does not helpmuch 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 multi-plicative factor to (21)

nn(D) =N0√

2πD lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (29)

Note that (29) is only normalized if N0 = 1.0 (cf. Section 3.4.2).Applying (6) to (29) yields

dN

dD=

N0√2πD lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (30)

Multiplying each side of (30) by D and substituting d lnD = D−1 dD leads to

dN

d lnD=

N0√2π lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (31)

The derivative in (31) is with respect to the logarithm of the diameter. The change in the inde-pendent variable of differentiation defines a new distribution which could be written nn(lnD) todistinguish it from the normal linear distribution nn(D) (6). However, the nomenclature nn(lnD)could be misinterpreted. We follow Seinfeld and Pandis (1997) and denote logarithmically-defineddistributions with a superscript e on the distribution that re-inforces the use of lnD as the indepen-dent variable

nen(lnD) ≡ nn(lnD) ≡ dN

d lnD(32)

The SI units of nn(D) (6) and nen(lnD) (32) are [# m−3 m−1] and [# m−3], respectively.

Remote sensing applications often retrieve columnar distributions rather than volumetric dis-tributions. The columnar number distribution nc

n(D), for example, is simply the vertical integral

18 The almucantar radiances are radiance measurements in a circle of equal scattering angle centered in a planeabout the Sun, i.e., radiance measurements at known forward scattering phase function angles.


of the particle number distribution nn(D),

ncn(D) ≡ dN c

0

dD=

∫ z=∞

z=0

nn(D, z) dz = same (33a)

ncx(D) ≡ dAc

0

dD=

∫ z=∞

z=0

nx(D, z) dz =

∫ z=∞

z=0

π

4D2nn(D, z) dz (33b)

ncs(D) ≡ dSc

0

dD=

∫ z=∞

z=0

ns(D, z) dz =

∫ z=∞

z=0

πD2nn(D, z) dz (33c)

ncv(D) ≡ dV c

0

dD=

∫ z=∞

z=0

nv(D, z) dz =

∫ z=∞

z=0

π

6D3nn(D, z) dz (33d)

ncm(D) ≡ dM c

0

dD=

∫ z=∞

z=0

nm(D, z) dz =

∫ z=∞

z=0

π

6ρD3nn(D, z) dz (33e)

SI units of the columnar distributions ncx for x = n, x, s, v,m (33) are one less “per meter” than the

corresponding volumetric distributions, e.g., nv and ncv are in [m3 m−3 m−1] and [m3 m−2 m−1],

respectively. This is because of integration over the vertical coordinate.Combining (33) with (31) leads to

ne,cn (lnD) ≡ dN c

0

d lnD=

N c0√

2π lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (34a)

ne,cx (lnD) ≡ dAc

0

d lnD=

√π

2

N c0D

2

4 lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (34b)

ne,cs (lnD) ≡ dSc

0

d lnD=

√π

2

N c0D

2

lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (34c)

ne,cv (lnD) ≡ dV c

0

d lnD=

√π

2

N c0D

3

6 lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (34d)

ne,cm (lnD) ≡ dM c

0

d lnD=

√π

2

ρN c0D

3

6 lnσg

exp

−1

2

(ln(D/Dn)

lnσg

)2 (34e)

These logarithmic columnar (vertically integrated) distributions (34) are one less “per meter” thanthe corresponding linear columnar distributions (33), e.g., nc

v and ne,cv are in [m3 m−2 m−1] and

[m3 m−2], respectively. In order for the area under the curve to be proportional to the integrateddistributions, logarithmic distributions should be plotted on semi-log axes, e.g., horizontal axiswith logarithmic size D and vertical axis with linearly spaced values of ne

v(lnD) (Seinfeld andPandis, 1997, p. 415).

Measurements (or retrievals such as AERONET) are usually reported in historical units thatcan be counted rather than in pure SI. 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 volumeper 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 (lnD) the verti-cally integrated logarithmic volume distribution (34d), the logarithmic derivative of the columnarvolume V c

0 .

3.3.4 Variance

According to (15), the variance σ2D of the lognormal distribution (21) is

σ2D =

1√2π lnσg

∫ ∞0

1

Dexp

−1

2

(ln(D/Dn)

lnσg

)2 (D − D)2 dD (35)

3.3.5 Non-standard terminology

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 param-eters. However, their terminology is inconsistent with their definitions. Distributions computedaccording to their definitions are not lognormal distributions. Dubovik et al. (2002a) Equation A1(their p. 606) defines the mean logarithmic radius rv of the volume distribution which they confus-ingly name the volume median radius rv. Dubovik et al. (2002a) Equation A2 (their p. 606) definesthe standard deviation of the logarithm of the volume distribution. This differs from the geometricstandard deviation σg of a lognormal distribution. The correct parameters of a lognormal distribu-tion (21) are rn and σg (or σg ≡ lnσg) For a lognormal volume path distribution ne,c

v (lnD) (34d)the appropriate parameters are rv and σg (or σg ≡ lnσg), not rv and

√σ2r (35). Dubovik et al.

(2002a) Equation 1 (their p. 593) is the correct form for ne,cv (lnD) (34d).

3.3.6 Bounded Distribution

The statistical properties of a bounded lognormal distribution are expressed in terms of the errorfunction (§5.2). The cumulative concentration bounded by Dmax is given by applying (2) to (21)

N(D < Dmax) =N0√

2π lnσg

∫ Dmax

0

1

Dexp

−1

2

(ln(D/Dn)

lnσg

)2 dD (36)

We make the change of variable z = (lnD − ln Dn)/√

2 ln σg

z = (lnD − ln Dn)/√

2 ln σg

D = Dne√

2 z lnσg

= Dnσ√

2 zg

dz = (√

2D lnσg)−1 dD

dD =√

2 ln σgDne√

2 z lnσg dz

=√

2 ln σgDnσ√

2 zg dz (37)


which maps D ∈ (0, Dmax) into z ∈ (−∞, lnDmax − ln Dn)/√

2 lnσg). In terms of z we obtain

N(D < Dmax) =N0√

2π lnσg

∫ (lnDmax−ln Dn)/√

2 lnσg

−∞

1

Dne√

2 z lnσge−z

2 √2 lnσgDne

√2 z lnσg dz

=N0√

π


2 lnσg

−∞e−z

2

dz

=N0√

π

(∫ 0

−∞e−z

2

dz +


2 lnσg

0

e−z2

dz

)

=N0

2

(2√π

∫ +∞

0

e−z2

dz +2√π


2 lnσg

0

e−z2

dz

)

=N0

2

[erf(∞) + erf

(ln(Dmax/Dn)√

2 ln σg

)]

=N0

2+N0

2erf

(ln(Dmax/Dn)√

2 ln σg

)(38)

where we have used the properties of the error function (§5.2). The same procedure can be per-formed to compute the cumulative concentration of particles smaller than Dmin. When N(D <Dmin) is subtracted from (38) we obtain the truncated concentration (4)

N(Dmin, Dmax) =N0

2

[erf

(ln(Dmax/Dn)√

2 ln σg

)− erf

(ln(Dmin/Dn)√

2 ln σg

)](39)

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

nx(D) =π

4D2nn(D) (40a)

ns(D) = πD2nn(D) (40b)

nv(D) =π

6D3nn(D) (40c)

nm(D) =π

6ρD3nn(D) (40d)

so that we may simply substitute Dn = Dv, for example, in (39) and we obtain

V (Dmin, Dmax) =N0

2

[erf

(ln(Dmax/Dv)√

2 lnσg

)− erf

(ln(Dmin/Dv)√

2 lnσg

)](41)

3.3.7 Statistics of Bounded Distributions

All of the relations given in Table 1 may be re-expressed in terms of truncated lognormal distribu-tions, 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

g∗ =

∫ Dmax

Dmin

Dp∗(D) dD (42)

Note the domain of integration, D ∈ (Dmin, Dmax), reflects the fact that we are considering abounded distribution. The superscript ∗ indicates that the average statistic refers to a truncateddistribution and reminds us that g∗ 6= g. Defining a closed form expression for p∗(D) requiressome consideration. This truncated distribution hasN∗0 defined by (39), and is completely specifiedon D ∈ (0,∞) by

p∗(D) =

0 , 0 < D < Dmin

N(Dmin, Dmax) p(D)/N0 , Dmin ≤ D ≤ Dmax

0 , Dmax < D <∞(43)

The difficulty is that the three parameters of the lognormal distribution, Dn, σg, and N0 are definedin terms of an untruncated distribution. Using (39) we can write

p∗(D) =1

N∗0nn(D)N∗0 = N(Dmin, Dmax) (44)

If we think of p∗ order to be properly normalized to unity, note that (fxm) Thus when we speakof truncated distributions it is important to keep in mind that the parameters Dn, σg, and N0 referto the untruncated distribution.

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

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

D =

∫ Dmax

Dmin

Dp(D) dx (45)

3.3.8 Overlapping Distributions

Consider the problem of distributing I independent and possibly overlapping distributions of par-ticles into J independent and possibly overlapping distributions of particles. To reify the problemwe call the I bins the source bins (these bins represent the parent size distributions in mineral dustsource 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 sinkbins and we know the fraction of the total mass to distribute which resides in each source distribu-tion, Mi. The problem is to determine matrices of overlap factors Ni,j and Mi,j which determinewhat 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

I∑i=1

Mi =I∑i=1

Ni = 1 (46)

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

Nj =I∑i=1

Ni,jNi

= N0

I∑i=1

Ni,jNi (47)

Mj =I∑i=1

Mi,jMi

= M0

I∑i=1

Mi,jMi (48)

Using (39) and (46) we find

Ni,j =1

2

[erf

(ln(Dmax,j/Dn,i)√

2 ln σg,i

)− erf

(ln(Dmin,j/Dn,i)√

2 ln σg,i

)](49)

Mi,j =1

2

[erf

(ln(Dmax,j/Dv,i)√

2 lnσg,i

)− erf

(ln(Dmin,j/Dv,i)√

2 lnσg,i

)](50)

fxm: The mathematical derivation appears correct but the overlap factor appears to asymptote to0.5 rather than to 1.0 for Dmax � Dn � Dmin.

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

M(D) =i=I∑i=1

Mi(D)

3.3.9 Median Diameter

Substituting D = Dn into (38) we obtain

N(D < Dn) =N0

2(51)

This proves that Dn is the median diameter (5). The lognormal distribution is the only distributionknown (to us) which is most naturally expressed in terms of its median diameter.

3.3.10 Mode Diameter

The mode is the most frequently occuring value of a distribution. The mode diameter or modaldiameter of the number distribution nn(D) is the diameter Dn that satisfies

dnn(D)

dD

∣∣∣∣D=Dn

= 0 (52)


Applying condition (52) to (21) proves that the median and modal diameters are identical forlognormal distributions

Dn = Dn (53)

The number, surface, volume, and mass distributions are all lognormal if any one is. Therefore(53) implies Ds = Ds, and Dv = Dv.

3.3.11 Multimodal Distributions

Realistic particle size distributions may be expressed as an appropriately weighted sum of individ-ual modes.

nn(D) =I∑i=1

nin(D) (54)

where nin(D) is the number distribution of the ith component mode19. Such particle size distribu-tions are called multimodal istributions because they contain one maximum for each componentdistribution. Generalizing (1), the total number concentration becomes

N0 =I∑i=1

∫ ∞0

nin(D) dD

=I∑i=1

N i0 (55)

where N i0 is the total number concentration of the ith component mode.

The median diameter of a multimodal distribution is obtained by following the logic of (36)–(39). The number of particles smaller than a given size is

N(D < Dmax) =I∑i=1

N i0

2+N i

0

2erf

(ln(Dmax/D

in)√

2 ln σig

)(56)

(57)

For the median particle size, Dmax ≡ Dn, and we can move the unknown Dn to the LHS yielding

I∑i=1

N i0

2+N i

0

2erf

(ln(Dn/D

in)√

2 ln σig

)=

N0

2

I∑i=1

N i0 erf

(ln(Dn/D

in)√

2 ln σig

)= 0 (58)

where we have used N0 =∑I

i Ni0. Obtaining Dn for a multimodal distribution requires numeri-

cally solving (58) given the N i0, Di

n, and σig.

19Throughout this section the i superscript represents an index of the component mode, not an exponent.

3.4 Higher Moments 21

3.4 Higher MomentsIt is often useful to compute higher moments of the number distribution. Each factor of the inde-pendent variable weighting the number distribution function nn(D) in the integrand of (14) countsas a moment. The kth moment of nn(D) is

F (k) =

∫ ∞0

nn(D)Dk dD (59)

The statistical properties of higher moments of the lognormal size distribution may be obtainedby direct integration of (59).

F (k) =N0√

2π lnσg

∫ ∞0

1

Dexp

−1

2

(ln(D/Dn)

lnσg

)2Dk dD

=N0√

2π lnσg

∫ ∞0

Dk−1 exp

−1

2

(ln(D/Dn)

lnσg

)2 dD (60)

We make the same change of variable z = (lnD − ln Dn)/√

2 ln σg as in (37). This maps D ∈(0,+∞) into z ∈ (−∞,+∞). In terms of z we obtain

F (k) =N0√

2π lnσg

∫ +∞

−∞(Dne

√2 z lnσg)k−1e−z

2√2 ln σgDne

√2 z lnσg dz

=N0√

π

∫ +∞

−∞(Dne

√2 z lnσg)ke−z

2

dz

=N0D

kn√

π

∫ +∞

−∞e√

2kz lnσge−z2

dz

=N0D

kn√

π

∫ +∞

−∞e−z

2+√

2kz lnσg dz

=N0D

kn√

π

√π exp

(2k2 ln2 σg

4

)= N0D

kn exp(1

2k2 ln2 σg) (61)

where we have used (74) with α = 1 and β =√

2k lnσg.Applying the formula (61) to the first five moments of the lognormal distribution function we

obtain

F (0) =

∫ ∞0

nn(D) dD = N0 = N0 = N0

F (1) =

∫ ∞0

nn(D)D dD = N0Dn exp(12

ln2 σg) = D0 = N0Dn

F (2) =

∫ ∞0

nn(D)D2 dD = N0D2n exp(2 ln2 σg) =

S0

π= N0D

2s

F (3) =

∫ ∞0

nn(D)D3 dD = N0D3n exp(9

2ln2 σg) =

6V0

π= N0D

3v

F (4) =

∫ ∞0

nn(D)D4 dD = N0D4n exp(8 ln2 σg)

(62)


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 meritare ratios of consecutive moments. For example, the volume-weighted diameter Dv is computedby weighted each diameter by the volume of particles at that diameter and then normalizing by thetotal volume of all particles.

Dv =

∫ ∞0

Dπ

6D3nn(D) dD

/∫ ∞0

π

6D3nn(D) dD

=

∫ ∞0

D4nn(D) dD

/∫ ∞0

D3nn(D) dD

= F (4)/F (3)

=N0D

4n exp(8 ln2 σg)

N0D3n exp(9

2ln2 σg)

= Dn exp(72

ln2 σg) (63)

The surface-weighted diameter Ds is defined analogously to Dv. Ds is more often known byits other name, the effective diameter (twice the effective radius). The term “effective” refers tothe light extinction properties of the distribution. Light impinging on a particle distribution is, inthe 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 (63), the effective diameter is

Ds = F (3)/F (2)

=N0D

3n exp(9

2ln2 σg)

N0D2n exp(2 ln2 σg)

= Dn exp(52

ln2 σg) (64)

Moment-weighted diameters, such as the volume-weighted diameter Dv (63), characterize dis-perse distributions. A disperse mass distribution nm(D) behaves most like a monodisperse dis-tribution with all mass residing at D = Dv. Due to approximations, physical operators maybe 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 be-ing modeled. For example, Dv best represents the gravitational sedimentation of a distribution ofparticles. On the other hand, Ds (64) best represents the scattering cross-section of a distributionof particles.

3.4.1 Aspherical Particles

The useful relation (??) is a property of the lognormal distribution itself, rather than the particleshape. A lognormal distribution of aspherical particles also obeys (??). Important measurableproperties of most convex aspherical habits may be represented by a constant times the kth momentF (k) of the distribution. For example, the surface area Sh [m2] and volume Vh [m3] of hexagonalprisms are given by (??)–(??). To be consistent with the diameter-centric expressions in Table 1,

3.4 Higher Moments 23

we introduce Dh, the hexagonal prism diameter. Adopting the convention that Dh ≡ 2a, thefull-width of the basal face, we obtain

Sh =

(3√

3

4+ 3Γ

)D2

h (65)

Vh =3√

3

8ΓD3

h (66)

The functional forms for Sh and Vh consist of constants multiplying the diameter’s second andthird moments, respectively. The surface area (πD2) and volume (πD3/6) of spheres have thesame form. Therefore the higher moments of aspherical particle distributions must be the sameas spherical particle distributions modulo the leading constant expressions. Inserting Sh and Vh

into (65), (66), and (??) leads to analytic expressions for the total surface area S0,h [m2 m−3] andvolume V0,h [m3 m−3] of a lognormal distribution of hexagonal prisms:

S0,h =

(3Γ +

3√

3

4

)D2

n,h exp(2σg2) (67)

V0,h =3√

3

8ΓD3

n,h exp(9σg2/2) (68)

The total concentrationN0,V/S of equivalent V/S-spheres corresponding to a known distributionof hexagonal prisms must be computed numerically unless the size dependence of the aspect ratioΓ(D) takes an analytic form. In the simplest case, one can imagine or assume distributions ofhexagonal prisms with constant aspect ratio, i.e, Γ 6= Γ(D). In this idealized case, the ratioNV/S/Nh (??) is constant throughout the distribution. Then the analytic number concentration ofequivalent V/S-spheres is simply NV/S/Nh times the analytic number concentration of hexagonalprisms which is presumably known directly from the lognormal size distribution parameters (cf.Table 1).

3.4.2 Normalization

We show that (21) is normalized by considering

nn(D) =CnD

exp

−1

2

(ln(D/Dn)

lnσg

)2 (69)

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

y = lnD − ln Dn

D = Dney

dy = D−1 dD

dD = Dney dy (70)

24 4 IMPLEMENTATION IN NCAR MODELS

This transformation maps D ∈ (0,+∞) into y ∈ (−∞,+∞). In terms of y, the normalizationcondition (7) becomes∫ +∞

−∞

Cn

Dn exp yexp

[−1

2

(y

lnσg

)2]Dn expy dy = 1

∫ +∞

−∞Cn exp

[−1

2

(y

lnσg

)2]

dy = 1

Next we change variables to z = y/ lnσg

z = y/ lnσg

y = z lnσg

dz = (lnσg)−1 dy

dy = lnσg dz (71)

This transformation does not change the limits of integration and we obtain∫ +∞

−∞Cn exp

(−z2

2

)lnσg dz = 1

Cn√

2π lnσg = 1

Cn =1√

2π lnσg

(72)

In the above we used the well-known normalization property of the Gaussian distribution function,∫ +∞−∞ e−x

2/2 dx =√

2π (73).

4 Implementation in NCAR modelsThe discussion thus far has centered on the theoretical considerations of size distributions. Inpractice, these ideas must be implemented in computer codes which model, e.g., Mie scatteringparameters or thermodynamic growth of aerosol populations. This section describes how theseideas have been implemented in the NCAR-Dust and Mie models.

4.1 NCAR-Dust ModelThe NCAR-Dust model uses as input a time invariant dataset of surface soil size distribution. Thetwo such datasets currently used are from Webb et al. (1993) and from IBIS (Foley, 1998). TheWebb 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 thesesoil types. To accomplish this, the partitioning scheme assumes a size distribution for the sourcesoil of the deflated particles. Table 4 lists the lognormal distribution parameters associated withthe surface soil texture data of Webb et al. (1993) and of Foley (1998). The dust model is a sizeresolving aerosol model. Thus, overlap factors are computed to determine the fraction of eachparent size type which is mobilized into each atmospheric dust size bin during a deflation event.

4.1 NCAR-Dust Model 25

Soil Texture Dn σg Description

Sand SandSilt SiltClay Clay

Soil Texture Dn σg DescriptionSand SandSilt SiltClay Clay

Table 4: Source size distribution associated with surface soil texture data of Webb et al. (1993) andof Foley (1998).

26 4 IMPLEMENTATION IN NCAR MODELS

4.2 Mie Scattering ModelThis section documents the Mie scattering code mie. mie is box model intended to provide ex-act simulations of microphysical processes for the purpose of parameterization into larger scalemodels. mie provides instantaneous and equilibrium decriptions of many processes ranging fromsurface 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 (bound-ary and initial conditions) of the environment. Given these, the program solves for the associatedrates of change and unknown variables.

There is no time-stepping loop primarily because mie generates an extraordinary amount ofinformation about the instantaneous state. Time-stepping this environment in a box-model-likeformat 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 in-put switches which control its behavior. This section describes how some of these switches arecommonly used to control fundamental properties of the microphysical environment. A completereference table for these switches, there default values, and dimensional units, is presented in Ap-pendix 5.3.

The heart of mie is an aerosol size distribution. Most users will wish to initialize this sizedistribution to a particular type of aerosol, and to a particular shape. This is accomplished with thecmp aer and psd typ keywords. The linearity, range, and resolution of the grid on which theanalytic size distribution is discretized are controlled by the sz grd, sz mnm, sz mxm, sz nbrswitches, respectively. Compute size distribution characteristics of a lognormal distribution

mie -dbg -no_mie --psd_typ=lognormal --sz_grd=log --sz_mnm=0.01 \--sz_mxm=10.0 --sz_nbr=300 --rds_nma=0.4 --gsd_anl=2.2mie -dbg -no_mie --psd_typ=lognormal --sz_grd=log --sz_mnm=1.0 \--sz_mxm=10.0 --sz_nbr=25 --rds_nma=2.0 --gsd_anl=2.2

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

# 1. Lognormal distribution with mass median diameter 3.5 um, GSD = 2.0mie -no_mie --psd_typ=lognormal --sz_grd=log --sz_nbr=1000 \--sz_mnm=0.005 --sz_mxm=50.0 --dmt_vma=3.5 --gsd_anl=2.0# 2. Extract median and weighted analytic moments of diameterncks -H -v dmt_vwa,dmt_vma,dmt_swa,dmt_sma,dmt_nwa,dmt_nma ${DATA}/mie/mie.nc# 3. Extract median and weighted resolved moments of diameterncks -H -v dmt_vwr,dmt_vmr,dmt_swr,dmt_smr,dmt_nwr,dmt_nmr ${DATA}/mie/mie.nc# 4. Extract median and weighted analytic moments of diameter

27

ncks -H -v rds_vwa,rds_vma,rds_swa,rds_sma,rds_nwa,rds_nma ${DATA}/mie/mie.nc# 5. Extract median and weighted resolved moments of diameterncks -H -v rds_vwr,rds_vmr,rds_swr,rds_smr,rds_nwr,rds_nmr ${DATA}/mie/mie.nc# 6. Extract number, surface area, and volume distributions at specific sizesncks -H -C -F -u -v dst,dst_rds,dst_sfc,dst_vlm -d sz,1.0e-6 ${DATA}/mie/mie.nc

4.2.3 Generating Properties for Multi-Bin

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

The psd.pl batch script calls mie repeatedly in a loop over particle bin. As input, psd.placcepts 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 dis-cretized into only 25 sub-bins.

˜/dst/psd.pl --dbg=1 --CCM_SW --ftn_fxd --psd_nbr=4 --spc_idx_sng={01,02,03,04} \--sz_mnm={0.05,0.5,1.25,2.5} --sz_mxm={0.5,1.25,2.5,5.0} --sz_nbr={200,25,25,25} \--dmt_vma_dfl=3.5 > ${DATA}/dst/mie/psd_CCM_SW.txt.v3 2>&1 &

5 Appendix

5.1 Properties of GaussiansThe area under a Gaussian distribution may be expressed in closed form for infinite domains.This result can be obtained (IIRC) by transforming to polar coordinates in the complex planex = r(cos θ + i sin θ). ∫ +∞

−∞e−x

2/2 dx =√

2π (73)

This is a special case of a more general result∫ +∞

−∞exp(−αx2 − βx) dx =

√π

αexp

(β2

4α

)where α > 0 (74)

To obtain this result, complete the square under the integrand, change variables to y = x + β/2α,and then apply (73). Substituting α = 1/2 and β = 0 into (74) yields (73).

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

erf(z) =2√π

∫ z

0

e−x2

dx (75)

28 5 APPENDIX

Using (73) and the symmetry of a Gaussian curve, it is simple to show that the error function isbounded by the limits erf(0) = 0 and erf(∞) = 1. Thus erf(z) is the cumulative probability func-tion for a normally distributed variable z (fxm: True??). Most compilers implement erf(x) as anintrinsic function. Thus erf(x) is used to compute areas bounded by finite lognormal distributions(§3.3.6).

5.3 Command Line Switches for mie CodeTable 5 summarizes all of the command line arguments available to control the behavior of the mieprogram. This is a summary only—it is impractical to think that written documentation could everyconvey the exact meaning of all the switches20. The most frequently used switches are describedin Section 4.2.1. The only way to learn the full meaning of the more obscure switches is to readthe source code itself.

20Perhaps the most useful way to begin to contribute to this FACT would be to systematize and extend the docu-mentation of command line switches

http://dust.ess.uci.edu/facts

5.3 Command Line Switches for mie Code 29

Tabl

e5:

Com

man

dL

ine

Switc

hesf

ormie

code

2122

Switc

hPu

rpos

eD

efau

ltU

nits

Boo

lean

flags

--abcflg

Alp

habe

tize

outp

utw

ithncks

true

Flag

--absnclwkmdmflg

Abs

orbi

ngin

clus

ion

inw

eakl

y-ab

sorb

ing

sphe

refa

lse

Flag

--bchflg

Bat

chbe

havi

orfa

lse

Flag

--coatflg

Ass

ume

coat

edsp

here

sfa

lse

Flag

--drvrdsnmaflg

Der

ive

rds

nma

from

bin

boun

dari

esfa

lse

Flag

--fdgflg

Tune

the

extin

ctio

nof

apa

rtic

ular

band

fals

eFl

ag--hxgflg

Asp

heri

calp

artic

les

are

hexa

gona

lpri

sms

true

Flag

--vtsflg

App

lyeq

ual-

V/S

appr

oxim

atio

nfo

rasp

heri

cal

optic

alpr

oper

ties

fals

eFl

ag

--ftnfxdflg

Fort

ran

fixed

form

atfa

lse

Flag

--hrzflg

Prin

tsiz

e-re

solv

edop

tical

prop

ertie

sat

debu

gw

avel

engt

hfa

lse

Flag

--mcaflg

Mul

ti-co

mpo

nent

aero

solw

ithef

fect

ive

med

ium

appr

oxim

atio

nfa

lse

Flag

--mieflg

Perf

orm

Mie

scat

teri

ngca

lcul

atio

ntr

ueFl

ag--noabcflg

Setabcflg

tofalse

Flag

--nobchflg

Setbchflg

tofalse

Flag

--nohrzflg

Sethrzflg

tofalse

Flag

--nomieflg

Setmieflg

tofalse

Flag

--nowrnntpflg

Setwrnntpflg

tofalse

Flag

--wrnntpflg

Prin

tWA

RN

ING

sfr

omntpvec()

true

Flag

Var

iabl

es--RHlqd

Rel

ativ

ehu

mid

ityw

/r/t

liqui

dw

ater

0.8

Frac

tion

--asprathxgdfl

Hex

agon

alpr

ism

aspe

ctra

tio1.

0Fr

actio

n

30 5 APPENDIX

Tabl

e5:

(con

tinue

d)

Switc

hPu

rpos

eD

efau

ltU

nits

--aspratlpsdfl

Elli

psoi

dala

spec

trat

io1.

0Fr

actio

n--bndSWLW

Bou

ndar

ybe

twee

nSW

and

LWw

eigh

ting

5.0×

10−

6m

--bndnbr

Num

bero

fsub

-ban

dspe

rout

putb

and

1N

umbe

r--cmpcor

Com

posi

tion

ofco

re“a

ir”

Stri

ng--cmpmdm

Com

posi

tion

ofm

ediu

m“a

ir”

Stri

ng--cmpmnt

Com

posi

tion

ofm

antle

“air

”St

ring

--cmpmtx

Com

posi

tion

ofm

atri

x“a

ir”

Stri

ng--cmpncl

Com

posi

tion

ofin

clus

ion

“air

”St

ring

--cmpprt

Com

posi

tion

ofpa

rtic

le“s

ahar

andu

st”

Stri

ng--cncnbranldfl

Num

berc

once

ntra

tion

anal

ytic

,def

ault

1.0

#m−

3

--cncnbrpcpanl

Num

berc

once

ntra

tion

anal

ytic

,rai

ndro

p1.

0#

m−

3

--cpvfoo

Intr

insi

cco

mpu

tatio

nalp

reci

sion

tem

pora

ryva

riab

le0.

0Fr

actio

n

--dbglvl

Deb

uggi

ngle

vel

0In

dex

--dmnnbrmax

Max

imum

num

bero

fdim

ensi

ons

allo

wed

insi

ngle

vari

able

inou

tput

file

2N

umbe

r

--dmnfrc

Frac

tald

imen

sion

ality

ofin

clus

ions

3.0

Frac

tion

--dmnrcd

Rec

ord

dim

ensi

onna

me

“”St

ring

--dmtdtc

Dia

met

erof

dete

ctor

0.00

1m

--dmtnmamcr

Num

berm

edia

nan

alyt

icdi

amet

ercmdlndfl

µm

--dmtpcpnmamcr

Dia

met

ernu

mbe

rmed

ian

anal

ytic

,rai

ndro

p,m

icro

ns10

00.0

µm

--dmtswamcr

Surf

ace

area

wei

ghte

dm

ean

diam

eter

anal

ytic

cmdlndfl

µm

--dmtvmamcr

Volu

me

med

ian

diam

eter

anal

ytic

cmdlndfl

µm

--dnscor

Den

sity

ofco

re0.

0kg

m−

3


Tabl

e5:

(con

tinue

d)

Switc

hPu

rpos

eD

efau

ltU

nits

--dnsmdm

Den

sity

ofm

ediu

m0.

0kg

m−

3

--dnsmnt

Den

sity

ofm

antle

0.0

kgm−

3

--dnsmtx

Den

sity

ofm

atri

x0.

0kg

m−

3

--dnsncl

Den

sity

ofin

clus

ion

0.0

kgm−

3

--dnsprt

Den

sity

ofpa

rtic

le0.

0kg

m−

3

--doy

Day

ofye

ar[1

.0..3

67.0

)13

5.0

day

--drcdat

Dat

adi

rect

ory

/data/zender/aca

Stri

ng--drcin

Inpu

tdir

ecto

ry${HOME}/nco/data

Stri

ng--drcout

Out

putd

irec

tory

${HOME}/c++

Stri

ng--dsddbgmcr

Deb

uggi

ngsi

zefo

rrai

ndro

ps10

00.0

µm

--dsdmnmmcr

Min

imum

diam

eter

inra

indr

opdi

stri

butio

n99

9.0

µm

--dsdmxmmcr

Max

imum

diam

eter

inra

indr

opdi

stri

butio

n10

01.0

µm

--dsdnbr

Num

bero

frai

ndro

psi

zebi

ns1

Num

ber

--fdgidx

Ban

dto

tune

byfd

gva

l0

Inde

x--fdgval

Tuni

ngfa

ctor

fora

llba

nds

1.0

Frac

tion

--flerr

File

fore

rror

mes

sage

s“c

err”

Stri

ng--flidxrfrcor

File

orfu

nctio

nfo

rref

ract

ive

indi

ces

ofco

re“”

Stri

ng--flidxrfrmdm

File

orfu

nctio

nfo

rref

ract

ive

indi

ces

ofm

ediu

m“”

Stri

ng--flidxrfrmnt

File

orfu

nctio

nfo

rref

ract

ive

indi

ces

ofm

antle

“”St

ring

--flidxrfrmtx

File

orfu

nctio

nfo

rref

ract

ive

indi

ces

ofm

atri

x“”

Stri

ng--flidxrfrncl

File

orfu

nctio

nfo

rref

ract

ive

indi

ces

ofin

clus

ion

“”St

ring

--flidxrfrprt

File

orfu

nctio

nfo

rref

ract

ive

indi

ces

ofpa

rtic

le“”

Stri

ng--flslrspc

File

orfu

nctio

nfo

rsol

arsp

ectr

um“”

Stri

ng--fltfoo

Intr

insi

cflo

atte

mpo

rary

vari

able

0.0

Frac

tion

32 5 APPENDIX

Tabl

e5:

(con

tinue

d)

Switc

hPu

rpos

eD

efau

ltU

nits

--flxLWdwnsfc

Lon

gwav

edo

wnw

ellin

gflu

xat

surf

ace

0.0

Wm−

2

--flxSWnetgnd

Sola

rflux

abso

rbed

bygr

ound

450.

0W

m−

2

--flxSWnetvgt

Sola

rflux

abso

rbed

byve

geta

tion

0.0

Wm−

2

--flxfrcdrcsfccmdln

Surf

ace

inso

latio

nfr

actio

nin

dire

ctbe

am0.

85Fr

actio

n--flxvlmpcprsl

Prec

ipita

tion

volu

me

flux,

reso

lved

−1.

0m

3m−

2s−

1

--gsdanldfl

Geo

met

ric

stan

dard

devi

atio

n,de

faul

t2.

0Fr

actio

n--gsdpcpanl

Geo

met

ric

stan

dard

devi

atio

n,ra

indr

op1.

86Fr

actio

n--hgtmdp

Mid

laye

rhei

ghta

bove

surf

ace

95.0

m--hgtrfr

Ref

eren

cehe

ight

(i.e

.,10

m)a

twhi

chsu

rfac

ew

inds

are

eval

uate

dfo

rdus

tmob

iliza

tion

10.0

m

--hgtzpddpscmdln

Zer

opl

ane

disp

lace

men

thei

ght

cmdlndfl

m--hgtzpdmbl

Zer

opl

ane

disp

lace

men

thei

ghtf

orer

odib

lesu

rfac

es0.

0m

--idxrfrcorusr

Ref

ract

ive

inde

xof

core

1.0

+0.

0iC

ompl

ex--idxrfrmdmusr

Ref

ract

ive

inde

xof

med

ium

1.0

+0.

0iC

ompl

ex--idxrfrmntusr

Ref

ract

ive

inde

xof

man

tle1.

33+

0.0i

Com

plex

--idxrfrmtxusr

Ref

ract

ive

inde

xof

mat

rix

1.0

+0.

0iC

ompl

ex--idxrfrnclusr

Ref

ract

ive

inde

xof

incl

usio

n1.

0+

0.0i

Com

plex

--idxrfrprtusr

Ref

ract

ive

inde

xof

part

icle

1.33

+0.

0iC

ompl

ex--latdgr

Lat

itude

40.0

◦

--lblsng

Lin

e-by

-lin

ete

st“C

O2”

Stri

ng--lgnnbr

Num

bero

fter

ms

inL

egen

dre

expa

nsio

nof

phas

efu

nctio

n8

Num

ber

--lndfrcdry

Dry

land

frac

tion

1.0

Frac

tion

--mmwprt

Mea

nm

olec

ular

wei

ght

0.0

kgm

ol−

1


Tabl

e5:

(con

tinue

d)

Switc

hPu

rpos

eD

efau

ltU

nits

--mnolngdpscmdln

Mon

in-O

bukh

ovle

ngth

cmdlndfl

m--mssfrccly

Mas

sfr

actio

ncl

ay0.

19Fr

actio

n--mssfrcsnd

Mas

sfr

actio

nsa

nd0.

777

Frac

tion

--nglnbr

Num

bero

fang

les

inM

ieco

mpu

tatio

n11

Num

ber

--oro

Oro

grap

hy:o

cean

=0.0

,lan

d=1.

0,se

aic

e=2.

01.

0Fr

actio

n--pnttypidx

Plan

ttyp

ein

dex

14In

dex

--prsmdp

Env

iron

men

talp

ress

ure

1008

25.0

Pa--prsntf

Env

iron

men

tals

urfa

cepr

essu

reprsSTP

Pa--psdtyp

Part

icle

size

dist

ribu

tion

type

“log

norm

al”

Stri

ng--qH2Ovpr

Spec

ific

hum

idity

cmdlndfl

kgkg−

1

--rdsffcgmmmcr

Eff

ectiv

era

dius

ofG

amm

adi

stri

butio

n50

.0µ

m--rdsnmamcr

Num

berm

edia

nan

alyt

icra

dius

0.29

86µ

m--rdsswamcr

Surf

ace

area

wei

ghte

dm

ean

radi

usan

alyt

iccmdlndfl

µm

--rdsvmamcr

Volu

me

med

ian

radi

usan

alyt

iccmdlndfl

µm

--rghmmndpscmdln

Rou

ghne

ssle

ngth

mom

entu

mcmdlndfl

m--rghmmnicestd

Rou

ghne

ssle

ngth

over

sea

ice

0.00

05m

--rghmmnmbl

Rou

ghne

ssle

ngth

mom

entu

mfo

rero

dibl

esu

rfac

es10

0.0×

10−

6m

--rghmmnsmt

Smoo

thro

ughn

ess

leng

th10.0×

10−

6m

--rflgnddff

Diff

use

refle

ctan

ceof

grou

nd(b

enea

thsn

ow)

0.20

Frac

tion

--sfctyp

LSM

surf

ace

type

[0..2

8]2

Inde

x--slftsttyp

Self

-tes

ttyp

e“B

oH83

”St

ring

--slrcst

Sola

rcon

stan

t13

67.0

Wm−

2

--slrspckey

Sola

rspe

ctru

mst

ring

“LaN

68”

Stri

ng

34 5 APPENDIX

Tabl

e5:

(con

tinue

d)

Switc

hPu

rpos

eD

efau

ltU

nits

--slrzennglcos

Cos

ine

sola

rzen

ithan

gle

1.0

Frac

tion

--slvsng

Mie

solv

erto

use

“Wis

79”

Stri

ng--snwhgtlqd

Equ

ival

entl

iqui

dw

ater

snow

dept

h0.

0m

--soityp

LSM

soil

type

[1..5

]1

Inde

x--spcheatprt

Spec

ific

heat

capa

city

0.0

Jkg−

1K−

1

--spcabbsng

Spec

ies

abbr

evia

tion

forF

ortr

anda

ta“f

oo”

Stri

ng--spcidxsng

Spec

ies

inde

xfo

rFor

tran

data

“foo

”St

ring

--ssalbcmdln

Sing

lesc

atte

ring

albe

do1.

0Fr

actio

n--szdbgmcr

Deb

uggi

ngsi

ze1.

0µ

m--szgrdsng

Type

ofsi

zegr

id“l

ogar

ithm

ic”

Stri

ng--szmnmmcr

Min

imum

size

indi

stri

butio

n0.

9µ

m--szmxmmcr

Max

imum

size

indi

stri

butio

n1.

1µ

m--sznbr

Num

bero

fpar

ticle

size

bins

1N

umbe

r--szprmrsn

Size

para

met

erre

solu

tion

0.1

Frac

tion

--thrnbr

Thr

ead

num

ber

0N

umbe

r--tmdlt

Tim

este

p12

00.0

s--tptbbdwgt

Bla

ckbo

dyte

mpe

ratu

reof

radi

atio

n27

3.15

K--tptgnd

Gro

und

tem

pera

ture

300.

0K

--tptice

Ice

tem

pera

ture

tptfrzpnt

K--tptmdp

Env

iron

men

talt

empe

ratu

re30

0.0

K--tptprt

Part

icle

tem

pera

ture

273.

15K

--tptsoi

Soil

tem

pera

ture

297.

0K

--tptsst

Sea

surf

ace

tem

pera

ture

300.

0K

--tptvgt

Veg

etat

ion

tem

pera

ture

300.

0K


Tabl

e5:

(con

tinue

d)

Switc

hPu

rpos

eD

efau

ltU

nits

--tstsng

Nam

eof

test

tope

rfor

m(htg

,lbl

,nc

,nsz

,psdntgdgn

)“”

Stri

ng

--varffcgmm

Eff

ectiv

eva

rian

ceof

Gam

ma

dist

ribu

tion

1.0

Frac

tion

--vlmfrcmntl

Frac

tion

ofvo

lum

ein

man

tle0.

5Fr

actio

n--vmrCO2

Volu

me

mix

ing

ratio

ofC

O2

355.

0×

10−

6m

olec

ule

mol

ecul

e−1

--vmrHNO3gas

Volu

me

mix

ing

ratio

ofga

seou

sH

NO

30.

05×

10−

9m

olec

ule

mol

ecul

e−1

--vwcsfc

Volu

met

ric

wat

erco

nten

t0.

03m

3m−

3

--wblshp

Wei

bull

dist

ribu

tion

shap

epa

ram

eter

2.4

Frac

tion

--wndfrcdpscmdln

Fric

tion

spee

dcmdlndfl

ms−

1

--wndmrdmdp

Surf

ace

laye

rmer

idio

nalw

ind

spee

d0.

0m

s−1

--wndznlmdp

Surf

ace

laye

rzon

alw

ind

spee

d10

.0m

s−1

--wvldbgmcr

Deb

uggi

ngw

avel

engt

h0.

50µ

m--wvlgrdsng

Type

ofw

avel

engt

hgr

id“r

egul

ar”

Stri

ng--wvldltmcr

Ban

dwid

th0.

1µ

m--wvlmdpmcr

Mid

poin

twav

elen

gth

cmdlndfl

µm

--wvlmnmmcr

Min

imum

wav

elen

gth

0.45

µm

--wvlmxmmcr

Max

imum

wav

elen

gth

0.55

µm

--wvlnbr

Num

bero

fout

putw

avel

engt

hba

nds

1N

umbe

r--wvndltxcm

Ban

dwid

th1.

0cm−

1

--wvnmdpxcm

Mid

poin

twav

enum

ber

cmdlndfl

cm−

1

--wvnmnmxcm

Min

imum

wav

enum

ber

1818

2cm−

1

--wvnmxmxcm

Max

imum

wav

enum

ber

2222

2cm−

1

--wvnnbr

Num

bero

fout

putw

aven

umbe

rban

ds1

Num

ber

--xptdsc

Exp

erim

entd

escr

iptio

n“”

Stri

ng

36 5 APPENDIX

Table 6 summarizes the fields output by SWNB.


Tabl

e6:

SWN

BO

utpu

tFie

lds23

Nam

e(s)

Purp

ose

Uni

ts

absbbSAS

Bro

adba

ndab

sorp

tanc

eof

surf

ace-

atm

osph

ere

syst

emfr

actio

n

absbbatm

Bro

adba

ndab

sorp

tanc

eof

surf

ace

frac

tion

absbbsfc

Bro

adba

ndab

sorp

tanc

eof

atm

osph

ere

frac

tion

absnstSAS

FSB

Rab

sorp

tanc

eof

surf

ace-

atm

osph

ere

syst

emfr

actio

nabsnstatm

FSB

Rab

sorp

tanc

eof

surf

ace

frac

tion

absnstsfc

FSB

Rab

sorp

tanc

eof

atm

osph

ere

frac

tion

absspcSAS

Spec

tral

abso

rpta

nce

ofsu

rfac

e-at

mos

pher

esy

stem

frac

tion

absspcatm

Spec

tral

abso

rpta

nce

ofat

mos

pher

efr

actio

nabsspcsfc

Spec

tral

abso

rpta

nce

ofsu

rfac

efr

actio

nalbsfc

Spec

ified

Lam

bert

ian

surf

ace

albe

dofr

actio

naltcldbtm

Hig

hest

inte

rfac

ebe

neat

hal

lclo

uds

inco

lum

nm

eter

altcldthick

Thi

ckne

ssof

regi

onco

ntai

ning

allc

loud

sm

eter

altntf

Inte

rfac

eal

titud

em

eter

alt

Alti

tude

met

erazidgr

Azi

mut

hala

ngle

(deg

rees

)de

gree

azi

Azi

mut

hala

ngle

(rad

ians

)ra

dian

bnd

Mid

poin

twav

elen

gth

met

erflxabsatmrdr

Flux

abso

rbed

inat

mos

pher

eat

long

erw

avel

engt

hsW

m−

2

flxbbabsatm

Bro

adba

ndflu

xab

sorb

edby

atm

osph

eric

colu

mn

only

Wm−

2

flxbbabssfc

Bro

adba

ndflu

xab

sorb

edby

surf

ace

only

Wm−

2

38 5 APPENDIX

Tabl

e6:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

flxbbabsttl

Bro

adba

ndflu

xab

sorb

edby

surf

ace-

atm

osph

ere

syst

emW

m−

2

flxbbabs

Bro

adba

ndflu

xab

sorb

edby

laye

rW

m−

2

flxbbdwnTOA

Bro

adba

ndin

com

ing

flux

atTO

A(t

otal

inso

latio

n)W

m−

2

flxbbdwndff

Diff

use

dow

nwel

ling

broa

dban

dflu

xW

m−

2

flxbbdwndrc

Dir

ectd

ownw

ellin

gbr

oadb

and

flux

Wm−

2

flxbbdwnsfc

Bro

adba

nddo

wnw

ellin

gflu

xat

surf

ace

Wm−

2

flxbbdwn

Tota

ldow

nwel

ling

broa

dban

dflu

x(d

irec

t+di

ffus

e)W

m−

2

flxbbnet

Net

broa

dban

dflu

x(d

ownw

ellin

g−up

wel

ling)

Wm−

2

flxbbup

Upw

ellin

gbr

oadb

and

flux

Wm−

2

flxnstabsatm

FSB

Rflu

xab

sorb

edby

atm

osph

eric

colu

mn

only

Wm−

2

flxnstabssfc

FSB

Rflu

xab

sorb

edby

surf

ace

only

Wm−

2

flxnstabsttl

FSB

Rflu

xab

sorb

edby

surf

ace-

atm

osph

ere

syst

emW

m−

2

flxnstabs

FSB

Rflu

xab

sorb

edby

laye

rW

m−

2

flxnstdwnTOA

FSB

Rin

com

ing

flux

atTO

A(t

otal

inso

latio

n)W

m−

2

flxnstdwnsfc

FSB

Rdo

wnw

ellin

gflu

xat

surf

ace

Wm−

2

flxnstdwn

Tota

ldow

nwel

ling

FSB

Rflu

x(d

irec

t+di

ffus

e)W

m−

2

flxnstnet

Net

FSB

Rflu

x(d

ownw

ellin

g−up

wel

ling)

Wm−

2

flxnstup

Upw

ellin

gFS

BR

flux

Wm−

2

flxslrfrc

Frac

tion

ofso

larfl

uxfr

actio

nflxspcabsSAS

Spec

tral

flux

abso

rbed

bysu

rfac

e-at

mos

pher

esy

stem

Wm−

2m−

1


Tabl

e6:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

flxspcabsatm

Spec

tral

flux

abso

rbed

byat

mos

pher

icco

lum

non

lyW

m−

2m−

1

flxspcabssfc

Spec

tral

flux

abso

rbed

bysu

rfac

eon

lyW

m−

2m−

1

flxspcabs

Spec

tral

flux

abso

rbed

byla

yer

Wm−

2m−

1

flxspcactphtTOA

Spec

tral

actin

icph

oton

flux

atTO

A#

m−

2s−

1m−

1

flxspcactphtsfc

Spec

tral

actin

icph

oton

flux

atsu

rfac

e#

m−

2s−

1m−

1

flxspcdwnTOA

Spec

tral

sola

rins

olat

ion

atTO

AW

m−

2m−

1

flxspcdwndff

Spec

tral

diff

use

dow

nwel

ling

flux

Wm−

2m−

1

flxspcdwndrc

Spec

tral

dire

ctdo

wnw

ellin

gflu

xW

m−

2m−

1

flxspcdwnsfc

Spec

tral

sola

rins

olat

ion

atsu

rfac

eW

m−

2m−

1

flxspcdwn

Spec

tral

dow

nwel

ling

flux

Wm−

2m−

1

flxspcphtdwnsfc

Spec

tral

phot

onflu

xdo

wnw

ellin

gat

surf

ace

#m−

2s−

1m−

1

flxspcup

Spec

tral

upw

ellin

gflu

xW

m−

2m−

1

frcicettl

Frac

tion

ofco

lum

nco

nden

sate

that

isic

efr

actio

nhtgratebb

Bro

adba

ndhe

atin

gra

teK

s−1

jNO2

Phot

olys

isra

tefo

rNO

2+

hν−→

O(3

P)

+N

Os−

1

jspcNO2sfc

Spec

tral

phot

olys

isra

teat

sfc

for

NO

2+

hν−→

O(3

P)

+N

Os−

1m−

1

latdgr

Lat

itude

(deg

rees

)de

gree

lcltimehr

Loc

alda

yho

urho

urlclyrday

Day

ofye

arin

loca

ltim

eda

ylevp

Inte

rfac

epr

essu

repa

scal

lev

Lay

erpr

essu

repa

scal

mpcCWP

Tota

lcol

umn

Con

dens

edW

ater

Path

kgm−

2

40 5 APPENDIX

Tabl

e6:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

nrgpht

Ene

rgy

ofph

oton

atba

ndce

nter

joul

eph

oton

-1ntnbbaa

Bro

adba

ndaz

imut

hally

aver

aged

inte

nsity

Wm−

2sr−

1

ntnbbmean

Bro

adba

ndm

ean

inte

nsity

Wm−

2sr−

1

ntnspcaandrsfc

Spec

tral

inte

nsity

ofna

dirr

adia

tion

atsu

rfac

eW

m−

2m−

1sr−

1

ntnspcaandr

Spec

tral

inte

nsity

ofna

dirr

adia

tion

Wm−

2m−

1sr−

1

ntnspcaasfc

Spec

tral

inte

nsity

ofra

diat

ion

atsu

rfac

eW

m−

2m−

1sr−

1

ntnspcaazensfc

Spec

tral

inte

nsity

ofze

nith

radi

atio

nat

surf

ace

Wm−

2m−

1sr−

1

ntnspcaazen

Spec

tral

inte

nsity

ofze

nith

radi

atio

nW

m−

2m−

1sr−

1

ntnspcchn

Full

spec

tral

inte

nsity

ofpa

rtic

ular

band

Wm−

2m−

1sr−

1

ntnspcmean

Spec

tral

mea

nin

tens

ityW

m−

2m−

1sr−

1

odacspcaer

Aer

osol

abso

rptio

nop

tical

dept

hto

surf

ace

frac

tion

odacspcbga

Bac

kgro

und

aero

sola

bsor

ptio

nop

tical

dept

hto

surf

ace

frac

tion

odacspcice

Liq

uid

wat

erab

sorp

tion

optic

alde

pth

tosu

rfac

efr

actio

nodacspclqd

Ice

wat

erab

sorp

tion

optic

alde

pth

tosu

rfac

efr

actio

nodalobsaer

Lay

erae

roso

labs

orpt

ion

optic

alde

pth

frac

tion

odalobsbga

Lay

erba

ckgr

ound

aero

sola

bsor

ptio

nop

tical

dept

hfr

actio

n

odslobsaer

Lay

erae

roso

lsca

tteri

ngop

tical

dept

hfr

actio

nodslobsbga

Lay

erba

ckgr

ound

aero

sols

catte

ring

optic

alde

pth

frac

tion

odxcobsaer

Col

umn

aero

sole

xtin

ctio

nop

tical

dept

hfr

actio

nodxcobsbga

Col

umn

back

grou

ndae

roso

lext

inct

ion

optic

alde

pth

frac

tion

odxcspcCO2

CO

2op

tical

dept

hto

surf

ace

frac

tion

odxcspcH2OH2O

H2O

dim

erop

tical

dept

hto

surf

ace

frac

tion


Tabl

e6:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

odxcspcH2O

H2O

optic

alde

pth

tosu

rfac

efr

actio

nodxcspcNO2

NO

2op

tical

dept

hto

surf

ace

frac

tion

odxcspcO2N2

O2N

2op

tical

dept

hto

surf

ace

frac

tion

odxcspcO2O2

O2O

2op

tical

dept

hto

surf

ace

frac

tion

odxcspcO2

O2

optic

alde

pth

tosu

rfac

efr

actio

nodxcspcO3

O3

optic

alde

pth

tosu

rfac

efr

actio

nodxcspcOH

OH

optic

alde

pth

tosu

rfac

efr

actio

nodxcspcRay

Ray

leig

hsc

atte

ring

optic

alde

pth

tosu

rfac

efr

actio

nodxcspcaer

Aer

osol

extin

ctio

nop

tical

dept

hto

surf

ace

frac

tion

odxcspcbga

Bac

kgro

und

aero

sole

xtin

ctio

nop

tical

dept

hto

surf

ace

frac

tion

odxcspcice

Ice

wat

erex

tinct

ion

optic

alde

pth

tosu

rfac

efr

actio

nodxcspclqd

Liq

uid

wat

erex

tinct

ion

optic

alde

pth

tosu

rfac

efr

actio

nodxcspcttl

Tota

lext

inct

ion

optic

alde

pth

tosu

rfac

efr

actio

nodxlobsaer

Lay

erae

roso

lext

inct

ion

optic

alde

pth

frac

tion

odxlobsbga

Lay

erba

ckgr

ound

aero

sole

xtin

ctio

nop

tical

dept

hfr

actio

n

plrcos

Cos

ine

pola

rang

le(d

egre

es)

frac

tion

plrdgr

Pola

rang

le(d

egre

es)

degr

eeplr

Pola

rang

le(r

adia

ns)

radi

anrflbbSAS

Bro

adba

ndal

bedo

ofen

tire

surf

ace-

atm

osph

ere

syst

emfr

actio

n

rflbbsfc

Bro

adba

ndal

bedo

ofsu

rfac

efr

actio

nrflnstSAS

FSB

Ral

bedo

ofen

tire

surf

ace-

atm

osph

ere

syst

emfr

actio

nrflnstsfc

FSB

Ral

bedo

ofsu

rfac

efr

actio

n

42 5 APPENDIX

Tabl

e6:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

rflspcSAS

Spec

tral

plan

etar

yflu

xre

flect

ance

frac

tion

slrzennglcos

Cos

ine

sola

rzen

ithan

gle

frac

tion

tauprs

Opt

ical

leve

l(pr

essu

re)

pasc

altau

Opt

ical

leve

l(op

tical

dept

h)fr

actio

ntptntf

Inte

rfac

ete

mpe

ratu

reke

lvin

tpt

Lay

erTe

mpe

ratu

reke

lvin

trnbbatm

Bro

adba

ndtr

ansm

issi

onof

atm

osph

eric

colu

mn

frac

tion

trnnstatm

FSB

Rtr

ansm

issi

onof

atm

osph

eric

colu

mn

frac

tion

trnspcatmCO2

Col

umn

tran

smis

sion

due

toC

O2

abso

rptio

nfr

actio

ntrnspcatmH2OH2O

Col

umn

tran

smis

sion

due

toH

2Odi

mer

abso

rptio

nfr

actio

n

trnspcatmH2O

Col

umn

tran

smis

sion

due

toH

2Oab

sorp

tion

frac

tion

trnspcatmNO2

Col

umn

tran

smis

sion

due

toN

O2ab

sorp

tion

frac

tion

trnspcatmO2N2

Col

umn

tran

smis

sion

due

toO

2-N

2ab

sorp

tion

frac

tion

trnspcatmO2O2

Col

umn

tran

smis

sion

due

toO

2-O

2ab

sorp

tion

frac

tion

trnspcatmO2

Col

umn

tran

smis

sion

due

toO

2ab

sorp

tion

frac

tion

trnspcatmO3

Col

umn

tran

smis

sion

due

toO

3ab

sorp

tion

frac

tion

trnspcatmOH

Col

umn

tran

smis

sion

due

toO

Hab

sorp

tion

frac

tion

trnspcatmRay

Col

umn

tran

smis

sion

due

toR

ayle

igh

scat

teri

ngfr

actio

ntrnspcatmaer

Col

umn

tran

smis

sion

due

toae

roso

lext

inct

ion

frac

tion

trnspcatmbga

Col

umn

tran

smis

sion

due

toba

ckgr

ound

aero

sol

extin

ctio

nfr

actio

n

trnspcatmice

Col

umn

tran

smis

sion

due

toic

eex

tinct

ion

frac

tion

trnspcatmlqd

Col

umn

tran

smis

sion

due

toliq

uid

extin

ctio

nfr

actio

n


Tabl

e6:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

trnspcatmttl

Spec

tral

flux

tran

smis

sion

ofen

tire

colu

mn

frac

tion

wvlctr

Mid

poin

twav

elen

gth

inba

ndm

eter

wvldlt

Wid

thof

band

met

erwvlgrd

Wav

elen

gth

grid

met

erwvlmax

Max

imum

wav

elen

gth

inba

ndm

eter

wvlmin

Min

imum

wav

elen

gth

inba

ndm

eter

wvlobsaer

Wav

elen

gth

ofae

roso

lopt

ical

dept

hsp

ecifi

catio

nm

eter

wvlobsbga

Wav

elen

gth

ofba

ckgr

ound

aero

solo

ptic

alde

pth

spec

ifica

tion

met

er

wvnctr

Mid

poin

twav

enum

beri

nba

ndce

ntim

eter

-1wvndlt

Ban

dwid

thin

wav

enum

bers

cent

imet

er-1

wvnmax

Max

imum

wav

enum

beri

nba

ndce

ntim

eter

-1wvnmin

Min

imum

wav

enum

beri

nba

ndce

ntim

eter

-1

44 5 APPENDIX

Table 7 summarizes the fields output by CLM.


Tabl

e7:

CL

MO

utpu

tFie

lds24

Nam

e(s)

Purp

ose

Uni

ts

CO2vmrclm

Car

bon

Dio

xide

volu

me

mix

ing

ratio

frac

tion

N2Ovmrclm

Nitr

ous

Oxi

devo

lum

em

ixin

gra

tiofr

actio

nCH4vmrclm

Met

hane

volu

me

mix

ing

ratio

frac

tion

CFC11vmrclm

CFC

11vo

lum

em

ixin

gra

tiofr

actio

nCFC12vmrclm

CFC

12vo

lum

em

ixin

gra

tiofr

actio

nRHice

Rel

ativ

ehu

mid

ityw

/r/t

ice

frac

tion

RH

Rel

ativ

ehu

mid

ityfr

actio

nRHlqd

Rel

ativ

ehu

mid

ityw

/r/t

liqui

dfr

actio

nalbsfcNIRdrc

Alb

edo

forN

IRra

diat

ion

atst

rong

zeni

than

gles

frac

tion

albsfcNIRdff

Alb

edo

forN

IRra

diat

ion

atw

eak

zeni

than

gles

frac

tion

albsfc

Pres

crib

edsu

rfac

eal

bedo

frac

tion

albsfcvsbdrc

Alb

edo

forv

isib

lera

diat

ion

atst

rong

zeni

than

gles

frac

tion

albsfcvsbdff

Alb

edo

forv

isib

lera

diat

ion

atw

eak

zeni

than

gles

frac

tion

altcldbtm

Hig

hest

inte

rfac

ebe

neat

hal

lclo

uds

inco

lum

nm

eter

altcldmid

Alti

tude

atm

idpo

into

fall

clou

dsin

colu

mn

met

eraltcldthick

Thi

ckne

ssof

regi

onco

ntai

ning

allc

loud

sm

eter

altcldtop

Low

esti

nter

face

abov

eal

lclo

uds

inco

lum

nm

eter

altdlt

Lay

eral

titud

eth

ickn

ess

met

eralt

Alti

tude

met

eraltntf

Inte

rfac

eal

titud

em

eter

cldfrc

Clo

udfr

actio

nfr

actio

ncncCO2

CO

2co

ncen

trat

ion

mol

ecul

em−

3

cncCH4

CH

4co

ncen

trat

ion

mol

ecul

em−

3

46 5 APPENDIX

Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

cncN2O

N2O

conc

entr

atio

nm

olec

ule

m−

3

cncCFC11

CFC

11co

ncen

trat

ion

mol

ecul

em−

3

cncCFC12

CFC

12co

ncen

trat

ion

mol

ecul

em−

3

cncH2OH2O

H2O

dim

erco

ncen

trat

ion

mol

ecul

em−

3

cncH2O

H2O

conc

entr

atio

nm

olec

ule

m−

3

cncN2

N2

conc

entr

atio

nm

olec

ule

m−

3

cncNO2

NO

2co

ncen

trat

ion

mol

ecul

em−

3

cncO2O2

O2O

2co

ncen

trat

ion

mol

ecul

em−

3

cncO2cncN2

O2

num

berc

once

ntra

tion

times

N2

num

ber

conc

entr

atio

nm

olec

ule2

m−

6

cncO2cncO2

O2

num

berc

once

ntra

tion

squa

red

mol

ecul

e2m−

6

cncO2

O2

conc

entr

atio

nm

olec

ule

m−

3

cncO2nplN2clm

Col

umn

tota

lO2

num

berc

once

ntra

tion

times

N2

num

berp

ath

mol

ecul

e2m−

5

cncO2nplN2

O2

num

berc

once

ntra

tion

times

N2

num

berp

ath

mol

ecul

e2m−

5

cncO2nplO2clm

Col

umn

tota

lO2

num

berc

once

ntra

tion

times

O2

num

berp

ath

mol

ecul

e2m−

5

cncO2nplO2clmfrc

Frac

tion

ofco

lum

nto

talO

2-O

2at

orab

ove

each

laye

rfr

actio

n

cncO2nplO2

O2

num

berc

once

ntra

tion

times

O2

num

berp

ath

mol

ecul

e2m−

5

cncO3

O3

conc

entr

atio

n#

m−

3

cncOH

OH

conc

entr

atio

n#

m−

3

cncdryair

Dry

conc

entr

atio

n#

m−

3

cncmstair

Moi

stai

rcon

cent

ratio

n#

m−

3

dnsCO2

Den

sity

ofC

O2

kgm−

3


Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

dnsCH4

Den

sity

ofC

H4

kgm−

3

dnsN2O

Den

sity

ofN

2Okg

m−

3

dnsCFC11

Den

sity

ofC

FC11

kgm−

3

dnsCFC12

Den

sity

ofC

FC12

kgm−

3

dnsH2OH2O

Den

sity

ofH

20H

2Okg

m−

3

dnsH2O

Den

sity

ofH

2Okg

m−

3

dnsN2

Den

sity

ofN

2kg

m−

3

dnsNO2

Den

sity

ofN

O2

kgm−

3

dnsO2O2

Den

sity

ofO

2-O

2kg

m−

3

dnsO2dnsN2

O2

mas

sco

ncen

trat

ion

times

N2

mas

sco

ncen

trat

ion

kg2

m−

6

dnsO2dnsO2

O2

mas

sco

ncen

trat

ion

squa

red

kg2

m−

6

dnsO2

Den

sity

ofO

2kg

m−

3

dnsO2mplN2clm

Col

umn

tota

lO2

mas

sco

ncen

trat

ion

times

N2

mas

spa

thkg

2m−

5

dnsO2mplN2

O2

mas

sco

ncen

trat

ion

times

N2

mas

spa

thkg

2m−

5

dnsO2mplO2clm

Col

umn

tota

lO2

mas

sco

ncen

trat

ion

times

O2

mas

spa

thkg

2m−

5

dnsO2mplO2

O2

mas

sco

ncen

trat

ion

times

O2

mas

spa

thkg

2m−

5

dnsO3

Den

sity

ofO

3kg

m−

3

dnsOH

Den

sity

ofO

Hkg

m−

3

dnsaer

Aer

osol

dens

itykg

m−

3

dnsbga

Bac

kgro

und

aero

sold

ensi

tykg

m−

3

dnsdryair

Den

sity

ofdr

yai

rkg

m−

3

dnsmstair

Den

sity

ofm

oist

air

kgm−

3

48 5 APPENDIX

Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

eqntimesec

foo

seco

ndextcffmssaer

Aer

osol

mas

sex

tinct

ion

coef

ficie

ntm

2kg−

1

extcffmssbga

Bac

kgro

und

aero

solm

ass

extin

ctio

nco

effic

ient

m2

kg−

1

frcice

Frac

tion

ofco

nden

sate

that

isic

efr

actio

nfrcicettl

Frac

tion

ofco

lum

nco

nden

sate

that

isic

efr

actio

nfrcstrzennglsfc

Surf

ace

frac

tion

ofst

rong

zeni

than

gle

depe

nden

cefr

actio

n

gascstmstair

Spec

ific

gas

cons

tant

form

oist

air

joul

eki

logr

am-1

kelv

in-1

gmtday

foo

day

gmtdoy

foo

day

gmthr

foo

hour

gmtmnt

foo

min

ute

gmtmth

foo

mon

thgmtsec

foo

seco

ndgmtydy

foo

day

gmtyr

foo

year

grv

Gra

vity

met

erse

cond

-2oro

Oro

grap

hyfla

gfla

glatcos

Cos

ine

ofla

titud

efr

actio

nlatdgr

Lat

itude

(deg

rees

)de

gree

lat

Lat

itude

(rad

ians

)ra

dian

lcltimehr

Loc

alda

yho

urho

urlclyrday

Day

ofye

arin

loca

ltim

eda

ylev

Lay

erpr

essu

repa

scal


Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

levp

Inte

rfac

epr

essu

repa

scal

lmtday

foo

day

lmtdoy

foo

day

lmthr

foo

hour

lmtmnt

foo

min

ute

lmtmth

foo

mon

thlmtsec

foo

seco

ndlmtydy

foo

day

lmtyr

foo

year

londgr

foo

degr

eelon

foo

radi

anlonsec

foo

seco

ndltstday

foo

day

ltstdoy

foo

day

ltsthr

foo

hour

ltstmnt

foo

min

ute

ltstmth

foo

mon

thltstsec

foo

seco

ndltstydy

foo

day

ltstyr

foo

year

mmwmstair

Mea

nm

olec

ular

wei

ghto

fmoi

stai

rki

logr

amm

ole-

1mpcCO2

Mas

spa

thof

CO

2in

colu

mn

kgm−

2

mpcCH4

Mas

spa

thof

CH

4in

colu

mn

kgm−

2

mpcN2O

Mas

spa

thof

N2O

inco

lum

nkg

m−

2

50 5 APPENDIX

Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

mpcCFC11

Mas

spa

thof

CFC

11in

colu

mn

kgm−

2

mpcCFC12

Mas

spa

thof

CFC

12in

colu

mn

kgm−

2

mpcCWP

Tota

lcol

umn

Con

dens

edW

ater

Path

kgm−

2

mpcH2OH2O

Mas

spa

thof

H2O

dim

erin

colu

mn

kgm−

2

mpcH2O

Mas

spa

thof

H2O

inco

lum

nkg

m−

2

mpcIWP

Tota

lcol

umn

Ice

Wat

erPa

thkg

m−

2

mpcLWP

Tota

lcol

umn

Liq

uid

Wat

erPa

thkg

m−

2

mpcN2

Mas

spa

thof

N2

inco

lum

nkg

m−

2

mpcNO2

Mas

spa

thof

NO

2in

colu

mn

kgm−

2

mpcO2O2

Mas

spa

thof

O2-

O2

inco

lum

nkg

m−

2

mpcO2

Mas

spa

thof

O2

inco

lum

nkg

m−

2

mpcO3DU

Mas

spa

thof

O3

inco

lum

nD

obso

nmpcO3

Mas

spa

thof

O3

inco

lum

nkg

m−

2

mpcOH

Mas

spa

thof

OH

inco

lum

nkg

m−

2

mpcaer

Tota

lcol

umn

mas

spa

thof

aero

sol

kgm−

2

mpcbga

Tota

lcol

umn

mas

spa

thof

back

grou

ndae

roso

lkg

m−

2

mpcdryair

Mas

spa

thof

dry

airi

nco

lum

nkg

m−

2

mpcmstair

Mas

spa

thof

moi

stai

rin

colu

mn

kgm−

2

mplCO2

Mas

spa

thof

CO

2in

laye

rkg

m−

2

mplCH4

Mas

spa

thof

CH

4in

laye

rkg

m−

2

mplN2O

Mas

spa

thof

N2O

inla

yer

kgm−

2

mplCFC11

Mas

spa

thof

CFC

11in

laye

rkg

m−

2

mplCFC12

Mas

spa

thof

CFC

12in

laye

rkg

m−

2

mplCWP

Lay

erC

onde

nsed

Wat

erPa

thkg

m−

2


Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

mplH2OH2O

Mas

spa

thof

H2O

dim

erin

laye

rkg

m−

2

mplH2O

Mas

spa

thof

H2O

inla

yer

kgm−

2

mplIWP

Lay

erIc

eW

ater

Path

kgm−

2

mplLWP

Lay

erL

iqui

dW

ater

Path

kgm−

2

mplN2

Mas

spa

thof

N2

inla

yer

kg2

m−

5

mplNO2

Mas

spa

thof

NO

2in

laye

rkg

m−

2

mplO2O2

Mas

spa

thof

O2-

O2

inla

yer

kgm−

2

mplO2

Mas

spa

thof

O2

inla

yer

kg2

m−

5

mplO3

Mas

spa

thof

O3

inla

yer

kgm−

2

mplOH

Mas

spa

thof

OH

inla

yer

kgm−

2

mplaer

Lay

erm

ass

path

ofae

roso

lkg

m−

2

mplbga

Lay

erm

ass

path

ofae

roso

lkg

m−

2

mpldryair

Mas

spa

thof

dry

airi

nla

yer

kgm−

2

mplmstair

Mas

spa

thof

moi

stai

rin

laye

rkg

m−

2

npcCO2

Col

umn

num

berp

ath

ofC

O2

mol

ecul

em−

2

npcCH4

Col

umn

num

berp

ath

ofC

H4

mol

ecul

em−

2

npcN2O

Col

umn

num

berp

ath

ofN

2Om

olec

ule

m−

2

npcCFC11

Col

umn

num

berp

ath

ofC

FC11

mol

ecul

em−

2

npcCFC12

Col

umn

num

berp

ath

ofC

FC12

mol

ecul

em−

2

npcH2OH2O

Col

umn

num

berp

ath

ofH

2Odi

mer

mol

ecul

em−

2

npcH2O

Col

umn

num

berp

ath

ofH

2Om

olec

ule

m−

2

npcN2

Col

umn

num

berp

ath

ofO

2m

olec

ule

m−

2

npcNO2

Col

umn

num

berp

ath

ofN

O2

mol

ecul

em−

2

npcO2O2

Col

umn

num

berp

ath

ofO

2O2

mol

ecul

em−

2

52 5 APPENDIX

Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

npcO2

Col

umn

num

berp

ath

ofO

2m

olec

ule

m−

2

npcO3

Col

umn

num

berp

ath

ofO

3m

olec

ule

m−

2

npcOH

Col

umn

num

berp

ath

ofO

Hm

olec

ule

m−

2

npcdryair

Col

umn

num

berp

ath

ofdr

yai

rm

olec

ule

m−

2

npcmstair

Col

umn

num

berp

ath

ofm

oist

air

mol

ecul

em−

2

nplCO2

Num

berp

ath

ofC

O2

inla

yer

mol

ecul

em−

2

nplCH4

Num

berp

ath

ofC

H4

inla

yer

mol

ecul

em−

2

nplN2O

Num

berp

ath

ofN

2Oin

laye

rm

olec

ule

m−

2

nplCFC11

Num

berp

ath

ofC

FC11

inla

yer

mol

ecul

em−

2

nplCFC12

Num

berp

ath

ofC

FC12

inla

yer

mol

ecul

em−

2

nplH2OH2O

Num

berp

ath

ofH

2Odi

mer

inla

yer

mol

ecul

em−

2

nplH2O

Num

berp

ath

ofH

2Oin

laye

rm

olec

ule

m−

2

nplN2

Num

berp

ath

ofN

2in

laye

rm

olec

ule2

m−

5

nplNO2

Num

berp

ath

ofN

O2in

laye

rm

olec

ule

m−

2

nplO2O2

Num

berp

ath

ofO

2-O

2in

laye

rm

olec

ule

m−

2

nplO2

Num

berp

ath

ofO

2in

laye

rm

olec

ule2

m−

5

nplO3

Num

berp

ath

ofO

3in

laye

rm

olec

ule

m−

2

nplOH

Num

berp

ath

ofO

Hin

laye

rm

olec

ule

m−

2

npldryair

Num

berp

ath

ofdr

yai

rin

laye

rm

olec

ule

m−

2

nplmstair

Num

berp

ath

ofm

oist

airi

nla

yer

mol

ecul

em−

2

odxcobsaer

Col

umn

aero

sole

xtin

ctio

nop

tical

dept

hfr

actio

nodxcobsbga

Col

umn

back

grou

ndae

roso

lext

inct

ion

optic

alde

pth

frac

tion

odxlobsaer

Lay

erae

roso

lext

inct

ion

optic

alde

pth

frac

tion


Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

odxlobsbga

Lay

erba

ckgr

ound

aero

sole

xtin

ctio

nop

tical

dept

hfr

actio

n

oneDfoo

pprCO2

Part

ialp

ress

ure

ofC

O2

pasc

alpprCH4

Part

ialp

ress

ure

ofC

H4

pasc

alpprN2O

Part

ialp

ress

ure

ofN

2Opa

scal

pprCFC11

Part

ialp

ress

ure

ofC

FC11

pasc

alpprCFC12

Part

ialp

ress

ure

ofC

FC12

pasc

alpprH2OH2O

Part

ialp

ress

ure

ofH

2Odi

mer

pasc

alpprH2O

Part

ialp

ress

ure

ofH

2Opa

scal

pprN2

Part

ialp

ress

ure

ofN

2pa

scal

pprNO2

Part

ialp

ress

ure

ofN

O2

pasc

alpprO2O2

Part

ialp

ress

ure

ofO

2O2

pasc

alpprO2

Part

ialp

ress

ure

ofO

2pa

scal

pprO3

Part

ialp

ress

ure

ofO

3pa

scal

pprOH

Part

ialp

ress

ure

ofO

Hpa

scal

pprdryair

Part

ialp

ress

ure

ofdr

yai

rpa

scal

prscldbtm

Hig

hest

inte

rfac

ebe

neat

hal

lclo

uds

inco

lum

npa

scal

prscldmid

Pres

sure

atm

idpo

into

fall

clou

dsin

colu

mn

pasc

alprscldthick

Thi

ckne

ssof

regi

onco

ntai

ning

allc

loud

sm

eter

prscldtop

Low

esti

nter

face

abov

eal

lclo

uds

inco

lum

npa

scal

prsdlt

Lay

erpr

essu

reth

ickn

ess

pasc

alprs

Pres

sure

pasc

alprsntf

Inte

rfac

epr

essu

repa

scal

54 5 APPENDIX

Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

prssfc

Surf

ace

pres

sure

pasc

alqCO2

Mas

sm

ixin

gra

tioof

CO

2kg

kg−

1

qCH4

Mas

sm

ixin

gra

tioof

CH

4kg

kg−

1

qN2O

Mas

sm

ixin

gra

tioof

N2O

kgkg−

1

qCFC11

Mas

sm

ixin

gra

tioof

CFC

11kg

kg−

1

qCFC12

Mas

sm

ixin

gra

tioof

CFC

12kg

kg−

1

qH2OH2O

Wat

erva

pord

imer

mas

sm

ixin

gra

tiokg

kg−

1

qH2OH2OrcpqH2O

Rat

ioof

dim

erm

mrt

om

onom

erm

mr

frac

tion

qH2O

Wat

erva

porm

ass

mix

ing

ratio

frac

tion

qN2

Mas

sm

ixin

gra

tioof

N2

kgkg−

1

qNO2

Mas

sm

ixin

gra

tioof

NO

2kg

kg−

1

qO2O2

Ozo

nem

ass

mix

ing

ratio

kgkg−

1

qO2

Mas

sm

ixin

gra

tioof

O2

kgkg−

1

qO3

Ozo

nem

ass

mix

ing

ratio

kgkg−

1

qOH

Mas

sm

ixin

gra

tioof

OH

kgkg−

1

qstH2Oice

Satu

ratio

nm

ixin

gra

tiow

/r/t

ice

kgkg−

1

qstH2Olqd

Satu

ratio

nm

ixin

gra

tiow

/r/t

liqui

dkg

kg−

1

rCO2

Dry

-mas

sm

ixin

gra

tio(r

)ofC

O2

kgkg−

1

rCH4

Dry

-mas

sm

ixin

gra

tio(r

)ofC

H4

kgkg−

1

rN2O

Dry

-mas

sm

ixin

gra

tio(r

)ofN

2Okg

kg−

1

rCFC11

Dry

-mas

sm

ixin

gra

tio(r

)ofC

FC11

kgkg−

1

rCFC12

Dry

-mas

sm

ixin

gra

tio(r

)ofC

FC12

kgkg−

1

rH2OH2O

Dry

-mas

sm

ixin

gra

tio(r

)ofH

2Odi

mer

kgkg−

1

rH2O

Dry

-mas

sm

ixin

gra

tio(r

)ofH

2Okg

kg−

1


Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

rN2

Dry

-mas

sm

ixin

gra

tio(r

)ofN

2kg

kg−

1

rNO2

Dry

-mas

sm

ixin

gra

tio(r

)ofN

O2

kgkg−

1

rO2O2

Dry

-mas

sm

ixin

gra

tio(r

)ofO

2O2

kgkg−

1

rO2

Dry

-mas

sm

ixin

gra

tio(r

)ofO

2kg

kg−

1

rO3

Dry

-mas

sm

ixin

gra

tio(r

)ofO

3kg

kg−

1

rOH

Dry

-mas

sm

ixin

gra

tio(r

)ofO

Hkg

kg−

1

rdsfctice

Eff

ectiv

era

dius

ofic

ecr

ysta

lsm

icro

nrdsfctlqd

Eff

ectiv

era

dius

ofliq

uid

drop

lets

mic

ron

rghlen

Aer

odyn

amic

roug

hnes

sle

ngth

met

ersclhgt

Loc

alsc

ale

heig

htm

eter

sfcems

Surf

ace

emis

sivi

tyfr

actio

nslrazidgr

Sola

razi

mut

han

gle

degr

eeslrcrdgmmdgr

foo

degr

eeslrcst

Sola

rcon

stan

tW

m−

2

slrdcldgr

Sola

rdec

linat

ion

degr

eeslrdmtdgr

Dia

met

erof

sola

rdis

cde

gree

slrdstau

Ear

th-s

undi

stan

ceas

tron

omic

alun

itsslrelvdgr

Sola

rele

vatio

nde

gree

slrflxTOA

Sola

rflux

atTO

AW

m−

2

slrflxnrmTOA

Sola

rcon

stan

tcor

rect

edfo

rorb

italp

ositi

onW

m−

2

slrhrngldgr

Sola

rhou

rang

lede

gree

slrrfrngldgr

Sola

rref

ract

ion

angl

ede

gree

slrrgtascdgr

Sola

rrig

htas

cens

ion

degr

eeslrzennglcos

Cos

ine

sola

rzen

ithan

gle

frac

tion

56 5 APPENDIX

Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

slrzenngldgr

Sola

rzen

ithan

gle

inde

gree

sde

gree

slrzenngl

Sola

rzen

ithan

gle

radi

ansnowdepth

Snow

dept

hm

eter

spcheatmstair

Spec

ific

heat

atco

nsta

ntpr

essu

reof

moi

stai

rjo

ule

kilo

gram

-1ke

lvin

-1timelmt

Seco

nds

betw

een

1969

and

LM

Tof

sim

ulat

ion

seco

ndtimeltst

Seco

nds

betw

een

1969

and

LTST

ofsi

mul

atio

nse

cond

timeunix

Seco

nds

betw

een

1969

and

GM

Tof

sim

ulat

ion

seco

ndtptcls

Lay

erte

mpe

ratu

re(C

elsi

us)

cels

ius

tptclsntf

Inte

rfac

ete

mpe

ratu

re(C

elsi

us)

cels

ius

tpt

Lay

erTe

mpe

ratu

reke

lvin

tptntf

Inte

rfac

ete

mpe

ratu

reke

lvin

tptsfc

Tem

pera

ture

ofai

rin

cont

actw

ithsu

rfac

eke

lvin

tptskn

Tem

pera

ture

ofsu

rfac

eke

lvin

tptvrt

Vir

tual

tem

pera

ture

kelv

invmrCO2

Volu

me

mix

ing

ratio

ofC

O2

num

bern

umbe

r-1

vmrCH4

Volu

me

mix

ing

ratio

ofC

H4

num

bern

umbe

r-1

vmrN2O

Volu

me

mix

ing

ratio

ofN

2Onu

mbe

rnum

ber-

1vmrCFC11

Volu

me

mix

ing

ratio

ofC

FC11

num

bern

umbe

r-1

vmrCFC12

Volu

me

mix

ing

ratio

ofC

FC12

num

bern

umbe

r-1

vmrH2OH2O

Volu

me

mix

ing

ratio

ofH

2Odi

mer

num

bern

umbe

r-1

vmrH2O

Volu

me

mix

ing

ratio

ofH

2Onu

mbe

rnum

ber-

1vmrN2

Volu

me

mix

ing

ratio

ofN

2nu

mbe

rnum

ber-

1vmrNO2

Volu

me

mix

ing

ratio

ofN

O2

num

bern

umbe

r-1

vmrO2O2

Volu

me

mix

ing

ratio

ofO

2O2

num

bern

umbe

r-1


Tabl

e7:

(con

tinue

d)

Nam

e(s)

Purp

ose

Uni

ts

vmrO2

Volu

me

mix

ing

ratio

ofO

2nu

mbe

rnum

ber-

1vmrO3

Volu

me

mix

ing

ratio

ofO

3nu

mbe

rnum

ber-

1vmrOH

Volu

me

mix

ing

ratio

ofO

Hnu

mbe

rnum

ber-

1wvlobsaer

Wav

elen

gth

ofae

roso

lopt

ical

dept

hsp

ecifi

catio

nm

eter

wvlobsbga

Wav

elen

gth

ofba

ckgr

ound

aero

solo

ptic

alde

pth

spec

ifica

tion

met

er

xntfac

Ecc

entr

icity

fact

orfr

actio

n

58 BIBLIOGRAPHY

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IndexCFC11 vmr clm, 45CFC12 vmr clm, 45CH4 vmr clm, 45CO2 vmr clm, 45N2O vmr clm, 45RH ice, 45RH lqd, 29, 45RH, 45abc flg, 29abs bb SAS, 37abs bb atm, 37abs bb sfc, 37abs ncl wk mdm flg, 29abs nst SAS, 37abs nst atm, 37abs nst sfc, 37abs spc SAS, 37abs spc atm, 37abs spc sfc, 37alb sfc NIR dff, 45alb sfc NIR drc, 45alb sfc vsb dff, 45alb sfc vsb drc, 45alb sfc, 37, 45alt cld btm, 37, 45alt cld mid, 45alt cld thick, 37, 45alt cld top, 45alt dlt, 45alt ntf, 37, 45alt, 37, 45asp rat hxg dfl, 29asp rat lps dfl, 30azi dgr, 37azi, 37bch flg, 29bnd SW LW, 30bnd nbr, 30bnd, 37cld frc, 45cmp aer, 26cmp cor, 30cmp mdm, 30

cmp mnt, 30cmp mtx, 30cmp ncl, 30cmp prt, 30cnc CFC11, 46cnc CFC12, 46cnc CH4, 45cnc CO2, 45cnc H2OH2O, 46cnc H2O, 46cnc N2O, 46cnc N2, 46cnc NO2, 46cnc O2O2, 46cnc O2 cnc N2, 46cnc O2 cnc O2, 46cnc O2 npl N2 clm, 46cnc O2 npl N2, 46cnc O2 npl O2 clm frc, 46cnc O2 npl O2 clm, 46cnc O2 npl O2, 46cnc O2, 46cnc O3, 46cnc OH, 46cnc dry air, 46cnc mst air, 46cnc nbr anl dfl, 30cnc nbr pcp anl, 30coat flg, 29cpv foo, 30dbg lvl, 30dmn frc, 30dmn nbr max, 30dmn rcd, 30dmt dtc, 30dmt nma mcr, 30dmt pcp nma mcr, 30dmt swa mcr, 30dmt vma mcr, 30dns CFC11, 47dns CFC12, 47dns CH4, 47dns CO2, 46

61

62 INDEX

dns H2OH2O, 47dns H2O, 47dns N2O, 47dns N2, 47dns NO2, 47dns O2O2, 47dns O2 dns N2, 47dns O2 dns O2, 47dns O2 mpl N2 clm, 47dns O2 mpl N2, 47dns O2 mpl O2 clm, 47dns O2 mpl O2, 47dns O2, 47dns O3, 47dns OH, 47dns aer, 47dns bga, 47dns cor, 30dns dry air, 47dns mdm, 31dns mnt, 31dns mst air, 47dns mtx, 31dns ncl, 31dns prt, 31doy, 31drc dat, 31drc in, 31drc out, 31drv rds nma flg, 29dsd dbg mcr, 31dsd mnm mcr, 31dsd mxm mcr, 31dsd nbr, 31eqn time sec, 48ext cff mss aer, 48ext cff mss bga, 48fdg flg, 29fdg idx, 31fdg val, 31fl err, 31fl idx rfr cor, 31fl idx rfr mdm, 31fl idx rfr mnt, 31fl idx rfr mtx, 31

fl idx rfr ncl, 31fl idx rfr prt, 31fl slr spc, 31flt foo, 31flx LW dwn sfc, 32flx SW net gnd, 32flx SW net vgt, 32flx abs atm rdr, 37flx bb abs atm, 37flx bb abs sfc, 37flx bb abs ttl, 38flx bb abs, 38flx bb dwn TOA, 38flx bb dwn dff, 38flx bb dwn drc, 38flx bb dwn sfc, 38flx bb dwn, 38flx bb net, 38flx bb up, 38flx frc drc sfc cmd ln, 32flx nst abs atm, 38flx nst abs sfc, 38flx nst abs ttl, 38flx nst abs, 38flx nst dwn TOA, 38flx nst dwn sfc, 38flx nst dwn, 38flx nst net, 38flx nst up, 38flx slr frc, 38flx spc abs SAS, 38flx spc abs atm, 39flx spc abs sfc, 39flx spc abs, 39flx spc act pht TOA, 39flx spc act pht sfc, 39flx spc dwn TOA, 39flx spc dwn dff, 39flx spc dwn drc, 39flx spc dwn sfc, 39flx spc dwn, 39flx spc pht dwn sfc, 39flx spc up, 39flx vlm pcp rsl, 32frc ice ttl, 39, 48

INDEX 63

frc ice, 48frc str zen ngl sfc, 48ftn fxd flg, 29gas cst mst air, 48gmt day, 48gmt doy, 48gmt hr, 48gmt mnt, 48gmt mth, 48gmt sec, 48gmt ydy, 48gmt yr, 48grv, 48gsd anl dfl, 32gsd pcp anl, 32hgt mdp, 32hgt rfr, 32hgt zpd dps cmd ln, 32hgt zpd mbl, 32hrz flg, 29htg rate bb, 39htg, 35hxg flg, 29idx rfr cor usr, 32idx rfr mdm usr, 32idx rfr mnt usr, 32idx rfr mtx usr, 32idx rfr ncl usr, 32idx rfr prt usr, 32j NO2, 39j spc NO2 sfc, 39lat cos, 48lat dgr, 32, 39, 48lat, 48lbl sng, 32lbl, 35lcl time hr, 39, 48lcl yr day, 39, 48levp, 39, 49lev, 39, 48lgn nbr, 32lmt day, 49lmt doy, 49lmt hr, 49lmt mnt, 49

lmt mth, 49lmt sec, 49lmt ydy, 49lmt yr, 49lnd frc dry, 32lon dgr, 49lon sec, 49lon, 49ltst day, 49ltst doy, 49ltst hr, 49ltst mnt, 49ltst mth, 49ltst sec, 49ltst ydy, 49ltst yr, 49mca flg, 29mie flg, 29mie, 5, 26, 28mmw mst air, 49mmw prt, 32mno lng dps cmd ln, 33mpc CFC11, 50mpc CFC12, 50mpc CH4, 49mpc CO2, 49mpc CWP, 39, 50mpc H2OH2O, 50mpc H2O, 50mpc IWP, 50mpc LWP, 50mpc N2O, 49mpc N2, 50mpc NO2, 50mpc O2O2, 50mpc O2, 50mpc O3 DU, 50mpc O3, 50mpc OH, 50mpc aer, 50mpc bga, 50mpc dry air, 50mpc mst air, 50mpl CFC11, 50mpl CFC12, 50

64 INDEX

mpl CH4, 50mpl CO2, 50mpl CWP, 50mpl H2OH2O, 51mpl H2O, 51mpl IWP, 51mpl LWP, 51mpl N2O, 50mpl N2, 51mpl NO2, 51mpl O2O2, 51mpl O2, 51mpl O3, 51mpl OH, 51mpl aer, 51mpl bga, 51mpl dry air, 51mpl mst air, 51mss frc cly, 33mss frc snd, 33ncks, 29nc, 35ngl nbr, 33no abc flg, 29no bch flg, 29no hrz flg, 29no mie flg, 29no wrn ntp flg, 29npc CFC11, 51npc CFC12, 51npc CH4, 51npc CO2, 51npc H2OH2O, 51npc H2O, 51npc N2O, 51npc N2, 51npc NO2, 51npc O2O2, 51npc O2, 52npc O3, 52npc OH, 52npc dry air, 52npc mst air, 52npl CFC11, 52npl CFC12, 52

npl CH4, 52npl CO2, 52npl H2OH2O, 52npl H2O, 52npl N2O, 52npl N2, 52npl NO2, 52npl O2O2, 52npl O2, 52npl O3, 52npl OH, 52npl dry air, 52npl mst air, 52nrg pht, 40nsz, 35ntn bb aa, 40ntn bb mean, 40ntn spc aa ndr sfc, 40ntn spc aa ndr, 40ntn spc aa sfc, 40ntn spc aa zen sfc, 40ntn spc aa zen, 40ntn spc chn, 40ntn spc mean, 40odac spc aer, 40odac spc bga, 40odac spc ice, 40odac spc lqd, 40odal obs aer, 40odal obs bga, 40odsl obs aer, 40odsl obs bga, 40odxc obs aer, 40, 52odxc obs bga, 40, 52odxc spc CO2, 40odxc spc H2OH2O, 40odxc spc H2O, 41odxc spc NO2, 41odxc spc O2N2, 41odxc spc O2O2, 41odxc spc O2, 41odxc spc O3, 41odxc spc OH, 41odxc spc Ray, 41odxc spc aer, 41

INDEX 65

odxc spc bga, 41odxc spc ice, 41odxc spc lqd, 41odxc spc ttl, 41odxl obs aer, 41, 52odxl obs bga, 41, 53oneD foo, 53oro, 33, 48plr cos, 41plr dgr, 41plr, 41pnt typ idx, 33ppr CFC11, 53ppr CFC12, 53ppr CH4, 53ppr CO2, 53ppr H2OH2O, 53ppr H2O, 53ppr N2O, 53ppr N2, 53ppr NO2, 53ppr O2O2, 53ppr O2, 53ppr O3, 53ppr OH, 53ppr dry air, 53prs cld btm, 53prs cld mid, 53prs cld thick, 53prs cld top, 53prs dlt, 53prs mdp, 33prs ntf, 33, 53prs sfc, 54prs, 53psd.pl, 27psd ntg dgn, 35psd typ, 26, 33q CFC11, 54q CFC12, 54q CH4, 54q CO2, 54q H2OH2O rcp q H2O, 54q H2OH2O, 54q H2O vpr, 33

q H2O, 54q N2O, 54q N2, 54q NO2, 54q O2O2, 54q O2, 54q O3, 54q OH, 54qst H2O ice, 54qst H2O lqd, 54r CFC11, 54r CFC12, 54r CH4, 54r CO2, 54r H2OH2O, 54r H2O, 54r N2O, 54r N2, 55r NO2, 55r O2O2, 55r O2, 55r O3, 55r OH, 55rds fct ice, 55rds fct lqd, 55rds ffc gmm mcr, 33rds nma mcr, 33rds swa mcr, 33rds vma mcr, 33rfl bb SAS, 41rfl bb sfc, 41rfl gnd dff, 33rfl nst SAS, 41rfl nst sfc, 41rfl spc SAS, 42rgh len, 55rgh mmn dps cmd ln, 33rgh mmn ice std, 33rgh mmn mbl, 33rgh mmn smt, 33scl hgt, 55sfc ems, 55sfc typ, 33slf tst typ, 33slr azi dgr, 55

66 INDEX

slr crd gmm dgr, 55slr cst, 33, 55slr dcl dgr, 55slr dmt dgr, 55slr dst au, 55slr elv dgr, 55slr flx TOA, 55slr flx nrm TOA, 55slr hr ngl dgr, 55slr rfr ngl dgr, 55slr rgt asc dgr, 55slr spc key, 33slr zen ngl cos, 34, 42, 55slr zen ngl dgr, 56slr zen ngl, 56slv sng, 34snow depth, 56snw hgt lqd, 34soi typ, 34spc abb sng, 34spc heat mst air, 56spc heat prt, 34spc idx sng, 34ss alb cmd ln, 34sz dbg mcr, 34sz grd sng, 34sz grd, 26sz mnm mcr, 34sz mnm, 26sz mxm mcr, 34sz mxm, 26sz nbr, 26, 34sz prm rsn, 34tau prs, 42tau, 42thr nbr, 34time lmt, 56time ltst, 56time unix, 56tm dlt, 34tpt bbd wgt, 34tpt cls ntf, 56tpt cls, 56tpt gnd, 34tpt ice, 34

tpt mdp, 34tpt ntf, 42, 56tpt prt, 34tpt sfc, 56tpt skn, 56tpt soi, 34tpt sst, 34tpt vgt, 34tpt vrt, 56tpt, 42, 56trn bb atm, 42trn nst atm, 42trn spc atm CO2, 42trn spc atm H2OH2O, 42trn spc atm H2O, 42trn spc atm NO2, 42trn spc atm O2N2, 42trn spc atm O2O2, 42trn spc atm O2, 42trn spc atm O3, 42trn spc atm OH, 42trn spc atm Ray, 42trn spc atm aer, 42trn spc atm bga, 42trn spc atm ice, 42trn spc atm lqd, 42trn spc atm ttl, 43tst sng, 35var ffc gmm, 35vlm frc mntl, 35vmr CFC11, 56vmr CFC12, 56vmr CH4, 56vmr CO2, 35, 56vmr H2OH2O, 56vmr H2O, 56vmr HNO3 gas, 35vmr N2O, 56vmr N2, 56vmr NO2, 56vmr O2O2, 56vmr O2, 57vmr O3, 57vmr OH, 57vts flg, 29

INDEX 67

vwc sfc, 35wbl shp, 35wnd frc dps cmd ln, 35wnd mrd mdp, 35wnd znl mdp, 35wrn ntp flg, 29wvl ctr, 43wvl dbg mcr, 35wvl dlt mcr, 35wvl dlt, 43wvl grd sng, 35wvl grd, 43wvl max, 43wvl mdp mcr, 35wvl min, 43wvl mnm mcr, 35wvl mxm mcr, 35wvl nbr, 35wvl obs aer, 43, 57wvl obs bga, 43, 57wvn ctr, 43wvn dlt xcm, 35wvn dlt, 43wvn max, 43wvn mdp xcm, 35wvn min, 43wvn mnm xcm, 35wvn mxm xcm, 35wvn nbr, 35xnt fac, 57xpt dsc, 35

AERONET, 14almucantar, 14arithmetic mean size, 4aspherical particles, 3average size, 4

bounded distribution, 16

columnar volume, 16command line arguments, 28convention, 4cumulative concentration, 2, 7, 16

differential number concentration, 4

distribution function, 2dust emissions, 18

effective diameter, 22effective radius, 22effective size, 5effective variance, 5equivalent area spherical radius, 3equivalent diameter, 3equivalent radius, 3equivalent volume spherical radius, 3error function, 6, 16, 17, 27

fields, 36, 44

gamma distribution, 5geometric optics, 22geometric standard deviation, 6, 16gravitational sedimentation, 22

independent variable, 3input switches, 26

Legendre expansion, 32lognormal distribution, 6lognormal distribution function, 6lower bound concentration, 2

mass distribution, 19mdlsxn, 2mean size, 4mean value, 4median diameter, 19, 20median radius, 2Mie theory, 3mineral dust, 3, 18modal diameter, 19mode, 19mode diameter, 19, 19moment, 21monodisperse distribution, 22multimodal distribution, 20multimodal istributions, 20

nomenclature, 2normal distribution, 5normalization constant, 23

68 INDEX

number concentration, 22number distribution, 19number mean size, 4number-weighted mean size, 4

overlap factors, 18overlapping distributions, 18

PDF, 3Perl, 27PRIDE, 13probability density functions, 3

radiative transfer, 3

scattering cross-section, 22SI, 15sink bins, 18size distribution, 2source bins, 18source distributions, 18spectral density function, 2spherical particles, 4standard deviation, 5surface-weighted diameter, 22

truncated concentration, 2

variance, 5, 16volume-weighted diameter, 22

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Particle Size Distributions: Theory and Application to ...dust.ess.uci.edu/facts/psd/psd.pdf · 4 2 STATISTICS OF SIZE DISTRIBUTIONS and volume of ice crystals have been computed