Download Some Useful Formulae for Aerosol Size Distributions and

Some Useful Formulae for Aerosol Size
Distributions and Optical Properties
R.G. Grainger
1
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Acknowledgements
I would like to acknowledge helpful comments that improved this manuscript
from Dwaipayan Deb.
iii
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Contents
1 Describing Aerosol Size
1.1 Size Distributions . . . . . . . . . . . . . . . .
1.2 Mean, Mode, Median and Standard Deviation
1.3 Geometric Mean and Standard Deviation . .
1.4 Effective radius . . . . . . . . . . . . . . . . .
1.5 Area, Volume and Mass Distributions . . . .
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1
1
1
2
2
3
2 Normal and Logarithmic Normal Distributions
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . .
2.2 Properties of the Lognormal Distribution . . . .
2.2.1 Normal versus Lognormal . . . . . . . . .
2.2.2 Median and Mode . . . . . . . . . . . . .
2.2.3 Derivatives . . . . . . . . . . . . . . . . .
2.2.4 Moments . . . . . . . . . . . . . . . . . .
2.3 Log-Normal Distributions of Area and Volume .
2.4 A Different Basis State . . . . . . . . . . . . . . .
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4
4
5
5
7
7
8
9
13
3 Other Distributions
3.1 Gamma Distribution . . . . . . . . . .
3.2 Modified Gamma Distribution . . . . .
3.3 Inverse Modified Gamma Distribution
3.4 Regularized Power Law . . . . . . . .
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14
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15
16
17
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4 Modelling the Evolution of an Aerosol Size Distribution
19
5 Optical Properties
5.1 Volume Absorption, Scattering and
5.2 Back Scatter . . . . . . . . . . . .
5.3 Phase Function . . . . . . . . . . .
5.4 Single Scatter Albedo . . . . . . .
5.5 Asymmetry Parameter . . . . . . .
5.6 Integration . . . . . . . . . . . . .
5.7 Formulae for Practical Use . . . . .
5.8 Cloud Liquid Water Path . . . . .
5.9 Aerosol Mass . . . . . . . . . . . .
20
20
21
21
21
21
21
22
24
24
v
Extinction Coefficients .
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Note that this is a working draft. Comments that will be excluded from
the final text are indicated by
XXX
.
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
1
Describing Aerosol Size
A complete description of an ensemble of particles would describe the composition and geometry of each particle. Such an approach for atmospheric
aerosols whose concentrations can be ∼ 10, 000 particles per cm3 is impracticable in most cases. The simplest alternate approach is to use a statistical
description of the aerosol. This is assisted by the fact that small liquid
drops adopt a spherical shape so that for a chemically homogeneous aerosol
the problem becomes one of representing the number distribution of particle
radii. The size distribution can be represented in tabular form but it is usual
to adopt an analytic functional. The success of this approach hinges upon
the selection of of an appropriate size distribution function that approximates
the actual distribution. There is no a priori reason for assuming this can be
done.
1.1
Size Distributions
The distribution of particle sizes can be represented by a differential radius
number density distribution, n(r) which represents the number of particles
with radii between r and r + dr per unit volume, i.e.
N (r) =
Z
r+dr
r
n(r) dr.
(1)
Hence
dN (r)
.
dr
n(r) =
(2)
The total number of particles per unit volume, N0 , is then given by
N0 =
1.2
Z
∞
0
n(r) dr.
(3)
Mean, Mode, Median and Standard Deviation
Generally a size distibution is characterised by its centre and by its spread.
The centre of a distribution can be represented by the
mean, µ0 defined by
µ0 =
R∞
1 Z∞
rn(r) dr
R∞
=
rn(r) dr
N0 0
0 n(r) dr
0
mode The mode is the peak (maximum value) of a size distribution.
1
(4)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
median The median is the ”middle” value of a data set, i.e. 50 % of particles
are smaller than the median (and so 50 % of particles are larger).
while the spread is captured through the standard deviation, σ0 , defined
σ02 =
1 Z∞
(r − µ0 )2 n(r) dr.
N0 0
(5)
More generally, the i-th raw moment of a distribution is defined as
Z
mi =
1.3
∞
ri n(r) dr,
0
(6)
Geometric Mean and Standard Deviation
The geometric mean µg of a set of n numbers {r1 , r2 , . . . , rn } is
√
µ g = n r1 × r2 × . . . × rn
(7)
The geometric mean can also be written
h
µg = exp ln
√
n
r1 × r2 × . . . × rn
i
= exp
Pn
ln ri
n
i=1
!
(8)
The geometric standard deviation σg or S is defined
s P
σg = S = exp 
n
i=1

(ln ri − ln µg )2 
n
(9)
that is, the geometric standard deviation is the exponential of σ, the standard
deviation of ln r. So ln(σg ) = ln(S) = σ.
1.4
Effective radius
The effective radius (or area-weighted mean radius) of an aerosol distribution,
re , is defined as the ratio of the third moment of the drop size distribution
to the second moment, i.e.
re
R∞
r3 n(r)dr
m3
0
.
= R∞
=
m2
r2 n(r)dr
(10)
0
The usefulness of effective radius comes from the fact that energy removed
from a light beam by a particle is proportional to the particle’s area (provided the radius of the particle is similar to or larger than larger than the
wavelength of the incident light).
2
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
1.5
Area, Volume and Mass Distributions
Analogous to the description of the distribution of particle number with radius it is also possible to describe particle area, volume or mass with equivalent expressions to Equations 1-3.
The distribution of particle area can be represented by a differential area
density distribution, a(r) which represents the area of particles whose radii
lie between r and r + dr per unit volume, i.e.
A(r) =
Z
r+dr
r
a(r) dr.
(11)
Hence
dA
.
dr
(12)
dA dN
= 4πr2 n(r).
dN dr
(13)
a(r) =
For spherical particles
a(r) =
The total particle area per unit volume, A0 , is then given by
A0 =
Z
∞
a(r) dr.
0
(14)
The distribution of particle volume can be represented by a differential
volume density distribution, a(r) which represents the volume contained in
particles whose radii lie between r and r + dr per unit volume, i.e.
V (r) =
Z
r+dr
r
v(r) dr.
(15)
Hence
dV
.
dr
(16)
4
dV dN
= πr3 n(r).
dN dr
3
(17)
v(r) =
For spherical particles
v(r) =
The total particle area per unit volume, V0 , is given by
V0 =
Z
∞
0
3
v(r) dr.
(18)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
The distribution of mass can be represented by a differential mass density
distribution, m(r) which represents the mass contained in particles with radii
between r and r + dr per unit volume, i.e.
M (r) =
Z
r+dr
r
m(r) dr.
(19)
Hence
dM
.
dr
(20)
dM dN
4
= πr3 ρn(r),
dN dr
3
(21)
m(r) =
For spherical particles
m(r) =
where ρ is the density of the aerosol material. The total particle mass per
unit volume, M0 , is
M0 =
2
2.1
Z
∞
0
m(r) dr.
(22)
Normal and Logarithmic Normal Distributions
Definitions
One particle distribution to consider adopting is the normal distribution
(r − µ0 )2
N0 1
,
exp −
n(r) = √
2σ02
2π σ0
#
"
(23)
where µ0 is the mean and σ0 is the standard deviation of the distribution.
The size of particles in an aerosol generally covers several orders of magnitude. As a result the normal distribution fit of measured particle sizes often
has a very large standard deviation. Another drawback of the normal distribution is that it allows negative radii. Aerosol distributions are much better
represented by a normal distribution of the logarithm of the particle radii.
Letting l = ln(r) we have
N0 1
(l − µ)2
dN (l)
=√
exp −
nl (l) =
dl
2σ 2
2π σ
"
4
#
(24)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
where the mean, µ, and the standard deviation, σ, of l = ln(r) are defined
µ =
R∞
lnl (l) dl
1 Z∞
=
lnl (l) dl
N0 −∞
−∞ nl (l) dl
(25)
1 Z∞
(l − µ)2 nl (l) dl
N0 −∞
(26)
R−∞
∞
σ2 =
It is common for the lognormal distribution to be expressed in terms of the
radius. Noting that
1
dl
=
dr
r
(27)
then in terms of radius rather than log radius we have
dN (r)
dN (l) dl
dl
N0 1 1
(ln r − µ)2
n(r) =
=
= nl (l) = √
exp −
dr
dl dr
dr
2σ 2
2π σ r
"
#
(28)
It is also common to express the spread of the distribution using the
geometric standard deviation, S. From this definition S must be greater or
equal to one otherwise the log-normal standard deviation is negative. When
S is one the distribution is monodisperse. Typical aerosol distributions have
S values in the range 1.5 - 2.0.
The lognormal distribution appears in the atmospheric literature using
any of combination of rm or µ and σ or S with perhaps the commonest being
N0 1 1
(ln r − ln rm )2
√
n(r) =
exp −
2 ln2 (S)
2π ln(S) r
"
#
(29)
Be particularly careful about σ and S whose definitions are sometimes reversed!
2.2
2.2.1
Properties of the Lognormal Distribution
Normal versus Lognormal
Figure 1 shows the particle number density per unit log(radius) where the
distribution is Gaussian. Also shown in the figure is the particle number
density distribution plotted per unit radius. In performing the transform the
area (ie. total number of particles) is conserved and the median, mean or
mode in log space is the natural logarithm of the median radius, rm , in linear
space.
5
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Figure 1: Log-normal distribution with parameters N0 = 1, µ = −1 and
σ = 0.4 plotted in log space (top panel) and linear space (bottom panel).
Indicated are the distribution mode, median and mean in log and linear space
respectively.
6
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
It is possible to show the geometric mean, rg , of the radius is the same
as the median in linear space, i.e.
1
rg = (r1 × r2 × . . . × rn ) n
(30)
1
ln(r1 ) + ln(r2 ) + . . . + ln(rn )
(31)
⇒ ln(rg ) = ln (r1 × r2 × . . . × rn ) n =
n
which has the continuous form
1 Z∞
ln(rg ) =
lnl (l) dl = µ = ln(rm )
N0 −∞
⇔ rg = rm
2.2.2
(32)
(33)
Median and Mode
It can be shown that the mode of the log-normal distribution, rM , is related
to the median. Let
A
N0 1 1
(ln r − ln rm )2
= exp [B]
n(r) = √
exp −
(34)
2
r
2 ln (S)
2π ln(S) r
N0 1
(ln r − ln rm )2
(ln r − ln rm )
dB
where A = √
, B=−
=−
and
.
2
dr
2 ln (S)
ln2 (S)r
2π ln(S)
#
"
then
dn(r)
A
= − 2 exp [B] +
dr
r
A
= − 2 exp [B] −
r
dB
A
exp [B]
r
dr
A
(ln r − ln rm )
exp [B]
r2
ln2 (S)
(35)
(36)
Setting the left hand side to zero so r becomes rM
A
A
(ln rM − ln rm )
exp [B] − 2 exp [B]
2
rM
rM
ln2 (S)
(ln rM − ln rm )
−1 =
ln2 (S)
ln rM = ln rm − ln2 (S) = ln(rm ) − σ 2
0 = −
2.2.3
(37)
(38)
(39)
Derivatives
The first and second derivatives of Equation 24 are
nl
dnl
=
dN0
N0
(40)
7
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
dnl
dl
dnl
dσ
2
d nl
dN02
d 2 nl
dl2
(l − µ)
nl
2
#
"σ
1
(l − µ)2
= nl
−
σ3
σ
= −
(41)
(42)
= 0
= nl
d 2 nl
=
dσ 2
(43)
"
l
(l − µ)2
− 2
4
σ
σ
"
 (l − µ)2
nl 
σ3
#
1
−
σ
(44)
#2

1
(l − µ)2
+
−3
σ4
σ2 
(45)
If using Equation 29 the derivatives are :
dn
n
=
dN0
N0
"
#
n
(ln r − ln rm )
dn
= − 1+
dr
r
ln2 (S)
"
#
n
(ln r − ln rm )2
dn
=
−1
dS
S ln(S)
ln2 (S)
(46)
(47)
(48)
The second derivatives are:
d2 n
= 0
(49)
dN02
"
#
d2 n
3 ln r − 3 ln rm − 1 (ln r − ln rm )2
2
= n 2+
(50)
+
dr2
r
r2 ln2 (S)
r2 ln4 (S)
"
#
(ln r − ln rm )4
(ln r − ln rm )2 (ln r − ln rm )2
2
1
d2 n
= n
−5
−
+ 2 2
+
dS 2
S 2 ln6 (S)
S 2 ln4 (S)
S 2 ln3 (S)
S ln (S) S 2 ln(S)
(51)
2.2.4
Moments
The i-th raw moment of a lognormal distribution is given by
mi
i2 σ 2
= N0 exp iµ +
.
2
!
(52)
The first three moments are
m1 =
Z
∞
0
1
1 2
1
ln S
rn(r) dr = N0 exp µ + σ 2 = N0 rm exp σ 2 ≡ N0 rm exp
2
2
2
8
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
m2 =
m3 =
Z
Z
∞
0
∞
0
2
2
r2 n(r) dr = N0 exp 2µ + 2σ 2 = N0 rm
exp 2σ 2 ≡ N0 rm
exp 2 ln2 S
9
9 2
9
3
3
exp σ 2 ≡ N0 rm
exp
ln S
r n(r) dr = N0 exp 3µ + σ 2 = N0 rm
2
2
2
3
The mean radius, the surface area density and the volume density of a lognormal distribution are given by
1 Z∞
1
1 2
1
mean =
m1 =
N0 exp µ + σ ,
rn(r) dr =
N0 0
N0
N0
2
1 2
1 2
= rm exp σ ≡ rm exp
ln S ,
2
2
area =
Z
∞
0
4πr2 n(r) dr = 4πm2 = 4πN0 exp 2µ + 2σ 2 ,
2
2
= 4πN0 rm
exp 2σ 2 ≡ 4πN0 rm
exp 2 ln2 S ,
and
(53)
(54)
(55)
(56)
4
4
9
4 3
πr n(r)dr = πm3 = πN0 exp 3µ + σ 2 , (57)
volume =
3
3
3
2
0
4
4
9
9
3
3
πN0 rm
exp σ 2 ≡ πN0 rm
exp
ln2 S .
(58)
=
3
2
3
2
Z
∞
For a lognormal distribution the effective radius is
re
2.3
exp 3µ + 29 σ 2
m3
5
= exp µ + σ 2 ,
=
=
4
2
m2
2
exp 2µ + 2 σ
5
5 2
= rm exp σ 2 ≡ rm exp
ln S .
2
2
(59)
(60)
Log-Normal Distributions of Area and Volume
The log-normal area density distribution is
(ln(r) − µa )2
A0 1 1
exp −
a(r) = √
2σa2
2π σa r
"
#
(61)
where A0 is the total aerosol surface area per unit volume, µa is the radius
of the median area and σa the geometric standard deviation. These constants can all be related to the description of a log-normal number density
distribution starting from
N0 1 1
(ln r − µ)2
a(r) = 4πr n(r) = 4πr √
exp −
2σ 2
2π σ r
2
"
2
9
#
(62)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Making the substitution r2 = exp(2 ln r) and completing the square gives
h
i
N0 1 1
(ln r − (µ + 2σ 2 ))
a(r) = 4π √
exp 2µ + 2σ 2 exp −
2σ 2
2π σ r
"
2#
(63)
Equating with Equations 61 and 62 gives
A 1 1
(ln(r) − µa )2
√0
exp −
2σa2
2π σa r
"
#
h
i
N0 1 1
(ln r − (µ + 2σ 2 ))
exp 2µ + 2σ 2 exp −
= 4π √
2σ 2
2π σ r
"
2#
(64)
which is true if
h
i
4πN0
A0
exp 2µ + 2σ 2
=
σa
σ
(65)
and
(ln(r) − µa )2
(ln r − (µ + 2σ 2 ))
=
2σa2
2σ 2
2
(66)
From Equation 55
A0 = 4πN0 exp 2µ + 2σ 2
(67)
which gives σa = σ when inserted into Equation 65. This shows that is the
number density distribution is log-normal then the surface area density distribution is log-normal with the same geometric standard deviation. Applying
this result to Equation 66 gives
µa = µ + 2σ 2 .
(68)
This states that the area median radius is greater than the median radius.
Equivalent expressions can be calculated for a volume density distribution,
v(r) defined in Section 1.5. The log-normal volume density distribution is
(ln(r) − µv )2
V0 1 1
exp −
v(r) = √
2σv2
2π σv r
"
#
(69)
where V0 is the total aerosol volume per unit volume, µv is the radius of
the median volume and σv the geometric standard deviation. These constants can all be related to the description of a log-normal number density
distribution starting from
4 3
4
N0 1 1
(ln r − µ)2
v(r) =
πr n(r) = πr3 √
exp −
3
3
2σ 2
2π σ r
"
10
#
(70)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Making the substitution r3 = exp(3 ln r) and completing the square gives
4 N0 1 1
(ln r − µ)2 − 6σ 2 ln r
π√
exp −
v(r) =
3
2σ 2
2π σ r
"
#
(71)
Complete the square
h
i
4 N0 1 1
(ln r − (µ + 3σ 2 ))
v(r) =
π√
exp 3µ + 4.5σ 2 exp −
3
2σ 2
2π σ r
"
2#
(72)
Equating with Equations 69 and 72 gives
V 1 1
(ln(r) − µv )2
√0
exp −
2σv2
2π σv r
"
#
h
i
(ln r − (µ + 3σ 2 ))
4 N0 1 1
π√
exp 3µ + 4.5σ 2 exp −
=
3
2σ 2
2π σ r
"
2#
(73)
which is true if
h
i
V0
4 N0
exp 3µ + 4.5σ 2
= π
σv
3 σ
(74)
and
(ln(r) − µv )2
(ln r − (µ + 3σ 2 ))
=
2σv2
2σ 2
2
(75)
From Equation 57
V0 =
4
πN0 exp 3µ + 4.5σ 2
3
(76)
which gives σv = σ when inserted into Equation 74. This shows that is the
number density distribution is log-normal then the volume density distribution is log-normal with the same geometric standard deviation. Applying
this result to Equation 75 gives
µv = µ + 3σ 2 .
(77)
They relationships area and volume density log-normal parameters and the
number size distribution parameters are summarised in Table 1.
Lastly Table 2 shows derived values for a log-normal distributions over a
range of rm and with S = 1.5.
11
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Table 1: Relationships area and volume density log-normal parameters and
the number size distribution parameters, N0 , µ and σ.
Distribution
Parameters
Area density
Volume density
A0
µa
σa
V0
µv
σv
Relation to Number
Density Parameters
= 4πN0 exp (2µ + 2σ 2 )
= µ + 2σ 2
=σ
= 34 πN0 exp (3µ + 4.5σ 2 )
= µ + 3σ 2
=σ
Table 2: Values derived for a log-normal number density size distribution
with S = 1.5.
Median
Radius
Mean
Radius
Effective
Radius
µm
0.1
0.2
0.5
1
2
5
10
20
50
100
µm
0.1
0.3
0.6
1.3
2.5
6.4
13
25
64
127
µm
0.3
0.7
1.7
3.3
6.6
17
33
67
166
332
12
Area
Median
Radius
µm
0.3
0.5
1.3
2.6
5.2
13.
26
52
131
261
Volume
Median
Radius
µm
0.4
0.8
2.1
4.2
8.5
21
42
85
211
423
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
2.4
A Different Basis State
While the shape of the lognormal distribution is controlled by rm and S an
alternative is re and v = V /N0 . If these parameters are adopted rm and S
can be recovered as follows:
S =
rm =
v

u
3v
u
t ln 4πre3 

exp 
 −

3
3v
4π
(5/6)
13
1
(3/2)
re
(78)
(79)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
3
Other Distributions
3.1
Gamma Distribution
Figure 2: Normalised gamma distribution.
The gamma distribution is given by Twomey (1977) as
n(r) = ar exp (−br) ,
(80)
where a and b are positive constants. The mode of the distribution occurs
where r = b−1 and it falls off slowly on the small radius side and exponentially
on the large radius side. The i-th moment of the gamma distribution is given
by
mi = ab−2−i Γ (2 + i) ,
= ab−2−i (1 + i)!.
(81)
(82)
If the constant a is used to denote the total number density then the normalised distribution (see Figure 2) can be expressed
n(r) = ab2 r exp (−br) ,
14
(83)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
which has moments defined by
mi = ab−i (1 + i)!.
3.2
(84)
Modified Gamma Distribution
Figure 3: Normalised modified gamma distribution (α = 2, b = 100, γ = 2)
and gamma distribution (b = 10).
The modified gamma distribution is given by Deirmendjian (1969) as
n(r) = arα exp (−brγ ) .
(85)
The four constants a, α, b, γ are positive and real and α is an integer. The
mode of the distribution occurs where r =
distribution are1
mi
1
α
bγ
1/γ
and the moments of this
!
α+1+i
a − α+1+i
.
b γ Γ
=
γ
γ
See 3 · 478/1 of Gradshteyn and Ryzhik (1994).
15
(86)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Hence the integral of the modified gamma distribution is
Z
!
a − α+1
α+1
b γ Γ
.
n(r) dr =
γ
γ
∞
0
(87)
In order that the first parameter, a, is the total aerosol concentration it is
convenient to define the normalized modified gamma distribution (see Figure 3) as
n(r) = a
rα exp (−brγ )
1 −
b
γ
α+1
γ
Γ
α+1
γ
,
(88)
where a, α, b, γ are positive and real but α is no longer constrained to be an
integer. The moments of this distribution are given by
mi = ab
3.3
− γi
α+i+1
γ
α+1
Γ γ
Γ
.
(89)
Inverse Modified Gamma Distribution
The inverse modified gamma distribution is defined by Deepak (1982) as
n(r) = ar−α exp −br−γ .
(90)
−1/γ
α
and it falls off
The mode of the distribution occurs where r = bγ
slowly on the large radius side and exponentially on the small radius side.
The moments are defined by
mi
!
α−1−i
a − α−1−i
=
b γ Γ
.
γ
γ
(91)
The normalized inverse modified gamma distribution can be defined
n(r) = a
r−α exp (−br−γ )
1 −
b
γ
α−1
γ
Γ
α−1
γ
.
(92)
The moments of this distribution are given by
i
mi = ab γ
16
α−1−i
γ
α−1
Γ γ
Γ
.
(93)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Figure 4: Normalised inverse modified gamma distribution (α = 2, b =
0.01, γ = 2) and gamma distribution(b − 10).
3.4
Regularized Power Law
The regularized power law is defined by Deepak (1982) as
n(r) = abα−2 h
rα−1
1+
α iγ ,
r
b
(94)
where the positive constants a, b, α, γ mainly effect the number density, the
mode radius, the positive gradient and the negative gradient respectively.
The mode radius is given by
α−1
r = b
1 + α(γ − 1)
!1/α
,
(95)
and the moments by
mi = a
bi Γ(1 + i/α)Γ(γ − 1 − i/α)
.
α
Γ(γ)
17
(96)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Hence the normalised distribution is
rα−1
α i γ .
n(r) = aαγbα−2 h
1 + rb
18
(97)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
4
Modelling the Evolution of an Aerosol Size
Distribution
For retrieval purposes it is necessary to describe the evolution of an aerosol
size distribution. Consider the case where an aerosol size distribution is
described by three modes which are parametrized by a mode radius, rm,i and
a spread, σi . We wish to alter the mixing ratios, µi , of each of the modes to
achieve a given effective radius re . How do we do this?
Firstly calculate the effective radius of each of the modes according to
re,i = rm,i exp
If re ≤ re,1
5 2
ln Si .
2

(98)

1
r / exp 25 ln2 S1 
 e,i


.
then µ =  0  and rm = 
rm,2


0
rm,3




rm,1
0






rm,2
Similarly if re ≥ re,3 then µ =  0  and rm = 
.
2
5
1
re,i / exp 2 ln S3
If re,1 < re < re,3 then µ1 and is estimated by linearly interpolating
between [0,1] as a function of re i.e.


re − re,1
re,3 − re,1
µ1 =
(99)
We now have two equations
µ1 + µ2 + µ3 =(100)
1
3
µ1 rm,1
exp
9
2
2
ln S1 +
3
µ2 rm,2
exp
9
2
2
ln S2 +
3
µ3 rm,3
exp
9
2
ln2 S3
2
2
2
exp 2 ln2 S1 + µ2 rm,2
µ1 rm,1
exp 2 ln2 S2 + µ3 rm,3
exp 2 ln2 S3
=(101)
re
and two unknowns i.e. µ2 and µ3 . The second equation is simplified by
substitution i.e.
Aµ1 + Bµ2 + Cµ3
= re
(102)
Dµ1 + Eµ2 + F µ3
and the two equations solved to give
re E − B − µ1 (A − B + re (E − D))
C − B + re (E − F )
re F − C − µ1 (A − C + re (F − D))
=
B − C + re (F − E)
µ2 =
(103)
µ3
(104)
19
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
5
5.1
Optical Properties
Volume Absorption, Scattering and Extinction Coefficients
The volume absorption coefficient, β abs (λ, r), the volume scattering coefficient, β sca (λ, r), and the volume extinction coefficient, β ext (λ, r), represent
the energy removed from a beam per unit distance by absorption, scattering,
and by both absorption and scattering. For a monodisperse aerosol they are
calculated from

abs
abs
2 abs

 β (λ, r) = σ (λ, r)N (r) = πr Q (λ, r)N (r),
monodisperse only


β sca (λ, r) = σ sca (λ, r)N (r) = πr2 Qsca (λ, r)N (r),
(105)
β ext (λ, r) = σ ext (λ, r)N (r) = πr2 Qext (λ, r)N (r),
where N (r) is the number of particles per unit volume at some radius, r. The
absorption cross section, σ ext (λ, r), the scattering cross section, σ ext (λ, r),
and the extinction cross section, σ ext (λ, r), are determined from the extinction efficiency factor, Qext (λ, r), extinction efficiency factor, Qext (λ, r), extinction efficiency factor, Qext (λ, r), respectively.
For a collection of particles, the volume coefficients are given by
β abs (λ) =
β sca (λ) =
β ext (λ) =
Z
Z
Z
∞
0
∞
0
∞
0
σ abs (λ, r)n(r) dr =
σ sca (λ, r)n(r) dr =
σ ext (λ, r)n(r) dr =
Z
∞
Z 0∞
0
Z
∞
0
πr2 Qabs (λ, r)n(r) dr,
(106)
πr2 Qsca (λ, r)n(r) dr,
(107)
πr2 Qext (λ, r)n(r) dr,
(108)
where n(r) represents the number of particles with radii between r and r +dr
per unit volume. It is also useful to define the quantities per particle i.e.
β abs (λ)
,
N0
0
0
Z ∞
.Z ∞
β sca (λ)
sca
sca
σ̄ (λ) =
,
n(r) dr =
σ n(r) dr
N0
0
0
Z ∞
.Z ∞
β ext (λ)
ext
ext
σ̄ (λ) =
n(r) dr =
σ n(r) dr
,
N0
0
0
σ̄ abs (λ) =
Z
∞
σ abs n(r) dr
.Z
∞
n(r) dr =
(109)
(110)
(111)
where σ̄ abs (λ), σ̄ sca (λ) and σ̄ ext (λ) are the mean absorption cross section, the
mean scattering cross section and the mean extinction cross section respectively.
20
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
5.2
Back Scatter
to be done
5.3
Phase Function
The phase function represents the redistribution of the scattered energy.
For a collection of particles, the phase function is given by
1 Z
P (λ, θ) =
5.4
β sca
∞
0
πr2 Qsca (λ, r)P (λ, r, θ)n(r) dr.
(112)
Single Scatter Albedo
The single scatter albedo is the ratio of the energy scattered from a particle
to that intercepted by the particle. Hence
β sca (λ)
.
β ext (λ)
ω(λ) =
5.5
(113)
Asymmetry Parameter
The asymmetry parameter is the average cosine of the scattering angle,
weighted by the intensity of the scattered light as a function of angle. It
has the value 1 for perfect forward scattering, 0 for isotropic scattering and
-1 for perfect backscatter.
1 Z
g =
5.6
β sca
∞
0
πr2 Qsca (λ, r)g(λ, r)n(r) dr
(114)
Integration
The integration of an optical properties over size is usually reduced from
the interval r = [0, ∞] to r = [rl , ru ] as n(r) → 0 as r → 0 and r → ∞.
Numerically an integral over particle size becomes
Z
ru
rl
f (r)n(r) dr =
n
X
wi f (ri )
(115)
i=1
where wi are the weights at discrete values of radius, ri .
For a log normal size distribution the integrals are
β ext (λ) =
N0
σ
r
π
2
Z
ru
rl

rQext (λ, r) exp −
21
1
2
!2 
ln r − ln rm 
dr
σ
(116)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
n
πX

=
N0
σ
r
=
n
X
wi′ Qext (λ, ri )
2
i=1
wi ri Qext (λ, ri ) exp −
1
2
!2 
ln ri − ln rm 
σ
i=1
n
X
β abs (λ) =
wi′ Qabs (λ, ri )
i=1
n
X
β sca (λ) =
wi′ Qsca (λ, ri )
i=1
N0
P (λ, θ) =
σ
r
N0
=
σ
r

1
π 1 Z ru sca
rQ (λ, r)P (λ, r, θ) exp −
sca
2β
2
rl

n
π 1 X
1
wi ri Qsca (λ, ri )P (λ, ri , θ) exp −
sca
2 β i=1
2
n
1 X
=
β sca
!2 
 dr
ln ri − ln rm
σ
!2 
wi′ Qsca (λ, ri )P (λ, ri , θ)
i=1
N0
g =
σ
r
N0
=
σ
r
β sca

π 1 Z ∞ sca
1
rQ (λ, r)g(λ, r) exp −
sca
2β
2
0
ln r − ln rm
σ

n
1
π 1 X
wi ri Qsca (λ, ri )g(λ, ri ) exp −
sca
2 β i=1
2
n
1 X
=
ln r − ln rm
σ
!2 
 dr
ln ri − ln rm
σ
wi′ Qsca (λ, ri )g(λ, ri )
!2 

i=1
where
N0
=
σ
wi′
5.7
r

π
1
ri exp −
2
2
ln ri − ln rm
σ
!2 
 wi
(117)
Formulae for Practical Use
As part of the retrieval process it is helpful to have analytic expression for the
partial derivatives of β ext (Equation 116) with respect to the size distribution
parameters (N0 , rm , σ).
∂β ext
∂N0
∂β ext
∂rm
=
=
1
σ
r
π
2
N0
rm σ 3
Z
∞
0
r
π
2
"
(ln r − ln rm )2
rQext (λ, r) exp −
2σ 2
Z
∞
0
(ln r − ln rm )rQ
ext
22
"
#
dr,
(ln r − ln rm )2
(λ, r) exp −
2σ 2
(118)
#
dr, (119)

Some Useful Formulae for Aerosol Size Distributions and Optical Properties
∂β ext
∂σ
N0
= − 2
σ
r
π
2
Z
N0
+
σ
r
π
2
Z
N0
σ2
π
2
Z
∞
=
r
"
(ln r − ln rm )2
rQext (λ, r) exp −
2σ 2
∞
0
dr
"
(ln r − ln rm )2 ext
(ln r − ln rm )2
rQ
(λ,
r)
exp
−
σ3
2σ 2
∞
0
"
0
#
#
#
dr,
"
(ln r − ln rm )2
(ln r − ln rm )2
ext
−
1
rQ
(λ,
r)
exp
−
σ2
2σ 2
#
dr.
(120)
To linearise the retrieval it is better to retrieve T (= ln N0 ) rather than N0 .
In addition to limit the values of rm and σ to positive quantities it is better to
retrieve lm (= ln rm ) and G (= ln σ). In terms of these new variables volume
extinction coefficient for a log normal size distribution is
β
ext
exp T
(λ) =
exp G
r
#
"
π Z ∞ 2l ext
(l − lm )2
dl.
e Q (l, λ) exp −
2 −∞
2 exp(2G)
(121)
The partial derivatives of the transformed parameters (Equation 121) are
∂β ext
∂T
=
exp T
exp G
r
π
2
∂β ext
∂lm
=
∂β ext
∂G
exp T
= −
exp(G)
exp T
exp(3G)
exp T
+
exp G
=
exp T
exp(G)
Z
r
∞
2l
e Q
−∞
π
2
Z
r
π
2
π
2
Z
π
2
Z
r
r
ext
"
∞
2l
−∞
Z
(l − lm )e Q
∞
2l
e Q
−∞
∞
−∞
∞
−∞
#
(l − lm )2
(l, λ) exp −
dl,
2 exp(2G)
ext
ext
(122)
#
"
(l − lm )2
dl,
(l, λ) exp −
2 exp(2G)
"
(123)
#
(l − lm )2
(l, λ) exp −
dl
2 exp(2G)
#
"
(l − lm )2 2l ext
(l − lm )2
dl,
e Q (l, λ) exp −
exp(2G)
2 exp(2G)
"
#
"
#
(l − lm )2
(l − lm )2
− 1 e2l Qext (l, λ) exp −
dl.
exp(2G)
2 exp(2G)
(124)
23
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
5.8
Cloud Liquid Water Path
The mass l of liquid per unit area in a cloud with a homogeneous size distribution is given by
l = ρ
Z
∞
0
4 3
πr n(r) dr × z
3
(125)
where ρ is the density of the cloud material (water or ice) and z is the vertical
distance through the cloud. The liquid water path is usually expressed as
g m−2 . Note that
τ = β ext × z
(126)
4
πr3 n(r) dr
3
β ext
(127)
so
l = ρτ
R∞
0
∞ 3
r n(r) dr
4
πρτ R ∞ 20 ext
.
=
3
(λ, r)n(r) dr
0 πr Q
R
(128)
For drops large with respect to wavelength we assume Qext (λ, r) = 2. Hence
∞ 3
r n(r) dr
2
4
ρτ R0∞ 2
= ρτ re
l =
6
3
0 r n(r) dr
R
(129)
So for a water cloud (ρ = 1 g cm−3 ) of τ = 10, re = 15 µm we get
l =
2
× 1 × 10 × 15 g cm−3 µm = 100 g m−2
3
(130)
While for an ice cloud (-45 ◦ C, ρ = 0.920 g cm−3 ) of τ = 1, re = 25 µm we
get
l =
2
× 0.92 × 1 × 25 g cm−3 µm = 15 g m−2
3
(131)
.
5.9
Aerosol Mass
Consider the measurement of optical depth and effective radius made by a
imaging instrument. How can this be related to the mass of aerosol present
in the atmosphere? Consider a volume observed by the instrument whose
24
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
area is A. If ρ is the density of the aerosol and Z is the height of this volume
then the total mass of aerosol, M , in the box is given by
M = ρ×v×N ×A×Z
(132)
where N is the number of particles per unit volume and v is the average
volume of each particle. If we divide both sides by A we obtain the mass per
unit area m, i.e.
m = ρ×v×N ×Z
(133)
This formula can re-expressed in terms of more familiar optical measurements
of the volume. First note that the optical depth is related to the β ext by
τ = β ext × Z
(134)
so that
m =
ρ×v×N ×τ
ρ×v×τ
=
ext
β
σ̄ ext
(135)
For a given size distribution n(r) the average volume of each particle is
v =
R∞
0
4
πr3 n(r) dr
3
N
(136)
so that the mass per unit area is given by
ρτ Z ∞ 4 3
m =
πr n(r) dr
β ext 0 3
(137)
The important thing to note here is that N disappears explicitly from the
equation.
In terms of re the mass per unit area is given by
R∞ 2
r n(r) dr
4ρτ
4πρτ Z ∞ 3
R0∞
r
n(r)
dr
×
=
re
m =
2
3β ext 0
3Q̃ext
0 r n(r) dr
25
(138)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
Substance
Density (g cm−3 )
Reference
Ice
Volcanic ash
Water
0.92
2.42±0.79
1
Bayhurst, Wohletz and Mason (1994)
Table 3: Density of some materials that form aerosols.
which uses an area weighted extinction efficiency
Q̃
ext
∞ ext
β ext
σ n(r) dr
= R∞ 2
= R0∞ 2
=
π 0 r n(r) dr
0 πr n(r) dr
R
R∞
0
πr2 Qext n(r) dr
2
0 πr n(r) dr
R∞
(139)
If the size distribution is much larger than the wavelength then Qext → 2
and Q̃ext ≈ 2 and Equation 129 becomes identical to Equation 138.
If the aerosol size distribution is log-normal with number density N0 ,
mode radius rm and spread σ then the integral in Equation 137 can be
completed analytically i.e.
ρτ 4
9
3
m =
πN0 rm
exp σ 2
ext
β 3
2
(140)
Typically ρ is in g cm−3 , N0 is in cm−3 , rm is in µm, and β ext is in km−1 so
that the units of m are
g 1
3
1
g
cm3 cm3 µm =
10−18 m3 103 m = 10−3 g m−2 (141)
1
−6
3
−6
3
10 m 10 m
km
Table 3 list the bulk density of some aerosol components.
If the effective radius, re , is known rather than rm then we can use the
relationship between re and rm
re
5
= rm exp σ 2 ,
2
(142)
to get
4ρτ πN0 3
4ρτ πN0 3
15 2
9 2
2
exp
=
m=
r
exp
−
σ
σ
r
exp
−3σ
. (143)
3β ext e
2
2
3β ext e
Equating this expression to Equation 129 gives
Q̃ext =
β ext
.
πN0 re2 exp (−3σ 2 )
26
(144)
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
which is true for a log-normal distribution.
For a multi-mode lognormal distribution where the ith model is parameterised by Ni , ri , σi and density ρi we have
τ ×ρ×N ×v
τ ni=1 ρi Ni vi
m =
=
Pn
ext
β ext
i=1 Ni σ̄i
P
(145)
where σ̄iext is the extinction cross section per particle for the ith mode. Remember the volume per particle for the ith mode is
9
4 3
πri exp σi2
3
2
4
3
i=1 ρi Ni 3 πri exp
vi =
(146)
Hence
m = τ
Pn
4
πτ
=
3
Pn
i=1
Pn
i=1
Ni σ̄iext
ρi Ni ri3 exp
Pn
i=1
27
9 2
σ
2 i
Ni σ̄iext
(147)
(148)
9 2
σ
2 i
Some Useful Formulae for Aerosol Size Distributions and Optical Properties
References
Bayhurst, G. K., Wohletz, K. H. and Mason, A. S. (1994). A method for characterizing volcanic ash from the December 15, 1989 eruption of Redoubt
volcano, Alaska, in T. J. Casadevall (ed.), Volcanic Ash and Aviation
Safety, Proc. 1st Int. Symp. on Volcanic Ash and Aviation Safety, USGS
Bulletin 2047, Washington, pp. 13–17.
Deepak, A. (ed.) (1982). Atmospheric Aerosols, Spectrum Press, Hampton,
Virginia.
Deirmendjian, D. (1969). Electromagnetic scattering on spherical polydispersions, Elsevier, New York.
Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of integrals, series, and
products, Academic Press, London.
Twomey, S. (1977). Atmospheric aerosols, Elsevier, Amsterdam, The Netherlands.
28