Some Useful Formulae for Aerosol Size Distributions and Optical Properties

Home , Geometric mean

R.G. Grainger 16 November 2020

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.

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Some Useful Formulae for Aerosol Size Distributions and Optical Properties

Contents

1 Describing Aerosol Size 1 1.1 SizeDistribution ...... 1 1.2 Moments, Mean, Mode, Median and Standard Deviation ...... 2 1.3 GeometricMeanandStandardDeviation ...... 2 1.4 Eﬀectiveradius ...... 3 1.5 Area,VolumeandMassDistributions ...... 3

2 Normal and Logarithmic Normal Distributions 5 2.1 Deﬁnitions...... 5 2.2 Propertiesofthe Lognormal Distribution ...... 6 2.2.1 NormalversusLognormal ...... 6 2.2.2 MedianandMode ...... 8 2.2.3 Derivatives ...... 8 2.2.4 Moments ...... 9 2.3 Log-Normal Distributions of Area and Volume ...... 10 2.4 ADiﬀerentBasisState...... 12

3 Other Distributions 14 3.1 GammaDistribution ...... 14 3.1.1 Moments ...... 14 3.2 ModiﬁedGammaDistribution...... 15 3.2.1 Moments ...... 16 3.3 InverseModiﬁedGammaDistribution ...... 16 3.4 RegularizedPowerLaw ...... 17 3.4.1 Moments ...... 18

4 Modelling the Evolution of an Aerosol Size Distribution 19

5 Optical Properties 20 5.1 Volume Absorption, Scattering and Extinction Coeﬃcients..... 20 5.2 BackScatter ...... 21 5.3 PhaseFunction ...... 21 5.4 SingleScatterAlbedo ...... 21 5.5 AsymmetryParameter...... 21 5.6 Integration ...... 21 5.7 FormulaeforPracticalUse...... 22 5.8 CloudLiquidWaterPath ...... 24 5.9 AerosolMass ...... 24

v 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 com- position and geometry of each particle. Such an approach for atmospheric aerosols whose concentrations can be 10, 000 particles per cm3 is imprac- ∼ ticable 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 Distribution A measured distribution of particle sizes can be described by a histogram of the number of particles per unit volume within defined size bins. By making the bin size tend to zero a continuous function is formed called the radius number density distribution N(r) which represents the number of particles with radii between r and r + dr per unit volume. The difficulty with this representation is that is is unmeasurable! This is because there are an infinite number of radius values so that the probability of the radius being any one specific value is zero. The problem is avoided by using the differential radius number density distribution n(r) defined by

dN(r) n(r) = . (1) dr The same equation can be written in integral form as

r+dr dN(r) = n(r) dr. (2) Zr

The total number of particles per unit volume, N0, is given by ∞ N0 = n(r) dr. (3) Z0

1 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

1.2 Moments, Mean, Mode, Median and Standard De- viation The moments of a size distribution are simple metrics to express the shape th 1 of the distribution. The i moment mi of n(r) is deﬁned

∞ 1 i mi = (r c) n(r) dr (4) N0 Z0 − where c is some constant. The raw moment is the moment where c = 0. The moments about the mean are called central moments. Generally a size distribution is characterised by its centre and by its spread. The centre of a distribution can be represented by the mean, µ0 deﬁned by

∞ ∞ 0 rn(r) dr 1 µ0 = ∞ = rn(r) dr. (5) R 0 n(r) dr N0 Z0 The mean is the ﬁrst rawR moment of a size distribution. mode The mode is the peak (maximum value) of a size distribution. 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).

2 while the spread is captured through the variance, σ0, deﬁned ∞ 2 1 2 σ0 = (r µ0) n(r) dr. (6) N0 Z0 − The variance is the second central moment of the size distribution. Finally, the standard deviation, σ0, of a distribution is the square root of the variance.

1.3 Geometric Mean and Standard Deviation The geometric mean µ of a set of n numbers r ,r ,...,r is g { 1 2 n} µ = √n r r ... r (7) g 1 × 2 × × n The geometric mean can also be written

n n i=1 ln ri µg = exp ln √r1 r2 ... rn = exp (8) × × × n ! h i P 1 Many texts omit the normalising term before the integral assuming that N0 = 1.

2 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

The geometric standard deviation σg is deﬁned

n 2 i=1 (ln ri ln µg) σg = exp − (9) sP n    that is, the geometric standard deviation is the exponential of σ, the standard deviation of ln r. So ln(σg)= σ. Many publications use the symbol S rather than σg for the geometric standard deviation.

1.4 Effective radius The mean along with the other size distribution metrics can be helpful in describing a particle distribution, but by far the most useful for optical measurements is the effective radius. The advantage of using the effective radius, re, 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, the wavelength of the incident light). Weighting each radius by πr2n(r) gives

∞ rπr2 n(r)dr 0 re = R∞ . (10) πr2 n(r)dr 0 R This makes clear why the effective radius is sometimes called the area- weighted mean radius. More often the effective radius is defined as the ratio of the third moment of the drop size distribution to the second moment, i.e.

∞ r3 n(r)dr m3 0 re = = R∞ . (11) m2 r2 n(r)dr 0 R Although this is identical mathematically it misses why the eﬀective radius is such useful metric and physically descriptive in radiative transfer.

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 2and3.

3 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

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.

r+dr A(r) = a(r) dr. (12) Zr Hence dA a(r) = . (13) dr For spherical particles dA dN a(r) = =4πr2n(r). (14) dN dr

The total particle area per unit volume, A0, is then given by ∞ A0 = a(r) dr. (15) Z0 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.

r+dr V (r) = v(r) dr. (16) Zr Hence dV v(r) = . (17) dr For spherical particles dV dN 4 v(r) = = πr3n(r). (18) dN dr 3

The total particle area per unit volume, V0, is given by ∞ V0 = v(r) dr. (19) Z0 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.

r+dr M(r) = m(r) dr. (20) Zr 4 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

Hence dM m(r) = . (21) dr For spherical particles dM dN 4 m(r) = = πr3ρn(r), (22) dN dr 3 where ρ is the density of the aerosol material. The total particle mass per unit volume, M0, is ∞ M0 = m(r) dr. (23) Z0 2 Normal and Logarithmic Normal Distribu- tions

2.1 Deﬁnitions One particle distribution to consider adopting is the normal distribution

2 N0 1 (r µ0) n(r) = exp − 2 , (24) √2π σ0 "− 2σ0 # 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 magni- tude. 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

2 dN(l) N0 1 (l µ) nl(l) = = exp − (25) dl √2π σ "− 2σ2 # where the mean, µ, and the standard deviation, σ, of l = ln(r) are deﬁned

∞ ∞ −∞ lnl(l) dl 1 µ = ∞ = lnl(l) dl (26) −∞ R −∞ nl(l) dl N0 Z R ∞ 2 1 2 σ = (l µ) nl(l) dl (27) −∞ N0 Z − 5 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

It is common for the lognormal distribution to be expressed in terms of the radius. Noting that dl 1 = (28) dr r then in terms of radius rather than log radius we have 2 dN(r) dN(l) dl dl N0 1 1 (ln r µ) n(r) = = = nl(l) = exp − dr dl dr dr √2π σ r "− 2σ2 # (29) 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 2 N0 1 1 (ln r ln rm) n(r) = exp −2 (30) √2π ln(S) r "− 2ln (S) # Be particularly careful about σ and S whose deﬁnitions are sometimes re- versed!

2.2 Properties of the Lognormal Distribution 2.2.1 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. It is possible to show the geometric mean, rg, of the radius is the same as the median in linear space, i.e. 1 r = (r r ... r ) n (31) g 1 × 2 × × n 1 ln(r1)+ln(r2)+ ... + ln(rn) ln(r ) = ln(r r ... r ) n = (32) ⇒ g 1 × 2 × × n n which has the continuous form 1 ∞ ln(rg) = lnl(l) dl = µ = ln(rm) (33) −∞ N0 Z r = r (34) ⇔ g m 6 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.

7 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

2.2.2 Median and Mode

It can be shown that the mode of the log-normal distribution, rM, is related to the median. Let

2 N0 1 1 (ln r ln rm) A n(r) = exp −2 = exp[B] (35) √2π ln(S) r "− 2ln (S) # r N 1 (ln r ln r )2 dB (ln r ln r ) where A = 0 , B = − m and = − m . √2π ln(S) − 2ln2(S) dr − ln2(S)r then dn(r) A A dB = exp[B]+ exp[B] (36) dr −r2 r dr A A (ln r ln r ) = exp[B] exp[B] − m (37) −r2 − r2 ln2(S)

Setting the left hand side to zero so r becomes rM

A A (ln rM ln rm) 0 = 2 exp[B] 2 exp[B] 2− (38) −rM − rM ln (S) (ln r ln r ) 1 = M − m (39) − ln2(S) ln r = ln r ln2(S) = ln(r ) σ2 (40) M m − m − 2.2.3 Derivatives The ﬁrst and second derivatives of Equation 25 are dn n l = l (41) dN0 N0 dn (l µ) l = − n (42) dl − σ2 l 2 dnl (l µ) 1 = nl − (43) dσ " σ3 − σ # 2 d nl 2 = 0 (44) dN0 2 2 d nl (l µ) l = nl − (45) dl2 " σ4 − σ2 # 2 2 2 2 d nl (l µ) 1 (l µ) 1 = nl − 3 − + (46) dσ2 " σ3 − σ # − σ4 σ2      8 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

If using Equation 30 the derivatives are : dn n = (47) dN0 N0 dn n (ln r ln rm) = 1+ −2 (48) dr − r " ln (S) # 2 dn n (ln r ln rm) = −2 1 (49) dS S ln(S) " ln (S) − #

The second derivatives are: d2n 2 = 0 (50) dN0 2 2 d n 2 3ln r 3ln rm 1 (ln r ln rm) = n + − 2 − + − 4 (51) dr2 "r2 r2 ln (S) r2 ln (S) # 2 4 2 2 d n (ln r ln rm) (ln r ln rm) (ln r ln rm) 2 1 = n − 6 5 − 4 − 3 + 2 + dS2 " S2 ln (S) − S2 ln (S) − S2 ln (S) S2 ln (S) S2 ln(S)# (52)

2.2.4 Moments The i-th raw moment of a lognormal distribution is given by

i2σ2 mi = N0 exp iµ + . (53) 2 !

The ﬁrst three moments are ∞ 1 2 1 2 1 2 m1 = rn(r) dr = N0 exp µ + σ = N0rm exp σ N0rm exp ln S 0 2 2 ≡ 2 Z ∞ 2 2 2 2 2 2 m2 = r n(r) dr = N0 exp 2µ +2σ = N0rm exp 2σ N0rm exp 2ln S 0 ≡ Z ∞ 3 9 2 3 9 2 3 9 2 m3 = r n(r) dr = N0 exp 3µ + σ = N0rm exp σ N0rm exp ln S Z0 2 2 ≡ 2 The mean radius, the surface area density and the volume density of a log- normal distribution are given by

∞ 1 1 1 1 2 mean = rn(r) dr = m1 = N0 exp µ + σ , (54) N0 Z0 N0 N0 2 1 2 1 2 = rm exp σ rm exp ln S , (55) 2 ≡ 2

9 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

∞ 2 2 area = 4πr n(r) dr =4πm2 =4πN0 exp 2µ +2σ , (56) 0 Z = 4πN r2 exp 2σ2 4πN r2 exp 2ln2 S , (57) 0 m ≡ 0 m and ∞ 4 3 4 4 9 2 volume = πr n(r)dr = πm3 = πN0 exp 3µ + σ , (58) Z0 3 3 3 2 4 3 9 2 4 3 9 2 = πN0rm exp σ πN0rm exp ln S . (59) 3 2 ≡ 3 2 For a lognormal distribution the eﬀective radius is

9 m exp 3µ + σ2 5 r = 3 = 2 = exp µ + σ2 , (60) e 4 m2 2 2 exp 2µ + 2 σ 5 2 5 2 = rm exp σ rm exp ln S . (61) 2 ≡ 2 2.3 Log-Normal Distributions of Area and Volume The log-normal area density distribution is

2 A0 1 1 (ln(r) µa) a(r) = exp −2 (62) √2π σa r "− 2σa # 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 N 1 1 (ln r µ)2 a(r) = 4πr2n(r)=4πr2 0 exp − (63) √2π σ r "− 2σ2 # Making the substitution r2 = exp(2ln r) and completing the square gives

N 1 1 (ln r (µ +2σ2))2 a(r) = 4π 0 exp 2µ +2σ2 exp − (64) √2π σ r "− 2σ2 # h i Equating with Equations 62 and 63 gives

2 A0 1 1 (ln(r) µa) exp −2 √2π σa r "− 2σa # N 1 1 (ln r (µ +2σ2))2 = 4π 0 exp 2µ +2σ2 exp − (65) √2π σ r "− 2σ2 # h i 10 Some Useful Formulae for Aerosol Size Distributions and Optical Properties which is true if A 4πN 0 = 0 exp 2µ +2σ2 (66) σ σ a h i and

2 2 2 (ln(r) µa) (ln r (µ +2σ )) −2 = − 2 (67) 2σa 2σ From Equation 56

2 A0 = 4πN0 exp 2µ +2σ (68) which gives σa = σ when inserted into Equation 66. 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 67 gives

2 µa = µ +2σ . (69)

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

2 V0 1 1 (ln(r) µv) v(r) = exp −2 (70) √2π σv r "− 2σv # 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 4 N 1 1 (ln r µ)2 v(r) = πr3n(r)= πr3 0 exp − (71) 3 3 √2π σ r "− 2σ2 #

Making the substitution r3 = exp(3ln r) and completing the square gives

4 N 1 1 (ln r µ)2 6σ2 ln r v(r) = π 0 exp − − (72) 3 √2π σ r "− 2σ2 # Complete the square

4 N 1 1 (ln r (µ +3σ2))2 v(r) = π 0 exp 3µ +4.5σ2 exp − (73) 3 √2π σ r "− 2σ2 # h i 11 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

Equating with Equations 70 and 73 gives 2 V0 1 1 (ln(r) µv) exp −2 √2π σv r "− 2σv # 4 N 1 1 (ln r (µ +3σ2))2 = π 0 exp 3µ +4.5σ2 exp − (74) 3 √2π σ r "− 2σ2 # h i which is true if V 4 N 0 = π 0 exp 3µ +4.5σ2 (75) σ 3 σ v h i and 2 2 2 (ln(r) µv) (ln r (µ +3σ )) −2 = − 2 (76) 2σv 2σ From Equation 58 4 V = πN exp 3µ +4.5σ2 (77) 0 3 0 which gives σv = σ when inserted into Equation 75. 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 76 gives

2 µv = µ +3σ . (78) 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.

2.4 A Diﬀerent Basis State While the shape of the lognormal distribution is controlled its median radius, rm, and its geometric standard deviation, S. An alternative is the eﬀective radius, re, and volume per particle, v = V/N0. If these parameters are adopted rm and S can be recovered as follows:

3v ln 3 4πre S = exp v  (79) u u− 3 t    3v (5/6) 1  rm = (3/2) (80) 4π re

12 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 Relation to Number Density Parameters 2 A0 =4πN0 exp(2µ +2σ ) 2 Area density µa = µ +2σ σa = σ 4 2 V0 = 3 πN0 exp(3µ +4.5σ ) 2 Volume density µv = µ +3σ σv = σ

Table 2: Values derived for a log-normal number density size distribution with S =1.5.

Median Mean Eﬀective Area Volume Radius Radius Radius Median Median Radius Radius µm µm µm µm µm 0.1 0.1 0.3 0.3 0.4 0.2 0.3 0.7 0.5 0.8 0.5 0.6 1.7 1.3 2.1 1 1.3 3.3 2.6 4.2 2 2.5 6.6 5.2 8.5 5 6.4 17 13. 21 10 13 33 26 42 20 25 67 52 85 50 64 166 131 211 100 127 332 261 423

13 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) , (81) − 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 (see Figure 2) .

3.1.1 Moments The i-th moment of the gamma distribution is given by −2−i −2−i mi = ab Γ(2+ i)= ab (1 + i)!. (82) If the constant a is used to denote the total number density then the normalised distribution can be expressed n(r) = ab2r exp( br) , (83) − 14 Some Useful Formulae for Aerosol Size Distributions and Optical Properties which has moments deﬁned by

−i mi = ab (1 + i)!. (84) The ﬁrst three moments are 2a m = ab−1 (1+1)! = 1 b 6a m = ab−2 (1+2)! = 2 b2 24a m = ab−3 (1+3)! = 3 b3 problem with normalization above

3.2 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) − 15 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

The four constants a,α,b,γ are positive and real and α is an integer. The 1/γ α mode of the distribution occurs where r = bγ . 3.2.1 Moments The moments of thisa modiﬁed gamma distribution are2

a − α+1+i α +1+ i mi = b γ Γ . (86) γ γ !

Hence the integral of the modiﬁed gamma distribution is

∞ a − α+1 α +1 n(r) dr = b γ Γ . (87) Z0 γ γ ! In order that the first parameter, a, is the total aerosol concentration it is convenient to define the normalized modified gamma distribution (see Fig- ure 3) as

rα exp( brγ) n(r) = a − α+1 − , (88) 1 γ α+1 γ b Γ γ 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

α+i+1 − i Γ γ m = ab γ . (89) i α+1 Γ γ 3.3 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 The mode of the distribution occurs where r = bγ and it falls oﬀ slowly on the large radius side and exponentially on the small radius side. The moments are deﬁned by

a − α−1−i α 1 i mi = b γ Γ − − . (91) γ γ !

2See 3 478/1 of Gradshteyn & Ryzhik (1994). · 16 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). −

The normalized inverse modiﬁed gamma distribution can be deﬁned r−α exp( br−γ) n(r) = a − α−1 − − . (92) 1 γ α 1 γ b Γ γ The moments of this distribution are given by

α−1−i i Γ γ m = ab γ . (93) i α−1 Γ γ 3.4 Regularized Power Law The regularized power law is deﬁned by Deepak (1982) as

α−1 α−2 r n(r) = ab α γ , (94) r 1+ b h i where the positive constants a,b,α,γ mainly eﬀect the number density, the mode radius, the positive gradient and the negative gradient respectively.

17 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

The mode radius is given by

α 1 1/α r = b − , (95) 1+ α(γ 1)! − and the moments by

bi Γ(1 + i/α)Γ(γ 1 i/α) m = a − − . (96) i α Γ(γ)

Hence the distribution normalised so that the total number of particles is a is

α−1 α−2 r n(r) = aαγb α γ . (97) r 1+ b h i 3.4.1 Moments The ﬁrst three moments are b Γ(1+1/α)Γ(γ 1 1/α) m = a − − 1 α Γ(γ) b2 Γ(1+2/α)Γ(γ 1 2/α) m = a − − 2 α Γ(γ) b3 Γ(1+3/α)Γ(γ 1 3/α) m = a − − 3 α Γ(γ)

above formulae need to be checked and possibly simpliﬁed The mean radius, the surface area density and the volume density of a regularized power law distribution are given by For a regularized power law distribution the eﬀective radius is

b3 Γ(1+3/α)Γ(γ−1−3/α) m a Γ(1+3/α)Γ(γ 1 3/α) r = 3 = α Γ(γ) = b , (98) e b2 Γ(1+2/α)Γ(γ−1−2/α) − − m2 a Γ(1+2/α)Γ(γ 1 2/α) α Γ(γ) − −

18 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

5 2 re,i = rm,i exp ln Si . (99) 2 5 2 1 re,i/ exp 2 ln S1 If re re,1 then µ =  0  and rm =  rm, 2  . ≤ 0    rm,3      0  rm,1 Similarly if re re,3 then µ =  0  and rm =  rm,2 . ≥ 1 r / exp 5 ln2 S    e,i 2 3      If re,1 < re < re,3 then µ1 and is estimated by linearly interpolating  between [0,1] as a function of re i.e. r r µ = e − e,1 (100) 1 r r e,3 − e,1 We now have two equations

µ1 + µ2 + µ3 =(101) 1 3 9 2 3 9 2 3 9 2 µ1rm,1 exp 2 ln S1 + µ2rm,2 exp 2 ln S2 + µ3rm,3 exp 2 ln S3 =(102)re 2 2 2 2 2 2 µ1rm,1 exp 2ln S1 + µ2rm,2 exp 2ln S2 + µ3rm,3 exp 2ln S3 and two unknowns i.e. µ2 and µ3. The second equation is simpliﬁed by substitution i.e.

Aµ1 + Bµ2 + Cµ3 = re (103) Dµ1 + Eµ2 + Fµ3 and the two equations solved to give r E B µ (A B + r (E D)) µ = e − − 1 − e − (104) 2 C B + r (E F ) − e − r F C µ (A C + r (F D)) µ = e − − 1 − e − (105) 3 B C + r (F E) − e −

19 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

5 Optical Properties

5.1 Volume Absorption, Scattering and Extinction Co- efficients The volume absorption coefficient, βabs(λ,r), the volume scattering coefficient, βsca(λ,r), and the volume extinction coefficient, βext(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 βabs(λ,r)= σabs(λ,r)N(r)= πr2Qabs(λ,r)N(r), sca sca 2 sca monodisperse only  β (λ,r)= σ (λ,r)N(r)= πr Q (λ,r)N(r(106)),  βext(λ,r)= σext(λ,r)N(r)= πr2Qext(λ,r)N(r),  where N(r) is the number of particles per unit volume at some radius, r. The absorption cross section, σext(λ,r), the scattering cross section, σext(λ,r), and the extinction cross section, σext(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 ∞ ∞ βabs(λ) = σabs(λ,r)n(r) dr = πr2Qabs(λ,r)n(r) dr, (107) 0 0 Z ∞ Z ∞ βsca(λ) = σsca(λ,r)n(r) dr = πr2Qsca(λ,r)n(r) dr, (108) 0 0 Z ∞ Z ∞ βext(λ) = σext(λ,r)n(r) dr = πr2Qext(λ,r)n(r) dr, (109) Z0 Z0 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.

∞ ∞ βabs(λ) σ¯abs(λ)= σabsn(r) dr n(r) dr = , (110) 0 0 N Z . Z 0 ∞ ∞ βsca(λ) σ¯sca(λ)= σscan(r) dr n(r) dr = , (111) 0 0 N Z . Z 0 ∞ ∞ βext(λ) σ¯ext(λ)= σextn(r) dr n(r) dr = , (112) 0 0 N Z . Z 0 whereσ ¯abs(λ),σ ¯sca(λ) andσ ¯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 2 sca P (λ,θ) = sca πr Q (λ,r)P (λ,r,θ)n(r) dr. (113) β Z0 5.4 Single Scatter Albedo The single scatter albedo is the ratio of the energy scattered from a particle to that intercepted by the particle. Hence βsca(λ) ω(λ) = . (114) βext(λ)

5.5 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 2 sca g = sca πr Q (λ,r)g(λ,r)n(r) dr (115) β Z0 5.6 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→ → → ∞

ru n f(r)n(r) dr = wif(ri) (116) rl Z Xi=1 where wi are the weights at discrete values of radius, ri. For a log normal size distribution the integrals are

2 N π ru 1 ln r ln r βext(λ) = 0 rQext(λ,r)exp − m dr (117)   σ r 2 Zrl −2 σ !   21 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

N π n 1 ln r ln r 2 = 0 w r Qext(λ,r )exp i − m i i i   σ r 2 i=1 −2 σ ! X n   ′ ext = wiQ (λ,ri) i=1 Xn abs ′ abs β (λ) = wiQ (λ,ri) i=1 Xn sca ′ sca β (λ) = wiQ (λ,ri) Xi=1 2 N π 1 ru 1 ln r ln r P (λ,θ) = 0 rQsca(λ,r)P (λ,r,θ)exp − m dr sca   σ r 2 β Zrl −2 σ !   N π 1 n 1 ln r ln r 2 = 0 w r Qsca(λ,r )P (λ,r ,θ)exp i − m sca i i i i   σ r 2 β i=1 −2 σ ! X 1 n   = w′Qsca(λ,r )P (λ,r ,θ) βsca i i i Xi=1 N π 1 ∞ 1 ln r ln r 2 g = 0 rQsca(λ,r)g(λ,r)exp − m dr sca   σ r 2 β Z0 −2 σ !   N π 1 n 1 ln r ln r 2 = 0 w r Qsca(λ,r )g(λ,r )exp i − m sca i i i i   σ r 2 β i=1 −2 σ ! X 1 n   = w′Qsca(λ,r )g(λ,r ) βsca i i i Xi=1 where

2 ′ N π 1 ln r ln r w = 0 r exp i − m w (118) i i   i σ r 2 −2 σ !   5.7 Formulae for Practical Use As part of the retrieval process it is helpful to have analytic expression for the partial derivatives of βext (Equation 117) with respect to the size distribution parameters (N0, rm, σ).

∂βext 1 π ∞ (ln r ln r )2 = rQext(λ,r)exp − m dr, (119) ∂N σ 2 − 2σ2 0 r Z0 " # ∂βext N π ∞ (ln r ln r )2 = 0 (ln r ln r )rQext(λ,r)exp − m dr, (120) ∂r r σ3 2 − m − 2σ2 m m r Z0 " #

22 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

∂βext N π ∞ (ln r ln r )2 = 0 rQext(λ,r)exp − m dr ∂σ − σ2 2 − 2σ2 r Z0 " # N π ∞ (ln r ln r )2 (ln r ln r )2 + 0 − m rQext(λ,r)exp − m dr, σ 2 σ3 − 2σ2 r Z0 " # N π ∞ (ln r ln r )2 (ln r ln r )2 = 0 − m 1 rQext(λ,r)exp − m dr. σ2 2 σ2 − − 2σ2 r Z0 " # " # (121)

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

exp T π ∞ (l l )2 βext(λ)= e2lQext(l,λ)exp − m dl. (122) −∞ exp Gr 2 Z "−2 exp(2G)# The partial derivatives of the transformed parameters (Equation 122) are

∂βext exp T π ∞ (l l )2 = e2lQext(l,λ)exp − m dl, (123) ∂T exp G 2 −∞ −2 exp(2G) r Z " # ext ∞ 2 ∂β exp T π 2l ext (l lm) = (l lm)e Q (l,λ)exp − dl, (124) ∂l exp(3G) 2 −∞ − −2 exp(2G) m r Z " # ∂βext exp T π ∞ (l l )2 = e2lQext(l,λ)exp − m dl ∂G −exp(G) 2 −∞ −2 exp(2G) r Z " # exp T π ∞ (l l )2 (l l )2 + − m e2lQext(l,λ)exp − m dl, exp G 2 −∞ exp(2G) −2 exp(2G) r Z " # exp T π ∞ (l l )2 (l l )2 = − m 1 e2lQext(l,λ)exp − m dl. exp(G) 2 −∞ exp(2G) − −2 exp(2G) r Z " # " # (125)

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 ∞ 4 l = ρ πr3n(r) dr z (126) Z0 3 × 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 gm−2. Note that

τ = βext z (127) × so ∞ 4 πr3n(r) dr l = ρτ 0 3 (128) R βext ∞ 3 4 0 r n(r) dr = πρτ ∞ 2 ext . (129) 3 0 πrR Q (λ,r)n(r) dr R For drops large with respect to wavelength we assume Qext(λ,r) = 2. Hence

∞ 3 4 0 r n(r) dr 2 l = ρτ ∞ 2 = ρτre (130) 6 R0 r n(r) dr 3

R−3 So for a water cloud (ρ =1gcm ) of τ = 10, re = 15 µm we get 2 l = 1 10 15 g cm−3µm=100 gm−2 (131) 3 × × ×

◦ −3 While for an ice cloud (-45 C, ρ = 0.920 g cm ) of τ = 1, re = 25 µm we get 2 l = 0.92 1 25 g cm−3µm=15gm−2 (132) 3 × × × .

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 (133) × × × × 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 (134) × × × 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 βext by τ = βext Z (135) × so that ρ v N τ m = × × × (136) βext For a given size distribution n(r) the average volume of each particle is ∞ 4 πr3n(r) dr v = 0 3 (137) R N so that the mass per unit area is given by ∞ ρτ 4 3 m = ext πr n(r) dr (138) β Z0 3 The important thing to note here is that N disappears explicitly from the equation. For optical measurement it is more common to know the eﬀective radius rather than the full size distribution. In terms of re the mass per unit area is given by ∞ ∞ 2 4πρτ 3 0 r n(r) dr 4ρτ m = r n(r) dr ∞ = re (139) ext 2 ˜ext 3β Z0 × R0 r n(r) dr 3Q R 25 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

Substance Density (g cm−3) Reference

Ice 0.92 Volcanic ash 2.42 0.79 Bayhurst,Wohletz&Mason(1994) Water 1 ±

Table 3: Density of some materials that form aerosols. which uses an area weighted extinction eﬃciency

ext ∞ ext ∞ 2 ext ˜ext β 0 σ n(r) dr 0 πr Q n(r) dr Q = ∞ 2 = ∞ 2 = ∞ 2 (140) π 0 r n(r) dr R0 πr n(r) dr R 0 πr n(r) dr If the particlesR with the size distributionR are mostlyR much larger than the wavelength then Qext 2 and Q˜ext 2. With this assumption Equation 130 becomes identical to Equation→ 139 whose≈ derivation made the same approx- imation. If the aerosol size distribution is log-normal with number density N0, mode radius rm and spread σ then the integral in Equation 138 can be completed analytically i.e.

ρτ 4 3 9 2 m = ext πN0rm exp σ (141) β 3 2 −3 −3 ext −1 Typically ρ is in gcm , N0 is in cm , rm is in µm, and β is in km so that the units of m are

g 1 3 cm3 cm3 µm g 1 −18 3 3 −3 −2 1 = −6 3 −6 3 10 m 10 m=10 g m (142) km 10 m 10 m 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

5 2 re = rm exp σ , (143) 2 to get 4ρτπN 15 9 4ρτπN m = 0 r3 exp σ2 exp σ2 = 0 r3 exp 3σ2 . (144) 3βext e − 2 2 3βext e − Equating this expression to Equation 130 gives βext Q˜ext = . (145) πN r2 exp( 3σ2) 0 e − 26 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 parame- terised by Ni,ri,σi and density ρi we have

n τ ρ N v τ i=1 ρiNivi m = × ×ext × = n ext (146) β Pi=1 Niσ¯i

ext P th whereσ ¯i is the extinction cross section per particle for the i mode. Re- member the volume per particle for the ithmode is

4 3 9 2 vi = πri exp σi (147) 3 2 Hence

n 4 3 9 2 i=1 ρiNi 3 πri exp 2 σi m = τ n ext (148) P i=1 Niσ¯i n 3 9 2 4 i=1PρiNiri exp 2 σi = πτ n ext (149) 3 P i=1 Niσ¯i P

27 Some Useful Formulae for Aerosol Size Distributions and Optical Properties

References

Bayhurst, G. K., Wohletz, K. H. & Mason, A. S. (1994). A method for char- acterizing 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 polydisper- sions, Elsevier, New York.

Gradshteyn, I. S. & Ryzhik, I. M. (1994). Table of integrals, series, and products, Academic Press, London.

Twomey, S. (1977). Atmospheric aerosols, Elsevier, Amsterdam, The Nether- lands.

28 Index mean, 2 median, 2 mode, 2 moment, 2 size distribution, 1 standard deviation, 2 variance, 2