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Presentation Slides
for
Chapter 13
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
jacobson@stanford.edu
March 29, 2005
Sizes of Atmospheric Constituents

Mode              Diameter (mm)   Number (#/cm3)
Gas molecules         0.0005      2.45x1019
Aerosol particles
Small              < 0.2       103-106
Medium             0.2-2       1-104
Large              1-100       <1 - 10
Hydrometeor particles
Fog drops          10-20       1-1000
Cloud drops        10-200      1-1000
Drizzle            200-1000    0.01-1
Raindrops          1000-8000   0.001-0.01

Table 13.1
Particles and Size Distributions
Particle
Agglomerations of molecules in the liquid and / or solid
phases, suspended in air. Includes aerosol particles, fog drops,
cloud drops, and raindrops

Example 13.1. - Idealized particle size distribution
10,000 particles of radius between 0.05 and 0.5 mm
100 particles of radius between 0.5 and 5.0 mm
10 particles of radius between 5.0 and 50 mm

Example 13.2. Number of size bins needs to be limited
105 grid cells
100 size bins
100 components per size bin
--> 109 words = 8 gigabytes to store concentration
Volume Ratio Size Structure
Volume of particles in one size bin                    (13.1)

i  Vrati 1

(13.2)
i1
i  1Vrat

Volume-diameter relationship for spherical particles

i  di3 6
Volume Ratio Size Structure
Variation in particle sizes with the volume ratio size structure

Fig. 13.1
Volume Ratio Size Structure
Volume ratio of adjacent size bins                               (13.3)
1 N B 1               3 N B 1
 N B                 dN B   
Vrat                              
 1                    d1    

Example 13.3.

d1         = 0.01 mm

dNB        = 1000 mm
NB         = 30 size bins
--->   Vrat       = 3.29
Volume Ratio Size Structure
Number of size bins                                (13.4)


ln  d N d1
 B
 
3 

NB  1 
ln Vrat
Example 13.4.
d1      = 0.01 mm
dNB    = 1000 mm
Vrat   =4
--->   NB     = 26 size bins
Vrat   =2
--->   NB     = 51 size bins
Volume Ratio Size Structure
Average volume in a size bin                        (13.5)
1
2

i  i,hi  i,lo    
Relationship between high- and low-edge volume      (13.6)

i,hi  Vrati,lo

Substitute (13.6) into (13.5) --> low edge volume   (13.7)
2i
i,lo 
1  Vrat
Volume Ratio Size Structure
Volume width of a size bin                             (13.8)

2i1   2i      2i Vrat  1
i  i,hi  i,lo                 
1 Vrat 1 Vrat     1  Vrat

Diameter width of a size bin                           (13.9)

           
13
6 1 3 1 3
              13         1 3 Vrat  1
di  di,hi  di,lo         i,hi  i,lo  di 2
                                      13
1  Vrat 
Particle Concentrations
Number concentration in a size bin                (13.10)
vi
ni 
i
Number concentration in a size distribution       (13.11)
NB
ND       ni
i1
Volume concentration in a size bin                (13.12)
NV
vi     v q,i
q1
Surface area concentration in a size bin          (13.13)
2         2
ai  ni 4ri  ni di
Particle Concentrations
Mass concentration in a size bin                      (13.14)
NV            NV                    NV
mi    m q,i  cm qvq,i  cm p,i vq,i  cmp,ivi
q1           q 1                  q 1

Volume-averaged mass density (g cm-3) of particle of size i
(13.15)
NV
 vi,q q 
q1
 p,i  N
V
 v i,q
q 1
Particle Concentrations
Example 13.5
mq,i = 3.0 mg m-3 for water
mq,i = 2.0 mg m-3 for sulfate
di      = 0.5 mm
q      = 1.0 g cm-3 for water
q      = 1.83 g cm-3 for sulfate
--->   v q,i   = 3 x 10-12 cm3 cm-3 for water
--->   v q,i   = 1.09 x 10-12 cm3 cm-3 for sulfate
--->   mi      = 5.0 mg m-3
--->   vi      = 4.09 x 10-12 cm3 cm-3
--->   i      = 6.54 x 10-14 cm3
--->    ni     = 62.5 partic. cm-3
--->    ai     = 4.8 x 10-7 cm2 cm-3
Lognormal Distribution
Bell-curve distribution on a log scale

Geometric mean diameter
50% of area under a lognormal curve lies below it

Geometric standard deviation
68% of area under a lognormal curve lies between +/-1 one
geometric standard deviation around the mean diameter
Lognormal Distribution
Describes particle concentration versus size
2
10
/ d log10 Dpp
) / d log
10

101

100
-3
dv (mm33cm-3)
dv m m cm

-1
10
-2
10
(

D    D
-3             1    2
10
0.001        0.01        0.1      1
Particle diameter (D, mm)
p
Fig. 13.2a
Lognormal Distribution
The lognormal curve drawn on a linear scale

2
/ d log10 D p
p
10
) / d log
10

101

100
dv (mm3 cm-3)
dv ( m3 cm-3

-1
10
-2
10
m

10-3
0       0.05        0.1       0.15
Particle diameter (D, mm)
p
Fig. 13.2b
Lognormal Parameters From Data
Low-pressure impactor -- 7 size cuts
0.05   - 0.075 mm                0.5 - 1.0 mm
0.075 - 0.12 mm                  1.0 - 2.0 mm
0.12   - 0.26 mm                 2.0- 4.0 mm
0.26   - 0.5 mm
Lognormal Parameters From Data
Natural log of geometric mean mass diameter        (13.16)

7
 m j ln d j 
1
ln DM 
ML
j1

Total mass concentration of particles (mg m-3)
7
ML      mj
j 1
Lognormal Parameters From Data
Natural log of geometric mean volume diameter             (13.17)
7
 v j ln d j 
1
ln DV 
VL
j 1
Total volume concentration of particles (cm3 cm-3)
7
mj
VL    vj                        vj 
c m j
j 1
Lognormal Parameters From Data

Natural log of geometric mean area diameter                (13.18)
7
 a j ln d j 
1
ln DA 
AL
j1
Total area concentration of particles (cm2 cm-3)
7                               3m j
AL    a j                     aj 
c m j rj
j1
Lognormal Parameters From Data
Natural log of geometric mean number diameter              (13.19)

1 7
ln DN 
NL

 n j ln d j    
j1

Total number concentration of particles (partic. cm-3)
7
mj
NL    n j                      nj 
cm j  j
j1
Lognormal Parameters From Data
Natural log of geometric standard deviation           (13.20)

1 7       2 d j          1 7       2 d j 
ln g 
ML
 m j ln D           VL
 v j ln D 
j 1       M             j 1       V 

1 7       2 d j         1 7      2 d j 

AL
 a j ln D       
NL
 n jln D 
j1         A           j1       N 
Redistribute With Lognormal Parameter
Redistribute mass concentration in model size bin     (13.21)

M Ldi          ln2 di DM 
mi                exp         2

di 2 ln g       2 ln  g 
                
Redistribute volume concentration                      (13.22)
VLdi          ln 2 di DV 
vi               exp        2

di 2 ln g       2 ln  g 
              
Redistribute area concentration                       (13.23)

A L di       ln2 di DA 
ai              exp
       2

di 2 ln g     2 ln  g 
             
Redistribute With Lognormal Parameter
Redistribute number concentration                   (13.24)

N L di      ln2  i D N 
d
ni              exp
       2

di 2 ln g     2 ln  g 
              

Exact volume concentration in a mode                (13.25)

            
               3      9 2 
VL   vd dd       ndd dd  6 D N exp2 ln  gN L
3
6                               
0            0
Lognormal Modes
Number (partic. cm-3), area (cm2 cm-3), and volume (cm3 cm-3)
concentrations distributed lognormally
(x=n, a, v)

5
dx / d log10 Dp (x=n,a,v)

10
D
N
3
10
DA
p

n
D
dx/d log

1
10
10

D
-1                          V
10           a
v
-3
10
0.001       0.01            0.1     1
Particle diameter (D, mm)
p            Fig. 13.3
Lognormal Param. for Cont. Particles

Nucleation   Accumulation   Coarse
Parameter             Mode          Mode        Mode
g                    1.7           2.03        2.15
NL (particles cm-3)   7.7x104       1.3x104     4.2
DN (mm)               0.013         0.069       0.97
AL (mm2 cm-3)         74            535         41
DA (mm)               0.023         0.19        3.1
VL (mm3 cm-3)         0.33          22          29
DV (mm)               0.031         0.31        5.7

Table 13.2
Size distribution at Claremont, California, on the morning of
August 27, 1987
6
10                                                              300
2   -3
105                     da (mm cm )/d log D
10 p                    250
4
10
3      -3        200
10
3                  -3
dn (No. cm )                dv (mm cm )
/dlog D                     /d log D               150
102              10 p                              10 p

101                                                             100
0                                                             50
10
10-1                                                            0
0.01             0.1             1                     10
Particle diameter (D, mm)                     Fig. 13.4
p
Marshall-Palmer Distribution
Raindrop number concentration between di and di+di (13.30)

ni  di n 0 e  r di

din0   = value of ni at di = 0
n0      = 8.0 x 10-6 partic. cm-3 mm-1
r      = x Rmm-1
R       = rainfall rate (1-25 mm hr-1)

Total number concentration and liquid water content

6           4
nT  n0 r                 w L  10          w n0 r
Marshall-Palmer Distribution
Example 13.6.

R        = 5 mm hr-1
di       = 1 mm
di+di = 2 mm
--->   ni       = 0.00043 partic. cm-3
--->   nT       = 0.0027 partic. cm-3
--->   wL       = 0.34 g m-3
Modified Gamma Distribution
Number concentration (partic. cm-3) of drops in size bin i (13.30)

          g 

g    g  ri  
ni  ri Ag ri exp
  g rc,g  
           
Modified Gamma Distribution
Parameters
Liquid   Number
Cloud Type          Ag       g    g      rcg     Water     Conc.
(mm)    Conten    (partic.
t      cm-3)
(g m-3)
Stratocumulus base    0.2823     5.0   1.19   5.33     0.141      100
Stratocumulus top     0.19779    2.0   2.46   10.19    0.796      100
Stratus base          0.97923    5.0   1.05   4.70     0.114      100
Stratus top           0.38180    3.0   1.3    6.75     0.379      100
Nimbostratus base     0.08061    5.0   1.24   6.41     0.235      100
Nimbostratus top      1.0969     1.0   2.41   9.67     1.034      100
Cumulus congestus     0.5481     4.0   1.0     6.0     0.297      100
Light rain           4.97x10-8   2.0   0.5    70.0     1.17       0.01
Table 13.3
Modified Gamma Distribution
Example 13.7.
Find number concentration of droplets between 14 and 16 mm
in radius at base of a stratus cloud

--->   ri     = 15 mm
--->   ri    = 2 mm
--->   ni     = 0.1506 partic. cm-3
Full-Stationary Size Structure
Average single-particle volume in size bin (i) stays constant.
When growth occurs, number concentration in bin (ni) changes.

• Covers wide range in diameter space with few bins
• Nucleation, emissions, transport treated realistically

• When growth occurs, information about the original
composition of the growing particle is lost.
• Growth leads to numerical diffusion
Full-Stationary Size Structure
Demonstration of a problem with the full-stationary size bin structure

Fig. 13.5
Full-Moving Structure
Number concentration (ni) of particles in a size bin does not change
during growth; instead, single-particle volume (i) changes.

• Core volume preserved during growth
• No numerical diffusion during growth

• Nucleation, emissions, transport treated unrealistically
• Reordering of size bins required for coagulation
Full-Moving Structure
Preservation of aerosol material upon growth and evaporation in a
moving structure

Fig. 13.6
Full-Moving Structure
Particle size bin reordering for coagulation

Fig. 13.7
Quasistationary Structure
Single-particle volumes change during growth like with full-moving
structure but are fit back onto a full-stationary grid each time step.

• Similar to those of full stationary structure
• Very numerically diffusive
Quasistationary Structure
After growth, particles in bin i have volume i’, which lies
between volumes of bins j and k

 j     k
i
Partition volume of i between bins j and k while conserving
particle number concentration                         (13.32)
ni  n j  nk
and particle volume concentration                        (13.33)

ni   n j  j  nk k
i
Solution to this set of two equations and two unknowns (13.34)
k  
                               
   j
i
n j  ni       i                     nk  ni
k  j                              k   j
Moving-Center Structure
Single-particle volume (i) varies between i,hi and i,lo during
growth, but i,hi, i,lo, and di remain fixed.

• Covers wide range in diameter space with few bins
• Little numerical diffusion during growth
• Nucleation, emission, transport treated realistically

• When growth occurs, information about the original
composition of the growing particle is lost
Moving-Center Structure
Comparison of moving-center, full-moving, and quasistationary size
structures during water growth onto aerosol particles to form cloud
drops.
dv (mm cm ) / d log10 p p

Moving-center
10 D D

7
10
Full-moving
) /d log

105
Quasi-
stationary
103
-3
-3
dv m m3 cm

Initial
( 3

101

10-1
0.1       1              10       100
Particle diameter (D, mm)
p             Fig. 13.8

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