1. No category

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
[email protected]
March 29, 2005
Sizes of Atmospheric Constituents
Mode
Diameter (mm)
Gas molecules
0.0005
Aerosol particles
Small
< 0.2
Medium
0.2-2
Large
1-100
Hydrometeor particles
Fog drops
10-20
Cloud drops
10-200
Drizzle
200-1000
Raindrops
1000-8000
Number (#/cm3)
2.45x1019
103-106
1-104
<1 - 10
1-1000
1-1000
0.01-1
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
1 N B 1
N B 
Vrat  


 1 
Example 13.3.
--->
d1
= 0.01 mm
dNB
= 1000 mm
NB
= 30 size bins
Vrat
= 3.29
(13.3)
3 N B 1
dN B 
 

d
 1 
Volume Ratio Size Structure
Number of size bins


ln  d N B d1

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
 
3 
(13.4)
Volume Ratio Size Structure
Average volume in a size bin

1
i  i,hi  i,lo
2
(13.5)

Relationship between high- and low-edge volume
(13.6)
i,hi  Vrat i,lo
Substitute (13.6) into (13.5) --> low edge volume
2i
i,lo 
1  Vrat
(13.7)
Volume Ratio Size Structure
Volume width of a size bin
(13.8)
2i Vrat  1
2i1
2i
i  i,hi  i,lo 


1 Vrat 1 Vrat
1  Vrat
Diameter width of a size bin
(13.9)


13
13
6
Vrat  1


13
13
1
3
di  di,hi  di,lo    i,hi  i,lo  di 2
13
 
1

V
 rat 
Particle Concentrations
Number concentration in a size bin
vi
ni 
i
(13.10)
Number concentration in a size distribution
(13.11)
NB
 ni
ND 
Volume concentration in a size bin
vi 
i1
(13.12)
NV
 v q,i
q1
Surface area concentration in a size bin
ai  ni 4ri2  ni di2
(13.13)
Particle Concentrations
Mass concentration in a size bin
mi 
(13.14)
NV
NV
NV
q1
q 1
q 1
 m q,i  cm   q vq,i  cm  p,i  v q,i  c m  p,i vi
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
= 1.0 g cm-3 for water
= 1.83 g cm-3 for sulfate
v q,i
v q,i
mi
vi
i
ni
ai
= 3 x 10-12 cm3 cm-3 for water
= 1.09 x 10-12 cm3 cm-3 for sulfate
q
q
--->
--->
--->
--->
--->
--->
--->
= 5.0 mg m-3
= 4.09 x 10-12 cm3 cm-3
= 6.54 x 10-14 cm3
= 62.5 partic. cm-3
= 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
cm
( m33cm
-3-3)
m
dvdv
(mm
p
) / d log
Dp
/ d log1010D
Describes particle concentration versus size
10
2
10 1
10 0
10-1
10-2
10-3
0.001
D1
D2
0.01
0.1
Particle diameter (D p , mm)
1
Fig. 13.2a
Lognormal Distribution
p
) / d log
-3
3 cm
3
-3
dv
(
dv (mm
D
m m cm ) / d log10
D
10 p
The lognormal curve drawn on a linear scale
10
2
10 1
10 0
10-1
10-2
10-3
0
0.05
0.1
0.15
Particle diameter (D p , mm)
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
1
ln DM 
ML
7
 m j ln d j 
j1
Total mass concentration of particles (mg m-3)
ML 
7
 mj
j 1
(13.16)
Lognormal Parameters From Data
Natural log of geometric mean volume diameter
1 7
ln DV 
v j ln d j

VL

j 1

Total volume concentration of particles (cm3 cm-3)
VL 
7
vj
j 1
vj 
mj
cm j
(13.17)
Lognormal Parameters From Data
Natural log of geometric mean area diameter
1 7
ln DA 
a j ln d j

AL

(13.18)

j1
Total area concentration of particles (cm2 cm-3)
AL 
7
a j
j1
aj 
3m j
c m j rj
Lognormal Parameters From Data
Natural log of geometric mean number diameter
1 7
ln DN 
n j ln d j

NL


j1
Total number concentration of particles (partic. cm-3)
NL 
7
n j
j1
nj 
mj
cm j  j
(13.19)
Lognormal Parameters From Data
Natural log of geometric standard deviation
ln g 

1 7 
2 d j 
m j ln




ML
DM 
j 1 
1 7 
2 d j 
a j ln



AL
D
A 
j1

(13.20)
1 7 
2 d j 
v j ln



VL
D
V 
j 1 
1 7 
2 d j 
n j ln



NL
D
N 
j1
Redistribute With Lognormal Parameter
Redistribute mass concentration in model size bin
(13.21)


2 d D


ln
M Ldi
i
M

mi 
exp 
2
di 2 ln  g

 2 ln  g 

Redistribute volume concentration
(13.22)
2 d D 

ln
VLdi
i V

vi 
exp 
2
di 2 ln  g

 2 ln  g 



Redistribute area concentration
(13.23)

2 d D

ln
A L di
i
A

ai 
exp 
2
di 2 ln  g

 2 ln  g



Redistribute With Lognormal Parameter
Redistribute number concentration
(13.24)

2 d D

ln
N L di
i
N
ni 
exp 
2
di 2  ln  g

2
ln
g

Exact volume concentration in a mode


0
0

VL  vd dd 
6





(13.25)
 3
9 2 


nd d dd  D N exp
ln  g N L
2

6
3
Lognormal Modes
10 5
D
10
N
3
D
p
(x=n, a, v)
dx/d log
Dp (x=n,a,v)
dx / d log1010D
Number (partic. cm-3), area (cm2 cm-3), and volume (cm3 cm-3)
concentrations distributed lognormally
10
1
10-1
10
-3
0.001
n
A
DV
a
v
0.01
0.1
Particle diameter (D p , mm)
1
Fig. 13.3
Lognormal Param. for Cont. Particles
Nucleation
Parameter
Mode
g
1.7
NL (particles cm-3)
7.7x104
DN (mm)
0.013
AL (mm2 cm-3)
74
DA (mm)
0.023
VL (mm3 cm-3)
0.33
DV (mm)
0.031
Accumulation
Mode
2.03
1.3x104
0.069
535
0.19
22
0.31
Coarse
Mode
2.15
4.2
0.97
41
3.1
29
5.7
Table 13.2
Quadramodal Size Distribution
Size distribution at Claremont, California, on the morning of
August 27, 1987
6
10
105
2
300
-3
da (mm cm )/d log D
250
10 p
4
10
103
2
10
-3
dn (No. cm )
/dlog D
3
200
-3
dv (mm cm )
/d log D
10 p
150
10 p
101
0
10
100
10 -1
0
50
0.01
0.1
1
Particle diameter (D p , mm)
10
Fig. 13.4
Marshall-Palmer Distribution
Raindrop number concentration between di and di+di (13.30)
ni  di n 0e r di
din0
n0
r
R
= value of ni at di = 0
= 8.0 x 10-6 partic. cm-3 mm-1
= x Rmm-1
= rainfall rate (1-25 mm hr-1)
Total number concentration and liquid water content
nT  n0  r
w L  106 w n0 4r
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 
r
 c,g  

Modified Gamma Distribution
Parameters
Cloud Type
Ag
g
g
(mm)
Liquid
Water
Conten
t
(g m-3)
Numb er
Conc.
(partic.
cm-3)
rcg
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
4.97x10-8
2.0
0.5
70.0
1.17
0.01
Light rain
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
ri
ni
= 15 mm
= 2 mm
= 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.
Advantages:
• Covers wide range in diameter space with few bins
• Nucleation, emissions, transport treated realistically
Disadvantages:
• 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.
Advantages:
• Core volume preserved during growth
• No numerical diffusion during growth
Disadvantages:
• 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.
Advantages and Disadvantages:
• 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  
i  k
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 
i  n j  j  nk k
Solution to this set of two equations and two unknowns (13.34)
k  
i
n j  ni
k  j

ij
nk  ni
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.
Advantages:
• Covers wide range in diameter space with few bins
• Little numerical diffusion during growth
• Nucleation, emission, transport treated realistically
Disadvantages:
• When growth occurs, information about the original
composition of the growing particle is lost
Moving-Center Structure
dv m
( m33 cm-3-3
) /d log
dv (mm cm ) / d log1010DpDp
Comparison of moving-center, full-moving, and quasistationary size
structures during water growth onto aerosol particles to form cloud
drops.
10
7
Moving-center
Full-moving
105
103
10
Quasistationary
Initial
1
10 -1
0.1
1
10
Particle diameter (D , mm)
p
100
Fig. 13.8