Presentation Slides for
Chapter 13of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
March 29, 2005
Sizes of Atmospheric Constituents
Table 13.1
Mode Diameter (m) Number (#/cm3)Gas molecules 0.0005 2.45x1019
Aerosol particlesSmall < 0.2 103-106
Medium 0.2-2 1-104
Large 1-100 <1 - 10Hydrometeor particles
Fog drops 10-20 1-1000Cloud drops 10-200 1-1000Drizzle 200-1000 0.01-1Raindrops 1000-8000 0.001-0.01
Particles and Size DistributionsParticle
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 distribution10,000 particles of radius between 0.05 and 0.5 m100 particles of radius between 0.5 and 5.0 m10 particles of radius between 5.0 and 50 m
Example 13.2. Number of size bins needs to be limited105 grid cells100 size bins 100 components per size bin --> 109 words = 8 gigabytes to store concentration
Volume Ratio Size StructureVolume of particles in one size bin (13.1)
(13.2)
Volume-diameter relationship for spherical particles
υi =Vratυi −1
υi =υ1Vrati−1
υi =πdi3 6
Volume Ratio Size Structure
Fig. 13.1
Variation in particle sizes with the volume ratio size structure
υ1 Vrat x υ 1 Vrat x υ 2 Vrat x υ i-1
i=1 i=2 i=3 i
Volume Ratio Size StructureVolume ratio of adjacent size bins (13.3)
Example 13.3. d1 = 0.01 m
= 1000 m
NB = 30 size bins
---> Vrat = 3.29
Vrat =υNBυ1
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 1 NB−1( )
=dNBd1
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 3 NB−1( )
dNB
Volume Ratio Size StructureNumber of size bins (13.4)
Example 13.4. d1 = 0.01 m
= 1000 m
Vrat = 4
---> NB = 26 size bins Vrat = 2
---> NB = 51 size bins
NB =1+ln dNB d1( )
3⎡ ⎣ ⎢
⎤ ⎦ ⎥
lnVrat
dNB
Volume Ratio Size StructureAverage volume in a size bin (13.5)
Relationship between high- and low-edge volume (13.6)
Substitute (13.6) into (13.5) --> low edge volume (13.7)
υi =12 υi,hi +υi,lo( )
υi,hi =Vratυi,lo
υi,lo = 2υi1+Vrat
Volume Ratio Size StructureVolume width of a size bin (13.8)
Diameter width of a size bin (13.9)
Δυi =υi,hi −υi,lo = 2υi+11+Vrat
− 2υi1+Vrat
=2υi Vrat−1( )1+Vrat
Δdi =di,hi −di,lo = 6π
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 13
υi,hi13 −υi,lo
13( )=di213 Vrat13−1
1+Vrat( )13
Particle ConcentrationsNumber concentration in a size bin (13.10)
Volume concentration in a size bin (13.12)
Number concentration in a size distribution (13.11)
Surface area concentration in a size bin (13.13)
ni = viυi
ND = nii=1
NB∑
vi = vq,iq=1
NV∑
ai =ni 4πri2 =niπdi2
Particle ConcentrationsMass concentration in a size bin (13.14)
Volume-averaged mass density (g cm-3) of particle of size i (13.15)
mi = mq,iq=1
NV∑ =cm ρqvq,iq=1
NV∑ =cmρp,i vq,iq=1
NV∑ =cmρp,ivi
ρp,i =vi,qρq( )
q=1
NV∑
vi,qq=1
NV∑
Particle ConcentrationsExample 13.5
= 3.0 g m-3 for water
---> = 5.0 g 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
= 2.0 g m-3 for sulfatedi = 0.5 m = 1.0 g cm-3 for water = 1.83 g cm-3 for sulfate
---> = 3 x 10-12 cm3 cm-3 for water---> = 1.09 x 10-12 cm3 cm-3 for sulfate
mq,imq,i
ρqρqvq,ivq,imiviυiniai
Lognormal DistributionBell-curve distribution on a log scale
Geometric mean diameter50% of area under a lognormal curve lies below it
Geometric standard deviation68% of area under a lognormal curve lies between +/-1 one geometric standard deviation around the mean diameter
Lognormal Distribution
Fig. 13.2a
10
-3
10
-2
10
-1
10
0
10
1
10
2
0.001 0.01 0.1 1
dv (
3
c
-3
) / d log
10
D
p
Particle diaeter (D
p
, )
D
2
D
1dv (
m3 c
m-3) /
d lo
g 10 D
p
Describes particle concentration versus size
Lognormal Distribution
Fig. 13.2b
The lognormal curve drawn on a linear scale
10
-3
10
-2
10
-1
10
0
10
1
10
2
0 0.05 0.1 0.15
dv (
3
c
-3
) / d log
10
D
p
Particle diaeter (D
p
, )
dv (
m3 c
m-3) /
d lo
g 10 D
p
Lognormal Parameters From DataLow-pressure impactor -- 7 size cuts
0.05 - 0.075 m 0.5 - 1.0 m
0.075 - 0.12 m 1.0 - 2.0 m
0.12 - 0.26 m 2.0- 4.0 m0.26 - 0.5 m
Lognormal Parameters From DataNatural log of geometric mean mass diameter (13.16)
lnD M = 1ML
mj lndj( )j=1
7∑
Total mass concentration of particles (g m-3)
ML = mjj =1
7∑
Lognormal Parameters From Data
Natural log of geometric mean volume diameter (13.17)
Total volume concentration of particles (cm3 cm-3)
lnD V = 1VL
vj lndj( )j =1
7∑
VL = vjj =1
7∑ vj =
mjcmρj
Lognormal Parameters From Data
Natural log of geometric mean area diameter (13.18)
Total area concentration of particles (cm2 cm-3)
lnD A = 1AL
aj lndj( )j=1
7∑
A L = a jj=1
7∑ a j =
3mjcmρj rj
Lognormal Parameters From DataNatural log of geometric mean number diameter (13.19)
lnD N = 1NL
nj lndj( )j=1
7∑
Total number concentration of particles (partic. cm-3)
N L = njj=1
7∑ nj =
mjcmρj υ j
Lognormal Parameters From Data
Natural log of geometric standard deviation (13.20)
lnσg = 1ML
mjln2 djD M
⎛ ⎝ ⎜
⎞ ⎠ ⎟
j =1
7∑ = 1
VLvj ln2 dj
D V
⎛ ⎝ ⎜
⎞ ⎠ ⎟
j =1
7∑
= 1A L
aj ln2 djD A
⎛ ⎝ ⎜
⎞ ⎠ ⎟
j=1
7∑ = 1
N Lnjln2 dj
D N
⎛ ⎝ ⎜
⎞ ⎠ ⎟
j=1
7∑
Redistribute With Lognormal ParameterRedistribute mass concentration in model size bin (13.21)
Redistribute volume concentration (13.22)
Redistribute area concentration (13.23)
mi = MLΔdidi 2π lnσg
exp−ln2 di D M( )
2ln2σg
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
vi = VLΔdidi 2π lnσg
exp−ln2 di D V( )
2ln2σg
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ai = ALΔdidi 2π lnσg
exp−ln2 di D A( )
2ln2σg
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Redistribute With Lognormal ParameterRedistribute number concentration (13.24)
Exact volume concentration in a mode (13.25)
ni = NLΔdidi 2π lnσg
exp−ln2 di D N( )
2ln2σg
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
VL = vddd0
∞∫ =π
6 ndd3dd0
∞∫ =π
6 D N3 exp 9
2 ln2σg⎛ ⎝ ⎜ ⎞
⎠ ⎟ N L
Lognormal Modes
Fig. 13.3
Number (partic. cm-3), area (cm2 cm-3), and volume (cm3 cm-3) concentrations distributed lognormally
10
-3
10
-1
10
1
10
3
10
5
0.001 0.01 0.1 1
dx/d log
10
D
p
(x=n, a, v)
Particle diameter (D
p
, )
D
V
D
N
D
An
v
a
dx /
d lo
g 10 D
p (x
=n,a
,v)
Lognormal Param. for Cont. Particles
Table 13.2
Nucleation Accumulation CoarseParameter Mode Mode Modeg 1.7 2.03 2.15NL (particles cm-3) 7.7x104 1.3x104 4.2DN (m) 0.013 0.069 0.97AL (m2 cm-3) 74 535 41DA (m) 0.023 0.19 3.1VL (m3 cm-3) 0.33 22 29DV (m) 0.031 0.31 5.7
Quadramodal Size DistributionSize distribution at Claremont, California, on the morning of August 27,
1987
Fig. 13.4
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
0
50
100
150
200
250
300
0.01 0.1 1 10
Particle diameter (D
p
, )
dn (No. c
-3
)
/dlog
10
D
p
da (
2
c
-3
)/d log
10
D
p
dv (
3
c
-3
)
/d log
10
D
p
Marshall-Palmer DistributionRaindrop number concentration between di and di+Ddi (13.30)
Ddin0 = value of ni at di = 0n0 = 8.0 x 10-6 partic. cm-3 m-1
lr = .1x10-3 R-0.21 m-1
R = rainfall rate (1-25 mm hr-1)
Total number concentration and liquid water content
ni =Δdin0e−λrdi
nT =n0 λr wL =10−6ρwπn0 λr4
Marshall-Palmer DistributionExample 13.6.
R = 5 mm hr-1
di = 1 mm
di+Ddi = 2 mm
---> ni = 0.00043 partic. cm-3
---> nT = 0.0027 partic. cm -3
---> wL = 0.34 g m-3
Modified Gamma DistributionNumber concentration (partic. cm-3) of drops in size bin i (13.30)
ni =Δri Agriαg exp−
αgγg
rirc,g
⎛ ⎝ ⎜ ⎜
⎞ ⎠ ⎟ ⎟
γg⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Modified Gamma Distribution Parameters
Table 13.3
Cloud Type Ag
ag
gg
rcg
()
LiqυidWaterConten
t(g -3)
Nυb erConc.(partic.c -3)
Stratocυυlυ bae 0.2823 5.0 1.19 5.33 0.11 100
Stratocυυlυ top 0.19779 2.0 2.6 10.19 0.796 100
Stratυ bae 0.97923 5.0 1.05 .70 0.11 100
Stratυ top 0.38180 3.0 1.3 6.75 0.379 100
Ni botratυ bae 0.08061 5.0 1.2 6.1 0.235 100
Ni botratυ top 1.0969 1.0 2.1 9.67 1.03 100
Cυυlυ congetυ 0.581 .0 1.0 6.0 0.297 100
Light rain .97x10-8 2.0 0.5 70.0 1.17 0.01
Modified Gamma DistributionExample 13.7.
Find number concentration of droplets between 14 and 16 m in radius at base of a stratus cloud
---> ri = 15 m---> Dri = 2 m---> ni = 0.1506 partic. cm-3
Full-Stationary Size StructureAverage single-particle volume in size bin (υi) 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 StructureDemonstration of a problem with the full-stationary size bin structure
Fig. 13.5
Full-Moving StructureNumber concentration (ni) of particles in a size bin does not change during growth; instead, single-particle
volume (υi) 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 StructurePreservation of aerosol material upon growth and evaporation in a moving
structure
Fig. 13.6
Full-Moving StructureParticle size bin reordering for coagulation
Fig. 13.7
A
B
Quasistationary StructureSingle-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
Partition volume of i between bins j and k while conserving particle number concentration(13.32)
and particle volume concentration (13.33)
Solution to this set of two equations and two unknowns (13.34)
ni =Δnj +Δnk
ni ′ υ i =Δnj υ j +Δnkυk
Δnj =niυk − ′ υ iυk −υ j
Δnk =ni′ υ i −υ j
υk −υ j
After growth, particles in bin i have volume υi’, which lies between volumes of bins j and k
υ j ≤ ′ υ i <υk
Moving-Center StructureSingle-particle volume (υi) varies between υi,hi and υi,lo during growth, but υi,hi, υi,lo, and dυi
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 StructureComparison of moving-center, full-moving, and quasistationary size structures
during water growth onto aerosol particles to form cloud drops.
Fig. 13.8
10
-1
10
1
10
3
10
5
10
7
0.1 1 10 100
dv (
3
c
-3
) /d log
10
D
p
Particle diaeter (D
p
, )
Initial
Fυll-oving
Qυai-
tationary
Moving-center
dv (
m3 c
m-3) /
d lo
g 10 D
p
