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REPRESENTING AEROSOL DYNAMICS AND PROPERTIES IN ATMOSPHERIC
CHEMICAL TRANSPORT MODELS BY THE METHOD OF MOMENTS
Stephen E. Schwartz, Robert McGraw, Carmen M. Benkovitz, and Douglas L. Wright
Upton, New York, U.S.A.
Symposium on Environmental Chemistry in Multiphasic Systems in honor of
Michael R. Hoffmann
ACS Awardee for Creative Advances in Environmental Science and Technology
American Chemical Society
221st National Meeting
Division of Environmental Chemistry
San Diego CA April 1-5, 2001
OUTLINE
• Introduction: Importance of atmospheric aerosols, key properties
• Moments of the aerosol particle size distribution
• Obtaining aerosol properties from the moments
• The Method of Moments (MOM) and the Quadrature Method ofMoments (QMOM) for calculating aerosol evolution
• Evaluations of QMOM
• Application of QMOM in a large scale chemical transport model
• Summary
IMPORTANCE OF ATMOSPHERIC AEROSOLS
• Human health - impairment via inhalation
• Light scattering and absorption - visibility, climate change
• Heterogeneous reactions - stratosphere, troposphere
• Modification of cloud physical properties - hydrology and climate
• Modification of fog, cloud, and precipitation composition
Understanding and describing the role of aerosols in these phenomenarequire size- and composition-dependent treatment of chemical andphysical processes involving aerosols.
This descriptive capability must be represented in atmospherictransport and transformation models on a variety of scales fromregional to global.
REPRESENTATIONS OF AEROSOLSIZE DISTRIBUTION
Whitby, Atmos. Environ., 1978
SIZE MATTERSLight scattering efficiency of ammonium sulfate vs. radius
1 0
8
6
4
2
0
Sca
tte
rin
g
Eff
icie
ncy
, m
2/g
(SO
42
- )
0 . 0 1 0 . 1 1Radius, µm
5
Data of Ouimette and Flagan, Atmos. Environ., 1982
KEY AEROSOL OPTICAL PROPERTIES
Aerosol optical thickness, τ aerosol
SURFACE
ATMOSPHERE
Etoa
Esfcθ
E
Esfc
toameas= −exp( / cos )τ θ
τ ττ τ
aerosol meas
Rayleigh gas abs
=− +( )
τ aerosol depends on aerosol loading and properties.
Ångström exponent, α
α τ= − d
d wavelength
loglog
aerosol
0.160.140.120.10
0.08
Opt
ical
dep
th
4 5 6 7 8 91000Wavelength, nm
α = – slope
α is largest for smallest particle size.
Aerosol radiative forcing
Forcing = Flux(aerosol) - Flux(no aerosol)
Flux may be at top of atmosphere or surface.
SIZE DEPENDENCE OF AEROSOL OPTICALDEPTH AND TOA RADIATIVE FORCING
Ammonium Sulfate, 10 mg (sulfate) m-2 at 80% RH
Cloud-free Sky, Surface Reflectance 0.15
6
5
4
3
2
1
0
– (F
orci
ng),
W m
-2
3 4 5 6 7 8 90.1
2 3 4 5 6 7 8 91
Radius, µm
Global Average TOA Forcing As Function Particle Radius
20
15
10
5
0
– (F
orci
ng),
W m
-2
89°
78°
37°
64°
7°
SZATop-of-Atmosphere Forcing
As Function of Solar Zenith AngleAnd Particle Radius
0.20
0.15
0.10
0.05
0.00
Opt
ical
Dep
th
Optical DepthAs Function of Particle Radius
MONTHLY AVERAGE AEROSOL JUNE 1997
Optical Thickness at 865 nm
Ångström Exponent
Polder/Adeos CNES/NASDA LOA/LSCE
MOMENTS OF THE PARTICLE SIZE DISTRIBUTION
µkkr
dN
drdr≡ ∞
∫ ( )0
(not normalized)
Moment Physical Interpretation Unit
µ0 Particle number concentration cm-3
µ1 Total radius per unit volume cm cm-3
µ2 ( )4 1π − × Area per unit volume cm2 cm-3
µ3 ( )43
1π − × Volume per unit volume cm3 cm-3
Particle Size Distribution
3
2
1
0Dis
trib
uti
on
fu
nc
tio
n f
(r)
0.40.30.20.10
Radius r, µm
reffravg
0.70.60.5
Average radius: ravg = µ1/µ0.
Effective radius: reff = µ3/µ2.
λ = 0.55 µm
0.01 0.1 1.0 10Radius, µm
n = 1.5r
nia 0b 0.001c 0.01d 0.1e 1f 10
Qs
0
1
2
3
4
5
EVALUATION OF AEROSOL PROPERTIESAerosol properties are integrals of kernel function over size distribution
P r f r dr=∞
∫ σ ( ) ( )0
Property (P) Kernel (σ)
Particle number densitySurface area densityVolume concentration
1r2
r3
Optical properties Optical kernelsscattering r2 × Qs →forcingÅngström exponenteffective radius µ3/µ2
Cloud activationPM 2.5
Step functionH(r-r0)
Concentration atspecific radius r0
Delta functionδ(r-r0)
AEROSOL PHYSICAL PROPERTIESHOW TO OBTAIN FROM MOMENTS?
Aerosol physical or optical properties are an integralover the size distribution, requiring integrals like
P r f r dr=∞
∫ σ ( ) ( )0
where the kernel function σ ( )r describes the propertyof interest.
Most integration (quadrature) methods require that theintegrand be known as some set of points, for exampleSimpson's rule:
In our case σ ( )r is known, but the distribution functionf r( ) is not known, only a few of its moments.
CALCULATING AEROSOL PROPERTIESBY GAUSSIAN QUADRATURES
Problem: How to evaluate integrals over the aerosolsize distribution when only the lower-order momentsof the distribution are known???
Solution: Gaussian quadrature
σ σ( ) ( ) ( )r f r dr r wii
N
i≈=
∞ ∑∫1
0
Here σ ( )r is the known kernel function and f r( ) is theunknown size distribution.
The N abcissas { ri} and N weights { wi } aredetermined from 2N moments of f r( ) by inversion of
µkk
ik
i
N
ir f r dr r w≡ ==
∞ ∑∫ ( )1
0 k = 0, 1, …, 2N-1.
DISTRIBUTIONS USED IN TEST OF RETRIEVALOF OPTICAL PROPERTIES FROM MOMENTS
0.001 0.010 0.100 1.000101
102
103
104
105
106
f(r)1
3
21
27
28
22
23
0.010 0.100
102
103
104
6
78
1218
2019
11
0.010 0.100
102
103
104
f(r)
13
2
4
17
16
915
r (µm) 0.010 0.100 1.000100
101
102
103
5
24
2625
14
10
x100x100
x10
r (µm)
r (µm) r (µm)
f(r)
f(r)
D. L. Wright, Jr., J. Aerosol Sci., 2000
SKILL OF GAUSSIAN QUADRATURESTO OBTAIN AEROSOL OPTICAL PROPERTIES
FOR 28 TEST DISTRIBUTIONSIndex of refraction n = 1.55 - 0i
0 10 20 300.1
1
10
100
Ext
inct
ion
Coe
ffici
ent,
Kex
t
0 10 20 300.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Kex
t, G
Q3
/ K
ext,
exac
t
Test distribution number
Exact
3-Point quadrature
633 nm
633 nm
Solar spectrum
Test distribution number
, µm
2 cm-3
D. L. Wright, Jr., J. Aerosol Sci., 2000
THE MOMENTS DO NOT UNIQUELY DEFINETHE DISTRIBUTION
Question: How many identical moments do the followingdistributions have in common?
16
14
12
10
8
6
4
2
0
f(r)
0.40.30.20.10.0Radius r, µm
Answer: Infinitely many! These distributions have identicalmoments for all non-negative integer values of k.
µkkr f r dr≡ ∞
∫ ( )0
Despite the differences in the distributions, the opticalproperties are virtually identical!
50
40
30
20
10
0
Pha
se F
unct
ion
18013590450Scattering angle, degrees
λ = 550 nmn = 1.4-0i
McGraw et al., J. Aerosol Sci. (1998)
SKILL OF MIDAS METHOD(Multi Isomomental Distribution Aerosol Simulator)TO OBTAIN AEROSOL OPTICAL PROPERTIES
FOR 28 TEST DISTRIBUTIONSIndex of refraction n = 1.55 - 0i
Kex
t, M
IDA
S /
Kex
t, ex
act
0 10 20 30
0 10 20 300.930.940.950.960.970.980.991.001.011.021.03
0.1
1
10
100
Ext
inct
ion
Coe
ffici
ent,
Kex
t, µm
2 cm-3
Test distribution number
Test distribution number
ExactMIDAS, lognormalMIDAS, modified gamma
633 nm
MIDAS, lognormal, 633 nm
MIDAS, modified gamma, 633 nm
MIDAS, modified gamma, solar
D. L. Wright, Jr., J. Aerosol Sci., 2000
AEROSOL DYNAMICS BY THEMETHOD OF MOMENTS
The method of moments is an approach todescribing aerosol properties and dynamics in terms ofthe moments µk of the radial number size distributionf r( ).
µkk
r f r dr= ∞∫0 ( )
Aerosol properties (e.g., light scattering coefficient)can be accurately represented as simple functions oflow order moments.
Aerosol dynamics can be represented by growth laws(differential equations) in the moments.
The moments advect and mix just like chemicalspecies--they are conserved and additive.
Hence representing aerosol properties and dynamics in3-D transport models is equivalent to representing asmall number of additional chemical species -- the loworder moments.
METHOD OF MOMENTSHeuristic Description
Consider accretion of monomer by existing aerosol.
This can be considered a reaction between monomer(m) and aerosol surface area (A)
m A
kmA
+ → slightly larger distribution
Rate =
Aerosol surface area density is
A r f r dr= ∫ =∞ 4 420 2π πµ( )
So accretion of monomer by existing aerosol is areaction between monomer and second moment.
METHOD OF MOMENTSGeneral Description
Consider the growth law for particles of radius r:
dr
dtr r ( ) = φ( )
The corresponding moment evolution equation is:
1 10k
d
dtr r f r drk
kµ φ= −∞∫ ( ) ( )
Closure of the moment evolution equations requires thegrowth law to be of the form:
φ( )r a br= +
where a and b are independent of r. In this casemoment evolution is evaluated as:
1 1
1
1
k
d
dtr r f r dr
a r f r dr b r f r dr
a b
kk
k k
k k
µ φ
µ µ
= ∫
= ∫ + ∫= +
−
−
−
( ) ( )
( ) ( )
This case includes free-molecular growth, φ( )r a= .
QUADRATURE METHOD OF MOMENTS
The requirement that the growth law be of the form
φ( )r a br= +
is not generally satisfied.
The quadrature method of moments evaluates themoment evolution equation
d
dtk r r f r drk
kµ φ= −∞∫ 10
( ) ( )
by Gaussian quadratures:
d
dtk r r wk i
k
ii iµ φ≅ −
=∑ 1
1
3( )
This approach is completely general and highlyaccurate.
R. McGraw, Aerosol Sci. Technol. (1997)
COMPARISON OF EXACT AND QUADRATURE MOMENT EVOLUTION
2.0
1.5
1.0
0.5
0.0
NO
RM
ALI
ZE
D D
EN
SIT
Y
43210
PARTICLE RADIUS, µm
INITIAL
FINAL DISTRIBUTION (EXACT)
7
6
5
µ 1
20151050
TIME(s)
5000
4000
3000
µ 4
500
400
300µ 3
60x103
50
40
30
µ 5
Exact QMOM
60
50
40
µ 2
% E
RR
OR
[(Q
MO
M -
EX
AC
T)/
EX
AC
T]
-0.20
-0.15
-0.10
-0.05
0.00
20151050
µ1
TIME(s)
-0.04
-0.03
-0.02
-0.01
0.00
µ3
-0.015
-0.010
-0.005
0.000
µ5
Initial size distribution: Normalized Khrgian-Mazin distribution with mean particle radius of 5 µm.
Diffusion controlled growth by accretion of water vapor for 20 s at 278 K and fixed saturation of 101%.
McGraw, Aerosol Sci. Technol. (1997)
CONDITIONS OF TESTS FOR SKILLOF QUADRATURE METHOD OF MOMENTS
QMOM was extensively compared in box-model calculationsto results obtained by discrete integration of the PSD.
Initial aerosol Initial concentrations Test case Dry deposition Cloud type
N0
cm-3
rg
µm
σg [H2SO4]0
mol cm-3
[SO2]0
mol cm-3
1-3 none Cumulus 0 0 0 0 0
4-6 none Cumulus 100 0.01 2.0 0 0
7-9 none Stratiform 100 0.01 2.0 0 0
10-12 none Stratiform 100 0.01 2.0 5.0x10-15 1.0x10-12
13-15 none Cumulus 100 0.01 2.0 5.0x10-15 1.0x10-12
16-18 none Cumulus 0 0 0 5.0x10-15 1.0x10-12
19-21 to land (W = 5.0 m/s) Cumulus 0 0 0 5.0x10-15 1.0x10-12
22-24 to ocean (W = 10 m/s) Stratiform 100 0.01 2.0 5.0x10-15 1.0x10-12
25-27 none Stratiform 100 0.10 2.0 5.0x10-15 1.0x10-12
28-30 none Stratiform 1000 0.10 2.0 5.0x10-15 1.0x10-12
number
Each aerosol test case was run for three sets of meteorological conditions.Wright, Kasibhatla, McGraw and Schwartz, JGR, in press, 2001
DISTRIBUTIONS IN TESTS OF SKILL OFQUADRATURE METHOD OF MOMENTS
Initial Final
Wright, Kasibhatla, McGraw and Schwartz, JGR, in press, 2001
MOMENT EVOLUTION AND ERRORS INBOX MODEL TESTS OF SKILL OF
QUADRATURE METHOD OF MOMENTSComparison of QMOM vs. discrete integration of PSD
10-6
10-4
10-2
100
102
104
µ k, µ
mk
cm-3
302520151050
9
10-6
10-4
10-2
100
102
104
µ k, µ
mk
cm-3
3
Time, hr
-80
-60
-40
-20
0
20
40
60
Fra
ctio
nal D
iffer
ence
, %
302520151050
9
-80
-60
-40
-20
0
20
40
60
Fra
ctio
nal D
iffer
ence
, % 3
Time, hr
-80
-60
-40
-20
0
20
40
60
Fra
ctio
nal D
iffer
ence
, %
6 10-6
10-4
10-2
100
102
104
µ k, µ
mk
cm-3
6
k 0 1 2 3 4 5
Cloud Cloud
k
0
1
2
3
4
5
k
0
1
2
3
4
5
k
0
1
2
3
4
5
QMOMQMOM Discrete
Departures are associated mainly with cloud events.
Some of the departures are attributable to errors in discreteintegration.
Wright, Kasibhatla, McGraw and Schwartz, JGR, in press, 2001
COMPARISON OF CLOUD DROPLETNUMBER CONCENTRATION BETWEEN
MOMENT AND DISCRETE MODELS
Discrete Model
Moment Model
Wright, Kasibhatla, McGraw and Schwartz, JGR, in press, 2001
Aerosol Chemical Transport Model GChM-O
MomentsMicrophysical PropertiesOptical Properties
METEOROLOGY
Winds
Clouds
Precipitation
ECMWF
1.125°× 1.125°15 levels
TRANSPORT
Bott/Easter
EMISSIONS
Sulfur species
Primary Sulfate
DMS
Seasalt aerosol
Smith, O'Dowd
DRY DEPOSITION
Time, Locationdependent
Size Resolved
WET DEPOSITION
CHEMISTRY Gas Phase
SO2
AEROSOL MICROPHYSICSNucleationCoagulationGrowth (clear air, cloud)
Method of Moments - McGraw
SO2 + H 2O2SO2 + O 3
→ H2SO4→ H2SO4
Global Chemistry Model Driven by Observation-Derived Meteorological Data
DMSSO2
MSASO4
2-
SO2 + OHDMS + OH
→ H2SO4
HO2 +HO2 → H2O2
Aqueous Phase
MODEL DOMAIN AND GRID CELL STRUCTURE1.125˚ latitude and longitude; 15 terrain-following vertical levels
61 × 180 × 15 = 164,700 grid cells
10
20
30
40
50
60
Cel
l Coo
rdin
ate
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
Cell Coordinate
-140 -120 -100 -80 -60 -40 -20 0 20 40 60
Longitude
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Latit
ude
MODELING EVOLUTION OF AEROSOL LOADING AND PROPERTIES
Moment-based representation of aerosol microphysicsDay Log10 N Log10 µ3 Mean Radius
1
3
7
10
Based on Wright, McGraw, Benkovitz, and Schwartz, GRL, 2000
AEROSOL PROPERTIES FROM SUBHEMISPHERIC SULFATETRANSPORT AND TRANSFORMATION MODEL
835 nm October 14-31, 1986
Aerosolopticaldepth
Ångströmexponent
Average Sulfate + Sea Salt Aerosol
3
2
1
0
-1
1
0.1
0.01
0.001
0.0001
APPROACHES TO REPRESENTING AEROSOL SIZEDISTRIBUTION AND ITS EVOLUTION IN MODELS
Conventional Moment Methods
Size distributionrepresented by...
Number of particles in binsas function of radius Ni(ri)
Moments, µ = ∫kkr N r dr( )
Aerosolevolutionrepresented by...
Differential equations inNi(ri)
Differential equations inlow-order moments,k = 0 ... 5
Aerosolpropertiesrepresented by...
Direct integration overkernel function,σ σ= ∫ ( ) ( )r N r dr
Gaussian quadrature,σ σ= ∑w ri i( );radii ri and weights widetermined from moments
Advantages Explicit knowledge of sizedistribution
Compact representation
Issues Numerical diffusionbetween bins; manyvariables Ni(ri) required
Closure of the set ofmoment equations; non-uniqueness of distributions
SUMMARY
There is a need for accurate and efficient representation of aerosolmicrophysical properties in chemical transport models.
The Method of Moments (MOM) meets this need. It is orders ofmagnitude more efficient than discrete aerosol models and highlyaccurate.
MOM does not provide the particle size distribution, but it tells you(almost) everything else you want to know about the aerosol.
MOM is beginning to see application in atmospheric models byseveral groups.
MOM is also being applied to combustion aerosols. It has manypotential applications.
For further information see www.ecd.bnl.gov/steve/schwartz.html