representing aerosol dynamics and properties ...etoa esfc θ e e sfc toa =−exp( / cos )τθmeas...

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


, µm

2 cm-3


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



ExactMIDAS, lognormalMIDAS, modified gamma

633 nm

MIDAS, lognormal, 633 nm

MIDAS, modified gamma, 633 nm

MIDAS, modified gamma, solar


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



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.


COMPARISON OF CLOUD DROPLETNUMBER CONCENTRATION BETWEEN

MOMENT AND DISCRETE MODELS

Discrete Model

Moment Model


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

representing aerosol dynamics and properties ...etoa esfc θ e e sfc toa =−exp( / cos )τθmeas...

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