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Atmospheric Diffusion Modeling
Karl B. Schnelle, Jr.
VanderbiltUniversity
I. UseofModelsandRelationshiptoAtmospheric VII. CalculatedConcentrationandAveragingTime
Pollution VIII. AtmosphericConcentrations
II. AtmosphericTurbulence withDiffusionCeilings
III. GaussianPlumeModelforContinuousSources IX. MultipleSourcesandReceptors
IV. DiffusionCoefficients X. DesigningaChimneytoLimit
V. PlumeRise Ground-LevelConcentrations
VI. AccidentalReleasesofMaterial XI. TheNewModels
totheAtmosphere
GLOSSARY representationofthecrosswindandverticalconcentra-
tiondistributionofadiffusingsubstance.
Atmosphericturbulence Randomvelocityandpressure Inversion Positive temperature gradient or increase in
fluctuationsoccurringintheatmosphere. temperature with elevation, resulting in adverse con-
Deterministicmodel Mathematicalmodelinwhicheach ditionsfordispersionofpollutants.
variableandparametercanbeassignedadefinitefixed Isopleth Linesofconstantconcentrationinadiagramor
number. plotofamathematicalmodelofconcentrationversus
Diffusion ceiling Stable layer of air above the mixed distance.
layer, which restricts the vertical motion of a diffus- Lapserate Decreaseintemperaturewithheight;thatis,
ingsubstance. anegativetemperaturegradient.
Diffusion coefficient Parameter of the Gaussian mathe- Mixingdepth Thicknessoftheturbulentregionnextto
maticalmodelwhichmustbedeterminedbymeasure- the ground, in which atmospheric properties are well
ment.Alsocalledthesigmavalue. mixed.
Dosage Concentration–time relationship that is related Plumerise Distanceasmokeplumerisesaboveitsemis-
to the effects that a polluting substance has on a re- sionpointduetotheinherentmomentumandbuoyancy
ceptor. oftheplume.
Evolutionary process Stochastic process in which the Richardsonnumber Dimensionlessratiodescribingthe
probabilitydistributionisafunctionoftime. relative importance of convection and the turbulence
Gaussianmodel Mathematicalmodelthatischaracteris- generatedbymechanicalshear.
ticallybell-shaped.TheGaussianmodelisthecommon Stability of the atmosphere Term describing the
679
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680 AtmosphericDiffusionModeling
dynamiccharacteristicofturbulenceintheatmosphere. TheU.S.EnvironmentalProtectionAgency(EPA)hasde-
Stationary process Stochastic process in which the veloped a set of computer codes based on the Gaussian
probability distribution function is not a function of model which carry out the calculations needed for regu-
time. latorypurposes.ThesemodelsareavailablefromtheAp-
Stochasticmodel Modelinwhichtheprincipleofuncer- plied Modeling Research Branch, Office of Air Quality
taintyisintroduced.Variablesandparameterscanonly Planning and Standards, U.S. Environmental Protection
beassignedaprobabilityoflyingbetweenarangeof Agency,ResearchTrianglePark,NorthCarolina27711.
values.
Temperature gradient Change of temperature with
height.Itisnormallynegativeintheloweratmosphere; I. USE OF MODELS AND RELATIONSHIP
thatis,thetemperaturedecreaseswithheightundernor- TO ATMOSPHERIC POLLUTION
malatmosphericconditions.
To make a calculation, the source of pollution must be
defined and the geographic relation between the source
ANATMOSPHERICDIFFUSIONMODELisamathe- andthereceptorofpollutionmustbeknown.Furthermore,
maticalexpressionrelatingtheemissionofmaterialinto themeansoftransportbetweenthesourceandthereceptor
theatmospheretothedownwindambientconcentrationof mustbeunderstood.Thusthesource–transport–receptor
thematerial.Theheartofthematteristoestimatethecon- trilogymustbequantitativelydefinedtomakethedesired
centrationofapollutantataparticularreceptorpointby computation.
calculationfromsomebasicinformationaboutthesource
ofthepollutantandthemeterologicalconditions.
A. TheSource
Deterministic, statistically regressive, stochastic mod-
els,andphysicalrepresentationsinwatertanksandwind Weather affects many kinds of pollution sources. For
tunnels have been developed. Solutions to the determin- example,onacolddaymorefuelwillbeusedforspace
istic models have been analytical and numerical, but the heating. Hot weather brings on a greater use of space
complexitiesofanalyticalsolutionaresogreatthatonly cooling, which produces a greater demand on electricity
a few relatively simple cases have been solved. Numer- production.
ical solutions of the more complex situations have been Defining the source is difficult in most cases. One
carriedoutbutrequireagreatamountofcomputertime. mustconsiderfirstwhetheritismobileorstationaryand
Progress appears to be most likely for the deterministic whetherthemethodofemissionisfromapointoraline,or
models.However,forthepresentthestochasticallybased moregenerallyfromanarea.Thenitschemicalandphys-
Gaussian type model is the most useful in modeling for icalpropertiesmustbedetermined,mostappropriatelyby
regulatorycontrolofpollutants. samplingandanalysis,butthisisnotalwayspossible.One
AlgorithmsbasedontheGaussianmodelformthebasis mustturntoestimationorperhapstoamassbalanceover
ofmodelsdevelopedforshortaveragingtimesof24hror the whole emission process to determine the amount of
less and for long-time averages up to a year. The short- material lost as pollutant. The major factors required to
termalgorithmsrequirehourlymeteorologicaldata,while describethesourceare
thelong-termalgorithmsrequiremeteorologicaldataina
frequencydistributionform.Algorithmsareavailablefor 1. Composition,concentration,anddensity;
singleandmultiplesourcesaswellassingleandmultiple 2. Velocityofemission;
receptorsituations.Onageographicalscale,effectiveal- 3. Temperatureofemission;
gorithmshavebeendevisedfordistancesupto10–20km 4. Diameterofemittingstackorpipe;and
forbothurbanandruralsituations.Long-rangealgorithms 5. Effectiveheightofemission.
areavailablebutarenotaseffectiveasthosefortheshorter
distance.Basedonacombinationoftheseconditions,the Fromthesedatatheflowrateofthetotalemissionstream
Gaussianplumemodelcanprovideatareceptoreither andofthepollutantinquestioncanbecalculated.
The source problem is further complicated when the
1. theconcentrationofanairpollutantaveragedover emissionisinstantaneousinsteadofacontinuousstream.
timeand/orspace,or Mass balances become impractical in such a case. Mea-
2. acumulativefrequencydistributionofconcentration suringthefactorsmentionedpreviouslymaybecomeim-
exceededduringaselectedtimeperiod. practicalaswell.Grabsampleswithsubsequentlaboratory
analysismightbeusedtodetermineconcentration.Ifflow
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AtmosphericDiffusionModeling 681
ratecanbemeasuredorestimated,aswell,thenthetotal critical factor in determining plume rise. As the upward
emissioncanbecalculated. plume momentum is spent, further plume rise is depen-
dentupontheplumedensity.Plumesthatareheavierthan
B. TheReceptor airwilltendtosink,whilethosewithadensitylessthan
thatofairwillcontinuetoriseuntilthebuoyancyeffectis
Inmostcaseslegislationandsubsequentregulationswill
spent.Whentheatmospherictemperatureincreaseswith
determine the ambient concentrations of pollutant to
altitude,aninversionissaidtoexist.Lossofplumebuoy-
whichthereceptorislimited.Airqualitycriteriadelineate
ancy tends to occur more quickly in an inversion. Thus,
theeffectsofairpollutionandarescientificallydetermined
the plume may cease to rise at a lower altitude, and be
dosage–responserelationships.Therelationshipsspecify
trappedbytheinversion.
thereactionofthereceptorortheeffectswhentherecep-
Many formulas have been devised to relate the chim-
tor is exposed to a particular level of concentration for
neyandthemeteorologicalparameterstotheplumerise.
varying periods of time. Air quality standards are based
The most commonly used model, due to Briggs, will be
on air quality criteria and set forth the concentration for
discussedinalatersection.Theplumerisethatiscalcu-
agivenaveragingtime.Thus,theobjectiveforacalcula-
latedfromthemodelisaddedtotheactualheightofthe
tionistodetermineifanemissionwillresultinambient
chimneyandistermedtheeffectivesourceheight.Itisthis
concentrationsthatmeetairqualitystandards.
heightthatisusedintheconcentrationpredictionmodel.
Usually, in addition to the receptor, the locus of the
pointofmaximumconcentration,orthecontourenclosing
2. BulkTransportofthePollutant
an area of maximum concentration, and the value of the
concentrationassociatedwiththelocusorcontourshould Pollutantstraveldownwindatthemeanwindspeed.Spec-
bedetermined.Theshorttimeaveragesthatareconsidered ification of the wind speed must be based on data usu-
inregulationsareusually3min,15min,1hr,3hr,or24hr. allytakenatweatherstationsseparatedbylargedistances.
Longertimeaveragesareoneweek,onemonth,aseason, Sincewindvelocityanddirectionarestronglyaffectedby
orayear. thesurfaceconditions,thenatureofthesurface,predom-
inanttopologicfeaturessuchashillsandvalleys,andthe
C. Transport presenceoflakes,rivers,andbuildings,theexactpathof
pollutantflowisdifficulttodetermine.Furthermore,wind
Understandingtransportbeginswiththethreeprimaryfac-
patternsvaryintime,forexample,fromdaytonight.The
torsthataffectthemixingactionoftheatmosphere:radi-
Gaussianconcentrationmodeldoesnottakeintoaccount
ationfromthesunanditseffectatthesurfaceoftheearth,
windspeedvariationwithaltitude,andonlyinafewcases
rotationoftheearth,andtheterrainortopographyandthe
aretherealgorithmstoaccountforthevariationintopogra-
natureofthesurfaceitself.Thesefactorsarethesubjects
phy.Forthefuture,progressinthisareawillcomethrough
ofbasicmeteorology.
numericalsolutionsofthedeterministicmodels.
Thewayinwhichatmosphericcharacteristicsaffectthe
concentrationofairpollutantsaftertheyleavethesource
canbeviewedinthreestages: 3. DispersionofthePollutants
Dispersion of the pollutant depends on the mean wind
Effectiveemissionheight
speedandatmosphericturbulence,describedinmorede-
Bulktransportofthepollutants
tailinthenextsection.Dispersionofaplumefromacon-
Dispersionofthepollutants
tinuouselevatedsourceincreaseswithincreasingsurface
roughnessandwithincreasingupwardconvectiveaircur-
1. TheEffectiveEmissionHeight
rents. Thus a clear summer day produces the best me-
Afterahotorbuoyanteffluentleavesaproperlydesigned teorological conditions for dispersion, and a cold winter
source,suchasachimney,itkeepsonrising.Thehigher morningwithastronginversionresultsintheworstcon-
theplumegoes,thelowerwillbetheresultantgroundlevel ditionsfordispersion.
concentration.Themomentumofthegasesrisingupthe
chimneyinitiallyforcesthesegasesintotheatmosphere.
D. UsesoftheModels
Thismomentumisproportionaltothestackgasvelocity.
However,velocitycannotsustaintheriseofthegasesaf- Atmosphericdiffusionmodelshavebeenputtoavariety
tertheyleavethechimneyandencounterthewind,which ofscientificandregulatoryuses.Primarilythemodelsare
willcausetheplumetobend.Thusmeanwindspeedisa usedtoestimatetheatmosphericconcentrationfieldinthe
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682 Atmospheric Diffusion Modeling
absence of monitored data. In this case, the model can be a clear days. It is these convective eddies or currents that re-
part of an alert system serving to signal when air pollution sult in the vertical air motions that birds and glider pilots
potential is high and requiring interaction between control use to climb into the upper air without the expenditure of
agenciesandemitters.Themodelscanservetolocateareas other energy.
of expected high concentration for correlation with health Mechanical eddies result from the shearing forces pro-
effects. Real-time models can serve to guide officials in duced when the wind blows over the surface of the earth.
cases of nuclear or industrial accidents or chemical spills. At ground level the wind speed is zero, and it reaches a
Here the direction of the spreading cloud and areas of maximum usually at many thousands of meters above the
critical concentration can be calculated. After an accident, surface.Mechanicalturbulenceincreaseswithincreasing
models can be used in a posteriori analysis to initiate wind speed and is greater over rough surfaces than over
control improvements. smoothsurfaces.Terrainroughnesscanbecharacterized
A current popular use for atmospheric diffusion models byaroughnesslengthz whichisproportionaltothesize
0
is in air quality impact analysis. The models serve as the oftheeddiesthatcanexistamongtheroughnesselements.
heart of the plan for new source reviews and the prevention Theroughnesslengthisrelativelysmalliftheroughness
of significant deterioration of air quality. Here the mod- elementsareclosetogether.Forsmoothsand,z isabout
0
els are used to calculate the amount of emission control 0.10 cm. Over cities, z can increase to several meters.
0
required to meet ambient air quality standards. The mod- (Typicalvaluesofz are listed later in Table V.)
0
els can be employed in preconstruction evaluation of sites
for the location of new industries. Models have also been
A. AdiabaticLapseRateand
used in monitoring network design and control technology
PotentialTemperature
evaluation.
Lapserateistherateoftemperaturedecreasewithheight
intheatmosphere.Ifweconsiderthehypotheticalcaseof
II. ATMOSPHERIC TURBULENCE a bubble of air rising through a dry atmosphere with no
mixingorheatexchangebetweenthebubbleanditsenvi-
Diffusion in the atmosphere is dependent upon the mean ronment,thebubblewillexpandandcool.Thecoolingis
wind speed and the characteristics of atmospheric turbu- saidtobeatthedryadiabaticlapserate(DALR).Thusthe
lence. These factors determine the stability characteristics DALRcanbedefinedasthenegativeofthetemperature
of the atmosphere. Turbulence consists of horizontal and gradient that is established as dry air expands adiabati-
vertical eddies that mix the pollutant with the air surround- cally while ascending. It should be noted that the term
ingit.Thus,inasmokeplume,theturbulencedecreasesthe temperaturegradienthastheoppositealgebraicsignfrom
concentration of the pollutants in the plume and increases the lapse rate. Both temperature gradient and lapse rate
the concentration in the surrounding air. When turbulence areusedwhendiscussingthetemperaturestructureofthe
is strong, the pollutants are dispersed more rapidly. Strong atmosphere.
turbulence exists in an unstable atmosphere in which ver- To calculate the temperature gradient for a reversible
tical motion is enhanced. Maximum instability occurs in adiabatic expansion of dry air, the ideal gas equation of
the summer on a clear sunny day in the early afternoon. stateiscombinedwiththehydrostaticequationofmotion
Conversely, when turbulence is weak, a very stable at- andintegrated.Thefollowingequationresults,
mosphere is present. This condition would be most pro- DALR=λ=−dT/dz =(g/g )(γ −1)/Rγ (1)
c
nounced on a clear winter day, early in the morning just
as dawn breaks. The diffusion parameters in the models whereg/gc istheratioofaccelerationduetogravitytoa
describing turbulent mixing are quantitatively specified in conversionconstant,λ,theratioofspecificheats, R,the
relation to the stability condition; therefore, stability, sur- idealgasconstant,T,theabsolutetemperature,andz,the
face roughness, and wind conditions must be explicitly altitude.
defined. For dry air with a molecular weight of 29, γ = 1.41,
Turbulent eddies are formed in the atmosphere by con- g/g =1.0gmf/gm,and R =84.78(gmf/cm2liter)
c gmol◦C
vection and by geologic and manmade structures. Con- λ=0.995◦C/100m (2)
vection occurs when the air is heated from below by the
warm surface of the earth and the buildings and pavement Airsaturatedwithwatervaporreleasesheatasitcools,and
covering it. Whenever the temperature decreases rapidly liquid water condenses. Thus the adiabatic temperature
with height, convection is most pronounced and may per- decreaseisgreaterthanthatgivenbyEq.(1).Thewetadi-
sist up to many hundreds of meters above the surface on abatictemperaturegradientrangesfromabout−0.9◦Cper
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AtmosphericDiffusionModeling 683
100minthepolarregiontoabout−0.4◦Cper100min TABLEI RichardsonNumberandStability
thetropics.
Richardson
The potential temperature is defined as the tempera- Classification number Comment
tureresultingwhendryairisbroughtadiabaticallyfrom
itsinitialstatetoastandardpressureof1000mb(mb= Stable Ri>0.25 Noverticalmixing,windsweak,
stronginversion,mechanical
millibar).Foranadiabaticexpansionofanidealgas:
turbulencedampened,negligible
T =T(P /P)γ−1/γ (3) spreadingofsmokeplume
0 0 Stable 0<Ri<0.25 Mechanicalturbulenceweakened
bystablestratification
where T is the temperature at the standard pressure P .
If P =10000 mb, then T =θ, the potential temperature0. Neutral Ri=0 Mechanicalturbulenceonly
0 0 Unstable −0.03<Ri<0 Mechanicalturbulenceand
Substitutingfor T ,takingthelogarithmofbothsidesof
0 convection
theequation,anddifferentiatingwithrespecttoz,onecan
Unstable Ri<−0.04 Convectionpredominant,winds
derivethefollowingequation:
weak,strongverticalmotion,
(cid:2) (cid:3) (cid:4)(cid:3) (cid:4)(cid:5) smokerapidlyspreading
1dθ 1 dT γ −1 g/g
= + c (4) verticallyandhorizontally
θ dz T dz γ R
ThelasttermcanbeidentifiedasDALRorλ.Forchanges
dispersionofasmokeplumedecreases,approachingneu-
inpressureof200mborless,θ isapproximatelyequalto tral stability where (∂θ/∂z)=0. Finally, as the Richard-
T within5%error.Thusthepotentialtemperaturegradient
son number becomes positive, vertical mixing ceases,
canbeapproximatedby
andmechanicalturbulenceisdampened.Theatmosphere
(cid:2)(cid:3) (cid:4) (cid:5)
(cid:7)θ dT becomes stably stratified, and very little vertical disper-
= +λ (5) sion of a smoke plume occurs. Table I summarizes these
(cid:7)z dz
actual conditions.
B. TheRichardsonNumberandStability
C. StabilityClassificationSchemes
Therelativeimportanceofconvectionandtheturbulence
1. Pasquill–GiffordStabilityClassification
generated by mechanical shear can be judged by the
Richardsonnumber.IndefiningtheRichardsonnumber,it As a simplified measure of stability, Gifford modified a
isconvenientfirsttodefineastabilityparameters,which systemofstabilityclassificationbaseduponsuggestions
willbeusedlaterinmodelingsmokeplumereleases, by Pasquill at the British Meteorological Office. In this
(cid:3) (cid:4) classificationitisassumedthatstabilityinthelayersnear
g (cid:7)θ thegroundisdependentonnetradiationasanindication
s = (6)
T (cid:7)z ofconvectiveeddiesandonwindspeedasanindicationof
mechanicaleddies.Insolation(incomingradiation)with-
Theterms canbethoughtofasbeingproportionaltothe
outcloudsduringthedayisdependentonsolaraltitude,
rate at which stability suppresses the generation of tur-
whichisafunctionoftimeofdayandyear.Whenclouds
bulence. Turbulence is also being generated by mechan-
arepresent,theextentoftheircoverageandthicknessde-
ical shear forces at a rate proportional to (∂u¯/∂z)2. The
creasestheincomingandoutgoingradiation.Daytimein-
Richardsonnumberistheratioofthesetwoprocesses.
solationmustbemodifiedaccordingtotheexistingcloud
s g (∂θ/∂z) coverandceilingheight.Atnight,theradiationcondition
Ri= = (7) is judged solely on cloud cover. Six stability categories
(∂u¯/∂z)2 T (∂u¯/∂z)2
are defined in Table II.
The Richardson number is a turbulence indicator and
also an index of stability. Meteorologists classify atmo-
2. Turner’sStabilityClassification
sphericstabilityinthesurfacelayerasunstable,neutral,
and stable. Strongly negative Richardson numbers indi- TurnerhastakenthePasquill–Giffordworkandproduced
cate that convection predominates, winds are weak, and astabilityclassificationthatisbasedonhourlymeteoro-
thereisastrongverticalmotioncharacteristicofanunsta- logicalobservationstakenatweatherbureaustations.This
ble atmosphere. Smoke leaving a source spreads rapidly stabilityclassificationhasbeenmadecompletelyobjective
verticallyandhorizontally.Asmechanicalturbulencein- sothatanelectroniccomputercanbeusedtocomputesta-
creases,theRichardsonnumberapproacheszero,andthe bility.Therearesevenstabilityclasses:
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684 AtmosphericDiffusionModeling
TABLEII Pasquill–GiffordStabilityCategories
Night
Surfacewind
(measuredat10m) Day Thinlyovercast
insolation or≥4/8 ≤3/8
(m/sec) (mph) Strong moderate Slight cloudinessa cloudinessa
<2 4.5 A A–B B — —
2–3 4.5–6.7 A–B B C E —
3–5 6.7–11.2 B B–C C D E
5–6 11.2–13.4 C C–D D D D
6 13.4 C D D D D
aThedegreeofcloudinessisdefinedasthatfractionofskyabovethelocalapparenthorizonthatiscovered
byclouds.
NOTES:
1.Insolationistherateofradiationfromthesunreceivedperunitofearth’ssurface.
2.Stronginsolationcorrespondstosunnymid-dayinsummer.Slightinsolationcorrespondstosimilarconditions
inmid-winter.
3.ForA–B,B–C,andsoforth,valuestaketheaverageofAandBvalues.
4.Nightreferstotheperiodfromonehourbeforesunsettoonehourafterdawn.
5.Regardlessofwindspeed,theneutralcategoryDshouldbeassumedforovercastconditionsduringdayornight
andforanyskyconditionsduringthehourprecedingorfollowingnight.
1mph=0.4470m/sec
1m/sec=2.237mph
A—extremelyunstable D—neutral
B—moderatelyunstable E—slightlystable
C—slightlyunstable F—moderatelystable
1. Extremelyunstable tiondirectedtowardtheground;anindexof−2isgiven
2. Unstable to the highest net negative radiation, which is radiation
3. Slightlyunstable directedawayfromtheground.Anunstableconditionoc-
4. Neutral curswithhighpositivenetradiationandlowwindspeed;
5. Slightlystable neutralconditionwithcloudyskiesorhighwindspeeds;
6. Stable stableconditionswithhighnegativenetradiationandlight
7. Extremelystable winds.Thenetradiationindexisdeterminedinreference
to Table IV and the following procedure.
Stabilityclassisdeterminedasafunctionofwindspeed
and the net radiation index and is listed in Table III. The 1. Ifthetotalcloudcoveris10/10andtheceilingisless
netradiationindexrangesfrom4to−2.Anindexof4is
than 7000 ft, use net radiation index equal to 0
giventothehighestpositivenetradiation,whichisradia-
(whetherdayornight).
2. Fornighttime(betweensunsetandsunrise):
TABLEIII StabilityClassAsaFunctionofNetRadiation
andWindSpeed a. Iftotalcloudcover≤4/10,usenetradiationindex
equalto−2.
Wind Netradiationindex b. Iftotalcloudcover>4/10,usenetradiationindex
speed
(knots) 4 3 2 1 0 −1 −2 equalto−1.
0,1 1 1 2 3 4 6 7
TABLEIV InsolationAsaFunctionofSolar
2,3 1 2 2 3 4 6 7 Altitude
4,5 1 2 3 4 4 5 6
Insolation
6 2 2 3 4 4 5 6
Solaraltitude(a) Insolation classnumber
7 2 2 3 4 4 4 5
8,9 2 3 3 4 4 4 5 60◦<a Strong 4
10 3 3 4 4 4 4 5 35◦<a<60◦ Moderate 3
11 3 3 4 4 4 4 4 15◦<a≤35◦ Slight 2
≥12 3 4 4 4 4 4 4 a≤15◦ Weak 1
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AtmosphericDiffusionModeling 685
3. Fordaytime: The height of the lowest inversion can also be used
a. Determinetheinsolationclassnumberasafunc- to describe the thickness of the planetary boundary
tion of solar altitude from Table IV. layer. In the daytime, temperature typically decreases
b. If total cloud cover ≤5/10, use the net radiation with height within the planetary boundary layer and
index in Table III corresponding to the insolation increases with height above it. Any region in which the
classnumber. temperature increases with height is called an inversion.
c. Ifcloudcover>5/10,modifytheinsolationclass In an inversion, turbulence is suppressed. Thus in the
numberbyfollowingthesesixsteps. daytime the mixed layer can be approximated by the
(1) Ceiling<7000ft,subtract2. heightofthelowestinversion.
(2) Ceiling≤7000ftbut<16,000ft,subtract1. Onclearnightswithweakwinds,infraredradiationes-
(3) If total cloud cover equals 10/10, subtract 1. tablishesaground-basedinversion.Onlythebottomofthe
(This applies only to ceilings ≥7000 ft since cases with boundarylayerbecomesturbulentundertheseconditions.
10/10 coverage below 7000 ft are considered in item 1 Thustheheightsoftheinversionandthemixedlayerare
above.) considerablydifferent.
(4) Ifinsolationclassnumberhasnotbeenmodi-
fiedbysteps(1),(2),or(3),assumemodifiedclassnumber
1. NeutralConditions
equaltoinsolationclassnumber.
(5) Ifmodifiedinsolationclassnumberislessthan Withinthemixedlayer,theEarth’srotationinfluencesthe
1,letitequal1. vertical wind shear and therefore the intensity of turbu-
(6) Use the net radiation index in Table III corre- lence. Thus, under neutral conditions, the depth of the
spondingtothemodifiedinsolationclassnumber. mixedlayershouldbeproportionaltothesurfacefriction
velocityu∗andtheCoriolisparameter f,
(cid:6)
In urban areas a large amount of heat is retained in u∗ = τ/ρ (8)
thebuildingsandpavementafterthesungoesdown.This f =2(cid:14)sinφ (9)
heatisreradiatedatnight.Duringthedaythesurfacesof
where τ is surface stress, ρ air density, φ latitude, and
anurbanareaaremorereflectiveandbecomehotter,thus
(cid:14)=7.29×10−5 sec−1, the rate of the earth’s rotation.
producingmoreconvectiveeddies.Forthesereasons,con-
Thethicknesscanthenbeestimatedas
vectiveturbulenceinanurbanareaisnotasinsignificant
as it is in the rural areas, and urban areas are rarely as h (cid:11)0.2u∗ (10)
f
stable.Thus,itispossibletocombinestabilitycategories
(6)and(7)—or(5),(6),and(7)—intoonecategorywhen Sinceu∗increaseswithwindspeedandsurfaceroughness,
h increaseswiththesetwoparametersaswell.
usingTurner’sclassificationforurbanmodeling.
2. UnstableConditions
D. ThePlanetaryBoundaryLayer
During the day the depth of the mixed layer grows as a
andItsThickness
functionofthesurfaceheatflux H.Iftheinitialdaytime
The earth’s surface exerts a drag on the atmosphere that surfacetemperatureisT0andthetemperatureattimetisT
resultsinanairflowthatissimilartothataroundasphere afteranadiabatictemperatureprofilehasbeenestablished,
inawindtunnel.Thisdraginfluenceswindspeedandthe thenconservationofheatenergyrequires:
(cid:7)
mixingofatmospheric-bornesubstancesuptoaheightof t C ρh(T −T )
as much as 2 km. In the daytime this region is typically Hdt = p 0 (11)
2
turbulent.Atnight,withweakwinds,thethicknessofthe t0
whereC isthespecificheatofair.
turbulent layer can be as low as a few tens of meters. p
From this we can calculate the depth of the mixed
Theregioniscalledtheplanetaryboundarylayerandisa
layer noting that the ratio (T −T )/h is equivalent to
regioninwhichtheatmosphereexperiencessurfaceeffects 0
λ−dT/dz:
throughverticalexchangeofmomentum,heat,andmass
(cid:8) (cid:9) (cid:10)
intheformofmoisture. 2 t Hdt 1/2
Thethicknessoftheplanetaryboundarylayerischar- h = t0 (12)
C ρ(λ−dT/dz)
acterizedbythethicknessoftheturbulentregionnextto p
theground.Sinceatmosphericpropertiesarewellmixed Equation (12) is a simplified expression which can
bytheturbulencewithinthelayer,thelayerissometimes be affected by atmospheric phenomena that cause the
calledthemixingdepth. inversionlayertochange.
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686 AtmosphericDiffusionModeling
3. StableConditions possibleandtakenasindependentofheight.ThevalueofL
dependsonlyonu∗andHandisabout−10monstrongly
Thethicknessofthelowestcontinuouslymixedturbulent
convective days, −100 m on windy days with some so-
layercanbeestimatedfrom
(cid:11) larheating,andapproachesinfinityinpurelymechanical
h (cid:11)0.4 u∗L (13) turbulence.Sinceheatflux H isdownwindatnight, L is
f positiveandsmallinlight-wind,stableconditions.
AccordingtotheMonin–Obukhovhypothesis,various
whereListheMonin–Obukhovlengthtobedefinedinthe
statisticsofatmosphericparameters,whennormalizedby
nextsection.Thisequationhashadrelativelylittlesuccess
properpowersofu∗andL,areuniversalfunctionsofz/L.
incorrelatingmeasuredvalues.
They define a nondimensional wind shear S, which is a
functionofz/L,
4. Holzworth’sMethod (cid:3) (cid:4)
k z∂u¯ z
Mixing depth has been determined by Holzworth of the S = a =φ (15)
u∗ ∂z m L
U.S.EnvironmentalProtectionAgencyinreferencetothe
morningandafternoon.Themorningmixingheightiscal-
1. NeutralAir
culatedastheheightabovegroundatwhichthedryadia-
baticextensionofthemorningminimumsurfacetemper- For neutral air, that is, in the case of purely mechanical
ature plus 5◦C intersects the vertical temperature profile turbulence,S=1.0and
observedat12:00GreenwichMeanTime.Theafternoon (cid:3) (cid:4)
mixingheightiscalculatedinthesamewayexceptthatthe u¯ = u∗ ln z z ≥z (16)
0
maximum surface temperature observed between 12:00 k z
a 0
and16:00localstandardtimeissubstitutedforthemini-
mumtemperatureplus5◦C.Acompilationofthemixing theclassicallogarithmicwindprofile.Hereka isthevon
Karmanconstant.Thevalueofthisconstantvariesfrom
depthdatahasbeenpreparedbytheU.S.Environmental
0.35–0.43;usuallyavalueof0.40isusedforthecasein
ProtectionAgency.
whichmeasurementshavenotbeenmade.
The term z is the value of z at which u¯ vanishes. It
0
E. SurfaceLayerWindStructure
representsturbulenteddysizeatthesurfaceandisamea-
There is no precise definition of surface layer. Typically sureofroughness.Thusz0istermedtheroughnesslength.
the vertical fluxes of momentum, heat, and moisture are Table V lists typical values of z0andu∗.
large at the surface and decrease to zero at the top of
the planetary boundary layer. It would be well to make 2. UnstableandStableAir
a definition of surface layer such that the fluxes remain
The relations below have been established for unstable
constant. However, a change of 10% influx is tolerable.
andstableair:
Assuming that decrease in flux is linear with height, a
feasibledefinitionofsurfacelayeristherefore10%ofthe
mixingdepth. TABLEV Valuesofz0andu∗forUseinVertical
Thenearconstancyoffluxesinthesurfacelayerimplies WindSpeedProfiles
thatwinddirectiondoesnotchangewithheight.Thusthe Typeofsurface z0incm u∗inm/seca
mean wind can be described by u¯ only, the mean wind
speed.Meanvariablesinthesurfacelayerarealsofunc- Smoothmudflats,ice 0.001 0.16
tionsonlyofheight,andthevariationwithheightiscon- Smoothsnow 0.005 0.19
trolled by the surface stress, the vertical heat flux at the Smoothsea 0.02 0.22
surface,andtheterrainroughness. Leveldesert 0.03 0.23
Becausebothmechanicalandconvectiveforcesdeter- Snowsurface,lawngrass 0.1 0.26
to1.0cmhigh
mine turbulence, Monin and Obukhov proposed a simi-
Lawn,grassto5cm 1–2 0.38–0.43
larity theory that introduced two scaling parameters, the
Lawn,grassto60cm 4–9 0.51–0.65
frictionvelocityu∗andthelength L,where
Fullygrownrootcrops 14 0.75
L =−ρCpTu3∗ (14) Pastureland 20 0.87
kgH Suburbanhousing 60 1.66
Forests,cities 100 2.89
withupwardheatflux H takenaspositive.Notethat L is
computed from measurements as close to the surface as aForu¯at2.0m = 5.0m/sec,ka=0.40.
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AtmosphericDiffusionModeling 687
(cid:3) (cid:4) (cid:3) (cid:4)
z z 1/4 TABLEVI EstimatesofPower pinVelocityProfile
φ = 1−15 (unstable) (17) Equation
m
L L
(cid:3) (cid:4) (cid:3) (cid:4) Stabilityclass
z z
φ = 1+5 (stable) (18)
m
L L A B C D E F
Equation(15)canthenbeintegratedtogive Urban 0.15 0.15 0.20 0.25 0.30 0.30
(cid:2) (cid:3) (cid:4) (cid:3) (cid:4)(cid:5) Rural 0.07 0.07 0.10 0.15 0.35 0.55
u¯ = u∗ ln z −ψ z (19)
m
k z L
a 0
where z should be interpreted as the average height of
where
(cid:3) (cid:4) (cid:7) thelayerinvolved.Forthecaseofneutralconditionswith
ψ z = z/L[1−φ (ζ)]dζ (20) strongwindsthisequationreducesto
m L m ζ
0 p =1/ln(z/z ) (26)
0
ApplyingEqs.(17)and(18)tothisintegralresultsinthe
Thegeometricmeanvalueofz overthelayertobedealt
followingequationsforthevelocitydistributioninunsta-
with is then more appropriate. For a windy day with
bleandstableflow:
(cid:12) (cid:3) (cid:4) (cid:2) (cid:3) (cid:4)(cid:5) some sunshine L=−100 m, with z =20 cm, a value
0
u¯ = u∗ ln z −2ln 1 1+ 1 appropriateforgrasslandswithsometrees,z=100mand
ka z0 2 φm zm=10m,and p=0.20withmechanicalturbulenceand
(cid:2)(cid:3) (cid:4)(cid:5)
0.145 when convection is added. It is apparent that con-
1 1
− ln 1+ vectioncanhaveasignificanteffectonthevalueof p.
2 φ2
m Valuesofphavebeenfoundthatrangefrom0.02–0.87.
(cid:3) (cid:4)(cid:13)
1 π Table VI lists values of p that can be used successfully up
+2tan−1 φ − 2 (unstable) (21) toaheightof200m.Thistablereflectstheeffectofsurface
m
(cid:2) (cid:3) (cid:4) (cid:3) (cid:4)(cid:5) conditionsbypresentingvaluesof p forurbanandrural
u¯ = u∗ ln z +5 z (stable) (22) areas.StabilityeffectsaregivenaccordingtothePasquill–
k z L Giffordclasses.
a 0
Itshouldbenotedthatforneutralconditions,theintegralof
Eq.(20)iszero,andEq.(19)reducestoEq.(16).Further- III. GAUSSIAN PLUME MODEL FOR
more,Eq.(19)canbesolvedforthefrictionvelocityu∗, CONTINUOUS SOURCES
k u¯
u∗ = ln(z/z )−a ψ (z/L) (23) Thephysicalpictureofasmokeplumecanbedeveloped
0 m by considering first a puff of smoke emitted as a point
Thus Eq. (23), with appropriate measurements, can be source. The puff is made up of a gas or small particles
used to determine u∗. Equation (23) shows that u∗ whichfollowthedirectionofthewindwiththespeedof
increaseswithwindspeedu¯ androughness z atagiven thewind.Smalleddiescausedilutionandexpansionofthe
0
heightz. puffaboutthecenterlineofitspathbypumpingfreshair
intothepuff.Largeeddiesbuffetthepuffaboutandtrans-
port it downwind. Linking together an infinity of puffs
3. ThePowerLawfortheWindProfile
results-intheformationofcontinuousreleasefromapoint.
It is common engineering practice to describe the wind An instantaneous photograph of a continuous plume
profilewithapowerlaw would show violent fluctuations and a meandering flow.
(cid:3) (cid:4) Figure 1 shows an instantaneous plume compared with
u¯ z p
= (24) time-averaged plumes. The 10-min average shows a
u¯m zm smoothed concentration profile; the 2-hr average is the
whereu¯ istheaveragewindspeedatheightz,u¯ average samebuthasareducedmaximumandismorespreadout.
m
wind speed measured at height z , and p power whose The easiest to describe mathematically is the time-
m
value is dependent upon stability conditions and surface averagedcase,whichappearsasaGaussiancurve.
roughness.Itcanbeshownthat The Gaussian plume model for atmospheric diffusion
has emerged as the most commonly used mathemati-
φ (z/L)
p = m (25) caltechniquefordispersioncalculationsfromcontinuous
ln(z/z0)−ψm(z/L) sources.Thereareanumberoffactorsinitsfavor.
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688 AtmosphericDiffusionModeling
FIGURE1 Comparisonofaninstantaneousplumewithatime-averagedplume.
1. Itproducesresultsthatagreewithexperimentaldata A. TheStatisticalApproachtotheTurbulent
aswellasanymodel. DiffusionProcess
2. Itcanbeobtainedasananalyticalsolutiontothe
Thedispersionofanensembleofsmallparticlesorcluster
deterministicmodelforconstantwindspeedand
ofmoleculesinspacemusttakeplacethroughtherandom
diffusioncoefficient.
motionsoftheensemble.Inthisrandomorstochasticpro-
3. Sincetheturbulentmotionoftheatmosphereis
cess, the variables are usually functions of time, and the
essentiallyarandomprocess,aGaussianformatis
valueofthevariablescanbespecifiedonlyintermsofa
suggestedtodescribethenatureoftheresultantaction
probabilitydistribution.
oftheatmosphere.
There are two ways to look at a cloud of diffusing
4. Otherconcentrationdistributionswouldfitthedata
molecules:theEulerianviewpointortheLagrangianview-
equallywell,buttheGaussianisnotmathematically
point. In the Eulerian viewpoint, the diffusion equations
complex,anditisconceptuallysatisfying.
wouldbederivedfromaconsiderationofconcentrations
andfluxatafixedpointinspace.Suchquantitiesareeasily
Therearealsoanumberoflimitationsthatreducethe observable.Itispossibletoarriveatthesameresultsfrom
effectivenessoftheGaussianmodelforgeneraluse.Wind theLagrangianviewpoint,whichfocusesonthehistoryof
shearisnotconsidered.Allmeteorologicalfactorsareas- the random movements of the diffusing material. Statis-
sumed to remain constant over space and over the aver- ticalpropertiesoftherandommotionswouldhavetobe
aging time used for the calculation. Calm winds are ex- mathematicallydescribedtoproduceusefulresults.
cluded,andbulktransportofthepollutantalongthemean TheEulerianapproachtodiffusioncanberegardedas
wind direction is assumed to be much greater than dif- relating to the ensemble average concentration field of
fusioninthemeanwinddirection.Notransient-response thediffusingmolecules.Inturbulentdiffusion,onemust
case can be handled by the Gaussian plume model, and distinguish between the ensemble average concentration
chemical and physical transformations of the plume can fieldandaninstantaneouslyobservableone.Thekinemat-
behandledonlyinaverysimplemanner.Itisassumedthat ics of diffusing particle movements leads to elucidation
theplumetravelsonflatterrain,oratbest,severalsimple of certain properties of the turbulent diffusion process.
terraincorrectionsaremade,andtheplumeisreflectedoff Theseconsiderationsareofprimeimportanceinarriving
thegroundandanydiffusionceilingimposed,suchasan at a satisfactory understanding of the complex turbulent
inversion. phenomenon.TheresultsoftheLagrangiantreatmentof
Someoftheselimitationscanbeovercomebytheuse therandommotioncomplementstheinformationderived
ofothermodeltypes.Forexample,predictionsofdown- fromtheEuleriantreatment.
wind concentrations from instantaneous releases or puff
diffusioncanbetreatedbyreferencetosimilaritytheory.
1. Taylor’sTheorem
Shear fields can be introduced in the numerical solution
ofthedeterministicmodel,andthestatisticaltheorycan Torelatetheensembleaveragefieldtothekinematicsof
assistindevelopingtheGaussianmodelitself. thediffusingparticlemovements,weturntoprobabilistic
