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A.R.A.P Report No. 725

PC-SCIPUFF Version 1.3Technical Documentation

R. Ian Sykes, et al.

Titan CorporationARAP GroupP.O. Box 2229Princeton, NJ 08543-2229

December 2000

Technical Report

CONTRACT No. DNA 001-95-C-0180

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SUMMARY

Version 1.3 of the Second-order Closure Integrated Puff model, SCIPUFF, isdescribed. SCIPUFF is an atmospheric dispersion model with a wide range ofapplication. The turbulent diffusion parameterization is based on second-orderturbulence closure theory, which relates the dispersion rate to velocity fluctuationstatistics. In addition to the average concentration value, the closure model provides aprediction of the statistical variance in the concentration field resulting from the randomfluctuations in the wind field. The variance is used to estimate a probability distributionfor the predicted value.

SCIPUFF uses a collection of Gaussian puffs to represent an arbitrary three-dimensional, time-dependent concentration field, and incorporates an efficient scheme forsplitting and merging puffs. Wind shear effects are accurately modeled, and puffs aresplit when they grow too large for single point meteorology to be representative. Thesetechniques allow the puff model to describe complex flow effects on dispersion, such asterrain-driven circulations.

SCIPUFF has been developed with a flexible interface, to describe many types ofsource geometries and material properties. Solid particles, liquid droplets, and gaseousmaterials are represented, with both primary and secondary evaporation mechanisms thatproduce vapor puffs as the droplets evaporate in the air or after deposition on the ground.Precipitation washout effects are also included for particles and droplets. SCIPUFFdescribes dynamic effects of buoyant rise due to thermal release or lighter-than-airmaterials, and also the effects of a dense cloud near the ground surface.

The model also uses several types of meteorological input, including surface andupper air observations or three-dimensional grid data. Planetary boundary layerturbulence is represented explicitly in terms of surface heat flux and shear stress usingparameterized profile shapes.

A number of enhancements and improvements to Version 1.0 are described.These include the capability to estimate the uncertainties in the meteorological inputs,due to forecast or extrapolation errors (see Sections 9.4 and 12.4). A model for flashingliquid/aerosol effects associated with the instantaneous release of volatile liquids stored

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under pressure is included (see Section 6.3). Casualty tables are available for nuclearweapon releases, using a world-wide population database (see Section 9.5). Otherenhancements include a generalized description of the radiation dose from nuclearfacility releases (Section 8.2), and a vegetative/urban canopy model for transport anddispersion (Section 11.4).

The technical basis of SCIPUFF and the revised structure of the input and outputfiles are described in this document. A section describing a number of test calculationsand comparison with observational data is also included (see Section 14).

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TABLE OF CONTENTS

Section PageSUMMARY .................................................................................................... III

1. INTRODUCTION ....................................................................................... 1

2. TRANSPORT AND DIFFUSION ............................................................... 3

2.1 PUFF MOMENT EQUATIONS. ..................................................... 32.2 CONCENTRATION FLUCTUATION VARIANCE. ...................... 12

2.3 SURFACE DEPOSITION................................................................. 252.4 SURFACE INTEGRALS. ................................................................ 26

3. NUMERICAL TECHNIQUES .................................................................... 283.1 PUFF SPLITTING SCHEME. ......................................................... 29

3.2 PUFF MERGING SCHEME. ........................................................... 33

3.3 PUFF INVERSION CAPPING. ....................................................... 37

3.4 FINITE-DIFFERENCE SCHEMES. ................................................ 38

3.5 ADAPTIVE TIME STEPS............................................................... 41

3.6 STATIC PUFFS............................................................................... 433.7 ADAPTIVE SURFACE GRIDS....................................................... 46

4. PARTICLE MATERIALS........................................................................... 48

4.1 PARTICLE MATERIAL PROPERTIES.......................................... 48

4.2 PARTICLE GRAVITATIONAL SETTLING. .................................. 48

4.3 DRY DEPOSITION......................................................................... 51

4.4 PRECIPITATION WASHOUT........................................................ 564.5 EFFECTS ON DIFFUSION. ............................................................ 61

5. LIQUID MATERIALS................................................................................ 635.1 LIQUID SPECIFICATIONS............................................................ 635.2 PRIMARY EVAPORATION........................................................... 645.3 SECONDARY EVAPORATION...................................................... 71

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TABLE OF CONTENTS (Continued)

Section Page6. DYNAMIC EFFECTS................................................................................. 77

6.1 BUOYANCY AND MOMENTUM RISE. ....................................... 776.2 DENSE GAS EFFECTS................................................................... 83

6.3 FLASHING LIQUID/AEROSOL EFFECTS.................................... 917. SOURCE SPECIFICATION........................................................................ 98

7.1 CONTINUOUS SOURCES. ............................................................ 987.2 INSTANTANEOUS SOURCES. ..................................................... 99

7.3 LIQUID POOL EVAPORATION. ................................................... 1007.4 FLASHING LIQUID/AEROSOL SOURCES. ................................. 102

8. SPECIAL MATERIALS.............................................................................. 104

8.1 ACTIVITY DECAY. ....................................................................... 104

8.2 NUCLEAR WEAPON MATERIALS. ............................................. 1058.3 NUCLEAR FACILITY MATERIALS. ............................................ 107

9. MODEL OUTPUT ...................................................................................... 1129.1 LOCAL CONCENTRATION VALUES. ......................................... 112

9.2 PROBABILISTIC CALCULATION................................................ 1149.3 POPULATION/AREA EXPOSURE. ............................................... 115

9.4 HAZARD AREAS. .......................................................................... 1179.5 CASUALTY OUTPUT. ................................................................... 118

10. METEOROLOGY SPECIFICATION........................................................ 124

10.1 BACKGROUND............................................................................ 124

10.2 OBSERVATIONAL INPUT. ......................................................... 124

10.3 GRIDDED INPUT. ........................................................................ 129

10.4 MEAN FIELD INTERPOLATION. ............................................... 130

10.5 MASS-CONSISTENCY. ............................................................... 131

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TABLE OF CONTENTS (Continued)

Section Page11. PLANETARY BOUNDARY LAYER....................................................... 134

11.1 GOVERNING PARAMETERS. .................................................... 134

11.2 SPECIFYING PBL PARAMETERS. ............................................. 137

11.3 PBL MEAN PROFILES................................................................. 147

11.4 VEGETATIVE/URBAN CANOPY PROFILES............................. 149

12. TURBULENCE SPECIFICATION ........................................................... 15212.1 PLANETARY BOUNDARY LAYER. .......................................... 152

12.2 VEGETATIVE/URBAN CANOPY LAYER.................................. 15412.3 MESOSCALE/SYNOPTIC SCALE............................................... 155

12.4 METEOROLOGICAL UNCERTAINTIES.................................... 15813. FILE FORMATS....................................................................................... 164

13.1 INPUT FILES. ............................................................................... 16413.2 METEOROLOGY INPUT. ............................................................ 185

13.3 OUTPUT FILES. ........................................................................... 201

14. MODEL TEST CALCULATIONS............................................................ ERROR! BOOKMARK NOT14.1 SHORT RANGE DIFFUSION....................................................... ERROR! BOOKMARK NOT14.2 LONG RANGE DIFFUSION......................................................... ERROR! BOOKMARK NOT DEFI14.3 DYNAMIC RISE EFFECTS. ......................................................... ERROR! BOOKMARK NOT14.4 DENSE GAS EFFECTS................................................................. ERROR! BOOKMARK NOT

15. REFERENCES.......................................................................................... 236

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FIGURES

Figure Page

2-1 Normalized fluctuation dissipation timescale as a function ofdimensionless time for a shear-distorted Gaussian puff. The solid line isthe analytic solution (2.50). Dashed line is the model prediction from(2.51) and (2.52)............................................................................................. 18

3-1 Maximum dimensionless error, s, from splitting a single Gaussian puffas a function of separation distance,

s. The error is relative to themaximum value in the original Gaussian, and the separation distance ofthe puffs after splitting is relative to the original Gaussian spread. Resultsare shown for both one-dimensional and two-dimensional splits. .................... 32

3-2 Schematic illustration of puff reflection for a split below the groundsurface............................................................................................................ 32

3-3 Schematic illustration of the adaptive multi-grid for locating puffs. Gridcell numbers are represented as C-n, and two levels of refinement areshown. See text for a description of the cell storage rules .............................. 35

3-4 Schematic illustration of the use of static puffs to represent a continuoussource. The shaded puff is stored as the release description at the end ofthe static phase ............................................................................................... 44

4-1 Scavenging coefficient versus particle diameter. Solid lines denote theSeinfeld (1986) model described above and the dashed lines denote themodel of Nieto et al. (1994) (with a Marshall and Palmer raindrop sizedistribution). The precipitation rate (

rP ) is equal to 0.5 and 25 mmh1 for

drizzle and heavy rain, respectively. ............................................................... 59

4-2 Scavenging coefficient versus particle diameter. Solid lines denote themodel implemented in SCIPUFF and the dashed lines denote the modelof Horn et al. (1988). The precipitation rate (

rP ) is equal to 0.5 and 10

mmh1 for drizzle and moderate rain, respectively .......................................... 60

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FIGURES (Continued)

Figure Page

5-1 Terminal velocity, Sherwood (Sh), and Nusselt (Nu) numbers for agentVX. Solid line is SCIPUFF algorithm, short dash lines are fromNUSSE4. Left column (a) is at 100m altitude, column (b) is at 10km ............ 67

5-2 Evolution of droplet diameter (D), altitude (H), fall velocity (V) anddroplet temperature (T) for GB droplets released at a height of 10km withdifferent initial diameters. Solid line is SCIPUFF, dashed line isNUSSE4......................................................................................................... 68

6-1 Schematic illustration of slumping dense cloud. ............................................. 85

6-2 Simplified velocity field representation for a dense cloud. .............................. 85

11-1 Schematic illustration of the idealized velocity profile in the planetaryboundary layer................................................................................................ 149

11-2 Canopy velocity profile shapes from (11.47) and (11.48) compared withthe experimental data of Cionco (1972). ......................................................... 151

12-1 Turbulence profiles within the canopy layer for different values of thecanopy parameter, c . (a) horizontal turbulence (shear drivencomponent), (b) vertical turbulence component. ............................................. 156

12-2 Normalized error variance growth for forecast and persistenceassumptions.................................................................................................... 160

131 Sample PC-SCIPUFF input file ...................................................................... 165

13-2 Sample PC-SCIPUFF release scenario file...................................................... 173

13-3 Sample PC-SCIPUFF meteorology scenario file............................................. 176

13-4 Example PC-SCIPUFF sampler location file .................................................. 179

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FIGURES (Continued)

Figure Page

13-5 General structure of a RAD file. Keywords are in upper case; lower caseitalics indicate user-specified input. Brackets denote optional input ............... 182

13-6 Header structure for a meteorology observation file. Records andvariables in brackets are optional or required only if nvarp is given ............... 185

13-7 General structure of numerical data in a meteorological observation file.(a) Without nvarp specified; (b) with nvarp specified ................................... 193

13-8 Fortran pseudo-code for reading a formatted MEDOC input file..................... 195

13-9 General structure of an HPAC gridded meteorology file ................................. 197

13-10 General structure of a terrain file .................................................................... 199

13-11 Fortran statements to write a surface output file header (HPAC3.2 only) ........ 202

13-12 Fortran statements to write a surface output file timebreak ............................. 203

13-13 Fortran statements to calculate field values from the surface output file.......... 205

13-14 Example of a sampler output file .................................................................... 209

14-1 Comparison between SCIPUFF predictions (solid lines) and the PGTdispersion curves for categories A through F .................................................. 211

14-2 Comparison between the observed maximum concentrations andSCIPUFF predictions for the passive releases in the Model Data Archive.Various experiments are indicated by the letter codes ..................................... 214

14-3 Comparison between SCIPUFF predictions and the observational data ofWeil et al. (1993)............................................................................................ 215

14-4 Comparison between SCIPUFF predictions and the observational data ofMikkelsen et al. (1988)................................................................................... 216

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FIGURES (Continued)

Figure Page

14-5 Comparison between SCIPUFF predictions and the observational data ofHgstrm (1964) ............................................................................................ 217

14-6 Comparison between SCIPUFF predictions of mean concentration andthe observational data from CONFLUX 1994................................................. 219

14-7 Comparison between SCIPUFF predictions of standard deviation to meanconcentration ratio and the observational data of CONFLUX 1994................. 220

14-8 Comparison of mean concentration between SCIPUFF predictions andETEX data (Mosca et al., 1997). Contours are 0.01, 0.02, 0.05, 0.1, 0.2,0.5, 1, 2, 5, 10, 20, and 50 ngm3 .................................................................... 222

14-9 Comparison between 3-hr average concentrations from SCIPUFFpredictions (dashed) and ETEX data (solid line) at 9 selected locations(Mosca et al., 1997). Concentration is in ngm3 ............................................. 223

14-10 Comparison between total integrated dose from SCIPUFF predictions andETEX data (solid line) at all locations (Mosca et al., 1997). Dose is in ngs m3. (a) compares each station, (b) shows ordered data, unmatched bylocation. ......................................................................................................... 226

14-11 Momentum (non-buoyant) jet centerline height, z , as a function ofdownstream distance for a range of R. Symbols are the data of Gordier(1959) - from Hirst (1971), solid lines are the SCIPUFF predictions............... 228

14-12 Comparison between SCIPUFF predictions of momentum jet centerlineheights for a range of R (solid lines) and the one-third law, (13.3), withtwo values of (short dashes, = 0.6; long dashes, = 0.5)........................... 228

14-13 Comparison between SCIPUFF predictions for buoyancy-dominated jetcenterline heights (solid lines) and theory given by (13.5) (dashed lines).Case A: R = 10, F U Fb a m

2 4 0 01= . ; Case B: R = 10, F U Fb a m2 4 1= ; Case

C: R = 1, F U Fb a m2 4 1= ................................................................................. 229

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FIGURES (Continued)

Figure Page

14-14 Comparison of SCIPUFF buoyancy-dominated jet centerline heights in auniformly stratified background (solid lines) with (13.6) (dashed lines).For all cases, R = 10. Case A: F U Fb a m

2 4 0 01= . , F N Ub a4 0 00572= . ;

Case B: F U Fb a m2 4 1= ,F N Ub a

4 0 00572= . ; Case C: F U Fb a m2 4 1= ,F N Ub a

4 0181= . ............................................................................................. 230

14-15 Centroid height of a light bubble released into a neutral quiescentbackground. Symbols are the data of Mantrom and Haigh (1973), solidlines are the SCIPUFF predictions. The Reynolds number for theexperiments are based on the initial bubble diameter and the terminalvelocity of a corresponding non-entraining sphere. Case A is for a singlepuff release with a Gaussian spread of D0/2; Case B is for a releasecomprised of 136 puffs representing a uniform distribution of mass withina spherical bubble of diameter D0. (a) Early time behavior; (b) expandedscale showing late time behavior .................................................................... 232

14-16 Comparison between the observed maximum concentrations andSCIPUFF predictions for the dense gas releases in the Model DataArchive. Various experiments are indicated by the letter codes...................... 234

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TABLES

Table Page

4-1 Precipitation rates in mm/hr associated with precipitation types...................... 61

11-1 Suggested values for surface roughness (Saucier, 1987). ................................ 139

11-2 Relationship between stability index, PGT stability class, Monin-Obukhov length, L, and an assumed boundary layer depth, zi, if notspecified as input............................................................................................ 140

11-3 Suggested values for surface albedo as a function of land-use and season(Paine, 1987). ................................................................................................. 143

11-4 Suggested values for Bowen ratio as a function of land-use and season(Paine, 1987). ................................................................................................. 143

131 Description of recognized keywords and associated input for RAD files. ........ 183

132 Valid keywords and associated input strings for PROFILE files. .................... 186

133 Meteorological variables and their units recognized in SCIPUFF. .................. 189

134 Valid landuse indices and associated landuse types for terrain files................. 200

141 Model performance statistics for SCIPUFF prediction of the ETEXsampler concentrations for different time periods. .......................................... 225

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SECTION 1

INTRODUCTION

SCIPUFF is a Lagrangian transport and diffusion model for atmosphericdispersion applications. The acronym SCIPUFF stands for Second-order ClosureIntegrated PUFF and describes two basic aspects of the model. First, the numericaltechnique employed to solve the dispersion model equations is the Gaussian puff method(Bass, 1980) in which a collection of three-dimensional puffs is used to represent anarbitrary time-dependent concentration field. Second, the turbulent diffusionparameterization used in SCIPUFF is based on the second-order turbulence closuretheories of Donaldson (1973) and Lewellen (1977), providing a direct connectionbetween measurable velocity statistics and the predicted dispersion rates.

The Lagrangian puff methodology affords a number of advantages foratmospheric dispersion applications from localized sources. The Lagrangian schemeavoids the artificial diffusion problems inherent in any Eulerian advection scheme, andallows an accurate treatment of the wide range of length scales as a plume or cloud growsfrom a small source size and spreads onto larger atmospheric scales. This range mayextend from a few meters up to continental or global scales of thousands of kilometers.In addition, the puff method provides a very robust prediction under coarse resolutionconditions, giving a flexible model for rapid assessment when detailed results are notrequired. The model is highly efficient for multiscale dispersion problems, since puffscan be merged as they grow and resolution is therefore adapted to each stage of thediffusion process.

The efficiency of SCIPUFF has been improved by the implementation of adaptivetime stepping and output grids. Each puff uses a time step appropriate for resolving itslocal evolution rate, so that the multiscale range can be accurately described in the timedomain without using a small step for the entire calculation. The output spatial fields arealso computed on an adaptive grid, avoiding the need for the user to specify gridinformation and providing a complete description of the concentration field within thecomputational constraints under most conditions.

The generality of the turbulence closure relations provides a dispersionrepresentation for arbitrary conditions. Empirical models based on specific dispersion

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data are limited in their range of application, but the fundamental relationship betweenthe turbulent diffusion and the velocity fluctuation statistics is applicable for a muchwider range. Our understanding of the daytime planetary boundary layer velocityfluctuations provides reliable input for the second-order closure description of dispersionfor these conditions. For larger scales and upper atmosphere stable conditions, theturbulence description is based on climatological information, but the closure frameworkis in place to accept improvement as our understanding of these regimes improves. Theclosure model has been applied on local scales up to 50km range (Sykes et al., 1988) andalso on continental scales up to 3000km range (Sykes et al., 1993c).

The second-order closure model also provides the probabilistic feature ofSCIPUFF through the prediction of the concentration fluctuation variance. In addition togiving a mean value for the concentration field, SCIPUFF provides a quantitative valuefor the random variation in the concentration value due to the stochastic nature of theturbulent diffusion process. This uncertainty estimate is used to provide a probabilisticdescription of the dispersion result, and gives a quantitative characterization of thereliability of the prediction. For many dispersion calculations, the prediction is inherentlyuncertain due to a lack of detailed knowledge of the wind field and a probabilisticdescription is the only meaningful approach.

The following sections will describe the technical basis of the SCIPUFFprediction, as well as the treatment of additional physical phenomena, such as particledeposition rates. The current model can treat multiple materials, including gases,particles, and liquid droplets, and multiple sources, both continuous plumes andinstantaneous puffs.

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SECTION 2

TRANSPORT AND DIFFUSION

2.1 PUFF MOMENT EQUATIONS.

2.1.1 Gaussian Moment Definition.

SCIPUFF uses a Gaussian puff representation for the concentration field of adispersing contaminant. In this section, we will restrict discussion to gaseous materials;solid particles and liquid materials are discussed in Sections 4 and 5. A three-dimensional Gaussian is completely described by its spatial integral moments up tosecond-order, and can be written in the form

cQDet

x x x xij i i j jx

=

3 2 1 21

212/ /( ) exppi (2.1)

For general atmospheric dispersion problems, we must also consider the effects of theground surface and the capping inversion at the top of the planetary boundary layer, whichare usually represented as reflective surfaces. The reflection description for thegeneralized Gaussian will be described in Section 9.1. For our present discussion, werestrict attention to the moment equations, which are independent of the shape assumption.

Using an angle bracket to denote an integral over all space, the spatial moments in(2.1) are defined asZeroth moment - mass

Q c= (2.2)First moment - centroid

Q x c xi i= (2.3)

Second moment - spread

Q c x x x xij i i j j = b g c h (2.4)

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Note that this generalized tensor definition of the moments is completelyindependent of any specific coordinate system, and can therefore describe an arbitrarilyoriented Gaussian shape. The full moment description requires 10 numbers to representthe mass, the 3 centroid coordinates, and 6 spread moments since ij is a symmetricsecond-rank tensor. The moments describe the unreflected Gaussian shape, but localconcentration values must account for surface reflection as described in Section 9.1. Theevaluation of the ambient meteorological variables should also strictly use the truecentroid location of the reflected puff, but this involves complex calculations. Wetherefore simply use the larger of z and 33 as the height (above the ground surface) atwhich the meteorological field variables, such as velocity and turbulence, are evaluated.

The specific Gaussian variation (2.1) applies to an individual puff, but in generalthe local concentration field will be composed of a sum of contributions from a numberof such puffs. The details of the concentration calculation are described in Section 9.1.In this section, we describe the transport and diffusion model equations for the individualpuff moments.

2.1.2 Moment Transport Equations.

The advection-diffusion equation for a scalar quantity in an incompressible flowfield can be written as

( ) 2ii

cu c k c S

t x

+ = +

(2.5)

where u ti x,a f is the turbulent velocity field, k is the molecular diffusivity, and S representsthe source terms. Surface deposition, described in Section 2.3, is an example of a sourceterm, but other terms associated with different materials are described in Sections 4 and 5.

The atmospheric velocity field is generally turbulent, so we use the Reynoldsaveraging technique to define a mean and a turbulent fluctuation value. Denoting themean by an overbar and the fluctuation by a prime, we have u u u= + , and a similardecomposition can be applied to the concentration, c. The Reynolds averagedconservation equation for the mean scalar concentration is thus

( ) ( ) 2i ii i

cu c u c k c S

t x x

+ = + +

(2.6)

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where u ci is the turbulent concentration flux.

At this point, the averaging in the overbar involves an unspecified ensemble ofvelocity fields. In some cases, the ensemble may be considered to include theconventional planetary boundary layer turbulence only, and the mean wind, u , is definedin the usual sense. The dispersion framework developed here is more general, however,and can include larger scale variability, or may be more restrictive and ignore meanderingmotions. The specification of the turbulent fluctuation values will be discussed morefully in Section 12.

Neglecting source terms, S, for the present, (2.6) is readily integrated over thespatial dimensions to give conservation of mass

dQdt

= 0 (2.7)

where Q now represents the integrated mean concentration, c . Note that (2.7) applies toan inert conserved tracer, with no loss or transformation processes; these effects will bedescribed later.

Equations for the higher spatial moments of c are obtained similarly, but requiresome assumption about the spatial variation of the velocity field. The simplestassumption is a constant velocity based on the value at the puff centroid, but this neglectsany effect of shear. We therefore use a linear representation for the local velocity field

( ) ( ) ( ) ii i j jj

uu u x x

x

= +

x x (2.8)

where the velocity gradient is also evaluated at the centroid location.

Multiplying (2.6) by xi and integrating by parts gives

iiiu cu cd x

dt Q Q

= + (2.9)

where the first term on the right-hand side represents transport by the mean wind and thesecond term is the turbulent drift. The latter arises from correlation between the velocityfluctuations and the concentration fluctuations. Substituting for u from (2.8) andassuming any symmetric spatial distribution for c, the velocity gradient term vanishes togive

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dxdt

uu c

Qi

ii

= +

xa f (2.10)

The second-moment equation is obtained by multiplying (2.6) by x xi j , where = x x xi i i , and integrating by parts to give

i j j iij j iik jk

k k

x u c x u cd u udt x x Q Q

= + + +

(2.11)

where the first two terms represent the shear distortion of the puff and the second twoterms represents the turbulent diffusion. The pairs of terms are required for theij-symmetry of the second moment tensor.

This completes the specification of the moment equations in the absence ofexternal source terms, S, but we have introduced turbulent flux moments in (2.10) and(2.11). This is the essence of the turbulence closure problem, in which the Reynoldsaverage operator always introduces higher-order fluctuation correlations. The turbulentfluctuations must be modeled empirically at some level and we next discuss theturbulence closure technique used in SCIPUFF.

2.1.3 Turbulence Closure Diffusion Model.

2.1.3.1 Second-order Closure Framework. The diffusion model in SCIPUFF is based onsecond-order turbulence closure, which provides a transport equation for the second-order fluctuation terms. First-order closure prescribes the turbulent fluxes in terms of thelocal mean gradients using an empirical turbulent diffusivity, but a more general relationcan be obtained from a higher-order closure. It is not our intention to review turbulenceclosure theory here, we only provide the basic model description; the interested readercan find detailed discussions in the literature, e.g., Mellor and Herring (1973), Launder etal. (1975), Lewellen (1977).

The equations for the puff moments involve the turbulent flux of concentration,and a rigorous conservation equation can be derived for this quantity from the scalar andmomentum equations. Neglecting the molecular diffusion terms, the flux transportequation can be written in the form

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0

i ii j i i j j i j

j j j j i

u gc pu c u u c u u u c u u c c c

t x x x x x T + = +

(2.12)

This equation involves higher-order terms, such as the triple correlation and the pressurecorrelation, which must be modeled empirically. We use the model of Lewellen (1977) toform a closed equation for the turbulent flux, giving

t

u c ux

u c u uc

xu c

u

x xv q

xu c

A q u c gT

c

i jj

i i jj

ji

j jc

ji

ii

+ = + F

HGI

KJ

+

0

(2.13)

where A = 0.75 and vc = 0.3 are empirical model constants. The turbulent velocity scale,q, is defined as q u ui i

2= , and

is the turbulent length scale; these quantities will bediscussed more fully below. The buoyancy term uses the Boussinesq approximation, andg gi = 0 0, ,a f is the gravitational acceleration, T0 is the reference temperature, and is thepotential temperature fluctuation.

The general equation (2.13) for the turbulent fluxes can be integrated spatially toprovide transport equations for the flux moments in (2.10) and (2.11). However, themultiple tensor indices give a large number of correlations to consider in the general case.We therefore introduce some restrictions for the atmospheric dispersion cases that allowus to neglect many of the correlations. First, we only consider the vertical component ofthe turbulent drift, u ci , since advection by the mean wind will generally dominate inthe horizontal directions. Second, the only off-diagonal component of the flux momenttensor, x u ci j , to be considered will be the symmetric horizontal term

X x u c x u c12 1 2 2 1= + (2.14)

In general, the off-diagonal terms represent puff distortions due to velocity covariances.Such covariances are usually generated by wind shear, which will dominate the distortionprocess through the mean shear terms in (2.11). The horizontal distortion is retained forthe special case of large-scale horizontal fluctuations, as described in Section 12.3. Withthese restrictions, we now describe the horizontal and vertical diffusion representations.

2.1.3.2 Horizontal Diffusion. Equations for the two diagonal horizontal moments arederived directly from (2.13) as

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ddt

x u c Qu A q x u c = 2

(2.15)

andddt

y v c Qv A q y v c = 2

(2.16)

where we have used the component values for the velocity and position vectors, i.e., = x x y zi , ,a f and = u u v wi , ,a f .

Under steady homogeneous conditions, (2.16) gives a very simple prediction forthe effective horizontal diffusivity and lateral spread. The diffusivity in the y-directioncan be defined as

Ky v c

Qy =

(2.17)

Then, if K y = 0 at t = 0, the solution for K y is

K v ty = 2 1 exp

(2.18)where the horizontal timescale = Aqa f . The solution for the lateral spread, y

222= ,

is then

y v t t2

02 22= + + exp

(2.19)where 0 is the initial spread.

The solution (2.19) is consistent with Taylor's (1921) exact analytic result forturbulent diffusion in homogeneous stationary conditions, with an exponential form forthe Lagrangian velocity autocorrelation function and an integral timescale of .

The above discussion uses a single length scale, , to describe the horizontalturbulence spectrum, but this is insufficient for a proper description of the atmosphericspectrum. The wide range of length scales demands a more complicated treatment, sinceeach length scale is associated with a distinct timescale. We consider three distincthorizontal turbulence components, representing shear-driven turbulence, buoyancy-driven turbulence, and a large-scale component representing mesoscale or synoptic scalevelocity fluctuations. The three components will be denoted by subscripts S, B, and L,respectively. The specification of the three components is discussed in Section 12. The

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three components are assumed to be statistically independent, so that the correlations cansimply be summed, giving

= + + y v c y v c y v c y v cS B L

(2.20)

and a similar relation for the x-component.

We further assume that the boundary layer components are isotropic (in thehorizontal plane), so that we only require a single correlation variable for the S and Bterms. Buoyancy generation is clearly isotropic in the horizontal, but the shearcomponent implies a preferred direction. It is also well known that the streamwiseturbulence intensity is larger than the lateral component. However, in the presence of amean wind shear, we expect the explicit velocity gradient terms in (2.11) to dominate thespread in the streamwise direction and we therefore simplify the turbulencerepresentation using the lateral component only.

Evolution equations for the horizontal velocity correlation integrals are exactlylike (2.16), except that the turbulent velocity and length scales represent the specificcomponent, i.e.,

ddt

y v c Q v A q y v cS S

S

SS

= 2

(2.21)

Similar equations are used for the B and L components. The total correlation integrals arethen written as

= + + x u c y v c y v c x u cS B L

(2.22)

= + + y v c y v c y v c y v cS B L

(2.23)

The large scale component is not assumed to be isotropic, and therefore requiresboth x- and y-components, and also the off-diagonal X12 defined by (2.14), which satisfies

ddt

X Q u v A q XL LL

12 122= (2.24)

Details of the specification of the velocity fluctuations and length scales areprovided in Section 12.

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10

2.1.3.3 Vertical Diffusion. The vertical diffusion parameterization is more complicatedthan the horizontal for two reasons. First, the inhomogeneity is more pronounced in thevertical direction, so that vertical gradients cannot be ignored, and second, the buoyancyforces introduce an additional phenomenon in the vertical direction. Dealing first withthe first moment of the vertical flux, z w c , integration of the full second-order closureexpression (2.13) gives

ddt

z w c Q w A q z w c gT

z cV

V

= + 2

0 (2.25)

where the subscript V denotes the vertical component for the turbulence velocity andscale.

A similar equation can be derived for the temperature correlation, using the modelequations of Lewellen (1977), as

ddt

z c Q w bs q z c ddz

z w cV

V

= 2

(2.26)

where b = 0.125 and s = 1.8 are turbulence model constants.

Equations (2.25) and (2.26) represent the full second-order closure model for thevertical flux, but give oscillatory solutions for stable temperature gradients and can leadto negative values of the diffusivity. A more robust scheme is obtained by relaxing thevertical flux moment toward its equilibrium value on the appropriate timescale, similar tothe horizontal equation (2.16). The equilibrium (steady-state) solution for the verticaldiffusivity, defined as

z

z w cK Q

= (2.27)

is obtained from (2.25) as

KAq

wgT

Gzeq V

Vzeq

= +F

HGI

KJ 2

0

(2.28)

where

Gz c

Qz =

(2.29)

Using a similar assumption for the temperature correlation, we obtain

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11

2eq eqVz z

V

G w Kbsq z

=

(2.30)

so that the equilibrium diffusivity is given by

2

02

20

2

112

V

Veq Vz

V V

V

g ww

T q bsK

Aq gAbs T zq

+

=

+

(2.31)

A simple equation for the vertical flux moment that satisfies the equilibriumequation and evolves on the proper timescale is therefore

ddt

z w c Aq QK z w cV

Vzeq

=

(2.32)

and this is the model equation employed in SCIPUFF. The use of the late-timeequilibrium value (2.31) in (2.32) over-estimates the early time growth of the diffusivityunder convective conditions, however, since the buoyancy term in (2.25) develops on thesame turbulence time scale as the diffusivity itself. We therefore approximate theequilibrium diffusivity in (2.32) using (2.28) but with a scaled value for Gzeq . Thebuoyancy term is scaled by the ratio of Kz to Kz

eq, so that the growth of the buoyancy

correlation is represented as being proportional to the growth of the diffusivity.

The zeroth moment of the vertical flux, the turbulent vertical drift, is important inregions with vertical gradients of the turbulent correlations. The role of the drift term inthe model is to account for non-uniformity in the vertical diffusivity, and the equilibriumdrift velocity must be equal to the gradient of the equilibrium diffusivity. We thereforedefine a simplified equation for the flux moment as

eqV z

V

q Kdw c A Q w c

dt z

=

(2.33)

It is emphasized that (2.33) is required for consistency with the simplifieddiffusivity moment equation (2.32), and is not directly derived from the second-orderclosure model. The main purpose of (2.33) is to maintain a uniform concentrationdistribution across a well-mixed layer with when puffs are splitting, as described inSection 3.1. Vertical variations in the turbulence fields lead to differential diffusion rates,which can distort an initially uniform concentration profile unless the drift term is

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12

included. A similar effect is found in stochastic particle dispersion models as discussedby Thomson (1987).

2.2 CONCENTRATION FLUCTUATION VARIANCE.

2.2.1 Concentration Variance Equation.

The probabilistic aspect of the SCIPUFF dispersion prediction is based on thesecond-order closure model for the concentration fluctuation variance. The dispersion ofany species in a turbulent velocity field is a random process since the turbulentfluctuations are effectively chaotic and cannot be measured or predicted in detail. Thescalar concentration is therefore a stochastic quantity, with a probability distribution thatdepends on the distribution of velocity fluctuations. Traditional deterministic estimatesof atmospheric dispersion only provide a single concentration value as a function ofspace and time, and this corresponds to the mean value, c , for some definition of thestatistical ensemble. The mean value is the first moment of the probability distribution,and contains no information about the statistical variability in the prediction. Highermoments are required to give a quantitative description of the variability. Theprobabilistic aspect of the SCIPUFF dispersion prediction is based on a transportequation for the statistical variance in the concentration value, that is the second momentof the probability distribution. The transport equation is based on the same second-orderturbulence closure theory described in Section 2.1 for the mean dispersion rates. Themean and the variance are then used to provide a probabilistic prediction using aparameterized probability density function, described in Section 9.3.

The concentration fluctuation variance equation can be obtained from the scalarmass conservation equation (2.5) in the form

( ) ( ) 22 2 2 2 22 2i i iii i i

cc cu c u c u c k k c

xt x x x

+ = + (2.34)

where the first three terms on the right-hand side are identified as turbulent production,turbulent diffusion, and turbulent dissipation, respectively. The last term is directmolecular diffusion of the scalar variance, and is negligible for the small values ofmolecular diffusivity relative to atmospheric dispersion scales. However, it is important

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13

to note that the only dissipation mechanism is molecular, as is evident from considerationof the total scalar variance equation

( ) ( ) 22 2 2 2 22i iii i

ccu c u c k k c

xt x x

+ = + (2.35)

where it can be seen that all the terms except dissipation are in flux form.

For very high values of the Schmidt number, i.e., very small values of , themolecular terms in both equations are effectively equal to the turbulent scalar dissipation

2

2ci

ck

x

= (2.36)

which is modeled using a dissipation timescale as

c c

c=

2

(2.37)

From the above, it is clear that the statistical variance in concentration due torandom fluctuations in the velocity field is controlled by the turbulent flux and turbulentdissipation rate. SCIPUFF employs closure models for these terms to provide aprediction of the scalar fluctuation variance. The critical aspect of the prediction lies inthe determination of the dissipation timescale, c , associated with the turbulent eddiesresponsible for the dispersion, and the recognition that large scale eddies can simplymeander the complete concentration field without producing a reduction in the variance.

2.2.2 Scalar Fluctuation Dissipation Timescale.

The key aspect of the scalar variance prediction is the dissipation timescale.Proper characterization of the dissipation timescale allows an accurate prediction of theconcentration fluctuation variance, as has been demonstrated in comparison withlaboratory data (Sykes, Lewellen, and Parker, 1986) and also with large-scaleatmospheric dispersion observations (Sykes et al., 1993c). The second-order closuremodel for the dissipation rate was originally developed by Sykes et al. (1984) using thelaboratory data of Fackrell and Robins (1982). A fundamental discovery in that studywas that the scalar dissipation scales were an internal property of the scalar field itself,rather than being determined exclusively by the velocity fluctuations. Essentially,

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14

velocity eddies with scales larger than the scalar plume or cloud will meander the entirescalar distribution but not cause any nonlinear cascade of scalar fluctuation variance ontosmaller scales. The turbulent cascade process is only driven by eddies with scales similarto the instantaneous plume size. This is not the case for persistent mean wind shear,which can distort a cloud by means of the continued stretching, and this phenomenon isdiscussed separately in Section 2.2.3.

The general form for the scalar dissipation time scale is1

c

c

c

q=

(2.38)

where qc and c represent the velocity and length scales for the scalar fluctuations. Thevelocity scale is obtained from a simplified description of the turbulent kinetic energyspectrum, and the length scale is predicted from a transport equation. Before discussingthe details of these parameterizations, however, we introduce a generalization of thedissipation model to account for the anisotropy between horizontal and vertical directions,and the differences between plume and puff meandering.

The scalar variance dissipation model introduced by Sykes et al. (1984, 1986)uses a single estimate of the velocity and length scales to define the dissipation timescalein (2.38). This is appropriate for the case of plume dispersion in two dimensions withnear-isotropic turbulence. Under the more general conditions of larger scale horizontaldispersion and arbitrary sources, we require a characterization of the different dissipationrates associated with each direction. The vertical direction must be distinguished, and wealso need two horizontal scales. The horizontal information is needed to distinguishbetween plume-type sources, where the diffusion occurs in two spatial dimensions, andpuff-type sources, which diffuse in all three dimensions. As part of the concentrationfluctuation prediction, we therefore require three length scales in addition to thefluctuation variance. The internal scales are denoted as c , cH , and cV for the twohorizontal and vertical directions, respectively.

The initial conditions for the internal length scales will be described as part of thesource definition in Section 7. In this section, we describe the dissipation length and timescale parameterization. The two scales, c and cV , are used to characterize thehorizontal and vertical concentration fluctuation length scales. The second horizontalscale is used to determine the dissipation timescale only. We use these two concentration

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15

length scales to estimate the dissipation velocity scales, using a simplified Kolmogorovspectrum assumption to determine the appropriate energy for each of the ambientturbulence populations. Thus, the horizontal velocity scale for the concentrationfluctuations is modeled as

q q q qc cL cB cS2 2 2 2

= + + (2.39)

where

q u vcL L LL

c

c

L

2 2 22 23

= + FHG

IKJ

FHG

IKJ

F

HGG

I

KJJe j min ,

(2.40)

q u vcB B BB

c

c

B

2 2 22 2 3

= + FHG

IKJ

FHG

IKJ

F

HGG

I

KJJe j min ,

(2.41)

q u v wcS S S SS

c

c

S

2 2 2 22 23

= + + FHG

IKJ

FHG

IKJ

F

HGG

I

KJJe j min ,

(2.42)

and the vertical dissipation fluctuation velocity scale is defined as the smaller of the twoestimates

q qcV VVB

cV

cV

VB

( ) min ,113

=

F

HGI

KJF

HGG

I

KJJ

(2.43)

qK

bscVzeq

cV

( )2=

(2.44)

where the second estimate accounts for the reduced dissipation under stable conditions.The turbulence model constants, b and s, were introduced in the previous section and takethe values 0.125 and 1.8, respectively (Lewellen, 1977). The equilibrium verticaldiffusivity, Kz

eq, is defined in (2.28).

Using the two velocity scales defined above, the scalar dissipation timescale isthen modeled as

1 c

c

c

c

cH

cV

cV

bs q q q= + +F

HGI

KJ (2.45)

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16

The internal fluctuation scales grow with the turbulent diffusion, and the internalvelocity scale gives a measure of the growth rate. The model for the two horizontalinternal length scales is

ddt

ddt

q q qc cH cL cL cB cB cS cS

= = + +122 2 2 2 2 2 c h (2.46)

where cL = 0.25 for c L< , otherwise cL = 0.098, and similarly for the other twocomponents.

The vertical scale growth is represented by

ddt

q q q

bsq q qcV cV cV cV

cV cV cV

=

0 introduces a factor, fR, in the integral I , where

f erf z z e erf z

e erf z e erf z

Rg

g g

= + +

+ +

12 33 33

33 33

d ie j c h

d i d i

(2.71)

Here,

z z zR

z z zR

z z zR R

g R z zR

g R z zR

g R R z zR R

= +

= +

= + +

= + FHG

IKJ

= + FHG

IKJ

= + + +F

HGIKJ

d i

d i

d i

d i

d i

d i d i

2

2

2

4

4

4

33

33

33

33

33

33

(2.72)

and

3333

1

1=

c h(2.73)

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22

In principle, the interaction calculation described above should be made for allpossible pairs of puffs. This is clearly impractical for a large number of puffs and isusually unnecessary since many pairs will be separated in space and the overlap integralwill be negligible. SCIPUFF uses a three-dimensional grid with a multi-level adaptivetechnique in the horizontal plane to determine near-neighbor puffs. The technique isdescribed in Section 3.2 in connection with the merging process. The multi-leveltechnique allocates each puff to a horizontal grid cell based on the puff size. Interactionsare only calculated for puffs in neighboring cells on this grid. Since puff growth islimited by the splitting process described in Section 3.1, the size of a puff is constrainedto be less than the grid dimensions. Contributions to the overlap integral from puffs moreremote than the neighbor cells are therefore small.

The overlap integral is only calculated for puffs on the same horizontal grid, sothat the interaction between puffs with different horizontal sizes is neglected. This isequivalent to an assumption of no correlation between such puffs, since the total varianceis calculated as a sum of the contributions from each puff. If the variance is computedwithout interaction, then a simple sum of variances implies uncorrelated fluctuations.This is a reasonable assumption for puffs with different size, since they must havedifferent transport histories and are generally uncorrelated.

We also note that SCIPUFF can calculate multiple species and/or particle sizegroups in a single computation and each puff is assigned a specific type descriptor todesignate its material properties. Interactions are only computed for puffs of the sametype, and correlations between different types are not calculated. Any subsequentcombination sum of material types or particle size groups must estimate a cross-correlation if the variance is required. For the special case of multiple particle size bins,two variance calculations are stored so that statistics for the total material concentrationsare available. SCIPUFF accounts for interactions between all puffs in the same sizegroup, and also between all puffs of the same material and any size group. This providesa variance calculation for the concentration of the individual size group, and also avariance for the total concentration obtained as the sum over all size bins.

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23

2.2.5 Conditional Average Dispersion.

The second-order closure framework of SCIPUFF provides a prediction of thestatistical variance in the concentration value, as described above. This variance gives ameasure of the uncertainty in the concentration prediction and depends on the velocityfluctuation statistics used to determine the dispersion rates. Under some circumstances, ameaningful quasi-deterministic prediction can be obtained by restricting the definition ofthe turbulent velocity fluctuations. By ignoring the large-scale meandering motions inthe turbulence, a conditional average result can be generated in which the uncertainty inthe plume/cloud location is ignored. It must be recognized that the conditional averagesimply neglects the meandering component of the turbulence and provides a prediction ofthe small-scale diffusion effects.

The conditional average is obtained by specifying an averaging time, Tav. Thistime scale is converted into a length scale, av, using the local wind speed, and used torestrict the velocity variance to the diffusive range of scales. The averaging time shouldbe thought of as a sampling time, but can also be considered to be the release duration ifthis is shorter than the sample time. Thus, an instantaneous measurement or aninstantaneous release both imply a zero averaging time.

A reduction factor is applied to the velocity variances if the length scale of thevelocity fluctuations is larger than the conditional average scale, which is defined as thelarger of

av and the instantaneous cloud scale. Turbulent motions on scales smaller thanthe instantaneous cloud are always diffusive in character, so this part of the spectrummust be included in the conditional average. When the instantaneous cloud is smallerthan the conditional average scale, the averaging time determines the range of scales. Asimple power law reduction, consistent with the Kolmogorov inertial range behavior ofthe energy spectrum, is assumed. Thus the boundary layer turbulence used in thedispersion model equations in Section 2.1.3 is defined as

=

u uBT BHT

HB

2 22 3

(2.74)

=

v vBT BHT

HB

2 22 3

(2.75)

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24

=

w wBT BVT

VB

2 22 3

(2.76)

=

w wBT BVT

VB

4 3

(2.77)

where the conditional filter scales are

HT HB c av= min , max ,

(2.78)

VT VB cV av= min , max ,

(2.79)

The new velocity correlations and length scales, defined in (2.74)-(2.79), are thenused in the dispersion model equations. A similar procedure is applied to the large-scalevelocity correlations and length scale.

The conditional average length scale, av, is defined in terms of the averagingtime scale as

av avV T= 0 03. (2.80)

where V is the local velocity scale, which includes both the mean velocity and theturbulence. Thus

V u u ui i i2 2

= + (2.81)

and the conditional boundary layer component uses the boundary layer turbulence. Thelength scale for the large-scale component uses the large-scale velocity variance in (2.81).

The conditional averaging representation is described in detail and compared withfield data by Sykes and Gabruk (1997); the field data comparisons are included in Section14 of this document.

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25

2.3 SURFACE DEPOSITION.

2.3.1 SCIPUFF Implementation.

Surface deposition of the gaseous material introduces a source term into theconcentration equation (2.5) and therefore modifies the puff moment equations. A simpleconstant deposition velocity, vD, is used, so the puff mass conservation equation (2.7)becomes

dQdt

Fs= (2.82)

where the mass flux at the surface is defined as

F v c dAs Dz

=

=

z0

(2.83)

Other deposition effects for particles and liquid droplets are discussed in Sections 4 and 5.Integrating the mean surface concentration for the generalized Gaussian puff gives

c dA QDD

z x y x yz H

s s s s

=

z = + + +0

332

112

222

1222

pi exp

(2.84)

where ij is defined in (2.62) as half the inverse of the puff moment tensor ij , D is thedeterminant of ij , and

D

xz

D

y zD

H

s

H

s

H

=

=

=

11 22 122

13 22 23 12

23 11 13 12

b g

b g

(2.85)

The expression in (2.84) includes a factor of 2 to account for the surface reflection, asdiscussed in Section 9.1.

The flux term in (2.82) can be used to define a deposition time scale

ss

QF

= , (2.86)

which is then used to decay all the other concentration-weighted puff moments, such asx u ci j .

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26

2.4 SURFACE INTEGRALS.

2.4.1 Surface Dose or Dosage Integration.

The integrated surface dose is one of the optional output choices, and is defined as

Tt

x y t c x y t dt, , , , ,a f a f= z 00

(2.87)

This quantity is accumulated on an adaptive grid as described in Section 3.7.

The probabilistic aspect of the SCIPUFF prediction is also incorporated into thesurface integral fields through a calculation of the variance of the integral, T2 . From(2.87), we obtain the equation for the mean-square fluctuation integral as

= z z Tt t

t dt dt c t c t20 0

a f a f a f (2.88)

which involves a two-time correlation for the concentration field. We have omitted thespatial coordinates in (2.88) since the time variation is the only concern in the presentdiscussion.

We make the assumption that the time correlation is exponential, i.e.,

( ) ( ) ( )2 expc

t tc t c t c t

T

=

(2.89)

where Tc is the integral time scale for the concentration fluctuations. Substituting in(2.88), and assuming that the time integration covers the range of the autocorrelationfunction, we obtain

= z Tt

ct dt c t T t2 2

0

2a f a f a f (2.90)

The estimation of the integral timescale for the surface concentration fluctuations,Tc, is currently based on the simplified meandering plume analyses of Gifford (1959) andSykes (1984). These analyses show that the effect of intermittency is to reduce theintegral time scale dramatically from the Eulerian wind fluctuation time scale and also tointroduce a logarithmic correction factor. The intermittency is measured by theconcentration fluctuation intensity, and Sykes (1984) suggests the approximation

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27

T T cc

c

ccP E= +

F

HG

I

KJ

2

2

2

21 2ln (2.91)

for the meandering plume case, where TE is the Eulerian velocity time scale.

For the meandering plume model, the fluctuation intensity can be related to theconcentration fluctuation scale, c , and the scale of the ambient turbulent velocityfluctuations. The horizontal scale of the turbulence, y , is estimated using an energy-weighted average of the three horizontal components. Thus

( ) ( ) ( )2 2 2 2 2 2 22

L L L B B B S S S S

yT

u v u v u v w

q

+ + + + + + = (2.92)

where qT2 is the total turbulent energy. A generalized approximation for the intensity is

then given by

Icy

c cH

=

F

HGI

KJ

132

(2.93)

Using this estimate of the intensity, the time scale (2.91) can be written as

TTI

IcP

E

c

c=

+lnln1

2b g

a f(2.94)

for c y . For c y> , we use T TcP E c y= , and the proportionality constants in(2.94) are chosen to match the two estimates at c y= . We estimate the Eulerian scaleof the turbulence as

2 2 20.7 yE

H

Tu v q

=

+ +(2.95)

The estimate in (2.94) is appropriate for continuous plumes, but instantaneoussources may have shorter time scales due to the short duration of passage. We thereforedefine a "time-of-passage" time scale as

2 2 20.7 cHcH

H

Tu v q

=

+ +(2.96)

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28

using the second horizontal concentration fluctuation scale. This time scale will be largefor a continuous source, since cH represents the streamwise length scale. The correlationtime scale is defined as the smaller of the two estimates, i.e.,

T T Tc cP cH= min ,b g (2.97)

The variance contribution is accumulated on the same surface grid as the mean,using the Gaussian shape of the mean concentration, so that

( ) ( )2

2 2 2T T cc

t t t c tT Q

+ = + (2.98)

In addition, the effective length scale of the fluctuations is calculated using thevariance as a weighting factor applied to c .

2.4.2 Surface Deposition Integration.

The mass deposited on the surface is computed in a very similar way to thesurface dose. In this case, the deposition is defined as

D Dt

x y t v c x y t dt, , , , ,a f a f= z 00

(2.99)

where vD is the total deposition velocity, including gravitational settling for particles ordroplets and turbulent dry deposition.

The mean and variance of the deposition are calculated exactly the same as thedose but with the extra factor of the deposition velocity. Conservation of total mass isensured by estimating the deposited mass as the actual loss of mass from the puff in thetime step t.

SECTION 3

NUMERICAL TECHNIQUES

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29

3.1 PUFF SPLITTING SCHEME.

The puff moment evolution equations given in Section 2.1 generally increase thesize of the puff through turbulent diffusion and elongation along the direction of the windshear. As the puff grows, the local representation of the turbulence and velocity fieldsusing the puff centroid location becomes increasingly inaccurate. When themeteorological fields are inhomogeneous, the accuracy of the calculation can only bemaintained by splitting puffs into smaller components that can sample the variations inthe meteorology explicitly. A grid-based method for splitting puffs was presented bySykes and Henn (1992), where the moment method of Egan and Mahoney (1972) wasextended to include some shear effects. In this scheme, grid cells can be thought of aseach containing a Lagrangian puff. As the puffs move and spread into neighboring gridcells in a time step, the masses are redistributed so as to maintain a single puff withineach cell. We wish to avoid any numerical grid in the current method, however, so theredistribution cannot be based on the rectangular grid cells.

The three diagonal moments, , give the effective length of the puff along eachcoordinate direction, and we choose to split a puff in the x-direction, say, when 11

2> H . Here H represents the limit for the horizontal spread and should be chosenso that the linearization of the velocity and turbulence fields is valid for a Gaussian withsmaller spread. Similar splitting criteria apply for the other two coordinate directions,using H in the y-direction and V in the z-direction. We wish to represent the originalGaussian puff with several smaller, overlapping puffs that conserve all the puff momentsand only change local concentration values by a small amount. For a split in the x-direction, the original puff is replaced by two smaller puffs as follows. The new centroidlocations are displaced by a fraction, r, of the puff spread in the x-direction, and by adistance proportional to the off-diagonal moment in the other two coordinate directions.Thus

x x r1 1 11( ) = (3.1)

x x r2 212

11

( )

= (3.2)

x x r3 313

11

( )

= (3.3)

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30

where = {1, 2} corresponds to the plus and minus sign, respectively. The diagonalmoments for the new puffs are obtained using the following relations

11 1121( ) = r

(3.4)

22 22

2122

11 22

1( ) = FHG

IKJ

r (3.5)

33 33

2132

11 331( ) =

F

HGI

KJr (3.6)

and the new off-diagonal moments are

12 1221( ) = r

(3.7)

13 1321( ) = r

(3.8)

23 23

212 13

11

( )=

r (3.9)

Conservation of all puff moments is ensured by this procedure, and the momentsof the new puffs are all reduced and are also realizable. The realizability constraintsinvolve the Schwartz inequality between the diagonal and off-diagonal moments, e.g., 12

211 22 and the strictly positive value for the determinant. These properties can be

verified directly from the above relations.

The generalized puff description contains other information in addition to themoments discussed above. The additional variables fall into two categories, either aconserved puff integral property (similar to the puff mass) or a puff value property (suchas turbulence length scale). Integral properties are simply divided equally between thetwo new puffs, and the value properties are assigned equally to both.

The reduction in puff size is controlled by the parameter r, but a larger reductionin puff size reduces the amount of overlap between the new puffs and gives a poorerrepresentation of the original Gaussian shape. For a puff with diagonal moment tensor,the splitting scheme described by (3.1)-(3.9) creates a pair of Gaussian puffs with adimensionless separation

s

r

r= =

21 2

(3.10)

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31

where is the distance between the new centroid positions and = 111

is the newspread in the x-direction.

In general, the difference in concentration distribution resulting from the splittingoperation is difficult to specify for an arbitrary Gaussian, but the simple spherical puffwith diagonal moments 2 provides an idealized case. Figure 3-1 shows the maximumdifference between the original Gaussian and the sum of the two smaller Gaussians aftersplitting in one-dimension, i.e., x-direction only, and also for a two-dimensional split inboth x and y, which creates 4 puffs. The difference is given relative to the maximumconcentration value in the original Gaussian, and is plotted as a function of thedimensionless separation, s. It is clear that the one-dimensional case provides betteroverlap between the new puffs and smaller differences for the same separation. Amaximum local concentration change of 20% requires a dimensionless separation of lessthan 2.25 for a one-dimensional split, but must be less than 1.8 for two dimensions. Inpractice, there will usually be additional overlap from other puffs and we have found thatr = 0.75. i.e., s = 2.25, gives the optimum size reduction with acceptable overlap. Indiffusive applications, the puff size increases after splitting and reduces the effectiveseparation and overlap errors.

The vertical displacement involved in splitting a puff can cause creation of puffsbelow the ground surface, so a reflection condition must be imposed in these cases. Thereflection adjustment is applied if the newly created puff is below the ground. In thiscase, the puff centroid is relocated by reflecting the vertical coordinate and positioningthe centroid along the original displacement line, as illustrated in Figure 3-2.

A similar condition is imposed at the boundary layer capping inversion, z = zi, toprevent diffusion of material through the interface from below. The inversion reflectionis applied when a puff within the PBL splits and creates a new puff with z zi> . In thiscase, the puff will be reflected back along the line joining its centroid to the original puffcentroid.

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32

0 0.5 1.0 1.5 2.52.00

0.1

0.2

0.3

0.4

0.5

s

1-d split

2-d split

sFigure 3-1. Maximum dimensionless error, s, from splitting a single Gaussian puff as a

function of separation distance,

s. The error is relative to the maximumconcentration value in the original Gaussian, and the separation distance ofthe puffs after splitting is relative to the original Gaussian spread. Resultsare shown for both one-dimensional and two-dimensional splits.

Originalpuffcentroid

Splitpuff-1centroid

Splitpuff-2centroid

Reflectedpuff-1centroid

h

h

Figure 3-2. Schematic illustration of puff reflection for a split below the ground surface.

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33

3.2 PUFF MERGING SCHEME.

Unfortunately, the splitting process can rapidly lead to excessive numbers of puffsunless some form of merging is employed to reduce the numbers. In general, splitting thepuff distribution will create many overlapping puffs that can be merged together as asingle Gaussian. The merging rules are actually much simpler than the splitting rules,since the only requirement is moment conservation. Thus, if superscripts 1 and 2represent two puffs deemed sufficiently close to each other to merge, then they can becombined as follows:

Q Q Q= +( ) ( )1 2 (3.11)Q x Q x Q xi i i= +( ) ( ) ( ) ( )1 1 2 2 (3.12)

( ) ( )(1) (1) (1) (1) (2) (2) (2) (2)ij ij i j ij i jQ Q x x Q x x = + + + (3.13)where ( ) ( )i i ix x x

= .

Other puff variables are combined according to their type. Integral properties arecombined, like the puff mass in (3.11), as a sum of the two values. Value properties aremass weighted and combined like the puff centroid, (3.12).

The difficulty associated with merging lies in deciding which puff pairs areeligible. In general, if we have N puffs, there are N2 possible pair combinations and itrapidly becomes impractical to consider all such pairs when N is large. Some techniqueshave been suggested for ordering randomly located, multi-dimensional data with theobjective of determining nearest neighbors, (e.g., Boris 1986) but we have chosen to usean adaptive multi-grid location scheme.

The basic concept of the grid location scheme is to assign each puff to a particulargrid box, then a search for near-neighbors can be carried out by searching over theneighboring grid cells. There are two technical difficulties that arise in the application ofthis concept. First, the grid cell size should be determined by the size of the puff itself, sothat a puff overlaps only a finite number of neighboring grid cells. Second, we mustaccount for an arbitrary number of puffs within each grid cell. Thus, we cannot simplyallocate storage for a fixed grid with a fixed number of puffs in each cell.

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The first problem is solved by means of an adaptive grid in the horizontal plane.In atmospheric cases, the vertical resolution is restricted by the stable stratification, so afixed grid size of V is used. However, the horizontal scale varies widely so an initialcoarse grid is defined with size 0 =

H. Each cell of this coarse grid can be individuallyrefined to create 4 additional cells with size 1 =

0/2. The new cells are simply added tothe end of the list of cells, and a pointer from the 0-cell contains the location of the firstof the 4 refined cells. This procedure can be continued as far as computer storage allows.

This successive refinement procedure has the advantage that all coarser levels ofthe grid are available in addition to the most refined level. Each cell of the grid storestwo integer variables, the pointer for the subsequent refinement (if it exists) and thenumber identifier of the puff located in that cell. The choice of refinement level forlocating a puff on the adaptive grid is determined by the largest of the horizontalmoments, i.e., the grid size n =

0/2n such that

n n< 4 1 max (3.14)where max

2= max 11,22( ) . The factor of 4 ensures that n 2 max , so non-neighbor

cells are separated by at least 2 max and their neglect in the overlap calculation of Section2.2.4 is justified.

A schematic illustration of the puff allocation is shown in Figure 3-3, where asingle puff with index number is located in cell number 12 (denoted by C12). The puffis allocated to the second level of grid refinement, and the grid system generated by thispuff is also shown. The grid cells are indicated by the C-number and the two numbersstored for each cell are given in parentheses below. The first number is the first grid cellof a refined block of four cells, and is zero for no refinement. The second number is theindex of the puff contained in this cell, and a zero indicates no puff in the cell. Whenboth numbers are zero for a cell, we have omitted the number pair in some cases. Thus,all cells except C12 contain no puffs and show the second number as zero. Cell 4 showsthe refinement to the first level and points to cell 5, which is further refined and points tocell 9. Clearly, a large number of puffs will produce a much more complicated grid andindex structure, but this simple adaptive refinement provides an efficient location schemefor locating arbitrary collections of puffs.

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35

In locating the puffs on the grid, the index number of the first puff placed in aparticular cell is stored in the grid cell storage location, but if other puffs share the samecell then their indices must be saved elsewhere. Unlimited numbers of puffs can beaccommodated by using a linked list; each puff carries a pointer to designate the nextpuff in the cell. Thus, if puff number j is found to share a grid cell with a previousoccupant, say puff number i, then the list pointer in puff i is set to point to puff j. Inpractice, the entire list for the cell must be scanned and the new puff added to the end ofthe list. However, since the cells are adapted to the puff size it is highly unlikely that asingle cell will contain more than a few puffs.

1

0 2

puff-C3(0,0)

C4(5,0)

C1(0,0)

C2(0,0)

C5(9,0)

C6

C7 C8

C9

C12(0, )

C10

C11

Figure 3-3. Schematic illustration of the adaptive multi-grid for locating puffs. Grid cellnumbers are represented as C-n, and two levels of refinement are shown.See text for a description of the cell storage rules.

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36

The use of the adaptive grid technique in conjunction with the linked list providesa very efficient method for sorting a large number of puffs with arbitrary locations andmoments, giving a compact list of puff pairs to test for possible merging. We note that asearch over the list in each grid cell finds most of the candidate pairs, although puffsallocated to different cells but lying very close to each other across the boundary will bemissed. In practical terms, it is more efficient to miss some merges rather than scanmultiple cells. The actual merging of a candidate pair is based on the overlap integral ofthe two Gaussian functions, which can be written in the form

ID

A= 12

32

1212

pia fb gexp (3.15)

where D12 is a combination of the two determinants and the exponential argument dependson both puff moments and their separation. This integral is computed as part of theconcentration fluctuation variance calculation (see Section 2.2). For two identical,spherical Gaussians with separation distance, and diagonal moments, 2 , the argumentcan be written very simply as 2 212 4A = . A pair is merged if A12 < m2 4, so that mis a measure of the centroid separation relative to the size of the two puffs. We note thatthe separation is not a direct indication that the two puffs are nearly coincident, since theycould be of different size or shape. However, the multigrid sorting scheme ensures thatthe pair have similar overall size and it is unlikely that two such puffs will arrive at thesame spatial position with very different shear histories. A more reliable merge criterioncan be developed to account for the shape differences as well as the centroid separation,but we have not found this to be necessary yet.

As discussed in connection with the splitting process, the merging criteriondetermines the extent of puff overlap, since a small value for m will maintain moreoverlap before merging a pair of puffs. It is important to ensure consistency between thesplit and merge criteria, however, in the sense that a newly created split pair should notsatisfy the merge criterion. In practice, we maintain a distinct gap between the twocriteria, and a reasonable value for m is found to be 1.73 in conjunction with the splitcriterion, r = 0.75, i.e., s = 2.25.

In addition to the overlap criterion, merging is restricted to puffs of the samematerial type and particle size bin, and position relative to the boundary layer cap, zi.Puffs within the mixed layer, z < zi , are not merged with puffs outside the mixed layer

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37

since this could result in artificially diffusing material across the capping inversion. Thepuffs are also required to have similar concentration fluctuation scales, c, so that puffsof significantly different size cannot be merged even if they are spatially coincident.

3.3 PUFF INVERSION CAPPING.

Under daytime mixing conditions, the atmospheric boundary layer is typicallycapped by a temperature inversion, which provides a limit on the vertical extent of theturbulent layer. The depth of the mixed layer is denoted by zi, and the specification ofthis inversion height is described in Section 11.2.4. Although there is entrainment acrossthis layer, the flux through the inversion is relatively small and the vertical diffusivitiesare also usually much smaller in the free atmosphere above the boundary layer. Thesimplest assumption is therefore to approximate the inversion as an impenetrable surfaceand apply a perfect reflection condition for the puff concentration distribution.

The level at which the reflection is applied cannot simply be defined as the localinversion height, zi, since the inversion can move up or down as the turbulent layer growsduring the daytime or collapses in the evening. If the capping level were reduced, thenthe local concentrations would increase as the mass became compressed into a thinnerlayer, and this would violate the mass conservation law. We therefore associate acapping height, zc, with each puff and adjust the capping level in response to the localboundary layer changes experienced by the puff. A capping height of zero is used toindicate no reflection.

We first define a capped boundary layer as one in which the surface heat flux isnon-negative and the overlying atmosphere is stable. .i.e., a positive potential temperaturegradient. The specification of the surface heat flux is described in Section 11.2. Uponrelease of a puff, zc is initialized as

zz

c

i=RST

if the boundary layer is capped0 otherwise

so that zero indicates no cap. At each timestep the puff cap is modified according to thefollowing rules:

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Case A: Boundary layer capped, and z zi

This implies the puff centroid lies inside a capped boundary layer, and if the puffis effectively contained within the layer then z zc i= . Containment is defined by thecondition

z zz i+ 2

If the puff is not contained then it extends beyond the current boundary layer. In this casea cap must still be defined since the vertical diffusivity is calculated at the puff centroidwithin the boundary layer and upward growth of the puff at this rate is inappropriate. Ifthe puff was previously capped at zc , then the new cap is set to the larger of zc and zi. Ifthe puff was not capped, then the new cap is defined as

z zc z= + 2

Case B: All other situations

This includes puffs above the boundary layer or uncapped boundary layers, suchas stable conditions. If the puff is already uncapped then it remains in that state,otherwise a check is made to test whether the cap should be moved or completelyremoved. If the puff is sufficiently below its cap, i.e., z zz c+ 2 , then the cap isremoved. Otherwise, the new cap is defined as

z z w tc cold

w= + +( )

.0 01b g

The cap is allowed to move upward due to advection by the mean vertical velocitycomponent, and also grows due to turbulent diffusion by the vertical velocity fluctuations.The 1% factor on the vertical velocity standard deviation is included to allow the cap todissipate slowly under stable nocturnal conditions.

3.4 FINITE-DIFFERENCE SCHEMES.

The ordinary differential equations for the puff moments and properties aregenerally solved using a simple, explicit forward time-difference. However, all lineardamping terms are represented implicitly, so that an equation of the form

ddt

F = (3.16)

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39

is represented numerically as t t t

tF t t t t

+ = +

a f a f a f a f a f (3.17)

i.e.

t tt t F t

t t+ =

+

+

a f

a f a f

a f1(3.18)

The puff mass, turbulence moments, and mean-square concentration integrals canall be expressed in the general form (3.16).

The puff centroid is advanced using the second-order Adams-Bashforth scheme

x t t x tt

u t u t ti i i i+ = + a f a f a f a f2

3 (3.19)

This technique gives accurate trajectory calculations for idealized velocity fields such asthe solid-body rotation flow, and avoids the numerical diffusion produced by thetrajectory divergence errors of a first-order scheme.

The puff second-moment tensor requires special treatment, since it is important tomaintain a realizable tensor with a positive determinant. The effective volume of thegeneralized Gaussian described by the six independent moments, ij, is proportional tothe square root of the determinant D = Det( ), and this quantity is used to determine localconcentrations and interaction terms. The volume of a fluid parcel is conserved in anincompressible flow, and in the absence of diffusion, conservation of the determinant isimplied by (2.11) if the velocity field is solenoidal. It is very important to conserve thedeterminant numerically, and we now describe a simple scheme with this property.

The evolution of ij over one timestep is computed using a sequence of steps,with the intermediate results from step m denoted as

( ) ijm

. The initial step, with m = 0,accounts for the diagonal velocity gradients using the relation

(0) ( ) expn

uut

x x

= +

(3.20)

where the Greek subscript implies no summation. Here, t is the numerical timestep andthe superscript n denotes the value from the previous timestep, i.e., after n timesteps. The

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vanishing velocity divergence ensures conservation of D after this step. The six off-diagonal velocity gradients are then applied sequentially in pairs using the representativescheme

(0) (0) (0)11 12 22

(1)11

2

2

1

u ut t

y yu v

ty x

+ +

=

(3.21)

(0) (0) (0)22 12 11

(1)22

2

2

1

v vt t

x x

u vt

y x

+ +

=

(3.22)

(0) (0) (0) 2 (0)12 22 11 12

(1)12

2

1

u v u vt t t

y x y xu v

ty x

+ + +

=

(3.23)

(0) (0)13 23

(1)13

2

1

ut

yu v

ty x

+

=

(3.24)

( )( ) ( )

231

230

130

21=

+

tv

x

tu

yv

x

(3.25)

and the remaining diagonal moment, 33 , is unaffected by these two components of thevelocity gradient. The remaining two off-diagonal velocity gradient pairs are included ina similar manner by cycling the indices and velocity components, 1 2 3 , and theadvanced time level is finally obtained as ( ) ij

3. This scheme is more complicated than the

single gradient component used in earlier versions of SCIPUFF, but it reduces the errorsarising from sequential application of the shear components and still preserves the valueof the determinant exactly. The relations (3.21)-(3.25) are strictly only first-order accuratein time, but the conservation of the determinant is a more important property than theaccuracy for long-term stability of the scheme.

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3.5 ADAPTIVE TIME STEPS.

The description of dispersion over a wide range of spatial scales, from a smallsource up to hundreds of kilometers for example, also involves a wide range of timescales. Small spatial scales are usually associated with short time scales, so close to alocalized source an accurate description of the dispersion process requires a small timestep. As the plume or cloud spreads onto larger scales, however, the time scales will alsoincrease. The discrete Lagrangian framework of the Gaussian puff model allows anefficient treatment of the time dependence since puffs can be advanced individually usingan appropriate step for each one. The only interaction between puffs is through theoverlap integrals in the interaction terms, and these are the only terms that require specialconsideration when using multiple time steps.

A large time step, tL, is defined for the calculation. This time step must resolvethe meteorological changes and any other time dependence in externally specifiedparameters, such as source variations. The large time step is successively halved toobtain appropriate puff time steps, and the multiple time steps are forced to bring all thepuffs into synchronization at intervals of tL. External parameters are updated only onthe large time step. Within a large time step, however, an individual puff uses its ownlocally determined step to integrate for the period tL.

At the beginning of a large time step, the list of puffs is scanned to determine thesmallest step required for this period. This gives the number of small internal steps forthe period, which must be of the form 2M since the local steps are obtained by successi