soil mixing depth after atmospheric deposition. i. model development and validation

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Atmospheric Environment 45 (2011) 4133e4140

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Atmospheric Environment

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Soil mixing depth after atmospheric deposition. I. Model developmentand validation

Peter Drivas a,*, Teresa Bowers a, Robert Yamartino b

aGradient, 20 University Road, Cambridge, MA 02138, USAb Integrals Unlimited, 509 Chandler’s Wharf, Portland, ME 04101, USA

a r t i c l e i n f o

Article history:Received 6 December 2010Received in revised form15 April 2011Accepted 9 May 2011

Keywords:Soil mixing depthAtmospheric depositionMathematical modelingDiffusion theoryEffective diffusion coefficient

* Corresponding author. Tel.: þ1 617 395 5000.E-mail address: [email protected] (P. Driv

1352-2310/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.atmosenv.2011.05.029

a b s t r a c t

Knowledge of a soil mixing depth, or the migration depth of various pollutants in soil, is necessary toassess the soil chemical concentration resulting from atmospheric deposition of a specific air emissionsource. A mathematical model has been developed that describes the depth and time behavior of thesoil concentration of an inert chemical after atmospheric deposition on surfaces. The soil mixing modelis based on one-dimensional diffusion theory, and analytic solutions have been derived for the cases of:(1) instantaneous surface deposition; (2) continuous surface deposition; and (3) a finite period ofcontinuous surface deposition, followed by a deposition-free time period. Comparisons of the modelwith measured soil depth profiles resulting from atmospheric deposition showed good agreement forlead, cesium, and dioxins. The best-fit effective diffusion coefficients in undisturbed soils varied from0.5 cm2 yr�1 to 2 cm2 yr�1. The soil mixing depth was found to be a strong function of the atmosphericdeposition time period. Calculated soil mixing depths in undisturbed soils were 2 cm after one year,5 cm after five years, and 10 cm after 20 years of continuous atmospheric deposition on the soilsurface.

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1. Introduction

Particulates released from an air emission sourcewill deposit onnearby soil surfaces as a function of particle size, particle density,meteorological parameters, and surface conditions. Many airdispersion models, such as the widely-used AERMOD and CALPUFFmodels recommended by the U.S. Environmental ProtectionAgency (USEPA, 2005a), can predict the atmospheric depositionrate from a specific emission source on surfaces in terms of massper unit time per unit surface area (e.g., g yr�1 cm�2). Even inundisturbed soils, the chemicals deposited onto the surface willmix downward into the soil column over time, as a result of variousprocesses. These processes include physical mixing (e.g., throughthe freeze-thaw cycle), chemical processes (e.g., dissolution at thesurface and adsorption to soil particles at some depth), and bio-logical processes (e.g., plant rooting and mixing by earthworms andother bioturbators).

As a result of these mixing processes, chemicals deposited onthe surface of undisturbed soils are slowly distributed downwardthrough the soil column as a function of time. Knowledge of the

as).

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depth over which soil mixing occurs is necessary to assess a soilchemical concentration resulting from atmospheric depositionof a specific air emission source over time. The calculation ofa soil concentration resulting from atmospheric deposition,without any removal or degradation of the chemical in soil, is verystraightforward:

Cs ¼ QTzd

(1)

where:

Cs¼ soil chemical concentration (g cm�3)Q¼ surface atmospheric deposition rate (g yr�1 cm�2)T¼ time period of deposition (yr)zd¼ soil mixing depth (cm)

The soil concentration Cs as calculated in Eq. (1) represents anaverage over the assumed soil mixing depth, and in any validationstudy, should be compared with measured soil data averaged overa similar depth. Since Eq. (1) does not contain any removal mech-anisms, predicted soil concentrations will continue to increase overtime, which will be valid only for highly immobile chemicals.Amore complex version of Eq. (1) has been developed that includeschemical removal or degradation from soil by incorporating a soil

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P. Drivas et al. / Atmospheric Environment 45 (2011) 4133e41404134

loss parameter (USEPA, 2005b; Barton et al., 2010). The soil lossparameter as defined by USEPA (2005b) is primarily applicable toorganic compounds and can consist of a combination of fiveseparate removal mechanisms: biotic and abiotic degradation; soilerosion; surface runoff; leaching; and volatilization. The atmo-spheric deposition rate (Q), which can be a combination of dry andwet deposition, is a standard output from many air models such asAERMOD or CALPUFF, and the deposition time period (T) forindustrial sources is usually well known. However, an appropriatesoil mixing depth (zd) is more uncertain. Intuitively, one wouldexpect an increasing mixing depth in soil with increasing time. Forhuman health risk assessments, the U.S. Environmental ProtectionAgency recommends 2 cm as a mixing depth for untilled soil to usewith calculated surface deposition rates from an air model (USEPA,2005b), but does not consider any time dependence of the 2-cmmixing depth. This paper shows that soil mixing depths can bemuch larger, and depend strongly on the atmospheric depositiontime period.

Several technical papers have measured soil depth profiles ofnuclear testing fallout of 137Cs and naturally occurring fallout of210Pb (Barisic et al., 1999; Blagoeva and Zikovsky, 1995; Doeringet al., 2006; He and Walling, 1997; Miller et al., 1990;VandenBygaart et al., 1999); the 210Pb soil profiles are typicallypresented as “excess” 210Pb, after correction for natural background210Pb from radon decay. The more recent 137Cs deposition from theChernobyl incident in 1986 has been monitored by Rosen et al.(1999) as a function of depth in Sweden soils from one to nineyears after the incident. Fernandez et al. (2008) measured indus-trial soil lead vs. depth profiles near the site of a zinc smeltercomplex in France that operated from 1900 to the early 1960s, andCernik et al. (1994) measured zinc and copper soil profiles neara brass smelter in Switzerland. Brzuzy and Hites (1995) havemeasured polychlorinated dibenzo-p-dioxins and dibenzofurans asa function of soil depth in Michigan. In general, the measured soilconcentrations from surface deposition are found to migratedownward with increasing time, and to decrease with depth in anapproximately exponential manner. Several empirical equations,which typically employ a decreasing exponential term with depth,have been developed to approximate the measured soil concen-tration vs. depth profiles (Barisic et al., 1999; Blagoeva and Zikovsky,1995; Miller et al., 1990).

The measured soil depth profiles have also been modeled byone-dimensional diffusion theory, using an effective diffusioncoefficient to approximate the soil mixing processes in undisturbedsoil (Cernik et al., 1994; He and Walling, 1997; Kaste et al., 2007).Based on diffusion equation solutions from Lindstrom and Boersma(1971) that contained a vertical velocity term, He and Walling(1997) derived effective diffusion coefficients for UK soils of0.4e0.5 cm2 yr�1, which were applicable for both 137Cs and 210Pb.Kaste et al. (2007), using a numerical solution of a similaradvective-diffusion equation, found best-fit effective diffusioncoefficients for 210Pb of 0.2 cm2 yr�1 in New England soils,1 cm2 yr�1 in Australian soils, and 2 cm2 yr�1 in Marin County, CAgrasslands.

2. Model development

Our objective was to derive analytic diffusion equation solutionsfor soil concentrations as a function of time for several realisticatmospheric deposition scenarios, and to compare these solutionswith measured data. The one-dimensional diffusion equation wassolved for the cases of: (1) instantaneous surface deposition; (2)continuous surface deposition; and (3) a finite period of continuoussurface deposition, followed by a deposition-free time period. Thebasic assumption in this analysis is that the mixing processes in soil

after surface deposition can be represented by the classic one-dimensional diffusion equation,

vCsvt

¼ Deffv2Csvz2

(2)

where:

Cs¼ soil chemical concentration (g cm�3)Deff¼ effective diffusion coefficient (cm2 yr�1)z¼ depth (cm), with a surface at z¼ 0 and increasing z withdeptht¼ time (yr)

2.1. Soil concentration from instantaneous surface deposition

The solution to Eq. (2) for a semi-infinite solid with an instan-taneous surface deposition source, M0 (g cm�2), applied at t¼ 0over the surface at z¼ 0, is a simple expression for the case ofa constant diffusion coefficient. The soil concentration as a functionof depth and time (t> 0) is (Crank, 1975):

Csðz; tÞ ¼ M0ffiffiffiffiffiffiffiffiffiffiffiffiffiffipDeff t

q exp

�z2

4Deff t

!(3)

where:

M0¼ instantaneous surfacedepositionmassperunit area (g cm�2)

2.2. Depth-averaged soil concentration from instantaneous surfacedeposition

The average soil concentration over any specific depth interval iscalculated by integrating Eq. (3) over a given depth interval fromz¼ L1 to z¼ L2 (L2> L1):

Cs;ave ¼

Zz¼L2

z¼L1

Csdz

Zz¼L2

z¼L1

dz

¼ M0

ðL2�L1ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffipDeff t

q Zz¼L2

z¼L1

"exp

�z2

4Deff t

!#dz (4)

The solution of Eq. (4) for the depth-averaged soil concentrationbetween z¼ L1 and z¼ L2 (L2> L1) for an instantaneous surfacedeposition source becomes:

Cs;ave ¼ M0

ðL2 � L1Þ

264erf

0B@ L2

2ffiffiffiffiffiffiffiffiffiffiDeff t

q1CA� erf

0B@ L1


q1CA375 (5)

where:

erf(x)¼ error function.

2.3. Soil concentration from continuous surface deposition

The solution for a continuous surface deposition source fora semi-infinite solid is derived by integrating the instantaneoussolution over time. For a constant surface deposition rate, Q(g yr�1 cm�2), applied at t� 0 over the surface z¼ 0, the solution forsoil concentration as a function of depth and time (t> 0) waspresented by Carslaw and Jaeger (1959) as:

P. Drivas et al. / Atmospheric Environment 45 (2011) 4133e4140 4135

Csðz; tÞ ¼ 2QDeff

264

ffiffiffiffiffiffiffiffiffiffiDeff tp

rexp

�z2

4Deff t

!� z2erfc

0B@ z


q1CA375 (6)

where:

Q¼ continuous surface deposition rate per unit area (g yr�1 cm�2)erfc(x)¼ complementary error function, erfc(x)¼ 1� erf(x)

It should be noted that the solution in Eq. (6) assumes no soilloss or depletion mechanisms, so the soil concentration alwaysincreases with total time of deposition. However, because ofmathematical difficulties, a numerical solution would be necessaryif a soil removal term were included.

2.4. Depth-averaged soil concentration from continuous deposition

The average soil concentration over any depth interval (L2� L1)is calculated by integrating Eq. (6) over the depth interval fromz¼ L1 to z¼ L2 (L2> L1), and is calculated as:

Csðz; tÞ ¼

ZL2L1

CS dz

ZL2L1

dz

¼ 2QDeff ðL2 � L1Þ

ZL2L1

264

ffiffiffiffiffiffiffiffiffiffiDeff tp

rexp

�z2

4Deff t

!

� z2erfc

0B@ z


q1CA375 dz ð7Þ

The latter of the terms in Eq. (7) is evaluated with the aid of theintegral expression:

Zx$erfcðxÞ dx ¼ x2erfcðxÞ

2þ erf ðxÞ

4� x$exp

��x2�

2ffiffiffiffip

p (8)

The solution of Eq. (7) for the depth-averaged soil concentrationfrom z¼ L1 to z¼ L2 then becomes:

Cs;ave ¼�

Q$tL2 � L1

�$

8><>:264

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2

pDeff t

sexp

�L224Deff t

!þ erf

0B@ L2


q1CA

�

L222Deff t

!erfc

0B@ L2


q1CA375�

264

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiL1

pDeff t

sexp

�L214Deff t

!

þ erf

0B@ L1


q1CA�

L21

2Deff t

!erfc

0B@ L1


q1CA3759>=>; ð9Þ

2.5. Soil concentration after a finite period of continuous deposition

Another realistic scenario is one where continuous surfacedeposition occurs over a finite period, from t¼ 0 to t¼ T, and thenends, as the air emission source ceases to operate. Such a solution isalso useful for superimposing solutions to determine soil concen-trations vs. depth for a scenario with reduced emissions andatmospheric deposition after a given period. From t¼ 0 to t¼ T(i.e., while the source is operating), the solutions for continuoussurface deposition are represented by Eqs. (6) and (9) above.

To compute the soil concentration behavior for times, t, greaterthan the stop time, T, the approach of Lindstrom and Boersma

(1971) results in difficult convolution integrals. A more directapproach is to begin with a continuous deposition solution basedon an integral of the instantaneous source solution in Eq. (3):

Csðz; tÞ ¼Zt0m0

dt0$Qðt0ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipDeff ðt � t0Þ

q exp

�z2

4Deff ðt � t0Þ

!(10)

where the upper integration, t0m, is simply t for the continuoussource, and becomes T for the source after ceasing emissions attime t0 ¼ T. The depth-averaged concentration from the surface toa depth L is then simply,

Cs;aveðtÞ ¼ 1L

Zt0m0

dt0$Qðt0Þ$erf

0B@ L

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeff ðt � t0Þ

q1CA (11)

Changing from variable t0 to s, where s ¼ l=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeff $ðt � t0Þ

q, and

dt0 ¼ ðl2=2Deff Þ$ds=s3, and using l¼ z for Eq. (10) and l¼ L forEq. (11) then facilitates solution of both integrals via integral tablesand integration by parts.

The resulting analytic solution for the soil concentration asa function of depth and time following a finite period, T, ofcontinuous deposition is, for times t> T:

Csðz; tÞ ¼ Q$zDeff

$

"exp

��s2L�

ffiffiffiffip

p$sL

þerf ðsLÞ�exp

��s2U�

ffiffiffiffip

p$sU

�erf ðsUÞ#

(12)

where: sL ¼ z=2ffiffiffiffiffiffiffiffiffiffiffiffiDeff $t

qand sU ¼ z=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeff $ðt�TÞ

q.

2.6. Depth-averaged soil concentration after a finite periodof continuous deposition

The corresponding analytic solution for the depth-averaged soilconcentration from z¼ 0 to z¼ L after a finite period, T, of contin-uous deposition is, for times t> T:

Cs;aveðtÞ ¼ Q$L2$Deff

$

"exp

��s2L�

ffiffiffiffip

p$sL

þ 1þ 1

2$s2L

!$erf ðsLÞ

� exp��s2U

�ffiffiffiffip

p$sU

� 1þ 1

2$s2U

!$erf ðsUÞ

#(13)

where now: sL ¼ L=2ffiffiffiffiffiffiffiffiffiffiffiffiDeff $t

qand sU ¼ L=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeff $ðt � TÞ

q.

The case of determining the average soil concentration over anyspecific depth interval from z¼ L1 to a deeper depth z¼ L2 (L2> L1),for the time period after a source stops operating, is a simple exten-sion of Eq. (13). The resulting analytic solution for the average soilconcentration betweendepths L1 and L2 (L2> L1) as a function of timefollowing afinite period, T, of continuousdeposition is, for times t> T:

Cs; aveðtÞ ¼ Q2$ðL2 � L1Þ$Deff

$

(L22$

"exp

��s2L2�

ffiffiffiffip

p$sL2

þ 1þ 1

2$s2L2

!$erf ðsL2Þ �

exp��s2U2

�ffiffiffiffip

p$sU2

� 1þ 1

2$s2U2

!$erf ðsU2Þ

#� L21$

"exp

��s2L1�

ffiffiffiffip

p$sL1

þ 1þ 1

2$s2L1

!$erf ðsL1Þ �

exp��s2U1

�ffiffiffiffip

p$sU1

� 1þ 1

2$s2U1

!$erf ðsU1Þ

#)ð14Þ


where: sL2 ¼ L2=2ffiffiffiffiffiffiffiffiffiffiffiffiDeff $t

q, sU2 ¼ L2=2


q, sL1 ¼

L1=2ffiffiffiffiffiffiffiffiffiffiffiffiDeff $t

q, and sU1 ¼ L1=2


qWe believe that Eqs. (12) through (14) represent new analytic

results. Note also that if one sets t¼ T in Eqs. (12) and (13), thensU/N, and Eqs. (12) and (13) revert to the continuous depositionsolutions in Eqs. (6) and (9), respectively.

Fig. 2. Comparison of Hille, Sweden cesium data with theory, Deff¼ 1 cm2 yr�1.

3. Model comparisons with measured soil depth profile data

3.1. Model comparison with cesium atmospheric deposition

Theweapons testing fallout, which had a sharp peak in 1963 (Heand Walling, 1997; Warneke et al., 2002), and the 1986 Chernobylincident represent approximately instantaneous sources of atmo-spheric deposition, and these 137Cs data can be used to derive aneffective diffusion coefficient for soil mixing over time from aninstantaneous surface source, using Eq. (5). Although nuclearweapons testing occurred over several years in the 1950s and1960s, there was a significant increase in 137Cs fallout during onlya few years in the early 1960s (Warneke et al., 2002), and this“spike” in 137Cs fallout circa 1963 can be considered as approxi-mately instantaneous over a 35-year time period. Rosen et al.(1999) measured 137Cs soil concentrations at eight differentundisturbed soil locations in Sweden from one to nine years afterthe Chernobyl incident in 1986. Since the soil types were differentin the Rosen et al. (1999) study, each of the eight sites was analyzedindividually to determine a potential range of the effective diffusioncoefficient in different soil types. Additionally, the VandenBygaartet al. (1999) study, which measured weapons testing 137Cs falloutin Ontario, Canada, was also analyzed to determine an effectivediffusion coefficient in a North American soil.

The 137Cs data measurements in Rosen et al. (1999) andVandenBygaart et al. (1999) typically reported averaged 137Cs soilconcentrations in 1 or 2 cm “slices” of a 25-cm deep soil coresample. Eq. (5) was used to directly compare with the 137Cs data atdifferent depth intervals for each of the eight measurement sites inSweden (Rosen et al., 1999) and the Ontario, Canada site(VandenBygaart et al., 1999). Although it is possible that earlierweapons testing 137Cs fallout may affect the more recent Chernobyl137Cs data, the Rosen et al. (1999) data, reproduced in Figs. 1through 8, do not show anomalous data at deeper depths thatmight be due to earlier weapons testing, indicating that the

Fig. 1. Comparison of Skogsvallen, Sweden cesium data with theory, Deff¼ 0.5cm2 yr�1.

relatively high Chernobyl input of 137Cs in 1986 predominated overany residual 137Cs concentrations from weapons testing.

For each location, a range of different values of the effectivediffusion coefficient were tested using Eq. (5) to determine anempirical “best-fit” theoretical curve to the data. The best-fit valuesof Deff were calculated by examining a range of Deff values from 0.25to 4 cm2 yr�1, at discrete increments of 0.25 cm2 yr�1. The timesafter the instantaneous deposition source varied from one to nineyears in the Rosen et al. (1999) data. For the VandenBygaart et al.(1999) data, which were taken in 1998, we assumed 35 years asan appropriate time after an instantaneous deposition source,based on the sharp peak in weapons testing fallout in 1963.

Figs. 1 through 8 present the measured data vs. theorycomparison, using the instantaneous deposition solution in Eq. (5),for each of the eight different locations in Sweden from Rosen et al.(1999). Although each location had a different soil type, the best-fiteffective diffusion coefficients (Deff) did not vary greatly. As shownin Figs. 1 through 8, the range of best-fit Deff for the Sweden datawas from 0.5 cm2 yr�1 to 1 cm2 yr�1. The measured data vs. theorycomparison in Figs. 1 through 8 also demonstrates that the diffu-sionmodel, which assumes a constantDeff, can predict the observedslowing down of the diffusion rate between 1 and 8 years afterChernobyl. In fact, all of the diffusionmodels presented in Section 2,which assume a constant Deff, predict that the diffusion rate willslow downwith time. Themodel predictions are entirely consistent

Fig. 3. Comparison of Mojsjovik, Sweden cesium data with theory, Deff¼ 0.75 cm2 yr�1.

Fig. 4. Comparison of Trodje, Sweden cesium data with theory, Deff¼ 0.5 cm2 yr�1. Fig. 6. Comparison of StoraBlasjon, Sweden cesium data with theory, Deff¼ 0.75cm2 yr�1.


with the Rosen et al. (1999) finding that “Migration rates were inthe range of 0.5e1.0 cmyr�1 for the first year and thereafter0.2e0.6 cmyr�1”. Likewise, Fig. 9 shows the measured data vs.theory comparison in Ontario, Canada from VandenBygaart et al.(1999). The best-fit Deff for the Ontario data after 35 years was0.75 cm2 yr�1, similar to the most typical value from the Swedendata.

3.2. Model comparison with lead atmospheric deposition from anindustrial source

The analytic solutions derived for a continuous source operatingfor a finite time period can be compared with the soil lead data, asa function of depth, taken by Fernandez et al. (2008). These dataweremeasured at a location in France near the site of a zinc smeltercomplex that operated from 1900 to the early 1960s, thus providinga data set directly applicable to Eqs. (13) and (14), with a depositionsource operating for approximately 62 years (i.e., T¼ 62 yr), fol-lowed by a time period with no deposition for about 44 years (i.e.,total time t¼ 106 yr). Soil lead measurements were made at 5 cmintervals in an undisturbed permanent pasture area, and anthro-pogenic lead concentrations were distinguished from naturallyoccurring lead levels by a Pb/La ratio technique. Although Semlaliet al. (2004) imply that significant anthropogenic soil lead inputsthrough atmospheric deposition were made by emission sources

Fig. 5. Comparison of Ramvik, Sweden cesium data with theory, Deff¼ 0.75 cm2 yr�1.

before 1928, the Fernandez et al. (2008) data, after adjustment toshow only anthropogenic soil lead, do not show any anomalousdata at deeper depths that might indicate earlier atmospheric leadinputs. Based on the anthropogenic soil lead concentrations,Fernandez et al. (2008) determined the total lead deposition interms of g cm�2 over the 62-year lifetime of the galvanizing plant(i.e., defining the source deposition rate, Q). This allowed a directcalculation using Eq. (14) of predicted soil lead concentrations at5 cm depth intervals, for comparison with the measured soil leaddata.

Using the model parameters derived from Fernandez et al.(2008) for the permanent pasture area of T¼ 62 yr, t¼ 106 yr, andQ¼ 3.39�10�5 g yr�1 cm�2, values of Cs,ave (g cm�3) vs. depth werecalculated using Eq. (14). A constant atmospheric deposition sourceover a 62-year time period was assumed, due to lack of moredetailed source operation information. Had more detailed sourceemission data been available, the model comparison could havebeen donemore accurately by using a superposition of solutions fordifferent times and different atmospheric deposition rates. Todirectly compare with the Fernandez et al. data, the theoreticalCs,ave calculations were converted to units of mg kg�1 by dividing bythe measured bulk soil density of 1.47 g cm�3. The comparison oftheory vs. data for the undisturbed permanent pasture area isshown in Fig. 10 for two values of the effective diffusion coefficient.

Fig. 7. Comparison of Hammarstrand, Sweden cesium data with theory, Deff¼ 0.75cm2 yr�1.

Fig. 8. Comparison of Blomhojden, Sweden cesium data with theory, Deff¼ 0.5cm2 yr�1.

Fig. 10. Comparison of measured soil lead data in France with Eq. (14), assuminga source operating for 62 years followed by 44 years of no deposition, with Deff valuesof 0.75 and 2 cm2 yr�1.


As shown in Fig. 10, an effective diffusion coefficient (Deff) of2 cm2 yr�1

fits the Fernandez et al. (2008) soil lead data relativelywell, while themore typicalDeff value of 0.75 cm2 yr�1 derived fromother studies predicts less vertical mixing than observed. As dis-cussed by Fernandez et al. (2008), a possible explanation forincreased soil mixing is the large amount of earthworm activity inthe permanent pasture area. If the uncertainty estimates in theFernandez et al. (2008) deposition flux of 21�4 gm�2 are consid-ered, the corresponding best-fit values of Deff would have anuncertainty of 2� 0.25 cm2 yr�1.

3.3. Model comparison with industrial dioxin atmosphericdeposition

Brzuzy and Hites (1995) have measured concentrations of poly-chlorinated dibenzo-p-dioxins and dibenzofurans (PCDD/F) asa function of soil depth at undisturbed locations in Michigan (twosites in Upper Michigan and one site in Lower Michigan). Since thePCDD/F concentrations represent long-term industrial deposition,Brzuzy and Hites (1995) assumed a 60-year period of atmosphericdeposition and, based on the total mass of PCDD/F in soil, calculatedaverage PCDD/F atmospheric deposition rates over a 60-year periodof 264 ng yr�1m�2 for theUpperMichigan sites and663 ng yr�1m�2

for the Lower Michigan site. Since a continuous atmospheric depo-sition rate (Q) has been defined in this study, the continuous soil

Fig. 9. Comparison of Ontario, Canada cesium data with theory, Deff¼ 0.75 cm2 yr�1.

mixing model in Eq. (6) can be used to directly compare with themeasured PCDD/F soil concentrations vs. depth.

Assuming a 60-year time period of continuous atmosphericdeposition at the average deposition rates developed by Brzuzy andHites (1995), Eq. (6) was used to compare with the measuredPCDD/F soil depth profile data at the Michigan locations. Fig. 11presents the model vs. data comparison at the two sites in UpperMichigan, while Fig. 12 shows the comparison at the one location inLower Michigan. In these comparisons, a typical soil density of1.5 g cm�3 was assumed to directly compare with the measuredPCDD/F soil concentrations in units of pg gsoil�1. As shown in Fig. 11,a best-fit Deff of 1 cm2 yr�1 shows reasonable agreement with oneof the Upper Michigan sites (Shingleton, MI), but the other location(Grayling, MI) does not fit the theory well, due to an approximatelyconstant PCDD/F soil concentration between 10 and 25 cm depth.The comparison in Fig. 12 with the Lower Michigan location (Ver-ona, MI) shows good agreement of the measured PCDD/F soil datawith the theory, using a relatively high Deff of 2 cm2 yr�1.

4. Soil mixing depth vs. atmospheric deposition time

The analytic solution for a continuous surface deposition sourcewas used to calculate appropriate soil mixing depths as a function

Fig. 11. Comparison of Upper Michigan PCDD/F data with theory, Deff¼ 1 cm2 yr�1.

Fig. 12. Comparison of Lower Michigan PCDD/F data with theory, Deff¼ 2 cm2 yr�1. Fig. 13. Cumulative soil mass vs. depth calculated for a continuous surface depositionsource using Eq. (15) and Deff¼ 0.75 cm2 yr�1.

Table 1Soil mixing depths for continuous surface deposition times,Deff¼ 0.75 cm2 yr�1.

Deposition time (yr) Soil mixing depth (cm)

0.1 0.70.2 1.00.5 1.60.75 1.91 2.22 3.13 3.84 4.45 5.07.5 6.310 7.015 8.620 9.930 12.140 14.050 15.775 19.2100 22.2


of the atmospheric deposition time. In Eq. (9), the mass (m) per unitarea averaged over a depth from 0 to L is L$Cs,ave, and the total massdeposited on the surface (mT) per unit area is Q$t. Thus, Eq. (9) canbe easily rearranged to represent the cumulative mass fraction(m/mT) for a continuous surface deposition source as a function oftime and depth,

mmT

¼

2640B@ Lffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pDeff tq

1CAexp

�L2

4Deff t

!þ erf

0B@ L


q1CA

�

L2

2Deff t

!erfc

0B@ L


q1CA375 (15)

Fig. 13 presents calculations using Eq. (15) and Deff¼0.75 cm2 yr�1 for the case of continuous surface deposition forperiods of one, five, and 20 years. As shown in Fig. 13, soil mixingcan occur to depths of approximately 2 cm after one year, 5 cm afterfive years, and 10 cm after 20 years of continuous surface deposi-tion. Examination of the cumulative curves in Fig. 13 suggests thata cumulative mass of approximately 98% can reasonably definea soil mixing depth for continuous surface deposition. Table 1presents soil mixing depths, calculated using Eq. (15) andDeff¼ 0.75 cm2 yr�1 with a 98% cumulative mass, for differentcontinuous atmospheric deposition times.

5. Discussion

A Deff range from 0.5 to 2 cm2 yr�1 has been derived for undis-turbed soils for the relatively immobile chemicals lead, cesium, anddioxins. This range of best-fit effective diffusion coefficients ofa factor of four may reflect differences in soil properties or differ-ences in physical mixing processes. For example, the relatively highderived Deff value of 2 cm2 yr�1 in France may result from a largeamount of earthworm activity, as discussed by Fernandez et al.(2008). The derived Deff range compares reasonably well to theHe and Walling (1997) and Kaste et al. (2007) effective diffusioncoefficient estimates that ranged from 0.2 to 2 cm2 yr�1 for lead andcesium in various types of soil. Because those two studiesaccounted for soil mixing using two adjustable parameters, aneffective diffusion coefficient and a vertical velocity, somewhatlower effective diffusion coefficients would be expected, sincesome of the downward transport in soil would be caused by thevertical velocity term. The one relatively high effective diffusioncoefficient of 2 cm2 yr�1 estimated for Marin County, CA by Kaste

et al. (2007) was attributed to very active bioturbation and phys-ical erosion for that area.

While diffusion in combination with advection has beenassessed in other studies, the mathematical difficulties requireeither highly complex equations for simplified situations (He andWalling, 1997) or numerical solutions (Kaste et al., 2007), andthese solutions need to fit two adjustable parameters e an effectivediffusion coefficient and a vertical advection velocity. The purposeof this paper was to develop a reasonably accurate, analytic solutionto a practical problem in atmospheric science e the time depen-dence of a soil mixing depth of chemicals after atmospheric depo-sition, using an effective diffusion coefficient as the only adjustableparameter. To make the “diffusion-only” approximation as accurateas possible, the chemicals that were chosen for consideration andvalidation in our paper (cesium, lead, and dioxins) are highlyimmobile in soil and relatively unaffected by water advection.

The good comparison with data over a wide variety of soil typesand locations, which results in only a factor of four difference in Deff,demonstrates the feasibility of using the “diffusion-only” conceptfor the relatively immobile chemicals studied in our paper.However, there will be some situations where the “diffusion-only”concept in our paper may not be the predominant mechanism forchemical migration in soil. For example, our model is likely


inappropriate for more mobile and soluble chemicals such as zinc,where water advection, soil moisture, and meteorological param-eters such as the Bowen ratio may be important. Also, the model islikely inappropriatewhen considering atmospheric deposition overinhomogeneous soils that have two or more layers near the surfacewith different physical properties (e.g., dissimilar porosity, granu-larity, moisture content, or organic carbon content).

As shown in many of the longer-term (6e35 years) graphicalcomparisons, the measurement data typically show a decreasedconcentration in soil very close to the surface, with the highestconcentration in soil slightly below the surface. Because thechemicals chosen for this study (cesium, lead, and dioxins) are allhighly immobile in soil, this phenomenon could be caused by long-term surface erosion (e.g., wind-blown dust), or surface runoffwhere the topmost layer of soil is removed during rainstorms.

Based on the calculated soil depth results in Table 1, which useda relatively low effective diffusion coefficient of 0.75 cm2 yr�1, a soilmixing depth of 2 cm from continuous surface deposition, which isrecommended for calculating soil concentrations for human healthrisk assessments (USEPA, 2005b), appears overly conservative forcontinuous deposition times longer than one year. The measureddata and the soil mixing equations clearly show increasing soilmixing depths with increasing atmospheric deposition times (e.g., asoil mixing depth of 10 cm after 20 years of continuous deposition).When used for calculating a soil concentration due to continuoussurface deposition from an air emission source, the soil mixingdepth should be adjusted for the appropriate time period ofatmospheric deposition.

The realistic scenario of relatively constant surface depositionfrom a specific source over a given period, followed by a period ofreduced surface deposition from the same source, can be assessedby straightforward superposition of the analytic solutions inEqs. (9) and (13). A forthcoming Part II of this paper will describea practical application of the soil mixingmodel, using superpositionof the solutions, to estimate the chemical recontamination rates ofremediated soils, for a specific source with decreasing air emissionsand atmospheric deposition over time.

Acknowledgments

We thank the Doe Run Company for providing financial supportfor portions of this work.

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soil mixing depth after atmospheric deposition. i. model development and validation

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