Soil mixing depth after atmospheric deposition. I. Model development and validation

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    Soil mixing depthAtmospheric depositionMathematical modeling

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    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


    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

    * Corresponding author. Tel.: 1 617 395 5000.

    Contents lists availab

    Atmospheric E


    Atmospheric Environment 45 (2011) 4133e4140E-mail address: (P. Drivas).dispersion 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 specic emission source on surfaces in terms of massper unit time per unit surface area (e.g., g yr1 cm2). 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-


    Cs QTzd(1)


    Cs soil chemical concentration (g cm3)Q surface atmospheric deposition rate (g yr1 cm2)T time period of deposition (yr)z soil mixing depth (cm)Diffusion theoryEffective diffusion coefcient

    1. Introduction

    Particulates released from an air enearby soil surfaces as a function ofmeteorological parameters, and s1352-2310/$ e see front matter 2011 Elsevier Ltd.doi:10.1016/j.atmosenv.2011.05.029continuous surface deposition, followed by a deposition-free time period. Comparisons of the modelwith measured soil depth proles resulting from atmospheric deposition showed good agreement forlead, cesium, and dioxins. The best-t effective diffusion coefcients in undisturbed soils varied from0.5 cm2 yr1 to 2 cm2 yr1. 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 ve years, and 10 cm after 20 years of continuous atmospheric deposition on the soilsurface.

    2011 Elsevier Ltd. All rights reserved.

    n sourcewill deposit onle size, particle density,conditions. Many air

    depth over which soil mixing occurs is necessary to assess a soilchemical concentration resulting from atmospheric depositionof a specic 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 veryKeywords:

    (1) instantaneous surface deposition; (2) continuous surface deposition; and (3) a nite period ofAccepted 9 May 2011soil 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:Soil mixing depth after atmospheric depand validation

    Peter Drivas a,*, Teresa Bowers a, Robert YamartinoaGradient, 20 University Road, Cambridge, MA 02138, USAb Integrals Unlimited, 509 Chandlers 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 2011

    a b s t r a c t

    Knowledge of a soil mixinassess the soil chemical cosource. A mathematical m

    journal homepage: www.eAll rights reserved.sition. I. Model development

    epth, or the migration depth of various pollutants in soil, is necessary tontration resulting from atmospheric deposition of a specic air emissionl has been developed that describes the depth and time behavior of the

    le at ScienceDirect


    vier .com/locate/atmosenv

  • nvirloss parameter (USEPA, 2005b; Barton et al., 2010). The soil lossparameter as dened by USEPA (2005b) is primarily applicable toorganic compounds and can consist of a combination of veseparate 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 proles 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 proles 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 proles 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 proles 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 proles (Barisic et al., 1999; Blagoeva and Zikovsky,1995; Miller et al., 1990).

    The measured soil depth proles have also been modeled byone-dimensional diffusion theory, using an effective diffusioncoefcient 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 coefcients for UK soils of0.4e0.5 cm2 yr1, which were applicable for both 137Cs and 210Pb.Kaste et al. (2007), using a numerical solution of a similaradvective-diffusion equation, found best-t effective diffusioncoefcients for 210Pb of 0.2 cm2 yr1 in New England soils,1 cm2 yr1 in Australian soils, and 2 cm2 yr1 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 nite period of continuoussurface deposition, followed by a deposition-free time period. The

    P. Drivas et al. / Atmospheric E4134basic assumption in this analysis is that the mixing processes in soilafter surface deposition can be represented by the classic one-dimensional diffusion equation,





    Cs soil chemical concentration (g cm3)Deff effective diffusion coefcient (cm2 yr1)z depth (cm), with a surface at z 0 and increasing z withdeptht time (yr)

    2.1. Soil concentration from instantaneous surface deposition

    The solution to Eq. (2) for a semi-innite solid with an instan-taneous surface deposition source, M0 (g cm2), applied at t 0over the surface at z 0, is a simple expression for the case ofa constant diffusion coefcient. The soil concentration as a functionof depth and time (t> 0) is (Crank, 1975):

    Csz; t M0pDeff t

    q exp z24Deff t



    M0 instantaneous surfacedepositionmassperunit area (g cm2)

    2.2. Depth-averaged soil concentration from instantaneous surfacedeposition

    The average soil concentration over any specic depth interval iscalculated by integrating Eq. (3) over a given depth interval fromz L1 to z L2 (L2> L1):







    pDeff t

    q ZzL2



    z24Deff t

    !#dz (4)

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

    Cs;ave M0L2 L1


    0B@ L22

    Deff t

    q1CA erf

    0B@ L12

    Deff t

    q1CA375 (5)


    erf(x) error function.

    2.3. Soil concentration from continuous surface deposition

    The solution for a continuous surface deposition source fora semi-innite solid is derived by integrating the instantaneoussolution over time. For a constant surface deposition rate, Q(g yr1 cm2), applied at t 0 over the surface z 0, the solution forsoil concentration as a function of depth and time (t> 0) was

    onment 45 (2011) 4133e4140presented by Carslaw and Jaeger (1959) as:

  • nvirCsz; t 2QDeff


    Deff tp


    z24Deff t

    ! z2erfc

    0B@ z2

    Deff t

    q1CA375 (6)


    Q continuous surface deposition rate per unit area (g yr1 cm2)erfc(x) complementary error function, erfc(x) 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 difculties, 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 fromz L1 to z L2 (L2> L1), and is calculated as:

    Csz; t


    CS dz



    2QDeff L2 L1



    Deff tp


    z24Deff t



    0B@ z2

    Deff t

    q1CA375 dz 7

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

    Zx$erfcx dx x


    erf x4



    p (8)

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


    Q$tL2 L1




    pDeff t


    L224Deff t

    ! erf

    0B@ L22

    Deff t


    L222Deff t


    0B@ L22

    Deff t




    pDeff t


    L214Deff t



    0B@ L12

    Deff t



    2Deff t


    0B@ L12

    Deff t

    q1CA3759>=>; 9

    2.5. Soil concentration after a nite period of continuous deposition

    Another realistic scenario is one where continuous surfacedeposition occurs over a nite period, from t 0 to t 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 t 0 to t T(i.e., while the source is operating), the solutions for continuoussurface deposition are represented by Eqs. (6) and (9) above.

    P. Drivas et al. / Atmospheric ETo compute the soil concentration behavior for times, t, greaterthan the stop time, T, the approach of Lindstrom and Boersma(1971) results in difcult convolution integrals. A more directapproach is to begin with a continuous deposition solution basedon an integral of the instantaneous source solution in Eq. (3):

    Csz; t Zt0m0

    dt0$Qt0pDeff t t0

    q exp

    z24Deff t t0


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

    Cs;avet 1LZt0m0


    0B@ L2

    Deff t t0

    q1CA (11)

    Changing from variable t0 to s, where s l=2Deff $t t0

    q, and

    dt0 l2=2Deff $ds=s3, and using l z for Eq. (10) and l 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 nite period, T, ofcontinuous deposition is, for times t> T:

    Csz; t Q$zDeff$


    s2L p


    erf sLexp



    erf sU#


    where: sL z=2Deff $t

    qand sU z=2

    Deff $tT


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

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

    Cs;avet Q$L2$Deff$




    1 1


    !$erf sL



    1 1


    !$erf sU


    where now: sL L=2Deff $t

    qand sU L=2

    Deff $t T


    The case of determining the average soil concentration over anyspecic depth interval from z L1 to a deeper depth z 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 anite period, T, of continuousdeposition is, for times t> T:

    Cs; avet Q2$L2 L1$Deff$





    1 1


    !$erf sL2



    1 1


    !$erf sU2

    # L21$




    1 1


    !$erf sL1



    1 12

    !$erf sU1


    onment 45 (2011) 4133e4140 41352$sU1

  • where: sL2 L2=2Deff $t

    q, sU2 L2=2

    Deff $t T

    q, sL1

    L1=2Deff $t

    q, and sU1 L1=2

    Deff $t T

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

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

    3. Model comparisons with measured soil depth prole 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 coefcient for soil mixing over time from an

    slow downwith time. Themodel predictions are entirely consistent

    Fig. 2. Comparison of Hille, Sweden cesium data with theory, Deff 1 cm2 yr1.

    P. Drivas et al. / Atmospheric Environment 45 (2011) 4133e41404136instantaneous surface source, using Eq. (5). Although nuclearweapons testing occurred over several years in the 1950s and1960s, there was a signicant increase in 137Cs fallout during onlya few years in the early 1960s (Warneke et al., 2002), and thisspike 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 diffusioncoefcient 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 coefcient 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 theFig. 1. Comparison of Skogsvallen, Sweden cesium data with theory, Deff 0.5cm2 yr1.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 coefcient were tested using Eq. (5) to determine anempirical best-t theoretical curve to the data. The best-t valuesof Deffwere calculated by examining a range of Deff values from 0.25to 4 cm2 yr1, at discrete increments of 0.25 cm2 yr1. 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-teffective diffusion coefcients (Deff) did not vary greatly. As shownin Figs. 1 through 8, the range of best-t Deff for the Sweden datawas from 0.5 cm2 yr1 to 1 cm2 yr1. 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 willFig. 3. Comparison of Mojsjovik, Sweden cesium data with theory, Deff 0.75 cm2 yr1.

  • theory vs. data for the undisturbed permanent pasture area is

    Fig. 4. Comparison of Trodje, Sweden cesium data with theory, Deff 0.5 cm2 yr1. Fig. 6. Comparison of StoraBlasjon, Sweden cesium data with theory, Deff 0.752 1

    P. Drivas et al. / Atmospheric Environment 45 (2011) 4133e4140 4137with the Rosen et al. (1999) nding that Migration rates were inthe range of 0.5e1.0 cmyr1 for the rst year and thereafter0.2e0.6 cmyr1. Likewise, Fig. 9 shows the measured data vs.theory comparison in Ontario, Canada from VandenBygaart et al.(1999). The best-t Deff for the Ontario data after 35 years was0.75 cm2 yr1, 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 nite 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., T 62 yr), fol-lowed by a time period with no deposition for about 44 years (i.e.,total time t 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 signicant anthropogenic soil lead inputs

    through atmospheric deposition were made by emission sources

    Fig. 5. Comparison of Ramvik, Sweden cesium data with theory, Deff 0.75 cm2 yr1.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 cm2 over the 62-year lifetime of the galvanizing plant(i.e., dening 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 T 62 yr, t 106 yr, andQ 3.39105 g yr1 cm2, values of Cs,ave (g cm3) 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 kg1 by dividing bythe measured bulk soil density of 1.47 g cm3. The comparison of

    cm yr .shown in Fig. 10 for two values of the effective diffusion coefcient.

    Fig. 7. Comparison of Hammarstrand, Sweden cesium data with theory, Deff 0.75cm2 yr1.

  • measured PCDD/F soil concentrations vs. depth.

    Fig. 10. Comparison of measured soil lead data in France with Eq. (14), assuming

    P. Drivas et al. / Atmospheric Environment 45 (2011) 4133e41404138As shown in Fig. 10, an effective diffusion coefcient (Deff) of2 cm2 yr1 ts the Fernandez et al. (2008) soil lead data relativelywell, while themore typicalDeff value of 0.75 cm2 yr1 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 ux of 214 gm2 are consid-ered, the corresponding best-t values of Deff would have anuncertainty of 2 0.25 cm2 yr1.

    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 period

    1 2 1 2

    Fig. 8. Comparison of Blomhojden, Sweden cesium data with theory, Deff 0.5cm2 yr1.of 264 ng yr m for theUpperMichigan sites and663 ng yr mfor the Lower Michigan site. Since a continuous atmospheric depo-sition rate (Q) has been dened in this study, the continuous soil

    Fig. 9. Comparison of Ontario, Canada cesium data with theory, Deff 0.75 cm2 yr1.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 prole 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 cm3 was assumed to directly compare with the measuredPCDD/F soil concentrations in units of pg gsoil1. As shown in Fig. 11,a best-t Deff of 1 cm2 yr1 shows reasonable agreement with oneof the Upper Michigan sites (Shingleton, MI), but the other location(Grayling, MI) does not t 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 yr1.

    4. Soil mixing depth vs. atmospheric deposition timemixing model in Eq. (6) can be used to directly compare with the

    a source operating for 62 years followed by 44 years of no deposition, with Deff valuesof 0.75 and 2 cm2 yr1.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, Deff 1 cm2 yr1.

  • chemical migration in soil. For example, our model is likely

    Fig. 13. Cumulative soil mass vs. depth calculated for a continuous surface depositionsource using Eq. (15) and Deff 0.75 cm2 yr1.

    nvironment 45 (2011) 4133e4140 4139of 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,


    2640B@ L

    pDeff tq


    L24Deff t

    ! erf

    0B@ L2

    Deff t



    2Deff t


    0B@ L2

    Deff t

    q1CA375 (15)

    Fig. 13 presents calculations using Eq. (15) and Deff0.75 cm2 yr1 for the case of continuous surface deposition forperiods of one, ve, and 20 years. As shown in Fig. 13, soil mixingcan occur to depths of approximately 2 cm after one year, 5 cm afterve 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 denea soil mixing depth for continuous surface deposition. Table 1presents soil mixing depths, calculated using Eq. (15) andDeff 0.75 cm2 yr1 with a 98% cumulative mass, for different

    Fig. 12. Comparison of Lower Michigan PCDD/F data with theory, Deff 2 cm2 yr1.

    P. Drivas et al. / Atmospheric Econtinuous atmospheric deposition times.

    5. Discussion

    A Deff range from 0.5 to 2 cm2 yr1 has been derived for undis-turbed soils for the relatively immobile chemicals lead, cesium, anddioxins. This range of best-t effective diffusion coefcients ofa factor of four may reect differences in soil properties or differ-ences in physical mixing processes. For example, the relatively highderived Deff value of 2 cm2 yr1 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 diffusioncoefcient estimates that ranged from 0.2 to 2 cm2 yr1 for lead andcesium in various types of soil. Because those two studiesaccounted for soil mixing using two adjustable parameters, aneffective diffusion coefcient and a vertical velocity, somewhatlower effective diffusion coefcients would be expected, sincesome of the downward transport in soil would be caused by thevertical velocity term. The one relatively high effective diffusioncoefcient of 2 cm2 yr1 estimated for Marin County, CA by Kasteet 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 difculties requireeither highly complex equations for simplied situations (He andWalling, 1997) or numerical solutions (Kaste et al., 2007), andthese solutions need to t two adjustable parameters e an effectivediffusion coefcient 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 coefcient 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-onlyconcept in our paper may not be the predominant mechanism forTable 1Soil mixing depths for continuous surface deposition times,Deff 0.75 cm2 yr1.

    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

  • 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 coefcient of 0.75 cm2 yr1, 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 measured

    surface deposition from an air emission source, the soil mixing


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    The realistic scenario of relatively constant surface depositionfrom a specic 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 specic source with decreasing air emissionsand atmospheric deposition over time.


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    Soil mixing depth after atmospheric deposition. I. Model development and validation1 Introduction2 Model development2.1 Soil concentration from instantaneous surface deposition2.2 Depth-averaged soil concentration from instantaneous surface deposition2.3 Soil concentration from continuous surface deposition2.4 Depth-averaged soil concentration from continuous deposition2.5 Soil concentration after a finite period of continuous deposition2.6 Depth-averaged soil concentration after a finite period of continuous deposition

    3 Model comparisons with measured soil depth profile data3.1 Model comparison with cesium atmospheric deposition3.2 Model comparison with lead atmospheric deposition from an industrial source3.3 Model comparison with industrial dioxin atmospheric deposition

    4 Soil mixing depth vs. atmospheric deposition time5 Discussion Acknowledgments References


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