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

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<ul><li><p>ob</p><p>Soil mixing depthAtmospheric depositionMathematical modeling</p><p>g dnceode</p><p>missioparticurface</p><p>logical processes (e.g., plant rooting and mixing by earthworms andother bioturbators).</p><p>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</p><p>d</p><p>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</p><p>* Corresponding author. Tel.: 1 617 395 5000.</p><p>Contents lists availab</p><p>Atmospheric E</p><p>lse</p><p>Atmospheric Environment 45 (2011) 4133e4140E-mail address: pdrivas@gradientcorp.com (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-</p><p>straightforward:</p><p>Cs QTzd(1)</p><p>where:</p><p>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</p><p>1. Introduction</p><p>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.</p><p> 2011 Elsevier Ltd. All rights reserved.</p><p>n sourcewill deposit onle size, particle density,conditions. Many air</p><p>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:</p><p>(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</p><p>Peter Drivas a,*, Teresa Bowers a, Robert YamartinoaGradient, 20 University Road, Cambridge, MA 02138, USAb Integrals Unlimited, 509 Chandlers Wharf, Portland, ME 04101, USA</p><p>a r t i c l e i n f o</p><p>Article history:Received 6 December 2010Received in revised form15 April 2011</p><p>a b s t r a c t</p><p>Knowledge of a soil mixinassess the soil chemical cosource. A mathematical m</p><p>journal homepage: www.eAll rights reserved.sition. I. Model development</p><p>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</p><p>le at ScienceDirect</p><p>nvironment</p><p>vier .com/locate/atmosenv</p></li><li><p>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.</p><p>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).</p><p>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.</p><p>2. Model development</p><p>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><p>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,</p><p>vCsvt</p><p> Deffv2Csvz2</p><p>(2)</p><p>where:</p><p>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)</p><p>2.1. Soil concentration from instantaneous surface deposition</p><p>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):</p><p>Csz; t M0pDeff t</p><p>q exp z24Deff t</p><p>!(3)</p><p>where:</p><p>M0 instantaneous surfacedepositionmassperunit area (g cm2)</p><p>2.2. Depth-averaged soil concentration from instantaneous surfacedeposition</p><p>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):</p><p>Cs;ave </p><p>ZzL2zL1</p><p>Csdz</p><p>ZzL2zL1</p><p>dz</p><p> M0L2L1</p><p>pDeff t</p><p>q ZzL2</p><p>zL1</p><p>"exp</p><p> z24Deff t</p><p>!#dz (4)</p><p>The solution of Eq. (4) for the depth-averaged soil concentrationbetween z L1 and z L2 (L2> L1) for an instantaneous surfacedeposition source becomes:</p><p>Cs;ave M0L2 L1</p><p>264erf</p><p>0B@ L22</p><p>Deff t</p><p>q1CA erf</p><p>0B@ L12</p><p>Deff t</p><p>q1CA375 (5)</p><p>where:</p><p>erf(x) error function.</p><p>2.3. Soil concentration from continuous surface deposition</p><p>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</p><p>onment 45 (2011) 4133e4140presented by Carslaw and Jaeger (1959) as:</p></li><li><p>nvirCsz; t 2QDeff</p><p>264</p><p>Deff tp</p><p>rexp</p><p> z24Deff t</p><p>! z2erfc</p><p>0B@ z2</p><p>Deff t</p><p>q1CA375 (6)</p><p>where:</p><p>Q continuous surface deposition rate per unit area (g yr1 cm2)erfc(x) complementary error function, erfc(x) 1 erf(x)</p><p>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.</p><p>2.4. Depth-averaged soil concentration from continuous deposition</p><p>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:</p><p>Csz; t </p><p>ZL2L1</p><p>CS dz</p><p>ZL2L1</p><p>dz</p><p> 2QDeff L2 L1</p><p>ZL2L1</p><p>264</p><p>Deff tp</p><p>rexp</p><p> z24Deff t</p><p>!</p><p> z2erfc</p><p>0B@ z2</p><p>Deff t</p><p>q1CA375 dz 7</p><p>The latter of the terms in Eq. (7) is evaluated with the aid of theintegral expression:</p><p>Zx$erfcx dx x</p><p>2erfcx2</p><p> erf x4</p><p> x$expx2</p><p>2p</p><p>p (8)</p><p>The solution of Eq. (7) for the depth-averaged soil concentrationfrom z L1 to z L2 then becomes:</p><p>Cs;ave </p><p>Q$tL2 L1</p><p>$</p><p>8>:264</p><p>L2</p><p>pDeff t</p><p>sexp</p><p> L224Deff t</p><p>! erf</p><p>0B@ L22</p><p>Deff t</p><p>q1CA</p><p>L222Deff t</p><p>!erfc</p><p>0B@ L22</p><p>Deff t</p><p>q1CA375</p><p>264</p><p>L1</p><p>pDeff t</p><p>sexp</p><p> L214Deff t</p><p>!</p><p> erf</p><p>0B@ L12</p><p>Deff t</p><p>q1CA</p><p> L21</p><p>2Deff t</p><p>!erfc</p><p>0B@ L12</p><p>Deff t</p><p>q1CA3759>=>; 9</p><p>2.5. Soil concentration after a nite period of continuous deposition</p><p>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><p>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):</p><p>Csz; t Zt0m0</p><p>dt0$Qt0pDeff t t0</p><p>q exp </p><p>z24Deff t t0</p><p>!(10)</p><p>where the upper integration, t0m, is simply t for the continuoussource, and becomes T for the source after ceasing emissions attime t0 T. The depth-averaged concentration from the surface toa depth L is then simply,</p><p>Cs;avet 1LZt0m0</p><p>dt0$Qt0$erf</p><p>0B@ L2</p><p>Deff t t0</p><p>q1CA (11)</p><p>Changing from variable t0 to s, where s l=2Deff $t t0</p><p>q, and</p><p>dt0 l2=2Deff $ds=s3, and using l z for Eq. (10) and l L forEq. (11) then facilitates solution of both integrals via integral tablesand integration by parts.</p><p>The resulting analytic solution for the soil concentration asa function of depth and time following a nite period, T, ofcontinuous deposition is, for times t> T:</p><p>Csz; t Q$zDeff$</p><p>"exp</p><p>s2L p</p><p>p$sL</p><p>erf sLexp</p><p>s2Up</p><p>p$sU</p><p>erf sU#</p><p>(12)</p><p>where: sL z=2Deff $t</p><p>qand sU z=2</p><p>Deff $tT</p><p>q.</p><p>2.6. Depth-averaged soil concentration after a nite periodof continuous deposition</p><p>The corresponding analytic solution for the depth-averaged soilconcentration from z 0 to z L after a nite period, T, of contin-uous deposition is, for times t> T:</p><p>Cs;avet Q$L2$Deff$</p><p>"exp</p><p>s2Lp</p><p>p$sL</p><p> 1 1</p><p>2$s2L</p><p>!$erf sL</p><p> exps2Up</p><p>p$sU</p><p> 1 1</p><p>2$s2U</p><p>!$erf sU</p><p>#(13)</p><p>where now: sL L=2Deff $t</p><p>qand sU L=2</p><p>Deff $t T</p><p>q.</p><p>The case of determining the average soil concentration over anyspecic depth interval from z L1 to a deeper depth z L2 (L2> L1),for the time period after a source stops operating, is a simple exten-sion of Eq. (13). The resulting analytic solution for the average soilconcentration betweendepths L1 and L2 (L2> L1) as a function of timefollowing anite period, T, of continuousdeposition is, for times t> T:</p><p>Cs; avet Q2$L2 L1$Deff$</p><p>(L22$</p><p>"exp</p><p>s2L2p</p><p>p$sL2</p><p> 1 1</p><p>2$s2L2</p><p>!$erf sL2 </p><p>exps2U2p</p><p>p$sU2</p><p> 1 1</p><p>2$s2U2</p><p>!$erf sU2</p><p># L21$</p><p>"exp</p><p>s2L1p</p><p>p$sL1</p><p> 1 1</p><p>2$s2L1</p><p>!$erf sL1 </p><p>exps2U1p</p><p>p$sU1</p><p> 1 12</p><p>!$erf sU1</p><p>#)14</p><p>onment 45 (2011) 4133e4140 41352$sU1</p></li><li><p>where: sL2 L2=2Deff $t</p><p>q, sU2 L2=2</p><p>Deff $t T</p><p>q, sL1 </p><p>L1=2Deff $t</p><p>q, and sU1 L1=2</p><p>Deff $t T</p><p>qWe believe that Eqs. (12) through (14) represent new analytic</p><p>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.</p><p>3. Model comparisons with measured soil depth prole data</p><p>3.1. Model comparison with cesium atmospheric deposition</p><p>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</p><p>slow downwith time. Themodel predictions are entirely consistent</p><p>Fig. 2. Comparison of Hille, Sweden cesium data with theory, Deff 1 cm2 yr1.</p><p>P. Drivas et al. / Atmospheric Environment 45 (2011) 4133e41404136instantaneous surface source, using Eq. (5). Although nuclearweapons testing oc...</p></li></ul>

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