A ²¹⁰Pb_ex mass balance model in cultivated soils in consideration of the radionuclide diffusion

A 210 Pb ex mass balance model in cultivated soils in consideration of the radionuclide diffusion 1 2 corresp first-author Corresponding author, yunqi768@163.com Corresponding author, yunqi768@163.com yunqi768@163.com 3 3 Sichuan Agriculture University, Chengdu611830, China Sichuan Agriculture University, Chengdu611830, China Shandong Provincial Key Laboratory of Water and Soil Conservation & Environment Protection, Linyi University, Linyi276000, China Shandong Provincial Key Laboratory of Water and Soil Conservation & Environment Protection, Linyi University, Linyi276000, China Institute of Mountain Hazards and Environment, Chinese Academy of Sciences, Chengdu610041, China Institute of Mountain Hazards and Environment, Chinese Academy of Sciences, Chengdu610041, China The existing 210 Pb ex mass balance models for the assessment of cultivated soil erosion are based on an assumption that 210 Pb ex is quite evenly mixed within the plough layer. However, the amount of 210 Pb ex distributed in the soils below the plough depth, like a downward tail in the lower part of the 210 Pb ex profile, has been largely ignored. In fact, after the initial cultivation of undisturbed soils, 210 Pb ex will diffuse downward from plough layer to the plough pan layer due to the concentration gradient. Assuming 210 Pb ex inventory is constant, the depth distribution in the two layers of the cultivated soils will achieve a steady state after continuous cultivation for 10.37 years, when 210 Pb ex is evenly distributed in the soils of the plough layer with an exponential concentration decline with depth in the soils of the plough pan layer, and the 210 Pb ex concentration at any depth will be invariable with time. The work reported in this paper attempts to explain the formation of the 210 Pb ex tail in the soil profile below the plough depth by theoretical derivation of the 210 Pb ex depth distribution process in the two layers of the cultivated soils, propose a 210 Pb ex mass balance model considering 210 Pb ex diffusion based on the existing model, and discuss the influence of the 210 Pb ex tail to the existing model. Cultivated soils 210 Pb ex depth distribution Diffusion Mass balance model Soil erosion I. INTRODUCTION 1 Because of temporal and spatial limitations associated with the application of 137 Cs technique to estimating rates of soil erosion and sediment redistribution [ 1 1 - 4 4 1-4 ], the use of 210 Pb ex measurement has attracted increasing attention, as an alternative approach [ 5 5 - 9 9 5-9 ]. Unlike 137 Cs, which is an artificial radionuclide that is released into the environment as a result of atmospheric test of nuclear weapons, 210 Pb is a natural product of the 238 U decay series, derived from the decay of gaseous 222 Rn, the daughter of 226 Ra [ 10 10 , 11 11 ]. 226 Ra is found naturally in most soils and rocks and the generated 210 Pb is in equilibrium with its parent. A small quantity of 222 Rn diffuses upward from the soil, introduces 210 Pb into the atmosphere and provides an input of this radionuclide to surface soils and sediments, which is not in equilibrium with its parent 226 Ra. This fallout component is termed unsupported or excess 210 Pb ( 210 Pb ex ) [ 12 12 , 13 13 ]. Like 137 Cs, 210 Pb ex is shown to have a strong affinity for soil and sediment particles. Upon reaching the soil surface, 210 Pb ex is quickly and strongly adsorbed by the surface soil, and the subsequent redistribution within the landscape will reflect the movement of soil and sediment particles associated with soil erosion and sediment transport processes [ 14 14 ]. Unlike the time-dependent fallout of 137 Cs, the atmospheric fallout of 210 Pb ex is essentially constant for its natural origin [ 15 15 ], and the 210 Pb ex inventory in soils can be invariable because the subsequent fallout provides an input to surface soils that is in equilibrium with the radioactive decay of the existing 210 Pb ex in the soils. In the reported works, the profiles of soils from undisturbed land show that the maximum concentrations of 210 Pb ex occur at the surface horizon and decline exponentially with depth ( Fig. 1(a), 1(c) Fig. 1(a), 1(c) ), while in the profiles of cultivated soils the 210 Pb ex concentration is almost uniform throughout the plough layer as a result of mixing caused by tillage ( Fig. 1(b), 1(d) Fig. 1(b), 1(d) ). For the cultivated soils, it is also obvious that an amount of 210 Pb ex distributed in the soils below the plough depth, like a downward tail in the lower part of the profile ( Fig. 1(b), 1(d) Fig. 1(b), 1(d) ). The 210 Pb ex tail in the profile demonstrates that the maximum 210 Pb ex concentrations occur at the bottom of plough layer and decline exponentially with depth. However, the amount of 210 Pb ex distributed below the plough depth has been largely ignored to date, suggesting that 210 Pb ex is quite evenly distributed and restricted to the plough layer. The existing 210 Pb ex mass balance models for the assessment of cultivated soil losses have similarly not taken into account the amount of 210 Pb ex distributed below the plough depth. Walling and He [ 16 16 ] developed a mass balance model for estimating erosion rates on cultivated soils from 210 Pb ex measurement. Based on this mass balance model, and ignoring the freshly deposited 210 Pb ex removed by erosion and influence of the particle size correction factor, Zhang et al . proposed a simplified 210 Pb ex mass balance model in the steady state for estimating erosion rates on cultivated soils [ 18 18 ], which can be expressed as: <math display="block" id="M1"><mrow><mi>I</mi><mo>−</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>+</mo><mfrac><mrow><mi>Δ</mi><mi>H</mi></mrow><mi>H</mi></mfrac><mo stretchy="false">)</mo><mi>A</mi><mo>=</mo><mn>0</mn><mn>,</mn></mrow></math> where A is the 210 Pb ex inventory (mBq/cm 2 ), λ is the decay constant of 210 Pb ex (/yr), Δ H is the erosion rate (cm/yr) and H is the plough depth (cm). The 210 Pb ex loss proportion ( l ) relative to the local reference inventory ( A ref , in mBq/cm 2 ) for an eroding site can be expressed as: <math display="block" id="M2"><mrow><mi>l</mi><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>A</mi><mrow><mtext>ref</mtext></mrow></msub><mo>−</mo><mi>A</mi><mo stretchy="false">)</mo><mo>/</mo><msub><mi>A</mi><mrow><mtext>ref</mtext></mrow></msub><mo stretchy="false">]</mo><mo>×</mo><mn>100</mn><mi>%</mi><mn>.</mn></mrow></math> Assuming the 210 Pb ex deposition flux is constant, the erosion rate can be calculated as Eq. (3) according to the relationship between the deposition flux and the reference inventory ( I = λA ref ). <math display="block" id="M3"><mrow><mi>Δ</mi><mi>H</mi><mo>=</mo><mfrac><mrow><mi>λ</mi><mi>H</mi><mi>l</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>l</mi></mrow></mfrac><mn>.</mn></mrow></math> So, the existing mass balance models (existing model for short) are based on the assumption that 210 Pb ex is quite evenly distributed within the plough depth. Nevertheless, neglecting 210 Pb ex distributed below the plough depth, more or less, would consequentially influence the application of existing model to soil erosion estimation. So, it is needed to know the influence level, and whether or not the existing model is applicable to soil erosion estimation as usual. In this paper, we attempt to explain the formation of 210 Pb ex tail in the lower part of profile by theoretical derivation of the 210 Pb ex depth distribution process in the plough and plough pan layers of the cultivated soils, propose a 210 Pb ex mass balance model in consideration of the radionuclide diffusion (revision model for short), and discuss the influence of the 210 Pb ex distributed below the plough depth on the application of existing model to erosion estimation. II. 210 Pb ex DISTRIBUTION PROCESSES IN CULTIVATED SOILS 1 A. Mechanism and process analysis 2 In fact, 210 Pb ex is quite evenly mixed by tillage within the plough layer, and there is little 210 Pb ex distributed below the plough depth at the beginning of cultivation, because the plough depth is usually no less than the measured depth range of 210 Pb ex distribution in the undisturbed soils before tillage. However, this initial depth distribution state is unsteady, and the radionuclide moves downward from the plough layer to the plough pan layer. Radionuclide downward movements in soil include diffusion and migration processes [ 19 19 ]. 210 Pb ex diffusion in soil is the process of radionuclide movement that is caused by the concentration gradient in the ionic or molecular form, which means that the radionuclide is only able to diffuse from a high concentration to a low concentration. In 210 Pb ex migration in soil, its downward movement is caused by transportation via media, such as adsorbed solid particles and dissolved fluids, which means that the migration is independent of the concentration gradient, and the radionuclide is able to move either from a high concentration to a low concentration or vice versa. For a soil profile in which the maximum radionuclide occurs at the top horizon and declines monotonically with depth, diffusion can be considered the predominant radionuclide movement process over the profile, while the radionuclide migration processes can be neglected [ 20 20 - 24 24 20-24 ]. Thus, it can be assumed that the diffusion process dominates the formation of the 210 Pb ex tail in the profile of the plough pan layer. At the beginning of cultivation, 210 Pb ex is quite evenly distributed within the plough layer; after the initial cultivation, the 210 Pb ex diffuses from the plough layer to the plough pan layer due to the concentration gradient; the 210 Pb ex inventory of the plough layer decreases while that of plough pan layer increases; and the 210 Pb ex tail in the profile of the cultivated soils progressively comes into being closely below the interface between the two layers ( Fig. 1(b), 1(d) Fig. 1(b), 1(d) ). Assuming the 210 Pb ex inventory is constant, the 210 Pb ex concentration at any soil depth will be invariable after a period of diffusion, and the 210 Pb ex depth distribution in the two layers of cultivated soils will achieve a steady state. B. The depth distribution in the unsteady state 2 Assuming no soil erosion or deposition occurs, when 210 Pb ex is evenly mixed within the plough layer by the initial cultivation of undisturbed soils. Subsequently, the 210 Pb ex starts moving from the plough layer to the plough pan layer. Taking the diffusion process into account, the 210 Pb ex movement in the soils of plough pan layer follows Fick’s second law of diffusion [ 25 25 ], which is: <math display="block" id="M4"><mrow><mi>D</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mrow><mo>(</mo><mrow><mfrac><mrow><mo>∂</mo><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><mo>∂</mo><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>−</mo><mi>λ</mi><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mn>,</mn></mrow></math> where D is the effective diffusion coefficient (cm 2 /yr), C pp is the 210 Pb ex concentration in the soils of the plough pan layer (mBq/cm 3 ), z is the depth downward from the interface between the plough and plough pan layers (cm), and t is the time elapsed since 210 Pb ex was well mixed within the plough layer (yr). When z → 0, the 210 Pb ex concentration of the interface soils is equal to the value of the 210 Pb ex concentration at any depth of the plough layer. By diffusion, the total 210 Pb ex ( A , in mBq/cm 2 ) is redistributed between the plough and plough pan layers, which means: <math display="block" id="M5"><mrow><mi>A</mi><mo>=</mo><msub><mi>A</mi><mtext>p</mtext></msub><mo>+</mo><msub><mi>A</mi><mrow><mtext>pp</mtext></mrow></msub><mn>,</mn></mrow></math> where A p and A pp are the 210 Pb ex inventory of the plough layer and plough pan layer, respectively. The 210 Pb ex inventory in the soils of the plough layer can be calculated as: <math display="block" id="M6"><mrow><msub><mi>A</mi><mtext>p</mtext></msub><mo>=</mo><mi>H</mi><msub><mi>C</mi><mtext>p</mtext></msub><mn>.</mn></mrow></math> Assuming the plough pan layer can be treated as a semi-infinite homogeneous porous medium, the process of 210 Pb ex diffusion from the plough layer to the plough pan layer can be considered as the infinite thin layer radionuclide resource diffusing to a one-dimensional semi-infinite medium. Thus, according to the Gauss solution of the Fick’s second law [ 26 26 ], by combining Eqs. (4), (5) and (6), the 210 Pb ex concentration variation with time in the soils of the plough layer can be expressed as: <math display="block" id="M7"><mrow><msub><mi>C</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo>/</mo><mo stretchy="false">[</mo><mi>H</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi>D</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mn>.</mn></mrow></math> And the variation of the 210 Pb ex concentration with time and depth in the soils of the plough pan layer can be expressed as: <math display="block" id="M8"><mrow><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>A</mi><mrow><mi>H</mi><mo>+</mo><msqrt><mrow><mi>D</mi><mi>t</mi></mrow></msqrt></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>D</mi><mi>t</mi></mrow></mfrac></mrow></msup><mn>.</mn></mrow></math> By integrating the two sides of Eq. (8) over depth z , the proportion of the 210 Pb ex inventory of the plough pan layer to the total 210 Pb ex inventory in the cultivated soil, namely the 210 Pb ex diffusion proportion, can be expressed as: <math display="block" id="M9"><mrow><mfrac><mrow><msub><mi>A</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mi>A</mi></mfrac><mo>=</mo><mfrac><mrow><msqrt><mrow><mi>D</mi><mi>t</mi></mrow></msqrt></mrow><mrow><mi>H</mi><mo>+</mo><msqrt><mrow><mi>D</mi><mi>t</mi></mrow></msqrt></mrow></mfrac><mo>×</mo><mn>100</mn><mi>%</mi><mn>.</mn></mrow></math> The above theoretical derivation, namely Eqs. (4)–(9), can also apply to the 137 Cs depth distribution process in cultivated soils. However, unlike 137 Cs, the 210 Pb ex depth distribution in the two layers of the cultivated soils finally achieves a steady state after a period of diffusion, assuming the 210 Pb ex deposition is constant over time. Before the steady state, the 210 Pb ex concentration at any depth of the two layers varies with time. If the effective diffusion coefficient D is known (which can be obtained by experiments), Eq. (7) can be used to simulate the 210 Pb ex concentration variation with time in the soils of the plough layer, and Eq. (8) can be used to simulate the 210 Pb ex concentration variation with time and depth in the plough pan layer. C. The depth distribution in the steady state 2 After achieving a steady state, the 210 Pb ex concentration at any depth of the cultivated soils is invariable with time, viz. ∂C pp ( z , t ) / ∂t = 0, and Eq. (4) can be written as: <math display="block" id="M10"><mrow><mi>D</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mrow><mo>(</mo><mrow><mfrac><mrow><mo>∂</mo><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mi>λ</mi><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.</mn></mrow></math> When z → 0, C pp(0) is the 210 Pb ex concentration in the interface soils between the plough and the plough pan layer, which represents the 210 Pb ex concentration at any depth of the plough layer or the 210 Pb ex concentration in the top soils of the plough pan layer. Based on this boundary condition, Eq. (10) can be solved as follows: <math display="block" id="M11"><mrow><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mtext>p</mtext></msub><msup><mi>e</mi><mrow><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>/</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mi>z</mi></mrow></msup><mn>.</mn></mrow></math> Integrating the two sides of Eq. (11) over the depth ( z ), the following can be derived: <math display="block" id="M12"><mrow><msub><mi>A</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mtext>p</mtext></msub><msup><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>/</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mn>.</mn></mrow></math> By combining Eqs. (5), (6), (11) and (12), the 210 Pb ex concentration in the soils of the plough layer in the steady state can be expressed as: <math display="block" id="M13"><mrow><msub><mi>C</mi><mtext>p</mtext></msub><mo>=</mo><mfrac><mi>A</mi><mrow><mi>H</mi><mo>+</mo><msqrt><mrow><mi>D</mi><mo>/</mo><mi>λ</mi></mrow></msqrt></mrow></mfrac><mn>.</mn></mrow></math> And the variation of the 210 Pb ex concentration with depth in the plough pan layer in the steady state can be expressed as: <math display="block" id="M14"><mrow><msub><mi>C</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>A</mi><mrow><mi>H</mi><mo>+</mo><msqrt><mrow><mi>D</mi><mo>/</mo><mi>λ</mi></mrow></msqrt></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>/</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mi>z</mi></mrow></msup><mn>.</mn></mrow></math> By combining Eqs. (9), (12) and (13), the time elapsed from initial cultivation to the steady state can be calculated as: <math display="block" id="M15"><mrow><msub><mi>t</mi><mrow><mtext>steady</mtext></mrow></msub><mo>=</mo><mn>1</mn><mo>/</mo><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>10.37</mn><mi>y</mi><mi>r</mi><mn>.</mn></mrow></math> And the diffusion proportion in the steady state can be expressed as: <math display="block" id="M16"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mi>k</mi></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msub><mi>A</mi><mrow><mtext>pp</mtext></mrow></msub></mrow><mi>A</mi></mfrac><mo>×</mo><mn>100</mn><mi>%</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msqrt><mrow><mi>D</mi><mo>/</mo><mi>λ</mi></mrow></msqrt></mrow><mrow><mi>H</mi><mo>+</mo><msqrt><mrow><mi>D</mi><mo>/</mo><mi>λ</mi></mrow></msqrt></mrow></mfrac><mo>×</mo><mn>100</mn><mi>%</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where k is the diffusion proportion in the steady state, which can be obtained by experiments. From Eq. (16), the effective diffusion coefficient of 210 Pb ex is: <math display="block" id="M17"><mrow><mi>D</mi><mo>=</mo><mi>λ</mi><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>k</mi><mi>H</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>k</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mn>.</mn></mrow></math> Equations (13) and (14) can be used to simulate the 210 Pb ex depth distribution in the steady state in the soil profile for the two layers of the cultivated land. A basic assumption of this theoretical derivation is that no erosion or deposition occurs. For an eroding soil, the 210 Pb ex inventory would be invariable assuming the erosion rate is constant for over 100 years. Then the 210 Pb ex depth distribution will also be in steady state, and Eqs. (10)–(17) can be applicable as usual. True, 210 Pb ex downward diffusion below the plough depth might be partially offset by soil erosion at the surface, causing the base of the plough layer to be displaced downwards, and some 210 Pb ex that has diffused downwards would return to the plough layer. However, the 210 Pb ex distributed below the plough depth, more or less, is a matter of fact. In the cultivated purple soil of the Sichuan Hilly Basin in China ( Fig. 1(d) Fig. 1(d) ), for example, ca. 37% of the downwards diffused 210 Pb ex , according to Eq. (8), returned to the plough layer at the first year of diffusion, assuming 210 Pb ex is initially quite evenly distributed within the plough depth and the soil erosion rate is 0.5 cm/yr (the real erosion rates are usually less than this value [ 17 17 ]). In a word, 210 Pb ex diffusion in cultivated soils is a matter of fact in spite of the usual erosion. And the proportion would decrease with year and eventually tend to constant. Then, the 210 Pb ex distribution would achieve the steady state with a 210 Pb ex tail in the profile below the plough depth. D. The simulated 210 Pb ex diffusion process in a soil core from the Yimeng mountains area 2 To know the 210 Pb ex depth distribution in cultivated soil in reality, a typical 210 Pb ex profile for a core from the cultivated cinnamon soils of the Yimeng mountains area in China was introduced in this study [ 27 27 ], based on which the 210 Pb ex diffusion process could be simulated for the core. The Yimeng Mountains area (ca. 17180 km 2 ) is located in the central south of Shandong Province, China (34;22; to 36;13;N, 117;24; to 119;11;E), with elevations within the range of 150–1165 m. The area belongs to the continental monsoon climate area of the warm temperate zone, and the cinnamon soils developed from limestone weathering crusts are wildly distributed. The core was collected from the fields situated in flat areas at the top of ploughed fields, where the disturbance except for cultivation over the past 100 years can be negligible. The profile ( Fig. 2(a) Fig. 2(a) ) reflects the steady state of 210 Pb ex depth distribution because the soils have not been disturbed except for cultivation in nearly ca. 100 years. From the profile and the field surveys it can be determined that, the plough depth ( H ) is ca. 20 cm because the 210 Pb ex concentrations in soils of the upper ∼ 20 cm are relatively uniform, suggesting the 210 Pb ex is mixed by tillage within the uppermost depth of ∼ 20 cm, which is also in agreement with our field surveys; the 210 Pb ex diffusion proportion ( k ) is 32%, which is the ratio of the 210 Pb ex inventory below the plough depth to the total inventory in the profile; the effective diffusion coefficient ( D ) is 2.72 cm 2 /yr, which can be calculated using Eq. (17). Equations (7) and (8) can simulate the 210 Pb ex redistribution process in the plough and plough pan layer of cinnamon soils after the initial cultivation before the steady state (e.g. t = 2, 4, 6, and 8 yr); Eqs. (13) and (14) can simulate the stable 210 Pb ex depth distribution in the two layers of the cultivated cinnamon soils ( t 10.37 yr) ( Fig. 2(b) Fig. 2(b) ). III. MASS BALANCE MODEL IN CONSIDERATION OF THE DIFFUSION 1 A. Model derivation 2 Considering 210 Pb ex depth distribution in the cultivated land in view of the radionuclide diffusion from the plough to the plough pan layer, Eq. (1) can be rewritten as: <math display="block" id="M18"><mrow><mi>I</mi><mo>−</mo><mi>λ</mi><mi>A</mi><mo>−</mo><mi>Δ</mi><mi>H</mi><msub><mi>A</mi><mtext>p</mtext></msub><mo>/</mo><mi>H</mi><mo>=</mo><mn>0</mn><mn>.</mn></mrow></math> By combining Eqs. (2), (6), (13), (17) and (18), the erosion rate of the cultivated soil can be expressed as Eq. (19) based on the relationship between the 210 Pb ex deposition flux and the reference inventory ( I = λA ref ). <math display="block" id="M19"><mrow><mi>Δ</mi><mi>H</mi><mo>=</mo><mfrac><mrow><mi>λ</mi><mi>H</mi><mi>l</mi></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> Therefore, considering the 210 Pb ex diffusion from the plough layer to the plough pan layer, the erosion rates can be estimated from Eq. (19). By ignoring the influence 210 Pb ex diffusion ( k =0), Eq. (19) can be reduced to Eq. (3). B. Comparison of the revision and existing models 2 As above, the difference between the revision and existing model lies in whether to take into account the amount of 210 Pb ex distributed below the plough depth. To compare the two models, let Δ H 1 and Δ H 2 be the erosion rates estimated using Eqs. (3) and (19), respectively, and we have: <math display="block" id="M20"><mrow><mi>Δ</mi><msub><mi>H</mi><mn>1</mn></msub><mo>=</mo><mi>λ</mi><mi>H</mi><mi>l</mi><mo>/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M21"><mrow><mi>Δ</mi><msub><mi>H</mi><mn>2</mn></msub><mo>=</mo><mi>λ</mi><mi>H</mi><mi>l</mi><mo>/</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mrow></math> It is clear that Δ H 1 < Δ H 2 , and the difference ratio ( d ) of Δ H 1 and Δ H 2 is: <math display="block" id="M22"><mrow><mi>d</mi><mo>=</mo><mo stretchy="false">(</mo><mi>Δ</mi><msub><mi>H</mi><mn>2</mn></msub><mo>−</mo><mi>Δ</mi><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>/</mo><mi>Δ</mi><msub><mi>H</mi><mn>2</mn></msub><mn>,</mn></mrow></math> By combining Eqs. (20)–(22), one has: <math display="block" id="M23"><mrow><mi>d</mi><mo>=</mo><mi>k</mi><mn>.</mn></mrow></math> So, the difference ratio can be represented by the 210 Pb ex diffusion proportion in the steady state of the 210 Pb ex depth distribution, and the difference increases with the proportion of diffusion. If the 210 Pb ex diffusion proportion ( k ) and loss proportion ( l ) are known, Eq. (19) can be used to estimate the erosion rates. These parameters can be derived from the experiments. Because the k is determined by the 210 Pb ex profile measurement, the accuracy of k is related to the accuracy of experiment, e.g. the study site selection, sampling method, the sampling section increments, laboratory measurement and etc.. Moreover, the recognition of the plough depth, according to the shape of the profile in conjunction with field survey, is also important to the accuracy of k . IV. DISCUSSION AND CONCLUSION 1 As above, the erosion rates estimated using the revision model are higher than those estimated using the existing model by comparison. In other words, the existing model underestimates the erosion rate of the cultivated soils to some extent because of ignoring the amount of 210 Pb ex distributed below the plough depth. Nevertheless, the existing model involves the superiorities including compact form, fewer parameters, and convenience for application. In contrast, the revision model involves the limitations including complicated form and more parameters for application. Eq. (23) suggests that the underestimation of soil erosion from the existing model is as well less than 10% when the 210 Pb ex diffusion proportion ( k ) is lower than 10%, which could be negligible. In other words, the existing model is as usual applicable to soil erosion estimation when the diffusion proportion is lower than 10%. When the diffusion proportion ranges between 10%–15%, the existing model is also suitable if the underestimation is acceptable in some practical applications. However, the underestimation could not be neglected when the diffusion proportion is greater than 15%, and the revision model would be more applicable to erosion estimation than the existing model. No matter what the diffusion proportion is, the revision model is universal because the existing model is merely a case of the revision model (when k =0). In a word, the application scope of revision model is universal, especially for the locations where the diffusion proportions in the cores are greater than 15%. Equation (17) indicates that the diffusion proportion is related to the diffusion coefficient ( D ) and the plough depth ( H ), viz. the diffusion proportion increases with the diffusion coefficient while decreases with the plough depth. The diffusion coefficients and the plough depths vary in different areas with different soil types and tillage systems. In the reported works, the diffusion proportions are usually less than 10%, e.g. a typical range of 5%–7% at the study sites in UK [ 16 16 , 19 19 ] and southern Italy [ 28 28 , 29 29 ]. In these cases, the diffusion proportion can be negligible, and the existing model is applicable as before. For cultivated brown and cinnamon soils of the Yimeng mountains area in China, however, the diffusion proportions are greater than 15% [ 27 27 ], so the revision model is more applicable to erosion estimation by comparison. 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