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TECHNICAL REPORT
ECOLOGICAL RISK ASSESSMENT

Implementation of Sulfate Adsorption in the SAFE Model

Department of Chemical Engineering II, Lund University, P.O. Box 124, S-221 00 Lund, Sweden

Corresponding author (tamon{at}niaes.affrc.go.jp)

Received for publication November 12, 1999.

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
An SO^2-₄ adsorption submodel has been implemented in the dynamic soil chemistry model SAFE. The submodel calculates pH-dependent SO^2-₄ and H⁺ adsorption to the soil, as well as the net surface charge development due to uneven adsorption of SO^2-₄ and H⁺, using the empirical equations derived from an electrostatic model (Extended Constant Capacitance Model, ECCM) of SO^2-₄ adsorption. The resulting new SAFE model was applied on a roof experiment plot in the Norway spruce [Picea abies (L.) H. Karst.] stand at Solling, Germany, where atmospheric S and N deposition was artificially reduced by the roof construction. The model performance was compared with the previous versions that used a pH-independent Freudlich model of SO^2-₄ adsorption or assumed no SO^2-₄ adsorption. With the ECCM-based SO^2-₄ adsorption submodel, SAFE simulated soil solution SO^2-₄ concentration and base saturation better, in comparison with measured data, than with the previous SO^2-₄ adsorption formulations. Through the model application, also, need of additional improvement was suggested, such as calibration of mass transfer coefficients.

Abbreviations: ANC, acid neutralizing capacity • ECCM, Extended Constant Capacitance Model

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
IN the modern industrialized era, terrestrial ecosystems are subject to effects of various anthropogenic environmental pollutants, such as acid depositions, ozone gas, and toxic organic and inorganic substances. Mathematical models are one of the potent means to assess the potential influence of the pollution on the ecosystems and human beings and to build the optimal strategies of pollutant abatement and remediation.

Aiming at the simulation of the responses of terrestrial ecosystems against acid depositions, a considerable number of dynamic soil chemistry models have been developed in recent years, including MAGIC (Cosby et al., 1985), ILWAS (Gherini et al., 1985), MIDAS (Holmberg et al., 1989), SMART (De Vries et al., 1989), SAFE (Warfvinge et al., 1993), RESAM (De Vries et al., 1994), and PnET-CN/CHESS (Postek et al., 1995). These models have been mostly designed for and applied on European and North American ecosystems, but a drastic increase in SO_x emissions is expected for Asian countries until the first half of the twenty-first century (Stockholm Environment Institute, 1998). Consequently, application of the dynamic soil chemistry models on Asian ecosystems will be required to assess the environmental effects of increased acid depositions.

Concerning the soil acidification–recovery processes, one of the distinct features of Asian ecosystems is the extensive occurrence of SO^2-₄ adsorbing soils, such as Andisols, in which SO^2-₄ adsorption capacity has a high correlation with the Al content in allophane and imogolite (Fumoto et al., 1996a,b). The molar H⁺ to SO^2-₄ adsorption ratio on soils and minerals is generally less than 2.0 and dependent on pH (Karltun, 1997; Persson and Lövgren, 1996; He et al., 1996), and the resulting net negative charge may affect the base cation dynamics in the soil. However, few of the current dynamic soil chemistry models simulate the pH dependency of SO^2-₄ and H⁺ adsorption. Wesselink et al. (1994) modeled pH-dependent SO^2-₄ adsorption on an acid forest soil in Solling, Germany, by assuming pH dependency of Freundlich adsorption parameters, but the molar H⁺ to SO^2-₄ adsorption ratio was still fixed at 2.0. Gustafsson (1995) developed an empirical model to predict pH-dependent SO^2-₄ adsorption on Bs horizons of northern Scandinavian soils for which the molar H⁺ to SO^2-₄ adsorption ratio could be simplified as 2.0. His empirical model, however, has not yet been integrated into a dynamic soil chemistry model. To date, therefore, we have not had any dynamic soil chemistry model that has a function to explicitly calculate pH-dependent SO^2-₄ and H⁺ adsorption, including a varying H⁺ to SO^2-₄ adsorption ratio.

For the reason above, a research project has been conducted to build an efficient SO^2-₄ adsorption submodel into the SAFE model. In the previous study (Fumoto and Sverdrup, 2000), the authors have shown that an electrostatic model (ECCM) is valid for the SO^2-₄ and H⁺ adsorption on Andisols. In addition, they found that the calculated SO^2-₄ and H⁺ adsorption can be reproduced by two simple empirical equations. This study reports the implementation of the SO^2-₄ adsorption submodel in the SAFE model and its application to Solling experimental forest, Germany. Although the current SAFE model modification is aimed at application to Asian sites, it is applied to the European site in this study because (i) the mineral weathering submodel for Asian soils is under development and (ii) SAFE requires historical time series of atmospheric deposition and nutrient uptake as the input, but reliable data and models for reconstructing the time series at Asian sites are in preparation for the time being. Solling was chosen for the application site also for the following reasons. (i) Since it is a long-running intensive experimental forest, extensive background data are available for model parameterization and evaluation. (ii) The soil at this site contains substantial amounts of amorphous Al and Fe oxides and adsorbs considerable amounts of SO^2-₄ (Tiktak et al., 1995). (iii) A roof experiment was conducted since 1992 to artificially reduce S and N deposition in one plot (clean rain plot), to experimentally examine the ecosystem response to reduced acidity load. SAFE was previously applied on the clean rain plot (Walse et al., 1998), and an innegligible difference was left between measured and simulated soil solution SO^2-₄ concentrations during the roof experiment.

To clarify the effect of different SO^2-₄ adsorption formulations on the model performance, SAFE was run using three SO^2-₄ adsorption formulations separately, that is, no SO^2-₄ adsorption, a pH-independent Freundlich model, and the ECCM-based submodel developed in this study.

The aim of this study is to (i) present how the ECCM-based SO^2-₄ adsorption submodel was implemented in the SAFE model, (ii) clarify the effect of different SO^2-₄ adsorption formulations on the model performance, and (iii) discuss the merits and shortcomings of the SAFE model as revealed through the application to Solling.

	MODEL DESCRIPTION

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Sulfate Adsorption Model
The ECCM (Persson and Lövgren, 1996) assumes two complexation planes on the SO^2-₄ adsorbent (oxide mineral) surface (i.e., the inner plane [0-plane] and the outer plane [ß-plane]), and allocates protonation of the surface hydroxyl group on the 0-plane and SO^2-₄ complexation on the ß-plane, as

[1]

[2]

This model calculates the electrostatic interactions between H⁺ and SO^2-₄ complexation and the surface potentials, depending on the following parameters concerning the electrical and chemical properties of the SO^2-₄ adsorbent surface: [MOH]_tot (mol kg^-1), the content of surface hydroxyl groups per unit mass of soil; S (m² kg^-1), the surface area of SO^2-₄ adsorbent per unit mass of the soil; C₁ (C V^-1 m^-2), the capacitance of the layer between the 0-plane and ß-plane; C₂ (C V^-1 m^-2), the capacitance of the layer between the ß-plane and the bulk solution; K_OH2, the intrinsic equilibrium constant of Eq. [1]; K_SO4, the intrinsic equilibrium constant of Eq. [2]. In the calculation, S, C₁, and C₂ can be treated as two independent parameters, S x C₁ and S x C₂, because they appear in the model only as these two products.

Fumoto and Sverdrup (2000) also found that the SO^2-₄ and H⁺ adsorption, simulated by the ECCM, can be reproduced by the following empirical equations:

[3]

[4]

where Ê_SO4 and Ê_H (kmol_c kg^-1) are the SO^2-₄ and H⁺ adsorption, respectively; and q, m, n, a, b, c, and d are the soil-specific fitting parameters. The values of these parameters are determined statistically (e.g., by the least square method) to fit the SO^2-₄ and H⁺ adsorption calculated by the ECCM. As Eq. [3] and [4] are much simpler than the ECCM, they were used in the SO^2-₄ adsorption submodel of SAFE in this study.

Brief Description of the SAFE Model
SAFE is a dynamic, process-oriented model to simulate the soil acidification–recovery process resulting from atmospheric depositions. Since the detailed description of SAFE is found in Warfvinge et al. (1993), only a brief description of the key features is given here.

Mass Balance Equation for Soil Solution Chemistry
The soil profile is divided into several layers according to the natural soil stratification, and each soil layer is treated as a perfectly mixed tank reactor. According to this treatment, temporal development of each ion concentration in the soil solution is described by the mass balance equation:

[5]

where z (m) is the height of the soil layer; (m m^-3) is the volumetric water content; Q (m s^-1) is the downward water flux from the soil layer; Q₀ (m s^-1) is the downward water flux from the upper soil layer; [·] (kmol m^-3) is the concentration in soil solution; [·]₀ (kmol m^-3) is the concentration in soil solution of the upper layer; r^w, r^x, r^u, and r^nm (kmol m^-2 s^-1) are the ion fluxes due to mineral weathering, ion exchange, nutrient uptake, and net mineralization, respectively. It depends on the individual ion species which of these fluxes is included in the calculation.

Acid Neutralizing Capacity
The key variable in the soil solution equilibrium system is the acid neutralizing capacity (ANC), which is defined as (Reuss and Johnsson, 1985):

[6]

where R^- stands for organic acid anions. As each constituent of ANC is treated as a function of [H⁺], [ANC] is a function of a single independent variable [H⁺], and can be expressed such as = h. According to this treatment, every flux of the ANC constituent is regarded as a flux of ANC. From the charge balance, also, the following relationship holds (Reuss and Johnsson, 1985):

[7]

In other words, the ANC can be calculated from the balance of these cations and anions.

Diffusion-Controlled Cation Exchange
In SAFE, the nutrient base cations (Ca²⁺, Mg²⁺, and K⁺) are lumped into divalent base cations (BC²⁺), and Na⁺ is treated separately. One mole of K⁺ is counted as 0.5 mole of divalent base cations. The rate of cation exchange reaction is assumed to be controlled by the transport of base cations between the bulk soil solution and the cation exchanger surface, as:

[8]

where E_BC (kmol_c kg^-1) is the amount of exchangeable base cations, k_BC (m³ kmol^-1 s^-1) is the mass transfer coefficient of base cations to the cation exchanger surface, CEC (kmol_c kg^-1) is the cation exchange capacity, and [BC²⁺]_S (kmol m^-3) is the base cation concentration at the cation exchanger surface. E_BC is assumed to be in equilibrium with the surface concentrations of base cations and H⁺ according to the Gapon cation exchange equation. Each primed symbol, such as E^'_BC, denotes the time derivative of each variable hereinafter. k_BC consists as:

[9]

where D_BC (m² s^-1) is the lumped diffusion coefficient of base cations, l_BC (m) is the characteristic diffusion distance to the cation exchanger surface, and _BC (kmol m^-2) is the exchange site density on the cation exchanger surface. Evaluation of k_BC is discussed in a separate section later.

Weathering Rates based on Kinetic Rate Laws
The flux of base cations and Na⁺ due to mineral weathering, R_w (kmol_c m^-2 s^-1), is calculated as the sum of weathering rates of individual minerals according to the following equation:

[10]

where (unitless) is the degree of water saturation, A_exp (m² m^-3) is the exposed surface area of mineral matrix per unit volume of soil, R_i (kmol_c m^-2 s^-1) is the weathering rate of mineral i per unit surface area of the mineral, and x_i (unitless) is the fraction of mineral i in the exposed surface area of mineral matrix. The weathering rate of each mineral, R_i, is calculated as a function of the concentrations of H⁺, inorganic Al, base cations and DOC, the partial pressure of CO₂, and the temperature. The fluxes of individual base cations and Na⁺ are calculated assuming stoichiometric dissolution of each mineral.

Calculation Initiation and Calibration
As seen in Eq. [8], the SAFE model includes the cation concentrations at the cation exchanger surface as the variables, although their initial values are generally difficult to determine. Therefore, the calculation of SAFE is initiated in the preindustrialization era, when the ecosystem is assumed to be in a steady state, and consequently each ion concentration is the same in the bulk soil solution and at the cation exchanger surface. The initial steady-state ion concentrations are calculated by a separate routine, according to the mass balance equations and historical atmospheric deposition and nutrient uptake data. Consequently, time series of atmospheric deposition and nutrient uptake are required for calculation, and they are reconstructed based on available records and specific estimation methods. Alveteg et al. (1998) describe the methodology to reconstruct the time series of atmospheric deposition and nutrient uptake in detail. The calibration of the SAFE model is achieved by varying the initial base saturation of each soil layer to minimize the error sum between the measured and simulated recent base saturation.

Implementation of the Sulfate Adsorption Submodel into SAFE
Sulfate adsorption was implemented into the SAFE model according to the following assumptions. (i) Sulfate adsorption is a diffusion-controlled process, as is the base cation exchange. (ii) Adsorbed SO^2-₄ and H⁺ are in equilibrium with SO^2-₄ and H⁺ in the liquid phase at the adsorbent surface, and determined by Eq. [3] and [4]. (iii) Net negative charge on the SO^2-₄ adsorbent surface is neutralized by electrostatic adsorption of base cations, Na⁺, and inorganic Al cations, whereas net positive charge on the SO^2-₄ adsorbent surface is neutralized by electrostatic adsorption of NO^-₃ and Cl^-. (iv) The electrostatic adsorption of base cations, Na⁺, NO^-₃, and Cl^- to the SO^2-₄ adsorbent surface is also a diffusion-controlled process.

A detailed mathematical description is given in the appendix.

	MODEL APPLICATION

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Site Description
Solling Experimental Forest is situated in Germany (51°40' N, 9°30' E) on a plateau at a height of 505 m above sea level, in a large area of old beech (Fagus spp.) and spruce (Picea spp.) forests. The soil is a dystochrept (USDA) developed on triassic sandstone bedrock. Due to its location and the relatively high elevation, it has been exposed to long-range transported air pollutants, the sources of which are for the most part the Rhein–Ruhr industrial region to the west and the industrial centers of eastern Germany and the Czech Republic to the southeast. A substantial decrease in S deposition (ca. 60%) was observed in the area between 1975 and 1995. The F1 plot in the Norway spruce forest of this site has been intensively monitored since 1969. Detailed descriptions of climate, soil characteristics, hydrology, and deposition data are given by Bredemeier et al. (1995) and Tiktak et al. (1995).

In this study, the SAFE model was applied on the D1 plots of the roof experiment project, which has been performed near the F1 plot since 1992, including premanipulation measurements since 1989. The project includes four plots in a 66-yr-old (as of 1999) Norway spruce stand: a control plot without the roof (D0), a control plot with the roof (D2), a clean rain manipulation experiment plot (D1), and a drought experiment plot (D3). In the D1 plot, the actual throughfall was collected by the roof above the forest floor and replaced with synthetic throughfall, which had a composition roughly corresponding to the preindustrial conditions. The reduction in S and N in the synthetic throughfall was 85 and 90%, respectively, compared with the 1990 levels. The roof experiment is described in detail by Beier and Rasmussen (1993), Blanck et al. (1993), and Bredemeier et al. (1993).

Model Parameterization
Soil Layer Specific Data, Atmospheric Deposition, and Nutrient Uptake
Figures 1 and 2 show the time series of atmospheric deposition and nutrient uptake, respectively, used in the model simulation. Broadly speaking, the time series of atmospheric deposition was reconstructed by scaling the historical deposition trend to the present-day values measured during 1990 through 1991. Trends in historical S deposition were taken from Mylona (1993) and calculation with the RAINS model (Alcamo et al., 1990). Historical deposition of NH⁺₄ was estimated on the basis of Asman et al. (1988) in combination with RAINS model calculations. Records and estimates of historical NO^-₃ deposition are scarce, thus the historical NO^-₃ deposition included in the simulation before 1960 are very rough estimates. Historical deposition of base cations was estimated in several steps. First, the sea salt deposition was calculated and assumed to be constant over time. Second, all other present-day base cation deposition was assumed to be of anthropogenic origin and to follow the same trend as S. Between 1975 and 1989 (for S, between 1969 and 1989), measured deposition trends at the F1 plot (Bredemeier et al., 1995) were used for the reconstruction of deposition time series. After the commencement of the roof experiment in 1992, the controlled composition of synthetic throughfall was given as the atmospheric deposition.

View larger version (32K): [in this window] [in a new window]   Fig. 1. The time series of atmospheric depositions at the Solling D1 plot used in the simulation

View larger version (15K): [in this window] [in a new window]   Fig. 2. The time series of net base cation and nitrogen uptake at the Solling D1 plot used in the simulation

Sulfate Adsorption Parameters
Wesselink et al. (1994) reported the SO^2-₄ adsorption isotherm and its pH dependency for the fourth soil layer at the F1 plot. Sulfate adsorption isotherms at single pH values were reported for the upper three layers. These data were used to determine the parameters concerning SO^2-₄ adsorption in this study.

For the fourth layer, the parameters in the electrostatic SO^2-₄ adsorption model (ECCM) were optimized by the trial and error method to fit the SO^2-₄ adsorption data. The content of surface hydroxyl groups ([MOH]_tot) was estimated as 2.3 x 10^-2 mol kg^-1 from the specific surface area of the soil (Table 1) and the surface hydroxyl group density of 8.0 nm^-2 reported for a B horizon of a podzol (Karltun, 1997). When the ECCM was applied on the SO^2-₄ adsorption data, however, a satisfactory fit could not be achieved unless log K^intr_OH2 was given unrealistically small values such as 0.0. One of the possible explanations for this is that some necessary reaction is lacking in the ECCM. Persson and Lövgren (1996) found protonation of SO^2-₄ adsorbed to the goethite surface:

[11]

that had not originally been included in the ECCM in this study. Therefore, the ECCM was modified to include the reaction of Eq. [11] with an intrinsic equilibrium constant K^intr_SO4H. The parameters were optimized as S x C₁ = 9.2 x 10³ C V^-1 kg^-1, S x C₂ = 1.2 x 10³ C V^-1 kg^-1, log K^intr_OH2 = 5.5, log K^intr_SO4 = 10.95, and K^intr_SO4H = 12.5. Figure 3 shows the SO^2-₄ adsorption isotherms and the relationship between SO^2-₄ and H⁺ adsorption calculated by the ECCM. Since the pH dependency of H⁺ adsorption was relatively small, the empirical equations for SO^2-₄ and H⁺ adsorption (kmol_c kg^-1) were determined by the least square method as:

[12]

[13]

View larger version (22K): [in this window] [in a new window]   Fig. 3. The SO2-4 adsorption isotherms (upper graph) and relationships between SO2-4 and H+ adsorption (lower graph) of the fourth layer soil at Solling, calculated by the Extended Constant Capacitance Model (ECCM) for a liquid phase SO2-4 concentration range of 0.1 to 3.0 mmol L-1

Mass Transfer Coefficients
As seen in Eq. [9] and [A2], the mass transfer coefficients k_BC and _SO4 depend on the characteristic diffusion length to cation exchanger or SO^2-₄adsorbent surface, the magnitudes of which are difficult to assess explicitly. However, Warfvinge and Sverdrup (1989) succeeded in simulating the limestone dissolution rate in a lysimeter soil with a mathematical model that included the similar diffusion-controlled cation exchange process, using the k_BC values from 0.69 x 10^-4 to 25.4 x 10^-4 m³ kmol^-1 s^-1 depending on the limestone particle size. Referring to those values, k_BC was assigned 1.0 x 10^-4 m³ kmol^-1 s^-1 in this study. _SO4 was assigned 1.38 x 10^-4 m³ kmol^-1 s^-1, based on the fact that the diffusion coefficient of SO^2-₄ is 1.38 times that of Ca²⁺, the major base cation in soil solution. This is apparently an excessive simplification, because there is no guarantee that the values of l_BCBC and l_SO4SO4 (in Eq. [9] and [A2]) are of the same magnitude. However, even such an approach would fulfill, at least in part, the major objective of this study to show the effect of modeling of SO^2-₄ adsorption on the performance of SAFE model.

Calculation
The calculation was initiated with the assumed steady state in 1790 and continued up to 2020. The model was calibrated by adjusting the initial base saturation to minimize the error sum between the base saturation simulated and measured (extracted with 1 M NH₄Cl) in 1968, 1973, 1978, 1983, and 1991 at the F1 plot (Tiktak et al., 1995) for each soil layer. Base cations adsorbed to the net negative charge on the SO^2-₄ adsorbent surface were added to the calculated base saturation, because the measured exchangeable base cation includes the base cations on the SO^2-₄ adsorbent surface. On the other hand, base cations on the SO^2-₄ adsorbent surface are excluded from the calculation of the Gapon cation exchange constant. The calculated soil solution pH, SO^2-₄, and base cation concentrations in the second and fourth layers were compared with the yearly median of measured data at depths of 20 and 70 cm, respectively, from 1990 to 1994. In order to clarify the effect of different SO^2-₄ adsorption formulations, other simulations were performed using the pH-independent Freundlich model, or fixing SO^2-₄ and H⁺ adsorption at zero in each soil layer.

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Simulation at the 20-cm Depth
At the 20-cm depth, difference in SO^2-₄ adsorption formulation did not make a significant difference in the simulation of soil solution chemistry (Fig. 4–7) , as expected from the fact that the soil above this depth had only a small SO^2-₄ adsorption capacity (i.e., the content of surface hydroxyl groups), 2.1 mol m^-2 (from Tables 1 and 2). The measured SO^2-₄ concentration sharply declined after 1992, following the S deposition reduction by the roof (Fig. 6). Soil solution pH was reproduced successfully from 1990 to 1994 (Fig. 5), and SO^2-₄ concentration was also reproduced fairly well (Fig. 6). On the other hand, variation in base saturation was not followed by the model (Fig. 4), and base cation concentration was overestimated from 1990 to 1992 (Fig. 7), suggesting possible shortcomings in the SAFE model. These points are discussed later. The comparison between the different SO^2-₄ adsorption formulations cannot be made upon these results.

View larger version (25K): [in this window] [in a new window]   Fig. 4. Simulated base saturation of the second layer (20 cm deep) and fourth layer (70 cm deep) soils at the Solling D1 plot

View larger version (31K): [in this window] [in a new window]   Fig. 7. Simulated soil solution base cation concentration at the depths of 20 and 70 cm at the Solling D1 plot

View larger version (29K): [in this window] [in a new window]   Fig. 5. Simulated soil solution pH at the depths of 20 and 70 cm at the Solling D1 plot

View larger version (31K): [in this window] [in a new window]   Fig. 6. Simulated soil solution SO2-4 concentration at the depths of 20 and 70 cm at the Solling D1 plot

View larger version (31K): [in this window] [in a new window]   Fig. 8. Simulated SO2-4 and H+ adsorption in the second layer (20 cm deep) and fourth layer (70 cm deep) soils at the Solling D1 plot

Base Saturation and Soil Solution Base Cations
Simulated base cation depletion in the 1970s and 1980s was faster than the measurement, but the closest fit was achieved when the ECCM-based SO^2-₄ adsorption submodel was used (Fig. 4). This may suggest the fairness of the ECCM-based SO^2-₄ adsorption submodel to include the base cations adsorbed on the SO^2-₄ adsorbent surface in base saturation calculation.

The discrepancy between measured and calculated base saturation may have been caused by misestimation of the mass transfer coefficient of base cations (k_BC). This parameter directly affects the base cation exchange rate, in the SAFE formulation, but was hardly measurable and given a single guessed value for all soil layers. Such a coefficient seems likely to vary from soil layer to layer; it is also suggested by the fact that the base saturation variation was underestimated by the model in the second layer, while overestimated in the fourth layer (Fig. 4). Calibration of the mass transfer coefficient, as well as the initial base saturation, may be needed to improve the predicting ability of the SAFE model.

SAFE overestimated the base cation concentration at 70 cm from 1990 to 1993, regardless of SO^2-₄ adsorption formulation, as it did at 20 cm from 1990 to 1992. Possible causes for this include overestimation of mineral weathering rate and underestimation of base cation uptake. The overestimation of mineral weathering could be countermeasured by improving the process description in the model, but underestimation of base cation uptake could be controlled by destructive biomass sampling only, since SAFE requires nutrient uptake merely as input data. In this sense, SAFE is vulnerable to inaccuracy in nutrient uptake data.

	CONCLUSIONS

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The SO^2-₄ adsorption submodel, which was derived from the ECCM of SO^2-₄ adsorption, has been implemented in the SAFE model to simulate the acidification–recovery processes of SO^2-₄ adsorbing soils.

When applied on a roof experiment plot at Solling, Germany, the ECCM-based SO^2-₄ adsorption submodel simulated soil solution SO^2-₄ concentration and base saturation better than the previous SO^2-₄ adsorption formulations, such as the pH-independent Freudlich model.

SAFE may need additional improvements, such as calibration of mass transfer coefficients.

Mathematical Description of the Implementation of Sulfate Adsorption into SAFE
Common Governing Equations
Sulfate adsorption is assumed as a diffusion-controlled process, and the time derivative of SO^2-₄ adsorption is given as:

[A1]

where _SO4 (m³ kmol^-1 s^-1) is the mass transfer coefficient of SO^2-₄ to the SO^2-₄ adsorbent surface, [MOH]_tot (kmol kg^-1) is the surface hydroxyl group content of the soil, and _S (kmol m^-3) is the SO^2-₄ concentration at the SO^2-₄ adsorbent surface. _SO4 consists as:

[A2]

where D_SO4 (m² s^-1) is the diffusion coefficient of SO^2-₄, l_SO4 (m) is the characteristic diffusion distance to the SO^2-₄ adsorbent surface, and _SO4 (kmol m^-2) is the hydroxyl group density on the SO^2-₄ adsorbent surface. The mass transfer coefficients of other ions to the SO^2-₄ adsorbent surface are proportional to their diffusion coefficients. Evaluation of _SO4 is discussed in a separate section later.

From the mass balance of SO^2-₄:

[A3]

where is the bulk soil density (kg m^-3). As stated earlier, SO^2-₄ and H⁺ adsorptions are represented as:

[A4]

and

[A5]

where [H⁺]_S and pH_S are the H⁺ concentration and pH at the SO^2-₄ adsorbent surface, respectively. Differentiation of Eq. [A4] and [A5] with respect to time yields:

[A6]

and

[A7]

Equations [A1], [A3], [A6], and [A7] are the governing equations that do not depend on the net surface charge of SO^2-₄ adsorbent.

Governing Equations for Negative Surface Charge
When the net surface charge of the SO^2-₄ adsorbent is negative (i.e., E_SO4 > E_H), NO^-₃ and Cl^- are not adsorbed to the soil. Thus, equilibrium holds between their surface and the bulk solution concentrations, that is:

[A8]

and

[A9]

The mass balance equations become:

[A10]

and

[A11]

On the other hand, base cations, Na⁺, and inorganic Al cations will be adsorbed electrostatically to neutralize the negative surface charge. When u and v denote the fractions of the negative charge neutralized by base cations and Na⁺, respectively, base cation and Na⁺ adsorption is:

[A12]

and

[A13]

where Ê_BC and Ê_Na (kmol_c kg^-1) are the base cations and Na⁺ adsorbed to the net negative surface charge of SO^2-₄ adsorbent. The present model assumes that u and v are identical to the charge fractions of base cations and Na⁺ in the liquid phase. Thus,

[A14]

and

[A15]

where Ê_Al (kmol_c kg^-1) is the amount of inorganic Al cations adsorbed to the negative charge on the SO^2-₄ adsorbent surface, k_Gibb is the conditional gibbsite solubility, k_AlOH is the equilibrium constant of AlOH²⁺, and k_Al(OH)2 is the equilibrium constant of Al⁺₂. Since base cation and Na⁺ adsorption is controlled by diffusion between the surface and the bulk solution:

[A16]

[A17]

where _BC and _Na (m³ kmol^-1 s^-1) are the mass transfer coefficients of base cations and Na⁺ to the SO^2-₄ adsorbent surface, respectively. Thus, the mass balance equations of base cations and Na⁺ become:

[A18]

[A19]

On the other hand, differentiation of Eq. [A14] and [A15] with respect to time yields:

[A20]

and

[A21]

From the charge balance,

[A22]

From the mass balance equation of ANC and Eq. [A22],

[A23]

From Eq. [7], [A8], and [A9],

[A24]

Since [ANC]_S is a function of [H⁺]_S denoted by h([H⁺]_S), the equation above becomes:

[A25]

Differentiation with respect to time yields:

[A26]

In addition to the common governing equations, Eq. [A10], [A11], [A16] through [A23], and [A26] apply as the governing equations.

Governing Equations for Positive Net Surface Charge
When the net surface charge is positive, base cations and Na⁺ are not adsorbed to the SO^2-₄ adsorbent. Thus, equilibrium holds between their concentrations at the surface and in the bulk solution, that is:

[A27]

[A28]

Consequently, the mass balance equations of base cations and Na⁺ become:

[A29]

[A30]

On the other hand, NO^-₃ and Cl^- will be adsorbed to neutralize the positive surface charge. When the fraction of positive charge neutralized by NO^-₃ is denoted by w:

[A31]

where Ê_NO3 (kmol_c kg^-1) is the amount of NO^-₃ adsorbed to the positive surface charge. In the present model, w is assumed to be identical to the charge fraction of NO^-₃ in the liquid phase, that is:

[A32]

where Ê_Cl (kmol_c kg^-1) is the amount of Cl^- adsorbed to the positive surface charge. Adsorption rates of NO^-₃ and Cl^- are controlled by diffusion of these anions between the surface and bulk solution, as:

[A33]

and

[A34]

where _NO3 and _Cl (m³ kmol^-1 s^-1) are the mass transfer coefficients of NO^-₃ and Cl^- to the SO^2-₄ adsorbent surface, respectively. From the charge balance:

[A35]

On the other hand, differentiation of Eq. [A31] with respect to time yields:

[A36]

The mass balances of NO^-₃ and Cl^- are described as:

[A37]

and

[A38]

The mass balance of ANC is:

[A39]

The difference in ANC between the bulk solution and the surface becomes equivalent to the difference in SO^2-₄, NO^-₃, and Cl^- concentrations, that is:

[A40]

Since _S = h, the equation above becomes:

[A41]

Differentiation with respect to time yields:

[A42]

In addition to the common governing equations, Eq. [A29], [A30], [A33] through [A39], and [A42] apply as the governing equations.

Governing Equations at the Point where the Net Surface Charge is Neutral
At the point where the net surface charge is neutral, none of base cations, Na⁺, NO^-₃, or Cl^- are adsorbed to the SO^2-₄ adsorbent surface. Therefore, equilibrium holds between their concentrations in the bulk solution and at the SO^2-₄ adsorbent surface. To calculate the temporal development from this point, the following three cases have to be considered.

(i) When a negative net surface charge develops due to adsorption of more SO^2-₄ than H⁺, base cations, Na⁺, and Al species should be adsorbed to neutralize the negative surface charge, in the fractions determined by their concentrations in the liquid phase. Thus,

[A43]

[A44]

and

[A45]

Note that [BC²⁺]_S and [Na⁺]_S are equal to [BC²⁺] and [Na⁺], respectively. In this case, the difference in ANC between the SO^2-₄ adsorbent surface and the bulk solution is equivalent to the difference in SO^2-₄ concentrations. Thus,

[A46]

The mass balance equations of ANC, base cations, and Na⁺ become:

[A47]

[A48]

and

[A49]

When Eq. [A43] through [A49] and the common governing equations have the solution with Ê^'_SO4 > Ê^'_H, negative net surface charge develops. On this case, the mass balance equations of NO^-₃ and Cl^- become:

[A50]

and

[A51]

(ii) When positive net surface charge develops due to adsorption of less SO^2-₄ than H⁺, the mass balance equation of ANC becomes:

[A52]

Also in this case, the difference in ANC between the SO^2-₄ adsorbent surface and the bulk solution becomes equivalent to the difference in SO^2-₄ concentrations. Thus,

[A53]

When Eq. [A52], [A53], and the common governing equations have the solution with Ê^'_SO4 < Ê^'_H, positive net surface charge develops. Then, the time derivatives of NO^-₃ and Cl^- adsorption are:

[A54]

[A55]

The mass balance equations of base cations, Na⁺, NO^-₃, and Cl^- become:

[A56]

[A57]

[A58]

and

[A59]

(iii) If Eq. [A52], [A53], and the common governing equations have the solution with Ê^'_SO4 = Ê^'_H, the net surface charge stays neutral. Then, consequently:

[A60]

[A61]

[A62]

[A63]

and

[A64]

Numerical Solution
The computer program of the SAFE model was coded in Fortran. The governing equations of the system are differential-algebraic equations of the state variables, which include a set of implicit ordinary differential equations (ODEs). In the numerical solution, the set of implicit ODEs were first solved by the Gaussian elimination for the time derivatives, in terms of the state variables, to obtain a set of explicit ODEs. Then, the explicit ODEs were solved by the Runge–Kutta method.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
T. Fumoto, present address: National Institute of Agro-Environmental Sciences, Kannondai 3-1-1, Tsukuba, 305-8604 Japan.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION MODEL DESCRIPTION MODEL APPLICATION RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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