A mathematical model describes silicate mineral weathering processes in modern soils located in the boreal coniferous region of northern Europe. The process model results demonstrate a stabilizing biological feedback mechanism between atmospheric CO2 levels and silicate weathering rates as is generally postulated for atmospheric evolution. The process model feedback response agrees within a factor of 2 of that calculated by a weathering feedback function of the type generally employed in global geochemical carbon cycle models of the Earth's Phanerozoic CO2 history. Sensitivity analysis of parameter values in the process model provides insight into the key mechanisms that influence the strength of the biological feedback to weathering. First, the process model accounts for the alkalinity released by weathering, whereby its acceleration stabilizes pH at values that are higher than expected. Although the process model yields faster weathering with increasing temperature, because of activation energy effects on mineral dissolution kinetics at warmer temperature, the mineral dissolution rate laws utilized in the process model also result in lower dissolution rates at higher pH values. Hence, as dissolution rates increase under warmer conditions, more alkalinity is released by the weathering reaction, helping maintain higher pH values thus stabilizing the weathering rate. Second, the process model yields a relatively low sensitivity of soil pH to increasing plant productivity. This is due to more rapid decomposition of dissolved organic carbon (DOC) under warmer conditions. Because DOC fluxes strongly influence the soil water proton balance and pH, this increased decomposition rate dampens the feedback between productivity and weathering. The process model is most sensitive to parameters reflecting soil structure; depth, porosity, and water content. This suggests that the role of biota to influence these characteristics of the weathering profile is as important, if not more important, than the role of biota to influence mineral dissolution rates through changes in soil water chemistry. This process-modeling approach to quantify the biological weathering feedback to atmospheric CO2 demonstrates the potential for a far more mechanistic description of weathering feedback in simulations of the global geochemical carbon cycle.
 Earth's atmospheric concentration of the greenhouse gas CO2 is principally determined over millions of years by the balance between its supply from volcanoes and metamorphic degassing, and removal by the chemical weathering of calcium and magnesium silicate rocks [Berner, 1992]. Climatic control is introduced by a negative feedback loop whereby the rate of weathering of silicate minerals, and therefore the strength of the long-term CO2 sink, increases with higher temperatures and runoff [Walker et al., 1981; Berner et al., 1983]. Geochemical carbon cycle models predicting Earth's Phanerozoic CO2 history incorporate the direct effect of temperature on silicate weathering rates on the basis of (1) empirical relations between temperature and riverine concentrations of dissolved ions liberated by weathering [Berner et al., 1983; Berner, 1991], (2) high-temperature laboratory experiments [Lagache, 1965] extrapolated to low temperatures [Walker et al., 1981; Marshall et al., 1988; Schwartzman and Volk, 1989; Francois and Walker, 1992], and (3) Arrhenius formulations that relate the weathering rate constant to the activation energy of the mineral dissolution reaction [Brady and Carroll, 1994; Berner, 1994]. The effect of increased runoff on weathering rates is based on an empirical relation between runoff rate and riverine concentration of solutes originating from mineral dissolution, combined with a derived relation for runoff as a function of temperature from general circulation climate models.
 In addition to runoff and temperature, the evolutionary appearance and spread of vascular land plants is widely held to have introduced a potent biotic regulator of CO2 and climate in the long term as plant activities generally accelerate the weathering of silicate rocks [Volk, 1987, 1989; Marshall et al., 1988; Schwartzman and Volk, 1989; Berner, 1991, 1994]. Consequently, vegetation feedbacks on the long-term carbon cycle via silicate weathering through a range of different mechanisms [Berner et al., 2003; Beerling and Berner, 2005] are an important feature to represent in geochemical models. Indeed the effects of CO2, temperature and runoff alone yield a weathering feedback effect that is too weak, resulting in predictions of unrealistically high CO2 levels through the Phanerozoic [Berner, 1991]. To resolve this discrepancy, a wide range of geochemical carbon cycle models assume weathering rates to be proportional to the CO2 fertilization of plant productivity [Volk, 1987; Marshall et al., 1988; Berner, 1991, 1994; Berner and Kothavala, 2001; Wallmann, 2001; Bergman et al., 2004]. The underlying rationale for assuming this is, broadly, that vegetation productivity increases (1) the secretion of organic acids and chelators into soils by roots and their associated network of mycorrhizal fungi to obtain nutrient elements (principally P, K, Ca and Mg) from silicate minerals [Berner et al., 2003], (2) soil carbon dioxide partial pressure (PCO2, thus depressing pH) as a result of root and mycorrhizal respiration and microbial decomposition of organic matter, and (3) creation and retention of reactive mineral surfaces through mechanical cleavage by root growth and mycorrhizal biomass, and by anchoring colloidal soil particles against erosion to permit greater retention of soil water. The increased production of organic acids, elevated soil PCO2 and decreased pH can all accelerate the weathering of silicate minerals through abiotic mineral dissolution mechanisms that occur in terrestrial ecosystems [Andrews and Schlesinger, 2001; Williams et al., 2003].
 Here we develop, from first principles, a process model of soil weathering to quantify the impact of biological processes on mineral dissolution rates. Our aim is to clarify and critically assess the linked biological, chemical and hydrological processes that are proposed in global geochemical models to affect soil mineral weathering rates in response to changes in atmospheric CO2 content and climate. The model links biological productivity and respiration, hydrology, soil water chemistry and mineral dissolution in a modern soil that is assumed to be at steady state. The conceptual model is translated into a mathematical formulation through application of element mass and flux balance, reaction stoichiometry for mass transformation, chemical thermodynamics for phase and aqueous species equilibria, and kinetic mass action for irreversible reactions [e.g., Furrer et al., 1989]. Model parameter values are largely derived from independent laboratory and field process measurements and from runoff chemistry data for a well-documented catchment in central Sweden (60 °N) [see Maxe, 1994] and tested against similar data sets from a second Swedish catchment. Sections 1–3 in Text S1 list all process equations, parameter values and site specific input data.
 The rationale for this approach is the need to quantitatively assess the mechanistic basis for biological feedback to weathering that is postulated in the global models. Specifically, this process model formalizes and quantifies the proposed effects of biota on weathering through the role of organic acids, soil CO2 levels and reactive mineral surface area in kinetic mechanisms for mineral dissolution. In these simulations, modern forests are assumed to be in steady state with respect to nutrient cycling and we do not simulate long-term denudation of nutrient ions from catchments due to large episodic events such as landslip, floods or fires. The assumption of steady state is thus an inherent limitation, but is used here as a computationally tractable first step. It is envisioned that the mechanistic understanding developed with this modeling approach will eventually contribute to dynamic process models that are more complete and potentially usable as computational submodels within global geochemical cycle simulations.
2. Process-Based Modeling
2.1. General Approach
 We used the steady state soil chemistry model of Furrer et al.  calibrated and tested with site data from well-studied boreal forest catchments in central Sweden. The model considers slow mineral dissolution kinetics, described by empirical rate laws that are consistent with mechanisms and rate expressions based on rigorous chemical kinetic theory, and fast reversible reactions that describe geochemical equilibria using chemical thermodynamics (see supporting information (Figures S1–S8 and Text S1)). The model outputs soil water composition and the flux of solutes out of the soil system, including the aggregate flux of base cations (Na+, K+, Ca2+, Mg2+) as proxies for mineral weathering. The rate of cation export is referred to as the weathering flux, or more simply hereafter, as “weathering” (W, molar equivalents ha−1 yr−1). The relative change in weathering, compared to the present-day value (Wo) is a dimensionless variable (WR = W/Wo). We refer to WR as the “weathering feedback strength.”
 Application of the model proceeded through four stages: (1) definition of site specific parameters (i.e., mineralogy, hydrology, atmospheric deposition), (2) calibration and testing of model parameters against existing site data, (3) application to a benchmark simulation to test the effect of increasing atmospheric PCO2 and climate change on individual soil processes, and (4) sensitivity analysis for systematic variation of parameter values to assess the relative significance of key components of the weathering system.
 These changes in parameter values are selected to reflect variation of specific factors such as runoff rate, mean air temperature and biological productivity and their contribution to the weathering feedback strength at local scale. The results of the benchmark simulation were also compared with those calculated from the weathering feedback function in a global geochemical carbon cycle model [Berner, 1994, equation 34]. In this comparison, the same mean global Earth surface temperature, resulting from increasing atmospheric CO2 levels, was applied to both the process model and the weathering feedback function.
2.2. Model Description
 The soil model consists of three completely mixed soil solution compartments. Compartment A corresponds to the O and E layers at the Risfallet catchment (Figure 1a), while compartments B and C correspond to the B and C layers at the catchment (Figure 1b). The rate of infiltration, i.e., the downward water flow, in each compartment is modeled on the basis of existing data on the rate of evapotranspiration from each soil layer at the site [Jönsson, 1994]. The infiltration to compartment A is the average of the annual precipitation reduced by the evapotranspiration from the layer. The infiltration to compartment B (or C) is the average of the outflow from compartment A (or B) and the outflow from compartment A (or B) reduced by the evapotranspiration from the B (or C) layer. The discharge from compartment C is conceptualized as groundwater discharge to the stream, and is approximately equal to the average runoff from the catchment based on stream gauging.
 All compartments include both equilibrium and kinetic processes. Soil processes (Figure 2) occur within an idealized network of homogenous soil compartments connected by the route of water through the soil profile at Risfallet (Figure 3). Cation exchange reactions are not included since they do not contribute to a continuing flux of ions from an open system at steady state [Furrer et al., 1990].
2.3. Model Parameters
 This section describes the rationale and equations used to simulate the effect of increasing atmospheric CO2 on: (1) air and soil temperature as it affects river runoff and weathering rate constants, (2) the biological productivity, and (3) the dissolved organic carbon (DOC) decomposition rate. Section 5 in Text S1 lists the equations used to calculate the impact of increasing atmospheric CO2 levels on model parameter values Increases in atmospheric CO2 cause a general warming of the lower troposphere and a cooling of the stratosphere via the greenhouse effect. The greenhouse gas equation (equation 2 in Text S1) was used to calculate the increase in average annual Earth surface temperature as a function of increasing atmospheric CO2 levels. The temperature sensitivity coefficient Γ = 2.88 was based on an approximately 4°C increase in mean global surface temperature for a fourfold increase in atmospheric PCO2, from the climate simulation results of Manabe and Stouffer . Furthermore, because the equation was used to explore impacts on a modern soil, the effect of changing solar radiation flux over the Earth's geological history was neglected. The simulated weathering feedback response of the soil model to these changes in Earth surface temperature were compared directly with the weathering feedback function (equation 34 in the work of Berner , with Γ = 2.88 and Ws = 0). For both the calculations of the weathering feedback response and the weathering feedback function, an activation energy value of 15 kcal mole−1 was used to calculate the temperature effect on silicate mineral weathering using the Arrhenius equation (equation 3a in Text S1), in the benchmark simulation.
2.3.1. Amplitude of Seasonal Temperature Increases
 Warming is generally predicted to be greatest at high latitudes, due to a decreasing snow and ice albedo feedback, and weakest in the tropics [Manabe and Stouffer, 1980; Manabe and Bryan, 1985; Intergovernmental Panel on Climate Change (IPCC), 2001, 2007]. Furthermore, high-latitude warming by increased atmospheric PCO2 varies with season, being largest in early winter and smallest in summer, whereas low latitudes warming depends less upon season. Consequently, high latitudes in a high-CO2 world are likely to experience reduced seasonal amplitude of temperature variations.
 General circulation climate model calculations [Manabe and Stouffer, 1980] predict that a fourfold increase of atmospheric CO2 results in an approximate 10°C and 4°C warming of the surface air temperature in February and August, respectively, over continental landmasses between 60 and 65°N. Applied on the present-day seasonal temperature variation at Risfallet, this unevenly distributed warming over the year decreases the seasonal amplitude by around 3°C. Boer et al.  also calculated a decrease in seasonal amplitude of 2–3°C in Sweden, due to a doubling of the atmospheric CO2 level. Combining this result with that of Manabe and Stouffer , we define a linear relation between atmospheric PCO2 (P) and seasonal amplitude (a) in the interval of P/Po = 2–4 (equation (1)).
where the subscript o signifies the present-day amplitude and CO2 partial pressure.
2.3.2. Damping Depth
 The damping depth determines the extent to which surface air temperature amplitude decreases with increasing soil depth and is related to volumetric heat capacity, C (J m−3 °C) and thermal conductivity, kh (W m−1 °C) of the soil. These two parameters determine the ability of the soil to store and transport heat. Soil water content has a major impact on its heat capacity. The thermal conductivity shows large variations between the different components of the soil, being highest for pure minerals such as quartz and lowest for air. Air or water connects the different soil particles and so enables a heat transfer between adjacent particles. Given that the thermal conductivity of water is about 100 times greater than that of air, the soil water content also has a large impact on thermal conductivity. We may expect the soil damping depth to decrease with increasing soil water content, which is assumed to increase with runoff.
 Simulated hydrologic responses to a doubling in the concentration of atmosphere CO2 predict a fairly similar increase in evaporation rate over all latitudes but with increases in precipitation being largest at high latitudes and minimal at low latitudes [Manabe and Stouffer, 1980; IPCC, 2001, 2007]. The changes in terrestrial water balance produce latitudinally dependent changes in annual mean runoff, with runoff increasing at high latitudes and decreasing at low latitudes. To assess changes in runoff rate with rising CO2, we extrapolated these results up to 10× the present atmospheric level (PAL). The calculated maximum runoff increase for a fourfold increase in PCO2 was ca. 150 mm yr−1, while the corresponding decrease at low latitudes is about 110 mm yr−1.
2.3.4. Biological Productivity
 The physiological response of plant individuals to CO2 fertilization is relatively well understood [Bazzaz, 1990; Long et al., 2004; Ainsworth and Long, 2005]. The response of natural ecosystems to elevated CO2 levels is, however, difficult to assess [Bazzaz, 1990; Mooney et al., 1991; Long et al., 2004; Ainsworth and Long, 2005]. Water stress, nutrient availability and temperature affect the response of natural ecosystems to elevated PCO2, and vary to the degree that these factors set the limits on plant growth [Millard et al., 2007]. The maximum biomass productivity increase, expressed as a ratio relative to that of today (BPmax) on the ecosystem level, due to elevated CO2 levels, ranges in the literature from BPmax = 1–1.7 [Rotmans, 1990; Bazzaz, 1990; Beerling and Woodward, 2001]. In the GEOCARBII model of the global geochemical carbon cycle, a BPmax = 2 is used in combination with the assumption that, globally, only 35% of all land plants respond to CO2 fertilization [Berner, 1994]. This corresponds to a net global BPmax of 1.3, whereas Volk  argues that a BPmax of 2 is a reasonable value.
2.3.5. Dissolved Organic Carbon
 Stoichiometric uptake of inorganic nutrients occurs during biological production (equation (2)) while the reverse of this reaction corresponds to nutrient release during biomass decomposition. Rates of production and decomposition are set equal, reflecting an ecosystem at steady state.
A portion of the assimilated carbon in biomass (equation (3)) is, however, not completely oxidized during decomposition. This effect results in the formation of dissolved organic carbon (DOC, as moles C) and organic ligands represented as oxalate (C2O42−) and an associated H+ production (equation (4)).
consumes an equal amount of protons, i.e., produces alkalinity, and closes the biological carbon and proton cycles. If decomposition is so fast that the oxidation of the dissolved forms of C is completed within the residence time of the soil water, no net soil water acidification results. If oxidation of the dissolved carbon is not completed, then DOC and the organic ligands will be exported with the runoff, resulting in a net acidification of the soil. Thus, the flow rate of the soil water sets the residence time, while the rate of decomposition of dissolved Carbon set against residence time dictates the extent of acidification. This assumes that the pKa of the organic ligand is below the reference pH value that is used to define the level of alkalinity. Furthermore, if the reference pH value is selected as the endpoint of the acidimetric titration of the carbonic acid system, then the addition or loss of dissolved CO2 has no effect on calculated alkalinity levels in the model.
 Because the rates of most biological processes increase with temperature, thermal effects on microbial decomposition of DOC, described by the zero-order rate constant kDOC, are considered. We model this relation using a Q10 function in which Q10 represents the rate increase for every 10°C increase. Reported Q10 values for soil respiration range from 0.77 to 4.3 [Raich and Schlesinger, 1992; Kwon and Schnoor, 1994] with a median value of 2.4 [Raich and Schlesinger, 1992]. The chemical formula collectively representing DOC and the organic ligands (H2C2O4), is hereafter written as H2Org.
2.3.6. Regolith Depth, Soil Water Content, and Mineral Surface Area
 The weathering rate in each layer in the soil profile is represented by the expression:
where Rw = weathering rate per unit area of catchment (mol dm−2 s−1) h = depth of soil layer (dm) Ar = reactive surface area of mineral grains in the bulk soil (dm2 dm−3 soil solution) θ = soil water content (dm3 soil solution dm−3 soil volume) kw = weathering rate constant as stoichiometric mineral dissolution rate normalized to the mineral grain surface area (mol dm−2 s−1).
 The temperature variation of the weathering rate constants are described with an Arrhenius function. Weathering is also directly proportional to the three parameters which reflect the physical structure of the soil matrix (h, Ar and θ). All three parameters contribute to the ratio between the reactive mineral surface area and the volume of soil water, i.e., the total amount of wetted reactive mineral surface (Ar). The surface area of the mineral matrix depends on the particle geometry and size distribution, while soil water content reflects total porosity and the degree of hydraulic saturation. These factors are affected by soil erosion, but are also dependent on the collective biological influence of mycorrhiza and roots, as well as the accumulation of biomass, litter and humus.
 We assume that the soil acts as a linear tensiometer, i.e., that runoff is proportional to the water content. In addition, under elevated PCO2, an expected effect of CO2 fertilization on land plants is a more efficient use of water, with a suppression of plant transpiration by CO2-induced stomatal closure resulting in a net increase in soil water content and/or continental runoff. This effect has already been observed with the CO2 increase during the twentieth century [Gedney et al., 2006].
3. Model Calibration and Evaluation
 Calibration was achieved with runoff chemistry from the Risfallet catchment (60° 21′N, 16° 14′E) located in central Sweden in the northern boreal coniferous region. The catchment consists of gneissic granites and covers 45 ha of young pine forest, drained by a single stream. The dominating soil type is a podzol overlying a coarse sandy till extending to the bedrock located at depth of approximately 7 m. The distinct soil horizons have thicknesses of 6, 6, 53 and 35 cm for the O, E, B and C layers, respectively (Figure 1b). The present weathering rate is calculated to be 370 Eq ha−1 yr−1 [Maxe, 1994]. Additional details on model calibration methods and results are presented in section 4 of Text S1.
 The simulated water chemistry with the calibrated parameters for the outflow from the C layer is similar to the observed stream water runoff chemistry at Risfallet (Table 1); all ions other than calcium fall within the range of measured values and are generally close to the mean measured value. However, the measured calcium concentration is approximately double the model values and this is likely due to the uncertainty in the calcium content of hornblende.
Table 1. Results From Model Calibration at the Risfallet Sitea
Comparison of model predictions and field measurements for stream water chemistry (μEq dm−3). Field data from Maxe .
Composition of the outflow water from the C layer.
Mean composition of stream water during November 1986 to December 1990.
Range of stream water chemistry values measured in the field.
 The agreement between the modeled and observed stream water composition is noteworthy given that the rates of weathering for each Ca-, Na-, K- and Mg- silicate mineral were not individually calibrated. Only the total reactive surface area of the soil was calibrated to arrive at the correct total cation flux (section 4 of Text S1).
 The calibrated model was tested by transferring the parameter values obtained at Risfallet to a second catchment at Stubbetorp in Sweden. This was to test if any gross errors were evident, and to confirm parameter values obtained by calibration. The physical dimensions of the catchment and soil profile as well as average rainfall and soil mineralogy at Stubbetorp were implemented in the model.
 The predicted soil water composition in the B layer compares very well with that measured in the field and the simulated water chemistry for the outflow from the C layer compares very well with that of the stream runoff (Figure S4). All ion concentrations calculated by the model fall within the range of observed values for the stream, and are generally close to the average measured values. We conclude that the model provides a reasonable representation of weathering processes in a modern soil. The capability of the model to reproduce the stream water chemistry at both catchments is particularly good given the relatively small data requirements (base cation export, soil PCO2 and [DOC]) to calibrate the three parameters for each soil layer.
4. Results and Discussion
4.1. Weathering Feedback Response and Sensitivity to Model Parameter Values
 Model simulation of weathering in the Risfallet catchment with increasing atmospheric PCO2 yields a qualitatively similar weathering feedback response to that calculated with the weathering feedback function (Figure 4). The total weathering flux calculated by the soil process model for the calibrated model, applied to the Risfallet catchment, was 370 Eq ha−1 yr−1 which corresponds to the baseline condition at the present-day atmospheric CO2 level. The process model results in significantly lower weathering feedback response, compared to that calculated by the weathering feedback function, to the cumulative effects of temperature, runoff and biological productivity (Figure 5). The model simulation results show that when the effect of increasing soil temperature on biomass and DOC decomposition rates is included, the weathering feedback strength actually decreases slightly.
 Results of a parameter sensitivity analysis (Table 2) help identify the relative impact of individual mechanisms that give rise to the weathering feedback. Furthermore, comparison with the strength of the weathering feedback function sheds light on which processes most strongly contribute to the feedback at global scale. Weathering responds to changes in activation energy of mineral dissolution kinetics, changes in runoff rate, changes in biological productivity and changes in total reacting mineral surface area (Figure 6a). The model is relatively insensitive to PCO2-induced changes in runoff and BPmax but has greater sensitivity to Ea and total wetted area. Comparison with the weathering feedback function in Figure 6 shows that even prescribing an unrealistically high activation energy of 35 kcal mol−1 for silicate mineral dissolution is insufficient to account for the difference between models. Assessment of literature data shows that the range of reported activation energies for silicate minerals is 10–26 kcal mole−1 [Brady, 1991; Velbel, 1993; Brady and Carroll, 1994; Dorn and Brady, 1995].
Table 2. Parameter Values for the Sensitivity Analysis of the Risfallet Catchment Simulationsa
Range of Values Tested
The benchmark simulation is used as reference. Ar, reactive surface area of mineral grains in the bulk soil; Ea, activation energy; d, soil damping depth; BPmax maximum biomass productivity; Q10, rate increase for every 10°C increase.
Ar (m2 m−3 bulk soil)
Ea (kcal mole−1)
Runoff (mm a−1)
Q10 (applied on kDOC)
 Model results are most sensitive to the value of reacting surface area; demonstrating that underestimating this parameter by around 30% in the process model could account for any differences with the global feedback strength. Since this parameter is known to be relatively uncertain at field scale, the model simulation results do indeed strongly support the biological weathering feedback invoked in the global models. Inspection of equation (6) shows that weathering rate is proportional to, and thus equally sensitive to, the reacting surface area of minerals, the soil water content and the depth of soil. These results emphasize the important role of biota to enhance the reactive surface within the soil profile in a number of ways [Berner et al., 2003]. Increased biological productivity increases the belowground organic carbon flux and the development of plant roots and associated mycorrhizal networks. Roots alone provide tensile strength to the soil which inhibits mechanical erosion and allows the soil profile to develop. Mycorrhiza can penetrate the mineral matrix and thus increase porosity. Litter and humus accumulation can also stimulate microbial activity, and also the activity of the soil fauna which increase porosity through bioturbation.
 The process model results arise from simulation of chemical dissolution of minerals by the soil aqueous solution and the impact of biota on the solution composition. Direct uptake of nutrients from minerals to plant biomass is gaining recognition as a potential mineral dissolution pathway [Banfield et al., 1999; Leake et al., 2008; Balogh-Brunstad et al., 2008; Taylor et al., 2009]. Additionally, there is a possibility of extremely intense, localized biological activity that is not represented in processes that are averaged at catchment scale. The very dense mats of symbiotic mycorrhiza observed in association with Douglas fir stands in the Pacific Northwest [Griffiths et al., 1991] exemplify a potentially intense biological weathering environment. Soils within the mats were dramatically enriched (compared to nearby soil uncolonized by mats) in reactive solutes for mineral dissolution, such as protons and oxalic acid, and were also highly enriched in exchangeable cations [Griffiths et al., 1994] and soluble Al, Fe and Mn. If weathering hot spots do occur and potentially dominate the flux of inorganic nutrients from the soil minerals to the biota, this would not necessarily be captured in soil solution simulations that are described as occurring uniformly within a catchment soil system.
4.2. Individual Influences of Temperature, Runoff, and Biological Processes
4.2.1. Effect of Temperature on Mineral Dissolution Rates
 Temperature directly affects the rate constants, kw, for mineral dissolution through the Arrhenius relation. The contribution due to the effect of hydrolysis of the mineral structure, to the total mineral dissolution rate, is determined by a zero-order rate equation (Rate = kaq). For this reaction, an increase in temperature results in a proportional increase in kaq (equation (7)) and weathering rate. However, our model yields a ratio of 0.78:1 between the dissolution rate constant and the flux of base cations from the soil.
 Why does the process model deviate from the 1:1 proportionality between dissolution rate constants and weathering? The answer is that the rate of silicate weathering is due, in part, to the mass action kinetic effect of protons to accelerate weathering at lower pH values. The relative change in the proton-promoted weathering rate is dependent on the net change in soil water proton activity according to equation (8).
 An overall increase in weathering results in increased alkalinity production that mitigates the drop in pH. Inspection of equation (8) shows that this would cause the relative impact of proton-promoted weathering to decrease. Hence, even if rate constants increase with temperature, the effect on mineral dissolution rates is less than proportional. This lack of 1:1 proportionality becomes more pronounced in soils with lower pH because the relative contribution of equation (8) compared to equation (7) is greater. To complicate this picture further, the rate dependence on DOC is also affected by pH through the speciation of organic acids which exhibit pH-dependent adsorption and thus kinetic reactivity toward mineral dissolution.
 Considering the effect of changing amplitude in the annual variation of temperature also affects the dissolution rate. Weathering increases with an increasing temperature amplitude caused by rising atmospheric PCO2 (Figure 6b). In the benchmark simulation (Figure 5), the temperature amplitude is 4°C lower than the present-day amplitude of 11.2°C at Risfallet due to the sixfold increase in atmospheric CO2. This effect weakens the temperature feedback, for example, a 4°C decrease in amplitude (a = 7.2°C) results in a WR = 1.56. These results suggest that the effect of seasonal variation in surface air temperature is significant for the weathering feedback strength.
 Neglecting the change in the seasonal amplitude of temperature with rising atmospheric PCO2 also introduces uncertainty into assigning the appropriate activation energy. The increase in mineral dissolution rate constants for a sixfold increase in PCO2 is the same when using either an activation energy of 15 kcal mole−1 and an amplitude increase of 4°C (a = 15.2°C), or a 25 kcal mole−1 activation energy and 4°C (a = 7.2°C) amplitude decrease (Figure 6b).
4.2.2. Effect of Runoff on Weathering
 The weathering feedback response increases with increasing runoff rate (Figure 6c) and exhibits a sensitivity to values of this parameter that lies between that of activation energy and seasonal variation in temperature amplitude. We assume that the soil solution is always greatly undersaturated with respect to the primary silicate minerals. Under these conditions dissolution is kinetically controlled and the rate is not influenced by the degree of saturation. In such circumstances, the concentration of base cations (C), i.e., the weathering reaction products, in the runoff is expected to vary inversely proportional with the runoff rate (R) (equation (9), k′ is an empirical constant).
Given that weathering is the product of C and R, the weathering rate (W) is, in this case, independent of the runoff rate (equation (10)).
However, C and R are not inversely proportional in the process model, and we do see a dependence of weathering on runoff rate. The process model results show that Na+ and K+ concentrations are proportional to R−0.7, whereas the concentrations of Ca2+ and Mg2+ are proportional to R−0.9. This is because runoff correlates with the flow rate of percolating soil water. An increase in runoff decreases the residence time of the percolating water in each soil layer, decreasing the contact time between mineral and water. If mineral weathering is the dominant source of alkalinity, it and the associated pH will decrease. As weathering rate (and thus alkalinity production rate) increases with decreasing pH, the pH drop stabilizes at a steady state value which is lower than before the increase in runoff (see Figure S8).
 The decrease in pH is also associated with a relative increase in the organic carbon species [HOrg−], caused by increased protonation of the organic ligand (Org2−) in this pH range. Because HOrg− is formulated as the reactive species, the DOC-promoted weathering also increases. In other words, in a soil where mineral weathering is the dominant alkalinity source, an increase in flow rate of the percolating soil water has an acidifying effect. Whether this results in a net increase in weathering rate or not, depends on the contribution of proton-promoted and DOC-promoted weathering to the total weathering rate.
 We conclude that the residence time of the soil water directly influences the amount of alkalinity provided by weathering. Soil water residence time depends on the flow rate and correlates inversely with the runoff. Therefore increases in runoff rate result in a lower residence time, decreased alkalinity input from mineral dissolution and a net acidification of the soil. The result is an increase in weathering fate which stabilizes the drop in pH to an extent that further depends on the mineralogy.
4.2.3. Effect of Biological Processes on Weathering
 Atmospheric PCO2 fertilization affects biological productivity and the accumulation of DOC through several routes. First, when the rate of biological production increases, the concentration of DOC also increases. However, since microbial activity also increases with increasing temperature, there is an effect to increase the rate constant for DOC decomposition (kDOC) which impacts the soil solution proton balance, pH and thus the weathering rate.
Figure 6d illustrates this point by modeling changes in WR as a function BPmax for two different cases. In case 1, the rate of DOC decomposition is constant and equal to the present-day rate, corresponding to a Q10 of 1. Here, the effect is not very strong; that is, it requires a value for BPmax of around 16 to yield a weathering feedback response comparable to that of the weathering feedback function. In case 2, we consider the effect of temperature on the rate of DOC decomposition by applying a value of Q10 = 3. This clearly dampens the impact of CO2 fertilization on the weathering feedback response. This suggests that the rate of DOC decomposition is critical for the net effect of increased biomass productivity on the weathering feedback response.
4.2.4. Impact of Biota on Soil pH
 There are significant differences in pH, PCO2, [DOC] (Table 3) and weathering (Figure 6d) as a function of BPmax for Cases 1 (kDOC = constant) and 2 (Q10 = 3 for kDOC). The prescribed increase in biological productivity is equal in the two cases, while kDOC is greater in case 2. Results for both cases show that the greater the biological productivity, the lower the soil pH and the greater the soil PCO2 and DOC concentration. Note that in Case 1, the lower pH values and higher weathering rates are associated with equal or lower PCO2 values, compared to Case 2. This result contradicts the following two statements that are frequently made regarding soil PCO2, pH and weathering: (1) Soil PCO2 controls the soil water pH and hence (2) the greater the soil PCO2 the lower the pH and therefore the greater the weathering rate.
Table 3. Calculated Soil pH, PCO2, and DOC in the B Layer for Cases 1 and 2 as a Function of BPmaxa
pH, Case 1
pH, Case 2
PCO2, Case 1
PCO2, Case 2
PCO2 expressed as percent of the atmospheric pressure. DOC concentration in 10−6 mol dm−3. Case 1 is a Q10 = 1 applied on the DOC decomposition rate constant, and Case 2 is a Q10 = 3 applied on the DOC decomposition rate constant.
[H2Org], Case 1
[H2Org], Case 2
[HOrg−], Case 1
[HOrg−], Case 2
 Soil water pH and PCO2 are connected through the CO2/H2O system equilibrium reactions (equations (11)–(13)).
and the alkalinity [Alk] or acidity [Acy] balance (equation (14)).
This definition of alkalinity is consistent with a reference proton level based on the acidimetric titration endpoint for carbonic acid, where [H+] = [HCO3−]. Equation (11) shows that increases or decreases in CO2(g) levels will result in equimolar changes in [H+] and [HCO3−], hence no net impact on [Alk] or [Acy] as defined by equation (14).
 The pH value is uniquely determined for fixed values of alkalinity and PCO2. These values are in turn determined by the relative rates of processes which produce and consume alkalinity and CO2(g). Because the concentration of both CO2(g) and alkalinity affect pH, PCO2 is the sole control of pH only if the alkalinity stays constant.
 It is well established that biological processes have a large impact on both the alkalinity and CO2(g) budgets in soils [Schnoor and Stumm, 1985]. Root respiration and decomposition of biomass and DOC are all sources of soil CO2. Increased biological productivity therefore results in an increased soil PCO2. If soil solutions are undersaturated with respect to carbonate mineral phases, the only sinks for soil CO2 are dissolved export with runoff and gas diffusion to the atmosphere. The rate of CO2 degassing can be described by a linear mass transfer law, and is proportional to soil PCO2. Thus, the increase of soil PCO2 from increased biological productivity drives greater dissolved and diffusion fluxes which limit the PCO2 increase.
 The dominant sources of alkalinity are weathering reactions and decomposition of biomass (the reverse of equation (3)) and DOC (equation (6)). Alternatively, biomass production consumes alkalinity (equation (3)). If the rate of biomass production is greater than the rate of DOC decomposition, DOC will accumulate. Because protons are consumed during decomposition of DOC, the accumulation is associated with a net acidification (see section 2.3.5).
 The results in Table 3 can now be understood in terms of the relative rates of these biological processes. An increase in biological productivity can decrease soil pH, and hence increase the weathering rate, by a combination of two different mechanisms. The first is increased soil PCO2 due to increased respiration. The second is accumulation of DOC. The combination of acidification due to DOC accumulation, and the effect of increased PCO2, together determine the magnitude of the drop in pH. Thus, more rapid decomposition of DOC mitigates the drop in pH by releasing more alkalinity.
 These results illustrate that there is no simple relation between biological productivity, soil PCO2, pH and weathering rates. This is due to the high sensitivity of soil alkalinity and pH to biological processes, which is explained by the large magnitude of the H+ fluxes associated with them.
4.2.5. Contribution of Organic Acids to the Biological Weathering Feedback
 Current weathering theories propose that organic acids have an important impact on weathering rates in soils and for the biologically mediated weathering feedback [Schwartzman and Volk, 1989; Brady, 1991; Berner, 1992; Brady and Carroll, 1994]. However, our model simulations do not strongly support this hypothesis. By considering each individual kinetic rate law in our model (Figure S3), the contribution of DOC-promoted weathering is only 2% of the total weathering flux for the Risfallet catchment (results not shown).
 From Figure 6a, our model results indicate that variations in pH and reactive mineral surface area in different soil horizons, are important factors limiting the impact of organic acids on weathering. In our model, the singly protonated ion, HOrg− is the reactive aqueous species accelerating mineral weathering. We assign to it the same acidity and stability constants as for the protonation (and complexation) reactions of hydrogen oxalate ion, represented by equation (15).
 Over the pH range of the soil profile at Risfallet, pH 4.3 in the A horizon to pH 5.7 in the C horizon, the activity of HOrg− decreases due to the dominance of the unprotonated species Org2− at pH values above the pKa2 value of 4.3. Because accumulation of DOC results in net acidification (see section 4.2.4), we expect the highest rates of DOC-promoted weathering to occur in the organic (A) horizon where total DOC concentrations are highest, while pH values are low, to favor the formation of HOrg−. In the B horizon, lower total DOC concentration and higher pH results in much lower concentrations of the reactive species, HOrg−. The effect of organic acids on the weathering flux is therefore relatively unimportant in the B horizon, and contributes even less to weathering in the C horizon.
 The amount of reactive mineral surface area in each soil horizon also influences the relative contribution of DOC-promoted weathering to the total weathering flux. Although the concentration of HOrg− is highest in the organic A horizon at the Risfallet catchment, the specific mineral surface area and the depth of the soil layer are much less than in the B and C horizons (Tables 1 and 2). This explains why the effect of organic acids on weathering is negligible for the total weathering flux from the soil profile but important in the A horizon. Dissolved Al ions are also complexed by organic ligands, thus preventing the ligand from adsorbing on the mineral surface. This competition is greatest in the A horizon where the highest dissolved Al concentrations are encountered, and may reduce the impact of organic acids on weathering. These results suggest that although the concentration of soil organic acids, and hence potential weathering-promoting effect is high, the actual effect on weathering rates can be very limited. Consequently, ligand-promoted weathering by DOC may have only a minimal role in the biologically mediated weathering feedback. One caveat is that the process model averages processes over catchment scale. As noted above, an alternative explanation for intense weathering by organic acids is their highly localized, intense production that is not captured by the model.
4.2.6. Vadose and Groundwater Zone
 The annual variation of the groundwater level at the Risfallet catchment typically ranges between depths of 1 and 3.5 m, with an average value of 2.5 m [Maxe, 1994]. We further considered the contribution of the vadose zone and groundwater, between 1 and 2.5 m depth, by introducing a fourth soil layer compartment connected to the C layer compartment. All parameters, except the reactive mineral surface area, were set equal to those of the C layer. The reactive surface area was calibrated to give the groundwater pH measured at the field site (pH 6.2 at 2.7 m) [Maxe, 1994]. This resulted in a calibrated surface area of 4 × 103 m2 m−3 and a weathering of base cations equal to 1120 Eq ha−1 yr−1 down to 2.5 m. It is reasonable to assume for the Risfallet soil that the annual weathering flux between the surface and 1 m (370 Eq ha−1 yr−1) represents the average weathering per meter of depth in the till and, therefore, that the weathering rate in the soil profile down to 2.5 m would be 925 Eq ha−1 yr−1 which corresponds well with the model result. Figure 6e compares the weathering feedback strength, based on the total weathering flux down to 2.5 m, with that down to 1 m. The consideration of weathering in the vadose zone and groundwater results in a slightly stronger feedback than when omitted, but the effect is not dramatic. As expected, there is a proportionally greater weathering flux under current atmospheric PCO2, when including a greater depth of regolith. However, increasing atmospheric PCO2 does not significantly increase the contribution of weathering in the deeper regolith, to the total weathering flux; that is, the deeper regolith contributes to the total weathering, but not significantly to the weathering feedback response. One explanation for the observed small increase in weathering feedback response when including the deeper regolioth is that the deeper weathering profile is also where higher pH values occur, due to greater residence time and alkalinity production from weathering. With elevated PCO2 levels, higher pH values lead to a more pronounced impact of carbonate ions to adsorb and accelerate weathering rates in an analogous effect as organic acids at low pH [Bruno et al., 1992; Berg and Banwart, 2000].
4.2.7. Impact of Soil Structure on the Weathering Feedback
Figure 6f illustrates the response of weathering to changes in the total amount of reactive mineral surface area that is in contact with soil water (total wetted Ar). The parameters soil water content (θ), reactive surface area of the mineral matrix (Ar), and soil layer depth (h) all contribute to this parameter. The feedback strength is much more sensitive to these parameters, than to any of the others previously assessed. The two curves represent weathering down to 1 m and down to 2.5 m, respectively. When considering weathering to a greater depth, the sensitivity to this parameter is only slightly greater.
 The strong sensitivity to this parameter is an important result. This suggests that biotic influences on these key characteristics of the soil profile can play a dominant role in the weathering feedback strength. As discussed above, greater biological productivity is plausibly associated with greater mycorrhiza and root propagation, and accumulation of litter and humus, all of which contribute to production of reactive surface area and retention of soil water.
 The soil process model provides a “snapshot” of weathering processes in an ecosystem that is assumed to be at steady state; that is, runoff, temperature and biological productivity are assumed to remain constant for a given soil profile. This is also the same approach taken within a single time step for global geochemical cycle models. The main difference, however, is that global models assume steady state over time steps of perhaps 10 million years, after which parameter values are updated using the relationships describing climatic changes with atmospheric CO2. The average effect of episodic, extreme events such as floods, fires and landslip, can be assumed at steady state when averaged over such geological time scales. This is not the case for the time scale of the observations (decades) for the modern soils used here to develop the process model. Importantly, even with these limitations, the results of the process model and the weathering feedback function agree quantitatively within a factor of 2. Detailed analysis of the effect of biota on the mechanisms of weathering feedback show that the impact of productivity and biomass decomposition on soil acidity and pH are crucial. Also, because of the large sensitivity of the feedback strength to parameters reflecting soil structure, the role of biota to influence the structure of the soil profile is also very important. Furthermore, attention should be given to the possibility of locally intense biological impacts on weathering that are not captured in process models when averaging over catchment (and larger) scales.
 A. Berg was supported by the Swedish Environmental Research Council. We gratefully acknowledge funding from the U.K. Natural Environment Research Council (NERC) for collaboration between S. A. Banwart (grant NE/C521001/1) and D. J. Beerling (grant NE/E015190/1).