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TECHNICAL REPORTS

Landscape and Watershed Processes

Mathematical Modeling of Phosphorus Losses from Land Application of Hog and Cattle Manure

^a Department of Renewable Resources, University of Alberta, Edmonton, AB, Canada T6G 2E3
^b Plant Industry Division, Alberta Agriculture, Food and Rural Development, Edmonton, AB, Canada T6H 4P2

^* Corresponding author (robert.grant{at}ualberta.ca).

Received for publication November 22, 2002.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT PHOSPHORUS RUNOFF EXPERIMENT MODEL APPLICATION RESULTS DISCUSSION REFERENCES

 
Mathematical models may provide a means to estimate phosphorus (P) losses from land application of manure. Phosphorus losses typically occur during brief episodes of runoff and erosion. Models must be able to simulate P losses during these episodes by representing the basic chemical, physical, and biological processes by which these losses occur. The mathematical model ecosys combines dynamic distributed flow of solutes and nonsolutes through runoff and erosion with convective–dispersive transport of solutes, and both biologically and thermodynamically driven transformations between solutes and nonsolutes. This model was tested against P lost in runoff, erosion, and leachate measured during 90 min of controlled rainfall at 65 mm h^–1 on soils from six sites at which different rates of manure had been applied over the previous 3 to 6 yr. Transport and transformation kinetics in the model enabled it to simulate changes of dissolved inorganic phosphorus (DIP) in runoff from >1.0 to <0.05 mg L^–1 and changes of total phosphorus (TP) in sediment from 15 to 3 mg L^–1 measured during controlled rainfall on soils with diverse P contents. Results from 60-yr model runs using these kinetics with different application rates of cattle manure indicated that (i) a positive interaction exists between annual rainfall and application rate on P losses and (ii) rates greater than 30 Mg ha^–1 yr^–1 would cause TP concentrations in water leaving the site to rise above acceptable limits. The interaction between rainfall and rate suggests that P losses from manure application at any site should be assessed under the upper range of likely rainfall intensities.

Abbreviations: DIP, dissolved inorganic phosphorus in runoff • LP, organic + inorganic phosphorus in leachate • TP, total phosphorus in sediment

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT PHOSPHORUS RUNOFF EXPERIMENT MODEL APPLICATION RESULTS DISCUSSION REFERENCES

 
THE LAND APPLICATION of manure from expanding livestock production in Alberta has raised concerns about water quality. Many of these concerns focus on phosphorus (P) because ratios of P to N and C in manure are larger than those found in soils and taken up by plants. Phosphorus contents of surface water in areas of intensive manure application have been found to exceed water quality guidelines for the protection of aquatic life (Canada-Alberta Environmentally Sustainable Agriculture Council, 1998). There is a need therefore to establish application rates that enable disposal of livestock manure while avoiding P accumulation in surrounding waters.

Most P losses from land application of manure occur as dissolved organic and inorganic P in runoff, and as solid organic, adsorbed, or precipitated P in sediment. The concentration of dissolved P in runoff rises with repeated manure applications because the ability of soil to adsorb and precipitate added P declines as adsorption sites are occupied and coprecipitates are depleted. The concentration of solid P in sediment also rises with manure application because manure contributes to organic and inorganic P stocks at the soil surface. Attempts to relate the concentration of dissolved P in runoff to concentrations of P in soil are frequently based on P extracted according to existing soil test protocols (e.g., Daniel et al., 1993). However, P losses will also be affected by climatic, topographic, agronomic, and edaphic factors that determine runoff and erosion (Sharpley et al., 1996), requiring a broader approach to estimating these losses.

One emerging approach to estimating P losses from manure and other soil amendments is the use of mathematical models that simulate the key processes (adsorption, precipitation, mineralization, plant uptake, solute transport, runoff, and erosion) by which these losses are controlled. However, many such models function at temporal scales (typically daily) larger than those at which P losses occur (typically hourly or less), and are therefore forced to use simplifying assumptions to represent key processes. In some cases, key processes such as mineralization, adsorption, or leaching are omitted from the models (e.g., Zhang et al., 1995). Although these models have simulated P losses at monthly and annual time scales (e.g., Pierson et al., 2001; Yoon et al., 1994), they may have a limited ability to simulate events with temporal scales of minutes to hours during which most P loss occurs (Edwards et al., 1994). Some more detailed models simulate key processes explicitly at time scales better suited to modeling P loss events. However, these models often represent in detail only a subset of the key processes that control P losses, such as water, sediment, and solute transport (e.g., Walter et al., 2001; Wang et al., 1996). The modeling of P transfer between solid and soluble phases during these events is usually based on sorption isotherms that do not represent the mechanisms of P transformation in soil. Because these isotherms are fitted from soil-specific data, their parameters are not unique (Schoumans and Groenendijk, 2000) and hence are unlikely to be generally applicable (Mansell et al., 1985). These more detailed models are well adapted to simulating individual P loss events, but may not be well adapted to simulating long-term P loss under different land management practices unless all key processes affecting P losses are represented at similar levels of detail.

Because long-term P losses are the sum of several P loss events, models used to simulate long-term P losses must be capable of simulating individual P loss events over long periods of time. Such models must have three key attributes: (i) they must function at the time scales at which these events occur (minutes to hours), (ii) they must use basic theory to represent a comprehensive set of the processes by which P losses are controlled so that P loss events may be simulated accurately over several years, and (iii) all parameters used to model these processes should be evaluated from independent experiments to ensure generality of model application. We propose to combine these three attributes in the ecosystem model ecosys (Grant, 2001) by coupling P transport processes (runoff, erosion, solute movement, leaching), solved on a time step of 1 min, with P transformation processes, both biological (microbial mineralization–immobilization, root uptake) and geochemical (adsorption–desorption, precipitation–dissolution), solved on a time step of 1 h. We propose to model biological transformations from basic theory of microbial ecology and to model geochemical transformations from thermodynamic equilibria of specific P reactions. Values for all parameters used to model these processes are taken from published experiments conducted independently of the model.

The ability of this coupled model of P transport and transformation to simulate individual P loss events from manured soils was first tested against dissolved and total P in runoff, sediment, and leachate measured during controlled rainfall applied to several soils with different manure management histories. The ability of this same model to simulate long-term P losses as the total of individual P loss events was investigated by modeling P losses during 60 yr of different manure application rates on one of the soils tested under controlled rainfall. The model's ability to simulate longer-term P losses will be tested later against measurements of dissolved and total P from manured watersheds using this model in three-dimensional mode.

	MODEL DEVELOPMENT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT PHOSPHORUS RUNOFF EXPERIMENT MODEL APPLICATION RESULTS DISCUSSION REFERENCES

 
Inorganic Transformations
Precipitation–Dissolution
Precipitation–dissolution of P is simulated by concurrently solving the equilibrium reactions listed in Eq. [A1] to [A9] in the Appendix below for each spatial element defined by its north–south, east–west, and vertical position within a landscape. Each of these reactions is solved by calculating a flux Q such that:

[1]

where A, B, and C are soluble products with stoichiometric coefficients a, b, and c and where D and E are soluble reactants with stoichiometric coefficients d and e, all expressed as solution concentrations (mol m^–3). The terms in Eq. [1] represent solution activity coefficients calculated from ionic strength using the Davies relationship (Stumm and Morgan, 1970). Solubility constants K for these reactions were taken from Lindsay (1979) and Aggarwal et al. (1986).

Cation Exchange
Cation exchange is simulated by concurrently solving the Gapon equilibria listed in Eq. [A10] to [A15] below to calculate a flux Q for each spatial element such that:

[2]

and such that:

[3]

where A and B in Eq. [2] are cation pairs with valences a and b in solution (mol m^–3) and in adsorbed forms XA and XB (mol Mg^–1), and where terms represent solution activity coefficients. In Eq. [3], is the soil water content (m³ m^–3), n the number of exchangeable cation species N from Eq. [A10] to [A15], the soil density (Mg m^–3), and CEC the cation exchange capacity (mol_c Mg^–1). Gapon selectivity coefficients g used in Eq. [A10] to [A15] were taken from Reuss (1983) and Robbins et al. (1980).

Anion Exchange
Anion exchange is simulated by concurrently solving the adsorption equilibria listed in Eq. [A16] to [A20] below to calculate a flux Q for each spatial element such that:

[4]

and such that:

[5]

where A and B in Eq. [4] are anion pairs in solution (mol m^–3) and in adsorbed forms XA and XB (mol Mg^–1), terms represent solution activity coefficients, and ß terms represent surface activity coefficients calculated from surface potential according to Stumm et al. (1980). In Eq. [5], represents soil water content (m³ m^–3), n the number of exchangeable anion species N from Eq. [A16] to [A20], the soil density (Mg m^–3), and AEC the anion exchange capacity (mol_c Mg^–1). Intrinsic equilibrium constants c for protonation of the anion exchange sites (Eq. [A16], [A17]) are taken from Goldberg and Sposito (1984a)( 1984b). Intrinsic equilibrium constants c for phosphate exchange at these sites are calculated from Goldberg and Sposito (1984a)( HREF="#BIB8">1984b) to conform to the reactions (Eq. [A18]–[A20]) proposed by Ryden and Syers (1975).

Ion Pairs
Soluble ion pairing is calculated by concurrently solving the equilibrium reactions listed in Eq. [A22] to [A55] as described in Eq. [1]. Equilibrium constants in Eq. [A1] to [A55] remain unchanged for all model applications. Further details concerning the solutions to Eq. [A1] to [A55] are given in Grant and Heaney (1997).

Organic Transformations
Organic Matter Hydrolysis
Organic transformations in ecosys are based on four organic matter–microbe complexes: plant litterfall, animal manure, particulate organic matter, and humus, each of which consists of six organic states: substrate, soluble, sorbed, acetate (for methanogenesis), microbial biomass, and microbial residues, among which C, N, and P are transformed. Plant litterfall and animal manure are partitioned into carbohydrate, protein, cellulose, and lignin components, each of which is of differing vulnerability to hydrolysis by heterotrophic decomposers. Particulate organic matter and humus are also of differing vulnerability to hydrolysis.

The rate at which each component is hydrolyzed (D_Si,j,C) is a first-order function of the active biomass of obligately aerobic, facultatively anaerobic, and obligately anaerobic heterotrophic decomposers M_i,a,C associated in each organic matter–microbe complex (Eq. [A56], [A57]). These rates are affected by the temperatures and water contents of surface detritus and those of a spatially resolved soil profile (Grant and Rochette, 1994). Microbial biomass in ecosys is thus an active agent of organic matter transformation rather than a passive organic state as in models that use first-order kinetics. The rate at which each component is hydrolyzed is also a function of substrate concentration [S_i,C] that approaches first order at low concentrations, and zero order at high concentrations (Eq. [A58], [A59]). These rates are controlled by soil temperature through an Arrhenius function (Eq. [A60]) and by soil water content through its effect on aqueous microbial concentrations [M_i,a,C] (Eq. [A58], [A59]). Soil temperatures and water contents are calculated from surface energy balances and from heat and water transfer schemes through canopy–snow–residue–soil profiles. Release of P from hydrolysis of each component in each complex is determined by its P concentration (Eq. [A61], [A62]). Hydrolysis products are adsorbed or desorbed according to a power function of their soluble concentrations (Eq. [A63]–[A65]).

Microbial Growth, Respiration, and Decomposition
The concentration of soluble hydrolysis products [Q_i,C] drives C oxidation by each heterotrophic microbial population R_gi,C (Eq. [A66]), the total of which drives CO₂ emission from the soil surface. Heterotrophic oxidation rates may be constrained by temperature (Eq. [A66]), nutrients (Eq. [A67]), and O₂ (Eq. [A68]). Oxygen uptake R_O2i,C is driven by C oxidation (Eq. [A69]) and constrained by O₂ diffusivity (Eq. [A70]), so that C oxidation is coupled to O₂ reduction by all aerobic populations according to O₂ availability (Grant et al., 1993a, 1993b; Grant and Rochette, 1994). Carbon oxidation not coupled with O₂ is coupled with the sequential reduction of NO^–₃, NO^–₂, and N₂O by heterotrophic denitrifiers (Grant et al., 1993c, 1993d; Grant and Pattey, 1999) and with the reduction of organic C by fermenters and heterotrophic methanogens (Grant, 1998a; Grant and Roulet, 2002). In addition, autotrophic nitrifiers conduct NH⁺₄ oxidation and NO^–₃ production (Grant, 1994) and N₂O evolution (Grant, 1995), and autotrophic methanogens and methanotrophs conduct CH₄ production (Grant, 1998a) and oxidation (Grant, 1999).

All microbial populations in the model undergo maintenance respiration R_Mi,j,C (Eq. [A71], [A72]) and decomposition D_Mi,j,C (Eq. [A76], [A77]). Heterotrophic respiration in excess of maintenance requirements is used as growth respiration R_gi,C (Eq. [A73), the energy yield G of which determines heterotrophic and autotrophic growth yields, and hence substrate consumption (Eq. [A74]) and P uptake (Eq. [A75]). Changes in microbial biomass arise from differences between substrate consumption and maintenance + growth respiration + decomposition (Eq. [A78]). During these changes, microbial populations seek to maintain equilibrium ratios of biomass C to N to P by mineralizing or immobilizing NH⁺₄, NO^–₃, and H₂PO^–₄ (I_i,j,P in Eq. [A79]), thereby controlling solution concentrations of inorganic N and P. Changes in microbial P arise from differences in organic P uptake plus inorganic P immobilization and microbial P decomposition (Eq. [A80]). Labile and resistant components of microbial biomass are used to calculate active microbial biomass M_i,a,C (Eq. [A81]), which then drives C oxidation (Eq. [A56], [A57]).

Humification of Products from Residue Hydrolysis and Microbial Decomposition
Carbon and P products of litterfall and manure hydrolysis D_Si,j (Eq. [A56], [A61]) are transferred to the particulate organic matter complex at a rate H_Si,j that depends upon lignin hydrolysis and soil clay concentration (Eq. [A82]–[A85]) according to stoichiometry proposed by Shulten and Schnitzer (1997). Carbon and P products of microbial decomposition D_Mi,j (Eq. [A76], [A77]) are transferred to the humus complex at a rate H_Mi,j that depends on soil clay content (Eq. [A86], [A87]; Grant et al., 1993a, 1993b).

Plant Uptake
Phosphorus uptake is calculated for each plant species in a complex biome by solving for the aqueous concentrations of H₂PO^–₄ at both root and mycorrhizal surfaces in each rooted soil layer at which radial convective–dispersive transport from the soil solution equals active uptake by these surfaces. Active uptake is calculated from length densities and surface areas (Itoh and Barber, 1983) given by a root and mycorrhizal growth submodel (Grant, 1998b; Grant and Robertson, 1997). The products of P uptake are added to root and mycorrhizal storage pools from which they are combined with storage C when driven by growth respiration to form new plant biomass. The modeling of plant P uptake is described in more detail elsewhere (Grant, 1998b; Grant and Robertson, 1997).

Water Transport
Surface Flow
Surface flow is calculated as the product of runoff velocity v, depth of mobile surface water d, and width of flow paths L in west to east x and north to south y directions for each landscape position x,y (Eq. [A88]). Changes in the depth of surface water d_w arise from differences in surface flows among adjacent landscape positions. Runoff velocity is calculated in x and y directions for each x,y from the hydraulic radius R, from slope s, and from Manning's roughness coefficient z_r calculated from microtopographic roughness and particle size according to Morgan et al. (1998b). Slopes with easterly and southerly aspects have positive values (Eq. [A89a, A89b]), and those with westerly or northerly aspects have negative values (Eq. [A89c, A89d]). The depth of mobile surface water d in Eq. [A88] is the positive difference between depth of surface water d_w + ice d_i and the maximum depth of surface storage d_s, calculated from microtopographic roughness and slope according to Shaffer and Larson (1987). The value of d_w arises from the difference between rates of precipitation and infiltration (described in the next paragraph). The calculation of R in Eq. [A89] assumes overland flow through triangular channels (Schwab et al., 1996) (Eq. [A91]). The slopes s_x and s_y in Eq. [A89] are the elevational gradients of water surfaces calculated in x and y directions for each x,y from the sum of ground surface elevation E, d_s, and d (Eq. [A92]). Equations [A88] to [A92] thus implement the kinematic wave theory of overland water flow in which changes in horizontal flow plus changes in surface water depth equal the difference between rainfall and infiltration.

Subsurface Flow
Water fluxes (Q_w in Eq. [A93]) are the product of hydraulic conductances K' and water potential differences in west to east x, north to south y, and vertical z directions for each landscape position x,y,z. Water potentials are the sum of matric, osmotic (multiplied by a reflection coefficient), and gravitational components. Conductances are calculated from the geometric means of the hydraulic conductivities K (Green and Corey, 1971) of adjacent landscape positions in x, y, and z directions (Eq. [A94a], [A95a], and [A96a]) unless of one of the positions exceeds its air entry potential _e. In these cases conductance is calculated from hydraulic conductivity of the saturated position only (Eq. [A94b, A94c], [A95b, A95c], and [A96b, A96c]), and of the unsaturated position is calculated from a water content that excludes water added while of the first position of >_e, thereby simulating a wetting front. Water movement between adjacent positions thus alternates between Richard's and Green–Ampt flow depending upon vs. _e in each position. Water may also move through macropores driven by gravitational gradients and conductances calculated from Poiseuille–Hagen theory using set numbers and radii of macropore channels.

Solute Transport
Solute transport Q_r in x and y directions across the soil surface (Eq. [A97]) is calculated for each solute in Eq. [A1] to [A55] from surface water flow Q_r (Eq. [A88]) and surface solute concentration (Eq. [A98]). Solute transport Q_s in x, y, and z directions through the soil (Eq. [A99]) is calculated for each solute as the sum of convective (from subsurface water flow Q_w in Eq. [A93]) and dispersive–diffusive (Eq. [A100]) components. The concentrations of each solute by which transport is driven are generated from precipitation, ion exchange, and ion pairing reactions (Eq. [A1]–[A55]), and from production and consumption by roots, mycorrhizae (Grant, 1998b), and microbial communities (Eq. [A56]–[A87]).

Sediment Transport
Soil Detachment by Rainfall
The modeling of sediment transport in ecosys is based on that of Morgan et al. (1998a). Soil may be detached by kinetic energy of rainfall impact , calculated from intensity of direct rainfall (Eq. [A101]) and from effective height of indirect rainfall (Eq. [A102]). Direct and indirect rainfall are calculated from interception and storage of precipitation by leaf and stem surfaces based on their area indices. Kinetic energy of direct plus indirect rainfall is reduced by surface water depth d_w (Eq. [A90]). This energy is multiplied by soil detachability k, estimated from soil surface texture according to guide values in Morgan et al. (1998b), to calculate rate of soil detachment by rainfall D_r (Eq. [A103]).

Soil Detachment by Surface Flow
Soil detachment by surface flow D_f (Eq. [A104]) is modeled in x and y directions as the difference between current sediment concentration in surface water [S] and that at transport capacity [S_c], where positive differences indicate detachment and negative differences deposition. Detachment is constrained by soil cohesion J (Eq. [A105]) estimated from soil texture and organic C content according to guide values in Morgan et al. (1998b). Sediment concentration at transport capacity (Eq. [A106]) is calculated from unit stream power , which is the product of slope and flow velocity (Eq. [A107]), and its critical value _cr, using parameters based on median soil particle size according to Morgan et al. (1998a).

Sediment Transport
Sediment transport Q_e (Eq. [A108]) is the product of sediment concentration in surface water [S] (Eq. [A109]) and surface water flow Q_r in x and y directions (Eq. [A88]). Changes in surface sediment arise from soil detachment by rainfall plus soil detachment or deposition from surface flow, and from net sediment transport in x and y directions (Eq. [A110]). The transport of material in sediment (Eq. [A111]) is the product of sediment transport (Eq. [A108]) and the concentrations of precipitated plus exchangeable forms in the surface soil (Eq. [A1]–[A20]).

	PHOSPHORUS RUNOFF EXPERIMENT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT PHOSPHORUS RUNOFF EXPERIMENT MODEL APPLICATION RESULTS DISCUSSION REFERENCES

 
Composite 200-L soil samples were gathered during fall 2000 from the upper 10 cm of all replicate plots in each treatment of manure application trials conducted at six locations in central Alberta (Table 1). The soil was stored in waterproof, aerated containers until February 2001 when the soil was passed through a 1-cm sieve, mixed, sampled for extractable P content, and placed in a 50- x 95-cm stainless steel frames to a depth of 10 cm. These frames were designed to collect runoff and leachate during controlled rainfall events imposed by a rainfall simulator that consisted of two chambers each using a nozzle (Fulljet 1/2-HH-50WSQ; Spraying Systems, Wheaton, IL) operating 2.9 m above the target area. This nozzle, operating at 28 kPa at 3.0 m above a soil surface, produces rainfall-sized drops that fall within 2% of terminal velocity. Recent work has shown that this simulator produced raindrop distributions and velocities similar to those of natural rainfall. The average kinetic energy produced by this simulator (24.6 J m^–2 mm^–1) was about 85% of that measured in natural storms. Within each chamber, two frames were situated below the nozzle on tables with adjustable slopes, permitting up to four treatments to be rained on simultaneously. The nozzles were calibrated such that rainfall intensity was 65 mm h^–1 and coefficient of uniformity was 95% in each frame. Further information about the rainfall simulator is given in Wright et al. (2002).

	MODEL APPLICATION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT PHOSPHORUS RUNOFF EXPERIMENT MODEL APPLICATION RESULTS DISCUSSION REFERENCES

 
The model experiment consisted of three stages: (i) the management history of each manure treatment at each field site was simulated so that the state of each soil sample could be modeled at the time of the controlled rainfall event; (ii) the controlled rainfall event was simulated for a modeled soil sample from each manure treatment at each field site and model results for P losses in runoff, sediment, and leachate were tested with measurements; and (iii) the tested model was then used to predict long-term P losses from different rates of manure application at one of the field sites.

Stage 1: Simulation of Management History
Ecosys was provided with the soil (Table 2) and manure (Table 3) properties and with the crop rotations (Table 1) at each of the six sites included in the P runoff experiment described above. To simulate management history, the model was run from 1 Jan. 1995 (Calmar, Devon, Lacombe, Ponoka) or 1 Jan. 1998 (Ellerslie, Cooking Lake) to 31 Oct. 2000 for each manure treatment (Table 1) on a time step of 3 min, assuming constant surface boundary conditions during each hour. Surface boundary conditions for these runs were taken from daily climate data (maximum and minimum temperatures, solar radiation, relative humidity, windspeed, and precipitation) compiled for the ecodistrict within which each site was located as part of the Alberta Drought Program, and resolved into hourly values by ecosys. During the 3 yr (Ellerslie, Cooking Lake) or 6 yr (Calmar, Devon, Lacombe, Ponoka) that the model was run to simulate management history, P was added to the soils listed in Table 2 according to the manure application rates and schedules shown in Table 1 and the compositions in Table 3. Phosphorus was also removed from the soil during the model runs through plant uptake according to aqueous H₂PO^–₄ concentrations and root activity (Grant and Robertson, 1997), and through runoff, erosion, and leaching according to aqueous and total soil P concentrations.

Stage 2: Simulation of Controlled Rainfall Event
Comparisons of modeled vs. measured P after 3 or 6 yr of manure applications (Table 4) provided an indirect test of model performance, but better-constrained tests at smaller temporal scales were necessary to test model algorithms described above. To accomplish such tests, the upper 0.10 m of the soil modeled for each manure treatment at each site (Table 2) was used to simulate P losses during short-term controlled rainfall events. Subsurface irrigation was applied on 1 Apr. 2001 of each model run at 1 mm h^–1 for 18 h, after which the surface slope was set to 4° (7%) and rainfall was applied at 65 mm h^–1 for 2 h with both surface runoff and subsurface drainage fully enabled. Because soil was sieved and packed into the runoff frames used in the physical experiment, macroporosity was set to zero in the model. Model output for DIP and TP losses in runoff, sediment, and leachate from each manure treatment were compared with values measured during the controlled rainfall events described above. In the model, DIP in runoff and leachate was calculated as: DIP = + + + (soluble P from Eq. [A5]–[A9], [A18]–[A20], and [A47]–[A55]). The TP in runoff + sediment was calculated as: TP = DIP in runoff + AlPO_4(p) + FePO_4(p) + Ca(H₂PO₄)_2(p) + CaHPO_4(p) + Ca₅(PO₄)₃OH_(p) (precipitated P from Eq. [A5]–[A9]) + X-H₂PO₄ + X-HPO^–₄ (exchangeable P from Eq. [A18]–[A20]) + ^I_i=1 [Q_i,P + ^J_j=1 (S_i,j,P + Z_i,j,P + M_i,j,P)] (dissolved + solid organic P from Eq. [A56]–[A81]) in sediment. The term X denotes surface exchange site for cation or anion adsorption, while the term (p) is precipitated. The organic + inorganic phosphorus in leachate (LP) was calculated as: LP = DIP in leachate + ^I_i=1 Q_i,P (dissolved organic P) in leachate.

Stage 3: Predicting Long-Term Phosphorus Losses from Manure Application
To demonstrate model application, ecosys was then used to examine the long-term impact of manure applications on water quality by running the model for 60 yr under repeated sequences of 1995–2000 daily climate data with the manure treatments and cropping systems at Lacombe (Table 1). A key assumption used by ecosys to convert these data to hourly values was that daily rainfall occurred uniformly from 1500 to 1700 h. Model output for P loss was compared with current water quality guidelines to determine environmentally acceptable manure application rates for this site.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT PHOSPHORUS RUNOFF EXPERIMENT MODEL APPLICATION RESULTS DISCUSSION REFERENCES

 
Stage 1: Phosphorus Accumulation in Soil during Manure Application Trials
At the end of each 3- or 6-yr simulation of management history, exchangeable P in the model (X-H₂PO₄ + X-HPO^–₄ from Eq. [A18]–[A20]) was compared with exchangeable P estimated from modified Kelowna extractions conducted on soil from each manure treatment according to a relationship established by McKenzie et al. (1995) (Table 4). Three to six years of hog and cattle manure applications raised exchangeable P above control values, especially the large annual applications of cattle manure at Lacombe and Ponoka. These large applications caused exchangeable P modeled and measured at both Lacombe and Ponoka to exceed 0.25 of anion exchange capacity (Table 2). Ratios above this value were found by Breeuwsma and Silva (1992) to cause potentially unacceptable P movement into surface and ground waters. Rises in exchangeable P modeled for a given manure treatment were larger than those measured at Calmar, and smaller than those at Devon. Rises in exchangeable P modeled for manure treatments at Ellerslie, Cooking Lake, Lacombe, and Ponoka were close to those measured.

Stage 2: Phosphorus Loss in Runoff and Sediment during Controlled Rainfall
Model results from the controlled rainfall events on the soils at each site are presented below.

Calmar
In the model, simulated runoff (Eq. [A88]) began 5 min after the start of controlled rainfall and rose rapidly to near the rate of rainfall after 10 min. The time course of modeled runoff was similar to that of measured runoff during controlled rainfall in the hog and cattle manure treatments (Fig. 1a) , although the time course of runoff measured from the unmanured soil was atypically slower. Sediment loss in the model (Eq. [A108]) rose to 0.4 kg m^–2 h^–1 between 5 and 10 min after the start of rainfall, and stabilized thereafter as sediment concentrations equilibrated (Fig. 1b). Slightly lower sediment losses were modeled at higher manure application rates because resulting increases in organic C concentrations raised soil cohesion (Eq. [A105]). Concentrations of DIP modeled and measured in runoff remained <0.1 mg L^–1 during controlled rainfall on the unmanured soil, and <0.2 mg L^–1 on the manured soils (Fig. 1c). Higher DIP concentrations modeled in runoff from manured soils was caused by higher exchangeable P concentrations (Table 4) with which DIP was in equilibrium (Eq. [A18]–[A20]). Concentrations of TP in runoff (Eq. [A97]) and sediment (Eq. [A111]) were 12 to 15 mg L^–1 at the start of runoff but equilibrated at approximately 7.5 mg L^–1 after 5 min of runoff (Fig. 1d) once sediment flux had stabilized (Fig. 1b). Higher TP concentrations were modeled early in the rainfall event because soil detachment by rainfall impact (Eq. [A103]) contributed to surface sediment as soon as surface water appeared and before runoff began. This previously detached sediment caused higher sediment and hence TP concentrations to be modeled during the first few minutes of runoff. As surface water deepened with continued rainfall, soil detachment by rainfall impact declined and that by surface flow rose (Eq. [A104]). Higher TP concentrations modeled from the manured soils were attributed to higher exchangeable (Table 4), precipitated, and organic P concentrations. Differences between modeled and measured concentrations of DIP and TP were close to the standard errors of measured values during most of the rainfall event.

View larger version (33K): [in this window] [in a new window]   Fig. 1. (a) Runoff, (b) sediment loss, (c) dissolved inorganic phosphorus (DIP) concentration in runoff, and (d) total phosphorus (TP) loss in runoff and sediment, measured (symbols with vertical bars showing standard error) and modeled (lines) during 90 min of controlled rainfall on soils after 6 yr of different manure treatments (H, hog; C, cattle) at Calmar (see Table 1).

View larger version (34K): [in this window] [in a new window]   Fig. 2. (a) Runoff, (b) sediment loss, (c) dissolved inorganic phosphorus (DIP) concentration in runoff, and (d) total phosphorus (TP) loss in runoff and sediment, measured (symbols with vertical bars showing standard error) and modeled (lines) during 90 min of controlled rainfall on soils after 6 yr of different manure treatments (H, hog; C, cattle) at Devon (see Table 1).

View larger version (28K): [in this window] [in a new window]   Fig. 3. (a) Runoff, (b) sediment loss, (c) dissolved inorganic phosphorus (DIP) concentration in runoff, and (d) total phosphorus (TP) loss in runoff and sediment, measured (symbols with vertical bars showing standard error) and modeled (lines) during 90 min of controlled rainfall on soils after 3 yr of different manure treatments (H, hog; C, cattle) at Ellerslie (see Table 1).

View larger version (27K): [in this window] [in a new window]   Fig. 4. (a) Runoff, (b) sediment loss, (c) dissolved inorganic phosphorus (DIP) concentration in runoff, and (d) total phosphorus (TP) loss in runoff and sediment, measured (symbols with vertical bars showing standard error) and modeled (lines) during 90 min of controlled rainfall on soils after 3 yr of different manure treatments (H, hog; C, cattle) at Cooking Lake (see Table 1).

View larger version (32K): [in this window] [in a new window]   Fig. 5. (a) Runoff, (b) sediment loss, (c) dissolved inorganic phosphorus (DIP) concentration in runoff, and (d) total phosphorus (TP) loss in runoff and sediment, measured (symbols with vertical bars showing standard error) and modeled (lines) during 90 min of controlled rainfall on soils after 6 yr of different manure treatments (H, hog; C, cattle) at Lacombe (see Table 1).

View larger version (32K): [in this window] [in a new window]   Fig. 6. (a) Runoff, (b) sediment loss, (c) dissolved inorganic phosphorus (DIP) concentration in runoff, and (d) total phosphorus (TP) loss in runoff and sediment, measured (symbols with vertical bars showing standard error) and modeled (lines) during 90 min of controlled rainfall on soils after 6 yr of different manure treatments (H, hog; C, cattle) at Ponoka (see Table 1).

View larger version (42K): [in this window] [in a new window]   Fig. 7. (a) Mass and (b) concentration of total P in runoff and sediment modeled with annual applications of 0, 30, 60, 90, and 120 Mg ha–1 of cattle manure (Table 3) during 60 yr under repeated sequences of 1995–2000 climate data for the Lacombe ecodistrict (annual precipitation in 1995 = 435 mm, 1996 = 604 mm, 1997 = 534 mm, 1998 = 466 mm, 1999 = 450 mm, 2000 = 461 mm). Note logarithmic scale on vertical axis.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT PHOSPHORUS RUNOFF EXPERIMENT MODEL APPLICATION RESULTS DISCUSSION REFERENCES

The use of algorithms for thermodynamic equilibria to simulate geochemical P transformations (Eq. [A1]–[A55] using Eq. [1]–[ 5]) is a key difference between ecosys and other detailed models of P loss events. These algorithms were subjected to detailed testing during the controlled rainfall events in Stage 2 of the model experiment. Dynamic solutions to these thermodynamic equilibria allowed the model to simulate the declines from initial to equilibrium concentrations of DIP in runoff water that were measured during controlled rainfall events (Fig. 1c, 2c, 3c, 4c, 5c, and 6c). These equilibrium concentrations occurred in the model when rates of DIP loss in runoff (Eq. [A97]) equaled rates of P desorption (Eq. [A18]–[A20]) plus dissolution (Eq. [A5]–[A9]) in the surface soil layer (the upper 5 mm of the soil profile) plus rates of upward P solute transport (Eq. [A99]) from the soil layers below, sustained by rates of P desorption and dissolution in the lower layers. Modeled vs. measured equilibrium DIP concentrations during controlled rainfall were thus a test of the combined kinetics of transport and transformation in the model. These equilibrium concentrations rose with manure application rates because thermodynamic equilibria in the model allowed higher and to be maintained through transport and transformation processes in soils to which more manure had been applied (Table 4). During a series of controlled rainfall events, Sharpley (1995) also found that DIP concentrations in runoff declined toward equilibrium values that rose with manure P application rates similarly to those reported here.

Modeled DIP losses in runoff, calculated as the product of runoff rate (Fig. 1a, 2a, 3a, 4a, 5a, and 6a) and DIP concentrations (Fig. 1c, 2c, 3c, 4c, 5c, and 6c) approximated those measured at all sites except Devon where they were underestimated (Fig. 2). The DIP concentrations modeled from thermodynamic equilibria also contributed to modeled LP (Table 5), which rose only slightly with the lower manure application rates at Calmar, Devon, Cooking Lake, and Ellerslie, but rose sharply with the higher application rates at Lacombe and Ponoka. The higher DIP concentrations in runoff and leachate at Lacombe and Ponoka were modeled when the ratio of exchangeable P (Table 4) to anion exchange capacity (Table 2) exceeded 0.25, a ratio above which Breeuwsma and Silva (1992) found that P losses may cause adverse environmental impacts. The relationship between this ratio and DIP in runoff modeled and measured here was very similar to that found for several diverse soils by Sharpley (1995). For example, the ratios of approximately 0.1, approximately 0.3, and approximately 0.5 modeled and measured for the 0, 60, and 120 Mg ha^–1 yr^–1 treatments, respectively, at Lacombe and Ponoka (Table 4 vs. Table 2) were found by Sharpley (1995) to correspond to DIP concentrations in runoff of 0.25, 0.83, and 1.4 mg L^–1, respectively, close to those modeled and measured for these treatments early in the rainfall event (Fig. 5c and 6c). Changes in DIP concentrations modeled and measured during controlled rainfall indicate that the relationship between DIP concentrations and the ratio of exchangeable P to anion exchange capacity will vary with the intensity and duration of rainfall. Therefore the thermodynamic equilibria used in this model of P loss would benefit from testing under rainfall intensities and durations more typical of natural events.

Modeled and measured TP losses from sediment transport (Eq. [A108]) (Fig. 1d, 2d, 3d, 4d, 5d, and 6d) were an order of magnitude greater than those from DIP in runoff (Fig. 1c, 2c, 3c, 4c, 5c, and 6c), indicating that the accurate simulation of sediment loss (Fig. 1b, 2b, 3b, 4b, 5b, and 6b) is vital to that of P loss. Sediment loss in the model was sometimes lower than that measured, notably at Lacombe and Ponoka (Fig. 5b and 6b), but was frequently within standard errors of measured values at the other sites (Fig. 1b, 2b, 3b, and 4b). The underestimation at Lacombe and Ponoka could have been corrected by using the lower values of soil cohesion (Eq. [A105b]) for uncompacted soils given in Morgan et al. (1998b). High rates of manure application reduced sediment loss in the model by raising soil cohesion, although corresponding reductions in measured sediment loss were apparent only at Lacombe and Ponoka where manure C additions were largest (Fig. 5b and 6b). Phosphorus loss through sediment transport in the model (Eq. [A111]) also depended on sediment TP concentrations, which rose with manure application rates (Fig. 1d, 2d, 3d, 4d, 5d, and 6d). These concentrations caused modeled P losses through sediment transport to rise with manure application rate even though sediment transport declined (Fig. 1b, 2b, 3b, 4b, 5b, and 6b). These modeled rises were small except at Lacombe and Ponoka where manure application rates were highest. The effect of manure application on measured TP losses was variable. Soil transformations during several months of storage in 200-L containers (depth = 1 m) could have affected measured results.

Results from Stage 2 of the model experiment showed that DIP and TP losses were caused by short-term runoff and sediment transport events that particularly occur during intense rainfall events when soil infiltration rates are exceeded. Annual P losses modeled in Stage 3 were the sum of several short-term P losses driven by such events. These losses therefore rose sharply when high manure application rates coincided with high rainfall (Fig. 7). These rises suggest that estimates of long-term manure application rates designed to avoid adverse environmental impacts should be made under highest likely rates of rainfall rather than under long-term average rainfall. For example, if TP concentrations of <1 mg L^–1 need to be maintained in surface and subsurface water leaving the Lacombe site, then application of cattle manure with the composition given in Table 3 would have to be limited to 30 Mg ha^–1 yr^–1. Applications of 60 Mg ha^–1 yr^–1 would cause this concentration to be consistently exceeded within 10 yr, while higher applications would cause this concentration to be exceeded immediately. Applications of 60 Mg ha^–1 yr^–1 at Lacombe would cause P losses during high rainfall years to exceed 0.87 g P m^–2 yr^–1 (the target for P loss in the Netherlands by 2010) after 30 yr. The impacts of these P losses on surface and subsurface water quality could be assessed by testing and then using ecosys in three-dimensional mode to simulate P loading rates at a watershed scale. Such assessments are currently in progress.

The use of detailed ecosystem models such as ecosys to estimate P losses from manure applications addresses the need identified by Sharpley (1995) for a comprehensive approach that integrates soil P levels with variability in runoff and erosion caused by climatic, topographic, edaphic, and agronomic factors. Such models can also be used concurrently to assess other adverse environmental impacts of manure application, such as emissions of N₂O (Grant and Pattey, 1999, 2003) and CH₄ (Grant, 1998a, 1999; Grant and Roulet, 2002), needed in comprehensive studies of manure in agricultural ecosystems.

APPENDIX

Inorganic Transformations

Transformation Notes Value Equation Precipitation–dissolution Al(OH)3(p) (Al3+) + 3(OH–) amorphous Al(OH)3 –33.0 [A1] Fe(OH)3(p) (Fe3+) + 3(OH–) soil Fe –39.3 [A2] CaCO3(p) (Ca2+) + calcite –9.28 [A3] CaSO4(p) (Ca2+) + gypsum –4.64 [A4] AlPO4(p) (Al3+) + variscite –22.1 [A5] FePO4(p) (Fe3+) + strengite –26.4 [A6] Ca(H2PO4)2(p) (Ca2+) + 2 monocalcium phosphate –1.15 [A7] CaHPO4(p) (Ca2+) + monetite –6.92 [A8] Ca5(PO4)3OH(p) 5(Ca2+) + 3 + (OH–) hydroxyapatite –58.2 [A9] Cation exchange X-Ca + 2 2X-NH4 + (Ca2+) 1.00 [A10] 3X-Ca + 2(Al3+) 2X-Al + 3(Ca2+) 1.00 [A11] X-Ca + (Mg2+) X-Mg + (Ca2+) 0.60 [A12] X-Ca + 2(Na+) 2X-Na + (Ca2+) 0.16 [A13] X-Ca + 2(K+) 2X-K + (Ca2+) 3.00 [A14] X-Ca + 2(H+) 2X-H + (Ca2+) 1.00 [A15] Anion exchange X-OH+2 X-OH + (H+) –7.35 [A16] X-OH X-O– + (H+) –8.95 [A17] X-H2PO4 + H2O X-OH+2 + –2.80 [A18] X-H2PO4 + (OH–) X-OH + 4.20 [A19] X-HPO–4+ (OH–) X-OH + 2.60 [A20] X-COOH X-COO– + (H+) –5.00 [A21] Ion pairs (NH3)g + (H+) –9.24 [A22] H2O (H+) + (OH–) –14.3 [A23] (CO2)g + H2O (H+) + –6.42 [A24] (H+) + –10.4 [A25] (AlOH2+) (Al3+) + (OH–) –9.06 [A26] (Al(OH)2+) (AlOH2+) + (OH–) –10.7 [A27] (Al(OH)2+) + (OH–) –5.70 [A28] + (OH–) –5.10 [A29] (Al3+) + –3.80 [A30] (Fe3+) + (OH–) –12.1 [A31] + (OH–) –10.8 [A32] + (OH–) –6.94 [A33] + (OH–) –5.84 [A34] (Fe3+) + –4.15 [A35] (CaOH+) (Ca2+) + (OH–) –1.90 [A36] (Ca2+) + –4.38 [A37] (Ca2+) + –1.87 [A38] (Ca2+) + –2.92 [A39] (MgOH+) (Mg2+) + (OH–) –3.15 [A40] (Mg2+) + –3.52 [A41] (Mg2+) + –1.17 [A42] (Mg2+) + –2.68 [A43] (Na+) + –3.35 [A44] (Na+) + –0.48 [A45] (K+) + –1.30 [A46] (H3PO4) (H+) + –2.15 [A47] (H+) + –7.20 [A48] (H+) + –12.4 [A49] (Fe3+) + –5.43 [A50] (Fe3+) + –10.9 [A51] (Ca2+) + –1.40 [A52] (Ca2+) + –2.74 [A53] (Ca2+) + –6.46 [A54] (Mg2+) + –2.91 [A55] Organic transformations Organic matter hydrolysis DSi,j,C = D'Si,j,CMi,a,Cftg [A56] DZi,j,C = D'Zi,j,CMi,a,Cftg [A57] D'Si,j,C = {DSi,j,C[Si,C]}/{[Si,C] + KmD(1.0 + [Mi,a,C]/KiD)} [A58] D'Zi,j,C = {DZj,C[Zi,C]}/{[Zi,C] + KmD(1.0 + [Mi,a,C]/KiD)} [A59] ftg = Tl{exp[B – Ha/(RTl)]}/{1 + exp[(Hdl – STl)/(RTl)] + exp[(STl – Hdh)/(RTl)]} [A60] DSi,j,P = DSi,j,C(Si,j,P/Si,j,C) [A61] DZi,j,P = DZi,j,C(Zi,j,P/Zi,j,C) [A62] Yi,C = kts(aFs[Qi,C]b – Xi,C) [A63] Yi,P = Yi,C(Qi,P/Qi,C) ([Qi,C]b < Xi,C) [A64] Yi,P = Yi,C(Xi,P/Xi,C) ([Qi,C]b > Xi,C) [A65] Microbial growth, respiration, and decomposition Rhi,C = Mi,a,C{R'hC[Qi,C]}/{(KmQC + [Qi,C])}ftg [A66] R'hi,C = Rhi,C[CP/(CP + KCP)] [A67] Rhi,C = R'hi,C(RO2i,C/R'O2i,C) [A68] R'O2i,C = 2.67R'hi,C [A69] RO2i,C = R'O2i,C[O2m]/([O2m] + KO2h) = 4nMi,a,CDsO2[rmrw/(rw – rm)]([O2s] – [O2m] [A70] RMi,j,C = RmMi,j,Nftm [A71] ftm = exp[y(Tl – 303.16)] [A72] Rgi,C = Rhi,C – RMi,j,C [A73] Ui,C = Rgi,C(1 + G/Em) [A74] Ui,P = Ui,CQi,P/Qi,C [A75] DMi,j,C= DMi,jMi,j,Cftg [A76] DMi,j,P = DMi,jMi,j,Pftg [A77] Mi,j,C/t = FjUi,C – FjRhi,C – DMi,j,C [Rhi,C > RMi,j,C] [A78a] Mi,j,C/t = FjUi,C – RMi,j,C – DMi,j,C [Rhi,C < RMi,j,C] [A78b] Mi,C/t = Mi,j,C/t [A78c] Ii,j,P = (Mi,j,CCPj – Mi,j,P) (Ii,j,P < 0) [A79a] Ii,j,P = / (Ii,j,P > 0) [A79b] Mi,j,P/t = FjUi,P + Ii,j,P – DMi,j,P [A80] Mi,a,C = Mi,l,C + Mi,r,CFr/Fl [A81] Humification of products from residue hydrolysis and microbial decomposition HSi,j=lignin,C = DSi,j=lignin,CFh [A82] HSi,j=lignin,P = DSi,j=lignin,PFh [A83] HSi,jlignin,C = HSi,j=lignin,CLhj [A84] HSi,jlignin,P = HSi,jlignin,CSi,P/Si,C [A85] HMi,j,C = DMi,j,CFh [A86] HMi,j,P = HMi,j,CMi,j,P/Mi,j,C [A87] Water transport Surface flow Qr,x(x,y) = vx(x,y)dx,yLy(x,y) [A88a] Qr,y(x,y) = vy(x,y)dx,yLx(x,y) [A88b] (dw(x,y)Ax,y)/t = Qr,x(x,y) – Qr,x+1(x,y) + Qr,y(x,y) – Qr,y+1(x,y) [A88c] vx = R0.67s0.5x/zr [(E + ds + d)x,y > (E + ds + d)x+1,y] [A89a] vx = R0.67s0.5y/zr [(E + ds + d)x,y > (E + ds + d)x,y+1] [A89b] vy = –R0.67s0.5x/zr [(E + ds + d)x,y < (E + ds + d)x+1,y] [A89c] vy = –R0.67s0.5y/zr [(E + ds + d)x,y < (E + ds + d)x,y+1] [A89d] dx,y = max(0,dw(x,y) + di(x,y) – ds(x,y))dw(x,y)/(dw(x,y) + di(x,y)) [(E + ds + d)x,y > (E + ds + d)x+1,y] [A90a] or [(E + ds + d)x,y > (E + ds + d)x,y+1] [A90b] dx,y = max(0,dw(x+1,y) + di(x+1,y) – ds(x+1,y))dw(x+1,y)/(dw(x+1,y) + di(x+1,y)) [(E + ds + d)x,y < (E + ds + d)x+1,y] [A90c] dx,y = max(0,dw(x,y+1) + di(x,y+1) – ds(x,y+1))dw(x,y+1)/(dw(x,y+1) + di(x,y+1)) [(E + ds + d)x,y < (E + ds + d)x,y+1] [A90d] R = srd/ + 1)0.5] [A91] sx(x,y) = 2abs[(E + ds + d)x,y – (E + ds + d)x+1,y]/(Lx(x,y) + Lx(x+1,y)) [A92a] sy(x,y) = 2abs[(E + ds + d)x,y – (E + ds + d)x,y+1]/(Ly(x,y) + Ly(x,y+1)) [A92b] Subsurface flow Qw,x = K'x(x,y,z – x+1,y,z) [A93a] Qw,y = K'y(x,y,z – x,y+1,z) [A93b] Qw,z = K'z(x,y,z – x,y,z+1) [A93c] wx,y,z/t = Qw,x(x,y) – Qw,x+1(x,y) + Qw,y(x,y) – Qw,y+1(x,y) + Qw,z(x,y) – Qw,z+1(x,y) [A93d] K'x = 2Kx,y,zKx+1,y,z/(Kx,y,zLx,(x+1,y,z) + Kx+1,y,zLx,(x,y,z)) [x,y,z < e(x,y,z); x+1,y,z < e(x+1,y,z)] [A94a] or [x,y,z > e(x,y,z); x+1,y,z > e(x+1,y,z)] = 2Kx,y,z/(Lx(x+1,y,z) + Lx(x,y,z)) [x,y,z > e(x,y,z); x+1,y,z < e(x+1,y,z)] [A94b] = 2Kx+1,y,z/(Lx(x+1,y,z) + Lx(x,y,z)) [x,y,z < e(x,y,z); x+1,y,z > e(x+1,y,z)] [A94c] K'y = 2Kx,y,zKx,y+1,z/(Kx,y,zLy(x,y+1,z) + Kx,y+1,zLy(x,y,z)) [x,y,z < e(x,y,z); x,y+1,z < e(x,y+1,z)] [A95a] or [x,y,z > e(x,y,z); x,y+1,z > e(x,y+1,z)] = 2Kx,y,z/(Ly(x,y+1,z) + Ly(x,y,z)) [x,y,z > e(x,y,z); x,y+1,z < e(x,y+1,z)] [A95b] = 2Kx,y+1,z/(Ly(x,y+1,z) + Ly(x,y,z)) [x,y,z < e(x,y,z); x,y+1,z > e(x,y+1,z)] [A95c] K'z = 2Kx,y,zKx,y,z+1/(Kx,y,zLz(x,y,z+1) + Kx,y,z+1Lz(x,y,z)) [x,y,z < e(x,y,z); x,y,z+1 < e(x,y,z+1)] [A96a] or [x,y,z > e(x,y,z); x,y,z+1 > e(x,y,z+1)] = 2Kx,y,z/(Lz(x,y,z+1) + Lz(x,y,z)) [x,y,z > e(x,y,z); x,y,z+1 < e(x,y,z+1)] [A96b] = 2Kx,y,z+1/(Lz(x,y,z+1) + Lz(x,y,z)) [x,y,z < e(x,y,z); x,y,z+1 > e(x,y,z+1)] [A96c] Solute transport Surface flow Qr,x(x,y) = Qr,x(x,y)s[s]x,y,1 [A97a] Qr,y(x,y) = Qr,y(x,y)s[s]x,y,1 [A97b] [s]x,y,1 = s(x,y,1)/(wx,y,1 + dx,yLx(x,y)Ly(x,y)) [A98] Subsurface flow Qs,x(x,y,z) = Qw,x(x,y,z)[s]x,y,z + 2Ds,x(x,y,z)([s]x,y,z – [s]x+1,y,z)/(Lx(x+1,y,z) + Lx(x,y,z)) [A99a] Qs,y(x,y,z) = Qw,y(x,y,z)[s]x,y,z + 2Ds,y(x,y,z)([s]x,y,z – [s]x,y+1,z)/(Ly(x,y+1,z) + Ly(x,y,z)) [A99b] Qs,z(x,y,z) = Qw,z(x,y,z)[s]x,y,z + 2Ds,z(x,y,z)([s]x,y,z – [s]x,y,z+1)/(Lz(x,y,z+1) + Lz(x,y,z)) [A99c] Ds,x(x,y,z) = |Qw,x(x,y,z)| + D'sftas(x,y,z)0.5(w(x,y,z) + w(x+1,y,z)) [A100a] Ds,y(x,y,z) = |Qw,y(x,y,z)| + D'sftas(x,y,z)0.5(w(x,y,z) + w(x,y,z+1)) [A100b] Ds,z(x,y,z) = |Qw,z(x,y,z)| + D'sftas(x,y,z)0.5(w(x,y,z) + w(x,y,z+1)) [A100c] Sediment transport Soil detachment by rainfall d(x,y) = 8.95 + 8.44 log(Pd(x,y)) [A101] i(x,y) = max – 5.87) [A102] Dr(x,y) = kx,y(d(x,y)Pd(x,y) + i(x,y)Pi(x,y)) exp(–bdw(x,y))(1.0 – Fn) (dw > 0) [A103] Soil detachment by surface flow Df,x(x,y) = ßx,yLy(x,y)vs([Sc,x(x,y)] – [Sx,y]) [A104a] Df,y(x,y) = ßx,yLx(x,y)vs([Sc,y(x,y)] – [Sx,y]) [A104b] ßx,y = 1.0 ([Sc,x(x,y)] < [Sx,y]) or ([Sc,y(x,y)] < [Sx,y]) [A105a] ßx,y = 0.79 exp(–0.85Jx,y) ([Sc,x(x,y)] > [Sx,y]) or ([Sc,y(x,y)] > [Sx,y]) [A105b] [Sc,x(x,y)] = max[0.0, c(x(x,y) – cr)] [A106a] [Sc,y(x,y)] = max[0.0, c(y(x,y) – cr)] [A106b] x(x,y) = sx(x,y)vx(x,y) [A107a] y(x,y) = sy(x,y)vy(x,y) [A107b] Sediment transport Qe,x(x,y) = [Sx,y]Qr,x(x,y) [A108a] Qe,y(x,y) = [Sx,y]Qr,y(x,y) [A108b] [Sx,y] = Sx,y/(dx,yLx(x,y)Ly(x,y)) [A109] Sx,y/t = Dr(x,y) + Df,x(x,y) + Df,y(x,y) + Qe,x(x,y) – Qe,x+1(x,y) + Qe,y(x,y) – Qe,y+1(x,y) [A110] Qe,x(x,y) = Qe,x(x,y)(pr[pr]x,y,1 + x[x]x,y,1 + ij[o(i,j)]x,y,1) [A111a] Qe,y(x,y) = Qe,y(x,y)(pr[pr]x,y,1 + x[x]x,y,1 + ij[o(i,j)]x,y,1) [A111b] Variable Definition Unit Relevant equation Value a total substrate + residue C = ([Si,j,C] + [Zi,j,C]) g C Mg–1 [A63] A area of landscape position [A88] B parameter such that ftg = 1.0 at Tl = 303.15 K [A60] 17.24 b Freundlich exponent for sorption isotherm [A63] 0.85 b parameter that attenuates kinetic energy of precipitation   with surface water depth m–1 [A103] CP ratio of Mi,a,P to Mi,a,C g N g C–1 [A67] CPj maximum ratio of Mi,j,P to Mi,j,C maintained by Mi,j,C g P g C–1 [A79] 0.022 and 0.013 for j = labile and   resistant, respectively d depth of mobile surface water m [A88, A89, A90, A91,   A92, A98, A109] Df,x net soil detachment by runoff in x direction g m–2 h–1 [A104a, A110] Df,y net soil detachment by runoff in y direction g m–2 h–1 [A104b, A110] di depth of surface ice m [A90] DMi,j specific decomposition rate of Mi,j,C at 30°C g C g C–1 h–1 [A76, A77] 0.0125 and 0.00035 for j = labile and   resistant, respectively DMi,j,C decomposition rate of Mi,j,C g C m–2 h–1 [A76, A78, A86] DMi,j,P decomposition rate of Mi,j,P g P m–2 h–1 [A77, A80] Dr soil detachment by rainfall impact g m–2 h–1 [A103, A110] ds maximum depth of surface water storage m [A89, A90, A92] DsO2 aqueous dispersivity–diffusivity of O2 during microbial   uptake in soil m2 h–1 [A70] DSi,j,C decomposition rate of Si,j,C by Mi,a,C g C m–2 h–1 [A56, A61, A82] DSi,j,C specific decomposition rate of Si,j,C by Mi,a,C at 30°C   and saturating [Si,C] g C g C–1 h–1 [A58] 1.00, 1.00, 0.15, and 0.025 for i =   manure and j = protein, carbohydrate,   cellulose, and lignin, respectively DSi,j,P decomposition rate of Si,j,P by Mi,a,C g P m–2 h–1 [A61, A83] Ds,x aqueous dispersivity–diffusivity of solute during   transport in x direction m2 h–1 [A99a, A100a] Ds,y aqueous dispersivity–diffusivity of solute during   transport in y direction m2 h–1 [A99b, A100b] Ds,z aqueous dispersivity–diffusivity of solute during   transport in z direction m2 h–1 [A99c, A100c] dw depth of surface water m [A88, A90, A103] DZi,j,C decomposition rate of Zi,j,C by Mi,a,C g C m–2 h–1 [A81, A86] DZi,j,P decomposition rate of Zi,j,P by Mi,a,C g P m–2 h–1 [A62] DZj,C specific decomposition rate of Zi,j,C by Mi,a,C at 30°C   and saturating [Zi,C] g C g C–1 h–1 [A59] 0.25 and 0.10 for j = labile and   resistant, respectively D'Si,j,C specific decomposition rate of Si,j,C by Mi,a,C at 30°C g C g C–1 h–1 [A56, A58] D'Si,j,C specific decomposition rate of Zi,j,C by Mi,a,C at 30°C g C g C–1 h–1 [A57, A59] D's aqueous diffusivity of solute in water at 30°C m2 h–1 [A100] H2PO–4 = 3.0 x 10–6 E surface elevation m [A89, A90, A92] Em energy requirement for growth of Mi,a,C kJ g C–1 [A74] 25 Fh fraction of products from lignin hydrolysis and microbial   decomposition that are humified (function of clay   and OC content) [A82, A83, A86] Fl fraction of microbial growth allocated to labile   component Mi,l,C [A78, A80, A81] 0.55 Fn fraction of soil surface that is non-erodible [A103] Fr fraction of microbial growth allocated to resistant   component Mi,r,C [A78, A80, A81] 0.45 Fs equilibrium ratio between Qi,C and Hi,C [A63] fta temperature function for aqueous diffusivity dimensionless [A100] ftg temperature function for growth respiration dimensionless [A111, A56, A66, A76,   A77] ftm temperature function for maintenance respiration dimensionless [A71, A72] concentration of H2PO–4 in soil solution g P m–3 [A79] Ha energy of activation kJ mol–1 [A60] 57.5 Hc canopy height m [A102] Hdh energy of high temperature deactivation kJ mol–1 [A60] 220 Hdl energy of low temperature deactivation kJ mol–1 [A60] 190 HMi,j,C transfer of microbial C decomposition products to humus g C m m–2 h–1 [A86, A87] HMi,j,P transfer of microbial P decomposition products to humus g P m–2 h–1 [A87] HSi,j,C transfer of C hydrolysis products to particulate OM g C m–2 h–1 [A82, A83, A84, A85] HSi,j,P transfer of P hydrolysis products to particulate OM g P m–2 h–1 [A83, A85] i (subscript) substrate–microbe complex (plant litterfall, animal   manure, particulate OM, humus) I number of substrate classes I [A111] Id intensity of direct rainfall mm h–1 [A101] Ii,j,P mineralization (Ii,j,P < 0) or immobilization (Ii,j,P > 0)   of P by Mi,j,C g P m–2 h–1 [A80, A79] j (subscript) structural or kinetic components within each complex J number of components j in class I [A111] J soil cohesion kPa [A105b] K hydraulic conductivity m2 MPa–1 h–1 [A94, A95, A96] k soil detachability g J–1 [A103] KCP ratio of Mi,a,P to Mi,a,C at which Ui,C = 1/2 U'i,C [A67] 2.2 x 10–4 KiSi,C inhibition constant for Mi,a,C on Si,C g C Mg–1 [A58] KiZi,C inhibition constant for Mi,a,C on Zi,C g C Mg–1 [A59] KmD Michaelis–Menten constant for DSi,j,C g C Mg–1 [A58, A59] 25 KmQC Michaelis–Menten constant for Rgi,C on [Qi,C] g C m–3 [A66] 36 KO2h Michaelis–Menten constant for reduction of O2s   by heterotrophs g O2 m–3 [A70] 0.032 KP Michaelis–Menten constant for microbial uptake   of solution P g P m–3 [A79] 1.0 kts equilibrium rate constant for sorption h–1 [A63] 0.01 K' hydraulic conductance in x, y or z directions m MPa–1 h–1 [A93, A94, A95, A96] Lhj ratio of nonlignin to lignin components in humified   hydrolysis products [A84] 0.10, 0.05, and 0.05 for j = protein,   carbohydrate, and cellulose,   respectively Lx length of landscape element x,y in x direction m [A88a, A92a, A94, A99a,   A98, A104b, A109] 0.50 Ly length of landscape element x,y in y direction m [A88b, A92b, A95, A99b,   A98, A104a, A109] 0.95 Lz length of landscape element x,y in z direction m [A96, A99c] from soil file Mi,a,C active microbial C associated with (Si,j,C + Zi,j,C) g C m–2 [A56, A57, A66, A70,   A81] [Mi,a,C] concentration of Mi,a,C in soil water g C m–3 [A58, A59] Mi,j,C C content of microbial biomass Mi,j g C m–2 [A76, A78, A79, A81,   A87] Mi,j,N N content of microbial biomass Mi,j g N m–2 [A71] Mi,j,P P content of microbial biomass Mi,j g P m–2 [A77, A80, A79, A87] Mi,l,C labile microbial C (j = l) associated with (Si,j,C + Si,j,C) g C m–2 [A80] Mi,r,C resistant microbial C (j = r) associated with    (Si,j,C + Si,j,C) g C m–2 [A80] [O2m] O2 concentration at heterotrophic microsites g O2 m–3 [A70] [O2s] O2 concentration in soil solution g O2 m–3 [A70] Pd direct rainfall mm h–1 [A101, A103] Pi indirect rainfall mm h–1 [A103] Qe,x sediment flux in x direction g m–2 h–1 [A108a, A110, A111a] Qe,y sediment flux in y direction g m–2 h–1 [A108b, A110, A111b] Qe,x sediment flux of in x direction g m–2 h–1 [A111a] Qe,y sediment flux of in y direction g m–2 h–1 [A111b] Qi,C hydrolysis products of (DSi,j,C + DZi,j,C) g C m–2 [A63, A75] [Qi,C] solution concentration of Qi,C g C Mg–1 [A63, A66] Qi,P hydrolysis products of (DSi,j,P + DZi,j,P) g P m–2 [A64, A75] Qr,x surface flow in x direction m3 m–2 h–1 [A88a, A97a, A108a] Qr,y surface flow in y direction m3 m–2 h–1 [A88b, A97b, A108b] Qr,x surface flow of in x direction g m–2 h–1 [A97a] Qr,y surface flow of in y direction g m–2 h–1 [A97b] Qs,x aqueous transport of solute in x direction g m–2 h–1 [A99a] Qs,y aqueous transport of solute in y direction g m–2 h–1 [A99b] Qs,z aqueous transport of solute in z direction g m–2 h–1 [A99c] Qw,x subsurface water flow in x direction m3 m–2 h–1 [A93a, A99a, A100a] Qw,y subsurface water flow in y direction m3 m–2 h–1 [A93b, A99b, A100b] Qw,z subsurface water flow in z direction m3 m–2 h–1 [A93c, A99c, A100c] R gas constant J mol–1 K–1 [A60] 8.3143 R ratio of cross-sectional area to perimeter of surface flow m [A89, A91] Rgi,C growth respiration of Mi,a,C on Qi,C under nonlimiting   O2 and nutrients g C g C–1 h–1 [A73] Rhi,C specific heterotrophic respiration of Mi,a,C on Qi,C under   nonlimiting O2 and nutrients g C g C–1 h–1 [A66, A67] Rhi,C heterotrophic respiration of Mi,a,C under ambient O2 g C m–2 h–1 [A68, A74, A73, A73,   A78] Rm specific maintenance respiration at 30°C g C g N–1 h–1 [A71] 0.016 rm radius of heterotrophic microsite m [A70] 1.0 x 10–6 RMi,j,C maintenance respiration by Mi,j,C g C m–2 h–1 [A71, A73, A78] RO2i,C O2 uptake by Mi,a,C under ambient O2 g m–2 h–1 [A68, A70] rw radius of Rm + water film at current water content m [A70] R'hC specific heterotrophic respiration under nonlimiting [Qi,C],   O2, nutrients, and 30°C g C g C–1 h–1 [A66] 0.2 R'hi,C heterotrophic respiration of Mi,a,C on Qi,C under   nonlimiting O2 g C m–2 h–1 [A67, A68, A69] R'O2i,C O2 uptake by Mi,a,C under nonlimiting O2 g m–2 h–1 [A68, A69, A70] S change in entropy J mol–1 K–1 [A60] 710 [Sc,x(x,y)] sediment concentration capacity for transport   in x direction g m–3 [A104a, A105, A106a] [Sc,y(x,y)] sediment concentration capacity for transport   in y direction g m–3 [A104b, A105, A106b] [Si,c] concentration of Si,j,C in soil g C Mg–1 [A58] Si,j,C mass of solid or sorbed organic C in soil g C m–2 [A61, A84] Si,j,P mass of solid or sorbed organic P in soil g P m–2 [A61, A84] sr slope of channel sides during surface flow m m–1 [A91] 1 sx slope in x direction m m–1 [A89a, A89c, A92, A107a] Sx,y sediment g m–2 [A109, A110] [Sx,y] sediment concentration g m–3 [A104, A105, A109, A108] sy slope in y direction m m–1 [A89b, A89d, A92, A107b] Tl soil temperature of layer l K [A60, A72] Ui,C uptake of Qi,C by Mi,a,C under limiting nutrient availability g C m–2 h–1 [A74, A75, A78] Ui,P uptake of Qi,P by Mi,a,C under limiting nutrient availability g P m–2 h–1 [A75, A80] V volume of landscape element m3 [A93] vs settling velocity m h–1 [A104] 0.025 vx velocity of surface flow in x direction m h–1 [A88a, A89a, A107a] vy velocity of surface flow in y direction m h–1 [A88b, A89b, A107b] w soil water mass m3 m–2 [A93, A98] x (subscript) landscape position increasing from west to east Xi,C adsorbed C hydrolysis products g C Mg–1 [A63, A65] Xi,P adsorbed P hydrolysis products g P Mg–1 [A65] y (subscript) landscape position increasing from north to south y selected to give a Q10 for ftm of 2 [A72] 0.069 Yi,C sorption of C hydrolysis products g C m–2 h–1 [A63, A64, A65] Yi,P sorption of P hydrolysis products g P m–2 h–1 [A64, A65] z (subscript) landscape position increasing with depth [Zi,C] concentration of Zi,j,C in soil g C Mg–1 [A59] Zi,j,C mass of microbial residue C in soil g C m–2 [A62] Zi,j,P mass of microbial residue P in soil g P m–2 [A62] zr Manning's roughness coefficient m–1/3 h [A89] 0.01 ß soil detachment efficiency [A104, A105] s mass of element as inorganic or organic solute from   Eq. [A1] to [A55] and [A56] to [A87] g m–2 [A98] [o] concentration of element as organic solid from   Eq. [A56] to [A87] g Mg–1 [A111] [p] concentration of element as precipitate from   Eq. [A1] to [A55] g Mg–1 [A111] [s] concentration of element as inorganic or organic solute   from Eq. [A1] to [A55] and [A56] to [A87] g m–3 [A97, A98, A99] [x] concentration of element as exchangeable ion from   Eq. [A1] to [A55] g Mg–1 [A111] G energy yield from aerobic oxidation of Qi,C kJ g C–1 [A74] 37.5 empirical parameter derived from mean particle size that   relates [Sc,x(x,y)] to x(x,y) [A106] w soil water content m3 m–3 [A93, A100] d kinetic energy of direct rainfall at soil surface J m–2 mm–1 [A101, A103] i kinetic energy of indirect rainfall at soil surface J m–2 mm–1 [A102, A103] dispersion coefficient m [A100] 0.04 s tortuosity coefficient for aqueous diffusion [A100] soil water potential MPa [A93, A94, A95, A96] e air entry potential MPa [A94, A95, A96] cr critical stream power cm s–1 [A106] 0.4 x stream power in x direction cm s–1 [A106a, A107a] y stream power in y direction cm s–1 [A106b, A107b]