Erosion is the wearing down of a landscape over time. It includes the detachment, transport, and deposition of soil particles by the erosive forces of raindrops and surface flow of water.

A land area contains rills and channels. Raindrop impact can detach soil particles on unprotected land surfaces between rills and initiate transport of these particles to the rills. From the small rills, the particles move to larger rills, then into ephemeral channels and then into continuously flowing rivers. Entrainment and deposition of particles can occur at any point along the path. When erosion occurs without human influence, it is called geologic erosion. Accelerated erosion occurs when human activity increases the rate of erosion.

Erosion is a matter of concern to watershed and natural resource managers. Two of the main reasons reservoirs are built are water supply and flood control. Erosion upstream of a reservoir deposits sediment in the bottom of the reservoir which lowers the reservoir’s water-holding capacity and consequently its usefulness for both of these purposes. The soil surface is the part of the soil profile highest in organic matter and nutrients. Organic matter forms complexes with soil particles so that erosion of the soil particles will also remove nutrients. Excessive erosion can deplete soil reserves of nitrogen and phosphorus needed by plants to grow and extreme erosion can degrade the soil to the point that it is unable to support plant life. If erosion is severe and widespread enough, the water balance of a watershed can be altered—remember that most water is lost from a watershed via evapotranspiration.

Erosion caused by rainfall and runoff is computed with the Modified Universal Soil Loss Equation (MUSLE) (Williams, 1975). MUSLE is a modified version of the Universal Soil Loss Equation (USLE) developed by Wischmeier and Smith (1965, 1978).

USLE predicts average annual gross erosion as a function of rainfall energy. In MUSLE, the rainfall energy factor is replaced with a runoff factor. This improves the sediment yield prediction, eliminates the need for delivery ratios, and allows the equation to be applied to individual storm events. Sediment yield prediction is improved because runoff is a function of antecedent moisture condition as well as rainfall energy. Delivery ratios (the sediment yield at any point along the channel divided by the source erosion above that point) are required by the USLE because the rainfall factor represents energy used in detachment only. Delivery ratios are not needed with MUSLE because the runoff factor represents energy used in detaching and transporting sediment.

Some soils erode more easily than others even when all other factors are the same. This difference is termed soil erodibility and is caused by the properties of the soil itself. Wischmeier and Smith (1978) define the soil erodibility factor as the soil loss rate per erosion index unit for a specified soil as measured on a unit plot. A unit plot is 22.1-m (72.6-ft) long, with a uniform length-wise slope of 9-percent, in continuous fallow, tilled up and down the slope. Continuous fallow is defined as land that has been tilled and kept free of vegetation for more than 2 years. The units for the USLE soil erodibility factor in MUSLE are numerically equivalent to the traditional English units of 0.01 (ton acre hr)/(acre ft-ton inch).

Wischmeier and Smith (1978) noted that a soil type usually becomes less erodible with decrease in silt fraction, regardless of whether the corresponding increase is in the sand fraction or clay fraction.

Direct measurement of the erodibility factor is time consuming and costly. Wischmeier et al. (1971) developed a general equation to calculate the soil erodibility factor when the silt and very fine sand content makes up less than 70% of the soil particle size distribution.

$K_{USLE}=\frac{0.00021*M^{1.14}*(12-OM)+3.25*(c_{soilstr}-2)+2.5*(c_{perm}-3)}{100}$ 4:1.1.2

where $K_{USLE}$ is the soil erodibility factor, $M$ is the particle-size parameter, $OM$ is the percent organic matter (%), $c_{soilstr}$ is the soil structure code used in soil classification, and $c_{perm}$ is the profile permeability class.

The particle-size parameter, $M$, is calculated

$M=(m_{silt}+m_{vfs})*(100-m_c)$ 4:1.1.3

where $m_{silt}$ is the percent $silt$ content (0.002-0.05 mm diameter particles), $m_{vfs}$ is the percent very fine sand content (0.05-0.10 mm diameter particles), and $m_c$ is the percent clay content (< 0.002 mm diameter particles).

The percent organic matter content, $OM$, of a layer can be calculated:

$OM=1.72*orgC$ 4:1.1.4

where $orgC$ is the percent organic carbon content of the layer (%).

Soil structure refers to the aggregation of primary soil particles into compound particles which are separated from adjoining aggregates by surfaces of weakness. An individual natural soil aggregate is called a ped. Field description of soil structure notes the shape and arrangement of peds, the size of peds, and the distinctness and durability of visible peds. USDA Soil Survey terminology for structure consists of separate sets of terms defining each of these three qualities. Shape and arrangement of peds are designated as type of soil structure; size of peds as class; and degree of distinctness as grade.

The soil-structure codes for equation 4:1.1.2 are defined by the type and class of soil structure present in the layer. There are four primary types of structure:

-Platy, with particles arranged around a plane, generally horizontal

-Prismlike, with particles arranged around a verticle line and bounded by relatively flat vertical surfaces

-Blocklike or polyhedral, with particles arranged around a point and bounded by flat or rounded surfaces which are casts of the molds formed by the faces of surrounding peds

-Spheroidal or polyhedral, with particles arranged around a point and bounded by curved or very irregular surfaces that are not accomodated to the adjoining aggregates

Each of the last three types has two subtypes:

-Prismlike Prismatic: without rounded upper ends Columnar: with rounded caps

-Blocklike Angular Blocky: bounded by planes intersecting at relatively sharp angles Subangular Blocky: having mixed rounded and plane faces with vertices mostly rounded

-Spheroidal Granular: relatively non-porous Crumb: very porous

1. very fine granular

2.fine granular

3.medium or coarse granular

4.blocky, platy, prismlike or massive

1.rapid (> 150 mm/hr)

2.moderate to rapid (50-150 mm/hr)

3.moderate (15-50 mm/hr)

4.slow to moderate (5-15 mm/hr)

5.slow (1-5 mm/hr)

6.very slow (< 1 mm/hr)

Williams (1995) proposed an alternative equation:

The topographic factor, $LS_{USLE}$, is the expected ratio of soil loss per unit area from a field slope to that from a 22.1-m length of uniform 9 percent slope under otherwise identical conditions. The topographic factor is calculated:

$LS_{USLE}=(\frac{L_{hill}}{22.1})^m*(65.41*sin^2(\alpha_{hill})+4.56*sin\alpha_{hill}+0.065)$ 4:1.1.12

where $L_{hill}$ is the slope length ($m$), $m$ is the exponential term, and $\alpha_{hill}$ is the angle of the slope. The exponential term, $m$, is calculated:

$m=0.6*(1-exp[-35.835*slp])$ 4:1.1.13

where $slp$ is the slope of the HRU expressed as rise over run (m/m). The relationship between $\alpha_{hill}$ and $slp$ is:

$slp=tan\alpha_{hill}$ 4:1.1.14

Transport of sediment, nutrients, and pesticides from land areas to water bodies is a consequence of weathering that acts on landforms. Soil and water conservation planning requires knowledge of the relations between factors that cause loss of soil and water and those that help to reduce such losses.

The following five chapters review the methodology used by SWAT+ to simulate erosion processes.

The erosive power of rain and runoff will be less when snow cover is present than when there is no snow cover. During periods when snow is present in an HRU, SWAT+ modifies the sediment yield using the following relationship:

4:1.3.1

where is the sediment yield on a given day (metric tons), is the sediment yield calculated with MUSLE (metric tons), and is the water content of the snow cover (mm HO).

The USLE cover and management factor, CUSLE, is defined as the ratio of soil loss from land cropped under specified conditions to the corresponding loss from clean-tilled, continuous fallow (Wischmeier and Smith, 1978). The plant canopy affects erosion by reducing the effective rainfall energy of intercepted raindrops. Water drops falling from the canopy may regain appreciable velocity but it will be less than the terminal velocity of free-falling raindrops. The average fall height of drops from the canopy and the density of the canopy will determine the reduction in rainfall energy expended at the soil surface. A given percentage of residue on the soil surface is more effective that the same percentage of canopy cover. Residue intercepts falling raindrops so near the surface that drops regain no fall velocity. Residue also obstructs runoff flow, reducing its velocity and transport capacity.

Because plant cover varies during the growth cycle of the plant, SWAT+ updates daily using the equation:

4:1.1.10

where is the minimum value for the cover and management factor for the land cover, and is the amount of residue on the soil surface (kg/ha).

The minimum factor can be estimated from a known average annual factor using the following equation (Arnold and Williams, 1995):

4:1.1.11

where is the minimum factor for the land cover and is the average annual factor for the land cover.

The support practice factor, , is defined as the ratio of soil loss with a specific support practice to the corresponding loss with up-and-down slope culture. Support practices include contour tillage, stripcropping on the contour, and terrace systems. Stabilized waterways for the disposal of excess rainfall are a necessary part of each of these practices.

Contour tillage and planting provides almost complete protection against erosion from storms of low to moderate intensity, but little or no protection against occasional severe storms that cause extensive breakovers of contoured rows. Contouring is most effective on slopes of 3 to 8 percent. Values for and slope-length limits for contour support practices are given in Table 4:1-2.

Stripcropping is a practice in which contoured strips of sod are alternated with equal-width strips of row crop or small grain. Recommended values for contour stripcropping are given in Table 4:1-3.

Terraces are a series of horizontal ridges made in a hillside. There are several types of terraces. Broadbase terraces are constructed on gently sloping land and the channel and ridge are cropped the same as the interterrace area. The steep backslope terrace, where the backslope is in sod, is most common on steeper land. Impoundment terraces are terraces with underground outlets.

Terraces divide the slope of the hill into segments equal to the horizontal terrace interval. With terracing, the slope length is the terrace interval. For broadbase terraces, the horizontal terrace interval is the distance from the center of the ridge to the center of the channel for the terrace below. The horizontal terrace interval for steep backslope terraces is the distance from the point where cultivation begins at the base of the ridge to the base of the frontslope of the terrace below.

For comparative purposes, SWAT+ prints out sediment loadings calculated with USLE. These values are not used by the model, they are for comparison only. The universal soil loss equation (Williams, 1995) is:

4:1.2.1

where is the sediment yield on a given day (metric tons/ha), is the rainfall erosion index (0.017 m-metric ton cm/(m hr)), is the USLE soil erodibility factor (0.013 metric ton m hr/(m-metric ton cm)), is the USLE cover and management factor, is the USLE support practice factor, is the USLE topographic factor and is the coarse fragment factor. The factors other than are discussed in the preceding sections.

The transport of nutrients from land areas into streams and water bodies is a normal result of soil weathering and erosion processes. However, excessive loading of nutrients into streams and water bodies will accelerate eutrophication and render the water unfit for human consumption. This chapter reviews the algorithms governing movement of mineral and organic forms of nitrogen and phosphorus from land areas to the stream network.

Most soil minerals are negatively charged at normal pH and the net interaction with anions such as nitrate is a repulsion from particle surfaces. This repulsion is termed negative adsorption or anion exclusion.

Anions are excluded from the area immediately adjacent to mineral surfaces due to preferential attraction of cations to these sites. This process has a direct impact on the transport of anions through the soil for it effectively excludes anions from the slowest moving portion of the soil water volume found closest to the charged particle surfaces (Jury et al, 1991). In effect, the net pathway of the anion through the soil is shorter than it would be if all the soil water had to be used (Thomas and McMahon, 1972).

Nitrate may be transported with surface runoff, lateral flow or percolation. To calculate the amount of nitrate moved with the water, the concentration of nitrate in the mobile water is calculated. This concentration is then multiplied by the volume of water moving in each pathway to obtain the mass of nitrate lost from the soil layer.

The concentration of nitrate in the mobile water fraction is calculated:

$conc_{NO3,mobile}=\frac{NO3_{ly}*(1-exp[\frac{-w_{mobile}}{(1-\theta_e)*SAT_{ly}}])}{w_{mobile}}$ 4:2.1.2

where $conc_{NO3,mobile}$ is the concentration of nitrate in the mobile water for a given layer (kg N/mm H$_2$O), $NO3_{ly}$ is the amount of nitrate in the layer (kg N/ha), $w_{mobile}$ is the amount of mobile water in the layer (mm H$_2$O), $\theta_e$ is the fraction of porosity from which anions are excluded, and $SAT_{ly}$ is the saturated water content of the soil layer (mm H$_2$O). The amount of mobile water in the layer is the amount of water lost by surface runoff, lateral flow or percolation:

$w_{mobile}=Q_{surf}+Q_{lat,ly}+w_{perc,ly}$ for top 10 mm 4:2.1.3

$w_{mobile}=Q_{lat,ly}+w_{perc,ly}$ for lower soil layers 4:2.1.4

where $w_{mobile}$ is the amount of mobile water in the layer (mm H$_2$O), $Q_{surf}$ is the surface runoff generated on a given day (mm H$_2$O), $Q_{lat,ly}$ is the water discharged from the layer by lateral flow (mm H$_2$O), and $w_{perc,ly}$ is the amount of water percolating to the underlying soil layer on a given day (mm H$_2$O). Surface runoff is allowed to interact with and transport nutrients from the top 10 mm of soil.

Nitrate removed in surface runoff is calculated:

$NO3_{surf}=\beta_{NO3}*conc_{NO3,mobile}*Q_{surf}$ 4:2.1.5

where $NO3_{surf}$ is the nitrate removed in surface runoff (kg N/ha), $\beta_{NO3}$ is the nitrate percolation coefficient, $conc_{NO3,mobile}$ is the concentration of nitrate in the mobile water for the top 10 mm of soil (kg N/mm H$_2$O), and $Q_{surf}$ is the surface runoff generated on a given day (mm H$_2$O). The nitrate percolation coefficient allows the user to set the concentration of nitrate in surface runoff to a fraction of the concentration in percolate.

Nitrate removed in lateral flow is calculated:

$NO3_{lat,ly}=\beta_{NO3}*conc_{NO3,mobile}*Q_{lat,ly}$ for top 10 mm 4:2.1.6

$NO3_{lat,ly}=conc_{NO3,mobile}*Q_{lat,ly}$ for lower layers 4:2.1.7

where $NO3_{lat,ly}$ is the nitrate removed in lateral flow from a layer (kg N/ha), $\beta_{NO3}$ is the nitrate percolation coefficient, $conc_{NO3,mobile}$ is the concentration of nitrate in the mobile water for the layer (kg N/mm H$_2$O), and $Q_{lat,ly}$ is the water discharged from the layer by lateral flow (mm H$_2$O).

Nitrate moved to the underlying layer by percolation is calculated:

$NO3_{perc,ly}=conc_{NO3,mobile}*w_{perc,ly}$ 4:2.1.8

where $NO3_{perc,ly}$ is the nitrate moved to the underlying layer by percolation (kg N/ha), $conc_{NO3,mobile}$ is the concentration of nitrate in the mobile water for the layer (kg N/mm H$_2$O), and $w_{perc,ly}$ is the amount of water percolating to the underlying soil layer on a given day (mm H$_2$O).

Table 4:2-1: SWAT+ input variables that pertain to nitrate transport.

In large subbasins with a time of concentration greater than 1 day, only a portion of the surface runoff will reach the main channel on the day it is generated. SWAT+ incorporates a surface runoff storage feature to lag a portion of the surface runoff release to the main channel. Sediment in the surface runoff is lagged as well.

Once the sediment load in surface runoff is calculated, the amount of sediment released to the main channel is calculated:

$sed=(sed'+sed_{stor,i-1})*(1-exp[\frac{-surlag}{t_{conc}}])$ 4:1.4.1

where $sed$ is the amount of sediment discharged to the main channel on a given day (metric tons), $sed'$ is the amount of sediment load generated in the HRU on a given day (metric tons), $sed_{stor,i-1}$ is the sediment stored or lagged from the previous day (metric tons), $surlag$ is the surface runoff lag coefficient, and $t_{conc}$ is the time of concentration for the HRU (hrs).

The expression $(1-exp[\frac{-surlag}{t_{conc}}])$ in equation 4:1.4.1 represents the fraction of the total available sediment that will be allowed to enter the reach on any one day.

Figure 4:1-1 plots values for this expression at different values for $surlag$ and $t_{conc}$.

Note that for a given time of concentration, as $surlag$ decreases in value more sediment is held in storage.

Table 4:1-7: SWAT+ input variables that pertain to sediment lag calculations.

SWAT+ allows the lateral and groundwater flow to contribute sediment to the main channel. The amount of sediment contributed by lateral and groundwater flow is calculated:

$sed_{lat}=\frac{(Q_{lat}+Q_{gw})*area_{hru}*conc_{sed}}{1000}$ 4:1.5.1

where $sed_{lat}$ is the sediment loading in lateral and groundwater flow (metric tons), $Q_{lat}$ is the lateral flow for a given day (mm H$_2$O), $Q_{gw}$ is the groundwater flow for a given day (mm H$_2$O), $area_{hru}$ is the area of the HRU (km$^2$), and $conc_{sed}$ is the concentration of sediment in lateral and groundwater flow (mg/L).

Table 4:1-8: SWAT+ input variables that pertain to sediment lag calculations.

Organic N attached to soil particles may be transported by surface runoff to the main channel. This form of nitrogen is associated with the sediment loading from the HRU and changes in sediment loading will be reflected in the organic nitrogen loading. The amount of organic nitrogen transported with sediment to the stream is calculated with a loading function developed by McElroy et al. (1976) and modified by Williams and Hann (1978).

4:2.2.1

where is the amount of organic nitrogen transported to the main channel in surface runoff (kg N/ha), is the concentration of organic nitrogen in the top 10 mm (g N/ metric ton soil), is the sediment yield on a given day (metric tons), is the HRU area (ha), and is the nitrogen enrichment ratio.

The concentration of organic nitrogen in the soil surface layer, , is calculated:

4:2.2.2

where is nitrogen in the fresh organic pool in the top 10mm (kg N/ha), is nitrogen in the stable organic pool (kg N/ha), is nitrogen in the active organic pool in the top 10 mm (kg N/ha), is the bulk density of the first soil layer (Mg/m), and is the depth of the soil surface layer (10 mm).

As surface runoff flows over the soil surface, part of the water’s energy is used to pick up and transport soil particles. The smaller particles weigh less and are more easily transported than coarser particles. When the particle size distribution of the transported sediment is compared to that of the soil surface layer, the sediment load to the main channel has a greater proportion of clay sized particles. In other words, the sediment load is enriched in clay particles. Organic nitrogen in the soil is attached primarily to colloidal (clay) particles, so the sediment load will also contain a greater proportion or concentration of organic N than that found in the soil surface layer.

The enrichment ratio is defined as the ratio of the concentration of organic nitrogen transported with the sediment to the concentration in the soil surface layer. SWAT+ will calculate an enrichment ratio for each storm event, or allow the user to define a particular enrichment ratio for organic nitrogen that is used for all storms during the simulation. To calculate the enrichment ratio, SWAT+ uses a relationship described by Menzel (1980) in which the enrichment ratio is logarithmically related to sediment concentration. The equation used to calculate the nitrogen enrichment ratio, , for each storm event is:

4:2.2.3

where is the concentration of sediment in surface runoff (Mg sed/m HO). The concentration of sediment in surface runoff is calculated:

4:2.2.4

where is the sediment yield on a given day (metric tons), is the HRU area (ha), and is the amount of surface runoff on a given day (mm HO).

Table 4:2-2: SWAT+ input variables that pertain to organic N loading.

The enrichment ratio is defined as the ratio of the concentration of phosphorus transported with the sediment to the concentration of phosphorus in the soil surface layer. SWAT+ will calculate an enrichment ratio for each storm event, or allow the user to define a particular enrichment ratio for phosphorus attached to sediment that is used for all storms during the simulation. To calculate the enrichment ratio, SWAT+ uses a relationship described by Menzel (1980) in which the enrichment ratio is logarithmically related to sediment concentration. The equation used to calculate the phosphorus enrichment ratio, , for each storm event is:

4:2.4.3

where is the concentration of sediment in surface runoff (Mg /m HO). The concentration of sediment in surface runoff is calculated:

4:2.4.4

where is the sediment yield on a given day (metric tons), is the HRU area (ha), and is the amount of surface runoff on a given day (mm HO).

Table 4:2-4: SWAT+ input variables that pertain to loading of P attached to sediment.

Organic and mineral P attached to soil particles may be transported by surface runoff to the main channel. This form of phosphorus is associated with the sediment loading from the HRU and changes in sediment loading will be reflected in the loading of these forms of phosphorus. The amount of phosphorus transported with sediment to the stream is calculated with a loading function developed by McElroy et al. (1976) and modified by Williams and Hann (1978).

4:2.4.1

where is the amount of phosphorus transported with sediment to the main channel in surface runoff (kg P/ha), is the concentration of phosphorus attached to sediment in the top 10 mm (g P/ metric ton soil), is the sediment yield on a given day (metric tons), is the HRU area (ha), and is the phosphorus enrichment ratio.

The concentration of phosphorus attached to sediment in the soil surface layer, , is calculated:

4:2.4.2

where is the amount of phosphorus in the active mineral pool in the top 10 mm (kg P/ha), is the amount of phosphorus in the stable mineral pool in the top 10 mm (kg P/ha), is the amount of phosphorus in humic organic pool in the top 10 mm (kg P/ha), is the amount of phosphorus in the fresh organic pool in the top 10 mm (kg P/ha), is the bulk density of the first soil layer (Mg/m), and is the depth of the soil surface layer (10 mm).

The coarse fragment factor is calculated:

4:1.1.15

where rock is the percent rock in the first soil layer (%).

Table 4:1-5: SWAT input variables that pertain to sediment yield.

As surface runoff flows over the soil surface, part of the water’s energy is used to pick up and transport soil particles. The smaller particles weigh less and are more easily transported than coarser particles. When the particle size distribution of the transported sediment is compared to that of the soil surface layer, the sediment load to the main channel has a greater proportion of clay sized particles. In other words, the sediment load is enriched in clay particles. The sorbed phase of pesticide in the soil is attached primarily to colloidal (clay) particles, so the sediment load will also contain a greater proportion or concentration of pesticide than that found in the soil surface layer.

The enrichment ratio is defined as the ratio of the concentration of sorbed pesticide transported with the sediment to the concentration in the soil surface layer. SWAT+ will calculate an enrichment ratio for each storm event, or allow the user to define a particular enrichment ratio for sorbed pesticide that is used for all storms during the simulation. To calculate the enrichment ratio, SWAT+ uses a relationship described by Menzel (1980) in which the enrichment ratio is logarithmically related to sediment concentration. The equation used to calculate the pesticide enrichment ratio, , for each storm event is:

4:3.3.5

where is the concentration of sediment in surface runoff (Mg sed/m HO). The concentration of sediment in surface runoff is calculated:

4:3.3.6

where is the sediment yield on a given day (metric tons), is the HRU area (ha), and is the amount of surface runoff on a given day (mm HO).

Table 4:3-3: SWAT+ input variables that pertain to sorbed pesticide loading.

The transport of pesticide from land areas into streams and water bodies is a result of soil weathering and erosion processes. Excessive loading of pesticides in streams and water bodies can produce toxic conditions that harm aquatic life and render the water unfit for human consumption. This chapter reviews the algorithms governing movement of soluble and sorbed forms of pesticide from land areas to the stream network. Pesticide transport algorithms in SWAT+ were taken from EPIC (Williams, 1995).

The transport of pathogenic bacteria from land areas into streams and water bodies is a matter of concern in some watersheds. Excessive loading of bacteria into streams and water bodies could potentially contaminate drinking water and cause outbreaks of infection among the human population using the water. This chapter reviews the algorithms governing movement of bacteria from land areas to the stream network.

Pesticide attached to soil particles may be transported by surface runoff to the main channel. This phase of pesticide is associated with the sediment loading from the HRU and changes in sediment loading will impact the loading of sorbed pesticide. The amount of pesticide transported with sediment to the stream is calculated with a loading function developed by McElroy et al. (1976) and modified by Williams and Hann (1978).

$pst_{sed}=0.001*C_{solidphase}*\frac{sed}{area_{hru}}*\varepsilon_{pst:sed}$ 4:3.3.1

where $pst_{sed}$ is the amount of sorbed pesticide transported to the main channel in surface runoff (kg $pst$/ha), $C_{solidphase}$ is the concentration of pesticide on sediment in the top 10 mm (g $pst$/ metric ton soil), sed is the sediment yield on a given day (metric tons), $area_{hru}$ is the HRU area (ha), and $\varepsilon_{pst:sed}$ is the pesticide enrichment ratio.

The total amount of pesticide in the soil layer is the sum of the adsorbed and dissolved phases:

$pst_{s,ly}=0.01*(C_{solution}*SAT_{ly}+C_{solidphase}*\rho_b*depth_{ly})$ 4:3.3.2

where $pst_{s,ly}$ is the amount of pesticide in the soil layer (kg $pst$/ha), $C_{solution}$ is the pesticide concentration in solution (mg/L or g/ton), $SAT_{ly}$ is the amount of water in the soil layer at saturation (mm H$_2$O), $C_{solidphase}$ is the concentration of the pesticide sorbed to the solid phase (mg/kg or g/ton), $\rho_b$ is the bulk density of the soil layer (Mg/m$^3$), and $depth_{ly}$ is the depth of the soil layer (mm). Rearranging equation 4:3.1.1 to solve for $C_{solution}$ and substituting into equation 4:3.3.2 yields:

$pst_{s,ly}=0.01*(\frac{C_{solidphase}}{K_p}*SAT_{ly}+C_{solidphase}*\rho_b*depth_{ly})$ 4:3.3.3

which rearranges to

$C_{solidphase}=\frac{100*K_p*pst_{s,ly}}{(SAT_{ly}+K_p*\rho_b*depth_{ly})}$ 4:3.3.4

where $C_{solidphase}$ is the concentration of the pesticide sorbed to the solid phase (mg/kg or g/ton), $K_p$ is the soil adsorption coefficient ((mg/kg)/(mg/L) or $m^3$/ton) $pst_{s,ly}$ is the amount of pesticide in the soil layer (kg $pst$/ha), $SAT_{ly}$ is the amount of water in the soil layer at saturation (mm H$_2$O), $\rho_b$ is the bulk density of the soil layer (Mg/m$^3$), and $depth_{ly}$ is the depth of the soil layer (mm).

Suspended algal biomass is assumed to be directly proportional to chlorophyll $a$. Therefore, the algal biomass loading to the stream can be estimated as the chlorophyll $a$ loading from the land area. Cluis et al. (1988) developed a relationship between the nutrient enrichment index (total N: total P), chlorophyll $a$, and algal growth potential in the North Yamaska River, Canada.

$(AGP+chla)*v_{surf}=f*(\frac{TN}{TP})^g$ 4:5.1.1

where $AGP$ is the algal growth potential (mg/L), $chla$ is the chlorophyll $a$ concentration in the surface runoff ($\mu g$/L), $v_{surf}$ is the surface runoff flow rate (m$^3$/s), $TN$ is the total Kjeldahl nitrogen load (kmoles), $TP$ is the total phosphorus load (kmoles), $f$ is a coefficient and $g$ is an exponent.

The chlorophyll $a$ concentration in surface runoff is calculated in SWAT+ using a simplified version of Cluis et al.’s exponential function (1988):

$chla=0$ if $(v_{surf}<10^{-5}m^3/s)$or $(TP$ and $TN <10^{-6})$ 4:5.1.2

$chla =\frac{0.5*10^{2.7}}{v_{surf}}$ if $v_{surf}>10^{-5}m^3/s$ or ($TP$ and $TN>10^{-6}$) 4:5.1.3

$chla=\frac{0.5*10^{0.5}}{v_{surf}}$ if $v_{surf}>10^{-5}m^3/s$, $TP<10^{-6}$ and $TN>10^{-6}$ 4:5.1.4

Bacteria attached to soil particles may be transported by surface runoff to the main channel. This bacteria is associated with the sediment loading from the HRU and changes in sediment loading will be reflected in the loading of this form of bacteria. The amount of bacteria transported with sediment to the stream is calculated with a loading function developed by McElroy et al. (1976) and modified by Williams and Hann (1978) for nutrients.

$bact_{lp,sed}=0.0001*conc_{sedlpbact}*\frac{sed}{area_{hru}}*\varepsilon_{bact:sed}$ 4:4.2.1

$bact_{p,sed}=0.0001*conc_{sedpbact}*\frac{sed}{area_{hru}}*\varepsilon_{bact:sed}$ 4:4.2.2

where $bact_{lp,sed}$ is the amount of less persistent bacteria transported with sediment in surface runoff (#cfu/m$^2$), $bact_{p,sed}$ is the amount of persistent bacteria transported with sediment in surface runoff (#cfu/m$^2$), $conc_{sedlpbact}$ is the concentration of less persistent bacteria attached to sediment in the top 10 mm (# cfu/ metric ton soil), $conc_{sedpbact}$ is the concentration of persistent bacteria attached to sediment in the top 10 mm (# cfu/ metric ton soil), $sed$ is the sediment yield on a given day (metric tons), $area_{hru}$ is the HRU area (ha), and $\varepsilon_{bact:sed}$ is the bacteria enrichment ratio.

The concentration of bacteria attached to sediment in the soil surface layer is calculated:

$conc_{sedlpbact}=1000*\frac{bact_{lp,sorb}}{\rho_b*depth_{surf}}$ 4:4.2.3

$conc_{sedpbact}=1000*\frac{bact_{p,sorb}}{\rho_b*depth_{surf}}$ 4:4.2.4

where $bact_{lpsorb}$ is the amount of less persistent bacteria sorbed to the soil (#cfu/m$^2$), $bact_{psorb}$ is the amount of persistent bacteria sorbed to the soil (#cfu/m$^2$), $\rho_b$ is the bulk density of the first soil layer (Mg/m$^3$), and $depth_{surf}$ is the depth of the soil surface layer (10 mm).

The value of $EI_{USLE}$ for a given rainstorm is the product, total storm energy times the maximum 30 minute intensity. The storm energy indicates the volume of rainfall and runoff while the 30 minute intensity indicates the prolonged peak rates of detachment and runoff.

$EI_{USLE}=E_{storm}*I_{30}$ 4:1.2.2

where $EI_{USLE}$ is the rainfall erosion index (0.017 m-metric ton cm/(m$^2$ hr)), $E_{storm}$ is the total storm energy (0.0017 m-metric ton/m$^2$), and $I_{30}$ is the maximum 30-minute intensity (mm/hr).

The energy of a rainstorm is a function of the amount of rain and of all the storm’s component intensities. Because rainfall is provided to the model in daily totals, an assumption must be made about variation in rainfall intensity. The rainfall intensity variation with time is assumed to be exponentially distributed:

$i_t=i_{mx}*exp(-\frac{t}{k_i})$ 4:1.2.3

where $i_t$ is the rainfall intensity at time $t$ (mm/hr), $i_{mx}$ is the maximum rainfall intensity (mm/hr), $t$ is the time (hr), and $k_i$ is the decay constant for rainfall intensity (hr).

The USLE energy equation is

$E_{storm}=\Delta R_{day}*(12.1+8.9*log_{10}[\frac{\Delta R_{day}}{\Delta t}])$ 4:1.2.4

where $\Delta R_{day}$ is the amount of rainfall during the time interval (mm H$_2$O), and $\Delta t$ is the time interval (hr). This equation may be expressed analytically as:

$E_{storm}=12.1\int_0^{\infty}i_t dt+8.9\int_0^{\infty} i_t log_{10} i_tdt$ 4:1.2.5

Combining equation 4:1.2.5 and 4:1.2.3 and integrating gives the equation for estimating daily rainfall energy:

$E_{storm}=\frac{R_{day}}{1000}*(12.1+8.9*(log_{10}[i_{mx}]-0.434))$ 4:1.2.6

where $R_{day}$ is the amount of precipitation falling on a given day (mm H$_2$O), and $i_{mx}$ is the maximum rainfall intensity (mm/hr). To compute the maximum rainfall intensity, $i_{mx}$, equation 4:1.2.3 is integrated to give

$R_{day}=i_{mx}*k_i$ 4:1.2.7

and

$R_t=R_{day}*(1-exp[-\frac{t}{k_i}])$ 4:1.2.8

where $R_{day}$ is the amount of precipitation falling on a given day (mm H$_2$O), $i_{mx}$ is the maximum rainfall intensity (mm/hr), $k_i$ is the decay constant for rainfall intensity (hr), $R_t$ is the amount of rain falling during a time interval (mm H$_2$O), and $t$ is the time interval (hr). The maximum half-hour rainfall for the precipitation event is known:

$R_{0.5}=\alpha_{0.5}*R_{day}$ 4:1.2.9

where $R_{0.5}$ is the maximum half-hour rainfall (mm H$_2$O), $\alpha_{0.5}$ is the maximum half-hour rainfall expressed as a fraction of daily rainfall, and $R_{day}$ is the amount of precipitation falling on a given day (mm H$_2$O). Calculation of $\alpha_{0.5}$ is reviewed in Chapter 1:2 and Chapter 1:3. Substituting equation 4:1.2.9 and 4:1.2.7 into 4:1.2.8 and solving for the maximum intensity gives:

$i_{mx}=-2*R_{day}*1n(1-\alpha_{0.5})$ 4:1.2.10

where $i_{mx}$ is the maximum rainfall intensity (mm/hr), $R_{day}$ is the amount of precipitation falling on a given day (mm H$_2$O), and $\alpha_{0.5}$ is the maximum half-hour rainfall expressed as a fraction of daily rainfall.

The maximum 30 minute intensity is calculated:

$I_{30}=2*\alpha_{0.5}*R_{day}$ 4:1.2.11

where $I_{30}$ is the maximum 30-minute intensity (mm/hr), $\alpha_{0.5}$ is the maximum half-hour rainfall expressed as a fraction of daily rainfall, and $R_{day}$ is the amount of precipitation falling on a given day (mm H$_2$O).

Table 4:1-6: SWAT+ input variables that pertain to USLE sediment yield.

Pesticide in the soluble phase may be transported with surface runoff, lateral flow or percolation. The change in the amount of pesticide contained in a soil layer due to transport in solution with flow is a function of time, concentration and amount of flow:

$\frac{dpst_{s,ly}}{dt}=0.01*C_{solution}*w_{mobile}$ 4:3.2.1

where $pst_{s,ly}$ is the amount of pesticide in the soil layer (kg pst/ha), $C_{solution}$ is the pesticide concentration in solution (mg/L or g/ton), and $w_{mobile}$ is the amount of mobile water on a given day (mm H$_2$O). The amount of mobile water in the layer is the amount of water lost by surface runoff, lateral flow or percolation:

$w_{mobile}=Q_{surf}+Q_{lat,surf}+w_{perc,surf}$ for top 10 mm 4:3.2.2

$w_{mobile}=Q_{lat,ly}+w_{perc,ly}$ for lower soil layers 4:3.2.3

where $w_{mobile}$ is the amount of mobile water in the layer (mm H$_2$O), $Q_{surf}$ is the surface runoff generated on a given day (mm H$_2$O), $Q_{lat,ly}$ is the water discharged from the layer by lateral flow (mm H$_2$O), and $w_{perc,ly}$ is the amount of water percolating to the underlying soil layer on a given day (mm H$_2$O).

The total amount of pesticide in the soil layer is the sum of the adsorbed and dissolved phases:

$pst_{s,ly}=0.01*(C_{solution}*SAT_{ly}+C_{solidphase}*\rho_b*depth_{ly})$ 4:3.2.4

where $pst_{s,ly}$ is the amount of pesticide in the soil layer (kg pst/ha), $C_{solution}$ is the pesticide concentration in solution (mg/L or g/ton), $SAT_{ly}$ is the amount of water in the soil layer at saturation (mm H$_2$O), $C_{solidphase}$ is the concentration of the pesticide sorbed to the solid phase (mg/kg or g/ton), $\rho_b$ is the bulk density of the soil layer (Mg/m$^3$), and $depth_{ly}$ is the depth of the soil layer (mm). Rearranging equation 4:3.1.1 to solve for $C_{solidphase}$ and substituting into equation 4:3.2.4 yields:

$pst_{s,ly}=0.01*(C_{solution}*SAT_{ly}+C_{solution}*K_p*\rho_b*depth_{ly})$ 4:3.2.5

which rearranges to

$C_{solution}=\frac{pst_{s,ly}}{0.01*(SAT_{ly}+K_p*\rho_b*depth_{ly})}$ 4:3.2.6

Combining equation 4:3.2.6 with equation 4:3.2.1 yields

$\frac{dpst_{s,ly}}{dt}=\frac{pst_{s,ly}*w_{mobile}}{(SAT_{ly}+K_p*\rho_b*depth_{ly})}$ 4:3.2.7

Integration of equation 4:3.2.7 gives

$pst_{s,ly,t}=pst_{s,ly,o}*exp[\frac{-w_{mobile}}{(SAT_{ly}+K_p*\rho_b*depth_{ly})}]$ 4:3.2.8

where $pst_{s,ly,t}$ is the amount of pesticide in the soil layer at time t (kg $pst$t/ha), $pst_{s,ly,o}$ is the initial amount of pesticide in the soil layer (kg $pst$/ha), $w_{mobile}$ is the amount of mobile water in the layer (mm H$_2$O), $SAT_{ly}$ is the amount of water in the soil layer at saturation (mm H$_2$O), $K_p$ is the soil adsorption coefficient ((mg/kg)/(mg/L)), $\rho_b$ is the bulk density of the soil layer (Mg/m$^3$), and $depth_{ly}$ is the depth of the soil layer (mm).

To obtain the amount of pesticide removed in solution with the flow, the final amount of pesticide is subtracted from the initial amount of pesticide:

$pst_{flow}=pst_{s,ly,o}*(1-exp[\frac{-w_{mobile}}{(SAT_{ly}+K_p*\rho_b*depth_{ly})}])$ 4:3.2.9

where $pst_{flow}$ is the amount of pesticide removed in the flow (kg pst/ha) and all other terms were previously defined.

For the top 10 mm that interacts with surface runoff, the pesticide concentration in the mobile water is calculated:

$conc_{pst,flow}=min{[pst_{flow}/[w_{perc,surf}+\beta_{pst}(Q_{surf}+Q_{lat,surf})]], pst_{sol}/100}.$ 4:3.2.10

while for lower layers

$conc_{pst,flow}=min[{[pst_{flow}/w_{mobile}],}pst_{sol}/100.]$ 4:3.2.11

where $conc_{pst,flow}$ is the concentration of pesticide in the mobile water (kg $pst$/ha-mm H$_2$O), $pst_{flow}$ is the amount of pesticide removed in the flow (kg $pst$/ha), $\beta_{pst}$ is the pesticide percolation coefficient, $Q_{surf}$ is the surface runoff generated on a given day (mm H$_2$O), $Q_{lat,ly}$ is the water discharged from the layer by lateral flow (mm H$_2$O), $w_{perc,ly}$ is the amount of water percolating to the underlying soil layer on a given day (mm H$_2$O), $w_{mobile}$ is the amount of mobile water in the layer (mm H$_2$O), and $pst_{sol}$ is the solubility of the pesticide in water (mg/L).

Pesticide moved to the underlying layer by percolation is calculated:

$pst_{perc,ly}=conc_{pst,flow}*w_{perc,ly}$ 4:3.2.12

where $pst_{perc,ly}$ is the pesticide moved to the underlying layer by percolation (kg $pst$/ha), $conc_{pst,flow}$ is the concentration of pesticide in the mobile water for the layer (kg $pst$/mm H$_2$O), and $w_{perc,ly}$ is the amount of water percolating to the underlying soil layer on a given day (mm H$_2$O).

Pesticide removed in lateral flow is calculated:

$pst_{lat,surf}=\beta_{pst}*conc_{pst,flow}*Q_{lat,surf}$ for top 10 mm 4:3.2.13

$pst_{lat,ly}=conc_{pst,flow}*Q_{lat,ly}$ for lower layers 4:3.2.14

where $pst_{lat,ly}$ is the pesticide removed in lateral flow from a layer (kg $pst$/ha), $\beta_{pst}$ is the pesticide percolation coefficient, $conc_{pst,flow}$ is the concentration of pesticide in the mobile water for the layer (kg $pst$/mm H$_2$O), and $Q_{lat,ly}$ is the water discharged from the layer by lateral flow (mm H$_2$O). The pesticide percolation coefficient allows the user to set the concentration of pesticide in runoff and lateral flow from the top 10 mm to a fraction of the concentration in percolate.