The plant growth component of SWAT+ is a simplified version of the EPIC plant growth model. As in EPIC, phenological plant development is based on daily accumulated heat units, potential biomass is based on a method developed by Monteith, a harvest index is used to calculate yield, and plant growth can be inhibited by temperature, water, nitrogen, or phosphorus stress. Portions of the EPIC plant growth model that were not incorporated into SWAT+ include detailed root growth, micronutrient cycling and toxicity responses, and the simultaneous growth of multiple plant species in the same HRU.

The growth cycle of a plant is controlled by plant attributes summarized in the plant growth database and by the timing of operations listed in the management file. This chapter reviews the heat unit theory used to regulate the growth cycle of plants. Chapter 6:1 focuses on the impact of user inputs in management operations on the growth and development of plants.

Temperature is one of the most important factors governing plant growth. Each plant has its own temperature range, i.e. its minimum, optimum, and maximum for growth. For any plant, a minimum or base temperature must be reached before any growth will take place. Above the base temperature, the higher the temperature the more rapid the growth rate of the plant. Once the optimum temperature is exceeded the growth rate will begin to slow until a maximum temperature is reached at which growth ceases.

In the 1920s and 1930s, canning factories were searching for ways to time the planting of sweet peas so that there would be a steady flow of peas at the peak of perfection to the factory. Crops planted at weekly intervals in the early spring would sometimes come to maturity with only a 1- or 2-day differential while at other times there was a 6- to 8-day differential (Boswell, 1926; 1929). A heat unit theory was suggested (Boswell, 1926; Magoon and Culpepper, 1932) that was revised and successfully applied (Barnard, 1948; Phillips, 1950) by canning companies to determine when plantings should be made to ensure a steady harvest of peas with no “bunching” or “breaks”.

The heat unit theory postulates that plants have heat requirements that can be quantified and linked to time to maturity. Because a plant will not grow when the mean temperature falls below its base temperature, the only portion of the mean daily temperature that contributes towards the plant’s development is the amount that exceeds the base temperature. To measure the total heat requirements of a plant, the accumulation of daily mean air temperatures above the plant’s base temperature is recorded over the period of the plant’s growth and expressed in terms of heat units. For example, assume sweet peas are growing with a base temperature of 5°C. If the mean temperature on a given day is 20°C, the heat units accumulated on that day are 20 – 5 = 15 heat units. Knowing the planting date, maturity date, base temperature and mean daily temperatures, the total number of heat units required to bring a crop to maturity can be calculated.

The heat index used by SWAT+ is a direct summation index. Each degree of the daily mean temperature above the base temperature is one heat unit. This method assumes that the rate of growth is directly proportional to the increase in temperature. It is important to keep in mind that the heat unit theory without a high temperature cutoff does not account for the impact of harmful high temperatures. SWAT+ assumes that all heat above the base temperature accelerates crop growth and development.

The mean daily temperature during 1992 for Greenfield, Indiana is plotted in Figure 5:1-1 along with the base temperature for corn (8°C). Crop growth will only occur on those days where the mean daily temperature exceeds the base temperature. The heat unit accumulation for a given day is calculated with the equation:

When calculating the potential heat units for a plant, the number of days to reach maturity must be known. For most crops, these numbers have been quantified and are easily accessible. For other plants, such as forest or range, the time that the plants begin to develop buds should be used as the beginning of the growing season and the time that the plant seeds reach maturation is the end of the growing season. For the Greenfield Indiana example, a 120 day corn hybrid was planted on May 15. Summing daily heat unit values, the total heat units required to bring the corn to maturity was 1456.

The amount of daily solar radiation intercepted by the leaf area of the plant is calculated using Beer’s law (Monsi and Saeki, 1953):

$H_{phosyn}=0.5*H_{day}*(1-exp(-k_{\Box}*LAI))$ 5:2.1.1

where $H_{phosyn}$ is the amount of intercepted photosynthetically active radiation on a given day (MJ m$^{-2}$), $H_{day}$ is the incident total solar (MJ m$^{-2}$), $0.5*H_{day}$ is the incident photosynthetically active radiation (MJ m$^{-2}$), $k_{\Box}$ is the light extinction coefficient, and $LAI$ is the leaf area index.

Photosynthetically active radiation is radiation with a wavelength between 400 and 700 mm (McCree, 1972). Direct solar beam radiation contains roughly 45% photosynthetically active radiation while diffuse radiation contains around 60% photosynthetically active radiation (Monteith, 1972; Ross, 1975). The fraction of photosynthetically active radiation will vary from day to day with variation in overcast conditions but studies in Europe and Israel indicate that 50% is a representative mean value (Monteith, 1972; Szeicz, 1974; Stanhill and Fuchs, 1977).

Radiation-use efficiency is the amount of dry biomass produced per unit intercepted solar radiation. The radiation-use efficiency is defined in the plant growth database and is assumed to be independent of the plant’s growth stage. The maximum increase in biomass on a given day that will result from the intercepted photosynthetically active radiation is estimated (Monteith, 1977):

$\Delta bio=RUE*H_{phosyn}$ 5:2.1.2

where $\Delta bio$ is the potential increase in total plant biomass on a given day (kg/ha), $RUE$ is the radiation-use efficiency of the plant (kg/ha⋅(MJ/m$^2$)$^{-1}$ or 10$^{-1}$ g/MJ), and $H_{phosyn}$ is the amount of intercepted photosynthetically active radiation on a given day (MJ m$^{-2}$). Equation 5:2.1.2 assumes that the photosynthetic rate of a canopy is a linear function of radiant energy.

The total biomass on a given day, $d$, is calculated as:

$bio=\sum_{i=1}^{d}\Delta bio_i$ 5:2.1.3

where $bio$ is the total plant biomass on a given day (kg ha$^{-1}$), and $\Delta bio_i$ is the increase in total plant biomass on day $i$ (kg/ha).

SWAT+ categorizes plants into seven different types: warm season annual legume, cold season annual legume, perennial legume, warm season annual, cold season annual, perennial and trees. The differences between the different plant types, as modeled by SWAT+, are as follows:

1. warm season annual legume:

-simulate nitrogen fixation

-root depth varies during growing season due to root growth

2. cold season annual legume:

-simulate nitrogen fixation

-root depth varies during growing season due to root growth

-fall-planted land covers will go dormant when daylength is less than

the threshold daylength

3.perennial legume:

-simulate nitrogen fixation

-root depth always equal to the maximum allowed for the plant species and soil

-plant goes dormant when daylength is less than the threshold daylength

4.warm season annual:

-root depth varies during growing season due to root growth

5.cold season annual:

-root depth varies during growing season due to root growth

-fall-planted land covers will go dormant when daylength is less than the threshold

daylength

6.perennial:

-root depth always equal to the maximum allowed for the plant species and soil

-plant goes dormant when daylength is less than the threshold daylength

7. trees:

-root depth always equal to the maximum allowed for the plant species and soil

-partitions new growth between leaves/needles and woody growth

-growth in a given year will vary depending on the age of the tree relative to the

number of years required for the tree to full development/maturity

-plant goes dormant when daylength is less than the threshold daylength

Table 5:1-3: SWAT+ input variables that pertain to plant type.

SWAT+ assumes trees, perennials and cool season annuals can go dormant as the daylength nears the shortest or minimum daylength for the year. During dormancy, plants do not grow.

The beginning and end of dormancy are defined by a threshold daylength. The threshold daylength is calculated:

$T_{DL,thr}=T_{DL,mn}+t_{dorm}$ 5:1.2.1

where $T_{DL,thr}$ is the threshold daylength to initiate dormancy (hrs), $T_{DL,mn}$ is the minimum daylength for the watershed during the year (hrs), and tdorm is the dormancy threshold (hrs). When the daylength becomes shorter than $T_{DL,thr}$ in the fall, plants other than warm season annuals that are growing in the watershed will enter dormancy. The plants come out of dormancy once the daylength exceeds $T_{DL,thr}$ in the spring.

The dormancy threshold, $t_{dorm}$, varies with latitude.

$t_{dorm}=1.0$ if $\phi >$40 º N or S 5:1.2.2

$t_{dorm}=\frac{\phi - 20}{20}$ if 20 º N or S $\le \phi \le$ 40 º N or S 5:1.2.3

$t_{dorm}=0.0$ if $\phi <$20 º N or S 5:1.2.4

where $t_{dorm}$ is the dormancy threshold used to compare actual daylength to minimum daylength (hrs) and $\phi$ is the latitude expressed as a positive value (degrees).

At the beginning of the dormant period for trees, a fraction of the biomass is converted to residue and the leaf area index for the tree species is set to the minimum value allowed (both the fraction of the biomass converted to residue and the minimum LAI are defined in the plant growth database). At the beginning of the dormant period for perennials, 10% of the biomass is converted to residue and the leaf area index for the species is set to the minimum value allowed. For cool season annuals, none of the biomass is converted to residue.

Table 5:1-2: SWAT+ input variables that pertain to dormancy.

With annuals and perennials, the plants are able to reach full maturity within a single calendar year. With trees, a number of years are needed for a plant to transition from a seedling to a sapling to a fully-developed tree. The parameters in the plant growth database related to radiation-use efficiency represent the annual growth for a fully-developed tree. The heat units to maturity input in the management file is also used to simulate growth within a single year, defining for trees and perennials the period within a year bounded by the development of buds at the beginning of the annual growing season and the maturation of plant seeds at the end of the growing season.

To simulate the smaller amount of biomass accumulation seen in seedlings/saplings, tree growth within a single year is limited to a fixed amount determined by the age of the tree relative to the number of years for the tree species to reach full development. Parameters in the plant growth database define the total number of years for trees to reach full development as well as the biomass of a fully-developed tree. Until the trees in an HRU reach full development, the amount of biomass they can accumulate in a single year is limited to:

5:2.1.9

where is the amount of biomass a tree can accumulate in a single year (kg/ha), is the current age of the tree (years), is the number of years for the tree species to reach full development (years), is the biomass of a fully developed tree stand for the specific tree species (metric tons/ha), and 1000 is a conversion factor.

Once the total growth in biomass in a year, , reaches the annual limit, , no more growth occurs until the next year when a new annual limit is calculated. When a tree stand has reached its biomass limit in a year, the increase in plant biomass for a day,, is set to 0.

The amount of total plant biomass partitioned to the root system is 30-50% in seedlings and decreases to 5-20% in mature plants (Jones, 1985). SWAT+ varies the fraction of total biomass in roots from 0.40 at emergence to 0.20 at maturity. The daily root biomass fraction is calculated with the equation:

5:2.1.21

where is the fraction of total biomass partitioned to roots on a given day in the growing season, and is the fraction of potential heat units accumulated for the plant on a given day in the growing season.

Calculation of root depth varies according to plant type. SWAT+ assumes perennials and trees have roots down to the maximum rooting depth defined for the soil throughout the growing season:

5:2.1.22

where is the depth of root development in the soil on a given day (mm), and is the maximum depth for root development in the soil (mm). The simulated root depth for annuals varies linearly from 10.0 mm at the beginning of the growing season to the maximum rooting depth at = 0.40 using the equation:

if 5:2.1.23

if 5:2.1.24

where is the depth of root development in the soil on a given day (mm), is the fraction of potential heat units accumulated for the plant on a given day in the growing season, and is the maximum depth for root development in the soil (mm). The maximum rooting depth is defined by comparing the maximum potential rooting depth for the plant from the plant growth database (RDMX in crop.dat), and the maximum potential rooting depth for the soil from the soil input file (SOL_ZMX in .sol—if no value is provided for this variable the model will set it to the deepest depth specified for the soil profile). The shallower of these two depths is the value used for .

Plant growth is modeled by simulating leaf area development, light interception and conversion of intercepted light into biomass assuming a plant species-specific radiation-use efficiency.

Plant maturity is reached when the fraction of potential heat units accumulated, $fr_{PHU}$, is equal to 1.00. Once maturity is reached, the plant ceases to transpire and take up water and nutrients. Simulated plant biomass remains stable until the plant is harvested or killed via a management operation.

Table 5:2-1: SWAT+ input variables that pertain to optimal plant growth.

The change in canopy height and leaf area for annuals and perennials through the growing season as modeled by SWAT+ is illustrated using parameters for Alamo Switchgrass in Figures 5:2-1 and 5:2-2.

In the initial period of plant growth, canopy height and leaf area development are controlled by the optimal leaf area development curve:

$fr_{LAImx}=\frac{fr_{PHU}}{fr_{PHU}+exp(\Box_1 - \Box_2 * fr_{PHU})}$ 5:2.1.10

where $fr_{LAImx}$ is the fraction of the plant’s maximum leaf area index corresponding to a given fraction of potential heat units for the plant, $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on a given day in the growing season, and $\Box_1$and $\Box_2$ are shape coefficients. The fraction of potential heat units accumulated by a given date is calculated:

$fr_{PHU}=\frac{\sum_{i=1}^d HU_i}{PHU}$ 5:2.1.11

where $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on day d in the growing season, $HU$ is the heat units accumulated on day $i$ (heat units), and $PHU$ is the total potential heat units for the plant (heat units).

The shape coefficients are calculated by solving equation 5:2.1.10 using two known points ($fr_{LAI,1}$,$fr_{PHU,1}$) and ($fr_{LAI,2}$,$fr_{PHU,2}$):

$\Box_1=1n[\frac{fr_{PHU,1}}{fr_{LAI,1}}-fr_{PHU,1}]+\Box_2*fr_{PHU,1}$ 5:2.1.12

$\Box_2=\frac{(1n[\frac{fr_{PHU,1}}{fr_{LAI,1}}-fr_{PHU,1}]-1n[\frac{fr_{PHU,2}}{fr_{LAI,2}}-fr_{PHU,2}])}{fr_{PHU,2}-fr_{PHU,1}}$ 5:2.1.13

where $\Box_1$ is the first shape coefficient, $\Box_2$ is the second shape coefficient, $fr_{PHU,1}$ is the fraction of the growing season (i.e. fraction of total potential heat units) corresponding to the 1st point on the optimal leaf area development curve, $fr_{LAI,1}$ is the fraction of the maximum plant leaf area index (i.e. fraction of $LAI_{mx}$) corresponding to the 1st point on the optimal leaf area development curve, $fr_{PHU,2}$ is the fraction of the growing season corresponding to the 2nd point on the optimal leaf area development curve, and $fr_{LAI,2}$ is the fraction of the maximum plant leaf area index corresponding to the 2nd point on the optimal leaf area development curve.

The canopy height on a given day is calculated:

$h_c=h_{c,mx}*\sqrt{fr_{LAImx}}$ 5:2.1.14

where $h_c$ is the canopy height for a given day (m), $h_{c,mx}$ is the plant’s maximum canopy height (m), and $fr_{LAI,mx}$ is the fraction of the plant’s maximum leaf area index corresponding to a given fraction of potential heat units for the plant. As can be seen from Figure 5:2-1, once the maximum canopy height is reached, $h_c$ will remain constant until the plant is killed.

For tree stands, the canopy height varies from year to year rather than day to day:

$h_c=h_{c,mx}*(\frac{yr_{cur}}{yr_{fulldev}})$ 5:2.1.15

where $h_c$ is the canopy height for a given day (m), $h_{c,mx}$ is the plant’s maximum canopy height (m), $yr_{cur}$ is the age of the tree (years), and $yr_{fulldev}$ is the number of years for the tree species to reach full development (years).

The amount of canopy cover is expressed as the leaf area index. For annuals and perennials, the leaf area added on day $i$ is calculated:

$\Delta LAI_i=(fr_{LAImx,i}-fr_{LAImx,i-1})*LAI_{mx}*(1-exp(5*(LAI_{i-1}-LAI_{mx})))$

5:2.1.16

while for trees, the leaf area added on day $i$ is calculated:

$\Delta LAI_i=(fr_{LAImx,i}-fr_{LAImx,i-1})*(\frac{yr_{cur}}{yr_{fulldev}})*LAI_{mx}*(1-exp(5*(LAI_{i-1}-(\frac{yr_{cur}}{yr_{fulldev}})*LAI_{mx})))$

5:2.1.17

The total leaf area index is calculated:

$LAI_i=LAI_{i-1}+\Delta LAI_{i}$ 5:2.1.18

where $\Delta LAI_i$ is the leaf area added on day $i$, $LAI_i$ and $LAI_{i-1}$ are the leaf area indices for day $i$ and $i-1$ respectively, $fr_{LAImx,i}$ and $fr_{LAImx,i-1}$ are the fraction of the plant’s maximum leaf area index calculated with equation 5:2.1.10 for day $i$ and $i-1$,$LAI_{mx}$ is the maximum leaf area index for the plant, $yr_{cur}$ is the age of the tree (years), and $yr_{fulldev}$ is the number of years for the tree species to reach full development (years).

Leaf area index is defined as the area of green leaf per unit area of land (Watson, 1947). As shown in Figure 5:2-2, once the maximum leaf area index is reached, $LAI$ will remain constant until leaf senescence begins to exceed leaf growth. Once leaf senescence becomes the dominant growth process, the leaf area index for annuals and perrenials is calculated:

$LAI=LAI_{mx}*\frac{(1-fr_{PHU})}{(1-fr_{PHU,sen})}$ $fr_{PHU}>fr_{PHU,sen}$ 5:2.1.19

while for trees, the calculation is

$LAI=(\frac{yr_{cur}}{yr_{fulldev}})*LAI_{mx}*\frac{(1-fr_{PHU})}{(1-fr_{PHU,sen})}$ $fr_{PHU}>fr_{PHU,sen}$ 5:2.1.20

where $LAI$ is the leaf area index for a given day, $LAI_{mx}$ is the maximum leaf area index, $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on a given day in the growing season, $fr_{PHU,sen}$ is the fraction of growing season ($PHU$) at which senescence becomes the dominant growth process, $yr_{cur}$ is the number of years of development the tree has accrued (years), and $yr_{fulldev}$ is the number of years for the tree species to reach full development (years).

Once the potential water uptake has been modified for soil water conditions, the actual amount of water uptake from the soil layer is calculated:

5:2.2.7

where is the actual water uptake for layer (mm HO), is the amount of water in the soil layer on a given day (mm HO), and is the water content of layer at wilting point (mm HO). The total water uptake for the day is calculated:

5:2.2.8

where is the total plant water uptake for the day (mm HO), is the actual water uptake for layer (mm HO), and n is the number of layers in the soil profile. The total plant water uptake for the day calculated with equation 5:2.2.8 is also the actual amount of transpiration that occurs on the day.

5:2.2.9

where is the actual amount of transpiration on a given day (mm HO) and is the total plant water uptake for the day (mm HO).

Table 5:2-2: SWAT+ input variables that pertain to plant water uptake.

Radiation-use efficiency is sensitive to variations in atmospheric $CO_2$ concentrations and equations have been incorporated into SWAT+ to modify the default radiation-use efficiency values in the plant database for climate change studies. The relationship used to adjust the radiation-use efficiency for effects of elevated $CO_2$ is (Stockle et al., 1992):

$RUE=\frac{100*CO_2}{CO_2+exp(r_1-r_2*CO_2)}$ 5:2.1.4

where $RUE$ is the radiation-use efficiency of the plant (kg/ha⋅(MJ/m$^2$)$^{-1}$ or 10$^{-1}$ g/MJ), $CO_2$ is the concentration of carbon dioxide in the atmosphere (ppmv), and $r_1$ and $r_2$ are shape coefficients.

The shape coefficients are calculated by solving equation 5:2.1.4 using two known points ($RUE_{amb}$, $CO_{2amb}$) and ($RUE_{hi}$, $CO_{2hi}$):

$r1=1n[\frac{CO_{2amb}}{(0.01*RUE_{amb})}-CO_{2amb}]+r_2*CO_{2amb}$ 5:2.1.5

$r_2=\frac{(1n[\frac{CO_{2amb}}{(0.01*RUE_{amb})}-CO_{2amb}]-1n[\frac{CO_{2hi}}{(0.01*RUE_{hi})}-CO_{2hi}])}{CO_{2hi}-CO_{2amb}}$ 5:2.1.6

where $r1$ is the first shape coefficient, $r2$ is the second shape coefficient, $CO_{2amb}$ is the ambient atmospheric $CO_2$ concentration (ppmv), $RUE_{amb}$ is the radiation-use efficiency of the plant at ambient atmospheric $CO_2$ concentration (kg/ha⋅(MJ/m$^2$)$^{-1}$ or 10$^{-1}$ g/MJ), $CO_{2hi}$ is an elevated atmospheric $CO_2$ concentration (ppmv), $RUE_{hi}$ is the radiation-use efficiency of the plant at the elevated atmospheric $CO_2$ concentration, $CO_{2hi}$, (kg/ha⋅(MJ/m$^2$)$^{-1}$ or 10$^{-1}$ g/MJ). Equation 5:2.1.4 was developed when the ambient atmospheric $CO_2$ concentration was 330 ppmv and is valid for carbon dioxide concentrations in the range 330-660 ppmv. Even though the ambient atmospheric concentration of carbon dioxide is now higher than 330 ppmv, this value is still used in the calculation. If the $CO_2$ concentration used in the simulation is less than 330 ppmv, the model defines RUE = $RUE_{amb}$.

Stockle and Kiniry (1990) have shown that a plant’s radiation-use efficiency is affected by vapor pressure deficit. For a plant, a threshold vapor pressure deficit is defined at which the plant’s radiation-use efficiency begins to drop in response to the vapor pressure deficit. The adjusted radiation-use efficiency is calculated:

$RUE=RUE_{vpd=1}-\Delta rue_{dcl}*(vpd-vpd_{thr})$ if $vpd>vpd_{thr}$ 5:2.1.7

$RUE=RUE_{vpd=1}$ if $vpd \le vpd_{thr}$ 5:2.1.8

where $RUE$ is the radiation-use efficiency adjusted for vapor pressure deficit (kg/ha⋅(MJ/m$^2$)$^{-1}$ or 10$^{-1}$ g/MJ), $RUE_{vpd=1}$ is the radiation-use efficiency for the plant at a vapor pressure deficit of 1 kPa (kg/ha⋅(MJ/m$^2$)$^{-1}$ or 10$^{-1}$ g/MJ), $\Delta rue_{dcl}$ is the rate of decline in radiation-use efficiency per unit increase in vapor pressure deficit (kg/ha⋅(MJ/m$^2$)$^{-1}$⋅kPa$^{-1}$ or (10$^{-1}$ g/MJ)⋅kPa$^{-1}$), $vpd$ is the vapor pressure deficit (kPa), and $vpd_{thr}$ is the threshold vapor pressure deficit above which a plant will exhibit reduced radiation-use efficiency (kPa). The radiation-use efficiency value reported for the plant in the plant growth database, $RUE_{amb}$, or adjusted for elevated carbon dioxide levels (equation 5:2.1.4) is the value used for $RUE_{vpd=1}$. The threshold vapor pressure deficit for reduced radiation-use efficiency is assumed to be 1.0 kPa for all plants ($vpd_{thr}=1.0$).

The radiation-use efficiency is never allowed to fall below 27% of $RUE_{amb}$. This minimum value was based on field observations (Kiniry, personal communication, 2001).

For each day of simulation, potential plant growth, i.e. plant growth under ideal growing conditions, is calculated. Ideal growing conditions consist of adequate water and nutrient supply and a favorable climate. Differences in growth between plant species are defined by the parameters contained in the plant growth database.

Plant nitrogen uptake is controlled by the plant nitrogen equation. The plant nitrogen equation calculates the fraction of nitrogen in the plant biomass as a function of growth stage given optimal growing conditions.

$fr_N=(fr_{N,1}-fr_{N,3})*[1-\frac{fr_{PHU}}{fr_{PHU}+exp(n_1-n_2*fr_{PHU})}]+fr_{N,3}$ 5:2.3.1

where $fr_N$ is the fraction of nitrogen in the plant biomass on a given day, $fr_{N,1}$ is the normal fraction of nitrogen in the plant biomass at emergence, $fr_{N,3}$ is the normal fraction of nitrogen in the plant biomass at maturity, $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on a given day in the growing season, and $n_1$ and $n_2$ are shape coefficients.

The shape coefficients are calculated by solving equation 5:2.3.1 using two known points ($fr_{N,2}$, $fr_{PHU,50\%}$ ) and ($fr_{N,3}$, $fr_{PHU,100\%}$):

$n_1=1n[\frac{fr_{PHU,50\%}}{(1-\frac{(fr_{N,2}-fr_{N,3})}{(fr_{N,1}-fr_{N,3})})}-fr_{PHU,50\%}]+n_2*fr_{PHU,50\%}$ 5:2.3.2

$n_2=\frac{(1n[\frac{fr_{PHU,50\%}}{(1-\frac{(fr_{N,2}-fr_{N,3})}{(fr_{N,1}-fr_{N,3})})}-fr_{PHU,50\%}]-1n[\frac{fr_{PHU,100\%}}{(1-\frac{(fr_{N,\sim3}-fr_{N,3})}{(fr_{N,1}-fr_{N,3})})}-fr_{PHU,100\%}])}{fr_{PHU,100\%}-fr_{PHU,50\%}}$ 5:2.3.3

where $n_1$ is the first shape coefficient, $n_2$ is the second shape coefficient, $fr_{N,1}$ is the normal fraction of nitrogen in the plant biomass at emergence, $fr_{N,2}$ is the normal fraction of nitrogen in the plant biomass at 50% maturity, $fr_{N,3}$ is the normal fraction of nitrogen in the plant biomass at maturity, $fr_{N,\sim 3}$ is the normal fraction of nitrogen in the plant biomass near maturity, $fr_{PHU,50\%}$ is the fraction of potential heat units accumulated for the plant at 50% maturity ($fr_{PHU,50\%}$=0.5), and $fr_{PHU,100\%}$ is the fraction of potential heat units accumulated for the plant at maturity ($fr_{PHU,100\%}$=1.0). The normal fraction of nitrogen in the plant biomass near maturity ($fr_{N,\sim 3}$) is used in equation 5:2.3.3 to ensure that the denominator term $(1-\frac{(fr_{N,\sim3}-fr_{N,3})}{(fr_{N,1}-fr_{N,3})})$ does not equal 1. The model assumes $(fr_{N,\sim 3}-fr_{N,3})=0.00001$

To determine the mass of nitrogen that should be stored in the plant biomass on a given day, the nitrogen fraction is multiplied by the total plant biomass:

$bio_{N,opt}=fr_N*bio$ 5:2.3.4

where $bio_{N,opt}$ is the optimal mass of nitrogen stored in plant material for the current growth stage (kg N/ha), $fr_N$ is the optimal fraction of nitrogen in the plant biomass for the current growth stage, and $bio$ is the total plant biomass on a given day (kg ha$^{-1}$).

Originally, SWAT+ calculated the plant nitrogen demand for a given day by taking the difference between the nitrogen content of the plant biomass expected for the plant’s growth stage and the actual nitrogen content $N_{up}=bio_{N,opt}-bio_N$. This method was found to calculate an excessive nitrogen demand immediately after a cutting (i.e. harvest operation). The equation used to calculate plant nitrogen demand is now

$N_{up}=Min \begin{cases} bio_{N,opt}-bio_N \\ 4*fr_{N,3}* \Delta bio \end {cases}$ 5:2.3.5

where $N_{up}$ is the potential nitrogen uptake (kg N/ha), $bio_{N,opt}$ is the optimal mass of nitrogen stored in plant material for the current growth stage (kg N/ha), $bio_N$ is the actual mass of nitrogen stored in plant material (kg N/ha), $fr_{N,3}$ is the normal fraction of nitrogen in the plant biomass at maturity, and $\Delta bio$ is the potential increase in total plant biomass on a given day (kg/ha).

The depth distribution of nitrogen uptake is calculated with the function:

$N_{up,z}=\frac{N_{up}}{[1-exp(-\beta_n)]}*[1-exp(-\beta_n*\frac{z}{z_{root}})]$ 5:2.3.6

where $N_{up,z}$ is the potential nitrogen uptake from the soil surface to depth $z$ (kg N/ha), $N_{up}$ is the potential nitrogen uptake (kg N/ha), $\beta_n$ is the nitrogen uptake distribution parameter, $z$ is the depth from the soil surface (mm), and $z_{root}$ is the depth of root development in the soil (mm). Note that equation 5:2.3.6 is similar in form to the depth distribution for water uptake described by equation 5:2.2.1. The potential nitrogen uptake for a soil layer is calculated by solving equation 5:2.3.6 for the depth at the upper and lower boundaries of the soil layer and taking the difference.

$N_{up,ly}=N_{up,zl}-N_{up,zu}$ 5:2.3.7

where $N_{up,ly}$ is the potential nitrogen uptake for layer $ly$ (kg N/ha), $N_{up,zl}$ is the potential nitrogen uptake from the soil surface to the lower boundary of the soil layer (kg N/ha), and $N_{up,zu}$ is the potential nitrogen uptake from the soil surface to the upper boundary of the soil layer (kg N/ha).

Root density is greatest near the surface, and nitrogen uptake in the upper portion of the soil will be greater than in the lower portion. The depth distribution of nitrogen uptake is controlled by $\beta_n$, the nitrogen uptake distribution parameter, a variable users are allowed to adjust. Figure 5:2-4 illustrates nitrogen uptake as a function of depth for four different uptake distribution parameter values.

Nitrogen removed from the soil by plants is taken from the nitrate pool. The importance of the nitrogen uptake distribution parameter lies in its control over the maximum amount of nitrate removed from the upper layers. Because the top 10 mm of the soil profile interacts with surface runoff, the nitrogen uptake distribution parameter will influence the amount of nitrate available for transport in surface runoff. The model allows lower layers in the root zone to fully compensate for lack of nitrate in the upper layers, so there should not be significant changes in nitrogen stress with variation in the value used for $\beta_n$.

The actual amount if nitrogen removed from a soil layer is calculated:

$N_{actualup,ly}=min\lfloor N_{up,ly} +N_{demand},NO3_{ly}\rfloor$ 5:2.3.8

where $N_{actualup,ly}$ is the actual nitrogen uptake for layer $ly$ (kg N/ha), $N_{up,ly}$ is the potential nitrogen uptake for layer $ly$ (kg N/ha), $N_{demand}$ is the nitrogen uptake demand not met by overlying soil layers (kg N/ha), and $NO3_{ly}$ is the nitrate content of soil layer $ly$ (kg NO$_3$-N/ha).

The potential water uptake from the soil surface to any depth in the root zone is estimated with the function:

$w_{up,z}=\frac{E_t}{[1-exp(-\beta_w)]}*[1-exp(-\beta_w*\frac{z}{z_{root}})]$ 5:2.2.1

where $w_{up,z}$ is the potential water uptake from the soil surface to a specified depth, $z$, on a given day (mm H$_2$O), $E_t$ is the maximum plant transpiration on a given day (mm H$_2$O), $\beta_w$ is the water-use distribution parameter, $z$ is the depth from the soil surface (mm), and $z_{root}$ is the depth of root development in the soil (mm). The potential water uptake from any soil layer can be calculated by solving equation 5:2.2.1 for the depth at the top and bottom of the soil layer and taking the difference.

$w_{up,ly}=w_{up,zl}-w_{up,zu}$ 5:2.2.2

where $w_{up,ly}$ is the potential water uptake for layer $ly$ (mm H$_2$O), $w_{up,zl}$ is the potential water uptake for the profile to the lower boundary of the soil layer (mm H$_2$O), and $w_{up,zu}$ is the potential water uptake for the profile to the upper boundary of the soil layer (mm H$_2$O).

Since root density is greatest near the soil surface and decreases with depth, the water uptake from the upper layers is assumed to be much greater than that in the lower layers. The water-use distribution parameter, $\beta_w$, is set to 10 in SWAT+. With this value, 50% of the water uptake will occur in the upper 6% of the root zone. Figure 5:2-3 graphically displays the uptake of water at different depths in the root zone.

The amount of water uptake that occurs on a given day is a function of the amount of water required by the plant for transpiration, $E_t$, and the amount of water available in the soil, $SW$. Equations 5:2.2.1 and 5:2.2.2 calculate potential water uptake solely as a function of water demand for transpiration and the depth distribution defined in equation 5:2.2.1. SWAT+ modifies the initial potential water uptake from a given soil layer to reflect soil water availability in the following ways.

As the heat unit theory was proven to be a reliable predictor of harvest dates for all types of crops, it was adapted by researchers for prediction of the timing of other plant development stages such as flowering (Cross and Zuber, 1972). The successful adaptation of heat units to predict the timing of plant stages has subsequently led to the use of heat units to schedule management operations.

SWAT+ allows management operations to be scheduled by day or by fraction of potential heat units. For each operation the model checks to see if a month and day has been specified for timing of the operation. If this information is provided, SWAT+ will perform the operation on that month and day. If the month and day are not specified, the model requires a fraction of potential heat units to be specified. As a general rule, if exact dates are available for scheduling operations, these dates should be used.

Scheduling by heat units allows the model to time operations as a function of temperature. This method of timing is useful for several situations. When very large watersheds are being simulated where the climate in one portion of the watershed is different enough from the climate in another section of the watershed to affect timing of operations, heat unit scheduling may be beneficial. By using heat unit scheduling, only one generic management file has to be made for a given land use. This generic set of operations can then be used wherever the land use is found in the watershed. Also, in areas where the climate can vary greatly from year to year, heat unit scheduling will allow the model to adjust the timing of operations to the weather conditions for each year.

While scheduling by heat units is convenient, there are some negatives to using this type of scheduling that users need to take into consideration. In the real world, applications of fertilizer or pesticide are generally not scheduled on a rainy day. However when applications are scheduled by heat units, the user has no knowledge of whether or not the heat unit fraction that triggers the application will occur on a day with rainfall or not. If they do coincide, there will be a significant amount of the applied material transported with surface runoff (assuming runoff is generated on that day), much higher than if the application took place even one day prior to the rainfall event.

To schedule by heat units, the timing of the operations are expressed as fractions of the potential heat units for the plant or fraction of maturity. Let us use the following example for corn in Indiana.

The number of heat units accumulated for the different operation timings is calculated by summing the heat units for every day starting with the planting date (May 15) and ending with the day the operation takes place. To calculate the fraction of $PHU$ at which the operation takes place, the heat units accumulated is divided by the $PHU$ for the crop (1456).

Note that the fraction of $PHU$ for the harvest operation is 1.16. The fraction is greater than 1.0 because corn is allowed to dry down prior to harvesting. The model will simulate plant growth until the crop reaches maturity (where maturity is defined as $PHU$ = 1456). From that point on, plants will not transpire or take up nutrients and water. They will stand in the HRU until converted to residue or harvested.

While the operations after planting have been scheduled by fraction of $PHU$, operations—including planting—which occur during periods when no crop is growing must still be scheduled. To schedule these operations, SWAT+ keeps track of a second heat index where heat units are summed over the entire year using $T_{base}$ = 0°C. This heat index is solely a function of the climate and is termed the base zero heat index. For the base zero index, the heat units accumulated on a given day are:

$HU_0=\overline T_{av}$ when $\overline T_{av} >$0°C 5:1.1.3

where $HU_0$ is the number of base zero heat units accumulated on a given day (heat units), and $\overline T_{av}$ is the mean daily temperature (°C). The total number of heat units for the year is calculated:

$PHU_0=\sum_{d=1}^{365} HU_0$ 5:1.1.4

where $PHU_0$ is the total base zero heat units (heat units), $HU_0$ is the number of base zero heat units accumulated on day $d$ where $d=1$ on January 1 and 365 on December 31. Unlike the plant $PHU$ which must be provided by the user, $PHU_0$ is the average calculated by SWAT+ using long-term weather data provided in the .wgn file.

For the example watershed in Indiana, $PHU_0$= 4050. The heat unit fractions for the remaining operations are calculated using this value for potential heat units.

As stated previously, SWAT+ always keeps track of base zero heat units. The base zero heat unit scheduling is used any time there are no plants growing in the HRU (before and including the plant operation and after the kill operation). Once plant growth is initiated, the model switches to plant heat unit scheduling until the plant is killed.

The following heat unit fractions have been found to provide reasonable timings for the specified operations:

Table 5:1-1: SWAT+ input variables that pertain to heat units.

Plant growth may be reduced due to extreme temperatures, and insufficient water, nitrogen or phosphorus. The amount of stress for each of these four parameters is calculated on a daily basis using the equations summarized in the following sections.

Actual growth varies from potential growth due to extreme temperatures, water deficiencies and nutrient deficiencies. This chapter reviews growth constraints as well as overrides that the user may implement to ignore growth constraints.

Temperature stress is a function of the daily average air temperature and the optimal temperature for plant growth. Near the optimal temperature the plant will not experience temperature stress. However as the air temperature diverges from the optimal the plant will begin to experience stress. The equations used to determine temperature stress are:

$tstrs=1$ when $\overline T_{av} \le T_{base}$ 5:3.1.2

$tstrs=1-exp[\frac{-0.1054*(T_{opt}-\overline T_{av})^2}{(\overline T_{av}-T_{base})^2}]$ when $T_{base}<\overline T_{av} \le T_{opt}$ 5:3.1.3

$tstrs=1-exp[\frac{-0.1054*(T_{opt}-\overline T_{av})^2}{(2*T_{opt}-\overline T_{av}-T_{base})^2}]$ when $T_{opt}<\overline T_{av}\le 2*T_{opt}-T_{base}$ 5:3.1.4

$tstrs=1$ when $\overline T_{av} > 2*T_{opt}-T_{base}$ 5:3.1.5

where $tstrs$ is the temperature stress for a given day expressed as a fraction of optimal plant growth,$\overline T_{av}$is the mean air temperature for day (°C), $T_{base}$ is the plant’s base or minimum temperature for growth (°C), and $T_{opt}$ is the plant’s optimal temperature for growth (°C). Figure 5:3-1 illustrates the impact of mean daily air temperature on plant growth for a plant with a base temperature of 0°C and an optimal temperature of 15°C.

If upper layers in the soil profile do not contain enough water to meet the potential water uptake calculated with equation 5:2.2.2, users may allow lower layers to compensate. The equation used to calculate the adjusted potential water uptake is:

$w'_{up,ly}=w_{up,ly}+w_{demand}*epco$ 5:2.2.3

where $w'_{up,ly}$is the adjusted potential water uptake for layer $ly$ (mm H$_2$O), $w_{up,ly}$ is the potential water uptake for layer $ly$ calculated with equation 5:2.2.2 (mm H$_2$O), $w_{demand}$ is the water uptake demand not met by overlying soil layers (mm H$_2$O), and $epco$ is the plant uptake compensation factor. The plant uptake compensation factor can range from 0.01 to 1.00 and is set by the user. As $epco$ approaches 1.0, the model allows more of the water uptake demand to be met by lower layers in the soil. As $epco$ approaches 0.0, the model allows less variation from the depth distribution described by equation 5:2.2.1 to take place.

As the water content of the soil decreases, the water in the soil is held more and more tightly by the soil particles and it becomes increasingly difficult for the plant to extract water from the soil. To reflect the decrease in the efficiency of the plant in extracting water from dryer soils, the potential water uptake is modified using the following equations:

$w''_{up,ly}=w'_{up,ly}*exp[5*(\frac{SW_{ly}}{(.25*AWC_{ly})}-1)]$ when $SW_{ly}< (.25*AWC_{ly})$ 5:2.2.4

$w''_{up,ly}=w'_{up,ly}$ when $SW_{ly} \ge (.25*AWC_{ly})$ 5:2.2.5

where $w''_{up,ly}$is the potential water uptake adjusted for initial soil water content(mm H$_2$O), $w'_{up,ly}$ is the adjusted potential water uptake for layer $ly$ (mm H$_2$O), $SW_{ly}$ is the amount of water in the soil layer on a given day (mm H$_2$O), and $AWC_{ly}$ is the available water capacity for layer $ly$ (mm H$_2$O). The available water capacity is calculated:

$AWC_{ly}=FC_{ly}-WP_{ly}$ 5:2.2.6

where $AWC_{ly}$ is the available water capacity for layer $ly$ (mm H$_2$O), $FC_{ly}$ is the water content of layer $ly$ at field capacity (mm H$_2$O), and $WP_{ly}$ is the water content of layer $ly$ at wilting point (mm H$_2$O).

SWAT+ monitors plant uptake of nitrogen and phosphorus.

When a harvest or harvest/kill operation is performed, a portion of the plant biomass is removed from the HRU as yield. The nutrients and plant material contained in the yield are assumed to be lost from the system (i.e. the watershed) and will not be added to residue and organic nutrient pools in the soil with the remainder of the plant material. In contrast, a kill operation converts all biomass to residue.

The fraction of the above-ground plant dry biomass removed as dry economic yield is called the harvest index. For the majority of crops, the harvest index will be between 0.0 and 1.0. However, plants whose roots are harvested, such as sweet potatoes, may have a harvest index greater than 1.0.

The economic yield of most commercial crops is the reproductive portion of the plant. Decades of crop breeding have lead to cultivars and hybrids having maximized harvest indices. Often, the harvest index is relatively stable across a range of environmental conditions.

SWAT+ calculates harvest index each day of the plant’s growing season using the relationship:

$HI=HI_{opt}*\frac{100*fr_{PHU}}{(100*fr_{PHU}+exp[11.1-10*fr_{PHU}])}$ 5:2.4.1