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).

Plant maturity is reached when the fraction of potential heat units accumulated, , 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.

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:

$bio_{annual}=1000*(\frac{yr_{cur}}{yr_{fulldev}})*bio_{fulldev}$ 5:2.1.9

where $bio_{annual}$ is the amount of biomass a tree can accumulate in a single year (kg/ha), $yr_{cur}$ is the current age of the tree (years), $yr_{fulldev}$ is the number of years for the tree species to reach full development (years), $bio_{fulldev}$ 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, $bio$, reaches the annual limit, $bio_{fulldev}$, 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,$\Delta bio_i$, 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:

$fr_{root}=0.40-0.20*fr_{PHU}$ 5:2.1.21

where $fr_{root}$ is the fraction of total biomass partitioned to roots on a given day in the growing season, and $fr_{PHU}$ 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:

$z_{root}=z_{root,mx}$ 5:2.1.22

where $z_{root}$ is the depth of root development in the soil on a given day (mm), and $z_{root,mx}$ 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 $fr_{PHU}$= 0.40 using the equation:

$z_{root}=2.5*fr_{PHU}*z_{root,mx}$ if $fr_{PHU} \le 0.40$ 5:2.1.23

$z_{root}=z_{root,mx}$ if $fr_{PHU} > 0.40$ 5:2.1.24

where $z_{root}$ is the depth of root development in the soil on a given day (mm), $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on a given day in the growing season, and $z_{root,mx}$ 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 $z_{root,mx}$.

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).

Radiation-use efficiency is sensitive to variations in atmospheric 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 is (Stockle et al., 1992):

5:2.1.4

where is the radiation-use efficiency of the plant (kg/ha⋅(MJ/m) or 10 g/MJ), is the concentration of carbon dioxide in the atmosphere (ppmv), and and are shape coefficients.

The shape coefficients are calculated by solving equation 5:2.1.4 using two known points (, ) and (, ):

5:2.1.5

5:2.1.6

where is the first shape coefficient, is the second shape coefficient, is the ambient atmospheric concentration (ppmv), is the radiation-use efficiency of the plant at ambient atmospheric concentration (kg/ha⋅(MJ/m) or 10 g/MJ), is an elevated atmospheric concentration (ppmv), is the radiation-use efficiency of the plant at the elevated atmospheric concentration, , (kg/ha⋅(MJ/m) or 10 g/MJ). Equation 5:2.1.4 was developed when the ambient atmospheric 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 concentration used in the simulation is less than 330 ppmv, the model defines RUE = .

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:

if 5:2.1.7

if 5:2.1.8

where is the radiation-use efficiency adjusted for vapor pressure deficit (kg/ha⋅(MJ/m) or 10 g/MJ), is the radiation-use efficiency for the plant at a vapor pressure deficit of 1 kPa (kg/ha⋅(MJ/m) or 10 g/MJ), is the rate of decline in radiation-use efficiency per unit increase in vapor pressure deficit (kg/ha⋅(MJ/m)⋅kPa or (10 g/MJ)⋅kPa), is the vapor pressure deficit (kPa), and 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, , or adjusted for elevated carbon dioxide levels (equation 5:2.1.4) is the value used for . The threshold vapor pressure deficit for reduced radiation-use efficiency is assumed to be 1.0 kPa for all plants ().

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

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.