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).

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.

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).