# Biomass accumulation (Photosynthesis) The daily biomass accumulation ($\Delta Q$) corresponds to dry-matter above-ground biomass, and is calculated as a potential biomass accumulation resulting from radiation interception ($\Delta Q_{r}$, Equation \@ref(eq:BiomassProduction)) that is limited by soil water deficiency ($\Delta Q_{w}$, Equation \@ref(eq:WaterStressBiomassProduction-1)). ## Potential biomass accumulation from radiation use efficiency\label{subsec:Radiation-limited-biomass} The radiation-limited dry-biomass accumulation ($\Delta Q_{r}$) is calculated by the intercepted radiation ($I$), radiation use efficiency ($RUE$), diffuse factor ($f_{d}$, \autoref{par:Diffuse-factor}), stress factor ($f_{s}$, Equation \@ref(eq:StressFactor4Photosynthesis)) and carbon dioxide factor ($f_{c}$, Equation \@ref(eq:CO2Factor4Photosynthesis)). \begin{equation} \Delta Q_{r}=I\times RUE\times f_{d}\times f_{s}\times f_{c} (\#eq:BiomassProduction) \end{equation} where $f_{d}$, $f_{s}$ and $f_{c}$ are defined in the wheat.xml file. In the current version of APSIM-Wheat, only `Leaf` produces photosynthate. Diffuse factor ($f_{d}$) equals to 1 (\autoref{par:Diffuse-factor}), so that Equation \@ref(eq:BiomassProduction) can be: \begin{equation} \Delta Q_{r}=I\times RUE\times f_{s}\times f_{c} (\#eq:BiomassProduction2) \end{equation} ### Radiation interception Radiation interception is calculated from the leaf area index (LAI, m$^{2}$ m$^{-2}$) and the extinction coefficient (\textit{k}) \citep{monsi2005onthe}. \begin{equation} I=I_{0}(1-\exp(-k\times LAI\times f_{h})/f_{h}) (\#eq:RadiationInterception) \end{equation} where $I_{0}$ is the total radiation at the top of the canopy (MJ) which is directly imported from weather records; $f_{h}$ is light interception modified to give hedge-row effect with skip row. $f_{h}$ could be calculated based on the canopy width, but is not used in the current version of APSIM (i.e. $f_{h}$ = 1). So, Equation \@ref(eq:RadiationInterception) is reduced to. \begin{equation} I=I_{0}(1-\exp(-k\times LAI)) (\#eq:RadiationInterception-1) \end{equation} Extinction coefficient ($k$) varies with row spacing, \begin{equation} k=h_{e}(W_{r}) \end{equation} where $W_{r}$ is the row spacing which is specified by the user (in the APSIM interface, the .sim or .apsim file); $h_{e}$ is a function of rowing spacing which is defined for both green leaf and dead leaves by parameters `x_row_spacing`, `y_extinct_coef` in the wheat.xml file (Fig. \@ref(fig:wdRowExtinct)) and is linearly interpolated by APSIM. In the current version of APSIM-Wheat, no impact of row spacing is considered (Fig. \@ref(fig:wdRowExtinct)) ```{r wdRowExtinct,fig.cap='Values of extinction coefficient for different row spacings.' } p <- wdVisXY(wheat_xml, "x_row_spacing", c("y_extinct_coef", 'y_extinct_coef_dead'), xlab = 'Row spacing (mm)', ylab = 'Extinction coefficient (k)', keylab = c('Green leaf', 'Dead leaf'), keypos = c(0.9, 0.5)) print(p) ``` ### Radiation use efficiency $RUE$ (g MJ$^{\text{-1}}$) is a function of growth stages which is defined by parameters `x_stage_rue` and `y_rue` in wheat.xml (Fig. \@ref(fig:wdRUE)) and linearly interpolated by APSIM. In the current version of APSIM-Wheat, $RUE$ equal to 1.24 from emergence to the end of grain-filling and does not vary as a function of daily incident radiation as in the model NWHEAT. ```{r wdRUE,fig.cap='Radiation use efficiency (RUE) for different growth stages.' } p <- wdVisXY(wheat_xml, "x_stage_rue", "y_rue", xlab = 'Stage code', ylab = 'RUE') print(p) ``` \subsubsection{Stress factor (Temperature, nitrogen, phosphorus (not applied), oxygen (not applied))} Actual daily radiation-limited biomass accumulation can be reduced by a stress factor ($f_{s}$, Equation \@ref(eq:BiomassProduction) and Equation \@ref(eq:BiomassProduction2)). This stress factor is the minimum value of a temperature factor ($f_{T,\ photo}$, Equation \@ref(eq:TemStressPhoto)), a nitrogen factor ($f_{N\ photo}$, Equation \@ref(eq:NStressPhoto)), a phosphorus factor ($f_{P\ photo}$) and an oxygen factor ($f_{O\ photo}$). \begin{equation} f_{s}=\min(f_{T,\ photo},\ f_{N,\ photo},\ f_{P,\ photo},\ f_{O,\ photo}) (\#eq:StressFactor4Photosynthesis) \end{equation} No phosphorus stress $f_{P,\,photo}$ and oxygen stress $f_{O,\,photo}$ are applied in the current version of APSIM-Wheat. So, Equation \@ref(eq:StressFactor4Photosynthesis) is reduced to \begin{equation} f_{s}=\min(f_{T,\ photo},\ f_{N,\ photo}) (\#eq:StressFactor4Photosynthesis2) \end{equation} \paragraph{The temperature factor} $f_{T,\ photo}$ is a function of the daily mean temperature and is defined by parameters `x_ave_temp` and `y_stress_photo` in the wheat.xml (Fig. \@ref(fig:wdTemperatureFactorOnPhoto)). Values are linearly interpolated by APSIM. The temperature stress is applied from sowing to harvest. \begin{equation} f_{T,\ photo}=h_{T,\ photo}(\frac{T_{max}+T_{min}}{2}) (\#eq:TemStressPhoto) \end{equation} ```{r wdTemperatureFactorOnPhoto,fig.cap='Temperature factor in response to mean daily temperature.' } p <- wdVisXY(wheat_xml, "x_ave_temp", "y_stress_photo", xlab = expression(paste("Mean daily temperature", ~"("*degree*"C)")), ylab = expression(Temperature~factor~(f[T]))) print(p) ``` \paragraph{The nitrogen factor} $f_{N,\,photo}$ is determined by the difference between leaf nitrogen concentration and leaf minimum and critical nitrogen concentration. \begin{equation} f_{N,\,photo}=R_{N,\,photo}\sum_{leaf}\frac{C_{N}-C_{N,\,min}}{C_{N,\,crit}-C_{N,\,min}} (\#eq:NStressPhoto0) \end{equation} where $C_{N}$ is the nitrogen concentration of `Leaf` parts; $R_{N,\,expan}$ is multiplier for nitrogen deficit effect on phenology which is specified by `N_fact_photo` in the wheat.xml and default value is 1.5. \paragraph{The CO$_{\text{2}}$ factor} For C3 plants (like wheat), the CO$_{\text{2}}$ factor of APSIM is calculated by a function of environmental CO$_{\text{2}}$ concentration ($C$, ppm) and daily mean temperature ($T_{mean}$) as published by \citet{reyenga1999modelling} \begin{equation} f_{c}=\frac{(C-C_{i})(350+2C_{i})}{(C+2C_{i})(350-C_{i})} (\#eq:CO2Factor4Photosynthesis) \end{equation} where $C_{i}$ is the temperature dependent CO$_{\text{2}}$ compensation point (ppm) and is derived from the following function. \begin{equation} C_{i}=\frac{163-T_{mean}}{5-0.1T_{mean}} \end{equation} ```{r wdCardonDioxideFactor,fig.cap='CO$_{2}$ factor in response to the CO$_{2}$ level ($C$) for different mean air temperatures.' } p <- wdCarbonDioxideFactor() print(p) ``` \paragraph{Diffuse factor (not used in the current version)\label{par:Diffuse-factor}} The daily diffuse fraction was calculated using the functions suggested by \citet{roderick1999estimating}: \begin{equation} \begin{cases} \frac{R_{d}}{R_{s}}=Y_{0} & \qquad for\:\frac{R_{s}}{R_{o}}\leq X_{0}\\ \frac{R_{d}}{R_{s}}=A_{0}+A_{1}\frac{R_{s}}{R_{o}} & \qquad for\:X_{0}<\frac{R_{s}}{R_{o}}\leq X_{1}\\ \frac{R_{d}}{R_{s}}=Y_{1} & \qquad for\:\frac{R_{s}}{R_{o}}>X_{1} \end{cases} (\#eq:DiffuseFraction) \end{equation} where \begin{equation} \begin{array}{c} A_{0}=Y_{1}-A_{1}X_{1}\\ A_{1}=\frac{Y_{1}-Y_{0}}{X_{1}-X_{0}} \end{array} \end{equation} where $R_{o}$ is the daily extra-terrestrial solar irradiance (i.e. top of the atmosphere); $R_{d}$ and $R_{s}$ are the daily diffuse and global solar irradiance at the surface, respectively. $X_{0}$, $X_{1}$, $Y_{0}$ and $Y_{1}$ are four empirical parameters. \begin{equation} \begin{array}{l} X_{0}=0.26,\qquad Y_{0}=0.96,\qquad Y_{1}=0.05,\;and\\ X_{1}=0.80-0.0017|\varphi|+0.000044|\varphi|^{2} \end{array} \end{equation} where $\varphi$ is latitude. $R_{o}$ is derived from this function \begin{equation} R_{0}=\frac{86400\times1360\times(\varpi\times\sin(\varphi)\times\sin(\theta)+\cos(\varphi)\times\cos(\theta)\times\sin(\varpi_{0}))}{1000000\pi} \end{equation} where $\varpi_{0}$ is the time of sunrise and sunset, which derives from any solar declination ($\theta$) and latitude ($\varphi$) in terms of local solar time when sunrise and sunset actually occur (\url{http://en.wikipedia.org/wiki/Sunrise_equation}) \begin{equation} \varpi_{0}=\arccos(-\tan(\varphi)\tan(\theta)) \end{equation} Solar declination ($\theta$) can be calculated by \begin{equation} \theta=23.45\sin(\frac{2\pi}{365.25}(N-82.25)) \end{equation} where $N$ is day of year. $f_{d}$ is calculated by a function of the diffuse fraction which is not implemented in current wheat module, (i.e. $f_{d}$ = 1). ## Actual daily biomass accumulation The actual daily biomass accumulation ($\Delta Q$) results from water limitation applied on the potential radiation-driven biomass accumulation ($\Delta Q_{r}$). This water-limited biomass ($\Delta Q_{w}$) is a function of the ratio between the daily water uptake ($W_{u}$, Equation \@ref(eq:WaterUpdate)) and demand ($W_{d}$, Equation \@ref(eq:soilWaterDemand-1)) capped by \begin{equation} \Delta Q_{w}=\Delta Q_{r}f_{w,\,photo}=\Delta Q_{r}\frac{W_{u}}{W_{d}} (\#eq:WaterStressBiomassProduction-1) \end{equation} where $f_{w,\,photo}$ is the water stress factor affecting photosynthesis (Equation \@ref(eq:swstressphoto)); $W_{u}$ is the actual daily water uptake from the root system (which corresponds to the soil water supply ($W_{s}$) capped by $W_{d}$), $W_{d}$ is the soil water demand of Leaf and Head parts (\autoref{sec:Crop-Water-Relations}). When the soil water is non-limiting ($f_{w,\,photo}$ = 1, i.e. $W_{d}\geq W_{s}$), biomass accumulation is limited by the radiation ($\Delta Q=\Delta Q_{r}$, Equation \@ref(eq:actualBiomassProduction)). When the soil water is limiting, biomass accumulation is limited by water supply ($\Delta Q=\Delta Q_{w}$). The water demand ($W_{d}$, in mm) corresponds to the amount of water the crop would have transpired in the absence of soil water constraint, and is calculated from the potential biomass accumulation from RUE ($\Delta Q_{r}$, Equation \@ref(eq:BiomassProduction)). Following \citet{sinclair1986waterand}, transpiration demand is modeled as a function of the current day's crop growth rate, estimated by the potential biomass accumulation associated with intercepted radiation ($\Delta Q_{r}$, see Equation \@ref(eq:BiomassProduction)), divided by the transpiration efficiency. \begin{equation} W_{d}=\frac{\Delta Q_{r}-R}{TE} (\#eq:soilWaterDemand-1) \end{equation} where $R$ is respiration rate and equals to zero in the current version of APSIM-Wheat, $TE$ is transpiration efficiency (Equation \@ref(eq:TranspirationEfficiency)). See \autoref{sec:Crop-Water-Relations} for more details about water demand and supply. \medskip The daily biomass accumulation ($\Delta Q$) corresponds to dry matter above ground biomass is limited by the radiation interception ($\Delta Q_{r}$, Equation \@ref(eq:BiomassProduction)) or by soil water deficiency ($\Delta Q_{w}$, Equation \@ref(eq:WaterStressBiomassProduction)), so that daily biomass accumulation can be expressed as: \begin{equation} \Delta Q=\begin{cases} \Delta Q_{r}\qquad & W_{u}=W_{d}\\ \Delta Q_{w}\qquad & W_{u}