%% LyX 2.3.5.2 created this file. For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[english]{article} \usepackage{courier} \usepackage[latin9]{inputenc} \usepackage{geometry} \geometry{verbose,tmargin=2cm,bmargin=2cm,lmargin=2cm,rmargin=2cm} \usepackage{babel} \usepackage{array} \usepackage{verbatim} \usepackage{longtable} \usepackage{url} \usepackage{amsmath} \usepackage{graphicx} \usepackage[authoryear]{natbib} \usepackage{subscript} \usepackage[unicode=true,pdfusetitle, bookmarks=true,bookmarksnumbered=false,bookmarksopen=false, breaklinks=true,pdfborder={0 0 0},pdfborderstyle={},backref=false,colorlinks=false] {hyperref} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. %% Because html converters don't know tabularnewline \providecommand{\tabularnewline}{\\} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage[font={small,it}]{caption} \usepackage{float} \usepackage{pdflscape} \makeatother \begin{document} <>= source('Rcode/wdVisXY.R') source('Rcode/wdFunctions.R') opts_chunk$set(fig.path='figure/', fig.align='center', fig.show='hold', cache=FALSE, echo=FALSE, warning=FALSE, message=FALSE, dev = 'pdf',dpi=300, fig.pos='H',fig.width=4,fig.height=3) options(replace.assign=TRUE,width=90) library(lattice) trellis_default <- trellis.par.get() trellis_default$fontsize$text <- 8 trellis_default$fontsize$points <- 8 trellis.par.set(trellis_default) library(XML) wheat_xml <- xmlInternalTreeParse('wheat.xml') @ \title{The APSIM-Wheat Module (7.5 R3008)} \maketitle This documentation is compiled from the source codes and internal documents of APSIM-Wheat module by Bangyou Zheng (bangyou.zheng@csiro.au), Karine Chenu (karine.chenu@uq.edu.au), Alastair Doherty (alastair.doherty@daff.qld.gov.au) and Scott Chapman (scott.chapman@csiro.au). \tableofcontents{} \newpage{} \section{Scope of the APSIM-Wheat module} The APSIM-Wheat module simulates the wheat growth and development of a wheat crop in a daily time-step on an area basis (per square meter, not per single plant). In this module, the wheat crop Wheat growth and development responds to weather (radiation, temperature), soil water and soil nitrogen, and management practices. The wheat module returns information on its soil water and nitrogen uptake to the soil water and nitrogen modules on a daily basis for reset of these systems. Information on crop cover is also provided to the water balance module for calculation of evaporation rates and runoff. Wheat stover and root residues are 'passed' from wheat to the surface residue and soil nitrogen modules, respectively at the harvest of the wheat crop. The approaches used in modeling crop processes balance the need for a comprehensive description of the observed variation in crop performance over diverse production environments and the need to avoid reductionist approaches of ever-greater complexity with large numbers of parameters that are difficult to measure. A list of the module outputs is provided in the \textquoteleft Wheat module output\textquoteright s section below. Basically the module simulates phenological development, leaf area growth expansion, biomass and N concentration of different crop components (\texttt{Leaf}, \texttt{Stem}, \texttt{Root} and \texttt{Grain}) on a daily basis. It also predicts grain size and grain number. \section{APSIM-Wheat history} APSIM-Wheat has been developed from a combination of the approaches used in previous APSIM wheat modules:\citet{asseng1998useof,asseng1998performance,wang2003thenew,meinke1997improving,meinke1998improving}. The current version of the model is implemented within the APSIM Plant model framework which is currently used for other crops such as grain legumes and canola. Most of the model constants (species-specific) and parameters (cultivar specific) are externalized from the code (wheat.xml file). \section{Phenology} There are 11 phases in APSIM-Wheat module (\autoref{fig:PhenologWheatModule}). The timing of each phase (except from sowing to germination, which is driven by sowing depth and thermal time) is determined by the accumulation of thermal time ($TT$) adjusted for other factors which vary with the phase considered (e.g. vernalisation, photoperiod, N). The length of each phase is determined by a fixed thermal time (\textquoteleft thermal time target\textquoteright ), which is specified by \textquotedblleft tt\_\textless phase\_name\textgreater\textquotedblright{} in wheat.xml. Most parameters of thermal time targets are cultivar-specific. \subsection{Thermal time calculation\label{par:Thermal-time}} The daily thermal time ($\Delta TT$) is calculated from the daily average of maximum and minimum crown temperatures, and is adjusted by genetic and environmental factors. Hence, the duration of phases between emergence and floral initiation is adjusted for photoperiod and vernalisation, using the cultivar-specific parameters ``photoperiod factor'' ($f_{D}$, \autoref{eq:PhotoperiodFactor}) and ``vernalisation factor'' ($f_{V}$, \autoref{eq:VernalisationFactor}). Other environmental factors include soil water stress ($f_{W,\,pheno}$, \autoref{eq:SoilWaterStress}), nitrogen stress ($f_{N,\,pheno}$, \autoref{eq:NitrogenStress}) and phosphorus stress ($f_{P,\,pheno}$, \autoref{sub:Phosphorus-stress}) in all phases except from Sowing to Emergence (See details below), but they are all parametrized to have to effect in the current released APSIM-Wheat. All factors are bound from 0 to 1. \begin{figure}[h] \begin{centering} \includegraphics[width=17cm]{figure/wdWheatPhenology} \par\end{centering} \caption{\label{fig:PhenologWheatModule}Phenology in the APSIM\_Wheat module. Targets are expressed in adjusted thermal time (\autoref{eq:CumThermalTime2}) and are cultivar-specific parameters. The values given for the reference genotype Hartog. } \end{figure} Crown temperatures are simulated according to the original routines in CERES-Wheat and correspond to air temperatures for non-freezing temperatures.\textbf{ }The maximum and minimum crown temperatures ($T_{cmax}$ and $T_{cmin}$) are calculated according to the maximum and minimum air temperature ($T_{max}$ and $T_{min}$), respectively. \begin{equation} T_{cmax}=\begin{cases} 2+T_{max}(0.4+0.0018(H_{snow}-15)^{2}) & \quad T_{max}<0\\ T_{max} & \quad T_{max}\geq0 \end{cases}\label{eq:CrownMaxTemperature} \end{equation} \begin{equation} T_{cmin}=\begin{cases} 2+T_{min}(0.4+0.0018(H_{snow}-15)^{2}) & \quad T_{min}<0\\ T_{min} & \quad T_{min}\geq0 \end{cases}\label{eq:CrownMinTemperature} \end{equation} where $H_{snow}$ is the snow depth (cm). The default value of $H_{snow}$ is set to zero in the source codes (\autoref{fig:wdCrownTemperature}). For more detail information about \autoref{eq:CrownMaxTemperature} and \autoref{eq:CrownMinTemperature}, please see the function\textbf{ }\texttt{CWVernalPhase::vernalisation} in the APSIM code. <>= p <- wdCrownTemperature() print(p) @ The daily crown mean temperature ($T_{c}$) is calculated by the maximum ($T_{cmax}$) and minimum ($T_{cmin}$) crown temperature. \begin{equation} T_{c}=\frac{T_{cmax}+T_{cmin}}{2} \end{equation} Daily thermal time ($\Delta TT$) is calculated based on daily mean crown temperature, using three cardinal temperatures (\autoref{fig:wdThermalTime}). The default values of the cardinal temperatures and relative thermal time are specified by x\_temp (0, 26, 34) and y\_tt (0, 26, 0), respectively, in the wheat.xml (\autoref{fig:wdThermalTime}). Other crop modules in APSIM calculate thermal time every 3 hours. \begin{equation} \Delta TT=\begin{cases} T_{c} & \quad034 \end{cases}\label{eq:thermaltime} \end{equation} <>= p <- wdVisXY(wheat_xml, "x_temp", "y_tt", xlab = expression(paste("Crown Temperature", ~"("*degree*"C)")), ylab = expression(paste("Thermal Time", ~"("*degree*"Cd)"))) print(p) @ For each phenological stage, the daily thermal time ($TT^{\prime}$) is summed from the start of phase and can be reduced by photoperiod ($f_{D}$, \autoref{eq:PhotoperiodFactor}) and vernalisation factor ($f_{V}$, \autoref{eq:VernalisationFactor}) and also dependent on environmental factors (photoperiod and temperature). The environmental factors include soil water stress ($f_{W,\,pheno}$, \autoref{eq:SoilWaterStress}), nitrogen stress ($f_{N,\,pheno}$, \autoref{eq:NitrogenStress}) and phosphorus stress ($f_{P,\,pheno}$, \autoref{sub:Phosphorus-stress}). The next phenological stage occurs when this adjusted thermal time ($TT^{\prime}$ in \autoref{eq:CumThermalTime}) reaches the ``target thermal time'' for the stage considered \autoref{fig:PhenologWheatModule}. \begin{equation} TT^{\prime}=\sum[\Delta TT\times\min(f_{D},\;f_{V})\times\min(f_{W,\,pheno},\:f_{N,\,pheno},\;f_{P,\,pheno})]\label{eq:CumThermalTime} \end{equation} In the current released version, soil water, nitrogen and phosphorus stresses have no effect on phenological development (i.e. parameters $f_{W,\,pheno}=f_{P,\,pheno}=1$ \autoref{eq:SoilWaterStress}, and $f_{N,\,pheno}$ has values typically above 1 \autoref{eq:NitrogenStress}). So, \autoref{eq:CumThermalTime} is reduced to \begin{equation} TT^{\prime}=\sum[\Delta TT\times\min(f_{D},\;f_{V})]\label{eq:CumThermalTime2} \end{equation} In the output variables of wheat module, $TT^{\prime}$ from the start of each phase is named as ``ttafter\textless phasename\textgreater ''. For example, the output variable ``ttaftersowing'' is not the actual thermal time after sowing, but the thermal time adjusted for genetic and environmental factors. \subsection{Sowing-germination phase} The seed germination is determined by soil water availability in the seeded layer (specified by \texttt{pesw\_germ} with default value 0 mm). The crop will die if germination has not occurred before a certain period, defined by \texttt{days\_germ\_limit} in wheat.xml, which has a default value of 40 d. \subsection{Germination-emergence phase} The germination to emergence phase includes an effect of the depth of sowing ($D_{seed}$) on the thermal time target. The phase is comprised of an initial period of fixed thermal time during which shoot elongation is slow (the \textquotedblleft lag\textquotedblright{} phase, $T_{lag}$) and a linear period, where the rate of shoot elongation ($r_{e}$, C d mm$^{-1}$) towards the soil surface is linearly related to air temperature. Then, the period of emergence ($T_{emer}$) is calculated by \begin{equation} T_{emer}=T_{lag}+r_{e}D_{seed}\label{eq:Emergence} \end{equation} The crop will die if emergence has not occurred before a certain period, defined by \texttt{tt\_emerg\_limit} in wheat.xml, which has a default value of 300$^{\circ}\text{C}$ d. Most studies on seedling germination have simply recorded the accumulated thermal time between germination and 50\% emergence from a given sowing depth. For the purposes of model parametrization the value of $T_{lag}$ (\texttt{shoot\_lag}) has been assumed to be around 40 $^{\circ}\text{C}$ d, while $r_{e}$ (\texttt{shoot\_rate}) has been derived from studies where thermal time to emergence was measured and where sowing depth was known and it is set to 1.5 $^{\circ}\text{C}$ d per mm. This means that at a sowing depth of 40 mm emergence occurs 100$^{\circ}\text{C}$ d after germination ($40+1.5\times40$). There is the capability of increasing the time taken to reach emergence due to a dry soil layer in which the seed is germinating, through the relationship between \texttt{fasw\_emerg} and \texttt{rel\_emerg\_rate}. Currently this effect is \textquotedblleft turned off\textquotedblright{} in the Wheat.xml file. \subsection{Photoperiod impact on phenology\label{subsec:Photoperiod}} Photoperiod is calculated from day of year and latitude using standard astronomical equations accounting for civil twilight using the parameter twilight, which is assumed to be -6$^{\circ}$ (civil twilight) in wheat.xml. Twilight is defined as the interval between sunrise or sunset and the time when the true center of the sun is 6$^{\circ}$ below the horizon. Other crop modules of APSIM have used -2.2$^{\circ}$ as twilight parameters. In APSIM, the photoperiod affects phenology between emergence and floral initiation (\autoref{fig:PhenologWheatModule}). During this period, thermal time is affected by a photoperiod factor ($f_{D}$ in \autoref{eq:CumThermalTime} and \autoref{eq:CumThermalTime2}) that is calculated by \begin{equation} f_{D}=1-0.002R_{p}(20-L_{P})^{2}\label{eq:PhotoperiodFactor} \end{equation} where $L_{P}$ is the day length (h), $R_{P}$ is the sensitivities to photoperiod which is cultivar-specific and is specified by \texttt{photop\_sens} in wheat.xml. The default value of $R_{P}$ is 3 (\autoref{fig:wdPhotoperiod}). <>= p <- wdPhotoPeriod() print(p) @ \subsection{Vernalisation impact on phenology} In APSIM, vernalisation effects phenology between emergence and floral initiation (\autoref{fig:PhenologWheatModule}). During this period, thermal time is affected by a vernalisation factor ($f_{V}$ in \autoref{eq:CumThermalTime} and \autoref{eq:CumThermalTime2}). Vernalisation is simulated from daily average crown temperature ($T_{c}$), daily maximum ($T_{max}$) and minimum ($T_{min}$) temperatures using the original CERES approach (\autoref{fig:wdVernalisation}). \begin{equation} \Delta V=\min(1.4-0.0778T_{c},\:0.5+13.44\frac{T_{c}}{(T_{max}-T_{min}+3)^{2}})\quad\text{when, }T_{max}<30\,{}^{\circ}\text{C}\:\text{and}\,T_{min}<15\,{}^{\circ}\text{C} \end{equation} <>= p <- wdVernalisation() print(p) @ Devernalisation can occur if daily $T_{max}$ is above 30 $^{\circ}\text{C}$ and the total vernalisation ($V$) is less than 10 (\autoref{fig:wdDevernalisation}). \begin{equation} \Delta V_{d}=\min(0.5(T_{max}-30),\:V)\quad\text{when, }T_{max}>30\,{}^{\circ}\text{C}\;\text{and}\;V<10 \end{equation} <>= p <- wdDevernalisation() print(p) @ The total vernalisation ($V$) is calculated by summing daily vernalisation and devernalisation from Germination to Floral initiation (Composite phase \texttt{Vernalisation} in \autoref{fig:PhenologWheatModule}). \begin{equation} V=\sum(\Delta V-\Delta V_{d}) \end{equation} However, the vernalisation factor ($f_{v}$) is calculated just from Emergence to Floral initiation (Composite phases \textbf{eme2ej} in Fig. \ref{fig:PhenologWheatModule}). \begin{equation} f_{V}=1-(0.0054545R_{V}+0.0003)\times(50-V)\label{eq:VernalisationFactor} \end{equation} where $R_{V}$ is the sensitivities to vernalisation, which is cultivar-specific and is specified by \texttt{vern\_sens} in wheat.xml. The default value of $R_{V}$ is 1.5 (\autoref{fig:wdVernalisationFactor}) <>= p <- wdVernalisationFactor() print(p) @ \section{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 ($\Delta Q_{r}$, \autoref{eq:BiomassProduction}) that is limited by soil water deficiency ($\Delta Q_{w}$, \autoref{eq:WaterStressBiomassProduction-1}). \subsection{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 ($f_{s}$, \autoref{eq:StressFactor4Photosynthesis}) and carbon dioxide factor ($f_{c}$, \autoref{eq:CO2Factor4Photosynthesis}). \begin{equation} \Delta Q_{r}=I\times RUE\times f_{d}\times f_{s}\times f_{c}\label{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 \texttt{Leaf} produces photosynthate. Diffuse factor ($f_{d}$) equals to 1 (\autoref{par:Diffuse-factor}), so that \autoref{eq:BiomassProduction} can be: \begin{equation} \Delta Q_{r}=I\times RUE\times f_{s}\times f_{c}\label{eq:BiomassProduction2} \end{equation} \subsubsection{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})\label{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, \autoref{eq:RadiationInterception} is reduced to. \begin{equation} I=I_{0}(1-\exp(-k\times LAI))\label{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 \texttt{x\_row\_spacing}, \texttt{y\_extinct\_coef} in the wheat.xml file (\autoref{fig:wdRowExtinct}) and is linearly interpolated by APSIM. In the current version of APSIM-Wheat, no impact of row spacing is considered (\autoref{fig:wdRowExtinct}) <>= 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) @ \subsubsection{Radiation use efficiency} $RUE$ (g MJ$^{\text{-1}}$) is a function of growth stages which is defined by parameters \texttt{x\_stage\_rue} and \texttt{y\_rue} in wheat.xml (\autoref{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. <>= 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 ($f_{s}$, \autoref{eq:BiomassProduction} and \autoref{eq:BiomassProduction2}). This stress factor is the minimum value of a temperature factor ($f_{T,\ photo}$, \autoref{eq:TemStressPhoto}), a nitrogen factor ($f_{N\ photo}$, \autoref{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})\label{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, \autoref{eq:StressFactor4Photosynthesis} is reduced to \begin{equation} f_{s}=\min(f_{T,\ photo},\ f_{N,\ photo})\label{eq:StressFactor4Photosynthesis2} \end{equation} \paragraph{The temperature factor} $f_{T,\ photo}$ is a function of the daily mean temperature and is defined by parameters \texttt{x\_ave\_temp} and \texttt{y\_stress\_photo} in the wheat.xml (\autoref{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})\label{eq:TemStressPhoto} \end{equation} <>= 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}}\label{eq:NStressPhoto0} \end{equation} where $C_{N}$ is the nitrogen concentration of \texttt{Leaf} parts; $R_{N,\,expan}$ is multiplier for nitrogen deficit effect on phenology which is specified by \texttt{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})}\label{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} <>= 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}\label{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). \subsection{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 ($W_{u}$, \autoref{eq:WaterUpdate}) and demand ($W_{d}$, \autoref{eq:soilWaterDemand-1}) capped by \begin{equation} \Delta Q_{w}=\Delta Q_{r}f_{w,\,photo}=\Delta Q_{r}\frac{W_{u}}{W_{d}}\label{eq:WaterStressBiomassProduction-1} \end{equation} where $f_{w,\,photo}$ is the water stress factor affecting photosynthesis (\autoref{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 ($\Delta Q=\Delta Q_{r}$, \autoref{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 ($\Delta Q_{r}$, \autoref{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 \autoref{eq:BiomassProduction}), divided by the transpiration efficiency. \begin{equation} W_{d}=\frac{\Delta Q_{r}-R}{TE}\label{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 (\autoref{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 ($\Delta Q_{r}$, \autoref{eq:BiomassProduction}) or by soil water deficiency ($\Delta Q_{w}$, \autoref{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}>= p <- wdVisXY(wheat_xml, "x_stage_no_partition", "y_ratio_root_shoot", xlab = "Stage", ylab = "Ratio of root and shoot") print(p) @ \subsection{Biomass partitioning to \texttt{Head} (\texttt{Pod}, \texttt{Meal} and \texttt{Oil }(not applicable in this version))} Then all or part of available biomass ($\Delta Q$) are partitioned into \texttt{Heads} according to total demand of \texttt{Heads} (\texttt{Meal}, \texttt{Oil} and \texttt{Pod}). \texttt{Meal} and \texttt{Pod} demands are calculated by \autoref{eq:MealDemand} and \autoref{eq:GrainDemand}. \texttt{Oil} demand always equals to zero in the current version of the APSIM-Wheat module. Biomass directly partitioned in \texttt{Pod} or \texttt{Grain} is considered as structural and cannot be re-translocated, however the biomass providing from re-translocation is accumulated as non-structural biomass. The \texttt{Pod} non-structural biomass can then be re-translocated into \texttt{Grain} (See \autoref{sub:Re-translocation}). \begin{equation} \begin{array}{c} \Delta Q_{head}=\min(\Delta Q,\,D_{grain}+D_{pod})\\ \Delta Q_{grain}=\frac{D_{g}}{D_{head}}\Delta A_{head}\\ \Delta Q_{pod}=\frac{D_{p}}{D_{head}}\Delta A_{head} \end{array}\label{eq:HeadBiomass} \end{equation} where $\Delta Q_{head}$ is the daily available biomass for \texttt{Head}, $D_{head}$, $D_{grain}$and $D_{pod}$ are demands for \texttt{Head}, \texttt{Grain} and \texttt{Pod}, respectively (see \autoref{sub:Grain-(meal)-demand} and \autoref{sub:Pod-demand}). $\Delta Q_{grain}$ and $\Delta Q_{pod}$ are biomass increment of \texttt{Grain} and \texttt{Pod}, respectively. \subsection{Biomass partitioning to \texttt{Leaf}} Then, the remaining biomass (after the partitioning to the \texttt{Heads}) are partitioned into \texttt{Leaf} based on a stage dependent function (\autoref{fig:wdFractionLeaf}). \texttt{Leaf} biomass is considered as structural and thus cannot be re-mobilised. \begin{equation} \Delta Q_{leaf}=(\Delta Q-\Delta Q_{head})\times F_{leaf} \end{equation} where $\Delta Q_{leaf}$ is the daily increment in \texttt{Leaf} biomass; and $F_{leaf}$ is the fraction of available biomass partitioned to the leaf, which is defined by \texttt{x\_stage\_no\_partition} and \texttt{y\_frac\_leaf} in wheat.xml (\autoref{fig:wdFractionLeaf}). <>= p <- wdVisXY(wheat_xml, "x_stage_no_partition", "y_frac_leaf", xlab = "Stage", ylab = "Fraction of leaf") print(p) @ \subsection{Biomass partitioning to \texttt{Stem}} Finally, the whole remaining biomass (if any) are partitioned into \texttt{Stem} (\autoref{fig:BiomassPartition}). Until the stage ``start of grain filling'', 65\% of this biomass is distributed to structural biomass (\autoref{fig:wdStemGrowthStructuralFractionStage}), while remaining 35\% is allocated in un-structural biomass. Afterwards, all new biomass allocated to \texttt{Stem} is for non-structural biomass (which can re-mobilised). \begin{equation} \Delta Q_{stem}=\Delta Q-\Delta Q_{head}-\Delta Q_{leaf} \end{equation} \begin{equation} \Delta Q_{stem.\,structural}=\Delta Q_{stem}\times h_{structual} \end{equation} \begin{equation} \Delta Q_{stem.\,non-structural}=\Delta Q_{stem}\times(1-h_{structual})\label{eq:StemNonStructural} \end{equation} where $\Delta Q_{stem}$ is the daily increment in \texttt{Stem} biomass; $\Delta Q_{stem.\,structural}$ is the structural biomass of \texttt{Stem}; $\Delta Q_{stem.\,non-structural}$ is the non-structural biomass of \texttt{Stem}; and $h_{structual}$ is the fraction of \texttt{Stem} biomass distributed to structural biomass which depends on the growth stage (S). $h_{structual}$ is specified by \texttt{stemGrowthStructuralFraction} and \texttt{stemGrowthStructuralFractionStage} in wheat.xml, with a default value of 0.65 before beginning of grain filling and 0 after. <>= p <- wdStemGrowthStructuralFraction() print(p) @ \subsection{Re-translocation\label{subsec:Re-translocation}} If the supply in assimilate (daily biomass increase) is insufficient to meet \texttt{Grain} demand, then re-translocation may occur to meet the shortfall (\autoref{fig:BiomassPartition}). The biomass re-translocation first occurs from the \texttt{Stem} non-structural biomass. From the start of grain filling, the wheat module allows a total re-translocation of up to 20\% of \texttt{Stem} biomass per day. If required, biomass can then be re-translocated from the \texttt{Pod} non-structural biomass. The re-translocated biomass is used to fulfill the \texttt{Grain} and \texttt{Pod} demands (\autoref{sub:Grain-(meal)-demand} and \autoref{sub:Pod-demand}) and is accumulated as non-structural biomass. \begin{equation} D_{diff,\,head}=(D_{grain}-\text{\ensuremath{\Delta}}Q_{grain})+(D_{pod}-\Delta Q_{pod}) \end{equation} where $D_{diff,\,head}$ is the unfulfilled demand from the plant, $D_{grain}$ and $D_{pod}$ are the demands from \texttt{Grain} and \texttt{Pod} (\autoref{sub:Grain-(meal)-demand} and \autoref{sub:Pod-demand}), and $\text{\ensuremath{\Delta}}Q_{grain}$ and $\Delta Q_{pod}$ are the daily increments in biomass accumulated to \texttt{Grain} and \texttt{Pod} (before re-translocation; \autoref{eq:HeadBiomass}). \begin{equation} \Delta Q_{retrans,\,stem}=\min(D_{diff},\,Q_{stem.\,non-structural}\times20\%) \end{equation} where $\Delta Q_{retrans,\,stem}$ is the dry biomass re-translocated from \texttt{Stem}, and $Q_{stem.\,non-structural}$ is the non-structural part of the \texttt{Stem} biomass (\autoref{eq:StemNonStructural}). \begin{equation} D_{diff,\,head}=D_{diff,\,head}-\Delta Q_{retrans} \end{equation} where $D_{dff,\,head}$ is updated value of the unfulfilled demand from the head. \begin{equation} \Delta Q_{retrans,\,pod}=\min(D_{diff,\,head},\,Q_{pod,\,non-structural}) \end{equation} where $\Delta Q_{retrans,\,pod}$ from pod is the dry biomass re-translocated from \texttt{Pod}, and $Q_{pod,\,non-structural}$ is the non-structural part of the \texttt{Pod} biomass. \begin{equation} D_{dff,\,head}=D_{diff,\,head}-\Delta Q_{retrans,\,pod} \end{equation} where $D_{dff,\,head}$ is updated value of the unfulfilled demand from the head. \begin{equation} \Delta Q_{retrans}=\Delta Q_{retrans,\,stem}+\Delta Q_{retrans,\,pod} \end{equation} where $\Delta Q_{retrans}$ is re-translocated biomass within the plant. \begin{equation} \Delta Q_{grain.\,non-structural=}\Delta Q_{retrans\,to\,grain}=\frac{D_{diff,\,grain}}{D_{diff,\,head}}\Delta Q_{retrans} \end{equation} \begin{equation} \Delta Q_{retrans\,to\,pod}=\frac{D_{diff,\,pod}}{D_{diff,\,head}}\Delta Q_{retrans} \end{equation} \begin{equation} \Delta Q_{pod.\,non-structural=}\Delta Q_{retrans\,to\,pod}-\Delta Q_{retrans,\,pod} \end{equation} where $\Delta Q_{grain.\,non-structural}$ and $\Delta Q_{pod.\,non-structural}$ are the daily increment in the non-structural part of \texttt{Grain} and \texttt{Pod} biomass; $\Delta Q_{retrans\,to\,grain}$ and $\Delta Q_{retrans\,to\,pod}$ to pod are the daily biomass re-translocated to \texttt{Grain} and \texttt{Pod}; $D_{diff,\,grain}$ and $D_{diff,\,pod}$ are the unfulfilled demand of \texttt{Grain} and \texttt{Pod}, which are calculated as ($D_{grain}-\Delta Q_{grain}$) and ($D_{pod}-\Delta Q_{pod}$), respectively. \section{Head development} \subsection{Grain number} The number of grains per plant ($N_{g}$) is determined by the stem weight at anthesis. \begin{equation} N_{g}=R_{g}W_{s} \end{equation} where $W_{s}$ is the stem dry weight at anthesis, $R_{g}$ is the grain number per gram stem which is specified by \texttt{grain\_per\_gram\_stem} in wheat.xml, with default value at 25 grain g\textsuperscript{-1}. \subsection{\texttt{Grain} (\texttt{Meal}) demand\label{subsec:Grain-(meal)-demand}} The \texttt{Grain} demand (or \texttt{Meal} demand, $D_{g}$) is calculated in the growth phase \texttt{postflowering} (from flowering to end of grain filling \autoref{fig:PhenologWheatModule}). $D_{g}$ equals to 0 before flowering. \begin{equation} D_{g}=N_{g}R_{p}h_{g}(T_{mean})f_{N,\,grain}\label{eq:MealDemand} \end{equation} where $N_{g}$ is the grain number, $R_{p}$ is the potential rate of grain filling (0.0010 grain\textsuperscript{-1} d\textsuperscript{-1} from flowering to start of grain filling (\autoref{fig:PhenologWheatModule}); 0.0020 grain\textsuperscript{-1} d\textsuperscript{-1} during grain filling (\autoref{fig:PhenologWheatModule})), $h_{g}(T_{mean})$ is a function of daily mean temperature which affects the rate of grain filling (0-1) and is defined by parameters \texttt{x\_temp\_grainfill} and \texttt{y\_rel\_grainfill} in wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdTempGrainFill}). $f_{N,\,grain}$ is a nitrogen factor to grain filling. \begin{equation} f_{N,\,grain}=\frac{h_{N,\ poten}}{h_{N,\ min}}h_{N,\,grain}\sum_{stem,\,leaf}\frac{C_{N}-C_{N,\,min}}{C_{N,\,crit}\times f_{c,\,N}-C_{N,\,min}}\qquad(0\leq f_{N,\,fill}\leq1) \end{equation} where $h_{N,\ poten}$ is the potential rate of grain filling which is specified by \texttt{potential\_grain\_n\_filling\_rate} in wheat.xml and has a default value of 0.000055 g grain\textsuperscript{-1} d\textsuperscript{-1}; $h_{N,\ min}$ is the minimum rate of grain filling which is specified by \texttt{minimum\_grain\_n\_filling\_rate} in wheat.xml and has a default value of 0.000015 g grain\textsuperscript{-1} d\textsuperscript{-1}; $h_{N,\,grain}$ is a multiplier for nitrogen deficit effect on grain, which is specified by \texttt{n\_fact\_grain} in wheat.xml and has a default value of 1; $C_{N}$ is the nitrogen concentration of \texttt{Stem} or \texttt{Leaf} parts; $C_{N,\,crit}$ and $C_{N,\,min}$ are critical and minimum nitrogen concentration, respectively, for \texttt{Stem} and \texttt{Leaf} parts. $C_{N,\,crit}$ and $C_{N,\,min}$ are functions of growth stage and nitrogen concentration which is defined by parameters \texttt{x\_stage\_code}, \texttt{y\_n\_conc\_min\_leaf}, \texttt{y\_n\_conc\_crit\_leaf}, \texttt{y\_n\_conc\_min\_stem}, \texttt{y\_n\_conc\_crit\_stem} in wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdNitrogenConcentration}); and $f_{c,\,N}$ is a factor with a value of 1 (i.e. no impact) for Stem, and is depending on CO\textsubscript{2} for \texttt{Leaf} (\autoref{fig:wbCO2CritLeaf}). <>= p <- wdVisXY(wheat_xml, "x_temp_grainfill", "y_rel_grainfill", xlab = expression(paste("Daily mean temperature", ~"("*degree*"C)")), ylab = 'Factor affecting the rate of grain filling') print(p) @ <>= p <- wdVisXY(wheat_xml, "x_co2_nconc_modifier", "y_co2_nconc_modifier", xlab = 'CO2 concentration', ylab = 'Critical nitrogen concentration of Leaf') print(p) @ Finally, \texttt{Grain} demand is limited by the maximum grain size (corresponding to $D_{gm}$) \begin{equation} \begin{array}{c} D_{g}=\min(D_{g},\,D_{gm})\\ D_{gm}=N_{g}S_{gm}-Q_{meal}\qquad(D_{gm}\geq0) \end{array} \end{equation} where $N_{g}$ is the grain number; $Q_{meal}$ is the dry weight of \texttt{Meal} part (i.e. the \texttt{Grains}); $S_{gm}$ is the maximum grain size which is specified by max\_grain\_size in wheat.xml and is a cultivar-specific parameter with 0.04 g for default value. \subsection{\texttt{Pod} demand\label{subsec:Pod-demand}} \texttt{Pod} demand ($D_{p}$) is calculated by \texttt{Grain} demand ($D_{g}$, \autoref{eq:MealDemand}) or daily biomass accumulation ($\Delta Q$, \autoref{eq:actualBiomassProduction}) \begin{equation} D_{p}=\begin{array}{c} D_{g}h_{p}(S)\qquad D_{g}\text{>0}\\ \Delta Qh_{p}(S)\qquad D_{g}\text{=0} \end{array}\label{eq:GrainDemand} \end{equation} where $h_{p}(S)$ is a function of the growth stage ($S$) and of the \texttt{Pod} demand fraction of $D_{g}$ or $\Delta Q$. $h_{p}(S)$ is defined by parameters \texttt{x\_stage\_no\_partition} and \texttt{y\_frac\_pod} in wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdFractionOfPod}). <>= p <- wdVisXY(wheat_xml, "x_stage_no_partition", "y_frac_pod", xlab = "Stage codes", ylab = "Pod demand fraction of grain demand \n or daily biomass accumulation") print(p) @ \section{Leaf and node appearance and crop leaf area} In the current version of APSIM-Wheat, wheat plants are assumed to be uniclum (i.e. with a single stem), meaning that tillering is not simulated \textit{per se}. While a node corresponds to a phytomer on the main stem, it actually represents all the phytomers that appear simultaneously on different tillers (i.e. cohort of leaves) in the real world. \subsection{Node number} \subsubsection{Potential node appearance rate} At emergence (\autoref{fig:PhenologWheatModule}), a number of initial leaves are specified by \texttt{leaf\_no\_at\_emerg,} with a default value of 2. The initial number of nodes is the same as the initial number of leaves. During the tiller formation phase (i.e. up to 'Harvest rips', \autoref{fig:PhenologWheatModule}), nodes appear at a thermal time interval (the equivalent of a phyllochron for leaf appearance, $P_{n}$) that depends on the node number of the main stem ($n_{d}$, i.e. total number of nodes of the plant) at days after sowing ($d,$ days). \begin{equation} P_{n}=h_{P}(n_{d})\label{eq:phyllochron} \end{equation} where the function $h_{P}(n_{d})$ is defined by parameters \texttt{x\_node\_no\_app} and \texttt{y\_node\_app\_rate} in wheat.xml and is linearly interpolated by APSIM. In the current version of APSIM-Wheat, $P_{n}$ is set to 95 $^{\circ}\text{C}$ d, meaning that the 'node phyllochron' is supposed to be constant (\autoref{fig:wdPhyllochron}). No effect from water and N stress on leaf appearance is accounted for. <>= p <- wdVisXY(wheat_xml, "x_node_no_app", "y_node_app_rate", xlab = "Node number at main stem", ylab = expression(paste("'Node Phyllochron", ~"("*degree*"Cd)'"))) print(p) @ \subsubsection{Potential node number (daily increase)} The potential daily increase in the node number of this unique stem ($\Delta n_{d,\,p}$) is calculated by the daily thermal time (\autoref{fig:wdThermalTime}) and the 'node phyllochron', and occurs during the tiller formation phase (\autoref{fig:PhenologWheatModule}). \begin{equation} \Delta n_{d,\,p}=\frac{\Delta TT_{d}}{P_{n}}\label{eq:PotentialNodeNumber} \end{equation} where $\Delta TT_{d}$ is the thermal time ($^{\circ}\text{C}$d) at day $d$ (\autoref{fig:wdThermalTime} and \autoref{eq:thermaltime}). \subsection{Leaf number} \subsubsection{Potential leaf number (daily increase)} In the current version of APSIM-Wheat, all leaves appeared from a main and unique stem. The potential leaf number of each node is defined by a function ($h_{l}(n_{d})$) of node ($n_{d}$) number of day $d$ (or 'node position'; $n_{d}$) (\autoref{fig:wdTillerNumberByNode} and \autoref{eq:LeafExpansionStress}). $h_{l}(n_{d})$ is specified by parameters \texttt{x\_node\_no\_leaf} and \texttt{y\_leaves\_per\_node} in wheat.xml and linearly interpolated by APSIM. At day $d$, the leaf number of the current node $n_{d}$ nodes ($N_{n,\,d,\,p}$) is determined by the potential leaf number $d-1$ for the past $n_{d-1}$ nodes ($N_{n,\,d-1}$) and environmental stresses. \begin{equation} N_{d,\,p}=\min[N_{n,\,d-1},\;h_{l}(n_{d-1})]+[h_{l}(n_{d-1}+\Delta n_{d,\,p})-h_{l}(n_{d-1})]\times f_{S,\,expan}\label{eq:PotentialNodeNumberDaily} \end{equation} where $n_{d-1}$ is the node number at $d-1$ days after sowing, $\Delta n_{d,\,p}$ is the potential daily increase of node number (\autoref{eq:PotentialNodeNumber}), $f_{S,\,expan}$ is the environmental stresses for canopy expansion. \begin{equation} f_{S,\,expan}=\min\{[\min(f_{N,\,expan},\;f_{p,\,expan})]^{2},\;f_{w,\,expan}\}\label{eq:LeafExpansionStress} \end{equation} where $f_{N,\,expan}$, $f_{p,\,expan}$ and $f_{w,\,expan}$ are the nitrogen, phosphorus and soil water stress for canopy expansion, respectively, which is explained in \autoref{sub:Phosphorus-stress} and \autoref{eq:WaterStressLeafExpansion}, respectively. The potential daily increase in leaf number for the whole plant is calculated based on the potential increase for the current node and the potential increase in node number ($\Delta n_{d,\,p}$, \autoref{eq:PotentialNodeNumber}) as follows. \begin{equation} \Delta N_{d,\,p}=N_{n,\,d}\times\Delta n_{d,\,p} \end{equation} <>= p <- wdVisXY(wheat_xml, "x_node_no_leaf", "y_leaves_per_node", xlab = "Node number on the main stem", ylab = "Number of leaves per node") print(p) @ \subsubsection{Actual leaf number (daily increase)} The increase in actual leaf number ($\Delta N_{d,\ LAI}$) is calculated in relation to the fraction between the actual and stressed increase of leaf area index, as follow: \begin{equation} \Delta N_{d,\,LAI}=\Delta N_{d,\,p}\times h_{LAI}(\frac{\Delta\text{LAI}_{d}}{\Delta\text{LAI}_{d,\,s}})\label{eq:ActualLeafNumber} \end{equation} where $h_{LAI}$ is a function between the fraction of leaf area index and the fraction of leaf number which is defined by parameters \hyperlink{x_lai_ratio}{x\_lai\_ratio} and \hyperlink{y_leaf_no_frac}{y\_leaf\_no\_frac} in the wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdLAINodeNumber}). <>= p <- wdVisXY(wheat_xml, "x_lai_ratio", "y_leaf_no_frac", xlab = 'Fraction of leaf area index', ylab = 'Fraction of leaf number') print(p) @ \section{Leaf area expansion} \subsection{Actual leaf area (daily increase)} At emergence (\autoref{fig:PhenologWheatModule}), an initial leaf area is specified for each plant by \texttt{initial\_tpla}, with a default value of 200 mm\textsuperscript{2} plant\textsuperscript{-1}. During the tiller formation phase (\autoref{fig:PhenologWheatModule}), the daily increase in leaf area index ($\Delta\text{LAI}_{d}$) is the minimum between \textquoteleft stressed\textquoteright{} leaf area index ($\Delta\text{LAI}_{d,\,s}$) and the carbon-limited leaf area index ($\Delta\text{LAI}_{d,\,c}$). \begin{equation} \Delta\text{LAI}_{d}=\min(\Delta\text{LAI}_{d,\,s},\;\Delta\text{LAI}_{d,\,c}) \end{equation} \subsection{\textquotedblleft Stressed\textquotedblright{} leaf area} During the tiller formation phase, the ``stressed'' daily increase in leaf area ($\Delta LAI_{d,s}$) is calculated as the potential increase in LAI reduced by environmental factors. \begin{equation} \Delta\text{LAI}_{d,\,s}=\Delta\text{LAI}_{d,\,p}\times\min(f_{w,\;expan},\,f_{N,\,expan},\,f_{P,\,expan})\label{eq:StressLeafArea} \end{equation} where $f_{N,\,expan}$, $f_{p,\,expan}$ and $f_{w,\,expan}$ are the nitrogen, phosphorus and soil water stress factors concerning canopy expansion, respectively (\autoref{eq:NStressLeafExpansion}, \autoref{sub:Phosphorus-stress} and \autoref{eq:WaterStressLeafExpansion}). The potential daily increase of leaf area ($\Delta\text{LAI}_{d,\,p}$) is calculated by the potential daily increase in leaf number and leaf size. \begin{equation} \Delta\text{LAI}_{d,\,p}=\Delta N_{d,\,p}\times L_{n}\times D_{p} \end{equation} where $\Delta N_{d,\,p}$ is the potential increase in leaf number (for the whole plant), $D_{p}$ is the plant population, and $L_{n}$ is the potential leaf area for leaves of the ``current'' node (this corresponds to the new potential leaf area produced by the different tillers in the real world) and depends on the node number on the main and unique stem considered by APSIM-Wheat. \begin{equation} L_{n}=h_{ls}(n_{d}+n_{0}) \end{equation} where $n_{0}$ is the growing leaf number in the sheath (\texttt{node\_no\_correction} in wheat.xml) and equals to 2 as default value. The function $h_{ls}(n_{d})$ is defined by parameters \texttt{x\_node\_no} and \texttt{y\_leaf\_size} in wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdLeafSizeByNode}). <>= p <- wdVisXY(wheat_xml, "x_node_no", "y_leaf_size ", xlab = "Main stem node number", ylab = expression(paste("Potential leaf area per node", ~"("*mm^2*")"))) print(p) @ \subsection{Carbon-limited leaf area} Leaf area related to carbon production is calculated by the increase in leaf dry weight ($\Delta Q_{leaf}$ \autoref{eq:actualBiomassProduction}) and the maximum specific leaf area ($\text{SLA}_{max}$), which is related to leaf area index (LAI). \begin{equation} \Delta\text{LAI}_{d,\,c}=\Delta Q_{leaf}\times\text{SLA}_{max} \end{equation} \begin{equation} \text{SLA}{}_{max}=h_{SLA}(\text{LAI}) \end{equation} This function is defined by parameters \texttt{x\_lai} and \texttt{y\_sla\_max} in wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdSLA}). <>= p <- wdVisXY(wheat_xml, "x_lai", "y_sla_max", xlab = expression(paste("Leaf area index", ~"("*mm^2*" "*mm^{-2}*")")), ylab = expression(paste("Maximum specific leaf area", ~"("*mm^2*" "*g^{-1}*")"))) print(p) @ \section{Root growth and distribution} \subsection{Root depth growth} Between germination and start of grain filling (\autoref{fig:PhenologWheatModule}), the increase in root depth ($\Delta D_{r}$) is a daily rate multiplied by a number of factors. Daily root depth growth ($\Delta D_{r}$) is calculated by root depth growth rate ($R_{r}$), temperature factor ($f_{rt}$), soil water factor ($f_{rw}$), and soil water available factor ($f_{rwa}$) and root exploration factor ($\text{XF}(i)$). \begin{equation} \Delta D_{r}=R_{r}\times f_{rt}\times\min(f_{rw},\;f_{rwa})\text{\ensuremath{\times}XF}(i)\label{eq:rootDepthGrowth} \end{equation} where $i$ is the soil layer number in which root tips are growing. Root depth growth rate is a function of growth stage, which is defined by parameters \texttt{stage\_code\_list} and \texttt{root\_depth\_rate} in the wheat.xml and is linearly interpolated by APSIM (\autoref{fig:wdRootGrowthRate}). <>= p <- wdVisXY(wheat_xml, "stage_code_list", "root_depth_rate", xlab = "Stage codes", ylab = "Root depth growth rate (mm/d)") print(p) @ The temperature factor ($f_{rt}$) is calculated by daily mean temperature. \begin{equation} f_{rt}=h_{rt}(\frac{T_{max}+T_{min}}{2})\label{eq:RootGrowthTemperature} \end{equation} where $h_{rt}$ is a function of factor of temperature on root length and daily mean temperature and is defined by parameters \texttt{x\_temp\_root\_advance} and \texttt{y\_rel\_root\_advance} in the wheat.xml which is linearly interpolated by APSIM (\autoref{fig:wdTempRootFactor}). <>= p <- wdVisXY(wheat_xml, "x_temp_root_advance", "y_rel_root_advance", xlab = expression(paste("Mean daily temperature", ~"("*degree*"C)")), ylab = "Temperature factor on root length") print(p) @ The soil water factor ($f_{rw}$) is calculated by soil water stresses of photosynthesis ($f_{w,\,photo}$, \autoref{eq:swstressphoto}). \begin{equation} f_{rw}=h_{rw}(f_{w,\,photo}) \end{equation} where $h_{rw}$ is a function of soil-water factor affecting root depth growth in response to soil water stress for photosynthesis. This function is defined by parameters \texttt{x\_ws\_root} and \texttt{y\_ws\_root\_fac}, which are linearly interpolated by APSIM. The default value of $f_{rw}$ is 1, i.e. there is no soil water stress on root depth growth in current APSIM-Wheat. The soil water available factor ($f_{rwa}$) is calculated by fraction of available soil water. \begin{equation} f_{rwa}=h_{rwa}(\text{FASW})\label{eq:Soilwateravailablefactor} \end{equation} where $h_{rwa}$ is a function of the fraction of available soil water (FASW) is defined in wheat.xml by parameters \texttt{x\_sw\_ratio} and \texttt{y\_sw\_fac\_root} which is linearly interpolated by APSIM (\autoref{fig:wdWaterAvaiOnRoot}). <>= p <- wdVisXY(wheat_xml, "x_sw_ratio", "y_sw_fac_root", xlab = "Fraction of available soil water", ylab = "Stress factor for root depth growth") print(p) @ The fraction of available soil water (FASW) is calculated by a fraction of root dpeth in soil layer $i$ ($D_{r}(i)$) and depth of soil layer $i$ ($D_{s}(i)$), and FASW at layer $i+1$ and $i$. \begin{equation} \text{FASW}=\frac{D_{r}(i)}{D_{s}(i)}\text{FASW}(i+1)+(1-\frac{D_{r}(i)}{D_{s}(i)})\text{FASW}(i) \end{equation} where $\text{FASW}(i)$ is the fraction of available soil water in soil layer $i$. $D_{r}(i)$ is the root depth within the deepest soil layer ($i$) where roots are present , $D_{s}(i)$ is the thickness of this layer $i$, and \begin{equation} \text{FASW}(i)=\frac{\text{SW}(i)-\text{LL}(i)}{\text{DUL}(i)-\text{LL}(i)} \end{equation} where $\text{SW}(i)$ is the soil water content at layer $i$ (mm), $\text{LL}(i)$ is the lower limit of plant-extractable soil water in layer $i$ (mm), $\text{DUL}(i)$ is drained upper limit soil water content in soil layer $i$ (mm). $\text{XF}(i)$, $\text{SW}(i)$, $\text{LL}(i)$ and $\text{DUL}(i)$ are specified at the soil module of APSIM simulation files. Finally, \autoref{eq:rootDepthGrowth} is reduced to this function. \begin{equation} \Delta D_{r}=R_{r}\times f_{rt}\times f_{rwa}\text{\ensuremath{\times}XF}(i)\label{eq:rootDepthGrowth-1} \end{equation} Overall, root depth is constrained by the soil profile depth. The optimum root expansion rate is 30 mm d\textsuperscript{-1} (\autoref{fig:wdRootGrowthRate}). This can be limited by supra- or sub-optimal mean air temperatures (\autoref{fig:wdTempRootFactor}). Dry soil can slow root depth progression if the soil water content is less than 25\% of the extractable soil water (drained upper limit - lower limit) in the layers they are about to reach (\autoref{fig:wdWaterAvaiOnRoot}). The increase of root depth through a layer can also be reduced by knowing soil constraints (soil compression) through the use of the 0-1 parameter XF, which is input for each soil layer. Root depth is used by APSIM to calculate soil available water (e.g \autoref{sec:Crop-Water-Relations}). \subsection{Root length} Daily root length growth is calculated by daily growth of \texttt{Root} biomass ($\Delta Q_{root}$, \autoref{eq:RootBiomass}) and specific root length ($\text{SRL}$, defined by \texttt{specific\_root\_length} in wheat.xml with a default value of 105000 mm g\textsuperscript{-1}). \begin{equation} \Delta L_{r}=\Delta Q_{root}\times\text{SRL} \end{equation} The daily root length growth ($\Delta L_{r}$) is distributed to each soil layer $i$ according to root depth and soil water availability in soil layer $i$. \begin{equation} \Delta D_{r}(i)=\frac{f_{rl}(i)}{\sum_{j=1}^{N}f_{rl}(j)} \end{equation} where $f_{rl}(i)$ is a factor of root length growth in soil layer $i$. \begin{equation} f_{rl}(i)=f_{rwa}\times f_{b}(i)\text{\ensuremath{\times}XF}(i)\times\frac{D_{s}(i)}{D_{r}}\, \end{equation} where $\Delta L_{r}(i)$ is the daily root length growth for soil layer $i$, $D_{s}(i)$ is the depth of the soil layer $i$, $D_{r}$ is total root depth from the previous day, $\text{XF}(i)$ is root exploration factor in soil layer $i$, $f_{rwa}$ is soil water available factor (\autoref{eq:Soilwateravailablefactor}), \textbf{$f_{b}(i)$ }is branch factor at layer $i$. \begin{equation} f_{b}(i)=h_{b}(\frac{L_{r}(i)}{D_{p}D_{s}(i)}) \end{equation} where $L_{r}(i)$ is the root length in soil layer $i$, $D_{p}$ is plant population, $h_{b}$ is a function for branch factor that is defined by parameters \texttt{x\_plant\_rld} and \texttt{y\_rel\_root\_rate} in the wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdRootBranching}). <>= p <- wdVisXY(wheat_xml, "x_plant_rld", "y_rel_root_rate", xlab = "Root branching (mm/mm3/plant)", ylab = "Root branching factor") print(p) @ Root length has no effect on other traits in the current version of APSIM-Wheat. It is just used by the root senescence routine. \section{Senescence} \subsection{Leaf number senescence} The leaf senescence phase begins 40\% between floral initiation and end of juvenile, and ends at harvest ripe (\autoref{fig:PhenologWheatModule}), at which stage, all green leaves are dead. During leaf senescence phase (\autoref{fig:PhenologWheatModule}), leaf number senescence is calculated by daily thermal time ($\Delta TT$, \autoref{eq:thermaltime}) as follows: \begin{equation} \Delta N_{d,\,sen}=\Delta TT\times\frac{f_{sen,\,l}\times N_{d}}{r_{sen,\,l}} \end{equation} where $N_{d}$ is the total leaf number; $f_{sen,\,l}$ is the fraction of the total leaf number senescing per main stem node and specified by \texttt{fr\_lf\_sen\_rate} in wheat.xml (default value 0.035); $r_{sen,\,l}$ is the rate of node senescence on main stem and specified by \texttt{node\_sen\_rate} in wheat.xml (default value 60.0 $^{\circ}$Cd node\textsuperscript{-1}). \subsection{Leaf area senescence} There are five causes of leaf senescence: age ($\text{\ensuremath{\Delta}LAI}_{sen,\,age}$), water stress ($\text{\ensuremath{\Delta}LAI}_{sen,\,sw}$), light intensity ($\text{\ensuremath{\Delta}LAI}_{sen,\,light}$), frost ($\text{\ensuremath{\Delta}LAI}_{sen,\,frost}$) and heat ($\text{\ensuremath{\Delta}LAI}_{sen,\,heat}$). The maximum of these causes is the day's total leaf area index senescence. \begin{equation} \text{\ensuremath{\Delta}LAI}_{sen}=\max(\text{\ensuremath{\Delta}LAI}_{sen,\,age},\;\text{\ensuremath{\Delta}LAI}_{sen,\,sw},\;\Delta\text{LAI}_{sen,\,light},\;\text{\ensuremath{\Delta}LAI}_{sen,\,frost},\;\text{\ensuremath{\Delta}LAI}_{sen,\,heat}) \end{equation} Leaf area senescence caused by age corresponds to the leaf area of the number of leaves senesced ($\Delta N_{d,\,sen}$) from the lowest leaf position. Leaf area senescence caused by soil water ($\text{\ensuremath{\Delta}LAI}_{sen,\,sw}$) is calculated as follows. \begin{equation} \text{\ensuremath{\Delta}LAI}_{sen,\,sw}=k_{sen,\,sw}\times(1-f_{sw,\,photo})\times\text{LAI} \end{equation} where $k_{sen,\,sw}$ is the slope of the linear equation relating to soil water stress to leaf senescence rate and is specified by \texttt{sen\_rate\_water} in wheat.xml (default value 0.10); $f_{sw,\,photo}$ is soil water stress for photosynthesis (\autoref{eq:swstressphoto}); LAI is the leaf area index. Leaf area senescence caused by light intensity ($\text{\ensuremath{\Delta}LAI}_{sen,\,light}$) is calculated as follows: \begin{equation} \text{\ensuremath{\Delta}LAI}_{sen,\,light}=k_{sen,\,light}\times(\text{LAI}-\text{LAI}_{c,\,light})\times\text{LAI}\quad\text{LAI}>\text{LAI}_{c,\,light}\label{eq:SensLight} \end{equation} where $k_{sen,\,light}$ is sensitivity of leaf area senescence to shading and is specified by \texttt{sen\_light\_slope} in wheat.xml (default value 0.002); $\text{LAI}_{c,\,light}$ is the critical LAI when shading is starting to cause leaf area senescence and is specified by \texttt{lai\_sen\_light} in wheat.xml (default value 7). The leaf area senescence caused by frost is a ratio of LAI. \begin{equation} \text{\ensuremath{\Delta}LAI}_{sen,\,frost}=k_{sen,\,frost}\text{\ensuremath{\times}LAI}\label{eq:SensFrost} \end{equation} where $k_{sen,\,frost}$ is a function of daily minimum temperature and is defined by parameters \texttt{x\_temp\_senescence} and \texttt{y\_senescence\_fac} in wheat.xml, which are linearly interpolated by APSIM. The default value of $k_{sen,\,frost}$ is zero, i.e. there is no frost stress in leaf area in the current APSIM-Wheat module. Senescence by heat calculation has been added in APSIM 7.5. The leaf area senescence by heat is a ratio of LAI \citep{asseng2011theimpact}. \begin{equation} \text{\ensuremath{\Delta}LAI}_{sen,\,heat}=k_{sen,\,heat}\times\text{LAI}\label{eq:SensHeat} \end{equation} where $k_{sen,\,heat}$ is a function of daily maximum temperature which is defined by parameters \texttt{x\_maxt\_senescence} and \texttt{y\_heatsenescence\_fac} in wheat.xml which are linearly interpolated by APSIM. <>= p <- wdVisXY(wheat_xml, "x_maxt_senescence", "y_heatsenescence_fac", xlab = expression(paste("Maximum temperature", ~"("*degree*"C)")), ylab = "Senescence fraction of LAI") print(p) @ The total leaf area of plant must be more than the minimum plant area (\texttt{min\_tpla}), which has default value 5 mm$^{\text{2}}$ plant$^{\text{-1}}$. When some leaves are senesced, only a small amount of nitrogen is retained in the senesced leaf, the rest is made available for re-translocation included into the \texttt{Stem} N pool (\autoref{sub:NitrogenPartitioningAndRetranslocation}). The concentration of nitrogen in senesced material is specified in wheat.xml. \subsection{Biomass senescence} Leaf biomass senescence $\Delta Q_{sl}$ is the ratio of leaf area senescence ($\text{\ensuremath{\Delta}LAI}_{sen}$) with total the green LAI at the time considered (LAI). \begin{equation} \Delta Q_{sl}=\Delta Q_{l}\frac{\text{\ensuremath{\text{\ensuremath{\Delta}LAI}_{sen}}}}{\text{LAI}} \end{equation} where $\Delta Q_{l}$ is the daily increase of leaf biomass. \subsection{Root senescence} A rate of 0.5\% of root biomass and root length is senesced each day and detaches immediately being sent to the soil nitrogen module and distributed as fresh organic matter in the profile. \begin{equation} \Delta Q_{sen,\,root}=\Delta Q_{root}\times f_{sen,\,root} \end{equation} where $\Delta Q_{sen,\,root}$ is the daily \texttt{Root} senesced biomass, and $f_{sen,\,root}$ is the fraction of senesced root biomass, which is defined in \texttt{x\_dm\_sen\_frac\_root} and \texttt{y\_dm\_sen\_frac\_root} in wheat.xml (\autoref{fig:wdRootSens}) <>= p <- wdVisXY(wheat_xml, "x_dm_sen_frac_root", "y_dm_sen_frac_root", xlab = 'Fraction of material senescence', ylab = "Senescence fraction of Root biomass") print(p) @ \begin{equation} \Delta L_{sen,\,root}=\Delta Q_{sen,\,root}\times\text{SRL} \end{equation} where $\Delta L_{sen,\,root}$ is the daily root length senescence, and SRL is the specific root length. Root senescence occurs in each of the soil layers where roots are present, as a proportion of the total root length. \begin{equation} \Delta L_{sen,\,root}(i)=\Delta L_{sen,\,root}\times\frac{L_{r}(i)}{\sum_{j=1}^{i}L_{r}(j)} \end{equation} where $L_{sen,\,root}(i)$ is the root length senescence in soil layer $i$, $L_{r}(i)$ is root length in layer $i$, and $\sum_{j=1}^{i}L_{r}(j)$is the total root length for all the layers where root are present. \section{Crop Water Relations\label{sec:Crop-Water-Relations}} \subsection{Crop water demand\label{subsec:Crop-water-demand}} Following \citet{sinclair1986waterand}, transpiration demand is modeled as a function of the current day's potential crop growth rate, estimated by the potential biomass accumulation associated with intercepted radiation ($\Delta Q_{r}$, see \autoref{eq:BiomassProduction}), divided by the transpiration efficiency. \begin{equation} W_{d}=\frac{\Delta Q_{r}-R}{TE}\label{eq:soilWaterDemand} \end{equation} where $R$ is respiration rate and equal to zero in the current version of APSIM-Wheat, $TE$ is transpiration efficiency. $TE$ is related to the daylight averaged vapour pressure deficit ($VPD$, \autoref{eq:VPD}) and a multiple of CO\textsubscript{2} factor \citep{reyenga1999modelling}. \begin{equation} TE=f_{c,\,TE}\frac{f_{TE}}{VPD}\label{eq:TranspirationEfficiency} \end{equation} where $f_{c,\,TE}$ is the CO\textsubscript{2} factor for transpiration efficiency, which is a function of carbon dioxide concentration and is defined by parameters \texttt{x\_co2\_te\_modifier} and \texttt{y\_co2\_te\_modifier} in wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdCO2TE}). $f_{c,\,TE}$ linearly increases from 1 to 1.37 when CO\textsubscript{2} concentration increases from 350 ppm to 700 ppm \citep{reyenga1999modelling}. $f_{TE}$ is the coefficient of transpiration efficiency, which values are defined in wheat.xml by parameters \texttt{transp\_eff\_cf} in wheat.xml for the different growth stages and are linearly interpolated by APSIM (\autoref{fig:wdCoefficientOfTE}). <>= p <- wdVisXY(wheat_xml, "x_co2_te_modifier", "y_co2_te_modifier", xlab = "Carbon dioxide concentration", ylab = "Transpiration efficiency factor") print(p) @ <>= p <- wdVisXY(wheat_xml, "stage_code", "transp_eff_cf", xlab = "Stage", ylab = "Coefficient of \ntranspiration efficiency") print(p) @ $VPD$ is the vapour pressure deficit, which is estimated using the method proposed by \citet{tanner1983efficient} and only requires daily maximum and minimum temperatures. \begin{equation} VPD=f_{v}[6.1078\times\exp(\frac{17.269\times T_{max}}{237.3+T_{max}})-6.1078\times\exp(\frac{17.269\times T_{min}}{237.3+T_{min}})]\label{eq:VPD} \end{equation} In this method, it is assumed that the air is saturated at the minimum temperature. The saturated vapour pressure is calculated at both the maximum and minimum temperatures, and the default vapour pressure deficit for the day is taken as 75\% ($f_{v}$, defined by \texttt{svp\_fract} in wheat.xml) of the difference between these two vapour pressures. Crop water demand is capped to below a given multiple of potential ET (taken as Priestly-Taylor Eo from the water balance module) as specified by \hyperlink{eo_crop_factor_default}{eo\_crop\_factor\_default} in the wheat.xml file (default value 1.5). This limits water use to reasonable values on days with high VPD or in more arid environments. \subsection{Potential and actual extractable soil water} Potential and actual extractable soil water is the sum of root water contents available to the crop from each profile layer occupied by roots. If roots are only partially through a layer available soil water is scaled to the portion that contains roots. Potential extractable soil water ($\text{ESW}{}_{p}$) is the difference between drained upper limit soil water content (DUL) and lower limit of plant-extractable soil water (LL) for each soil layer. The actual extractable soil water ($esw_{a}$) is the difference between the soil water content (SW) and lower limit of plant-extractable soil water (LL) for each soil layer. \begin{equation} \begin{array}{c} \text{ESW}_{p}(i)=\text{DUL}(i)-\text{LL}(i)\\ \text{ESW}a(i)=\text{SW}(i)-\text{LL}(i)\\ \text{ESW}_{p}=\sum_{i=1}^{I}[\text{DUL}(i)-\text{LL}(i)]\\ \text{ESW}_{a}=\sum_{i=1}^{I}[\text{SW}(i)-\text{LL}(i)] \end{array}\label{eq:SoilWaterESW} \end{equation} where $i$ indicates soil layers (where roots are present), and $I$ indicates the deepest soil water of root presented. Similar variables are calculated for the entire soil profile (i.e. roots may not occupy all the layers). \begin{equation} \begin{array}{c} \text{PAWC}=\sum_{i}^{N_{s}}[\text{DUL}(i)-\text{LL}(i)]\\ \text{ESW}=\sum_{i}^{N_{s}}[\text{SW}(i)-\text{LL}(i)] \end{array} \end{equation} where $i$ indicates soil layers, $N_{s}$ indicates the number of soil layers, and PAWC is the plant available water capacity. \subsection{Crop water supply, i.e. potential soil water uptake} The APSIM-Wheat module can be coupled to either the SWIM2 module (see module documentation) or the SOILWAT2 module (default). When the APSIM-Wheat module is coupled to APSIM-SOILWAT2, potential soil water uptake (or water supply, $W_{s}$) is calculated using the approach first advocated by Monteith (1986). Crop water supply is considered as the sum of potential root water uptake from each profile layer occupied by root. If roots are only partially through a layer available soil water is scaled to the portion that contains roots. The potential rate of extraction in a layer is calculated using a rate constant (KL) as actual extractable soil water. The KL defines the fraction of available water able to be extracted per day. The KL factor is empirically derived, incorporating both plant and soil factors which limit rate of water uptake. Root water extraction values (KL) must be defined for each combination of crop species and soil type. \begin{equation} \begin{array}{c} \begin{array}{cc} W_{s}(i) & =\text{KL}(i)[\text{SW}(i)-\text{LL}(i)]\qquad\qquad if\,i\leq I-1\\ & =\frac{D_{r}(i)}{D_{s}(i)}\text{KL}(i)[\text{SW}(i)-\text{LL}(i)]\qquad\qquad if\,i=I \end{array}\\ W_{s}=\sum_{i=1}^{I}W_{s}(i) \end{array}\label{eq:WaterSupply} \end{equation} where $i$ is the soil layer, $I$ is the deepest soil layer where roots are present, $W_{s}(i)$ is the water supply available from layer $i$, $W_{s}$ is the crop water supply, $\text{SW}(i)$ is the soil water content in layer $i$, $\text{LL}(i)$ is the lower limit of plant-extractable soil water in layer $i$, $\text{KL}(i)$ is the root water extraction values in layer $i$, $D_{r}(i)$ is the root depth within the soil layer ($i$) where roots are present, and $D_{s}(i)$ is the thickness of this layer $i$. \subsection{Actual soil water uptake} The actual rate of water uptake is the lesser of the potential soil water supply ($W_{s}$, \autoref{eq:WaterSupply}) and the soil water demand ($W_{d}$, \autoref{eq:soilWaterDemand}), which is determining whether biomass production is limited by radiation or water uptake (\autoref{eq:actualBiomassProduction}) \begin{equation} W_{u}=\min(W_{d},\,W_{s})\label{eq:WaterUpdate} \end{equation} If the potential soil water supply (accessible by the roots) exceeds the crop water demand, then the actual soil water uptake ($W_{u}$) is removed from the occupied layers in proportion to the values of potential root water uptake in each layer. If the computed soil water supply from the profile is less than the demand then, and the actual root water uptake from a layer is equal to the computed potential uptake. If there are not soil water supply and demand, soil water update equals to zero. \begin{equation} \begin{array}{c} \Delta W_{s}(i)=-W_{s}(i)\times\frac{W_{d}}{W_{s}}\qquad if\;W_{s}W_{d}\\ \Delta W_{s}(i)=0\qquad if\;W_{s}=W_{d}=0 \end{array} \end{equation} where $\Delta W_{s}(i)$ is the daily change in soil water content at layer $i$ (where roots are present), and $W_{s}(i)$ is the water supply available from layer $i$ (\autoref{eq:WaterSupply}) . \subsection{Soil water stresses affecting plant growth} Soil water deficit factors are calculated to simulate the effects of water stress on different plant growth-and-development processes. Three water deficit factors are calculated which correspond to four plant processes, each having different sensitivity to water stress i.e. photosynthesis, leaf expansion, and phenology. Each of these factors is capped between 0 and 1, where the value of 0 corresponds to a complete stress, while 1 corresponds to no stress. Leaf expansion is considered more sensitive to stress than photosynthesis, while soil water has no impact on crop phenology in the current APSIM-Wheat version. \subsubsection{Phenology} Soil water stress of phenology is determined by the soil water deficiency. \begin{equation} f_{W,\,pheno}=h_{w,\,pheno}(\frac{esw_{a}}{esw_{p}})\label{eq:SoilWaterStress} \end{equation} where $esw_{a}$ is the actual extractable soil water in root layers, $esw_{p}$ is the potential extractable soil water in root layers. $h_{w,\,pheno}$ is a function of soil water available ratio and soil water stress, which is defined by parameters \texttt{x\_sw\_avail\_ratio} and \texttt{y\_swdef\_pheno} (default value 1) in wheat.xml and linearly interpolated by APSIM. In the current version of APSIM-Wheat module, no soil water stress for phenology is applied (\autoref{fig:wdSoilWaterStressPhenology}). The soil water stress of phenology for flowering (\texttt{x\_sw\_avail\_ratio\_flowering} and \texttt{y\_swdef\_pheno\_flowering}) and grain filling (\texttt{x\_sw\_avail\_ratio\_start\_grain\_fill} and \texttt{y\_swdef\_pheno\_start\_grain\_fill}) phases are calculated in the source code, but don't have influence on the phenology of wheat in the current APSIM-Wheat version (default value of 1). <>= p <- wdVisXY(wheat_xml, "x_sw_avail_ratio", "y_swdef_pheno ", xlab = "Ratio of available soil water", ylab = "Soil water stress \nof phenology") print(p) @ \subsubsection{Photosynthesis} Soil water stress of biomass accumulation ($f_{w,\,photo}$) is calculated as follows. \begin{equation} f_{w,\,photo}=\frac{W_{u}}{W_{d}}\label{eq:swstressphoto} \end{equation} where $W_{u}$ is the total daily water uptake from root system (\autoref{eq:WaterUpdate}), $W_{d}$ is the soil water demand of \texttt{Leaf} and \texttt{Head} parts (\autoref{eq:soilWaterDemand}). Finally, the potential biomass production (radiation-limited$\Delta Q$) can limit by water uptake ($f_{w,\,photo}<1$, i.e. when $W_{u}>= p <- wdVisXY(wheat_xml, "x_sw_demand_ratio", "y_swdef_leaf", xlab = "Soil water supply:demand ratio", ylab = "Soil water stress of expansion") print(p) @ \subsection{KL factor} APSIM 7.5 introduces a modifying factor on KL (rate of maximum daily water uptake per day) where there is an excess of chloride concentration (Cl), exchangeable sodium percentage (ESP), or electrical conductivity (EC) properties in the soil \citep{hochman2007simulating}. The KL modifier is optional and triggered by setting the ModifyKL parameter to \textquoteleft yes\textquoteright . When the KL modifier is activated, KL values are modified for each layer, by factors (concerning Cl, ESP, EC; \autoref{fig:wdKLFactoring}) applied to default KL values. The modifiers are calculated using one of the limiting factors in order of preference (Cl, ESP, EC), i.e. KL is modified only if there are no soil parameters for Cl. The parameters in the wheat.xml that control this mechanism are ClA, CLB, ESPA, ESPB, ECA, ECB (slope and intercept of linear relationship for Cl, ESP and EC). <>= p <- wdKLFactoring(wheat_xml) print(p) @ \section{Nitrogen} The nitrogen stress phase begins before 30\% floral initiation to finish at the 'harvest ripe' phase (\autoref{fig:PhenologWheatModule}), which are defined by \texttt{n\_stress} in wheat.xml. \subsection{Nitrogen supply} Ammonium ($\text{NH}_{4}^{+}$) is not taken up in wheat as wheat.xml parameter knh4 (constant for NH\textsubscript{4} extraction) is equal to 0. The model uses a simplified formulation for nitrate $\text{NO}_{3}^{-}$ uptake somewhat similar in structure to that employed in water uptake. During the nitrogen stress phase (\autoref{fig:PhenologWheatModule}), nitrogen supply for soil layer $i$ ($N_{s}(i)$, g m\textsuperscript{-2}) is calculated as follows: \begin{equation} N_{s}(i)=K_{NO3}N(i)[N(i)\frac{1000}{\text{BD}(i)D_{s}(i)}]\frac{\text{\text{ESW}}_{a}(i)}{\text{ESW}_{p}(i)} \end{equation} where $K_{NO3}$ is a constant of extractable soil nitrogen, which is defined by \texttt{kno3} with default value 0.02; $N(i)$ is the $\text{NO}_{3}^{-}$concentration in soil layer $i$ (g m\textsuperscript{-2}); $\text{BD}(i)$ is the bulk density of soil layer $i$ (g cm\textsuperscript{-3}); $D_{s}(i)$ is the depth of soil layer $i$ (cm); $\text{ESW}_{a}(i)$ is the actual extractable soil water in soil layer $i$ (\autoref{eq:SoilWaterESW}); $\text{ESW}{}_{p}(i)$ is the potential extractable soil water in soil layer $i$ (\autoref{eq:SoilWaterESW}). During non-nitrogen stress phase (\autoref{fig:PhenologWheatModule}), wheat could access to all available nitrogen. \begin{equation} N_{s}(i)=N(i)\frac{1000}{\text{BD}(i)D_{s}(i)}\label{eq:NitrogenSupply} \end{equation} The values of $N_{s}(i)$ for each layer of root presented are summed to get a total potential nitrogen uptake (or crop N supply, $N_{s}$) and then each layer $N_{s}(i)$ is scaled by maximum total nitrogen uptake ($N_{s,\,max}$), which is defined by \texttt{total\_n\_uptake\_max} with default value 0.6 g m\textsuperscript{-2}. \begin{equation} N_{s}'(i)=N_{s}(i)\frac{N_{s,\,max}}{N_{s}} \end{equation} where $N_{s}'(i)$ is the actual nitrogen uptake in the layer $i$. \subsection{Nitrogen demand} Total wheat nitrogen demand is the sum of the N demand in all parts (i.e. \texttt{Leaf}, \texttt{Stem}, and \texttt{Pod}). Wheat has a defined minimum ($C_{N,\,min}$), critical ($C_{N,\,crit}$) and maximum ($C_{N,\,max}$) nitrogen concentration for all plant parts (\autoref{fig:wdNitrogenConcentration}). These concentration limits change with phenological stages (\autoref{fig:wdNitrogenConcentration}). And they are defined by parameters \texttt{x\_stage\_code}, \texttt{y\_n\_conc\_min\_leaf}, \texttt{y\_n\_conc\_crit\_leaf}, \texttt{y\_n\_conc\_max\_leaf}, \texttt{y\_n\_conc\_min\_stem}, \texttt{y\_n\_conc\_crit\_stem}, \texttt{y\_n\_conc\_max\_stem, y\_n\_conc\_min\_pod, y\_n\_conc\_crit\_pod, y\_n\_conc\_max\_pod} in wheat.xml and linearly interpolated by APSIM . Physiologically, minimum nitrogen concentration ($C_{N,\,min}$) corresponds to the structural N required for the plant structure, and which cannot be re-translocated. Critical nitrogen concentration ($C_{N,\,crit}$) corresponds to the minimum concentration of N that plant parts will attempt to maintain (it drives the \textquoteleft N demand\textquoteright{} of the part), and maximum nitrogen concentration ($C_{N,\,max}$) reflects to the capacity of the part to accumulate the extra available N (i.e. fulfilling more than its \textquoteleft demand\textquoteright ) up to a this maximum threshold N. <>= p <- wdNitrogenConcentration() print(p$pod, position = c(0, 0, 1, 0.35), more = TRUE) print(p$stem, position = c(0, 0.31, 1, 0.68), more = TRUE) print(p$leaf, position = c(0, 0.65, 1, 1)) @ \subsubsection{Nitrogen demand of \texttt{Grain}} \texttt{Grain} nitrogen demand starts at anthesis and is calculated from grain number, thermal time and a potential grain nitrogen filling rate (g grain\textsuperscript{-1} $^{\circ}$Cd\textsuperscript{-1}). \begin{equation} N_{D,\;grain}=N_{g}\,R_{N,\,poten,}\,f_{N,\;grain}\,h_{grain}(T)\label{eq:NitrogenDemand} \end{equation} where $N_{g}$ is the grain number, $R_{N,\,poten,}$ is the potential nitrogen filling rate, which is defined by parameter \texttt{potential\_grain\_n\_filling\_rate} in wheat.xml with default value 0.000055 g grain\textsuperscript{-1} d\textsuperscript{-1}. $f_{N,\;grain}$ is the nitrogen factor of grain filling (\autoref{eq:NStressFilling}). $h_{grain}(T)$ is a function of daily mean temperature ($T$) to influence of grain filling (\autoref{fig:wdNitrogenTem}). <>= p <- wdVisXY(wheat_xml, "x_temp_grain_n_fill", "y_rel_grain_n_fill", xlab = expression(paste("Daily mean temperature", ~"("*degree*"C)")), ylab = 'Temperature factor to nitrogen demand of grain') print(p) @ \subsubsection{Nitrogen demand of other parts} Demand of nitrogen in each part (except Grain) attempts to maintain nitrogen at the critical (non-stressed) level. Nitrogen demand on any day is the sum of the demands from the pre-existing biomass of each part required to reach critical nitrogen content, plus the nitrogen required to maintain critical nitrogen concentrations in that day's produced biomass. For each plant part (\texttt{Leaf}, \texttt{Stem}, and \texttt{Pod}) the nitrogen demand is given by: \begin{equation} N_{D,\;crit}=\frac{\Delta Q_{part}C_{N,\,crit}}{f_{w,\,photo}}+f_{n}(C_{N,\,crit}-C_{N,\,part})\qquad if\:C_{N,\,crit}>C_{N,\,part}\;\&\;Q_{part}>0 \end{equation} \begin{equation} N_{D,\;max}=\frac{\Delta Q_{part}C_{N,\,max}}{f_{w,\,photo}}+f_{n}(C_{N,\,max}-C_{N,\,part})\qquad if\:C_{N,\,max}>C_{N,\,part}\;\&\;Q_{part}>0 \end{equation} where $\Delta Q_{part}$ is the growth dry weight of parts, $Q_{part}$ is the green (i.e. not senesced) dry weight of parts, $f_{w,\,photo}$ is soil water stress of biomass accumulation (\autoref{eq:swstressphoto}); $C_{N,\,part}$ is the nitrogen concentration of parts; $f_{n}$ is defined by parameter \texttt{n\_deficit\_uptake\_fraction} in wheat.xml with default value 0.0001. $C_{N,\,crit}$ and $C_{N,\,max}$ are the N concentration critic and maximal of the parts, respectively (\autoref{fig:wdNitrogenConcentration}).\textbf{ }$N_{D,\;crit}$ and $N_{D,\;max}$ equal to 0, if $Q_{part}=0$. \subsection{\label{subsec:NitrogenPartitioningAndRetranslocation}Nitrogen uptake, partitioning and re-translocation} \subsubsection{Nitrogen concentrations in wheat parts} The N concentration in Leaf is calculated as follows: \begin{equation} C_{N,\,leaf}=N_{leaf}/Q_{leaf} \end{equation} \subsubsection{Nitrogen uptake} Daily total nitrogen uptake ($N_{u}$) is the lesser of N demand ($N_{d}$, \autoref{eq:NitrogenDemand}) and N supply $N_{s}$, \autoref{eq:NitrogenSupply}). \begin{equation} N_{u}=\text{min}(N_{d},\;N_{s}) \end{equation} \subsubsection{Nitrogen translocation} Daily total nitrogen uptake is distributed to the plant parts in proportion to their individual demands. \subsubsection{Nitrogen re-translocation } If there is insufficient nitrogen supplied from senescing material and soil nitrogen uptake, Grain nitrogen demand is met by re-translocating nitrogen from other plant parts. Nitrogen is available for re-translocation from un-senesced leaves and stems until they reach their defined minimum nitrogen concentration. No N re-translocation is attributed to other parts than \texttt{Grain}. \subsection{Nitrogen stresses} \subsubsection{Phenology} Nitrogen stress on phenology (via $f_{N,\,pheno}$ in \autoref{eq:CumThermalTime}) is determined by the difference between organ nitrogen concentration and organ minimum and critical nitrogen concentration. \begin{equation} f_{N,\,pheno}=h_{N,\,pheno}\sum_{stem,\,leaf}\frac{C_{N}-C_{N,\,min}}{C_{N,\,crit}\times f_{c,\,N}-C_{N,\,min}}\label{eq:NitrogenStress} \end{equation} where $C_{N}$ is the nitrogen concentration of \texttt{Stem} or \texttt{Leaf} parts; $h_{N,\,pheno}$ is multiple for nitrogen deficit effect on phenology which is specified by \texttt{N\_fact\_pheno} in the wheat.xml and default value is 100; $C_{N,\,crit}$ and $C_{N,\,min}$ are the N concentration critic and minimal of the parts, respectively (\autoref{fig:wdNitrogenConcentration}); and $f_{c,\,N}$ is a factor with a value of 1 (i.e. no impact) for Stem, and is depending on CO\textsubscript{2} for \texttt{Leaf} (\autoref{fig:wbCO2CritLeaf}). The nitrogen stress on phenology is used in the calculation of the \textquoteleft adjusted\textquoteright{} thermal time (\autoref{eq:CumThermalTime}). However, In the current version of APSIM-Wheat module, the default parameters are applied for no nitrogen water stress for phenology. \subsubsection{Biomass accumulation} Nitrogen stress on biomass accumulation (via $f_{N,\,photo}$ in \autoref{eq:StressFactor4Photosynthesis}) is determined by the difference between leaf nitrogen concentration and leaf minimum and critical nitrogen concentration. \begin{equation} f_{N,\,photo}=h_{N,\,photo}\sum_{leaf}\frac{C_{N}-C_{N,\,min}}{C_{N,\,crit}\times f_{c,\,N}-C_{N,\,min}}\label{eq:NStressPhoto} \end{equation} where $C_{N}$ is the nitrogen concentration of \texttt{Leaf} parts; $h_{N,\,photo}$ is multiplier for nitrogen deficit effect on photosynthesis which is specified by \texttt{N\_fact\_photo} in the wheat.xml and default value is 1.5; $C_{N,\,crit}$ and $C_{N,\,min}$ are the N concentration critic and minimal of the parts, respectively (\autoref{fig:wdNitrogenConcentration}); and $f_{c,\,N}$ is a factor with a value of 1 (i.e. no impact) for Stem, and is depending on CO\textsubscript{2} for \texttt{Leaf} (\autoref{fig:wbCO2CritLeaf}). The nitrogen stress on biomass accumulation affects the radiation-limited biomass accumulation ($\Delta Q_{r}$, \autoref{eq:actualBiomassProduction}). \subsubsection{Leaf appearance and expansion (i.e. leaf number and LAI)} Nitrogen stress on leaf appearance and expansion (via $f_{N,\,expan}$ in \autoref{eq:LeafExpansionStress}) is determined by the difference between leaf nitrogen concentration and leaf minimum and critical nitrogen concentration. \begin{equation} f_{N,\,expan}=h_{N,\,expan}\sum_{leaf}\frac{C_{N}-C_{N,\,min}}{C_{N,\,crit}\times f_{c,\,N}-C_{N,\,min}}\label{eq:NStressLeafExpansion} \end{equation} where $C_{N}$ is the nitrogen concentration of \texttt{Leaf} parts; $h_{N,\,expan}$ is multiplier for nitrogen deficit effect on expansion which is specified by \texttt{N\_fact\_expansion} in the wheat.xml (default value 1); $C_{N,\,crit}$ and $C_{N,\,min}$ are the N concentration critic and minimal of the parts, respectively (\autoref{fig:wdNitrogenConcentration}); and $f_{c,\,N}$ is a factor with a value of 1 (i.e. no impact) for Stem, and is depending on CO\textsubscript{2} for \texttt{Leaf} (\autoref{fig:wbCO2CritLeaf}). The nitrogen stress on leaf appearance and expansion affects the potential leaf number ($N_{d,\,pot}$; \autoref{eq:PotentialNodeNumberDaily}) and the stressed leaf area index ($\Delta\text{LAI}_{d,\,s}$, \autoref{eq:StressLeafArea}). \subsubsection{Grain filling (biomass and nitrogen demand of grain)} Nitrogen stress on grain filling affects the biomass demand of \texttt{Grain} (via $f_{N,\,grain}$ in \autoref{eq:MealDemand}) and the N demand of \texttt{Grain} (\autoref{eq:NitrogenDemand}). The nitrogen factor $f_{N,\,grain}$ (that impacts N demand of grain) is determined by the difference between organ nitrogen concentration and organ minimum and critical nitrogen concentration as follows:. \begin{equation} f_{N,\,grain}=\frac{h_{N,\ poten}}{h_{N,\ min}}h_{N,\,grain}\sum_{stem,\,leaf}\frac{C_{N}-C_{N,\,min}}{C_{N,\,crit}\times f_{c,\,N}-C_{N,\,min}}\qquad(0\leq f_{N,\,fill}\leq1)\label{eq:NStressFilling} \end{equation} where $h_{N,\ poten}$ is the potential rate of grain filling which is specified by \texttt{potential\_grain\_n\_filling\_rate} in wheat.xml and has a default value of 0.000055 g grain\textsuperscript{-1} d\textsuperscript{-1}; $h_{N,\ min}$ is the minimum rate of grain filling which is specified by \texttt{minimum\_grain\_n\_filling\_rate} in wheat.xml and has a default value of 0.000015 g grain\textsuperscript{-1} d\textsuperscript{-1}; $h_{N,\,grain}$ is a multiplier for nitrogen deficit effect on grain, which is specified by \texttt{n\_fact\_grain} in wheat.xml and has a default value of 1; $C_{N}$ is the nitrogen concentration of \texttt{Stem} or \texttt{Leaf} parts; $C_{N,\,crit}$ and $C_{N,\,min}$ are critical and minimum nitrogen concentration, respectively, for \texttt{Stem} and \texttt{Leaf} parts. $C_{N,\,crit}$ and $C_{N,\,min}$ are functions of growth stage and nitrogen concentration which is defined by parameters \texttt{x\_stage\_code}, \texttt{y\_n\_conc\_min\_leaf}, \texttt{y\_n\_conc\_crit\_leaf}, \texttt{y\_n\_conc\_min\_stem}, \texttt{y\_n\_conc\_crit\_stem} in wheat.xml and linearly interpolated by APSIM (\autoref{fig:wdNitrogenConcentration}); and $f_{c,\,N}$ is a factor with a value of 1 (i.e. no impact) for Stem, and is depending on CO\textsubscript{2} for \texttt{Leaf} (\autoref{fig:wbCO2CritLeaf}). \section{Phosphorus\label{subsec:Phosphorus-stress}} In the current version of APSIM-Wheat module, no phosphorus stress $f_{P,\,pheno}=1$ is applied in the soil system through parameter labile\_p in the source codes. \section{Temperature } As mentioned in previous sections, the temperature affects: \begin{itemize} \item crop phenology via the thermal time ($\Delta TT$; \autoref{eq:thermaltime}) and crop vernalisation ($f_{V}$; \autoref{eq:VernalisationFactor}), and via crop emergence (\autoref{eq:Emergence}), \item root depth growth ($f_{rt}$; \autoref{eq:RootGrowthTemperature}; \autoref{fig:wdTempRootFactor}), \item radiation-limited biomass accumulation ($\Delta Q_{r}$; \autoref{eq:BiomassProduction}) via a stress factor ($f_{s}$), which depends on a temperature factor ($f_{T,photo}$; \autoref{eq:TemStressPhoto}), \item CO\textsubscript{2} effect on biomass accumulation via a temperature effect on the CO\textsubscript{2} compensation point $(C_{i}$; \autoref{eq:CO2Factor4Photosynthesis}; \autoref{fig:wdCardonDioxideFactor}), \item LAI senescence under minimum and maximum temperature ($\Delta LAI_{sen,\,frost}$, $\Delta LAI_{sen,\,heat}$; \autoref{eq:SensFrost} and \autoref{eq:SensHeat}), \item biomass demand of \texttt{Grain} ($D_{g}$) and the rate of grain filling (\autoref{eq:MealDemand}; \autoref{fig:PhenologWheatModule}), \item N demand of \texttt{Grain} (\autoref{eq:NitrogenDemand}; \autoref{fig:wdNitrogenTem}), \item VPD calculation (\autoref{eq:VPD}). \end{itemize} \section{Light } Light photoperiod is calculated as detailed in \autoref{sub:Photoperiod}. Photoperiod affects wheat phenology. Light intensity and photoperiod also have an effect on diffuse light fraction (\autoref{par:Diffuse-factor}), so that it could impact the diffuse factor ($f_{d}$; \autoref{eq:BiomassProduction}; \autoref{sub:Radiation-limited-biomass}) and reduce the radiation-limited biomass accumulation ($\Delta Q_{r}$; \autoref{sub:Radiation-limited-biomass}). However, in the current APSIM-Wheat, the diffuse factor equals to 1 (i.e. no impact of diffuse light on biomass production). Light intensity affects \begin{itemize} \item radiation-limited biomass accumulation ($\Delta Q_{r}$; \autoref{sub:Radiation-limited-biomass}) via the radiation interception ($I$; \autoref{eq:RadiationInterception}), which depends on the incoming radiation ($I_{0}$) and on a light-interception factor ($f_{h}$ ) based on the canopy width. However, this canopy factor has no impact in the current version of APSIM-Wheat ($f_{h}$ = 1), \item LAI senescence under low light condition ($\Delta LAI_{sen,\,light}$; \autoref{eq:SensLight}). \end{itemize} \section{CO\protect\textsubscript{2}} As mentioned in previous sections, CO\textsubscript{2} concentration affects: \begin{itemize} \item radiation-limited biomass accumulation ($\Delta Q_{r}$; \autoref{sub:Radiation-limited-biomass}) via a CO\textsubscript{2} factor affecting the RUE ($f_{c}$; \autoref{eq:CO2Factor4Photosynthesis}), \item transpiration efficiency ($TE$, \autoref{eq:TranspirationEfficiency}) via another CO\textsubscript{2} factor ($f_{c,\,TE}$; \autoref{fig:wdCO2TE}), \item N critic concentration of the leaves (\autoref{eq:NitrogenStress}, \autoref{eq:NStressPhoto}, \autoref{eq:NStressLeafExpansion}, \autoref{eq:NStressFilling} and \autoref{fig:wbCO2CritLeaf}). \end{itemize} \section{Vapour pressure deficit (VPD) } The vapour pressure deficit (VPD) is calculated as presented in \autoref{eq:VPD}. VPD affects the transpiration efficiency (\autoref{eq:TranspirationEfficiency}) and thus the crop water demand (\autoref{eq:soilWaterDemand}). \bibliographystyle{elsart-harv} \bibliography{wd} \newpage{} \setcounter{figure}{0} \renewcommand{\thefigure}{A.\arabic{figure}} \setcounter{table}{0} \renewcommand{\thetable}{A.\arabic{table}} \begin{landscape} \appendix \section{Parameter list of wheat module} \begin{center} \begin{longtable}[c]{>{\raggedright}p{0.3\columnwidth}>{\raggedright}p{0.1\columnwidth}>{\raggedright}p{0.1\columnwidth}>{\raggedright}p{0.45\columnwidth}} \hline Variables & Units & Default Value & Description\tabularnewline \hline \endhead \multicolumn{4}{l}{\textbf{Phenology}}\tabularnewline \hypertarget{tt_}{tt\_}, (\hypertarget{tt_emergence}{tt\_emergence}, \hypertarget{tt_end_of_juvenile}{tt\_end\_of\_juvenile},\hypertarget{tt_floral_initiation}{tt\_floral\_initiation}, \hypertarget{tt_flowering}{tt\_flowering}, \hypertarget{tt_start_grain_fill}{tt\_start\_grain\_fill}, \hypertarget{tt_end_grain_fill}{tt\_end\_grain\_fill}, \hypertarget{tt_maturity}{tt\_maturity}, \hypertarget{tt_end_crop}{tt\_end\_crop}, \hypertarget{tt_harvest_ripe}{tt\_harvest\_ripe}) & $^{\circ}\text{C}$ & \autoref{fig:PhenologWheatModule} & The thermal time target for all phases\tabularnewline \hypertarget{xtemp}{x\_temp}, \hypertarget{ytt}{y\_tt} & $^{\circ}\text{C}$, $^{\circ}\text{C}$d & \autoref{fig:wdThermalTime} & The function between cardinal temperature and effective thermal time.\tabularnewline \hypertarget{pesw_germ}{pesw\_germ} & mm mm$^{\text{-1}}$ & 0 & Plant extractable soil water in seedling layer inadequate for germination\tabularnewline \hypertarget{x_node_no_leaf}{x\_node\_no\_leaf}, \hypertarget{y_leaves_per_node}{y\_leaves\_per\_node} & node rank in main stem & \autoref{fig:wdTillerNumberByNode} & The function to define the potential new tiller number \tabularnewline \hypertarget{shoot_lag}{shoot\_lag} & $^{\circ}\text{C}$d & 40 & Time lag before linear coleoptile growth starts\tabularnewline \hypertarget{shoot_rate}{shoot\_rate} & $^{\circ}\text{C}$d mm$^{\text{-1}}$ & 1.5 & Growing deg day increase with depth for coleoptile\tabularnewline \hypertarget{fasw_emerg}{fasw\_emerg} & {[}{]} & 0.0 1.0 & Fraction of available soil water\tabularnewline \hypertarget{rel_emerg_rate}{rel\_emerg\_rate} & {[}{]} & 1.0 1.0 & Stress factor for thermal time calculation between germination and emergence\tabularnewline \hypertarget{tt_emergence}{tt\_emergence} & $^{\circ}\text{C}$d & 1 & The thermal time for seed emergence\tabularnewline \hypertarget{tt_end_of_juvenile}{tt\_end\_of\_juvenile} & $^{\circ}\text{C}$d & 400 & The potential period from end of juvenile stage to terminal spikelet stage% \begin{comment} May be from emergence \end{comment} \tabularnewline \hypertarget{twilight}{twilight} & $^{\circ}$ & -6.0 & Twilight is defined as the interval between sunrise or sunset and the time when the true\tabularnewline \hypertarget{photop_sens}{photop\_sens} & {[}{]} & 3 & Sensitivities to photoperiod\tabularnewline \hypertarget{vern_sens}{vern\_sens} & {[}{]} & 1.5 & Sensitivities to vernalisation\tabularnewline \hypertarget{N_fact_pheno}{N\_fact\_pheno} & {[}{]} & 100 & Multiplier for N deficit effect on phenology\tabularnewline & & & \tabularnewline \textbf{Biomass production} & & & \tabularnewline \hypertarget{x_stage_rue}{x\_stage\_rue} & {[}{]} & 1 2 3 4 5 6 7 8 9 10 11 & Numeric code for phenological stages\tabularnewline \hypertarget{y_rue}{y\_rue} & g MJ$^{\text{-1}}$ & 0 0 1.24 1.24 1.24 1.24 1.24 1.24 0.00 0.00 0 & The radiation use efficiency for each phenological stage\tabularnewline \hypertarget{sen_rate_water}{sen\_rate\_water} & {[}{]} & 0.10 & slope in linear equation relating soil water stress during photosynthesis to leaf senescence rate\tabularnewline \hypertarget{sen_light_slope}{sen\_light\_slope} & {[}{]} & 0.002 & sensitivity of leaf area senescence to shading\tabularnewline \hypertarget{lai_sen_light}{lai\_sen\_light} & m$^{\text{2}}$ m$^{\text{-2}}$ & 7.0 & induced senescence occurs by shading\tabularnewline \hypertarget{x_sw_avail_ratio}{x\_sw\_avail\_ratio}, \hypertarget{y_swdef_pheno}{y\_swdef\_pheno} & {[}{]}, {[}{]} & \autoref{fig:wdSoilWaterStressPhenology} & The function between available soil water ratio and soil water stress of phenology.\tabularnewline \hypertarget{x_sw_avail_ratio_flowering}{x\_sw\_avail\_ratio\_flowering}, \hypertarget{y_swdef_pheno_flowering}{y\_swdef\_pheno\_flowering} & {[}{]}, {[}{]} & \autoref{fig:wdSoilWaterStressPhenology} & The function between available soil water ratio and soil water stress of phenology for flowering phase.\tabularnewline \hypertarget{x_sw_avail_ratio_start_grain_fill}{x\_sw\_avail\_ratio\_start\_grain\_fill}, \hypertarget{y_swdef_pheno_start_grain_fill}{y\_swdef\_pheno\_start\_grain\_fill} & {[}{]}, {[}{]} & \autoref{fig:wdSoilWaterStressPhenology} & The function between available soil water ratio and soil water stress of phenology for grain filling phase.\tabularnewline \hypertarget{x_stage_code}{x\_stage\_code}, \hypertarget{y_n_conc_min_leaf}{y\_n\_conc\_min\_leaf}, \hypertarget{y_n_conc_crit_leaf}{y\_n\_conc\_crit\_leaf} \hypertarget{y_n_conc_min_stem}{y\_n\_conc\_min\_stem}, \hypertarget{y_n_conc_crit_stem}{y\_n\_conc\_crit\_stem} & & & The function between growth stage and minimum can critical nitrogen concentration.\tabularnewline \hypertarget{x_row_spacing}{x\_row\_spacing} & mm & 200 350 1000 & \tabularnewline \hypertarget{y_extinct_coef}{y\_extinct\_coef} & {[}{]} & 0.50 0.50 0.50 & \tabularnewline & & & \tabularnewline \textbf{Leaf growth} & & & \tabularnewline \hypertarget{leaf_no_at_emerg}{leaf\_no\_at\_emerg} & {[}{]} & 2 & Leaf number at emergence\tabularnewline \hypertarget{initial_tpla}{initial\_tpla} & mm$^{\text{2}}$ plant$^{\text{-1}}$ & 200 & Initial leaf area per plant\tabularnewline \hypertarget{node_no_correction}{node\_no\_correction} & {[}{]} & 2 & The node number correction\tabularnewline \hypertarget{min_tpla}{min\_tpla} & mm$^{\text{2}}$ plant$^{\text{-1}}$ & 5.0 & Lower limit of total leaf area per plant\tabularnewline \hypertarget{x_lai}{x\_lai}, \hypertarget{y_sla_max}{y\_sla\_max} & mm$^{\text{2}}$ mm$^{-2}$, mm$^{\text{2}}$ g$^{\text{-1}}$ & \autoref{fig:wdSLA} & The function between leaf area index and specific leaf area.\tabularnewline \hypertarget{x_lai_ratio}{x\_lai\_ratio}, \hypertarget{y_leaf_no_frac}{y\_leaf\_no\_frac} & {[}{]}, {[}{]} & \autoref{fig:wdLAINodeNumber} & The function between fraction of leaf area index and fraction of node number.\tabularnewline \hypertarget{fr_lf_sen_rate}{fr\_lf\_sen\_rate} & {[}{]} & 0.035 & Fraction of total leaf number senescing per main stem node\tabularnewline \hypertarget{node_sen_rate}{node\_sen\_rate} & $^{\circ}\text{C}$d node$^{\text{-1}}$ & 60.0 & Rate of node senescence on main stem\tabularnewline \hypertarget{x_node_no}{x\_node\_no}, \hypertarget{y_leaf_size}{y\_leaf\_size} & node rank in main stem, mm$^{\text{2}}$ & & The leaf size as a function of leaf number\tabularnewline \hypertarget{leaf_no_pot_option}{leaf\_no\_pot\_option} & {[}{]} & 2 & The option to calculate the potential leaf number. The option 2 is for wheat.\tabularnewline \hypertarget{x_sw_demand_ratio}{x\_sw\_demand\_ratio}, \hypertarget{y_swdef_leaf}{y\_swdef\_leaf} & {[}{]}, {[}{]} & & The function between supply of soil water and water stress for leaf expansion.\tabularnewline \hypertarget{N_fact_expansion}{N\_fact\_expansion} & {[}{]}, & 1 & Multiplier for N deficit effect on leaf expansion\tabularnewline & & & \tabularnewline \hline \end{longtable} \par\end{center} \end{landscape} \end{document}