apsim-classic-wheat-doc/WheatDocumentation.Rnw

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\begin{document}
<<setup, echo=FALSE, warning=FALSE, message=FALSE>>= 

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 ([email protected]),
Karine Chenu ([email protected]), Alastair Doherty ([email protected])
and Scott Chapman ([email protected]).

\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.

<<wdCrownTemperature,fig.cap='Crown temperature ($T_{c}$) in response to air temperature ($T$) for different snow depth ($H_{snow}$) in APSIM-Wheat. In the released APSIM version, $H_{snow}$ equals zero cm.'>>=  
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} & \quad0<T_{c}\leq26\\
\frac{26}{8}(34-T_{c}) & \quad26<T_{c}\leq34\\
0 & \quad T_{c}\leq0\;\text{or}\;T_{c}>34
\end{cases}\label{eq:thermaltime}
\end{equation}

<<wdThermalTime,fig.cap='Daily thermal time ($\\Delta TT$) in response to daily crown temperature ($T_{c}$) in APSIM-Wheat.'>>=  
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}).

<<wdPhotoperiod,fig.cap='Relationship between photoperiod factor ($f_{D}$) and day length ($L_{P}$) with different sensitivities to photoperiod ($R_{p}$). The default value of $R_{P}$ is 3.'>>=  
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}

<<wdVernalisation,fig.cap='Relationship between vernalisation ($\\Delta V$) and maximum ($T_{max}$) and minimum ($T_{min}$) temperature.'>>= 
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}

<<wdDevernalisation,fig.cap='Relationship between devernalisation ($\\Delta V_{d}$) and maximum temperature ($T_{max}$) when the total vernalisation ($V$) is less than 10.'>>= 
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}) 

<<wdVernalisationFactor,fig.width=4,fig.height=4,fig.cap='Relationship between cumulated vernalisation ($V$) and vernalisation factor ($f_{V}$) and for different sensitivities to vernalisation ($R_{V}$). The default value of $R_{V}$ is 1.5.'>>= 
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})

<<wdRowExtinct,fig.cap='Values of extinction coefficient for different row spacings.'>>=  
p <- wdVisXY(wheat_xml, 
		"x_row_spacing", 
		c("y_extinct_coef",
			'y_extinct_coef_dead'),
		xlab = 'Row spacing (mm)',
		ylab = 'Extinction coefficient (k)',
		keylab = c('Green leaf', 'Dead leaf'),
		keypos = c(0.9, 0.5))  
print(p)
@

\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.

<<wdRUE,fig.cap='Radiation use efficiency (RUE) for different growth stages.'>>=  
p <- wdVisXY(wheat_xml, 
		"x_stage_rue", "y_rue",
		xlab = 'Stage code',
		ylab = 'RUE') 
print(p)
@

\subsubsection{Stress factor (Temperature, nitrogen, phosphorus (not applied), oxygen
(not applied))}

Actual daily radiation-limited biomass accumulation can be reduced
by a stress factor ($f_{s}$, \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}

<<wdTemperatureFactorOnPhoto,fig.cap='Temperature factor in response to mean daily temperature.'>>=  
p <- wdVisXY(wheat_xml, 
		"x_ave_temp", "y_stress_photo",
		xlab = expression(paste("Mean daily temperature", ~"("*degree*"C)")),
		ylab = expression(Temperature~factor~(f[T])))
print(p)
@

\paragraph{The nitrogen factor}

$f_{N,\,photo}$ is determined by the difference between leaf nitrogen
concentration and leaf minimum and critical nitrogen concentration.

\begin{equation}
f_{N,\,photo}=R_{N,\,photo}\sum_{leaf}\frac{C_{N}-C_{N,\,min}}{C_{N,\,crit}-C_{N,\,min}}\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}

<<wdCardonDioxideFactor,fig.cap='CO$_{2}$ factor in response to the CO$_{2}$ level ($C$) for different mean air temperatures.'>>=  
p <- wdCarbonDioxideFactor()
print(p)
@

\paragraph{Diffuse factor (not used in the current version)\label{par:Diffuse-factor}}

The daily diffuse fraction was calculated using the functions suggested
by \citet{roderick1999estimating}:

\begin{equation}
\begin{cases}
\frac{R_{d}}{R_{s}}=Y_{0} & \qquad for\:\frac{R_{s}}{R_{o}}\leq X_{0}\\
\frac{R_{d}}{R_{s}}=A_{0}+A_{1}\frac{R_{s}}{R_{o}} & \qquad for\:X_{0}<\frac{R_{s}}{R_{o}}\leq X_{1}\\
\frac{R_{d}}{R_{s}}=Y_{1} & \qquad for\:\frac{R_{s}}{R_{o}}>X_{1}
\end{cases}\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}<W_{d}
\end{cases}\label{eq:actualBiomassProduction}
\end{equation}
where $W_{s}$ is water supply, $W_{d}$ is the soil water demand
from the shoot, limited by radiation interception (\autoref{sub:Crop-water-demand}).
In the current APSIM-Wheat, $W_{d}$ is actually only directly affected
by the soil water demand of the leaf (\autoref{sub:Crop-water-demand}).
$W_{u}$ and $W_{d}$ are calculated by soil module of APSIM.

\section{Biomass partitioning and re-translocation}

\subsection{Biomass partitioning}

In the wheat module, wheat is divided into four components or parts:
\texttt{Root}, \texttt{Heat}, \texttt{Leaf} and \texttt{Stem} (\autoref{fig:WheatClassStructure}),
and is derived from a more generic plant module (meaning that it has
some parts not used or has a terminology, better adapted to other
crops). \texttt{Leaf} includes only leaf blades. \texttt{Stem} is
defined in a functional rather than a morphological manner and includes
plant stems, leaf sheaths and stem-like petioles (not applicable for
wheat). \texttt{Head} is divided into \texttt{Grain} and \texttt{Pod}
(which correspond to spike without the grain). Then grain are separated
into \texttt{Meal} and \texttt{Oil} (not used). The structure of wheat
parts is shown in \autoref{fig:WheatClassStructure}. 

\begin{figure}[h]
\begin{centering}
\includegraphics[height=6cm]{figure/wdBiomassPartition}
\par\end{centering}
\caption{\label{fig:WheatClassStructure}The hierarchical structure of wheat
parts. Texts in the parentheses are classes of parts. The gray box
indicates a plant part not used in wheat.}
\end{figure}

On the day of emergence, biomass in plant parts (\texttt{Root}, \texttt{Head},
\texttt{Leaf}, \texttt{Stem}, \texttt{Pod}, \texttt{Meal} and \texttt{Oil})
are initialized by \texttt{root\_dm\_init} (set at 0.01 g plant\textsuperscript{-1}
in the wheat.xml file), \texttt{leaf\_dm\_init} (0.003 g plant\textsuperscript{-1}),
\texttt{stem\_dm\_init} (0.0016 g plant\textsuperscript{-1}), \texttt{pod\_dm\_init}
(0 g plant\textsuperscript{-1}), \texttt{meal\_dm\_init} (0 g plant\textsuperscript{-1}),
\texttt{oil\_dm\_init} (0 g plant\textsuperscript{-1}), respectively.
Daily biomass production (\autoref{eq:actualBiomassProduction}) is
then partitioned to different plant parts in different ratios that
vary with crop stage. Overall, Root biomass are calculated with a
shoot:root ratio from the above-ground biomass ($\Delta Q$; \autoref{fig:BiomassPartition}).
Then the above-ground biomass are partitioned into the different plant
parts hierarchically, with biomass being attributed first to \texttt{Head},
then \texttt{Leaf} and finally \texttt{Stem}. This means that all
parts might not have the biomass demand satisfied if the biomass production
is limited. 

\begin{figure}[h]
\begin{centering}
\includegraphics[height=4.5cm]{figure/wdBiomassPartitioning}
\par\end{centering}
\caption{\label{fig:BiomassPartition}Biomass partition rules in the APSIM-Wheat
module. Texts in the parentheses are partitioning methods of different
organ types. The above-ground biomass ($\Delta Q$) is used to calculate
\texttt{Root} biomass based on a shoot:root ratio, and is then partition
to (1) \texttt{Head} based on the demand from \texttt{Pod} and \texttt{Grain},
and then (2) \texttt{Leaf }(proportion of the remaining biomass),
and (3) \texttt{Stem}. Re-translocation occurs during grain filling,
when the biomass accumulation doesn't satisfy \texttt{Head} demand.
Biomass from \texttt{Stem} and \texttt{Pod} are then used to satisfy
the \texttt{Head} demand (\texttt{Pod} and \texttt{Grain}).}
\end{figure}


\subsection{Biomass partitioning to \texttt{Root}}

Firstly, some biomass are allocated to the root as a ratio of daily
available biomass ($\Delta Q$, \autoref{eq:BiomassProduction}).
The so-called 'magic' fraction of biomass going to \texttt{Root} is
calculated from a stage-dependent function, but is independent on
pedo-climatic factors (\autoref{fig:wdroothootRatio}). All biomass
in the \texttt{Root} is considered as structural fraction, meaning
that it cannot be re-translocated to other parts later on.

\begin{equation}
\Delta Q_{root}=\Delta Q\times R_{Root:Shoot}\label{eq:RootBiomass}
\end{equation}
where $\Delta Q_{root}$ is the daily increment in \texttt{Root} biomass;
and $R_{Root:Shoot}$ is the ratio root:shoot biomass, which is defined
by x\_sta\texttt{x\_stage\_no\_partition} and \texttt{y\_ratio\_root\_shoot}
in wheat.xml (\autoref{fig:wdroothootRatio}). 

\texttt{(}which is specified in wheat.xml ) 

<<wdroothootRatio,fig.cap='Relationship between ratio of root and shoot and growth stage.'>>=  
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}). 

<<wdFractionLeaf,fig.cap='Relationship between fraction of leafLeaf  and growth stage.'>>=  
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. 

<<wdStemGrowthStructuralFractionStage,fig.cap='Relationship between fraction of structural and unstructural biomass in Stem.'>>=  
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}).

<<wdTempGrainFill,fig.cap='Response of the factor affecting the rate of grain filling in regards to daily mean temperature.'>>=  
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)
@

<<wbCO2CritLeaf,fig.cap='The CO2 modifier for critical nitrogen concentration of Leaf.'>>=  
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}).

<<wdFractionOfPod,fig.cap='Pod demand over the stages (fraction of Grain demand or of  daily biomass accumulation).'>>=  
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.

<<wdPhyllochron,fig.cap="Relationship function ($h_{p}(n_{d})$) between 'node phyllochron' ($P_{n}$) and the node number at main stem ($n_{d}$).">>=  
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}

<<wdTillerNumberByNode,fig.cap='Number of leaves per node as a function of the number of nodes on the main stem and unique stem considered in APSIM-Wheat ($n_{d}$). This relation corresponds the function $h_{l}(n_{d})$.'>>=  
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}).

<<wdLAINodeNumber,fig.cap='Relationship between fraction of leaf area index and fraction of leaf number.'>>=  
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}).

<<wdLeafSizeByNode,fig.cap='Leaf area per node ($L_{n}$) in regards to the main stem node number $n_{0} + n_{d}$.'>>=  
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}).

<<wdSLA,fig.cap='Relationship between maximum specific leaf area and leaf area index.'>>=  
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}). 

<<wdRootGrowthRate,fig.cap='Relationship between root depth growth rate ($R_{r}$) and growth stages.'>>=  
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}). 

<<wdTempRootFactor,fig.cap='Relationship ($h_{rt}$) between temperature factor on root length and daily mean temperature.'>>=  
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}). 

<<wdWaterAvaiOnRoot,fig.cap='Available soil water fraction ($f_{rwa}$) in response to the fraction of available soil water (FASW).'>>=  
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}). 

<<wdRootBranching,fig.cap='Root branching factor in response to root branching.'>>=  
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. 

<<wdHeatSenescence,fig.cap='Fraction of senescence of leaf area index ($k_{sen,\\,heat}$) in response to maximum temperature.'>>=  
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})

<<wdRootSens,fig.cap='Fraction of senescence of root biomass.'>>=  
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}). 

<<wdCO2TE,fig.cap='Relationship between factor of carbon dioxide for transpiration efficiency ($f_{c,\\, TE}$) and CO2 concentration.'>>=  
p <- wdVisXY(wheat_xml, 
		"x_co2_te_modifier", "y_co2_te_modifier",
		xlab = "Carbon dioxide concentration",
		ylab = "Transpiration efficiency factor")
print(p)
@

<<wdCoefficientOfTE,fig.cap='Change in the coefficient of transpiration efficiency with growth stages.'>>=  
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)=-W_{s}(i)\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).

<<wdSoilWaterStressPhenology,fig.cap='Relationship between soil water stress factor affecting phenology ($f_{W,\\, pheno}$) and the ratio of available soil water ($\\frac{esw_{a}}{esw_{p}}$).'>>=  
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 $W_{u}<W_{d}$),
or not (when $f_{w,\,photo}=1$, i.e. when $W_{u}=W_{d}$)

\begin{equation}
\Delta Q_{w}=\Delta Q_{r}f_{w,\,photo}=\Delta Q_{r}\frac{W_{u}}{W_{d}}\label{eq:WaterStressBiomassProduction}
\end{equation}

$f_{w,photo}$ also affect the senescence of the leaves.

\subsubsection{Leaf expansion}

Soil water stress of leaf expansion is determined by the deficit of
soil water. 

\begin{equation}
f_{W,\,expan}=h_{w,\,expan}(\frac{W_{u}}{W_{d}})\label{eq:WaterStressLeafExpansion}
\end{equation}
where $W_{u}$ is the crop water uptake (\autoref{eq:WaterUpdate}),
$W_{d}$ is the crop water demand (\autoref{eq:soilWaterDemand}).
$h_{w,\,expan}$ is a function of soil water content and stress, and
is defined by parameters \texttt{x\_sw\_demand\_ratio} and \texttt{y\_swdef\_leaf}
in the wheat.xml, which is linearly interpolated by APSIM (\autoref{fig:wdWaterStress4LeafExpansion}). 

<<wdWaterStress4LeafExpansion,fig.cap='Relationship between the soil water stress factor affecting expansion ($f_{W,\\, expan}$) and supply:demand ratio ($\\frac{W_{e}}{W_{d}}$).'>>=  
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).

<<wdKLFactoring,fig.height=6,fig.cap='The KL factor in response to chloride concentration (Cl mg kg$^{-1}$, Exchangeable sodium percentage (ESP, \\%) and soil electrical conductivity (EC, dS m$^{-1}$.'>>=  
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.

<<wdNitrogenConcentration,fig.height=6,fig.cap='Relationship between maximum, critical, minimum nitrogen concentration and growth stages for the different plant parts (Leaf, Stem and Pod). Parameters are defined by defined by parameters x\\_stage\\_code, y\\_n\\_conc\\_min\\_leaf, y\\_n\\_critonc\\_crit\\_leaf, y\\_n\\_conc\\_max\\_leaf, y\\_n\\_conc\\_min\\_stem, y\\_n\\_critonc\\_crit\\_stem, y\\_n\\_critonc\\_max\\_stem in wheat.xml.'>>=  
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}).

<<wdNitrogenTem,fig.cap='Relationship between nitrogen demand of Grain and daily mean temperature.'>>=  
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}). 

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\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_<phase_name>}{tt\_<phase\_name>}, (\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}

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