# 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.


## Node number



### Potential node appearance rate


At emergence (Fig. \@ref(fig:PhenologWheatModule)), a number of initial
leaves are specified by `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', Fig. \@ref(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}) (\#eq:phyllochron)

\end{equation}

where the function $h_{P}(n_{d})$ is defined by parameters `x_node_no_app`
and `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 (Fig. \@ref(fig:wdPhyllochron)). No effect from water
and N stress on leaf appearance is accounted for.


```{r 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)

```



### 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 (Fig. \@ref(fig:wdThermalTime))
and the 'node phyllochron', and occurs during the tiller formation
phase (Fig. \@ref(fig:PhenologWheatModule)). 

\begin{equation}

\Delta n_{d,\,p}=\frac{\Delta TT_{d}}{P_{n}} (\#eq:PotentialNodeNumber)

\end{equation}

where $\Delta TT_{d}$ is the thermal time ($^{\circ}\text{C}$d)
at day $d$ (Fig. \@ref(fig:wdThermalTime) and Equation \@ref(eq:thermaltime)).



## Leaf number



### 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'; $n_{d}$) (Fig. \@ref(fig:wdTillerNumberByNode)
and Equation \@ref(eq:LeafExpansionStress)). $h_{l}(n_{d})$ is specified
by parameters `x_node_no_leaf` and `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} (\#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 (Equation \@ref(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}\} (\#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  Section @ref(Phosphorus-stress)
and Equation \@ref(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 ($\Delta n_{d,\,p}$, Equation \@ref(eq:PotentialNodeNumber))
as follows.


\begin{equation}

\Delta N_{d,\,p}=N_{n,\,d}\times\Delta n_{d,\,p}

\end{equation}



```{r 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)

```



### 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}}) (\#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 (Fig. \@ref(fig:wdLAINodeNumber)).


```{r 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)

```