# Root growth and distribution



## Root depth growth


Between germination and start of grain filling (Fig. \@ref(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) (\#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 `stage_code_list` and `root_depth_rate`
in the wheat.xml and is linearly interpolated by APSIM (Fig. \@ref(fig:wdRootGrowthRate)). 


```{r 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}) (\#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 `x_temp_root_advance`
and `y_rel_root_advance` in the wheat.xml which is linearly
interpolated by APSIM (Fig. \@ref(fig:wdTempRootFactor)). 


```{r 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 ($f_{w,\,photo}$, Equation \@ref(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 `x_ws_root` and `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}) (\#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 `x_sw_ratio`
and `y_sw_fac_root` which is linearly interpolated by APSIM
(Fig. \@ref(fig:wdWaterAvaiOnRoot)). 


```{r 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, Equation \@ref(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) (\#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} (Fig. \@ref(fig:wdRootGrowthRate)).
This can be limited by supra- or sub-optimal mean air temperatures
(Fig. \@ref(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 (Fig. \@ref(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}).


## Root length


Daily root length growth is calculated by daily growth of `Root`
biomass ($\Delta Q_{root}$, Equation \@ref(eq:RootBiomass)) and specific
root length ($\text{SRL}$, defined by `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 (Equation \@ref(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 `x_plant_rld` and `y_rel_root_rate`
in the wheat.xml and linearly interpolated by APSIM (Fig. \@ref(fig:wdRootBranching)). 


```{r 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.