Growth of a tree

Interim photosynthesis

Based on the incoming irradiance on top of a tree $I_{ind}$ (see section Competition and environmental limitations), organic dry matter is produced via gross photosynthesis. In this section, the interim photosynthesis is calculated without reduction due to limited soil water availability and temperature effects.

The interim gross photosynthesis $P_{ind}$ of an individual is modelled using the approach of Thornley & Johnson 1990. It is based on the single-leaf photosynthesis modelled by a Michaelis-Menten function – a typical saturation function describing the relation between the radiation $I_{leaf}$ available on top of a leaf and its gross photosynthetic rate $P_{leaf}$: \begin{equation} P_{leaf}(I_{leaf}) = \frac{\alpha \cdot I_{leaf} \cdot p_{max}}{\alpha \cdot I_{leaf} + p_{max}}, \label{P_leaf} \tag{75} \end{equation} where $\alpha$ is the quantum efficiency, also known as the initial slope of the type-specific light response curve, $I_{leaf}$ is the incoming irradiance on top of the surface of a single leaf within the individual's crown and $p_{max}$ is the maximum leaf gross photosynthetic rate.

To obtain the incoming irradiance on top of the surface of a single leaf $I_{leaf}$, the available irradiance $I_{ind}$ on top of the entire individual has to be modified: \begin{equation} I_{leaf}(L) =\frac{k}{1-m} ~ I_{ind} \cdot e^{-k\cdot L}, \label{I_leaf} \end{equation} where $k$ [-] is the type-specific light extinction coefficient, $m$ [-] represents the transmission coefficient and $I_{ind}$ denotes the available incoming irradiance on top of a tree.

The first part $\frac{k}{1-m} ~ I_{ind}$ in the eqn. above is correcting the incoming irradiance in order to obtain those parts that can be absorbed by a leaf. The second part $e^{-k\cdot L}$ in the eqn. above accounts for self-shading within the individual's crown. As the leaves of an individual are assumed to be homogeneously distributed within its crown, some leaves will be shaded by higher ones within the crown. In this process, $L=0$ represents the top of the individual and $L=LAI$ represents the bottom of the individual's crown with $LAI$ being its leaf area index (see section Geometry).

To obtain the interim gross photosynthetic rate of a tree per year $P_{ind}$, the single-leaf photosynthesis of eqn. (\ref{P_leaf}) is integrated over the individual’s leaf area index $LAI$ (see section Geometry): \begin{equation} P_{ind} = \int_{0}^{LAI}P_{leaf}(I_{leaf}(L))dL.\ \label{P_tree} \end{equation}

The integration results in the interim photosynthesis of a tree per year (Thornley & Johnson 1990): \begin{equation} P_{ind} = \frac{p_{max}}{k} \cdot ln \frac{\alpha ~k~I_{ind} + p_{max}(1-m)}{\alpha ~k ~ I_{ind} ~ e^{-k \cdot LAI} + p_{max} (1-m)}. \label{P_tree_integral} \end{equation}

To convert the interim photosynthesis $P_{ind}$ from $[µmol_{CO_2}/m² s]$ to $[t_{ODM}/y]$, $P_{ind}$ has to be multiplied by the individual’s crown area $C_A$ (see section Geometry), the type-specific photosynthetically active period $\varphi_{act}$ and finally a conversion factor $c_{odm}$: \begin{equation} P_{ind} \cdot C_A \cdot 60 \cdot 60 \cdot l_{day} \cdot \varphi_{act} \cdot \varphi_{odm}, \label{GPP_tree} \tag{79} \end{equation} where the multiplication by $60 \cdot 60$ accounts for the conversion from seconds to hours. The factor $l_{day}$ [h] represents the mean day length during the vegetation period $\varphi_{act}$ [d] (see section Competition and environmental limitations). The conversion factor $\varphi_{odm} = 0.63 \cdot 44 \cdot 10^{-12}$ includes the molar mass of $CO_2$, the conversion from $g$ to $t$ and the conversion from $CO_2$ to organic dry mass $ODM$ (Larcher 2001).

Gross primary production

The gross primary production $GPP$ of a tree is calculated from the interim photosynthesis $P_{ind}$ [$t_{ODM}/y$] (see section Interim photosynthesis): \begin{equation} GPP = P_{ind} ~ \varphi_{T} ~\varphi_{W}, \label{GPP_tree2} \end{equation} where $\varphi_{W}$ denotes the reduction factor accounting for limited soil water and $\varphi_{T}$ represents the limitation factor of air temperature effect. Both factors range between 0 and 1 and thus, only reducing $GPP$ in times of unfavorable conditions (see section Competition and environmental limitations).

Biomass increment of a tree

Gross primary production $GPP$ of the eqn. above is first used for the maintenance of the already existing aboveground biomass of a tree. Costs for maintenance are modelled as biomass losses in terms of maintenance respiration $R_m$ [$t_{ODM}/y$]. The remaining productivity ($GPP-R_m$) is then available for growth of new aboveground biomass. Costs for the production of new structural tissue are modelled also as biomass losses in terms of growth respiration. This results in the net productivity $\Delta B$ (Dislich et al. 2009): \begin{equation} \Delta B = (1-r_g) ~(GPP - R_m) , \label{Binc} \end{equation} where $r_g$ [-] represents a constant parameter describing the fraction of ($GPP-R_m$) attributed to growth respiration. In contrast, maintenance respiration $R_m$ is modelled proportionally to the already existing aboveground biomass of a tree (see section Maintenance respiration).

Maintenance respiration

The maintenance respiration $R_{m}$ of a tree is calculated inversely by rearranging the eqn. above: \begin{equation} R_m = GPP - \frac{\Delta B}{1 - r_g}. \label{RM} \end{equation}

Maintenance respiration $R_m$ is further modelled proportional to the already existing aboveground biomass $B$ [$t_{ODM}$] of an individual: \begin{equation} R_{m} = \kappa_T \cdot r_{m} \cdot B,
\label{R_main} \end{equation} where $r_{m}$ denotes the maintenance respiration rate [$\frac{1}{y}$] and $\kappa_{T}$ represents a limitation factor dependent on air temperature (see section Competition and environmental limitations).

Combining the last two eqations above and arranging in terms of the respiration rate $r_m$ results in: \begin{equation} r_{m} = \frac{1}{B \cdot \kappa_T} \cdot \left(GPP - \frac{\Delta B}{1 - R_g}\right). \label{rmrate} \tag{84} \end{equation}

In FORMIND 3.0 we have two different approaches of calculating the maintenance respiration rate based on the eqn. above:

  • Optimal approach (no limitation)
  • Observation-based approach

In the following, we describe both approaches in greater detail.

Optimal approach (most frequently used)

The maintenance respiration rate $r_{m}$ of the eqn. above is calculated using the assumption of full resource availability. That is, it is assumed that full resource availability (i.e. no limitation by shading, soil water or air temperature) results in the observed maxima of field measurements of stem diameter increments: \begin{equation} r_{m} = \frac{1}{B} \cdot \left(P_{ind}(I_0) - \frac{B(D+g(D))-B}{(1-R_g)}\right),
\label{R_mainsimple} \end{equation} where this equation can be obtained by substituting in eqn. (\ref{rmrate}) (i) $\kappa_T$ by 1, (ii) $GPP$ by the gross productivity under full resource availability $P_{ind}(I_0)$ (see eqn. \ref{GPP_tree}) with $I_0$ as the full available incoming irradiance and (iii) $\Delta B$ by the biomass increment derived from the maximum stem diameter increment under full resource availability $D+g(D)$ using the individual's geometry (see section Geometry). See section Maximum diameter growth curve for different modelling approaches of the maximum diameter growth curve $g(D)$.

This approach is proposed when climate data at the time of field measurements are not available.

Observation-based approach

In this approach, the maintenance respiration rate $r_{m}$ is calculated including those climatic conditions, that were observed during the field measurements of stem diameter increments. The correspondence of environmental factors (see section Competition and environmental limitations) to these climatic conditions during the observations is indicated by $\check{\cdot}$.

\begin{equation} r_{m} = \frac{1}{B} \cdot \left(GPP(\check{I_{ind}},\check{\varphi_{act}},\check{\varphi_T},\check{\varphi_W}) - \frac{B(D+g(D))-B}{(1-R_g)} \right),
\label{R_mainref} \end{equation} This equation can be obtained by substituting in eqn. (\ref{rmrate}) (i) $\kappa_T$ by 1, (ii) $GPP$ by the gross productivity under the climate during observations $GPP(\check{I_{ind}},\check{\varphi_{act}},\check{\varphi_T},\check{\varphi_W})$ and (iii) $\Delta B$ by the biomass increment derived from the maximum stem diameter increment using the individual's geometry $D+g(D)$ (see section Geometry). See section Maximum diameter growth curve for different modelling approaches of the maximum diameter growth curve $g(D)$.

This approach is proposed when climate data are available at the time field data on stem diameter increments were measured. In general, diameter increments are determined based on the difference of stem diameter measurements between two dates. For this time period climate data would be needed on which the limitation factors $\check{I_{ind}}$, $\check{\varphi_{act}}$, $\check{\varphi_T}$ and $\check{\varphi_W}$ of the eqn. above can be calculated as described in section Competition and environmental limitations.

Maximum diameter growth curve

In the field, diameter increments can be determined by calculating the differences between two measurements of the stem diameter per tree (at two distinct observation dates). The increments are then usually plotted against the measured stem diameter of the first observation date to get an impression of how much a tree of stem diameter $D$ is able to increase (see Fig below for an example).

Alternative Text
Illustration of a measured diameter growth curve. Points represent illustrative measurements. The solid line represents a growth function to the maximum values of the measurements. Dotted lines show important characteristics that would be needed for the first approach.

Such point clouds as illustrated in the Fig. above can be described by functional relationships. Please note that you have to adjust the increments according to a time step of 1 year. That means, if there is a period of e.g. 5 years between both observation dates of stem diameter measurements, you would have to correct the increments with respect to the smaller time scale.

In FORMIND 3.0 we have two different approaches of determining the diameter growth function $g(D)$:

  • Calculation of coefficients from curve characteristics
  • Coefficients as input parameters

Below, we describe both approaches in more detail.

Calculation of coefficients from curve characteristics

Only a few information of the measured diameter increment curve need to be derived:

  • maximum diameter increment $\Delta D_{max}$ [m/y]
  • stem diameter $D_{\Delta D_{max}}$ [$% ~ \text{of}~ D_{max}$], which reaches $\Delta D_{max}$
  • maximum diameter increment $\Delta D_{D_{min}}$ [$% ~ \text{of}~ \Delta D_{max}$] of the smallest possible tree (with $D = D_{min}$)
  • maximum diameter increment $\Delta D_{D_{max}}$ [$% ~ \text{of}~ \Delta D_{max}$] of the biggest possible tree (with $D = D_{max}$)

Based on these characteristics, the coefficients of the growth function $g(D)$ can be calculated explicitly. Two different approaches are available:

  • Polynomial approach
  • Chanter approach

In the following, we show for both functional approaches of $g(D)$ the calculation of their coefficients.

Polynomial approach

The polynomial approach describes the growth function $g(D)$ as a third order polynomial: \begin{equation} g(D)=a_0 + a_1 \cdot D + a_2 \cdot D^2 + a_3 \cdot D^3, \label{growth} \end{equation} where $a_0$,$a_1$,$a_2$ and $a_3$ are the type-specific coefficients, which are calculated as follows: \begin{eqnarray*} a_3 &=& \frac{(x_0 \cdot x_1 + (\Delta D_{D_{min}} - \Delta D_{D_{max}}) \cdot \Delta D_{max} \cdot x_2)}{x_3+x_4+x_5+x_6+x_7}\\ a_2 &=& \frac{(x_0 - a_3 \cdot x_8)}{(2 \cdot D_{min} \cdot (D_{\Delta D_{max}} \cdot D_{max}) - (D_{\Delta D_{max}} \cdot D_{max})^2 - D_{min}^2)}\\ a_1 &=& -3 \cdot a_3 \cdot (D_{\Delta D_{max}} \cdot D_{max})^2 - 2 \cdot a_2 \cdot (D_{\Delta D_{max}} \cdot D_{max}) \\ a_0 &=& \Delta D_{D_{min}} \cdot \Delta D_{max} - a_3 \cdot D_{min}^3 - a_2 \cdot D_{min}^2 - a_1\cdot D_{min}\ \end{eqnarray*} with \begin{eqnarray*} x_0 &=& \Delta D_{max}-\Delta D_{D_{min}} \cdot \Delta D_{max}\\ x_1 &=& 2 \cdot (D_{\Delta D_{max}} \cdot D_{max}) \cdot (D_{min} - D_{max}) - D_{min}^2 + D_{max}^2\\ x_2 &=& 2 \cdot (D_{\Delta D_{max}} \cdot D_{max}) \cdot (D_{min} - (D_{\Delta D_{max}} \cdot D_{max})) - D_{min}^2 + (D_{\Delta D_{max}} \cdot D_{max})^2\\ x_3 &=& (D_{\Delta D_{max}} \cdot D_{max})^4 \cdot (D_{max} - D_{min})\\ x_4 &=& 2 \cdot (D_{\Delta D_{max}} \cdot D_{max})^3 \cdot (D_{min}^2 - D_{max}^2)\\ x_5 &=& (D_{\Delta D_{max}} \cdot D_{max})^2 \cdot (5 \cdot D_{min}^3 + 3 \cdot D_{min} \cdot D_{max}^2 - 3 \cdot D_{max} \cdot D_{min}^2 + D_{max}^3)\\ x_6 &=& 2 \cdot (D_{\Delta D_{max}} \cdot D_{max}) \cdot (D_{max} \cdot D_{min}^3 - D_{min} \cdot D_{max}^3)\\ x_7 &=& D_{max}^3 \cdot D_{min}^2 - D_{max}^2 \cdot D_{min}^3 + D_{min}^4 - D_{min}^5\\ x_8 &=& 3 \cdot D_{min} \cdot (D_{\Delta D_{max}} \cdot D_{max})^2 - 2 \cdot (D_{\Delta D_{max}} \cdot D_{max})^3 - D_{min}^3\ \ \end{eqnarray*}

Chanter approach

This approach describes the growth function $g(D)$ as follows: \begin{equation} g(D)=a_0 \cdot D \cdot \left(1 - \frac{D}{D_{max}}\right) \cdot e^{-a_1 \cdot D}, \end{equation} where $a_0$ and $a_1$ are the type-specific coefficients, which are calculated by: \begin{eqnarray*} a_0 &=& \frac{e^{\frac{D_{max} - 2 \cdot (D_{\Delta D_{max}} \cdot D_{max})}{D_{max} - (D_{\Delta D_{max}} \cdot D_{max})}} \cdot D_{max} \cdot \Delta D_{max}}{(D_{max} - (D_{\Delta D_{max}} \cdot D_{max})) \cdot (D_{\Delta D_{max}} \cdot D_{max})}\\ a_1 &=& \frac{D_{max} - 2 \cdot (D_{\Delta D_{max}} \cdot D_{max})}{D_{max} \cdot (D_{\Delta D_{max}} \cdot D_{max}) - (D_{\Delta D_{max}} \cdot D_{max})^2},\\ \end{eqnarray*} where $D_{max}$ is calculated out of maximum height (see section Maximum Values).

Coefficients as input parameter

For this approach, the coefficients of the corresponding growth function $g(D)$ are input parameter already known prior to the start of the simulation. The following three different functional approaches of $g(D)$ are implemented:

Weibull approach

The growth function $g(D)$ is described by a Weibull function of: \begin{equation} g(D) = a_0 \cdot a_1 \cdot a_2 \cdot (a_1 \cdot D)^{a_2 - 1} \cdot e^{-(a_1 \cdot D)^{a_2}}, \end{equation} where $a_0$, $a_1$ and $a_2$ are the type-specific coefficients.

Richards approach

The growth function $g(D)$ is described by: \begin{equation} g(D) = a_0 \cdot a_1 \cdot a_2 \cdot e^{-a_1 \cdot D} \cdot \left(1 - e^{-a_1 \cdot D}\right)^{a_2 - 1}, \end{equation} where $a_0$, $a_1$ and $a_2$ are the type-specific coefficients.

Chanter approach (most freqeuntly used)

The growth function $g(D)$ is described by: \begin{equation} g(D) = a_0 \cdot D \cdot \left(1 - \frac{D}{D_{max}}\right) \cdot e^{-a_1 \cdot D}, \end{equation} where $a_0$ and $a_1$ are the type-specific coefficients.

Please note, when determining the type-specific coefficients prior to the start of the simulation, that the curve represents growth under full resource availability. That means, not all measurements should be fitted, but only the maximum diameter increments (see Fischer 2010 p. 55 for an example).