Modeling Canopy Photosynthesis

Summary

Canopy photosynthesis models (CPMs) calculate canopy photosynthetic rate as a sum of leaf photosynthetic rate. Here we focus on one-dimensional CPMs and show that simulated rates of canopy photosynthesis vary depending on whether multiple layers or a monolayer are considered and on whether direct and diffuse light sources are considered. We discuss how canopy photosynthetic rates vary depending on plant traits, which can differ within and among species; canopy photosynthetic rates are sensitive to leaf area index, light extinction coefficient, leaf photosynthetic capacity (photosynthetic nitrogen use efficiency), and nitrogen allocation between leaves. CPMs can predict exchange rates not only for carbon but also for water and energy. The predicted rates are consistent with observations. Finally, we describe how CPMs have been utilized for vegetation and global studies.

1 I. Introduction

By definition, canopy photosynthesis is the sum of the photosynthetic rates of all leaves in the canopy. The complexity of canopy photosynthesis was first described by Boysen-Jensen (1932), who demonstrated that light dependence of canopy photosynthesis differs from that of leaves isolated from the canopy (see also Hirose 2005). It differs because leaves are exposed to different environmental conditions depending on their position in the canopy, and have different morphological and physiological traits depending on their environment and ontogeny, as has been discussed in previous Chaps. 1, 2, 3, 4, 5 (Goudriaan 2016; Gutschick 2016; Hikosaka et al. 2016; Niinemets 2016; Pons 2016). Current canopy photosynthesis models (CPMs) have incorporated these issues and photosynthetic performance of each leaf in the canopy, and permit the estimation of canopy gas exchange rate, such that predicted values are close to observations. Here we review the development of CPMs, focusing mainly on one-dimensional models and how they show the dependence of canopy photosynthetic rates on environmental variables and on plant or canopy traits. We also highlight how CPMs play important roles in terrestrial carbon cycle models and dynamic vegetation models and how they have contributed to the understanding and projection of global carbon balance.

2 II. Advances in Canopy Photosynthesis Models

The first CPM was developed by Monsi and Saeki (1953). They assessed canopy structure using a stratified clipping method and determined vertical profiles of leaf and light distribution. They found that light distribution can be described by an exponential function like the Beer–Lambert law, and the slope (extinction coefficient, k) can differ among stands depending mainly on leaf angle . A rectangular hyperbola was used to express the light-response curve of photosynthesis , and the canopy photosynthetic rate was calculated as the sum of leaf photosynthetic rates. This model successfully included the essential parts of canopy photosynthesis in a mathematical manner.

Models of light distribution were further developed, for instance, by separate distributions of direct and diffuse light , solar angle, and leaf angle distributions (de Wit 1965; Gourdriaan 1977, see Chap. 1, 2016).

Leaf canopies are characterized by vertical gradients in leaf photosynthetic properties (Saeki 1959; see Chap. 4, Niinemets 2016). However, until the early 1980s, such gradients were ignored in CPMs (i.e., all leaves in the canopy were assumed to have the same characteristics), owing mainly to limitation in computation abilities. To incorporate such variation, Hirose and Werger (1987a) introduced a two-step process first determining the relationships between the parameters of the light-response curve of photosynthesis (maximum rate, respiration rate, initial slope, and convexity of the curve), and leaf nitrogen content and then calculating the leaf N distribution in the canopy.

For leaf photosynthesis , earlier models incorporated only light as an environmental variable. Farquhar et al. (1980) developed a biochemical model of leaf photosynthesis in which CO₂ assimilation rates are expressed as the function of light, CO₂ concentration, and temperature. Ball et al. (1987) proposed an empirical model of stomatal conductance as a function of air humidity, CO₂ concentration, and photosynthetic rate. Combining these models permits the estimation of gas exchange rates under fluctuating environmental conditions (Harley and Tenhunen 1991; Harley et al. 1992; Baldocchi 1994; Harley and Baldocchi 1995; see Chap. 3, Hikosaka et al. 2016).

During the 1990s, several CPMs that incorporated light distribution, leaf property gradient, and leaf physiology were developed (e.g., MAESTRO: Wang and Jarvis 1990). Some of them incorporated heat fluxes (e.g., CANOAK: Baldocchi and Harley 1995). The predicted gas exchange rate was strongly correlated with the rate measured by the eddy covariance method (Baldocchi and Harley 1995). Nowadays, we can accurately predict canopy gas exchange if canopy characteristics and environmental variables are given.

Alternative efforts have been made to describe canopy photosynthesis using simpler models that minimize calculation time and the need for parameterization. The simplest model expresses canopy productivity as the product of light use efficiency (or radiation use efficiency), interception efficiency and solar radiation (Monteith 1972).

Farquhar (1989) showed that an equation describing whole-leaf photosynthesis has the same form as one for individual chloroplasts across a leaf, provided the distribution of chloroplast photosynthetic capacity is in proportion to the profile of absorbed irradiance and that the shape of the response to irradiance is identical in all layers. This led to a new generation of big-leaf model s (BLMs; see Sect. III.B) for canopy photosynthesis. BLMs treat the canopy as a layer of one big leaf. Some studies have simply applied a leaf photosynthesis model to calculation of canopy photosynthetic rates (e.g., Lloyd et al. 1995). However, most BLMs did not separate direct from diffuse light in the canopy. de Pury and Farquhar (1997) developed a single layered model that separately accounts for beam and diffuse lights , and is as accurate as, but simpler than, multi-layer models . Several ecosystem models use the model of de Pury and Farquhar (1997) for calculating canopy photosynthesis.

CPMs have been incorporated into terrestrial ecosystem models (Table 9.2), which play important roles for understanding the present processes and in making future projections under global environmental change. For example, terrestrial carbon cycle models, such as the Vegetation Integrated SImulator for Trace gases (VISIT: Ito et al. 2005) and the Biosphere Model integrating Eco-physiology and Mechanistic approaches using Satellite data (BEAMS: Sasai et al. 2005), use canopy models to estimate ecosystem carbon uptake from the atmosphere. Similarly, several dynamic vegetation models, such as the Lund–Potsdam–Jena (LPJ: Sitch et al. 2003) and the Organized Carbon and Hydrology in Dynamic Ecosystems (ORCHIDEE: Krinner et al. 2005) models, include canopy models but in a relatively simple manner. Because these vegetation canopy models are used to estimate carbon and water budgets of ecosystems, their uncertainty eventually influences estimation of future ecosystem responses and feedback to global change.

CPMs have also been incorporated into crop models (van Ittersum et al. 2003). Simple models used relationships between radiation and crop growth based on the light use efficiency (e.g., LINTUL: Spitters and Schapendonk 1990). Some models are a mechanistic models that consider CO₂ assimilation and respiration as a function of environmental conditions (e.g., SUCROS: Goudriaan and van Laar 1994). Recent models consider the spatiotemporal dynamics of growth and development of plants, where complex three-dimensional structures of individual plants are combined with physiological mechanisms (functional–structural plant model s , FSPM; see Chap. 8, Evers 2016).

3 III. Models of One-Dimensional Canopy Photosynthesis

3.1 A. Multi-layer Model

In most multi-layer models , the canopy comprises many horizontal layers. Light from top to bottom of the canopy is modeled , and absorbed light by leaves in each layer is calculated as described in Chap. 1 (Goudriaan 2016). In earlier models, light availability was assumed to be identical among leaves within each layer, whereas more advanced models consider sunlit and shaded leaves separately (Fig. 9.1). Sunlit leaves receive both direct and diffuse light , whereas shaded leaves receive only diffuse light. In general, leaf angle is assumed to be identical, or mean values are used, but some models consider leaves with different angles separately (e.g., Anten and Hirose 2003). In earlier models, leaf photosynthetic traits were assumed to be identical among leaves, whereas recent models incorporate vertical distributions of photosynthetic capacity or nitrogen content as an exponential or linear functions of canopy depth (normally expressed as cumulative leaf area index from the top of the canopy). In some models, differences in leaf traits among individuals or species are taken into account (Anten and Werger 1996; Hikosaka et al. 1999; Anten and Hirose 2003; See Chap. 14, Anten and Bastiaans 2016). Canopy photosynthetic rate is calculated as the sum of photosynthetic rates of leaves (Fig. 9.1).

Fig. 9.1

Principle of the canopy photosynthesis model . Diffuse light diminishes in intensity with canopy depth, whereas direct light diminishes in area with canopy depth. Sunlit leaves receive direct and diffuse light whereas shaded leaves receive only diffuse light. CO₂ assimilation rates are calculated for sunlit and shaded leaves in each layer

3.2 B. Big-Leaf Model

BLMs treat the canopy as if it were a single-layer leaf. There are several types of BLMs. The simplest one applies a leaf photosynthesis model to the canopy (Amthor 1994; Lloyd et al. 1995), with assimilation rates expressed per unit ground area instead of per unit leaf area. This model is used when data of environmental dependence of stand CO₂ exchange are available for parameter calibration, by which model estimates are adjusted to observations.

A slightly more elaborate BLM derives canopy photosynthesis A _t as an integral of photosynthesis from top leaves with some assumptions:

$$ {A}_t={A}_o\frac{1- \exp \left(- kL\right)}{k} $$

(9.1)

where A _o denotes the photosynthetic rate of the top leaf in the canopy. The term 1 − exp(−kL) is the fraction of the incident light that is absorbed by the canopy. This equation assumes the photosynthetic capacity (light-saturated rate of photosynthesis , A _max) of a leaf to be proportional to the relative light availability that it receives; for example, if a leaf receives only 50 % of the light, its A _max is also 50 % of A _max of top leaves. It further assumes other environmental factors (such as humidity and temperature) to be identical among layers (See Box 9.1 for how this equation is derived). The environmental response of A _o can be simulated by the biochemical model of the photosynthesis as shown in Chap. 3 (Hikosaka et al. 2016).

Box 9.1 Derivation of Big-Leaf Model

Here we apply a rectangular hyperbola for gross leaf photosynthesis A.

$$ A=\frac{A_{\max}\phi {I}_c}{A_{\max }+\phi {I}_c} $$

(B9.1.1)

where A _max, ϕ, I _c are light-saturated rate, the initial slope and the absorbed light by the leaf, respectively.

Light extinction is described by Beer’s law.

$$ I={I}_o \exp \left(- kL\right)={I}_oq $$

(B9.1.2)

where I _o , k, L and q is the light intensity above the canopy, light extinction coefficient , cumulative leaf area index from the top, and relative light intensity. Ignoring light scattering by leaves, I _c is derived from the differentiate of I.

$$ {I}_c={I}_ok \exp \left(- kL\right) $$

(B9.1.3)

Here we assume that A _max of a leaf is proportional to q.

$$ {A}_{\max }={A}_{o \max }q $$

(B9.1.4)

where A _omax is A _max of the top leaf. Canopy photosynthesis A _t is give as the integral of leaf photosynthesis.

$$ {A}_t={\displaystyle {\int}_0^{L_{\max }}A\kern0.5em dL} $$

(B9.1.5)

Substituting Eqs. B9.2.2 and B9.2.5 to Eq. B9.2.6, A _t is derived as follows

$$ \begin{array}{l}{A}_t={\displaystyle {\int}_0^{L_{\max }}\frac{A_{o \max }q\phi {I}_c}{A_{o \max }q+\phi {I}_c} dL}\hfill \\ {}={\displaystyle {\int}_0^{L_{\max }}\frac{A_{o \max } \exp \left(- kL\right)\phi k{I}_o \exp \left(- kL\right)}{A_{o \max } \exp \left(- kL\right)+\phi k{I}_o \exp \left(- kL\right)} dL}\hfill \\ {}=\frac{A_{o \max}\phi {I}_{oc}}{A_{o \max }+\phi {I}_{oc}}{\displaystyle {\int}_0^{L_{\max }} \exp \left(- kL\right) dL}\hfill \\ {}={A}_o\frac{1- \exp \left(- kL\right)}{k}\hfill \end{array} $$

(B9.1.6)

where A _o is A of the top leaf.

3.3 C. Sun–Shade Model

One of the shortcomings of BLMs is that they consider only average light level at each layer and ignore direct and diffuse light , which have different canopy-transfer and photosynthetic properties. Given that the light-response curve of photosynthesis is concave, an increase in light intensity increases photosynthesis in low light but not in high light. Thus, a difference in frequency of light intensity affects photosynthetic rates even when average light intensities are similar. de Pury and Farquhar (1997) developed an adjustment to the above-mentioned BLM that considered the difference between direct (beam) and diffuse light and scattering within the canopy. The model divides the canopy into two components: sunlit and shaded leaves. Shaded leaves receive only diffuse light, whereas sunlit leaves receive both direct and diffuse light. Canopy photosynthesis is calculated as the sum of photosynthesis of sunlit and shade leaves.

3.4 D. Comparison of Calculated Rates Between Canopy Photosynthesis Models

Here we use five CPMs.

3.4.1 1. Multi-layer Model Under Direct–Diffuse Light (MDDM)

In this model , solar geometry and photosynthetically active photon flux density (PFD) above the canopy were modeled as described in Box 1.1 of Chap. 1 (Goudriaan 2016) (equations are shown in Box 9.2; Eqs. B9.2.1, B9.2.2, B9.2.3, B9.2.4 and B9.2.5). The canopy comprised multiple layers (in this example, 20 layers) in which leaves were randomly distributed. Leaves received direct PFD I _c,dir, diffuse PFD I _c,dif, and scattered direct PFD I _c,sca. An example is shown in Fig. 9.2. I _c,dir was constant across the layers (Eq. B9.2.7), but the fraction of leaves receiving direct PFD decreased with canopy depth (Eq. B9.2.6). I _c,dif decreased with canopy depth (Eq. B9.2.8). I _c,sca was calculated as the difference between light interception by black (no scattering ) and actual leaves (Eq. B9.2.9). Sunlit leaves received I _c,dir, I _c,dif, and I _c,sca whereas shade leaves received I _c,dif, and I _c,sca (Eqs. B9.2.10 and B9.2.11). The light extinction coefficient for direct light was calculated as a function of leaf angle and solar angle (Anten 1997; Kamiyama et al. 2010; Eqs. B9.2.12, B9.2.13 and B9.2.14). The light extinction coefficient for diffuse light k _dif was assumed to depend on the leaf inclination angle: 0.5, 0.7, and 0.9 when this angle was 75°, 45°, and 15°, respectively. For simplicity, leaf angle was assumed to be constant within the canopy. PFD absorption was calculated assuming that it was identical among sunlit leaves and among shade leaves within each layer (Eqns B9.2.6′, B9.2.8′, and B9.2.9′).

Fig. 9.2

An example of a vertical profile of photosynthetically active photon flux density (PFD) . Assumptions include 1200 on a cloudless vernal equinox day at the equator in and leaf angle of 45°. A canopy with leaf area index of 5 was divided into 20 layers and absorbed PFD was calculated. I _c,dir is absorbed PFD of direct light per unit sunlit leaf area. I _c,dif and I _c,sca denote absorbed PFD of diffuse and scattered direct light per unit total leaf area in each layer, respectively. MSM (diffuse-light model ) denotes the absorbed PFD per leaf area when all the light above the canopy is assumed to be diffuse light (see text). Fraction is the fraction of sunlit area in each layer

Box 9.2 Equations Used in the Models

See Table 9.1 for abbreviations and units.

• 1. Solar geometry (see Chap. 1 Goudriaan 2016)

  $$ \sin \delta =-\frac{23.44\pi }{180} \cos \frac{2\pi \left({t}_{\mathrm{day}}+10\right)}{365.24} $$

  (B9.2.1)
  $$ \sin \beta = \sin \lambda \sin \delta + \cos \lambda \cos \delta \cos \frac{2\pi \left({t}_h-12\right)}{24} $$

  (B9.2.2)

• 2. Multi-layer model under direct-diffuse light (MDDM)

  • 2.1 PFD above the canopy

    $$ {I}_{\mathrm{dir}}={a}^m{I}_e \sin \beta $$

    (B9.2.3)
    $$ m=\frac{P}{P_o \sin \beta } $$

    (B9.2.4)
    $$ {I}_{\mathrm{dif}}={f}_a\left(1-{a}^m\right){I}_e \sin \beta $$

    (B9.2.5)

  • 2.2 Light absorption at depth L (see Chap. 1 Goudriaan 2016)

    $$ {f}_{\mathrm{su}}= \exp \left(-{k}_{\mathrm{dir}}^{\prime }L\right) $$

    (B9.2.6)
    $$ {I}_{c,\mathrm{d}\mathrm{i}\mathrm{r}}=\left(1-\sigma \right){k}_{\mathrm{dir}}^{\prime }{I}_{\mathrm{dir}} $$

    (B9.2.7)
    $$ {I}_{c,dif}=\left(1-{\rho}_{dif}\right){k}_{dif}{I}_{dif,0} \exp \left(-{k}_{dif}L\right) $$

    (B9.2.8)
    $$ {I}_{c,\mathrm{s}\mathrm{c}\mathrm{a}}={I}_{\mathrm{dir}}\left(1-{\rho}_{\mathrm{dir}}\right){k}_{\mathrm{dir}} \exp \left(-{k}_{\mathrm{dir}}L\right)-{I}_{\mathrm{dir}}\left(1-\sigma \right){k}_{\mathrm{dir}}^{\prime } \exp \left(-{k}_{\mathrm{dir}}^{\prime }L\right) $$

    (B9.2.9)
    $$ {I}_{c,\mathrm{s}\mathrm{u}}={I}_{c,\mathrm{d}\mathrm{i}\mathrm{r}}+{I}_{c,\mathrm{d}\mathrm{i}\mathrm{f}}+{I}_{c,\mathrm{s}\mathrm{c}\mathrm{a}} $$

    (B9.2.10)
    $$ {I}_{c,\mathrm{s}\mathrm{h}}={I}_{c,\mathrm{d}\mathrm{i}\mathrm{f}}+{I}_{c,\mathrm{s}\mathrm{c}\mathrm{a}} $$

    (B9.2.11)
    $$ {k}_{\mathrm{dir}}=\frac{O_{\mathrm{av}}}{ \sin \beta } $$

    (B9.2.12)
    $$ \begin{array}{ll}{O}_{\mathrm{av}}= \sin \beta \cos \alpha \hfill & \upbeta >\upalpha \hfill \end{array} $$

    (B9.2.13a)
    $$ \begin{array}{lllll}{O}_{\mathrm{av}}=\frac{2}{\pi}\left( \sin \beta\ \cos \alpha\ \arcsin \frac{ \tan \beta }{ \tan \alpha}\right.\hfill \\ {}\left.\kern3.5em ,+,\sqrt{{ \sin}^2\alpha -{\sin}^2\beta}\right) & \alpha >\beta \end{array} $$

    (B9.2.13b)
    $$ {k}_{\mathrm{dir}}={k}_{\mathrm{dir}}^{\prime}\sqrt{1-\sigma } $$

    (B9.2.14a)
    $$ {k}_{\mathrm{dif}}={k}_{\mathrm{dif}}^{\prime}\sqrt{1-\sigma } $$

    (B9.2.14b)

  • 2.3 Nitrogen content at depth L

    $$ {N}_l=\frac{k_b{N}_t \exp \left(-{k}_bL\right)}{1- \exp \left(-{k}_b{L}_t\right)}+{N}_b $$

    (B9.2.15)

  • 2.4 Light absorption and nitrogen content per unit leaf area in a layer between Ln+1 and Ln

    $$ {f}_{\mathrm{su}}=\frac{ \exp \left(-{k}_{\mathrm{dir}}^{\prime }{L}_n\right)- \exp \left(-{k}_{\mathrm{dir}}^{\prime }{L}_{n+1}\right)}{k_{\mathrm{dir}}^{\prime}\left({L}_{n+1}-{L}_n\right)} $$

    (B9.2.6′)
    $$ \begin{array}{l}{I}_{c,\mathrm{d}\mathrm{i}\mathrm{f}}=\left(1-{\rho}_{\mathrm{dif}}\right){I}_{\mathrm{dif},0}\hfill \\ {}\frac{ \exp \left(-{k}_{\mathrm{dif}}{L}_n\right)- \exp \left(-{k}_{\mathrm{dif}}{L}_{n+1}\right)}{L_{n+1}-{L}_n}\hfill \end{array} $$

    (B9.2.8′)
    $$ \begin{array}{l}{I}_{c,\mathrm{s}\mathrm{c}\mathrm{a}}={I}_{\mathrm{dir}}\hfill \\ {}\kern1em \frac{\begin{array}{l}\left(1-{\rho}_{\mathrm{dir}}\right)\left[ \exp \left(-{k}_{\mathrm{dir}}{L}_n\right)- \exp \left(-{k}_{\mathrm{dir}}{L}_{n+1}\right)\right]-\\ {}\left(1-\sigma \right)\left[ \exp \left(-{k}_{\mathrm{dir}}^{\prime }{L}_n\right)- \exp \left(-{k}_{\mathrm{dir}}^{\prime }{L}_{n+1}\right)\right]\end{array}}{L_{n+1}-{L}_n}\hfill \end{array} $$

    (B9.2.9′)
    $$ \begin{array}{l}{N}_l = \left({N}_o - {N}_b\right)\\ {}\kern2.6em \frac{ \exp \left(-{k}_b{L}_n\right)- \exp \left(-{k}_b{L}_{n+1}\right)}{k_b\left({L}_{n+1}-{L}_n\right)} + {N}_b\end{array} $$

    (B9.2.15′)

    where ${N}_o=\frac{k_b{N}_t}{1- \exp \left(-{k}_b{L}_t\right)}+{N}_b$.

  • 2.5 Gas exchange (see Chap. 3)

    $$ {\theta}_{cj}{A}^2-A\left({A}_c+{A}_j\right)+{A}_c{A}_j=0 $$

    (B9.2.16)
    $$ {A}_c=\frac{V_{c \max}\left({C}_i-{\Gamma}^{*}\right)}{C_i+{K}_c\left(1+O/{K}_o\right)}-{R}_d $$

    (B9.2.17)
    $$ {A}_j=\frac{J\left({C}_i-{\Gamma}^{*}\right)}{4{C}_i+8{\Gamma}^{*}}-{R}_d $$

    (B9.2.18)
    $$ J=\frac{\phi_j{I}_c+{J}_{\max }-\sqrt{\begin{array}{l}{\left({\phi}_j{I}_c+{J}_{\max}\right)}^2-4{\phi}_j{I}_c{J}_{\max }{\theta}_j\end{array}}}{2{\theta}_j} $$

    (B9.2.19)
    $$ {V}_{c \max }={V}_{c \max 25} \exp \left[\frac{E_{aV}\left({T}_k-298\right)}{298\boldsymbol{R}{T}_k}\right] $$

    (B9.2.20)
    $$ {R}_d={R}_{d25} \exp \left[\frac{E_{aR}\left({T}_k-298\right)}{298\boldsymbol{R}{T}_k}\right] $$

    (B9.2.21)
    $$ {J}_{\max 25}=\frac{\begin{array}{l}{J}_{\max 25} \exp \kern-0.1em \left[\frac{E_{aJ}\left({T}_k-298\right)}{298\boldsymbol{R}{T}_k}\right]\kern-0.4em \left[\kern-0.1em 1+ \exp \left(\frac{298\Delta S-{H}_d}{298\boldsymbol{R}}\right)\kern-0.4em \right]\end{array}}{1+ \exp \left(\frac{T_k\Delta S-{H}_d}{\boldsymbol{R}{T}_k}\right)} $$

    (B9.2.22)
    $$ {V}_{c \max 25}={a}_V\left({N}_l-{N}_b\right) $$

    (B9.2.23)
    $$ {J}_{\max 25}={a}_J{V}_{c \max 25} $$

    (B9.2.24)
    $$ {R}_{25}={a}_R{V}_{c \max 25} $$

    (B9.2.25)
    $$ {A}_t={\displaystyle \sum A} $$

    (B9.2.26)

• 3. Multi-layer model with simple light extinction (MSM)

  • 3.1 PFD above the canopy

    $$ {I}_{\mathrm{dir}}=0 $$

    (B9.2.3′)
    $$ {I}_{\mathrm{dif}}={f}_a\left(1-{a}^m\right){I}_e \sin \beta +{a}^m{I}_e \sin \beta $$

    (B9.2.5′)

    Other equations are same as those in MDDM

• 4. Big-leaf model 2 (BLM2; de Pury and Farquhar 1997 with some modifications)

  PFD above the canopy is modeled as in MDDM

  • 4.1 Light absorption by the canopy

    $$ \begin{array}{l}{I}_t={\displaystyle {\int}_0^{L_t}{I}_c dL}={I}_{\mathrm{dir}}\left(1-{\rho}_{\mathrm{dir}}\right)\hfill \\ {}\kern1.8em \left\{1- \exp \left(-{k}_{\mathrm{dir}}{L}_t\right)\right\}+{I}_{\mathrm{dif}}\left(1-{\rho}_{\mathrm{dif}}\right)\hfill \\ {}\kern1.8em \left\{1- \exp \left(-{k}_{\mathrm{dif}}{L}_t\right)\right\}\hfill \end{array} $$

    (B9.2.27)

  • 4.2 Canopy gas exchange

    $$ {\theta}_{cj}{A_t}^2-{A}_t\left({A}_{ct}+{A}_{jt}\right)+{A}_{ct}{A}_{jt}=0 $$

    (B9.2.16′)
    $$ {A}_{ct}=\frac{V_{c \max t}\left({C}_i-{\Gamma}^{*}\right)}{C_i+{K}_c\left(1+O/{K}_o\right)}-{R}_{dt} $$

    (B9.2.17′)
    $$ {A}_{jt}=\frac{J_t\left({C}_i-{\Gamma}^{*}\right)}{4{C}_i+8{\Gamma}^{*}}-{R}_{dt} $$

    (B9.2.18′)
    $$ {J}_t=\frac{\phi_j{I}_t+{J}_{\max t}-\sqrt{\begin{array}{l}{\left({\phi}_j{I}_t+{J}_{\max t}\right)}^2-4{\phi}_j{I}_t{J}_{\max t}{\theta}_j\end{array}}}{2{\theta}_j} $$

    (B9.2.19′)
    $$ {V}_{c \max t25}={a}_V\left({N}_t-{L}_t{N}_b\right) $$

    (B9.2.23′)
    $$ {J}_{\max t25}={a}_J{V}_{c \max t25} $$

    (B9.2.24′)
    $$ {R}_{dt25}={a}_R{V}_{c \max t25} $$

    (B9.2.25′)

• 5. Sun–shade BLM (SSM, de Pury and Farquhar 1997 with some modifications)

  PFD above the canopy is modeled as in MDDM

  • 5.1 LAI of sunlit leaves

    $$ {L}_{\mathrm{su}}={\displaystyle {\int}_0^{L_t}{f}_{\mathrm{su}} dL=\frac{1- \exp \left(-{k}_{\mathrm{dir}}^{\prime }{L}_t\right)}{k_{\mathrm{dir}}^{\prime }}} $$

    (B9.2.28)

    Light absorption by the canopy is defined by B9.2.27.

  • 5.2 Light absorption of sunlit leaves

    $$ \begin{array}{l}{I}_{t,\mathrm{s}\mathrm{u}}={\displaystyle {\int}_0^{L_t}{I}_{c,\mathrm{s}\mathrm{u}}{f}_{\mathrm{su}} dL}={\displaystyle {\int}_0^{L_t}{I}_{c,\mathrm{d}\mathrm{i}\mathrm{r}}{f}_{\mathrm{su}} dL}\hfill \\ {}\kern2.3em +{\displaystyle {\int}_0^{L_t}{I}_{c,\mathrm{d}\mathrm{i}\mathrm{f}}{f}_{\mathrm{su}} dL}+{\displaystyle {\int}_0^{L_t}{I}_{c,\mathrm{s}\mathrm{c}\mathrm{a}}{f}_{\mathrm{su}} dL}\hfill \end{array} $$

    (B9.2.29)
    $$ \begin{array}{l}{\displaystyle {\int}_0^{L_t}}{I}_{c,\mathrm{d}\mathrm{i}\mathrm{r}}{f}_{\mathrm{su}} dL=\\ {}\kern1em {I}_{\mathrm{dir}}\left(1-\sigma \right)\left[1- \exp \left(-{k}_{\mathrm{dir}}^{\prime }{L}_t\right)\right]\end{array} $$

    (B9.2.29a)
    $$ \begin{array}{l}{\displaystyle {\int}_0^{L_t}}{I}_{c,\mathrm{d}\mathrm{i}\mathrm{f}}{f}_{\mathrm{su}} dL={I}_{\mathrm{dif}}\left(1-{\rho}_{\mathrm{dif}}\right)\\ {}\kern1em {k}_{\mathrm{dif}}\frac{1- \exp \left[- \exp \left(-{k}_{\mathrm{dif}}{L}_t-{k}_{\mathrm{dir}}^{\prime }{L}_t\right)\right]}{k_{\mathrm{dif}}+{k}_{\mathrm{dir}}^{\prime }}\end{array} $$

    (B9.2.29b)
    $$ \begin{array}{l}{\displaystyle {\int}_0^{L_t}{I}_{c,\mathrm{s}\mathrm{c}\mathrm{a}}{f}_{\mathrm{su}} dL}={I}_{\mathrm{dir}}\left(1-{\rho}_{\mathrm{dir}}\right)\hfill \\ {}\kern1em {k}_{\mathrm{dir}}\frac{1- \exp \left[- \exp \left(-{k}_{\mathrm{dir}}{L}_t-{k}_{\mathrm{dir}}^{\prime }{L}_t\right)\right]}{k_{\mathrm{dir}}+{k}_{\mathrm{dir}}^{\prime }}\hfill \\ {}\kern1em -{I}_{\mathrm{dir}}\left(1-\sigma \right)\frac{1- \exp \left(-2{k}_{\mathrm{dir}}^{\prime }{L}_t\right)}{2}\hfill \end{array} $$

    (B9.2.29c)

  • 5.3 Light absorption of shaded leaves

    $$ {I}_{t,\mathrm{s}\mathrm{h}}={I}_t-{I}_{t,\mathrm{s}\mathrm{u}} $$

    (B9.2.30)

  • 5.4 Total canopy photosynthetic capacity is calculated by B9.2.23′

    Photosynthetic capacity of sunlit leaves

    $$ \begin{array}{l}{V}_{c \max t25,su}={\displaystyle {\int}_0^{L_t}{V}_{c \max t25}{f}_{su} dL}\hfill \\ {}\kern1em ={a}_V\left({N}_o-{N}_b\right)\frac{1- \exp \left(-{k}_b{L}_t-{k}_{\mathrm{dir}}^{\prime }{L}_t\right)}{k_b+{k}_{\mathrm{dir}}^{\prime }}\hfill \end{array} $$

    (B9.2.31)

  • 5.5 Photosynthetic capacity of shaded leaves

    $$ {V}_{c \max t,sh}={V}_{c \max t}-{V}_{c \max t,su} $$

    (B9.2.32)

  • 5.6 Canopy photosynthesis

    $$ {A}_t={A}_{t,\mathrm{s}\mathrm{u}}+{A}_{t,\mathrm{s}\mathrm{h}} $$

    (B9.2.33)

    where A _t,su and A _t,sh are calculated using V _cmaxt,su and V _cmaxt,sh, respectively, as A _t in BLM2.

Table 9.1 Abbreviations, units, and values used in the models

3.4.2 2. Multi-layer Model With Simple Light Extinction (MSM)

In this model we estimated the effect of not separating direct and diffuse light on canopy photosynthesis . To this end, PFD above and within the canopy was assumed to be diffuse light (without direct light ). Total PFD above the canopy was the same as that in the MDDM. Other variables were same as for MDDM.

3.4.3 3. Big-Leaf Model 1 (BLM1)

In this model , we calculated photosynthesis of top leaves A ₀ using PFD above the canopy (I _c,dir + I _c,dir,o ) and nitrogen content of top leaves (N _o ; see Eq. B9.2.15). A _t was derived by Eq. 9.1.

3.4.4 4. Big-Leaf Model 2 (BLM2)

The BLM2 is a type of modified BLM described by de Pury and Farquhar (1997). In this model , the variation in nitrogen content in the canopy is taken into account. PFD absorbed by the canopy on a ground-area basis was calculated by Eq. B9.2.27. Canopy photosynthetic capacity V _cmaxt was calculated from total canopy nitrogen content N _t (B9.2.23′) and other rates were simply obtained as that for leaves (Eqs. B9.2.24′ and B9.2.25′). A _t was calculated in the same way for leaf photosynthetic rate (Eqs. B9.2.16′ and B9.2.19′).

3.4.5 5. Sun–Shade Big-Leaf Model (SSM)

This model was developed by de Pury and Farquhar (1997). The sunlit fraction of leaf area index (LAI) is given by Eq. B9.2.28. Total PFD absorbed by sunlit leaves comprises direct, diffuse, and scattered direct light (Eq. B9.2.29). Total PAR absorbed by shaded leaves was obtained as the difference between PFD absorbed by the canopy and PFD absorbed by sunlit leaves (Eq. 9.2.30). Photosynthetic capacity of sunlit leaves expressed per unit ground area, V _cmaxt,su, was calculated as the integral of V _cmax of sunlit leaves, which was calculated as a function of leaf nitrogen content (Eq. 9.2.31). Photosynthetic capacity of shaded leaves was calculated as the difference between canopy photosynthetic capacity and the photosynthetic capacity of sunlit leaves (Eq. B9.2.32).

In calculation of all these models , CO₂ assimilation rates were calculated based on the biochemical model of Farquhar et al. (1980) as shown in Eqs. B9.2.16, B9.2.17, B9.2.18, B9.2.19, B9.2.20′, B9.2.21 and B9.2.22. Maximal carboxylation (V _cmax), maximal electron transport (J _max), and respiration rates (R _d ) were assumed as linear functions of leaf nitrogen content per unit leaf area (N _l ; Eqs. B9.2.23′, B9.2.24′ and B9.2.25′). Most parameters of leaf gas exchange were taken from data obtained from a wheat canopy (de Pury and Farquhar 1997). To evaluate the effects of interspecific variation in leaf traits , we adopted higher, middle, and lower values of the V _cmax to photosynthetic nitrogen ratio (a _V ; 1.16, 0.58, and 0.29), corresponding roughly to PNUE of herbaceous, deciduous tree, and evergreen tree species (Hikosaka and Shigeno 2009; see Chap. 3, Hikosaka et al. 2016 for details of the gas exchange model).

The distribution of leaf photosynthetic nitrogen content was described with an exponential function (Eq. B9.2.15) and the slope (nitrogen distribution coefficient, k _b ) was assumed to be half of the light extinction coefficient for diffuse light (Anten et al. 2000; K. Hikosaka, unpublished data; discussed below). However, as described in Box 9.1, BLM1 assumes that A _max is proportional to the relative PFD . This proportionality is achieved when the value of k _b is identical to that of the light extinction coefficient. Thus, BLM1 implicitly assumes that nitrogen distribution is steeper than that in other models .

The model simulation was performed for the vernal equinox day at the equator in which no cloud in the sky was assumed. Leaf temperature and intercellular CO₂ partial pressure were fixed at 21 °C and 25 Pa, respectively, meaning that the effect of stomatal limitation was not considered (see Sect. V for incorporating stomatal functions). Canopy photosynthetic rate was calculated every 30 min from dawn (0600) to dusk (1800) and daily carbon gain was obtained by the trapezoidal rule. The nighttime respiration rate is assumed to be twice as high as the day respiration rate (see Chap. 3, Hikosaka et al. 2016).

Figure 9.3 shows the light-response curves of canopy photosynthetic rate per unit ground area. BLM1 and BLM2 simulated a convex curve in which the rate was saturated at lower irradiance because this light response was identical to that in leaves. The difference in the light-saturated rate of canopy photosynthesis between BLM1 and BLM2 resulted from the difference in canopy nitrogen content. As mentioned above, BLM1 assumed a steeper nitrogen distribution than the other models , but the nitrogen content of top leaves was identical in all models so that total canopy nitrogen content was lower in BLM1 than in other models, leading to its lower canopy photosynthetic capacity.

Fig. 9.3

Light-response curves of the canopy photosynthetic rate in five canopy photosynthesis models . MDDM, multi-layer model under direct–diffuse light ; MSM, Multi-layer model with simple light extinction; BLM1 and BLM2, Big-leaf model 1 and 2; SSM, sun–shade big-leaf model . See text for detailed explanation of each model. Different levels of photon active radiation above the canopy assume the temporal PFD change from morning to noon in the cloudless vernal equinox day at the equator. Leaf angle was 45° in all models. Canopy nitrogen per unit ground area (N _t) was 400 mmol m⁻² except for BLM1. In BLM1, leaf nitrogen content of the top layer (N _o ) was identical to other models. The ratio of V _cmax to leaf photosynthetic nitrogen content (a _V ) was 1.16, 0.58, and 0.29 × 10⁻³ s⁻¹ in a, b, and c, respectively

The light response of the multi-layer models (MDDM and MSM) was less convex than that of the BLMs. This difference arose because light saturation of photosynthesis was not synchronized among layers; photosynthesis in lower leaves was not saturated, whereas in upper leaves it was saturated (see Terashima and Saeki 1985). Canopy photosynthetic rate at high irradiance was higher in MSM than in MDDM. Furthermore, canopy photosynthetic rate in MDDM increased gradually, even at very high irradiance. These trends occurred because photosynthesis of all leaves was nearly light-saturated in MSM, but photosynthesis of shaded leaves in lower layers was not saturated in MDDM.

The light response of canopy photosynthesis in SSM was quite similar to that in MDDM. Give that SSM is a single-layer model , the computation effort is much lower than that in multi-layer models . Thus, SSM is a more accurate and useful morel.

The variation in the light-response of canopy photosynthesis among the five models was large when the V _cmax to photosynthetic nitrogen ratio a _V was high, but it was diminished when the ratio was small (Fig. 9.3). This is because most leaves were saturated at relatively lower irradiance when a _V is low. Therefore, the error in BLMs associated with ignoring direct light , may increase with the photosynthetic nitrogen-use efficiency of the plants considered.

4 IV. Effect of Canopy Traits on Canopy Photosynthesis

In addition to the effects of environmental factors, canopy photosynthesis of a given vegetation stand depends on the characteristics of the canopy. CPMs can analyze the dependencies of canopy photosynthesis on such characteristics as total leaf area index , light extinction, total leaf nitrogen in the canopy, nitrogen distribution among leaf layers, and leaf photosynthesis, including environmental response and nitrogen use (a _V ). Environmental response of leaf photosynthesis is described in Chap. 3 (Hikosaka et al. 2016). The decrease in a _V results in a decrease in the maximum rate of canopy photosynthesis (Fig. 9.3). Here we analyze effect of other variables on canopy photosynthesis using MDDM.

Figure 9.4a and b shows the light-response curve of the photosynthetic rates of canopies with different leaf area indices. In Figure 9.4a, b, the mean leaf nitrogen content per leaf area N _l and total leaf nitrogen content in the canopy per unit ground area N _t were constant, respectively. When the mean N _l was constant, the canopy photosynthetic rate was higher in a canopy with higher LAI irrespective of light (Fig. 9.4a). This difference occurs mainly because of the greater light absorption at higher LAI. In contrast, when N _t was constant, the effect of LAI on canopy photosynthesis is not simple; the photosynthetic rate of canopy with lower LAI was higher at high irradiance , but lower at low irradiance, compared with a canopy with a higher LAI (Fig. 9.4b). This complex result is owed to a trade-off between leaf area and leaf nitrogen content. Give that total nitrogen content (N _t ) is assumed to be fixed, mean leaf nitrogen content per area (N _l ) decreases with increasing LAI. Under high light, increasing N _l is advantageous because leaves have higher photosynthetic capacity. Under low light, increasing N _l is not beneficial because photosynthesis is limited by light rather than by nitrogen (see also Anten et al. 1995b). Decreasing LAI decreases light absorption, which lowers canopy photosynthesis. Consequently, when nitrogen is limited, extremely high or low LAI is disadvantageous. Fig. 9.5 shows simulation results of daily carbon gain of a canopy in which both LAI and canopy nitrogen were altered. When canopy nitrogen content was fixed, there was an optimal LAI that maximize s daily carbon gain (Anten et al. 1995b; Hirose et al. 1997). The optimal LAI increased with increasing canopy nitrogen content (Anten et al. 2004). Optimal LAI is discussed in Chap. 13 (Anten 2016).

Fig. 9.4

Light-response curves of the canopy photosynthetic rate in the multi-layer model under direct–diffuse light (MDDM). (a) Leaf area index (LAI) was different with constant mean leaf nitrogen content per unit leaf area (N _l  = 80 mmol m⁻²); (b) LAI was different with constant canopy nitrogen content per unit ground area (N _t  = 400 mmol m⁻²); and (c) leaf angle was different. In a and b, leaf angle was 45°. In c, LAI and N _t was 5 m² m⁻² and 400 mmol m⁻², respectively. The ratio of V _cmax to leaf photosynthetic nitrogen content (a _V ) was 1.16 × 10⁻³ s⁻¹ in every case

Fig. 9.5

Daily carbon gain as a function of leaf area index and canopy nitrogen content in a canopy with leaf angle of 15° (a) or 75° (b) in the multi-layer model under direct–diffuse light (MDDM). The ratio of V _cmax to leaf photosynthetic nitrogen content was 1.16 × 10⁻³ s⁻¹

Figure 9.4c shows light-response curves of photosynthetic rate in canopies with different leaf angle s . The canopy with more-horizontal leaves (15°) had higher photosynthetic rates at lower irradiance but lower rates at higher irradiance. The higher rates occurs because more-horizontal leaves absorb more light under the same irradiance, owing to the higher associated k values. However, as the light increases, horizontal leaves that are exposed to light become light-saturated and this saturation dampens the response of canopy photosynthesis to light. In contrast, vertical leaves let more light through to lower canopy layers where it is efficiently used for photosynthesis. Thus, light is more homogenously used by different layers in the canopy. Compared at the same N _t , the optimal LAI was greater in a canopy with more vertical leaves (Fig. 9.5; Saeki 1960; Anten et al. 1995b).

Compared at the same N _t value, daily carbon gain of the canopy was similar between different leaf angle s (Fig. 9.5). This result is inconsistent with simulation results of earlier models . For example, Saeki (1960) showed that maximal daily carbon gain was lower in canopies with higher k values (see also Hirose 2005). The result of Saeki (1960) was consistent with experimental results. For example, Tanaka (1972) manipulated the leaf inclination angle of a rice stand and found that canopy photosynthetic rate at high irradiance decreased with decreasing leaf angle (see also Monsi et al. 1973). Why does a canopy with horizontal leaves have low photosynthetic capacity in earlier studies? As mentioned above, earlier CPMs assumed a constant value for photosynthetic capacity. Thus, in a canopy with higher k values, photosynthesis of the topmost leaves is light-saturated, and absorbed light is not efficiently used for photosynthesis. Why then is canopy carbon gain similar between horizontal and vertical leaves in the present simulation? This is probably because we did not assume any constraint on increasing N _l or photosynthetic capacity. In this model , canopies with more horizontal leaves have steeper nitrogen distributions (higher k _b ), leading to a higher N _l and associated light-saturated photosynthesis in upper leaves than those with more vertical leaves. However, the convexity of light-response curve of photosynthesis (θ, θ _j , or θ _cj ), one of the other characteristics of the light-response curve of photosynthesis, often decreases with increasing N _l (Hirose and Werger 1987a; Hikosaka et al. 1999), which was ignored in the present model. Furthermore, increasing photosynthetic capacity of a leaf may be constrained by the availability of other resources such as water; if an increase in V _cmax is not accompanied by a proportionate increase in stomatal and mesophyll conductance , CO₂ concentration at the carboxylation site decreases. This response may lead to a smaller increase in photosynthetic capacity with an increase in leaf nitrogen content, and thus to a saturating relationship between photosynthetic capacity and leaf nitrogen content. To increase stomatal conductance , additional biomass may need to be invested in the root system to improve hydraulics. Otherwise, plants may suffer from a risk of water deficit in the leaf canopy (such as by embolism ; see Chap. 7, Woodruff et al. 2016). Therefore, having very high leaf photosynthetic rates may be costly. If water supply is saved, higher N content may be wasteful. Such constraints on increasing photosynthetic capacity lead to an upper limit of leaf nitrogen content, and horizontal leaves may be unable to achieve high canopy photosynthesis.

Figure 9.6 shows the effect of nitrogen distribution on daily carbon gain in as simulated by the MDDM and MSM (Hikosaka 2014). Open arrows denote the value of k _b (nitrogen distribution coefficient) observed in actual canopies (half of the light extinction coefficient ; Anten et al. 2000, K. Hikosaka et al., unpublished data). Dependence of daily carbon gain on k _b was considerably different between MDDM and MSM. In MSM, daily carbon gain was maximized when the value of k _b was the same as that of the light extinction coefficient. When k _b was optimal in MSM, nitrogen content was proportional to light availability (Anten et al. 1995a; see Chap. 13, Anten 2016 for details). In contrast, optimal k _b in MDDM was much higher than that in MSM. MSM assumes that light availability is identical among leaves within the same canopy layer. Thus, plants can allocate an appropriate amount of nitrogen to leaves so that the nitrogen content is proportional to light availability. In contrast, under direct–diffuse light conditions, some leaves receive both direct and diffuse light, whereas others receive only diffuse light. Furthermore, light availability of a leaf fluctuates depending on the solar position and other factors, leading to optimal nitrogen allocation being different from that shown in MSM.

Fig. 9.6

Effects of nitrogen distribution on the daily carbon gain of the canopy. Continuous and broken lines denote carbon gain under direct–diffuse light and under diffuse light only, respectively, plotted against the nitrogen distribution coefficient (k _b ). The dotted line describes leaf nitrogen content at the top layer. Closed and open circles show carbon gain of the canopy under direct–diffuse light and under diffuse light only, respectively, where nitrogen distribution is optimized under direct–diffuse light. The cross shows nitrogen content at the top layer in the canopy optimized under direct–diffuse light. Open arrows show carbon gain at actual nitrogen distribution, which is assumed to be half of k _dif. The closed arrow shows carbon gain when nitrogen distribution is optimized under diffuse light only. Redrawn from Hikosaka (2014)

5 V. Canopy Photosynthesis Models with Heat Exchange

CPMs described above can simulate CO₂ exchange rates as a function of environmental variables. However, these models need leaf temperature and intercellular CO₂ partial pressure C _i (or stomatal conductance ) as input data (note that simulations in Sect. IV used fixed leaf temperature and C _i ). Furthermore, leaf temperature generally differs from air temperature as a result of energy exchanges in the surroundings. For this reason, a complete canopy photosynthesis model needs to incorporate energy balance s .

The surface energy balance for both an individual leaf surface and a vegetated land surface is:

$$ {R}_{\mathrm{S}\_\mathrm{ab}}+{R}_{\mathrm{L}\_\mathrm{ab}}-{R}_{\mathrm{L}\_\mathrm{out}}=H+\lambda E+G $$

(9.2)

where R _{S_ab} and R _{L_ab} are total solar and thermal radiation absorbed by leaves and bulk vegetation surface, respectively, R _{L_out} is outgoing thermal radiation from a surface, H and E are sensible heat and water vapor fluxes from this surface, respectively, λ is the latent heat of vaporization of water, and G is the heat flux into thermal storage depending on surface temperature (T _c) in a way that can be determined precisely only by solving the heat diffusion equation. At the leaf-scale, R _{S_ab} and R _{L_ab} are the computation results from the formulation of radiative transfer within the canopy (see Kumagai et al. 2006). In contrast, on land surface-scale, R _{S_ab} is calculated as (1 − a _l ) × R _S , where a _l is the albedo (the reflection of solar radiation) and R _S is solar radiation, and incident thermal radiation can be substituted for R _{L_ab} . R _{L_out} is expressed as a function of T _c according to the Stefan–Boltzmann law at the leaf-scale. Two-sided evolution and probability of no contact within a given canopy layer must be considered (see Kumagai et al. 2006).

When we consider heat balance in each canopy layer, H and λE depend on T _k , and from a leaf surface, giving:

$$ H=2\Delta L{m}_a{c}_p{g}_H\left({T}_k-{T}_a\right) $$

(9.3)

$$ \lambda E=\lambda \Delta L{m}_e{g}_E\frac{e_{\mathrm{sat}}\left({T}_{\mathrm{k}}\right)-e}{P} $$

(9.4)

where ΔL is the LAI in a given canopy layer, m _a and m _e are the molecular weights of air and water, respectively, c _p is the specific heat of air at constant pressure, g _H and g _E are the leaf-scale heat and the vapor conductances, respectively, T _k and T _a are the leaf and air temperatures, e _sat(T _k ) and e are the saturation vapor pressures at T _k , which can be represented by a function of T _k , and atmospheric water vapor pressure , respectively, and P is the atmospheric pressure.

Equations 9.2, 9.3, and 9.4 are to be solved for one unknown, individual leaf-scale T _k . As may be seen, the T _k is among the most critical factors in computing both biological and physical aspects in the atmosphere-leaf exchange models . For example, the T _k influences a computation result of leaf-scale photosynthetic rate (A) via computation of biocatalytic reactions in the photosynthesis model , and thus, controls the stomatal conductance (see Chap. 3, Hikosaka et al. 2016). Also, T _k modulates the turbulent heat and moisture transfer by controlling thermal convection above the leaf surface, resulting in alteration of the leaf boundary layer conductance . Because H and λE in Eq. 9.2 are altered through these processes, the energy balance model (Eq. 9.2) with the flux models (Eqs. 9.3 and 9.4), the photosynthesis coupled with stomatal conductance model, and the boundary layer conductance model taking into consideration the effects of both forced and free convection (see Campbell and Norman 1998), must be solved simultaneously for all unknowns by numerical iteration.

When we consider the whole vegetation, fluxes from a vegetation surface are given by:

$$ H=\rho {c}_p{g}_{Ht}\left({T}_k-{T}_a\right) $$

(9.5)

$$ \lambda E=\frac{c_p\rho }{\gamma }{g}_{Et}\left({e}_{sat}\left({T}_k\right)-e\right) $$

(9.6)

where ρ is the density of air, γ is the psychrometric constant, T _k is canopy-scale leaf temperature and g _Ht and g _Et are the canopy-scale heat and the vapor conductances, respectively. Note that the reciprocal of the total conductance is the sum of the reciprocals of the component conductances, namely the stomatal and boundary layer conductance s.

Equations 9.2, 9.5, and 9.6 are formulations from a BLM (see Sect. III.B) to be solved for one unknown, canopy-scale T _k . Practically, the T _k is the canopy surface temperature denoting radiative temperature above the canopy, which can be derived using observed upward long-wave radiation and inversion of the Stefan–Boltzmann equation. Thus, note that the canopy-scale T _k cannot be related to the leaf-scale T _k by a simple equation. However, as with the leaf-scale T _k , the canopy-scale T _k alters a computation result of canopy-scale A and stomatal conductance , resulting in modifications of surface energy partitioning (i.e., H and λE) and atmospheric stability above the canopy.

Hence, we can define local feedbacks between the T _k formation and the atmosphere-land fluxes including a carbon flux such as A. Among the feedbacks, this section discusses “aerodynamic feedback,” impacts of atmospheric convective motion induced by the T _k formation on the fluxes using canopy-scale theory (Raupach 1998). It should also be noted that when the leaf-scale theory is used, formulations describing fluxes within atmospheric surface layers (see Chap. 10, Kumagai 2016) are almost the same between the two theories. As mentioned above, g _Et (and also the canopy-scale CO₂ conductance, g _Ct ) can be represented as:

$$ \frac{1}{g_{Et}}=\frac{1}{g_{at}}+\frac{1}{g_{st}} $$

(9.7)

where g _at is the aerodynamic conductance, and g _st is the canopy stomatal conductance , whose formulation is described in Chap. 2 (Gutschick 2016). For computing g _Ct , g _at and g _st should be adjusted to be appropriate for the CO₂ transfer. g _at is usually expressed using Monin-Obukhov similarity theory (Garratt 1992):

$$ {g}_{at}=\frac{\kappa^2u}{\chi_H\left(z-d,{z}_{0H},l\right){\chi}_M\left(z-d,\ {z}_{0M},l\right)} $$

(9.8)

where κ is the von Karman constant, u is wind velocity measured at a height of z, χ _H and χ _M are dimensionless temperature and velocity profiles, respectively, d is the zero-plane displacement, z _0H and z _0M are the roughness lengths for heat and momentum, respectively, and l is the Monin-Obukhov length. The aerodynamic feedback denotes the modulations of turbulent heat and moisture transfer by alteration of g _at through atmospheric stability (l) and thereby through the surface energy balance (Eq. 9.2), feeding back on Eq. 9.2 itself (Raupach 1998).

Raupach (1998) incorporated the convective boundary layer (see Chap. 10, Kumagai 2016) slab model (McNaughton and Spriggs 1986) into the surface energy balance model (Eq. 9.2) and investigated the impact of aerodynamic feedback on computations of water vapor and CO₂ fluxes (Fig. 9.7). Here, the effect of soil water stress on photosynthetic rate (s _W values in Fig. 9.7) was also considered. Without aerodynamic feedback, surface temperature reaches up to around 40 °C even under the moistest conditions, and the high surface temperature reduces the magnitudes of water vapor and CO₂ fluxes through the effect of stomatal closure in the hottest part of the day. When aerodynamic feedback is considered, turbulent heat and moisture transfer enhanced by unstable stratification induces a cooling effect on the surface, resulting in attenuating the tendency to heat-induced stomatal closure. Thus, assessing and describing surface temperature formation and the atmospheric convective motion above the surface are necessary for building a canopy exchange model.

Fig. 9.7

Temporal variations in water vapor (F_E) and CO₂ (F_C) fluxes, and surface temperature (T_S) computed by the surface energy balance model incorporated into the convective boundary layer slab model without (left panels) and with (right panels) aerodynamic feedback. Different curves represent different values of water stress parameter s_W, which can vary between 0 (highly water-stressed) and 1 (least water-stressed). Redrawn from Raupach (1998 with modification)

6 VI. Validation

6.1 A. Plant Growth and Model Prediction

Given that assimilated carbon that is not respired is utilized for plant growth, we can test the validity of predicted gas exchange rates by comparing them with plant growth rates. Previous studies have shown that the estimated canopy carbon exchange rate is closely related to plant growth rate. For example, Hirose et al. (1997) established stands of annual plants under two CO₂ concentrations in greenhouses with natural sunlight. Although they did not determine the respiration rates of stems and roots, canopy photosynthetic rates estimated with MSM (multilayer model using simple light extinction) were significantly correlated with stand growth rates. Borjigidai et al. (2009) established stands of Chenopodium album under two CO₂ concentrations using open-top chambers. Growth rates were estimated using biomass and dead parts of plants harvested four times during the growing season. Canopy photosynthesis was estimated using MSM with environmental variables determined near the open-top chambers and respiration rates of stems and roots were determined. The estimated carbon balance was not only strongly correlated but also showed a 1:1 relationship with stand growth (Fig. 9.8a), suggesting that CPMs predict quantitatively correct rates of CO₂ exchange rates. See also Chap. 12 (Ohtsuka et al. 2016) for the case of a forest ecosystem.

Fig. 9.8

Comparison between plant growth and estimated carbon balance. (a) Growth rate per unit ground area of Chenopodium album stands as a function of estimated daily carbon balance (canopy photosynthesis minus respiration ). Stands were established at two CO₂ concentrations (open and closed circle denotes ambient and 700 μmol mol⁻¹, respectively). The plant density was 93 m⁻² and top leaves of all individuals were exposed to the canopy top. The whole plant body was harvested four times and the growth rate for each interval was determined. Carbon balance was calculated as a mean of the interval. See Borjigidai et al. (2009) for details. (b) Growth rate of individuals in C. album stands as a function of estimated daily carbon gain. Stands were established at two CO₂ concentrations (open and closed circles show ambient and 700 μmol mol⁻¹, respectively). The plant density was 400 m⁻² and there was a large variation in plant size. Aboveground mass of each individual in the census quadrat was estimated in the first census using the allometric relationship of length and diameter of stems and aboveground mass of harvested individuals from another quadrat. In the second census aboveground part of individuals in the census quadrat was estimated. Carbon gain was estimated as average for leaves of each individual in the two census (allocation of assimilates to stems and roots was not considered). See Hikosaka et al. (2003) for details

CPMs are useful for estimation of carbon exchange of individuals in a plant stand (See Chap. 14, Anten and Bastiaans 2016 for their principle). Hikosaka et al. (1999, 2003) estimated growth rates of aboveground part of individuals in a dense stand of annual plants using allometric relationships between size and mass of individual plants. The calculated plant growth rates were strongly correlated with leaf daily carbon gain of individuals (Fig. 9.8b), suggesting that CO₂ exchange rates can also be estimated correctly even at an individual level.

6.2 B. Eddy Covariance and Model Prediction

Measurements of CO₂ flux by the eddy covariance method allow us to examine terrestrial carbon cycle models in terms of canopy- and ecosystem-scale net carbon budget at fine temporal resolutions (see Chap. 10, Kumagai 2016). The eddy-covariance method directly measures net ecosystem CO₂ exchange (NEE), which can be separated into photosynthetic and respiratory components on the basis of the nighttime temperature–NEE (respiration only) relationship (Reichstein et al. 2005). Thus, it is possible to compare gross primary production (GPP, which is essentially same as the gross canopy photosynthesis ), ecosystem respiration (RE), and NEE (= GPP − RE) between model estimations and flux measurements at typically 30-min time steps. Until the mid-1990s, it was impossible to evaluate NEE directly; in most cases, field-measured net primary production (NPP) and carbon stock data at annual time steps were used for model validation. Development of the flux measurement method allowed model validation in a novel and more accurate manner. At present, flux measurement sites constitute a worldwide network, called FLUXNET (Baldocchi et al. 2001); 732 sites as of July 2015. Since the establishment of the first tower site in Harvard Forest, U.S., in 1992, more than 20 years of records have accumulated and are accessible to researchers, allowing us to explore not only micrometeorological but also ecological aspects of CO₂ fluxes (Baldocchi 2008).

Although increasing amounts of flux measurement data are available, several limitations should be considered when model validation is performed with these data. First, there are some biases and errors associated with measurements by the eddy covariance method, especially during nighttime and in mountainous areas, owing to hilly terrain. The fundamental micrometeorological theory on which the eddy covariance method is based was developed for a sufficiently turbulent condition over a flat surface and cannot adequately consider the effect of advection (transport by air mass flow). Second, the quality of flux data depends heavily on the methods used for bias correction, data selection, and gap-filling. For example, Papale et al. (2006) showed that threshold friction velocity of wind (u*, an index of atmospheric turbulence) is one of the critically important factors for data selection and quality. Third, the spatial scale usually differs between the ecosystem model and the flux measurement. In general, the upwind area contributing to the flux measured by instruments (in micrometeorology, this area is called a footprint) depends on wind condition and covers up to a few square kilometers (km²), whereas ecosystem models are often applied to broader areas. In particular, the spatial resolution of global terrestrial ecosystem models is typically hundreds to thousands of km² (e.g., 0.5° × 0.5°, a typical global-model grid size, covers about 3000 km² on the equator) often containing diverse land cover types. In addition, the footprint of flux measurement varies with wind direction. Accordingly, when flux data are used for model validation, care about such data limitations and scale-gaps is essential.

During the last 10 years, more and more studies have used flux measurement data for validation of terrestrial carbon cycle models including CPMs. Sitch et al. (2003) validated the LPJ dynamic global vegetation model at six sites in Europe. Krinner et al. (2005) validated the ORCHIDEE model at 28 sites in Europe, not only for NEE but also for energy exchange fluxes. Ito et al. (2005) and Sasai et al. (2005) applied the VISIT and the BEAMS models, respectively, to the Takayama flux measurement site in Japan; Fig. 9.9 shows examples of the site-scale validation using flux measurement data.

Fig. 9.9

Comparison of net CO₂ exchange (NEE) between eddy-covariance measurement and model estimation at the Takayama site. A process-based terrestrial ecosystem model, VISIT (Ito et al. 2006), was applied to the site, using the sun–shade canopy scheme of de Pury and Farquhar (1997). (Left) Time-series of 30-min step simulation in mid-summer of 2002, and (right) 1:1 comparison for daily NEE in 1999 and 2000. Note that seasonal change in leaf photosynthetic parameters is included in an empirical manner

Recently, flux measurement data have been used for model validation in other ways as well, especially as benchmarking data for model intercomparison. Several studies have compared ecosystem models at multiple sites (e.g., Kramer et al. 2002; Morales et al. 2005; Schwalm et al. 2010; Richardson et al. 2012; Ichii et al. 2013). These studies indicated that the present models worked poorly in several regions; for example, Morales et al. (2005) showed that most models failed in simulating CO₂ fluxes at Mediterranean sites, where water stress is a key factor. Also, these comparison studies are effective for identifying key processes. For example, Richardson et al. (2012) suggested that the present ecosystem models have difficulty in simulating leaf phenology.

These validations show that the present models have improved in capturing canopy photosynthesis in various ecosystems. Incorporating biochemical photosynthesis models and canopy radiation models (mentioned above) into ecosystem models make them highly mechanistic, allowing researchers to interpret observational data from an ecophysiological point of view and to conduct various sensitivity analyses to identify key parameters. However, it is also apparent that uncertainties remain in the present models, as implied by model intercomparison studies. As shown by the canopy-model comparison (Fig. 9.3), differences in model structure, parameter values, and assumptions can result in remarkably different simulation results. To improve simulation credibility, we need further refinement of process models including CPMs and broad-scale validation using data from multiple sites.

7 VII. Application of Canopy Photosynthesis Models to Larger Scales

Because quantification of primary productivity of the biosphere has received much interest from ecologists and geochemists, many researchers have attempted to evaluate global GPP and NPP (Ito 2011). For example, the International Biological Programme (IBP, 1965–1972) collected a large number of field data of NPP from various ecosystems and estimated global total NPP. During the IBP period, empirical models (statistical regression) were used; for example, the Miami model estimates annual NPP of any terrestrial ecosystems as a function of annual mean temperature and annual precipitation (Lieth 1975). Although these models captured the geographic variability of mean annual NPP well, they were unable to simulate seasonal and interannual variability and environmental impacts, such as for land-use change.

During the last few decades, global environmental issues have gained increasing awareness from the general public and scientific community as one of the urgent issues. In particular, temporal variability and spatial heterogeneity of the carbon cycle, including terrestrial CO₂ uptake has received attention from many researchers. Accordingly, canopy or vegetation models have been used at large spatial scales, including the global scale. Table 9.2 summarizes the canopy parameterization approaches used in several global terrestrial ecosystem models. Additionally, several recent models have adopted individual-based approaches to simulating vegetation dynamics (e.g., Levy et al. 2004; Sato et al. 2007) in conjunction with some photosynthetic scheme. Because observational data of ecosystem properties are quite limited at these scales, the models are expected to work at as little input of a priori information as possible. As mentioned above, empirical models have been widely used to estimate terrestrial primary productivity. During the early period of global studies (the 1980s and early 1990s), only a few datasets of global climatology and land cover were available. In the 1990s, many different kinds of global terrestrial models were developed in accordance with the increase of global datasets.

Table 9.2 Summary of canopy schemes in several global terrestrial ecosystem models

In particular, global satellite remote sensing data became available, enabling us to evaluate vegetation activity at broader scales. Importantly, Monteith (1977) developed a fundamental relationship between canopy-observed solar energy and vegetation productivity; the conversion coefficient was termed light-use efficiency (LUE ; carbon exchange per unit absorbed light). Using the satellite-derived vegetation absorption of solar energy (PFD ) and the LUE principle, it was possible to estimate NPP by remote sensing. Since 1982, continuous monitoring data of global terrestrial vegetation are available such as NOAA-AVHRR, Terra/Aqua-MODIS, and SPOT-VEGETATION for the purposes of various analytical and modeling studies. Time-series of vegetation indices (e.g., NDVI , Normalized Difference Vegetation Index) have revealed seasonal and interannual variation of vegetation activity (Myneni et al. 1997; Nemani et al. 2003). The vegetation indices are also useful for characterizing the photosynthetic properties of vegetation stands. For example, Sellers (1985) estimated the fraction of canopy-absorbed PFD from the simple ratio of visible red to near-infrared reflectance. Subsequently, Potter et al. (1993) adopted this approach and developed a new terrestrial ecosystem model , the Carnegie-Ames-Stanford Approach (CASA). Similar methodology was employed by several models (e.g., Ruimy et al. 1996; Goetz et al. 1999). These models are simple and reasonably capture the present vegetation productivity. However, they have several shortcomings: (1) these models are driven by satellite-observed data and so are not applicable for future projections; (2) it is difficult to include ecophysiological findings to improve this kind of model, although Sasai et al. (2005) has developed a mechanistic satellite-driven model. More recently, new sensors, such as synthetic aperture radar and lidar, are used to assess canopy structure from the space.

To improve future projections, process-based models of terrestrial ecosystems are effective because these models consider ecophysiological factors such as different environmental responsiveness between C₃ and C₄ plants. Several process-based models have been developed on the basis of stand-scale carbon cycle models and tested using field data. Running and Hunt (1993) developed a global model , the Biome-BGC, on the basis of the Forest-BGC model developed for pine forest studies in Montana, U.S. Similarly, Ito and Oikawa (2002) developed the Simulation model of Carbon cYCle in Land Ecosystem (Sim-CYCLE) on the basis of a carbon cycle model that was developed for tropical forest studies in Pasoh, Malaysia. Figure 9.10 shows the global annual NPP and its water-use efficiency estimated by VISIT (Ito and Inatomi 2012), developed from Sim-CYCLE. Earlier global terrestrial models adopted the “big-leaf” canopy approach for simplicity, and it is notable that these models are able to predict LAI and estimate impacts of environmental change. For global application, these models should estimate leaf phenology in deciduous forests and grasslands, which is determined by temperature, water, and radiation (for example, day-length) conditions. Many models include some phenological scheme, in which leaf seasonal display and shedding occur on the basis of cumulative temperatures above/below certain threshold temperatures. The “big-leaf” scheme in earlier process-based models considers vertical attenuation of PFD within canopy but in most cases neglects the difference between direct and diffuse radiation as discussed earlier (see Sect. III.B. BLM). Using the Monsi–Saeki theory, leaf-level photosynthesis is integrated to canopy-level gross primary production (GPP), considering environmental factors (ambient CO₂, temperature, and moisture) in empirical but ecophysiological ways. For example, the leaf modules estimate stomatal conductance using several semi-empirical models (e.g., Ball et al. 1987; Leuning 1995; see Chap. 3, Hikosaka et al. 2016), which respond directly to atmospheric humidity and CO₂ concentration. A similar approach was adopted by land-surface parameterization schemes (such as the second version of Simple Biosphere [SiB2]; Sellers et al. 1997) used in climate models, which need to estimate surface energy and gas exchange including stomatal regulation. In most cases, the temperature and moisture limitations were included by developing empirical scholar functions (i.e., multipliers; from zero under severe conditions to one under standard condition) for maximum photosynthetic rate. Because terrestrial models differ in canopy integration (for example, assumption of light attenuation coefficient) and environmental functions, their simulation results are not always consistent, implying estimation uncertainty.

Fig. 9.10

Global distribution of (a) net primary production (NPP) of terrestrial ecosystems and (b) water-use efficiency (WUE, carbon assimilation per unit later loss) in 1995–2004, estimated by VISIT model (Ito and Inatomi 2012)

Since the late 1990s, advances in CPMs have made it easier to conduct simulations of global terrestrial production and the carbon cycle. On the other hand, model validations in comparison with observation data have indicated that there remain large uncertainties and insufficiencies in the present models . One evident example is the anomalous CO₂ uptake after the huge eruption of Mt. Pinatubo, Philippines, in June 1991. After the eruption, a massive amount of volcanic ash was ejected into the atmosphere, as far as to the stratosphere, resulting in unusual scattering of solar radiation . This event was associated with anomalous cooling of Earth’s surface by about 0.5 °C, affecting terrestrial ecosystems including croplands. Simultaneously, the atmospheric CO₂ growth rate slowed down notably, but the mechanism causing this slowdown has not been clear. Several model studies implied a reduction of respiratory emissions due to cooling, but it was insufficient to explain the phenomenon fully. Gu et al. (2003) pointed out that net CO₂ uptake of the Harvard Forest increased apparently after the Mt. Pinatubo eruption and proposed a hypothesis that increase of diffuse radiation after the event enhanced photosynthetic CO₂ uptake by the vegetation canopy at broad scale (Roderick et al. 2001). Because the “big-leaf” canopy scheme could not evaluate the effect of different radiation components (direct and diffuse radiation), the anomalous event enhanced use of the more mechanistic canopy radiation transfer and photosynthetic scheme to adequately capture the interannual variability of terrestrial CO₂ budget. Also, Mercado et al. (2009) implied that the diffuse-radiation fraction would increase as a result of human-emitted aerosols, indicating the importance of an improved canopy scheme for global terrestrial ecosystem models. Accordingly, several recent terrestrial models employ sun–shade canopy schemes (SSM) to include biochemical photosynthetic responses and canopy absorption of direct–diffuse light as discussed in Sect. III.

More than 30 terrestrial ecosystem models applicable to global scale have been developed. They are used for simulating not only current states but also past and future changes in response to environmental change. Process-based models are expected to work reasonably well under different conditions because they take account of ecophysiological factors (such as CO₂ fertilization effects on photosynthesis ) determining ecosystem responsiveness. For example, Melillo et al. (1993) estimated that global NPP would increase by about 20 % under doubled atmospheric CO₂ and climate change condition using the Terrestrial Ecosystem Model (TEM). More recently, Friend et al. (2014) assessed the future change in vegetation carbon budget on the basis of simulation results of seven terrestrial ecosystem models.

As shown in Table 9.2, current terrestrial models adopt different canopy schemes in terms of complexity and environmental responsiveness. Another important feature is the inclusion of nitrogen effects on canopy photosynthesis . Using the model sensitivity analysis, Friend (2001) found that realistic nitrogen allocation should be taken into account for improving model simulations, in comparison with classical “big-leaf” models. This finding is consistent with field-scale ecophysiological and modeling studies (e.g., Hirose and Werger 1987b; Hikosaka 2014). The current models differ in approach and complexity in parameterization of canopy processes (Table 9.2), leading to considerable intermodel difference in estimated results. For example, Cramer et al. (1999) compared global terrestrial NPP estimated by 17 models and found that the results ranged from 39.9 to 80.5 Pg C year⁻¹ (Pg = 10¹⁵ g), even using common input climate data and simulation protocols. Such estimation uncertainty has not been reduced until recently. In the Multi-scale Terrestrial Model Intercomparison Project (MsTMIP; Huntzinger et al. 2013), it was found that 10 terrestrial models differ in estimates of global terrestrial NPP from 36 to 67 Pg C year⁻¹ at the present time. A similar range of variability was also found among the terrestrial models embedded in Earth System Models (Todd-Brown et al. 2013), which are used for climate projection. Apparently, such uncertainty exerts serious influences on future projections under global environmental change, including the climatic feedback by the terrestrial biosphere through CO₂ exchange. Further studies are needed to improve vegetation canopy models and to reduce estimation uncertainty.

8 VIII. Conclusion

Current CPMs can predict land–atmosphere exchange rates of carbon, water, and energy almost correctly. Not only detailed models but also simplified models, some of which provide quite accurate predictions, have been developed. Scaling up from leaf to canopy level contributes to the understanding of mechanisms that cause variation in canopy exchange rates of gas and energy. CPMs also contribute to future projections of responses in ecosystem functions to future global climate change. For accurate prediction, however, we need detailed information of plant traits such as LAI , light extinction coefficient , leaf biochemical characteristics, and vertical variation in leaf traits, all of which vary considerably between species and as a function of environmental factors. Since part of environmental responses of such plant traits are not fully known, some projections involve large uncertainties. The increasing amounts of flux measurement data, ecophysiological findings, and the improvement of data–model fusion, especially in a collaborative manner, will bring new and deeper insights and eventually allow advanced prediction.