## 1. Introduction

Canopy models that link the terrestrial biosphere to atmosphere can be categorized as either multilayer or big-leaf models. A multilayer model integrates the fluxes from each layer to give the total flux (Wang and Jarvis 1990; Leuning et al. 1995); while the big-leaf approach maps properties of the whole canopy onto a single leaf to calculate the flux (e.g., Sellers et al. 1996a; Bonan 1996; Dickinson et al. 1998). These methods necessarily use different parameterizations for the nonlinear relationships that govern assimilation and transpiration. The multilayer models can use parameters that are measured at the leaf level. The big-leaf models require parameters at the canopy level that cannot be measured directly, nor defined as the arithmetic mean of leaf-level parameters because of nonlinearity (McNaughton 1994; Wang et al. 2001). Rather, they require some plausible assumption about the vertical profile of leaf properties such as the Rubisco enzyme. This is commonly determined through the hypothesis that the limiting rate of carbon uptake *V*_{max} should vary with canopy depth as the photosynthetically active radiation (PAR; e.g., Sellers et al. 1992). With such an assumption, canopy properties, such as *V*_{max} and potential electron transport rate *J*_{max} can be related to the corresponding measurable properties of the leaves at the canopy top.

Big-leaf models have been extensively used in land surface climate modeling (Sellers et al. 1996a; Bonan 1996; Dickinson et al. 1998; Dai et al. 2003). They require fewer parameters and are more economical in computer time than the multilayer models. They treat the canopy as single big leaf with the fluxes of leaf energy, water, and CO_{2} calculated by coupled equations. The models of Bonan (1996) and Dickinson et al. (1998) treat the assimilation and stomatal conductance of sunlit leaves separately from that of shaded leaves but use a single leaf temperature.

Wang and Leuning (1998) developed a more comprehensive two-leaf approach with the canopy described by variables and parameters that represent the bulk properties of all sunlit or shaded leaves, using the Goudriaan and van Larr (1994) radiation model to estimate the total amount of radiation in visible, near-infrared, and thermal bands. Conductances of the stomata and leaf boundary layer are also integrated over all the sunlit and shaded leaves separately. They compared the fluxes of CO_{2}, water vapor, and sensible heat as predicted by their two-leaf model with those of a multilayered model, and found no significant differences in their predictions. The two-leaf model has recently been refined to further improve the agreement in the predicted fluxes between the two-leaf model and a multilayered canopy model (Wang 2000). Wang (2003) showed that the fluxes predicted by the two-leaf model with Goudriaan and van Larr's radiation are very similar (relative differences <5%) to those by the two-leaf model using the two-stream approximation (Sellers 1985). Wang and Leuning (1998) derived the physiological and physical properties for both sunlit and shaded leaves similar to the approach of Sellers et al. (1992, 1996a), but considered sunlit and shaded leaves separately. Other differences between Wang and Leuning's two-leaf model and Seller's one big-leaf model are use of a different stomatal model, allowing for decoupled vegetation (see Raupach 1995).

The leaf biochemical processes depend on PAR and leaf temperature. Direct sunshine heats leaves more than the scattered light in the shade, and hence sunlit leaves can be several degrees warmer than shaded leaves under sunny and dry conditions. Leaves that are most frequently shaded are thinner than those in the sun. Both the epidermal and mesophyll tissues are thicker in sunlit leaves, and the number of cells per unit leaf volume is higher. Morpho-anatomical and physiological adaptations allow the chestnut to optimize its use of the limited radiant energy available, but shading greatly reduces productivity (Boardman 1977). The net photosynthetic rate *A*_{n} of sunlit leaves is relatively high due to light saturation but there is a drastic reduction in *A*_{n} with the low light levels of shaded leaves. If such differences in physiological properties of leaves and the differences in PAR and temperatures between sunlit and shaded leaves are neglected, the estimates of photosynthesis and energy fluxes for the canopy may be significantly in error. Photosynthesis of shaded leaves has an essentially linear response to absorbed PAR, while photosynthesis of sunlit leaves is often light saturated and so independent of absorbed PAR. Hence the averaging of PAR in each of these two classes of leaves, sunlit and shaded, is appropriate and should introduce little error in the final predicted canopy photosynthesis. This concept (Sinclair et al. 1976) has been introduced into land surface modeling by de Pury and Farquhar (1997), Wang and Leuning (1998).

Leaves change their position with wind and the sun during the day causing the surfaces in direct sun (sunflecks) to move around. Leaves require on the order of 3–20 min for opening stomata and 12–35 min for closing stomata (Woods and Turner 1971; Sellers 1985). Leaves change their temperatures over similar time scales or shorter. Thus, over the typical model time step, it is appropriate to assume leaf stomatal function and temperatures have equilibrated to light levels as assumed in the separation into sunlit and shaded fractions.

We have developed a submodel for assimilation and stomatal conductance with two big leaves that treats the leaf temperatures and fluxes (energy, water, and CO_{2}) separately for sunlit and shaded fractions of canopy. The submodel formulations and parameters are described in section 2. The submodel was incorporated into the Common Land Model (CLM) and run for two forest sites for 2 yr with the observed meteorological forcings. The simulated fluxes and leaf temperatures were evaluated with the observations and compared with what were obtained by CLM with a single big-leaf submodel and CLM with a National Center for Atmospheric Research (NCAR) Land Surface Model (LSM) scheme, as presented in section 4.

## 2. Model description

Canopy photosynthesis is modeled by equations describing the fluxes of both CO_{2} and water vapor at the leaf level, and some method of scaling from leaf to canopy. Leaf-level responses require some form of parameterization of both the biochemistry within chloroplasts and the stomatal response to the environment. Most canopy gas exchange models employ the leaf biochemical model of photosynthesis by Farquhar et al. (1980) and some form of empirical stomatal response (Ball 1988; Leuning et al. 1995). In this section, we give the expressions used for the fraction of sunlit and shaded leaves, energy balance, radiative transfer, the photosynthesis and stomatal conductance for sunlit and shaded leaves separately, and their scaling up from leaf to canopy.

### a. Fraction of sunlit/shaded leaves

*x,*

*f*

_{Sun}, and

*f*

_{Sha}, are calculated from

*x*is the cumulative leaf area index measured downwards from the top of the canopy,

*k*

_{b}is the direct beam extinction coefficient of the canopy and

*k*

_{b}

*G*

*μ*

*μ,*

*G*(

*μ*) =

*ϕ*

_{1}+

*ϕ*

_{2}

*μ,*

*ϕ*

_{1}= 0.5 − 0.633

*χ*− 0.33

*χ*

^{2},

*ϕ*

_{2}= 0.877(1 − 2

*ϕ*

_{1}), and

*χ*is an empirical parameter related to the leaf angle distribution, varying from −1 to 1 (1 for horizontal leaves, −1 for vertical leaves, and 0 for a spherical leaf angle distribution), and

*μ*is the cosine of solar zenith angle. The leaf area index of sunlit and shaded fraction of canopy are calculated by integrating (1):

This division into sunlit and shaded leaves is important in scaling canopy processes, because the sunlit leaves will receive a much higher light flux density than shaded leaves under sunny conditions. The possible importance of such leaf orientation effects as discussed by Sinclair et al. (1976) and Norman (1979) is not considered here. Figure 1 calculates the variation of the sunlit fraction of a canopy with solar angle *θ.* The sunlit fraction decreases significantly for *θ* > 30° and reaches constant values for leaf area index greater than 5. The *L*_{Sun} calculated by Eq. (3) is in reasonable agreement with the measurements by Kucharik et al. (1998): typically, *L*_{Sun} in aspen (*L*_{AI} = 3.3) range between 0.8–1.0 at a 70° sun angle and 1.1–1.6 at a 30° sun angle. The canopy keeps a considerable percentage of leaves in shade at all solar angles with a minimum fraction *F*_{Sha} = 20% for *L*_{AI} = 1 to *F*_{Sha} = 60% for *L*_{AI} = 5.

### b. Equations of leaf temperatures and fluxes

#### 1) Leaf temperature of sunlit/shaded fraction of canopy

*j*” denotes the sunlit (

*j*= 1) and shaded (

*j*= 2) fraction of canopy, respectively; [ ]

_{j=1}and [ ]

_{j=2}denote the integration for sunlit and shaded fractions of canopy, respectively, that is,

^{LAI}

_{0}

*f*

_{Sun}

*dx*and

^{LAI}

_{0}

*f*

_{Sha}

*dx*, but for temperatures and partial pressures, they mean the bulk (averaged) values. The value

*C*

_{c}is the canopy heat capacity (J m

^{−2}K

^{−1}) and is assumed to be negligible,

*I*

_{s}is the summed net solar radiation absorbed by sunlit and shaded fraction of canopy (W m

^{−2}),

*I*

_{ir}is the net longwave radiation absorbed by sunlit/shaded fractions of canopy (W m

^{−2}),

*H*

_{c}is the sensible heat flux from foliages to canopy air (W m

^{−2}),

*E*

_{c}is the moisture fluxes (transpiration and evaporation of intercepted water) from leaves to canopy air (kg m

^{−2}s

^{−1}). The equations are

*ρ*and

*c*

_{p}is the density and specific heat of air (kg m

^{−3}, J kg

^{−1}K

^{−1}), respectively;

*δ*

_{1}and

*δ*

_{2}is the step functions for sunlit and shaded leaves, respectively, and 1 for positive argument and 0 for negative argument of {

*q*

_{sat}([

*T*

_{l}]) −

*q*

_{af}};

*S*

_{AI}is the dead leaf or stem area index;

*f*

_{wet}is fraction of wet leaves;

*r*

_{b}is the averaged (bulk) leaf boundary layer resistance (s m

^{−1});

*r*

_{s}is the averaged leaf stomatal resistance (s m

^{−1});

*q*

_{sat}(

*T*

_{l}) is the saturated specific humidity (kg kg

^{−1}) at temperature

*T*

_{l};

*T*

_{l}is the leaf temperature of canopy. Here,

*T*

_{af}and

*q*

_{af}are the temperature and specific humidity of canopy space air, respectively, (K, kg kg

^{−1}).

*H*

_{c}]

_{j=1}, [

*H*

_{c}]

_{j=2}and from the ground

*H*

_{g}must be balanced by heat flux to atmosphere

*H*

_{a}, that is,

*H*

_{a}

*H*

_{g}

*H*

_{c}

_{j=1}

*H*

_{c}

_{j=2}

*E*

_{a}balances the flux from foliage [

*E*

_{c}]

_{j=1}and [

*E*

_{c}]

_{j=2}, and the flux from the ground,

*E*

_{g}, that is,

*E*

_{a}

*E*

_{g}

*E*

_{c}

_{j=1}

*E*

_{c}

_{j=2}

*T*

_{af}and

*q*

_{af}(or

*e*

_{a}) are updated by (8) and (9) within the iterative procedure for leaf temperature calculations, respectively. The numerical solution of Eq. (5) is described in appendix C.

N.B., the equations for *r*_{b}, *E*_{a}, *E*_{g}, *H*_{a}, and *H*_{g} are exactly the same as that in the CLM (http://climate.eas.gatech.edu/dickinson).

#### 2) CO_{2} balance

_{2}flux budget within the canopy can be described by a CO

_{2}concentration conservation equation:

_{2}storage (

*C*

_{CO2}

*R*

_{soil}is the CO

_{2}flux from soil surface to canopy air;

*R*

_{p}is the nonleaf plant respiration, which was taken as zero in this study. Here, [

*A*

_{n}]

_{j}is the net CO

_{2}assimilation of canopy by sunlit (

*j*= 1) or shaded leaves (

*j*= 2), and can be written as

*A*

_{n}

_{j}

*g*

_{b}

_{j}

*c*

_{a}

*c*

_{s}

_{j}

*F*

_{c}is the CO

_{2}flux from canopy to atmosphere, and can be written

*F*

_{c}

*g*

_{a}

*c*

_{a}

*c*

_{m}

*c*

_{s},

*c*

_{a}, and

*c*

_{m}are the mixing ratio of CO

_{2}in air at leaf surface, canopy air space and atmosphere reference height, respectively. Here,

*g*

_{a}is aerodynamic conductance of water vapor between reference height and canopy air space, [

*g*

_{b}]

_{j}is the bulk leaf boundary conductance (

*j*= 1 for sunlit and

*j*= 2 for shaded).

### c. Radiation absorption

#### 1) Solar radiation

The two-stream approximation for the radiation transfer within a canopy (Dickinson 1983; Sellers et al. 1985) is used to describe the interception, reflection, transmission, and absorption of radiation by vegetation and soil using standard parameterizations for soil and snow albedos (Dickinson et al. 1993). The solutions for singularity points, not previously considered, are provided in appendix A.

*I*

_{lb}, scattered direct-beam

*I*

_{lbs}, and diffuse radiation

*I*

_{ld}. That is,

*I*

_{lbs}and diffuse radiation

*I*

_{ld}:

*x*(per unit leaf area index)—excluding scattering,

*I*

_{lb}, is obtained from the exponential light profile with the complementary transmission as

*I*

_{lb}

*ω*

*k*

_{b}

*e*

^{−kbx}

*I*

_{b0}

*x*,

*I*

_{lbs}, is given by

*I*

^{↑}

_{dir}

*I*

^{↓}

_{dir}

*x*in the canopy without scattering; the second term is the change of net upward of scattered direct-beam radiation flux. Here,

*I*

_{b0}is the incident direct-beam radiation above the canopy,

*I*

_{d0}is the incident diffuse radiation above the canopy. The total amount of incident diffuse and scattered diffuse radiation absorbed by canopy depth

*x,*

*I*

_{ld}, is given by

*I*

^{↑}

_{dif}

*I*

^{↓}

_{dif}

*I*

_{s}

*I*

_{s}

_{j=1}

*I*

_{s}

_{j=2}

Figure 2 shows how the PAR absorptions by sunlit and shaded fractions of canopy vary with leaf area index. It is clear that the near maximum absorption of PAR as calculated by Eqs. (13a)–(13b) are achieved in the range of 4–5. Also, the effect of solar incident pattern (direct/diffuse) is significant, it is important to discriminate between the direct and diffuse components of solar incident. Sunlit leaves absorb the major portion of both the direct and diffuse component of solar visible fluxes, only a very low portion of direct beam is absorbed by shaded leaves (<5%). Thus the PAR absorption of direct beam by shaded leaves commonly has been ignored, such as in the NCAR LSM (Bonan 1996). The sum of the direct and diffuse terms add up to 0.96 and 0.95, respectively. The remaining radiation has contributed to the albedo.

#### 2) Longwave radiation

*L*

_{a}is the incident thermal infrared radiation,

*δ*

_{t}is the fraction of longwave radiation absorbed by canopy and

*δ*

_{t}= 1 −

*e*

^{−LAI}

*σ*is the Stefan–Boltzmann constant (W m

^{−2}K

^{−4}),

*T*

_{g}is the ground temperature (K).

### d. Canopy photosynthesis and stomatal conductance

#### 1) Leaf photosynthesis and stomatal conductance

This paper addresses how temperatures for sunlit and shaded fractions of canopy can be distinguished and photosynthesis aggregated from leaf to canopy. We use the leaf photosynthesis–stomatal conductance model [second version of Simple Biosphere Model (SiB2)] of Sellers et al. (1996a) that implemented the photosynthesis model of Farquhar et al. (1980), as formulated by Collatz et al. (1992) and relating stomatal conductance to carbon assimilation according to Ball (1988). This approach is summarized with modifications in Table 1. The symbols and definitions of parameters are listed in Table 2. Three modifications were made in 1) electron transport rate *J,* 2) soil water stress on maximum catalytic capacity of Rubico *V*_{max}, and 3) potential electron transport rate *J*_{max}.

*V*

_{m}is the maximum catalytic capacity of Rubisco at saturating levels of Ribulose bisphosphate (RuBP) and intercellular partial pressure of CO

_{2}, (mol m

^{−2}s

^{−1}), which varies with leaf temperature and soil water:

*V*

_{m}

*V*

_{max}

*f*

_{T}

*T*

_{l}

*f*

_{w}

*θ*

*V*

_{max}is the

*V*

_{m}in the presence of saturating soil water and at temperature 25°C, and is assumed to be a function of leaf nitrogen as over 20% of leaf nitrogen is invested in Rubisco. The temperature dependence

*f*

_{T}(

*T*

_{l}) of

*V*

_{m}are calculated as follows:

*J*is the electron transport rate for a given absorbed photon radiation

*I*

_{s}, and related to the minimum of the potential maximum (light saturated) electron transport rate 0.25

*J*

_{max}and the PAR absorbed by photosystem II, ε

*I*

_{s}. Usually,

*J*is described by a nonrectangular hyperbolic function:

*βJ*

^{2}− (ε

*I*

_{s}+ 0.25

*J*

_{max})

*J*+ 0.25ε

*I*

_{s}

*J*

_{max}= 0 (Smith 1937; Farquhar and Wong 1984; Wang and Polglase 1995) to smooth the transition. In SiB2 (Sellers et al. 1992, 1996a), it was only related to the PAR absorption with no maximum limit. To permit

*J*to be integrable analytically over the canopy leaf area, the simpler relation is taken in this model, that is,

*J*

*I*

_{s}

*J*

_{m}

*I*

_{s}refers to the PAR absorbed by the leaf (mol m

^{−2}s

^{−1}) and ε is the quantum yield of electron transport;

*J*

_{m}is potential electron transport rate (mol m

^{−2}s

^{−1}), which varies with leaf temperature and soil water (Tezara et al. 1999),

*J*

_{m}

*J*

_{max}

*f*

_{T}

*T*

_{l}

*f*

_{w}

*θ*

*f*

_{w}(

*θ*), is newly implemented to reduce

*J*

_{max}as done for

*V*

_{max}, when soil water is limiting (Wang et al. 2001). The temperature dependence

*f*

_{T}(

*T*

_{l}) of

*J*

_{m}is adopted from de Pury and Farquhar (1997) as

*H*= 220 × 10

^{3}J mol

^{−1}as a curvature parameter for

*J*

_{max};

*R*= 8.314 J mol

^{−1}K

^{−1}, the universal gas constant;

*S*= 710 J K

^{−1}mol

^{−1}, the electron-transport temperature response parameter;

*E*

_{a}= 37 000 J mol

^{−1}, the activation energy.

*n*is the total number of soil layers, subscript

*j*is the number of soil layer. The value

*f*

_{root,j}is the root fraction within soil layer

*j,*

*ψ*

_{max}is the maximum value of soil matrix potential before leaves wilt (−1.5 × 10

^{5}mm),

*ψ*

_{fc}is the soil matrix potential at filed capacity, the factor (

*ψ*

_{max}−

*ψ*)/(

*ψ*

_{max}−

*ψ*

_{fc}) ranges from 0 at the permanent wilting point to 1 at field capacity.

#### 2) Scaling up from leaf to canopy

The Eqs. (19)–(27) are applied to a single leaf with known physiological, physical properties, and forcing conditions. The next step is to integrate these equations to describe the canopy photosynthesis [*A*_{n}]_{j} and stomata conductance [*g*_{s}]_{j}.

*V*

_{max}could be correlated with a measurement of leaf nitrogen concentration

*N*

_{1}. A linear relationship between

*V*

_{max}and

*N*

_{1}is commonly assumed (Field 1983; Leuning et al. 1991; Harley et al. 1992) with a residual leaf nitrogen content

*N*

_{b}(a threshold value of leaf nitrogen content below which there is no photosynthesis, when

*V*

_{max}= 0), that is,

*V*

_{max}=

*χ*

_{n}(

*N*

_{1}−

*N*

_{b}), in which

*χ*

_{n}is the ratio of Rubisco capacity to leaf nitrogen, could be used to relate

*V*

_{max}(from leaf photosynthesis measurements) to measurements of

*N*

_{1}. The vertical profile of leaf nitrogen has been modeled as decreasing exponentially with cumulative relative leaf area index,

*x,*from the top of the canopy (Hirose and Werger 1987; Leuning et al. 1995; de Pury and Farquhar 1997), that is,

*N*

_{1}(

*x*) = (

*N*

_{0}−

*N*

_{b}) exp(−

*k*

_{n}

*x*) +

*N*

_{b}, where

*N*

_{0}is the nominal leaf nitrogen content at the top of the canopy,

*k*

_{n}is the coefficient of leaf nitrogen allocation. Hence, the Rubisco capacity

*V*

_{max}is given by

*V*

_{max}

*V*

_{cmax}

*k*

_{n}

*x*

*V*

_{cmax}is for leaves at the top of the canopy at 25°C, and

*V*

_{cmax}=

*χ*

_{n}(

*N*

_{0}−

*N*

_{b}).

*J*

_{max}, within the canopy was assumed to be proportional to exp(−

*k*

^{*}

_{d,1}

*x*) (Wang and Polglase 1995; Wang and Leuning 1998), that is, it decreases exponentially from the top to the bottom of the canopy, that is,

*J*

_{max}

*J*

_{cmax}

*k*

^{*}

_{d,1}

*x*

*k*

^{*}

_{d,1}

*J*

_{cmax}is for leaves at the top of the canopy at 25°C, the relationship of

*V*

_{cmax}and

*J*

_{cmax}has been widely investigated by Wohlfahrt et al. (1999) who found that a fixed value of

*J*

_{cmax}/

*V*

_{cmax}= 2.1 could be applied to a wide variety of leaves.

The equations for leaf-level photosynthesis and stomatal conductance can now be integrated over the depth of the canopy to yield bulk canopy values of [*A*]_{j} and [*g*_{s}]_{j} for sunlit and shaded fractions of canopy individually. This is done by the following:

Common (canopy average or bulk) [

*c*_{l}]_{j}, [*c*_{s}]_{j}, [*e*_{s}]_{j}and leaf temperature [*T*_{l}]_{j}for sunlit/shaded leaves, respectively, are assumed.- Equation (35) is inserted into (21), (23), and (24) via (28); and (21), (23), and (24) are integrated over the depth of the canopy for sunlit and shaded fractions of canopy. Essentially, these integrations are only needed for the
*V*_{max}, and photosynthetic capacity of the sunlit and shaded leaf-fractions of the canopy are calculated as the integrals, respectively, - Equation (36) is inserted into (22) via (31), and integrated over the depth of the canopy for sunlit and shaded fractions of canopy. The integration for absorbed PAR was given by (13), and the integrations for potential electron transport
*J*_{m}over sunlit and shaded leaf-fractions of the canopy are calculated as the integrals, respectively, - For avoiding abrupt transition, the assimilation rate terms are described by combining into two quadratic equations, which are then solved for the smaller roots:The canopy photosynthesis curvature factors
*β*_{cj}and*β*_{ps}are assumed to be 0.877 and 0.99. The net assimilations of leaves are*A*_{n}_{j}*A*_{l}*R*_{d}_{j} - The integrals of Eq. (25) are written bywhere
*b** =*bf*_{w}(*θ*) × leaf area index of sunlit/shaded fraction of canopy. Equation (42) was converted to a quadratic equation of [*g*_{s}]_{j}, and solved for the larger roots.

_{2}([

*c*

_{l}]

_{j}and [

*c*

_{s}]

_{j}) and leaf surface relative humidity [

*h*

_{s}]

_{j}= [

*e*

_{s}]

_{j}/[

*e*

_{i}]

_{j}are linked to conditions in the canopy air space through the canopy stomatal conductance [

*g*

_{s}]

_{j}, the bulk canopy boundary layer conductance [

*g*

_{b}]

_{j}, the net flux of CO

_{2}[

*A*

_{n}]

_{j}[Eq. (27)], and the transpiration [Eq. (26)], and are given by

*g*

_{b}]

_{j}is the bulk leaf boundary conductance, [

*g*

_{s}]

_{j}is the canopy stomata conductance, [

*e*

_{i}]

_{j}is water vapor pressure at interior leaf and taken as the saturated value (temperature

*T*

_{l}dependence only).

*g*

_{s}]

_{j}was converted from mol m

^{−2}s

^{−1}to m s

^{−1}before used by equations (7a) and (7b):

The success of scaling up photosynthesis largely depends on biochemical properties in one-layered canopy models. The maximum Rubisco activity *V*_{max} was assumed to be linearly related to the leaf nitrogen, and that the nitrogen allocation was assumed to decline exponentially with cumulative *L*_{AI}. The potential electron transport *J*_{max} has been found to be closely correlated with *V*_{max} (Wullschleger 1993; Wohlfahrt et al. 1999; Medlyn et al. 1999), and its vertical profile has also been assumed to parallel to that of light (Wang and Polglase 1995; Wang and Leuning 1998). Based on these assumptions, the canopy values of [*V*_{max}] and [*J*_{max}] can be obtained by integrating from leaf to canopy, that is, Eqs. (37) and (38). Figure 3 shows the unstressed photosynthetic capacity of canopy, [*V*_{max}] and [*J*_{max}], over a range of leaf area index values. Here, [*V*_{max}] and [*J*_{max}] for sunlit leaves saturate at lower values of *L*_{AI}(∼2) than that for shaded leaves (*L*_{AI} ∼ 5). However, [*J*_{max}] depends strongly on the assumed value of *k*_{n}. Both sunlit and shaded leaves have higher values of [*V*_{max}] at the lower value of *k*_{n} = 0.5. Thus, with a more rapid decrease of leaf nitrogen concentration within the canopy, Rubisco activity is more likely to be limiting for the same top leaves.

For the assumed parameters, Fig. 4 shows that the shaded fraction of the canopy is always electron-transport-limited (i.e., [*w*_{e}]_{j=2} < [*w*_{c}]_{j=2}) and that the sunlit leaves are usually Rubisco-limited (i.e., [*w*_{c}]_{j=1} < [*w*_{e}]_{j=1}), except when the incident PAR is lower than 150 W m^{−2}.

Figure 5 shows the increase of assimilation in changing from very clear sky to totally diffuse radiation. The rate of assimilation saturates at lower values of PAR with a higher diffuse fraction. Hence, it has been suggested that diffuse radiation alone enhanced noontime photosynthesis of deciduous forests by 23% in 1992 and 8% in 1993 under cloudless conditions after the 1991 Mount Pinatubo eruption (Gu et al. 2003). Evidently, it is important to be able to distinguish between the direct and diffuse components. Although such data is usually not available in observations, it is readily generated by climate models.

## 3. Implementation with the common land model

The Common Land Model (CLM) has been developed for community use by a grass-roots collaboration of scientists who have an interest in making a general land model available for public use and further development (Dai et al. 2003). Its major model characteristics include: enough unevenly spaced layers to adequately represent soil temperature and soil moisture, and a multilayer parameterization of snow processes; an explicit treatment of the mass of liquid water and ice water and their phase change within the snow and soil system; a runoff parameterization following the Topography-based Runoff Prediction Model (TOPMODEL) concept; a canopy photosynthesis–conductance model that describes the simultaneous transfer of CO_{2} and water vapor into and out of vegetation; a tiled treatment of subgrid fraction of energy and water balance. The CLM has been extensively evaluated in offline mode and by coupling runs with atmospheric models.

Its submodel of photosynthesis–stomatal conductance of CLM was directly adopted from the NCAR LSM (Bonan 1996), in which the [*A*_{n}] and [*g*_{s}] were calculated for the sunlit and shaded leaves using average absorbed PAR partitioned to sunlit and shaded leaves, but in which the calculation is applied a common energy budget equation and a common leaf temperature for sunlit and shaded leaves.

The original CLM code, has been modified to include a two-stream approximation scheme (Sellers 1985) (see appendix A) for a treatment of singularities. The subroutine for leaf temperature was rewritten for a nonlinear system of energy budget equations by the quasi-Newton–Raphson iteration method (see appendix C), the sequence of iteration calculations are listed as follows:

aerodynamic resistance and bulk boundary layer resistance of leaves; heat conductance for air, leaf and ground;

photosynthesis and stomatal conductance for sunlit fraction of canopy;

photosynthesis and stomatal conductance for shaded fraction of canopy;

leaf temperatures based on the energy balance equations;

fluxes: CO

_{2}, transpiration, evaporation, and sensible heat from leaves; andevaluation of stability-dependent variables using Monin–Obukhov length for next iteration.

## 4. Offline simulation with CLM

The runs were carried out for two forest sites using the observed meteorological forcing. For each case, the atmospheric forcing data for the first year are used for model spinup with soil water initialized at full saturation, both leaf and soil temperatures at the air temperature, and both snow depth and snow age at zero. The CLM were run with (i) two-big-leaf submodel (CLM 2L); (ii) one-big-leaf submodel (CLM 1L, appendix B); and (iii) NCAR LSM (*A*_{n} − *g*_{s}) scheme (i.e., CLM), respectively. The calculated CO_{2} flux, sensible heat flux, latent heat flux, canopy photosynthesis rate, canopy stomatal conductance, and the leaf temperatures were compared with the observations.

### a. Evergreen broadleaf forest (Reserva Jaru, Roddonia, Brazil)

The forcing data for the year 1992–93 were taken from the continued measurement by the automatic weather station at the top of the tower (52 m) at an undisturbed tropical rain forest at Reserva Jaru, Brazil (10.083°S, 61.916°W). The measurements of the CO_{2} flux, sensible and latent heat fluxes are for 8 days during the dry season (September 1992) and for 48 days during wet season (May–June 1993). An eddy covariance system mounted 15 m above the 30-m-tall forest was used to measure the fluxes over an area of about 1 km^{2} (Grace et al. 1995). The same vegetation parameters are assumed for all three runs, the biome-dependent parameters are listed in Table 3. The soil respiration was taken as a constant 0.22 *μ*mol m^{−2} s^{−1} in CLM 2L and CLM 1L runs.

Figures 6 and 7 show the averaged diurnal cycles of CO_{2} flux, sensible and latent heat fluxes (above the canopy), CO_{2} assimilation rate, canopy stomatal conductance and leaf temperatures over the dry season (8 days, September 1992) and the wet season (48 days, May– June 1993), respectively, by models and measurements. Positive value here for CO_{2} flux indicates a net downward flux into the canopy. The observed CO_{2} flux above the canopy shows net uptake during daylight and a nighttime efflux. A feature of two periods was the large spike of CO_{2} leaving in the early morning. Photosynthesis was maximal in the first part of the morning, well before 1600 local time the canopy was typically in a state of carbon balance; during the night, respiration was 1–20 *μ*mol m^{−2} s^{−1}. Except for the feature of CO_{2} efflux in the early morning, CLM 1L and CLM 2L could reproduce most features in the measured fluxes. As expected, CLM 1L predicted higher values of CO_{2} uptake, stomatal conductance, and hence related latent heat flux than did CLM 2L during the daytime. Bulk stomatal conductance predicted by CLM 1L and CLM 2L approach the peak value in the early morning, falling as the day progressed, which will match that obtained from measured evapotranspiration rates (Grace et al. 1996). Also as expected, CLM 2L predicted higher sunlit leaf temperature [*T*_{l}]_{j=1} than the shaded [*T*_{l}]_{j=2}, the difference of the peak values are 3.4°C in the wet season and 4.3°C in the dry season. The leaf temperature by CLM 1L and by CLM are within the [*T*_{l}]_{j=1} and [*T*_{l}]_{j=2} and very close to [*T*_{l}]_{j=2}. Compared with CLM 1L and CLM 2L, the CLM shows performances that are very similar to that of CLM 1L.

All three runs failed to reproduce the observed spikes of CO_{2} efflux above the canopy in the early morning, which were associated with the onset of turbulent conditions that followed calm nights. Typically, the natural ventilation of the canopy is poor at night, and respired CO_{2} accumulates near the ground to over 500 ppmv (Grace et al. 1995). This disagreement might be from either. 1) inappropriate formulation of the plant or soil respiration; or 2) our neglect of storage or lateral transport of the CO_{2} storage within the canopy or soil storage. The turbulence model used in the CLM and the overlying atmosphere should be as appropriate for other scalar transports as that of moisture.

There are not much differences in fluxes and temperatures between dry and wet season (Figs. 6 and 7) in both the observed and the simulated. There is good evidence to suggest that trees rely on deep roots in the dry seasons (Nepstad et al. 1994; Zeng et al. 1998; Dai et al. 2003).

Table 4 presented the comparison of mean values of fluxes and temperature over two full years by three runs. All numbers are as expected, that is, higher assimilation and latent heat flux, lower sensible heat flux and skin temperatures by CLM 1L compared to CLM.

### b. Evergreen conifer forest (Thompson, Manitoba, Canada)

The site is near the northern edge of the boreal forest, in the zone of discontinuous permafrost (55.88°N, 98.48°W). The forcing data for 2-yr period of 18 January 1994–17 January 1996 were the average of two neighbor sites of the BOREAS Northern Study Area, which were quality checked by A. K. Betts. The eddy-correlation fluxes for CO_{2} and H_{2}O were collected at a 30-m-tall tower that extends well above the canopy height of approximately 10-m-tall 120-year-old black spruce (Goulden et al. 1997). The same vegetation parameters are assumed for all three runs, the biome-dependent parameters for all three runs are listed in Table 3. The leaf and soil respirations in CLM 2L and CLM 1L runs are the fitted temperature-related equations from Lavigne et al. (1997): *R*_{d} = 0.26 exp{0.07(*T*_{L} − 283.16)}, soil respiration: *R*_{soil} = 0.6 exp{0.119(*T*_{g} − 273.16)}.

Figure 8 shows, as an example, the averaged diurnal cycles of CO_{2} flux, sensible and latent heat fluxes (above the canopy), CO_{2} assimilation rate, canopy stomatal conductance, and leaf temperatures over a period with the longest continuous recorded and high-quality observation (1 June–30 September 1994), by models and measurements. CLM 1L and CLM 2L provided a consistent performance as they did for the Amazon rainforest. CLM predicted lower assimilation rate and stomatal conductance, and hence, lower latent heat flux (Fig. 8 and Table 5), which are not consistent with that for the case of a tropical broadleaf forest. The underlying causes are unknown. But we quote the suspicion of Farquhar et al. (2001), “there is a need to introduce penumbral effects, especially for coniferous species.”

## 5. Summary and discussion

Stomatal conductance exerts a major control on canopy transpiration. Its functioning is closely linked to the assimilation of carbon by leaves. Hence carbon assimilation models are used to parameterize stomatal conductance for climate models. The two-big-leaf model presented here improves previous such modeling by Collatz et al. (1991, 1992), Sellers et al. (1992, 1996a), Bonan (1996), de Pury and Farquhar (1997), Dickinson et al. (1998), and Dai et al. (2003) through the following:

separate treatments of sunlit and shaded leaves;

separate calculation of radiation absorption for sunlit and shaded leaves;

separate calculation of leaf temperatures, stomatal conductance, and fluxes (water, CO

_{2}, and sensible heat) for sunlit and shaded fractions of canopy;inclusion of exponential profiles for distribution of canopy photosynthetic capacity, and maximum potential electron transport rate;

use of soil water limitation for both maximum rates of RuBP carboxylation and electron transport;

separation of scaling up from leaf to canopy for sunlit and shaded fractions separately; the parameters of the canopy model have equivalent definitions and relationships between the leaf and canopy scales; and

coupled solution of the energy balance equations for sunlit and shaded leaf temperatures.

The functioning of these parameterizations has been tested in two 2-yr offline simulations with CLM. The simulations were reasonably successful in predicting the diurnal variation in leaf temperatures, canopy conductance, and fluxes (energy, CO_{2}, and water vapor). Consequences of altering the distribution of leaf nitrogen (Fig. 3), the density of incident of PAR (Fig. 4), and the fractions of diffuse and direct-beam radiation (Fig. 5) on canopy photosynthesis were also explored. This type of sensitivity analysis allows examination of the relative importance of various parameters of the model, and hence, provides insight into the behavior of real plant canopies.

CLM 1L, which does not account for the difference in the radiation absorption and the difference of leaf temperature between sunlit leaves and shaded leaves, was found to give higher canopy fluxes (CO_{2} and water vapor) than the CLM 2L. Vertical variation in leaf area index, fraction of diffuse beam radiation, and the leaf nitrogen content, preclude an accurate treatment of canopy photosynthesis with the one-leaf model (de Pury and Farquhar 1997). The two-big-leaf model presented here allows within-canopy profiles of the important leaf properties for carbon assimilation, and hence gives an accurate biochemically based model of photosynthesis.

CLM, which does not account for the difference of leaf temperature between sunlit leaves and shaded leaves, the varying of photosynthetic capacity with leaf area index *L*_{AI}, and the *N* distribution with canopy depth, was found to overestimate canopy fluxes (CO_{2} and water vapor) for a tropical rainforest and underestimate them for the boreal conifer forest. CLM did use a fixed value of *V*_{max} at 25°C (i.e., include the nitrogen limitation in *V*_{max}, and does not allow the changes with *L*_{AI}), the *V*_{max} fitted at *L*_{AI} = 4 was 20% smaller than that at *L*_{AI} = 2 and even smaller at higher leaf area indices (at *k*_{n} = 0.5). For the plant with constant *L*_{AI} and uniform *N* distribution with canopy depth, the errors may be less important. However, it may be significant when *L*_{AI} is varying, such as in models that predict the response of vegetation to climate change.

Important aspects of canopy modeling not addressed here are: dependence of *V*_{max} on Rubisco, hence nitrogen cycling (Dickinson et al. 2002); plant and soil respiration; canopy turbulent transport of mass and energy.

## Acknowledgments

This work has been supported by funding agencies including DOE to Georgia Institute of Technology, NASA through the IDS Program Grant (429-81-22; 428-81-22), and NSFC of China under Grant 40225013. We would like to acknowledge the assistance and helpful discussions from Alan Betts. We would also like to thank Niall Hanan and an anonymous reviewer for their thorough and constructive review. The ABRACOS data were collected under the ABRACOS project and made available by the Institute of Hydrology (United Kingdom) and the Instituto Nacional de Pesquisas Espaciais (Brazil); the BOREAS meteorological data were prepared by Alan Betts from the BOREAS CD-ROM, and the fluxes data were made available by the Steven C. Wofsy Group at Harvard University.

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## APPENDIX A

### The Supplementaries to Two-Stream Approximation Radiative Transfer Model of Sellers (1985)

*I*

^{↑},

*I*

^{↓}are the upward and downward diffuse radiative fluxes, respectively, normalized by the incident flux;

*μ*is the cosine of the zenith angle of incident beam;

*G*(

*μ*) is the projected area of phytoelements in direction

*μ*;

*μ*

*K*is the optical depth of the direct beam per unit leaf area, and

*K*=

*G*(

*μ*)/

*μ*;

*β,*

*β*

_{0}are the upscatter parameters for diffuse and direct beams;

*ω*is the scattering coefficient of phytoelements;

*μ*is given by

*G*

*μ*

*ϕ*

_{1}

*ϕ*

_{2}

*μ,*

*ϕ*

_{1}= 0.5 − 0.633

*χ*− 0.33

*χ*

^{2},

*ϕ*

_{2}= 0.877(1 − 2

*ϕ*

_{1}), and

*χ*is an empirical parameter related to the leaf angle distribution. The average inverse diffuse optical depth per unit leaf area is

*ϕ*

_{1}≠ 0 and

*ϕ*

_{2}≠ 0. Actually,

*ϕ*

_{1}or

*ϕ*

_{2}could be zero, the integral (A4a) does not work for these cases. We provide supplementary solutions as follows:

*σ*≠ 0. We provide the solution for the singularity at

*σ*= 0,

## APPENDIX B

### One-Big-Leaf Model

^{LAI}

_{0}

*dx,*the sensible heat flux

*H*

_{c}and LE

_{c}latent heat flux from foliages to canopy air are given, respectively,

*r*

_{s}is the averaged leaf stomatal resistance for sunlit and shaded. The radiation absorbed by leaves of the canopy are given by,

_{2}flux budget within canopy can be described by the CO

_{2}concentration conservation equation:

*A*

_{n}−

*g*

_{s}] is similar to the two-big-leaf submodel, but the integrations of Eqs. (19)–(27) over canopy are for single big-leaf. The canopy values of

*V*

_{max}and

*J*

_{max}are given by

## APPENDIX C

### Numerical Scheme for Leaf Temperatures

*n*denotes the number of the iteration step,

*δ*, here, is the difference in iteration step

*n*+ 1 and

*n*. The convergent criterions for iteration are

## APPENDIX D

### List of Symbols Used in Paper

*A*Leaf assimilation rates (mol m^{−2}s^{−1})*A*_{n}Net leaf assimilation (mol m^{−2}s^{−1})*b*Minimum stomatal conductance (mol m^{−2}s^{−1})*c*_{a},*c*_{s},*c*_{i}Partial pressure of CO_{2}in canopy air, at the leaf surface and interior leaf, respectively (pa)*c*_{m}Partial pressure of CO_{2}in atmosphere reference height (pa)*c*_{p}Specific heat of air (J kg^{−1}K^{−1})*E*_{a}Flux of water from canopy air*e*_{a},*e*_{s},*e*_{i}Partial pressure of H_{2}O in canopy air space, at the leaf surface and inside the leaf, respectively (pa)*E*_{c}Water vapor flux (including transpiration and interception loss) from leaves to canopy air (kg m^{−2}s^{−1})*E*_{g}Water vapor flux from the ground*E*_{tr}Transpiration (kg m^{−2}s^{−1})*f*_{d}Leaf respiration factor*F*_{c}CO_{2}flux from canopy to atmosphere*f*_{root}Root fraction within soil layer*F*_{Sha}Fraction of shaded leaves of the canopy*f*_{sha}Fraction of shaded leaves at canopy depth*F*_{Sun}Fraction of sunlit leaves of the canopy*f*_{sun}Fraction of sunlit leaves at canopy depth*x**f*_{T}(*T*_{l}) Temperature dependence of*V*_{m}or*J*_{max}*f*_{w}(*θ*) Soil moisture dependence of*V*_{m}or*J*_{max}*f*_{wet}Fraction of wetted leaves of the canopy*g*_{a}Aerodynamic conductance of water vapor between reference height and canopy air space*g*_{b}Canopy stomatal conductance*g*_{s}Bulk canopy boundary layer conductance*g*_{l}Leaf boundary conductance (mol m^{−2}s^{−1})*g*_{s}Leaf stomatal conductance (mol m^{−2}s^{−1})*G*(*μ*) Projected area of phytoelements in direction of the sun*H*_{a}Sensible heat flux from canopy to atmosphere (W m^{−2})*H*_{c}Sensible heat flux from foliages to canopy air (W m^{−2})*H*_{g}Sensible heat flux from the ground (W m^{−2})*h*_{s}Relative humidity at leaf surface*I*_{b0}Incident direct beam (W m^{−2})*I*_{d0}Incident diffuse beam (W m^{−2}) Downward diffuse fluxes of diffuse incident solar multiscattered by leaves and ground, normalized by the incident diffuse solar flux*I*^{↓}_{dif} Upward diffuse fluxes of diffuse incident solar multiscattered by leaves and ground, normalized by the incident diffuse solar flux*I*^{↑}_{dif} Downward diffuse fluxes of the direct incident solar scattered by leaves and ground normalized by the incident direct solar flux*I*^{↓}_{dir} Upward diffuse fluxes of the direct incident solar scattered by leaves and ground, normalized by the incident direct solar flux*I*^{↑}_{dir}*I*_{ir}Net absorbed thermal radiation for sunlit leaves (W m^{−2})*I*_{lb}Absorbed direct incident beam—without scattering (W m^{−2})*I*_{lbs}Absorbed diffusion radiation scattered by leaves for direct incident beam (W m^{−2})*I*_{ld}Absorbed diffuse incident beam (W m^{−2})*I*_{s}Net solar radiation absorbed by sunlit/shaded fraction of canopy (W m^{−2})*J*_{m}Potential electron transport (mol m^{−2}s^{−1})*J*Electron transport rate (mol m^{−2}s^{−1})*J*_{cmax}Leaves at the top of the canopy at 25°C (mol m^{−2}s^{−1})*J*_{max}Maximum (light-saturated) electron transport rate (mol m^{−2}s^{−1})*k*_{b}Direct-beam solar extinction coefficient of canopy*K*_{c}Rubisco Michaels–Menten constant for CO_{2}(pa) Extinction coefficients for diffuse PAR*k*^{*}_{d,1}*k*_{n}Coefficient of leaf nitrogen allocation within canopy*K*_{o}Rubisco inhibition constant for oxygen (pa)*L*_{a}Incident thermal infrared radiation (mol m^{−2}s^{−1})*L*_{AI}Leaf area index (m^{2}m^{−2})*L*_{sha}Leaf area index of shaded fraction of canopy (m^{2}m^{−2})*L*_{Sun}Leaf area index of sunlit fraction of canopy (m^{2}m^{−2})*m*Stomatal slope factor*N*Leaf nitrogen*N*_{0}Nominal leaf nitrogen content at the top of the canopy*N*_{I}Leaf nitrogen concentration*N*_{b}A residual leaf nitrogen content (a threshold value of leaf nitrogen content below which there is no photosynthesis) when*V*_{max}= 0*O*_{2}Partial pressure of O_{2}in leaf interior (pa)*p*Atmospheric pressure at surface (pa)*Q*_{10}*Q*_{10}temperature coefficient*q*_{af}Specific humidity of canopy space air (kg kg^{−1})*q*_{sat}(*T*_{l}) Saturated specific humidity (kg kg^{−1}) at temperature*T*_{l}*R*Universal gas constant, 8.314 J mol^{−1}K^{−1}*r*_{b}Leaf boundary layer resistance (s m^{−1})*R*_{d}Dark respiration rate (mol m^{−2}s^{−1})*R*_{p}Nonleaf plant respiration (mol m^{−2}s^{−1})*r*_{s}Leaf stomatal resistance (s m^{−1})*R*_{soil}CO_{2}flux from soil surface to canopy air*S*Rubisco specificity for CO_{2}relative to oxygen (pa)*s*_{1}Slope of high temperature inhibition function (K^{−1})*s*_{2}One-half point of high temperature inhibition function (K)*s*_{3}Slope of low temperature inhibition function (K^{−1})*s*_{4}One-half point of low temperature inhibition function (K)*s*_{5}Slope of high temperature inhibition function (leaf respiration) (K^{−1})*s*_{6}One-half point of high temperature inhibition function (leaf respiration) (K)*S*_{AI}Dead leaf or stem area index*T*_{af}Temperature of canopy space air (K)*T*_{g}Ground temperature (K)*T*_{l}Leaf temperatures*V*_{cmax}Maximum rubisco capacity at top canopy at 25°C per leaf area (mol m^{−2}s^{−1})*V*_{m}Maximum catalytic capacity of Rubisco at saturating levels of Ribulose bisphosphate (RuBP) and intercellular partial pressure of CO_{2}, (mol m^{−2}s^{−1})*V*_{max}Maximum rate of carboxylation*w*_{c}Rubisco (leaf enzyme) limited rate of assimilation (mol m^{−2}s^{−1})*w*_{e}Light-limited rate of assimilation (mol m^{−2}s^{−1})*w*_{s}Carbon compound export limitation (C_{3}plants), or PEP-carboxylase (C_{4}) limitation on photosynthesis (mol m^{−2}s^{−1})*μ*Cosine of solar zenith angle*ρ*Density of air (kg m^{−3})*θ*Solar zenith angle, soil volumetric content*σ*Stefan–Boltzmann constant (W m^{−2}K^{−4})*χ*Empirical parameter related to the leaf angle distributionε Intrinsic quantum yield epsilon (mol mol

^{−1})Γ* CO

_{2}compensation point (pa)*δ*_{1},*δ*_{2}Step functions for sunlit and shaded leaves*δ*_{t}Fraction of longwave absorbed by canopy*ψ*_{fc}Soil matrix potential at filled capacity (mm)*ψ*_{max}Soil matrix potential before leaf desiccation (mm)*χ*_{n}Ratio of Rubisco capacity to leaf nitrogen

Equations of the leaf photosynthesis and stomatal conductance model

Biome physiological parameters

List of biome-dependent parameters used in the model for tropical evergreen broadleaf trees (TEBT) and northern old black spruce (NOBS). The *V*_{cmax} for TEBT is from Lloyd et al. (1995); *V*_{cmax} and ε for NOBS are from Dang et al. (1998) and Chen et al. (2000); other biochemical parameters are from Sellers et al. (1996b). Here, *L*_{AI} for TEBT is from Dai et al. (2003); for NOBS from Chen et al. (1997). The values bracketed are canopy averaged [*V*_{max} ] used by CLM

Comparison of fluxes and temperatures simulated by CLM 2L, CLM 1L, and CLM. Mean values are 2-yr averages from 1 Jan 1992 to 31 Dec 1993 at Amazon rainforest site

Comparison of fluxes and temperatures simulated by CLM 2L, CLM 1L, and CLM. Mean values are 2-yr averages from 18 Jan 1994 to 17 Jan 1996 at an old black spruce site of BOREAS/NOBS research