A refinement to the two-leaf model for calculating canopy photosynthesis

Agricultural and Forest Meteorology 101 (2000) 143–150

A refinement to the two-leaf model for calculating canopy photosynthesis Ying Ping Wang ∗ CSIRO Atmospheric Research, Private Bag No. 1, Aspendale, Victoria 3195, Australia Received 24 February 1999; received in revised form 21 November 1999; accepted 23 November 1999

Abstract The two-leaf model developed by Wang and Leuning (1998) [A two-leaf model for canopy conductance, photosynthesis and partitioning of available energy I. Model description and comparison with a multi-layered model. Agric. For. Meterol. 91, 81–111] usually overestimates the hourly photosynthesis, latent heat flux and canopy conductance of sunlit leaves, and underestimates the hourly sensible heat fluxes compared to a more detailed multi-layer model. In all cases errors on average were <5%. Here I present a refinement to the two-leaf model that reduces discrepancies between the two-leaf and multi-layered models to <3% while increasing by less than 1% the computing time required when the two-leaf model is used as the surface scheme in a climate model. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Canopy modelling; Photosynthesis; Two-leaf canopy model; Surface scheme; Latent; Sensible; Conductance

1. Introduction Wang and Leuning (1998) published a two-leaf model for calculating hourly canopy conductance, photosynthesis, latent and sensible heat fluxes. The canopy is separated into two big leaves: one sunlit and the other shaded in the two-leaf model. Equations for conductance, photosynthesis and energy exchange were developed for two big leaves separately. The idea of representing the canopy as two big leaves was introduced by Sinclair et al. (1976). By comparing the two-leaf model with a multi-layered model of Leuning et al. (1995), Wang and Leuning (1998) found that the two-leaf model overestimated hourly canopy photosynthesis, conductance and latent heat fluxes but underestimated the hourly sensible ∗ Fax: +61-3-9239-4444. E-mail address: [email protected] (Y.P. Wang).

heat fluxes. The differences were typically <5%, and were mainly due to small overestimates of photosynthesis of sunlit leaves at intermediate irradiances (800–1200 ␮mol m−2 s−1 ). I present here some refinement to the two-leaf model in calculating hourly canopy photosynthesis which reduces the discrepancies between the two-leaf and multi-layer models to <3%, with only a very small increase in computing time.

2. Theory In the two-leaf model, photosynthesis of the two big leaves is assumed to be limited by either Rubisco carboxylation or RuBP regeneration. This is usually a good approximation for the shaded leaves but not for the sunlit leaves. As shown in Fig. 1a for a canopy with constant leaf inclination angle, the photosynthesis of

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Fig. 1. (a) Rubisco-limited (solid curve) or RuBP-limited (dashed curve) photosynthesis of sunlit leaves within the canopy, the cumulative leaf area index at which photosynthesis of sunlit leaves is co-limited, ξ 1 is also indicated, and (b) the modelled response of the photosynthesis of sunlit leaves using the multi-layered model of Leuning et al. (1995) (solid curve) or two-leaf model of Wang and Leuning (1998) without the refinement (open circle) or with the refinement for Case I (filled circle). Canopy leaf area index is 5 and the leaf inclination angle is 57◦ from the horizontal. The zenith angle of the sun is 0. The beam fraction of incident PAR, fb is calculated as fb = min {0, (I1 − 300)/I1 }, where I1 is the flux density of incident PAR above the canopy. The non-rectangular PAR response of the potential electron transport rate was used with curvature parameter θ=0.70. Parameter values used are: C=245 ␮mol mol−1 , 0 ∗ =30 ␮mol mol−1 ; Km =585 ␮mol, α=0.385, kn =0.6; vcmax and jmax decrease exponentially within the canopy in proportional to exp(−kn ξ ), where ξ is the cumulative leaf area index from the canopy top, where the values of vcmax and jmax are 100 and 210 ␮mol m−2 s−1 , respectively.

sunlit leaves is limited by RuBP regeneration in the upper part of the canopy but by Rubisco carboxylation in the lower part of the canopy. This occurs because photosynthetic capacity of the leaves decreases with the cumulative leaf area index from the top of the canopy (ξ ) faster than the amount of photosynthetically active radiation (PAR) absorbed by the leaves within the canopy. The two-leaf model does not

take account of the variation in the processes limiting the photosynthesis of sunlit leaves with canopy depth, and therefore overestimates the photosynthesis of sunlit leaves as compared with the multi-layered model (see Fig. 1b). The discrepancies between the two-leaf (Wang and Leuning, 1998) and multi-layer models (Leuning et al., 1995) can be significantly reduced if we can calculate the Rubisco-limited and RuBP-limited photosynthetic rates of sunlit leaves separately, as was done in the multi-layered model. The amount of PAR absorbed by leaves within a horizontally homogeneous canopy depends on two structural properties of the canopy: the cumulative leaf area index above the leaf and the leaf angle distribution. In the two-leaf model, the total amount of absorbed PAR and the integrated photosynthetic properties of sunlit or shaded leaves were used in the calculation, the approach is mathematically equivalent to calculating the photosynthesis using the mean amount of absorbed PAR and mean photosynthetic properties separately for sunlit and shaded leaves. It is known that leaf photosynthesis is nonlinearly related to the absorbed PAR, and the two-leaf model does not account for the variation of absorbed PAR within the sunlit leaves and will therefore overestimate the photosynthesis of the sunlit leaves as compared with the multi-layered canopy model. To overcome the problem in the two-leaf model, we need to further separate the sunlit leaves into Rubisco-limited and RuBP-limited leaves, and calculate their photosynthesis separately. In the following, I present the theory of colimitation of leaf photosynthesis. I will then apply the theory to refine the calculation of canopy photosynthesis in the two-leaf model for two cases. I also discuss the implementation of the refinement into the two-leaf model of canopy conductance, photosynthesis and partitioning of available energy as presented by Wang and Leuning (1998). In presenting the theory, I use the notation of Wang and Leuning (1998).

3. Colimitation of leaf photosynthesis In the following, we assume that the colimitation of leaf photosynthesis occurs at a point on the photosynthetic light response curve. This is not strictly accurate (Sage, 1990). Colimitation of leaf photosynthesis

Y.P. Wang / Agricultural and Forest Meteorology 101 (2000) 143–150

usually occurs over a range of light flux densities, as chloroplasts in a leaf receive different light flux densities and have different photosynthetic characteristics, some chloroplasts are light limited whilst others are light saturated. When the incident light flux density on the leaf surface increases, fewer chloroplasts become light-limited, photosynthesis of leaf is still colimited until all the chloroplasts are light-saturated. As the transition from RuBP-limiting to Rubisco-limiting at individual leaf scale usually occurs over quite a narrow range of light flux densities, we consider this approximation reasonable. When Rubisco activity limits the rate of leaf photosynthesis (Af,c ), the model of Farquhar et al. (1980) states that Af,c

vc max (C − 0 ∗ ) = − rd (C + Km )

(1)

where ν cmax is the maximum carboxylation rate, C is intercellular CO2 concentration, Km is a Michaelis–Menten coefficient, and 0 ∗ is the CO2 compensation point in the absence of non-photorespiratory respiration, rd . When the rate of RuBP regeneration limits leaf photosynthesis, the photosynthetic rate (Af ,j ) is given by Af,j =

0.25j (C − 0 ∗ ) − rd (C + 20 ∗ )

(2)

where j is the potential electron transport rate, which is calculated using the non-rectangular light response function (Farquhar and Wong, 1984) θj 2 − (αq + jmax )j + αqjmax = 0

(3)

where θ is a curvature parameter, α is the initial slope of the electron-transport rate at low light, q is the absorbed PAR, and jmax is the maximum rate of potential electron transport. Leaf photosynthesis is co-limited when the potential rate of RuBP regeneration equals the Rubisco activity and thus 0.25j ∗ (C − 0 ∗ ) vc max (C − 0 ∗ ) = (C + Km ) (C + 20 ∗ )

(4)

where j∗ is the potential electron transport rate when leaf photosynthesis is co-limited, and is given by 4(C + 20 ∗ ) vc max ≡ βvc max . j∗ = (C + Km )

(5)

145

The parameter β is defined as β≡

4(C + 20 ∗ ) . (C + Km )

(6)

Leaf photosynthesis is limited by Rubisco activity if j>j∗ , or by RuBP regeneration otherwise. From Eqs. (3) and (5), the flux density of absorbed PAR (q∗ ) when j is equal to j∗ is given by q ∗ = λvcmax where λ and b are defined as β(θβ − b) λ≡ α(β − b) b≡

jmax vc max

(7)

(8) (9)

Leaf photosynthesis is limited by Rubisco activity if q>q∗ , or is limited by RuBP regeneration otherwise. Eq. (7) in another form was also presented by de Pury and Farquhar (1997). I will make use of Eq. (7) for separating the Rubisco-limited from RuBP-limited leaves and present the formula for calculating their photosynthesis separately within two canopies for the canopy model as presented by Wang and Leuning (1998). Case I. (The amount of absorbed PAR does not vary within sunlit or shaded leaves but is different between sunlit and shaded leaves at a given canopy depth). For Case I, we ignore the variation of the amount of absorbed PAR with leaf angle at a given canopy depth. If the absorption of scattered and diffuse PAR is ignored, the average amount of PAR absorbed by the sunlit leaves within the canopy, q1,1 , is given by q1,1 = Ib,1 kb (1 − ωf,1 )

(10)

where Ib,1 is the direct beam PAR at the top of the canopy, kb is the extinction coefficient for black leaves (Goudriaan and van Laar, 1994), and ωf,1 is the leaf scattering coefficient for PAR. If the absorption of scattered PAR is ignored, the average amount of PAR absorbed by the shaded leaves within the canopy, q2,1 is given by ∗ ∗ (1 − ρtd,1 )exp(−kd,1 ξ) q2,1 (ξ ) = Id,1 kd,1

(11)

where Id,1 is the diffuse PAR at the top of the canopy, ∗ is the effective extinction coefficient for diffuse kd,1 PAR, and ρ td,1 is canopy reflectance for diffuse PAR.

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Y.P. Wang / Agricultural and Forest Meteorology 101 (2000) 143–150

I also assume that vcmax decreases exponentially with ξ , i.e. 0 exp(−kn ξ ) vcmax (ξ ) = vcmax

(12)

0 is the maximum carboxylation rate of where vcmax the leaves at the top of the canopy and kn is the nitrogen distribution coefficient. A similar relation is assumed for the decrease of jmax with ξ because Wullschleger (1993) found that jmax /vcmax ≈ 2 in a survey of 109 species of plants. The value of ξ at which photosynthesis changes from being limited by RuBP regeneration to being limited by Rubisco carboxylation capacity can be obtained by substituting Eqs. (10) and (12) into Eq. (7), to give   0   log λvcmax − log kb Ib,1 (1 − ωf,1 ) (13) ξ1 = kn

for sunlit leaves.By substituting Eqs. (11) and (12) into Eq. (7), we obtain i h  0  ∗ (1 − ρ log λvcmax − log Id,1 kd,1 td,1 ) (14) ξ2 = ∗ kn − kd,1 for shaded leaves. Photosynthesis of sunlit leaves is always limited by Rubisco activity if ξ 1 <0, and by RuBP regeneration if ξ 1 >L, where L is the total leaf area index. Only when 0<ξ 1 kd,1 ∗ If kn < kd,1 and ξ 2 >0, photosynthesis of the shaded leaves is always limited by RuBP regeneration, and by Rubisco activity if ξ 2 >L. Under natural conditions, the numerator of Eq. (14) is usually positive and thus ξ 2 >0 only when ∗ . Therefore transition from RuBP-limitation kn > kd,1 to Rubisco-limitation of the photosynthesis of the shaded leaves is possible only when vcmax decreases with ξ faster than the absorbed PAR of the shaded leaves. On the other hand, the optimal theory put forward by Field (1983) suggests that allocation of leaf nitrogen within the canopy should be proportional to the absorbed PAR at ξ , but most field data seems to suggest that kn is usually less than ∗ (Hirose and Werger, 1994). Under these conkd,1 ditions, the photosynthesis of shaded leaves within

the canopy is usually limited by RuBP regeneration, the formulation of Wang and Leuning (1998) (WL’s formulation for short thereafter) for photosynthesis of shaded leaves agrees within 1% with the results from the multi-layered model (results not shown here). The true values of ξ 1 and ξ 2 will be somewhat smaller than those given by Eqs. (13) and (14), respectively, if the absorption of the scattered PAR is accounted for. For sunlit leaves, ξ 1 is likely to be within 0 and L only at high incident PAR flux density when the absorption of scattered and diffuse PAR only accounts for a relatively small fraction of the absorbed PAR. Therefore a small error (<1%) is caused by using Eq. (13) to calculate the transition point between the two rate limiting processes for photosynthesis. For shaded leaves absorption of the scattered beam PAR within a dense canopy under clear sky conditions can exceed the absorption of the diffuse PAR, and errors in using Eq. (14) can be quite large. However, photosynthesis of shaded leaves is limited by absorbed PAR ∗ , and the WL’s formulation still if kn is less than kd,1 provides a good approximation to the correct rate of photosynthesis. In the following, I only discuss the application of the refinement to the calculation of photosynthesis, conductance and the energy balance of the sunlit leaves. This refinement does not account for the variation of PAR absorbed by the leaves with leaf angle, therefore does not improve the agreement with the multi-layered model as much as the refinement for Case II, which is only applicable to a canopy with spherical leaf angle distribution. When 0<ξ 1
(15)

where Ac,1a and Ac,1b are the photosynthetic rates of the RuBP-limited and Rubisco-limited sunlit leaves within the canopy, respectively, and are calculated using the WL’s formulation in the two-leaf model except that the rate of potential electron transport is now calculated using the non-rectangular response function. The amount of PAR absorbed by the Rubisco-limited sunlit leaves within the canopy, Q1b,1 is

Y.P. Wang / Agricultural and Forest Meteorology 101 (2000) 143–150

Z Q1b,1 =

L

ξ

regeneration. The total Rubisco carboxylation capacity (Vcmax,1b ) and potential rate of RuBP regeneration (Jmax,1b ) of the Rubisco-limited sunlit leaves are

(q1,1 + q2,1 )dξ

1  ∗ ∗ φ ξ1 , kd,1 + kb = Id,1 (1 − ρtd,1 )kd,1  ∗ ∗ φ ξ1 , kb,1 + kb +Ib,1 (1 − ρtb,1 )kb,1

+Ib,1 (1 − ωf,1 )kb [φ {ξ1 , kb } − φ {ξ1 , 2kb }] (16)

Vcmax,1b Jmax,1b = 0 0 vcmax jmax Z L Z = exp(−kb ξ ) ζ1∗

The bulk photosynthetic parameters of the Rubiscolimited sunlit leaves within the canopy are 0 φ {ξ1 , kn + kb } Vcmax,1b = vcmax

Jmax,1b =

0 jcmax φ {ξ1 , kn

+ kb }

(18)

x

The amount of absorbed PAR (Q1a,1 ) and bulk photosynthetic parameters (Jmax,1a and Vcmax,1a ) for the RuBP-limited sunlit leaves within the canopy are: Q1a,1 = Q1,1 − Q1b,1

(20)

Jmax,1a = Jmax,1 − Jmax,1b

(21)

Vcmax,1a = Vcmax,1 − Vcmax,1b

(22)

Case II. (Spherical leaf angle distribution). If the absorption of scattered and transmitted diffuse PAR by the sunlit leaves is ignored, the PAR absorbed by a sunlit leaf within the canopy, q1,1 , is given by (23)

where γ is the angle between the direction of incident direct beam radiation and the normal to the leaf surface. Assuming the leaf angle distribution is spherical, photosynthesis of the leaf is colimited by Rubisco carboxylation and RuBP regeneration when γ =γ ∗ , and  0  ∗ −1 λvcmax exp(−kn ξ ) . (24) γ = cos (1 − ωf,1 )Ib,1 When γ = γ ∗ , photosynthesis of the leaf is limited by Rubisco carboxylation, otherwise by RuBP

γ∗

×φ(ξ1∗ , kb + 2kn ) and ξ1∗ =

sin γ exp(−kn ξ )dγ dξ

0

= φ(ξ1∗ , kb + kn ) −

(17)

0 is the maximum rate of potential electron where jmax transport of the leaf at the top of the canopy and function φ is defined as Z L exp(−kξ )dξ. (19) φ(x, k) =

q1,1 = (1 − ωf,1 )Ib,1 cos γ

147

0 λvcmax (1 − ωf,1 )Ib,1

(25)



0 log λvcmax − log (1 − ωf,1 )Ib,1 . kn

(26)

The total amount of absorbed PAR by the Rubisco-limited sunlit leaves, Q1b,1 is Z L Z γ∗  Q1b,1 = exp(−kb ξ ) sin γ (1 − ωf,1 )Ib,1 cos γ ζ1∗



0

+q2,1 dγ dξ


(1 − ωf,1 )Ib,1 φ ξ1∗ , kb = 2 # 2  0 λvcmax ∗ φ(ξ1 , kb + 2kn ) − (1 − ωf,1 )Ib,1  ∗ ∗ ) +Id,1 kd,1 (1 − ρtd,1 ) φ(ξ1∗ , kb + kd,1 

0 λvcmax ∗ φ(ξ1∗ , kb + kn + kd,1 ) (1 − ωf,1 )Ib,1  ∗ ∗ ) +Ib,1 kb,1 (1 − ρtb,1 ) φ(ξ1∗ , kb + kb,1

−



0 λvcmax ∗ φ(ξ1∗ , kb + kn + kb,1 ) − (1 − ωf,1 )Ib,1  −Ib,1 kb (1 − ωf,1 ) φ(ξ1∗ , 2kb )

 0 λvcmax ∗ φ(ξ1 , 2kb + kn ) . − (1 − ωf,1 )Ib,1

(27)

The total rate of potential maximal electron transport rate (Jmax,1a ) and the amount of absorbed PAR (Q1a,1 ) by the RuBP-limited sunlit leaves are Jmax,1a = Jmax,1 − Jmax,1b

(28)

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Y.P. Wang / Agricultural and Forest Meteorology 101 (2000) 143–150

Q1a,1 = Q1,1 − Q1b,1 .

(29)

The bulk parameters, Vcmax,1a , Vcmax,1b , Jmax,1a and Jmax,1b and the total amount of absorbed PAR, Q1a,1 and Q1b,1 can be used in the WL’s formulation (Wang and Leuning, 1998) for estimating the photosynthesis of RuBP-limited (Ac,1a ) and Rubisco-limited (Ac,1b ) sunlit leaves. For a canopy with non-spherical leaf angle distribution, simple analytic solutions are not possible for the bulk parameters and the amount of absorbed PAR by the Rubisco-limited and RuBP-limited sunlit leaves. In this case, the refinement for Case I should be used in the two-leaf model.

(31)

1Ec,1 = 1Gs,1 Ds,1

(32)

1Hc,1 =

∗ Gh,1 (Rn,1

− λEc,1 − λ1Ec,1 )

Gh,1 + Gr,1

(33)

By applying these refinements to the solution of the combined model, I found that the discrepancies between the two-leaf model and the multi-layered is reduced by 2%, and the computing time only increases by less than 1% when the two-leaf model is used as the surface scheme in a climate model.

5. Results

4. Implementation of the refinement in the combined model of Wang and Leuning (1998) As intercellar CO2 concentration of the big sunlit leaves is part of the solution to the combined model for conductance, photosynthesis and leaf energy balance of Wang and Leuning (1998), the refinement as discussed earlier should be incorporated into the combined model. This will increase the model complexity and computing time. However, I found that little error (<1%) usually results from applying the refinement after the solution of the combined model is obtained. This assumes that the solution to the combined model for intercellular CO2 concentration, and values at the leaf surface for CO2 concentration, temperature and water vapour pressure deficit are not affected significantly by the refinement. Given the refinement usually reduces the predicted photosynthesis of sunlit leaves by up to 5%, I consider that the assumption is reasonable. The difference in the calculated photosynthetic rate of the big sunlit leaf after and before the refinement is applied, 1Ac,1 is calculated as 1Ac,1 = A∗c,1 − Ac,1

Gs,1 1Ac,1 A∗c,1

1Gs,1 =

(30)

where Ac,1 is the photosynthetic rate of the big sunlit leaf calculated using the WL’s formulation without the refinement. After correcting the calculated photosynthesis of sunlit leaves, corrections must also be made to the estimated conductance, latent and sensible heat fluxes of the big sunlit leaf. These are calculated as:

The WL’s formulation for calculating canopy photosynthesis requires as input meteorological data, including the incident beam and diffuse PAR, plus the values of several parameters, θ , β, b and α. Among them, β depends on intercellular CO2 concentration and consequently on the environmental conditions; α is constant (Farquhar et al., 1980), b has also been found to be quite conservative (Wullschleger, 1993; Leuning, 1997), while the curvature parameter θ varies with leaf structure (Terashima and Saeki, 1985) and growth habitat (Leverenz, 1988). Using typical parameter values, the WL’s formulation predicts exactly the same initial slope and asymptote of the canopy light response as the multi-layered model, but overestimates the canopy photosynthesis at intermediate light flux densities for a canopy in which the leaf inclination angle is constant (Fig. 1). The overestimation depends on the all photosynthetic parameters of the leaf, and particularly θ . By separating the Rubisco-limited from RuBP-limited sunlit leaves, the calculated photosynthesis of sunlit leaves by the two-leaf model with refinement for Case I agrees with the multilayered model within ±2%. Fig. 2 shows that the distribution of leaf area with the absorbed PAR at different canopy depth, leaf angle distribution within the canopy is spherical. All sunlit leaves are limited by RuBP regeneration at the canopy top (ξ =0), but most sunlit leaves are Rubisco-limited at the bottom (ξ =5). The calculated photosynthetic rates of all sunlit leaves are 20.9, 23.8, 22.6 and 21.5 ␮mol m−2 s−1 by the multi-layered

Y.P. Wang / Agricultural and Forest Meteorology 101 (2000) 143–150

Fig. 2. Areal fraction of sunlit leaves absorbing different PAR flux density at different canopy depth (black curve). The number above each curve is the cumulative leaf area index from the canopy top to the depth within the canopy, and the grey curve is the value of q∗ at which leaf photosynthesis is colimited within the canopy. The leaf angle distribution is spherical and total canopy leaf area index is 5. The incident PAR flux density above the canopy is 800 ␮mol m−2 s−1 .

canopy model, two-leaf model without the refinement, two-leaf model with the refinement for Case I and for Case II, respectively. For Case I, the inclination angle of all leaves within the canopy is 57◦ from the horizontal, the mean leaf angle for the spherical leaf angle distribution. Compared with the multi-layered canopy model, the two-leaf model without the refinements overestimates the photosynthesis of sunlit leaves by 14%, as compared with only 3% for the two-leaf model with the refinement for Case II. If the variation of absorbed PAR by sunlit leaves with leaf angle at a given canopy depth is ignored, the calculated photosynthetic rate of sunlit leaves by the multi-layered canopy model is 22.2 ␮mol m−2 s−1 , and only differs from the estimate by the refined two-leaf model (Case I) by less than 2%. The refinement for Case II is only applicable to spherical leaf angle distribution, the often assumed distribution when data are not available. An analytic solution is not possible for a more general leaf angle distribution, such as the ellipsoidal leaf angle distribution (Campbell, 1986). For non-spherical leaf angle distribution, the refinement for Case I should be used in the two-leaf model to reduce the overestimation of the photosynthesis of the sunlit leaves. Fig. 3 compares the integrated photosynthesis of the sunlit leaves calculated using the multi-layer model of Leuning et al. (1995) and the WL’s formulation, with or without the refinements developed in this

149

Fig. 3. Modelled response of the photosynthesis of sunlit leaves to incident PAR flux density above the canopy using the multi-layered model of Leuning et al. (1995) (solid curve), or using the two-leaf model of Wang and Leuning (1998) without the refinement (open circle) or with the refinement for Case I (grey triangle) or with the refinement for Case II (filled circle). Values of all parameters are the same as for Fig. 1 except θ. For case II and multi-layered canopy model, spherical leaf angle distribution is used.

paper, leaf angle distribution within the canopy is assumed to be spherical. Calculations were performed using two extreme values of the curvature parameter: θ =0 corresponding to the rectangular hyperbola used in Wang and Leuning (1998), and θ =1, the so-called Blackman function. The results show that at θ =0 the uncorrected WL’s formulation overestimates photosynthesis at incident PAR above 500 ␮mol m−2 s−1 by up to 15%, the two-leaf model with the refinement for Case I or II reduces the overestimation to less than 9 and 5%, respectively, as compared with the multi-layered canopy model. Therefore the estimate of the photosynthesis of sunlit leaves is substantially improved by including the refinements described earlier. For θ =1, the refinements also improve the calculated photosynthesis of sunlit leaves by the twoleaf model by similar amount, as compared with the estimates by the multi-layered model, but when the incident PAR is less than 500 ␮mol m−2 s−1 or greater than 1200 ␮mol m−2 s−1 , the WL’s formula introduces little error to calculated photosynthesis, this is because the transition from RuBP-limitation to Rubisco limitation is more abrupt with increasing incident PAR flux density.

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I also repeated the comparison between the multi-layered model and the two-leaf model presented by Wang and Leuning (1998) but now with the above refinement included. I found that discrepancies between the two-leaf with the refinement for Case II and the multi-layered models were reduced to <3% in the simulated canopy conductances, and the fluxes of CO2 , H2 O and sensible heat, while increasing the computing time by <1% when the big-leaf model was used as the surface scheme in a climate model (results not shown here).

6. Conclusion The two-leaf formulation for calculating canopy photosynthesis has been refined by integrating the Rubisco-limited and RuBP-limited photosynthesis of sunlit leaves separately. In a comparison with a more detailed multi-layered model, the refinement in the WL’s formulation reduces discrepancies in canopy conductances and the fluxes of CO2 , water vapour and heat to <3%, while increasing the computing time by <1% when the two-leaf model is used in a land surface scheme.

Acknowledgements I thank Dr. Ray Leuning and Professor Graham Farquhar for their constructive comments. References Campbell, G.S., 1986. Extinction coefficients for radiation in plant canopies calculated using an ellipsoidal inclination angle distribution. Agric. For. Meteorol. 36, 317–321.

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