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. Equa-tions for conductance, photosynthesis and energy exchange were developed for two big leaves sepa-rately. 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 photo-synthesis of sunlit leaves at intermediate irradiances (800–1200mmol m−2s−1). I present here some
re-finement to the two-leaf model in calculating hourly canopy photosynthesis which reduces the discrepan-cies 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
Fig. 1. (a) Rubisco-limited (solid curve) or RuBP-limited (dashed curve) photosynthesis of sunlit leaves within the canopy, the cu-mulative leaf area index at which photosynthesis of sunlit leaves is co-limited,ξ1is 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, fbis calcu-lated asfb=min{0, (I1−300)/I1}, where I1is the flux density of incident PAR above the canopy. The non-rectangular PAR response of the potential electron transport rate was used with curvature pa-rameterθ=0.70. Parameter values used are: C=245mmol mol−1,
Ŵ∗=30mmol mol−1; K
m=585mmol,α=0.385, kn=0.6; vcmaxand jmax decrease exponentially within the canopy in proportional to exp(−knξ), where ξ is the cumulative leaf area index from the canopy top, where the values of vcmax and jmax are 100 and 210mmol m−2s−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 photosyntheti-cally active radiation (PAR) absorbed by the leaves within the canopy. The two-leaf model does not
take account of the variation in the processes limit-ing 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 hor-izontally homogeneous canopy depends on two struc-tural 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 ap-proach 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 pho-tosynthesis is nonlinearly related to the absorbed PAR, and the two-leaf model does not account for the varia-tion of absorbed PAR within the sunlit leaves and will therefore overestimate the photosynthesis of the sun-lit 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 calcu-late their photosynthesis separately. In the following, I present the theory of colimitation of leaf photosyn-thesis. I will then apply the theory to refine the calcu-lation 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 con-ductance, 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
usually occurs over a range of light flux densities, as chloroplasts in a leaf receive different light flux densi-ties 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 be-come light-limited, photosynthesis of leaf is still col-imited 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 pho-tosynthesis (Af,c), the model of Farquhar et al. (1980)
states that
Af,c=
vc max(C−Ŵ∗)
(C+Km)
−rd (1)
where νcmax is the maximum carboxylation rate, C is intercellular CO2 concentration, Km is a
Michaelis–Menten coefficient, andŴ∗is the CO2
com-pensation 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−Ŵ∗)
(C+2Ŵ∗) −rd (2)
where j is the potential electron transport rate, which is calculated using the non-rectangular light response function (Farquhar and Wong, 1984)
θj2−(αq+jmax)j+αqjmax=0 (3)
whereθis a curvature parameter,αis the initial slope of the electron-transport rate at low light, q is the ab-sorbed PAR, and jmaxis 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
vc max(C−Ŵ∗)
(C+Km)
=0.25j
∗(C−Ŵ∗)
(C+2Ŵ∗) (4)
where j∗is the potential electron transport rate when leaf photosynthesis is co-limited, and is given by
j∗=4(C+2Ŵ ∗)
(C+Km)
vc max ≡βvc max. (5)
The parameterβ is defined as
β≡ 4(C+2Ŵ ∗)
(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 (7)
whereλand b are defined as
λ≡ β(θβ−b)
α(β−b) (8)
b≡ jmax
vc max
(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 photosyn-thesis 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,1kb(1−ωf,1) (10)
where Ib,1 is the direct beam PAR at the top of the
canopy, kbis the extinction coefficient for black leaves
(Goudriaan and van Laar, 1994), andωf,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
q2,1(ξ )=Id,1kd∗,1(1−ρtd,1)exp(−kd∗,1ξ ) (11)
where Id,1is the diffuse PAR at the top of the canopy,
I also assume that vcmax decreases exponentially
withξ, i.e.
vcmax(ξ )=vcmax0 exp(−knξ ) (12)
wherevcmax0 is the maximum carboxylation rate of the leaves at the top of the canopy and knis the nitrogen
distribution coefficient. A similar relation is assumed for the decrease of jmaxwithξ because Wullschleger
(1993) found thatjmax/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 ob-tained by substituting Eqs. (10) and (12) into Eq. (7), to give
for sunlit leaves.By substituting Eqs. (11) and (12) into Eq. (7), we obtain
Photosynthesis of sunlit leaves is always limited by Rubisco activity ifξ1<0, and by RuBP regeneration if
ξ1>L, where L is the total leaf area index. Only when
0<ξ1<L, is there a transition from a limitation due
to RuBP regeneration to a limitation due to Rubisco activity at some level within the canopy. Similar ar-guments apply for shaded leaves andξ2ifkn> kd∗,1.
Ifkn < kd∗,1 andξ2>0, photosynthesis of the shaded
leaves is always limited by RuBP regeneration, and by Rubisco activity ifξ2>L.
Under natural conditions, the numerator of Eq. (14) is usually positive and thus ξ2>0 only when
kn> kd∗,1. Therefore transition from RuBP-limitation
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 pro-portional to the absorbed PAR at ξ, but most field data seems to suggest that kn is usually less than
kd∗,1 (Hirose and Werger, 1994). Under these con-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 ξ1 and ξ2 will be somewhat
smaller than those given by Eqs. (13) and (14), re-spectively, if the absorption of the scattered PAR is ac-counted for. For sunlit leaves,ξ1is likely to be within
0 and L only at high incident PAR flux density when the absorption of scattered and diffuse PAR only ac-counts for a relatively small fraction of the absorbed PAR. Therefore a small error (<1%) is caused by us-ing 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, photo-synthesis of shaded leaves is limited by absorbed PAR if kn is less thankd∗,1, and the WL’s formulation still
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, con-ductance 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<ξ1<L, photosynthesis of the sunlit leaves
within the canopy,A∗c,1is calculated as
A∗c,1=Ac,1a+Ac,1b (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.
Q1b,1=
The bulk photosynthetic parameters of the Rubisco-limited sunlit leaves within the canopy are
Vcmax,1b=vcmax0 φ{ξ1, kn+kb} (17)
Jmax,1b=jcmax0 φ{ξ1, kn+kb} (18)
wherejmax0 is the maximum rate of potential electron transport of the leaf at the top of the canopy and func-tionφis defined as
φ (x, k)=
Z L
x
exp(−kξ )dξ. (19)
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
ab-sorption 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
q1,1=(1−ωf,1)Ib,1cosγ (23)
where γ is the angle between the direction of inci-dent 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 car-boxylation and RuBP regeneration whenγ=γ∗, and
γ∗=cos−1 ited by Rubisco carboxylation, otherwise by RuBP
regeneration. The total Rubisco carboxylation capac-ity (Vcmax,1b) and potential rate of RuBP regeneration
(Jmax,1b) of the Rubisco-limited sunlit leaves are
Vcmax,1b
The total amount of absorbed PAR by the Rubisco-limited sunlit leaves, Q1b,1is
Q1b,1=
The total rate of potential maximal electron trans-port rate (Jmax,1a) and the amount of absorbed PAR
(Q1a,1) by the RuBP-limited sunlit leaves are
Q1a,1=Q1,1−Q1b,1. (29)
The bulk parameters, Vcmax,1a, Vcmax,1b, Jmax,1aand Jmax,1b and the total amount of absorbed PAR, Q1a,1
and Q1b,1can 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 distribu-tion, 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.
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 bal-ance of Wang and Leuning (1998), the refinement as discussed earlier should be incorporated into the com-bined 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,1is calculated as
1Ac,1=A∗c,1−Ac,1 (30)
where Ac,1is 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 es-timated conductance, latent and sensible heat fluxes of the big sunlit leaf. These are calculated as:
1Gs,1=
Gs,11Ac,1
A∗c,1 (31)
1Ec,1=1Gs,1Ds,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
The WL’s formulation for calculating canopy pho-tosynthesis requires as input meteorological data, including the incident beam and diffuse PAR, plus the values of several parameters,θ,β, b andα. 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 parameterθvaries with leaf structure (Terashima and Saeki, 1985) and growth habitat (Leverenz, 1988). Using typical param-eter values, the WL’s formulation predicts exactly the same initial slope and asymptote of the canopy light response as the multi-layered model, but overesti-mates the canopy photosynthesis at intermediate light flux densities for a canopy in which the leaf inclina-tion angle is constant (Fig. 1). The overestimainclina-tion de-pends on the all photosynthetic parameters of the leaf, and particularlyθ. 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 multi-layered model within±2%.
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 800mmol m−2s−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 hori-zontal, 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 pho-tosynthetic rate of sunlit leaves by the multi-layered canopy model is 22.2mmol m−2s−1, and only
dif-fers 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 distribu-tion, 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 refine-ment 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
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 exceptθ. For case II and multi-layered canopy model, spherical leaf angle distribution is used.
paper, leaf angle distribution within the canopy is as-sumed 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), andθ=1, the so-called Blackman function. The results show that atθ=0 the uncorrected WL’s formulation overestimates photo-synthesis at incident PAR above 500mmol m−2s−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 two-leaf model by similar amount, as compared with the estimates by the multi-layered model, but when the incident PAR is less than 500mmol m−2s−1 or
greater than 1200mmol m−2s−1, the WL’s formula
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 be-tween 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, H2O 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 Far-quhar for their constructive comments.
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