A numerical issue in calculating the coupled carbon and water fluxes in a climate model

Authors


Abstract

[1] The Community Land Model (CLM) uses a fixed-point iteration approach to solve the coupled photosynthesis and stomatal conductance model (Ags). Here we demonstrate that this approach does not converge in its iterative calculation of gross primary production (GPP) and transpiration in large portions of land surface, because the coupled Agsmodel does not always comply with a condition (the fixed-point theorem) required by the fixed-point approach for convergence. This iteration fails more frequently in some regions of the world than in others, leading to regionally varying uncertainty and global biases in the estimated carbon and water fluxes. Moreover, CLM applies an artificial constraint to the water vapor pressure of canopy air in its calculations ofAgs, with an intention to prevent the ‘numerical instability’ arising from the fixed-point approach. Our results show that this constraint reduces but does not prevent the occurrence of nonconvergence. Since this constraint is artificial, it can bias GPP and transpiration simulations. We then propose a Newton-Raphson iteration scheme to replace the fixed-point approach and show that this new approach can ensure convergence, does not require an artificial constraint on the atmospheric water vapor pressure, and is computationally efficient. On the other hand, the default fixed-point treatment in CLM leads to a ∼2.7 PgCyr−1overestimation of GPP globally but with much higher regional biases (∼27%). We suggest that the current fixed-point treatment in CLM be replaced with the Newton-Raphson approach and that the artificial constraint on the atmospheric water vapor pressure be removed.

1. Introduction

[2] Climate models are now used to predict future oceanic and terrestrial carbon storage and hence how much will remain in the atmosphere and contribute to global warming [e.g., Dickinson, 2012; Friedlingstein et al., 2006; Fung et al., 2005; Sarmiento et al., 2010]. Carbon enters the terrestrial ecosystem through leaf stomata, which also transpire water extracted from soil. Therefore, a coupled photosynthesis and stomatal conductance (inverse of stomatal resistance) model is often used within a general land surface modeling framework to simulate terrestrial gross primary production (GPP) and transpiration at various time and spatial scales [e.g., Bonan et al., 2011; Dai et al., 2004; Sellers et al., 1996]. GPP is the initial step of terrestrial carbon cycling [Beer et al., 2010; Zhao and Running, 2010]. It fuels the growth of plants, which in turn provide organic matter to soil [Dickinson, 2012]. Transpiration is a key component of evapotranspiration (ET) that by balancing absorbed solar radiation determines land surface temperature and other climate features [Dickinson et al., 2002]. Thus a robust solution of the coupled photosynthesis (A) and stomatal conductance (gs) process (denoted as Ags) is needed for accurate predictions of terrestrial carbon and water fluxes in climate models.

[3] A widely used framework of the coupled process of the CO2 and water vapor flow through the stomata was first implemented by Collatz et al. [1991], a variant version of which now is a component of the Community Land Model (CLM) [Oleson et al., 2010]. This Ags model encompasses a suite of nonlinear equations that in general must be solved iteratively, although under certain simplified conditions, analytical solutions are possible [Baldocchi, 1994].

[4] Various iteration methods are available for solving such nonlinear equations, with their own strengths and limitations for specific applications. Choosing an appropriate such scheme is of particular importance when it is applied for global simulations. We emphasize this point for a couple of reasons. First, in a global modeling framework, there are 103–106 land grids (depending on the simulation resolution), each of which can consist of mixed patches of various plant types that are forced by different environmental conditions. These vegetation and climatic conditions in turn strongly affect the behavior of the nonlinear equations involved and therefore the validity of their numerical treatment. Second, iterative calculations must be computationally inexpensive for global applications. However, we are not aware of any previous effort to examine these numerical issues in depth for the Ags model, although the coupling between carbon and water vapor fluxes is at the core of land surface modeling.

[5] We have found that the iteration approach used in CLM to solve the Agsmodel (and likely other such models of similar heritage) does not converge, hence introduces errors to the modeled carbon and water fluxes. We examine why this nonconvergence occurs and introduce a more robust iteration framework to remedy this numerical problem. This new approach is based on an in-depth analysis of the numerical behavior of theAgs model. We demonstrate that our new treatment corrects substantial biases in the GPP and transpiration simulated by the latest released version of CLM, i.e., CLM4 [Oleson et al., 2010].

2. The Numerical Treatment for the Ags Model in CLM4 Produces Large Regions of Nonconvergence

2.1. Characteristics of Iterative Solutions for the Coupled Nonlinear Equations in CLM4

[6] The carbon and water fluxes must be determined together with the constraint of canopy energy conservation, because the key parameters for photosynthesis and evapotranspiration (ET) strongly depend on leaf temperature. Models use iterative approaches to directly solve for two unknowns: leaf temperature (Tv) and the CO2 partial pressure inside the leaf intercellular space (ci). Other variables are calculated as functions of these two unknowns either during the iterative procedure or after the two unknowns have been solved.

[7] Consider two equations with the variables x and y to be determined, i.e., f(x,y) = 0 and g(x,y) = 0. In particular, f is the Ags model with x as ci, and g is the leaf temperature calculation with y as Tv. Currently in CLM4, these two equations are solved by nesting the Ags model into the Tv calculation, where the outer loop solves g(x,y) = 0 for y assuming a fixed x, while the inner loop solves f(x,y) = 0 for x keeping y fixed. Furthermore, the two loops use different iteration schemes, i.e., the inner loop fuses a fixed-point (FP) iteration approach independent of the outer loop whereas the outer loopguses a Newton-Raphson iteration inherited from the earlier Biosphere-Atmosphere Transfer Scheme (BATS) [e.g.,Dickinson et al., 1993, equation (84); Oleson et al., 2010, equation (5.116)]. Here we focus on the inner loop only, since the fixed-point iteration can fail to attain a solution for theAgs model and hence simulate carbon and water fluxes with questionable accuracy.

2.2. Formulations and Numerical Implementations of the Ags Model in CLM4

[8] The CLM4 uses a modified version of the Farquhar model [Farquhar et al., 1980; Collatz et al., 1991] to calculate photosynthesis for C3 plants and the Collatz et al. [1992] model for C4 plants. The photosynthesis A is expressed as a function of ci as follows:

display math

where wcis the Rubisco-limited carboxylation rate;wjis the light- or RuBP regeneration-limited carboxylation rate; andweis the export-limited (C3) or the CO2-limited rate of carboxylation (C4). For simplicity, wc, wj and we and thus A are all shown as functions of ci only; but they actually depend on a suite of parameters and environmental variables as well. For example, wc is determined by the maximum Rubisco carboxylation rate Vcmax, a key photosynthetic parameter that is limited by soil dryness βt, where 0 ≤ βt ≤ 1 [see Oleson et al., 2010, equation (8.13)]. Detailed formulations of wc, wj and we are documented elsewhere [Oleson et al., 2010]; here we only list key equations of direct relevance to our study.

[9] The CLM4 applies the Ball-Berry model [Ball et al., 1987; Collatz et al., 1991] to calculate stomatal resistance rs, which requires A as an input. Through incorporating CO2 diffusion via leaf boundary layer, rs is solved with the following quadratic equation [also Oleson et al., 2010, equation (8.27)],

display math

Here, mis a plant functional type (PFT)-dependent parameter;Patm the atmospheric pressure; cs the CO2 partial pressure at the leaf surface; b the minimum stomatal conductance; rb the leaf boundary layer resistance; ei the saturation vapor pressure inside the leaf; ea the constrained water vapor pressure of canopy air. We emphasize that CLM4 imposes an arbitrary lower limit on the parameter ea, which strongly affects the effectiveness of the fixed-point iteration (section 3). In CLM4, ea is determined by

display math

where ea is the actual water vapor pressure of canopy air; α(=0.25 for C3 or = 0.4 for C4 plants) is an arbitrary coefficient for the constraint on water vapor pressure enforced by CLM4. The rationale for this constraint is to “prevent numerical instability in the iterative stomatal resistance calculation” [Oleson et al., 2010, p. 172]. The lower limit is higher for C4 plants “because C4 plants are not as sensitive to vapor pressure as C3 plants” [Oleson et al., 2010, p. 172]. Later we examine the consequences of this artificial constraint for the calculated carbon and water fluxes when fixed-point iteration is used.

[10] The diffusion of CO2 through stomata forms the final independent equation needed to close the coupled Ags model:

display math

The set of equations equations (1), (2), and (4) is iterated over the variable ci given an initial guess, that is, ci = 0.7·ca for C3 or ci = 0.4·ca for C4, where ca is the atmospheric CO2 partial pressure [Oleson et al., 2010]. This current procedure in CLM4 is a typical implementation of the fixed-point iteration (details to be given insection 3). Its iteration of ci proceeds three times, without checking whether a convergence is achieved.

2.3. Demonstration of the Numerical Problems in the Calculated Carbon and Water Fluxes With Global Simulations

[11] Here we use global simulations to demonstrate that the fixed-point treatment for theAgsin the default CLM4 produces large regions of nonconvergence of GPP and transpiration over the global land surface. This diagnosis strategy is simple: we increased the maximum number of iterations N from 3 to 20 with a one-step increment for each simulation while keeping the fixed-point configurations as default in CLM4 (denoted as FP_wC,Table 1). Simulations were performed from the year 1948 to 2004 in the offline mode driven by the atmospheric forcing data of Qian et al. [2006]. The analyses were conducted for boreal summer, i.e., JJA (note this simulation and analysis protocol was also applied to the simulations with our proposed new iteration approach in section 5). For clarity, only results for the year 1948 are described here; results for other years are similar and given in the auxiliary material (Figure S1).

Table 1. Summary of Model Simulations
SimulationDescription
FP_wCfixed-point scheme with the constraint toea (the default CLM4 scheme)
FP_woCfixed-point scheme without the constraint toea
NR_wCNewton-Raphson scheme with the constraint toea
NR_woCNewton-Raphson scheme without the constraint toea

[12] Figure 1ashows clear differences in the JJA-mean GPP over North America and Eurasian between the default 3-step treatment and that with an additional iteration. This inter-step difference is pronounced in western United States, where a ∼200 g C m−2 yr−1(∼20% if using the default 3-step simulation as reference, also seeTable 2) difference in GPP is observed. Evidently, the fixed-point iteration suffers from widespread nonconvergence since the difference between two adjacent iteration steps would be minimal if the iterative calculation had converged. Furthermore, this failure of convergence is not a matter of the maximum number of iteration steps N, because the difference in GPP between two consecutive steps does not attenuate with increasing number of iterations in general (Figures 1b–1g and Table 2). Instead of convergence, a regular oscillation pattern exists in the modeled GPP with fixed-point iteration (Figures 1b–1g). Similar patterns are also found in the simulated transpiration (results not shown).

Figure 1.

Differences of 1948 JJA-mean GPP (g C m−2 yr−1) between two successive iteration steps for FP_wC: (a) 3rd and 4th, (b) 4th and 5th, (c) 5th and 6th, (d) 6th and 7th, (e) 7th and 8th, (f) 18th and 19th, and (g) 19th and 20th iteration step, respectively (the iteration steps are denoted as N3, N4, ···, and N20). Figures 1b–1g are the subsets of the United States highlighted by the box region in Figure 1a. Here the year 1948 is chosen for demonstration of nonconvergence arising from fixed-point approach, and is also used in the following relevant figures for consistency.

Table 2. Iteratively Calculated JJA-Mean GPP, Transpiration, and Sensible Heat Fluxes Averaged for Western United States (30∼50°N, 130∼105°W) and for Eastern Amazon (20S∼EQ, 60∼40°W)a
SimulationIteration StepWestern U.S.Eastern Amazon
GPPTranspirationSensible HeatGPPTranspirationSensible Heat
  • a

    GPP is in g C m−2 yr−1 while transpiration and sensible heat fluxes are in W m−2. For the fixed-point experiments, the regional average for the 3rd, 4th, 19th, and 20th steps are shown to demonstrate that the non-convergence does not attenuate with increasing number of iteration steps; while convergence can be achieved at the 20th iteration step for the Newton-Raphson approach.

FP_wCN3993228321425855
 N4787178621405855
 N19987228321405855
 N20790178621405855
FP_woCN3757158721695756
 N41034228320665557
 N19770158721605756
 N201029228320585457
NR_wCN19830188521405855
 N20830188521405855
NR_woCN19778158720625457
 N20778158720625457

[13] As stated above, CLM4 imposes a prior constraint on the water vapor pressure of canopy air, which artificially modifies the environmental conditions and hence their impact on the Ags calculation. We designed another set of simulation experiments (denoted as FP_woC, Table 1) to investigate the effect of this constraint on iterative calculations of GPP and transpiration. In these simulations, the lower limit on ea was removed such that equation (3) turns into math formula, while all other formulations and the fixed-point approach remain unchanged from the default.

[14] Figure 2 shows that the removal of the constraint on ea substantially degrades the convergence of simulated GPP and transpiration. Compared to Figure 1a, more regions show nonconvergence of GPP, e.g., eastern Amazon, southern Africa, Australia, and a large portion of Eurasia (Figure 2a). In western U.S., the magnitude of disparity between the 3th and 4th iteration is larger than that with this constraint. A concomitant degradation is also found in canopy transpiration, because of its strong coupling with photosynthesis (Figure 2b).

Figure 2.

Differences of JJA-mean (1948) (a) GPP (g C m−2 yr−1) and (b) canopy transpiration (W m−2) between the 3rd and the 4th iteration step for FP_woC. Figure 2a is similar to Figure 1a but for FP_woC.

[15] These findings demonstrate that the fixed-point iteration approach employed by the CLM4 fails to solve the coupledAgs model for substantial regions of the land surface on Earth.

3. Why the Iteration Approach in CLM4 Fails: A Mathematical Perspective

3.1. The Theorem of Fixed-Point Iteration

[16] The fixed-point iteration is the simplest numerical approach to solve nonlinear equations. Any equation can be expressed in the fixed-point form as,

display math

The root of equation (5) is commonly called the fixed point of the iterative function h(x). This form is appealing because it allows a very straightforward iterative procedure:

display math

Geometrically, the root of equation (5) is the intersection between the diagonal line y = x and the curve y = h(x) in a Cartesian coordinate system. Although the fixed-point iteration is easy to apply, its success critically depends on the shape ofh(x). The fixed-point iteration does not converge if the absolute value of the derivative ofh(x) with respect to x is greater than one, i.e., |h′(x)| > 1, over the domain around the root of x.This is the well-known fixed-point theorem and its violation leads to failure of convergence [Burden and Faires, 2011].

3.2. Numerical Behaviors of the Fixed-Point Solution to theAgs Model

[17] Equation (4)as implemented in CLM4 is a typical form of fixed-point iteration, whereci is considered as the input xand the right-hand side represents the iterative functionh:

display math

Whether equation (7)can be solved effectively in a global simulation using the fixed-point approach depends on its mathematical properties, as determined by calculatingA and rs from equations (1) and (2), hence h(ci) from equation (7) as a function of ci, and examining the shape of the resulting h(ci) curve. This process is different from the iterative calculation of ci in that it does not repeat the iterative loop. It is equivalent to examining how h(x) changes with x in equation (5). Only the photosynthesis model for C3 plants is examined here since they are the dominant vegetation types globally.

[18] Figure 3 shows the geometrical shape of h(ci) and the effect of model parameters ea on the iterative calculation of ci. A couple of common characteristics of h(ci) curves are seen. First, the h(ci) curve is flat when ci is below the CO2 compensation point Γ*. This occurs because in CLM4, wc and wj and hence A are set to be zero if ci is less than Γ* (for details, see Oleson et al. [2010]). As a consequence, h(ci) = ca (note both ci and h(ci) are confined to the range of 0 to ca) as shown in Figure 3a. Second, as ci increases from Γ*, h(ci) decreases initially with a very steep slope with |h′(ci)| > 1; only as ci further increases does the slope become gentler (|h′(ci)| < 1). The specific geometry of the h(ci) curve depends on model parameters in equations (1) and (2). As one parameter varies, the absolute value of the derivative of h(ci) can equal or exceed one and so violate the fixed-point theorem.

Figure 3.

Typical convergence scenarios of the fixed-point iteration functionh(ci) (equation (7)) for the coupled photosynthesis-stomatal conductance model (C3) as formulated in CLM4. These scenarios are demonstrated with representative values of model parameters and varying values of the constrained water vapor pressure of canopy air ea (equation (3)). Both the x- andy-axis are confined to the range from 0 to the atmospheric CO2 partial pressure ca, thus the maximum values of ci and h(ci) are ca. (a) The impact of different values of ea on the shape of h(ci) and its derivative at the root, i.e., h′(c*i); the iterative generation of ci for different scenarios in Figure 3a, identified with curve colors: (b) ci converges within 3 steps of iteration (solid green), (c) ci converges but beyond 3 steps (solid purple), (d) a nonconvergence case with ci alternating between two constant values in the range of 0 and ca (solid blue), and (e) a nonconvergence scenario similar to Figure 3d but with ci taking the exact value of 0 and ca in alternation (solid red). The four scenarios corresponds to different values of ea and hence the derivative of h(ci). The dash yellow curve in Figure 3a represents the boundary from convergence to nonconvergence, occurring at the root h′(c*i) = −1, the criterion of the fixed-point theorem. The area below this boundary is shaded, corresponding to the nonconvergence region, whereh′(c*i) < −1. The iterative solution c*i is the intersection of the h(ci) curve and the diagonal line y = x (solid gray), i.e., the root of h(ci) = ci. The sequences of ci,k (the subscripts k is the iteration step) are marked on the h(ci) curves, with black dash lines indicating the iteration loop. For clarity, only the first three ci,k are shown. The initial guesses ci,1 have the same value for all scenarios, i.e., ci,1 = 0.7·ca. The insets in (b)–(e) show the ci (Pa, the y axis) as a function of iteration step (from 1 to 20, the x axis) for each scenario. The vertical gray lines in the insets of Figures 3b and 3c highlight the step at which the convergence is achieved, that is 3 steps for Figure 3b and 10 steps for Figure 3c, respectively. Model parameters used to generate these curves are: the absorbed photosynthetically active radiation ϕ = 2000 μmol photon m−2 s−1, the maximum Rubisco carboxylation rate Vcmax = 40 μmol m−2 s−1, the CO2 compensation point Γ* = 2 Pa, the composite Michaelis-Menten constantKco = 100 Pa (Kco = Kc · (1 + O / Ko)), leaf boundary layer resistance rb = 1.0 × 10−6 s m2 μmol−1, and the atmospheric CO2 partial pressure ca is 28 Pa.

[19] Here we show that the shape of h(ci) is very sensitive to the constrained water vapor pressure of canopy air eaand hence it is the most important parameter determining whether the fixed-point approach converges. Aseadecreases, the fixed-point iteration switches from convergence to nonconvergence. The transition boundary occurs ath′(ci) = −1 with ci at the root c*i (Figure 3a). Figure 3ashows four typical convergence scenarios for fixed-point iteration given an identical initial guessci,1 (=0.7·ca, i.e., the CLM4 setting for C3 physiology; the subscript number represents the iteration step). Scenario I (Figure 3b) demonstrates the effectiveness of the fixed-point iteration when it converges: the solutionc*i is achieved within 3 iteration steps because ci,k at all iteration steps fall in the region of the h(ci) curve where |h′(ci)| < 1. Scenario II (Figure 3c) also converges with the fixed-point approach, but it requires 10 steps to approach the root. This slower convergence rate is a consequence of a relatively large |h′(ci)| compared to that of Scenario I (i.e., closer to 1). Scenario III (Figure 3d) fails to converge at the root, because the iterative ci,k periodically falls onto the zone with |h′(ci)| > 1, i.e., the initial guess of ci,1 results in a value of ci,2 that falls onto the “steep declining zone” of h(ci); then ci,3 is a little bit larger than ci,1 and leads to an ci,4 on the same part of h(ci) as ci,2. The ci from the 3rd step onward oscillates between the values of ci,3 and ci,4. As a consequence, consecutive values of ci fluctuate around the root but never approach it, regardless of the maximum iteration steps N. Scenario IV (Figure 3e) is similar to Scenario III, but with the magnitude of the oscillation of ci larger than alternating between 0 and ca. Since ci,2 directly falls below Γ* (thus A is set to be zero), and then ci,3 equals to ca yielding ci,4 to be zero. Again, from this point, the iteration circles around but never reaches the root. These trajectories of ci with iteration steps shown in Figures 3d and 3eillustrate the consequences of the violation of the fixed-point theorem.

[20] Figure 4 shows that other parameters also affect the shape of h(ci) curves, but with a lesser degree of sensitivity by individually varying the maximum Rubisco carboxylation rate Vcmax, the absorbed photosynthetically active radiation ϕ, the composite Michaelis-Menten constantKco = (Kc · (1 + O / Ko), and the CO2 compensation point Γ*, while keeping unchanged from boundary curve in Figure 3a for all other input variables and parameters. Figure 4a shows that a larger Vcmax and Figure 4c that a smaller Kco produce a steeper h(ci) curve and thus failure of the convergence of the fixed-point iteration. Whenϕ is over 500 μmol photon m−2 s−1, the geometry of h(ci) is not sensitive to its variation, and hence ϕ has less impact on convergence (Figure 4b). The Γ* modifies the h(ci) shape in a different way (Figure 4d). It not only affects the steepness of the h(ci) curve, but also alters the width of the “flat zone” of the h(ci) curve. As Γ* increases, the derivative |h′(ci)| increases and the curve moves rightward and gradually enters the zone with |h′(ci)| > 1, leading to the nonconvergence of ci.

Figure 4.

Sensitivity of the h(ci) shape to key parameters in the coupled photosynthesis-stomatal conductance model (C3). The parameters investigated are: (a) the maximum Rubisco carboxylation rate Vcmax, (b) the absorbed photosynthetically active radiation ϕ, (c) the composite Michaelis-Menten constantKco (=Kc · (1 + O/Ko)), and (d) the CO2 compensation point G*. The solid red and solid green curves represent nonconvergence and convergence cases, respectively. The dash yellow curves represent the boundary between the convergence and nonconvergence regions, identical to that in Figure 3a. The shaded area in each plot corresponds to the nonconvergence region, which is below the yellow boundary in Figures 4a–4c but above it in Figure 4d. The solid gray line is the reference y = x. These h(ci) curves are produced by varying individual parameters in Figures 4a–4d, with all the other parameters fixed at the same value as the boundary curve. Note that the relative distributions of convergence and nonconvergence region are only meaningful to the particular parameter under consideration and are not comparable across plots (parameters) as they are entirely determined by how the concerned parameter affects the derivative of h(ci) with respect to ci near the root when all other parameters are fixed. For example, when all other parameters are fixed, increasing Γ*in Figure 4d tends to steepen the slope of the iteration curve, which changes the fixed-point iteration from convergence to nonconvergence and moves the iteration curve to the upper right corner in theh versus. ci domain. In contrast, decreasing Kco (Figure 4c) tends to have the same effect in terms of convergence but move the curve to the lower right corner.

[21] To summarize, the nonlinear equation represented by the Ags model (C3) does not always comply with the conditions required to ensure the convergence of the fixed-point iteration. The shape of the equation depends on the values of parameters and input environmental variables. In particular, the parametereamatters most for the effectiveness of the fixed-point approach in finding the root ofci, since ci is sensitive to rs (equation (7)), which in turn is quite sensitive to variations of ea. Furthermore, the eahas larger variations (strong diurnal cycle and seasonal variation) than other parameters involved, thus it contributes most to the numerical failure of fixed-point iteration in solving theAgs model.

3.3. The Underlying Causes for the Spatial Variation of the Failure of Fixed-Point Iteration in Global Simulations

[22] Across the land of the globe, the failure of fixed-point iteration does not occur randomly (Figures 1 and 2). The mathematical investigation of the shape of h(ci) provides some insights to the physical causes of the nonconvergence of the fixed-point approach in simulating carbon and water fluxes. It can be largely attributed to the joint effects of the relative humidity (RH) and air temperature in determining the water vapor pressure of canopy airea (Figure S2). We can see that the nonconvergence regions in Figure 2, e.g., the western U.S., eastern Amazon, southern Africa, northern Australia, and Eurasia, have a low RH (generally below 50%), which in turn decreases ea. Northern Africa has the lowest RH but does not show the nonconvergence problem, because the associated high air temperature and thus high saturation water vapor pressure offset the effect of low ea on ea and the convergence of iteration. In addition, the sparse vegetation leads to an extremely low productivity in this area. Apart from northern Africa, the spatial distribution of the nonconvergence (Figure 2) has a good correspondence with that of RH (Figure S2). Thus it is primarily RH that determines why the fixed-point scheme works for some grids but not for others.

[23] This high sensitivity of h(ci) to eaalso explains why the application of the prior constraint can improve the convergence of the fixed-point approach (see the difference betweenFigures 1 and 2): by maintaining a higher water vapor pressure it reduces the steepness of the curve h(ci) (Figures 3 and S4) and makes the iteration more likely to satisfy the fixed-point theorem.

4. An Alternative Iteration Framework: The Newton-Raphson Approach

[24] The Newton-Raphson (NR) iteration scheme has been applied in climate models, for example, to calculate leaf temperature from energy balance equations [Oleson et al., 2010] as mentioned earlier and to treat the complicated cloud microphysics [Neale et al., 2010]. Here we apply this strategy to solve the coupled Agsmodel. The standard form of Newton-Raphson iteration is [Burden and Faires, 2011]

display math

Again xk and xk+1 are the iterated variable at the kth and k + 1th step; j(x) is the iteration function to be zeroed; and j′(x) is the derivative of j(x) with respect to x.Starting with an initial guess, the Newton-Raphson scheme approximatesj(x) by its tangent line, the slope of which is j′(x). The x-intercept of the tangent line is taken as the guess of the root for the next iteration step. The iteration converges if the difference between two successive steps, i.e.,xk and xk+1, is smaller than a prescribed threshold that is close to zero. The converged point is the root of the nonlinear equation. Geometrically, the root of j(x) is the intersection between y = 0 and y = j(x). Unlike the fixed-point iteration, the absolute value of the derivative |j′(x)| does not need to be less than 1 for the Newton-Raphson approach to achieve convergence.

[25] To apply this new scheme to the coupled Ags model, we reorganize equation (7) to form the iteration function

display math

For global model simulations, calculation of j′(x) can be computationally expensive; here we use a simple and typical numerical approximation [e.g., Burden and Faires, 2011],

display math

The application of the Newton-Raphson scheme using this approximation proceeds as follows:

[26] 1. Assign an initial guess of ci;

[27] 2. Calculate A(ci) from ci according to the photosynthesis model equation (1);

[28] 3. Calculate rs(A) according to equation (2) from the A;

[29] 4. Calculate the function j(ci) from ci, A(ci) and rs(A) according to equation (9);

[30] 5. Calculate an intermediate A1(ci + j(ci)) according to the photosynthesis model equation (1);

[31] 6. Calculate an intermediate rs1(A1) according to equation (2);

[32] 7. Calculate an intermediate j1(ci + j(ci)) from ci + j(ci), and estimated A1(ci + j(ci)) and rs1(A1) according to equation (9);

[33] 8. Obtain the approximate derivative j′(ci) according to equation (10), where j1(ci + j(ci)) and j(ci) correspond to the two terms in the numerator while the latter also serves as the denominator;

[34] 9. Update ci using equation (8);

[35] 10. Exit the iterative loop and determine the final A and rs if the difference of ci between two successive steps satisfies a given convergence criterion or if the current iteration step exceeds the specified maximum number N; otherwise, repeat 2–9.

[36] The convergence criterion of ciis set here to be 0.001 Pa and the upper limit of iteration is N = 20. As demonstrated later, the Newton-Raphson scheme we used here rarely reaches this upper limit.

[37] In order to investigate the effectiveness of this new approach for the calculation of the Ags model, we first explored the mathematical properties of j(ci) curves in a similar manner to h(ci): introducing variations to the j(ci) curve through systematically changing ea. Figure 5demonstrates that this new approach has two benefits. First, for the two convergence scenarios illustrated for the fixed-point iteration (Figures 3b and 3c), the Newton-Raphson scheme has a much faster convergence rate, with its solution attained at the 2nd (Figure 5b) and 3rd (Figure 5c) iteration step for Scenario I and II, respectively. Second, this new scheme turns the nonconvergence scenarios of fixed-point (Figures 3d and 3e) into convergence cases (Figures 5d and 5e) within 5 and 7 iteration steps, respectively. Evidently, the proposed new scheme is a more effective way for solving the nonlinear equations of the Ags model, with both accuracy (or validity) and computational efficiency guaranteed. Its “validity” stems from the absence of a strict requirement for the derivative of j(ci). So the convergence does not depend on the model parameters. Its “efficiency” is a consequence of the mathematical properties of j(ci), i.e., the value for ci at a following step always moves toward the root (Figure 5e). All comparisons here used the same initial guess as the fixed-point approach to eliminate the possible contribution from “a better initial guess.”

Figure 5.

(a) The mathematical properties of the Newton-Raphson iteration functionj(ci) (equation (9)) for the coupled photosynthesis-stomatal conductance model (C3) as developed in this study. The curves displayed here correspond to those in Figure 3, using the same sets of model parameters. Here the roots of ciare achieved for all scenarios through the Newton-Raphson approach, regardless of the derivative of the iteration function. The iterative solutionc*i is the intersection of the j(ci) curve and the reference line y = 0 (solid gray), i.e., the root of j(ci) = 0. The sequences of ci,k are marked on the j(ci) curve in Figure 5e, with black dash lines indicating the iteration processes and with the arrows pointing to the approximated derivative of j(ci) at ci,1 and ci,2. The insets show the calculated ci (Pa, the y axis) at each iteration step (the x axis). The vertical gray lines in the insets of Figures 5b–5e highlight the step at which the convergence is achieved, that is at (b) 2 steps, (c) 3 steps, (d) 5 steps, and (e) 7 steps, respectively.

5. The Effectiveness of the Newton-Raphson Iteration Scheme for Global Applications

[38] We performed two sets of global simulations with a version of CLM4 that implements the new Newton-Raphson iteration approach: NR_wC, a simulation that keeps the constraint onea present, and NR_woC that removes this constraint (Table 1). These experiments follow the same protocols as FP_wC. Again, the year 1948 is presented only for demonstrating the superiority of this new iteration scheme; the historical simulations show similar patterns and are given in the auxiliary material (Figure S3). Here we used the simulations at the 20th step as reference since the prescribed maximum number of iteration is 20.

[39] We found that, with the artificial constraint, the simulated GPP is perfectly identical between the 19th and 20th steps (Figure 6a). In the western U.S., the convergence is almost achieved by the 6th step (Figure 6c). Further, after removing the constraint, the Newton-Raphson scheme can still ensure the convergence of GPP and transpiration.Figure 7 shows that with the removal of the constraint, the deviation of calculated GPP and transpiration from the reference attenuates with an increasing number of iterations. For example, GPP converges within 19 steps, while no visible disparities of transpiration from the reference are observed by the 15th step. Nevertheless the convergence speed is slower, compared to the simulations with the constraint. We found that setting an initial guess of ci slightly larger than the CO2 compensation point Γ*can increase the convergence rate, thus improve the iteration efficiency. This is because the Newton-Raphson approach uses gradient for calculations at the following iteration step, and thus favors a larger derivative (or steeper slope) of the iteration functionj(ci).

Figure 6.

Similar to Figure 1(with exception of Figure 6a), but for NR_wC. Here Figure 6a shows the differences of JJA-mean GPP (g C m−2 yr−1) between the 20th and 19th iteration step since N20 is the reference for the Newton-Raphson approach.

Figure 7.

Differences of JJA-mean (1948) (a–d) GPP (g C m−2 yr−1) and (e–h) canopy transpiration (W m−2) between the 20th and 5th, 10th, 15th, and 19th iteration step for NR_woC.

6. Comparison Between the Fixed-Point and Newton-Raphson Iteration Approaches

[40] We estimated the biases in modeled GPP and transpiration arising from the use of the fixed-point iteration scheme and the associated artificial constraint onea. This is achieved through comparing the default CLM4 simulation with the version of CLM4 that implements the Newton-Raphson scheme and discards the constraint.Figure 8shows that across the globe the default numerical treatment predominantly overestimates GPP and transpiration although over Mexico and along the southern-central edge of the Sahara, GPP and transpiration tend to be underestimated. Globally, a positive bias of ∼2.7 Pg C yr−1 in the simulated GPP is found. The western U.S. has the most noticeable nonconvergence problem, with 27% overestimation of GPP and 46% for transpiration (Table 2). Further, such overestimation can affect the energy portioning and hence bias the calculation of sensible heat fluxes (Table 2).

Figure 8.

Differences of JJA-mean (1948) (a) GPP (g C m−2 yr−1) and (b) canopy transpiration (W m−2) between the 3rd iteration step of FP_wC and the 20th step of NR_woC, indicating the bias of the default CLM4 FP_wC(N3) from the true estimates NR_woC(N20).

[41] The new approach proposed in this study not only remedies the nonconvergence problems, but also guarantees the computational efficiency for the convergent grids in the fixed-point framework.Figure 9ashows that the Newton-Raphson approach works equally well as the fixed-point scheme at the 3rd iteration step when the latter converges, a consequence of it sharing similar built-in mathematical features as the fixed-point scheme. However, the Newton-Raphson approach works under conditions that lead to failure of the fixed-point iteration (Figure 9b). Further, it is computationally more efficient than the fixed-point scheme.Figure 10shows the fraction of nonconvergent land grids as a function of iteration steps. It shows that ∼35% of grids cannot converge regardless of the number of iteration steps if fixed-point approach is used and the artificial constraint oneais removed. In contrast, over 90% of the global land grids by the 3rd step and almost all by 9th step can converge with the Newton-Raphson scheme, indicating its superiority.

Figure 9.

Evaluation of grid-scale GPP (g C m−2 yr−1) simulated by the fixed-point (FP) and by the Newton-Raphson (NR) iteration scheme at multiple iteration steps. (a) Comparison of the GPP simulation of NR_woC versus that of FP_woC. Both simulations are performed using the maximum iteration step N = 3. Each dot represents a grid cell on land. All land grids across the globe are displayed here, excluding those in Antarctic. The green dots, which are located around they = x line, represent grids where both the NR and FP schemes achieve convergence at the 3rd step. The red dots, which deviate from the y = x line, represent the nonconvergence grids. (b) The GPP simulation of NR_woC with N = 3 (purple, left ordinates) and with N = 19 (blue, right ordinates) versus that with N = 20 for all the nonconvergence grids in Figure 9a. The blues dots form a y = x line, indicating that the convergence of NR scheme can be achieved completely within 20 iteration steps. The purple dots are close to the y = xline, suggesting that the Newton-Raphson iteration with N = 3 can yield a GPP that closely approaches the true value, but more iterations are needed to ensure a full convergence.

Figure 10.

The fraction of nonconvergent grids versus the iteration step for FP_wC, FP_woC, NR_wC and NR_woC, respectively. The fraction is calculated as the number of nonconvergent grids relative to the total land grids (except Antarctic).

7. Discussion

[42] Our results show that the fixed-point scheme with three iteration steps in the current CLM4 cannot guarantee convergence of the coupledAgs model. This approach fails more frequently in some regions of the world (e.g., the western U.S.) than in others, leading to regionally varying uncertainty and global biases in estimated GPP and transpiration.

[43] A mathematical examination reveals that the nonconvergence arises because the behavior of the iterative function h(x) in CLM4 cannot always satisfy a strict mathematical condition, i.e., the fixed-point theorem. In particular, the effectiveness of the fixed-point iteration is quite sensitive toea, which modifies the shape or the derivative of h(x) and therefore affects whether or not the fixed-point theorem is violated. Theea in turn is strongly affected by RH, thus increasing air dryness relative to saturation vapor pressure may shift the h(x) of some grids from convergence to nonconvergence. A number of previous studies have reported that more severe and frequent drought may occur in the future [e.g., Sitch et al., 2008; Solomon et al., 2009], which if so may lead to a spatial expansion of the failure of the fixed-point iteration in a changing climate and so add more bias to the existing uncertainties for model predictions. Hence, the default fixed-point treatment in CLM4 is not adequate for reliable simulations of carbon and water fluxes and may jeopardize the future projections of flux variables for regional or global applications.

[44] The application of the prior constraint on eacan reduce but not eliminate the numerical problems caused by the fixed-point method. Essentially, this constraint sets an artificial limit to the atmospheric water vapor pressure deficit (VPD), one of the most important variables to which stomata respond, particularly under drought conditions [Gu et al., 2006]. It is known that the current CLM performs poorly in simulating ecosystem responses to drought [e.g., Sakaguchi et al., 2011]. This artificial constraint may have contributed to this poor performance. The application of this constraint stems from an attempt to fix the potential failure of the Ags calculation, viewed as ‘numerical instability’ by Oleson et al. [2010]. However, our analysis shows that this ‘instability’ is not inherent in the coupled Agsmodel but rather is a consequence of applying the fixed-point method under conditions where the mathematical requirement for convergence is not satisfied. Hence, the constraint on water vapor can implicitly turn an originally nonconvergent case into a convergent one by reducing the magnitude of |h′(x)|, leading to conditions that satisfy the fixed-point theorem (Figures 3 and S4). For example, nonconvergent grids emerge in eastern Amazon, southern Africa, and Australia with absence of this constraint, as seen in the differences between Figures 1 and 2. This “quick fix” of convergence may degrade the CLM performance under water stress, as it artificially modifies the air dryness. Hence, the bias arising from the nonconvergence may be small under normal conditions, but may be exacerbated during drought periods.

[45] This study is based on the latest released version of CLM4 [Oleson et al., 2010], but does not consider the revisions proposed by Bonan et al. [2011, 2012]. To our knowledge, these unreleased updates did not change the iteration scheme and thus did not fix the numerical issue reported in our study. The nonconvergence problem exists regardless of the new updates, since the fixed-point theorem is a strict mathematical restriction, which works only under certain conditions. Given the large variations of the parameters and forcing variables involved in the coupled carbon-water flux model and thus the geometry of the iterative function, the fixed-point approach is not sufficient for convergence for global or regional applications at any time scales.

[46] Our study shows that the Newton-Raphson approach can ensure the convergence of the coupledAgsmodel and thus produce numerically accurate estimates of GPP and transpiration, since the condition for the Newton-Raphson approach to converge is much more relaxed as compared to the fixed-point. Moreover, this new scheme performs well even if the prior constraint on the atmospheric vapor pressure is removed, indicating its effectiveness under drought conditions (Figure 7 and see historical runs in Figure S3), which could result in more frequent occurrences of nonconvergence if fixed-point approach is otherwise used. Further, this scheme is computationally efficient for global simulations: at the points where the fixed-point method succeeds, the Newton-Raphson iteration converges at a similar or even faster rate due to their similar built-in features; for those where the fixed-point fails, the convergence with the Newton-Raphson can generally be achieved within 10 steps although longer iterations may be needed for few isolated grids.

[47] The Farquhar photosynthesis model is a change point model, i.e., with discontinuity point at the transition of ci among wc, wj and we [Gu et al., 2010] (equation (1)). This has the potential to cause failures of any iterative approach, as such usually requires the iterative function and its derivative to be continuous near the root [Burden and Faires, 2011]. However, the convergence would not be affected as long as the root is not close to the change point. In all our simulations, we did not encounter any discontinuity problems. Bonan et al. [2011]has included the “co-limitation” treatment to smooth the change point (see their Table B1), which if accounted for should make our proposed Newton-Raphson method more effective.

[48] Would changing the initial guess for ci be a “simpler and quicker fix” for the nonconvergence issue than a new iteration scheme? This strategy is ineffective as long as ci falls on the “steep declining zone” of h(x), where the violation of fixed-point theorem occurs. On the other hand, tuning the initial guess may speed up the convergence rate of Newton-Raphson iteration if the new initialization happens to be close to the root. Speeding up of convergence can still be achieved if one locates the initial guess to be in the steep portion of the Newton–Raphson iteration function (Figure 5; e.g., with ci slightly above the CO2compensation point) as the Newton-Raphson approach uses gradient to determine the following iteration step. These effects, however, would not occur to fixed-point iteration if the new initial guess is also on the “steep declining” portion ofh(x), even if it is close to the root.

[49] For this present study we have not changed the double loop structure in the current CLM4 for solving leaf temperature Tv and carbon fluxes. The computational efficiency of CLM4 could be improved if this nested loop were replaced by simultaneously solving Tv and cithrough the Newton-Raphson approach. However, the substantial modification of the current CLM4 coding structure that would be required is beyond the scope of this present study. It is necessary to point out that the failure of convergence in the CLM4 is not caused by its nested loop structure, but rather by its use of the fixed-point iteration approach, since this approach is restricted by a mathematical theorem.

[50] The results shown in this study are based on the year 1948. However, the spatial distribution of the bias caused by the fixed-point iteration and the superiority of the Newton-Raphson approach are quite persistent, i.e., independent of the specific year chosen for the demonstration purpose (see historical runs inFigures S1 and S3).

[51] Different iterative methods are available for solving nonlinear equations. Each method has its strength and weakness and no method is universally valid and efficient. Thus, when choosing an iterative method to solve a given nonlinear equation, the numerical behavior of this nonlinear equation must be carefully analyzed so that an optimal method is selected. The fixed-point iteration has also been applied in other research areas, e.g., eddy covariance flux corrections [Aubinet et al., 2012], since it is the easiest and most straightforward way to be implemented. It is also possible that other land surface models may have employed a similar strategy to solve the coupled carbon and water flux model, since they share the same heritage as CLM. Our study appears to be the first attempt to discuss the numerical caveats of the fixed-point iteration in the context of land surface modeling, which could remind modelers to check the specific implementation of processes that require iteration, not only for CLM team, but also for other land surface modeling communities.

8. Conclusions

[52] The CLM4 uses a fixed-point iteration scheme with three iterative steps to solve the coupled carbon and water flux model. This approach cannot guarantee convergence in the iterative calculation for GPP and transpiration. It fails more frequently in some regions of the world (e.g., the western U.S.) than in others, leading to regionally varying uncertainty and global biases in estimated GPP and transpiration. The arbitrary constraint applied by CLM4 to the atmospheric water vapor pressure prior to the numerical implementation can partially cover the iteration problems of the fixed-point method; however, this is artificial and can cause biased estimates of carbon and water fluxes. Mathematically, we show that environmental factors modify the shape of the fixed-point iterative function and hence determine whether the conditions required for convergence (i.e., the fixed-point theorem) are satisfied. Because of this strong limitation, the fixed-point iteration scheme should not be used to solve the coupled carbon-water flux model, given the importance of this coupled model in land surface modeling. We propose a Newton-Raphson approach to replace the default fixed-point scheme. Our results show that the new approach can ensure convergence and is computationally efficient. Further it does not require the artificial constraint to the atmospheric water vapor pressure. We find that the default CLM4 treatment leads to a ∼2.7 Pg C yr−1overestimation of GPP globally, with regional bias being up to 27%. We thus suggest that the current fixed-point treatment in CLM4 be replaced with the Newton-Raphson approach and that the artificial constraint on the atmospheric water vapor pressure be removed.

Acknowledgments

[53] This study was carried out at University of Texas – Austin with support from National Science Foundation (ATM-0921898) and Department of Energy (DE-FG02-01ER63198), and also at Oak Ridge National Laboratory (ORNL). ORNL is managed by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725.

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