We present a generic flux limiter to account for mass limitations from an arbitrary number of substrates in a biogeochemical reaction network. The flux limiter is based on the observation that substrate (e.g., nitrogen, phosphorus) limitation in biogeochemical models can be represented as to ensure mass conservative and non-negative numerical solutions to the governing ordinary differential equations. Application of the flux limiter includes two steps: (1) formulation of the biogeochemical processes with a matrix of stoichiometric coefficients and (2) application of Liebig's law of the minimum using the dynamic stoichiometric relationship of the reactants. This approach contrasts with the ad hoc down-regulation approaches that are implemented in many existing models (such as CLM4.5 and the ACME (Accelerated Climate Modeling for Energy) Land Model (ALM)) of carbon and nutrient interactions, which are error prone when adding new processes, even for experienced modelers. Through an example implementation with a CENTURY-like decomposition model that includes carbon, nitrogen, and phosphorus, we show that our approach (1) produced almost identical results to that from the ad hoc down-regulation approaches under non-limiting nutrient conditions, (2) properly resolved the negative solutions under substrate-limited conditions where the simple clipping approach failed, (3) successfully avoided the potential conceptual ambiguities that are implied by those ad hoc down-regulation approaches. We expect our approach will make future biogeochemical models easier to improve and more robust.

Biogeochemical modeling has been one of the major themes in developing Earth system models (Hurrell et al., 2013), yet developing numerically robust and mathematically consistent biogeochemical (BGC) models has been challenging (Broekhuizen et al., 2008). In biogeochemical modeling, the systems of interest, such as terrestrial ecosystems, are often nutrient limited under a wide range of conditions (Vitousek and Howarth, 1991; Vitousek et al., 2010). Therefore, proper modeling of nutrient limitation is a prerequisite for credible predictions of carbon–climate feedbacks (Bouskill et al., 2014; Thomas et al., 2015). In the Earth system models (ESMs) joining phase 5 of the Coupled Model Intercomparison Project (CMIP5), only CLM-CN (Thornton et al., 2007) considered carbon and nitrogen interactions, although observations indicate nitrogen has significantly limited the terrestrial carbon sink (Arora et al., 2013). Further, many analyses indicate phosphorus is critical for improving carbon–climate feedback predictions (Vitousek et al., 2010; Yang et al., 2014; Wieder et al., 2015), and other nutrients (e.g., sulfur, potassium, molybdenum) may also be important (Schmidt et al., 2013; Moro et al., 2014). Therefore, we expect that as more processes are included in future biogeochemical models, more substrates will limit different biogeochemical processes under different conditions.

To develop numerically accurate biogeochemical models, it is important to develop a robust formulation of the biogeochemical processes, such that modelers can safely add or remove biogeochemical processes without degrading the numerical solution. This capability would allow users to focus only on deriving the governing ordinary differential equations (ODEs) of the biogeochemical processes. If the model uses a standard operator-splitting approach (as is common, e.g., Tang et al., 2013), which solves the transport and chemistry separately, then the numerical solver could resolve the numerical details, such as maintaining mass conservation and avoiding nonphysical values, without knowing the details of the ODEs.

Existing terrestrial biogeochemical models often describe substrate
limitation as occurring when the total available substrate cannot satisfy
the demand from all consuming fluxes over a particular time step. For
nitrogen limitation, many BGC models impose substrate limitation when the
total potential ecosystem nitrogen demand (i.e., demand in the absence of
nitrogen limitation; Thornton et al., 2007; Wang et al., 2010; Thomas et
al., 2015) exceeds the total available mineral nitrogen, provided nitrogen
from nitrogen fixers is supplied in mineral form over that time step and
contributions from organic nitrogen are assumed negligible (the latter of
which could be incorrect; see Chapin et al., 1993). However, this conceptual
model (which served as the basis for those ad hoc down-regulation
approaches) is not mathematically consistent with the ODE that governs
nitrogen limitation:

In this note, we show that, by ensuring mass conservation and non-negative solutions to the governing equations of a given biogeochemical model, it is possible to obtain a universal solution to the mass limitation for an arbitrary number of substrates. We organize the remainder of this paper as follows: Sect. 2 describes the technical details of our method, Sect. 3 presents an evaluation of the method based on a CENTURY-like organic matter decomposition model (Parton et al., 1988; Appendix A, Tables 1 and 2), and Sect. 4 summarizes our findings. Note that, even though our evaluation of the approach is based on a soil biogeochemical model, the approach is generic and could be applied to any biogeochemical models.

Our approach makes use of the reaction-based formulation of a biogeochemical
model (e.g., Reichert et al., 2001; Batstone et al., 2002; Fang et al.,
2013). Mathematically, for the

By defining reaction rate

We describe the generic model structure using the Peterson matrix form (e.g.,
Russell, 2006) as

We now separate

Several approaches (other than clipping) have been proposed to ensure
non-negative and mass conservative solutions to equations such as Eq. (5).
For instance, Sandu (2001) proposed two projection-based approaches to
post-correct the negative solution using the null space of

A list of biogeochemical reactions as represented in the CENTURY-like organic matter decomposition model (Appendix A). The decomposition is calculated as in Parton et al. (1988). Here we use CN to represent carbon to nitrogen ratio, and CP to represent carbon to phosphorus ratio. The subscript “min” designates mineral pool for a nutrient, such as nitrogen (N) and phosphorus (P).

Parameter values used in this study. These values are based on syntheses from Parton et al. (1988), Yang et al. (2014), and Zhu et al. (2016).

To propose a simple solution to ensure non-negative numerical solutions to
Eq. (5), we restrict our ODE integrator to the first order and apply a vector
of flux limiters that are dependent on the reactant stoichiometry

In our evaluation, we compared the performance of our new approach to the
mBBKS approach (Broekhuizen et al., 2008) and two ad hoc down-regulation
formulations derived based on the nitrogen limitation scheme in CLM4.5 (CLM-1
and CLM-2). During a particular numerical time step, CLM-1 assumes complete
independence between nutrient mobilizers and immobilizers, while CLM-2
assumes complete coupling between nutrient mobilizers and immobilizers (see
details in Appendix C). We analyzed scenarios where the organic matter
decomposition is (1) not nutrient limited (Case 1) and (2) nitrogen and
phosphorus limited (Case 2 and Case 3); the latter situations are where a
direct solution (without flux limitation) to Eq. (5) may produce negative
values, and clipping will be triggered in methods like ODE45. We evaluated
the difference between simulations for predicted mineral nitrogen N

Simulated decomposition dynamics for Case 1 in Table 3. In all panels, all results overlap each other.

In simulations for the decomposition of nutrient-sufficient organic matter (i.e., no nutrient limitation; Fig. 1), we found that our new approach (Fortran 90 code, Eq. 6), mBBKS, CLM-1, and CLM-2 predicted almost identical time series for the various pools when compared to that from ODE45, indicating that the four approaches are implemented correctly as benchmarked with ODE45.

However, for Case 2 (Fig. 2, Table 3), where both nitrogen and phosphorus are insufficient to support decomposition (because it has even fewer mineral nutrients available than the nutrient-limited Case 3), mBBKS failed to predict any organic matter decomposition after the mineral nutrients are consumed in the first few time steps and predicted that all decomposition pathways were phosphorus limited thereafter (cyan line in Fig. 2b). In contrast, the two ad hoc down-regulation approaches, CLM-1 and CLM-2, and our new approach all predicted visually identical time series of the different pools and correctly indicated that the decomposition of SOM pools (SOM1, SOM2, and SOM3 as derived from litter decomposition using the non-zero initial pools of mineral nutrients) released small amounts of mineral nutrients to support further litter organic matter decomposition (as can be inferred from Table 2, which shows that the decomposition of litter pools are all nutrient limited by stoichiometry). This response was missed by mBBKS, because it applied a single flux limiter to all decomposition pathways, preventing the release of nutrients from mineralizing pathways to support further decomposition. Besides mBBKS, ODE45 also failed to predict meaningful decomposition dynamics because, by clipping the derivatives of the negative to-be state variables to zero, it introduced artificial mass into some state variables during the integration. Specifically, ODE45 predicted the final total nitrogen and total phosphorus (including both mineral and organic pools) as 0.8066 gN and 0.0511 gP, as compared to the correct values of 0.4445 gN and 0.0175 gP, whereas CLM-1, CLM-2, and our new approach all conserved carbon, nitrogen, and phosphorus mass within the machine round-off error.

Simulated decomposition dynamics for Case 2 in
Table 3. Note that the ODE45 scheme (shown on the
right-hand

Initial conditions and integration length (#days) for the analyzed model simulations.

For Case 3 (Fig. 3, Table 3), where non-zero SOM pools were introduced to release nitrogen and phosphorus to support litter decomposition, mBBKS again predicted no visible decomposition because of its use of a single flux limiter to all fluxes (based on the nutrient-limited litter decomposition), even though the SOM decomposition should not be nutrient limited. ODE45 also failed for Case 3, and predicted very different time series for the various pools as compared to CLM-1, CLM-2, and our new approach. By day 300, ODE45 predicted the total nitrogen and total phosphorus (including both mineral and organic pools) as 3.2164 gN and 0.2338 gP as compared to their correct values of 3.1046 gN and 0.2273 gP, respectively.

Simulated decomposition dynamics for Case 3 in Table 3. In all panels, the result from CLM-1 overlaps with that from our new method.

Although we found very small differences between our new method, CLM-1, and CLM-2 in predicted decomposition dynamics for the three simple cases analyzed (Figs. 1, 2, and 3), we acknowledge that large differences should be expected when applying our new method and the two ad hoc down-regulation approaches CLM-1 and CLM-2 for modeling ecosystem dynamics because they define nutrient limitation differently (Fortran 90 code, Eq. 6, and Appendix C; Figs. 4 and 5). As one would infer from Eq. (1), mathematically, nutrient limitation occurs only when the state variable that represents a certain nutrient becomes negative if the reaction rates are not limited during a given numerical integration time step. However, (as we implemented in the Fortran 90 code, Eq. 6), this situation is equivalent to assuming that a released mineral nutrient from the mobilizers will be instantaneously available to all immobilizers that demand this nutrient. Although the existing mineral nutrient pool and the newly released mineral nutrients will be tapped proportionally by the immobilizers, this assumption may still be too strong if a given grid cell covers a too large spatial domain to support this assumption of homogeneity (Manzoni et al., 2008). CLM-1 and CLM-2 represent the two extremes of this coupling between mineral nutrient mobilizers (which release nutrients) and mineral nutrient immobilizers (which take up nutrients) in that CLM-1 assumes the mobilizers and immobilizers are completely independent during the calculation of mineral nutrient uptake, whereas CLM-2 assumes the nutrients released by mobilizers are first assimilated by immobilizers, and if there is additional demand, the remainder comes from the mineral nutrient pool (thus CLM-2 adopts an even stronger mobilizer and immobilizer coupling than our new approach). Indeed, the difference between CLM-1 and CLM-2 is already discernible for Case 3 (Fig. 4), and when the decomposition model is coupled with other nutrient consumers in an ecosystem model, one would potentially find very different predictions of carbon dynamics (see Case 4 in Fig. 5 as a model with slightly more complicated dynamics than Case 3). With slight modification, our new approach will allow a consistent representation of the coupling between mobilizers and immobilizers, including both the CLM-1 and CLM-2 assumptions regarding nutrient competition. This approach will provide a new tool to analyze prediction uncertainty from the ambiguity of defining the coupling strength between nutrient mobilizers and immobilizers.

Differences between simulated decomposition dynamics by CLM-1 and CLM-2 for Case 3.

Simulated decomposition dynamics for Case 4 in Table 3. CLM-1NP performs nitrogen down-regulation before phosphorus down-regulation, whereas CLM-1PN reverses the order. Similarly to CLM-1 (Appendix C), both CLM-1NP and CLM-1PN assume the nutrient mobilizer and immobilizer are independent within a numerical time step. In all panels, CLM-1PN predictions overlap with CLM-1 predictions.

Another advantage of our new approach, compared to the ad hoc down-regulation approaches (e.g., CLM-1 and CLM-2 discussed above), is that it can handle limitation from an arbitrary number of substrates, as long as the matrix of stoichiometric coefficients is formulated. In principle, any biogeochemical reaction can be formulated into reaction form (e.g., Fang et al., 2013); thus, our approach will avoid the ordering problem often encountered in those ad hoc approaches. In this context, the “ordering problem” refers to the situation that different answers are calculated depending on the order of nutrient limitation (e.g., resolving nitrogen limitation first, and then phosphorus limitation). For example, following the nutrient limitation definition in CLM-1, when nitrogen and phosphorus limitation are treated sequentially, the predicted decomposition dynamics differ significantly from when the opposite order is applied (CLM-1NP vs. CLM-1PN in Fig. 5). The implementation where nitrogen limitation occurs before phosphorus limitation (CLM-1NP, cyan circles in Fig. 5) predicted stronger litter decomposition than when phosphorus limitation is applied before nitrogen limitation (CLM-1PN, black dots that overlap with blue line in Fig. 5c). Analogously, in the current CLM4.5 soil biogeochemical formulation (Oleson et al., 2013), organic matter decomposition and methane oxidation are often limited by oxygen (Riley et al., 2011), and nitrogen limitation is imposed after accounting for oxygen limitation, which would potentially result in different predictions were nitrogen limitation imposed before oxygen limitation. This ordering issue could become more severe if more nutrients (e.g., phosphorus, sulfur) are to be introduced in future biogeochemical formulations, and our approach relieves numerical inaccuracies associated with this ordering ambiguity.

In this study, we proposed a generic law of the minimum-based flux limiter to handle substrate limitation in biogeochemical models. Evaluations indicate that our method could produce results as accurate as those produced by ad hoc down-regulation approaches implemented in existing biogeochemical models for simple decomposition dynamics that only include decomposers. Additionally, our new approach provides a way to resolve some conceptual ambiguities implied in those ad hoc down-regulation approaches. We expect our new approach to help the community to develop more robust and easier to maintain biogeochemical codes to better predict carbon–climate feedbacks.

The soil biogeochemical model used in this study adopts the form from the
CENTURY-model, which uses a turnover-pool-based formulation of soil organic
matter decomposition (Parton et al., 1988). The model includes three pools of
litter, one pool of coarse wood debris (CWD) and three pools of SOM. The
model calculates the non-nutrient-limited decomposition of a given organic
matter pool

For two vectors

The first ad hoc down-regulation approach (CLM-1) follows the implementation
of nitrogen down-regulation in CLM4.5 (Oleson et al., 2013), where all
nitrogen immobilization fluxes N

The second ad hoc down-regulation approach (CLM-2) is similar to the
first one, except that it first subtracts the mobilizing fluxes from the
immobilizing fluxes, such that

One can further define

For a certain time step

J. Y. Tang developed the theory, conducted the analyses, and co-wrote the paper. W. J. Riley discussed the analyses and co-wrote the paper.

This research was supported by the Director, Office of Science, Office of Biological and Environmental Research of the US Department of Energy under contract no. DE-AC02-05CH11231 as part of the Next-Generation Ecosystem Experiments (NGEE-Arctic) and the Accelerated Climate Model for Energy project in the Earth System Modeling program. We thank Niall Broekhuizen and an anonymous reviewer for their constructive comments, which improved the paper significantly. Edited by: U. Seibt