**Methods for assessment of models**
28 May 2019

**Methods for assessment of models** | 28 May 2019

# The quasi-equilibrium framework revisited: analyzing long-term CO_{2} enrichment responses in plant–soil models

_{2}enrichment responses in plant–soil models

Mingkai Jiang Sönke Zaehle Martin G. De Kauwe Anthony P. Walker Silvia Caldararu David S. Ellsworth and Belinda E. Medlyn

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**Mingkai Jiang et al.**Mingkai Jiang Sönke Zaehle Martin G. De Kauwe Anthony P. Walker Silvia Caldararu David S. Ellsworth and Belinda E. Medlyn

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^{1}Hawkesbury Institute for the Environment, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia^{2}Max Planck Institute of Biogeochemistry, Jena, Germany^{3}ARC Centre of Excellence for Climate Extremes, University of New South Wales, Sydney, NSW 2052, Australia^{4}Climate Change Research Center, University of New South Wales, Sydney, NSW 2052, Australia^{5}Environmental Sciences Division and Climate Change Science Institute, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

^{1}Hawkesbury Institute for the Environment, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia^{2}Max Planck Institute of Biogeochemistry, Jena, Germany^{3}ARC Centre of Excellence for Climate Extremes, University of New South Wales, Sydney, NSW 2052, Australia^{4}Climate Change Research Center, University of New South Wales, Sydney, NSW 2052, Australia^{5}Environmental Sciences Division and Climate Change Science Institute, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

**Correspondence**: Mingkai Jiang (m.jiang@westernsydney.edu.au)

**Correspondence**: Mingkai Jiang (m.jiang@westernsydney.edu.au)

Received: 18 Nov 2018 – Discussion started: 07 Dec 2018 – Revised: 01 May 2019 – Accepted: 14 May 2019 – Published: 28 May 2019

Elevated carbon dioxide (CO_{2}) can increase plant growth, but the
magnitude of this CO_{2} fertilization effect is modified by soil
nutrient availability. Predicting how nutrient availability affects plant
responses to elevated CO_{2} is a key consideration for ecosystem
models, and many modeling groups have moved to, or are moving towards,
incorporating nutrient limitation in their models. The choice of assumptions
to represent nutrient cycling processes has a major impact on model
predictions, but it can be difficult to attribute outcomes to specific
assumptions in complex ecosystem simulation models. Here we revisit the
quasi-equilibrium analytical framework introduced by Comins and
McMurtrie (1993) and explore the consequences of specific model assumptions
for ecosystem net primary productivity (NPP). We review the literature applying this framework to plant–soil
models and then analyze the effect of several new assumptions on predicted
plant responses to elevated CO_{2}. Examination of alternative
assumptions for plant nitrogen uptake showed that a linear function of the
mineral nitrogen pool or a linear function of the mineral nitrogen pool with
an additional saturating function of root biomass yield similar CO_{2}
responses at longer timescales (>5 years), suggesting that the added
complexity may not be needed when these are the timescales of interest. In
contrast, a saturating function of the mineral nitrogen pool with linear
dependency on root biomass yields no soil nutrient feedback on the
very-long-term (>500 years), near-equilibrium timescale, meaning that one
should expect the model to predict a full CO_{2} fertilization effect
on production. Secondly, we show that incorporating a priming effect on slow
soil organic matter decomposition attenuates the nutrient feedback effect on
production, leading to a strong medium-term (5–50 years) CO_{2}
response. Models incorporating this priming effect should thus predict a
strong and persistent CO_{2} fertilization effect over time. Thirdly,
we demonstrate that using a “potential NPP” approach to represent nutrient
limitation of growth yields a relatively small CO_{2} fertilization
effect across all timescales. Overall, our results highlight the
fact that the quasi-equilibrium
analytical framework is effective for evaluating both the consequences and
mechanisms through which different model assumptions affect predictions. To
help constrain predictions of the future terrestrial carbon sink, we
recommend the use of this framework to analyze likely outcomes of new model
assumptions before introducing them to complex model structures.

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_{2}enrichment responses in plant–soil models, Geosci. Model Dev., 12, 2069–2089, https://doi.org/10.5194/gmd-12-2069-2019, 2019.

Predicting how plants respond to atmospheric carbon dioxide (CO_{2})
enrichment (eCO_{2}) under nutrient limitation is fundamental for an
accurate estimate of the global terrestrial carbon (C) budget in response to
climate change. There is now ample evidence that the response of terrestrial
vegetation to eCO_{2} is modified by soil nutrient availability
(Fernández-Martínez et al., 2014; Norby et al., 2010; Reich and
Hobbie, 2012; Sigurdsson et al., 2013). Over the past decade, land surface
models have developed from C-only models to carbon–nitrogen (CN) models
(Gerber et al., 2010; Zaehle and Friend, 2010). The inclusion of CN
biogeochemistry has been shown to be essential to capture the reduction in
the CO_{2} fertilization effect with declining nutrient availability and
therefore its implications for climate change (Zaehle et al.,
2015). However, it has also been shown that models incorporating different
assumptions predict very different vegetation responses to eCO_{2}
(Lovenduski and Bonan, 2017; Medlyn et al., 2015). Careful examination of
model outputs has provided insight into the reasons for the different model
predictions (De Kauwe et al., 2014; Medlyn et al., 2016; Walker et al.,
2014, 2015; Zaehle et al., 2014), but it is generally
difficult to attribute outcomes to specific assumptions in these plant–soil
models that differ in structural complexity and process feedbacks
(Lovenduski and Bonan, 2017; Medlyn et al., 2015; Thomas et al., 2015).

Understanding the mechanisms underlying predictions of ecosystem carbon
cycle processes is fundamental for the validity of prediction across space
and time. Comins and McMurtrie (1993) developed an analytical
framework, the “quasi-equilibrium” approach, to make model predictions
traceable to their underlying mechanisms. The approach is based on the
two-timing approximation method (Ludwig et al., 1978) and makes
use of the fact that ecosystem models typically represent a series of pools
with different equilibration times. The method involves the following: (1) choosing a time
interval (*τ*) such that the model variables can be divided into
“fast” pools (which approach effective equilibrium at time *τ*) and
“slow” pools (which change only slightly at time *τ*); (2) holding the
slow pools constant and calculating the equilibria of the fast
pools (an effective equilibrium as this is not a true equilibrium of the
entire system); and (3) substituting the fast pool effective equilibria
into the original differential equations to give simplified differential
equations for the slow pools at time *τ*.

In a CN model, plant net primary production (NPP) can be estimated from two constraints based on equilibration of the C balance (the “photosynthetic constraint”) and the N balance (the “nitrogen recycling constraint”) (Comins and McMurtrie, 1993). Both constraints link NPP with leaf chemistry (i.e., N : C ratio) (derivation in Sect. 3.1). The simulated production occurs at the intersection of these two constraint curves (shown graphically in Fig. 1). To understand behavior on medium and long timescales (e.g., wood and slow and passive soil organic pools in Fig. 2; 20–200 years), one can assume that plant pools with shorter equilibration times in the model (e.g., foliage, fine-root, or active soil organic pools in Fig. 2) have reached quasi-equilibrium, and model dynamics are thus driven by the behavior of the longer-timescale pools.

The recent era of model development has seen some significant advances in
representing complex plant–soil interactions, but models still diverge in
future projections of CO_{2} fertilization effects on NPP (Friend et
al., 2014; Koven et al., 2015; Walker et al., 2015). A recent series of
multi-model intercomparison studies has demonstrated the importance of
understanding underlying response mechanisms in determining model response
to future climate change (Medlyn et al., 2015), but
this can be difficult to achieve in complex global models. The
quasi-equilibrium framework is a relatively simple but quantitative method
to examine the effect of different assumptions on model predictions. As
such, it complements more computationally expensive sensitivity analyses
and can be used as an effective tool to provide a priori evaluation of both the
consequence and mechanism through which different new model implementations
affect model predictions.

Here, by constructing a quasi-equilibrium framework based on the structure of
the Generic Decomposition And Yield
(G'DAY) model (Comins and McMurtrie, 1993), we evaluate the effects on plant
responses to eCO_{2} of some recently developed model assumptions
incorporated into ecosystem models, for example the Community Land Model
(CLM) (Oleson et al., 2004), the Community Atmosphere–Biosphere Land
Exchange (CABLE) model (Kowalczyk et al., 2006), the Lund–Potsdam–Jena
(LPJ) model (Smith et al., 2001), the JSBACH model (Goll et al., 2017b), and
the O-CN model (Zaehle et al., 2010). Specifically, we test how different
functions affecting plant N uptake influence NPP responses to eCO_{2}
at various quasi-equilibrium time steps. The present study is a continuation
of the series of quasi-equilibrium studies reviewed in Sect. 2, with a
general aim of helping researchers to understand the similarities and
differences of predictions made by different process-based models, as
demonstrated in Sect. 3.

Many of the assumptions currently being incorporated into CN models have
previously been explored using the quasi-equilibrium framework; here we
provide a brief literature review describing the outcomes of this work
(Table 1). Firstly, the flexibility of plant and soil stoichiometry has
recently been highlighted as a key assumption (Stocker et al., 2016;
Zaehle et al., 2014). A key finding from early papers applying the
quasi-equilibrium framework was that model assumptions about the flexibility
of the plant wood N : C ratio (Comins, 1994; Comins and McMurtrie, 1993;
Dewar and McMurtrie, 1996; Kirschbaum et al., 1994, 1998;
McMurtrie and Comins, 1996; Medlyn and Dewar, 1996) and soil N : C ratio
(McMurtrie and Comins, 1996; McMurtrie et al., 2001; Medlyn et al., 2000)
were critical determinants of the magnitude of the transient (10 to
>100 years) plant response to eCO_{2} (Fig. 1). Different to
the effect of foliar N : C ratio flexibility, which has an instantaneous
effect on photosynthesis, the flexibility of the wood N : C ratio controls the
flexibility of nutrient storage per unit biomass accumulated in the slow
turnover pool. Therefore, a constant wood N : C ratio, such as was assumed in
CLM4 (Thornton et al., 2007; Yang et al., 2009), means that effectively a
fixed amount of N is locked away from the active processes such as
photosynthesis on the timescale of the life span of the woody tissue. In
contrast, a flexible wood N : C ratio, such as was tested in O-CN
(Meyerholt and Zaehle, 2015), allows variable N storage in the
woody tissue and consequently more nutrients available for C uptake on the
transient timescale. Similarly, flexibility in the soil N : C ratio determines
the degree of the soil N cycle feedback (e.g., N immobilization and
mineralization) and therefore its effect on plant response to eCO_{2}. A
large response to eCO_{2} occurs when the soil N : C ratio is allowed to
vary, whereas there could be little or no response if the soil N : C ratio is
assumed to be inflexible (McMurtrie and Comins, 1996).

Changes in plant allocation with eCO_{2} are also a source of disagreement
among current models (De Kauwe et al., 2014). The quasi-equilibrium framework
has been used to investigate a number of different plant C allocation schemes
(Comins and McMurtrie, 1993; Kirschbaum et al., 1994; Medlyn and Dewar,
1996). For example, Medlyn and Dewar (1996) suggested that plant
long-term growth responses to eCO_{2} depend strongly on the extent to
which stem and foliage allocations are coupled. With no coupling (i.e., fixed
allocation of C and N to stemwood), plant growth was not responsive to
eCO_{2}; with linear coupling (i.e., allocation to stemwood proportional to
foliage allocation), a significant long-term increase in total growth
following eCO_{2} was found (Fig. S1 in the Supplement). The reason for
this is similar to the argument behind wood N : C ratio flexibility:
decreasing C allocation to wood decreases the rate of N removal per unit of C
invested in growth. In contrast, Kirschbaum et al. (1994) found that changes in allocation between different parts of a plant only
marginally changed the CO_{2} sensitivity of production at different
timescales. The fundamental difference between the two allocation schemes was
that Kirschbaum et al. (1994) assumed that the root
allocation coefficient was determined by a negative relationship with the
foliar N : C ratio, meaning that the increase in foliar N : C ratio would lead to
a decreased root allocation and increased wood and foliage allocation,
whereas Medlyn and Dewar (1996) investigated stem–foliage allocation
coupling without introducing a feedback via the foliar N : C ratio. The
comparison of the two allocation schemes is indicative of the underlying
causes of model prediction divergence in recent inter-model comparisons
(De Kauwe et al., 2014; Walker et al., 2015).

Another hypothesis currently being explored in models is the idea that
increased belowground allocation can enhance nutrient availability under
elevated CO_{2} (Dybzinski et al., 2014; Guenet et al., 2016).
Comins (1994) argued that the N deficit induced by CO_{2}
fertilization could be eliminated by the stimulation of N fixation. This
argument was explored in more detail by McMurtrie et al. (2000),
who assumed that eCO_{2} led to a shift in allocation from wood to root
exudation, which resulted in enhanced N fixation. They showed that, although
the increase in N fixation could induce a large eCO_{2} response in NPP
over the long term, a slight decrease in NPP was predicted over the
medium term. This decrease occurred because increased exudation at eCO_{2}
increased soil C input, causing increased soil N sequestration and lowering
the N available for plant uptake. Over the long term, however, both NPP and
C storage were greatly enhanced because the sustained small increase in N
input led to a significant build-up in total ecosystem N on this timescale.

The interaction between rising CO_{2} and warming under nutrient
limitation is of key importance for future simulations. Medlyn et al. (2000)
demonstrated that short-term plant responses to warming, such as
physiological acclimation, are overridden by the positive effects of warming
on soil nutrient availability in the medium to long term. Similarly,
McMurtrie et al. (2001) investigated how the flexibility of the soil N : C
ratio affects predictions of the future C sink under elevated temperature and
CO_{2}. They showed that assuming an inflexible soil N : C ratio
with elevated temperature would mean a release of nitrogen with enhanced
decomposition, leading to a large plant uptake of N to enhance growth. In
contrast, an inflexible soil N : C ratio would mean that the extra N
mineralized under elevated temperature is largely immobilized in the soil and
there is hence a smaller increase in C storage. This effect of soil N : C
stoichiometry on the response to warming is opposite to the effect on
eCO_{2} described above. Therefore, under a scenario in which both
temperature and CO_{2} increase, the C sink strength is relatively
insensitive to soil N : C variability, but the relative contributions of
temperature and CO_{2} to this sink differ under different soil
N : C ratio assumptions (McMurtrie et al., 2001). This outcome may explain
the results observed by Bonan and Levis (2010) when comparing coupled carbon
cycle–climate simulations. The Terrestrial Ecosystem Model (TEM; Sokolov et al., 2008)
and CLM (Thornton et al., 2009), which assumed inflexible stoichiometry, had
a large climate–carbon feedback but a small CO_{2}
concentration-carbon feedback, contrasting with the O-CN model (Zaehle et
al., 2010), which assumed flexible stoichiometry and had a small
climate–carbon feedback and a large CO_{2} concentration–carbon
feedback. Variations among models in this stoichiometric flexibility
assumption could also potentially explain the trade-off between CO_{2}
and temperature sensitivities observed by Huntzinger et al. (2017).

This section combines both methods and results together because equation
derivation is fundamental to the analytical and graphic interpretation of
model performance within the quasi-equilibrium framework. Below we first
describe the baseline simulation model and derivation of the
quasi-equilibrium constraints (Sect. 3.1); we then follow with analytical
evaluations of new model assumptions using the quasi-equilibrium framework
(Sect. 3.2). Within each subsection (Sect. 3.2.1 to 3.2.3), we first
provide key equations for each assumption and the derivation of the
quasi-equilibrium constraints with these new assumptions; we then provide our
graphic interpretations and analyses to understand the effect of the model
assumption on plant NPP responses to eCO_{2}.

More specifically, we tested alternative model assumptions for three processes that affect plant carbon–nitrogen cycling: (1) Sect. 3.2.1 evaluates different ways of representing plant N uptake, namely plant N uptake as a fixed fraction of mineral N pools, as a saturating function of the mineral N pool linearly depending on root biomass (Zaehle and Friend, 2010), or as a saturating function of root biomass linearly depending on the mineral N pool (McMurtrie et al., 2012); (2) Sect. 3.2.2 tests the effect the potential NPP approach that downregulates potential NPP to represent N limitation (Oleson et al., 2004); and (3) Sect. 3.2.3 evaluates root exudation and its effect on the soil organic matter decomposition rate (i.e., priming effect). The first two assumptions have been incorporated into some existing land surface model structures (e.g., CLM, CABLE, O-CN, LPJ), whereas the third is a framework proposed following the observation that models did not simulate some key characteristic observations of the DukeFACE experiment (Walker et al., 2015; Zaehle et al., 2014) and therefore could be of importance in addressing some model limitations in representing soil processes (van Groenigen et al., 2014; Zaehle et al., 2014). It is our purpose to demonstrate how one can use this analytical framework to provide an a priori and generalizable understanding of the likely impact of new model assumptions on model behavior without having to run a complex simulation model. Here we do not target specific ecosystems to parameterize the model but anticipate the analytical interpretation of the quasi-equilibrium framework to be of general applicability for woody-dominated ecosystems. One could potentially adopt the quasi-equilibrium approach to provide case-specific evaluations of model behavior against observations (e.g., constraining the likely range of wood N : C ratio flexibility).

## 3.1 Baseline model and derivation of the quasi-equilibrium constraints

Our baseline simulation model is similar in structure to G'DAY (Generic
Decomposition And Yield; Comins and
McMurtrie, 1993), a generic ecosystem model that simulates biogeochemical
processes (C, N, and H_{2}O) at daily or sub-daily time steps. A
simplified G'DAY model version that simulates plant–soil C–N interactions
at a weekly time step was developed for this study (Fig. 2). In G'DAY, plants
are represented by three stoichiometrically flexible pools: foliage, wood,
and roots. Each pool turns over at a fixed rate. Litter enters one of four
litter pools (metabolic and structural aboveground and belowground) and
decomposes at a rate dependent on the litter N : C ratio, soil moisture,
and temperature. Soil organic matter (SOM) is represented as active, slow,
and passive pools, which decay according to first-order decay functions with
different rate constants. Plants access nutrients from the mineral N pool,
which is an explicit pool supplied by SOM decomposition and an external
input, which is assumed to be constant, as a simplified representation of
fixation and atmospheric deposition.

The baseline simulation model further assumes the following: (1) gross primary
production (GPP) is a function of a light-use efficiency (LUE), which depends
on the foliar N : C ratio (*n*_{f}) and atmospheric CO_{2}
concentration (*C*_{a}) (Appendix A1); (2) carbon use efficiency (the
ratio NPP : GPP) is constant; (3) allocation of newly fixed carbon among
foliage (*a*_{f}), wood (*a*_{w}), and root (*a*_{r})
pools is constant; (4) foliage (*n*_{f}), wood (*n*_{w}), and
root N : C (*n*_{r}) ratios are flexible; (5) wood and root N : C
ratios are proportional to the foliar N : C ratio, with constants of
proportionality *r*_{w} and *r*_{r}, respectively; (6) a
constant proportion (*t*_{f}) of foliage N is retranslocated before
leaves senesce; (7) active, slow, and passive SOM pools have fixed N : C
ratios; and (8) an N uptake constant determines the plant N uptake rate.
Definitions of the parameters and forcing variables are summarized in Table 2.
For all simulations, the ambient CO_{2} concentration (aCO_{2}) was
set at 400 ppm and eCO_{2} at 800 ppm.

We now summarize the key derivation of the two quasi-equilibrium constraints,
the photosynthetic constraint, and the nutrient cycling constraint from our
baseline simulation model (details provided in Appendix A1 and A2). The
derivation follows Comins and McMurtrie (1993), which is further elaborated
in work by McMurtrie et al. (2000) and Medlyn and Dewar (1996) and
evaluated by Comins (1994). First, the
photosynthetic constraint is derived by assuming that the foliage C pool
(*C*_{f}) has equilibrated. Following the GPP and CUE assumptions (see
above) and the detailed derivations made in Appendix A1, there is an implicit
relationship between NPP and *n*_{f}:

where *I*_{0} is the incident radiation, *k* is the canopy light extinction
coefficient, and *σ* is the specific leaf area. This equation is the
photosynthetic constraint, which relates NPP to *n*_{f}.

Secondly, the nitrogen cycling constraint is derived by assuming that nitrogen inputs to and outputs from the equilibrated pools are equal. Based on the assumed residence times of the passive SOM (∼400 years), slow SOM (15 years), and woody biomass (50 years) pools, we can calculate the nutrient recycling constraint at three different timescales (conceptualized in Fig. 3): very long (VL, >500 years, all pools equilibrated), long (L, 100–500 years, all pools equilibrated except the passive pool), or medium (M, 5–50 years, all pools equilibrated except slow, passive, and wood pools). In the VL term, we have

where *N*_{in} is the total N input into the system, and *N*_{loss} is the
total N lost from the system via leaching and volatilization. Analytically,
with some assumptions about plant N uptake (Appendix A2), we can transform
Eq. (2) into a relationship between NPP and *n*_{f}, expressed as

where *l*_{n} is the fraction of N mineralization that is
lost, *a*_{f}; *a*_{w}
and *a*_{r} are the allocation coefficients for foliage, wood, and
roots, respectively, and *n*_{fl}, *n*_{w}, and *n*_{r}
are the N : C ratios for foliage litter, wood, and roots, respectively.
Since *n*_{w} and *n*_{r} are assumed proportional to
*n*_{f} (Table 2), the nutrient recycling constraint also links NPP
and *n*_{f}. The intersection with the photosynthetic constraint
yields the very-long-term equilibria of both NPP and *n*_{f}.
Similarly, we can write the nitrogen recycling constraint in the L term and M
term as a function between NPP and *n*_{f} (details explained in
Appendix A2). Their respective interaction with the photosynthetic constraint
yields the L-term and M-term equilibria points of both NPP and *n*_{f}
(Figs. 1 and 3). Essentially, at each timescale, there are two unknowns (NPP
and *n*_{f}) to be resolved via both the nitrogen recycling constraint
and the photosynthetic constraint equations. Based on this set of analytical
equations, one can evaluate how different assumptions affect the behavior of
the model quantitatively. Below, we describe how different new model
assumptions affect the predicted plant response to a doubling of the
CO_{2} concentration at various timescales.

## 3.2 Evaluations of new model assumptions based on the quasi-equilibrium framework

### 3.2.1 Explicit plant N uptake

We now move to considering new model assumptions. We first consider
different representations of plant N uptake. In the baseline model, the
mineral N pool (*N*_{min}) is implicit, as we assumed that all mineralized N
in the soil is either taken up by plants (*N*_{U}) or lost from the system
(*N*_{loss}). Here, we evaluate three alternative model representations
in which
plant N uptake depends on an explicit *N*_{min} pool and their effects on
plant responses to eCO_{2}. We consider plant N uptake as (1) a fixed
coefficient of the mineral N pool, (2) a saturating function of root biomass
and a linear function of the mineral N pool (McMurtrie et
al., 2012), and (3) a saturating function of the mineral N pool and a linear
function of root biomass. The last function has been incorporated into some
land surface models, for example, O-CN (Zaehle and Friend, 2010) and CLM
(Ghimire et al., 2016), while the first two have been incorporated
into G'DAY (Corbeels et al., 2005).

A mineral N pool was made explicit by specifying a constant coefficient
(*u*) to regulate the plant N uptake rate (i.e., $\cdot {N}_{\mathrm{U}}=u\phantom{\rule{0.125em}{0ex}}{N}_{\mathrm{min}}$). N lost from the system is a function of the mineral N pool
(*N*_{min}) regulated by a loss rate (*l*_{n, rate}, yr^{−1}). For
the VL-term equilibrium, we have *N*_{in}=*N*_{loss}, which means
${N}_{\mathrm{min}}=\frac{{N}_{\mathrm{in}}}{{l}_{\mathrm{n},\phantom{\rule{0.125em}{0ex}}\mathrm{rate}}}$, and hence

where *n*_{fl} is the foliage litter N : C ratio, which is proportional to
*n*_{f} (Table 2). At the VL equilibrium, we can rearrange the above
equation to relate NPP to *n*_{f}:

which indicates that the N cycling constraint for NPP is inversely dependent
on *n*_{f}.

The second function represents plant N uptake as a saturating function of
root biomass (*C*_{r}) and a linear function of the mineral N pool
(McMurtrie et al., 2012), expressed as

where *K*_{r} is a constant. At the VL equilibrium, we have ${N}_{\mathrm{in}}={N}_{\mathrm{loss}}={l}_{\mathrm{n},\phantom{\rule{0.125em}{0ex}}\mathrm{rate}}{N}_{\mathrm{min}}$ and ${C}_{\mathrm{r}}=\frac{\mathrm{NPP}\cdot {a}_{\mathrm{r}}}{{s}_{\mathrm{r}}}$, where *s*_{r} is the lifetime of the root. Substituting for
*C*_{r} in Eq. (6), we relate *N*_{U} to NPP:

Since *N*_{U} is also a function of NPP, we can rearrange and get

Comparing with Eq. (5), here NPP is also inversely dependent on *n*_{f} but
with an additional negative offset of $\frac{{K}_{\mathrm{r}}{s}_{\mathrm{r}}}{{a}_{\mathrm{r}}}$. The third
approach to represent N uptake (e.g., O-CN and CLM) expresses N uptake as a
saturating function of mineral N also linearly depending on root biomass
(Zaehle and Friend, 2010), according to

where *K* is a constant coefficient, and *V*_{max} is the maximum root N uptake
capacity simplified as a constant here. Since *N*_{U} is also a function
of NPP, we get

This equation sets a limit to possible values of *n*_{f}. In equilibrium, for
*N*_{min} to be nonzero, we need $\left({a}_{\mathrm{f}}{n}_{\mathrm{fl}}+{a}_{\mathrm{w}}{n}_{\mathrm{w}}+{a}_{\mathrm{r}}{n}_{\mathrm{r}}\right)<{V}_{max}\frac{{a}_{\mathrm{r}}}{{s}_{\mathrm{r}}}$. The N loss rate
is still proportional to the mineral N pool, so *N*_{loss} is given by

The above equation provides an *N*_{loss} term that no longer depends on NPP
but only on *n*_{f}. If the N leaching loss is the only system N loss, the
VL-term nutrient constraint no longer involves NPP, implying that the full
photosynthetic CO_{2} fertilization effect is realized. The L- and M-term
nutrient recycling constraints, however, are still NPP dependent due to
feedbacks from the slowly recycling wood and SOM pools (e.g., Eq. A11–A15).

The impacts of these alternative representations of N uptake are shown in
Fig. 4. First, the explicit consideration of the mineral N pool with a
fixed uptake constant (*u*) of 1 yr^{−1} has little impact on the
transient response to eCO_{2} when compared to the baseline model (Figs. 4a, 1a, Table 3). Varying *u* does not strongly (<5 %)
affect plant responses to CO_{2} fertilization at different time steps
(Fig. S2). This is because *u* is only a scaling factor of NPP, meaning it
affects NPP but not its response to eCO_{2} (Table 4), as depicted by Eq. (5).

Moreover, the approach that assumes N uptake as a saturating function of
root biomass linearly depending on the mineral P pool (McMurtrie et al., 2012) has comparable eCO_{2} effects on
production to the baseline and the fixed uptake coefficient models (Fig. 4b, Table 3). Essentially, if $\frac{{K}_{\mathrm{r}}{s}_{\mathrm{r}}}{{a}_{\mathrm{r}}}$ is small, we can
approximate NPP by $\frac{{N}_{\mathrm{in}}}{{l}_{\mathrm{n},\mathrm{rate}}\left({a}_{\mathrm{f}}{n}_{\mathrm{fl}}+{a}_{\mathrm{w}}{n}_{\mathrm{w}}+{a}_{\mathrm{r}}{n}_{\mathrm{r}}\right)}$, which shares a similar structure to the baseline and
fixed uptake coefficient models (Eqs. 8, 5, and A10). Furthermore,
Eq. (8) also depicts the fact that an increase in *a*_{r} should lead to higher NPP, and
an increase in *s*_{r} or *K*_{r} should lead to decreased NPP. However, these
predictions depend on assumptions of *l*_{n,rate} and *n*_{f}. If
*l*_{n,rate} or *n*_{f} is small, NPP would be relatively less sensitive to
*a*_{r},*K*_{r}, or *s*_{r}.

By comparison, representing N uptake as a saturating function of mineral N
linearly depending on root biomass (Ghimire et al., 2016; Zaehle and
Friend, 2010) no longer involves the VL-term nutrient recycling constraint
on production (Fig. 4c), which is predicted by Eq. (11). Actual VL-term NPP
is determined only by *n*_{f} along with the photosynthetic constraint, meaning
that the full CO_{2} fertilization effect on production is realized with
the increase in CO_{2}. The magnitudes of the CO_{2} fertilization
effect at other time steps are comparable to those of the baseline model
(Table 3) because the *N*_{loss} term is smaller than the *N*_{w},*N*_{Sp}, or
*N*_{Ss} terms, meaning it has a relatively smaller effect on NPP at
equilibrium. However, steeper nutrient recycling constraint curves are
observed (Fig. 4c), indicating a stronger sensitivity of the NPP response
to changes in *n*_{f}.

### 3.2.2 Potential NPP

In several vegetation models, including CLM-CN, CABLE, and JSBACH, potential (non-nutrient-limited) NPP is calculated from light, temperature, and water limitations. Actual NPP is then calculated by downregulating the potential NPP to match nutrient supply. Here we term this the potential NPP approach. We examine this assumption in the quasi-equilibrium framework following the implementation of this approach adopted in CLM-CN (Bonan and Levis, 2010; Thornton et al., 2007). The potential NPP is reduced if mineral N availability cannot match the demand from plant growth:

where *P*_{dem} is the plant N demand, and NPP_{pot} is the potential NPP
of the plant. Writing $\left({a}_{\mathrm{f}}{n}_{\mathrm{f}}+{a}_{\mathrm{w}}{n}_{\mathrm{w}}+{a}_{\mathrm{r}}{n}_{\mathrm{r}}\right)$
as *n*_{plant}, the whole-plant N : C ratio, and the whole-soil N : C ratio as
*n*_{soil}, we can calculate the immobilization N demand as

where *f* is the fraction of litter C that becomes soil C, *C*_{lit} is the
total litter C pool, and *s*_{t} is the turnover time of the litter pool.
Actual plant N uptake is expressed as

Actual NPP is expressed as

For the VL constraint, we have *N*_{in}=*N*_{loss}. We can calculate
NPP_{pot} as

For an actual NPP, we need to consider the immobilization demand. Rearranging the above, we get

This equation removes the NPP_{act} dependence on NPP_{pot}. It can
be shown that the fraction of ${P}_{\mathrm{dem}}/({I}_{\mathrm{dem}}+{P}_{\mathrm{dem}}$) depends only on
the N : C ratios and *f*, not on NPP_{pot}. This means that there will be no
eCO_{2} effect on NPP_{act}.

As shown in Fig. 5a, the potential NPP approach results in relatively flat
nutrient recycling constraint curves, suggesting that the CO_{2}
fertilization effect is only weakly influenced by soil N availability.
Despite a sharp instantaneous NPP response, CO_{2} fertilization effects
on NPP_{act} are small on the M-, L-, and VL-term timescales (Table 3). This
outcome can be understood from the governing equation for the nutrient
recycling constraint, which removes NPP_{act} dependence on NPP_{pot} (Eq. 17). Although in the first instance, the plant can increase its production,
over time the litter pool increases in size proportionally to NPP_{pot},
meaning that immobilization demand increases to match the increased plant
demand, which leads to no overall change in the relative demands from the
plant and the litter. This pattern is similar under alternative wood N : C
ratio assumptions (Fig. 5b, Table 3).

### 3.2.3 Root exudation to prime N mineralization

The priming effect is described as the
stimulation of the decomposition of native soil organic matter caused by
larger soil carbon input under eCO_{2} (van Groenigen et al., 2014).
Experimental studies suggest that this phenomenon is widespread and
persistent (Dijkstra and Cheng, 2007), but this process has not been
incorporated by most land surface models (Walker et al., 2015). Here we
introduce a novel framework to induce the priming effect on soil
decomposition and test its effect on plant production response to
eCO_{2} within the quasi-equilibrium framework.

To account for the effect of priming on decomposition of SOM, we first
introduce a coefficient to determine the fraction of root growth allocated
to exudates, *a*_{rhizo}. Here we assumed that the N : C ratio of
rhizodeposition is the same as the root N : C ratio. The coefficient
*a*_{rhizo} is estimated by a function dependent on foliar N : C:

where *n*_{ref} is a reference foliar N : C ratio to induce plant N stress
(0.04), and *a*_{0} and *a*_{1} are tuning coefficients (0.01 and 1,
respectively). Within the quasi-equilibrium framework, for the VL soil
constraint we now have

To introduce an effect of root exudation on the turnover rate of the slow SOM pool, rhizodeposition is transferred into the active SOM pool according to a microbial use efficiency parameter (${f}_{\mathrm{cue},\phantom{\rule{0.125em}{0ex}}\mathrm{rhizo}}=\mathrm{0.3}$). The extra allocation of NPP into the active SOM is therefore

The increased active SOM pool N demand is associated with the degradation rate of the slow SOM pool, expressed as

where *k*_{slow} is the original decomposition rate of the slow SOM pool,
and *k*_{m} is a sensitivity parameter. The decomposition rate of the slow
SOM pool affects *N*_{Rs}, the amount of N released from the slow SOM pools,
as

where *C*_{s} is the slow SOM pool, and Ω_{ss} and
Ω_{ps} represent the proportion of C released through the
decomposition of the slow and passive SOM pools that subsequently enters the slow
SOM pool, respectively.

Root exudation and the associated priming effect result in a strong M-term
plant response to eCO_{2} when compared to the baseline model (Fig. 6a
in comparison to Fig. 4a). In fact, the magnitude of the priming effect on
the M-term NPP response to eCO_{2} is comparable to its L- and VL-term NPP
responses, indicating a persistent eCO_{2} effect over time (Table 3). A
faster decomposition rate and therefore a smaller pool size of the slow SOM
pool are observed (Table 5). With a fixed wood N : C ratio assumption, the NPP
response to eCO_{2} is drastically reduced in the M term compared to
the model with a variable wood N : C assumption (Fig. 6b), but it is comparable
to its corresponding baseline fixed wood N : C model (Table 3). Varying
parameter coefficients (*a*_{0}, *a*_{1}, *f*_{cue, rhizo}, and *k*_{m})
affects the decomposition rates of the slow soil organic pool and hence could
lead to variation of the priming effect on M-term CO_{2} response (Fig. S3). Further experimental studies are needed to better constrain these
parameters. Adding root exudation without influencing the slow SOM pool
decomposition rate (Eq. 21) leads to a smaller predicted M-term CO_{2}
response than the model with the direct effect on the slow SOM pool.
However, it also leads to a higher predicted M-term CO_{2} response than
the baseline model (Fig. 7) because *a*_{r} and *n*_{r} affect the reburial
fraction of the slow SOM pool, as shown in McMurtrie et al. (2000). Finally, the model with a variable wood N : C assumption indicates
that there is no increase in NUE (Table 2) in the M term compared to its
L- and VL-term responses (Fig. 6c). In comparison, the fixed wood N : C
ratio assumption means that there is a decreased wood “quality” (reflected
via a decreased N : C ratio), and therefore faster decomposition of the slow SOM
pool does not release much extra N to support the M-term CO_{2} response,
leading to a significant rise of NUE in the M term (Fig. 6d).

## 4.1 Influence of alternative N uptake assumptions on predicted CO_{2} fertilization

The quasi-equilibrium analysis of the time-varying plant response to
eCO_{2} provides a quantitative framework to understand the relative
contributions of different model assumptions governing the supply of N to
plants in determining the magnitude of the CO_{2} fertilization effect.
Here, we evaluated how plant responses to eCO_{2} are affected by widely
used model assumptions relating to plant N uptake, soil decomposition, and
immobilization demand under alternative wood N–C coupling strategies
(variable and fixed wood N : C ratios). These assumptions have been adopted in
land surface models such as O-CN (Zaehle and Friend, 2010), CABLE
(Wang et al., 2007), LPJ-Guess N (Wårlind et
al., 2014), JASBACH-CNP (Goll et al., 2012), ORCHIDEE-CNP
(Goll et al., 2017a), and CLM4 (Thornton et al., 2007). In line with previous findings (Comins
and McMurtrie, 1993; Dewar and McMurtrie, 1996; Kirschbaum et al., 1998;
McMurtrie and Comins, 1996; Medlyn and Dewar, 1996), our results show that
assumptions related to wood stoichiometry have a very large impact on
estimates of plant responses to eCO_{2}. More specifically, models
incorporating a fixed wood N : C ratio consistently predicted smaller CO_{2}
fertilization effects on production than models using a variable N : C ratio
assumption (Table 3). Examples of models assuming constant (Thornton et
al., 2007; Weng and Luo, 2008) and variable (Zaehle and Friend,
2010) plant tissue stoichiometry are both evident in the literature, and
therefore, assuming that all other model structures and assumptions are similar,
prediction differences could potentially be attributed to the tissue
stoichiometric assumption incorporated into these models, as suggested in
some previous simulation studies (Medlyn et al., 2016,
2015; Meyerholt and Zaehle, 2015; Zaehle et al., 2014). Together with a more
appropriate representation of the trade-offs governing tissue C–N coupling
(Medlyn et al., 2015), further tissue biochemistry
data are necessary to constrain this fundamental aspect of ecosystem model
uncertainty (Thomas et al., 2015).

C–N coupled simulation models generally predict that the CO_{2}
fertilization effect on plant production is progressively constrained by soil
N availability over time: the progressive nitrogen limitation hypothesis (Luo
et al., 2004; Norby et al., 2010; Zaehle et al., 2014). Here we showed
similar temporal patterns in a model with different plant N uptake
assumptions (Fig. 4) and the potential NPP assumption (Fig. 5). In
particular, the progressive N limitation effect on NPP is shown as a
downregulated M-term CO_{2} response after the sharp instantaneous
CO_{2} fertilization effect on production is realized. However, the
model incorporating a priming effect of C on soil N availability with a
flexible wood N : C ratio assumption induced a strong M-term CO_{2}
response (13 % increase in NPP), thereby introducing a persistent
CO_{2} effect over time (Fig. 6a). This strong M-term CO_{2}
response is due to an enhanced decomposition rate of soil organic matter,
consistent with a series of recent observations and modeling studies (Finzi
et al., 2015; Guenet et al., 2018; Sulman et al., 2014; van Groenigen et al.,
2014). However, as a previous quasi-equilibrium study showed, a significant
increase in the M-term CO_{2} response can occur via changes in litter
quality into the slow SOM pool or increased N input into the system (McMurtrie et
al., 2000). Our study differs from McMurtrie et al. (2000) in that we
introduced an explicit effect of C priming on *k*_{slow} – the
decomposition rate of the slow SOM pool – via extra rhizodeposition (Eq. 21). As
such, a faster decomposition rate of slow SOM is observed (Table 5),
equivalent to adding extra N for mineralization to support the M-term
CO_{2} response (Fig. 6c). More complex models for N uptake,
incorporating a carbon cost for nitrogen acquisition, are being proposed
(Fisher et al., 2010; Ghimire et al., 2016; M. Shi et al., 2015); we suggest
that the likely effects of introducing these complex sets of assumptions into
large-scale models could usefully be explored with the quasi-equilibrium
framework.

Processes regulating progressive nitrogen limitation under eCO_{2}
were evaluated by Liang et al. (2016) based on a meta-analysis,
which bridged the gap between theory and observations. It was shown that the
expected diminished CO_{2} fertilization effect on plant growth was not
apparent at the ecosystem scale due to extra N supply through increased
biological N fixation and decreased leaching under eCO_{2}. Here, our
baseline assumption assumed fixed N input into the system, and therefore
plant-available N is progressively depleted through increased plant N
sequestration under eCO_{2}, as depicted by the progressive N limitation
hypothesis (Luo et al., 2004). A function that allows the N fixation parameter
to vary could provide further assessment of the tightness of the ecosystem N
cycle process and its impact on plant response to eCO_{2}. Furthermore,
given the significant role the wood N : C ratio plays in plant N sequestration,
matching the modeled range of wood tissue stoichiometry with observations can
provide an additional level of evaluation of model performance. Our study
provides a generalizable evaluation based on the assumption that the wood N : C
ratio, when allowed to vary in a model, is proportional to the leaf N : C ratio.
Case-specific, more realistic evaluations can be performed based on the
quasi-equilibrium framework to bridge models with observations.

A strong M term and persistent CO_{2} fertilization effects over time was
also found by some models in Walker et al. (2015), but without introducing a
priming effect. In models such as CLM, N losses from the system are
concentration dependent, and plant N uptake is a function of both N supply
and plant demand. Increased plant N demand in models in which N uptake is a
function of plant N demand reduces the soil solution N concentration and
therefore system N losses. This means that over time N can accumulate in the
system in response to eCO_{2} and sustain an eCO_{2} response. Here, our
quasi-equilibrium framework considers N lost as a fixed rate that depends
linearly on the mineral N pool, and the mineral N pool changes at different
equilibrium time points. For example, as shown in Table S1, the M-term N loss
rate is significantly reduced under eCO_{2} compared to the VL-term N
loss rate under aCO_{2}. This suggests a positive relationship between N
loss and NPP, as embedded in Eq. (4).

We also showed that the magnitude of the CO_{2} fertilization effect is
significantly reduced at all timescales when models incorporate the
potential NPP approach (Fig. 5). Among all model assumptions tested, the
potential NPP approach induced the smallest M- to VL-term responses (Table 3). It can be shown from equation derivation (Eq. 17) that the fraction
${P}_{\mathrm{dem}}/({P}_{\mathrm{dem}}+{I}_{\mathrm{dem}}$) depends only on the N : C ratios and *f* (fraction of
litter C become soil C), implying that models incorporating the potential
NPP assumption should show no response of NPP to CO_{2}. Both our study
and simulation-based studies showed small CO_{2} responses (Walker et
al., 2015; Zaehle et al., 2014), possibly because the timing of *P*_{dem} and
*I*_{dem} differs due to the fluctuating nature of GPP and N mineralization
at daily to seasonal time steps such that N is limiting at certain times of
the year but not at others. Additionally, models such as CLM have
volatilization losses (not leaching) that are reduced under eCO_{2}, which
may lead to production not limited by N availability, meaning that a full
CO_{2} fertilization effect may be realized. Finally, leaching is
simplified here and treated as a fixed fraction of the mineral N pool. In
models such as CLM or JASBACH, it is a function of the soil-soluble N
concentration, implying a dependency on litter quality (Zaehle et al., 2014).

## 4.2 Implications for probing model behaviors

Model–data intercomparisons have been shown as a viable means to investigate
how and why models differ in their predicted response to eCO_{2} (De
Kauwe et al., 2014; Walker et al., 2015; Zaehle et al., 2014). Models make
different predictions because they have different model structures
(Lombardozzi et al., 2015; Meyerholt et al., 2016; Shi et al., 2018; Xia
et al., 2013; Zhou et al., 2018), parameter uncertainties (Dietze et al.,
2014; Wang et al., 2011), response mechanisms (Medlyn et al., 2015), and numerical implementations
(Rogers et al., 2016). It is increasingly difficult
to diagnose model behaviors from the multitude of model assumptions
incorporated into the model. Furthermore, while it is true that the models
can be tuned to match observations within the domain of calibration, models
may make correct predictions but based on incorrect or simplified
assumptions (Medlyn et al., 2005, 2015; Walker et al.,
2015). As such, diagnosing model behaviors can be a challenging task in
complex plant–soil models. In this study, we showed that the effect of a
model assumption on plant response to eCO_{2} can be analytically
predicted by solving the photosynthetic and nutrient recycling
constraints together. This provides a constrained model framework to evaluate the
effect of individual model assumptions without having to run a full set of
sensitivity analyses, thereby providing an a priori understanding of the underlying
response mechanisms through which the effect is realized. We suggest that
before implementing a new function into the full structure of a plant–soil
model, one could use the quasi-equilibrium framework as a test bed to examine
the effect of the new assumption.

The quasi-equilibrium framework requires that additional model assumptions
be analytically solvable, which is increasingly not the case for complex
modeling structures. However, as we demonstrate here, studying the
behavior of a reduced-complexity model can nonetheless provide real insight
into model behavior. In some cases, the quasi-equilibrium framework can
highlight where additional complexity is not valuable. For example, here we
showed that adding complexity in the representation of plant N uptake did
not result in significantly different predictions of plant response to
eCO_{2}. Where the quasi-equilibrium framework indicates little effect of
more complex assumptions, there is a strong case for keeping simpler
assumptions in the model. However, we do acknowledge that the
quasi-equilibrium framework operates on timescales of >5 years;
where fine-scale temporal responses are important, the additional complexity
may be warranted.

The multiple-element limitation framework developed by Rastetter
and Shaver (1992) analytically evaluates the relationship between short-term
and long-term plant responses to eCO_{2} and nutrient availability under
different model assumptions. It was shown that there could be a marked
difference in the short-term and long-term ecosystem responses to eCO_{2}
(Rastetter et al., 1997; Rastetter and Shaver, 1992). More specifically,
Rastetter et al. (1997) showed that the ecosystem NPP
response to eCO_{2} appeared on several characteristic timescales: (1) there was an instantaneous increase in NPP, which results in an increased
vegetation C : N ratio; (2) on a timescale of a few years, the vegetation
responded to eCO_{2} by increasing uptake effort for available N through
increased allocation to fine roots; (3) on a timescale of decades, there was
a net movement of N from soil organic matter to vegetation, which enables
vegetation biomass to accumulate; and (4) on the timescale of centuries,
ecosystem responses were dominated by increases in total ecosystem N, which
enable organic matter to accumulate in both vegetation and soils. Both the
multiple-element limitation framework and the quasi-equilibrium framework
provide information about equilibrium responses. These approaches also
provide information about the degree to which the ecosystem replies to
internally recycled N vs. exchanges with external sources and sinks. The
multiple-element limitation framework also offers insight into the C–N
interaction that influences transient dynamics. These analytical frameworks
are both useful tools for making quantitative assessments of model
assumptions.

A related model assumption evaluation tool is the traceability framework, which decomposes complex models into various simplified component variables, such as ecosystem C storage capacity or residence time, and hence helps to identify structures and parameters that are uncertain among models (Z. Shi et al., 2015; Xia et al., 2013, 2012). Both the traceability and quasi-equilibrium frameworks provide analytical solutions to describe how and why model predictions diverge. The traceability framework decomposes complex simulations into a common set of component variables, explaining differences due to these variables. In contrast, quasi-equilibrium analysis investigates the impacts and behavior of a specific model assumption, which is more indicative of mechanisms and processes. Subsequently, one can relate the effect of a model assumption more mechanistically to the processes that govern the relationship between the plant N : C ratio and NPP, as depicted in Fig. 1, thereby facilitating efforts to reduce model uncertainties.

Models diverge in future projections of plant responses to increases in
CO_{2} because of the different assumptions that they make. Applying model
evaluation frameworks, such as the quasi-equilibrium framework, to attribute
these differences will not necessarily reduce multi-model prediction spread
in the short term (Lovenduski and Bonan, 2017). Many model
assumptions are still empirically derived, and there is a lack of
mechanistic and observational constraints on the effect size, meaning that
it is important to apply models incorporating diverse process
representations. However, use of the quasi-equilibrium framework can provide
crucial insights into why model predictions differ and thus help identify
the critical measurements that would allow us to discriminate among alternative
models. As such, it is an invaluable tool for model intercomparison and
benchmarking analysis. We recommend the use of this framework to analyze likely
outcomes of new model assumptions before introducing them to complex model
structures.

The code repository is publicly available via DOI https://doi.org/10.5281/zenodo.2574192 (Jiang et al., 2019).

Here we show how the baseline quasi-equilibrium framework is derived. Specifically, there are two analytical constraints that form the foundation of the quasi-equilibrium framework, namely the photosynthetic constraint and the nitrogen cycling constraint. The derivation follows Comins and McMurtrie (1993), which is further elaborated in work by McMurtrie et al. (2000) and Medlyn and Dewar (1996) and evaluated Comins (1994).

## A1 Photosynthetic constraint

Firstly, gross primary production (GPP) in the simulation mode is calculated
using a light-use efficiency approach named MATE (Model Any Terrestrial
Ecosystem) (McMurtrie et al., 2008; Medlyn et al., 2011; Sands, 1995), in
which absorbed photosynthetically active radiation is estimated from leaf
area index (*L*) using Beer's law and is then multiplied by a light-use
efficiency (LUE), which depends on the foliar N : C ratio (*n*_{f}) and
atmospheric CO_{2} concentration (*C*_{a}):

where *I*_{0} is the incident radiation, *k* is the canopy light extinction
coefficient, and *L* is leaf area index. The derivation of LUE for the MATE
is described in full by McMurtrie et al. (2008); our version differs
only in that the key parameters determining the photosynthetic rate follow
the empirical relationship with the foliar N : C ratio given by Walker et
al. (2014), and the expression for stomatal conductance follows Medlyn et
al. (2011).

In the quasi-equilibrium framework, the photosynthetic constraint is derived
by assuming that the foliage C pool (*C*_{f}) has equilibrated. That is, the
new foliage C production equals turnover, which is assumed to be a constant
fraction (*s*_{f}) of the pool:

where *a*_{f} is the allocation coefficient for foliage. From Eq. (A1), net
primary production is a function of the foliar N : C ratio and the foliage C
pool:

where *σ* is the specific leaf area. Combining the two equations above
leads to an implicit relationship between NPP and *n*_{f},

which is the photosynthetic constraint.

## A2 Nutrient recycling constraint

The nitrogen cycling constraint is derived by assuming that nitrogen inputs to and outputs from the equilibrated pools are equal. Based on the assumed residence times of the passive SOM (∼400 years), slow SOM (15 years), and woody biomass (50 years) pools, we can calculate the nutrient recycling constraint at three different timescales: very long (VL, >500 years, all pools equilibrated), long (L, 100–500 years, all pools equilibrated except the passive pool), or medium (M, 5–50 years, all pools equilibrated except slow, passive and wood pools).

In the VL term, we have

where *N*_{in} is the total N input into the system, and *N*_{loss} is the
total N lost from the system via leaching and volatilization. Following
Comins and McMurtrie (1993), the flux *N*_{in} is assumed to be a
constant. The total N loss term is proportional to the rate of N
mineralization (*N*_{m}), following

where *l*_{n} is the fraction of N mineralization that is lost. It is assumed
that mineralized N that is not lost is taken up by plants (*N*_{U}):

Combining with Eq. (A6), we have

The plant N uptake rate depends on production (NPP) and plant N : C ratios, according to

where *a*_{f}, *a*_{w}, and *a*_{r} are the allocation coefficients for
foliage, wood, and roots, respectively, and *n*_{fl}, *n*_{w}, and *n*_{r} are
the N : C ratios for foliage litter, wood, and roots, respectively. The foliage
litter N : C ratio (*n*_{fl}) is proportional to *n*_{f}, according to Table 2.
Combining Eq. (A9) with Eqs. (A5) and (A8), we obtain a function of NPP that can
be related to total N input, which is the nutrient recycling constraint in
the VL term, expressed as

Since *n*_{w} and *n*_{r} are assumed proportional to *n*_{f}, the nutrient
recycling constraint also links NPP and *n*_{f}. The intersection with the
photosynthetic constraint yields the very-long-term equilibria of both NPP
and *n*_{f}.

In the L term, we now have to consider N flows leaving and entering the passive SOM pool, which is no longer equilibrated:

where ${N}_{{\mathrm{R}}_{\mathrm{p}}}$ and ${N}_{{\mathrm{S}}_{\mathrm{p}}}$ are the release and sequestration of the passive SOM N pool, respectively. The release flux, ${N}_{{\mathrm{R}}_{\mathrm{p}}}$, can be assumed to be constant on the L-term timescale. The sequestration flux, ${N}_{{\mathrm{S}}_{\mathrm{p}}}$, can be calculated as a function of NPP. In G'DAY, as with most carbon–nitrogen coupled ecosystem models, carbon flows out of the soil pools are directly related to the pool size. As demonstrated by Comins and McMurtrie (1993), such soil models have the mathematical property of linearity, meaning that carbon flows out of the soil pools are proportional to the production input to the soil pool, or NPP. Furthermore, the litter input into the soil pools is assumed proportional to the foliar N : C ratio, with the consequence that N sequestered in the passive SOM is also related to the foliar N : C ratio. The sequestration flux into the passive soil pool (${N}_{{\mathrm{S}}_{\mathrm{p}}}$) can thus be written as

where *n*_{p} is the N : C ratio of the passive SOM pool, and
${\mathrm{\Omega}}_{{\mathrm{p}}_{\mathrm{f}}}$, ${\mathrm{\Omega}}_{{\mathrm{p}}_{\mathrm{w}}}$, and ${\mathrm{\Omega}}_{{\mathrm{p}}_{\mathrm{r}}}$ are
the burial coefficients for foliage, wood, and roots (the proportion of plant
carbon production that is ultimately buried in the passive pool),
respectively. The burial coefficients ${\mathrm{\Omega}}_{{\mathrm{p}}_{\mathrm{f}}}$,
${\mathrm{\Omega}}_{{\mathrm{p}}_{\mathrm{w}}}$, and ${\mathrm{\Omega}}_{{\mathrm{p}}_{\mathrm{r}}}$ depend on the N : C
ratios of foliage, wood, and root litter (detailed derivation in Comins and
McMurtrie, 1993). Combining and rearranging, we obtain the nutrient recycling
constraint in the L term as

Similarly, in the M term, we have

where ${N}_{{\mathrm{R}}_{\mathrm{s}}}$and ${N}_{{\mathrm{R}}_{\mathrm{w}}}$ are the N released from the slow SOM and wood pool, respectively, and ${N}_{{\mathrm{S}}_{\mathrm{s}}}$ and ${N}_{{\mathrm{S}}_{\mathrm{w}}}$ are the N stored in the slow SOM and wood pool, respectively (Medlyn et al., 2000). The nutrient recycling constraint in the M term can thus be derived as

where *n*_{s} is the slow SOM pool N : C ratio, and ${\mathrm{\Omega}}_{{\mathrm{s}}_{\mathrm{f}}}$ and ${\mathrm{\Omega}}_{{\mathrm{s}}_{\mathrm{r}}}$ are foliage and root C sequestration rate into the slow SOM pool,
respectively (Medlyn et al., 2000). The intersection between
the nitrogen recycling constraint and the photosynthetic constraint provides
an analytical solution to both NPP and *n*_{f} at different timescales, and we can
then interpret how changing model assumptions affect the predicted plant
responses to elevated CO_{2}.

The supplement related to this article is available online at: https://doi.org/10.5194/gmd-12-2069-2019-supplement.

BEM and MJ designed the study; MJ, BEM, and SZ performed the analyses; APW, MGDK, and SZ designed the priming effect equations; all authors contributed to results interpretation and paper writing.

The authors declare that they have no conflict of interest.

This paper builds heavily on ideas originally developed by Ross McMurtrie and Hugh Comins (now deceased). We would like to acknowledge their intellectual leadership and inspiration.

Sönke Zaehle and Silvia Caldararu were supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (QUINCY; grant no. 647204) and the German Academic Exchange Service (DAAD; project ID 57318796). David S. Ellsworth and Mingkai Jiang were also supported by the DAAD.

This paper was edited by David Lawrence and reviewed by two anonymous referees.

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