Assumptions of the C cycle models and validity of this analysis
This analysis is built upon Eq. (1), which represents the majority of
terrestrial C cycle models developed in the past decades (Manzoni and
Porporato, 2009; Sierra and Müller, 2015). These models have several
assumptions, which may influence the validity of this analysis. First, these
models assume that donor pools control C transfers among pools and
decomposition follows first-order decay functions (assumption 1). This
assumption is built upon observations from litter and SOC decomposition.
Analysis of data from nearly 300 studies of litter decomposition (Zhang et
al., 2008), about 500 studies of soil incubation (Schädel et al., 2014;
Xu et al., 2016), more than 100 studies of forest succession (Yang et al.,
2011), and restoration (Matamala et al., 2008) almost all suggests that the
first-order decay function captures macroscopic patterns of land C dynamics.
Even so, its biological, chemical, and physical underpinnings need more study
(Luo et al., 2016). This assumption has recently been challenged by a notion
that microbes are actively involved in decomposition processes. To describe
the active roles of microbes in organic C decomposition, a suite of nonlinear
microbial models has been proposed using Michaelis–Menten or reverse
Michaelis–Menten equations (Allison et al., 2010; Wieder et al., 2013). Those
nonlinear models exhibit unique behaviors of modeled systems, such as damped
oscillatory responses of soil C dynamics to small perturbations and
insensitivity of the equilibrium pool sizes of litter or soil carbon to
inputs (Li et al., 2014; Wang et al., 2014, 2016). Oscillations have been
documented for single enzymes at timescales between 10-4 and 10 s
(English et al., 2006; Goldbeter, 2013; Xie, 2013). Over longer timescales
with largely diverse mixtures of enzyme-substrate complexes in soil,
oscillations may be likely averaged out so that the first-order decay functions
may well approximate these average dynamics of organic matter decomposition
(Sierra and Müller, 2015).
Second, those models all assume that multiple pools can adequately
approximate transformation, decomposition, and stabilization of SOC in the
real world (assumption 2). The classic SOC model, CENTURY, uses three
conceptual pools, active, slow, and passive SOC, to represent SOC dynamics
(Parton et al., 1987). Several models define pools that correspond to
measurable SOC fractions to match experimental observation with modeling
analysis (Smith et al., 2002; Stewart et al., 2008). Carbon transformation in
soil over time has also been described by a partial differential function of
SOM quality (Bosatta and Ågren, 1991; Ågren and Bosatta, 1996). The
latter quality model describes the external inputs of C with certain quality,
C loss due to decomposition, and the internal transformations of the quality
of soil organic matter. It has been shown that multi-pool models can
approximate the partial differential function or continuous quality model as
the number of pools increases (Bolker et al., 1998; Sierra and Müller,
2015).
Assumption 3 is on partitioning coefficients of C input (i.e., elements in
vector B) and C transformation among plant, litter, and soil pools
(i.e., elements in the matrix Aξ(t)K).
Some of the terrestrial C cycle models assume that elements in vector
B and matrices A and K are constants. All the
factors or processes that vary with time are represented in the diagonal
matrix ξ(t). In the real world, C transformation is
influenced by environmental variables (e.g., temperature, moisture, oxygen,
N, phosphorus, and acidity varying with soil profile, space, and time),
litter quality (e.g., lignin, cellulose, N, or their relative content),
organomineral properties of SOC (e.g., complex chemical compounds,
aggregation, physiochemical binding and protection, reactions with inorganic,
reactive surfaces, and sorption), and microbial attributes (e.g., community
structure, functionality, priming, acclimation, and other physiological
adjustments) (Luo et al., 2016). It is not practical to incorporate all of
those factors and processes into one model. Only a subset of them is
explicitly expressed, while the majority is implicitly embedded in the C cycle
models. Empirical studies have suggested that temperature, moisture, litter
quality, and soil texture are primary factors that control C transformation
processes of decomposition and stabilization (Burke et al., 1989; Adair et
al., 2008; Zhang et al., 2008; Xu et al., 2012; Wang et al., 2013). Nitrogen
influences C cycle processes mainly through changes in photosynthetic C
input, C partitioning, and decomposition. It is yet to be identified how other
major factors and processes, such as microbial activities and organomineral
protection, regulate C transformation.
Assumption 4 is that terrestrial C cycle models use different response
functions (i.e., different ξ(t) in Eq. 1) to represent C
cycle responses to external variables. As temperature modifies almost all
processes in the C cycle, different formulations, including exponential,
Arrhenius, and optimal response functions, have been used to describe C cycle
responses to temperature changes in different models (Lloyd and Taylor, 1994;
Jones et al., 2005; Sierra and Müller, 2015). Different response
functions are used to connect C cycle processes with moisture, nutrient
availability, soil clay content, litter quality, and other factors. Different
formulations of response functions may result in substantially different
model projections (Exbrayat et al., 2013) but are unlikely to change basic dynamics
of the model behaviors.
Assumption 5 is that disturbance events are represented in models in
different ways (Grosse et al., 2011; West et al., 2011; Goetz et al., 2012;
Hicke et al., 2012). Fire, extreme drought, insect outbreaks, land
management, and land cover and land use change influence terrestrial C
dynamics via (1) altering rate processes, for example, gross primary
productivity (GPP), growth, tree mortality, or heterotrophic respiration;
(2) modifying microclimatic environments; or (3) transferring C from one pool to
another (e.g., from live to dead pools during storms or release to the
atmosphere with fire) (Kloster et al., 2010; Thonicke et al., 2010; Luo and
Weng, 2011; Prentice et al., 2011; Weng et al., 2012). Those disturbance
influences can be represented in terrestrial C cycle models through changes
in parameter values, environmental scalars, and/or discrete C transfers among
pools of Eq. (1) (Luo and Weng, 2011). While Eq. (1) does not explicitly
incorporate disturbances for their influences on land C cycle, Weng et
al. (2012) developed a disturbance regime model that combines Eq. (1) with
frequency distributions of disturbance severity and intervals to quantify net
biome exchanges.
The sixth assumption that those models make is that the lateral C fluxes
through erosion or local C drainage are negligible so that Eq. (1) can
approximate terrestrial C cycle over space. If soil erosion is substantial
enough to be modeled with horizontal movement of C, a third dimension should
be added in addition to two-dimensional transfers in classic models.
Our analysis on transient dynamics of terrestrial C cycle is valid unless
some of the assumptions are violated. Assumption 1 on the first-order decay
function of decomposition appears to be supported by thousands of datasets.
It is a burden on microbiologists to identify empirical evidence to support
the nonlinear microbial models. Assumption 2 may not affect the validity of
our analysis no matter how C pools are divided in the ecosystems. Our
analysis in this study is applicable no matter whether elements are
time-varying or constant in vector B and matrices A and
K as in assumption 3. Neither assumption 4 nor 5 would affect the
analysis in this study. The environmental scalar, ξ(t), as
related to assumption 4 can be any forms in the derived equations (e.g.,
Eq. 2). Disturbances of fire, land use, and extreme drought change rate
processes but do not alter the basic formulation of Eq. (1). If soil erosion
and lateral transportation of C become major research objectives, Eq. (1)
can no longer be analyzed to understand the mathematical foundation
underlying transient dynamics of terrestrial C cycle.
Carbon storage capacity
One of the two components this analysis introduces to understand transient
dynamics of terrestrial C storage is the C storage capacity (Eq. 2).
Olson (1963) is probably among the first who systematically analyzed C
storage dynamics at the forest floor as functions of litter production and
decomposition. He collected data of annual litter production and
approximately steady-state organic C storage at the forest floor, from which
decomposition rates were estimated for a variety of ecosystems from Ghana in
the tropics to alpine forests in California. Using the relationships among
litter production, decomposition, and C storage, Olson (1963) explored
several issues, such as decay without input, accumulation with continuous or
discrete annual litter fall, and adjustments in production and decay
parameters during forest succession. His analysis approximated the
steady-state C storage as the C input times the inverse of decomposition
(i.e., residence time). The steady-state C storage is also considered the
maximal amount of C that a forest can store.
This study is not only built upon Olson's analysis but also expands it in at
least two aspects. First, we similarly define the C storage capacity
(i.e., Eq. 5a and 5b). Those equations can be applied to a whole ecosystem
with multiple C pools, while Olson's analysis is for one C pool. Second,
Olson (1963) treated the C input and decomposition rate as yearly constants
at a given location even though they varied with locations. This study
considers both C input and rate of decomposition being time dependent. A
dynamical system with its input and parameters being time dependent
mathematically becomes a nonautonomous system (Kloeden and Rasmussen, 2011).
As terrestrial C cycle under global change is transient, we need to treat it
as a nonautonomous system to better understand the properties of transient
dynamics. Olson (1963) approximated the nonautonomous system at the yearly
timescale without global change so as to effectively understand properties
of steady-state C storage at the forest floor. In comparison, Eq. (5a)
and (5b) are not only more general but also essential for understanding
transient dynamics of the terrestrial C cycle in response to global change.
Under the transient dynamics, the C storage capacity as defined by Eq. (5a)
and (5b) still sets the maximal amount of C that one ecosystem can store at
time t. This capacity represents instantaneous responses of ecosystem C
cycle to external forcing via changes in both C input and residence time, and
thus varies within 1 day, over seasons of a year, and interannually over
longer timescales as forcings vary. The variation of the C storage capacity
can result from cyclic environmental changes (e.g., dial and seasonal
changes), directional global change (e.g., rising atmospheric CO2,
nitrogen deposition, altered precipitation, and warming), disturbance events,
disturbance regime shifts, and changing vegetation dynamics (Luo and Weng,
2011). Since the capacity sets the maximal amount of C storage (Fig. 2a), it is
a moving attractor toward which the current C storage chases. When the
capacity is larger than the C storage itself, C storage increases. Otherwise,
the C storage decreases.
Novel approaches to model evaluation and improvement
Our analysis of transient C cycle dynamics offers new approaches to
understand, evaluate, diagnose, and improve land C cycle models. We have
demonstrated that many global land C cycle models can be exactly represented
by the matrix equation (Eqs. 1 and 2) (i.e., physical emulators). As a
consequence, outputs of all those models can be placed into a three
dimensional (3-D) space (Eq. 7) to measure their differences. In addition,
components of land C cycle models are simulated in a mutually independent
fashion so that modeled C storage can be decomposed into traceable components
for traceability analysis. Moreover, the physical emulators computationally
enable data assimilation to constrain complex models.
Physical emulators of land C cycle models
We have developed matrix representations (i.e., physical emulators) of CABLE,
LPJ-GUESS, CLM3.5, CLM4.0, CLM4.5, BEPS, and TECO (Xia et al., 2013; Hararuk
et al., 2014; Ahlström et al., 2015; Chen et al., 2015). The emulators
can exactly replicate simulations of C pools and fluxes with their original
models when driven by a limited set of inputs from the full model (GPP, soil
temperature, and soil moisture) (Fig. 1b and c). However, the physical
emulators differ for different models since the elements of each matrix could be
differently parameterized or formulized in different models. Also, different
models usually have different pool-flux structures, leading to different
non-zero elements in the A matrix. Nonetheless, the physical
emulators make complex models analytically clear, and therefore give us a
way to understand the effects of forcing, model structures, and parameters on
modeled ecosystem processes. They greatly simplify the task of understanding
the dynamics of submodels and interactions between them. The emulators allow
us to analyze model results in the 3-D parameter space and the traceability
framework.
Parameter space of C cycle dynamics
Equation (7) indicates that transient dynamics of modeled C storage are
determined by three parameters: C input, residence time, and C storage
potential. The 3-D parameter space offers one novel approach to uncertainty
analysis of global C cycle models. As global land models incorporate more and
more processes to simulate C cycle responses to global change, it becomes
very difficult to understand or evaluate complex model behaviors. As such,
differences in model projections cannot be easily diagnosed and attributed to
their sources (Chatfield, 1995; Friedlingstein et al., 2006; Luo et al.,
2009). Equation (7) can help diagnose and evaluate complex models by placing
all modeling results within one common parameter space in spite of the fact
that individual global models may have tens or hundreds of parameters to
represent C cycle processes as affected by many abiotic and biotic factors
(Luo et al., 2016). The 3-D space can be used to measure how and how much the
models diverge.
Traceability analysis
The two terms on the right side of Eq. (2) can be decomposed into traceable
components (Xia et al., 2013) so as to identify sources of uncertainty in C
cycle model projections. Model intercomparison projects (MIPs) all illustrate
great spreads in projected land C sink dynamics across models (Todd-Brown et
al., 2013; Tian et al., 2015). It has been extremely challenging to attribute
the uncertainty to sources. Placing simulation results of a variety of C
cycle models within one common parameter space can measure how much the model
differences are in a common metric (Eq. 7). The measured differences can be
further attributed to sources in model structure, parameter, and forcing
fields with traceability analysis (Xia et al., 2013; Rafique et al., 2014;
Ahlström et al., 2015; Chen et al., 2015). The traceability analysis can also
be used to evaluate effectiveness of newly incorporated modules into
existing models, such as adding the N module on simulated C dynamics (Xia et
al., 2013) and locate the origin of model ensemble uncertainties to external
forcing vs. model structures and parameters (Ahlström et al., 2015).
Constrained estimates of terrestrial C sequestration
Traditionally, global land C sink is indirectly estimated from airborne
fraction of C emission and ocean uptake. Although many global land models
have been developed to estimate land C sequestration, a variety of MIPs
indicate that model predictions widely vary among them and do not fit
observations well (Schwalm et al., 2010; Luo et al., 2015; Tian et al.,
2015). Moreover, the prevailing practices in the modeling community,
unfortunately, may not lead to significant enhancements in our confidence on
model predictions. For example, incorporating an increasing number of
processes that influence the C cycle may represent the real-world phenomena
more realistically but makes the models more complex and less tractable. MIPs
have effectively revealed the extent of the differences between model
predictions (Schwalm et al., 2010; Keenan et al., 2012; De Kauwe et al.,
2013) but provide limited insights into sources of model differences (see
Medlyn et al., 2015). The physical emulators make data assimilation
computationally feasible for global C cycle models (Hararuk et al., 2014,
2015) and thus offer the possibility to generate independent yet constrained
estimates of global land C sequestration to be compared with the indirect
estimate from the airborne fraction of C emission and ocean uptake. With the
emulators, we can assimilate most of the C flux- and pool-related datasets
into those models to better constrain global land C sink dynamics.