Soils are a main component of the global carbon (C) cycle, storing ca.
2200 Pg of C in the top 100 cm according to recent estimates (Batjes,
2014). This soil C pool is dynamic and often exists in a non-equilibrium
state as the result of an imbalance between input and output C fluxes, in
which case it will act either as a C sink or as a C source over time. Changes in the
speed at which soil organisms decompose soil organic matter (SOM) and
mineralize soil organic carbon (SOC) into

It is well known that SOC mineralization and resulting

Future predictions of soil C dynamics require the use of mathematical models. Early soil C models and most still in use are based on first-order decay of multiple C pools, with temperature and moisture having general non-interactive effects on decay rates (Rodrigo et al., 1997). When appropriately calibrated, these models do well at simulating soil respiration fluxes of soils under relatively stable conditions. They are often developed to approximate long-term steady-state conditions under specific land uses. They are also capable of fitting long-term trends of soil C loss, such as data from long-term bare fallow where all litter input has stopped (Barré et al., 2010). However, they lack a theoretical basis justifying their basic assumptions of pool partitioning and decay mechanisms. They also generally need calibration for specific soil types or land cover types and often fail to properly simulate observed short- and mid-term variability in soil respiration. Some of the most relevant observations these models have failed to reproduce include changes (typically a dampening) of temperature sensitivities of decomposition over time (Hamdi et al., 2013), non-linear responses to soil moisture content (Borken and Matzner, 2009), and changes in decomposition rates in response to variations in concentrations of organic matter (Blagodatskaya and Kuzyakov, 2008). Such model shortcomings, which reflect missing or wrongly simulated processes, create a difficult-to-quantify uncertainty in global long-term predictions of soil C and its feedback to climate change. It is therefore unclear if first-order models can predict long-term changes in C stocks under more dynamic (and therefore realistic) environmental conditions.

Second-order models have a more realistic basic structure compared to conventional first-order models, since they simulate organic matter decomposition as a reaction between SOC and decomposers (i.e. a microbial or enzyme pool). This single but fundamental change in decomposition kinetics strongly affects predicted long-term changes in soil C, largely as a result of the dynamics of the decomposer pool, which itself can respond to temperature in a number of ways (Wutzler and Reichstein, 2008). Second-order models also lead to more complex dynamics of short- to mid-term soil respiration, with apparent temperature sensitivities that vary over time, more in line with many observations.

The temporal variability in the response of decomposition to moisture is most evident in the strong respiration pulses after dry soils are re-wetted, known as the Birch effect (Birch, 1958). But studies have shown that a successful simulation of such pulses requires the incorporation of additional mechanisms, namely the explicit representation of a bio-available C pool, such as dissolved organic matter (DOC), and a moisture regulation of decomposer's access to this pool that may differ from the moisture regulation on the decomposition reaction itself (Lawrence et al., 2009; Zhang et al., 2014).

The response of soil respiration to temperature and moisture is highly dynamic, both spatially and temporally (Hamdi et al., 2013; Moyano et al., 2012). Moisture and temperature interactions have been observed in a number of experimental studies (Craine and Gelderman, 2011; Rey et al., 2005; Suseela et al., 2012; Wickland and Neff, 2008), but neither consistent trends nor general explanatory theories have been identified. Improving our understanding of these interactions is a crucial step towards increasing confidence in models and important for interpreting modelling and experimental results (Crowther et al., 2016; Tang and Riley, 2014). Identifying the model structures and parameterizations that can best represent these interactive effects has been attempted by very few studies (Sierra et al., 2015, 2017).

The objectives of this study are to compare the ability of different soil C modelling approaches to reproduce temperature and moisture interactive effects on soil carbon fluxes and thus to gain insight into mechanisms underlying the observed responses. With the hypothesis that a more mechanistic model will be better capable of simulating such interactions, we compare different model structures, testing first-order, second-order, and Michaelis–Menten reaction kinetics in combination with an explicit simulation of diffusion fluxes. We then compare the best diffusion model with versions based on common empirical moisture relationships.

Measurements of the interaction effects of temperature and moisture on soil respiration fluxes were obtained by incubating a crop field soil at several fixed levels of soil moisture and variable levels of temperature over a period of ca. 6 months, as detailed in the following.

Soils from 0 to 20 cm depth were sampled at Versailles, France, from the “Le
Closeaux” experimental field plot, cultivated with wheat until 1992 and with
maize since 1993. Mean annual temperature and rainfall are 10

Sampled soils were thoroughly mixed, sieved at 2 mm, and stored moist at
4

All samples were brought to a pF of 4.2, corresponding to about 12.5 % volumetric moisture. Three replicate samples were then adjusted to each of the moisture levels (2 %, 5.5 %, 12.5 %, 16 %, 20 %, 23.5 %, 27 %, 30.5 %, 34 %, 38 %, and 45 %) by adding water or air drying. These values range from air-dry to saturated. Immediately thereafter, the plastic cylinders were put in 500 mL jars containing a small amount of water on the bottom (except for the 2 % and 5.5 % moisture) to prevent soil drying and equipped with a lid and a rubber septum for gas sampling. Because of the extremely low respiration rates, samples with 2 % moisture were placed in 125 mL jars containing 170 g of soil.

To minimize post-disturbance effects, samples were pre-incubated at
4

As shown in Fig. 1, the timing of temperature treatments was not equal for
all samples, with some temperature steps missing at low moisture levels. This
was partly due to the time required for

Graphical representation of the incubated soil samples showing the fixed levels of moisture content and the times at different temperatures.

We started with a basic soil C model with the following state variables: a
bio-unavailable polymeric C pool (

The rates of change of the model state variable were defined as

Diagram showing C pools and fluxes, as well as the points of diffusion limitations. Second-order decay may refer also to Michaelis–Menten reaction kinetics. Variations of this scheme were tested in this study.

The flux of

The value for

Diffusion fluxes depend on a concentration difference, a diffusivity term,
and the distance over which diffusion occurs (Manzoni et al., 2016). For the
purpose of modelling diffusion in soils, values of diffusivity and diffusion
distances are required that best average or represent the actual underlying
soil complexity. For practical purposes, we combined these two values into a
single calibrated parameter, a conductance (

Reaction rates (

Calibrated and non-calibrated parameters for all models are given in the
Supplement (Tables S1, S2, and S3). Whenever possible, fixed parameters as
well as lower and upper bounds for calibrated parameters (Table S1) were set
according to values reported in the literature (e.g. Hagerty et al., 2014; Li et
al., 2014; Price and Sowers, 2004). Equilibrium conditions were not assumed
at the start of the experimental procedure, as such a state is unlikely for
samples that have been processed and disturbed. Therefore, initial conditions
were obtained by also optimizing the fractions of initial carbon pool sizes
(

Parameter optimization was carried out in two steps. We first explored parameter spaces using a Latin hypercube of parameter values. For this we randomly selected unique parameter sets from a uniform distribution over each parameter range (R function randomLHS, package lhs; Stein, 1987) to obtain 30 000 parameter sets. Model costs were then obtained by running models with each set. In the second step we used the Nelder–Mead algorithm (as implemented in the function modFit in package FME of the R programming language; R Development Core Team, 2016; Soetaert and Petzoldt, 2010) with initial parameter values being the set from the previous step with the lowest model cost. For the cost calculations we used an error term (“err” argument to FME function modCost) to weight the residuals. The error was calculated as the normalized (0–1) standard deviation of measured values at each combination of temperature and moisture, with 0.1 added to avoid an unreasonable weighting of measurements with near-zero errors.

For a visual inspection of the model–data fits, we plotted both the measured
and the model relationship between soil respiration and moisture, soil
respiration and temperature, and apparent activation energy (

Different model versions with their weighted and unweighted root
mean squared errors (RMSEs, in units mg C kg soil

Models were named according to their decomposition kinetics followed by the
uptake kinetics and the moisture function, using the abbreviations:
1 for first order, 2 for second order, M for Michaelis–Menten,
M

A second model comparison was carried out to test the impact of different
approaches for modelling moisture effects. For this we modified the model
M2-dif (Table 1), removing diffusion fluxes and adding empirical moisture
functions. This consisted in removing all diffusion effects (so that

Equations for steady state were derived by setting the rate of change in the state variables to zero in Eqs. (1)–(5) (where the flux terms are replaced by their respective equations) and then solving for the state variables. This was performed in Python using the “SymPy” package (Meurer et al., 2017).

A sensitivity analysis was carried out on model parameters using the
“sensFun” function from the R package FME, which perturbs each parameter
individually by a small amount. We ran the model as above, i.e. simulating
the incubation and using daily output. Daily sensitivities were then averaged
to obtain an overall value. Sensitivity values were calculated for the

For model validation, we used soil respiration data from the study by Rey et
al. (2005), where a Mediterranean oak forest soil was incubated for 1 month
in a full factorial design at 100 %, 80 %, 60 %, 40 %, and
20 % of water holding capacity and at 30, 20, 10, and 4

The calibrated values for all models are shown in Table S3. Using different
reaction kinetics resulted in variations in model performance as measured by
RMSE (Table 1). Changes in RMSE were more sensitive to the kinetics of
decomposition (

Models M1-dif, M2-dif, and M

Model vs. measured accumulated

Replacing diffusion effects with empirical moisture scalars followed by
recalibration decreased model performance compared to a diffusion-based
model, both when using relative water saturation (M2-sat) and when using water potential
(M2-wp) functions (Table 1). Although empirical functions were able to
approximate the shape of the respiration–moisture relationship at
20

Relationship of soil respiration with volumetric soil moisture.
Results shown over three temperature levels (5, 20, 35

Figures 5 and 6 show the apparent temperature sensitivities fitted to
observations and modelled fluxes at different moisture levels and for two
temperature ranges, 5–20 and 20–35

Temperature sensitivities of respiration and decomposition fluxes,
showing activation energy (

Model steady-state equations are provided in the Supplement. For
20

Equivalent to Fig. 5 but showing observational data (R-obs) next to models with different moisture functions (R-M2-dif, R-M2-wp, and R-M2-sat).

Table 2 shows the averaged values from the sensitivity analysis done on the
model

Parameters of the model M2-diff, calibrated and non-calibrated, with
results of a sensitivity analysis (sens). Sens shows a relative measure of
the sensitivity of the model

Simulation of the incubated soil from the study of Rey et al. (2005) resulted
in a very high fit to the validation data after calibration of initial SOC
fractions and

The interaction often observed in the effects of temperature and moisture on the cycling of soil C is an indicator of the complex nature of soil systems. Such responses are often ignored, particularly by modellers trying to minimize model complexity and derive functions that are easy to parameterize, but also by experimentalists focusing on finding an invariable response to a single factor. But a careful consideration of the nature of soils suggests that interactions should be expected, something that becomes evident in multi-factorial experiments as well as in field measurements. Here we found clear interactive effects in our experimental observations, adding to the evidence that fixed empirical temperature and moisture scalars, as used in conventional soil C models, are inappropriate for simulating the variability often found in natural conditions.

Model vs. measured accumulated

Relationship of soil respiration with volumetric soil moisture shown for the model M2-dif and observations from the validation data (obs). Results are shown over four temperature levels.

Since the total amount of soil C was equal among samples and its relative
change in the 6 months of incubation was small, we expected that
second-order kinetics would do as well as Michaelis–Menten kinetics. But using
Michaelis–Menten increased the

Apparent activation energy (

The relative importance of different processes was also shown by the model parameter sensitivity values. It is perhaps not surprising that some of the highest values were related to diffusion and temperature, since these were the two factors that varied in our experiment. However, these factors also vary considerably in natural ecosystems and largely drive changes in decomposition rates. No strong correlations between the effects of different parameters were found, with most being below 0.6 (Fig. S4), thus giving a degree of confidence in the estimated values. While we did not obtain statistical confidence intervals, kernel density estimations (Figs. S5–S12) suggest differing degrees of likelihood for different parameters. Activation energies in particular showed narrow ranges of optimal values with a strong dependence on model structure.

Since optimizing all parameters against our data resulted in an

Model steady state or equilibrium is attained when the rate of change of all
state variables equals zero, reflecting the state towards which the system
will tend under invariant input and forcing conditions. Even though external
drivers are in constant change in natural systems, steady-state information
can indicate the approximate model behaviour under specific average
conditions. Results here showed that the model M2-dif gives realistic values in
the range of temperature for which it was calibrated but leads to
unrealistic values under colder conditions. In addition, the

It is important to point out that our approach was to use a simple model with
few processes and C pools and modify only those components we tested. This
allowed us to distinguish the effects of each modification and minimize
parameter identifiability problems arising from having too many parameters
with effects that may correlate. While this allowed us to focus on specific
processes, it also meant that important mechanisms were left out. Some of
these mechanisms are oxygen limitations in saturated conditions, leaching of

Unlike other calibrated parameters, the activation energy values for
microbial (

The difference between prescribed and observed temperature sensitivities may
be related to two factors. First, the apparent sensitivities do not represent
the instantaneous sensitivities dictated by the prescribed values but reflect
also the effects of other limiting factors that change with time. Pool sizes,
including

The model results described above are thus emergent effects leading to
apparent temperature sensitivities that vary in time but are based on
constant model parameter (

Decomposition, which was only modelled, consistently showed a lower apparent
temperature sensitivity than respiration, with a

Diffusion fluxes are a function of water content, diffusivity coefficients,
and pool concentrations. Different equations have been used to calculate
diffusion as a function of water content in soils (Hamamoto et al., 2010; Hu
and Wang, 2003). All these equations generally predict a strong positive
near-exponential effect of water content on diffusion. Following previous
studies (Manzoni et al., 2016), we chose the function from Hamamoto et
al. (2010). This equation allows for an adjustment of the percolation
threshold (

Diffusion regulations can be implemented either by simulating two separate
pools between which diffusion takes place or by determining the available
amount of a pool as a function of diffusivity (or conductance in our case) at
each time step. In our model we used a combination, simulating a diffusion
flux between enzyme pools and calculating how much

In our study especially, but also in the validation data, the moisture response tended to become less linear and have a larger plateau at higher temperatures. The mechanisms leading to such interactions are still unclear, but our model comparison indicates that solute diffusion limitations play a central role. The plateau behaviour, a decrease near saturation, and even near-linear responses all contrast with the near-exponential relationship between moisture and conductance given by Eq. (11) and with the fact that no oxygen limitations at high moisture levels were modelled. They may, however, result from a faster depletion of available carbon at high moisture levels and at high temperatures, driving down the accumulated fluxes over time.

While a low supply of

In models where decomposition and respiration are separated processes, these
fluxes can show different responses. This decoupling is especially evident
when diffusion limitations come into play. Plots of modelled fluxes against
temperature and moisture (Fig. S3) showed a different relationship when
comparing respiration and decomposition. Figure 10 shows modelled
decomposition against respiration (using M2-dif) as accumulated values, each
line being a sample at a different water content. Without any diffusion
limitation, the relationship follows a slope of ca. 0.3, determined by

Modelled decomposed vs. respired C shown as accumulated values over the entire simulated incubation, including temperature steps. Each line is a sample at a different moisture content level.

As the main mechanism linking water content with the movement of substrates, microbes, and enzymes, diffusion plays a central role in soil organic matter decomposition. We here showed that integrating it into models can significantly improve our understanding of soil C dynamics. Diffusion-based models were better at simulating the effects of moisture and improved the simulated temperature responses, thus allowing for a better interpretation of the observed temperature sensitivities. This and similar studies indicate that measured temperature sensitivities cannot be generalized or correctly interpreted without having a full understanding of the relevant mechanisms, their interactions, and the state of soil carbon and microbial pools.

We also found evidence that Michaelis–Menten kinetics plays an important
role in soil C dynamics, explaining the strong differences in temperature
sensitivities across temperature ranges. Our results are consistent with
relatively high activation energies for both the

Creating models that capture the variability in the response of C dynamics across different soils and at different levels of driving factors remains challenging. However, process-based models are of central importance for establishing confidence in C cycle predictions and soil–climate feedbacks. As seen here, the structure and process representation of models can be critical for simulating the complex response of soil C fluxes to combined changes in temperature and moisture. Diffusion as a moisture regulation of soil C fluxes has not been used in large scale predictions, which still rely on empirical scaling functions. Evidence of interactions seen in experiments and presented from a mechanistic model perspective indicates that these simpler approaches do not always hold. Further research should focus on more extensive validation and finding the relationships necessary for extending the application of models to diverse soil types.

All code and data used for this analysis are available
at

The supplement related to this article is available online at:

FEM developed the model, analysed the data, and wrote the paper. NV carried out the lab experiments and revised the paper. LM contributed to model optimization, discussions, and revisions.

The authors declare that they have no conflict of interest.

This study was partially supported by funds from the R2DS projects “Le carbone stable des sols: processus de stabilisation et vulnérabilité” and “Vulnérabilité du C stable et labile du sol aux changements climatiques: mise en évidence, facteurs explicatifs et intégration dans un modèle de biosphère spatialisée”.This open-access publication was funded by the University of Göttingen. Edited by: Andreas Ibrom Reviewed by: Thomas Wutzler and one anonymous referee