The theoretical framework is illustrated in Fig. 1. We start from the premise that heterotrophic respiration follows changes in soil moisture during drying (Rd) and that respiration pulses occur immediately following rewetting. As such, respiration pulses depend on both the soil moisture at the end of the dry period and the soil moisture increase caused by rainfall (Rr). The stochasticity of rainfall timing and amount determines a range of possible durations of dry spells and soil moisture increments when rainfall occurs. As a result, respiration can be regarded as a stochastic process. To statistically characterize the two types of respiration, the statistical properties of both soil moisture and soil moisture changes at rewetting are needed. These statistical properties are included in the probability density function (PDF) of soil moisture and the joint PDF of soil moisture and its increase at rewetting. Both distributions are derived in Sect. 2.1.1. The PDF of respiration rates during drying and respiration pulses at rewetting are derived in Sect. 2.1.2 and 2.1.3, respectively. All symbols are defined in Table 1.

Laboratory incubation data were used to parameterize the functions linking heterotrophic respiration to soil moisture. As expected, the respiration pulses at rewetting depend on both rewetting intensity (y) and pre-wetting soil moisture (xd), and this relation is well-characterized by Eq. (13) (Fig. 2). In Fig. 2, respiration pulses at rewetting are normalized by the amount of organic C in each soil to facilitate comparisons. However, after accounting for variations in organic C content, bulk density, and soil layer depth, the values of Rr,max and Rd,max per unit ground area are higher in the organic soils than in mineral soils (Table 2) and so is the ratio between Rr,max and Rd,max. The sensitivity parameter b shows milder variation across soils than the other parameters, with an average value b≈0.1. Based on this data analysis, in the following theoretical exploration we set parameter values intermediate between the extremes reported in Table 2 (i.e., Rr,max=5 gC m-2, Rd,max=1 gC m-2 d-1, and b=0.1). In addition, we explore how the contribution of respiration pulses varies between mineral vs. organic soils, using the average parameter values reported in Table 2.

Figure 3 shows two examples of the simulated trajectories of soil moisture and heterotrophic respiration, for contrasting climatic conditions (more frequent precipitation in the left panels than in the right panels). It is important to note that in this comparison across climatic conditions (and in the comparisons that follow), the maximum respiration Rr,max and Rd,max are fixed, while in reality they are likely proportional to soil organic C availability, which in turn is the result of a long-term and soil-moisture-dependent balance between C inputs from vegetation and respiration (this limitation is discussed in Sect. 4.2). Respiration rates during dry periods follow soil moisture changes, declining as soil dries and returning to higher levels at rewetting (Fig. 3b, f). In addition to this rewetting-induced restoration of high respiration rates, rewetting causes CO2 emission pulses, represented by vertical bars. Under the wetter climate (Fig. 3b), respiration pulses are more frequent than under the dry climate (Fig. 3f) because of the higher precipitation frequency. However, most of the respiration pulses are small because soil moisture increments at rewetting are often limited by the available soil pore space, and a relatively large fraction of precipitation is lost to runoff and deep percolation. In contrast, under dryer conditions, changes in soil moisture are large because on average soil moisture is low and the pore space is rarely filled up completely. As a result, the fewer respiration pulses can be larger under dry than under wet conditions.

The bottom panels in Fig. 3 show the PDF of respiration for the same two climatic conditions analyzed in the upper panels. While the PDF of Rr is positively skewed regardless of climate (but with heavier tails under dry conditions, Fig. 3c, g), the PDF of Rd is strongly affected by climatic conditions – the probability of high values for Rd is higher under wet conditions (negatively skewed PDF) and lower under dry conditions (positively skewed PDF, Fig. 3d, h). This pattern is caused by the prevalence of high soil moisture values in the wet climate scenario, which maintain relatively high Rd. Figure 3c, d, g, h also show that the theoretical PDF (Eqs. 11 and 18) matches perfectly to the distribution of the numerically simulated data. The shape of the theoretical PDF of Rd in Fig. 3h might seem incorrect, as it increases sharply at high respiration values. This increase is due to the flat derivative of the Rd–soil moisture relation (Eq. 7), which causes an asymptote in the PDF at Rd=Rd,max (Eq. 11). However, the area under this spike is vanishingly small when climatic conditions are dry as in the example of Fig. 3e–h, so that it is highly unlikely to have any respiration value around Rd,max.

Field data were used to test whether the hydrologic and soil respiration models could capture trends in the mean evapotranspiration and heterotrophic respiration along a precipitation gradient (Fig. 4). The trend of the mean evapotranspiration rate with increasing mean rainfall depth was captured reasonably well (Fig. 4a), considering that no formal calibration was conducted, and all parameters were estimated based on independent information. Similarly, the model correctly predicts the trend in soil moisture (not shown), but with an overestimation bias around 0.05–0.1 (in terms of normalized soil moisture x). This overestimation is expected, because soil moisture had been measured in the drier top 0.1 m of soil, while the model considers average soil moisture over a 0.2 m depth. Also the trend in total heterotrophic respiration is predicted correctly by the full model, which explains 77 % of the variance in the respiration data (black curve in Fig. 4b). Calibrating the two parameters of Eq. (13) and Rd,max would allow a better fit, but since the goal here is to provide a qualitative model validation and not a quantitative performance assessment, we deem the model suitable for the following theoretical analyses.

We also tested the simpler version of the model, in which respiration pulses only depend on the soil moisture increment. Without the effect of pre-wetting soil moisture, this version predicts higher mean respiration than the full model (red lines in Fig. 4b) and a higher contribution of rewetting respiration to the total heterotrophic respiration (red lines in Fig. 4c).

Figure 5 shows the predicted effect of precipitation regimes on heterotrophic respiration during drying and at rewetting (Fig. 5a, b) on the total heterotrophic respiration rate (Fig. 5c) and on the fraction of respiration contributed by rewetting pulses (Fig. 5d). As in Fig. 3, Rr,max and Rd,max are fixed to focus on the role of climatic conditions, so the patterns shown in Fig. 5 should be interpreted as changes of mean respiration rates along gradients of precipitation frequency (λ) and mean depth (α) for given soil organic C stocks. Because in this minimal model the mean precipitation rate is given by 〈P〉=αλ, precipitation can be increased by assuming more frequent rain events (i.e., increasing λ), larger events (i.e., increasing α), or both. Any of these changes increase mean respiration during drying and at rewetting (Fig. 5a, b). As 〈Rd〉 increases with precipitation more than 〈Rr*〉, the relative contribution of respiration pulses to the total respiration rate, 〈Rr*〉/〈Rt〉, tends to decrease from drier to wetter conditions, especially when rain events become more frequent (as opposed to more intense) (Fig. 5d). This pattern is caused by the relatively larger respiration pulses occurring when soils are dry and rewetting causes large soil moisture increments (compare examples in Fig. 3b and f). Moreover, the relative change of 〈Rr*〉/〈Rt〉 is smaller than the change in 〈Rd〉 or 〈Rr*〉 as precipitation regimes are varied.

Not only the mean respiration rates, but also the variability of both respiration rates during drying and respiration pulses at rewetting vary with hydroclimatic conditions (Fig. 6). The standard deviation of Rd exhibits maxima at intermediate α when λ is fixed and at intermediate λ when α is fixed (Fig. 6a). This pattern is due to a shift in the shape of the PDF of Rd when moving from dry to wet conditions. Under dry conditions, the PDF of Rd has relatively low variance and is negatively skewed (Fig. 3h); as conditions become wetter the PDF flattens and the variance increases, and finally under wet conditions the PDF transitions again to a low-variance but positively skewed PDF (Fig. 3d). In contrast, the PDF of Rr is always positively skewed, with variance decreasing with increasing rainfall frequency (Fig. 6b; compare examples in Fig. 3c and g). However, increasing α for fixed λ is predicted to increase the variance of Rr. The coefficients of variation (CV) of Rd and Rr vary less than the corresponding standard deviations and tend to decrease as conditions move from dry to wet (Fig. 6c, d). Specifically, the CV of Rd decreases with both increasing λ and increasing α. In contrast, the CV of Rr is nearly independent of α but decreases with increasing λ.

Results shown in Figs. 5 and 6 are based on average respiration model parameters; here, we explore how changing organic C content from mineral to organic soils affects the contribution of rewetting pulses to total soil heterotrophic respiration. We also focus on changes in respiration patterns along gradients of rainfall intensification, i.e., decreasing precipitation frequency λ while precipitation event depth α is increased and total precipitation is kept fixed (as along the white contour curves in Figs. 5 and 6). Figure 7 shows that rainfall intensification decreases 〈Rt〉 (Fig. 7a) but increases 〈Rr*〉/〈Rt〉 (Fig. 7b), regardless of soil organic C availability (black vs. gray curves) and total precipitation (dashed vs. solid curves). However, for a given total precipitation, organic soils (gray curves) exhibit both higher 〈Rt〉 and higher 〈Rr*〉/〈Rt〉 than mineral soils (black curves), due to their higher Rr,max (Table 2). As a result, in organic soils, the contribution of respiration pulses can be as high as 20 % of the total heterotrophic respiration, whereas in mineral soils it tends to be lower than 10 %. Moreover, in both soils, higher precipitation increases 〈Rt〉 while decreasing 〈Rr*〉/〈Rt〉 (compare solid vs. dashed curves).

Most biogeochemical models assume that heterotrophic respiration (and other processes) depend on a generic soil property φ following an empirical function fφ (Bauer et al., 2008; Moyano et al., 2013). As φ changes through time (e.g., soil moisture and temperature), the biogeochemical rate associated with φ also varies. Thus, the biogeochemical models use the function fφ to convert measured time series of soil moisture and other environmental variables into biogeochemical rates. The different approach we follow here consists in linking a known probability density of φ to the probability density of the function fφ to capture the propagation of the statistical properties of φ to fφ. This can be done by the derived distribution approach, as in Eq. (8). This approach has been used to investigate gaseous nitrogen emissions in response to soil moisture fluctuations (Ridolfi et al., 2003), but the only example studying soil heterotrophic respiration rate we are aware of focused on respiration responses to temperature fluctuations (Sierra et al., 2011). These approaches provide simple and mathematically elegant solutions but have so far been limited to the effect of a single driver of the biogeochemical flux of interest. The responses of heterotrophic respiration to changes in soil moisture are more complex because rewetting pulses depend on both soil moisture increment and pre-wetting soil moisture (Fig. 2), requiring the solution of a bivariate stochastic process. Thus, our approach – by accounting for both these effects – is more general and applicable along gradients where the statistical properties describing the precipitation regime vary significantly (Fig. 4).

A previous stochastic approach focused on the CO2 concentration in the pore space instead of respiration rates (Daly et al., 2008). Observations show that CO2 concentration increases rapidly after rainfall and then decreases following a negative exponential function. This dynamic can be described as a stochastic process where CO2 concentration is the random variable and precipitation represents the stochastic forcing (Daly et al., 2008). With this approach, the long-term mean CO2 concentration was found to depend on the average rainfall rate (λα), while the standard deviation of CO2 concentration depends on λα2. This indicates that rainfall intensity (in terms of mean event depth α) plays a more important role than rainfall frequency in driving the variability of soil CO2 concentration. Soil respiration was shown to be approximately proportional to CO2 concentration in the pore space over a broad range of concentrations (Daly et al., 2008), so that respiration statistics are also expected to scale with rainfall statistics in the same way as soil CO2 concentrations. This result is consistent with our finding that all components of heterotrophic respiration increase with both λ and α (Fig. 5).

Numerical process-based models have also been driven by randomly generated rainfall time series (e.g., Tang et al., 2019). These models do not allow analytical solutions for the respiration statistical properties to be found, but they offer insights into the individual processes affecting these properties. For scenarios of constant total rainfall and variable rain event frequency, Tang et al. (2019) found that rainfall intensification increased heterotrophic respiration in a semi-arid grassland, even though in their simulations soil organic C stocks also slightly increased due to higher plant productivity. This result differs from our finding that total heterotrophic respiration decreases with rainfall intensification (moving right to left along the curves in Fig. 7a) and was likely caused by how plant productivity and its feedback to soil organic C were modeled in their study.

Three model assumptions can alter the interpretation of our results: (i) that heterotrophic respiration pulses can be regarded as instantaneous, (ii) that the two parameters Rd,max and Rr,max are independent of climatic and vegetation conditions, and (iii) that hydroclimatic conditions are statistically stationary.

Respiration pulses are modeled as instantaneous events of CO2 emission with a given size (Sect. 2.1.3). While mathematically convenient, rewetting respiration pulses are known to last for a few days after the rewetting has ended. Indeed, when analyzing laboratory incubation data, the pulse size is generally calculated by integrating through time the respiration rates above the rate occurring at stable soil moisture. The integration window ranges between 2 and 3 d (e.g., Fischer, 2009). This simplified approach to separate the actual rewetting pulse from the respiration rate at stable soil moisture requires some caution when rainfall events are frequent. In that case, pulses would overlap rather than being distinct. Moreover, with frequent rainfall, respiration could be inhibited due to water logging (Moyano et al., 2013; Rubio and Detto, 2017), and no respiration pulse might occur. Thus, to avoid these issues, our equations should not be used in wet environments with λ>0.3 d-1.

We calculated the statistical properties of the heterotrophic respiration rate, but we did not consider the dynamics of the soil organic matter and plants that supply resources for microbial growth and respiration. Widely different precipitation amounts and distributions such as those depicted in Figs. 5 and 6 are associated with different plant communities, whose productivity increases along gradients of precipitation (Huxman et al., 2004; Luyssaert et al., 2007), providing litter and root exudates whose C is eventually stabilized into soil organic matter. Indeed, soil organic C stocks increase with increasing mean annual precipitation (Guo et al., 2006). Hence, soil organic matter probably varies along the axes of Figs. 5 and 6, which are instead interpreted here as purely climatic gradients. Such variations in organic matter content would affect the maximum respiration rate and pulse size, Rd,max and Rr,max (e.g., compare mineral and organic soils in Table 2). Because the mean respiration rates scale with the maximum rates (as apparent analytically from Eq. 21), it is reasonable to expect that higher organic matter content along precipitation gradients increases the sensitivity of respiration to changes in precipitation compared to predictions in Fig. 5. Indeed, even when keeping precipitation constant while varying the frequency and depth of precipitation events, the variations in total heterotrophic respiration are larger in organic soils than in mineral soils (Fig. 7a).

Moreover, soil C substrates might be depleted through multiple drying and rewetting events – a behavior we do not consider in the proposed statistically stationary model. While some experiments show sustained rewetting pulses (Miller et al., 2005; Xiang et al., 2008), others show reduced total heterotrophic respiration with increasing frequency of drying and rewetting, possibly due to substrate depletion (Shi and Marschner, 2014). To capture these dynamics, a more complex model describing the changes in substrate and microbial compartments would be needed (e.g., Brangarí et al., 2018; Lawrence et al., 2009; Tang et al., 2019) at the cost of losing the analytical tractability.

Our focus in this contribution is on heterotrophic respiration, but the data we used to parameterize the model are from laboratory studies without plants. Therefore, our heterotrophic respiration estimates neglect contributions from fresh C inputs from roots to the rhizosphere (Finzi et al., 2015; Kuzyakov and Gavrichkova, 2010). However, the timing of rhizodeposition depends on plant activity, which in turn depends on previous environmental conditions – differently from soil microbes that respond to soil moisture changes rapidly, plant responses integrate previous conditions, thereby partly decoupling root activity from current soil moisture. It is thus nontrivial to include rhizosphere processes in the current framework.

In addition to these limitations, our results should also be interpreted with caution when rainfall seasonality is important, because the assumption of stochastic stationarity (Sect. 2.1.1) may not be met, requiring the derivation of a different probability density function of soil moisture (e.g., Vico et al., 2017). Nevertheless, our results will still hold for parts of the year when the rainfall regime is relatively stable.

The axes of Figs. 5 and 6 can be interpreted in terms of changes in precipitation patterns caused by ongoing climatic changes. If rainfall in a semiarid or mesic environment increases (due to either more frequent or larger events), heterotrophic respiration also increases (Yan et al., 2014; Zhang et al., 2019) – this is not surprising as soils become on average wetter, removing water limitation and promoting microbial activity. These observations are consistent with our findings that the mean respiration pulse at rewetting and respiration during drying increases with increasing α, λ, or their product – i.e., total precipitation. However, the variability in respiration does not always change monotonically with increasing rainfall. Figure 6b shows that the standard deviation of the respiration pulses increases with more intense (higher α) and less frequent (lower λ) rainfall. In contrast, the standard deviation of the respiration rate during drying, Rd, peaks at intermediate α and λ and declines thereafter because the respiration response is flat and thus has higher variance at intermediate wetness (Eq. 7; Fig. 6a). Therefore, higher precipitation as driven by increasing α or λ is expected to increase the respiration pulses (Fig. 5b) and their variability (Fig. 6b), while decreasing their contribution to the total heterotrophic respiration (Fig. 5d).

It is perhaps more interesting to understand respiration responses to changes in rainfall patterns for given total rainfall amounts. When α and λ are changed simultaneously while keeping their product fixed (moving along the white curves in Figs. 5–6; or along the x axis in Fig. 7), the mean respiration pulse at rewetting and the standard deviations of both respiration components increase with more intermittent and intense rainfall events. In experimental rainfall manipulations that mimic the predicted climatic changes, increased variability in soil moisture associated with more intense but less frequent precipitation events decreases total soil respiration (Harper et al., 2005). This observation is consistent with our result that the mean total heterotrophic respiration decreases with rainfall intensification while maintaining a given mean precipitation rate (i.e., moving right to left along the curves in Fig. 7a). Our result is explained by the higher runoff and deep percolation losses predicted by the soil hydrologic model when precipitation events are large but rare (Rodriguez-Iturbe and Porporato, 2004). These water losses cause soil moisture to be on average lower as the precipitation regime becomes more intermittent – a pattern also confirmed empirically in rainfall manipulation experiments (Harper et al., 2005). Our approach neglects the lower plant C inputs and contributions to total soil respiration under a more intermittent precipitation regime (Harper et al., 2005), which further reduces the total (combined autotrophic and heterotrophic) soil respiration rate.

We also found that the contribution of rewetting pulses to the total heterotrophic respiration increases when rainfall becomes more intermittent and rainfall events larger (i.e., moving right to left along the curves in Fig. 7b). This result is consistent with observations in a temperate steppe (Yan et al., 2014). The rewetting pulse contribution is also larger in organic soils compared to mineral soils (gray vs. black curves in Fig. 7) – this effect is expected, because more C can be mobilized by drying and rewetting cycles in C-rich soils (Canarini et al., 2017). We can thus surmise that climatic changes causing longer dry period and more intense rainfall events (IPCC, 2012) will increase the role of pulse responses, including not only respiration but also nitrogen mineralization pulses that could release nitrogen at a time when plant uptake is low. In turn, this can cause a decoupling of nitrogen supply and demand, with possible negative consequences for ecosystem productivity (Augustine and McNaughton, 2004; Dijkstra et al., 2012).

Our findings are based on time-invariant relations between heterotrophic respiration and soil moisture, but temperature and other environmental conditions also affect microbial activity – in part directly and in part indirectly via rhizodeposition – raising the question of how our results could be impacted by other respiration-controlling factors. As a first approximation, temperature could be assumed to alter directly both respiration rates during drying and respiration pulses in a similar way. This implies that our results would hold even under fluctuating temperatures, at least during the growing season, when temperature variations are limited and precipitation can be described by a simple marked Poisson process (Sect. 2.1.1). However, a different modeling approach would be needed to quantify the mean heterotrophic respiration rate during seasons with frequent rainfall events, when respiration pulses are likely to be less important and anaerobic conditions (here neglected) could play a role. As the timescale expands from the growing season to the whole year, seasonal fluctuations in plant activity that delay the supply of C substrates to microbes will also play a role (Finzi et al., 2015; Kuzyakov and Gavrichkova, 2010), leading to a hierarchy of responses at multiple timescales – a more complex problem than the one addressed in this contribution.