Rainfall intensification increases the contribution of rewetting pulses to soil respiration

Soil drying and wetting cycles promote carbon (C) release through large heterotrophic respiration pulses at rewetting, known as ‘Birch’ effect. Empirical evidence shows that drier conditions before rewetting and larger changes in soil moisture at rewetting cause larger respiration pulses. Because soil moisture varies in response to rainfall, also these respiration pulses depend on the random timing and intensity of precipitation. In addition to rewetting pulses, heterotrophic respiration 15 continues during soil drying, eventually ceasing when soils are too dry to sustain microbial activity. The importance of respiration pulses in contributing to the overall soil respiration flux has been demonstrated empirically, but no theoretical investigation has so far evaluated how the relative contribution of these pulses may change along climatic gradients or as precipitation regimes shift in a given location. To fill this gap, we start by assuming that rewetting pulses and respiration rates during soil drying can be treated as random variables dependent on soil moisture fluctuations, and develop a stochastic model 20 for soil heterotrophic respiration rates that analytically links the statistical properties of respiration to those of precipitation. Model results show that both the mean rewetting pulse respiration and the mean respiration during drying increase with increasing mean precipitation. However, the contribution of respiration pulses to the total heterotrophic respiration increases with decreasing precipitation frequency and to a lesser degree with decreasing precipitation depth, leading to an overall higher contribution of respiration pulses under future more intermittent and intense precipitation. Moreover, the variability of both 25 components of soil respiration is also predicted to increase under these conditions. Therefore, our results suggest that with future more intermittent precipitation, respiration pulses and the associated nutrient release will intensify and become more variable, contributing more to soil biogeochemical cycling.


Introduction
Respiration pulses often occur after dry soils are wetted by rainfall or irrigation (Borken and Matzner, 2009;Daly et al., 2008; where Γ[•] and Γ [•,•] are the complete and incomplete gamma functions (defined in Table 1). The PDF of soil moisture is the basis to obtain the PDF of respiration during soil drying (Section 2.1.2).

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The last distribution needed to calculate the statistical properties of soil respiration pulses (Section 2.1.3) is the joint PDF of soil moisture at the end of a dry period and soil moisture increase due to precipitation events, denoted by , ( , ) (note that both y and xd are stochastic variables in this joint PDF). Thanks to the properties of the Poisson process, the PDF of soil moisture at the end of the dry period is equal to the PDF of soil moisture at a generic time (Cox and Miller, 2001); i.e., ( ) = ( ). Because precipitation does not depend on antecedent soil moisture conditions in this model, the PDF of soil 115 moisture at the end of a dry period is independent of the PDF of the subsequent precipitation event and soil moisture increase.

Respiration during soil drying
During a dry period, the heterotrophic respiration rate decreases in response to the gradual decrease in soil moisture, following a concave-downward trend (Manzoni et al., 2012;Moyano et al., 2012). Consistent with the hydrologic model setup, we 120 assume that the soil drains rapidly and hence does not remain under saturated conditions long enough to develop anoxic conditions. It is thus reasonable to assume that respiration declines between the soil field capacity (equivalent to s1 in this model) and a lower soil moisture threshold for microbial activity. This lower threshold corresponds to water potential levels around -15 MPa in sieved soil samples (Manzoni and Katul, 2014), but here we assume that respiration becomes much smaller than rates under well-watered conditions already at the plant wilting point sw; i.e., at a water potential of -1.5 MPa. This where Rd and Rd,max respectively denote the respiration rate during drying and the maximum respiration rate in the absence of rapid rewetting (i.e., Rd at x=1 or s=s1). Using other monotonic and concave-downward relations between respiration and soil moisture would not qualitatively alter the results.
In Eq. (1), soil moisture is a random variable, whose PDF follows Eq. (4). Therefore, Rd from Eq. (7) is also a random variable, which can be obtained from the PDF of soil moisture using the derived distribution approach (Kottegoda and Rosso, 1998), where on the right-hand side soil moisture is expressed as a function of Rd by inverting Eq. (7), In turn, Eq. (9) allows calculating the slope of the ( ) relation, which is also needed in Eq. (8), The PDF of Rd is thus obtained from Eq. (8)-(10) as https://doi.org/10.5194/bg-2020-95 Preprint. Discussion started: 6 April 2020 c Author(s) 2020. CC BY 4.0 License.
where the normalized respiration = , is introduced to simplify the notation. This PDF can now be used to calculate analytically the long-term mean of Rd, denoted by 〈 〉, where for convenience the parameter group = Γ − Γ , is defined. The standard deviation of Rd, denoted by , can not be obtained analytically, but it can be calculated through numerical integration of Eq. (11).

Respiration pulses at rewetting
Respiration pulses at rewetting are caused by mineralization of available C and microbial products at the end of the dry period, which in turn depend on how intense the rewetting event was. As a result of these processes, in a given soil, rewetting events 145 depend on both soil moisture before the rewetting xd, and the change in soil moisture y (Birch, 1958;Lado-Monserrat et al., 2014). This relation can be captured by the empirical function (justified and parameterized in Section 2.2.1), where Rr,max is the largest respiration pulse possible (achieved when y=1 and xd=0), and b is a parameter weighing the effect of antecedent soil moisture conditions. The last term in Eq. (13) is a Heaviside function limiting the relation between Rr and y to conditions in which soil moisture at most fills the available pore space (as in Eq. (3), [•] is equal to one only when < 150 1 − ). If before the rain event soil moisture is at the plant wilting point (xd=0) and the precipitation event is sufficient to reach s1 (i.e., = − = 1), the maximum respiration pulse is attained and Rr=Rr,max. Here, Rr represents an amount of C respired when the rewetting event occurs, so its dimensions differ from those of the respiration rate during drying, Rd; these two quantities are combined in the total respiration rate in Section 2.1.5.
Because both y and xd are random variables that follow the PDF of Eq. (6), also Rr should be regarded as a random variable 155 following its own PDF. Different from the PDF of Rd, which was obtained from the univariate PDF of soil moisture, the PDF of Rr has to be derived from the joint PDF of y and xd. The derived distribution approach can still be used, but it requires the determinant of the Jacobian matrix of the transformation from y and xd to Rr (Kottegoda and Rosso, 1998). To proceed, it is first convenient to introduce an auxiliary variable X=xd, which is used together with Eq. (13) to find the transformation from the original variables y and xd to Rr and X, where the inequality limits the soil-moisture increments as the Heaviside function in Eq. (13). Second, the system on the left of Eq. (14) is inverted to express the original variables as a function of the transformed variables (reported on the right of Eq. (14)), similar to the inversion done in Eq. (9). Third, we calculate the Jacobian matrix, and the determinant of the Jacobian, where as in Section 2.1.2 all the terms on the right-hand side only depend on X and Rr, and , is given by Eq. (6). Finally, to obtain the (marginal) PDF of Rr, the joint PDF in Eq. (17) is integrated over all possible values of X, where on the right hand side the normalized respiration pulse = , is introduced to simplify the notation, and as before . Due to the complexity of Eq. (18), the long-term mean and standard deviation of Rr, respectively denoted by 〈 〉 and , need to be obtained via numerical integration of the PDF of Rr.
where ( | ) is from Eq. (3). The ( ) is then obtained as, Thanks to the simplicity of Eq. (20), in this particular case the long-term mean and standard deviation of the respiration pulses are found analytically, Thus, when respiration pulses are simply proportional to the soil moisture change at rewetting, their mean only depends on the maximum pulse size Rr,max and the ratio of soil water storage capacity and mean precipitation depth (i.e., the parameter group 180 = ( ) ).

Combining respiration during soil drying and at rewetting
The total mean heterotrophic respiration rate is given by the sum of the mean respiration rate during soil drying 〈 〉 (Eq. (12); expressed in gC m -2 d -1 ) and the mean rate of respiration resulting from the sequence of rewetting pulses over the study period (denoted by 〈 * 〉 and also expressed in gC m -2 d -1 ). The 〈 * 〉 is calculated as the mean amount of respired carbon (〈 〉 from 185 Eq. (18), expressed in gC m -2 ) divided by the mean rainfall inter-arrival time, 1⁄ , The mean total heterotrophic respiration rate is then obtained as, https://doi.org/10.5194/bg-2020-95 Preprint. Discussion started: 6 April 2020 c Author(s) 2020. CC BY 4.0 License.
In what follows, the ratio of respiration pulse to total respiration (i.e., 〈 * 〉 〈 〉 ⁄ ) will also be considered, to evaluate the overall contribution of respiration pulses.

Data analysis
190 2.2.1. Laboratory incubation data for model calibration The phenomenological respiration models in Eq. (7) and (13)  Because respiration amounts and rates in these laboratory incubations were expressed respectively in µg g -1 and µg g -1 d -1 (or on a per unit soil organic C basis), units were converted to g m -2 and g m -2 d -1 using bulk densities and sampling depths reported moisture (not shown), but with an overestimation bias around 0.05-0.1 (in terms of normalized soil moisture x). This 265 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 Figure 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.

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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 Figure 4b), and higher contribution of rewetting respiration to the total heterotrophic respiration (red lines in Figure 4c).

3.4.
Dependence of respiration on rainfall statistical properties Figure 5 shows the predicted effect of precipitation regimes on heterotrophic respiration during drying and at rewetting ( Figure   275 5a, b), on the total respiration rate (Figure 5c), and on the fraction of total heterotrophic respiration contributed by rewetting pulses (Figure 5d). As in Figure 3, Rr,max and Rd,max are fixed to focus on the role of climatic conditions, so the patterns shown in Figure 5a-c should be interpreted as changes of mean respiration rates along gradients of precipitation frequency ( ) and mean depth ( ) for given soil organic C stocks. In contrast, results in Figure 5d can be generalized to soils with contrasting C stocks, because the relative contribution of respiration pulses to the total respiration is less dependent on the organic C 280 availability than the mean respiration rates.
Because in this minimal model the mean precipitation rate is given by 〈 〉 = , precipitation can be increased by assuming more frequent rain events (i.e., increasing ), deeper events (i.e., increasing ), or both. Any of these changes increase mean respiration during drying and at rewetting (Figure 5a, b). As 〈 〉 and 〈 * 〉 increase with precipitation, the relative contribution of respiration pulses to the total respiration rate, 〈 * 〉 〈 〉 ⁄ , tends to decrease from drier to wetter conditions, especially when 285 rain events become more frequent (as opposed to more intense) (Figure 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 Figure 3b and 3f). Moreover, the relative change of 〈 * 〉 〈 〉 ⁄ is smaller than the change in 〈 〉 or 〈 * 〉 as precipitation regimes are varied.
Not only the mean respiration rates vary with hydro-climatic conditions, but also the variability of both respiration rates during 290 drying and respiration pulses at rewetting ( Figure 6). The standard deviation of Rd exhibits maxima at intermediate when is fixed, and at intermediate when is fixed (Figure 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 ( Figure 3d); 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 (Figure 3h). In contrast, the PDF of Rr is always positively 295 skewed with variance decreasing with increasing wetness (Figure 6b; compare examples in Figure 3c and g). The decrease in variance occurs both when increasing and when increasing .
The coefficients of variation (CV) of Rd and Rr-which are expected to be less affected by variations in organic C availability along climatic gradients-vary less than the corresponding standard deviations and tend to decrease as conditions move from dry to wet (Figure 6c, d). Specifically, the CV of Rd decreases with both increasing and increasing . In contrast, the CV of more intermittent while maintaining a given mean precipitation rate (i.e., moving right to left along one of the white curves in https://doi.org/10.5194/bg-2020-95 Preprint. Discussion started: 6 April 2020 c Author(s) 2020. CC BY 4.0 License. Figure 5). Our result is explained by the concavity of the relation between respiration and soil moisture during drying ( Figure   1), which causes the early phase of soil drying after a large precipitation event to have a large effect on the mean respiration.
In other words, few large rainfall events that saturate the soil play a more important role than many small events. Moving left along the white curves of Figure 5, precipitation becomes less frequent, but the average precipitation depth increases, which 385 amplifies this effect causing 〈 〉 (Figure 5b), and also 〈 〉 (Figure 5c), to increase. However, our approach neglects the lower plant C inputs and contributions to total soil respiration under this more challenging precipitation regime (Harper et al., 2005), which likely explains the observed reduction in total (combined autotrophic and heterotrophic) 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 (again moving left along the white curves in Figures 5-6). This result is consistent 390 with observations in a temperate steppe (Yan et al., 2014). 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).

Conclusions
Heterotrophic respiration depends nonlinearly on soil moisture-not only does it follow soil moisture during a dry period, but it also responds rapidly to rewetting. These rewetting responses occur in the form of pulses of CO2 whose size increases with increasing soil moisture increment and decreasing pre-wetting soil moisture. We used this relation between respiration pulses and soil moisture to characterize analytically the statistical properties of respiration rates as a function of the statistical 400 properties of the rainfall events that drive the soil moisture changes. Consistent with empirical evidence, our model predicts that dryer climatic conditions (either lower rainfall depths or longer dry periods between two rain events) lower total heterotrophic respiration. More interestingly, we showed that the contribution of rewetting pulses to the total heterotrophic respiration increases in dryer climates, but also when the precipitation regimes shift towards more intermittent and intense events (even at constant total average rainfall). Therefore, our results suggest that the expected intensification of precipitation 405 will increase the role of rewetting respiration pulses in the ecosystem C budgets.

Data availability
All data used in this study are published and available in the original publications and linked supplementary materials (see Table 2 for references on the laboratory data and Section 2.2.2 for references on the field data).

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SM and GV conceptualized the study; SM, GV, and AP developed the theory; TF provided and discussed data; AC analysed data and prepared Fig. 2; SM prepared the other figures and drafted the manuscript; all authors discussed the study ideas, and read and commented the manuscript.
the fraction of the total heterotrophic respiration rate due to rewetting pulses 〈 * 〉 〈 〉 ⁄ (d). The white contour curves indicate combinations of and that generate different annual precipitation rates (〈 〉 = =0.25, 0.5, 1, and 2 m y - Figure 6: Effect of precipitation statistical properties (mean event frequency and depth ) on the standard deviations of heterotrophic respiration rates during dry periods (a) and respiration pulses at rewetting (b), and on the coefficients of variations of respiration rates during dry periods (c) and respiration pulses at rewetting (d).