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**Biogeosciences**
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**Research article**
10 Aug 2020

**Research article** | 10 Aug 2020

Rainfall intensification increases the contribution of rewetting pulses to soil heterotrophic respiration

^{1}Department of Physical Geography, Stockholm University, 10691 Stockholm, Sweden^{2}Bolin Centre for Climate Research, Stockholm University, 10691 Stockholm, Sweden^{3}Central Analytical Laboratory, Brandenburg University of Technology, Cottbus, Germany^{4}Department of Ecology, Evolution, and Marine Biology, University of California, Santa Barbara, USA^{5}Department of Civil and environmental Engineering, Princeton University, Princeton, USA^{6}Department of Crop Production Ecology, Swedish University of Agricultural Sciences, Uppsala, Sweden

^{1}Department of Physical Geography, Stockholm University, 10691 Stockholm, Sweden^{2}Bolin Centre for Climate Research, Stockholm University, 10691 Stockholm, Sweden^{3}Central Analytical Laboratory, Brandenburg University of Technology, Cottbus, Germany^{4}Department of Ecology, Evolution, and Marine Biology, University of California, Santa Barbara, USA^{5}Department of Civil and environmental Engineering, Princeton University, Princeton, USA^{6}Department of Crop Production Ecology, Swedish University of Agricultural Sciences, Uppsala, Sweden

**Correspondence**: Stefano Manzoni (stefano.manzoni@natgeo.su.se)

**Correspondence**: Stefano Manzoni (stefano.manzoni@natgeo.su.se)

Abstract

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Soil drying and wetting cycles promote carbon (C) release through large heterotrophic respiration pulses at rewetting, known as the “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, these respiration pulses also depend on the random timing and intensity of precipitation. In addition to rewetting pulses, heterotrophic respiration 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 heterotrophic 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 heterotrophic respiration rates during soil drying and pulses at rewetting can be treated as random variables dependent on soil moisture fluctuations, and we develop a stochastic model 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. Specifically, higher rainfall intermittency at constant total rainfall can increase the contribution of respiration pulses up to ∼10 % or 20 % of the total heterotrophic respiration in mineral and organic soils, respectively. Moreover, the variability of both components of soil heterotrophic respiration is also predicted to increase under these conditions. Therefore, with future more intermittent precipitation, respiration pulses and the associated nutrient release will intensify and become more variable, contributing more to soil biogeochemical cycling.

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Manzoni, S., Chakrawal, A., Fischer, T., Schimel, J. P., Porporato, A., and Vico, G.: Rainfall intensification increases the contribution of rewetting pulses to soil heterotrophic respiration, Biogeosciences, 17, 4007–4023, https://doi.org/10.5194/bg-17-4007-2020, 2020.

1 Introduction

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Heterotrophic respiration pulses often occur after dry soils are wetted by
rainfall or irrigation (Barnard
et al., 2020; Borken and Matzner, 2009; Canarini et al., 2017; Jarvis et
al., 2007; Kim et al., 2012). The respiration rates achieved at rewetting
can be much higher than the rates maintained under permanently moist
conditions, suggesting that the rewetting itself triggers a
disproportionally high CO_{2} production. Even if they are short-lived,
these pulses can contribute a significant fraction of the annual CO_{2}
release (Kim
et al., 2012; Li et al., 2004; Yan et al., 2014). Their occurrence had been
documented as long ago as Birch (1958) – for which the phenomenon has been
named the “Birch effect” – but they remain difficult to explain and predict.

Respiration pulses are larger when the change in soil moisture is larger and
when the soil was drier before rewetting, as shown by observations under
both laboratory (Birch,
1958; Fischer, 2009; Guo et al., 2014; Lado-Monserrat et al., 2014;
Schaeffer et al., 2017; Williams and Xia, 2009) and field conditions (Cable
et al., 2008; Carbone et al., 2011; Lopez-Ballesteros et al., 2016; Rubio
and Detto, 2017; Unger et al., 2010; Yan et al., 2014). Besides CO_{2}
displacement at rewetting, several mechanisms linked to microbial processes
have been postulated to explain these patterns (Barnard
et al., 2020; Canarini et al., 2017; Kim et al., 2012; Schimel et al.,
2007). It has been argued that cell lysis due to a rapid increase in water
potential and subsequent consumption of the dead cells may cause the pulse
(Bottner, 1985). Later measurements showed that little cell lysis
occurs but that intracellular materials (osmolytes) can be released at
rewetting, contributing to the respiration pulse
(Fierer and Schimel, 2003). However, in some soils
microbial cells become dormant during drying rather than accumulating
osmolytes (Boot et al., 2013). It is thus possible
that respiration pulses are triggered by a physical process associated with
the rewetting event – possibly reestablishment of hydrologic connectivity
between substrates and microorganisms
(Manzoni et al., 2016), or physical
disruption of soil aggregates releasing old organic matter
(Homyak et al., 2018). Indeed, there is a strong
correlation between the CO_{2} production after rewetting and the amount
of extractable organic C consumed, suggesting that extractable C accumulated
during the previous dry period could fuel the respiration pulse (Canarini
et al., 2017; Guo et al., 2014; Williams and Xia, 2009). It is likely that
multiple mechanisms work in concert, shifting their relative importance
under different conditions (Slessarev and
Schimel, 2020).

The focus on the processes causing respiration pulses resulted in extensive work conducted under idealized laboratory conditions, in which soil moisture changes were controlled, typically following a regular pattern of drying and wetting (Fierer and Schimel, 2002; Miller et al., 2005; Shi and Marschner, 2014, 2015; Xiang et al., 2008). However, soil moisture varies randomly due to the stochastic nature of rainfall events (Katul et al., 2007; Rodriguez-Iturbe and Porporato, 2004), and this temporal variability can either promote or decrease soil organic C storage depending on its effects on soil microbes (Lehmann et al., 2020). Two features of soil moisture dynamics are particularly important because they directly affect the intensity of a respiration pulse – the duration of dry periods and the soil moisture increment at rewetting. Therefore, experimental designs based on regular cycles of drying and wetting do not allow exploration of how the stochastic nature of soil moisture fluctuations may affect respiration pulses. Capturing the effect of these stochastic fluctuations can be important as climatic changes are altering rainfall patterns – often lengthening the duration of droughts and increasing the intensity of the (less frequent) rainfall events (IPCC, 2012).

To quantify how the long-term mean heterotrophic respiration varies as a function of statistical rainfall properties (duration of dry periods and intensity), we developed a stochastic soil moisture and respiration model, parameterized using available respiration data. Specifically, we ask – how does variability in rainfall translate into variability in respiration pulses? How does the long-term mean contribution of respiration pulses vary along climatic gradients? These questions are motivated by the hypothesis that respiration pulses contribute a larger proportion of soil heterotrophic respiration under climates with more intermittent and intense rainfall events, compared to climates in which soil moisture variations are mild. If that is the case, future climatic conditions characterized by longer droughts and more intense rainfall events are expected to increase the overall role of respiration pulses in ecosystem C budgets.

2 Methods

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The theoretical framework is illustrated in Fig. 1. We start from the
premise that heterotrophic respiration follows changes in soil moisture
during drying (*R*_{d}) 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 (*R*_{r}). 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.

Soil moisture varies in response to rainfall events and the subsequent loss of soil water by percolation below the rooting zone and evapotranspiration. The dynamics of soil moisture in the rooting zone (the most biogeochemically active soil layer) can be described by the mass balance equation (Laio et al., 2001; Rodriguez-Iturbe and Porporato, 2004),

$$\begin{array}{}\text{(1)}& n{Z}_{\mathrm{r}}{\displaystyle \frac{\mathrm{d}s}{\mathrm{d}t}}=P\left(t\right)-E\left(s\left(t\right)\right)-L\left(s\left(t\right),t\right),\end{array}$$

where *s* is the saturation level (i.e., the relative volumetric soil
moisture), *n* is the soil porosity, *Z*_{r} is the rooting depth, and *P*, *E*, and
*L* represent precipitation inputs, evapotranspiration rate, and the
combination of water losses due to percolation below the rooting zone and
surface runoff. Equation (1) is interpreted at the
daily timescale. Given our aim to describe the statistical properties of
respiration rather than the details of soil moisture dynamics, we simplify
the soil moisture mass balance equation to a form that is analytically
tractable. Thus, we assume that evapotranspiration is the dominant water
loss when soil moisture is lower than a threshold *s*_{l} (equivalent to the
soil field capacity), whereas runoff and deep percolation dominate above
this threshold. Also, runoff and percolation are assumed to occur rapidly
compared to the timescales of the soil dry-down (free drainage conditions),
so that, after a precipitation event that brings soil moisture above the
level *s*_{l}, soil moisture decreases instantaneously to *s*_{l}. For
simplicity, evapotranspiration is modeled as a linear function of soil
moisture (Porporato et al., 2004),

$$\begin{array}{}\text{(2)}& E={\displaystyle \frac{s-{s}_{\mathrm{w}}}{{s}_{\mathrm{l}}-{s}_{\mathrm{w}}}}{E}_{max}=x{E}_{max},\end{array}$$

where *E*_{max} is the maximum rate of evapotranspiration, *s*_{w} is the
plant wilting point (below which ET becomes negligible), and *s*_{l} is the
threshold above which runoff and percolation are dominant. In the second
equality, a normalized soil moisture denoted by *x* is introduced to further
simplify the notation. With these assumptions and definitions, *s* ranges
between *s*_{w} and *s*_{l}, while the normalized soil moisture varies between
0 and 1.

Precipitation is treated as a marked Poisson process with mean frequency
*λ* and rain-event depths exponentially distributed with mean *α*. At each rain event, soil moisture increases by an amount corresponding to
the rain event depth (normalized by *n**Z*_{r}), unless the depth exceeds the
available soil storage capacity (i.e., *n**Z*_{r}(*s*_{l}−*s*)).
Assuming that rainfall exceeding this capacity is routed to runoff, the PDF
of soil moisture increments due to a rain event, *y*, for a given soil moisture
at the end of the dry period, *x*_{d}, is given by
(Laio et al., 2001)

$$\begin{array}{}\text{(3)}& \begin{array}{rl}{p}_{y}\left(y\mathrm{|}{x}_{\mathrm{d}}\right)& =\mathit{\theta}\left[\left(\mathrm{1}-{x}_{\mathrm{d}}\right)-y\right]\mathit{\gamma}{e}^{-\mathit{\gamma}y}\\ & +\mathit{\delta}\left[y-\left(\mathrm{1}-{x}_{\mathrm{d}}\right)\right]{e}^{-\mathit{\gamma}\left(\mathrm{1}-{x}_{\mathrm{d}}\right)},\end{array}\end{array}$$

where *p*_{y}(*y*|*x*_{d}) is the PDF of *y* conditional on soil moisture
at the end of the dry period, *x*_{d}; *θ*[⋅] is the
Heaviside step function; *δ*[⋅] is the Dirac delta
function; and *γ* is a parameter group defined as $\mathit{\gamma}=\frac{n{Z}_{\mathrm{r}}\left({s}_{\mathrm{l}}-{s}_{\mathrm{w}}\right)}{\mathit{\alpha}}$ (*γ* can
be interpreted as the number of average rainfall events needed to replenish
the plant-available soil water). The first term on the right-hand side of
Eq. (3) represents the probability density of a
soil moisture increase *y* equal to the rainfall depth (*θ*[⋅] is equal to 1 for $y<\mathrm{1}-{x}_{\mathrm{d}}$; zero otherwise). The second
term represents the probability of a soil moisture increase from the value
*x*_{d} to the soil field capacity (*x*=1). This term is also referred to as
an “atom of probability” because *δ*[⋅] is equal to
zero for all soil moisture increments, except *y*=1-*x*_{d}, at which $\mathit{\delta}\left[\cdot \right]=\mathrm{\infty}$.

With this stochastic description of precipitation events and further assuming stochastic stationary conditions, the PDF of the normalized soil moisture driven by the dynamics in Eq. (1) can be obtained analytically and reads (Porporato et al., 2004)

$$\begin{array}{}\text{(4)}& {p}_{x}\left(x\right)={C}_{\mathrm{l}}{\displaystyle \frac{{e}^{-x\mathit{\gamma}}{x}^{-\mathrm{1}+\frac{\mathit{\lambda}}{\mathit{\eta}}}}{\mathit{\eta}}},\end{array}$$

where *η* is a parameter group defined as $\mathit{\eta}=\frac{{E}_{max}}{n{Z}_{\mathrm{r}}\left({s}_{\mathrm{l}}-{s}_{\mathrm{w}}\right)}$. *C*_{l} is a
normalization constant that guarantees that the area under *p*_{x}(x) between *x*=0 and 1 is 1,

$$\begin{array}{}\text{(5)}& {C}_{\mathrm{l}}={\displaystyle \frac{\mathit{\eta}{\mathit{\gamma}}^{\frac{\mathit{\lambda}}{\mathit{\eta}}}}{\mathrm{\Gamma}\left[\frac{\mathit{\lambda}}{\mathit{\eta}}\right]-\mathrm{\Gamma}\left[\frac{\mathit{\lambda}}{\mathit{\eta}},\mathit{\gamma}\right]}},\end{array}$$

where Γ[⋅] and $\mathrm{\Gamma}\left[\cdot ,\cdot \right]$ 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 (Sect. 2.1.2).

The last distribution needed to calculate the statistical properties of soil
respiration pulses (Sect. 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 ${p}_{y,{x}_{\mathrm{d}}}\left(y,{x}_{\mathrm{d}}\right)$ (note that both *y* and
*x*_{d} 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., ${p}_{{x}_{\mathrm{d}}}\left({x}_{\mathrm{d}}\right)={p}_{x}\left(x\right)$. Because precipitation does not depend on
antecedent soil moisture conditions in this model, the PDF of soil moisture
at the end of a dry period is independent of the PDF of the subsequent
precipitation event and soil moisture increase. Thus, the joint PDF of
*x*_{d} and *y* is given by the product of the PDFs of *x*_{d} (Eq. 4) and of *y* conditional to *x*_{d} (Eq. 3),

$$\begin{array}{}\text{(6)}& {p}_{y,{x}_{\mathrm{d}}}\left(y,{x}_{\mathrm{d}}\right)={p}_{{x}_{\mathrm{d}}}\left({x}_{\mathrm{d}}\right){p}_{y}\left(y\mathrm{|}{x}_{\mathrm{d}}\right).\end{array}$$

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 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 *s*_{l} 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
*s*_{w}, i.e., at a water potential of −1.5 MPa. This assumption is motivated
by the observation that in intact soil cores and under field conditions
respiration stops in wetter conditions than at −15 MPa
(e.g., −2.7 MPa; Carbone
et al., 2011). Moreover, this allows us to keep the parameter number to a
minimum, consistent with the minimal soil moisture balance model of Eqs. (1) and (2) and the
overall idealized representation of soil heterotrophic respiration. The
respiration decrease with a lower threshold *s*_{w} (corresponding to
*x*=0) can be captured by a parabolic relation,

$$\begin{array}{}\text{(7)}& {R}_{\mathrm{d}}={R}_{\mathrm{d},\mathrm{max}}\left(\mathrm{2}x-{x}^{\mathrm{2}}\right),\end{array}$$

where *R*_{d} denotes the respiration rate during drying, and
*R*_{d,max} is the maximum respiration rate in the absence of rapid
rewetting (i.e., *R*_{d} at *x*=1 or *s*=*s*_{l}). 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, *R*_{d} from
Eq. (7) is also a random variable, which can be
obtained from the PDF of soil moisture using the derived distribution
approach, also referred to as the Jacobian rule (Kottegoda and Rosso,
1998),

$$\begin{array}{}\text{(8)}& {p}_{{R}_{\mathrm{d}}}\left({R}_{\mathrm{d}}\right)={p}_{x}\left(x\left({R}_{\mathrm{d}}\right)\right)\left|{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}{R}_{\mathrm{d}}}}\right|,\end{array}$$

where on the right-hand side the PDF of soil moisture is evaluated at
moisture values corresponding to given respiration values. This is done by
inverting Eq. (7) and expressing *x* as a function of
*R*_{d},

$$\begin{array}{}\text{(9)}& x\left({R}_{\mathrm{d}}\right)=\mathrm{1}-\sqrt{\mathrm{1}-{\displaystyle \frac{{R}_{\mathrm{d}}}{{R}_{\mathrm{d},\mathrm{max}}}}}.\end{array}$$

We note that Eq. (7) is monotonic in the domain
$\mathrm{0}\le x\le \mathrm{1}$, which allows unambiguous definition of the inverse of
*R*_{d}(x). Had we used a nonmonotonic *R*_{d}(x)
function (e.g., for applications of this approach to soils experiencing long
saturation periods), the derived distribution approach would have required
splitting the *x* domain into two – one for each monotonic branch of
*R*_{d}(x). In turn, Eq. (9) allows the
calculation of the slope of the *x*(*R*_{d}) relation, which is also
needed in Eq. (8),

$$\begin{array}{}\text{(10)}& {\displaystyle \frac{\mathrm{d}x}{\mathrm{d}{R}_{\mathrm{d}}}}={\left(\mathrm{2}{R}_{\mathrm{d},\mathrm{max}}\sqrt{\mathrm{1}-{\displaystyle \frac{{R}_{\mathrm{d}}}{{R}_{\mathrm{d},\mathrm{max}}}}}\right)}^{-\mathrm{1}}.\end{array}$$

The PDF of *R*_{d} is thus obtained from Eqs. (8)–(10) as

$$\begin{array}{}\text{(11)}& {p}_{{R}_{\mathrm{d}}}\left({R}_{\mathrm{d}}\right)={C}_{\mathrm{l}}{\displaystyle \frac{{e}^{-\mathit{\gamma}\left(\mathrm{1}-\sqrt{\mathrm{1}-{r}_{\mathrm{d}}}\right)}{\left(\mathrm{1}-\sqrt{\mathrm{1}-{r}_{\mathrm{d}}}\right)}^{-\mathrm{1}+\frac{\mathit{\lambda}}{\mathit{\eta}}}}{\mathrm{2}\mathit{\eta}{R}_{\mathrm{d},\mathrm{max}}\sqrt{\mathrm{1}-{r}_{\mathrm{d}}}}},\end{array}$$

where the normalized respiration ${r}_{\mathrm{d}}=\frac{{R}_{\mathrm{d}}}{{R}_{\mathrm{d},\mathrm{max}}}$ is
introduced to simplify the notation. This PDF can now be used to analytically calculate
the long-term mean of *R*_{d}, denoted by
〈*R*_{d}〉,

$$\begin{array}{}\text{(12)}& \begin{array}{rl}\langle {R}_{\mathrm{d}}\rangle & ={\displaystyle \frac{{R}_{\mathrm{d},\mathrm{max}}{C}_{\mathrm{2}}}{{\mathit{\gamma}}^{\mathrm{2}}}}\left\{\mathrm{\Gamma}\left[\mathrm{2}+{\displaystyle \frac{\mathit{\lambda}}{\mathit{\eta}}},\mathit{\gamma}\right]-\mathrm{2}\mathit{\gamma}\mathrm{\Gamma}\left[\mathrm{1}+{\displaystyle \frac{\mathit{\lambda}}{\mathit{\eta}}},\mathit{\gamma}\right]\right.\\ & \left.-{\displaystyle \frac{\mathit{\lambda}\left(\mathit{\eta}-\mathrm{2}\mathit{\gamma}\mathit{\eta}+\mathit{\lambda}\right)}{{\mathit{\eta}}^{\mathrm{2}}}}\mathrm{\Gamma}\left[{\displaystyle \frac{\mathit{\lambda}}{\mathit{\eta}}}\right]\right\},\end{array}\end{array}$$

where for convenience the parameter group ${C}_{\mathrm{2}}={\left(\mathrm{\Gamma}\left[\frac{\mathit{\lambda}}{\mathit{\eta}}\right]-\mathrm{\Gamma}\left[\frac{\mathit{\lambda}}{\mathit{\eta}},\mathit{\gamma}\right]\right)}^{-\mathrm{1}}$ is defined. The standard deviation of
*R*_{d}, denoted by ${\mathit{\sigma}}_{{R}_{\mathrm{d}}}$, can not be obtained analytically, but
it can be calculated through numerical integration of Eq. (11).

Heterotrophic 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 depend on both soil moisture
before the rewetting *x*_{d} 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 Sect. 2.2.1)

$$\begin{array}{}\text{(13)}& {R}_{\mathrm{r}}={R}_{\mathrm{r},\mathrm{max}}{\displaystyle \frac{y}{\mathrm{1}+\frac{{x}_{\mathrm{d}}}{b}}}\mathit{\theta}\left[\left(\mathrm{1}-{x}_{\mathrm{d}}\right)-y\right],\end{array}$$

where *R*_{r,max} is the largest respiration pulse possible, which is
achieved when an initially dry soil reaches saturation, i.e., *y*=1 and
*x*_{d}=0. The parameter *b* accounts for the effect of antecedent soil
moisture conditions – for a given value of *x*_{d}, the respiration pulse
increases with increasing *b*. The last term in Eq. (13) is a Heaviside function limiting the relation
between *R*_{r} and *y* to conditions in which soil moisture at most fills the
available pore space (as in Eq. 3, *θ*[⋅] is equal to 1 only when $y<\mathrm{1}-{x}_{\mathrm{d}}$). If before the rain
event soil moisture is at the plant wilting point (*x*_{d}=0) and the
precipitation event is sufficient to reach *s*_{l} (i.e., $y=x-{x}_{\mathrm{d}}=\mathrm{1}$), the
maximum respiration pulse is attained and ${R}_{\mathrm{r}}={R}_{\mathrm{r},\mathrm{max}}$. Here,
*R*_{r} represents an amount of C respired when the rewetting event occurs,
so its dimensions differ from those of the respiration rate during drying,
*R*_{d}; these two quantities are combined in the total heterotrophic
respiration rate in Sect. 2.1.5.

Because both *y* and *x*_{d} are random variables that follow the PDF of Eq. (6), *R*_{r} should also be regarded as a random
variable following its own PDF. Different from the PDF of *R*_{d}, which was
obtained from the univariate PDF of soil moisture, the PDF of *R*_{r} has to
be derived from the joint PDF of *y* and *x*_{d}. The derived distribution
approach can still be used, but it requires the determinant of the Jacobian
matrix of the transformation from *y* and *x*_{d} to *R*_{r}
(Kottegoda and Rosso, 1998). To proceed, it is first convenient
to introduce an auxiliary variable *X*=*x*_{d}, which is used together with
Eq. (13) to find the transformation from the
original variables *y* and *x*_{d} to *R*_{r} and *X*,

$$\begin{array}{}\text{(14)}& \begin{array}{rl}& \left\{\begin{array}{c}X={x}_{\mathrm{d}}\\ {R}_{\mathrm{r}}={R}_{\mathrm{r},\mathrm{max}}\frac{y}{\mathrm{1}+\frac{{x}_{\mathrm{d}}}{b}}\end{array}\right.\Rightarrow \left\{\begin{array}{c}{x}_{\mathrm{d}}=X\\ y=\frac{{R}_{\mathrm{r}}}{{R}_{\mathrm{r},\mathrm{max}}}\left(\mathrm{1}+\frac{X}{b}\right)\end{array}\right.\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{for}\phantom{\rule{0.25em}{0ex}}y<\mathrm{1}-{x}_{\mathrm{d}},\end{array}\end{array}$$

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,

$$\begin{array}{}\text{(15)}& \mathbf{J}=\left[\begin{array}{cc}\frac{\partial {x}_{\mathrm{d}}}{\partial X}& \frac{\partial {x}_{\mathrm{d}}}{\partial {R}_{\mathrm{r}}}\\ \frac{\partial y}{\partial X}& \frac{\partial y}{\partial {R}_{\mathrm{r}}}\end{array}\right]=\left[\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \frac{{R}_{\mathrm{r}}}{{R}_{\mathrm{r},\mathrm{max}}}\frac{\mathrm{1}}{b}& \frac{\mathrm{1}}{{R}_{\mathrm{r},\mathrm{max}}}\left(\mathrm{1}+\frac{X}{b}\right)\end{array}\right],\end{array}$$

and the determinant of the Jacobian,

$$\begin{array}{}\text{(16)}& \left|\mathbf{J}\right|={\displaystyle \frac{\mathrm{1}}{{R}_{\mathrm{r},\mathrm{max}}}}\left(\mathrm{1}+{\displaystyle \frac{X}{b}}\right).\end{array}$$

Fourth, the joint PDF of the variables *X* and *R*_{r} is obtained using the
derived distribution approach,

$$\begin{array}{}\text{(17)}& {p}_{X,{R}_{\mathrm{r}}}\left(X,{R}_{\mathrm{r}}\right)={p}_{y,{x}_{\mathrm{d}}}\left(y\left(X,{R}_{\mathrm{r}}\right),{x}_{\mathrm{d}}\left(X,{R}_{\mathrm{r}}\right)\right)\left|\mathbf{J}\right|,\end{array}$$

where as in Sect. 2.1.2 all the terms on the
right-hand side only depend on *X* and *R*_{r}, and ${p}_{y,{x}_{\mathrm{d}}}$ is given by
Eq. (6). Finally, to obtain the (marginal) PDF of
*R*_{r}, the joint PDF in Eq. (17) is integrated
over all possible values of *X*,

$$\begin{array}{}\text{(18)}& \begin{array}{rl}{p}_{{R}_{\mathrm{r}}}\left({R}_{\mathrm{r}}\right)& =\underset{\mathrm{0}}{\overset{\mathrm{1}}{\int}}{p}_{X,{R}_{\mathrm{r}}}\left(X,{R}_{\mathrm{r}}\right)\mathrm{d}X\\ & ={\displaystyle \frac{{C}^{\prime}}{\left(b+{r}_{r}\right){R}_{\mathrm{r},\mathrm{max}}}}\mathit{\{}{e}^{-\mathit{\gamma}}{\left[{\displaystyle \frac{\mathit{\gamma}\left(\mathrm{1}-{r}_{r}\right)}{\mathrm{1}+\frac{{r}_{r}}{b}}}\right]}^{{\scriptscriptstyle \frac{\mathit{\lambda}}{\mathit{\eta}}}}{\displaystyle \frac{\mathrm{1}+b}{\mathrm{1}-{r}_{r}}}\\ & +{e}^{-\mathit{\gamma}{r}_{r}}{\left({\displaystyle \frac{\mathrm{1}}{\mathrm{1}+\frac{{r}_{r}}{b}}}\right)}^{{\scriptscriptstyle \frac{\mathit{\lambda}}{\mathit{\eta}}}}\left[\mathit{\gamma}\left(b+{r}_{r}\right)\left(\mathrm{\Gamma}\left[{\displaystyle \frac{\mathit{\lambda}}{\mathit{\eta}}}\right]\right.\right.\\ & \left.-\mathrm{\Gamma}\left[{\displaystyle \frac{\mathit{\lambda}}{\mathit{\eta}}},\mathit{\gamma}\left(\mathrm{1}-{r}_{r}\right)\right]\right)+\mathrm{\Gamma}\left[\mathrm{1}+{\displaystyle \frac{\mathit{\lambda}}{\mathit{\eta}}}\right]\\ & \left.-\mathrm{\Gamma}\left[\mathrm{1}+{\displaystyle \frac{\mathit{\lambda}}{\mathit{\eta}}},\mathit{\gamma}\left(\mathrm{1}-{r}_{r}\right)\right]\right]\mathit{\}},\end{array}\end{array}$$

where on the right-hand side the normalized respiration pulse
${r}_{\mathrm{r}}=\frac{{R}_{\mathrm{r}}}{{R}_{\mathrm{r},\mathrm{max}}}$ is introduced to simplify the notation, and
as before ${C}_{\mathrm{2}}={\left(\mathrm{\Gamma}\left[\frac{\mathit{\lambda}}{\mathit{\eta}}\right]-\mathrm{\Gamma}\left[\frac{\mathit{\lambda}}{\mathit{\eta}},\mathit{\gamma}\right]\right)}^{-\mathrm{1}}$. Due to the complexity of Eq. (18),
the long-term mean and standard deviation of *R*_{r}, respectively denoted by
〈*R*_{r}〉 and ${\mathit{\sigma}}_{{R}_{\mathrm{r}}}$, need to be obtained via numerical integration.

It is useful to consider respiration pulses that only depend on the soil
moisture increments; i.e., *b*≫*x*_{d}. In this case, Eq. (13) reduces to ${R}_{\mathrm{r}}=y{R}_{\mathrm{r},\mathrm{max}}$ (i.e., $y={R}_{\mathrm{r}}/{R}_{\mathrm{r},\mathrm{max}})$ – equivalent to always having a
completely dry soil before rewetting. Thanks to the simplicity of the
respiration pulse equation, ${p}_{{R}_{\mathrm{r}}}\left({R}_{\mathrm{r}}\right)$ can be obtained
as a derived distribution from the marginal PDF of the soil moisture changes
*y*,

$$\begin{array}{}\text{(19)}& {p}_{y}\left(y\right)=\underset{\mathrm{0}}{\overset{\mathrm{1}}{\int}}{p}_{y}\left(y\mathrm{|}{x}_{\mathrm{d}}\right)\mathrm{d}{x}_{\mathrm{d}}=\left[\mathrm{1}+\mathit{\gamma}\left(\mathrm{1}-y\right)\right]{e}^{-y\mathit{\gamma}},\end{array}$$

where *p*_{y}(*y*|*x*_{d}) is from Eq. (3). The ${p}_{{R}_{\mathrm{r}}}\left({R}_{\mathrm{r}}\right)$ is then
obtained as

$$\begin{array}{}\text{(20)}& \begin{array}{rl}{p}_{{R}_{\mathrm{r}}}\left({R}_{\mathrm{r}}\right)& ={p}_{y}\left(y\left({R}_{\mathrm{r}}\right)\right)\left|{\displaystyle \frac{\mathrm{d}y}{\mathrm{d}{R}_{\mathrm{r}}}}\right|\\ & ={\displaystyle \frac{\mathrm{1}+\mathit{\gamma}\left(\mathrm{1}-\frac{{R}_{\mathrm{r}}}{{R}_{\mathrm{r},\mathrm{max}}}\right)}{{R}_{\mathrm{r},\mathrm{max}}}}{e}^{-{\scriptscriptstyle \frac{\mathit{\gamma}{R}_{\mathrm{r}}}{{R}_{\mathrm{r},\mathrm{max}}}}}.\end{array}\end{array}$$

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,

$$\begin{array}{}\text{(21)}& {\displaystyle}\langle {R}_{\mathrm{r}}\rangle ={\displaystyle \frac{{R}_{\mathrm{r},\mathrm{max}}}{{\mathit{\gamma}}^{\mathrm{2}}}}\left({e}^{-\mathit{\gamma}}+\mathit{\gamma}-\mathrm{1}\right),\text{(22)}& {\displaystyle}{\mathit{\sigma}}_{{R}_{\mathrm{r}}}={\displaystyle \frac{{R}_{\mathrm{r},\mathrm{max}}}{{\mathit{\gamma}}^{\mathrm{2}}}}\sqrt{\left(\mathit{\gamma}-\mathrm{2}\right)\mathit{\gamma}-\mathrm{1}+\mathrm{2}{e}^{-\mathit{\gamma}}\left(\mathrm{1}+\mathit{\gamma}+{\mathit{\gamma}}^{\mathrm{2}}\right)-{e}^{-\mathrm{2}\mathit{\gamma}}}.\end{array}$$

Thus, when respiration pulses are simply proportional to the soil moisture
change at rewetting, their mean only depends on the maximum pulse size
*R*_{r,max} and the ratio of soil water storage capacity and mean
precipitation depth (i.e., the parameter group $\mathit{\gamma}=\frac{n{Z}_{\mathrm{r}}\left({s}_{\mathrm{l}}-{s}_{\mathrm{w}}\right)}{\mathit{\alpha}}$).

The total mean heterotrophic respiration rate is given by the sum of the
mean respiration rate during soil drying
〈*R*_{d}〉 (Eq. 12; expressed in grams of carbon per square meter per day) and
the mean rate of respiration resulting from the sequence of rewetting pulses
over the study period (denoted by
$\langle {R}_{\mathrm{r}}^{*}\rangle $ and also expressed in grams of carbon per square meter per day). The
$\langle {R}_{\mathrm{r}}^{*}\rangle $ is calculated as the mean amount of respired carbon
(〈*R*_{r}〉 from Eq. 18, expressed in grams of carbon per square meter) divided
by the mean rainfall inter-arrival time, 1∕*λ* (expressed in days),

$$\begin{array}{}\text{(23)}& \langle {R}_{\mathrm{r}}^{*}\rangle =\mathit{\lambda}\langle {R}_{\mathrm{r}}\rangle .\end{array}$$

The mean total heterotrophic respiration rate is then obtained as

$$\begin{array}{}\text{(24)}& \langle {R}_{\mathrm{t}}\rangle =\langle {R}_{\mathrm{d}}\rangle +\langle {R}_{\mathrm{r}}^{*}\rangle .\end{array}$$

In what follows, the ratio of respiration pulse to total respiration (i.e., $\langle {R}_{\mathrm{r}}^{*}\rangle /\langle {R}_{\mathrm{t}}\rangle $) will also be considered, to evaluate the overall contribution of respiration pulses.

The phenomenological respiration models in Eqs. (7)
and (13) require knowledge of three parameters: the
heterotrophic respiration rate at the soil field capacity (*R*_{d,max}), the
maximum respiration pulse size (*R*_{r,max}), and the sensitivity of the
respiration pulse to the initial soil moisture (*b*). To estimate these three
parameters, we selected datasets where both the soil moisture before
rewetting and the soil moisture increments were manipulated (Fischer,
2009; Guo et al., 2014; Lado-Monserrat et al., 2014). All data reported in
these three publications were used, except data from the litter-amended
soils in Lado-Monserrat et al. (2014) (we chose to focus on
“natural” conditions) and data from small (*y*<0.3) rewetting events
in Fischer (2009) (they exhibited small respiration peaks
despite nearly stable soil moisture). The reported respiration amounts at
rewetting were corrected to isolate the pulse size (*R*_{r}) from the
respiration that would have occurred at constant soil moisture (*R*_{d}).
This was done by calculating *R*_{d,max} from control soil samples kept
constantly wet (Guo et al., 2014)
or from the post-pulse respiration rate before soil moisture started to
decline in experiments where drying was allowed in all samples
(Lado-Monserrat et al., 2014).
In contrast, respiration pulses had already been isolated by Fischer (2009). The last step of the parameter estimation
involved fitting Eq. (13) to the data using a
nonlinear least-square algorithm (*fminunc* function in MATLAB, R2018b,
MathWorks, Inc.).

Because respiration amounts and rates in these laboratory incubations were expressed respectively in micrograms per gram and micrograms per gram per day (or on a per-unit soil organic C basis), units were converted to gram per square meter and gram per square meter per day using bulk densities and sampling depths reported in the original publications (results are shown in Table 2).

In addition to estimating the values of the three parameters in Eqs. (7) and (13), we validated
the results from the whole stochastic model by comparing the predicted
long-term mean heterotrophic respiration rates to observations along a
rainfall manipulation gradient in a semiarid steppe (Zhang
et al., 2017b, 2019). Briefly, the precipitation gradient was established by
excluding 30 % and 60 % of precipitation with rain shelters and by
increasing precipitation by 30 % and 60 % through irrigation. By design,
only precipitation amounts (not timing) were altered, resulting in five mean
rainfall depths *α*=2.6, 3.9, 5.1, 6.4, and 7.6 mm. Mean
evapotranspiration rates, soil moisture, and heterotrophic respiration rates
along the rainfall gradient were obtained from the published supplementary
materials in Zhang et al. (2019) or from
the Dryad dataset by Zhang et al. (2017a).
Hydrologic parameters that were not provided were estimated as follows. The
maximum evapotranspiration rate (assumed equal to the potential
evapotranspiration) and the mean rainfall frequency were estimated from
May–August CRU data at the rainfall manipulation site (${E}_{max}=\mathrm{4.3}$ mm d^{−1} and *λ*=0.41 d^{−1}). The soil at the site has a sandy loam
texture (Bingwei Zhang, personal communication, 2019), and soil properties were
obtained accordingly: *n*=0.42, *s*_{w}=0.11, *s*_{l}=0.52
(Table 2.1 in Rodriguez-Iturbe and Porporato, 2004). Finally,
the rooting depth *Z*_{r}=0.2 m was estimated as the soil depth above which
approximately 70 % of belowground productivity occurs, based on data from
Zhang et al. (2020).

Regarding the parameters of the rewetting respiration function (Eq. 13), we assumed ${R}_{\mathrm{d},\mathrm{max}}=\mathrm{2}$ gC m^{−2} d^{−1} and *b*=0.1. These values are deemed reasonable for mineral soils
based on Table 2 and accounting for a rooting depth about double the
sampling depth of the incubation experiments (which doubles the
*R*_{d,max} values in Table 2). Without specific information on respiration
pulse sizes, we let *R*_{r,max} vary over a wide range. Additionally, we
tested the simplified respiration model (Sect. 2.1.4), which does not require any assumption on
*b*, against the same total heterotrophic respiration dataset.

3 Results

Back to toptop
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 (*x*_{d}), 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 *R*_{r,max} and
*R*_{d,max} per unit ground area are higher in the organic soils than in
mineral soils (Table 2) and so is the ratio between *R*_{r,max} and
*R*_{d,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., ${R}_{\mathrm{r},\mathrm{max}}=\mathrm{5}$ gC m^{−2}, ${R}_{\mathrm{d},\mathrm{max}}=\mathrm{1}$ 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 *R*_{r,max} and
*R*_{d,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 CO_{2} 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 *R*_{r} is
positively skewed regardless of climate (but with heavier tails under dry
conditions, Fig. 3c, g), the PDF of *R*_{d} is strongly affected by
climatic conditions – the probability of high values for *R*_{d} 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 *R*_{d}. 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 *R*_{d} in
Fig. 3h might seem incorrect, as it increases sharply at high respiration
values. This increase is due to the flat derivative of the *R*_{d}–soil
moisture relation (Eq. 7), which causes an
asymptote in the PDF at ${R}_{\mathrm{d}}={R}_{\mathrm{d},\mathrm{max}}$ (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 *R*_{d,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 *R*_{d,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,
*R*_{r,max} and *R*_{d,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*〉=*α**λ*, 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
〈*R*_{d}〉 increases with precipitation more than
$\langle {R}_{\mathrm{r}}^{*}\rangle $, the relative contribution of respiration pulses to the total respiration
rate, $\langle {R}_{\mathrm{r}}^{*}\rangle /\langle {R}_{\mathrm{t}}\rangle $, 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
$\langle {R}_{\mathrm{r}}^{*}\rangle /\langle {R}_{\mathrm{t}}\rangle $ is smaller than the change in
〈*R*_{d}〉 or $\langle {R}_{\mathrm{r}}^{*}\rangle $ 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 *R*_{d} 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 *R*_{d} when moving from dry to wet
conditions. Under dry conditions, the PDF of *R*_{d} 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 *R*_{r} 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 *R*_{r}. The
coefficients of variation (CV) of *R*_{d} and *R*_{r} 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 *R*_{d} decreases
with both increasing *λ* and increasing *α*. In contrast, the
CV of *R*_{r} 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
〈*R*_{t}〉 (Fig. 7a) but increases
$\langle {R}_{\mathrm{r}}^{*}\rangle /\langle {R}_{\mathrm{t}}\rangle $ (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
〈*R*_{t}〉 and higher $\langle {R}_{\mathrm{r}}^{*}\rangle /\langle {R}_{\mathrm{t}}\rangle $ than mineral soils (black curves), due to their higher *R*_{r,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
〈*R*_{t}〉 while decreasing $\langle {R}_{\mathrm{r}}^{*}\rangle /\langle {R}_{\mathrm{t}}\rangle $ (compare solid vs. dashed curves).

4 Discussion

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Heterotrophic respiration fluctuates at multiple temporal scales in response to hydroclimatic variability (Messori et al., 2019; Rubio and Detto, 2017) – from interannual variations due to climatic anomalies and extreme events (Reichstein et al., 2013), to seasonal variations partly linked to plant activity (Zhang et al., 2018), to short-term fluctuations induced by soil drying and rewetting (Daly et al., 2009). Here we focus on respiration fluctuations during drying–wetting cycles and how they are affected by precipitation regimes. Differently from most other modeling approaches to describe these dynamics, we develop a probabilistic model with analytical solutions for the probability density function of respiration rate (discussed in Sect. 4.1). For the sake of analytical tractability, this model rests on important assumptions (Sect. 4.2), but despite its simplicity it has the potential to assess the effect of precipitation variability (and its expected changes) on heterotrophic respiration (Sect. 4.3).

Most biogeochemical models assume that heterotrophic respiration (and other
processes) depend on a generic soil property *φ* following an
empirical function *f*(φ) (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 *f*(φ) 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 *f*(φ) to capture the propagation of the statistical properties of
*φ* to *f*(φ). 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 CO_{2} concentration in the
pore space instead of respiration rates (Daly et al., 2008). Observations show
that CO_{2} concentration increases rapidly after rainfall and then
decreases following a negative exponential function. This dynamic can be
described as a stochastic process where CO_{2} concentration is the random
variable and precipitation represents the stochastic forcing
(Daly et al., 2008). With this
approach, the long-term mean CO_{2} concentration was found to depend on
the average rainfall rate (*λ**α*), while the standard deviation
of CO_{2} concentration depends on *λ**α*^{2}. 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 CO_{2} concentration. Soil respiration was shown to be approximately
proportional to CO_{2} 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 CO_{2} 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 *R*_{d,max} and *R*_{r,max} are independent of climatic and
vegetation conditions, and (iii) that hydroclimatic conditions are
statistically stationary.

Respiration pulses are modeled as instantaneous events of CO_{2} 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, *R*_{d,max} and *R*_{r,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,
*R*_{d}, 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.

5 Conclusions

Back to toptop
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 CO_{2} 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 analytically characterize the
statistical properties of respiration rates as a function of the statistical
properties of the rainfall events that drive 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 will increase the role of rewetting
respiration pulses in the ecosystem C budgets.

Data availability

Back to toptop
Data availability.

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

Author contributions

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Author contributions.

Stefano Manzoni and Giulia Vico conceptualized the study; Stefano Manzoni, Giulia Vico, and Amilcare Porporato developed the theory; TF provided and discussed data; Arjun Chakrawal analyzed data and prepared Fig. 2; Stefano Manzoni prepared the other figures and drafted the manuscript; all authors discussed the study ideas and read and commented on the manuscript.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank Antonio L. Lidón and Bingwei Zhang for sharing data and assisting in their interpretation and Thomas Wutzler and the two anonymous reviewers for their constructive comments.

Financial support

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Financial support.

This research has been supported by the Swedish Research Council Vetenskapsrådet (grant nos. 2016-04146 and 2016-04910) and the Swedish Research Council Formas (grant nos. 2018-00425 and 2018-00968).

The article processing charges for this open-access

publication were covered by Stockholm University.

Review statement

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Review statement.

This paper was edited by Frank Hagedorn and reviewed by two anonymous referees.

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Short summary

Carbon dioxide is produced by soil microbes through respiration, which is particularly fast when soils are moistened by rain. Will respiration increase with future more intense rains and longer dry spells? With a mathematical model, we show that wetter conditions increase respiration. In contrast, if rainfall totals stay the same, but rain comes all at once after long dry spells, the average respiration will not change, but the contribution of the respiration bursts after rain will increase.

Carbon dioxide is produced by soil microbes through respiration, which is particularly fast when...

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