Catchment mixing processes and travel time distributions

Transcription

1 WATER RESOURCES RESEARCH, VOL. 48, W05545, doi: /2011wr011160, 2012 Catchment mixing processes and travel time distributions Gianluca Botter 1 Received 19 July 2011; revised 10 April 2012; accepted 10 April 2012; published 24 May [1] This work focuses on the description and the use of the probability density functions (pdfs) of travel, residence and evapotranspiration times, which are comprehensive descriptors of the fate of rainfall water particles traveling through catchments, and provide key information on hydrologic flowpaths, partitioning of precipitation, circulation and turnover of pollutants. Exploiting some analytical solutions to the transport problem derived by Botter et al. (2011), this paper analyzes the features of travel, residence and evapotranspiration time pdfs resulting from different assumptions on the mixing processes occurring during streamflow formation and plant uptake (namely, complete mixing and translatory flow). The ensuing analytical solutions are analyzed through numerical Monte Carlo simulations of a stochastic model of soil moisture and streamflow dynamics. Travel and residence time pdfs are shown to be time-variant as they mirror the variability of the relevant hydrological fluxes. In particular, the temporal fluctuations of the mean residence time are shown to reflect rainfall dynamics, whereas the variability of the mean travel time is chiefly driven by streamflow dynamics, with lower frequency and higher amplitude fluctuations. Dry climates enhance the effect of the type of mixing on catchment transport features (e.g., mean travel times and seasonal dynamics of stream concentrations). The implications for the interpretation of tracer eriments are also discussed, showing through specific examples that models disregarding nonstationarity may significantly misestimate travel time pdfs. Citation: Botter, G. (2012), Catchment mixing processes and travel time distributions, Water Resour. Res., 48, W05545, doi: / 2011WR Introduction [2] The catchment travel time represents the time spent by a water particle traveling through a hillslope/catchment, from the time when it falls on the catchment surface as rainfall, to the passage across a suitable control section as streamflow. The catchment travel time is intrinsically a random variable due to heterogeneities of morphological, geological and hydrologic attributes characterizing the environments and processes involved in the terrestrial part of the water cycle, and it is thus defined by a probability density function (pdf) [e.g., Rinaldo and Marani, 1987; Dagan, 1989; Cvetkovic and Dagan, 1994; Beven, 2001; McGuire and McDonnell, 2006; Rinaldo et al., 2005, 2006a, 2006b; McDonnell et al., 2010]. The key difference of travel times with respect to residence and evapotranspiration times in catchment control volumes has been recently pointed out by McGuire and McDonnell [2006], McDonnell et al. [2010], Botter et al. [2010, 2011] and Rinaldo et al. [2011]. According to the above distinction, the catchment residence time distribution represents the probability density function of the ages of the water particles stored inside a catchment/hillslope at a 1 Dipartimento di Ingegneria Civile, Edile e Ambientale, Università di Padova, Padua, Italy. Corresponding author: G. Botter, Dipartimento di Ingegneria Civile, Edile e Ambientale, Università di Padova, via Loredan, 20 I-35131, Padua, Italy. (botter@idra.unipd.it) American Geophysical Union. All Rights Reserved /12/2011WR given time, while the evapotranspiration time pdf represents the distribution of the time taken from the entrance to the exit for the particles subtracted to the soil by evapotranspiration. The pdfs of travel, residence and evapotranspiration times are key to describe the partitioning of precipitation into green and blue water, and the storage and turnover of pollutants in catchments. Despite their practical and conceptual importance, and the impact on the travel time pdf, the residence and evapotranspiration time pdfs have received quite limited attention in the literature, probably due to the difficulties inherent in their erimental quantification. Recent erimental, modeling and theoretical studies [McGuire et al., 2007; Botter et al., 2010, 2011; Duffy, 2010; Van der Velde et al., 2010; Hrachowitz et al., 2010; Rinaldo et al., 2011] have shown that catchment travel, residence and evapotranspiration time pdfs are in general time-variant to reflect the variability of the underlying climate, rainfall and hydrologic conditions. In particular, Botter et al. [2011] derived a set of general analytical ressions for all the above distributions, which were ressed as a function of the temporal evolution of the underlying hydrologic fluxes and of the type of mixing (i.e., age selection process during uptake and streamflow production). In this paper, the general ressions derived by Botter et al. [2011], where the features of the underlying mixing processes were left unspecified, are detailed by analyzing travel, residence and evapotranspiration time pdfs emerging assuming two different types of mixing: complete mixing (i.e., both evapotranspiration and discharge sample the full distribution of ages available in the soil) and translatory flow (i.e., the soil is assumed to release at any time the oldest water particles W of15

2 contained in the control volume). The choice of these relatively simplified mixing schemes is functional to the objective of the paper, which is not aimed at assessing the ability of a specific model to reproduce the observed chemical response of real-world catchments. This operation may be indeed problematic due to parameter-identification and equifinality issues (e.g., Iorgulescu et al. [2005] and Figure 11) (let alone the need for long-term high-resolution data sets), and is thus deferred to forthcoming works. Conversely, the objectives of this paper are: (1) to study the origin of the time-variance of travel and residence time pdfs, and analyze the dependence of such time-variance on climate and hydrologic features, in particular by evidencing the major drivers of the mean travel/residence time variability; (2) to investigate the impact of the type of mixing on the features of the transport processes (e.g., marginal travel/residence time pdfs, chemical response of river basins) (3) to discuss the potential ability to identify time-variant travel, residence and evapotranspiration time pdfs from hydrochemical data, also in relation with the modelistic tools employed and with the type of measurements available. [3] It is worth emphasizing the relationship between this paper and some previous works using a framework similar to that adopted in this study [i.e., Botter et al., 2010; Rinaldo et al., 2011]. As per the methods, the set of solutions derived in the case of translatory flow are completely new, while the ressions of the travel time pdf corresponding to complete mixing was already obtained by Botter et al. [2010] (even though via a different approach). Such solution is here complemented with the ressions of the corresponding residence and evapotranspiration time pdfs. As per the objectives, the issue of the time variance of travel time pdf has been partially dealt with also by Rinaldo et al. [2011]. In the present paper, however, the role of climate conditions is lored more systematically, different types of mixing schemes are considered and compared, and the primary controls on the mean residence and travel times are singled out through numerical simulations and theoretical arguments. 2. Background [4] This paper seeks to analyze the probabilistic structure of the time spent by rainfall water particles to travel through a control volume V, which represents a hillslope, a catchment or a subcatchment of a river basin. The most relevant processes affecting the temporal evolution of the water storage in the control volume S(t) are assumed to be precipitation (J), evapotranspiration (ET) and soil drainage (Q) so as the water balance equation reads: dsðtþ ¼ JðtÞ ETðtÞ QðtÞ : (1) dt The transport processes affecting the movement of water within the catchment control volume are conceptualized by decomposing the rainfall input into a large number of water particles, each of which enters V through precipitation at a given injection time t i, spends a certain amount of time inside the catchment, and then exits the control volume (either via Q or ET) at a given exit time, t e. The travel time t T of a tagged water particle is defined as the difference between the time when a particle exits V as Q and the time at which the same particle has entered V as rainfall [Botter et al., 2010, 2011; Rinaldo et al., 2011]. Accordingly, the evapotranspiration time t ET is the difference between the time when the particle exits V as ET and the time at which the above particle enters V as rainfall. Moreover, the residence time t R of a tagged water particle contained within V at time t is defined as the age of such water particle, which is by definition the difference between the current time t and the injection time of the tagged particle: t R ¼ t t i ðt > t i Þ. [5] It has been shown [e.g., Botter et al., 2010; 2011; Rinaldo et al., 2011] that t i, t ET, t T, t e and t R can be thought of as interdependent random variables embedding the stochasticity of the transport processes taking place in river basins, which can be characterized by the correspondent (marginal and joint) probability density functions. In particular, this paper shall make reference to the following functions: (1) the pdfs of the travel times t T, conditional to all the possible injection times t i, p T ðt T ; t i Þ; (2) the pdfs of the travel times t T, conditional to all the possible exit times t e, p 0 T ðt T; t e Þ; (3) the pdfs of the evapotranspiration times t ET, conditional to all the possible injection times t i, p ET ðt ET ; t i Þ; (4) the pdfs of the evapotranspiration times t ET, conditional to all the possible exit times t e, p 0 ET ðt ET; t e Þ;(5)thetimedependent pdf of the residence time t R, evaluated at any possible observation time t, p RT ðt R ; tþ. The physical interpretation of these functions has been already clarified by e.g., Botter et al. [2011] and Rinaldo et al. [2011], but is briefly recalled in the following. [6] The travel time pdf conditional to the injection time represents the (normalized) mass flux ( Q ) produced by a uniform and instantaneous injection of a passive solute ( Q / p T ðt t 0 ; t 0 Þ, where t 0 is the time of the injection), and thus characterizes the solute loads released from a catchment at different time scales. The travel time pdf conditional to the exit time quantifies instead the relative importance of the chemical characteristics of past rainfall events to determine the current chemical composition of the flow for passive solutes [Niemi, 1977] and thus relates rainfall (C J ) and streamflow (C Q ) flux concentrations (C Q ðtþ ¼ R t 1 C Jðt i Þp 0 T ðt t i; tþdt i ). The function p 0 T thus indicates the persistency of contamination episodes in streamflow concentrations. Analogously, the evapotranspiration time pdf conditional to the injection time can be interpreted as the mass flux through ET which is produced by a (uniform) injection of a passive solute at time t i, and the evapotranspiration time pdf conditional to the exit time, p 0 ET ðt ET; t e Þ quantifies the relative importance of the chemical characteristics of past rainfall events to determine the current R chemical composition of the sap flow (C ET ðtþ ¼ t 1 C Jðt i Þp 0 ET ðt t i; tþdt i ). Finally, the residence time pdf represents the distribution of weights that need to be assigned to the solute concentration of past rainfall pulses to determine the average concentration of the water storage < C s ðtþ > (< C s ðtþ >¼ R t 1 C Jðt i Þp RT ðt t i ; tþdt i ). [7] The dependence of the functions p RT, p T (p ET ) and p 0 T (p 0 ET )ont, t i and t e reflects the temporal variability of the underlying transport features, and is thus referred to as time-variance or nonstationarity. 3. Analytical Expressions for Catchment Travel, Residence and Evapotranspiration Time pdfs [8] Botter et al. [2011] have demonstrated that the travel, residence and evapotranspiration time pdfs defining 2of15

3 catchment-scale transport processes can be analytically ressed in terms of the input/output fluxes, and of the type of mixing taking place during streamflow generation and plant uptake. The latter is ressed by means of the age functions! Q and! ET defined as the ratio between the travel/evapotranspiration time pdf conditional to the exit time, and the residence time pdf at the same time:! Q ðt T ; t e Þ¼ p0 T ðt T ; t e Þ p RT ðt T ; t e Þ (2)! ET ðt ET ; t e Þ¼ p0 ET ðt ET ; t e Þ p RT ðt ET ; t e Þ : (3) [9] The age functions ress the affinity of the two output fluxes (discharge and evapotranspiration) with respect to the ages available in the control volume, and hence describe the type of mixing taking place during streamflow production and plant uptake. In particular,! Q ðt T ; t e Þ defines the ratio between the number of water particles with an age in the interval (t T, t T þ dt T )sampledbyq at time t e and the amount of particles with the same age stored in the control volume at that time (and analogously for! ET ). The underlying travel, residence and evapotranspiration time pdfs can be thus consistently ressed as a function of S, J, ET, Q and! as: p RT ðt R ; tþ ¼ Jðt t RÞ Sðt t R Þ Z t Z t t t R JðxÞ dx QðxÞ½1! Q ðt R t þ x; xþš t t R Z t ETðxÞ½1! ET ðt R t þ x; xþš t t R dx dx ; p 0 T ðt T ; t e Þ¼ Jðt e t T Þ! Q ðt T ; t e Þ Sðt e t T Þ Z te QðxÞ½1! Q ðt T t e þ x; xþš JðxÞ dx t e t T Z te ETðxÞ½1! ET ðt T t e þ x; xþš dx ; t e t T p T ðt T ; t i Þ¼ Qðt i þ t T Þ! Q ðt T ; t T þ t i Þ Sðt i Þðt i Þ Z tiþt T QðxÞ½1! Q ðx t i ; xþš JðxÞ dx t i Z tiþt T ETðxÞ½1! ET ðx t i ; xþš dx ; t i p 0 ET ðt ET ; t e Þ¼ Jðt e t ET Þ! ET ðt ET ; t e Þ Sðt e t ET Þ Z te QðxÞ½1! Q ðt ET t e þ x; xþš JðxÞ dx t e t ET Z te ETðxÞ½1! ET ðt ET t e þ x; xþš dx ; t e t ET (4) (5) (6) (7) p ET ðt ET ; t i Þ¼ ETðt i þ t ET Þ! ET ðt ET ; t ET þ t i Þ Sðt i Þ½1 ðt i ÞŠ Z tiþt ET QðxÞ½1! Q ðx t i ; xþš JðxÞ dx t i Z tiþt ET ETðxÞ½1! ET ðx t i ; xþš dx ; t i where ðt i Þ¼Jðt i Þ 1R 1 0 p 0 T ðx; x þ t iþqðx þ t i Þdx is a partition coefficient defining the fraction of rainfall water particles injected at time t i and destined to leave V via Q [Botter et al., 2011; Van der Velde et al., 2010]. Equations (4) to (8) are derived on the basis of the only assumption represented by the definition of a catchment control volume and of the fluxes acting on it. In simple terms, the catchment is seen as a storage of water particles with different ages: some particles enter through rainfall with a zero age and progressively grow old. Meanwhile, other particles leave the system via ET and Q, with a certain age distribution which is a subset of the distribution of the ages available in the system at that time. As a result, the variability of the fluxes drives the variability of the distribution of the ages available in the control volume, which in turn controls the variability of the age distribution sampled by Q and ET (p 0 T and p0 ET ). This is the reason for which travel, residence and evapotranspiration time pdfs need to be mutually consistent, and consistent with the underlying input/output fluxes. [10] Equations (4) to (8) shed light on which are the processes responsible for generating the time-variance of the travel and residence time pdfs: the variability of the ratio between fluxes and stores and the time-variance of catchment mixing processes (embedded into the timevariance of the age functions! Q and! ET ). In this paper, we focus on the degree of time variance produced by time-variable J, ET and Q, by assuming time invariant age functions. Equations (4) to (8) suggest a double effect of time-variance induced by the variability of the ratios J/S, ET/S and Q/S: (1) a real-time effect, impacting with a similar strength all the travel/residence times, related to the multiplicative terms appearing before the onential functions; (2) a long-term effect, which becomes important only for large travel/residence times, keeping track of the variability of the integral of the ratios Q/S, J/S and ET/S calculated within different time windows. This long-term effect is in part modulated by the type of mixing processes taking place, provided that the integrands at the right-hand side of equations (4) (8) are suitably weighted through the age functions!. This suggests that the time-variance of travel and residence time pdfs is primarily controlled by the time-variability of the fluxes/stores, with the specific type of mixing acting as a second-order control. [11] In order to analyze the features of the travel and residence time pdfs, the type of mixing is specified by considering two different simplified scenarios: (1) the case where the output fluxes sample an ergodic mixture of ages available in the control volume (complete mixing); (2) the case where mixing is negligible and the water particles released via Q and/or ET at a given time are the oldest water particles contained within the soil at that time (translatory (8) 3of15

4 flow). These two end-member scenarios are representative of different degrees of mixing in catchment soils (complete mixing versus no mixing), and hence define the bonds of the set of cases which may be observed in real world contexts [see Botter et al., 2009], allowing for closed-form analytical solutions. It should be recognized that in real-world catchments, soil mixing processes are by far more complex than what predicted by these simple models. Nevertheless, for the reasons discussed above, these mixing schemes well serve the three general issues dealt with in the paper. Moreover, the analysis of numerical results obtained using more sophisticated mixing schemes (e.g., decreasing or increasing age functions) suggests that the behavior of travel and residence time pdfs evidenced here is quite general, and not restricted to the specific examples discussed in the paper. [12] In the derivation of the analytical solutions corresponding to the case of complete mixing and translatory flow, for the sake of simplicity, this paper also assumes that ET and Q behave similarly with respect to the selection of the ages available in the soil pool (i.e.,! Q ðt T ; tþ ¼! ET ðt ET ; tþ if t T ¼ t ET ) Complete Mixing [13] This subsection describes the case where subsurface environments can be treated as well-mixed systems from the viewpoint of the ages of the water particles leaving the control volume as streamflow and evapotranspiration (i.e., different soil regions have the same age distribution of the whole hillslope) [Botter et al., 2010]. The above simplification was proved to be quite robust through the comparison with observational data and detailed spatially distributed numerical simulations [see e.g., Botter et al., 2008a, 2009; Rinaldo et al., 2011]. The situation is representative of catchments where mixing of ages is enhanced as a result of heterogeneous flowpaths and high dispersion. The assumption of complete mixing is mathematically formulated by postulating that both ET and Q sample at every time step the whole distribution of ages available in the control volume at that time, leading to the following ressions of the age functions:! Q ðt T ; tþ ¼! ET ðt ET ; tþ 1 ; (9) which ress the identity between the residence time pdf at time t and the travel (evapotranspiration) time pdf conditional to the exit time t. [14] Under the above assumptions, equation (4) reduces to p RT ðt R ; tþ ¼ Jðt t RÞ Sðt t R Þ JðxÞ t t R dx Z t ; (10) which resses the residence time pdf in terms of the rainfall input and of the related storage. From a physical viewpoint, equation (10) resses the fact that, at time t, the probability to find in the storage water particles characterized by a residence time close to t R is proportional to the probability associated to the age t R 0 (Jðt t R Þ= Sðt t R Þ) evaluated at the past time t t R. According to equation (5), the travel time pdf conditional to a given exit time t e coincides with the residence time pdf, and can be thus written as: p 0 T ðt T ; t e Þ¼ Jðt e t T Þ Sðt e t T Þ JðxÞ t e t T dx ; (11) Z te which is the same as the evapotranspiration time pdf conditional to a given exit time, p 0 ET, provided that the subscript T is replaced by the subscript ET everywhere in the above equation. The ression of the function p ET is derived from equation (8) as: p ET ðt ET ; t i Þ¼ ETðt ET þ t i Þ ½1 ðt i ÞŠSðt i Þ Z tiþt ET t i JðxÞ dx : (12) Finally, the travel time pdf conditional to a given injection time t i can be obtained by means of equation (6) as p T ðt T ; t i Þ¼ Qðt T þ t i Þ Sðt i Þðt i Þ Z tiþt T t i JðxÞ dx ; (13) which can be proved to be equivalent to the ression derived by Botter et al. [2010] (equation (41)) once J is ressed in terms of Q, ET and ds/dt using equation (1) Old-Water-First Scheme (Translatory Flow) [15] This subsection describes the case where the water particles leaving the control volume via Q and/or ET are assumed to be the oldest water particles available in the control volume. This scheme is taken here as a surrogate of the cases where hillslope dispersion is limited and the mixing among different ages is reduced. In particular, it mimics a translatory, piston-like flow which may occur in a hillslope/catchment where transport processes in the vadosezone dominate and water velocities have a dominant vertical component [Rao et al., 1985; Harmann et al., 2011]. Therein, the incoming water push downward the older water particles, as in a First-In-Last-Out queue [see e.g., Horton and Hawkins, 1965; Hewlett and Hibbert, 1967; Botter et al., 2008a; McDonnell, 2009]. [16] According to the assumptions made, the pdfs of the travel and the evapotranspiration times, either conditional to a given injection time or to a given exit time are all Dirac delta distributions ðþ [Abramowitz and Stegun, 1996]. The major problem that needs to be addressed in this case is the determination of the time variable delay between the exit and entrance times of the water particles leaving V via Q or ET at a given time t e, which is denoted in the following as Tðt e Þ. Tðt e Þ represents the time spent within the system by the particles leaving the control volume at time t e (i.e., the travel time of the oldest water particles contained in the control volume at time t e ), and is determined by the following implicit ordinary differential equation (see Appendix A): dtðt e Þ ¼ 1 Qðt eþþetðt e Þ : (14) dt e Jðt e Tðt e ÞÞ 4of15

5 The travel time pdf conditional to a given exit time is then derived from equation (5) (after recognizing that the age function! Q is a Dirac delta function) as: p 0 T ðt T ; t e Þ¼ðt T Tðt e ÞÞ ðqðt e Þ > 0Þ : (15) Meanwhile, the evapotranspiration time pdf conditional to the exit time is: p 0 ET ðt ET ; t e Þ¼ðt T Tðt e ÞÞ ðetðt e Þ > 0Þ : (16) The following residence time pdf is then obtained from equation (4) (Appendix A): p RT ðt R ; tþ ¼ Jðt t RÞ H½TðtÞ t R Š ; (17) SðtÞ H being the unit step function. Equation (17) shows that the residence time pdf at time t depends on the temporal evolution of past rainfall events down to the time t TðtÞ, which corresponds to the (finite and time variable) memory displayed by the considered catchment. [17] Finally, the travel time pdf and the evapotranspiration time pdf conditional to a given injection time can be derived using their own definition as: p T ðt T ; t i Þ¼ðTðt i þ t T Þ t T Þ ðqðt i þ t T Þ > 0Þ (18) p ET ðt ET ; t i Þ¼ðTðt i þ t ET Þ t ET Þ ðetðt i þ t T Þ > 0Þ ; (19) which are, as ected, Dirac delta distributions. 4. Numerical Simulations [18] This section will discuss the features of the travel and residence time pdfs drawn from a series of numerical Monte Carlo simulations of a minimalist, stochastic model where soil water dynamics are driven by an intermittent rainfall input [e.g., Porporato et al., 2004; Settin et al., 2007; Botter et al., 2007, 2008b]. It will focus in particular on the intraannual variability of the travel and residence time distributions, neglecting the interannual variability produced by long-term drifts of hydrologic drivers. The results presented are based on a series of 50 years Monte Carlo simulations of equation (1), where the rainfall input is represented by a stationary marked Poisson process, while evapotranspiration and flow rates are nonlinear functions of the water storage S (see Appendix B for further details on the hydrologic model). Only the last 40 years of each simulation have been considered in the calculations, so as to eliminate the dependence on the initial conditions. [19] The results obtained using the well mixed scheme are summarized in the plots from Figures 1, 2, 3, and 4. Figures 1b, 1c, and 1d show the huge difference among the travel time pdfs conditional to three different injection times (namely t 1 ¼ 70 d, t 2 ¼ 160 d and t 3 ¼ 295 d), and highlights the effect of the time-variance on the transport dynamics emerging in catchment forced by intermittent rainfall inputs. Each travel time pdf conditional to the injection time is multimodal, with a number of more or less pronounced peaks whose magnitude is strongly influenced Figure 1. Travel time pdfs conditional to the injection time evaluated using a well mixed scheme. (a) temporal evolution of the streamflows Q(t); (b) travel time pdf conditional to the injection time t 1 (solid line) and ensemble average of the travel time pdfs conditional to the injection time calculated as a weighted average among all the possible injection times, dotted line); (c) same as Figure 1b but conditional to the injection time t 2 ; (d) same as Figure 1b but conditional to the injection time t 3. Model parameters used in this simulation are the following: n ¼ 0:4, Z r ¼ 100 cm, ET m ¼ 0:25 cmd 1, s w ¼ 0:2, s ¼ 0:7, K s ¼ 0:0015 cms 1, b ¼ 10, ¼ 0:3d 1 and ¼ 1:5 cm. by the hydrologic dynamics. Also shown (Figure 1a) is the temporal evolution of the streamflows Q, which evidences the correlation between the position of the spikes of the travel time pdfs and the occurrence time of the flood events. This may lead to both early peaks (Figure 1d) or to late peaks (Figure 1c) in the travel time pdf. The graph also reproduces the ensemble average of the travel time pdf conditional to the injection time (dotted line), which shows a remarkably regular behavior and resembles a gamma pdf. Interestingly, each travel time pdf conditional to the injection time may be close to the corresponding ensemble average (Figure 1b) or radically different from it (Figure 1d), depending on the specific injection time. Figure 2 shows the residence time distribution evaluated at the same three 5of15

6 Figure 2. Residence time pdfs evaluated at different times using a well mixed scheme. (a) temporal evolution of the rainfall input J(t); (b) travel time pdf at time t 1 (solid line) and ensemble average of the residence time pdfs calculated as a temporal average (dotted line); (c) same as Figure 2b but for the time t 2 ; (d) same as Figure 2b but for the time t 3. Model parameters are the same as used in Figure 1. times considered in Figure 1. One may notice that the three pdfs are all very different from the corresponding travel time pdfs shown in Figure 2, and also very different one from each other: in this case, all of them reflect the temporal evolution of the rainfall input (shown in Figure 2b) and are made of a series of pronounced spikes. Indeed, p RT ðt R ; tþ should be zero when Jðt t R Þ¼0 (equation (4)). Interestingly, all the residence time pdfs are also markedly different from their ensemble average (dotted line in Figure 2), which does not differ dramatically from the ensemble average of the travel time pdfs. The plots suggest a remarkable effect of the time variability of precipitation and flow rates on the shape of individual residence and travel time pdfs, either conditional to the exit time or the entrance time. [20] Figures 3 and 4 show the dependence of the mean travel time < t T >¼ R 1 0 t T p T ðt T ; t i Þdt T on the injection Figure 3. Temporal evolution of the mean travel and residence times under the well mixed assumption. (a) temporal evolution of the rainfall input J(t); (b) temporal evolution of the mean residence time in the control volume, < t R > (solid line) and long-term average of < t R > (dashed line) also shown on the left, (c) the pdf of the mean residence time; (d) mean travel time < t T > (solid line) as a function of the injection time t i, and long-term average of < t T > (dashed line) also shown on the left, (e) the pdf of the mean travel time conditional to the injection time; (f) temporal evolution of the stream flows Q(t). The parameters adopted here refer to a dry type climate regime, with ¼ 0:1 d 1 and ¼ 1 cm; the other model parameters are the same as those used in Figure 1. time t i and the temporal evolution of the mean residence time < t R >¼ R 1 0 t R p RT ðt R ; tþdt R (Figures 3b and 4b and Figures 3d and 4d, respectively) during a time window of about 3 years. Figures 3 and 4 consider two different climate scenarios which are representative of a relatively dry climate regime (rainfall frequency ¼ 0:1 d 1 and average rainfall depth in rainy days ¼ 1 cm, Figure 3), and of a wet type regime characterized by more frequent and intense rainfall events ( ¼ 0:3 d 1 and ¼ 2 cm, Figure 4). The mean annual rainfall corresponding to the above scenarios are 370 mm and 2200 mm, respectively. The dry regime may be well representative of the climate observed in Mediterranean catchments, while the wet regime can be considered to be representative of the most humid region of the globe. It is useful to relate the climate regimes lored in the present paper with the rainfall features of the erimental catchments were tracer measurements aimed at estimating travel times have been undertaken. Most of these catchments [see e.g., McGuire and McDonnell, 2006; Kirchner et al., 2010; Soulsby et al., 2010] are located in humid areas where the annual precipitation exceeds 1500 mm (e.g., Wales, Scotland, New Zealand, northwestern US), sometimes reaching 2500 mm. Seasonal trends (dry summers versus wet winters) 6of15

7 Figure 4. Temporal evolution of the mean travel and residence times under the well mixed assumption. (a) Temporal evolution od the rainfall input J(t); (b) temporal evolution of the mean residence time in the control volume, < t R > (solid line) and long-term average of < t R > (dashed line) also shown on the left, (c) the pdf of the mean residence time; (d) mean travel time < t T > (solid line) as a function of the injection time t i, and long-term average of < t T > (dashed line) also shown on the left (e) the pdf of the mean travel time conditional to the injection time; (f) temporal evolution of the stream flows Q(t). The parameters adopted here refer to a wet type climate regime, with ¼ 0:3 d 1 and ¼ 2 cm; the other model parameters are the same as those used in Figure 1. are detectable in most cases, but they are particularly evident in northwestern US. The rainfall frequency observed in these areas typically ranges from 0.3 to 0.5 d 1 (depending on the seasonandlocation),withanaveragerainfalldepthinrainy days typically below 1:5 cm. Hence, the wet regimes considered in this paper (see also Figures 5c and 5g) are as humid as the rainfall regimes of the regions where most erimental studies have been carried out. Less erimental information is instead available for relatively arid climate conditions. [21] Figures 3 and 4 also show the temporal evolution of the relevant water fluxes (namely, the rainfall input J and of the stream flows Q, Figures 3a and 4a and Figures 3f and 4f, respectively), and the pdfs of the mean travel and residence times quantifying the time-variance of travel and residence time pdfs. Figure 3 suggests that the stochastic fluctuations of < t R > closely reproduce those of the rainfall dynamics, with a linear increase of < t R > in between the events due to the aging of the storage, and a decrease of the mean residence time in correspondence of rainfall events (due to the infiltration into the soil of young water particles). The fluctuations of < t R > around the long-term mean are faster and less pronounced that those erienced by the mean travel time < t T >, leading to a narrow pdf of the mean residence time (Figure 3c). The dynamics of the mean travel time conditional to the injection time, instead, are eminently affected by the fluctuations of the streamflow Q. In fact, most of the particles injected via rainfall events producing high discharges are quickly released from the soil, while the particles injected within dry periods have larger mean travel times. As a result, the dynamics of the mean travel time show higher amplitude fluctuations, with a series of minima in correspondence of each peak flow. The related pdf is found to be relatively wide (Figure 3e), with a long-term average similar to the long-term average of the mean residence time. Figure 4 reinforces the same considerations drawn above on the correlations between J and < t R > (and between Q and < t T >), and suggests that under wet climate conditions the residence and travel times are much smaller than in dry regimes. Note that in this case the dynamics of J are much more similar to those of Q, leading to smaller differences between the underlying travel and residence time pdfs (Figures 4c and 4e). The control exerted by J and Q on the mean residence and travel times evidenced by Figures 3 and 4 is not conditional to the specific type of mixing considered in these plots, but emerges in the more general case. Analytically, it can be lained through the dependence of p RT and p T on the ratios J/S and Q/S evidenced by equations (4) and (6). Note also that Figures 3c and 3e and Figures 4c and 4e suggest that the degree of time variance is only weakly reduced under wet climate conditions. [22] The dependence of the magnitude and timing of the fluctuations of the travel time pdfs on the underlying climate regime (under the well mixed assumption) is further lored in Figure 5 which reports the dependence on t e of the mean and the coefficient of variation of the travel time pdf conditional to the exit time (namely < t 0 T > ðt eþ¼ R 1 0 t T p 0 T ðt T; t e Þdt T, and CV 0 t T ðt e Þ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ R 1 0 ðt T < t 0 T >Þ2 p 0 T ðt T; t e Þdt T Š=<t 0 T >) under four different climate scenarios obtained by varying the rainfall frequency and the potential evapotranspiration rate (Figures 5b, 5c, 5f, and 5g). In all the cases lored the coefficient of variation and the mean of the travel time pdf are negatively correlated, with higher values of CV 0 t T associated to lower values of < t 0 T >. In particular, during each rainfall event the mean travel time conditional to the exit time decreases due the low age of the rainfall input, while the coefficient of variation of the travel time pdf conditional to the exit time increases, due to the difference between the age of the storage and the zero age of the rainfall input. The plots suggest that the temporal fluctuations of the mean and the coefficient of variation of the travel time pdf conditional to the exit time are eminently related to the rainfall input, with reduced travel times and more frequent fluctuations associated to higher rainfall frequencies. Moreover, it is shown that the decrease of the mean travel time due to enhanced evapotranspiration is significant only under relatively dry regimes because otherwise Q ET. The plots shown in Figure 5 also suggest that wet climates slightly reduce but do not eliminate the time variance of the system (with coefficient of variation of the pdf of the mean travel times not far from unit also in wet regimes). This is due to the fact that under humid regimes rainfall and streamflow variability is reduced with 7of15

8 Figure 5. Temporal evolution of the mean travel time conditional to the exit time < t 0 T > (and corresponding pdf), and temporal evolution of the coefficient of variation of the travel time pdf conditional to the exit time, CV 0 ½t T Š, under different climate scenarios: (a and b) an arid climate ( ¼ 0:15 d 1 ) with a reduced evapotranspiration (ET m ¼ 0.05 cmd 1 ); (c and d) a relatively wet climate ( ¼ 0:45 d 1 ) with a reduced evapotranspiration (ET m ¼ 0.05 cmd 1 ); (e and f) an arid climate ( ¼ 0:15 d 1 ) with enhanced evapotranspiration (ET m ¼ 0.03 cmd 1 ); (g and h) a wet climate ( ¼ 0:45 d 1 ) with enhanced evapotranspiration (ET m ¼ 0.03 cmd 1 ). Figures 5b, 5c, 5f, and 5g show the temporal evolution of the mean travel time conditional to the exit time and of the coefficient of variation of the travel time pdf conditional to the exit time (properly normalized with respect to the corresponding long-term averages, and, whose numerical value in the four cases is also indicated). Figures 5a, 5d, 5e, and 5g show the pdf of the mean travel time, normalized with respect to the long-term average. respect to dry climates, but still significant [see e.g., Botter et al., 2007]. The major effect on the time variance obtained by increasing the rainfall frequency, instead, is a significant increase of the rate of change of the fluctuations of the properties of individual travel time pdfs, with increased occurrence rate of conditional pdfs characterized by properties similar to the correspondent marginal pdf (i.e., mean travel time and coefficient of variation similar R T to their corresponding long-term averages ¼ lim T!1 0 < t 0 T > ðt R T eþdt e =T and ¼ lim T!1 0 CV0 t T ðt e Þdt e =T). This is documented by the increase of the rate at which < t T > and CV 0 t T crosses the corresponding thresholds and (indicated by the dashed line). Figures 5a, 5d, 5e, and 5h also show the pdfs of the mean travel time conditional to the exit time in the four climatic regimes lored. The four plots, which have been graphically rescaled according to the corresponding values of, suggest that the effect of the climatic regime (rainfall and evapotranspiration) on the shape of the pdf of < t 0 T > is overall weak, the major effect being only a suitable variation of the long-term average,. [23] The Figures 6, 7, and 8 analyze the travel and residence time pdfs emerging when the old water first mixing scheme is employed. In particular, Figure 6 shows the residence time pdf in correspondence of three different times taken from a sample simulation of 600 days. As ected, the residence time pdf resembles the temporal evolution of the past rainfall events. As in the well mixed scheme, the residence time pdfs are very different one from each other, and all of them differ from the correspondent marginal distribution, indicated by the dotted line. The latter is shown to have an inflection point and a mean around 50 days and does not differ much from the marginal residence time pdf obtained assuming complete mixing (Figure 2). The travel time pdfs under the old water first assumption are Dirac delta distributions, and therefore they can be described by simply representing the temporal evolution of the center of such distributions (which also corresponds to their mean). This has been done during a sample simulation of 500 days in Figure 7, which shows that the temporal evolution of the mean travel time conditional to the injection time, < t T >, and the mean travel time conditional to the exit time, < t 0 T >. The plot also reproduces the temporal evolution of the rainfall input J and of the stream flows Q (Figures 7a and 7f), and the probability distributions of < t T > and < t 0 T > embedding the time variance of the system. Figure 7 shows that, in analogy to what observed for the well mixed scheme, the mean of the travel time pdf conditional to the exit time decreases abruptly right after each major flood, and increases almost linearly during time periods characterized by moderate or low-rainfall inputs. Conversely, the mean travel time conditional to the injection time grows right after each rainfall event and decreases almost linearly in between the events. Figure 7 also represent the entrance and exit times of two separate pulses, characterized by different travel times. As ected, the pdf of the mean travel time in the two cases are similar as the range of variability of < t T > and < t 0 T > is the same. 8of15

9 Figure 6. Residence time pdfs evaluated at different times using the old water first scheme. (a) Temporal evolution of the rainfall input J(t); (b) travel time pdf at time t 1 (solid line) and marginal residence time pdf, calculated as the ensemble average of the conditional pdfs (dotted line); (c) same as Figure 6b but for the time t 2 ; (d) same as Figure 6b but for the time t 3. Model parameters are the same as used in Figure 1. Figure 7. Mean travel time conditional to the exit time and conditional to the injection time according to the old water first scheme. (a) Temporal evolution of the rainfall input. Highlighted are the entrance times of two specific pulses labeled as 1 and 2. (b) Temporal evolution of the mean travel time as a function of the injection time; (c) pdf of the mean travel time calculated considering the distribution of individual mean travel times; (d) travel time as a function of the exit time; the plot emphasizes that the travel time conditional to a given exit time t e is equal to the travel time conditional the specific injection time t i ¼ t e Tðt e Þ. (e) same as Figure 7c but calculated by considering the distribution of travel times conditional to the exit time; (f) temporal evolution of stream flows; highlighted the times at which the pulses 1 and 2 leave the control volume. Model parameters are the same as used in Figure 1. [24] Figure 8 represents the temporal evolution of the mean residence time < t R >, and its pdf, during the same simulation shown in Figure 7. The plot emphasizes that the mean residence time is smaller than the corresponding mean travel time, making the distinction between residence and travel time pdfs a key feature of the transport processes. The temporal pattern of < t R > in this case leads to a reduced variability around the mean, if compared with that observed under the well mixed assumption. Indeed, the negative jumps in correspondence of the events are smaller, due to the smaller average residence times observed in the soil. Moreover, in between the events, the average residence time increases less then linearly because the aging of the stored particles is compensated by the release of particles much older than the average. [25] A direct comparison of the two mixing schemes lored in the paper is presented in Figures 9 and 10. In particular, Figure 9 shows the marginal travel time pdfs Figure 8. Temporal evolution of the mean residence time according to the old water first scheme. (a) Temporal evolution of the rainfall input; (b) temporal evolution of the mean catchment residence time; (c) pdf of the timedependent catchment mean residence time (also shown the long-term average as a dashed line). Model parameters are the same as used in Figure 7. 9of15

10 Figure 9. Comparison between the marginal travel time pdf < p T > evaluated using the two mixing schemes considered in this paper: complete mixing (dashed lined) and old water first (dotted line). The two panels refer to (a) wet and (b) dry climate conditions (model parameters are the same as used in Figure 3 and 4). (< p T >) calculated as the average of individual travel time pdfs conditional to the injection time: Z 1 T < p T >¼ lim p T ðt T ; t i Þdt i ; (20) T!1 T 0 under a relatively wet (Figure 9a) and a relatively dry (Figure 9b) climate regime. In the wet regime, the type of mixing schemes weakly impacts the marginal travel time pdf, and does not change the mean of the distribution. This is because the decrease of the age of the storage associated with the old-water first scheme is compensated by the preferential affinity of the flow process for the older ages. Conversely, in the dry regime, the two schemes are characterized not only by a different shape of the marginal travel time pdf (monotonic versus hump shaped), but also by a remarkably different mean. For the old-water-first scheme, indeed, the age of the particles leaving V via Q is much higher than the mean age of the storage, leading to a marked increase of the mean travel times (500 days versus 100 days). The effect of the mixing scheme on the chemical features of the streamflows is further investigated in Figure 10, where the temporal Figure 10. Comparison between the streamflow solute concentrations, C Q, obtained using the two mixing schemes considered in this paper: complete mixing (solid thick line) and old water first (dotted line). Also shown is the input rainfall concentration, C J (continuous thin line). All the concentrations are normalized with respect to the average rainfall concentration C 0. The two panels refer to (a) a wet and (b) a dry climate regimes, and the model parameters are the same as used in Figures 3 and 4. evolution of the (normalized) streamflow concentration C Q calculated from the travel time pdf p T 0 as discussed in section 2 (but see also equation (22)) are compared. In particular, this paper shows C Q ðtþ resulting from computing p 0 T according to the old-water first scheme (equation (15), thick dashed line) and to the well mixed scheme (equation (11), thick solid line). Figure 10 considers either both wet and dry climates, and assumes for the input concentration C J a sinusoidal behavior with an amplitude equal to the mean input concentration C 0, and a period of 1 year, to which is superimposed a uniform white noise with amplitude C 0 =2. Note that the period characterizing the major fluctuations of rainfall tracer concentrations may depend on the type of solutes considered but, for most natural tracers, is typically around 1 year to reflect the seasonal variability of climate-driven fractionation processes [De Walle et al., 1997; Perini et al., 2009]. The white noise embeds instead the short-term random variability of physical and chemical processes affecting solute rainfall composition (e.g., temperature, atmospheric composition). The synthetic input signal considered in this example is only taken as a surrogate of the typical behavior evidenced by 10 of 15

11 commonly employed tracers like chloride or d 18 O [see e.g., McGuire and McDonnell, 2006; McDonnell et al., 2010, Hrachowitz et al., 2010, Kirchner et al., 2010]. The plot shown in Figure 10a suggests that higher values of C Q are associated with the old water scheme (which simply redistributes in time the input concentration according to a timevariable delay), whereas the well mixed schemes ensures the damping of the input signal due to mixing of ages. Nevertheless, the seasonal evolution of the concentration is similar in the two cases (a byproduct of the similarity of the mean travel times). Figure 10b shows the same quantities reported in Figure 10a, but for a drier rainfall regime. In this case the chemical response of the simulated catchment is dramatically impacted by the underlying mixing scheme. When a well mixed scheme is used, the output signal C Q is strongly damped and delayed with respect to the input concentration C J, with limited seasonal dynamics induced by the huge mixing. When a old-water-first scheme is used, instead, the output signal shows larger fluctuations and the seasonal variability of the input concentration is preserved, being only stretched or compressed in time. These results indicate that the occurrence of preferential flow displacing old (or event) water and reducing the mixing of ages in soil states may lead to a notable increase the temporal fluctuations of the streamflows concentrations. 5. Inferring Travel, Residence and Evapotranspiration Time pdfs [26] Travel time pdfs are usually estimated by monitoring the release of suitable tracers through the hydrologic response of a catchment. Field studies are typically based on the following two types of eriments: (1) uniform and instantaneous injections of a known mass of a non native, conservative tracer (in this case streamflows Q(t) and streamflow flux concentrations C Q ðtþ are typically measured after the injection); and (2) the monitoring of passive scalars (e.g., isotopes) naturally and continuously introduced into the soil through rainfall (in the latter case, rainfall rates, streamflows and the corresponding flux concentrations are continuously measured during a suitable time window). [27] Instantaneous external injections of a passive tracer over a whole catchment are often difficult to be erimentally simulated due to the large spatial scales involved. Moreover, the result of such eriments may be dependent on the specific hydrologic conditions observed during and after the application. On the other hand, the mathematical description of the response to a pulse injection of solutes is relatively simple. The solute mass flux produced by a pulse injection is indeed fully described by the travel time pdf conditional to the time when the injection occurred (assumed for simplicity equal to zero in the following), p T ðt; t i ¼ 0Þ, which represents the memory of the solute mass discharge to a disturbance. The erimental monitoring of the mass flux needs to be characterized by high-frequency sampling (given the pronounced temporal variability of the output signal, see Figure 1), and by an adequate continuity in time, allowing for a proper quantification of the remobilization of old water particles occurring during each event [McGuire et al., 2007]. The travel time pdf conditional to a given (fixed) exit time, instead, cannot be directly inferred from concentration measurements after pulse solute injections. What can be actually observed through an instantaneous injection of a nonnative tracer at t i ¼ 0, is instead a subset of pdfs conditional to different exit times, p 0 Tðt; tþ, which represents the memory of the stream flux concentration to a disturbance applied at t ¼ 0, and will be thus termed concentration memory function. The concentration memory function differs in general from any p 0 T evaluated at a fixed exit time (except than under steady state conditions). For instance, all the concentration memory functions p 0 T ðt; t t iþ are not pdfs, even though they must decay to zero for large times. Hence, also under humid climates (Figures 4 and 5), a single injection eriment allows one to characterize only a single, specific catchment response, which differs in general from the responses to future or past disturbances. [28] The description of the response to a continuous input of solute occurring through the rainfall forcing (which includes most of the applications with isotopes or other natural tracers) is mathematically more involved, though being more informative of the transport processes taking place in catchments. The chemical features of the input signal in this case may strongly influence our ability to reconstruct the fluctuations in time of the travel time pdfs. This issue will be addressed by investigating the ability of a time invariant operator (say, f(t)) to properly approximate, once convoluted with the input rainfall concentration, the temporal evolution of the streamflow concentration, as postulated by the lumped convolution approach, where the conditional travel time pdf p 0 T is replaced by a time invariant function f ðþ to make easier its identification [e.g., Kirchner et al., 2000, 2010; McGuire and McDonnell,2006;Broxton et al., 2009]: C Q ðtþ ¼ Z t 1 C J ðt i Þf ðt t i Þdt i : (21) The critical point here is to note that the equality C Q ðtþ ¼ Z t 1 C J ðt i Þp 0 T ðt t i; tþdt i ¼ Z t 1 C J ðt i Þf ðt t i Þdt i : (22) (i.e., the ability of a linear and time invariant scheme to fit the output concentrations resulting from a time-variant and nonlinear system) does not guarantee the equivalence between f and p 0 T (for any fixed exit time), as long as p0 T is time variant. In other words, only because of the timevariance, the ability of a time invariant operator f in reproducing observed chemographs, at least from a mathematical viewpoint, may not be a sufficient condition to conclude that f is a good proxy of the travel time pdf. [29] More specifically, the time-invariant operator obtained by deconvolving output and input concentrations may be different not only from any single travel time pdf (due to the time variance), but also from the correspondent ensemble average and its shape depends on the features of the input concentration. To provide an example of this fact, in Figure 11 the streamflow composition obtained by means of the time-variant, well-mixed scheme (section 3.1) is compared to the chemical response that would be obtained in case rainfall and streamflows concentrations were linked via a time invariant relationship of the type shown by equation (21). Different types of rainfall concentrations C J have been lored by superimposing a white noise to a sinusoidal 11 of 15