We define tracer age as the time elapsed since the incoming tracer mass dissolved into the system's storage. The dissolution time coincides with the time of entry when the tracer is already in dissolved form or dissolves instantaneously, but it may differ when dissolution is delayed. Based on this definition, the tracer age distribution ρ←(T,t) is the distribution of tracer mass with respect to age T at time t. This concept applies to the tracer mass stored within the system (either in dissolved or sorbed form) and to that leaving the system.

The random sampling assumption introduced in Sect.  implies that all water age distributions are identical to the storage age distribution, and thus, the streamflow concentration equals the storage concentration. Under this assumption, it can be shown that tracer mass is also uniformly sampled, leading to equal mass age distributions across storage and fluxes: ρQ←(T,t)=ρET←(T,t)=ρR←(T,t)=ρS←(T,t). This relationship holds even in the presence of reactions that may alter tracer concentration during transport through the system.

We introduce a tracer age balance, which is analogous to the concept of water age balance, describing the evolution of tracer mass within the system as a function of both time and age. Under the random sampling assumption, the tracer age balance corresponding to the mass balance in Eq. () is given by: 13d[MS(t)ρS←(T,t)]dt=m˙I(t)δ(T)-m˙Q(t)ρS←(T,t)-m˙ET(t)ρS←(T,t)+m˙R(t)ρS←(T,t) with initial condition ρS←(T,t=0) = ρS0←(T) and null boundary condition. The input tracer mass flux m˙I is again associated with a Dirac delta function δ(T), as by definition the new tracer mass dissolving into the system has age zero. The mass age balance described by Eq. () may appear similar to the one introduced by , but it has an important structural difference in the way that age is tracked. While the mass age balance by addresses the mass (potentially of different ages) associated with a water parcel of a given age, our balance addresses directly the mass of a given age (regardless of the age of the water it is associated with). In many cases, these two approaches coincide in practice. However, they are notably different in case of dissolution processes: new dissolving mass is associated with (water of) multiple ages in the approach of , while it is only associated with an age of zero in our tracer-oriented balance.

Incorporating Eqs. ()–() into Eq. (), and moving the input mass to the boundary condition ρS←(T=0,t) = m˙I(t)/MS(t), Eq. () can be developed into the first-order partial differential equation: 14∂ρS←(T,t)∂t+∂ρS←(T,t)∂T=-Q(t)+α⋅ET(t)R⋅S(t)+k+dMSdtMS(t)ρS←(T,t) which can be solved analytically see Appendix and. The general solution is: 15ρQ←(T,t)=ρS←(T,t)=m˙I(t-T)MS(t)exp⁡-∫t-TtQ(τ)+α⋅ET(τ)R⋅S(τ)+kdτ Both the water TTD (Eq. ) and the tracer TTD (Eq. ) have an equivalent formula where the input fluxes rather than output fluxes appear inside the integral (see Appendix ). However, such formula does not allow one to see the model parameters explicitly and so it is not reported. Because the hydrologic and mass fluxes may vary over time, the mean tracer transit time will change over time as well, but the long-term mean tracer transit time mMTT‾ can be approximated through the mean of ρQ← at steady state: 16mMTT‾=R⋅S‾Q‾+α⋅ET‾+k⋅R⋅S‾ The mean transit time does not have much meaning in practical applications because it's very difficult to estimate reliably , but it's useful here for interpreting the analytical expressions.

To derive the formula for the mass breakthrough curve in streamflow (i.e. the fraction of the input tracer mass found in streamflow at subsequent lag times), we proceed as we did with the water breakthrough curve, by invoking the Niemi relationship (Eq. ) applied to the mass age distributions. We term θm˙Q the partitioning coefficient that quantifies the fraction of mass entering the system at time ti that will ultimately exit the system via streamflow. The function ρQ→(T,ti) denotes the “forward” TTD for tracer mass entering at time ti and leaving through streamflow (which is formally equal for all output fluxes in a randomly sampled system). The result for the mass normalized breakthrough curve is: 17mNBTC(T,ti)=θm˙Q(ti)ρQ→(T,ti)=Q(ti+T)R⋅S(ti+T)exp⁡-∫titi+TQ(τ)+α⋅ET(τ)R⋅S(τ)+kdτ Equations () and () provide a complete description of tracer transit times through the simplified hydrological system described here. They are the tracer counterparts to the water transit time distributions (Eqs.  and ) and as such they are the starting point to explore how tracer TTDs are different from water TTDs. While the more general case of non-randomly sampled systems could theoretically be investigated through the introduction of tracer mass-StorAge Selection functions (the tracer equivalent of water StorAge Selection functions), those would introduce substantial technical difficulty and are not necessary to pursue our goals, i.e. understanding the key principles governing the discrepancies between water and tracer transit times.

The water and mass age equations (Eqs.  and ) are first-order, linear, partial differential equations whose solutions (Eqs.  and ) are exponential distributions with time-variable exponents that depends primarily on the hydrologic fluxes and storage.

In the case of an ideal passive tracer – that is, one that does not undergo sorption, evapoconcentration or decay (R=1, α=1, k=0) – the breakthrough curves for water and tracer mass (Eqs.  and ) coincide. This is expected as tracer mass in our system moves transported by water. However, even if each mass application moves exactly like water, the age distributions (i.e., backward TTDs) of water and tracer are not the same because the input timeseries (or more precisely, the timeseries of the ratio between input and storage) are different. This is clearly visible when comparing Eqs. () and (), which only differ in the term in front of the exponential. As the age distribution in streamflow represents the contribution of past inputs to present streamflow, different input patterns will generate different distributions. Concrete examples with seasonal tracer input patterns are given in Sect. .

Another insight that emerges is that one does not need tracer data to compute the breakthrough curves mNBTC (Eq. ) nor the long-term mean transit time mMTT (Eq. ), as they are entirely determined by the hydrologic variables (Q, ET and S) and the transport parameters (R, α and k). The effect of such parameters can thus be investigated theoretically. The retardation factor R≥1, which is used to reduce the total tracer mass storage to the soluble mass storage, always appears as a multiplier of the water storage S and can be virtually interpreted as a storage magnifier. Thus, the effect of sorption is effectively equivalent to the effect of a larger storage: it will delay the tracer and result in longer tracer transit times. The evapoconcentration parameter α can be seen as a regulator directing mass to ET. When α<1, less mass will go to ET, resulting in tracer accumulation in storage and thus longer transit times. Conversely, when α>1 (net tracer extraction) the larger tracer output rate will result in a depleted tracer storage, with faster turnover and shorter transit times. Degradation can be seen just as an additional tracer output flux controlled by the kinetic rate constant k [T⁻¹]. Because of this additional exit pathways, which results again in a depletion of the storage mass and faster turnover, transit times through the system are reduced. Therefore, the outflowing mass will generally be younger when degradation is faster.

While the theoretical analysis of tracer TTDs proves already insightful, further explorations under realistic, time-variable conditions based on the numerical implementation described in Sect. are presented in Sect. –.

As shown theoretically in Sect. , if the input timeseries of water and passive tracers differ, these differences will be reflected in the age distributions. Using the numerical implementation, we illustrate these discrepancies in Fig. , by considering a seasonally variable input timeseries of passive tracer mass (R=1, α=1, k=0) to the system (Fig. d), with larger mass inputs in the summer and lower mass inputs in the winter. This seasonal variability may be representative of seasonal patterns in water stable isotope content or agricultural products application. While the overall time-averaged water and tracer TTDs appear similar (Fig. a), as they reflect approximately steady-state conditions, the individual (time-varying) TTDs vary considerably, as shown by the thin lines in Fig. a. Moreover, the input tracer seasonal cycle influences the TTD of tracer mass over different times of the year. During the winter months (Fig. b), when tracer inputs are low, most of the tracer mass leaving the system originates from inputs that occurred during previous high-input periods (i.e. in summer). This is reflected in the time-averaged tracer TTD which shows most of its weight around a transit time of approximately 200 d. In contrast, during summer (Fig. c), a large fraction of mass in or leaving the system will be relatively young, due to the high input rates of (necessarily young) tracer at that time. Because the water input fluxes in this study exhibit less seasonality than the tracer inputs, such patterns are not observed in the water TTDs. As a result, even for passive tracers, the transit times of water and tracer leaving the system differ under these conditions. These discrepancies imply that, although the convolution integral relating output to input tracer concentrations (Eq. 1) remains valid, it may also be reformulated explicitly in terms of tracer-specific transit time distributions.

Based on Eq. (), it is clear that tracer-specific characteristics influence the shape of tracer age distributions. To illustrate this effect, we compute the overall time-averaged age distributions by varying one parameter at a time, allowing us to isolate and visualize the influence of each parameter separately (Fig. ). As expected, increasing the retardation factor leads to longer transit times, as reflected by the flattening of the TTDs for higher retardation values in Fig. a, compared to the TTD of water. The increased sorption effectively slows the movement of tracer mass through the system. Tracer mass fluxes leaving through evapotranspiration have the potential to slow down (α<1) or accelerate (α>1) the tracer movement through the system compared to water (Fig. b). As explained in Sect. , this behavior arises from how the evapoconcentration factor influences the total mass stored in the system, ultimately affecting the tracer's turnover rates. Finally, while ET can both accelerate or decelerate the tracer's transport through the system, degradation consistently accelerates tracer transport relative to water, potentially by large factors depending on the decay rate associated with the tracer (Fig. c). Similar to the effect of high mass rates extracted by ET (α>1), this behavior is explained by the reduction in total tracer mass stored in the system under high decay rates, which increases the overall turnover and shortens transit times.

While age distributions reflect the distribution of transit times of tracer mass leaving the system at a given time–strongly shaped by the history of input time series–the tracer breakthrough curves in streamflow (mNBTC) describe the distribution of transit times for a tracer mass parcel that entered the system at a specific time. In addition, mNBTCs reflect the fraction of that tracer mass that ultimately reached streamflow at subsequent time steps. They are therefore influenced both by the overall velocity of tracer mass through the system and by how that mass is partitioned among the different outflow fluxes. To assess how different tracer characteristics influence the mNBTCs, we computed the mNBTC for every individual tracer mass input across the study period. Following the approach in Sect. , we varied each of the three tracer parameters separately, keeping the other two fixed at their passive-tracer values. Results for a single, arbitrary input time step are presented in Fig. , where panels (a)–(c) show the mNBTCs, and panels (d)–(f) display the corresponding cumulative mNBTCs for the same tracer mass injection, thereby illustrating the progressive recovery of the input mass in streamflow over time.

Individual, time-varying mNBTCs are typically irregular (Fig. a–c), reflecting the dynamic nature of the hydrological system considered here. As previously shown for water TTDs by , discharge plays a key role in shaping the tracer breakthrough curves, as it is a multiplier of the exponent in Eq. () and regulates the exported mass. A discharge timeseries is reported for comparison in Fig. a. The transport parameters affect tracer transport in intuitive ways. An increase in sorption (higher retardation factor R) visibly slows the release of tracer mass to the stream (Fig. a), as shown by the flattening of the mNBTC. This effect is solely due to a reduction in tracer velocity through the system as the mass partitioning among the different outputs is unchanged (because the other parameters are held at their passive-tracer settings). All curves shown in Fig. d will reach the same plateau of 0.41, corresponding to the partitioning term θm˙Q of the considered mass application, although this is not apparent in Fig. due to the cutoff at 400 d. In contrast, increasing the fraction of mass that is extracted by ET (α>1, Fig. b, e) accelerates turnover through the system but reduces the tracer recovery in streamflow, leading to a breakthrough curve that is always lower than that of the passive tracer. The effect of changing α only becomes relevant when ET fluxes are important. This is visible in Fig. b, e where summer starts around day 100. Finally, higher degradation rates (lower DT50) decrease tracer recovery in streamflow and shorten transit times due to faster mass turnover (Fig. c, f).

Tracer application (or tracer “labeling”) experiments are routinely used to investigate transport processes in soils. In some cases, contrasting breakthrough curves have been reported for different tracers e.g., likely due to differences in sorption, degradation, or interactions with root water uptake. The analytical breakthrough curves developed here, especially when arranged in parallel or in series to move beyond a single randomly sampled unit, are useful for the quantitative interpretation of empirical data from tracer experiments, particularly when multiple tracers are used.

While tracer characteristics influence their transit times through the system, they also affect the ultimate fate of the tracer mass entering the system. As described in Sect. , these characteristics shape the tracer breakthrough curves in streamflow, but more generally, they influence how mass is ultimately partitioned among the different output fluxes. A convenient way to explore this partitioning is to compute the partitioning coefficients. Because the system is dynamic, these partitioning coefficients will differ for each individual mass input. Accordingly, we compute the three partitioning coefficients (θm˙Q, θm˙ET, and θm˙R) corresponding to the three output mass fluxes, for every tracer mass input. The partitioning coefficients were computed by integrating the breakthrough curves for discharge, evapotranspiration and degradation fluxes over all transit times T (see formulas in Appendix ).

Figure presents box plots of the three partitioning coefficients computed over the entire time window, for four cases with different combinations of tracer-characteristic parameters. The spread of the partitioning coefficients in the box plots is primarily driven by variations in hydrological fluxes and storage, which may include a seasonal component, and is therefore largely influenced by the specific time series used in this study. In the first case (Fig. a), the parameter set is chosen such that the partitioning coefficients are approximately balanced, with the tracer behaving as a passive tracer in all respects except for its decay. In the second case (Fig. b), the retardation factor is increased, leading to a significant shift in tracer mass partitioning towards decay. This outcome reflects the fact that the combination of strong sorption and degradation is particularly effective at removing mass from the system via decay, at the expense of the two other output fluxes. This effect is also evident in the steady-state analytical expressions for partitioning coefficients (Appendix ), where the product of the decay rate k and the retardation factor R directly influences the partitioning toward reactive loss. The third case (Fig. c) represents a scenario with prominent evapoconcentration (i.e., α≪1). As a consequence, the mass partitioned toward evapotranspiration drops sharply, while partitioning toward both streamflow and decay increases. In the final case (Fig. d), the DT50 value is doubled, effectively halving the decay rate. As expected, this leads to a substantial decrease in the partitioning toward decayed mass, accompanied by increases in the fractions reaching streamflow and evapotranspiration. More generally, Fig. c–d illustrate that similar tracer recovery in streamflow can result from very different internal processes–high evapoconcentration in one case, and low degradation in the other.