## 3 Results and Discussion  
With Equations (2), (5), (8)-(10), and (16), we are in the position of evaluating the variance of outflow concentration by combining Equations (13) and (14). The evaluation of the integral in Equation (11) cannot be performed analytically for the case of \(\alpha\) in Equation (2) being a fractional number. However, to take the advantage of closed-form solutions for Equations (13) and (14), two cases are considered: (1) \(\alpha=1\) and (2) \(\alpha=2\). These expressions will be used to investigate the influence of various parameters on outflow concentration variability. It is expected that the concentration variability behavior for the case of \(\alpha\) being a fractional number will be qualitatively similar to that of being an integer, although not quantitatively. For the case of \(\alpha=1\), Equation (2) becomes  
\[g(t)=\frac{\eta_{3}}{\tau_{m}} \tag{22}\]  
where \(\eta_{3}=\exp(-t/\tau_{m})\). Based on Equations (8)-(10), (16), and (22), the evolutionary power spectrum for the outflow concentration process in Equation (14) is given by  
\[S_{CC}(t;\omega)=\sigma_{\xi}^{2}\frac{\lambda\tau_{C}^{2}}{\pi(1-\gamma_{2}) \mu_{1}\mu_{2}\mu_{3}}\{\phi_{1}+2[\phi_{2}+\phi_{3}\cos(\omega t)+\phi_{4} \rho\sin(\omega t)]\} \tag{23}\]  
which leads to the following expression for the variance of outflow concentration:  
\[\sigma_{C}^{2}(t)=\frac{\sigma_{\xi}^{2}\tau_{C}^{2}\gamma_{1}}{(1-\gamma_{2} )^{2}}\{\frac{T_{1}}{1+\gamma_{1}}+\frac{T_{2}}{1+\gamma_{2}}[\frac{1}{1+ \gamma_{1}}-\frac{\gamma_{2}^{2}}{\gamma_{1}+\gamma_{2}}]+2\frac{T_{3}}{(1- \gamma_{1}^{2})(1+\gamma_{2})}+2\frac{\gamma_{2}^{2}T_{4}}{(1+\gamma_{2})( \gamma_{1}^{2}-\gamma_{2}^{2})}\} \tag{24}\]  
In Equations (23) and (24), \(\gamma_{2}=\tau_{m}/\tau_{C}\), \(\mu_{3}=1+\tau_{m}^{2}\omega_{2}\), and expressions for \(\phi_{1}-\phi_{4}\) and \(T_{1}-T_{4}\) are given, respectively, in the Appendix A. For a quantitative evaluation of the analytical developments presented in this section, it is convenient to present the analytical results as the following dimensionless variables: \(t/\lambda\) (dimensionless time), \(\tau_{m}/\lambda\) (dimensionless catchment mean travel time), \(\tau_{C}/\lambda\) (dimensionless solute response time), \(\tau_{m}/\tau_{C}\) (a ratio of the catchment mean travel time to the solute response time), \(t/\tau_{C}\) (\(=(t/\lambda)/(\tau_{C}/\lambda)\), a ratio of the dimensionless time to the dimensionless solute response time), and \(\tau_{C}\omega\) (a dimensionless frequency). Note that dimensionless graphs provide much more information than when dimensions are included because it is possible to cover a wider range of the parameters. Figure 1 illustrates the effect of the inflow response time scale \(\tau_{C}\) on the variance of outflow concentration (Equation (24)) at various time scales. It is seen that the inflow with a larger \(\tau_{C}\) will lead to higher outflow concentration variability. An increase in \(\tau_{C}\) enhances the inflow concentration variability and, in turn, the outflow concentration variability. i.e., the temporal variability of outflow concentration persists longer for larger values of \(\tau_{C}\). The variance of outflow concentration represented as function of \(\tau_{m}\) is depicted in Figure 2 using Equation (24) for various values of \(\lambda/\tau_{C}\). The parameter \(\tau_{m}\) apparently plays the role in smoothing out the concentration variability. This can be attributed to the reduction in the persistence of outflow concentration perturbation with \(\tau_{m}\) which reduces the concentration variability. Here, the persistence is referred to the tendency for low values to be followed by low values and high values by high values. Additionally, from Figure 2, we see that for a fixed \(\tau_{m}\), an increase in \(\lambda\) results in higher variability of outflow concentration. A larger value of \(\lambda\) increases the persistence of inflow concentration and, in turn, the outflow concentration, leading to higher concentration variability.