## 2 Methods and Background
### Flow and Mixing Characteristics at a Confluence
#### 2.1.2 Transverse Mixing at a Confluence  
Under high-momentum-flux-ratio conditions, the shear plane caused by tributary inflows induces helical motion, as shown in Figure 1. This helical motion influencestransverse mixing in confluences, which can be divided into slow mixing or rapid mixing according to the hydraulic characteristics. The width of a solute cloud can be defined as six times the standard deviation of a concentration curve under the assumption of the Fickian mixing [39]. When the concentration changes are less than 5% throughout the channel width, transverse mixing is considered to be complete [40]. When the distance between the solute input source and transverse mixing completion point (\(L_{m}\)) is five to 10 times the width of the channel, it is considered to be rapid mixing [30]. In the case of slow mixing, the distance between the source and completion point is 100 times the channel width [31; 41]. The transverse concentration changes caused by tributary inflows can be evaluated using a standard deviation defined as  
\[\sigma_{r}=\frac{\sigma_{ds}}{\sigma_{us}} \tag{4}\]  
where \(\sigma_{r}\) is the dimensionless mixing metric, and \(\sigma_{us}\) and \(\sigma_{ds}\) are the standard deviations of transverse concentration variations in the upstream and downstream cross-sections, respectively. If the value of \(\sigma_{r}\) is one, then the concentration changes throughout the upstream and downstream widths of the channel are the same, and transverse mixing does not occur. In contrast, if the value of \(\sigma_{r}\) is zero, then transverse mixing is considered to be complete [41]. For a transient solute input from a tributary, transverse mixing can be analyzed by integrating time-varying concentration curves over time and converting them into steady-state conditions. The dosage curve is the time-integrated value of the concentration passing through one point, which can be calculated using the following equations [42]:  
\[\theta=\int_{0}^{\infty}C(x,\ y,\ t)dt \tag{5}\]  
\[\Theta=\int_{0}^{1}\theta(x,\ \eta)d\eta \tag{6}\]  
\[\eta=\frac{\int_{0}^{y}\sqrt{\overline{u}^{2}+\overline{v}^{2}}hyd}{Q}=\frac {q}{Q} \tag{7}\]  
where \(\theta\) is the time integration of the concentration passing a point, \(C\) is the depth-averaged concentration, \(\Theta\) is the total dosage in a cross-section, \(\eta\) is the dimensionless cumulative discharge, \(\overline{u}\) and \(\overline{v}\) are the depth-averaged velocity in the \(x\) and \(y\)-direction, respectively, \(q\) is the cumulative discharge, and \(h\) is the water depth. The transverse dispersion coefficient can be estimated from the dosage curves shown in Equation (5) using the moment method as  
\[D_{T}=\frac{U}{2}\frac{d\sigma_{\theta}^{2}}{dx} \tag{8}\]  
\[\sigma_{\theta}^{2}=\frac{\int_{0}^{W}\theta y^{2}dy}{\int_{0}^{W}\theta\ dy} \tag{9}\]  
where \(D_{T}\) is the transverse dispersion coefficient, \(\sigma_{\theta}^{2}\) is the variance of the dosage curves, and \(W\) is the channel width. The recirculating flow in a confluence can form a storage zone that traps the solute inflow from a tributary. Such a storage zone increases the residence time of solutes and generates a long-tailed concentration curve [43]. The mixing characteristics in rivers change as a result of these storage zones, which affects far-field mixing results.