A Two‐Dimensional Closed‐Form Analytical Solution for Heat Transport With Nonvertical Flow in Riparian Zones

Heat as a tracer has received increasing attention in recent decades and has become a popular and widely used tool for quantifying streambed fluxes, among many other applications. Nonvertical flow component is an important parameter for heat transport in riparian zones. However, previous investigations using heat as a tracer in the quantification of streambed fluxes frequently overlooked or simplified nonvertical flow component with erroneous presumptions. In this study, a novelty two‐dimensional closed‐form analytical solution for heat transport with vertical and nonvertical flow components and arbitrary initial and boundary conditions under losing conditions is presented for the first time using Green's function method. The new model is tested by finite‐element solution and stochastic modeling. The capabilities of the new analytical solution are demonstrated using field data. Results show that the nonvertical flow component has a significant impact on heat transport in streambed. The temperature difference between the streambed and the surface water increases as the nonvertical flow component increases, and the amplitude of the temperature curves decreases with increasing horizontal distance from the center of stream at the same depth due to the influences of nonvertical flow. Stochastic modeling shows that an increasing nonuniform flow can result in overestimation of vertical and nonvertical flow components near the bank of stream, and the new analytical model works well in a heterogeneous streambed when the variance of natural logarithm of the streambed hydraulic conductivity is less than 0.10 ( σlnK2≤0.10 ${\sigma }_{\mathrm{ln}K}^{2}\le 0.10$ ). The vertical and nonvertical flow components are more sensitive to temperature‐time data than the other parameters. The new analytical model is a significant advancement of the previous one‐dimensional analytical models which only considered vertical flow.