Water temperature for a river node is computed using simple mixing--a weighted average of the water temperatures in the inflows from upstream, tributaries, return flows, and groundwater inflows.
As water flows downstream, the water temperature can change due to gains of heat from net solar short-wave radiation and atmospheric long-wave radiation, and losses of heat due to conduction, convection and evaporation.
The volume for a reach is defined by its length and average cross sectional area, and the assumption of steady state during the time step. A heat balance equation is written for each reach on the river.
where the first term on the right-hand side is the upstream heat input to the stream segment with constant volume, V (m3) expressed as a relationship of flow, Qi (m3/time) and temperature, Ti at the upstream node. The second term is the net radiation input, Rn, to the control volume with density rho, and Cp the specific heat of water and H (m), the mean water depth of the stream segment. The third term is the atmospheric long-wave radiation into the control volume, with the Stefan-Boltzmann constant, Tair the air temperature (C), a, a coefficient to account for atmospheric attenuation and reflection and the air vapor pressure, eair. The fourth term is the heat leaving the control volume, while the fifth term is the long-wave radiation of the water that leaves the control. The sixth and seventh terms are the conduction of heat to the air and the removal of heat from the river due to evaporation. The terms f(u) and g(u) are wind functions, and D is the vapor pressure deficit. The temperature, Ti+1 is solved for the downstream node with a fourth-order Runge-Kutta and is the boundary condition temperature for the next reach (after mixing of any other inflows into the downstream node is considered).