## 2 Theoretical Model for Water Temperature Field
### Vertical Profile Temperature Calculation Model of Water  
When considering the influence of solar radiation, for horizontally homogeneous water bodies, the momentum, temperature, and salinity of any fluid parcel satisfy the following governing equations [41]:  
\[\frac{\partial u_{x}}{\partial t}=fu_{y}+\frac{\partial}{\partial z}[(k_{m}+v) \frac{\partial u_{x}}{\partial z}] \tag{24}\]  
\[\frac{\partial u_{x}}{\partial t}=fu_{y}+\frac{\partial}{\partial z}[(k_{m}+v) \frac{\partial u_{x}}{\partial z}] \tag{25}\]  
\[\frac{\partial T}{\partial t}=\frac{\partial}{\partial z}[(k_{t}+aa)\frac{ \partial T}{\partial t}]-\frac{\partial}{\partial z}(\frac{\psi_{ms}}{\rho c_{p} }) \tag{26}\]  
\[\frac{\partial S}{\partial t}=-\frac{\partial}{\partial z}[(k_{s}+\gamma) \frac{\partial S}{\partial z}] \tag{27}\]  
Among them, \(t\) represents time; \(u_{x}\) and \(u_{y}\) denote the velocity components of the fluid parcel along the \(x\) and \(y\) directions, respectively; \(f\) is the Coriolis parameter; \(k_{m}\), \(k_{t}\), and \(k_{s}\) are the eddy diffusion coefficients for momentum, temperature, and salinity, respectively; \(v\), \(aa\), and \(\gamma\) are the molecular diffusion coefficients for momentum, temperature and salinity, respectively; \(T\) and \(S\) represent the temperature and salinity of the fluid parcel, respectively; \(\rho\) is the density of the fluid parcel and \(c_{p}\) is the specific heat of water. In temperature-stratified water bodies, the formation of the thermocline results in relatively minor horizontal temperature variations. Even under extreme hydrological and wind disturbances, which may temporarily alter longitudinal and transverse gradients, horizontal temperature gradients dissipate rapidly due to thermal convection and conduction, restoring the water temperature to its vertical distribution state. Consequently, a one-dimensional vertical heat balance equation is employed to model the thermal distribution of the water body. By discretizing the water body into a series of horizontal microelements along the vertical axis and assuming negligible temperature variations within each microelement, heat transfer and diffusion are predominantly vertical. The heat balance equation for any horizontal microelement is thus simplified from Equation (28) to the following equation:  
\[\frac{\partial T}{\partial t}=\frac{1}{A_{s}}\frac{\partial}{\partial z}[(A_{s }d_{z})\frac{\partial T}{\partial t}]-\frac{1}{A_{s}}\frac{\partial}{\partial z }(\frac{A_{s}\psi_{ms}}{\rho c_{p}}) \tag{28}\]  
where \(A_{s}\) represents the area of the horizontal microelement layer, and \(d_{z}\) denotes the vertical diffusion coefficient. By substituting Equation (20) into Equation (28), the heat balance equation under varying solar radiation intensities is obtained. The equation is discretized using an implicit difference scheme, where the entire water body is divided into \(M\) horizontal microelement layers along the vertical direction. Each microelement layer has an absolute thickness of \(\Delta z\), and the time step for vertical temperature evolution is \(\Delta t\). For the \(j\)th horizontal microelement layer, the difference scheme is formulated as follows:  
\[\frac{\partial T}{\partial t}=\frac{T(j\Delta z,t+\Delta t)}{\Delta t} \tag{29}\]  
\[\frac{\partial^{2}T}{\partial z^{2}}=\frac{T((j+1)\Delta z,t+\Delta t)+T((j-1) \Delta z,t+\Delta t)-2T(j\Delta z,t+\Delta t)}{\Delta z^{2}} \tag{30}\]Substituting Equation (29) into Equation (30) yields the following:  
\[a_{j}T[(j-1)\Delta z,t+\Delta t]+b_{j}T[j\Delta z,t+\Delta t]+c_{j}[(j+1)\Delta z,t+\Delta t]=d_{j} \tag{31}\]  
where the following conditions are satisfied:  
\[ \begin{array}{l}a_{j}=-\frac{\Delta t}{\Delta z^{2}}d_{z};b_{j}=-\frac{ \Delta t}{\Delta z^{2}}d_{z};\\ c_{j}=1+2\frac{\Delta t}{\Delta z^{2}}d_{z};d_{j}=T(j\Delta z,t)-\frac{\Delta t }{\rho c_{p}}\frac{\partial\psi_{\rm lag}}{\partial z}\end{array}  \tag{32}\]  
When the water body reaches thermal equilibrium, the boundary conditions are expressed as follows:  
\[\left\{