## 2 Theoretical Model for Water Temperature Field
### Gas-Liquid Two-Phase Interface Heat Flux Calculation Model
Building on the characteristics of air-water interface interactions, [PERSON] established the interrelationships and calculation criteria for temperature and humidity roughness parameters, which satisfy the following equation:  
\[z_{qq}=z_{oT}=\text{min}(1.15\times 10^{-4},5.5\times 10^{-5}\times(\frac{u_{*} z_{o}}{v_{a}})^{-0.6}) \tag{19}\]  
Solar shortwave radiation serves as the primary driver for the formation of thermal stratification in water bodies. The shortwave radiation spectrum is predominantly concentrated in the 0.2-3 \(\upmu\)m range, exerting a decisive influence on the thermodynamic and dynamic structure of the upper water layer. Under clear weather conditions, approximately half of the solar radiation energy is absorbed within the upper 2 m of the water body. The region of significant warming propagates downward from the water surface, and wind-driven shear turbulence facilitates the mixing of heat into deeper layers of the water. For optically homogeneous water bodies, the solar radiation absorbed at different water depths \(\psi_{ns}\) is expressed as follows:  
\[\psi_{ns}(z)=\psi_{sun}(1-b)\left\{1-\frac{1}{z}\sum_{i=1}^{N}F_{i}\zeta_{i}[1- \exp(-\frac{z}{\zeta_{i}})]\right\} \tag{20}\]  
where \(\psi_{sun}\) denotes the intensity of solar shortwave radiation, \(b\) is the albedo, set to 0.07 [40], \(z\) represents the water depth, and \(N\) indicates the number of wavelength bands; \(F_{i}\) and \(\zeta_{i}\) correspond to the irradiance constant and wavelength irradiance, respectively. Based on the attenuation characteristics of solar spectral bands in water, [PERSON] [40] discretized the incident solar radiation flux using a 9-wavelength band spectrum. The values of the irradiance constants are provided in Table 1.  
Before reaching the Earth's surface, solar radiation is attenuated by the atmosphere. The solar radiation absorbed by the atmosphere is subsequently emitted as longwave radiation in the form of sky background radiation. The intensity of this radiation is influenced by air temperature and cloud cover in the sky, adhering to the Stefan-Boltzmann law:  
\[\psi_{\downarrow l}=\sigma\varepsilon_{a}(273.15+T_{a})(1-\gamma_{a}) \tag{21}\]  
where \(\sigma\) is the Stefan-Boltzmann constant, set as \(5.67\times 10^{-8}W/m^{2}K^{4}\); \(\gamma_{a}\) represents the longwave reflectivity; \(T_{a}\) denotes the air temperature near the water surface; and \(\varepsilon_{a}\) is the atmospheric emissivity, expressed as follows:  
\[\varepsilon_{a}=1-0.261\exp(-0.74\times 10^{-4}T_{a}^{2})(1+KC^{2}) \tag{22}\]  
where C is the cloud cover fraction, and the parameter \(K\) is the cloud height factor, with a mean value of 0.017. The radiation emitted by the water body back to the atmosphere \(\psi_{\uparrow l}\) also adheres to the Stefan-Boltzmann law, expressed as follows:  
\[\psi_{\uparrow l}=\sigma\varepsilon_{\rm w}(273.15+T_{sw})^{4} \tag{23}\]  
where \(\varepsilon_{\rm w}\) represents the longwave emissivity of the water body, taken as 0.97. \begin{table}
\begin{tabular}{c c c} \hline \(\lambda(\mu m)\) & \(F_{i}\) & \(\zeta_{i}(\mathbf{m})\) \\ \hline
0.2–0.6 & 0.237 & 34.8492 \\
0.6–0.9 & 0.360 & 2.2662 \\
0.9–1.2 & 0.179 & 3.149 \(\times\) 10\(-\)2 \\
1.2–1.5 & 0.087 & 5.483 \(\times\) 10\(-\)3 \\
1.5–1.8 & 0.080 & 8.317 \(\times\) 10\(-\)4 \\
1.8–2.1 & 0.0246 & 1.261 \(\times\) 10\(-\)4 \\
2.1–2.4 & 0.025 & 3.133 \(\times\) 10\(-\)4 \\
2.4–2.7 & 0.007 & 7.819 \(\times\) 10\(-\)5 \\
2.7–3.0 & 0.0004 & 1.443 \(\times\) 10\(-\)5 \\ \hline \end{tabular}
\end{table} 
Table 1: Solar irradiance constant at each band.