## 2 Theoretical Model for Water Temperature Field
### Gas-Liquid Two-Phase Interface Heat Flux Calculation Model  
The primary driver of vertical temperature stratification in water bodies is the heat exchange occurring at the air-water interface. This exchange facilitates a dynamic thermal equilibrium between the water body and the atmosphere through mechanisms such as molecular heat conduction and phase transitions. Within the water body, heat transfer between the upper and lower layers is achieved via heat conduction, convection, and mixing, ultimately establishing stable vertical temperature stratification. As depicted in Figure 2, heat exchange at the air-water interface and within the water body encompasses convective heat transfer, latent heat exchange, and radiative exchange. Radiative exchange includes solar shortwave radiation, atmospheric longwave radiation, and longwave back radiation from the water body. The net heat flux at the air-water interface \(\psi\) is expressed as follows [36]:  
\[\psi=\psi_{nl}+\psi_{s}+\psi_{l}-\psi_{ns} \tag{1}\]  
where \(\psi_{s}\) represents the sensible heat flux, \(\psi_{l}\) denotes the latent heat flux, \(\psi_{nl}\) is the net longwave radiation flux, calculated as \(\psi_{nl}=\psi_{\uparrow l}-\psi_{\downarrow l}\) (the difference between atmospheric longwave radiation \(\psi_{\downarrow l}\) and water body longwave back radiation \(\psi_{\uparrow l}\)), and \(\psi_{ns}\) signifies the solar shortwave radiation flux absorbed by the interfacial thin layer. Figure 1: Thermal stratification temperature profile of water. Figure 2: Schematic diagram of water–air interface and heat exchange in water body. The COARE3.0 (Coupled Ocean-Atmosphere Response Experiment) model is an advanced numerical framework specifically developed to investigate the intricate interactions between the ocean and the atmosphere [37]. This model is predominantly utilized for the precise computation of ocean surface fluxes, including heat, momentum, and moisture fluxes, and is extensively employed in the fields of marine meteorology and climate science. It rigorously implements the Monin-Obukhov similarity theory, a standard approach for near-surface meteorological measurements. Furthermore, the model incorporates sophisticated sub-models to account for the ocean's cool skin layer and the diurnal warm layer, enhancing its accuracy and applicability in diverse environmental conditions. For specific heat fluxes, the individual components are computed utilizing the air-sea flux model. The sensible heat flux and latent heat flux are mathematically defined as follows [37,38]:  
\[\psi_{s}=\rho_{a}c_{pa}\overline{w^{\prime}T^{\prime}}=-\rho_{a}c_{pa}u_{*}T_{*} \tag{2}\]  
\[\psi_{l}=\rho_{a}L_{c}\overline{w^{\prime}q^{\prime}}=-\rho_{a}L_{c}u_{*}q_{*} \tag{3}\]  
where  
\[u_{*}=\sqrt{-\tau/\rho_{a}} \tag{4}\]  
where \(\rho_{a}\) and \(c_{pa}\) denote air density and specific humidity, respectively; \(w,T,q^{\prime}\) represent the turbulent fluctuations of vertical wind speed, temperature, and water vapor mixing ratio, respectively; \(u_{*}\), \(T_{*}\), \(q_{*}\) are the MOS (Monin-Obukhov Similarity) scaling parameters for velocity, humidity, and temperature, respectively; \(L_{e}\) represents the latent heat of evaporation; and \(\tau\) signifies the air shear stress.