### 2.12.5.1. Introduction[¶](#introduction "Permalink to this headline")

The (optional-) snow, lake body (water and/or ice), soil, and bedrock system is unified for the lake temperature solution. The governing equation, similar to that for the snow-soil-bedrock system for vegetated land units (Chapter [2.6](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#rst-soil-and-snow-temperatures)), is

(2.12.30)[¶](#equation-12-25 "Permalink to this equation")\\\[\\tilde{c}\_{v} \\frac{\\partial T}{\\partial t} =\\frac{\\partial }{\\partial z} \\left(\\tau \\frac{\\partial T}{\\partial z} \\right)-\\frac{d\\phi }{dz}\\\]

where \\(\\tilde{c}\_{v}\\) is the volumetric heat capacity (J m\-3 K\-1), \\(t\\) is time (s), _T_ is the temperature (K), \\(\\tau\\) is the thermal conductivity (W m\-1 K\-1), and \\(\\phi\\) is the solar radiation (W m\-2) penetrating to depth _z_ (m). The system is discretized into _N_ layers, where

(2.12.31)[¶](#equation-12-26 "Permalink to this equation")\\\[N=n\_{sno} +N\_{levlak} +N\_{levgrnd} ,\\\]

\\(n\_{sno}\\) is the number of actively modeled snow layers at the current timestep (Chapter [2.8](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Snow_Hydrology/CLM50_Tech_Note_Snow_Hydrology.html#rst-snow-hydrology)), and \\(N\_{levgrnd}\\) is as for vegetated land units (Chapter [2.6](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#rst-soil-and-snow-temperatures)). Energy is conserved as

(2.12.32)[¶](#equation-12-27 "Permalink to this equation")\\\[\\frac{d}{dt} \\sum \_{j=1}^{N}\\left\[\\tilde{c}\_{v,j} (t)\\left(T\_{j} -T\_{f} \\right)+L\_{j} (t)\\right\] \\Delta z\_{j} =G+\\left(1-\\beta \\right)\\vec{S}\_{g}\\\]

where \\(\\tilde{c}\_{v,j} (t)\\)is the volumetric heat capacity of the _j_th layer (section [2.12.5.5](#radiation-penetration)), \\(L\_{j} (t)\\)is the latent heat of fusion per unit volume of the _j_th layer (proportional to the mass of liquid water present), and the right-hand side represents the net influx of energy to the lake system. Note that \\(\\tilde{c}\_{v,j} (t)\\) can only change due to phase change (except for changing snow layer mass, which, apart from energy required to melt snow, represents an untracked energy flux in the land model, along with advected energy associated with water flows in general), and this is restricted to occur at \\(T\_{j} =T\_{f}\\) in the snow-lake-soil system, allowing eq. to be precisely enforced and justifying the exclusion of \\(c\_{v,j}\\) from the time derivative in eq..

