Summary:

### Unified Lake Temperature Solution

The article discusses the unified system of snow, lake body (water and/or ice), soil, and bedrock for the lake temperature solution. The governing equation for this system is:

(2.12.30) \\tilde{c}_v \\frac{\\partial T}{\\partial t} = \\frac{\\partial}{\\partial z} (\\tau \\frac{\\partial T}{\\partial z}) - \\frac{d\\phi}{dz}

Where \\tilde{c}_v is the volumetric heat capacity, T is the temperature, \\tau is the thermal conductivity, and \\phi is the solar radiation penetrating to depth z.

The system is discretized into N layers, where N = n_sno + N_levlak + N_levgrnd, with n_sno being the number of actively modeled snow layers and N_levgrnd being the number of ground layers.

Energy is conserved in the system as described by equation (2.12.32):

\\frac{d}{dt} \\sum_{j=1}^{N}[\\tilde{c}_{v,j}(t)(T_j - T_f) + L_j(t)]\\Delta z_j = G + (1-\\beta)\\vec{S}_g

Where \\tilde{c}_{v,j}(t) is the volumetric heat capacity, L_j(t) is the latent heat of fusion per unit volume, and the right-hand side represents the net influx of energy to the lake system.

The article notes that \\tilde{c}_{v,j}(t) can only change due to phase change, and this is restricted to occur at T_j = T_f in the snow-lake-soil system.