Here is a concise summary of the provided article:

## Numerical Solution for Soil Temperature

The article outlines the numerical scheme used to solve for soil temperature in the Community Land Model (CLM).

Key points:

### Soil and Snow Discretization
- The soil column is discretized into 25 layers.
- The overlying snow pack is modeled with up to 12 layers, with the number and thickness of layers depending on the snow depth.

### Heat Flux Calculations
- The heat flux between layers is calculated using thermal conductivity at the layer interfaces.
- A Crank-Nicholson method is used to solve the energy balance equations for each layer.

### Boundary Conditions
- At the top, the heat flux from the atmosphere is accounted for, including solar radiation, longwave radiation, sensible heat, and latent heat.
- At the bottom, a zero heat flux boundary condition is applied.

### Surface Temperature Calculation
- The top layer temperature is adjusted to better match the analytical solution for diurnal heating.
- When surface water is present, the equations are modified to include the surface water temperature.

The article provides the detailed mathematical formulations for the numerical scheme, including the equations and coefficient definitions for the different soil, snow, and surface water layers.