Summary:

## Soil and Snow Temperatures

The article discusses the principles of heat conduction and the numerical solution for calculating soil, snow, and surface water temperatures.

### Heat Conduction Equation
The first law of heat conduction is expressed as:
$F = -\lambda \nabla T$
where $F$ is the heat flux, $\lambda$ is the thermal conductivity, and $\nabla T$ is the spatial temperature gradient.

In one-dimensional form, the equation becomes:
$F_z = -\lambda \frac{\partial T}{\partial z}$
where $z$ is the vertical direction, positive downward, and $F_z$ is positive upward.

To account for transient conditions, the principle of energy conservation is invoked:
$c\frac{\partial T}{\partial t} = -\frac{\partial F_z}{\partial z}$
where $c$ is the volumetric snow/soil heat capacity and $t$ is time.

Combining the equations yields the second law of heat conduction in one-dimensional form:
$c\frac{\partial T}{\partial t} = \frac{\partial}{\partial z} \left[\lambda \frac{\partial T}{\partial z}\right]$

### Numerical Solution
This equation is solved numerically to calculate the soil, snow, and surface water temperatures for a 25-layer soil column with up to twelve overlying snow layers and a single surface water layer. The boundary conditions include the heat flux from the atmosphere at the top and zero heat flux at the bottom of the soil column.

The temperature profile is first calculated without phase change and then readjusted for phase change.