One of the most important state variables in the soil physical processes is the soil temperature. It is calculated by a differential equation solved numerically by finite difference method. The equation describes the temperature flux along the soil profile in function of soil water content and soil mineral and organic components, according to Campbell (1984). The Campbell's work is based on the pioneer work of DeVries (1963).

The differential equation, time dependent, for soil temperature is:

 

 

where Ch is the specific heat of the soil (J m-3 K-1)  and λ is the thermal conductivity (W m-1K-1).

Soil volumetric specific heat can be estimated from soil water content and  porosity. Also thermal conductivity can be estimated from water content, bulk density and clay fraction.

 

Assuming soil depth not affecting thermal conductivity and capacity, soil temperature is computed as:

 

 

where D (m2 s-1) is thermal diffusivity.

 

Numerical solution of this equation needs to be in both depth and time. Also heat will be stored or taken from storage within the soil.

The energy balance equation for node i is:

 

 

 

where j is an index of time, i is an index of space, Ch is soil volumetric specific heat (J m-3 K-1), λ is the thermal conductivity (W m-1K-1), Z is the layer depth and Δt is the time increment (sec).

 

Then an appropriate mean temperature ranges from Tj to Tj+1:

 

 

where h is a weighting factor ranging from 0 to 1.

 

Boundary conditions

 The boundary condition at the bottom of the soil column is specified as remaining at some constant, measured temperature (_Tb, °C).

 The surface temperature value equals air temperature or can be calculated according to Parton (1984) functions.

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