1. Surface Energy Budget Q*=Net radiation QE=Latent Heat FluxQH QE Q* QG
Q*=Net radiation
QE=Latent Heat Flux
QH=Sensible Heat Flux
QG =Ground Heat Flux
Radiative fluxes are positive if directed toward the surface (gain of energy for the surface)
Non-Radiative fluxes are positive if directed away from the surface (loss of energy for the surface)

2. Soil heat flux is (usually) positive during the day (directed away from the surface, it is a loss), and negative during night (directed towards the surface, it is a gain).

3. Heat conduction in soil
Heat conduction: process of heat transfer from molecule to molecule by vibration QG
The flux density of heat conducted in the soil (QG) is proportional to the vertical gradient of soil temperature (TG). k is the thermal conductivity (units W m-1 oK-1 )

4. Some definitions
Heat= total amount of energy of molecules
Temperature = proportional to the average kinetic energy of the molecules
Specific heat (c) =quantity of heat required to raise temperature of unit mass by one degree Celsius.
Heat capacity (C) = quantity of heat required to raise temperature of unit volume by one degree Celsius.

5. The axis z is direct downward, so dT/dz is positive when it increases with depth, and negative when it decreases with depth T z T z

6. Thermal conductivity is the ability of the matter to conduce heat
Thermal conductivity is the ability of the matter to conduce heat. Mineral matter is good conductor, water intermediate, air very poor. Mineral soils have larger thermal conductivity than organic soils .
Mineral soil
Organic soil
Wet soils have larger thermal conductivity than dry soils (addition of water gives non linear increase in k)
Wet soil
Dry soil

7. Ds z z+Dz From the budget equation for a layer of soil
Assuming that energy fluxes are only function of depth (z), and that positive fluxes are downward.

8. then
m is the mass of soil in the layer
In differential form
If r (soil mass density, units kg m-3) and c (soil specific heat, units J kg-1 oK) are not function of z.
Fourier’s equation of heat conduction
Thermal diffusivity (units m2 s-1)

9. Thermal diffusivity is the ability to diffuse temperature waves.
Heat capacity is the product of the soil mass density and the soil specific heat. CS=cr

10. Thermal properties of soils
C=rc where r is soil density.
Change in heat content per unit volume is equal to DQ=CDT
Soil heat capacity (CS)
CS =Cm qm + Co qo + Cw qw + Ca qa
Where q is volume fraction occupied by mineral (m), organic (o), water (w), and air (a), and Cm,o,w,a is the heat capacity if the mineral, organic, water, and air constituents.

11. The daily and annual periodic forcing by the sun creates a surface temperature wave which propagates into the soil
To understand the physics of the phenomena, let study a simple case:
Thermal diffusivity constant
Sinusoidal surface temperature

12. Using this equation as Boundary Condition for the Fourier heat conduction equation, it can be shown that the solution is:
Phase leg of the wave
Amplitude at depth z
The wave amplitude decays exponentially with depth. It is e-1 (=0.37) of the surface value at
zD is called the damping depth
The damping depth increases with the thermal diffusivity

13. z=0
z=zD p/8
z=zD p/4
z=zD p/2
Dowward
Upward

14. The time lag for the wave maximum and minimum to reach lower depths is given by:
Where t1 and t2 are the times at which the wave maximum and minimum reaches depths z1 and z2
The maximum of ground heat flux is reached 1/8 of the cycle before the maximum temperature (3hrs in the daily cycle, and 1.5 months in the annual cycle)
The ratio between the maximum ground heat flux at the surface, and the amplitude of the temperature wave at the surface is:

15. Thermal admittance is the ability of soil surface to accept or release heat following a change in soil heat flux. Units are J m-2 oC-1 s-1/2
All other things being equal, soil with higher thermal admittance will have smaller surface temperature wave.
High thermal admittance (wet areas, clay, bare rocks)- relatively small temperature range
Low thermal admittance (dry, sandy organic soils), large daily temperature range

16. Temperature Ground Heat Fluxes Temperature tendency time=0 time=P/4
At z=3zD the wave amplitude is 5% of the surface value
Ground Heat Fluxes
Temperature tendency
Cooling
Heating