Soil Temperature Changes with Depth and Time
25
where Tave is the mean daily soil surface temperature, o is n/12, as in
Eq. (2.2), A(0) is the amplitude of the temperature fluctuations at the
surface (half of the peak-to-peak variation) and D is called the damping
depth. The "-8" in the sine function is a phase adjustment to the time
variable so that when time t = 8, the sine of the quantity in brackets is
zero at the surface (z = 0). We discuss computation of diurnal damping
depth in Ch. 8. It has a value around 0.1 m for moist, mineral soils, and
0.03 to 0.06 m for dry mineral soils and organic soils.
In many cases we are not interested in the time dependence of the soil
temperature, but would just like to know the range of temperatures at a
particular depth. It is known that the range of the sine function is - 1 to
1 so Eq. (2.4) gives the range of soil temperature variation as
where the + gives the maximum temperatures and the - the minimum.
Example 2.4. At what depth is the soil temperature within f 0.5" C of
the mean daily surface temperature if the temperature variation at the
surface (amplitude) is f 15" C?
Solution. The amplitude of the desired temperature variation is 0.5" C.
Rearranging Eq. (2.5) and taking the logarithm of both sides gives
If D = 12cm, then the depth for diurnal variations less than k0.5"C
would be 3.4 x 12 cm = 41 cm. Therefore a depth of at least 40 cm needs
to be dug to obtain a soil temperature measurement that is not influenced by the time of day the temperature is measured.
The annual soil temperature pattern is similar to the diurnal one, but
with a much lower frequency and a much larger damping depth. Equations (2.4) and (2.5) are used to describe the annual variation, but D is
around 2 m, and o is 2x1365 days.
While Eqs. (2.4) and (2.5) are u s e l l relationships for getting a general
idea of how soil temperature varies with depth and time, it is important to
remember their limitations. The thermal properties do vary with depth,
and the temperature variation at the surface is not necessarily sinusoidal.
Temperature variations over periods longer than a day or a year also have
an effect. In spite of these limitations, however, a lot can be learned from
this simple model.
Clearly, from Eq. (2.4), the value of the damping depth D is key to
predicting the penetration into the soil of a temperature variation at the
surface. Data such as that in Fig. 2.5 can be used to estimate D. Solving
Eq. (2.5) for T(z) - Tave and applying it at two depths permits solution
for D. If the amplitude of the temperature wave is T ( 2 , ) - Twe = A1 at
25
where Tave is the mean daily soil surface temperature, o is n/12, as in
Eq. (2.2), A(0) is the amplitude of the temperature fluctuations at the
surface (half of the peak-to-peak variation) and D is called the damping
depth. The "-8" in the sine function is a phase adjustment to the time
variable so that when time t = 8, the sine of the quantity in brackets is
zero at the surface (z = 0). We discuss computation of diurnal damping
depth in Ch. 8. It has a value around 0.1 m for moist, mineral soils, and
0.03 to 0.06 m for dry mineral soils and organic soils.
In many cases we are not interested in the time dependence of the soil
temperature, but would just like to know the range of temperatures at a
particular depth. It is known that the range of the sine function is - 1 to
1 so Eq. (2.4) gives the range of soil temperature variation as
where the + gives the maximum temperatures and the - the minimum.
Example 2.4. At what depth is the soil temperature within f 0.5" C of
the mean daily surface temperature if the temperature variation at the
surface (amplitude) is f 15" C?
Solution. The amplitude of the desired temperature variation is 0.5" C.
Rearranging Eq. (2.5) and taking the logarithm of both sides gives
If D = 12cm, then the depth for diurnal variations less than k0.5"C
would be 3.4 x 12 cm = 41 cm. Therefore a depth of at least 40 cm needs
to be dug to obtain a soil temperature measurement that is not influenced by the time of day the temperature is measured.
The annual soil temperature pattern is similar to the diurnal one, but
with a much lower frequency and a much larger damping depth. Equations (2.4) and (2.5) are used to describe the annual variation, but D is
around 2 m, and o is 2x1365 days.
While Eqs. (2.4) and (2.5) are u s e l l relationships for getting a general
idea of how soil temperature varies with depth and time, it is important to
remember their limitations. The thermal properties do vary with depth,
and the temperature variation at the surface is not necessarily sinusoidal.
Temperature variations over periods longer than a day or a year also have
an effect. In spite of these limitations, however, a lot can be learned from
this simple model.
Clearly, from Eq. (2.4), the value of the damping depth D is key to
predicting the penetration into the soil of a temperature variation at the
surface. Data such as that in Fig. 2.5 can be used to estimate D. Solving
Eq. (2.5) for T(z) - Tave and applying it at two depths permits solution
for D. If the amplitude of the temperature wave is T ( 2 , ) - Twe = A1 at