Soil Temperature Changes with Depth and Time

2.5 Soil Temperature Changes with Depth and Time

The temperature of the soil environment is also important to many liv- ing organisms. Figure 2.1 shows a typical range for diurnal temperature

Temperature

variation with depth in soil. The features to note are that the temperature extremes occur at the surface where radiant energy exchange occurs, and that the diurnal variation decreases rapidly with depth in the soil. Figure

2.5 also shows these relationships and gives additional insight into soil temperature variations. Here, temperatures measured at three depths are shown.

Note that the diurnal variation is approximately sinusoidal, that the am- plitude decreases rapidly with depth in the soil, and that the time of maxi- mum andminimum shifts with depth. At the surface, the time temperature is around

hours, as it is in the air. At deeper depths the times of the maxima and minima lag solar noon even farther, and at 30 to

40 cm, the maximum is about 12 hours after the maximum at the surface. We derive equations for heat flow and soil temperature later when we discuss conductive heat transfer. For the moment, we just give the equation which models temperatures in the soil when the temperature at the surface is known. This model assumes uniform soil properties throughout the

soil profile and a sinusoidally varying surface temperature. Given these assumptions, the following equation gives the temperature as a function of depth and time:

Time (hrs)

2.5. Hypothetical temperature variations in a uniform soil at the surface and two depths showing the attenuation of diurnal variations and the shift in

F I G U RE

maxima and minima with depth.

Soil Temperature Changes with Depth and Time

where is the mean daily soil surface temperature, is as in Eq.

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

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

C of the mean daily surface temperature if the temperature variation at the surface (amplitude) is

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 = then the depth for diurnal variations less than 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 influ- enced 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. Equa- tions (2.4) and (2.5) are used to describe the annual variation, but D is around 2 m, and is

days.

While Eqs. (2.4) and (2.5) are 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. 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

and applying it at two depths permits solution for D. If the amplitude of the temperature wave is T

Eq. (2.5) for

= at

Temperature

one depth and is T

at a second depth, then