Available Energy: Net Radiation and Soil Heat Flux - Transport in the Atmosphere-Vegetation-Soil Continuum - page 66

Equating Eqs. ( 2.47 ) and ( 2.48 ) yields a differential equation for z f (so-called Stefan
problem) with the solution:

( TT t

L

− ) λ

f 0 s

z = 2

(2.49) f

θρ

w f

Thus, the penetration of the freezing front has a square root dependence on time, as
well as on the temperature at the top of the soil. The square root dependence can be
understood as follows. If we assume the surface temperature to be ixed, the temper-
ature contrast between the surface and the freezing front is also ixed. However, as
the freezing front progresses downward, the distance over which this temperature
contrast occurs becomes larger: the temperature gradient decreases. Because of this
decreased temperature gradient less heat that results from freezing can be removed
from the soil and hence the freezing front progresses at a slower rate.
This square root dependence can be used in the empirical estimation of frost pen-
etration in soils using the freezing index I n (see, e.g., Riseborough et al., 2008 ). I n is
deined as follows:

n

∑

− ( ) I = T T (2.50)

n i f

i = 1

where T i is the daily mean air temperature (as an approximation of the surface tempera-
ture T 0 in Eq. ( 2.49 )). The summation is started when T i drops below the freezing point
of water (i.e., 0 ºC) for the irst time. Then the frost penetration is estimated as:

zaI n

f =−

(2.51)

where a is an empirical constant that has typical values of 0.03 to 0.06 m K -1/2 day -1/2 .
If we compare Eq. ( 2.49 ) with Eq. ( 2.51 ), we see that the constant a depends on soil
type (through λ s ) and soil moisture content θ . Furthermore, the constant a needs to
absorb all errors related to the approximation of T 0 by the air temperature. 6 The mean
effect of this approximation indeed can be taken into account in the value of the
empirical constant a . However, day-to-day variations in radiation, wind speed and
humidity, as well as the changes in snow cover, will cause a random modulation of
the relationship between air temperature and surface temperature. Those random luc-
tuations will decrease the predictive power of Eq. ( 2.51 ) on short time scales. Apart
from using Eq. ( 2.51 ) for the prediction of frost penetration, it can also be used to
monitor the removal of frost from the soil, if the summation in Eq. ( 2.50 ) is continued
after the air temperature has risen above 0 ºC again.

6 The values for a quoted above correspond to a saturated soil with a porosity of 0.4 and thermal conductivities of
0.7 and 2.5 W m -1 K -1 , respectively (if we neglect the error due to the use of the air temperature).