Geology Reference
In-Depth Information
5.8.2. The Stefan Equation
There are a number of equations that can be used to calculate the thickness of the active
layer. The most common is the Stefan equation:
Z
=
2
K
i
(5.11)
where
Z
is the thickness of the active layer (m or cm),
T
is the ground surface temperature
during the thaw season (°C),
K
is thermal conductivity of unfrozen soil (W/m K or
kcal/m °C h),
t
is the duration of the thawing season (day, hour, second), and
Qi
is
volumetric latent heat of fusion (kJ/m
3
).
Qi
is expressed as:
(
)
Qi
=
Lpd W
−
Wu
(5.12)
where
L
80 cal/g or 8 kcal/kg (latent heat of ice),
pd
is the dry density of soil (kg/m
3
),
W
is total moisture content (%), and
Wu
is unfrozen water content (%). Gold and
Lachenbruch (1973) provide a further approximation of active-layer thickness,
x
(cm),
with the following:
=
x
=απ
P
log e
AT
0
(5.13)
0
where
is soil thermal diffusivity,
P
is the period of the temperature cycle,
A
0
is surface
temperature amplitude, and
T
0
is mean annual surface temperature.
Typical values of thermal conductivity and thermal diffusivity are given in Table 5.1.
A simple example illustrates use of the Stefan equation. If mean surface temperature
during the thawing season is 6.5 °C and thaw duration is approximately 150 days (
α
3620
hours), and if one assumes the thermal conductivity of unfrozen soil (
K
) is 1.25 kcal m h,
dry density (
pd
) is 1334 kgm
3
, water content (
W
) is 18%, and unfrozen water content (
Wu
)
is 3%, the thickness of the active layer will approximate 1.92 m. It is important to note
that Equation (5.11) can also be used to calculate the depth of frost penetration. In this
case, time
t
is the duration of the freezing season (
T
∼
0 °C) and
K
represents the thermal
conductivity of frozen soil. The Stefan equation can also be used to calculate the previous
surface temperatures that would have been responsible for the formation, at depth, of a
thaw unconformity (see Chapters 7 and 11).
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5.8.3. Active-Layer Thermal Regime
Ground climates have been discussed in general in Chapter 3 within the context of
n
factors and the thermal offset. Here, two further aspects of the active-layer ground-
thermal regime are briefl y mentioned.
First, freezing and thawing of the active layer occurs either on a diurnal basis, as
in many temperate and subtropical regions, or on a seasonal basis, as in high latitudes.
It is important to note that thawing is one-sided, from the surface downwards. Autumn
freeze-back is a more complex process because in regions underlain by permafrost
freezing is two-sided, occurring both downwards from the surface and upwards from the
perennially-frozen ground beneath. Second, active-layer thermal regimes are closely
related to snow thickness and density. This has been illustrated earlier with data from
Churchill, Manitoba (see Table 5.5).
Other aspects of the ground-thermal regime are discussed elsewhere in this chapter
and in Chapters 4, 9, 10, and 15.
