Geoscience Reference
In-Depth Information
7.3.4 The timescale of conductive heat flow
Geological structures such as young mountain belts are not usually in thermal
equilibrium because the thermal conductivity of rock is so low that it takes many
millions of years to attain equilibrium. For example, consider the model rock
column with the geotherm shown as curve a in Fig. 7.3.Ifthe basal heat flow were
suddenly increased from 21
Wm
−
2
, the temperature of
the column would increase until the new equilibrium temperatures were attained
(curve d in Fig. 7.3). That this process is very slow can be illustrated by considering
a rock at depth 20 km. The initial temperature at 20 km would be 567
◦
C, and,
20 Ma after the basal heat flow increased, conduction would have raised the
temperature at 20 km to about 580
◦
C. Only after 100 Ma would the temperature
at 20 km be over 700
◦
C and close to the new equilibrium value of 734
◦
C. This
can be estimated quantitatively from Eq. (7.17):
10
−
3
10
−
3
×
to 42
×
∂
T
∂
t
=
κ
∂
2
T
∂
z
2
l
2
The
characteristic time
gives an indication of the amount of time nec-
essary for a change in temperature to propagate a distance of
l
in a medium
having therm
al
diffusivity
τ
=
/κ
κ
. Likewise, the
characteristic thermal diffusion dis-
=
√
κτ
tance
,
l
,gives an indication of the distance that changes in temperature
propagate during a time
.Togiveageological example, it would take many
tens of millions of years for thermal transfer from a subduction zone at 100 km
depth to have a significant effect on the temperatures at shallow depth if all heat
transfer were by conduction alone. Hence, melting and intrusion are important
mechanisms for heat transfer above subduction zones. As a second example, a
metamorphic belt caused by a deep-seated heat source is characterized by abun-
dant intrusions, often of mantle-derived material; this is the dominant factor in
transfer of heat to the surface. Magmatism occurs because large increases in
the deep heat flow cause large-scale melting at depth long before the heat can
penetrate very far towards the surface by conduction.
When a rock column is assembled by some process such as sedimentation,
overthrusting or intrusion, the initial temperature gradient is likely to be very
different from the equilibrium gradient. This should always be borne in mind
when evaluating thermal problems.
τ
7.3.5 Instantaneous cooling or heating
Assume that there is a semi-infinite solid with an upper surface at
z
=
0, no heat
generation (
A
=
0) and an initial temperature throughout the solid of
T
=
T
0
.For
t
>
0, let the surface be kept at temperature
T
=
0. We want to determine how
the interior of the solid cools with time.
