Geology Reference
In-Depth Information
heat production plays a minor role in these
phenomena, so that we can assume
H
D
0.
However, in this instance, the diffusion Eq.
(
12.13
) must be used to determine both the
distribution of temperature and its time evolution.
In this equation, the diffusivity ›
k
/
c
¡ has units
[m
2
s
-1
]. Therefore, if the temperature changes
over a characteristic time interval £, then such
a variation will
pro
pagate over a distance of
the order
L
p
›£, while a time £
L
2
/› is
required for a temperature change to propagate
over a distance
L
. Now we are going to face the
case of instantaneous heating or cooling of a
half-space. As shown by Turcotte and Schubert
(
2002
), the corresponding solution can be used
in the modelling of several important geological
problems. Let us assume that the temperature
is uniform along horizontal planes, so that heat
transfer occurs along the
z
direction only. In
this instance, heat conduction is described by
the one-dimensional version of the diffusion Eq.
(
12.13
):
@t
›
@
2
T
@T
@
z
2
D
0
(12.25)
This equation can be easily solved in a half-
space having uniform temperature
T
(
z
,
t
)
D
T
i
for
t
0 and whose surface is instantaneously heated
(or cooled) and maintained at a different constant
temperature
T
0
,sothat
T
(0,
t
)
D
T
0
for
t
> 0.
In this instance, for
T
0
>
T
i
heat is transferred
into the half-space and the internal temperature
increases, whereas for
T
0
<
T
i
the half-space
cools and its temperature decreases. An example
of the former situation is illustrated in Fig.
12.6
.
As shown in this figure, at any time
t
> 0wehave
the boundary condition:
lim
z
!1
T.
z
;t/
D
T
i
for any t>0 (12.26)
The diffusion Eq. (
12.25
) can be solved by
similarity introducing a dimensionless tempera-
ture ratio:
T
T
i
T
0
T
i
™
(12.27)
In terms of ™, the diffusion Eq. (
12.25
)as-
sumes the form:
Fig. 12.5
A continental lithosphere geotherm (
solid line
)
that fits pressure and temperature estimates of garnet peri-
dotite nodules from kimberlites (Redrawn from McKenzie
et al.
2005
)
@t
›
@
2
™
@
™
@
z
2
D
0
(12.28)
Fig. 12.6
Heating of a half-space by a sudden increase of the surface temperature
