Geoscience Reference
In-Depth Information
Forexample, imagine a dyke of width 2
w
and of infinite extent in the
y
and
z
directions. If we assume that there is no heat generation and that the dyke has an
initial temperature of
T
0
, and if we ignore latent heat of solidification, then the
differential equation to be solved is
∂
T
2
T
∂
z
2
∂
t
=
κ
∂
with initial conditions
(i)
T
=
T
0
at
t
=
0 for -
w
≤
x
≤
w
and
(ii)
T
=
0at
t
=
0 for
|
x
|
>
w
.
The solution of this equation which satisfies the initial conditions is
erf
w
−
x
erf
w
+
x
T
0
2
T
(
x
,
t
)
=
2
√
κ
+
2
√
κ
(7.37)
t
t
=
1m)and intruded at a temperature of 1000
◦
C
If the dyke were 2 m in width (
w
were 10
−
6
m
2
s
−
1
, then the temperature at the centre of the dyke would
be about 640
◦
C after one week, 340
◦
C after one month and only about 100
◦
C
after one year! Clearly, a small dyke cools very rapidly.
For the general case, the temperature in the dyke is about
T
0
/
and if
κ
2when
t
=
w
2
5w
2
. High temperatures outside the dyke are
confined to a narrow contact zone: at a distance
w
away from the edge of the dyke
the highest temperature reached is only about
T
0
/
/κ
and about
T
0
/
4when
t
=
/κ
4. Temperatures close to
T
0
/
2
are reached only within about
w
/
4ofthe edge of the dyke.
Example: periodic variation of surface temperature
Because the Earth's surface temperature is not constant but varies periodically
(daily, annually, ice ages), it is necessary to ensure that temperature measurements
are made deep enough that distortion due to these surface periodicities is minimal.
We can model this periodic contribution to the surface temperature as
T
0
e
i
ω
t
,where
ω
is 2
π
multiplied by the frequency of the temperature variation, i is the square root
of
−
1 and
T
0
is the maximum variation of the mean surface temperature. The
temperature
T
(
z
,
t
)isthen given by Eq. (7.13) (with
A
=
0) subject to the following
two boundary conditions:
(i)
T
(0,
t
)
=
T
0
e
i
ω
t
and
(ii)
T
(
z
,
t
)
→
0as
z
→∞
.
We can use the separation-of-variables technique to solve this problem. Let us
assume that the variables
z
and
t
can be separated and that the temperature can be
written as
T
(
z
,
t
)
=
V
(
z
)
W
(
t
)
(7.38)
This supposes that the periodic nature of the temperature variation is the same at all
depths as it is at the surface, but it allows the magnitude and phase of the variation
