Geology Reference
In-Depth Information
Fig. 13.12
Time
evolution of the
temperature field within
and around an infinite slab
with initial thermal
anomaly
T
and thickness
2
a
. Numbers close to the
curves are values of the
dimensionless parameter
›
t
/
a
2
(
1973
), Richter and Parsons (
1975
), Jarvis and
Peltier (
1982
), Bercovici et al. (
1989
), and
Tackley (
1998
). The typical streamline pattern
arising from these models is illustrated in
Fig.
13.11
. Here we will limit to do some simple
physical considerations about the conditions for
the onset of thermal convection in the upper
mantle. Let us consider a spherical blob of radius
a
and characterized by a temperature excess
or deficit
T
with respect to the surrounding
induces a density anomaly ¡
D
'¡
T
that
corresponds (by Archimedes' principle) to an
additional buoyancy force given by:
T.x;0/
D
f.x/
I1
<x<
C1
(13.142)
with the initial condition (
13.142
) is known as
Laplace's solution:
C
Z
f
x
0
e
.
xx
0
/
2
=
.
4t
/
dx
0
1
2
p
›t
T.x;t/
D
1
(13.143)
Usually, the integral in (
13.143
) is evaluated
numerically. However, when
f
(
x
) has the shape
of a square pulse like that of Fig.
13.12
,sothat
f
(
x
)
D
0for
j
x
j
>
a
, the integral reduces to a sum
of two error functions (Carslaw and Jaeger
1959
):
f
b
D
'¡T g
(13.140)
This body will sink or rise with increasing
velocity until the viscous drag is balanced by the
buoyancy (
13.140
). From this point on, the mo-
tion will proceed at constant velocity
v
s
given by:
2
T
erfc
a
x
C
erfc
a
C
x
1
T.x;t/
D
2
p
›t
2
p
›t
I
1
<x<
C1
(13.144)
The time evolution of the temperature field
predicted by (
13.144
) is illustrated in Fig.
13.12
.
We note that for ›
t
/
a
2
2
9
a
2
'¡T
v
s
D
ǜ
g
(13.141)
D
5 the slab is not anymore
thermically distinguishable from the surround-
ing region. Therefore, for
t
£
5
a
2
/› we can
say that there is temperature equilibration. For
example, for a 100 km thick slab sinking in
the asthenosphere (
a
D
50 km) we would have
equilibration in
340 Myrs. Now let us come
back to the anomalous blob that is rising or falling
at velocity
V
s
. During the time interval £,the
blob may travel a distance
L
D
v
s
£. Consequently,
the free rising or falling of the blob is possible
only for
L
a
, because if this condition is not
This is a form of
Stokes' law
and
v
s
is referred
to as the
terminal velocity
. Clearly, during its rise
or fall the sphere either transfers or adsorbs heat
by diffusion, thereby the thermal anomaly
T
cannot be a constant. To determine the velocity
of this process, let us consider the simple case of
an infinite slab with initial temperature anomaly
T
, placed in the region between
x
D
a
and
x
D
C
a
(Fig.
13.12
). This is only an example of how
the temperature can be distributed at time
t
D
0.
In general, we have:
