Geology Reference
In-Depth Information
Fig. 12.7
Distribution of temperature after instantaneous heating from 20 to 500
ı
C at the surface of a half-space after
10, 100, and 1,000 s. The assumed thermal diffusivity is ›D 1m
2
s
-1
LJ
LJ
LJ
LJ
z
D0
D
temperature variations are significant is known
as the
thermal boundary layer
(TBL). Turcotte
and Schubert (
2002
) define the thickness of this
region as the depth at which ™
D
0.1. Clearly,
such a depth changes with time insofar as the
half-space heats or cools after the initial surface
variation. By the definition (
12.27
), the condi-
tion ™
D
0.1 means that the actual variation of
temperature with respect to the initial value is
10 % of the temperature change at the half-
space surface. Substituting the value ™
D
0.1 into
(
12.40
) gives a corresponding value, ǜ
T
,forthe
similarity variable ǜ:
q
0
D
q.0;t/
D
k
@T
@
z
k.T
i
T
0
/
p
›t
(12.44)
12.4
Cooling of the Oceanic
Lithosphere
The solution (
12.41
) to the diffusion equation
can be adapted to describe the formation of the
oceanic crust by cooling of MORB produced
at a mid-ocean ridge after contact with oceanic
seawater. Furthermore, in Sect.
1.3
we have seen
that when fertile and wet asthenosphere melts at
a spreading ridge by adiabatic decompression,
the residual column of asthenospheric material
leaving the melting regime is also dragged hor-
izontally away from the ridge axis and cools by
conductive loss of heat. At any time, we can
divide this column into an upper part, where the
potential temperature is fallen below the astheno-
sphere
T
P
(
1,280
ı
C), and a lower hotter zone,
which has not yet lost a significant amount of
ǜ
T
erfc
1
.0:1/
Š
1:16
(12.42)
Therefore, the TBL thickness at time
t
will be
given by:
z
T
.t/
D
2ǜ
T
p
›t
Š
2:32
p
›t
(12.43)
The surface heat flux corresponding to the
cooling or heating law (
12.41
) can be easily
obtained by differentiation. We have:
