Geology Reference
In-Depth Information
the diffusivity › are independent from
T
,whichis
only approximately true (e.g., Hofmeister
1999
).
Therefore, more general equations of conductive
heat transfer can be written. In steady-state con-
ditions, the temperature does not change with
time, so that (
12.13
) assumes the form:
generated in a volume of slab
dV
having unit
surface area, thickness
dz
,andmass
dm
is given
by:
dQ
D
Hdm
D
¡Hd
z
(12.17)
Therefore, the one-dimensional version of the
stationary equation of conductive heat transfer
with sources now reads:
2
T
D
0
r
(12.14)
Consequently, in steady-state conditions and
in absence of sources the temperature field
satisfies Laplace's equation in
R
,sothatitis
harmonic. Accordingly, it will not have neither
maxima nor minima in
R
-
S
(
R
). For example,
in a problem where the temperature depends
only from the depth
z
, we would have
dT
/
dz
D
const
,sothat
T
would change linearly with
depth. In presence of heat sources, the diffusion
Eq. (
12.13
) must be generalized to take into
account of the local rate of heat production. Let
H
D
H
(
r
,
t
) be the rate of heat generation per unit
mass at location
r
and time
t
. In this instance, the
heat diffusion equation assumes the following
more general form:
k
d
2
T
d
z
2
C
¡H
D
0
(12.18)
A solution to this equation can be easily found
assuming a half-space with top boundary at
z
D
0. Let us assign the boundary conditions as
follows:
T
(0)
D
T
0
,
q
(0)
D
-
q
0
. In this case,
the solution to (
12.18
) is the following parabolic
function:
q
0
k
z
¡H
2k
z
2
T.
z
/
D
T
0
C
(12.19)
The function
T
D
T
(
z
) is called a
geotherm
.
In principle, it could be used to predict the varia-
tions of temperature within the continental litho-
sphere, granted that the rate of radiogenic heat
production
H
can be considered constant in the
crust and in the lithospheric mantle. Assuming
T
0
D
300 K,
q
0
D
60 mWm
-2
, ¡
D
3,300 kg m
-3
,
k
D
3Wm
-1
K
-1
,and
H
D
6
10
-11
Wkg
-1
gives
the continental geotherm shown in Fig.
12.3
.
This plot would fail to describe the distribution
of temperature in the asthenosphere, because it
predicts partial melting of the mantle peridotite
starting from
80 km depth. Conversely, it repre-
sents an acceptable approximation of the effective
geotherm in continental areas.
An improved conductive geotherm model can
be obtained taking into account that the pro-
duction of radiogenic heat is not constant but
decreases with depth. A good method is to con-
sider different crustal layers, each with constant
radiogenic heat production rate, and decrease
H
stepwise with depth. Alternatively, Turcotte and
Schubert (
2002
) argued that a good choice for the
function
H
D
H
(
z
)is:
@T
@t
›
r
H
c
D
0
2
T
(12.15)
Therefore, even in steady-state conditions, the
presence of heat sources implies that a tempera-
ture field depending only from
z
cannot increase
linearly with the depth. We can easily apply these
results to the simple case of a vertical heat flow
through a thin horizontal slab having thickness
dz
. In this instance, the net flux through the slab
is simply:
d
z
d
z
D
k
d
2
T
dq
dq
D
q.
z
C
d
z
/
q.
z
/
D
d
z
2
d
z
(12.16)
This expression implies that
dq
¤
0iff
T
is
not
a linear function of depth, so that its one-
dimensional Laplacian
d
2
T
/
dz
2
is not zero. In
this instance, by the conservation of energy, the
net flux through the lamina must be supplied by
internal sources of heat. Let
H
be the rate of heat
produced per unit mass within the slab. The heat
H.
z
/
D
H
0
e
z
=h
(12.20)