Geology Reference
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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)
 
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