Geology Reference
In-Depth Information
I
I
where the production of radiogenic heat in the
oceanic crust and in the core has been neglected.
The bulk Earth Urey ratio measures the relative
contribution of internal heat generation with re-
spect to the total surface heat flux:
ˆ.S/ D
q .n/dS D
q dS
S. R /
S. R /
D k I
r T dS
(12.8)
S . R /
q.H CC / C q.H m /
q CC C q OC
U b D
(12.5)
At any time, the total heat, Q ,in R is given
by:
A quantity that is more commonly used by
geophysicists in studies about the thermal history
of Earth is the convective Urey ratio ,which
Korenaga ( 2008 ) defines as the ratio of heat gen-
eration in the mantle over the mantle heat flux:
Q.t/ D Z
T.r;t/dV
(12.9)
R
where c is the specific heat and ¡ is the density
in R . Clearly, the rate of variation of Q must
coincide with the net incoming flow through
S ( R ), so that applying the divergence theorem
(see Appendix 1 )
q.H m /
q m
U c D
(12.6)
and
reversing
the
sign
we
Finally, it is possible to introduce an internal
heating ratio , I , as the ratio of surface heat flow
associated with mantle sources to the mantle heat
flux:
obtain:
Q.t/ D k I
r T dS D k Z
2 TdV
r
(12.10)
S . R /
R
q m q c
q m
I D
(12.7)
By ( 12.9 ), the rate of variation of Q can be also
written as follows:
Q.t/ D Z
R
Korenaga ( 2008 ) pointed out that while the U b
is probably 0.35, the convective Urey ratio U c
is estimated to be 0.2. Therefore, according to
this author only 20 % of the basal lithospheric
heating associated with mantle convection would
originate from radioactive decay, while 80 % of
q m would come from secular cooling. In the next
section, we shall determine the temperature field
in the continental crust starting from the surface
heat flow data and from an estimate of the radio-
genic heat produced in the Earth's interior.
@T
@t dV
(12.11)
Therefore, combining this
expression with
( 12.10 ) provides:
Z
@T
2 T dV D 0
@t k r
(12.12)
R
The fact that this integral vanishes for any
choice of the region R implies that the integrand
itself must be zero throughout R . Therefore,
introducing the thermal diffusivity k / c ¡
[m 2 s -1 ], we obtain the following equation of
conductive
12.2
Continental Geotherms
Let us consider the temperature distribution in a
homogeneous region R , bounded by the surface
S ( R ). According to the form ( 12.2 ) of Fourier's
law, the maximum heat flux q is a potential
field and the temperature itself is the associated
potential. Assuming that there are no sources of
heat in R , the total free flux of heat through S ( R )
will be given by:
heat
transfer
(or
heat
diffusion
equation ):
@T
@t r
2 T D 0
(12.13)
This equation holds, in the present form, in
the hypothesis that the thermal conductivity k and
 
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