Geology Reference
In-Depth Information
which might be the velocity of tectonic plates back then. We might also get its rate
of heat loss, and thus some basic notion of what level of tectonic activity might
have been occurring. In this section we will show how that calculation works in
a little more detail. Later we will look at potential complications, such as density
differences caused by melting.
To begin, let's relate the rate of change of heat content to the rate of change
of temperature. Recall, from Section 5.4, that the change in heat content,
H
,is
related to the change in temperature,
T
, through the specific heat,
C
P
:
H
=
MC
P
T ,
(9.2)
where
M
is the mass of the mantle. If these changes occur within a time interval
t
, then we can relate the
rates
of change:
T
t
=
1
MC
P
H
t
.
(9.3)
There are two kinds of heat input into the mantle: radioactive heating,
Q
R
,and
heat entering from the core,
Q
C
. Heat is being lost through the Earth's surface,
Q
S
.
Thus
T
t
=
1
MC
P
(
Q
R
+
Q
C
−
Q
S
)
.
(9.4)
This expresses the same relationships as Eq. (9.1).
To go further, we need detailed expressions for the heat inputs and loss. Most of
the radioactive heating of the mantle is due to four isotopes:
238
U,
235
U,
232
Th and
40
K. The decay of these isotopes is well determined, but the expressions involved
are a little messy, so they are given in Appendix B. The abundance ratios of the
isotopes are fairly well determined, but the absolute amounts are still debated, as
we will see later. For this reason, we define an adjustable ratio, the Urey ratio, as
the ratio of present heat generation in the Earth to present heat loss, i.e.
Ur
=
Q
R
/Q
S
=
MH
R
/Q
S
,
(9.5)
where
H
R
is the heat generation per unit mass given by Eq. (B.1) in Appendix B.
In the tradition of fluid-dynamical ratios like the Rayleigh number, the Urey ratio
is given a two-letter symbol, Ur, so the 'r' is not a subscript.
In Chapter 5 we derived an expression for the velocity of plates, Eq. (5.18),
and an expression for the heat flow out of the mantle, Eq. (5.22). If we leave
out the approximate numerical factor and use previous expressions for Ra and
q
c
,
Eq. (5.22) can be written as
gραT D
3
κμ
1
/
3
KT
D
q
S
=
,
(9.6)
