Geology Reference
In-Depth Information
Appendix B
Thermal evolution details
B.1 Heat generation
The heat generation per unit mass due to radioactive heating is given by
4
H
R
=
h
i
exp [
λ
i
(
t
E
−
Ur
U
e
t
)]
,
(B.1)
i
=
1
where Ur is the Urey ratio, defined by Eq. (9.5),
U
e
is the equivalent uranium
concentration (38 ng/g) that would yield the observed heat loss, the index
i
refers to the
isotopes
238
U,
235
U,
232
Th and
40
K,
h
i
is the heat production per unit mass of uranium,
λ
i
is the decay constant and
t
E
is the age of the Earth (4.5 Ga). Values for the required
parameters are given in Table B.1.
B.2 Parametrised thermal evolution model
The thermal evolution model in Figure 9.1 is a variation on the models reported in Davies
[65]. These models include the effects of the core crystallising to form the inner core.
There are many details of this process, and many parameters involved, which won't be
reproduced here, as they can be found in that paper. The other main parameters and
outputs of this model are given in Table B.2.
There is one feature of this solution worth noting, as it adds usefully to the examples in
Davies [65]. In the present calculation, the starting temperature of the core was increased
from 4000
◦
C to 4500
◦
C, relative to Case 8 of the paper. This increased the core heat loss
to within the range estimated in Chapter 7. It also increased the energy available within
the core to drive the magnetic dynamo. There are several apparently conflicting
constraints that the paper attempted to reconcile, one of them being the need for enough
energy to drive the dynamo, which is not well constrained but was estimated at the time
to be around 1 TW. Core energy dissipation and the inner core growth are included in
Figure B.1. The thermal dissipation starts above 1 TW and decreases, but after the inner
core starts to crystallise the dissipation driven by compositional convection adds to this
and boosts it back above 1 TW. The lowest total dissipation is 0.51 TW at the time the
inner core nucleates. This calculation more easily reconciles the various constraints than
those given in the paper above [65].
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