Geology Reference
In-Depth Information
where the subscript on
q
S
is to remind us that this is the surface heat loss per unit
area of the Earth's surface. Then
4π
R
E
q
S
,
Q
S
=
A
E
q
S
=
(9.7)
where
A
E
istheareaoftheEarth'ssurfaceand
R
E
is the radius of the Earth.
Temperature occurs explicitly twice in Eq. (9.6), and implicitly once, because the
viscosity is a strong function of temperature, as we saw in Chapter 4 (Eq. (4.9) and
Figure 4.4). The other quantities involved in
Q
S
are either constant or not likely
to vary much as the mantle temperature evolves. Thus, although these equations
look rather messy, the essential point is that they define how the heat loss varies
as the mantle temperature varies. Thus, referring back to the initial discussion, we
could estimate what the heat loss was one billion years ago, when the mantle was
probably a bit hotter than at present.
An analogous expression can be derived for the heat transported by mantle
plumes, though it's even messier. It can be found in
Dynamic Earth
[1]. The essential
points here are that it is a little less sensitive to temperature, and that the relevant
viscosity is the viscosity of the hottest mantle material, right next to the core.
That's because it has the lowest viscosity and is the most mobile material.
Thus we can get expressions for
Q
R
,
Q
C
and
Q
S
in Eq. (9.4) that depend on
the temperature of the mantle or on time. That means we can do the calculations
outlined at the start of this section, i.e. calculate the present rate of change of
temperature, estimate from that the mantle temperature at some previous time,
recalculate the heat terms and the rate of change of temperature, estimate the
temperature at an even earlier time, and so on. This, essentially, is what is done,
except it's done numerically and using much smaller time steps than one billion
years. There are well-documented clever ways of minimising errors in such a
calculation, but we don't need to go into all that. You can find everything you ever
wanted to know in
Numerical Recipes
[139], an excellent resource.
The temperature of the core can be calculated in a similar way. The heat loss
from the core is just
Q
C
, and this calculation assumes there is no radioactivity in
the core, though that is debated by some. Thus the relationship is like Eq. (9.4) with
only the heat loss term, and with parameters appropriate for the core. Both of these
calculations can also be run forwards in time, if you assume a starting temperature.
The result of one such calculation is shown in Figure 9.1. The calculation was
run forwards in time. The initial state is envisaged to be like that sketched in
Figure 7.3, curve (a), in which the mantle and the core have the same temperature
where they contact. Parameters have been adjusted to match observed present values
of the upper-mantle temperature (
T
U
), surface heat flow (
Q
S
) and plume heat flow
(
Q
C
). The lower-mantle temperature (
T
L
) is assumed to be 1000
◦
C higher than
the upper-mantle temperature at present. The core temperature (
T
C
) is assumed to
