Geology Reference
In-Depth Information
the fluid on the other side of the descending sheet will exert a similar resistance,
and the fluid will also resist the motion of the surface plate, because of the shearing
depicted by the horizontal arrows. In our idealised square box, these three forces
are the same. Thus we might estimate the total resisting forces as the remarkably
simple expression
F
R
=
6
μv.
(5.8)
We have now evaluated both terms in the force balance equation (5.6), so we get
−
gρ
0
αT Dd/
2
+
6
μv
=
0
,
which rearranges into
v
=
gρ
0
αT Dd/
12
μ.
(5.9)
Here we have a formula for the velocity of the sinking sheet, which in the square-
box approximation is the same as the velocity of the surface plate. We can evaluate
it using values already used, plus the observation (from seismology) that plates
are about 100 km thick, so we can use
d
=
100 km. We also need the depth of the
mantle,
D
3000 km, and we will use a viscosity that takes account of the fact that
the viscosity is greater in the lower mantle, i.e. 10
22
Pa s [41]. This gives a velocity
of 3
=
10
−
9
m/s, which is about 10 cm/yr.
Observed plate velocities range from about 2 to 12 cm/yr, with an average of
about 5 cm/yr. Our estimate is within a factor of 2 of the observed average. Thus
our initial deep ignorance (Are plate velocities millimetres per million years, or
kilometres per year, or something else even?) has been replaced with an estimate
that is better than an order of magnitude in accuracy, even though our estimate is
based on a very crude analysis. Again, this shows the value of even a very simple
physical analysis, provided reasonably representative values are used.
×
5.4 Heat conduction
We can actually do better than the simple theory of the previous section. We eval-
uated the thickness of the plate,
d
, by appealing to observations. But in convection
the thickness of the thermal boundary layer is determined internally, as part of the
process of convection, so the theory ought to determine
d
without any extra input.
We will extend our convection theory to do this in the next section, after we have
looked at heat conduction, which is also known as
thermal diffusion
.Thereare
several ideas to walk through, but bear with it because we get to a simple relation-
ship that is quite powerful because it can be applied to many situations to deduce
important things like timescales and depth scales.
