Geology Reference
In-Depth Information
difference
between volumes within the fluid. Because buoyant things go up, we
define buoyancy to be positive when the force is upwards. Thus the buoyancy of a
piece of ice of volume
V
in water is
B
=−
g
(
ρ
i
−
ρ
w
)
V,
(5.4)
where
g
is the acceleration due to gravity, and
ρ
i
and
ρ
w
are the densities of ice and
water, respectively (
V
is now the total volume of the ice, not the specific volume
used in the previous section).
A volume of warmer fluid will be less dense than its cooler surroundings and
therefore buoyant. In this case we can write, using Eq. (5.3),
B
=−
gρV
=
g
(
ρ
0
αT
)
V.
(5.5)
For example, suppose we approximate the buoyant column in Figure 5.1 as a
cylinder of height
h
50 km. Suppose it is 300
◦
C
hotter than the surrounding fluid. Then its buoyancy is
B
=
g
(
ρ
0
αT
)(π
r
2
h
)
=
2000 km and diameter
r
=
10
17
N
.
=
×
4
.
6
9.8 m/s
2
and other values as previously specified. The units
of force are newtons, N. This value of
B
is a large number, but it does not have
much meaning until it is related to other things.
Here I have used
g
=
5.3 Plate velocity - simple mechanical version
Figure 5.2(a) is a sketch of a system of tectonic plates, in cross-section, with the
middle plate subducting. Because the plate sinks, it displaces mantle material and
thus induces flow in the mantle. The plate is taken to have a thickness
d
and to
sink with velocity
v
. The mantle has an internal temperature
T
, a viscosity
μ
and a
depth
D
. The surface temperature is
T
0. If the plate is sinking because it is cold
and heavy, then this is an example of convection as it was defined above. In this
case there is a cold thermal boundary layer at the top of the fluid, and the active
component is a negatively buoyant sheet of cold material sinking away from the
thermal boundary layer, whereas in Figure 5.1 it is the buoyant rising column that
is the active component.
Figure 5.2(b) is a simplification of the situation in Figure 5.2(a). It may seem
crude, but we can use the same approach here as we used in estimating the viscosity
of the mantle in Chapter 4. I want to pose the question: 'How fast will the subducting
plate move?' If you knew nothing about plate tectonics, you would have little idea
of the answer. It might be kilometres per year, or millimetres per million years, or
almost anything. By using the very simplified situation in Figure 5.2(b) we can make
an estimate that will give us some idea of the answer. If we choose representative
=
