Geology Reference
In-Depth Information
which is not moving upwards. Thus the fluid on the axis rises through the head and
spreads out towards the 'equator' of the head.
A relatively simple theory developed by Griffiths and Campbell [68] predicts
that a plume head will start with a diameter of about 400 km near the bottom of
the mantle and grow to a diameter of about 1000 km near the top of the mantle.
The growth of the head is due both to new fluid flowing up the tail into the head
and to surrounding fluid being thermally entrained into the head as illustrated in
Figure 7.9. Simple theory, which will be presented below, also shows that a tail
diameter of around 100 km (in the upper mantle) allows a sufficient flow up the tail
to account for the buoyancies and heat flows inferred in the previous section.
To summarise this section, the hot thermal boundary layer that we expect at the
bottom of the mantle is likely to be unstable and to generate buoyant upwellings.
These upwellings are more likely to take the form of columns rather than vertical
sheets. The upwelling material will have a lower viscosity than the ambient mantle,
and this will result in the upwelling forming into a head-and-tail structure. The
roughly spherical head will start with a diameter of about 400 km and grow to
about 1000 km in diameter near the top of the mantle. The growth is due to thermal
entrainment of surrounding material into the plume head, as well as to more material
flowing up the plume tail. The tail diameter of about 100 km is sufficient to account
for observed plume tail flows.
Simplified versions of the required theoretical results will be given in the fol-
lowing subsections. The discussion so far will have conveyed the essential physics
of mantle plumes, but the simple theory shows that plumes are predicted by well-
quantified physics. The application of this physical understanding to observations
of the Earth is taken up again in Section 7.4.
7.3.2 The Rayleigh-Taylor instability
We can analyse the instability of a buoyant fluid layer using the same approach as
we used in Chapter 4 on flow and viscosity and in Chapter 5 on convection.
If the interface between the two layers of fluid in Figure 7.6 were perfectly
horizontal, the fluid would not move, even though the lower layer is buoyant. Of
course, nothing in nature is perfect, and there will be undulations in the interface,
depicted as a simple bulge in the sketch. Because the fluid in the bulge is less dense
than the fluid on either side of it, it will generate a buoyancy force,
B
, proportional
to the mass deficit in the bulge. The mass deficit is the density deficit times the
volume of the bulge. We will do the usual rough approximating here, so let's take
the volume to be simply
wh
, the width times the height of the bulge, which we will
assume to be quite small. Putting these things together, we get
B
=
gρwh.
(7.11)
