The plume mode of convection - Mantle Convection for Geologists - page 91

hand, thermal diffusion will tend to smooth out the bulge, with a timescale like τ D .
Which process wins will depend on the ratio of the timescales:

gρD 3 /κμ. τ D /τ RT = (7.18)

If this looks familiar, it should - it is the Rayleigh number, given by Eq. (5.19).
So, we have a new insight into what the Rayleigh number tells us. If τ D is much
smaller than τ RT , then thermal diffusion is rapid compared with the growth of the
layer instability, and we might expect that the instability will be inhibited. On the
other hand, if τ D is much larger than τ RT , thermal diffusion will be relatively slow,
and the instability might develop with little restraint.
It turns out that there is a threshold value of Ra below which no convection occurs.
The threshold or critical value, Ra c , varies with circumstance, but is typically about
1000. This fact was established by Lord Rayleigh, hence the number bears his
name. Convection with Ra just above Ra c takes the form of simple, regular cells,
similar to that shown in Figure 6.2. Such cells had been observed experimentally
by Benard, and this form of convection is known as Rayleigh-Benard convection.
It is the form of convection most commonly shown in textbooks, and Rayleigh's
analysis is often the starting point of discussions of convection (e.g. Turcotte and
Schubert [53]). However, convection in the Earth's mantle has Ra well above Ra c
and it would not occur in simple cells, even if the plates were not present, as you
can see from Figure 6.4.
With insight from Eq. (7.18), we can see that the reason convection does not
occur below a threshold value Ra c is that the initial instability cannot establish
itself faster than it is removed by thermal diffusion. Incidentally, we might have
expected the critical Rayleigh number to be more like 1 than 1000. The reason it is
so large is probably that we have used the time it takes for diffusion to act across
the whole fluid layer. However, diffusion may only need to act over a thin thermal
boundary layer, and if we used, for example, D/ 30 as our diffusion length scale,
then the critical ratio of timescales would be about 1.

7.3.3 Rate of flow up a plume tail
It is useful to consider flow up a plume tail in more detail, as we can get a more
physically based estimate of the radius of a plume tail. It will also help us to
understand how most plume tails can be relatively thin.
Figure 7.10(a) shows a parcel of fluid within a conduit of buoyant fluid. We will
assume that the fluid outside the conduit is not moving, so the velocity of the fluid
in the conduit will vary from a maximum in the centre of the conduit to zero at the