The plume mode of convection - Mantle Convection for Geologists - page 90

There are two more aspects of the Rayleigh-Taylor instability and its application
to a thermal boundary layer that can be analysed. One is that there is an optimum
scale (i.e. an optimum value of w ) that maximises the growth rate of the bulge. The
other is that if the bulge grows too slowly then thermal diffusion can wipe it out,
and the instability dies. I will not go through even the simple analyses of the first
point here. It can be found in Dynamic Earth [1], and more rigorous mathematical
treatments can be found in other texts, such as Turcotte and Schubert's Geodynamics
[53]. Here I will just describe the physics in words.
The growth rate given by Eqs (7.13)-(7.15) is larger for larger w .However,the
analysis is only valid for w<D , the total layer depth (Figure 7.6). When w is
larger than D , the dominant viscous resistance comes from horizontal flow, and the
more relevant velocity gradient is w/D . The result of this change is that, for large
w ,as w gets larger, the growth rate gets smaller, the opposite of the analysis for
small w . The growth rate is a maximum when w ∼ D , and the minimum growth

time is approximately

τ RT = μ/gρD, (7.16)

where the subscript RT denotes the Rayleigh-Taylor timescale. This establishes
that there is an optimum horizontal length scale or wavelength for which growth of
the instability is fastest. This not only gives us an indication of the size of expected
upwellings, but also tells us that there will not be flocks of tiny upwellings. The latter
idea has sometimes been invoked to argue that there may be many small plumes
that elude detection because they are small. However, the idea is fluid-dynamically
implausible.
If the lower fluid layer of Figure 7.6 is buoyant because it has a different
composition, then there is no more to say about its instability. If, however, it is
buoyant because it is hotter, then we can go beyond the basic Rayleigh-Taylor
analysis, because there is more physics to consider. Thermal diffusion will tend
to smooth out any temperature variations. In Section 5.4 we examined thermal
diffusion and found that the time taken for temperature to diffuse over a distance d
is proportional to d 2 . Equation (5.15a) expresses this as t d 2 /κ ,where κ is the

thermal diffusivity. The time for diffusion to act across the whole fluid layer, depth
D , will then be

=

D 2 /κ, τ D = (7.17)

where the subscript D denotes a thermal diffusion timescale.
We can now think about two physical processes that compete. Bulges in the
unstable fluid interface grow, and the timescale of that growth is τ RT . On the other