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Table 8.2 Possible Rayleigh numbers for the mantle
Thickness (km)
Rayleigh number, Ra
10 6
Upper mantle
670
10 7
Whole mantle
2900
6
×
Thus, as convection becomes the dominant mechanism of heat transport, the
Nusselt number increases (when Ra
Ra c , Nu
1; for the situation in the mantle
10 3 Ra c , Nu
with Ra
10).
In regions where upwelling occurs, such as beneath a mid-ocean-ridge axis,
heat is carried upwards, or advected, by the rising material. The thermal Peclet
number ( Pe t ), the ratio of convected to conducted heat, is another measure of the
relative importance of convective to conductive heat transport,
ul
κ
Pe t =
(8.36)
where u is the velocity at which the material is moving, l a length scale and
the
thermal diffusivity. If Pe t is much larger than unity, advection dominates; if Pe t
is much smaller than unity, conduction dominates. In the mantle Pe t is about 10 3 ,
showing that the heat is transported mainly by advection.
The dimensionless Prandtl number ,
κ
Pr = υ/κ
= Pe t / Re
(8.37)
indicates the relative importance of viscous forces in diffusing momentum of
the fluid compared with heat. The Prandtl number is a physical property of the
material and is independent of any flow. For the mantle with
10 18 m 2 s 1
υ
10 24 , demonstrating that the viscous response to any
perturbation is instantaneous compared with the thermal response.
Table 8.2 gives approximate values of Ra calculated for the mantle using
Eq. (8.32) with A
10 6 m 2 s 1 , Pr
and
κ
=
0 and assuming the following values: for the upper mantle
10 5
C 1 ,
10 6
m 2
s 1 , Q
10 2 Cm 1
α =
2
×
κ =
/
k
=
1.5
×
and
υ =
3
×
10 17 m 2 s 1 ; for the whole mantle
10 5 C 1 ,
10 6 m 2 s 1 ,
α =
1.4
×
κ =
2.5
×
10 17 m 2 s 1 and Q
10 3 Cm 1 . Similar values for the Rayleigh
υ =
4
×
/
k
=
7
×
number are obtained with Q
=
0 and the internal heat generation of the mantle
about 10 11 Wkg 1 .
It is clear from Table 8.2 that the Rayleigh number is much greater than the
critical Rayleigh number (
10 3 )irrespective of whether flow is considered to
occur throughout the whole mantle or to be separate in the upper and lower
mantle. Although the exact value of the Rayleigh number depends on the values
chosen for the properties of the mantle, it is clear that convection in the mantle
is vigorous. The most unrealistic assumption in these calculations is that of a
constant-viscosity mantle. It is possible that the viscosity of the mantle is highly
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