Geology Reference
In-Depth Information
if there were no convection, then the conducted heat flow,
q
c
, would be
q
c
=
KT/D.
The ratio of these quantities is just
D/d
, so from Eq. (5.20)
Ra
48
1
/
3
q
q
c
=
.
(5.22)
Thus we have another very compact expression.
You can see that, apart from numerical factors, the Rayleigh number contains
within it information about the boundary layer thickness, the velocity of convection
and the amount of heat transported by convection. For example, the larger is Ra,
the faster the convection will flow and the more heat it will transport. Perhaps
less obviously, Eq. (5.20) shows that the more vigorous convection will have a
thinner thermal boundary layer. That's because the faster fluid spends less time at
the surface and does not cool to as great a depth.
Two of the ratios we have just defined also have names. The ratio of velocities
in Eq. (5.21) is called the Peclet number, written Pe. The ratio of heat fluxes in
Eq. (5.22) is called the Nusselt number, written Nu. The ratio of thicknesses in
Eq. (5.20) has not been given a name. Thus we can summarise the above results
as
Ra
−
1
/
3
,
d/D
∼
Ra
2
/
3
,
=
∼
Pe
v/V
(5.23)
Ra
1
/
3
.
Nu
=
q/q
c
∼
10
6
.This
is regarded as a moderately high value, indicating reasonably vigorous convection.
This qualitative statement is based on the fact that there will not be any convection
unless Ra exceeds about 1000. We will look at this some more in Chapter 7, in the
context of mantle plumes. Suffice to say for now that at low Ra any upwelling that
begins to form, like that in Figure 5.1, is smoothed out by thermal diffusion before
it can become large enough to rise significantly. The disturbance in the fluid dies
away and the heat is transported by conduction through static fluid. So the mantle
value of Ra is well above the level at which convection can begin.
Suppose you want to set up a laboratory experiment to model convection in
the mantle. How do you know what tank size and fluid properties to use? Well, if
the tank experiment has a similar Rayleigh number to the mantle, then the flows
should have a similar vigour. Suppose your laboratory fluid has a density similar
to water, 1000 kg/m
3
, thermal diffusivity 10
−
7
m
2
/s, thermal expansion coefficient
5
With the values we have been using, the mantle value of Ra is 3.5
×
10
−
4
◦
C
−
1
×
and viscosity 5 Pa s. If you can impose a temperature difference of
