Geology Reference
In-Depth Information
Finally, the complete expression for the stream
function for the corner flow is:
§ D r
sary to start from the consideration that thermal
convection ultimately is driven by density varia-
tions arising from thermal expansion or contrac-
tion. Consequently, we will be concerned with the
problem to give a specific form to the equation
of state ( 13.19 ), which determines the relation-
ship between thermodynamic pressure, density,
and temperature. Unfortunately, solids lack an
equation of state that can be formulated on the
basis of theoretical arguments, similar to the well-
known ideal gas equation, PV D nRT . In general,
each solid has its own equation of state based
on empirical grounds, which specifies how the
density changes as a function of temperature and
pressure: ¡ D ¡( T , p ). However, in the Earth's
mantle there are also significant density changes
associated with phase transitions or, possibly,
with variations of chemical composition.
Despite these complications, most of the den-
sity variations in the Earth's mantle are due
to hydrostatic compression. Therefore, starting
from a hypothetical homogeneous mantle in hy-
drostatic equilibrium, where heat is transferred
only by conduction, it should be possible to find
an approximate solution of the equations that
describe the onset of thermal instabilities and the
formation of steady Rayleigh-Bénard rolls. Such
an approximation exists and is called Boussinesq
approximation . Although this method has been
widely used in mantle geodynamics studies, its
theoretical ground is often outlined using confus-
ing arguments, both in articles and topics. With
the objective to delimitate precisely the field of
0 C sin™ 0 sin ™ C v 0 .1 C cos™ 0 /
v 0 0
0 C sin ™ 0
0 C sin ™ 0 cos ™
v 0 sin ™ 0
sin ™
(13.97)
The streamlines of this function are shown
in Fig. 13.10 . It is interesting to note that both
components of the velocity field generated by the
stream function ( 13.97 ) are independent from r .
Lastly, we mention that a very similar solution
exists also in the case of non-Newtonian fluids
(Tovish et al. 1978 ).
13.6
Rayleigh-Bénard Convection
Rayleigh - Bénard convection ,or thermal convec-
tion , is a natural convection that occurs in a fluid
layer that is heated from below and cooled from
above. In these conditions, a regular pattern of
convection cells develops between the top and
the bottom of the layer, because the hot light
fluid near the lower boundary tends to ascend,
while cooled fluid near the top becomes denser
than the average and sinks. At the scale of the
upper mantle Richter and Parsons ( 1975 ), proved
that small-scale convective rolls superimpose the
large-scale upper mantle flow determining the
complex flow pattern illustrated in Fig. 13.11 .To
describe quantitatively this process, it is neces-
Fig. 13.11 Large scale
upper-mantle circulation
(blue lines) and
Rayleigh-Bénard rolls (in
red). The total flow pattern
in the upper mantle results
from superposition of these
two flows
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