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and the Prandtl number (Pr
is the
kinematic viscosity. Only when both of these are
the same in two geometrically similar situations
can the flow patterns be expected to be same. In
general, the numbers appropriate for the man-
tle cannot be duplicated in the laboratory. The
Prandtl
= ν/κ
) where
ν
perature drops across the layer with and without
convection or equivalently, the ratio of the half-
depth of the layer to the thermal boundary layer
thickness.
The
varies from
about 10 2 in liquid metals to 1 for most gases,
and slightly more than 1 for liquids such as water
and oil. Pr for the mantle is about 10 24 , which
means that the viscous response to a pertur-
bation is instantaneous relative to the thermal
response. If one changes a boundary condition, or
inserts a crack into a plate, the effect is felt imme-
diately by the whole system. Thermal perturba-
tions, however, take time to be felt. The square
root of Pr gives the ratio of the thicknesses of
the mechanical boundary layer (MBL) to the ther-
mal boundary layer (TBL). If the container, or the
mantle is smaller than this, then convection will
organize itself so as to have a small number of
upwellings and downwellings. The limiting case
is rotation of the whole fluid.
One cannot take one's experience with smoke
plumes in the atmosphere or hot plumes in boil-
ing pots of water and apply it to the mantle. One
must look at systems with comparable Rayleigh
and Prandtl numbers. Narrow plumes are char-
acteristic of high Rayleigh number, low Prandtl
number flows. The mantle is the opposite. There
are no large velocity gradients in high Pr fluids.
The Reynolds number is defined by Re
Prandtl
number,
Pr
= ν/κ
number
for
the
mantle
is
essentially
10 23 . Inertial forces can be ignored in
mantle convection.
The Rayleigh number,
infinite,
Ra
=
Gr Pr
TL 3
Ra
=
g
α
/νκ
is the ratio between thermal driving and viscous
dissipative forces, and is proportional to the tem-
perature difference,
T ,andthecubeofthe
scale of the system. It is a measure of the vigor
of convection due to thermally induced density
variations,
operat-
ing in a gravity field g . This is for a uniform fluid
layer of thickness d with a superadiabatic tem-
perature difference of
α
T , in a fluid of viscosity
ν
T maintained between
the top and the bottom. If the fluid is heated
internally, the
α
T term is replacd by the volu-
metric heat production. Convection will occur if
Ra exceeds a critical value of the order of 10 3 .For
large Ra the convection and heat transport are
rapid. The Rayleigh number depends on the scale
of convection as well as the physical properties.
If L istakentobethedepthofthemantle
one obtains very large Ra. However, if convection
is layered, the scale drops and Ra can become
very small. At high pressure, the combination of
properties in Ra also drives Ra down.
The Nusselt number gives the relative impor-
tance of convective heat transport compared with
the total heat flux:
α
v ,
for a fluid of kinematic viscosity v , flowing with
speed U past a body of size L . In aerodynam-
ics, it characterizes similarities between flows
with the same Reynolds number. Turbulence at
very high Reynolds numbers is expected to be
controlled by inertial effects (as viscosity passi-
vely smoothes out the smallest scales of motion),
as seen for flows around an aircraft or car. The
Reynolds number can also be written
=
UL
/
Nu
=
total heat flux across the layer/conducted
heat flux in the absence of convection
= QL / K T
Re
=
Pe
/
Pr
where Q is the rate of heat transfer per unit unit
area, and L and
For typical plate tectonic rates and dimen-
sions, Re
10 21 .ForRe
T are the length and tempera-
ture difference scales.
The Nusselt number is the ratio of the actual
heat flux to the flux that would occur in a purely
conducting regime (so it expresses the efficiency
of convection for enhancing heat transfer). For an
internally heated layer Nu is the ratio of the tem-
1 inertial effects are
negligible, and this is certainly true for the man-
tle. Re is important in aerodynamics and hydro-
dynamics but not in mantle dynamics. Inertia
can be ignored in plate tectonics. In free thermal
convection Re and the velocity are functions of
Pr and Ra; they are not independent parameters.
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