Geoscience Reference
In-Depth Information
Convection can be driven by heating from
below or within, or by cooling from above. The
usual case treated is where the convection is ini-
tiated by a vertical temperature gradient. When
the vertical increase of temperature is great
enough to overcome the pressure effect on den-
sity, the deeper material becomes buoyant and
rises. An adiabatic gradient simply expresses the
condition that the parcel of fluid retains the
same density contrast as it rises. Horizontal tem-
perature gradients can also initiate convection.
Convection can be driven in a tank of fluid where
the side-walls differ in temperature. There is no
critical Rayleigh number in this situation. Lateral
temperature gradients can be caused by the pres-
ence of continents or variations in lithosphere
thickness such as at fracture zones or between
oceans and continents. For high Ra and Pr most
of the temperature contrast occurs across narrow
boundary layers.
In the case of natural convection, velocity is
not imposed but is set by buoyancy effects. A cen-
tral issue is to find a relationship between a tem-
perature difference applied to the system and
the corresponding heat flux. Fundamental stud-
ies are often concerned with Rayleigh--Bénard
convection of a fluid layer heated from below
and cooled from above, and where temperature
is the only control on density. Natural systems
are not so ideal. At very high Ra, the velocity and
the transported heat flux are expected to become
independent of viscosity and heat conductivity,
which is reached in large-scale systems such as
the atmosphere.
The Gruneisen parameter can be regarded as
a nondimensional incompressibility
where
h
is the thickness of the convecting layer. If
h
is the thickness of the mantle Di is about 0.5.
One hesitates to assign someone's name to this
number, even a dead physicist. The dissipation
number divided by the Gruneisen ratio is the
ratio of the thickness of the convecting layer
to the density-scale height or
h
d
g
K
s
. When Di
is large, the assumption of incompressible flow
is not valid. Nevertheless, the incompressible
mass conservation equation is usually adopted
in mantle convection studies. Compression also
changes the physical properties of the mantle,
the Rayleigh number and the possibility of chem-
ical stratification.
The buoyancy ratio is
ρ/
B
=
ρ
c
/ρα
T
where
ρ
c
is the intrinsic chemical density con-
trast between layers. When this is small we
have purely thermal convection but when it is
large the dense components can no longer be
entrained and chemical layering results. Pressure
serves to decrease
and therefore to stabilize
chemically stratified convection. In discussions of
layered mantle convection
B
is the most impor-
tant parameter.
α
Rayleigh-like numbers
The first order questions of mantle dynamics
include:
(1) Why does Earth have plate tectonics?
(2) What controls the onset of plate tectonics, the
number, shape and sizes of the plates, the
locations of plate boundaries and the onset
of plate reorganization?
(3) What is the organizing principle for plate
tectonics; is it driven or organized from the
top or by the mantle? What, if anything, is
minimized
?
γ
=
α
K
s
/ρ
C
p
=
α
K
T
/ρ
C
s
It is important in compressible flow calcula-
tions. This effect is different from the effect of
compression on physical and thermal properties.
The
density
scale
height
in
a
convecting
Surprisingly, these are not the questions being
addressed by mantle geodynamicists or computer
simulations.
Marangoni or Bénard-Marangoni
con-
vection is controlled by a dimensionless number,
layer is
h
d
=
δ
z
/δ
ln
ρ
=
γ
C
p
/α
g
=
K
s
/ρ
g
where
z
is the radial (vertical) coordinate.
The dimensionless dissipation number, Di, is
Di
=
hg
α/
C
p
M
=
σ
TD
/ρνκ
