Geoscience Reference
In-Depth Information
vacancies, interstitials and dislocations and
their associated strain fields, which considerably
broadens the defects cross section. These 'static'
mechanisms generally become less important at
high temperature. Elastic strains in the crystal
scatter because of the strain dependence of the
elastic properties, a nonlinear or anharmonic
effect.
The effect of pressure on
D
can be written
∂
RT
K
T
ln
G
∂
ln
ρ
V
∗
−
T
=
∂
ln
G
∂
ln
ρ
1
G
∗
1
K
T
=
T
−
or
∂
ln
D
∂
∂
ln
G
∂
T
−
1
G
∗
RT
T
=−
Diffusion and viscosity
ln
ρ
ln
ρ
RT
we have
V
∗
decreasing from 4.3 to 2.3 cm
3
/mole with depth
in the lower mantle. This gives a decrease in dif-
fusivity, and an increase in viscosity, due to com-
pression, across the lower mantle. Phase changes,
chemical changes, and temperature also affect
these parameters.
For a typical value of 30 for
G
∗
/
Diffusion and viscosity are activated processes
and depend more strongly on temperature and
pressure than the properties discussed up to now.
The diffusion of atoms, the mobility of defects,
the creep of the mantle and seismic wave atten-
uation are all controlled by the diffusivity.
D
(
P
,
T
)
=
ζ
a
2
v
exp[
−
G
∗
(
P
,
T
)
/
RT
]
Viscosity
There are also effects on viscosity that depend
on composition, defects, stress and grain size.
Viscosities will tend to decrease across mantle
chemical discontinuities, because of the high
thermal gradient, unless the activation energies
are low for the dense phases. Even if the viscosi-
ties of two materials are the same at surface con-
ditions, the viscosity contrast at the boundary
depends on the integrated effects of
T
and
P
,and
the activation energies and volumes.
The combination of physical parameters that
enters into the Rayleigh number decreases
rapidly with compression. The decrease through
the mantle due to pressure, and the possibil-
ity of layered convection, may be of the order
of 10
6
to 10
7
. The increase due to temperature
mostly offsets this; there is a delicate balance
between temperature, pressure and stable stratifi-
cation. All things considered, the Rayleigh num-
ber of deep mantle layers may be as low as 10
4
.
The local Rayleigh number in thermal bound-
ary layers increases because of the dominance
of the thermal gradient over the pressure gra-
dient. Most convection calculations, and convec-
tive mixing calculations, use Rayleigh numbers
appropriate for whole-mantle convection and no
pressure effects. These can be high by many
orders of magnitude. The mantle is unlikely to
where
G
∗
is the Gibbs free energy of activation,
ζ
is a geometric factor and
v
is the attempt
frequency (an atomic vibrational frequency). The
Gibbs free energy is
G
∗
=
E
∗
+
PV
∗
−
TS
∗
where
E
∗
,
V
∗
and
S
∗
are activation energy, volume
and
entropy,
respectively.
The
diffusivity
can
therefore be written
D
o
exp
−
(
E
∗
+
PV
∗
)
/
RT
D
=
=
ζ
a
2
v
exp
S
∗
/
RT
D
o
Typical
D
o
values are in Table 21.4. The theory for
the volume dependence of
D
o
is similar to that
for thermal diffusivity,
C
v
.Itincreases
with depth but the variation is small, perhaps
an order of magnitude, compared to the effect
of the exponential term. The product of
K
L
times
viscosity is involved in the Rayleigh number, and
the above considerations show that the temper-
ature and pressure dependence of this product
depend mainly on the exponential terms.
The activation parameters are related to the
derivative of the rigidity (Keyes, 1963):
κ
=
K
L
/ρ
K
T
)
∂
ln
G
∂
ln
ρ
1
V
∗
/
G
∗
=
/
T
−
(
l
