Geoscience Reference
In-Depth Information
mantle minerals (see Table 18.5). The moduli can
then be corrected for the volume change using
equations given. The normalized parameters can
then be used in an equation of state to calculate
M
(
P
,
T
), for example, from finite strain. The nor-
malized form of the pressure derivatives can be
assumed to be either independent of temperature
or functions of
V
(
T
). Temperature is less effective
in causing variations in density and elastic prop-
erties in the deep mantle than in the shallow
mantle and the relationships between these vari-
ations are different from those observed in the
laboratory. Anharmonic effects also are predicted
to decrease rapidly with compression. Anhar-
monic contributions are particularly important
at high temperature and low pressure. Intrinsic
or anharmonic effects probably remain impor-
tantinthelowermantle,eveniftheydecrease
with pressure. Anelastic thermal effects on mod-
uli are also probably more important in the
upper mantle than at depth. Large velocity vari-
ations in the deep mantle are probably due to
phase changes, composition and melting, or the
presence of a liquid phase.
}
T
respectively, and this notation
will be used later.
For many minerals, and the lower mantle
K
s
}
T
and
G
{
{
(
∂
ln
K
S
/∂
ln
G
)
T
=
(
K
S
/
G
)
is a very good approximation.
Intrinsic and extrinsic temperature
effects
Temperature has several effects on elastic moduli;
intrinsic, extrinsic, anharmonic and anelastic.
The effects of temperature on the properties
of the mantle must be known for various geo-
physical calculations. Because the lower man-
tle is under simultaneous high pressure and
high temperature, it is not clear that the sim-
plifications that can be made in the classical
high-temperature limit are necessarily valid. For
example, the coefficient of thermal expansion,
which controls many of the anharmonic prop-
erties, increases with temperature but decreases
with pressure. At high temperature, the elas-
tic properties depend mainly on the volume
through the thermal expansion. At high pres-
sure, on the other hand, the intrinsic effects
of
Temperature and pressure
derivatives of elastic moduli
temperature
may
become
relatively
more
important.
The temperature derivatives of the elastic
moduli can be decomposed into extrinsic and
intrinsic components:
The pressure derivatives of the adiabatic bulk
modulus for halides and oxides generally fall in
the range 4.0 to 5.5. Rutiles are generally some-
what higher, 5 to 7. Oxides and silicates having
ions most pertinent to major mantle minerals
have a much smaller range, usually between 4.3
and 5.4. MgO has an unusually low value, 3.85.
The density derivative of the bulk modulus,
(
∂
ln
K
S
/∂
ln
ρ
)
T
=
(
∂
ln
K
S
/∂
ln
ρ
)
T
−
α
−
1
(
∂
ln
K
S
/∂
T
)
V
or
{
K
S
}
P
={
K
S
}
T
−{
K
S
}
V
for the adiabatic bulk modulus,
K
s
, and a similar
expression for the rigidity,
G
. Extrinsic and intrin-
sic effects are sometimes called 'volumetric' or
'quasi-harmonic' and 'anharmonic.' The anelastic
effect on moduli, treated later, is also important.
The intrinsic contribution is
K
S
)
K
S
(
∂
ln
K
S
/∂
ln
ρ
)
T
=
(
K
T
/
for mantle oxides and silicates that have been
measured usually fall between 4.3 and 5.4 with
MgO, 3.8, again being low.
The rigidity,
G
, has a much weaker volume or
density dependence. The parameter
(d ln
M
/
d
T
)
V
=
α
[(
δ
ln
M
/δ
ln
ρ
)
T
−
(
δ
ln
M
/δ
ln
ρ
)
P
]
(
∂
ln
G
/∂
ln
ρ
)
T
=
(
K
T
/
G
)
G
where
M
is
K
s
or
G
. There would be no intrinsic
or anharmonic effect if
generally falls between about 2.5 and 2.7. The
above dimensionless derivatives can be written
=
M
(
V
). The various terms in this equation require
α
=
0orif
M
(
V
,
T
)
