Geoscience Reference
In-Depth Information
may cause a large increase in lattice conductiv-
ity. Other pressure effects on physical properties --
viscosity, thermal expansion -- all go in the direc-
tion of suppressing thermal instabilities -- narrow
plumes -- at the core-mantle boundary.
The ratio
Table 21.2
Estimates of lattice
thermal diffusivity in the mantle
Depth
κ
(km)
(cm
2
/s)
K
L
decreases rapidly with depth
in the mantle, thereby decreasing the Rayleigh
number. Pressure also increases the viscosity,
an effect that further decreases the Rayleigh
number of the lower mantle. The net effect of
these pressure-induced changes in physical prop-
erties is to make convection sluggish in the lower
mantle, to decrease thermally induced buoyancy,
to increase the likelihood of chemical stratifica-
tion, and to increase the thickness of the ther-
mal boundary layer in D
, above the core--mantle
boundary.
The mechanism for transfer of thermal energy
is generally well understood in terms of lat-
tice vibrations, or high-frequency sound waves.
This is not enough, however, since thermal con-
ductivity would be infinite in an ideal har-
monic crystal. We must understand, in addition,
the mechanisms for scattering thermal energy
and for redistributing the energy among the
modes and frequencies in a crystal so that ther-
mal equilibrium can prevail. An understanding
of thermal 'resistivity,' therefore, requires an
understanding of higher order effects, including
anharmonicity.
Debye theory explains the thermal conductiv-
ity of dielectric or insulating solids in the follow-
ing way. The lattice vibrations can be resolved
into traveling waves that carry heat. Because
of anharmonicities the thermal fluctuations in
density lead to local fluctuations in the veloc-
ity of lattice waves, which are therefore scat-
tered. Simple lattice theory provides estimates of
specific heat and sound velocity and how they
vary with temperature and volume. The theory
of attenuation of lattice waves involves an under-
standing of how thermal equilibrium is attained
and how momentum is transferred among lattice
vibrations.
The thermal resistance is the result of
interchange of energy between lattice waves,
that is, scattering. Scattering can be caused
by static imperfections and anharmonicity.
Static imperfections include grain boundaries,
α/
10
−
3
50
5.9
×
10
−
3
150
3.0
×
10
−
3
300
2.9
×
10
−
3
400
4.7
×
650
7.5
×
10
−
3
1200
7.7
×
10
−
3
2400
8.1
×
10
−
3
2900
8.4
×
10
−
3
Horai and Simmons (1970).
collision time or a measure of the strength
and distribution of scatterers, velocities of sound
waves and the interatomic distances.
Both thermal conductivity and thermal
expansion depend on the anharmonicity of the
interatomic potential and therefore on dimen-
sionless measures of anharmonicity such as
γ
or
αγ
T
. The lattice or phonon conductivity is
K
L
=
C
V
Vl
/
3
Where
V
is the mean sound speed,
l
is the mean
free path, which depends on the interatomic dis-
tances and the isothermal bulk modulus,
K
T
. This
gives
[
δ
ln
K
L
/δ
ln
ρ
]
=
[
δ
ln
K
T
/δ
ln
ρ
]
−
2[
δ
ln
γ/δ
ln
ρ
]
+
γ
−
1
/
3
For lower-mantle properties this expression is
dominated by the bulk modulus term and the
variation of
K
L
with density is expected to be sim-
ilar to the variation of
K
T
.
The lattice conductivity decreases approx-
imately linearly with temperature, a well-
known result, but increases rapidly with den-
sity. The temperature effect dominates in the
shallow mantle, but pressure dominates in the
lower mantle. This has important implications
regarding the properties of thermal boundary
layers, the ability of the lower mantle to con-
duct heat from the core, and the convective
mode of the lower mantle. The spin-pairing and
post-perovskite transitions in the deep mantle
