Geoscience Reference
In-Depth Information
(Raj, 1975). In the initial elastic stage, sliding
occurs locally along the segments between the
obstacles, causing concentrations of stress at the
obstacles. In the later diffusional stage, sliding
occurs even where the obstacles exist, because
gaps are filled and overlaps smoothed out by the
diffusion of matter. Because the rate-limiting pro-
cess in this diffusional stage is the same as that
of diffusion creep, the characteristic frequency is
given by the Maxwell frequency calculated from
the diffusion-creep viscosity (Raj, 1975; Gribb &
Cooper, 1998; Morris & Jackson, 2009a,b). The
spectrum
X
GB
(
elastically accommodated grain boundary sliding
on the basis of the high normalized frequency
data obtained at low homologous temperatures.
The potential importance of anelastic relax-
ation, caused by the motions of dislocations,
has been demonstrated both experimentally
(Gueguen
et al
., 1981, 1989) and theoretically
(Karato & Spetzler, 1990). However, neither the
quantitative
relationship between relaxation
disl
ijkl
and dislocation density nor the
detailed form of
X
disl
(
strength
) has been clarified yet.
In the melt phase, the shear stress is relaxed
by the viscous flow within each pore. In addition,
the heterogeneity of melt pressure between neigh-
boring pores is relaxed by viscous flow (squirt
flow). The squirt flow is to be measured on a
much longer timescale than the flow within each
pore. When the pore shape and melt viscosity are
known, the relaxation strength and the timescale
of these processes can be estimated quantitatively
(e.g., O'Connell & Budiansky, 1977; Faul
et al
.,
2004). For the equilibrium geometry of pores
filled with basalt melt, the relaxation timescale
of the squirt flow is estimated to be much shorter
than the periods of seismic waves (Section 3.5.1).
Therefore, seismic waves are usually considered
in the relaxed state of these flows. In calculating
the elastic compliance tensor
S
e
ijkl
based on the
contiguity model, neither shear stress in the melt
phase nor melt pressure heterogeneity in REVwas
taken into account. This means that
S
e
ijkl
repre-
sents the relaxed state of the squirt flow. In this
case,
τ
) is expected to be a superposition
of the spectra corresponding to these two stages,
but the relative contributions of each stage to the
total relaxation strength is unknown.
In recent experimental studies undertaken with
fine-grained olivine aggregates at
f
τ
1-10
-
3
Hz
(Gribb & Cooper, 1998; Tan
et al
., 2001; Jackson
et al
., 2002), and with a fine-grained rock ana-
logue (borneol) at
f
=
10
−
4
Hz (McCarthy
et al
., 2011), it was found that the relaxation
spectra could be closely fitted by
X
=
10
−
0.3
,
and they were called the high-temperature
background. A large relaxation strength, ex-
hibited by the data, is roughly consistent with
the theoretical prediction of
0.2
∼
∝
τ
GB
ijkl
. McCarthy
et al
. (2011) showed that the spectra can be
universally scaled by the Maxwell frequency
f
M
, regardless of the particular experimental
conditions or material, and the mechanism
underlying these data is therefore diffusionally
accommodated grain boundary sliding. McCarthy
et al
. (2011) also showed that the normalized
experimental
melt
ijkl
X
melt
(
) in Equation (3.24) must not in-
clude the relaxations caused by the viscous flow
of the melt, and
τ
10
5
)
is considerably lower than the normalized
seismic frequency range (
f
frequency
range
(
f
/
f
M
<
)and
S
e
ijkl
have to be
defined consistently. It is worth noting that the
other two terms in (3.24), corresponding to grain
boundary sliding and the motions of dislocations,
are also expected to be influenced significantly by
the presence of a melt.
melt
ijkl
X
melt
(
τ
10
9
), and
this is because
f
M
in the mantle is much lower
than in the fine-grained laboratory samples.
Therefore, although a considerable amount of
data was obtained for the anelasticity related to
grain boundary sliding, it is not possible to decide
whether it is elastic or diffusional accommoda-
tion that is active with respect to the seismic
waves. More data are needed at higher normal-
ized frequencies. Recently, Jackson and Faul
(2010) discussed a possible contribution from the
10
6
/
f
M
=
−
3.6.2 Anelastic effects of melt, water,
temperature, and grain size
Seismic wave velocities are influenced by both
elasticity and anelasticity, and the latter is called
the ''anelastic effect.'' I briefly summarize here
