Geoscience Reference
In-Depth Information
derived using various approaches (e.g., Mavko,
1980; Cooper
et al
., 1989; Takei, 1998a,b, 2002;
Hammond & Humphreys, 2000; Takei & Holtz-
man, 2009a,b,c; Simpson
et al
., 2010a,b). In the
following, I will review the standard properties
and their uncertainties through a compilation of
these theoretical and experimental results.
compressibility ratio between solid and liquid
(
k
S
/
16) are very different from those in par-
tially molten peridotite (
k
f
=
1
.
ν
S
∼
0.25 and
k
S
/
k
f
∼
5
at P
1 GPa), a direct comparison can be made
for
V
S
/
=
V
S
0
, which is insensitive to
k
f
but sensitive to the melt geometry. Figure 3.8
shows the measured
V
S
/
ν
S
and
k
S
/
V
S
0
(symbols) together
with the theoretically predicted
V
S
/
V
S
0
(lines),
demonstrating that the measured velocities agree
well with the predictions of the contiguity model
with
A
3.5.1 Elasticity
By solving the elastic constitutive Equation (3.10)
together with the other governing equations of the
two-phase system, the shear (S) and longitudinal
(P) wave velocities can be obtained as follows
3, the tube model (Mavko, 1980) with
the tube shape parameter
=
2
.
0 (i.e., a triangular
cross section with three cusps), and the oblate
ε
=
N
/µ
S
ρ/ρ
S
V
S
V
S
=
spinel lherzolite 1GPa
borneol
+
melt (
θ =
35
°
)
borneol
1
(3.21a)
contiguity model
A
=
2.3
)
borneol
+
melt (
θ
= 17
°
)
ice
+
melt (
θ
= 22
°
K
eff
/
k
S
+
(4
γ/
3)
N
/µ
S
+
NaCl brine (
θ
~0
°
)
V
P
V
P
=
0.9
1
(3.21b)
·
ρ/ρ
S
+
4
γ/
3
tube model
ε = ∞
0.8
and
K
eff
k
S
=
oblate
spheroid
model
α =
0.05
k
S
)
2
K
b
k
S
+
(1
−
K
b
/
k
f
,
(3.21c)
0.7
0.1
1
−
φ
−
K
b
/
k
S
+
φ
k
S
/
α =
where
ρ
S
represent the average and solid
densities, respectively,
ρ
and
0.6
ε =
0
γ
=
µ
S
/
k
S
,and
k
f
the in-
trinsic bulk modulus of the liquid phase (e.g.,
Takei, 2002, 2009), and
V
S
0
and
V
P
0
the velocities
of the solid phase.
V
P
and
V
S
have been measured by using
ultrasonic waves. For example, the values of
V
P
and
V
S
in partially molten peridotite were
measured at 1GPa for several melt fractions
(Murase & Kushiro, 1979; Murase & Fukuyama,
1980). They reported that at 4% melting, melts
only occurred at triple point junctions.
V
P
and
V
S
have been measured for several melt
fractions (
0.5
0
0.05
0.1
0.15
Melt fraction,
φ
Fig. 3.8
Normalized shear wave velocity
V
S
/
V
S
0
versus melt fraction
,where
V
S
0
represents the shear
wave velocity of the solid phase. Symbols show the
experimental data for a partially molten peridotite
measured at 1 GPa (
f
φ
0.6-0.3 MHz; Murase &
Fukuyama, 1980), and a partially molten rock analogue
with various dihedral angles (borneol
=
+
melt,
f
=
0.2
MHz, Takei, 2000: ice
NaCl brine, Spetzler &
Anderson, 1968). These data were normalized to
V
S
0
at
each run temperature. Lines show the theoretical
predictions: the thick solid line shows the contiguity
model with
A
=
+
φ
=
0.03-0.11) and dihedral angles
35
◦
-17
◦
) in a texturally equilibrated partially
molten rock analogue (borneol
(
θ
=
+
melt; eutectic
43
◦
C) at ambient pressure (Takei,
2000). The equilibrium dihedral angle in this
analogue system is similar to that of partially
molten peridotite (
temperature
=
3 (see Equation (3.1) for the
definition of
A
); the dashed lines show the tube model
(Mavko, 1980) with
2
.
(circular cross section) and 0
(a triangular cross section with three cusps); the thin
solid lines show the oblate spheroid model (Berryman,
1980) with aspect ratios (
ε
=∞
40
◦
-20
◦
; Holness, 1997)
and can be controlled by temperature. Although
the Poisson's ratio in the solid (
θ
=
ν
S
=
0.37) and the
α
) of 0.05 and 0.1.
