Geology Reference
In-Depth Information
material, with an application to sea ice, is presented in
Golden
[1995].
cc jc
j
(3.93)
1
2
hi
hi
The two solutions of the quadratic equation (3.90) are
given by
1.
Spherical Inclusions
These inclusions approximate
the shape of brine pockets within frazil, granular, and
fine‐grained (transition) columnar ice structures. It also
suits the shape of air bubbles in MY ice. In this case,
AAA
x
b
b c
a
2
4
m
(3.94)
2
1
3
. The dielectric constant is an isotropic
(scalar) quantity. Equation (3.84) can be rewritten as
follows:
y
z
The solution with the positive sign was found to be
acceptable in the case of brine inclusions, while the solu-
tion with the negative sign was acceptable in the case of
air inclusions. The alternate solution resulted in negative
values of the dielectric constant. Therefore, the expres-
sions used to calculate the real and imaginary parts of
m
for FY ice are
i
h
13
V
for
V
01
.
(3.85)
mh
i
i
2
i
h
or
bd
a
m
1
1
(3.95)
2
i
h
(3.86)
3
V
for
V
01
.
mh
im
i
2
i
m
bd
a
(3.96)
m
2
2
The unknown parameter
m
appears on both sides of
equation (3.86). Hence the solution for
m
can be obtained
from the following quadratic equation:
2
where
d
=
d
1
+
jd
2
is the square root of the discriminant
complex term (
b
2
− 4
ac
).
2. Random Needle Inclusions
In this case
A
x
=
A
y
= 0.5
and
A
z
= 0. However, due to the randomness of the needle
orientation, the term involving
A
u
on the RHS of equa-
tion (3.84) is treated as being the arithmetic average from
using the three orthogonal components of
A
u
. Hence, the
dielectric constant of the mixture remains isotropic.
2
(3.87)
2
2
3
V
0
mmi
h
i
i
h
i h
a. Solution for the Case
V
i
≤ 0.1
Equation (3.85) can be
rationalized using the definition given in equation (3.84)
to produce the following expressions:
2
2
V
1
(3.88)
i
h
i
h
h
13
V
i
(3.97)
mh
i
2
2
mh
i
h
2
3
i
h
i
uabc
,,
1
A
i
1
u
*
2
9
(3.89)
ii h
The second term on the RHS is the average of the three
components of
A
u
. By considering * =
h
when
V
i
≤ 0.1and
* =
m
when
V
i
>0.1, equation (3.97) takes the form
m
2
2
2
i
h
i
b. Solution for the Case
V
i
> 0.1
Equation (3.87) can
be written as
V
.
(3.98)
i
i
h
5
for
V
0 1
mh
h
i
i
3
i
h
0
(3.90)
2
abC
m
m
and
where
a
,
b
, and
C
are complex coefficients. Their expres-
sions can be obtained by comparing equations (3.90) and
(3.87), taking into consideration equation (3.84):
V
(3.99)
i
i
h
5
for
V
0 1
.
mh
mi
i
3
i m
a. Solution for the Case
V
i
≤ 0.1
aa ja
2
20
j
(3.91)
1
V
B
A
h
i
i
i
(3.100)
mh
bb jb Vj
(3.92)
3
2
2
23 3
V
1
2
i
h
ii
i
i
i
h
i
i
