Geology Reference
In-Depth Information
ice with random frazil structure is noticeably less than
that from oriented frazil or columnar‐grained structure.
Multiyear ice, especially from the hummock, is associated
with higher values that are typically salt free.
Similarly,
τ
b
(
T
, 0) is the relaxation time of pure water.
This was obtained by
Stogryn
[1971] by fitting a polynomial
to the data reported by
Grant et al.
[1957]:
1
2
1 1109 10
.
10
3 824 10
.
12
T
3.6.1. Dielectric Constant of Brine
T
(3.78)
b
0
6 938 10
.
14
T
2
5 096 10
.
16
T
3
The dielectric constant of brine is a function of its tem-
perature
T
and normality
N
b
. The normality is related to
the brine salinity
S
b
by the following relation [
Klein and
Swift,
1977]:
The rest of the functions are given by the following
expressions:
2
10 39 2 378
.
.
N
0 683
.
N
b
b
(3.79)
25
,
NN
2
5
92
NS
1 707 10
.
1 205 10
.
S
4 058 10
.
S
b
b
b
3
2
4
0 135
.
N
10110
.
N
b
b
b
b
(3.71)
b
b
22
3
aN
10 0 255
.
.
N
51510
.
N
6 89
.
N
(3.80)
1
b
b
b
b
This equation is valid for NaCl solutions of salinities
smaller than 260‰. It is, therefore, applicable to sea ice.
Brine salinity is governed by ice temperature as presented
in equations (3.23) and (3.24). Equations to calculate the
complex dielectric constant of brine were developed by
Stogryn
[1971]. Its real and imaginary terms are given by
2
2
10 0 146 10
.
.
TN
48910
.
N
bTN
b
b
(3.81)
,
1
b
22
33
N
b
29710
.
N
5 64 10
.
b
2
5
cN
,
10 19610 8 08 10
30210392 10
.
. .
. .
N
b
17510658 10
1
b
5
5
b
0
b
2
(3.72)
N
(3.82)
b
b
2
b
.
5
.
6
12
f
b
b
0
b
b
2
f
(3.73)
3.6.2. Dielectric Mixing Models
b
b
2
f
2
12
f
0
b
The simplest approach to determine the dielectric con-
stant of sea ice is by using a linear model that accounts
for the volume fraction of the components of sea ice: the
pure ice crystals (host material) and the inclusions
(brine, air, or both). The shape of the inclusions is not
accounted for. If
h
and
i
are the complex dielectric con-
stant of the host and the inclusion
i
, which has volume
fractions
V
i
, then the dielectric constant of the sea ice
mixture
m
is defined by the following equation [
Ulaby
et al.,
1986]:
where
b
0
= low‐frequency (static) limit of
b
b
∞
= high‐frequency (optical) limit of
b
σ
b
= ionic conductivity of brine
f
= electromagnetic frequency (in GHz)
τ
b
= relaxation time of brine
0
= permittivity of free space (=8.854 × 10
−2
F/m)
and
b
∞
is independent of salinity [
Stogryn,
1971].
Therefore, it is equal to the optical limit of water permit-
tivity of 4.9. The terms
b
0
,
τ
b
, and
σ
b
are functions of
T
and
N
b
as given by the following expressions:
n
mh
V
1
(3.83)
i
i
h
i
TN TaN
,
,
0
(3.74)
b
0
b
b
0
1
b
The volume fraction of each component is obtained as
shown in section 3.4. The dielectric constant of air can
be assumed to be 1 and the dielectric constant of brine
can be calculated using the method suggested by
Stogryn
and Desargent
[1985]. This linear model will be referred to
in the rest of this section as the L‐model. It is simple, but
does not produce desired results.
A two‐phase dielectric mixing model developed by
Polder and Van Santen
[1946] and modified later by
de
Loor
[1968] is pursued in the following discussions and
will be referred to as the PVD model. The model is based on
the assumption that the dielectric is composed of a host
TN TbTN
,
,
0
,
(3.75)
b
b
b
0
1
b
TN
,
25
,
Nc N
,
(3.76)
b
b
b
b
1
b
where Δ = 25 −
T
(in °C) and
b
0
(
T
, 0) is the static dielectric
constant of brine with zero normality, i.e., of pure water.
This was determined by
Klein and Swift
[1977] using a
regression fit for the measurements conducted:
42
53
b
T
,
0 8 045 0 4147
.
.
T
6 295 10
.
T
1 075 10
.
T
0
(3.77)
