Geology Reference
In-Depth Information
Table 3.4
Symbols used in the equations to calculate volume
fractions of sea ice constituents.
Brine density (kg/m
3
) is related to brine salinity (‰)
through an empirical equation [
Cox and Weeks,
1975]:
Pure
Ice
Salt in
Brine
Solid
Salt
Bulk
Sea Ice
1000 0.
S
(3.22)
Brine
Air
b
b
Mass
m
pi
m
b
m
sb
m
ss
m
a
M
si
The two parameters
S
b
and
k
are unique functions of
temperature. The dependence of
S
b
on the sea ice tem-
perature
T
si
(in °C) is given by the following empirical
equations [
Assur,
1958] based on best fit results of meas-
ured data. Since sodium sulfate starts to precipitate at
−8.2 °C, then two equations are needed because it is dif-
ficult to develop a single equation for the full temperature
range:
Volume
V
pi
V
b
—
V
ss
V
a
V
si
Density
ρ
pi
ρ
b
—
ρ
ss
ρ
a
ρ
si
Salinity
—
S
b
—
—
—
S
si
m
m
S
sb
(3.16)
b
b
These two equations yield
2
S
1 725 18 756
.
.
T
0 396
.
T
for
mSV
bbb
(3.17)
b
si
si
sb
20
.
C
T
82
.
C
(3.23)
si
3.4.1. Brine Volume Fraction
2
S
57 041 929 162
82
.
.
T
.
T
for
b
si
si
Brine volume represents the amount of liquid in the
composite sea ice medium. Brine volume and salinity at
any point within the sea ice sheet are functions of local
ice temperature. Depending on the local temperature,
brine salinity can be determined using the sea ice phase
diagram (Figure 2.1). There are two approaches to deter-
mine brine volume. The first is by deriving an expression
based on equations (3.15)-(3.17) after incorporating an
expression for brine salinity. This approach is presented
in
Cox and Weeks
[1983]. The second is by using empiri-
cal equations that relate brine volume to ice tempera-
tures and salinity. The first approach is reiterated here
first. By combining equations (3.15) through (3.17) the
following expression for brine volume can be obtained:
.
C
22 9
.
C
(3.24)
Notz and Worster
[2009] presented an alternative
expression for brine salinity that can be used for the
temperature range between −22 and 0 °C:
2
3
S
214 886
.
T
.
T
0 017
.
T
(3.25)
b
si
si
si
This equation is accurate at lower temperatures. It fits
results from a model presented in the same reference
based on data presented in
Assur
[1958] with a maxi-
mum deviation of less than 5%. Given the expressions
of
S
b
[equations (3.23) and (3.24)] and
ρ
b
in [equation
(3.22)], the denominator in equation (3.21) can be com-
bined into one term,
F
1
(
T
), and the equation can take
the form
V
MS
S
1
si
si
(3.18)
b
1
k
bb
V
V
S
FT
where
k
is defined as
(3.26)
b
si
si
si
1
m
m
k
ss
(3.19)
The values of
F
1
(
T
) are given
Cox and Weeks
[1983]
.
Brine volume can also be expressed as a function of the
average ice salinity
S
si
and temperature
T
si
. A reasonable
estimate can be obtained using the equation presented in
Frankenstein and Garner
[1967] where
S
si
is in ‰:
sb
since
M
i
=
ρ
i
V
i
, then the brine volume fraction can be
obtained from the equation
V
V
S
S
1
(3.20)
b
si
si
1
k
3
V
10
S
0 532 49 185
.
.
/
T
(3.27)
si
bb
b
si
si
At temperatures above −15.2 °C, the presence of solid
salts can be neglected. Hence
k
= 0 in equation (3.20) and
the equation can be reduced to
where −0.5 °C ≥
T
si
≥ −22.9 °C.
A more accurate expression is presented in
Cox and
Weeks
[1983]:
V
V
S
S
si
(3.21)
b
si
VS FT
b
/
(3.28)
si
si
si
si
bb
