Geology Reference
In-Depth Information
impurities in the form of brine pockets in the spacing
between the dendrites and eventually into the sea ice vol-
ume as explained later. As a rule of thumb, the faster the
ice growth rate the more convoluted the interface will be
and hence more salts and gas solutes will be entrapped in
ice. On the contrary, a slower ice growth rate is associated
with a relatively planar interface and hence less entrap-
ment of the inclusions.
A simple sketch that clarifies the compositional super-
cooling is presented in Figure 2.19. It can also be used to
facilitate a qualitative description of the metamorphism
of the sea ice interface with the underlying water as well
as brine and air entrapment in sea ice.
Thermal equilibrium at the ice‐water interface stipu-
lates that the temperature of the ice must be equal to the
freezing temperature of the adjacent water. Obviously,
this temperature is colder than the seawater temperature
farther away from the ice‐water interface. Hence, heat
flows from the underlying warmer seawater, creating a
profile of increasing temperature as the distance from
the ice interface increases. Since salts are rejected dur-
ing ice growth, they start to spread into the underlying
water through diffusion and convection. In general,
the salt concentration at the ice interface is highest and
it decreases with the distance from the interface.
Consequently, the freezing temperature of the saline
water increases nonlinearly with depth. A composition-
ally supercooled layer will occur if the temperature gradi-
ent at the interface is less than that of freezing temperature
(also called liquidus temperature in the field of metal-
lurgy). Figure 2.19 shows this condition. A simple ana-
lytical derivation of this condition is presented in
Weeks
[2010] based on a number of earlier studies [
Elbaum,
1959;
Rutter and Chalmers,
1953;
Tiller et al.,
1953]. It is
summarized in the following paragraphs.
The salt concentration
C
at any point in the water
under the ice interface is determined by the propagation
of the interface. If the origin of a coordinate system is
chosen such that
x
= 0 is always at the interface, then the
interface propagation can be envisioned as stationary
while seawater is moving upward with velocity
V
, which is
in fact the velocity of propagation of the ice interface. If
a unit volume is considered in the underlying seawater,
then the amount of salt flowing into this volume due to
diffusion is
D
(
d
2
C
/
dx
2
), where
D
is the diffusion coeffi-
cient (m
2
/s), and the amount of salt flowing out of the
volume due to propagation of the interface is
V
(
dC
/
dx
).
These two quantities should be in balance assuming that
(1) mixing in the liquid under the ice interface is caused
only by diffusion (i.e., no convection current), and (2)
diffusion in the ice is neglected. At a steady state, which
implies that ∂
C
/∂
t
= 0, the balance of the two terms is
expressed by the following equation:
2
D
dC
dx
V
dC
dx
0
(2.23)
2
By applying the boundary conditions that
C
=
C
0
(the ini-
tial salt concentration far from the interface) at
x
= ∞
(where ∂
C
/∂
x
= 0), and
C
=
C
L
at x
= 0, then the steady
state solution of the above equation takes the form
1
K
K
Vx
D
CC
1
exp
(2.24)
0
Ice top
In this equation,
K
is defined as
C
0
/
C
L
and is called the
distribution coefficient.
The above formulations imply that a layer exists under
the ice interface that exhibits an exponential decrease in
C
. The freezing temperature of the saline water
T
f
under
the ice is related to
C
as illustrated in the phase diagram
of the binary mixture of NaCl and H
2
O in Figure 2.2.
The relation takes a linear form
Salinity increase Temperature increase
Ice bottom
Tangent to the
freezing temp. profile
Compositionally
supercooled
layer
Water
temperature
Seawater
Seawater
TTmC
f
(2.25)
Freezing
temperature
0
Salinity profile
where
T
0
is the freezing temperature of seawater at the ice
interface and
m
is the slope of the line as obtained from
the phase diagram. By substituting equation (2.23) into
(2.24), the freezing temperature profile can be written as
Figure 2.19
Schematic showing the ice‐water interface and
the salinity and temperature profiles in the underlying water.
Also shown is the freezing temperature profile that results
from the decrease of salinity with distance from the interface.
Supercooled water is created if the slope of the water tempera-
ture is higher than the slope of the freezing temperature curve
at the interface.
1
K
K
Vx
D
TTmC
1
exp
(2.26)
f
0
0
