Geology Reference
In-Depth Information
where T a and T w are the air and water temperature, respec-
tively, k is the effective heat transfer coefficient between
ice surface and atmosphere, and k i and k s are the thermal
conductivity of ice and snow, respectively. Substituting
this equation into the (one before it) yields
If ( T a ) N is the mean air temperature of the N th day from
the freezing‐up date, then equation (2.12) gives the ice
growth on that day:
kk
L
TT
kh kh
(
)
(2.13)
(
h
)
is
w
a N
iN
(
)
(
)
dh
dt LFTT k
1
h
k
h
k
i
i
s
i N
1
i sN
i
(
)
i
s
(2.7)
i
i
w
a
w
i
s
where ( h s ) N is the average snow thickness on the day under
consideration ( N ) and ( h i ) N − 1 is the ice thickness at the
end of the previous day.
The total thickness of ice over a number of days D can
then be written as the summation of ice growth per day:
By assuming a constant heat transfer coefficient k , the
integration of the above equation produces an expression
for the ice thickness:
1
1
2
k
k
kh
kh
2
k
L
k
k
h
2
i
1
i
s
h
i
1
i
r
i
i
s
ND
kk
L
ND
TT
kh kh
(
)
si
i
i
s
(2.14)
(2.8)
(
h
)
is
w
a N
(
TT
)
dt Fdt
iN
(
)
(
)
w
a
w
N
1
i N
1
i
s
i N
1
i sN
This equation can be used if time series of snow depth,
air temperature, and heat flux from the ocean are pro-
vided. A simplified form of the above equation can be
written under the following assumptions: (1) the ice sur-
face is snow free ( h s = 0), (2) the oceanic heat flux F w is
zero, and (3) the atmospheric temperature T a is stable and
equal to the ice surface temperature, therefore k → ∞:
Equation (2.14) can be rearranged to give
D
D
kk
L
TT
(
)
is
(
kh kh h
(
)
(
)
(
)
(2.15)
w
a N
s
i N
1
i sN i N
N
1
N
1
i
i
Growth of ice in Eclipse Sound near Pond Inlet, Baffin
Island, Canada, during the two winter seasons 1977-
1978  and 1978-1979 was determined from equation
(2.15) [ Sinha and Nakawo, 1981]. Results are shown in
Figure 2.14 as a function of accumulated degree‐days of
freezing. Dates given at the top of the figure indicate the
time of the season. Calculations were based on a constant
snow thickness of 11.4 cm and appropriate ice and snow
conductivities. The equation produces reasonable results
in general but underestimates ice thickness during the
early part of the season and overestimates it toward the
end. The use of measured snow improved the agreement
with observations during the early growth period, but not
during the late season. This could be attributed to increas-
ing solar radiation.
Equation (2.14) can be rewritten in an integral form to
relate the growth of ice in terms of accumulated degree‐
days of freezing:
2 (
kT T
L
)
2
h
i w a
t
(2.9)
i
i
i
Equation (2.9) can also be derived for predicting growth
in terms of accumulated freezing degree‐days, following
Sinha and Nakawo [1981], using the thermal balance
principle, which dictates that heat fluxes through ice
and snow layers must be equal, and they are both equal
to the heat released during ice freezing. Assuming, once
again, that there is no heat flux through the ice‐water
interface, then the increase of ice thickness Δ h i in time
Δ t can be written as
Lhk TT
h
t TT
h
w
b
b
a
t
(2.10)
i
i
i
i
i
s
where T b is the temperature of the snow base or the
snow‐ice interface (Figure 2.13). The second equality in
this equation gives
L
kk
D
D
(
TTdt
)
i
i
(
kh kh dh
)
(2.16)
wa
is si i
1
1
is
)
(2.11)
TkhT khTkhkh
b
(
)
/ (
i sw sia
s
i
i
s
For constant snow thickness, the above equation further
reduces to
Substituting T b in equation (2.10),
kk
L
TT
kh kh
L
Lh
k
D
h
is
w
a
t
(2.12)
2
(
TTdt
)
i
i
(
h
)
i
i
s
h
(2.17)
i
wa
i
i
2
k
1
i
i
si is
i
s
 
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