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however, care should be taken with parameterization of the thermal properties when the
ice approaches the melting stage.
A typical grid size in numerical models has been 5 cm. Both the upper and lower
surfaces move due to ice growth and melting that need to be considered in the top and
bottom grid cells. The upper boundary is strongly dependent on the atmospheric condi-
tions so that the surface temperature can be best solved with a coupled ice
atmospheric
boundary layer model (Cheng 2002). For a strongly stable atmospheric surface layer, the
heat transfer is tough to solve (Yang et al. 2013). The boundary conditions are given in
general form as
-
z ¼ h
0
:
dh
0
q
w
q
s0
P
q
q
0
E
þ
q
L
Q
0
þ j
@
T
@
z
; j
@
T
dt
¼
@
z
¼ Q
0
ð
4
:
61a
Þ
Q
w
þ j
@
T
@
z
z ¼ h
b
:
dh
b
dt
¼
q
L
;
T ¼ T
f
ð
4
:
61b
Þ
where
ˁ
0
is the density at the surface. The thickness of
snow increases due to snowfall, given as the water equivalent into the model and changed
to snow thickness using a
ˁ
s0
is the density of new snow and
fixed density of new snow, and decreases due to sublimation
and melting. The threshold value between snow and no-snow conditions is a model
parameter, e.g. 3 cm in Saloranta (2000).
Slush formation in the snow layer is technically straightforward to model. Flooding is
assumed to follow from Archimedes law, i.e. lake water is let to penetrate the ice as soon
as the ice surface level is beneath the lake water level. A threshold snow overload can be
speci
flooding event to begin. The proportion of snow crystals in water-saturated
snow is assumed to rise to 50 %. When negative freeboard conditions appear, the amount
of new slush is calculated from
ed for a
fl
@
h
sh
@
t
¼
1
q
w
þ q
s
q
sh
@
@
t
ð
q
w
V
s
B
Þ
ð
4
:
62
Þ
B ¼ h
i
ðq
w
q
i
Þþ
h
si
ðq
w
q
si
Þþ
h
sh
ðq
w
q
sh
Þ
The product gB is the buoyancy of ice, snow-ice and old slush layers. Liquid pre-
cipitation is obtained from the forcing assuming that precipitation is liquid when the air
temperature is above a given limit, e.g. T
a
>1
C. Melt water of snow comes from the
model directly. This approach for snow-ice formation was taken by Saloranta (2000), used
for lake ice by Lepp
°
ranta and Uusikivi (2002).
At the bottom of the ice sheet, the heat balance is determined by phase changes,
conductive
ä
flux of heat into the ice sheet and the transfer of heat from the water column
(see Sect.
4.2.2
). The heat
fl
fl
flux is formally written
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