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q
L
f
@m
if T ¼ 0
C0
m
1
if T
\
0
C
;
@
t
¼ Q
I
;
ð
Þ; m
¼ 0
;
ð
4
:
35d
Þ
@
@
t
Þ
¼Q
I
ð
T
0
C
Þ
ð
q
cT
ð
4
:
35e
Þ
where Q
I
= Q
I
(z, t) is the energy
flux into the ice interior. Equations (
4.35a
,
b
) give the
decrease of ice thickness, i.e. melting at the top and bottom surfaces. Equations (
4.35c
fl
e
)
are for the melting in the interior and present the evolution of the porosity and temper-
ature, conditioned so that porosity can be nonzero only at T =0
-
C and temperature can be
below 0
°
C only if
ν
= 0. Equations (
4.35a
,
b
) can be extended for brackish, saline or
hypersaline ice but then porosity and temperature change together at all temperatures
dictated by the phase diagram of the ice in question. Always when ice temperature
increases, the volume of liquid brine increases. For sea ice case the problem has been
largely investigated (see, e.g., Assur 1958; Cox and Weeks 1983; Lepp
°
ä
ranta and Man-
ninen 1988; Wadhams 2000; Weeks 2010).
Internal melting gives rise to structural defects. It starts from the crystal boundaries,
and therefore the crystal structure is revealed by to human eye. Consequently, melting
columnar-grained ice is called candled ice in common language. Once the porosity has
reached 0.3
oes and
blocks move with the wind, heat from the water body to ice increases strongly, and ice
disappears rapidly. Breakage of ice increases the surface-to-volume ratio of the ice mass
that also increases the melt rate. In the 1990s the author observed a 45-cm thick ice cover
to disappear in 2 days in Lake Tuusulanj
0.5, the ice cannot bear its own weight and breaks. Thereafter ice
fl
-
rvi, Finland; the ice was already porous to start
with, risky to walk on. The very fast break-up of ice has been sometimes phrased as
ä
'
ice
sinks overnight
(Humphreys 1934).
The physics of melting can be simpli
'
ed assuming that the temperature is at the
melting point across the ice sheet. Then there is no conduction through the ice but the ice
melts at the boundaries by the surface
fluxes and inside by the solar radiation. Melting of
ice can be examined as a process with two characteristics, ice thickness h and mean
porosity v:
fl
Þ
dh
q
L
f
1
m
ð
dt
¼
Q
0
þ
Q
w
ð
Þ
0
ð
:
Þ
4
36a
Q
T
0
q
L
f
h
d
m
dt
¼ 1
e
j
h
ð
4
:
36b
Þ
The first equation stands for the thickness of ice (distance between top and bottom
surfaces) and the second equation stands for internal melting; the net ice volume is
h
ð
1
v
Þ
first, then the ice. Internal deterioration starts
up only after snow has melted. Internal melting gives rise to structural defects and
. At the top surface the snow melts
finally
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