Geoscience Reference
In-Depth Information
snow it is typically around 0.5 (Sherstyankin 1975; Jakkila et al. 2009; Lepp
ranta et al.
2010). Bare, dry ice has an albedo of 0.5, and with increasing surface wetness and melt
ponds, the albedo decreases to 0.2
ä
0.3 during the melting season. When bare ice is thinner
than 30 cm, albedo also depends on the thickness of ice, approaching open water albedo
when thickness goes down to zero. This is due to much lower scattering of light in liquid
water beneath the ice than in the ice itself. Thin (10 cm) and bare ice has an albedo of
0.2
-
0.3, but when the ice thickness is more than 30 cm and the surface is dry, the albedo is
at the 0.5 reference level. In spring, when the snow on ice melts, the transparency of the
ice increases and the light level beneath ice becomes high. In open water conditions,
albedo is low (5
-
10 %) and stable.
-
Example 3.6
. The solar energy
fl
flux to a lake ice sheet can be expressed as (1
− ʱ
0
)q
0
,
where q = Q
s
/(
L
f
). The dimension of q
0
is length/time and can be interpreted as the upper
limit of the melt rate if other
ρ
fl
uxes sum to zero.
(ii)
If the albedo is constant, the total melt in time t is h
0
=(1
− ʱ
)q
0
t; for t =30d,
ʱ
0
= 0.5 and q
0
= 2 cm d
−
1
we have h
0
= 30 cm.
½
bq
0
t
2
;
(ii)
If the albedo decreases with time,
ʱ
=
ʱ
0
−
bt, b > 0, the additional melt is
for b = 0.3 month
−
1
, the melt increases to 39 cm.
(iii)
If also melt rate increases with time, q = q
0
+ ct, the additional melt to (ii) is
½
− ʱ
0
)ct
2
+ 1/3 bct
3
; for c = 2/30 cm d
−
2
(this means increasing the solar heating
from 2 to 4 cm d
−
1
(1
in 1 month) the melt increases to 60 cm.
The attenuation law of irradiance (Eq.
3.18
) integrates to
"
#
Z
z
E
d
z
; ðÞ
¼E
d
0
þ
; k
Kz
0
; k
Þ
dz
0
ð
Þ
exp
ð
0
"
#
ð
3
:
21
Þ
Z
z
¼ 1
r 0
; k
E
d
0
; k
Kz
0
; k
Þ
dz
0
½
ð
Þ
ð
Þ
exp
ð
0
where the notation 0
+
and 0
−
stand for just beneath and just above the surface, respec-
tively. For a layer with K independent of depth, the inverse K
−
1
is its optical thickness.
The in
uence of optically active substances on the attenuation is additive, i.e. we can write
the attenuation coef
fl
cient as the sum of the contributions from pure water, gas bubbles,
CDOM, particles and chlorophyll, respectively, as:
K ¼ K
w
þ
K
g
þ
K
y
þ
K
p
þ
K
c
:
ð
3
:
22
Þ
Irradiance measurements at two levels provide the average diffuse attenuation coeffi-
-
1m
−
1
; Arst et al. (2008) reported
cient of the layer between them. In congelation ice, K
*
values mainly between 0.5 and 2 m
−
1
in Finnish and Estonian lakes.
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