Proglacial Lakes - Freezing of Lakes and the Evolution of their Ice Cover - page 192

easily obtained from Eq. ( 6.6 ), if heat diffusion is ignored and a constant attenuation
coef

cient is assumed. Then, de ning the lake water body by mð z Þ [ m o , the lake depth

scale becomes

¼ 1
j

h 0 j
m 0

¼ 1
j

Q T t j

H

log log ð 6 : 11 Þ

m 0 q L f

1m − 1 ,

where h ′ is the meltwater production by the available radiation. For ʺ *

m o 1 = 3 , we have H

100 W m − 2 , t

Q T * .Atv > v o , the strength

of the ice becomes low and disappears, and the ice matrix changes into slush as observed.
Thus the lake depth in the present study region scales as the inverse of the light attenuation
coef

60 days, and 1 : 65 m

*

cient.
Light attenuation in ice is described by a linear law as shown in Sect. 3.4 (Eq. 3.18 ) ,
and the solution can be written as

0
@

1
A ¼

Z

z

dz 0

ðÞ ¼

ðÞ

j

ðÞ

ð j Þ ð : Þ

Q s z Q T exp Q T exp z 6 12

0

j

where is the mean attenuation coef cient in the surface layer of depth z. The optical

ʺ − 1 thickness of the lake body is ≈ 1 m. According to observations (Lepp ä ranta et al. 2013b),

½ -

warming of ice is strongest 1 m beneath the surface in the beginning of summer when the

melting of ice initiates. In the hard ice beneath the lake body, the attenuation is expected to
be larger for ice rich in sediments or gas bubbles but smaller for clear ice. Since the
attenuation of light in ice and water is wavelength-dependent (Sect. 3.4 ), for a more accur ate
solution of the light transfer the spectral distribution should be accounted for. However,
for simplicity, in the scaling analysis we can take a constant, representative attenuation
coef cient across the PAR band and depth.

The original diffusion equation can be examined in more detail for the warm-up of the
glacial surface layer and initialization of melting. For this, a steady state solution is
obtained for Eq. ( 6.6 ). Taking the both boundary conditions at the surface ( 6.7a ), we have
the solution:

Q k Q T

k

1
j

e j z Tz ðÞ ¼ T 0 z z ð 1 Þ ð 6 : 13 Þ

cation by the
depth-distributed solar heating. The scale of validity of this solution is up to the length of
the steady state conditions; the longer the boundary conditions hold, the deeper the
solution goes. For

This consists of a linear pro le from the surface heating plus a modi

ʺ → ∞ , sunlight is absorbed at the surface and linear temperature

pro le is nonlinear in the surface layer, and

when Q 0 < 0, a temperature maximum is obtained beneath the surface at the depth

le results. For a finite ʺ , the temperature pro