Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 118

Q 0 = k 0 + k 1 (T 0 − T a ) is used (see Sect. 4.1.5 ). The growth condition is that Q 0 < 0, and, by

the continuity of the heat

fl ow, we have

q L f dh

dt ¼ k T f T 0

¼ k 0 þ k 1 T a T 0 ½ ð Þ 0 ð 4 : 39 Þ

h

Elimination of the surface temperature from the latter equation gives

q L f dh

dt ¼ k T f T a b 0

; b 0 ¼ k 0

k 1 ; b 1 ¼ k

ð 4 : 40 Þ

h þ b 1 k 1

cient: b 0 accounting mainly
for the radiation losses and b 1 for the turbulent losses. By physics, b 1 ≥

There are two parameters in addition to the Stefan ' s coef

0 that adds to the
insulation of ice bottom from the atmosphere; and in practice, due to low solar radiation
level, b 0 ≤ 0 in the growth season that adds to the freezing-degree-days. The numerical

values of these parameters are b 0 * -

°

3 C, b 1 * 10 cm. With k 0 and k 1 constants the

solution is (see also Fig. 4.8 ):

q

h 2 ðÞþ a 2 S a ðÞ b 0 t þ b 1

h ðÞ ¼ ½ b 1 ð 4 : 41 Þ

Barnes (1928) and Zubov (1945) considered the situation k 0 = 0, where turbulence
transfers the heat away from the ice surface. The relative difference between the Stefan

' s

model and Eq. ( 4.40 ) is largest when ice thickness is small. When k 1 → ∞ , T 0 → T a and

the Stefan is growth law is obtained. This means that all heat conducted through the ice

can be moved away. The coef

'

0, and ignoring
solar radiation the ice growth is determined solely by the terrestrial radiation balance.
Then k 0 ≈

cient k 1 becomes small at the limit of U a →

50 W m − 2 and k 1 ≈ 4Wm − 2 C − 1 , and, consequently, b 0 ≈

° °

12.5 C and

b 1 ≈ 50 cm.

4.3.2.2 Influence of Snow Cover on Congelation Ice Growth
The presence of snow cover gives a major impact on ice growth. The heat conduction law
is as for ice (Eq. 4.24 ). Thermal inertia is ignored as in ice, and the temperature pro le

across snow and ice layers becomes piecewise linear. The system of Eq. ( 4.39 )is
expanded into

q L f dh

dt ¼ k T f T s

¼ k s T s T 0
h s

¼ k 0 þ k 1 T a T 0 ½ ð Þ 0 ð 4 : 42 Þ

h

where T s is the temperature at the snow-ice interface, k s is thermal conductivity of snow,
and h s is snow thickness. Here we have assumed that snow is not

fl flooded by lake water: a

h s . 3 h

suf (flooding leads to snow-ice

formation treated in the following section). Eliminating the temperatures T 0 and T s from
the system ( 4.42 ), we can obtain the differential equation:

cient condition is that ˁ s h s ≤ ( ˁ w − ˁ s )h or ( fl