Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 108

The upper boundary may lose ice due to melting (Q 0 > 0) or sublimation (E), while at
the lower boundary both freezing and melting take place. Congelation ice growth is
primarily a vertical process and a small-scale phenomenon, determined by the conduction
of heat through the ice. In the full melting stage, the temperature of lake ice approaches
the melting point and heat conduction. Then the boundary conditions ( 4.25a , b , c ) give the
melting at the top and bottom surfaces, while internal melting follows from Eq. ( 4.24 ). In
the limiting case, at the absence of conduction, ice temperature equals the melting point,
solar heating goes to latent heat of melting increasing the porosity:

@m

@ t ¼ Q T q L Ke Kz :

ð 4 : 26 Þ

However, due to the cool skin phenomenon, the surface temperature is mostly below
the melting point, and a fraction of solar radiation absorbed near the surface is conducted
back to the atmosphere. If QT T * 10 W m − 2 and K 1m − 1 , we have ∂ν / ∂ t 0.28 %

* *

day − 1 .
Snow in uences ice growth and decay due to isolation effects and snow-ice formation.

The evolution of snow properties in snow metamorphic processes causes further com-
plications. Next, the isolation is considered. For the mass and heat balance, a system of
equations corresponding to the ice layer (Eqs. 4.24 , 4.25 a, b, c) can be constructed. The
elevation of the snow surface (z =

fl

h s ) changes due to precipitation, sublimation, melting
and compression of snow, while at the snow-ice interface the continuity of heat

-

flow is
required. The heat conduction equation is formally the same, but the boundary conditions
are for the snow layer:

fl

z ¼ h s : T \ 0 C : dh s

q s
q w

Þ; k s @ T

dt ¼ c h s þ

ð P E

@ z ¼ Q 0

ð 4 : 27a Þ

T ¼ 0 C : dh s

q s
q w

Þ Max ð Q 0 ; 0 Þ
q s L f dt ¼ c h s þ

ð P E ð 4 : 27b Þ

k s @ T

@ z ¼ Min ð Q 0 ; 0 Þ
z ¼ 0 : k s @ T
@ z

z¼0 ¼ k @ T z¼0 þ

ð 4 : 27c Þ

@ z

where ˁ s is snow density, and k s is the thermal

conductivity of snow. Snow metamorphosis leads

ʳ is the snow compression coef cient,

first of all to packing, i.e. decrease of
thickness and increase of density. This has a major in

uence on the thermal conduction,
since k s ∝ ˁ 2 (see Sect. 3.2.3 ) . Consequently, we need an equation for the snow density in
ice models (Yen 1981; Lepp

fl

ä ranta 1983; Saloranta 2000):