Geoscience Reference
In-Depth Information
4.4.3 Time-Dependent Models
Time-dependent models solve the heat conduction equation of the snow and ice layers for
the evolution of temperature and thickness in a discrete grid or in a
finite element system.
Improvement as compared with quasi-steady models is the inclusion of thermal inertia that
allows high vertical resolution in the simulations. Congelation ice models are based on the
sea ice model of Maykut and Untersteiner (1971) and snow-ice models to Saloranta
(2000). Numerical time-dependent lake ice models have been used mostly after the year
2000 (Lepp
ranta 2009). For Finnish lakes, the sea ice model of Saloranta (2000) was
applied by Lepp
ä
ranta and Uusikivi (2002), and the Baltic Sea model of Launiainen and
Cheng (1998) and Cheng (2002) was applied by Yang et al. (2012).
The ice portion may have separate congelation ice and snow-ice layers, and the snow
portion may have dry snow and slush layers (slush is understood as snow saturated with
liquid water). At the upper boundary, the solar and terrestrial radiative
ä
fl
uxes and the
turbulent heat
fluxes are parameterized for weather data or taken from an atmospheric
model. The surface temperature is solved from the surface energy balance that couples the
ice sheet with the atmosphere (Eq. 4.58c ). Snow thickness and density are modelled from
precipitation, wind and temperature (see Eq. 4.57a , b ). Formation of snow-ice is a source
for the ice layer and sink for the snow layer. At the bottom of the ice sheet, the heat and
mass balance are controlled by freezing and melting and heat
fl
fl
flux from the water.
'
refers to the requirement that daily cycles in the temperature evolution can be resolved. The
time scale of heat diffusion in ice is
Time-dependent models have high vertical resolution (5
10 cm).
'
High resolution
-
L 2 /D=10 (L/m) 2 day, where D is the heat diffusion
τ *
coef
0.1 day, which is
approximately the time-step needed for daily cycles in numerical modelling.
The full model simulates the development of four distinct layers: snow, slush, snow-ice
and congelation ice. These layers interact in a dynamic way: snow accumulation creates
slush and snow-ice depending on the total thickness of ice, while the growth and decay of
congelation ice depend on the snow and slush conditions. Slush and snow-ice may form a
multiple layer structure with alternating layers of snow-ice and slush. The thickness of
snow decreases due to three different reasons: surface melting, compaction, and freezing
of slush into snow-ice.
The core of time-dependent lake ice thermodynamics models consists of the classical
1-d heat conduction equation:
cient and L is the vertical length scale. For L
0.1 m, we have
τ *
*
@
@ t
Þ ¼ @
@ z
j @ T
ð
q cT
@ z Q s ð z Þ
ð
4
:
60
Þ
The thermal properties of ice vary in different layers in the ice sheet. For freshwater ice,
this is a simple linear equation to solve numerically, and the main dif
culties are in the
treatment of the moving boundaries with mass and heat
fl
fluxes across. In saline ice,
 
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