Geoscience Reference
In-Depth Information
These models solve the full time-dependent heat conduction equation. The advantages are
that the temporal and vertical resolution can be taken in practice as high as desired, while
in quasi-steady models they are limited by the time-scale of thermal diffusion.
The ice growth modelling problem possesses a negative feedback to errors. The
background Stefan
is law implies that the squared ice thickness is proportional to the
freezing-degree-days S, and in the next step the sensitivity of model ice growth to tem-
perature is
'
d h a 2
h
d S
ð
:
Þ
4
57a
If the model ice thickness exceeds the observed level, the model ice grows then more
slowly and returns towards the reality, and vice versa. Ice melting problem, on the other
hand, possesses a positive feedback mechanism. In the next step, the sensitivity of ice
melting to albedo is
d h Q s da
ð
4
:
57b
Þ
When albedo begins to decrease the surface is transformed to absorb solar radiation
better and produce more and more liquid water and consequently lower and lower albedo.
Due to this reason the surface of a melting lake ice cover is patchy.
4.4.2 Quasi-steady Models
In the quasi-steady approach, the ice sheet is divided into n layers and the heat conduction
is solved through all layers simultaneously. Whether we have cooling or warming depends
on the boundary conditions, and the solution at each time step is iterated to satisfy the
continuity of the heat
fl
flux. The system of equations is:
T 0 \ T f : q L f dh
dt þ Q w ¼ k T f T n 1
þ q n 2 ¼ ¼k T 1 T 0
þ q 2
ð
:
Þ
4
58a
h n
h 1
T 0 ¼ T f : q L f dh
dt ¼ Q w þ Q 0 þ Q T
ð
Þ
ð
4
:
58b
Þ
k T 1 T 0
h 1
¼ F ð T a ; e a ; U a ; N ; Q s ; T 0 Þ
ð
4
:
58c
Þ
where q p '
s distribute the solar radiation penetrating into different ice layers. Equation
( 4.58c ) gives the surface boundary condition in general form; the exact formulation is up
to the modeller (see Sect. 4.1 ). The extremes are the full surface
fl
flux model or the
linearized form with own parameters
fixed for each time step.
 
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