Simple analytical models of glacier-climate interactions - by Prof. J ...

1. A mass-balance model
The processes governing the exchange of heat and mass between glacier surface and
atmosphere contain many nonlinear features, and one may well wonder whether an
analytical approach to glacier mass-balance modelling makes any sense. Let us have a
look.
We start by ignoring the daily cycle and assume that the daily loss of mass due to melting
is determined by a potential surface energy flux ψ. Moreover, we assume that this flux
has a sinusoidal shape over the year (period T):
ψ = A 0 + A 1 cos 2π t/T . (1.1)
The coefficients A 0 and A 1 contain both a temperature effect and a solar radiation effect.
We do not include a phase shift in time (which is irrelevant) and take A 1 > 0. In Fig. 1.1
eq. (1.1) is compared to the surface energy flux (daily mean values) as measured by an
automatic weather station (located on the snout of the Morteratschgletscher, Switzerland).
It is clear that eq. (1.1) does not work for the wintertime. However, this is not a problem
because in this period little melting takes place.
Ablation occurs when Ψ is positive, at a rate ψ/L m (L m is the latent heat of melting). Note
that there is never any ablation when A 0 < -A 1 , and there is ablation all year round when
A 0 > A 1 . To keep the model tractable accumulation has to be prescribed in a simple way.
We assume that the accumulation rate is constant when ψ < 0 and zero otherwise. Fig.
1.2 depicts the case when -A 1 < A 0 < A 1 . There is ablation from t = 0 onwards and it
will stop at time te, given by:
t e = T 2π arccos - A 0
A 1
. (1.2)
Fig. 1.1
300
200
total energy flux
sine fit
Morteratschgletscher
year 2000
100
(W m 2 )
0
-100
-200
0 50 100 150 200 250 300 350 day
3