Jökull - 01.07.2003, Page 8

de Ruyter de Wildt et al.
Turbulent fluxes
 and  are calculated with the bulk transfer
method (e.g., Munro, 1990). This method requires
values of windspeed, temperature and humidity at the
surface and at some height above the surface (usually
2 m) as input. The basis for the bulk method, Monin-
Obukhov similarity theory, is not strictly valid when
a low level wind maximum is present, as is the case
over sloping and melting glacier surfaces (Munro and
Davies, 1978). In spite of this, recent work (Denby
and Greuell, 2000) has shown that the bulk method
only slightly overestimates and. The roughness
length for momentum (
) has been reported to vary
considerably in space and time over the ice surface
of Breiðamerkurjökull, where values between 3 mm
and 6 cm were found (Smeets et al., 1999). The large
values were caused by ice hummocks up to almost 2
m in height, which developed during the melting sea-
son. The smallest values were measured before these
hummocks developed and these values are compara-
ble with those for smooth ice surfaces found in the
literature (Morris, 1989). Unfortunately, the afore-
mentioned kind of irregularities can hardly be mod-
eled and, moreover, do not arise in all ablation areas
of Vatnajökull. We therefore choose an intermediate
value of 5 mm for 
over ice. Denby and Greuell
(2000) remarked that the error in the calculated tur-
bulent heat fluxes due to an order of magnitude er-
ror in 
will be roughly 25%. We expect this to be
an upper limit for the error present in the calculated
turbulent heat fluxes. For snow surfaces, where these
problems literally and figuratively do not arise, we use
a value of 0.1 mm for dry snow and 2 mm for wet
snow. These values are often found for snow surfaces
(Morris, 1989). The roughness lengths for heat and
moisture are calculated from 
with the often-used
expressions of Andreas (1987).
Insulation
For most weather stations, the sum of observed net
radiation and turbulent fluxes (as computed from ob-
served 2 m variables) matches the energy that is re-
quired for the melt observed during the experiment
(Figure 4). At sites where tephra covered ice ap-
peared at the surface (I6, U8, U9 and R2) the sim-
ulated amount of melt is too high. This is proba-
bly caused by insulation of the underlying ice (e.g.,
Bozhinsky et al., 1986; Kirkbride, 1995), because in
the thermal channels of the NOAA satellite these parts
of Vatnajökull appear slightly warmer than the rest of
the ice cap surface. For the stations I6, U9 and R2 the
observed melt corresponds to about 80% of the melt
energy that was available when the surface was snow-
free. For station U8 there was tephra-covered ice at
the surface only during a few days of the 1996 ex-
periment and the effect of insulation is small here. A
reduction of melt by 20% is plausible, for this corre-
sponds to a tephra layer of a few cm (e.g., Kirkbride,
1995). We do not attempt to model this effect elab-
orately, because it concerns only a minor part of the
ablation area. Only on Dyngjujökull a significant part
of the ablation area is covered by tephra or other de-
bris (Figure 3). Furthermore, information needed to
do so (notably thickness and thermal conductivity of
the layer) is not available. We simply reduce the melt
by 20% when the albedo is 0.15 or lower.
RECONSTRUCTION OF THE MASS
BALANCE
We compute the mass balance ( ) as the annual sum
of ablation and solid precipitation:


 


   


 (3)
where
 is the ablation rate,  the latent heat
of melting of ice,  the solid precipitation rate and 
the latent heat of sublimation. The time step that we
use is 30 minutes and we let the balance year start on
September 21. We neglect refreezing of meltwater
and assume that the meltwater drains away instanta-
neously. Superimposed ice was not found anywhere
on the ice cap during field work in the years 1992–
1995 (Björnsson et al., 1998a). Furthermore, the re-
freezing process is only important when the winter
snowpack is cold (e.g., Colbeck, 1975) and the ab-
lation season is short (Greuell and Oerlemans, 1986).
Winters in Iceland are relatively mild and the ablation
season on Vatnajökull is long. At station U7 (1530 m
6 JÖKULL No. 52, 2003