Jökull - 01.12.1974, Page 18

and as = 300 mm/yr outside the geothermal
area (these estimates must be subject to later
revision). An ice flux of = 0.30 km3/yr into
the geothermal area plus the net surface bal-
ance rate of 0.20 km3/yr over the geothermal
area provide the 0.50 km3/yr of ice melted
subglacially by the geothermal heat. It is inte-
resting that the border of the geothermal area
is in a sense an equilibrium line for the water
basin. The model estimates that the ratio be-
tween the accumulation area and the ablation
area is about 2:1.
The ice flux into the geothermal area can
be compared to theoretical predictions, obtain-
ed from estimates of the ice velocity through a
vertical cross-section at the border -of the geo-
thermal area. The velocity of a glacier due to
deformation of the ice is given by
(8) us-ub = BTn-H‘
n + 1
r = Hj • pjg sin a
in which us and ub are the glacier surface ancl
bottom velocities, respectively. The symbol T
represents the shear stress at the glacier bottom,
Hj denotes the glacier thickness, and n and B
are constants in the ice flow law. Assumed
values for the constants are n = 4.2 and B =
0.29 yr-^bar-4-2 (Paterson, 1969). For an average
surface slope of a — 2 • 10~2 in the water basin
and an ice thickness Hj = 600 m one obtains
u — ub = 40 m/yr. For a glacier width of D =
10 km, the ice deformation flux is about 0.25
km3/yr which, added to a slightly smaller flux
due to sliding at the glacier bed, might predict
a total ice flux of q, = 0.4 km3/yr. This is in
good agreement with the ice flux of q^ =
0.30 km3/yr determined from the mass balance
model.
The water flux towarcl Grimsvötn
The flux of water toward the lake obtained
from the surface ablation and the bottom melt-
ing is given by equation (5) (the vertical drain-
age of water inside the lake is not considered).
If the lake area Aj is assumed to be 40 km2
(an overestimate), the following estimates are
obtained: the contribution from the surface
ablation is onty 0.06 km3/yr outside the geo-
thermal area (A — Ag = 200 km2, as = 300 mm/
yr) and 0.06 km3/yr from that part of the geo-
thermal area which is outside the lake (Ag —Aj
= 60 km2, as = Í000 mm/yr). There is no sur-
face drainage in the water basin. The entire
water flux of 0.12 km3/yr is transported
through interglacial waterways down to the
glacier bed or directly into the lake. Less than
0.01 km3/yr of this water is carried into the
lake by the moving ice. No moulins or large
channels are visible on the glacier surface. A
water table has been observed at about 35 m
depth. A water flux of 0.12 km3/yr seems to be
driven by a sloping water table through the
intergranular vein system. The flow is ap-
proximately liorizontal and the vertical cross
section is about 6 km2. The specific discharge
is about 20 m/yr. According to Nye and Frank
(1973) this flow would require a fractional
volume of veins of 5 • 10-3. The mean velocity
of the water would be 4 km/yr. The time taken
for the water to flow the 20 km from Bárclar-
bunga to Grímsvötn would be about 5 years.
The permeability of the glacier can be estim-
ated by assuming that Darcy’s law is valid.
Making the approximation that the hydraulic
gradient is equal to the glacier surface slope
(-2 • ÍO-2) one obtains a permeability of k = 6
darcy. This value corresponds to an aquifer of
very fine sand.
The meltwater contribution from the geo-
thermal area is about 0.30 km3/yr as the aver-
age subglacial ablation rate is ag = 5 m/yr
(melting 0.50 km3/yr inside an area of Ag =
100 km2). According to equations (1) and (2), a
uniform water sheet of the thickness of d =
4 • 10~3 m would be required to transport a
water flux of 0.30 km3/yr if the average glacier
surface slope is yhs = — 2 • 10-2 and the basin
width is D = 10-20 km. The concentration
time of the drainage basin would be about 3
days. These high values required for the water
sheet thickness might throw some doubt on
whether all the subglacial water actually could
flow in a water sheet. The alternative is that the
water mainly drains by a channef system (cf.
Nye, 1973; Röthlisberger, 1972).
The extent and flux density
of the geothermal area
A geothermal area of Ag = 100 km2, as sug-
gested in the present paper on the basis of the
mass balance study, would inclucle the entire
1 6 JÖKULL 24. ÁR