Jökull - 01.12.1974, Page 15

(2)
d2
V = Ufy" (~ APw)
in which pj and pw are the density of ice and
water, respectively, hs and hb are the elevation
of the glacier surface and the glacier bottom,
respectively, g is the acceleration of gravity, d
is the thickness of the water sheet, and tj is
the dynamic viscosity of water [r\ = 1.8- 10—2
poise). A map showing the negative pressure
gradient — Vpw (or the velocity vector if one
assumes a constant sheet thickness d) in the
topographical model is shown in Fig. 11. The
map is derived from the glacier surface and
bottom maps in Figs. 6 and 7. The map of nega-
tive pressure gradients defines a subglacial
drainage basin for sheet flow. The estimated
area of this subglacial drainage basin is 295
km2. The effect of the bottom topography on
the direction of subglacial water sheet flow is
almost negligible, except for a slight effect in a
narrow area along the watershed north-west and
east of Grímsvötn which causes water to flow
outward. Water flow in subglacial tunnels
might cross this watershed but in the present
study it is assumed that the subglacial drainage
basin is equal to the ice flow basin, which lias
an estimated area A = 300 km2.
A model for the water and energy balance
The terms in a water and energy balance
model for Grímsvötn, illustrated in Fig. 12, will
now be described. Consider a long-term model
of a closed water basin (A); there is no net base
runoff (groundwater) and the time scale con-
sidered is much larger than the concentration
tirne for the water drainage basin. In this
rnodel, the volurne of a jökulhlaup (V) can be
estimated by integrating the input rate of melt-
water to the lake (q) over the period At be-
tween the jökulhlaups,
(3) V = jqdt = j J(as + ab) dSdt + j f
At At A At A„
in which as is the glacier surface ablation rate,
ab is the subglacial ablation rate due to a
norrnal geothermal gradient (ab = 1 cm/yr), and
ag is the subglacial ablation rate in the geo-
thermal area Ag. The second integral on the
right hand side gives the meltwater volume
which has been melted by the geothermal area.
The energy flux of this geothermal area is there-
fore given by
(4) -^=JagdS
Ag
in which L is the latent heat of melting per
unit volume of ice (L = 300 MJ m~3). The total
water flux to the Grímsvötn lake is given as
(5) qw = f (+ + ab) dS + J (as + ab + ag) dS
A — Ag Ag — Ax
in which Aj is the lake area. The first integral
gives the water flux from outside the geotherm-
al area and the second integral gives the water
flux from that part of the geothermal area
which is situated outside the lake.
Assuming that the glacier is in a steady state,
the ice flux can be included in the mass bal-
ance model. Consider the mass balance of the
geothermal area. The influx from the area out-
side the geothermal area added to the net bal-
ance inside the geothermal area must equal the
ice meltecl by the geothermal area. Hence,
(6) J (b— ab) dS + j (b— aB) dS = jagdS,
A — A„ A_ A„
b = c — a„
in which b is the net balance rate and c is the
rate of accumulation of ice on the glacier sur-
face. Since the glacier is assumed to be in a
steady state there is a balance between the rate
of accumulation of ice over the entire drainage
basin and the flow of water into the lake
(?)
agdSdt
This model will now be tested on the Gríms-
vötn basin.
Mass balance data for the water basin.
Main results of the water balance model
No detailed mass balance measurements have
JÖKULL 24. ÁR 1 3