Jökull - 01.06.2000, Page 25

An ice-dammed lake in Jökulsárgil
melt rate and the closure rate are however, close to
each other at the lower end of the discharge range.
The closure rate estimated above is based on the
assumption that the strength of the ice tunnel is the
same as for a circular or semi-circular tunnel with
the same cross-sectional area fully enclosed in ice. It
is proposed that the assumption concerning conduit
cross-sectional geometry implicit in the modelling is
ill-founded, and could result in an over-estimate of
tunnel closure times.
DISCUSSION OF TUNNEL CLOSURE
DYNAMICS IN VIEW OF CONDITIONS
AT S ÓLHEIMAJÖKULL
Complications exist in the consideration of subglacial
tunnel dynamics and problems are inherent in some of
the modelling treatments of such conduits. Generally,
straight conduits of either a semi-circular or circul-
ar cross-sectional form are assumed for simplicity.
However, Hooke et al. (1990) state a belief that is
implied by other workers in glacial hydrology (e.g.
Lliboutry, 1983; Iken and Bindschadler, 1986); that
conduits at the glacier bed are broad and low in cross-
sectional form. This shape is believed to develop
through melting low on the conduit walls when water
flow is low. As melting is only offsetting overburd-
en closure on the lowest part of the conduit walls,
the tunnel roof will continue to be forced down und-
er overburden pressure. Hooke et al. (1990) also state
that these conduits are most likely to form when the
bed over which the water is flowing consists of soft
sediments that can be transported in water.
Usual assumptions made about the circular or
semi-circular tunnel cross-sectional form do not
seem to adequately represent the true nature of the
subglacial conduit at Sólheimajökull. Observations
made around both the northern and southern tunn-
el outlets indicate that the ice slopes back into the
portals and that the actual tunnel cross-section does
not readily conform to any particular shape. The tunn-
el portals tend towards a semi- circular cross-sectional
form, but at the back of the portals ice can be obser-
ved to be in contact with water almost at the level of
the bed. Thus the water passes through the glacier
in the form of a gravel-bedded river with ice very
closely above it. The tunnel therefore does not possess
a definable radius, but is more adequately described
as having height and width characteristics (Figure 3).
Thus the cross-sectional shape of the tunnel at Sól-
heimajökull probably does not conform to that used in
many models of ice conduit dynamics, and modelling
using the traditional assumptions of a semi-circular
or circular cross-sectional form could give rise to
inaccurate predictions.
Hooke et al. (1990) discuss a method of app-
roximating the shape of subglacial conduits by the
space between the chord of a circle and the arc su-
btended by that chord (Figure 4). If the width of the
tunnel remains constant, the factor that determines
the height/width ratio of the conduit is the angle 9.
When 9 = 180°, the height of the tunnel will be equal
to W/2; this dimension would be referred to as the
radius in unmodified models of tunnel dynamics. The
tunnel height decreases as 9 decreases. Therefore to
represent a very shallow tunnel with a broad cross-
sectional shape, 9 would be very small. Hooke et al.
(1990) state that, for an ice tunnel that is very broad
and low in cross-sectional form, conduit closure rates
could lie between limiting values derived by substitut-
ing the conduit radius in Nye’s (1953) theory with the
conduit height, followed by the half-width W/2. They
use the mean of the two values (i.e. (H + W/2)/2) to
predict subglacial conduit water pressures using Röt-
hlisberger’s (1972) model. Whilst Hooke et al. (1990)
suggest that H and W/2 are limiting values for the
replacement of the tunnel radius (R) in Nye’s (1953)
theory, they do not justify using a value that is the me-
an of H and W/2 and it is not even clear that the radius
W/2 is a true maximum for a very low tunnel. Nevert-
heless it seems clear from the qualitative observati-
ons of the tunnel geometry described above, that an
effective radius that represents the “strength” of the
ice roof over the tunnel could be much larger than the
equivalent radius of a circular or semi-circular tunnel
with the same cross-sectional area as the ice tunnel.
Equation (1) indicates that a tunnel with a larger ef-
fective radius would collapse at a more rapid rate th-
an a circular or a semi-circular tunnel with the same
cross-sectional area. Such an increase in the rate of
JÖKULL No. 48 23