Jökull - 01.12.1966, Page 14

broad, consisting largely of quenched lava in
a more or less fragmental state, Fig. 1. There
will then be a very effective insulation be-
tween the hot lava at the base and the ice on
top. The stage is now set for the sole contest
between lava pressure and the weight of the
ice cover.
It is sufficient to study a cross section per-
pendicular to the volcanic fissure, i.e. a verti-
cal slice of 1 cm thickness. The thickness of
the ice-cap be h, that of the dyke of fluid lava
filling the fissure is taken as 2 m, while
greater effective thickness will be considered
later.
The first question is the magma pressure at
the upper end of the dyke. The magma being
stagnant, this corresponds to the height to
which the magma could rise in a fissure. Of
this we have a fair idea. Whereas in many
volcanic areas the volcanoes rise to 5000 or
6000 m or even rnore, tlie maximum height
in Iceland is 2000 m. Mostly, the Icelandic
volcanoes do not surpass 1500 m, but a very
few go a little higher and a single one just
passes 2000 m. For such a very volcanic countrv
as Iceland this circumstance must be taken to
mean that the lava is incapable of rising higher
than about 2000 m.
This result is in keeping with the probably
shallow depth of magma under Iceland. Take
the depth of the magma as D, the maximum
height of rise as d, the density of the solid
crust as 3.0 and that of the fluid lava as 2.5
then we have 3 D = 2.5 (D -(- d), which gives
D = 10 km if d = 2 km, while the general
thermal gradient in Iceland would suggest
D 00 15 km. In a continent with D = 30 km
one gets d = 6 km and a reasonable change
of the densities does not alter this result
materially. We shall thus take d = 2000 m for
Iceland. This gives a pressure of 500 kg/cm2 at
the surface of our dyke at sea-level, or a total
force of 10s kg in our section. If the eruption
takes place at a height of 500 m above sea-
level, which is representative for much of the
interior of the country, the pressure would be
375 kg/cm2. The critical pressure of water
vapour is 205 kg/cm2 and has no effect as long
as we only consider the cross section of the
dyke.
We shall now study tlie lifting capabilitv of
the magmatic force in several steps.
1) The ice layer is so fractured that a 2 m
broad free slice of ice overlies the 2 m broad
dyke. Then, clearly, the slice will be lifted if
its height is less than 5000 m. This is a most
unrealistic case; it demands not only that dense
vertical fractures in the ice are almost exactlv
parallel to the volcanic fissure, but further that
there is no friction at the sides of the slices.
2) The glacier is in a sense plastic and yields
by flow to prolonged pressure. But for the
present inquiry this is unimportant as the dyke
would have consolidated to a great depth be-
fore any marked yield of the ice by flow had
occurred.
3) We consider the glacier to be compact
and without fractures; the yielding to the
magma pressure is entirely elastic up to the
breaking point.
c

fí L
K
Fig. 2. Lifting of a beam by force K, with
joints at A, B and C.
2. mynd. Lyfting bita með liðamótum í A,
B og C.
a. A solution is first approximated by postu-
lating joints at A, B, and C (Fig. 2). Then
the lifting force K is half the weight of the
ice between A and B. Putting tentatively 1 =
10 h, we have 10ð = i/2 • h/10 • 0.9 • 10 h-100,
where h is in metres. This gives h = 47 m,
which is the maximum thickness of ice that the
force K can lift by the given relation between
h and 1.
b. A better approximation is obtained bv
dropping the joint at C. Then K is 2/s instead
of 1/2 of the ice load.
c. Finally we use the theory of thin beams
and take the cross section of the ice-cap as a
very long beam lifted in point C by the force
2K, but resting on the ground farther away on
both sides, Fig. 3.
For u0 we get the expression
u0 = 1.16 • 104 • K4/h6, see Appendix.
168 JÖKULL