Jökull - 01.12.1957, Page 15

(31)

from Great Britain has been corrected for
erosion. The reduced heat flow may be sligthly
lower and more in accord with the global
average.
Surface heat flow and gravity anomalies due
to intrusives. In general, both conduction and
mass transport of heat contribute to the cool-
ing of intrusive bodies. The mass transport
may be due to magmatic convection within the
intrusive in its initial fluid phase. Further-
more, convection currents in the adjacent
ground water are possible.
The magmatic convection is unimportant
in the case of intrusives which are relatively
small compared to the depth. Convection of
the ground water depends on the permeability
of the country rock and is absent in rock of
negligible permeability. Therefore, both effects
may be disregarded at certain conditions.
The temperature field around an extensive
horizontal intrusive sill may in the first ap-
proximation be regarded as one-dimensional.
Furthermore, a relatively thin sill of constant
thickness will roughly be equivalent to the
instantaneous liberation of a constant amount
of heat at the depth of the sill. The outward
conduction of heat from the sill may then be
estimated on the following basis.
Suppose a semi-infinite solid of zero initial
temperature. According to Carslaw (10) the
temperature T at the depth x and the time t
due to the instantaneous liberation of a con-
stant amount of heat FI cal/cm2 at the depth
y and t:me zero is given by
T = -
H
2cp -y/ nat
E
(X - y)2
4at
(x + y)~al
4at (29)
where c is the specific heat by constant pres-
sure, p the density and a the diffusivity of
the material. These properties of the material
are assumed constant in space and time.
The temperature gradient at the surface is
found to be
(Tx) x ~ o
Hy
2kt jtat
e
_y2_
4at
(30)
where k is the heat conductivity.
The r.h.s. of equation (30) may be regarded
as a function of u = yjZ\/a.t. The function
f (u) = ue — u2
has a maximum f (u) = 0.43 at u = 1 /-y/ 2.
The maximum surface temperature gradient
is therefore obtained at y=1.4 y/ at and is
(Tx)x = 0 = 0.24-Jk (32)
This may be written in terms of the maximum
surface heat flow
Qmax = 0.24 H/t (33)
Furthermore, an integration of equation (30)
gives the amount of heat Q (t) which at the
time t has been conducted through the surface
of the semi-infinite solid.
Q(tj=Herfc(y/2V«) (34)
Hence, 32% of the heat have been conducted
out of the solid when the maximum surface
flow is reached.
In the case of the intrusive sill the quantity
H = md, where m is the sensible heat content
of the magma and d the thickness of the sill.
By a density contrast of Ap the gravity anomaly
of the sill is given by (Jakosky, (25)).
Ag = 4.2 • 10—4 Ap d (35)
where Ag is given in mgal and d in cm. For
d we may insert H/m and obtain
Ag = 4.2 • 10—4 Ap H/m (36)
The application of equation (33) and m = 103
cal/cm3 gives
Ag = 53 Ap Qmax t (37)
where Qmax is measured in microcal/cm2 sec
ancl t in 10° years.
Equation (37) gives approximately the rela-
tion between the gravity anomaly and the
maximum surface heat flow from an extensive
thin liorizontal intrusive sill which was formed
instantaneously at the time t before present.
The average depth of the sill giving the
maximum surface heat flow is y=1.4-^/at.
Actually, equation (37) is more general than
indicated by the above derivation. It may also
be applied to the case of the spherical intru-
sive emplaced at a depth which is great com-
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