Download A thermal study of the formation of oceanic crust

Geophys. J. R . astr. Soc. (1976)41,83-95
A thermal study of the formation of oceanic crust
N. J. Kusznir and M. H. P. Bott
Department of Geological
Sciences, South Road, Durham DHI 3LE
Received 1976 April 30; in original form February 26
Summary. The temperature distribution beneath an ocean ridge with a
magma chamber solidifying to form the crustal layer 3 has been investigated
by numerical methods. Assuming that water circulation rapidly cools layer 2
but does not penetrate deeper, it is found that a magma chamber forms provided that the half spreading rate exceeds about 0.45 cm y-' ;for slower rates
instantaneous dyke-like solidification occurs down to the Moho beneath the
ridge axis. Numerical models show that the width of the magma chamber
and the thickness of the dyke complex depend on half-spreading rate. The
width is about 20 km for 3 cm y-'. If there is significant crystal settling to
form cumulates, the chamber width is much reduced. The resulting heat
flow pattern is as follows: an outer zone where latent heat effects are negligible; a middle zone about twice as wide as the magma chamber where latent
heat significantly contributes to the heat loss; a narrow axial zone where
quenching of magma to form layer 2 provides the main heat loss. The computations support Cann's petrological model and suggest that the depth of
water penetration at the accretion boundary defines the layer 2-layer 3
boundary.
Introduction
According to plate tectonic theory, new oceanic lithosphere is formed at accretionary
boundaries beneath the axial regions of oceanic ridges from partially fused mantle material
rising from below. The residual solid fraction goes to form the mantle part of the lithosphere,
and the basaltic magma fraction rises to form new crust at the accretionary boundary. This
paper describes a thermal investigation of the solidification and cooling of the basalt magma
to form the new crust, based on the assumption that the crustal layer 2 is rapidly quenched
by water circulation, but that layer 3 is formed by slower cooling and solidification of a
magma chamber by thermal conduction. The shape and extent of the magma chamber is
investigated quantitatively by numerical methods for a variety of assumptions on spreading
rate and mode of solidification, and the associated isotherm distributions in the new oceanic
lithosphere and surface heat losses are evaluated.
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N. J. Kusznir and M.H. P.Bott
A series of petrological models for the formation of new oceanic crust have been proposed
by Cann (1968, 1970, 1974) in which layer 2 is formed of rapidly cooled pillow basalts and
layer 3 is formed of more slowly cooled gabbros and dolerites above and crystal cumulates
below. The models of Cann agree well with the geophysically determined oceanic crustal
structure and with the study of ophiolite suites (Gass, Smith & Vine 1975; Jackson, Green &
Moores 1875). The thermal investigations presented support Cann's models and add some
further insights into the processes which may occur at accretionary boundaries.
McKenzie (1 967) originally studied the temperature distribution in the spreading lithosphere of assumed constant thickness, neglecting latent heat effects. Parker & Oldenburg
(1973) and Oldenburg (1975) took latent heat into account by assuming that solidification
of the partially fused material takes place at the lithosphere-asthenosphere boundary. This
enabled them to compute the shape of this boundary, which was found to deepen away from
the ridge axis proportionally to the square root of spreading time. They did not take into
account the separation of a magma fraction to form the crust, and their computations predict too great a lithospheric thickness at distance from the ridge axis for a realistically-small
magma fraction. We have solved essentially the same problem as Parker & Oldenburg but we
have applied it to formation of a crustal magma chamber rather than to the lithosphere,
using finite difference rather than analytical methods. Our results are not sensitive to whether
the lithosphere has uniform or varying thickness.
Sleep (1975) has recently carried out a study similar to ours, in which he approximately
applies the latent heat effect as a boundary condition at the accretion boundary. Although
his formulation does not allow accurate delineation of the magma chamber, nevertheless he
reaches broad conclusions which are in close agreement with our own independent inferences.
The thermal model
In the steady state temperature frame relative to the ridge axis, the differential equation of
thermal equilibrium in a spreading oceanic lithosphere of uniform thermal properties,
neglecting latent heat effects and assuming symmetry about the ridge axis, is given by
where p is the density, u is the specific heat, k is the thermal conductivity, v, is the half
spreading rate and T is the temperature. The z-axis is taken to point vertically downwards at
the ridge axis and the x-axis to point horizontally away from the ridge axis along the upper
surface of the lithosphere. The analytical solution of this equation (McKenzie 1967) giving
the temperature at a point in the lithosphere is
where To is the temperature along the z-axis, 0°C is that along the x-axis, L is the thickness
of the lithosphere and p = uxLpa/k is assumed to be constant. The small adiabatic gradient
beneath the ridge axis is neglected in (2) but is hardly significant and can be added to the
solutions.
The presence of magma in the crust beneath the ridge axis requires inclusion of the latent
heat of solidification in a thermal study of oceanic crustal formation. For the purposes of
the thermal model, it is assumed that magma is restricted to the crust, because the under-
The formation of oceanic crust
85
lying upper mantle is left below its solidus on fractionation of the basalt liquid (Presnall
1969;Cann 1974). It is assumed that layer 2 is cooled rapidly at the ridge axis by circulation
of convecting sea water (Bodvarsson & Lowell 1972; Palmason 1973; Lister 1974), but that
layer 3 is cooled mainly by conduction alone. Thus the layer 2-layer 3 interface is assumed
to correspond to the lower limit of convecting sea water and the temperature of this boundary is assumed to be 0°C for the thermal model and it is taken as the x-axis.
The latent heat of solidification of a basalt melt may be introduced into the thermal
model in two ways:
(1) The latent heat may be released throughout the temperature interval of liquidus to
solidus(Hess 1960; Jaeger 1968). A convenient way of doing this is to introduce an effective
specific heat over the solidification temperature interval which incorporates the distributed
latent heat (Jaeger 1968), such that
where a is the actual specific heat, H is the latent heat, and TI and Ts are the liquidus and
solidus temperatures respectively. The introduction of the temperature dependent specific
heat, a'(T), results in the differential equation
which may be solved by numerical methods.
(2) Alternatively, the latent heat may be assumed to be released at the solidus temperature T,. The solidification may then be represented in the thermal analysis by a
I
T=T,
0
I
I
mantle
I
I
I
I
I
I
4
I
I
I
I
I
I
I
I
I
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L
T=
To
P i e 1. Thermal model of lithosphere including latent heat effect, with latent heat released at
86
N. J. Kusznir and M. H. P. Bott
differential equation applied to the solid-liquid boundary. If (Y is the angle made by the
liquid-solid interface with the positive x-axis, the differential equation is
where ux is the horizontal velocity of the phase boundary. For a steady state position of the
phase boundary with respect to the ridge axis, the velocity ux coincides with the halfspreading rate u, (Fig. 1). The assumption that the latent heat is released at the solidus
temperature implies a high thermal mobility for the magma. The magma probably convects
so that this assumption may be valid. It is necessary to incorporate into the latent heat a
term representing heat liberated in the magma on cooling from liquidus to solidus, thus
H ' = H + ( T l - T,)u.
(6)
Both methods of incorporating the latent heat into the thermal model are used in the
following sections. They probably represent the two extremes, the true situation lying
between them. The method of finite differences has been used, obviating the need to
assume that all the heat is released at the ridge axis as has been done by Sleep (1975).
In the subsequent computations a two-dimensional rectangular-shaped lithosphere 80-km
thick has been assumed. The lithosphere probably actually thins towards the ridge crest, but
the results will still apply as the thermal penetration is shallow here. A liquidus temperature
of To = 1125°C has been assumed to apply at the vertical boundary x = 0 beneath the ridge
crest and at the base of the lithosphere. A temperature of 0°C is assumed for the layer 2layer 3 boundary at z = 0. The vertical boundary remote from the ridge axis is assigned a
temperature-depth profile according to the analytical solution given by equation (2)
because at that distance the temperature perturbation of the latent heat effect is negligible.
The value of the solidus temperature T, is taken as ~ 0 0 0 ° C(Hess 1960; Jaeger 1968).
We assume that k = 2.5 W m-' "C-' (0.006 cal cm-'s-' "C-') (Clark & Ringwood 1964), H =
4.2 x lo5 J kg-' (100 cal g-') (Bottinga 1974), p = 3.0 g cmP3, and u = 1.09 x lo3 J kg-'
"C-' (0.26 cal g-' "C-I).
An approximate analytical model
The spreading rate has to exceed a critical value if a quasi-stationary magma chamber is to
form beneath the ridge axis. At lower spreading rates, the magma solidifies effectively
instantaneously in the form of dykes. This critical spreading rate can be approximately
estimated using analytical methods. If the horizontal temperature gradient at the idealized
accretionary boundary computed without reference to latent heat effects exceeds the
temperature gradient required to carry away the latent heat, then the magma will solidify
instantaneously at the accretion boundary. If not, then a magma chamber will form. This
phenomenon is investigated by calculating the horizontal temperature gradient from
equation (2) and comparing it with the rate of release of latent heat. A vertical solidification
boundary is assumed as this is the geometry of the molten region for slow spreading. It is
assumed that the latent heat is released at the solidus temperature Ts.
The ratt of release of latent heat per unit area is QL = u,Hp. The thermal gradient in the
absence of latent heat, from equation ( 2 ) is
The formation of oceanic crust
87
and the horizontal heat flow is
The solidification boundary will be stationary when AQ = Q, - QL = 0. The value of AQ
has been computed at 5-km depth as a function of x for various spreading rates (Fig. 2). The
critical half-spreading velocity below which solidification occurs instantaneously on intrusion
throughout the crust is calculated to be approximately 0.45 cm y-' . This suggests that for
very slow spreading rates below this critical value the oceanic crust is entirely formed of
dykes intruded from below and solidifying at the accretion boundary. For faster spreading
rates, a magma chamber would be expected to form within layer 3 provided that significant
cooling by water percolation does not penetrate below layer 2. Sleep (1975) has obtained a
similar result, but with a rather higher critical velocity.
0
1
Distance lkml
2
Figure 2. A Q (see text) as a function of distance from ridge axis and half spreading rate, computed at a
depth of 5 km within layer 3.
Solution by finite difference method
Where solidification occurs, the two-dimensional partial differential equation for temperature
in the region can best be solved by finite difference methods'(Cars1aw & Jaeger 1959). This
method if first applied to the use of an equivalent specific heat uf, as defined by equation
(3), as a method of including the latent heat of solidification in the thermal calculations.
Equation ( 4 ) with a temperature dependent specific heat has to be solved. As a first step, a
network of regularly-spaced points is constructed throughout the region; the horizontal and
vertical spacing of the grid points may be allowed to vary provided that the differential
equation can still be represented by a difference equation t o acceptable accuracy (Fig. 3).
The difference equation corresponding to equation (4) is
where a is the grid spacing. Similar relationships can be derived for the boundary where
aa
N. J. Kusznirand M. H. P. Bott
grid spacing changes. The temperature values on the boundary points of the grid must be
specified, and the values at interior points are then determined by relaxation involving
successive approximation. T o start the process, a first estimate of the temperature distribution is made using an approximate analytical solution or otherwise. It is then found that the
difference equation (8) is not exactly satisfied at the internal points. A correction ATo is
therefore applied to each internal grid point as given by
[
AT, = ?4a2 Ti + Tz + T3 + T.q - 4To
--pa'(T)
k
a'
(T3 - T i )
UX
242
(9)
The whole process is repeated for the whole body, and iterations are continuea until the
equilibrium state is reached when the residuals become negligible. For large values of ux, the
solution of equation ( 4 ) may become unstable using this method, but the instability may be
overcome by multiplying ATo of equation (9) by a relaxation factor of 0.1.
The alternative method of including solidification in the thermal calculations involves the
introduction of an auxiliary boundary representing the boundary between solid and liquid
crust. The solution must therefore satisfy the boundary differential equation (5). The
temperature of the auxiliary boundary is assumed to be that of the solidus T,. Elsewhere in
the solid lithosphere equation (1) applies.
The finite difference method can be extended to solve this problem by representing the
auxiliary boundary by a series of finite difference points or nodes. The nodes chosen are
those nearest to the position of the boundary of the finite difference grid, and are moved
from their regular position onto the boundary. The numerical solution of the problem is
then found by repeated iterations of a two part sequence, as follows:
(1) The temperature field in the solid part of the lithosphere is calculated as above.
The auxiliary boundary is f x e d during the solution.
(2) The vertical and horizontal temperature gradients are calculated for the position
midway between two boundary nodes. The differential equation for the auxiliary boundary
will not be satisfied exactly but will give rise to a residual specified by
k
# 0.
The residual R is used to perturbate the position of the auxiliary boundary in the horizontal
direction by a distance E given by
RAX
€=
( Z ) - R
The perturbation ?he is applied in turn to the boundary nodes, each node receiving two perturbations corresponding to the sections of the boundary immediately above and below it.
Having applied the perturbation to the auxiliary boundary, the temperature distribution
in the solid lithosphere is disturbed. The two part sequence of adjusting the temperatures of
the lithosphere and perturbing the auxiliary boundary is continued until the boundary ceases
to move significantly indicating convergence. The numerical calculations are greatly speeded
by using a good initial estimate. It has been found advantageous to multiply the boundary
perturbations by a factor of value 0.1 which ensures numerical convergence and also speeds
the solution.
The formation of oceanic crust
n
..
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..
..
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..
T=0
..
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..
Distance Ikml
..
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..
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.
..
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..
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...
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...
..
...
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...
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200
..
...
...............
80 . . . . . . . . . . . . . . .
.
.
T=T.
Figure
Diagrammatical representation o the finite difference gri- used in the numerical E - d a t i o n s .
g(x, z ) delined by equation (2).
The finite difference grid used for both types of solution employs a coarse node spacing
of 10 km at distance from the ridge axis, and a finer spacing of 0.25, 0.5 or 1.0 km in the
vicinity of the ridge crest where solidification occurs (Fig. 3). Convergence was obtained
within 600 iterations of the finite difference corrections for the whole body for both
methods. This reduced the maximum nodal temperature residuals to less than 0.loC and
using the second method the perturbations of the boundary were less than 10 m.
Temperature distribution and the shape of the molten region
The temperature distribution in the lithosphere and in detail near the ridge crest has been
calculated using the numerical techniques described above, for both thermal models and for a
range of spreading rates. Fig. 4 shows the isotherms for the distributed latent heat model
assuming a half spreading rate of 3 cm y-' . Analytically derived isotherms for the same
spreading rate but without the latent heat effect are shown for comparison. The greatest
difference occurs near the base of the crust below the ridge axis. At distance from the ridge
axis the two isotherm distributions become identical, this occurring at about 30-km distance
for the spreading rate of 3 cm y-'. Latent heat is effectively an extra heat source in the
0
0
Distance ( k m )
10
20
30
-E
Y
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5 5
Q
Q,
n
10
Figure 4. Isotherm distribution ("0in the oceanic crustal layer 3 and upper mantle at ridge axis and
environment, with distributed latent heat, assuming a spreading rate of 3 cm y-' and thermal properties
as stated in text. Isotherms for model neglecting latent heat are shown as dashed curves.
90
N J. Kusznir and M.H. P. Bott
Distance (kml
0
10
0
20
30
h
(u
n
10
Fkure 5 . Isotherm distribution ("C)in the oceanic crustal layer 3 and upper mantle near the ridge axis for
model with latent heat released at solidus, assuming a spreading rate of 3 cm y-'. Isotherms for model
neglecting latent heat are shown as dashed curves.
vicinity of the ridge axis and it causes both temperature and heat flow to be somewhat higher
in the central region when the effect is included, than when latent heat is neglected (McKenzie
1967; Sclater & Francheteau 1970; Haigh 1973).
Fig. 5 shows the isotherms for the thermal model with latent heat released entirely at the
solidus temperature. This model gives rise to a slightly more rapid release of heat consequent
on convection in the magma chamber, and fhe magma chamber is correspondingly slightly
narrower. Temperatures are correspondingly higher in the crust beneath the ridge axial
region. It is felt that the two models represent extremes with the true situation lying somewhere in between.
For both types of model, the solidus isotherm corresponds to the limits of the magma
chamber in the crust, and the crust-mantle boundary corresponds to the base of the
chamber. In Fig. 6, the position of the solidus isotherm is shown as a function of spreading
rate for both models. The width of the magma chamber is approximately proportional to
the spreading rate. The models where the latent heat is entirely released at the solidus are all
0
10
2 0 (km)
Figure 6. Positions of the solidus isotherm in the oceanic crustal layer 3 as a function of half spreading
rate (cm y-l); (a) latent heat distributed over liquidus-solidus interval; (b) latent heat released entirely
at the solidus.
The formation of oceanic crust
91
narrower than the corresponding distributed latent heat models. The effect of reducing the
thickness of the molten region at the ridge axis from 5 to 3 km has also been investigated
and it is found that the shape of the magma chamber does not change significantly except
that it is truncated at 3-km depth below the top of layer 3.
The computations show that layer 3 is on average more quickly cooled for slower spreading rates as Sleep (1975) also concluded, in agreement with the existence of a critical spreading rate below which instantaneous solidification occurs at the accretion boundary. The
upper part of the magma chamber cools more rapidly than the lower part. For slow spreading
rates of about 1 cm y-I, the upper part may cool instantaneously as a dyke while a magma
chamber of finte width forms below (Fig. 6). As slow cooling is required for crystal settling
within the magma chamber, these computations indicate that the cumulate layer at the base
of layer 3 may be thicker for faster spreading rates.
Hess (1960) suggested that the latent heat of solidification of basalt is distributed throughout the liquidus to solidus interval, 60 per cent being released above 1100°C and 40 per cent
below. This effect has been investigated, and is found to result in a small (10-20 per cent)
reduction in the width of the magma chamber.
The shape of the magma chamber is likely to be more significantly modified by the
accumulation of crystals settling onto the bottom of the chamber, possibly accompanied by
some piecemeal stoping. The distributed latent heat model (Figs 4 and 6) can be adapted to
include this effect. It is assumed that the ratio of solidified material forming the chamber
roof to that falling to the bottom is 1 :F. The thickness of crystal accumulate on the base of
the chamber is thus implicit from the thickness of solid crust above. To take account of this
effect, the latent heat must be adjusted to take account of the latent heat released by the
settled material, such that H’ = H(l + F ) . The effective specific heat between liquidus and
solidus is computed by replacing H by H’ in equation (3). The further assumption must be
made that the solidified material does not re-melt. The resulting shape of the magma chamber
has been computed for values of F equal to ?4, 1 and 2 as a function of spreading rate
(Fig.7). The effect of crystal settling is thus to make the magma chamber narrower and
shallower, as was qualitatively inferred by Sleep (1975).
Pattern of heat loss from the cooling oceanic lithosphere
Because of water circulation in layer 2 beneath ocean ridges, the pattern of heat loss in the
vicinity of ridges cannot be satisfactorily determined from present heat flow observations.
However, the pattern of heat loss can be computed theoretically for specified models, including the latent heat effect and the rapid quenching of layer 2 rocks near the ridge axis.
Thus the rate of heat loss from the upper surface of layer 3 has been computed for the models
in Figs 4 and 5, and is shown in Fig. 8.
Three heat flow regimes can thus be defined for the cooling of the oceanic lithosphere.
In an oufer zone, beyond about 30 km from the ridge axis in Fig. 8, the heat loss is effectively
indistinguishable from that of the simple c o o h g model without latent heat effects. Between
about 2 and 30km (depending on spreading rate), is a middle zone where heat loss is
strongly augmented by the solidification of the underlying magma chamber; this zone is
substantially wider than the magma chamber itself. An inner zone occurs within about 2 km
or less of the ridge axis, where the majority of the heat loss results from solidification and
cooling of the layer 2 rocks by water circulation.
The rate of heat loss by quenching of layer 2 rocks in the inner zone is given by
(Hp + upTo)tux where t is the thickness of the layer. Using the same thermal constants as
for layer 3 and taking p = 2.60 g cm-’, t = 1.5 km, u, = 3 cm y-’, the average heat loss
N J. Kusznir and M.H. P.Bott
92
5
0
0
O
L
10 IkmJ
. . . . 5. . . . . 10. (kmJ
. .
1
. 5. . . . . 10. .(kmJ
.
Fz2.0
5
I
- - - - - - - - - - - - - - - - - - -- - _ -
Figure 7. Position of the solidus in the oceanic crustal layer 3 as a function of half-spreading rate (cmy-')
and crystal settling factor F.
0
0
20
10
30
Dirtonce lkm)
Figure 8. Heat loss from layer 3 at the layer 2 contact as a function of distance from the ridge axis, for
various assumptions regarding solidification at a spreading rate of 3 cm y-'.
(a) latent heat neglected;
(b) latent heat released at solidus; (c) latent heat released uniformly throughout liquidus to solidus.
The formation of oceanic crust
93
within 2 km of the ridge axis caused by this effect and by the initial cooling and solidification
of layer 3 is computed to be approximately 6.3 W m-' (150pcal cm-'s-').
Discussion
The computations presented are based on the assumptions that the oceanic crustal layer 2 is
cooled rapidly by water circulation but that layer 3 solidifies from magma and then cools by
thermal conduction. A uniform spreading rate has been assumed and quasi-stationary
solutions to the heat flow equations within a reference frame defined by the ridge axis
have been obtained. A simple analytical approach indicates that a magma chamber within
layer 3 only forms if the half spreading rate exceeds about 0.5 cm y-l. At slower
spreading rates, solidification can occur instantaneously at the accretion boundary, so that
layer 3 would be formed predominantly of basalt and dolerite dykes.
-1 0
-5
0
5
10 Ikml
Figure 9. Model to show formation of oceanic crust.
Numerical solutions make it possible to determine the shape of the magma chamber and
the associated temperature distributions for half spreading rates above 0.5 cm y-'. The
width of the magma chamber is approximately directly proportional to the spreading rate,
and for a half spreading rate of 3 cm y-' the chamber is estimated to extend 10 km from
the ridge axis in the absence of crystal settling. The inclusion of crystal settling in the computations shows the strong dependence of the predicted shape and width on the proportion
of the magma settling as crystals onto the floor of the magma chamber. With increase of
crystal settling, the magma chamber becomes less wide and more flat topped, because it can
cool more rapidly through the thinner overlying roof. The models thus predict that layer 3
is a doleritic dyke complex at the top where it cools rapidly near the accretionary boundary,
becoming gabbroic below with slower cooling; the bottom part would be expected to consist of cumulates formed by progressive crystal settling (Fig. 9). This structure is fully compatible with the petrological model of Cann (1974). The numerical calculations also suggest
that for increasing spreading rate the thickness of doleritic dykes and fine grained gabbros at
the top of layer 3 decreases, but that the thickness of underlying coarse gabbros and cumulates increases.
Shallow seismic refraction studies in the median valley of the Mid-Atlantic Ridge (Whitmarsh 1975) have possibly detected a narrow zone of very low velocity material, while small
earthquakes in the crust at the ridge axis (Francis & Porter 1973) have been used to suggest a
shallow velocity inversion in the crust consistent with a magma chamber. Present evidence
does not yet positively confirm the existence of a magma chamber and search for it must
remain an important objective.
N. J. Kusznir and M.H. P.Bott
94
Theoretically, the existence or absence of a magma chamber for normal oceanic spreading
rates appears to depend on whether or not there is significant water circulation in layer 3 at
the accretionary boundary. The good agreement with the thermal models presented here
and Cann's petrological models with ophiolite and crustal structure studies is the best present
indication that a magma chamber does form and thus that water percolation is effectively
restricted to layer 2. This suggests that presence or absence of water percolation may cause
the physicaldistinctions between layer 2 and layer 3 rocks. If so, layer 2 rocks are dominantly
formed by quenched basalt magma rather than by lavas outflowing at the surface; such
rapid quenching might be expected to produce textures and structures resembling those of
submarine lavas rather than normal dykes even though the magma does not reach the seabed.
In contrast, layer 3 is formed by rocks of similar composition which cool by conduction and
are consequently more coarse grained and less porous, with dykes at the top and gabbros
and cumulates below.
The computations enable a realistic estimate of heat loss in the vicinity of the accretionary
boundary to be made, showing that the heat loss within a few tens of kilometres is substantially increased by the latent heat effect. On the other hand, the overall loss of heat
from the oceanic lithosphere is only increased by 5-10 per cent by the latent heat effect.
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