JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B7, 2332, doi:10.1029/2001JB001050, 2003
Quantitative modeling of granitic melt generation and segregation
in the continental crust
Matthew D. Jackson
Department of Earth Science and Engineering, Imperial College of Science, Technology and Medicine, London, UK
Michael J. Cheadle
Department of Geology and Geophysics, University of Wyoming, Laramie, Wyoming, USA
Michael P. Atherton
Department of Earth Sciences, The Jane Herdman Laboratories, University of Liverpool, Liverpool, UK
Received 23 August 2001; revised 23 August 2002; accepted 6 March 2003; published 9 July 2003.
[1] We present a new quantitative model of granitic (in a broad sense) melt generation and
segregation within the continental crust. We assume that melt generation is caused by the
intrusion of hot, mantle-derived basalt, and that segregation occurs by buoyancy-driven
flow along grain edges coupled with compaction of the partially molten source rock. We
solve numerically the coupled equations governing heating, melting, and melt migration in
the source rock, and cooling and crystallization in the underlying heat source. Our results
demonstrate that the spatial distribution and composition of the melt depends upon the
relative upward transport rates of heat and melt. If melt transport occurs more quickly than
heat transport, then melt accumulates near the top of the source region, until the rock
matrix disaggregates and a mobile magma forms. As the melt migrates upward, its
composition changes to resemble a smaller degree of melting of the source rock, because it
thermodynamically equilibrates with rock at progressively lower temperatures. We
demonstrate that this process of buoyancy-driven compaction coupled with local
thermodynamic equilibration can yield large volumes of mobile granitic magma from
basaltic and meta-basaltic (amphibolitic) protoliths over timescales ranging from 4000
years to 10 Myr. The thickness of basaltic magma required as a heat source ranges from
40 m to 3 km, which requires that the magma is emplaced over time as a series of sills,
concurrent with melt segregation. These findings differ from those of previous studies,
which have suggested that compaction operates too slowly to yield large volumes of
INDEX TERMS: 3230 Mathematical Geophysics: Numerical solutions; 8120
segregated granitic melt.
Tectonophysics: Dynamics of lithosphere and mantle—general; 8434 Volcanology: Magma migration;
KEYWORDS: granite, melt generation, melt segregation, compaction, lower crust
Citation: Jackson, M. D., M. J. Cheadle, and M. P. Atherton, Quantitative modeling of granitic melt generation and segregation in the
continental crust, J. Geophys. Res., 108(B7), 2332, doi:10.1029/2001JB001050, 2003.
1. Introduction
[2] Many granitic rocks (in a broad sense) originate as
partial melts of rocks in the mid- to lower-crust [e.g.,
Chappell, 1984; Atherton, 1993; Brown, 1994]. The melt
segregates from its partially molten host rock and accumulates to form a magma, which migrates upward through the
crust before being emplaced at higher structural levels [e.g.,
Miller et al., 1988; D’Lemos et al., 1992; Atherton, 1993;
Brown, 1994]. Melting is caused by the advection of heat
from the mantle to the crust, either by the intrusion of hot
mantle-derived magma, asthenospheric upwelling following
delamination, or a combination of these processes [e.g.,
Hodge, 1974; Bird, 1979; Huppert and Sparks, 1988; Kay
Copyright 2003 by the American Geophysical Union.
0148-0227/03/2001JB001050$09.00
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and Kay, 1991, 1993; Nelson, 1991; Bergantz, 1989;
Bergantz and Dawes, 1994]. During melting, the solid
fraction of the source rock (the restite) maintains an
interconnected matrix unless the melt volume fraction
reaches the Critical Melt Fraction (CMF), which is the
fraction at which the solid matrix disaggregates [Arzi,
1978; van der Molen and Paterson, 1979]. Geochemical
evidence suggests that the degree of melting required to
yield granitic partial melt fractions is small for most source
rock compositions, so static melt volume fractions are
significantly less than the CMF [e.g., Clemens, 1989,
1990; Vielzeuf et al., 1990; Beard and Lofgren, 1991;
Rushmer, 1991; Rapp and Watson, 1995]. Consequently,
the granitic partial melt fraction must flow relative to the
solid matrix and accumulate until it forms a magma, which
is mobile and which can migrate away from the source
region [e.g., Wickham, 1987; Brown, 1994; Sawyer, 1994,
3-1
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3-2
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
1996]. This process is termed segregation. The problem is
that the physical mechanism by which segregation occurs is
poorly understood. Note that we define ‘‘melt’’ to be crystal
free liquid, and ‘‘magma’’ to be melt plus suspended
crystals, although we recognize that, depending upon the
efficacy of the segregation process, a magma may contain a
negligibly small proportion of crystals.
[3] Melt segregation requires that the partially molten
rock matrix is permeable, that there is a pressure gradient to
drive melt flow relative to the matrix, and that there is space
available to accommodate the melt. All of the available
theoretical and experimental evidence suggests that the
matrix of most partially molten crustal source rocks will
be permeable even at low (<0.04) porosities [e.g., von
Bargen and Waff, 1986; Cheadle, 1989; Wolf and Wyllie,
1991, 1995; Vicenzi et al., 1988; Wark and Watson, 1998;
Lupulescu and Watson, 1999], and it is difficult to envisage
how segregation could otherwise occur. However, the
origin of both the pressure gradient required to drive flow,
and the space to accommodate the melt, is still poorly
understood. Several studies have suggested that if the
source rock is layered and the layers have different rheological properties, then tectonically driven deformation can
provide both a pressure gradient and an accommodation
space: Because the layers respond differently to deformation, pressure gradients form between them, and space is
created at dilatant sites such as boudins and fractures [e.g.,
Sawyer, 1991, 1994, 1996; Brown, 1994; Brown et al.,
1995]. However, these models are largely qualitative or
semiquantitative; no quantitative model has yet been developed which predicts the mobilization of large quantities
of granitic melt by deformation-enhanced melt segregation.
Moreover, they are based upon observations of exposed
migmatite terrains, yet it is not clear that migmatite terrains
represent the source regions of mobile granitic magmas:
migmatites can rarely be genetically linked to granitic
plutons at higher structural levels [Wickham, 1987], and
have been described as ‘‘failed’’ granites because the
granitic partial melt fraction has been retained in the source
region as leucosome [e.g., White and Chappell, 1990;
Clemens and Mawer, 1992].
[4] Other studies have suggested that melt migration is
driven by buoyancy, either along grain boundaries or
through a network of fractures [e.g., McKenzie, 1985;
Wickham, 1987; Petford, 1995; Petford and Koenders,
1998]. The problem with these models is that they do not
provide space to accommodate the melt. The flow behavior
of fractured oil reservoirs demonstrates that flow occurs
through fractures only if the fractures are connected to a
well, through which fluid is extracted from the reservoir
[e.g., Aguilera, 1995]. By analogy, in a partially molten
source region, flow will occur through fractures only if the
fractures provide a path out of the source region to higher
structural levels, where melt extracted from the source
region can be accommodated. Yet the work of Rubin
[1993, 1995] suggests that fractures filled with granitic melt
find it difficult to propagate out of a partially molten source
region unless the surrounding rock has been warmed, within
a localized region, by repeated dyke intrusion. Repeated
dyke intrusion requires an abundant source of segregated
melt to be available within the source region. Consequently,
we argue that fractures are unlikely to transport significant
quantities of melt during segregation, although we agree
that they may subsequently provide a mechanism for
segregated melt (magma) to ascend through the crust.
[5] If flow occurs along grain boundaries, then meltenhanced diffusional creep processes provide a mechanism
for changing the morphology of the grains [Pharr and
Ashby, 1983; Cooper and Kohlstedt, 1984, 1986; Karato
et al., 1986; Kohlstedt and Chopra, 1994], so the partially
molten source rock can compact in response to melt flow
[McKenzie, 1984]. Compaction provides space to accommodate the melt. However, applications of McKenzie’s
characteristic compaction length- and time-scales suggest
that compaction cannot accommodate significant quantities
of granitic melt in the crust within geologically reasonable
timescales [Wickham, 1987; Petford, 1995]. Yet these
length- and time-scales are derived from a model in which
the processes of melt generation and segregation are
decoupled [Richter and McKenzie, 1984]. In crustal melting
zones, melt generation and segregation are complimentary,
coupled processes which occur simultaneously; the results
of Jackson and Cheadle [1998] demonstrate that the interplay between heat and mass transport within a compacting
melting zone has a significant impact on the distribution and
compositional evolution of the melt. However, their model
is general and they did not apply it specifically to melting
within the crust.
[6] The aim of this paper is to present a new quantitative
model of granitic melt generation, coupled with buoyancydriven melt segregation and compaction, within the continental crust. Melting is assumed to be caused by the
intrusion of hot mantle-derived basaltic magma. We present
the conservation equations governing cooling and crystallization in the underlying basaltic heat source, and melting
coupled with buoyancy-driven melt segregation and compaction in the overlying source rock. To simplify the
problem, we assume that only heat is transported from the
cooling magma into the melting source rock; in reality, mass
transport may also occur as buoyant melt migrates upward
from the magma into the source rock.
[7] The model formulation is quite general; however, in
this paper we do not attempt to apply it to all granite source
rock compositions and tectonic settings. Rather, we investigate melting and melt segregation from homogenous
basaltic and meta-basaltic (amphibolite and eclogite) source
rocks in settings where high heat fluxes and thickened crust
promote partial melting. In continental arc environments,
partial melting of basalt has been invoked as the source of
Na-rich granitic batholiths which comprise the large Mesozoic Cordilleran complexes of the western Americas, Antartica, and New Zealand [e.g., Tepper et al., 1993; Leat et al.,
1995; Atherton and Petford, 1996; Petford and Atherton,
1996; Petford et al., 1996; Wareham et al., 1997; Muir et
al., 1998]. High heat fluxes and crustal thickening result
from basaltic underplating, which acts both as a heat source
for partial melting and as a material source for granitic melt
[e.g., Atherton and Petford, 1993, 1996; Tepper et al., 1993;
Petford and Atherton, 1996]. Basaltic and meta-basaltic
rocks are also classically regarded as protoliths for Archean
trondjhemite-tonalite-granodiorite (TTG) suites [e.g., Rapp
et al., 1991; Rapp and Watson, 1995; Martin, 1999] and our
results may contribute to the debate concerning their petrogenesis [Atherton and Petford, 1993; Martin, 1999].
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Figure 1. Model formulation: (a) initially, the temperature
in the region z > 0 is dictated by an initial temperature
gradient, the rock is at its solidus (Tsol) at z = 0, and there is
no melt present. At t = 0, a sill of thickness zsill, containing
basaltic magma at a uniform temperature Tsill, is instantaneously emplaced in the region z < 0. (b) Heat transport
from the magma in the region z < 0 to the rock in the region
z > 0 causes heating and melting of the rock and conjugate
cooling and crystallization of the magma.
[8] We solve the governing equations numerically, and
demonstrate that buoyancy-driven melt migration coupled
with local thermodynamic equilibration of the melt and
matrix during segregation can yield large volumes of mobile
granitic melt from a basaltic protolith over timescales ranging
from 4000 years to 10 Myr. The thickness of magma
required as a heat source ranges from 40 m to 3 km, which
implies that the magma is most likely emplaced over time as a
series of sills, concurrent with melt segregation in the
overlying source rock. Our model predictions are qualitatively similar, but quantitatively very different to those of
Fountain et al. [1989], who also presented a numerical model
of melting and buoyancy-driven melt migration in a compacting rock. This is principally because their expressions for
conservation of energy and mass are different to those
presented in this paper: they do not describe phase change
in the correct moving reference frame [Jackson and Cheadle,
1998]. Consequently, their model neglects melting and
freezing due to the migration of melt and matrix, which has
a significant impact on the solutions obtained.
[10] Before magma emplacement (t < 0), we assume that
the temperature profile in the solid rock is dictated by the
current lower crustal geotherm, the rock is at its solidus
(Tsol) at z = 0, and there is no melt present (Figure 1a).
These initial conditions are consistent with our application
of the model to tectonic settings where high heat fluxes and
thickened crust yield high lower crustal temperatures [e.g.,
Atherton and Petford, 1993; Tepper et al., 1993]. At time
t = 0, magma emplacement in the region z < 0 causes the
temperature at the contact (z = 0) to be increased from Tsol to
Tc (Figure 1b), which leads to heating and partial melting of
the rock in the region z > 0 and conjugate cooling and
crystallization of the magma in the region z < 0. We assume
that the thermodynamic properties of the basaltic source
rock and magma heat source are identical.
[11] At times t > 0, the model can be divided into three
distinct regions: solid rock in the region z > zsol, partially
molten rock in the region 0 z zsol, and magma in the
region z < 0. In the region z > zsol, heat transport occurs by
conduction only; mass and momentum transport are zero
because there is no melt present. In the region z < 0, heat
transport is also assumed to occur only by conduction,
based upon the work of Marsh [1989] and Bergantz and
Dawes [1994] which suggests that although the magma in
the sill convects, the transport of heat from the magma to
the adjacent country rock occurs ‘‘as if the magma were
cooling by conduction only’’ [Bergantz and Dawes, 1994].
Their findings are supported by observational data from
cooling basaltic lava lakes [e.g., Helz et al., 1989]. However, the assumption of conductive cooling is not valid if the
magma is superheated [Huppert and Sparks, 1988], so we
assume this is not the case. In the region 0 z zsol, heat,
mass, and momentum transport occur as buoyant melt is
generated and migrates upward along grain edges through
the solid rock matrix, and the matrix compacts or dilates in
response. These transport processes within a melting, deformable mush have been described in general terms by
Jackson and Cheadle [1998]. For simplicity, we follow their
approach and consider the problem in one dimension only
(1-D), invoking the Boussinesq approximation throughout
(i.e., assuming rm rs = r except in the buoyancy term).
2.1. Governing Equations
[12] The 1-D conservation equation governing the conductive transport of heat in the solid rock above the partially
molten rock (i.e., in the region z > zsol) may be written as
(see Table 1 for an explanation of the nomenclature)
2. Model Formulation
[9] Consider a region of homogenous, isotropic hydrated
basaltic rock in the lower crust, into which hot mantlederived basaltic magma, initially at temperature Tsill, is
intruded as a horizontal layer of thickness zsill (Figure 1).
Let the spatial origin of the model (z = 0) denote the position
of the magma/rock contact, and the temporal origin denote
the time at which the magma is ‘‘instantaneously’’ emplaced.
For convenience, we describe the layer of basaltic magma as
a sill, although we do not expect that all of the magma within
the sill is intruded as a single event; rather, we envisage that
intrusion is continuous over the timescale of melting and
melt segregation in the source rock, and eventually yields the
sill thickness initially assumed in the model.
3-3
kt
@2T
@T
¼ rcp
:
@z2
@t
ð1Þ
The 1-D conservation equations governing the transport of
heat, mass, and momentum in the partially molten rock (i.e.,
in the region 0 z zsol) may be written as [Jackson and
Cheadle, 1998]
kt
@2T
@T
¼ rcp
þL
@z2
@t
m;
@f
@
r ½ð1 fÞws ¼ m ;
@t
@z
2
4
@ ws
m ws
xs þ ms
¼ ð1 fÞðrs rm Þg m ;
@z2
k
3
r
ð2Þ
ð3Þ
ð4Þ
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Table 1. Nomenclature
Symbol
a
b
cp
ceff
g
kt
k
K
L
n
Ste
t
tseg
T
Tc
Tsol
Tliq
Tsill
Tgeo
wm
ws
z
zsol
z0
zsill
zsill
zmin
c
d
f
j
c
m
keff
mm
ms
vm
q
qseg
qc
qs
r
rs rm
w
t
xs
ygeo
Description
Units
matrix grain radius
constant in permeability relationship
specific heat capacity
effective specific heat capacity
acceleration due to gravity
thermal conductivity
permeability
characteristic permeability
latent heat
exponent in permeability relation
Stefan number
time
dimensionless segregation time
temperature
temperature at the contact (z = 0)
solidus temperature
liquidus temperature
initial temperature of sill
initial temperature gradient
melt velocity
matrix velocity
vertical Cartesian coordinate
position of solidus isotherm
position of dimensionless solidus isotherm (q = 0)
thickness of sill
dimensionless half thickness of sill
dimensionless sill half thickness required
for magma mobilization
ratio of thermal diffusivities in the partially
molten and solid rock
characteristic lengthscale
(McKenzie’s compaction length)
melt volume fraction (porosity)
initial degree of melting at the contact (z = 0)
rate of production of crystals
rate of production of melt
dimensionless effective thermal diffusivity
melt shear viscosity
matrix shear viscosity
degree of equilibrium melting
dimensionless temperature
(normalized degree of melting)
dimensionless segregation temperature
dimensionless contact temperature
dimensionless temperature at the center of the sill
density
matrix - melt density contrast
characteristic velocity scale
characteristic timescale
matrix bulk viscosity
dimensionless initial temperature gradient
m
none
J kg1 K1
J kg1 K1
m s2
W K1 m1
m2
m2
J kg1
none
none
s
none
K
K
K
K
K
K km1
m s1
m s1
m
m
none
m
none
none
none
m
¼r
@vm
@vm
þ rðwm þ ws Þ
;
@t
@z
k ¼ ba2 fn ;
kt
@2T
@T
¼ rcp
L c;
@z2
@t
none
none
kg m3 s1
kg m3 s1
none
Pa s
Pa s
none
none
none
none
none
kg m3
kg m3
m s1
s
Pa s
none
c
¼ r
@vm
:
@t
ð9Þ
The initial conditions are given by
T ð0; 0Þ ¼ Tsol ;
@
½T ð z; 0Þ
¼ Tgeo ;
@z
ð10aÞ
z > 0;
z < 0;
fð z; 0Þ ¼ vm ð z; 0Þ ¼ wm ð z; 0Þ ¼ ws ð z; 0Þ ¼ 0;
ð6Þ
ð7Þ
where equation (2) describes conservation of heat, equation
(3) describes conservation of mass, equation (4) describes
conservation of momentum, equation (5) relates the melt
and matrix velocities, equation (6) describes the rate of melt
production in the correct reference frame, in terms of the
ð10bÞ
ð10cÞ
ð10dÞ
where Tgeo is a linear initial temperature gradient. The
boundary conditions are given by
T ðzsol ; t Þ ¼ Tsol ;
ð11aÞ
@
½T ð z ! þ1; t Þ
! Tgeo ;
@z
ð11bÞ
@
½T ðzsill =2; t Þ
¼ 0;
@z
ð11cÞ
vm ðzsol ; t Þ ¼ 0;
ð11dÞ
wm ð0; t Þ ¼ ws ð0; tÞ ¼ wm ðzsol ; t Þ ¼ ws ðzsol ; t Þ ¼ 0:
ð5Þ
ð8Þ
where c denotes the rate of production of crystals (rate of
crystallization). If the degree of melting in the source rock
and the degree of crystallization in the magma are identical
functions of temperature, then the rate of crystallization may
be expressed as
T ð z; 0Þ ¼ Tsill ;
m
with the supplementary relations
fwm ¼ ð1 fÞws ;
equilibrium melt volume fraction (also termed the ‘‘degree
of melting’’), and equation (7) is a simple expression
relating the matrix porosity and permeability.
[13] Given that the thermodynamic properties of the
source rock and magma are assumed to be identical, the
1-D conservation equation governing the conductive transport of heat in the magma (i.e., in the region z < 0) may be
expressed as [Bergantz, 1989]
ð11eÞ
The boundary condition (11c) follows from the assumption
that conductive heat transport in the sill is symmetric about a
horizontal plane at its center. This leads to an underestimate
of the heat transported from the sill to the overlying rock.
2.2. Simplification and Nondimensionalization
of the Governing Equations
[14] Following the approach of Jackson and Cheadle
[1998], the degree of melting of the rock and the underlying
magma heat source will be described as a continuous, linear
function of temperature
vm ¼
T Tsol
Tliq Tsol
ð12Þ
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
3-5
The remaining variables are nondimensionalized by writing
[Jackson and Cheadle, 1998]
ðxs þ 4ms =3ÞK 1=2
with d ¼
;
mm
0
z ¼ z=d;
t0 ¼ t=t;
with t ¼
w0 ¼ w=w;
1
mm ðxs þ 4ms =3Þ 1=2
;
ð1 jÞðrs rm Þg
K
ð17Þ
with w ¼
k 0 ¼ k=K;
Figure 2. Empirically derived degree of melting (melt
volume fraction) as a function of temperature, at 8 kbar, for
a meta-basalt (amphibolite) source rock. Data from
Rushmer [1991].
and furthermore, latent heat will be assumed to be absorbed/
released linearly as melting/crystallization proceeds. This
approach is reasonable for hydrated basaltic and metabasaltic rocks in which melting occurs due to the breakdown of hydrous minerals such as amphiboles, under water
absent or undersaturated conditions [e.g., Clemens and
Vielzeuf, 1987; Beard and Lofgren, 1991; Rapp and Watson,
1995; Petford and Gallagher, 2001] (see also Figure 2). The
assumption of linear melting means that the dimensionless
temperature may be written as [Jackson and Cheadle, 1998]
T Tsol
T0 ¼
;
Tliq Tsol
q ¼ q=j;
0
f ¼ f=j:
with K ¼ ba2 jn ;
ð18Þ
ð19Þ
where the characteristic lengthscale d is identical to the
compaction length of McKenzie [1984, 1985]. Note that the
characteristic permeability is defined in terms of the initial
degree of melting at the contact (j).
[15] Substituting for the degree of melting (vm) in
equations (6) and (9), substituting for the rate of melt
production in equations (2) and (3), substituting for the
rate of crystallization in equation (8), substituting for the
permeability (k) in equation (4), substituting the scaled and
dimensionless variables in equations (15a) – (19), simplifying, and dropping primes, yields the dimensionless governing equations
@q
@2q
¼ ckeff 2 ;
@t
@z
z > z0 ;
ð20Þ
@f 1 @
@q
@q
½ð1 jfÞws þ þ ðwm þ ws Þ ;
¼
@t j @z
@t
@z
ð21Þ
@q
@2q
@q
¼ keff 2 Steðwm þ ws Þ ;
@t
@z
@z
ð22Þ
0 z z0 ;
@ 2 ws ws ð1 jfÞ
¼ nþ
;
@z2
f
ð1 jÞ
ð23Þ
jfwm ¼ ð1 jfÞws ;
ð24Þ
@q
@2q
¼ keff 2 ;
@t
@z
z < 0;
ð25Þ
with
kt t
;
rceff d2
ð26Þ
L
;
ceff Tliq Tsol
ð27Þ
L
;
Tliq Tsol
ð28Þ
keff ¼
ð14Þ
Ste ¼
The porosity (f) and dimensionless degree of melting (q)
may then be normalized by writing
0
K ð1 jÞðrs rm Þg
;
mm
ð13Þ
in which case it is numerically equivalent to the degree of
melting vm and will be represented by a new variable q; i.e.,
q = T0 = vm. We now wish to normalize the porosity (f) and
the dimensionless degree of melting (q). In previous models
of compaction, the porosity has typically been normalized
against a uniform ‘‘background’’ or ‘‘early formed’’
porosity upon which the solutions are superimposed [e.g.,
Richter and McKenzie, 1984; Barcilon and Richter, 1986;
Spiegelman, 1993; Wiggins and Speigelman, 1995]. However, in this model there is no background or early formed
porosity, so a new scaling factor is required. A convenient
factor is the initial degree of melting at the contact between
source rock and underlying magma (at t = z = 0 in Figure 1)
defined by [Jackson and Cheadle, 1998]
Tsill Tsol
:
j¼ 2 Tliq Tsol
ð16Þ
ceff ¼ cp þ
ð15aÞ
ð15bÞ
c¼1þ
L
:
cp Tliq Tsol
ð29Þ
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Note that equation (25) governing conductive heat transport
in the magma heat source is valid only if cooling of the
magma is always accompanied by crystallization (with
associated latent heat exchange), which is consistent with
our requirement that the magma is not superheated.
[16] In dimensionless terms, the initial conditions become
qð0; 0Þ ¼ 0;
@
½qð z; 0Þ
¼ ygeo ;
@z
qð z; 0Þ ¼ 2;
ð30aÞ
z > 0;
z < 0;
ð30bÞ
ð30cÞ
fð z; 0Þ ¼ wm ð z; 0Þ ¼ ws ð z; 0Þ ¼ 0;
ð30dÞ
ygeo ¼ d=j Tliq Tsol Tgeo ;
ð31Þ
and the boundary conditions become
@
½qðzsill ; t Þ
¼ 0;
@z
wm ð0; t Þ ¼ ws ð0; t Þ ¼ wm ðz0 ; tÞ ¼ ws ðz0 ; t Þ ¼ 0;
ð32aÞ
ð32bÞ
qðz0 ; t Þ ¼ 0;
ð32cÞ
@
½qð z ! þ1; t Þ
! ygeo ;
@z
ð32dÞ
where z0 denotes the dimensionless position of the q = 0
isotherm, and zsill denotes the dimensionless half thickness
of the underlying magma.
[17] Simplifying and nondimensionalizing the governing
equations has reduced them to a system of six equations
(equations (20) – (25)) governed by six externally prescribed
dimensionless parameters: the effective thermal diffusivity
in the partial melt zone and in the crystallizing sill (keff); the
Stefan number (Ste); the exponent in the permeability
relation (n); the initial degree of melting at the contact
(j); the ratio of the thermal diffusivity in the solid rock to
the effective thermal diffusivity in the partial melt zone (c);
and the initial dimensionless temperature gradient (ygeo).
We will explore the system in terms of these dimensionless
parameters. In the next section, we discuss their likely
values.
3. Dimensionless Parameters
[18] The values of the dimensionless governing parameters keff, ygeo, Ste, c, and j depend upon the dimensional
variables which appear in their defining equations (14),
(26), (27), (29), and (31). These variables are listed, along
with suitable values for basaltic source rocks, in Table 2,
and are discussed further in the Appendices. Substituting
this range of dimensional variables yields the values for the
dimensionless parameters given in Table 2. As we demonstrate in the next section, the key dimensionless parameter
governing melting and melt segregation is the effective
thermal diffusivity (keff). The predicted value of this parameter for a basaltic protolith is very uncertain. The key
variables which contribute to this uncertainty are the melt
shear viscosity (mm), the matrix bulk and shear viscosities
(xs, ms), and the characteristic matrix permeability (K).
These variables govern the values of the characteristic timeand lengthscales (t and d), which appear in the definition of
keff (equations (16), (17), and (26)).
[19] The characteristic matrix permeability is poorly constrained because it depends strongly upon the matrix grain
size (keff a2) and the initial degree of melting at the
contact (keff j3; j is used as the porosity scaling factor in
the defining equation (19)). Consequently, small uncertainties in each yield large uncertainties in permeability. It also
contains a scaling factor (b), which must be estimated on the
basis of very limited experimental and theoretical data
(Appendix A). Predicted values of the matrix grain size
vary by an order of magnitude, while values of the permeability scaling factor vary by nearly 2 orders of magnitude
(Table 2). The value of j depends upon the solidus,
liquidus, and initial temperatures of the protolith, the initial
temperature of the intruded magma, and the kinetics of heat
transfer in the magma. It cannot exceed 0.5 because the
thermodynamic properties of the source rock and magma
heat source are assumed to be identical, and the magma is
not superheated. It must be less than the CMF for the model
to be valid. The parameter j also corresponds to the
maximum degree of melting, and to yield reasonable
volumes of granitic melt, it cannot be very small; in this
paper we assume it approaches 0.5 and calculate the
characteristic permeability on this basis. Significantly
smaller values of j will yield only small volumes of granitic
melt which are unlikely to become mobilized.
[20] The other significant dimensionless parameter governing melting and melt segregation is the initial temperature gradient (ygeo), the predicted value of which is also
uncertain. This uncertainty results from the scaling of length
using the characteristic lengthscale (d) rather than from
uncertainty in the initial dimensional temperature gradient
(Tgeo).
[21] None of the governing dimensionless parameters j,
c, Ste, or ygeo, are independent of keff (equations (26) –
(29)). However, for fixed values of Ste and c, keff may still
vary over effectively its entire range; consequently, we
assume Ste and c are independent of keff. In contrast, Ste
and c cannot be assumed to be independent; however,
numerical experiments demonstrate that varying the values
of c and Ste has negligible effect on the results, and we fix
their values at c = 2.67 and Ste = 0.625. More significantly,
keff and ygeo cannot be assumed to be independent, and both
vary over a wide range of values. Substituting the dimensional parameters given in Table 2 yields the permissible
combinations of keff and ygeo shown in Table 3.
4. Results
[22] The governing equations (20) – (25) were approximated using finite difference schemes, following the ap-
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
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3-7
Table 2. Summary of the Parameters Used in the Model, With Suitable Values for Hydrated Basalt and Meta-Basalt (Amphibolite and
Eclogite) Source Rocks
Symbol
Description
Min.-Max. Values
kt
cp
L
Tliq-Tsol
Tgeo
r
rs rm
a
n
b
mm
ms
xs
CMF
j
keff
ygeo
Ste
c
thermal conductivitya
specific heat capacityb
latent heatc
liquidus-solidus intervald
initial temperature gradiente
densityf
matrix-melt density contrastg
matrix grain radiush
exponent in permeability relationi
constant in permeability relationj
melt shear viscosityk
matrix shear viscosityl
matrix bulk viscosityl
critical melt fractionm
degree of melting at the contactn
dimensionless effective thermal diffusivityo
dimensionless initial temperature gradientp
Stefan number
ratio of thermal diffusivities in the partial
melt zone and solid rock
1.5 – 3
1020 – 1220
400,000 – 600,000
200 – 400
15 – 25
2900 – 3100
400 – 700
5 104 – 5 103 (0.5 – 5 mm)
3
1/2500 – 1/50
103 – 106
1015 – 1019
1015 – 1019
0.4 – 0.65
0.5
105 – 10+8
106 – 1
0.45 – 0.75
1.82 – 3.94
Values Used in Figure 9
2
1200
600,000
300
20
3000
600
2.5 103 (2.5 mm)
3
1/500
104
1015 – 1019
1015 – 1019
0.5
0.5
0.356 – 35.6
1.67 103 – 0.167
0.625
2.67
Units
W K1m1
J kg1K1
J kg1
K
K km1
kg m3
kg m3
m
none
none
Pa s
Pa s
Pa s
none
none
none
none
none
none
a
Data from Clauser and Huenges [1995], valid over the temperature range 600 – 1300 K. Thermal conductivities vary by less than 25% over this
temperature range. The experimentally determined values presented by Murase and McBirney [1973] for basalt, andesite, and rhyolite melts also lie within
this range.
b
Calculated using data for individual minerals (albite/anorthite, diopside) from Robie et al. [1978], valid over the temperature range 600 – 1300 K.
Specific heat capacities of minerals vary by less than 20% over this temperature range, and by less than 25% between minerals and their melts [Richet and
Bottinga, 1986; Neuville et al., 1993].
c
Calculated using enthalpies of fusion of individual minerals (albite/anorthite, diopside) from Richet and Bottinga [1986].
d
Data from Clemens and Vielzeuf [1987], Beard and Lofgren [1991], Rushmer [1991], Rapp and Watson [1995], and Petford and Gallagher [2001].
e
Data from Pollack and Chapman [1977] and Uyeda and Watanabe [1970]. We apply the model to tectonic settings with high heat flow and
corresponding steep geothermal gradients (e.g., continental arc environments such as the western Cordillera of Peru).
f
Rock densities calculated using mineral modal analysis from Beard and Lofgren [1991], Rushmer [1991], and Rapp and Watson [1995], and density data
for individual minerals [Deer et al., 1992]. The values presented by Turcotte and Schubert [1982] lie within the range of calculated values.
g
Rock densities calculated using mineral modal analysis from Beard and Lofgren [1991], Rushmer [1991], and Rapp and Watson [1995]. Densities for
granitic melt compositions calculated using the model of Lange and Carmichael [1987], and compositional data from Beard and Lofgren [1991], Rushmer
[1991], Rapp and Watson [1995], and analysis of granitic dykes from the Rosses pluton, Co. Donegal, Eire. The experimentally determined values
presented by Murase and McBirney [1973] for rhyolite melt lie within the range of calculated values.
h
Typical grain size for high grade metamorphic (granulite facies) rocks, many of which are interpreted to be the residues left after the extraction of a
granitic partial melt fraction. Grainsize measurements from Vernon [1968], Spry [1969], Yardley et al. [1990], and sections of granulite facies rocks from the
Ivrea Zone, northwest Italy.
i
In the original derivation of the Blake-Kozeny-Carman permeability equation, n = 2 for a network of randomly oriented tubes of constant cross section,
while n = 3 for a bed of packed spheres [Scott and Stevenson, 1986]. The experimental results of Wark and Watson [1998] and Zhang et al. [1994], and the
theoretical data of Cheadle [1989] indicate that a value of n = 3 is the most suitable for both texturally equilibrated and unequilibrated rocks (see also Figure
A1).
j
See Appendix A.
k
Values for hydrated, crystal free granitic (in a broad sense) melts, from Shaw [1963], Scaillet et al. [1998], Baker and Vaillancourt [1995], and Clemens
and Petford [1999].
l
See Appendix B.
m
See Appendix C.
n
The degree of melting at the contact depends upon the solidus, liquidus, and initial temperatures of the crust, the initial temperature of the intruded
magma, and the kinetics of heat transfer in the magma. It cannot exceed 0.5 because the thermodynamic properties of the source rock and magma heat
source are assumed to be identical, and the magma is not superheated. It must be less than the CMF for the model to be valid. See text for details.
o
As the degree of melting at the contact tends toward zero, keff tends toward infinity. The maximum value of keff given in Table 1 is obtained assuming
that j = 0.5.
p
As the degree of melting at the contact tends toward zero, ygeo tends toward zero. The minimum value of ygeo given in Table 1 is obtained assuming that
j = 0.5.
proach of Jackson and Cheadle [1998], and solved numerically. We are interested in four key issues: (1) whether melt
will segregate; (2) the thickness of intruded basalt required
for segregation; (3) the time required for segregation; and
(4) the composition of the segregated melt. Our numerical
experiments were designed to investigate these issues in
terms of the dimensionless governing parameters.
4.1. Melt Segregation
[23] Figure 3 shows a selection of normalized spatial
porosity (melt volume fraction) and degree of melting
(dimensionless temperature) distributions after 30 time units
have elapsed. The most significant feature of the solutions is
the presence of a ‘‘porosity wave,’’ the amplitude of which
is particularly high for the solutions with keff = 1 and keff =
100 (Figures 3b– 3e). The leading wave may be trailed by
several smaller waves, the amplitude of which decreases
with increasing depth (e.g., Figure 3b) [see also Jackson
and Cheadle, 1998].
[24] The development of these porosity waves depends
upon the relative upward transport rates of heat and melt,
which is predominantly governed by the magnitude of keff
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Table 3. Range of Available Values for the Dimensionless Initial
Temperature Gradient ygeo, and the Dimensionless Effective
Diffusivity keff, Obtained Using Equations (26) and (31) and
Values of the Constituent Variables (xs, ms, etc.) From Table 1a
ygeo
keff
106
105
104
103
102
101
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
1
5
10
104
103
102
101
100
10+1
10+2
10+3
10+4
10+5
10+6
10+7
10+8
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
a
The dimensionless initial temperature gradient ygeo and the dimensionless effective diffusivity keff are not independent; for a given value of keff,
the range of available values for ygeo is restricted. ‘‘Yes’’ denotes values
that are available. The range of values of keff and ygeo shown in bold
denotes the range of values for which the formation of a mobile granitic
(in a broad sense) magma is predicted; see text for details.
[Jackson and Cheadle, 1998]. High amplitude porosity
waves are observed for keff in the range 102 to 104.
Porosity waves do not form for large values of keff (>104)
because these correspond to conditions which promote slow
melt transport and rapid heat transport; consequently, the
porosity distribution is similar to that obtained from purely
conductive heating with no melt migration [see Figure 3 of
Jackson and Cheadle, 1998]. As keff decreases, the rate of
melt transport increases relative to the rate of heat transport,
until the melt migrates upward more quickly than the top of
the partial melt zone (the position of the solidus isotherm),
at which the porosity and permeability fall to zero. The
upward migrating melt accumulates below the top of the
zone, the matrix dilates to accommodate it, and a porosity
wave forms (Figure 3). If the compaction rate exceeds the
melting rate in the region immediately below a porosity
wave, then the porosity decreases and a local restriction to
melt migration forms below which melt accumulates, leading to the formation of a trailing wave. High-amplitude
porosity waves do not form for small values of keff (<102)
because melt transport occurs so rapidly compared to heat
transport that freezing of the melt during ascent keeps the
porosity too low for a wave to form [Jackson and Cheadle,
1998].
[25] The porosity distribution also depends to a lesser
extent upon the value of the dimensionless initial temperature gradient ygeo; for a given value of keff the effect of
increasing the temperature gradient is to increase the amplitude of the leading porosity wave, decrease the spatial
extent of the partial melt zone, and move the position of the
porosity maximum upward relative to the top of the melt
zone (compare Figures 3c and 3e, and Figures 3b and 3d).
The extent of the melt zone decreases because the thermal
perturbation cannot propagate as far from the sill; this
decreases the distance over which the melt must migrate
before it reaches the top of the melt zone, thus enhancing
the accumulation of melt.
[26] If the amplitude of the leading porosity wave continuously increases with time, then the local melt volume
fraction will eventually exceed the CMF, in which case the
contiguity of the solid grains breaks down and a mobile
magma forms [Jackson and Cheadle, 1998]. Numerical
experiments to track the increase in porosity with time
suggest that if sufficient enthalpy is available, and if the
value of the CMF does not greatly exceed the initial degree
of melting at the contact (j 0.5), mobile magma formation is possible for keff approximately in the range 102 <
keff < 104 and ygeo approximately in the range 104 < ygeo
< 1 (Figure 4). This corresponds to the majority of available
values of keff and ygeo (Table 3). If the value of the CMF
greatly exceeds the initial degree of melting at the contact, it
takes longer for the amplitude of the porosity wave to reach
the CMF and the range of keff for which segregation is
predicted decreases slightly to 1 < keff < 104. The key
finding of this section is that, if sufficient enthalpy is made
available, melt segregation is predicted for the majority of
values of the key dimensionless parameter governing melting and melt migration: the effective thermal diffusivity keff.
We investigate the enthalpy required for segregation in the
following section.
4.2. Thickness of Basalt Heat Source (Sill) Required
for Melt Segregation
[27] If the contact temperature begins to fall before a
porosity wave has developed, the source rock may cool too
quickly for magma mobilization to occur. The timescale
over which the contact temperature begins to fall is primarily governed by the thickness of the basalt sill which
provides the enthalpy for melting, and the values of keff
and ygeo. For given values of keff and ygeo, we can estimate
the minimum dimensionless sill thickness required for
granitic melt mobilization. Figures 5a and 5c show the
minimum sill half thickness (zmin) as a function of keff,
for ygeo = 104 and 102, respectively, and for CMF 0.5.
In both cases, the minimum thickness increases with increasing keff. This is partly because the time required for
segregation varies as a function of keff (see following
section), and partly because the position of the porosity
maximum, and hence of incipient magma formation,
moves closer to the sill/rock contact [Jackson and Cheadle,
1998; see also Figure 3]. Figures 5b and 5d show the
corresponding dimensionless temperature at the sill/rock
contact (qc) and the center of the sill (qs), at the time of
granitic melt mobilization. In both cases, qc and qs increase
with increasing keff. The generally low dimensionless temperatures at the time of magma mobilization indicate that
mobilization can occur while the source region as a whole is
cooling.
4.3. Time Required for Melt Segregation
[28] We have found that the dimensionless thickness of
basalt heat source required for segregation increases as a
function of keff. Another important quantity is the time
required for segregation. We can estimate this by recording
the time required for the porosity at any point to reach the
CMF. Figure 6 shows the dimensionless time required for
segregation obtained using the minimum sill thickness
derived in the previous section, as a function of keff, with
CMF 0.5. Also shown is the segregation time obtained
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
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Figure 3. Normalized dimensionless spatial porosity (f) and temperature/degree of melting (q)
distributions in the sill (z < 0) and partially molten rock (z > 0), after 30 time units, with: (a) keff = 103,
ygeo = 1; (b) keff = 1, ygeo = 102; (c) keff = 1, ygeo = 104; (d) keff = 100, ygeo = 102; and (e) keff = 100,
ygeo = 104. In all cases, j = 0.5, Ste = 0.625, and c = 2.67. The half thickness of the sill is given by
zsill = 2(kefft)1/2; note that the initial dimensionless temperature of the sill is 2, and the initial
dimensionless contact temperature between the underlying sill and the overlying rock is 1. The
temperature profile is shown only in the upper half of the sill. Note that both ordinate and abscissa axis
scales differ between plots.
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Figure 4. Maximum normalized dimensionless porosity as a function of time, for (a) keff = 102, ygeo =
102 and 1; (b) keff = 1, ygeo = 102 and 1; and (c) keff = 102, ygeo = 102 and 104. In all cases, j = 0.5;
Ste = 0.625, and c = 2.67.
assuming a constant contact temperature; this is equivalent
to assuming an infinite sill thickness. Segregation times are
generally shortest for keff in the range 1 < keff < 104 for
ygeo = 104, and 1 < keff < 102 for ygeo = 102, and then
increase with both increasing and decreasing values of keff.
There is clearly a range of optimum values of keff for which
the dimensionless segregation time is minimized. Segregation times are similar regardless of whether the minimum or
infinite sill thickness is used, which suggests that, so long as
sufficient enthalpy is made available for segregation to
occur, the dynamics of melting and melt migration in the
source rock are not significantly affected by the thickness of
basaltic magma which is ultimately emplaced.
4.4. Predicted Melt Compositions
[29] We now investigate the composition of the segregated melt. If the melt and matrix remain in local thermodynamic equilibrium during melting and melt segregation, the
equilibrium composition of each may be predicted in terms
of the local temperature and bulk composition. Figure 7
shows a plot of dimensionless temperature at the point of
incipient magma formation for the same system shown in
Figure 6. This temperature controls the initial composition
of the segregated melt. In general, the temperature increases
with increasing keff because the position of the porosity
maximum, and hence of incipient magma formation, moves
closer to the sill/rock contact [Jackson and Cheadle, 1998;
see also Figure 3]. The anomalous decrease in temperature
with increasing keff observed in Figure 7b, for keff > 102,
occurs because the leading porosity wave bifurcates, and
with increasing keff the position of the porosity wave moves
closer to the top of the source region.
[30] If a suitable phase diagram was available, melt and
matrix compositions could be predicted directly. Indeed, if
the number of components were not too large, an alternative
method of handling mass conservation would be to track the
movement of components rather than phases, and to use the
phase diagram to determine the equilibrium porosity (melt
volume fraction) corresponding to the predicted local bulk
and phase compositions. Unfortunately, for the complex
rock types we are attempting to model, the number of
components is large and a suitable phase diagram is not
available. Consequently, we will estimate melt compositions
using the results of equilibrium melting experiments. The
compositions we predict will not be exactly correct as the
local bulk composition has changed, but the approach is
reasonable for major elements due to their close to eutectic
behavior.
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
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Figure 5. Dimensionless minimum sill half thickness (zmin) required for magma mobilization in the
overlying rock, and dimensionless sill temperature at the time of mobilization (contact qc and center qs),
against keff, for the case CMF j 0.5, with (a and b) ygeo = 104; (c and d) ygeo = 102. In all cases,
Ste = 0.625 and c = 2.67. The initial dimensionless temperature of the sill is 2, and the initial
dimensionless contact temperature between the underlying sill and the overlying rock is 1. Curves denote
best fit polynomials.
[31] Figure 8 shows empirically derived major element
melt compositions, as a function of temperature and degree
of melting, obtained during dehydration partial melting of
meta-basaltic rocks at various pressures [Beard and Lofgren, 1991; Rapp and Watson, 1995]. Regardless of the
source rock and pressure, the composition of the melt is
granitic (in a broad sense) for degrees of melting up to
0.4. For example, melting at high pressures, at which
garnet is a residual phase, yields small melt fractions which
are initially trondjhemitic in character, and change to
tonalitic-granodioritic as the degree of melting increases
[Rapp and Watson, 1995]. Melt compositions may be
estimated using Figure 8 in conjunction with the appropriate
dimensionless temperature curve. For example, consider
the melt composition in the leading porosity wave shown
in Figure 3c. The porosity wave is located at a dimensionless height of 16, and the melt at this location has a
dimensionless temperature of 0.08, which for j = 0.5
corresponds to a dimensional degree of melting of 0.04
(equation 15). If the source rock were amphibolite melting
at lower crustal depths and pressures, then the melt composition would be trondhjemitic (Figure 8). In like fashion,
it may be deduced from Figure 7b that the composition of
the segregated melt in a system with keff = 10 and ygeo =
104 corresponds to a dimensional degree of melting of
0.15 (a dimensionless temperature of 0.3) and would be
tonalitic-granodioritic.
[32] As the melt migrates upward through the partial
melt zone, its composition changes because it thermodynamically equilibrates with partially molten rock at progressively lower temperatures [Jackson and Cheadle,
1998]. For a given source rock composition, the composition of the melt therefore depends upon the position of the
melt relative to the position of the solidus isotherm (the top
of the partial melt zone). The melt in a porosity wave may
occupy a large volume fraction of the source rock, yet its
major element composition resemble only a small degree
of melting of the rock. This is important, as the degree of
melting required to yield granitic partial melt fractions is
typically small. Our results suggest that basaltic and meta-
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Figure 6. Dimensionless segregation time (tseg) against keff, obtained using the minimum sill thickness
presented in Figure 5, and an ‘‘infinite’’ sill thickness (constant contact temperature), for the case CMF j 0.5, with (a) ygeo = 104; (b) ygeo = 102. In all cases, Ste = 0.625 and c = 2.67. Curves denote best
fit polynomials. The segregation time is obtained by recording the time required for the porosity at any
point to reach the CMF.
basaltic protoliths will yield segregated melt compositions
ranging from granitic/trondjhemitic to tonalitic-granodioritic. Jackson and Cheadle [1998] argue that the assumption of
local thermodynamic equilibrium is justified in the lower
crust, but it remains one of the more stringent limits on the
validity of the model predictions.
5. Discussion
5.1. Melting and Buoyancy-Driven Compaction
in the Continental Crust
[33] We have investigated melting and buoyancy-driven
melt migration and compaction in the continental crust in
terms of six dimensionless parameters. We have found that
the spatial distribution of porosity (melt volume fraction)
within a partial melting zone depends upon the relative rates
of upward transport of heat and buoyant melt, which is
predominantly governed by the magnitude of a dimensionless parameter which we term the effective thermal diffusivity keff. This is the key parameter governing melt
segregation. So long as sufficient enthalpy is made available, then in melt zones characterized by values of keff
approximately in the range 102 < keff < 104, the rate of
melt transport is rapid compared to the rate of heat transport,
so the melt migrates upward faster than the position of the
solidus isotherm, which acts as a ‘‘lid’’ on top of the zone.
Consequently, the melt accumulates and a porosity wave
forms, the amplitude of which increases with time until the
Figure 7. Dimensionless segregation temperature (qseg) against keff, obtained using the minimum sill
thickness presented in Figure 5, and an ‘‘infinite’’ sill thickness (constant contact temperature), for the
case CMF j 0.5, with (a) ygeo = 104; (b) ygeo = 102. In all cases, Ste = 0.625 and c = 2.67.
Curves denote best fit polynomials. The segregation temperature is defined as the temperature at the
position of incipient magma formation.
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
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Figure 8. Melt composition in terms of component oxide mass fractions, as a function of the degree of
melting (vm) and temperature (T), during partial melting of amphibolite: (a) data from Beard and Lofgren
[1991], obtained at 1 kbar; (b) data from Rapp and Watson [1995], obtained at 16 kbar. In all cases, plain
lines plot on the left-hand ordinate axis; dashed lines plot on the right-hand ordinate axis.
local porosity exceeds the CMF. This yields a mobile
magma which may migrate away from the partial melt
zone. Because the melt has thermodynamically equilibrated
with cool solid matrix, its major element composition
resembles a small degree of melting of the source rock
despite occupying a large volume fraction of the rock. In
partially molten basaltic and meta-basaltic source rocks the
predicted composition of the melt fraction of the mobile
magma is granitic (in a broad sense).
[34] This progressive evolution of the melt composition
to reflect smaller degrees of melting of the source rock is
similar in some respects to the process of zone refining, in
which a melt body moving through rock by melting at the
leading edge and freezing at the trailing edge becomes
enriched in the more incompatible elements [Harris,
1957]. As yet, we have investigated the evolution only of
major elements; however, it is well known that melt-matrix
interaction during melt migration can give rise to characteristic trace element signatures [e.g., Richter, 1986; Navon
and Stolper, 1987; Kenyon, 1990]. Work is underway to
predict these signatures and use them as a diagnostic test.
5.2. Length- and Time-Scales of Melt Segregation
[35] We have predicted that melt segregation due to
buoyancy-driven compaction can occur in a partial melting
zone in the crust. However, predicting the length- and timescales over which segregation will occur is complicated
because of the large number of dimensional variables which
dictate the behavior of the system. It is also complicated by
the wide range of uncertainty in the values of some of these
variables, particularly the matrix bulk and shear viscosities
(xs and ms), the melt viscosity (mm), the grain size (a), and
the permeability constant (b) (Table 2). The dimensional
results are difficult to generalize and we have not attempted
to exhaustively investigate them in terms of the dimensional
governing variables. However, we have estimated the timescale of segregation and the lengthscale of intruded basaltic
magma required as a heat source. We choose a value of keff
and an associated value of ygeo, and calculate the dimensionless time and lengthscales for segregation (e.g., Figures
5 and 6). We then permute the values of the dimensional
variables which yield this value of keff, calculating the
dimensional time and lengthscales for each combination.
Repeating this for the full range of keff for which segregation is predicted (approximately 102 keff < 104) yields
segregation times ranging from 4000 years to 10 Myr.
These are rapid compared to previous estimates; for example, Wickham [1987] predicts that segregation due to buoyancy-driven compaction occurs at the kilometer scale over
timescales >108 years.
[36] The corresponding thickness of intruded basaltic
magma required for segregation ranges from 40 m to
3 km. This requires that the magma is emplaced over time
as a series of sills, concurrent with melt segregation in the
overlying source rock. Short segregation times indicate that
batches of granitic magma may be generated by only a few
underplating events, so repeated underplating will yield
numerous individual batches. Longer segregation times
indicate that repeated underplating is required to yield a
single, large batch of granitic magma. A pulsed magma
supply is consistent with the internal contacts and compositional zonation observed in many granitic bodies [Pitcher,
1979, 1993].
[37] We illustrate our findings by considering a few
specific dimensional results. Consider a partial melt zone
with matrix bulk and shear viscosities of xs ms 1015 Pa s,
melt viscosity of mm 104 Pa s, grain size of a 2.5 mm,
permeability constant of b 1/500, density contrast of (rs rm) 600 kg m3, initial melt fraction at the contact of j 0.5, and values of k, cp, L, (Tliq Tsol), Tgeo, and r given in
Table 2 (in the column labeled "Values Used in Figure 9").
Substituting these values into equations (14), (16), (17),
(19), and (26) yields a value of keff 36. The resulting
porosity distribution is shown in Figure 9a at the time of
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JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Figure 9. Porosity (f) and temperature (T) against vertical distance (z) at the time of incipient magma
formation, for (a) ms xs 1015 Pa s (keff = 35.6, ygeo = 1.67 103); (b) ms xs 1017 Pa s (keff =
3.56, ygeo = 1.67 102); (c) ms xs 1019 Pa s (keff = 0.356, ygeo = 0.167). See Table 2 for the values
of the other governing dimensional variables. Porosity (f) is plotted against the lower abscissa axis;
temperature (T) is plotted against the upper abscissa axis. The curve which denotes the temperature (T)
also denotes the degree of melting (vm) if read from the lower abscissa axis. The temperature profile in the
upper half of the sill only is shown; in both cases CMF j 0.5. In all cases, Ste = 0.625 and c = 2.67.
The solidus and liquidus temperatures of both source rock and magma heat source are 1160 and 1560 K,
respectively.
incipient magma formation (with CMF 0.5). The time
required for segregation is 20,000 years, the thickness of
the partial melt zone is 900 m and the sill thickness
required for segregation is 500 m. The melt in the leading
porosity wave has a composition corresponding to a degree
of melting of 0.14, which for the amphibolite shown in
Figure 8b, melting at 16 kbar, would be trondjhemitic.
Increasing the matrix bulk and shear viscosities yields
smaller values of keff, longer segregation times, and a
thicker partial melt zone at the time of incipient magma
formation (Figures 9b and 9c). Note how much cooler the
sill is in Figure 9c when compared to Figure 9a; the reason
the minimum sill thickness required for segregation
increases so slowly compared to the thickness of the partial
melt zone is that the melt segregates much closer to the top
of the source region for smaller values of keff. Consequently,
the sill can cool further before inhibiting melt segregation in
the overlying rock.
[38] Previous estimates of the time required for buoyancy-driven compaction and melt segregation in the crust
have been based on applications of the characteristic compaction time of McKenzie [1985, 1987].
th ¼
hmm f
k ð1 fÞ2 ðrs rm Þg
ð33Þ
in which the porosity f is equated with the degree of melting
of the source rock, and h denotes the source region thickness
[Wickham, 1987; Wolf and Wyllie, 1991, 1994; Petford,
1995]. Predicted segregation times are generally very large.
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
For example, consider the time required for melt segregation in the partial melt zone shown in Figure 9b. The model
presented in this paper yields a time of 170,000 years to
segregate a granitic melt fraction with a composition
corresponding to a small degree of melting of 0.02. In
contrast, setting f = 0.02, assuming a source region
thickness of h 1.8 km, and substituting for the values of
mm, a, b, and (rs rm) in equation (33), yields a time of
20 Myr.
5.3. Residues of Partial Melting
in the Lower Crust
[39] Many exposed lower crustal granulite terrains exhibit
geochemical and mineralogical characteristics which indicate that they may be residues left after the in situ extraction
of a granitic partial melt fraction, although the evidence is
rarely unambiguous [e.g., Fyfe, 1973; Clifford et al., 1981;
Clemens, 1989, 1990; Pin, 1990; Vielzeuf et al., 1990]. The
model presented in this paper predicts that the solid matrix
of the partially molten rock will not be disrupted during
segregation except in the localized region of magma mobilization, and will remain in the source region as a ‘‘restitic’’
residue. The mineralogy of the residue following partial
melting of basaltic and meta-basaltic protoliths corresponds
to that of many granulite terrains (pyroxene ± plagioclase,
quartz, olivine, garnet, magnetite, ilmenite) [Beard and
Lofgren, 1991; Rushmer, 1991; Rapp and Watson, 1995].
This residue may become gravitationally unstable and
undergo delamination, detaching from the crust and sinking
into the underlying mantle [e.g., Bird, 1979; Houseman et
al., 1981; Kay and Kay, 1991, 1993; Nelson, 1991]. Basaltic
underplating, segregation of a low density partial melt, and
delamination of the high density residue, are important
processes in the evolution of the continental crust and
crust-mantle recycling [e.g., Arndt and Goldstein, 1989;
Kay and Kay, 1991, 1993; Nelson, 1991].
[40] Many granitic rocks appear to contain restitic material which has been transported with the magma from the
source region to the emplacement level [e.g., White and
Chappell, 1977, 1990; Chappell, 1984; Chappell et al.,
1987; Pitcher, 1993]. The model predicts that the mobile
magma will initially contain a high proportion of restite, but
the amount which will become entrained within the magma
and subsequently leave the source region is poorly constrained, because it depends upon factors such as the rate at
which restite settles out from the magma, the time the
magma spends in the source region, and the ascent mechanism. If the restite settles quickly and the magma has a
long residency in the source region prior to ascent, then the
restite fraction could be very small. Alternatively, if the
restite settles slowly and the magma ascends soon after
segregating, then the restite fraction could be high. In the
following section, we argue that flow through dykes is the
most likely ascent mechanism; further work is required to
clarify whether dykes are likely to transport large volumes
of restite from the source region to the emplacement level.
5.4. Implications for Magma Ascent
[41] The magma which forms in the source region must
subsequently ascend through the crust to the emplacement
level. However, the ascent mechanism is still a source of
controversy; the orthodox view is that granitic magmas
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3 - 15
ascend through the crust as diapirs [e.g., Ramberg, 1970;
White and Chappell, 1977, 1990; Bateman, 1984; Chappell,
1984, Weinberg and Podladchikov, 1994; Weinburg, 1996],
yet magma ascent via diapirism is a slow process limited by
the high viscosity of the surrounding crust, and theoretical
studies indicate that diapirs cool and crystallize before
reaching upper crustal levels [e.g., Marsh, 1982; Mahon et
al., 1988]. An alternative is that granitic magmas ascend
through dykes, faults, or fractures in a manner analogous to
that of basaltic magmas [Clemens and Mawer, 1992; Petford
et al., 1993, 1994; Petford, 1996; Clemens et al., 1997;
Clemens, 1998]. In contrast to diapirism, magma ascent via
dykes is a rapid process limited only by the viscosity of the
magma [Petford et al., 1993, 1994; Petford, 1996].
[42] The ascent mechanism may ultimately be governed
by the rheology and geometry of the magma body which
forms in the source region. The results presented in this
paper are 1-D and cannot be used to quantitatively predict
the 3-D geometry of the magma body. Jackson and Cheadle
[1998] argue that in a homogeneous, isotropic rock heated
constantly from below, the 1-D results may be robust in 3-D
at early times, because the rate of upward migration of the
leading porosity wave is limited by the rate of upward
migration of the solidus isotherm; the lengthscale for flow is
constrained by the thermal lengthscale rather than the
compaction lengthscale. Connolly and Podladchikov
[1998] found that porosity waves are stabilized in 3-D
when deformation is controlled by thermally activated
creep; in this case, the lengthscale for flow is constrained
by the activation energy for deformation rather than the
compaction lengthscale. They concluded that lateral flow
may occur on greater lengthscales than previously thought.
In our model, the mobilized granitic magma would initially
form a ‘‘sill’’ like layer which may act as a reservoir from
which magma is extracted by dykes/fractures (Figure 10).
The results of Rubin [1993, 1995] indicate that ‘‘isolated,’’
granitic magma filled dykes propagating out of a partial
melt zone are rapidly halted by freezing of the magma at the
tip; however, the presence of a magma reservoir facilitates
‘‘repeated’’ dyke intrusion which warms the surrounding
rock and promotes the propagation of subsequent dykes.
Repeated, localized dyke intrusion would lead to the formation of a localized ascent zone, to which granitic magma
could migrate laterally through the ‘‘sill.’’ As the source
region will be laterally extensive, this migration of granitic
magma is likely to be significant. Such an ascent zone is
similar to that proposed by Ryan et al. [1981] and Ryan
[1988] for the ascent of basaltic magma beneath Kilauea
Volcano, Hawaii; moreover, granitic plutons are often
associated with granitic dyke swarms [e.g., Pitcher, 1979,
1993; Scaillet et al., 1995], which may represent the surface
manifestation of the ascent zone. At later times the magma
layer is likely to become unstable, and form a more complex
structure in 3-D. This may provide a further mechanism for
channeling the melt into a localized ascent zone.
6. Conclusions
[43] We have presented a new quantitative model of
granitic (in a broad sense) melt generation and segregation
within the continental crust, assuming that melt generation
is caused by the intrusion of hot mantle-derived basalt, and
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3 - 16
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
Figure 10. Schematic of a granite source region in the lower crust, which is produced by heating from
below following basaltic underplating. The thickness of the partial melt zone increases with time as heat
migrates from the underlying basalt magma into the overlying source rock. The porosity distribution in
the partial melt zone is governed by the relative upward transport rates of heat and melt; if melt migrates
upward more quickly than the top of the source region (defined by the position of the solidus isotherm)
then it accumulates at the top of the source region and a porosity wave forms. Eventually, this local
porosity exceeds the Critical Melt Fraction yielding a mobile magma. As the melt migrates upward its
major element composition evolves to resemble a smaller degree of melting of the source rock (granitic in
a broad sense for basaltic and meta-basaltic (amphibolite) protoliths). The accumulated granitic magma
may be tapped by dykes and ascend to higher crustal levels. As the source region will be laterally
extensive this magma migration to a localized ascent zone is likely to be significant.
that segregation occurs by buoyancy-driven flow along
grain edges coupled with compaction of the partially molten
source rock (Figure 10). We applied the model to tectonic
settings where high heat fluxes and/or thickened crust
promote partial melting, with basaltic underplating acting
as both a heat source for partial melting and as a material
source for granitic melt.
[44] Our results suggest that significant volumes of granitic partial melt may segregate from a basaltic protolith
within geologically reasonable timescales, and that underplating can sustain melting for the length of time required
for segregation and mobilization to occur. These findings
differ from those of previous studies, which have suggested
that compaction operates too slowly in the crust to yield
large volumes of segregated granitic melt. However, the
results of the model are sensitive to the values of the
variables which govern the dynamics of melt segregation;
in particular, the bulk and shear viscosities of the partially
molten protolith, the melt viscosity, and the grain size.
These are poorly constrained, and the limited experimental
and theoretical data has been used to estimate suitable
values (Table 2). The model predicts that melt segregation
and mobilization will occur in a basalt/amphibolite protolith
over 4000 years to 10 Myr, that underplate thicknesses
of 40 m to 3 km are required to provide sufficient heat,
and that the composition of the melt which becomes
mobilized ranges from granitic/trondjhemitic to tonaliticgranodioritic.
[45] These results suggest that partial melting of basaltic
underplate in continental arc environments may yield large
volumes of Na-rich granitic magma within reasonable timescales. This magma can ascend through the crust, perhaps
through dykes, faults, or fractures, and be emplaced to form
the Na-rich granitic batholiths found in the western Americas, Antartica, and New Zealand. The composition of the
predicted residue left in the lower crust corresponds to that
of many exposed granulite terrains; this residue may become gravitationally unstable and undergo delamination,
detaching from the crust and sinking into the underlying
mantle. These processes contribute to the evolution of the
continental crust and crust-mantle recycling.
[46] The model results also suggest a possible mechanism
for generating the large volumes of TTG rock found within
Archean crust [e.g., Jahn et al., 1981; Martin et al., 1983].
The origin of these rocks is controversial, with some
workers suggesting that they are formed by partial melting
of basalt within subducting oceanic crust and others suggesting that they are formed by partial melting of underplated basalt [e.g., Martin, 1999 and references therein]. As
yet no physical model has been presented to describe the
generation and segregation of Archean slab melts. Here we
suggest a mechanism for the formation of Archean TTG
melts through partial melting of underplated basalt.
[47] Despite its relatively simple nature, this model represents a significant advance on the qualitative and semiquantitative models of granitic melt generation, segregation,
and mobilization which have previously been developed,
and lays the foundation for more sophisticated models.
Appendix A: Characterization of Permeability
[48] The simple permeability-porosity equation used in
this paper (equation 7) is based upon the semiempirical
Blake-Kozeny-Carman equation, in which the constant b is
chosen to fit experimentally derived permeability-porosity
data for a given material [Bear, 1972; Dullien, 1979]. This
equation is a reasonable fit for permeability measurements
on both texturally equilibrated and unequilibrated systems,
so long as the percolation threshold fc is accounted for
[McKenzie, 1984; Cheadle, 1989; Wark and Watson, 1998]
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
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3 - 17
Figure A1. (a) Permeability-porosity relations for texturally equilibrated and unequilibrated aggregates
with a grain size (diameter) of 1 mm. The upper curve is given by the permeability-porosity equation (7)
with b = 1/50 and no percolation threshold, and denotes the best fit to the theoretical data of Cheadle
[1989], for a texturally equilibrated, monominerallic, fluid saturated mush of isotropic grains with a
dihedral angle of 50 and a grain size (diameter) of 1 mm. The percolation threshold fc in this system is
very close to zero [Cheadle, 1989]. The lower curves are given by equation (7) with b = 1/2500 and b =
1/100, and represent the permeability-porosity behavior of a texturally unequilibrated aggregate with a
percolation threshold of fc = 0.04. (b) (Inset) Experimentally determined permeability-porosity relations
for a texturally unequilibrated, hot pressed calcite aggregate with a grain size (diameter) of 1 – 30 mm
[Zhang et al., 1994], which was used to deduce the permeability-porosity relations for the texturally
unequilibrated aggregate shown in Figure A1a. The right-hand curve is given by equation (7) with b =
1/2500 and no percolation threshold, and denotes the best fit to the experimental data of Zhang et al.
[1994] for porosity in the range f > 0.06. The left-hand curve is given by equation (7) with the porosity
f replaced by the effective (connected) porosity (f fc), b = 1/100, and fc = 0.04, and denotes the
best fit to the experimental data of Zhang et al. [1994] for porosity in the range 0.04 < f < 0.06. When
fitting the curves to the experimental data, the mean grain size of the inequigranular calcite aggregate
was assumed to be 5 mm, based on the argument that the permeability of inequigranular materials is
governed by the small grains, because they reside in channels between large grains [Wark and Watson,
1998]. Axis labels are identical to those of plot in Figure A1a.
(Figure A1). Estimates of the percolation threshold vary:
Zhang et al. [1994] estimated a threshold of 0.04 in a
texturally unequilibrated calcite aggregate, Wark and Watson [1998] estimated thresholds ranging from 0.005 to
0.027 in texturally equilibrated quartz and calcite aggregates, while Wolf and Wyllie [1991] estimated a threshold
of <0.02 in a texturally unequilibrated, partially molten
hydrated basalt. Numerical experiments reveal that the
effect of a percolation threshold of 0.04 is negligible
on solutions of the governing equations for which magma
mobilization is observed, because the porosity at the
leading edge of the leading porosity wave does not fall
to the percolation threshold until late times, by which time
the wave is well developed. Based on this, all solutions
presented in this paper are obtained using the permeabilityporosity equation (7) over the entire porosity range. The
maximum and minimum values of b given in Table 2 are
based upon the permeability-porosity relations shown in
Figure A1, and the range of values include that proposed
by McKenzie [1984] of b = 1/1000, based upon permeability-porosity data obtained experimentally by Maaloe
and Scheie [1982].
Appendix B: Characterization of Matrix Shear
and Bulk Viscosities
[49] The rheology of a partially molten rock is primarily
governed by its mineralogy and grain size, and the strain
rate regime [Ranalli, 1987]. Large grain sizes and/or high
strain rates favor deformation by dislocation creep, while
small grain sizes and/or low strain rates favor deformation
by diffusion creep [Dell’Angelo et al., 1987; Dell’Angelo
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3 - 18
JACKSON ET AL.: GRANITIC MELT GENERATION AND SEGREGATION
and Tullis, 1988]. We assume that deformation in the
partially molten lower crust occurs by melt-enhanced diffusion creep, although the available data is very limited. The
experimental work of Kohlstedt and Chopra [1994] indicates that melt-enhanced diffusion creep governs the rheology of olivine aggregates containing 9% basaltic melt, at
temperatures of 1573 K, for strain rates of 1010 s1 and
a grain size (diameter) of 1 mm. Given that maximum strain
rates predicted by the model are 1013 – 1020 s1, up to 7
orders of magnitude less than the strain rate used by
Kohlstedt and Chopra [1994], the assumption of meltenhanced diffusion creep appears reasonable, although an
olivine-basalt system is clearly not an ideal analogue for
partially molten basalt/amphibolite.
[50] Melt-enhanced diffusion creep is a form of grain
boundary (Coble) creep, in which the microscale diffusion
rate of components along solid-solid grain boundaries
governs the macroscale creep rate [Pharr and Ashby,
1983; Cooper and Kohlstedt, 1984; Ranalli, 1987]. The
rheology of an aggregate undergoing steady state deformation by diffusion creep is Newtonian [Cooper and Kohlstedt, 1984; Ranalli, 1987]. For the simplest case of a single
component, equigranular aggregate deforming by grain
boundary creep, the shear viscosity may be expressed as
[Ranalli, 1987]
ms ¼
kb Ta3
;
141
Dg
ðB1Þ
where kb is Boltzmann’s constant (= 1.381 1023 J K1),
is the atomic volume of the material, and Dg is the
effective grain boundary diffusivity. If the aggregate
contains a small fraction (<0.1) of spherical pores, the bulk
viscosity may be expressed as [Arzt et al., 1983]
h i
3 f2=3 1 þ f2=3 þ 1 lnð1=fÞ kb Ta3
xs ¼ ð1 fÞ
:
54
Dg
1 f2=3
ðB2Þ
[51] Experimentally measured values of Dg are >1021
m2 s1 for calcium in anorthite at a temperature of 1273 K
and atmospheric pressure [Farver and Yund, 1995], but
have not been measured for other species or for the other
major minerals found in basalt/amphibolite. Using the value
of for anorthite [6.69 1028 m3; Lide and Frederiske,
1998], assuming a temperature of 1273 K, and assuming Dg
1021, equation (B1) yields a maximum shear viscosity
of 1018 Pa s for a grain size (radius) of 2.5 mm, and a
minimum shear viscosity of 1015 Pa s for a grain size
(radius) of 0.25 mm. Equation (B2) yields a maximum bulk
viscosity of 1019 Pa s for an aggregate with a grain size
(radius) of 2.5 mm and a porosity of f = 0.01, and a
minimum bulk viscosity of 1015 Pa s for an aggregate with
a grain size (radius) of 0.25 mm and a porosity of f = 0.1.
[52] It is not clear how these calculated viscosities relate
to the viscosities of partially molten basalt/amphibolite as
the diffusion controlled creep of polymineralic aggregates is
poorly understood. Ranalli [1987] states that when more
than one diffusing component is present, the slowest moving controls creep, yet Wheeler [1992] argues that components may mix and interact during diffusion, leading to
complex behavior which may significantly enhance diffusional creep rates. Neither of the equations for the shear and
bulk viscosity includes the effect of melt in the pore spaces,
which should enhance the creep rate [Cooper and Kohlstedt,
1984; Dell’Angelo et al., 1987], although during compaction the melt must be expelled from the pores and the effect
of this on the bulk viscosity is unknown [Dingwell et al.,
1993]. Moreover, the spherical pores assumed in the derivation of equation (B2) are not likely to be a good
approximation of the interconnected pores of a partially
molten rock [McKenzie, 1984]. However, in the absence of
experimental data for either the bulk and shear viscosities of
partially molten basaltic rock, our calculated range of shear
and bulk viscosities will be used to provide order of
magnitude estimates. The values of the shear and bulk
viscosities given in Table 2 are based upon these estimates.
Appendix C: Critical Melt Fraction
[53] The CMF is important because it dictates the porosity at which a partially molten rock disaggregates and forms
a mobile magma. If the rock is in textural equilibrium, then
the value of the CMF depends only upon the contiguity of
the matrix [Miller et al., 1988]. Contiguity is a quantitative
measure of the solid grain-grain interconnectivity, and
depends upon the fluid volume fraction, the microscopic
distribution of fluid throughout the mush, and the solid
grain size distribution. The contiguity of a texturally equilibrated, monomineralic, equigranular mush of isotropic
grains as a function of porosity and dihedral angle has been
derived theoretically by Cheadle [1989], and for appropriate
dihedral angles of 30– 60, yields values for the CMF of
0.37– 0.46. These represent ‘‘minimum’’ values, because at
porosities greater than the minimum energy porosity the
melt becomes inhomogeneously distributed and does not all
contribute to reducing the contiguity. For partially molten
rocks which are not in textural equilibrium, experimentally
and theoretically obtained estimates of the CMF range from
0.2 to 0.7 [e.g., Arzi, 1978; van der Molen and Paterson,
1979; Philpotts and Carroll, 1996]. The values of the CMF
given in Table 2 are based upon these estimates for both
equilibrated and unequilibrated rocks, with the exception of
the lowest estimate of 0.2 ± 0.1 presented by Arzi [1978],
which was obtained theoretically using simple arguments
based upon the Einstein-Roscoe equation for rigid particles
suspended in a viscous, Newtonian melt. This is a poor
approximation of a partially molten rock. Also, the highest
estimate of 0.7 obtained experimentally by Philpotts and
Carroll [1996] for a partially molten tholeiitic basalt has
been reduced to 0.65. They performed their experiments at
compressive stresses of the order of kilopascals, yet compressive stresses in the lower crust are of the order of
gigapascals, and it is unlikely that their sample would have
maintained its strength at such high melt fractions for
stresses of this magnitude.
[54] Acknowledgments. Peter Keleman, Mike Bickle, and Hikaru
Iwamori are thanked for their careful and constructive reviews. M. D. J.
was supported by NERC grant GT4/93/191/G.
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M. P. Atherton, Department of Earth Sciences, The Jane Herdman
Laboratories, University of Liverpool, Brownlow Street, Liverpool, L69
3BX, UK. ([email protected].)
M. J. Cheadle, Department of Geology and Geophysics, University of
Wyoming, Laramie, WY 82071-3006, USA. ([email protected].)
M. D. Jackson, Department of Earth Science and Engineering, Royal
School of Mines Building, Imperial College London, London SW7 2BP,
U.K. ([email protected].)