JOURNAL OF PETROLOGY
VOLUME 40
NUMBER 2
PAGES 241–254
1999
A Two-stage Thermal Evolution Model of
Magmas in Continental Crust
TAKEHIRO KOYAGUCHI1∗ AND KATSUYA KANEKO2
1
EARTHQUAKE RESEARCH INSTITUTE, UNIVERSITY OF TOKYO, 113-0032 TOKYO, JAPAN
2
CENTRAL RESEARCH INSTITUTE OF ELECTRIC POWER INDUSTRY, 1646 ABIKO, ABIKO-SHI, 270-1194 CHIBA, JAPAN
RECEIVED JULY 1, 1997; REVISED TYPESCRIPT ACCEPTED JUNE 26, 1998
When a basaltic magma is emplaced in a continental crust, a
silicic magma is generated by melting of the crust. The light silicic
magma forms a separate magma layer with little chemical interaction
with the underlying dense basaltic magma layer. Extensive melting
occurs at the boundary between the silicic magma and the crust
while the basalt acts a heat source. The mass and heat transfer at
the boundary between the silicic magma and the crust controls the
thermal evolution of the silicic magma. The thermal evolution of
the silicic magma after the basalt emplacement is divided into two
stages. In the first stage, the temperature in the silicic magma rises
above and then decays back to the melting temperature of the crust
on a short timescale (102 years). The results of fluid dynamics
experiments suggest that the silicic magma generally has a lower
melting temperature than the crust because of fractional crystallization
and mixing of partial melts during the first stage, and that it can
be effectively liquid at the end of the first stage. In the second stage,
the silicic magma cools slowly by heat conduction on a much longer
timescale (105 years). Petrological features of the magma in the
second stage are strongly constrained by petrological features of the
surrounding crust as well as those of the supplied magma itself; its
temperature remains at or just below the melting temperature of the
crust for a long time because of the slow cooling rate; its phenocryst
content reflects the difference in the melt fraction vs temperature
relationships between the magma and the crust. Judging from the
distinct cooling rate between the two stages, erupted magmas are
statistically more likely to reflect the characteristics of magmas in
the second stage.
INTRODUCTION
chamber; continental crust; crustal melting; magma
residence times; fluid dynamics experiments
When magma is emplaced into continental crust, it is
believed to be stored in a ‘magma chamber’ for a certain
period. Extensive studies have revealed how petrological
and geochemical features of magmas are controlled by
processes in magma chambers such as crystallization of
magma, melting of crustal materials, and mixing of
magmas (e.g. McBirney, 1980; Huppert & Turner, 1981;
Sparks et al., 1984; Koyaguchi & Blake, 1991). This paper
discusses how petrologic processes such as assimilation
and fractional crystallization affect the thermal evolution
and the residence times of magmas in magma chambers
in continental crust.
Some exploratory models for the residence time of a
magma chamber in continental crust have been proposed
on the basis of the physics of heat and mass transfer
(Spera, 1979; Huppert & Sparks, 1988a; Marsh, 1989).
To formulate the present problem it is helpful to introduce
the model by Huppert & Sparks (1988a), in which the
physics of crystallization and melting at the chamber roof
as well as the effects of thermal convection in the magma
are taken into account. According to their model, a
basalt emplaced into continental crust results in extensive
melting of the crustal materials at the chamber roof to
form a silicic magma. The temperature in the silicic
magma rises above and then decays back to the fusion
temperature of the crust (in general ‘the effective fusion
temperature’, the definition of which will be given in a
later section) on a short timescale of 101–103 years after
the basalt emplacement. Huppert & Sparks (1988a) also
pointed out that the silicic magma generated by melting
of the crust must be solidified when its temperature
reaches the fusion temperature of the crust, because the
silicic magma should have the same fusion temperature
∗Corresponding author. Telephone: 81-3-3812-2111, ext. 5755. Fax:
81-3-5802-3391. e-mail: [email protected]
 Oxford University Press 1999
KEY WORDS: magma
JOURNAL OF PETROLOGY
VOLUME 40
as the crust. This model, which implies that the residence
times of silicic magmas in the continental crust will be
short, is apparently in conflict with the model of long-lived
magma chambers with cooling times of over 105 years (e.g.
Spera, 1979), even though both models similarly took
into account the effects of heat transfer because of magma
convection and latent heat. These different predictions
of the residence times have led to divergent views of the
correct interpretation of geochemical and geological data
from a silicic magma system (e.g. Halliday et al., 1989;
Halliday, 1990; Mahood, 1990; Sparks et al., 1990). The
purpose of this paper is to propose another view, which,
we believe, reconciles the two hypotheses. We show that
the melting and crystallization processes are rapid as long
as the temperature of the silicic magma is higher than
the fusion temperature of the crust, and that this is the
key hypothesis of the Huppert & Sparks model.
To examine the conditions under which long-lived
magmas develop, we proceed by the following three
steps. First, we review some theories of mass and heat
transfer at crust–magma boundaries and show that the
above paradox about the magma residence times results
from differences in the boundary conditions in their
numerical models; a magma can be long lived if it has
a lower temperature than the fusion temperature of the
surrounding crustal rocks. Second, we show the results
of fluid dynamics experiments and numerical calculations
to investigate possible mechanisms that might cause a
magma to have a lower fusion temperature than the
surrounding crustal rocks. These analyses imply that the
thermal evolution of the crustal magma is generally
divided into two distinct stages: a first stage of rapid
cooling and a second stage of slow cooling. Third, petrological features of the long-lived magmas in the second
stage will be discussed from the viewpoint of the phase
equilibria of crustal materials.
HEAT AND MASS TRANSFER
BETWEEN MAGMA AND CRUST
The factors that govern timescales of melting and solidification of magmas are considered here on the basis
of simple theories of one-dimensional heat and mass
transfer. Melting and solidification processes of multicomponent materials such as crustal rocks and magmas
are complicated because they are composed of both
solid and liquid phases between the solidus and liquidus
temperatures. We start by describing some basic features
of mass and heat transfer in simple systems where magma
or crust has no melting interval but their melting temperatures are allowed to be different. We will discuss the
effect of the melting interval later.
The conditions at the boundary in the simple systems
can be divided into the following four cases according
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to whether there are phase changes at the solid–liquid
boundary and the occurrence of convective heat flux
from the liquid to the interface:
Case 1: moving boundary because of melting or crystallization with convective heat flux from the liquid region
to the interface;
Case 2: moving boundary with conductive heat flux from
the liquid region to the interface;
Case 3: stationary boundary with convective heat flux
from the liquid region to the interface;
Case 4: stationary boundary with conductive heat flux
from the liquid region to the interface.
Case 4 is the simplest problem of heat conduction, and
no further explanation is necessary here. The problems of
heat conduction including boundary migration caused
by phase changes (Case 2) are known as ‘Stefan Problems’,
solutions of which are widely applied to geological situations (e.g. Turcotte & Schubert, 1982). In this case, the
rate of boundary migration has to be calculated from an
energy balance involving the heat flux on both sides of
the boundary and the latent heat. The velocity of the
boundary towards the solid side, ȧ, is given by
ȧ=
Fl−Fs
qL
(1)
where Fl is the heat flux from the liquid to the interface,
Fs is the heat flux from the interface to the solid, q is the
density, and L is the latent heat. The rate of melting or
solidification is limited by the difference in the conductive
heat fluxes between the liquid and the solid media [Fl –
Fs in equation (1)] in this case.
In Case 1, the rate of melting or solidification at the
boundary is also determined by equation (1); however,
because the rate of heat release from the liquid to the
interface, Fl, is determined by the convective heat flux
in the liquid, ȧ can be much greater than in the case of
pure conduction. This situation represents, for example,
mass and heat transfer at the roof of a magma chamber
where the magma temperature is higher than the melting
temperature of the wall rock (Huppert & Sparks, 1988a).
When the liquid undergoes high Rayleigh number thermal convection, the heat flux is expressed (Turner, 1973)
as
Fl=kkl
ag
jlm
1/3
(Tl−Ti)4/3
(2)
where Tl and Ti are the temperatures of the liquid and
interface, respectively, a, kl, jl and m are the coefficients
of thermal expansion, heat conduction, thermal diffusivity
and the kinematic viscosity of the liquid, respectively, g
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THERMAL EVOLUTION OF MAGMAS
is the gravitational acceleration, and k is a constant. In
a quasi-steady state, the rate of melting can be solved
analytically (Huppert & Sparks, 1988b) to yield
ȧ=
Fl
q[c(Ti−Tx)+L]
(3)
where c is the specific heat and Tx is the temperature
of the solid far from the interface. Because the temperature at the interface is fixed at the melting temperature of the wall rocks in Case 1, rapid melting occurs
at the boundary at a rate limited by the convective heat
flux from the liquid region given by equation (2), until
the liquid temperature reaches the melting temperature
of the wall rocks [i.e. Tl – Ti ~ 0 in equation (2)]. On
the other hand, after the liquid temperature reaches the
melting temperature, the system will evolve in the manner
of Case 2 or Case 3, which will be introduced next.
In Case 3 no melting is allowed at the boundary, and
the rate of heat release from the liquid layer to the solid
is determined by the convective heat flux [Fl in equation
(2)] as in Case 1. This situation represents, for example,
the case where the melting temperature of the solid is
higher than the temperature of the liquid. The most
important difference from Case 1 is that the temperature
at the interface, Ti, is not fixed to the melting temperature
in Case 3. If the convective heat flux is initially greater
than the conductive heat flux in the solid, the temperature
at the interface will increase until the convective heat
flux in the liquid becomes equal to the conductive heat
flux in the solid [see the Appendix of Huppert & Sparks
(1988b) and Marsh (1989)]. Therefore, the heat flux
from the liquid to the solid is eventually limited by the
conductive heat flux in the solid.
Figure 1 shows the results of calculations of the cooling
time of a 1 km thick liquid layer for the above three
cases on the basis of the previously published formulae
[for detailed derivations, see Huppert & Sparks (1988a,
1988b) for Case 1, Turcotte & Schubert (1982) for Case
2, and the Appendix of Huppert & Sparks (1988b) for
Case 3]. Two distinct modes of cooling are recognized;
the timescale is as short as 102 years in Case 1, whereas
it is 104–105 years in the other cases. It is particularly
important that, even though convection occurs, the timescale of cooling in Case 3 is much closer to the purely
conductive cases (e.g. Case 2) and much longer than in
the case where convection and melting simultaneously
occur (Case 1). The two different hypotheses on the
residence times of crustal magma chambers (Spera, 1979;
Huppert & Sparks, 1988a) are therefore interpreted to
have resulted from different views on the appropriate
boundary conditions (i.e. Case 3 and Case 1, respectively).
The difference in timescales between Case 1 and the
other cases is explained by the difference in limiting
243
Fig. 1. Variation of liquid temperature with time after a contact of a
hot-liquid–cold-solid pair under various boundary conditions (Cases 1,
2, and 3; see text for their definitions) on the basis of one-dimensional
mass and heat transfer models. The average temperature in the liquid
is given for Case 2. The initial thickness and temperature of the liquid
layer are 1000 m and 1200°C, respectively. The circles, squares and
diamonds represent the results for initial solid temperatures of 150,
500, and 675°C, respectively. The melting temperature is assumed to
be 850°C for the calculations for Cases 1 and 2. Values of other
variables used are listed in Table 1.
processes. The rate of decrease in liquid temperature is
limited by heat loss because of wall rock conduction in
Cases 2, 3 and 4, whereas it is determined by the
increasing proportion of initially cold melts generated by
melting which convectively mix with the hot liquid region
in Case 1. It is reasonable, therefore, to define two modes
of mass and heat transfer in terms of the limiting processes;
hereafter referred to as ‘convection and melting mode
(C&M mode)’ and ‘conductive cooling mode (CC mode)’.
Whether a system evolves in C&M mode depends more
on whether the magma temperature is higher than the
melting temperatures of the wall rocks than on other
detailed physics.
To apply the above argument to the natural multicomponent system, the effects of melting interval should
Table 1: Values used for calculation in Fig. 1
Quantity
Value used
Unit
q
2·6 × 103 (s), 2·4 × 103 (l)
kg/m3
k
2·4 (s and l)
W/m per K
c
1·3 × 103 (s and l)
J/kg per K
m
1·0 × 102
m2/s
L
2·9 × 105
J/kg
a
5·0 × 10–5
per K
g
9·8
m/s2
k
0·1
The letters ‘s’ and ‘l’ represent solid and liquid, respectively.
JOURNAL OF PETROLOGY
VOLUME 40
be taken into account. In the context of mass and heat
transfer, we must specify the temperature at which a
material behaves as a liquid in a fluid mechanical sense
(in this paper we use the term ‘liquid’ in the fluid
mechanical sense and the term ‘melt’ as a phase equilibria
term). The concept of ‘effective fusion’ (e.g. Huppert &
Sparks, 1988a) is useful in describing the physical properties of such solid–liquid mixtures. When the temperature of a crustal material is only slightly higher than
its solidus temperature and the melt fraction is so low
that the crystalline phases form an interconnected framework, the crust can be regarded as partially molten
solid. At temperatures much higher than the solidus
temperature, the melt fraction is greater, and beyond a
critical narrow range of melt fractions the connectedness
of the crystalline framework is destroyed and the rock
behaves as liquid suspending crystals (i.e. magma). Experimental and theoretical studies indicate that a drastic
transition in the mechanical properties of a solid–liquid
mixture occurs at a narrow range of melt fraction between
30 and 50% (e.g. Marsh, 1981; Rutter & Neumann,
1995). We tentatively take a critical value of 50% melt
fraction in this study (Fig. 2). Because the melt fraction
of a crustal material with a given chemical composition
at a fixed pressure is a function of temperature alone,
we can define ‘effective fusion temperature (EFT)’ as a
function of chemical composition. Mode of heat and
mass transfer greatly changes at the EFT. For example,
in the case of melting and solidification at the roof of a
magma chamber, the heat and mass can be efficiently
conveyed by vigorous convection above the EFT, whereas
they migrate only by conduction and diffusion below it
(Huppert & Sparks, 1988a).
The relationships between melt fraction and temperature for the crustal materials must be specified to
determine their EFTs (Fig. 2). Huppert & Sparks (1988a)
assumed a single monotonic relationship between melt
fraction and temperature for each melting and crystallization event in most of their calculations (Fig. 2a). In
fact, this was one of the most critical assumptions which
led to the hypothesis of short residence times of silicic
magmas. A silicic magma generated by melting of a
crust crystallizes very quickly in C&M mode until its
temperature returns to the EFT of the crust. Because of
the above assumption, melt fraction and temperature of
the silicic magma change along a single melt fraction vs
temperature relationship, and so when the temperature
returns to the EFT the silicic magma is inevitably effectively solidified (Fig. 2a). However, the evolution of the
melt fraction vs temperature relationship during a melting
and crystallization event in the natural continental crust
is generally more complex. For example, if a crustal
material has a eutectic composition, then its melt fraction
vs temperature relationship is discontinuous and a magma
of up to 100% liquid can exist at its ‘EFT’ (Fig. 2b).
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Furthermore, the EFT of magma may become lower
than that of its surrounding crust because of shifts of
melt fraction vs temperature relationships as a result of
compositional changes during melting and crystallization
processes in C&M mode (Fig. 2c). In such cases magmas
can remain as liquids at or below the EFT of the
surrounding crust. The analyses in this section indicate
that, if a magma can remain as a liquid at the EFT of
the surrounding crust, subsequent cooling is a very slow
process (e.g. Case 2 or Case 3); in other words, the
magma is long lived.
Consequently, the evolution of the magma and the
continental crust after each input of a high-temperature
magma is conceptually divided into two stages. In the
first stage, a silicic magma is rapidly generated by melting
and as long as its temperature is above the EFT of the
crust it crystallizes in C&M mode. In the second stage,
the silicic magma is cooled slowly in CC mode. Whether
or not CC stage actually exists depends on whether the
magma has a melt fraction vs temperature relationship
such that the magma has a lower EFT than the surrounding crust (Fig. 2c). We outline below some examples
of petrological processes that result in the shifts of melt
fraction vs temperature relationships as in Fig. 2c. We
will also point out in a later section that the situation
that is qualitatively equivalent to the discontinuous melt
fraction vs temperature relationships as in Fig. 2b can
occur in a wide compositional range of hydrous crusts.
POSSIBLE MECHANISMS TO
GENERATE LONG-LIVED MAGMAS
After basalt emplacement, the light silicic magma generated by crustal melting forms a magma layer overlying
the dense basaltic magma layer. Extensive melting occurs
at the boundary between the silicic magma and the crust
while the basalt acts a heat source (Campbell & Turner,
1987; Huppert & Sparks, 1988a; see Fig. 3a). We focus
on the process at the boundary between the silicic magma
and the crust.
Selective assimilation (fluid dynamics
experiments)
Huppert & Sparks (1988a) assumed that the silicic magma
and the crust have the same chemical compositions, and
hence the same melt fraction vs temperature relationship,
throughout the melting process on the basis of the experimental results of roof melting. When melting occurs
at the roof of a magma chamber, the silicic magma
generated forms a separate layer with little chemical
interaction with the underlying basaltic magma because
of their large density contrast. Furthermore, although
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KOYAGUCHI AND KANEKO
THERMAL EVOLUTION OF MAGMAS
Fig. 2. Schematic diagram showing the definition of the effective fusion temperature (a) and possible melt fraction vs temperature relationships
in continental crust (a, b and c). The relationship may be a single monotonic function (a), a discontinuous one for eutectic compositions (b), or
it may shift because of a certain fractionation process during the C&M stage (c).
the crust is partially melted, evolved light interstitial melt
cannot segregate from the residual crystals at the roof
into the main silicic magma. The crustal material will
mix with the main silicic magma as a whole when it
becomes effectively liquid. In this case, the assumption
of the same composition between the crust and the silicic
magma may be justified unless the effect of crystal settling
in the magma is significant. On the other hand, when
melting occurs at the floor or the side-walls of a magma
chamber, evolved melts formed by partial melting may
segregate from the boundary and selectively mix with
the main silicic magma body (e.g. Woods, 1991; Kerr,
1994). The compositional fractionation because of the
floor or side-wall melting modifies the melt fraction
vs temperature relationship of the magma and affects
subsequent mass and heat transfer between the silicic
magma and the surrounding crust. We present some
results of fluid dynamics experiments to show how a
compositionally uniform solid–liquid pair fractionates by
floor and/or side-wall melting.
To simulate the melting behaviour at the floor and
side-wall of a magma chamber, experiments were conducted using aqueous solutions and solids in the NH4Cl–
H2O binary system (Fig. 3). The NH4Cl–H2O binary
system is a simple eutectic system and the eutectic temperature and composition are –15·4°C and 19·7 wt %
NH4Cl, respectively (Fig. 3c). A solid–liquid pair of a
uniform chemical composition of 28 wt % NH4Cl is used
as starting material. Under this condition of 28 wt %
NH4Cl, the density of the released liquid decreases along
the liquidus surface towards the eutectic composition
and the density of the liquid generated at the solidus
temperature is the minimum in the liquid layer (Fig. 3c).
This density and compositional relationship is qualitatively identical to the typical crust and silicic magma
pair with a uniform initial composition.
The apparatus used consists of a Perspex rectangular
tank (10 cm × 10 cm × 30 cm for the floor melting
experiments and 10 cm × 20 cm × 15 cm for the sidewall experiments) and thick foam plastic plates covering
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Fig. 3. A schematic sketch of floor melting and side-wall melting of a magma chamber in a continental crust (a), and the design of the analogue
experiments in the NH4Cl–H2O binary system (b, c). The contours in the phase diagram (c) show density of the liquid (g/cm3).
the tank for thermal insulation (Fig. 3b). The configurations of the experiments are designed to extract the
essence of the physical processes at the floor and sidewall and do not necessarily represent the actual configuration of a magma chamber itself. The experiments
were started with a solid layer (typically 18 cm deep for
the floor melting experiments and 10 cm thick for the
side-wall experiments) at temperatures slightly below the
solidus temperature (about –16°C). A dyed hot aqueous
solution of identical bulk composition at liquidus temperature (24°C) was rapidly poured and the temperatures
inside the tank were monitored at 20–26 fixed positions
and the compositions of liquid were measured at five
heights during the experiments. Other experiments with
various solid compositions (19·7 wt % and 73 wt %
NH4Cl) and temperatures (down to –45°C) were also
carried out for comparison and systematic quantitative
analyses, some results of which were reported elsewhere
(Koyaguchi & Kaneko, 1995; Kaneko, 1996; Kaneko &
Koyaguchi, 1996).
Figure 4 shows the results of the floor melting experiments. Immediately after the start of each run, a
mushy layer developed between the liquid and solid
layers by crystallization of NH4Cl, and the boundary
between the solid and mushy layers moved downwards
by partial melting (Fig. 4a and b). At the boundary
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KOYAGUCHI AND KANEKO
THERMAL EVOLUTION OF MAGMAS
Fig. 4. (a).
between the solid and mushy layers, the solid layer
partially melted to form eutectic liquid leaving 10%
residual NH4Cl solid at the eutectic temperature. Many
plumes rose from the mushy layer and resulted in vigorous
convective motions in the liquid region. Temperature
and composition of the liquid region were almost uniform
because of the vigorous convection in the earliest stage
and subsequently weak thermal and compositional gradients developed; the temperature rose and the NH4Cl
concentration decreased upwards (Fig. 4c). The average
NH4Cl content in the liquid region decreased with time
because of crystallization of NH4Cl crystals in the mushy
region and simultaneous mixing with partial melts generated at the solid–mush interface. On the other hand,
the bulk NH4Cl content of the partially molten zone
estimated from the mass balance between the liquid
region and the mushy layer increased with time (Fig. 4c).
The dyed liquid penetrated at a much higher rate than
is expected by diffusion alone to the level below the
original solid–liquid boundary, suggesting that there was
convective exchange between interstitial liquid of the
partially molten solid layer and the overlying liquid.
Figure 5 shows the results of the side-wall melting
experiments with a liquid–solid pair of uniform composition (28 wt % NH4Cl). A vertical mushy layer
developed between the liquid and solid layers by melting
and crystallization. It collapsed and slumped down from
time to time and a pile of NH4Cl crystals formed at the
bottom of the liquid region (Fig. 5). The interstitial melt
in the mushy layer mixed with the main liquid body
while the ‘mush avalanche’ was falling down. Many
plumes rose from the crystal pile at the bottom, which
resulted in vigorous convective motions in the liquid
region. Double diffusive convective layers (e.g. Turner,
1973) with stepwise thermal and compositional gradients
developed in the liquid region (Fig. 5). The NH4Cl
concentration in the uppermost convective layer decreased with time and it became as low as 21 wt %
NH4Cl. The initially dyed liquid layer penetrated into
the mushy layer also in the side-wall experiments, suggesting the presence of liquid exchange between the
partially molten solid layer and the liquid region. The
solution in the liquid region was dyed from the top
to the bottom of the tank throughout the experiment,
suggesting that partial melts mixed with the dyed liquid
while they were incorporated into the liquid region.
The experimental conditions and the natural system
are different in some respects, such as dimensions, viscosities of liquid, kinetics of crystallization and phase
equilibria of constituent materials. For example, the
efficient fractional crystallization and the melt segregation
from residual crystals seen in the experiments may be
largely due to the low viscosity of the solution. In spite
of these differences, there are at least two important
implications of the above experimental results. First, there
was considerable mixing of partial melts from the floor
and side-wall boundaries with the main liquid region.
Second, regardless of details of the processes, both mixing
of the partial melts at the floor and/or side-walls and
fractional crystallization increase the low melting point
component in the liquid region. This means that the
melt fraction vs temperature relationship of the magma
shifts in the direction which decreases its EFT compared
with the original solid wall rock (see Fig. 2c).
Fractional crystallization (numerical
analyses)
The effect of crystal settling plays a role regardless of the
presence or absence of the floor and/or side-wall melting,
because crystallization occurs inside the magma chamber,
even if melting occurs at the roof of the chamber alone
(see Huppert & Sparks, 1988a). As crystals settle, they
form a cumulate pile at the bottom of the chamber,
leaving a magma of higher melt fraction. This effect is
evaluated by a simple model in a binary system (Appendix). To separate the effect of fractional crystallization
from other complications such as mixing with partial
melts and eutectic melting, crystallization of a given mass
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Fig. 4. A photograph (a), a schematic diagram (b), and the evolution of gradient in NH4Cl content (c) of the floor melting experiments. The
NH4Cl contents in the mushy and partially molten layers are based on the mass balance assuming a linear concentration gradient in these
layers.
of an initially completely liquid magma along a linear
liquidus surface at a constant cooling rate is considered
here. Crystals are assumed to be homogeneously distributed by vigorous convection in the magma layer
because of high Rayleigh number, but to be separated
from the magma layer to the cumulate pile at the bottom
at the rate of its settling velocity (see Martin & Nokes,
1988). It is also assumed that the cumulate pile has the
assumed critical melt fraction bounding effective liquid
and solid (i.e. 50 vol. %) and that there is no chemical
interaction between the interstitial melt and the main
magma body after the melt is trapped in the cumulate
pile.
Bulk composition, melt fraction, and depth of the
magma layer evolve with time, as the crystals and the
less evolved interstitial melts are incorporated into the
cumulate pile. In the present context, the melt fraction
and the depth of the magma layer at the moment when
the temperature reaches the EFT of the crust are of
primary interest. These quantities depend on the ratio
of the settling timescale to that for the temperature to
reach the EFT of the crust (Fig. 6; see Appendix for
derivations). If the timescale of settling is much the smaller
of the two (i.e. the case of efficient crystal settling), a
crystal-poor magma of approximately 1/4 of the initial
depth would remain at the EFT of the crust. On the
other hand, if the temperature reaches the EFT of the
crust before efficient settling, a large amount of highly
porphyritic magma results. Crystals that are suspended
at this moment settle further while the temperature is
kept at or just below the EFT of the crust on a long
timescale. The depth of final, crystal-free magma layer
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THERMAL EVOLUTION OF MAGMAS
Fig. 7. Timescale of C&M stage [i.e. the timescale for (T – Te)/(T0 –
Te) to reach 0·05] and timescale of crystal settling (H/vs in the Appendix)
for magma body with 1 km thickness as a function of viscosity of
magma. Spherical crystals of 10–2 m in diameter are assumed in the
calculations of settling velocities. The other values used in the calculations are the same as in Fig. 1 (see Table 1).
Fig. 5. A photograph of the side-wall melting experiments.
inversely proportional to the rate of incorporation of the
cold surrounding crust, which is proportional to the
boundary velocity expressed by equation (3). The timescale of settling, on the other hand, is proportional to
viscosity, because of the form of Stokes’ law. The ratio
of the two timescales depends on the dimension of the
magma body only very weakly and hence we express it
in the form
r>Am2/3
Fig. 6. Melt fraction and depth of magma layer at the moment when
the temperature reaches the EFT of the crust, and depth of final,
crystal-free magma layer after complete crystal settling as a function
of ratios of the timescale of crystal settling to that for temperature to
reach the EFT of the crust [i.e. H/(vssF) in the Appendix]. The depth
of the magma layer is normalized such that initial thickness is unity.
after complete crystal settling decreases with the
increasing ratios of timescale of settling to that for temperature to reach the EFT of the crust (Fig. 6).
Although the simple situation considered in the Appendix is not identical to the situation where melting
occurs at the same time, it would be reasonable to
consider that the above timescale for the temperature to
reach the EFT of the crust roughly represents the timescale of the C&M stage. The results of numerical calculations indicate that both the timescale of the C&M
stage and that of crystal settling are approximately proportional to initial thickness of the magma layer, and
that they increase as magma viscosity increases (Fig. 7).
The timescale of the C&M stage is approximately proportional to the cube root of viscosity, because it is
(4)
where A is a constant. The ratio, r, is unity when the
viscosity is of order 104–105 Pa s (Fig. 7) for crystals of
10–2 m diameter under the conditions given in Table 1.
These results suggest that low-viscosity magma forms a
high-melt fraction magma, and high-viscosity magma
tends to form a highly porphyritic magma at the end of
the C&M stage.
The above experimental results and the numerical
analyses suggest that a silicic magma generated by melting
of crust inevitably has a lower EFT than the surrounding
crust at the end of the C&M stage as a result of selective
assimilation and/or fractional crystallization, and that
the two-stage thermal evolution is a natural consequence
after each input of a high-temperature magma in continental crust.
PETROLOGICAL IMPLICATIONS
There are obviously many factors that govern the evolution of magmas in continental crust, and no single
model can explain all the petrological features of observed
magmas. Nevertheless, the fact that the cooling rate of
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JOURNAL OF PETROLOGY
VOLUME 40
magma is two or three orders of magnitude greater in
the C&M stage than in the CC stage may statistically
reflect petrological features of erupted magmas. Marsh
(1981) has pointed out that the probability of a magma
being erupted can be closely related to the cooling rate
of the magma; for example, assuming that eruptions
sample magmas randomly in time, the chance of observing a magma within a temperature interval of rapid
cooling is smaller than that within a temperature interval
of slow cooling. This means that the chance of observing
magmas in the CC stage (we call them ‘long-lived magmas’ hereafter) may be greater than that of observing
magmas in the C&M stage. To test the present hypothesis
of the two-stage thermal evolution model from the viewpoint of petrology, petrological features which characterize long-lived magmas must be specified.
The characteristics of long-lived magmas are determined by two factors: (1) temperature is fixed at or
just below the EFT of the surrounding crust for a long
time because of the slow cooling rate during the CC
stage; (2) phenocryst content depends on the degree of
the shift of the melt fraction vs temperature relationships
during the C&M stage (see Fig. 2c). The shift of melt
fraction vs temperature relationships must be quantified
from the viewpoint of the phase equilibria of crustal
materials, although the fact that the magma has a lower
EFT than that of the surrounding crust is robust regardless
of the detail of the phase equilibria.
According to recent experimental studies (e.g. Wolf &
Wyllie, 1995), low-viscosity hydrous silicic melts generated by dehydration melting of hydrous minerals can
be efficiently segregated from the host, and these would
be the most probable candidate of the selective assimilant.
We have analysed the variation of the melt fraction vs
temperature relationships caused by mixing of hydrous
silicic melt with natural crustal materials using the
thermodynamic model ‘MELTS’ of Ghiorso & Sack
(1995). Melt fraction increases markedly around 700°C
with increasing mixing ratio of the hydrous silicic melt
(Fig. 8). The shifts of the melt fraction vs temperature
relationships result in decrease in phenocryst content in
magmas at the EFT of the crust (Fig. 9). The rate of
decrease in phenocryst content at the same mixing ratio
as the hydrous silicic melt depends on the bulk composition of the crust involved; it is greater for gabbroic
compositions compared with tonalitic or granodioritic
compositions (Fig. 9).
These features of the shifts of the melt fraction vs
temperature relationships and their dependence on the
bulk compositions are not artefacts of a particular thermodynamic model, but can be understood as common
features of multicomponent eutectic systems. The
amounts of melt around 700°C in Fig. 8 are basically
determined by the eutectic melting of quartz, orthoclase
and plagioclase for silicic compositions (e.g. Piwinskii &
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Fig. 8. Variation of melt fraction vs temperature relationships for a
granodioritic crust caused by mixing of a hydrous granitic melt (6 wt
% water) for mixing ratios from 10 to 50% on the basis of thermodynamics model ‘MELTS’ of Ghiorso & Sack (1995). The pressure is
assumed to be 200 MPa. The chemical compositions of the experimental
starting materials (Sample 766 with SiO2 70%, and Sample 705
with SiO2 77%) of Piwinskii & Wyllie (1968, 1970) are used for
the calculations, to cross-check the thermodynamics model and the
experimental results. Although there are slight discrepancies between
the two approaches, partly because of the lack of a thermodynamic
model of hornblende and its dehydration melting in MELTS code
(Ghiorso & Sack, 1995) and partly because of large errors in modal
analyses of the experimental products, the results agree fairly well
especially in the range of melt fraction >50%. Initial water content of
the crustal materials is assumed to be 0·5 wt %.
Fig. 9. Variation in phenocryst content of long-lived magmas at the
EFT of the crusts as a function of mass fraction of added hydrous
silicic melts on the basis of the thermodynamics model. The chemical
compositions of the experimental starting materials are used; these are
gabbro (SiO2 51%, High-Al basalt 82-66) of Sisson & Grove (1993),
tonalite (SiO2 62%, Sample 1213) of Piwinskii & Wyllie (1968), granodiorite (SiO2 70%, Sample 766) of Piwinskii & Wyllie (1968), and
granite (SiO2 77%, Sample 705) of Piwinskii & Wyllie (1970). Initial
water content of the crustal materials is assumed to be 0·5 wt %. The
depth of final, crystal-free magma layer normalized by the initial depth
is also shown on the vertical axis at the right-hand side.
Wyllie, 1968). Because the hydrous silicic melt has a
composition near the ternary minimum of the natural
KOYAGUCHI AND KANEKO
THERMAL EVOLUTION OF MAGMAS
mode at the eutectic temperature. During the CC stage
phenocrysts settle leaving a crystal-poor granitic magma
with the eutectic composition in the upper part of the
magma chamber.
Judging from the effects of the shift and the form of
melt fraction vs temperature relationships as well as
effect of crystallization during the C&M stage, possible
petrological features of long-lived magmas can be summarized as follows:
(1) Magmas with viscosities of less than 104–105 Pa s
(e.g. mafic magmas in a gabbroic crust or hydrous
magmas) can be crystal-poor long-lived magmas because
of crystal settling during the C&M stage.
(2) More than 10% of hydrous silicic melts or 1%
water is required as a selective assimilant to form a
crystal-poor long-lived magma with <25% crystals. The
rate of decrease in crystal content is less remarkable in
a crust of intermediate composition than in a gabbroic
crust.
(3) A crystal-poor long-lived magma at the eutectic
temperature forms by eutectic melting in hydrous granodioritic to granitic crusts with water content more than
a few weight per cent.
These results suggest a general tendency of the petrological features of long-lived magmas: they can be crystal
poor when they are mafic or granitic compositions, but
they tend to be porphyritic when they have intermediate
compositions. The observation that andesites tend to be
more porphyritic than basalts or rhyolites (Ewart, 1982)
may support the two-stage thermal evolution model and
the ubiquity of long-lived magmas as erupted magmas.
Fig. 10. Variation of melt fraction vs temperature relationships for a
granodioritic crust because of addition of water on the basis of the
thermodynamics model. The compositions used in the calculations are
same as in Fig. 8.
Fig. 11. Variation in phenocryst content of long-lived magmas at the
EFT of the crust as a function of added water content. Initial water
content of the crustal materials is assumed to be 0·5 wt %. The
compositions used in the calculations are same as in Fig. 9. The depth
of final, crystal-free magma layer normalized by the initial depth is
also shown on the vertical axis at the right-hand side.
system, the amount of melt formed by melting at the
eutectic temperature increases with the increasing mixing
ratio of the hydrous silicic melt. Melt fraction at a given
temperature for crustal materials also increases greatly
with increasing water content (Fig. 10). A long-lived
magma with <20 vol. % phenocrysts can form by mixing
of at most 1 wt % water in granitic or gabbroic crusts
(Fig. 11).
Figures 8 and 10 indicate that the magma is still
effectively liquid at the eutectic temperature in a hydrous
granodioritic to granitic crust with water content more
than a few weight per cent. The thermal evolution of
the magma in such crusts is similar to that of the crust
with the eutectic composition (see Fig. 2b) in the sense
that the magma slowly solidifies by losing heat in CC
Future problem
The hypothesis of the two-stage thermal evolution model
can be tested in the future by several different kinds of
observations and theoretical modelling. The shift of melt
fraction vs temperature relationship is controlled by melt
segregation of partial melts and vapour circulation around
a magma chamber. Observations and experimental works
on melt segregation in the crust [for a summary of the
recent progress, see Brown et al. (1995)], and studies on the
physics of melt segregation in partially molten materials or
the mushy layer at the floor and wall (e.g. Tait & Jaupart,
1989, 1992; Woods, 1991; Worster, 1991; Kerr, 1994)
will give constraints on the segregation process. Isotope
geochemistry of magma such as d18O and other geochemical studies on volcanic gases and fluids in geothermal areas will also be useful to test the contribution
of hydrothermal water as selective assimilant. The probability of observing long-lived magmas as erupted
magmas is dependent on timings of basaltic inputs and
eruptions. Chronological approaches such as those of
radioactive disequilibria of the U–Th series would give
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VOLUME 40
information on the timings of basalt inputs and eruptions.
In the present study, simplified models for mass and heat
transfer are applied to extract the fundamental physics;
however, more realistic models which take into account
the effects of fracturing of chamber wall and kinetics of
crystallization and melting (e.g. Bergantz, 1995; Hort,
1997) may be necessary to quantitatively compare the
model with these natural observations.
ACKNOWLEDGEMENTS
We thank Steve Sparks for his constructive comments
on an earlier version of the manuscript. Comments by
Steve Tait and two anonymous referees are also greatly
appreciated. K.K. was supported by the Japanese Society
for the Promotion of Science.
REFERENCES
Bergantz, G. W. (1995). Changing techniques and paradigms for the
evaluation of magmatic processes. Journal of Geophysical Research 100,
17603–17613.
Brown, M., Rushmer, T. & Sawyer, E. W. (1995). Introduction to
special section: mechanisms and consequences of melt segregation
from crustal protoliths. Journal of Geophysical Research 100, 15551–
15563.
Campbell, I. & Turner, J. S. (1987). A laboratory investigation of
assimilation at the top of a basaltic magma chamber. Journal of
Geology 95, 155–172.
Ewart, A. (1982). The mineralogy and petrology of Tertiary–Recent
orogenic volcanic rocks: with special reference to the andesitic–
basaltic compositional range. In: Thorpe, R. S. (ed.) Andesites: Orogenic
Andesites and Related Rocks. New York: John Wiley, pp. 25–95.
Ghiorso, M. S. & Sack, R. O. (1995). Chemical mass transfer in
magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid–solid
equilibria in magmatic systems at elevated temperatures and pressures. Contributions to Mineralogy and Petrology 119, 197–212.
Halliday, A. N. (1990). Reply to comment of R. S. J. Sparks, H. E.
Huppert & C. J. N. Wilson on ‘Evidence for long residence times
of rhyolitic magma in the Long Valley magmatic system: the isotopic
record in precaldera lavas of Glass Mountain’. Earth and Planetary
Science Letters 99, 390–394.
Halliday, A. N., Mahood, G. A., Holden, P., Metz, J. M., Dempster,
T. J. & Davidson, J. P. (1989). Evidence for long residence times of
rhyolitic magma in the Long Valley magmatic system: the isotopic
record in precaldera lavas of Glass Mountain. Earth and Planetary
Science Letters 94, 274–290.
Hort, M. (1997). Cooling and crystallization in sheet-like magma bodies
revisited. Journal of Volcanology and Geothermal Research 76, 297–317.
Huppert, H. E. & Sparks, R. S. J. (1988a). The generation of granitic
magmas by intrusion of basalt into continental crust. Journal of
Petrology 29, 599–624.
Huppert, H. E. & Sparks, R. S. J. (1988b). Melting the roof of a
chamber containing a hot, turbulently convecting fluid. Journal of
Fluid Mechanics 188, 107–131.
Huppert, H. E. & Turner, J. S. (1981). A laboratory model of a
replenished magma chamber. Earth and Planetary Science Letters 54,
144–152.
NUMBER 2
FEBRUARY 1999
Kaneko, K. (1996). The processes of the thermal and material evolution
of a magma system in a crust. Ph.D. Thesis, University of Tokyo.
Kaneko, K. & Koyaguchi, T. (1996). Evolution of magma system in
the crust—investigations based on thought experiments and analogue
experiments (in Japanese with English abstract). Memoirs of the Geological Society of Japan 46, 29–41.
Kerr, R. C. (1994). Melting driven by vigorous compositional convection. Journal of Fluid Mechanics 280, 255–285.
Koyaguchi, T. & Blake, S. (1991). Origin of mafic enclave: constraints
on the magma mixing model from fluid dynamics experiments. In:
Didier, J. & Barbarin, B. (eds) Enclaves and Granitic Petrology. Amsterdam: Elsevier, pp. 415–429.
Koyaguchi, T. & Kaneko, K. (1995). Thermal and petrological evolution of magma system. Abstracts of Todai Symposium ‘The Role of
Magmas in the Evolution of the Earth’. Tokyo: University of Tokyo,
pp. 36–37.
Mahood, G. A. (1990). Second reply to comment of R. S. J. Sparks,
H. E. Huppert & C. J. N. Wilson on ‘Evidence for long residence
times of rhyolitic magma in the Long Valley magmatic system: the
isotopic record in precaldera lavas of Glass Mountain’. Earth and
Planetary Science Letters 99, 395–399.
Marsh, B. D. (1981). On the crystallinity, probability of occurrences
and rheology of lava and magma. Contributions to Mineralogy and
Petrology 78, 85–98.
Marsh, B. D. (1989). On convection style and vigor in sheet-like magma
chambers. Journal of Petrology 30, 479–530.
Martin, D. & Nokes, R. (1988). Crystal settling in a vigorously convecting
magma chamber. Nature 332, 534–536.
McBirney, A. R. (1980). Mixing and unmixing of magmas. Journal of
Volcanology and Geothermal Research 7, 357–371.
Piwinskii, A. J. & Wyllie, P. J. (1968). Experimental studies of igneous
rock series: a zoned pluton in the Wallowa batholith, Oregon. Journal
of Geology 76, 205–234.
Piwinskii, A. J. & Wyllie, P. J. (1970). Experimental studies of igneous
rock series: felsic body suites from the Needle Point Pluton, Wallowa
batholith, Oregon. Journal of Geology 78, 52–76.
Rutter, E. H. & Neumann, D. H. K. (1995). Experimental deformation
of partially molten Westerly granite under fluid absent conditions,
with implications for the extraction of granitic magmas. Journal of
Geophysical Research 100, 15697–15751.
Sisson, T. W. & Grove, T. L. (1993). Experimental investigations of
the role of H2O in calc-alkaline differentiation and subduction zone
magmatism. Contributions to Mineralogy and Petrology 113, 143–166.
Sparks, R. S. J., Huppert, H. E. & Turner, J. S. (1984). The fluid
dynamics of evolving magma chambers. Philosophical Transactions of
the Royal Society of London, Series A 310, 511–534.
Sparks, R. S. J., Huppert, H. E. & Wilson, C. J. N. (1990). Comment
on ‘Evidence for long residence times of rhyolitic magma in the
Long Valley magmatic system: the isotopic record in precaldera
lavas of Glass Mountain’ by A. N. Halliday, G. A. Mahood,
P. Holden, J. M. Metz, T. J. Dempster, & J. P. Davidson. Earth and
Planetary Science Letters 99, 387–389.
Spera, F. J. (1979). Thermal evolution of plutons: a parameterized
approach. Science 207, 299–301.
Tait, S. & Jaupart, C. (1989). Compositional convection in viscous
melts. Nature 338, 571–574.
Tait, S. & Jaupart, C. (1992). Compositional convection in a reactive
crystalline mush and melt differentiation. Journal of Geophysical Research
97, 6735–6756.
Turcotte, D. L. & Schubert, G. (1982). Geodynamics: Applications of
Continuum Physics to Geological Problems. New York: John Wiley.
Turner, J. S. (1973). Buoyancy Effects in Fluids. Cambridge: Cambridge
University Press.
252
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Wolf, M. B. & Wyllie, P. J. (1995). Liquid segregation parameters from
amphibolite dehydration melting experiments. Journal of Geophysical
Research 100, 15611–15621.
Woods, A. W. (1991). Fluid mixing during melting. Physics of Fluids A3,
1393–1404.
Worster, M. G. (1991). Natural convection in a mushy layer. Journal
of Fluid Mechanics 224, 335–359.
Fig. A1. Fractional crystallization model in a binary system with a linear liquidus surface. (See text for explanation.)
APPENDIX: FRACTIONAL
CRYSTALLIZATION MODEL IN A
BINARY SYSTEM
Let us consider a material in a binary system with a
linear liquidus surface (Fig. A1), where compositional
scale is normalized, such that the solid composition is
C = 1 and the liquid composition at the EFT is C =
0. For simplicity it is assumed that the density difference
between the solid and liquid is small enough for values
of volume fraction to approximately represent those of
mass fraction. The initial bulk composition is, therefore,
C = φ, where φ is the critical melt fraction between
effective solid and liquid. We take φ = 0·5 in the
following calculations.
If the material of C = φ has initially a higher melt
fraction (say, u0) and the crystallization proceeds as
temperature decreases at a given cooling rate, then the
crystals settle to form a cumulate pile at the bottom and
the depth of ‘magma’ (i.e. the region in an effectively
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liquid state) decreases. Three more assumptions are made:
(1) the melt fraction of the cumulate pile has also the
critical value, φ; (2) the main magma body is homogenized; (3) no chemical interaction between the interstitial melt and the main magma body is allowed. Under
these assumptions, the rate of change in melt composition
is approximated as
u −φ
dC
=− 0
dt
u0sF
(A1)
where sF is the timescale to reach the EFT. The growth
rate of the cumulate pile is given by
1 1−u
dz
=−
vs
dt
H u−φ
(A2)
Fig. A2. Composition of magma that is incorporated into the cumulate
pile against the height from the bottom for various ratios of timescale
of settling to that for temperature to reach the EFT of the crust
[H/(vssF)].
where H is the total height, vs is the settling velocity of
crystals, z is the normalized depth of the magma (see
Fig. A1), and u is the melt fraction of the magma. The
mass balance at the front of the growing cumulate pile
can be expressed as
u dC
d(1−u) (1−u+uC)−(1−φ+φC) dz
=
−
dt
z(1−C)
dt 1−C dt
(A3)
where C is the melt composition. From equations (A1)–
(A3), we can numerically obtain the evolutions of composition of magma that is incorporated into the cumulate
pile (Fig. A2), phenocryst content, and the depth of
magma layer (Fig. A3). For a given initial condition of
u0, the behaviours of these evolutions depend on the
ratios of the timescale of settling to that for temperature
to reach the EFT [i.e. H/(vssF)] alone. To evaluate the
effects of fractional crystallization during the C&M stage,
the values at the moment when the temperature reaches
the EFT of the surrounding crust against varying ratios
of the two timescales are the most meaningful, and these
are shown in Fig. 6.
254
Fig. A3. The relationships between depth and melt fraction of magma
layer for various ratios of timescale of settling to that for temperature
to reach the EFT of the crust [H/(vssF)]. The depth of magma layer
starts at z = 1, decreases with time as crystallization proceeds, and
terminates at the position of circles when temperature reaches the EFT
of the crust.