This article appeared in a journal published by Elsevier. The attached
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling or
licensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of the
article (e.g. in Word or Tex form) to their personal website or
institutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies are
encouraged to visit:
http://www.elsevier.com/copyright
Author's personal copy
Tectonophysics 500 (2011) 34–49
Contents lists available at ScienceDirect
Tectonophysics
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t e c t o
Two models for the formation of magma reservoirs by small increments
Chloé Michaut a,⁎, Claude Jaupart b
a
b
Institut de Physique du Globe de Paris, 4, av. de Neptune, 94107 Saint-Maur des Fossés, France
Institut de Physique du Globe de Paris, 4, Place Jussieu, 75257 Paris Cédex 05, France
a r t i c l e
i n f o
Article history:
Received 7 November 2008
Received in revised form 22 June 2009
Accepted 19 August 2009
Available online 6 September 2009
Keywords:
Magma reservoirs
Crystallization kinetics
Thermal evolution
Plutons
Bimodal volcanism
Intrusion
Magma injection rate
Latent heat of crystallization
a b s t r a c t
It is now clear that magma reservoirs develop over long time intervals out of a large number of individual
intrusions. Evidence from some large volcanic deposits, including those of the Bishop Tuff, California, and
Fish Canyon Tuff, Colorado, indicates that such reservoirs may experience pervasive and rapid heating before
catastrophic eruptions. Injection of hot primitive melt in the reservoir is often invoked to explain these
observations, but this requires the rapid emplacement of a large magma volume (at least 25 km3 in less than
100 years for the Bishop Tuff) and efficient heat transfer through a thick cumulate pile. Within a magma
body that grows incrementally, temperatures rise progressively and may eventually lead to a permanent
volume of melt. Temperature and crystallization follow two completely different paths depending on the
thickness of individual intrusive sheets. In one limit, corresponding to thick sheets, crystallization proceeds
at equilibrium. Initial magma batches crystallize completely and a permanent body of melt develops
progressively once ambient temperatures in the complex rise above the solidus. In the other limit,
corresponding to thin sheets, crystallization is kinetically-controlled, such that initial intrusions do not
crystallize completely and preserve a glassy residue. Once temperatures within the growing magma pile
reach a certain threshold value, nucleation and growth of crystals get reactivated in the residual glass.
Crystallization then proceeds catastrophically in a positive feedback loop involving latent heat release and
temperature rise. Thus, an initial phase with no evolved melt present ends with the sudden formation of a
large volume of evolved melt in a crystal mush. After this initial phase, new intrusions get emplaced in
heated surroundings and follow a thermal path similar to that of the equilibrium case, with melts that
become increasingly more primitive with time. Evidence for kinetic controls on crystallization in the natural
environment is provided by the significant amounts of glass (up to 40%) that are found in thin isolated
basaltic sills and dykes. Owing to sluggish crystallization kinetics, more glass should be generated in
andesitic and dacitic magmas emplaced in similar conditions. Devitrification of such quantities of glass
accounts for the rapid heating of a thick sill complex.
The two models of magma reservoir formation involve different requirements: the intrusion thickness must
be smaller than a critical value in the kinetic model and the average magma input rate must exceed a
threshold value in the equilibrium model. For a given injection rate, a permanent magma reservoir forms in a
shorter time interval and over a smaller cumulative magma thickness in the kinetic model than in the
equilibrium one. The kinetic model is such that the amount of evolved melt generated increases with
decreasing injection rate, in contrast to the equilibrium model for which the opposite relationship holds.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Crustal magma reservoirs may feed eruptions over long time
intervals that exceed one million years (Halliday et al., 1989; Davies
et al., 1994; Simon and Reid, 2005). This has challenged our understanding of how such reservoirs form and evolve thermally. If a large
volume of magma is emplaced rapidly with little further melt input,
the bulk compositional and thermal evolution is monotonic. Convective motions in the melt are unavoidable, implying that cooling is
⁎ Corresponding author.
E-mail address: [email protected] (C. Michaut).
0040-1951/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.tecto.2009.08.019
effected rapidly, typically in less than a thousand years (Worster et al.,
1990; Sparks et al., 1990). One must therefore consider that magma
reservoirs grow in small increments, with initial magma batches
providing a heated crustal environment that allows later intrusions to
remain partially molten (Petford and Gallagher, 2001; Annen and
Sparks, 2002; Glazner et al., 2004; Annen et al., 2006). Detailed
geological and petrological observations show indeed that large
batholiths and plutons have developed out of sill and dyke complexes
(Sisson et al., 1996; McNulty et al., 1996; Coleman et al., 2004). On a
smaller scale, mixing and mingling figures, primitive magma enclaves
as well as cycles of cooling and heating recorded by crystal zoning
provide evidence for the repeated injection of primitive magma in the
same storage zone. Mahood (1990) has envisioned phases of cooling
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
and differentiation followed by the remelting of older cumulate
assemblages depending on the balance between cooling during
repose periods and heating due to recharge events.
In a magma body that grows by small increments, temperatures
rise progressively, so that the residual melt becomes increasingly
more primitive with time. In contradiction with this prediction,
several eruption deposits provide evidence for the rapid formation of
large volumes of evolved magma (Murphy et al., 2000; Bachmann and
Bergantz, 2003). Zoning in quartz phenocrysts from the Bishop Tuff
indicates that temperatures rose by about 100 °C in less than 100 yr
just before eruption (Wark et al., 2007). Similar heating events prior
to major eruptions have been documented in other volcanic systems
(Table 1).
Taken together, the different pieces of evidence available seem
contradictory. In a reservoir that has grown in small increments, rapid
rejuvenation of a thick cumulate pile requires a major change in the
intrusion sequence and the sudden emplacement of a large primitive
magma volume elsewhere. For example, emplacement of at least
25 km3 of primitive magma is required for the Bishop Tuff (Wark
et al., 2007). In that particular case, the injection process must have
lasted less than the duration of the heating phase, indicating that the
input rate was at least 0.25 km3 yr− 1, which is much larger than
known rates in active and fossil systems (Crisp, 1984; Fialko and
Simons, 2000). Furthermore, heating of an essentially solid cumulate
pile can only be achieved by conduction, which cannot be efficient
over more than a few tens of meters in such a short time. To overcome
these difficulties, Lipman et al. (1997) and Bachmann and Bergantz
(2006) have proposed that rejuvenation is achieved through gas
sparging, which involves pervasive gas flow due to the degassing of an
underlying magma body. Bachmann and Bergantz (2003) find that
about 3000 km3 of volatile rich basaltic magma is required to reheat
the Fish Canyon cumulate pile in this manner, implying the accumulation of at least 2 km of magma beneath it. Another requirement is
that the cumulate pile must be highly permeable.
Physical models of a reservoir that grows incrementally have been
developed by several authors (Petford and Gallagher, 2001; Annen and
Sparks, 2002; Glazner et al., 2004; Annen et al., 2006). Successive
injections progressively warm up the crust until temperatures become
large enough to sustain a melt body. Tens to hundreds of thousand
years must elapse before the formation of a permanent melt body and
the magma input rate must be larger than a threshold value of a few
centimeters per year per unit area (Annen et al., 2008), which is larger
than values that have been determined for a large number of
magmatic/plutonic systems (Crisp, 1984; Fialko and Simons, 2000).
One observation that is not accounted for by these models is a late and
rapid heating event. A recent model accounts for this peculiar feature.
According to this model, magma emplacement proceeds through
thin intrusive sheets which cool so rapidly that crystallization is
kinetically-controlled. In such conditions, early magma batches do not
crystallize completely and leave a substantial amount of residual glass.
With time, as the amount of magma emplaced increases, temperatures
35
rise in the reservoir, which eventually acts to reactivate the nucleation
and growth of crystals in the glassy interstitial parts of the cumulate
pile (Michaut and Jaupart, 2006). Such crystallization releases latent
heat at near solidus temperatures, which thermally rejuvenates the
magma pile and leads to a rather homogeneous crystal mush containing chemically evolved melt. In this model, thermal rejuvenation is
an intrinsic feature of the thermal sequence and proceeds fast because
it operates quasi simultaneously over a large magma thickness. This
can reconcile the two features of reservoir evolution invoked above,
with slow growth due to small magma increments leading to a late and
rapid heating/rejuvenation event. Like the other models, the kinetic
model has specific requirements because it only works if individual
magma additions are sufficiently thin.
In this paper, we reevaluate the kinetic model and compare it to
the equilibrium crystallization model of (Annen and Sparks, 2002;
Annen et al., 2006). We assess its validity using recently published
laboratory experiments (Pupier et al., 2008). We review field
observations of glassy residues in sills and dykes as well as data on
magma emplacement rates and intrusion thicknesses. One criticism of
the kinetic model is that field evidence for thin intrusive sheets is
difficult to find because chilled margins and contacts between intrusive units get obliterated in the interior of magma bodies (although
they can be found, e.g. (Wiebe et al., 2007)). In a similar fashion, the
other models for the formation of magma reservoirs have specific
requirements that are difficult to test in the field. For example, direct
physical evidence for pervasive gas sparging is lacking. Also, it is
currently impossible to verify that the average input rate in a fossil
magmatic system was indeed above the threshold value needed for a
permanent melt body. Thus, we resort to indirect evidence, i.e. predictable consequences, to assess the merits, likelihood and limitations
of the different models.
2. Crystallization kinetics
There is plenty of evidence for some kinetic control on crystallization in geological conditions. These include crystal size variations
away from chilled margins, as well as quench textures and crystal
morphologies that may be found even in the deep interior of plutons
(Moore and Lockwood, 1973; Tegner et al., 1993; Sisson et al., 1996).
In some cases, the order of appearance of certain mineral phases is
kinetically-controlled (Gibb, 1974). Such evidence has already been
reviewed by Brandeis et al. (1984) and Michaut and Jaupart (2006)
and need not be repeated here. A few kinetic data are available from
different types of measurements, either in the laboratory or in the
natural environment, but they do not allow construction of full kinetic
functions over a whole crystallization interval. Calculations demonstrate that the nucleation rate is the critical input (Brandeis and
Jaupart, 1987a). For our present purposes, what is important is how
crystallization kinetics can be included in a large-scale thermal model
and specifically how the nucleation and growth rates of crystals
depend on temperature and composition.
Table 1
Heating of magma reservoir before an eruption.
Eruption
Volume
Heating
Santorini, Greece,
Minoan rhyodacite
Soufriere Hills, Montserrat,
1995–1999 andesitic eruption
Ceboruco, Mexico,
Jala pumice eruption, 1000 yr
Mount Unzen, Japan,
1991–1995 rhyodacitic eruption
La Garita Caldera, Colorado
Fish Canyon Tuff, 28 Ma
Bishop tuff magma system
Campanian Ignimbrite, Italy
30 km3
35 to 85 °C
~ 0.3 km3 mixed dacite
(+ ~ 3 km3 rhyodacite)
5000 km
500 km3
200 km3
3
Timescales
Authors
Cottrell et al. (1999)
20 to 200 °C
3 yr
Murphy et al. (2000)
~ 70 °C
34 to 47 days
Chertkoff and Gardner (2004)
60 to 110 °C
Few weeks
Few months
Venezky and Rutherford (1999)
Nakamura (1999)
Bachmann et al. (2002)
<100 yr
~ 100 yr
Wark et al. (2007)
Pappalardo et al. (2008)
~ 40 °C
~100 °C
Author's personal copy
36
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
Table 2
Values for the physical properties.
Parameters and physical properties
Symbol
Value
Thermal conductivity
Heat capacity
Latent heat of crystallization
Magma density
Magma liquidus temperature
Initial silica content of magma
Country rock solidus temperature
Country rock liquidus temperature
Intrusion depth
k
Cp
L
ρ
TL
C0
Tsr
Tlr
Zs
2.5 W K− 1 m− 1
1.3 × 103 J kg− 1 K− 1
4.18 × 105 J kg− 1
2500 kg m− 3
1187 °C
47%
800 °C
1100 °C
5 km
Recent experiments shed important light on crystallization
kinetics and provide benchmark data for the theoretical model
(Pupier et al., 2008). For these experiments, the starting liquid
composition was that of a Fe-rich basalt thought to be the parental
liquid for the famous Skaergaard intrusion of Greenland. For the redox
conditions of the QFM buffer and at atmospheric pressure, the natural
order of crystallization is as follows: plagioclase (1175 °C), olivine
(1165 °C), clinopyroxene (1130 °C) and Fe–Ti oxides (1100 °C).
Experiments were conducted with two control parameters: the time
spent by the melt above the liquidus and the cooling rate. Cooling
rates ranged from 3 K h− 1 to 0.02 K h− 1, which are relevant to natural
conditions. Pupier et al. (2008) find that nucleation of the liquidus
phase critically depends on the time spent above the liquidus. The
longer this time is the less efficient nucleation is. Crystallization can in
fact be completely suppressed for lack of suitable nucleation sites.
Pupier et al. (2008) were not able to identify nucleation sites in their
experiments and proposed that they were microbubbles presumably
due to the sample preparation technique (a powder that is heated to
the liquidus). One expects that natural conditions favour heterogeneous nucleation due to the entrainment of xenoliths and perhaps
antecrysts from the deep magma source, but the fact remains that
crystallization efficiency is sensitive to the number of pre-existing
nucleation sites. Even when the liquidus phase crystallizes easily,
nucleation can be difficult for the other phases. In the experiments of
Pupier et al. (2008), the nucleation delay for the second phase on the
liquidus, olivine, is between 10 °C and 20 °C, and that for the third
phase, clinopyroxene, is even larger as it exceeds 30 °C. In fact, Pupier
et al. (2008) found that clinopyroxene is absent from almost all their
kinetic experiments. Nucleation delays probably depend on the
cooling conditions and specifically on the cooling rate, which is not
kept constant in the natural environment. Thus, the only way to assess
the true crystallization behaviour in geological conditions is to use
data on natural samples. Numerical calculations of kineticallycontrolled crystallization reproduce the observed variations of crystal
size away from dyke margins and can be used to infer nucleation and
growth rates from crystal size data (Brandeis and Jaupart, 1987a;
Spohn et al., 1988).
The basic principle behind kinetically-controlled crystallization is
encapsulated by the following equation for melt fraction U:
∂U
f ðT ⁎Þ
= −U
τk
∂t
ð1Þ
where τk is a characteristic time for crystallization. U, the noncrystallized fraction, depends on time and position within the
intrusion. T ⁎ = T/TL is the dimensionless temperature, with T and TL,
the liquidus temperature, in K. f(T ⁎) is an effective function for the
total crystallization rate which lumps together the nucleation and
growth rates (Fig. 1). For scaling purposes, this dimensionless
function is such that its maximum value is 1. In principle, one should
introduce a separate kinetic function for each phase but there are not
enough experimental data to allow this. Accordingly, we use a simple
function and test it against the data of Pupier et al. (2008). We discuss
Fig. 1. Effective kinetic function f for both nucleation and growth rates as a function of
K2
dimensionless temperature T ⁎ = T/TL with T and TL in K. f ðT ⁎Þ = CT ⁎exp −
2
T ⁎ðT ⁎−1Þ
h
i
K
exp − 3⁎ , with K2 = 10− 3, K3 = 30, and C such that max( f ) = 1. The small horizontal
T
arrow shows the nucleation delay of the initial liquidus phase. Also shown is the
temperature at which devitrification becomes significant.
independently the key ingredients that are required for kinetics to
play a significant role in natural emplacement conditions.
From the available kinetic data (summarized in Tables 3 and 4), we
have estimated that, to within a factor about 2, characteristic time τk
is about 106 s for common basaltic magmas (Michaut and Jaupart,
2006) and this is the value we have chosen for our calculations. If
dimensionless function f(T ⁎) is of the box-car type, such that it is 1 for
a large range of undercoolings, the melt fraction decreases to 37% in
time t = τk, or about 280 h for τk = 106 s. This is within the time-scale
of the laboratory studies of Pupier et al. (2008). We use below a more
appropriate kinetic function, which peaks at a dimensionless
temperature of 0.96, corresponding to an undercooling (TL − T) of
about 60 K for TL = 1460 K (Fig. 1). For a fixed cooling rate Γ, Eq. (1) is
easily integrated and compared to the experimental data (Fig. 2). As
expected from the preceding discussion, calculated values of the melt
fraction are smaller than the experimental ones when nucleation sites
are absent (i.e. for melts that have been kept above the liquidus for
more than 10 h). They are close to the experimental values for
heterogeneous nucleation and in fact underestimate the residual melt
fraction for the smallest cooling rate investigated (0.2 K h− 1). Overall,
calculated values are reasonably close to the experimental data.
For experiments at a constant cooling rate Γ, it is easy to evaluate
the influence of the two unknowns in the kinetic function, characteristic time τk and function f(T ⁎). Eq. (1) can be rewritten as
follows:
Ln½UðT ⁎Þ =
1 T⁎
∫ f ðuÞdu
Γτk 1
ð2Þ
where Γ is positive and melt fraction U is expressed as a function of
temperature (or, equivalently, undercooling). From this, the amount
of glass that is formed in an experiment, noted Ug, may be obtained by
setting dimensionless temperature T ⁎ at a very small value, for
example 0: the result is not sensitive to the exact value chosen
because the kinetic function drops to zero rapidly. Thus:
LnðUg Þ =
1 0
∫ f ðuÞdu
Γτk 1
ð3Þ
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
37
Table 3
Rates of nucleation (I) and growth (Y) in silicate melts.
Y (cm s− 1)
I (cm− 3 s-1)
Refs
Ym = 1 × 10− 5
Ym = 5 × 10− 5
Ym = 2 × 10− 6
/
/
/
Muncill and Lasaga (1987)
Muncill and Lasaga (1987)
Kirkpatrick et al. (1979)
10− 10–10-9
6 × 10− 11–10− 10
3 × 10− 11–2 × 10− 10
3 × 10− 9–8 × 10− 7
10− 9–2 × 10− 8
2 × 10− 11–5 × 10− 8
2 × 10− 11–4 × 10− 8
10− 2–1
3 × 10− 2
9 × 10− 7–6 × 10− 6
3 × 10− 4–3 × 10− 2
2 × 10− 4–3 × 10− 3
10− 3–4 × 10− 1
10− 3–8 × 10− 2
Kirkpatrick (1977)
Cashman and Marsh (1988)
Mangan (1990)
Burckard (2002)
Burckard (2002)
Oze and Winter (2005)
Oze and Winter (2005)
Laboratory experiments on natural systems
Plagioclase
Basalt
Pyroxene
Basalt
Plagioclase
Basalt
Plagioclase
Basalt
10− 11–10− 10
10− 9–10− 8
6 × 10− 11–3 × 10− 10
Ym = 10− 9–3 × 10− 10
10− 6–10− 4
10− 6–10− 4
107–1010
Burckard (2002)
Burckard (2002)
Pupier et al. (2008)
Pupier et al. (2008)
Theoretical calculations of crystal size variations
Opx and Plag.
Diabase Dykes
Ym ≈ 10− 7
Im ≈ 1
Brandeis and Jaupart (1987b)
Mineral
System
Laboratory experiments on synthetic systems
Ab–An
Plagioclase An30
Ab–An
Plagioclase An40
Ab–An
Plagioclase An50
Natural systems–natural conditions
Plagioclase
Basaltic
Plagioclase
Basaltic
Olivine
Basaltic
Plagioclase
Basaltic
Pyroxene
Basaltic
Plagioclase
Basaltic
Pyroxene
Basaltic
lava
lava
lava
lava
lava
lava
lava
lake in-situ
lake
lake
lake
lake
flow
flow
I and Y are average values measured in small samples at small undercoolings, Im and Ym are maximum rates over the whole crystallization interval.
This introduces the width of the kinetic function, noted ΔTc⁎, which
is related to the undercooling interval over which crystallization
occurs:
ΔTc⁎ = ∫0 f ðuÞdu
1
ð4Þ
Note that this parameter is dimensionless, such that ΔTc⁎ = ΔTc/TL.
Thus:
"
ΔT ⁎
Ug = exp − c
Γτk
#
ð5Þ
explained in Brandeis and Jaupart (1987b) and Michaut and Jaupart
(2006), available measurements of kinetic rates as well as simulations
of crystal size variations indicate that τk ≈ 106 s for many basalts.
More evolved melt compositions are associated with sluggish kinetics
and larger values of τk (Michaut and Jaupart, 2006) (Table 4).
The other parameter of importance is the temperature range over
which nucleation may occur. For the function f(T ⁎) chosen here, the
crystallization rate becomes entirely negligible at a dimensionless
temperature of about 0.65, or about 950 K for TL = 1460 K (Fig. 1). For
this function and this liquidus temperature, ΔTc ≈ 100 K. ΔTc
corresponds to the width of the kinetic function at mid-height and
is smaller than the whole crystallization interval, which may be
spread over a large temperature interval of ~500 K.
This shows that the amount of glass increases with increasing
cooling rate and increasing τk, as well as with decreasing ΔTc⁎. These
equations show that the melt fraction is very sensitive to characteristic time τk: changing the value of τk by a factor of 2 would lead to
unacceptable departures from the experimental data. We conclude
that our kinetic formulation is adequate for the ferro-basalt of Pupier
et al. (2008). For other magma compositions, the data are fewer. As
Table 4
Laboratory determinations of peak rates of nucleation (Im) and growth (Ym) in evolved
melts.
Mineral
System
Ym
(cm s− 1)
Im
(cm-3 s-1)
Refs
Plagioclase
Andesite + 6.4% H20†
1.7 × 10− 9
3.2 × 10− 2
Plagioclase
Granite (synthetic) +
3.5% H20
Granodiorite (synthetic) +
6.5% H20
Granodiorite (synthetic) +
12% H20
Granite (synthetic) +
3.5% H20
Granodiorite (synthetic) +
6.5% H20
Granodiorite (synthetic) +
12% H20
10− 6
‡
Couch et al.
(2003)
Swanson
(1977)
Swanson
(1977)
Swanson
(1977)
Swanson
(1977)
Swanson
(1977)
Swanson
(1977)
Plagioclase
Plagioclase
Alkali Fs
Alkali Fs
Alkali Fs
5 × 10
−7
‡
≈10− 8
‡
2 × 10− 7
‡
−7
‡
10
≈10
−8
‡
† Crystallization is induced by decompression. ‡ Not measured.
Fig. 2. Mass fraction of residual melt as a function of temperature calculated for cooling
rates equal to 0.2 °C/h and 3 °C/h, using (2) and kinetic function f (Fig. 1). For
comparison, results from different experiments from Pupier et al. (2008) are indicated
with different symbols. Experiments XP01 and XP04 were both conducted at a cooling
rate of 0.2 °C/h. Experiments XP06 and XP07 were conducted at a cooling rate of 3 °C/h.
Author's personal copy
38
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
3. Controls on the thickness of individual intrusive sheets
physical properties that are not known accurately, however, and it is
useful to evaluate natural observations.
The time-scale for conductive cooling of an intrusion of thickness d
is given by:
2
τd =
d
4κ
ð6Þ
where thermal diffusivity κ is typically ≈10− 6 m2 s− 1. This characteristic time must be compared to the kinetic time-scale τk. For τd/
τk ≪ 1, kinetic limitations do not allow large rates of nucleation and
growth and the melt does not crystallize completely, leaving a glassy
residue. The amount of residual glass decreases with increasing values
of τd/τk. In the other limit, such that τd/τk ≫ 1, kinetics are not important and crystallization proceeds at equilibrium. For τk = 106 s, the
ratio τd/τk is equal to 1 for d = 2 m. We now evaluate the controls on
the thickness of individual intrusions using both simple mechanical
models and observations on both active and fossil volcanic systems.
3.1. Mechanics of emplacement
For the geometry of a sill, solutions are available for the inflation of
a circular fracture fed from the center (Fialko et al., 2001a). For the
purposes of this discussion, we treat roof deformation in an approximate manner using the analytical solution for an elastic plate whose
edges are clamped, which is valid for small roof aspect ratios, i.e. for
small values of the depth to radius ratio of the magma sheet (Fialko
et al., 2001a). This simple solution is within a factor of 3 of that for the
circular crack for aspect ratios that are smaller than 0.5. We assume
a constant magma overpressure ΔP within the intrusive sheet. The
vertical displacement of the roof is largest at the axis, where it is equal
to:
2
w=
4
3 ð1−ν Þ ΔPR
16
E
h3
ð7Þ
where E is Young's modulus, ν Poisson's ratio, h the roof thickness (or
reservoir depth) and R the intrusion radius. The displacement amplitude is proportional to the magmatic overpressure and very sensitive
to the intrusion size (here radius R).
For illustration purposes, we use R = 10 km, h = 5 km and a typical
value of 4 × 1010 Pa for E/(1−ν2). The magma overpressure is unknown
in practice, but may be estimated using two different arguments. One
argument is that this overpressure is due to magma buoyancy and hence
cannot exceed the pressure difference at the top of a static magma
column extending from source to emplacement level. In reality, most of
the magma buoyancy is taken up by viscous head losses due to flow and
hence a static calculation provides an overestimate of magma overpressure. Taking into account the compressibility of silicate melt, the
average buoyancy of basalt in crustal rocks is about Δρ = 102 kg m− 3.
For a magma column extending over 20 km, as appropriate in an
extensional environment, the maximum static magma overpressure is
about 2 × 107 Pa. Another argument is that the magma overpressure
cannot exceed the tensile strength of upper crustal rocks, which is less
than 107 Pa (Rubin, 1995). We therefore consider that ΔP < 107 Pa,
which leads to w < 4 m for R = 10 km and h = 5 km. Note that the
displacement is smaller for smaller magma bodies with R < 10 km.
In this calculation, we have assumed a uniform magma overpressure over the whole roof area, and hence have neglected viscous
losses due to the horizontal flow of magma filling the cavity. In reality,
the magma overpressure must decrease away from the injection
point, so that the average overpressure applied to the roof region is
smaller than the above estimates. We conclude that it is difficult to
envision individual intrusive sheets that are more than a few meters
in thickness. The calculation depends on several parameters and
3.2. Deformation in active volcanic zones
Monitoring of ground deformation in active volcanic zones has
considerably improved recently due to the development of new
instruments and techniques such as satellite InSAR. Most cases investigated so far do not allow clear-cut conclusions on the geometrical
shapes of intrusions and data interpretation relies on either the Mogi
point-source or an inflating spherical body. The geometry of a sill has
been used to fit the data in very few instances. Fialko et al. (2001b)
have studied broad surface uplift in Socorro, New Mexico and
determined that a deep crustal source has been inflating there at a
rate of about 6 × 10− 3 km3 yr− 1 over a 30 yr interval between 1951
and 1981. Such a long period of inflation is obviously not consistent
with a sudden intrusion event and Fialko et al. (2001b) have proposed
that the observed deformation is due to crustal anatexis induced by
the intrusion of mafic magmas. More recently, InSAR images of an
intrusive episode at Eyjafjallajškull volcano, Iceland, indicate the
injection of a sill at a depth of 6.3 km (Pedersen and Sigmundsson,
2006). The sill thickness is about 1 m and the total intruded volume is
small (≈0.03 km3).
3.3. Sills, laccoliths and sill complexes
Several studies document the presence of individual intrusive
units which usually are a few meters thick (Table 5). We briefly
review some of the available evidence.
In Iceland, Gudmundsson (1995) has described sheet swarms with
an average thickness of about 1 m near extinct central volcanoes.
Many of these swarms are associated with large plutons which can be
interpreted as fossil magma reservoirs. The small Njardvik sill is wellexposed and consists of at least seven injections over a total thickness
of 20 m (Burchardt, 2008), which corresponds to an average sheet
thickness of less than 3 m.
In the Isle of Mull, Scotland, where there was a large central
volcano, one finds a large number of sills whose thicknesses typically
vary between 0.5 and 6 m (Preston et al., 1998). These sills involve a
range of magma compositions and were fed from a reservoir or
storage zone. They can be separated in two different groups
depending on their thickness and characteristics. Sills that are thicker
than ≈3 m fed fissure eruptions, are associated with large thermal
aureoles and lack chilled margins, indicating that they were kept
active for extended lengths of time (Holness and Humphreys, 2003).
Thus, they may not be representative of intrusions involved in the
build-up of a magma reservoir at depth.
In areas where there is no evidence for large volcanoes, magma
sheet thicknesses are also in the same range. Laccoliths are typically in
a range of 50 m–1 km thickness and are often found near individual
sills with thicknesses of a few meters (Corry, 1988). For example, one
finds 0.5–10 m thick diorite sills near 10–200 m thick laccoliths at
Henry Mountains, Utah (Johnson and Pollard, 1973). Detailed field
mapping shows that the latter are in fact made of several intrusive
sheets whose thickness is typically a few meters (Jackson and Pollard,
1988; Morgan et al., 2008). Dolerite sills that intrude Silurian sedimentary rocks in SW Connacht, Ireland, have thicknesses in the 2–7 m
range (Mohr, 1990). Amongst these, the thickest ones are clearly
made of several sheets.
At deeper structural levels in the crust, identification of individual
sheets is more difficult but has been made in several instances. 0.1 to
4 m thick units have been identified in the Onion Valley complex,
(Sisson et al., 1996). In the Lightning Creek complex, Queensland,
Australia, the thickness of intrusions ranges from 1 mm to a few
meters (Perring et al., 2000). The border phases of the McDoogle
pluton, Sierra Nevada, California, terminate via lit-par-lit injections of
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
39
Table 5
Sheeted sill complexes.
Location
Composition
Sill thickness
Refs
Sierra Nevada Batholith, Onion Valley, California
Upper sill complex
Lower sill complex
Hornblende gabbro
Hornblende gabbro
0.1–1.5 m
2–4 m
Sisson et al. (1996)
Everest region, eastern Nepal
Leucogranite
cm to several m
Viskupic and Bowring (2005)
Southeast Coast Plutonic Complex, British Columbia
Scuzzy pluton margins
Tonalite
cm to 100 m
Brown (2000)
Queensland, Australia
Lightning Creek sill complex
Quartzofeldspathic
1 mm to a few m
Perring et al. (2000)
McDoogle Pluton and adjacent plutons, Sierra Nevada, California
Sheeted intrusions
Mafic granodiorite
1 to 10 m
Mahan et al. (2003)
Njardvik, Northeast Iceland
Sill
Basalt
3 m on average
Burchardt (2008)
Loch Scridain, Isle of Mull, Scotland
Sill Complex
Tholeiite to rhyolite
1 to 14 m
(Preston et al., 1998;
Holness and Humphreys, 2003)
Icelandic cone-sheet swarms
Extinct central volcanoes
Regional dyke swarms
Geitafell volcano, SE
Thverartindur volcano
Gabbro
Gabbro
Gabbro
Gabbro
1 m on average
2 m on average
0.5 to 1 m
0.9 m on average
Gudmundsson (1995)
Gudmundsson (1995)
Gudmundsson and Brenner (2005)
Klausen (2004)
numerous sheets with a typical thickness of a few meters (Mahan
et al., 2003).
The fact that individual intrusive sheets are typically a few meters
thick is in good agreement with estimates of injection rates and repose
time intervals that have been obtained at a large number of volcanic
and magmatic systems (Crisp, 1984; White et al., 2006). Repose time
intervals between two eruptions are typically between 10 and 102 yr
when primitive lavas are involved (White et al., 2006). They are much
larger for intermediate to silicic compositions, between 104 to 106 yr,
presumably because the formation of chemically evolved melts
requires time for magma accumulation, differentiation and assimilation at depth. Repose time intervals between two eruptions of
primitive lavas provide estimates for the time separating individual
intrusive events. For time intervals between 10 and 100 yr and
eruption rates per unit area in the ~10− 3–10− 2 m yr− 1 range (Crisp,
1984), we find that the characteristic thickness of an individual
intrusive sheet is indeed about 1 m.
4. Two crystallization models
The thermal model has already been described in Michaut and
Jaupart (2006) and its main features are summarized in the
Appendices, where the governing equations are also given. The
thermal problem is solved in the vertical direction only because of the
small aspect ratios of sill complexes. Intrusive sheets of thickness d
are emplaced at a constant time interval τi, such that the average
magma input rate is Q = d/τi. The peculiar thermal evolution of such a
sill complex is recapitulated in detail in Appendix B, where we show
results that did not appear in Michaut and Jaupart (2006). Parameters
of the model are given in Table 2. Calculations were made for a simple
binary eutectic diagram and a simple kinetic function involving
specific choices for parameters that remain poorly constrained. Thus,
the results may not be very accurate and are only meant to illustrate
the important features of the kinetic model. We assess the
applicability of this model using observations on the amounts of
glass in natural intrusions and simple thermal balance arguments.
The importance of kinetic effects may be measured by dimensionless ratio τd/τk, which is very sensitive to the intrusion thickness. In
this section, we compare two cases with the same input rate
Q = 0.025 m yr− 1, one with 4 m thick intrusions (4 m every 160 yr),
for which kinetic effects are negligible, and another one where units
are only 1 m thick (1 m every 40 yr) and crystallize in a kineticallycontrolled regime. For the former case, we chose a thickness of 4 m
because thicker sills may develop convective instabilities upon
cooling and hence evolve in a different regime than the conductive
one studied here. For the latter case, the small thickness was chosen at
the lower end of the natural range in order to illustrate the kinetic
model in its most blatant form, with large amounts of residual glass.
4.1. Thick intrusions: equilibrium crystallization
4.1.1. Thermal evolution
In this case, magma injections are thick enough for crystallization
at thermal equilibrium (τd ≫ τk). Crystallization proceeds to completion even in the first sill emplaced in cold country rock and all latent
heat is released soon after emplacement. Temperatures increase
progressively with each new injection (Fig. 3). At t = τce = 110 kyr,
the solidus temperature is reached in a complex that extends over
2.75 km in height. At that time, there is still no residual melt and yet
another intrusion must occur for temperatures to rise above the
solidus. Subsequent evolution is slow and a permanent body of melt
builds up gradually, so that a large number of intrusions are required
to sustain a large magma reservoir. During the initial phase at
temperatures below the solidus, the amount of latent heat released is
maximum. Once temperatures exceed the solidus, crystallization does
not proceed to completion (Annen and Sparks, 2002) and latent heat
gets released in ever-diminishing amounts. In this second phase,
residual melts become progressively more primitive.
4.1.2. Conditions for the formation of a permanent magma reservoir
Here, “permanent” means that the storage zone sustains a volume
of melt for times longer than the interval between two intrusions. For
thick injections, a minimum input rate is required (Annen et al.,
2008). Each new intrusion acts to enhance the thermal anomaly that
builds up in the roof region. If injections are allowed to proceed
indefinitely, thermal steady-state conditions are achieved, such that
Author's personal copy
40
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
For larger rates of injection, the temperature at the top of the
storage zone eventually increases above the solidus. The approximate
thermal balance is:
Qρ½CP ðTl −Tmax Þ + LχðTmax Þ = k
Tmax −T0
ZS
ð11Þ
where χ(Tmax) is the fraction crystallized at temperature Tmax. This
equation shows that the amount of melt sustained in the storage
zones increases with input rate Q.
4.1.3. Time to reach the solidus
At temperatures below the solidus, the amount of heat lost to the
surroundings by each intrusion is equal to ρ(CPΔT + L) per unit area,
when T < Ts, with ΔT = Tl − T. The heat lost is evacuated by conduction
between two intrusions, i.e. in time τi. We assume that the maximum
temperature is at the top of the system, corresponding to sheets that
get emplaced at the top of the pile. Heat is transported through a
thermal boundary layer of thickness δ ≈ (κt)1/2. An approximate heat
balance at the roof for the cooling of one intrusion is:
Fig. 3. Maximum temperature evolution in two different sill complexes growing at a
rate of 0.025 m/yr. Dashed line, sills are 4 m thick and are injected every 160 yr. Solid
line, sills are 1 m thick and are injected every 40 yr. The solidus temperature is reached
after 67 kyr, i.e. for a sill complex of 1.67 km thickness in the kinetically-controlled
scenario (thin intrusive sheets), and after 110 kyr, i.e. for a sill complex of 2.75 km
thickness, in the equilibrium scenario (thick sheets).
the amount of heat brought by each injection is balanced by heat lost
by conduction through the roof rocks. The minimum magma input
rate Qc for a permanent melt body corresponds to such a steady-state
with the temperature at the top of the sill complex at the solidus and
can be estimated with a very simple thermal balance. As will be seen
later, the time required to reach the solidus exceeds 100,000 years.
For such a long time, the thermal halo extends over a few kilometers
and hence over a large fraction of the roof. We may therefore assume
that the conductive heat flux out of the reservoir is:
φT = k
TS −T0
ZS
ρd½CP ΔT + L≈k
T−T0
τi
ðκtÞ1 = 2
ð12Þ
where τi is the time until the next intrusion and T the roof temperature. Thus, T = Ts is achieved at time t = τce such that:
2
Ts −T0
−2
Q
τce ≈κ
ΔTs + L= CP
ð13Þ
where we have used Q = d/τi. This simple analysis fits very well the
results of the full numerical simulation (Fig. 4).
4.2. Thin intrusions
For the specific parameterization used in the present calculations,
crystallization is kinetically-limited for d < 4 m. In this case, it does not
go to completion and leaves a glassy residue in the solidified sheet.
ð8Þ
where TS is the solidus temperature, T0 ≈ 0 the surface temperature
and ZS the roof thickness, i.e. the reservoir depth. The total heat
released by one intrusion is the sum of sensible heat due to cooling
from the injection temperature Tl to the solidus temperature and
latent heat due to crystallization. Crystallization proceeds to completion and the thermal balance is:
Qc ρ½CP ðTl −Ts Þ + L = k
Ts −T0
ZS
ð9Þ
and hence:
Qc =
kðTs −T0 Þ
ZS ρ½CP ðTl −Ts Þ + L
ð10Þ
Using k = 2.5 W m− 1 K− 1, Tl = 1187 °C, Ts = 1000 °C, T0 = 0 °C, ρ =
2500 kg m− 3, CP = 1300 J kg− 1 K− 1 and ZS = 5000 m, L = 4.18 × 105 J/k,
the critical rate of injection is equal to 0.01 m yr− 1 per unit area, i.e. 4 m
every 400 yr, at the top of the range of geological input rates determined by Crisp (1984). This value is consistent with the numerical
results of Annen et al. (2008). One should note that this estimate
corresponds to thermal steady-state in the roof region and that more
heat is lost by the magma pile in early phases of intrusion, when the
thermal aureole in the roof region is thin. As a consequence, early
formation of a permanent magma reservoir requires magma input rates
that are larger than Qc.
Fig. 4. Time to reach the solidus or time for catastrophic devitrification as a function of
magma input rate Q. Circles: time for catastrophic devitrification τck for 1 m thick
injections; diamonds: time to reach the solidus τce for4m thick intrusions. Thin line:
time to reach the solidus calculated from Eq. (13) using TS = 1000 °C, TL = 1187 °C. Bold
line, time for catastrophic devitrification, i.e. time to reach temperature Tc at which
kinetics start to kick in, calculated from a similar thermal balance as Eq. (13), but with
no latent heat release. In both cases, the simple thermal balance allows a very good fit to
the full numerical results.
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
We present results for d = 1 m, which should be considered as a limit
case emphasizing the amount of glass that can accumulate in a frozen
magma pile.
4.2.1. Thermal and structural evolution
The basic evolution is the same as before, with temperatures
gradually rising in the sill complex (Fig. 3). Crystallization does not
proceed to completion for a large number of injections and a large
amount of uncrystallized magma accumulates in the storage zone.
With time, as temperatures increase, the thermal contrast between
a new injection and the older sheets decreases, which enhances
crystallization and hence latent heat release. Eventually, temperatures
become large enough to reactivate nucleation and growth. Because of
latent heat release, the magma pile has a peculiar thermal evolution
with the residual glass crystallizing whilst temperatures are rising.
Starting from low crystallization rates at large undercoolings (Fig. 1),
heating promotes faster nucleation and growth and hence large rates
of latent heat release. This process is such that temperatures rise very
rapidly and may be called runaway devitrification (see Appendix B).
The solidus is reached at t = τck = 67 kyr for Q = 0.025 m yr− 1,
43,000 years earlier than for 4-m thick intrusions (Fig. 3). At that
time, a transient reservoir of residual liquid in a crystal mush (with
10–25 wt.% liquid) is formed and the sill complex is much thinner
than in the equilibrium model.
Extensive crystallization of magma within the sill complex leads to
the formation of a large volume of very evolved and homogeneous melt.
In addition, rapid heating enhances melting of the encasing rocks. A
large volume of evolved melt from two different sources is therefore
made available for eruption. Fractional crystallization is rapid and
typically occurs over a few tens to a few hundred years, so that almost
no intermediate lavas can be sampled by eruption. On Fig. 5, melt
compositions are sampled every 40 yr at the maximum temperature in
the sill complex, just before a new intrusion event. Before the runaway
devitrification event, only primitive lavas associated with the latest
injection can be sampled at the surface. Due to devitrification, the silica
content of the non-crystallized part jumps from 57 to 75 wt.% in
40 years only. In this example, devitrification generates melt at the
eutectic.
The time to the devitrification event increases as the magma input
rate decreases (Fig. 4). As in the equilibrium model, this time is
approximately proportional to Q− 2 because the basic thermal balance
is essentially the same. For the kinetic function used here, devitrifi-
41
cation begins at temperatures below the solidus, and hence the time
to the runaway event is less than the time to reach the solidus in the
magma pile.
Dividing the volumes of the Fish Canyon and Bishop Tuff deposits
by the areas of the associated calderas, and accounting for the
presence of crystals in the Fish Canyon tuff, we estimate that pure
melt had accumulated over a cumulative thickness of at least 1 km in
both cases. By extrapolation of the curves in Fig. 11, these volumes are
achieved at magma input rates of ≥10− 3 m/yr. These curves can also
be deduced from a simple thermal argument which reproduces the
full numerical simulations, and hence are valid over a larger parameter range than that of the calculations. For our calculations, we have
used relatively high values for the solidus and liquidus temperatures
of country rock (800° and 1100 °C) and primitive magma (1000° to
1200 °C). Using lower values for one or the other, much larger
volumes of liquid would be obtained (see Appendix B).
If the deep magma source remains active after the catastrophic
devitrification event, temperatures continue to increase gradually
with each new magma injection. A key difference with the initial
phase is that temperatures are high enough for equilibrium crystallization. As temperatures rise, new injections crystallize less than in
the initial, kinetically-controlled, phase, and a permanent melt body
grows progressively, as in the equilibrium model. The composition of
the melt gradually evolves from highly differentiated to primitive,
spanning the whole compositional range if the magma source remains
active for long enough (Fig. 5).
5. Conditions for the rejuvenation of a thick magma pile
by devitrification
As explained above, the kinetic model relies on a single function
for the crystallization rate that lumps together the nucleation and
growth of several mineral phases. Laboratory kinetic data are not
sufficient to determine the full function over the whole crystallization
interval (Brandeis and Jaupart, 1987b), implying that our numerical
model may not be perfectly accurate. Furthermore, the calculations
were made for a basalt to take advantage of the relatively large data
set available and in particular of the experiments by Pupier et al.
(2008). Thus, they cannot be applied directly to other magmas. They
do illustrate, however, the basic physical principles for runaway
devitrification and allow estimates of the intrusive sheet thicknesses
that are required. We also note that crystallization kinetics seem to be
Fig. 5. Left: maximum temperature as a function of time for 1 m thick intrusions injected every 40 yr. Right: composition of the non-crystallized fraction at the depth where the
temperature is maximum, as a function of time. Until the catastrophic devitrification event, glasses are primitive in composition and hence lavas that can be sampled at the surface
are also primitive and associated with new injections at depth. As runaway devitrification proceeds, evolved liquids are formed rapidly.
Author's personal copy
42
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
more sluggish for more evolved melts, such as andesites for example
(Table 4). To evaluate the relevance of the devitrification mechanism
in natural conditions, we turn to geological observations and simple
thermal calculations.
5.1. Residual glass in mafic intrusions
Reduced to its essentials, the mechanism requires that enough
residual glass remains after cooling of the initial intrusions for
devitrification to elevate temperatures significantly. Data on the
amount of residual glass in a number of sills and dykes can be found in
many old papers aimed at documenting magmatic differentiation
trends. We list a few such determinations in the following, keeping in
mind that slow devitrification over large timescales may obliterate
interstitial glassy residues. Another important limitation is that all the
data pertain to high-level intrusions involving magmas which had
already started to degas, as shown by the presence of small vesicles.
Degassing generates undercooled melt and hence may trigger early
crystallization before emplacement. Thus, the data may well underestimate the amount of glass that can be generated at crustal depths
that are appropriate for magma reservoirs. Emplacement at deeper
crustal levels would occur in hotter country rock, but the thermal
contrast would still be large enough for glass generation (Michaut and
Jaupart, 2006).
For reference, we note that the base of the Eskdalemuir tholeiitic
dyke, in the Southern Uplands of Scotland, whose thickness is about
50 m, contains 42.2% glass (Elliott, 1956). Chilled margins invariably
contain much more glass than intrusion interiors and we focus on the
latter. The glass fraction is 25% within a 3 m-thick dolerite sill at
Dalmeny, Scotland, and 30% within a 15-m wide dolerite dyke at
Kinkell, Scotland (including chlorophaeite, a product of late glass
alteration) (Walker, 1930). Glass percentages from the central parts of
three other basaltic Tertiary Scottish dykes of unreported thicknesses
range from 6 to 21% (Walker, 1935). All samples analyzed contain
significant amounts of ilmenite and magnetite, some of which seem
be due to the late breakdown of the glass (Walker, 1935). Thus, the
amount of pristine glass may have been larger than the values quoted.
Reviewing data from a large number of basaltic samples, Walker
(1935) found that the amounts of glass and pyroxene are anticorrelated, which indicates that the glass was formed when pyroxene
was on the liquidus. This illustrates the importance of pyroxene in the
kinetically-controlled crystallization sequence, which had already
been emphasized by the experiments of Pupier et al., (2008).
The ~65 Ma Delakhari sill is part of the Deccan Trap intrusion
sequence and exhibits upper and lower chilled zones containing >12%
glass (Sen, 1980). Glass is present (about 5 to 10%) throughout the
whole intrusion, even though it is quite thick (~200 m). This large sill
is noteworthy because it fed eruptive fissures and hence was probably
active for an extended length of time, which is not favorable to the
formation of glass (Holness and Humphreys, 2003). Interestingly, the
pyroxene content is also negatively correlated with the glass content,
decreasing from ~ 32% at the center to ~ 27% at the margins.
5.2. Devitrification by clinopyroxene nucleation and growth
Several experimental studies illustrate how devitrification proceeds in basaltic glass samples that are heated in controlled conditions. Magnetite and pyroxene are the first mineral phases to appear
at temperatures which may be as low as 650 °C (Bandyopadhyay et al.,
1983). Yilmaz et al. (1996) have identified two exothermic peaks of
crystallization in basaltic glass at 788 °C and 845 °C, corresponding to
diopside and augite, respectively. Znidarsic-Pongrac and Kolar (1991)
report that the first phase to appear in diabase glass is diopside,
at 865 °C, and that crystallization proceeds from magnetite nuclei.
Observations on natural glassy basalts confirm that clinopyroxene is
indeed the first major phase to crystallize upon devitrification (Fowler
et al., 2002; Monecke et al., 2003).
Together with the Pupier et al. (2008) experiments, these laboratory studies emphasize the peculiar kinetics of pyroxene crystallization in both the cooling of a melt and the heating of a basaltic glass.
Such characteristics take on special significance in the light of the fact
that the amounts of glass and pyroxenes are negatively correlated in
tholeiite and dolerite sills.
5.3. Amount of glass required for a heating pulse
The magnitude of heating induced by devitrification can be
calculated without a complete thermal model. Devitrification generates heat locally and hence may raise temperatures over a thick pile
rapidly, with only thin thermal boundary layers developing at the top
and bottom, so that one may neglect heat loss to the exterior (see
Fig. 10). The temperature rise ΔT due to latent heat release by crystallization of a mass fraction ΔΦ of clinopyroxene is:
Cp ΔT≈LCpx ΔΦ
ð14Þ
The latent heat of crystallization of clinopyroxene LCpx is respectively
7.7 × 105 for clinoenstatite and 6.4 × 105 J/kg for diopside (Richet and
Bottinga, 1986). Using LCpx = 7 × 105 J/kg, Cp = 1200 J/kg/K, one finds
that a temperature rise in the 40–100 °C range, as inferred for the Bishop
Tuff and Fish Canyon Tuff for example (Bachmann and Bergantz, 2003;
Wark et al., 2007), can be achieved by the crystallization of 7 to 17%
glass. With only this amount of glass, however, devitrification consumes
all the uncrystallized volume available and no melt is left at the end. In
order to generate melt (see Fig. 10), an additional amount of glass is
required: to achieve a melt fraction in the 10–20% range, a total of about
20–30% interstitial glass is needed in the complex. This total amount of
glass is both expected and observed in thin intrusive magma sheets, as
shown above. Melting of roof rocks does not require more glass because
it is due to heat from the upper boundary layer of the complex that
undergoes complete crystallization (Fig. 10).
Experiments show that growth of clinopyroxene crystals from
heated basaltic glass occurs at the expense of smaller particles and
nuclei (Bandyopadhyay et al., 1983), suggesting that runaway devitrification may lead to pyroxene crystals that are larger than the
primary ones that formed during the initial cooling phase.
6. Discussion
6.1. Contrasting features of the two crystallization models
For each crystallization model, specific conditions must be met that
are difficult to verify in the natural environment. The equilibrium
model only works if the magma input rate exceeds a threshold value
which is outside the range of values that have been determined in a
large number of plutons and volcanic systems, whereas the kinetic one
requires that individual intrusive sheets are thinner than a limit value
of a few meters. In a similar fashion, the gas-sparging model for the
rejuvenation of thick cumulate piles relies on large open-space permeability allowing the flow of large quantities of gas. It is difficult to
establish which requirement in this list is met in geological conditions,
which makes model validation difficult. We may evaluate which
model is most appropriate, however, by comparing their respective
merits. Both crystallization models require that a minimum quantity of
magma must be accumulated before a permanent melt reservoir can
form. They differ greatly, however, in their other characteristics.
In the kinetic model, the large amount of latent heat that gets
released upon devitrification cannot be evacuated efficiently by
conduction. A large temperature rise ensues, leading to a significant
amount of evolved melt in a short time. In contrast, in the equilibrium
model, latent heat is released progressively, which allows efficient
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
heat loss through roof rocks and hence enhances cooling. Thus, the
solidus temperature is reached at a later time than in the kinetic
model. The difference of timing between the two models increases
with decreasing magma input rate (Fig. 4). As a consequence, for the
same magma input rate, a much larger magma volume is required to
reach the solidus in the equilibrium model than in the kineticallycontrolled one. For the examples in Fig. 3, the difference is significant:
with respect to the kinetic model, the equilibrium model requires an
additional magma thickness of >1 km.
In contrast to the equilibrium model, the kinetic one is insensitive
to vagaries of injection rate. This feature manifests itself in several
ways. Firstly, there is no critical input rate for the development of a
permanent melt reservoir. During the first phase of gradual
temperature increase, the amount of latent heat released increases
with time, which prevents the establishment of thermal steady-state
conditions in the roof region. Secondly, the amount of liquid formed
increases with the time between two intrusions, i.e. increases with
decreasing input rate (see Appendix B, Fig. 11). This is in sharp
contrast with the equilibrium model, which predicts the opposite
behaviour, i.e. that the volume of melt generated increases with the
input rate and decreases with the time separating two intrusions.
43
contrast across the unstable part of the boundary layer, is typically
several tens of degrees (Davaille and Jaupart, 1993). The Makaopuhi
data are consistent with ΔTe ≈ 60 K. Magma viscosity μ depends
strongly on composition and takes values from about 10 Pa s for mafic
melts to more than 108 Pa s for felsic crystal-rich magmas. The heat
flux at the top of the intrusion, ϕl, can be estimated from the general
scaling law:
ϕl =
Ra
Rac
1 = 3
ΔT
k e =
dc
2
4
2
αk ΔTe CP ρ g
μRac
!1 = 3
ð15Þ
where Rac ≈ 103 is the critical Rayleigh number for boundary layer
instability, α the coefficient of thermal expansion and dc the reservoir
thickness. Note that this heat flux is in fact independent of the intrusion thickness because it is determined by the breakdown of a thin
thermal boundary layer at the top of the melt body. The heat flux lost
by convection decreases from ~30 W m− 2 for μ = 106 Pa s, to 5 W m− 2
for μ = 2 × 108 Pa s (Fig. 6b).
Heat is supplied by a new magma addition to a sill complex at
average rate ϕb such that:
ϕb = ρQ ðCP ΔTb + LÞ
ð16Þ
6.2. Cooling regime of thick sills
The formation of a permanent melt body requires that magma
remains above the solidus between two successive intrusive events.
Convective instabilities should develop upon cooling in magma
sheets that are thicker than about 10 m (Worster et al., 1990; Davaille
and Jaupart, 1993). In such cases, therefore, cooling proceeds by
convection, which is much more efficient than conduction.
An unstable thermal boundary layer develops at the top of a liquid
body that gets cooled from above, which controls the rate of heat loss.
Magmas have temperature-dependent viscosity, implying that only
the lower part of this boundary layer breaks down, so that the upper,
cold and viscous part of this boundary layer remains stagnant and
convective breakdown only affects its lower part. The stagnant layer
includes both fully solidified magma and part of the melting interval,
as shown by theoretical analysis and direct observation in lava lakes
(Worster et al., 1990; Davaille and Jaupart, 1993). Scaling for variable
viscosity convection as well as temperature oscillations recorded in
the Makaopuhi lava lake, Hawaii, indicate that ΔTe, the temperature
where ΔTb =TL −T is the temperature drop that is achieved. Using
geological rates of injection per unit area derived from Crisp (1984) and
relevant values of the different parameters, we find that the convective
heat loss overwhelms the amount of heat brought by new intrusions,
even for very large viscosities (Fig. 6a and b). For a given input rate, the
thicker each intrusive sheet is, the larger the time between two
injections is, and hence the more advanced the cooling of each unit is. In
such conditions, a permanent melt body cannot be sustained. In this
sense therefore, the succession of small but frequent magma increments
which cool by conduction is in fact a more efficient heating mechanism
than the punctuated emplacement of large magma volumes.
From this discussion, we conclude that convection must be
prevented for the equilibrium model of reservoir formation to work,
which sets an upper limit of about 10 m for the thickness of intrusive
sheets. From our previous estimates of kinetic controls on nucleation
and growth, this model further requires that intrusive sheets are
thicker than ≈4 m. We conclude that the equilibrium model of
magma reservoir formation operates in a restricted thickness range.
Fig. 6. a) Heat flux released by magma injection as a function of intrusion rate Q, using ΔTb = 300 °C, ρ = 2500 kg/m3 and L = − 4.18 × 105 J kg− 1 in (16). b) Heat lost by convection as
a function of melt viscosity, using α = 5 × 10− 5 K− 1, ΔTe = 50 K in Eq. (15).
Author's personal copy
44
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
6.3. Crystal/melt separation
We have so far referred mostly to magma reservoirs that grow
from basaltic precursor melts, but the same basic principles also apply
to other magma compositions. In fact, we have pointed out that
kinetic limitations on crystallization may well be more stringent for
andesitic and dacitic melts. For the sake of the present discussion, we
rely on the Bishop Tuff system because it has been studied extensively
by a large number of authors and also because it shares many features
with other systems. The crystal content of the Bishop Tuff rhyolite was
generally small and increased as the eruption tapped deeper portions
of the reservoir. The large volumes of highly evolved melt were
dominantly produced by fractional crystallization (Michael, 1983;
Hildreth and Wilson, 2007), implying that an enormous repository of
cumulates was left behind in the upper crust. Trace element
concentrations vary by large amounts in this melt and delineate
strong vertical gradients in the upper part of the reservoir. Such lack of
homogenization led Hildreth and Wilson (2007) to postulate that
individual magma batches were extracted piecemeal from a thick
mush and accumulated at the top of the reservoir without mixing.
Using the ages of zircons, one of the earliest phase to crystallize, these
authors conclude that the reservoir system developed over at least
105 years. A model for the Bishop Tuff must address two questions:
how does evolved melt migrates out of the initial storage zone (where
it was generated by crystallization), and how can this melt remain
near its liquidus as it ponds beneath the cold roof region?
The two models of reservoir formation studied in this paper and
the mush rejuvenation model by gas sparging of (Bachmann and
Bergantz, 2003) do not specify how evolved melt gets separated from
a partially crystallized mushy pile. Model calculations for pervasive
porous flow involving compaction of the solid phase as well as local
crystal settling suggest small melt migration velocities in the 10− 1–
10− 2 m y− 1 range, implying that it takes 104–105 years to accumulate
evolved magma over 1 to 2 km (Bachmann and Bergantz, 2004).
These calculations rely on many unknowns and must be taken with
precaution, but they do suggest that extraction may proceed at rates
that are consistent with the large lifetime of the Bishop Tuff system.
More stringent constraints, however, come from mass and heat
balance arguments.
Regardless of the melt migration mechanism, a key problem is to
account for the preservation of a large volume of melt beneath the
reservoir roof. Hildreth and Wilson (2007) acknowledge this difficulty
and discuss various mechanisms to generate a stably stratified body
where convection does not operate. Even without convection,
however, cooling will affect a large volume of melt. Over the lifetime
of the Bishop Tuff system, which must be 105 years or more, conductive
cooling proceeds over a thickness of atleast 2 km, i.e. through the
whole volume of rhyolite that got erupted. This rules out the rapid
emplacement of a large liquid body at the top of the storage zone a long
time before eruption and sets the discussion back to the starting point.
For a melt body that grows incrementally against the roof, which is the
cooling interface, the problem is analogous to that for the whole
magma reservoir and we can use the same arguments as above, and
specifically the same thermal balance (Eq. (11)). Incoming melt
supplies heat that sustains losses through the roof. Evolved high-silica
rhyolite, however, crystallizes over a relatively narrow temperature
interval. For ≈10% crystallization, corresponding to the average crystal
content of the erupted Bishop Tuff magma, the temperature drop is
less than 20 °C and the minimum input rate is at least 0.1 m yr− 1
according to Eq. (11), so that the evolved melt body must have been
built up in less than 30,000 years. This is much shorter (by a factor of at
least 5) than the lifetime of the Bishop Tuff system (Hildreth and
Wilson, 2007). If the precursor melts were dacite or low-silica rhyolite,
they must have crystallized by more than 50% before extraction of
residual melt took place, and hence they must have been injected at
rates which were larger than 0.1 m yr− 1 by a factor of at least two and
possibly as much as five, depending on composition. Such rates of
magma production are undocumented.
An alternative mechanism for melt migration involves flow
through veins and channels in an almost rigid crystal mush, i.e.
through dyke-like fractures, which proceeds rapidly. Hildreth and
Wilson (2007) favour the latter mechanism because it is consistent
with piecemeal accumulation of rhyolitic melt at the top of the
reservoir. They comment that many granitoid plutons contain veins
and dike-like structures filled with evolved liquids and connected to
pockets with enhanced melt contents. As discussed in Appendix B, the
kinetic model predicts the generation of such pockets within the thick
mush zone upon devitrification. The phenomenon is due to the
competition between diffusive heat transport and kinetically-driven
latent heat release, and has been explained at length by Brandeis et al.
(1984). Furthermore, the large local density decrease within the mush
due to heating and volatile exsolution driven by crystallization leads
to the build-up of an overpressure. The temperature rise alone leads to
an overpressure of several MPa. The effect of gas exsolution is more
dramatic and likely leads to fracturing (Fig. 7) (see Appendix C for
details on the calculations). A key point, of course, is that melt and
melt pockets are generated shortly before eruption, which circumvents the thermal difficulty of maintaining a thick magma body at the
top of the reservoir for a large length of time.
6.4. Bimodal volcanism
Volcanic systems that exhibit a time-progression from primitive to
intermediate lavas over large time intervals followed by a voluminous
eruption of very evolved melt have been found in different geological
settings. Lipman (2007) has described this eruption pattern for the
recent calc-alkaline volcanic fields of the western United States. On
the whole, Icelandic volcanism is strongly bimodal, with approximately 85% basalt, 12% rhyolite and only 3% intermediate lavas
(Einarsson, 1994). Silicic volcanism is confined to central volcanoes
and is often associated with caldera formation. Recent petrological
studies seem to favour a crustal origin for the evolved melt but
Fig. 7. Overpressure evolution following a temperature increase and volatiles
exsolution driven by crystallization, for two different values of the initial fraction of
crystals Mci = 50 and 70%. Thick lines: solubility law for water in mafic magmas, i.e.
n = 0.7 and s = 6.8 × 10− 8; thin lines: solubility law for water in rhyolitic magmas, i.e.
n = 0.5 and s = 4.11 × 10− 6. Dashed lines: ΔT = 100 °C, solid lines ΔT = 0. We use Pi =
1.4 × 108 Pa, T = 1000 K, MH2O = 18 × 10− 3 kg mol− 1, R = 8.314 J/mol/K, L = 20 km,
h = 5 km, d = 1 km, E/(1 − ν 2 ) = 5 × 10 10 Pa, ρ c = 2800 kg/m 3 , ρ l = 2600 kg/m 3 ,
α = 3 × 10− 5 K− 1.
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
fractionation of a more primitive magma has also been invoked (Geist
and Larson, 1995).
Bimodal volcanism has been explained in various ways. MourtadaBonnefoi et al. (1999) attributed it to variations of physical properties
in a magma reservoir that evolves due to replenishment and eruption,
and Marsh (1981) to the sampling efficacy of eruptions. Geist and
Larson (1995) argued that bimodal volcanism is a feature of relatively
dry magma systems, such that water saturation and exsolution are
achieved very late in the differentiation trend. All these explanations
rely on the assumption of a large and well-mixed homogeneous melt
body, in contradiction with the very framework of the present study.
The kinetic model offers an alternative explanation that is consistent
with incremental reservoir growth and petrological constraints. In
this framework, evolved melts originate in part from partial melting of
country rock and in part by fractionation of the parent magma, with
proportions that depend on magma and rock compositions.
7. Conclusion
All the quantitative models that have been proposed for the
thermal evolution of large magma reservoirs suffer from similar
deficiencies, namely that they require rather extreme values for their
key parameters (thin intrusions for the kinetic model, very large and
sustained magma input rates for the equilibrium one, very large
permeability values for the gas sparging model). These quantitative
models provide insights into possible mechanisms and identify the
key variables, but must be treated with caution because they are not
yet at the level of sophistication that would allow detailed comparisons with data. Thus, most of the answers must be sought in field
observations, which is why we have made an extensive literature
search to document the three key features of the kinetic model: the
presence and amount of glass in intrusions, the thickness of individual
intrusive sheets and finally the behaviour of glass upon devitrification.
Our thermal model illustrates the likely long-term consequences
of sluggish crystallization kinetics in magma reservoirs that grow by
multiple thin injections. One of its key predictions is the occurrence of
a late heating phase due to devitrification of residual glass that has
accumulated over large lengths of time. Devitrification proceeds by a
burst of crystallization occurring as temperatures are rising, which is
consistent with deductions from zoning in quartz crystals from the
Bishop Tuff (Wark et al., 2007). Alternative explanations relying on
input from separate intrusive bodies have been put forward, but they
involve complicated heat and material transport processes as well as
major changes in the intrusion sequence (Wark et al., 2007). The
kinetic model involves only one magmatic system and only one
intrusion sequence and makes specific predictions that can be tested.
Laboratory data and natural observations point to pyroxene nucleation as a major factor in the evolution of a basaltic sill complex, both
in the initial cooling phase that follows emplacement and in the late
devitrification phase. Crystallization kinetics are more sluggish in
silicic magmas than in basaltic ones but are not documented in
sufficient detail for specific predictions.
The kinetic model can be reduced to a simple quantitative calculation independently of the full emplacement/crystallization sequence.
This allows direct use of field data to evaluate its potential. We find that
devitrification of about 20%–30% interstitial glass leads to rapid heating
and rejuvenation of a thick solidified sill complex. One key requirement
of the kinetic model is that the thickness of individual intrusive sheets
does not exceed about 4 m. This limitation is consistent with mechanical
constraints on magma emplacement, which are met more easily
with thin intrusions than with thick ones. The kinetic model has many
appealing features, but it relies on rather scant data on nucleation
and growth kinetic rates. Additional laboratory measurements would
allow more accurate model predictions. They would also allow us to
draw more information from the many disequilibrium features that
have been observed in cumulate igneous rocks.
45
Acknowledgements
We are grateful to Catherine Annen and an anonymous reviewer
for their useful comments and criticisms.
Appendix A. Model equations
Sills extend over large horizontal distances so that horizontal heat
transport can be neglected. Thus, we use the following heat equation
for both the sill complex and encasing rocks:
ρCP
∂T
∂2 T
∂U
= k 2 −ρL
∂t
∂t
∂z
ð17Þ
where T is temperature, z the vertical coordinate, k thermal conductivity, Cp heat capacity, L latent heat of crystallization and U the
melt fraction, see Table 2 for parameter values. In the country rock,
successive intrusions can lead to melting, which proceeds at equilibrium. In the sill complex, the non-crystallized fraction U depends on
crystallization kinetics.
Boundary conditions are as follows. The temperature is set to zero
at Earth's surface, and a fixed heat flux is imposed at the base of the
computational domain which is much larger than the magma storage
zone. Initially, the crust is in equilibrium with ageothermal gradient
equal to 15 °C/km. Each sheet is intruded instantaneously at depth ZS.
After each injection, the underlying pile is displaced downward, so
that the thickness of the roof rocks remains constant. We refer to
Michaut and Jaupart (2006) for more details and a discussion of other
emplacement sequences. Eq. (17) is solved using different expression
for melt fraction U in country rock and in the sill complex.
In country rock, melting proceeds at equilibrium and the melt
fraction depends only on the phase diagram, i.e. on temperature and
composition. Here we consider that the melt fraction is a linear
function of temperature over the melting interval:
U=
T−Tsr
Tlr −Tsr
ð18Þ
where Tsr = 800 °C and Tlr = 1100 °C are the solidus and liquidus
temperatures of country rock.
In the sill complex, the melt fraction is a function of the rates of
nucleation and growth which both depend on undercooling, as
specified in the main text. In kinetically-controlled conditions,
crystallization may proceed at temperatures that are below the
solidus, and hence the solidus temperature shown in Fig. 8 is used
only as a reference. The liquidus temperature varies as a function of
silica content in the melt (Fig. 8), and is given by, for 0.45 ≤ C ≤ 0.75:
Tl = 1000−
20
ðC−Ce Þ
3
ð19Þ
where C is the SiO2 content in wt.% in the melt, and Ce = 0.75 is the
eutectic composition. The balance in SiO2 content gives:
UC = C0 −Cs ð1−UÞ
ð20Þ
where C0 = 0.47 denotes the initial SiO2 content of the magma and
Cs = 0.4 is the SiO2 content in crystals.
For simplicity, we neglect changes in the physical properties of the
non-crystallized fraction involved in the formation of glass. The glass
transition occurs over a temperature interval that varies as a function
of composition, pressure and cooling rate, and hence cannot be described by a single temperature (Richet and Bottinga, 1986). However,
it is likely to be in the 600–900 °C range for the compositions of the
model: for instance it is reported at 630 °C for diopside and 600 °C for
obsidian (Richet and Bottinga, 1986; Sturkell and Sigmundsson,
Author's personal copy
46
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
Fig. 8. Simple phase diagram adopted for the model calculations. Solid lines: liquidus
and solidus temperatures as a function of composition in terms of SiO2 content in wt.%.
Dashed line: temperature for which the kinetic rate function f is maximum.
1995). Thus, the glass transition is crossed at the time of the sharp
temperature rise.
Appendix B. Thermal and compositional evolution for
thin intrusions
We summarize here the thermal and structural evolution of a sill
complex developing out of thin intrusions, such that crystallization is
kinetically-controlled. More details can be found in Michaut and
Jaupart (2006).
Thin injections do not crystallize completely and leave a glassy
residue. With new intrusions, temperatures increase gradually
(Fig. 9). Once the undercooling is small enough crystallization gets
reactivated in the complex. Latent heat released by crystallization
leads to larger temperatures and hence enhanced crystallization rates.
Starting from low temperatures, a temperature rise causes a positive
feedback between latent heat release and crystallization. This positive
feedback results in rapid heating associated with extensive crystallization and differentiation. The effect of devitrification becomes
significant (Figs. 3 and 9) when the increase in temperature due to
latent heat release becomes larger than ΔTs ~ 10 °C in Δts ~ 103 yr.
Using U = 0.5, such a heating rate is achieved for f(T*) ≥ CPΔTsτk/
ULΔts ~ 2 × 10− 6, i.e. for T* ~ 0.67 or T ~ 700 °C if TL = 1187 °C. This
threshold temperature is reached at time t = τck = 374.2 kyr for an
injection rate of 0.01 m yr− 1 (Fig. 9). In this scenario, a large
amount of latent heat is released in a short devitrification burst.
This burst is so rapid that heat cannot be evacuated by conduction
through the overlying rocks. Thus, the central part of the sill
complex, away from boundary layers at the top and bottom, heats
up at a quasi-uniform rate with a quasi-uniform temperature distribution. Thermal boundary layers with significant temperature
gradients develop at the upper and lower margins of the complex
and some of the latent heat released there gets evacuated to country
rock, inducing melting.
In the first phase of gradual temperature rise, temperature is
maximum at the top of the complex, where new magma gets
intruded. Once crystal nucleation and growth get reactivated in the
sill complex, runaway devitrification proceeds from top to bottom. For
an injection rate Q = 0.025 m/yr at t = 67400 yr, 400 yr after the
beginning of the heating pulse, devitrification has affected a thickness
of about 900 m (Fig. 10). Residual liquid remains in crystal mush
lenses containing 10 to 25 wt % liquid. On Fig. 10, two lenses of crystal
mush have been formed one after the other, in association with two
separate peaks in the temperature evolution (Fig. 3). The melt is very
evolved chemically and is thermally and compositionally homogeneous because it is at the eutectic. At that time, a cumulative thickness
of about 60 m of pure melt has been formed. More liquid is present in
the complex, however, because of melting in roof rocks. The amount
of melt formed by devitrification increases with the time between
injections, i.e. with decreasing injection rate (Fig. 11). Indeed, the
thickness of the central region, which melts and fractionates during
devitrification, increases with the time between two injections.
For very low input rates (~1–3 × 10− 3 m/yr), i.e. large replenishment
timescales (of the order of several hundreds of years to one thousand
years), the amount of liquid formed can represent over several
hundreds of meters per unit area, i.e. corresponding to melt volumes
involved in the Bishop Tuff and Fish Canyon Tuff. In our model, the
solidus temperatures of both evolved magma (1000 °C) and country
rock (1100 °C) are relatively high. More melt would be formed if these
temperatures were set to lower values.
Appendix C. Overpressure in the reservoir
Within the rejuvenated magma, overpressure must build up in
response to the decrease in density due to heating and volatile
exsolution driven by crystallization. To calculate the pressure
evolution as crystallization proceeds in the chamber, we modify the
model of Tait et al. (1989) to account for thermal expansion and for
the presence of crystals in the melt that formed at the time of magma
injection. We also consider country rock deformation in a geometrical
configuration that is appropriate for sills. Initial conditions are such
that the melt is saturated with dissolved water and there is no gas
phase. Mass conservation of liquid, crystals and gas are written as
follows:
Fig. 9. Evolution of the maximum temperature in a sill complex growing at a rate of 1 m
every 100 yr (i.e. 0.01 m/yr). At t = τck = 374200 yr, runaway devitrification proceeds
and a large reservoir of evolved liquid forms rapidly.
mc + ml + mg = M
ð21Þ
ml + mc = Mli + Mci
ð22Þ
md = xml
ð23Þ
mg + md = xi Mli
ð24Þ
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
47
Fig. 10. Solid lines: temperature as a function of depth in a sill complex formed at a rate of 1 m every 40 yr (0.025 m/yr), after 1685 injections, at t = 67400 yr, 400 yr after the
beginning of devitrification. Left: non-crystallized fraction (either glass or liquid depending on the temperature) as a function of depth, right, composition of the non-crystallized
fraction in terms of SiO2 content.
where M is the total mass (which is constant), Mli and Mci are the
initial masses of liquid and crystals, ml, mc and mg are the masses of
liquid, crystals and gas, md is the mass of water dissolved in the liquid,
x the mass fraction of dissolved water in the liquid. Suffix i stands
for initial values. The total volume V at pressure P and temperature T
is:
where ρc and ρl are the densities of the crystals and liquid. Changes
in liquid and crystal volumes due to temperature variations are given
by:
P−Pi
+ αðT−Ti Þ
Vl ðP; TÞ = Vli ðPi ; Ti Þ 1−
β
ð28Þ
VðP; TÞ = Vl ðP; TÞ + Vc ðP; TÞ + Vg ðP; TÞ
Vc ðP; TÞ = Vci ðPi ; Ti Þð1 + αðT−Ti ÞÞ
ð29Þ
ð25Þ
and the initial volume at pressure Pi and temperature Ti is:
Vi ðPi ; Ti Þ = Vli ðPi ; Ti Þ + Vci ðPi ; Ti Þ
ð26Þ
Using the mass conservation (22) at pressure Pi and temperature
Ti, we get:
Vli ðPi ; Ti Þ = Vl ðPi ; Ti Þ + ðVc ðPi ; Ti Þ−Vci ðPi ; Ti ÞÞ
ρc ðPi Þ
ρl ðPi Þ
ð27Þ
where β is the bulk modulus and we assume a thermal expansion
coefficient α = 3 × 10− 5 K− 1 for both liquid and crystals. Assuming
that the roof region behaves as a thin elastic plate, we calculate
the vertical deflection w(x) of a beam of length L and thickness h due
to an overpressure ΔP below. The plate is infinitely long in y, pinned
at its ends x = 0 and x = L. From Turcotte and Schubert (1982), we
have:
wðxÞ =
ΔPð1−ν2 Þ 2
2
x ðx−LÞ
2Eh3
ð30Þ
Author's personal copy
48
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
Fig. 11. Volume of liquid formed by runaway devitrification as a function of time
between intrusions, for 1 m thick magma sheets. Solid line: volume of liquid formed by
fractional crystallization, dashed line: volume of liquid formed by fractional
crystallization plus roof melting, using Trs = 800 °C and Trl = 1100 °C for roof rocks.
where E is Young's modulus and ν is Poisson's ratio. Integrating the
displacement over the plate length leads to the volume change ΔV as a
function of magma overpressure ΔP = P − Pi:
ΔV
V−Vi
ð1−ν2 ÞL4
ΔP
=
=
ΔP =
Vi
V
μ
60Eh3 d
ð31Þ
where μ = 60Eh3d/(1 − ν2)L4, with d the thickness of the mush layer.
Following the method of Tait et al. (1989), we obtain, by eliminating
V, Vi and Vl in Eq. (31):
1
1
+
−αΔT
Vg = Vli ðPi ; Ti Þ ðP−Pi Þ
μ
β
P−Pi ρc
− ð1 + αΔTÞ
+ Vci ðPi ; Ti Þ 1 +
μ
ρl
ρc
−1 ð1 + αΔTÞ
+ Vc ðPi ; Ti Þ
ρl
ð32Þ
Finally, we use the perfect gas law Vg = Mm RTP, where MH2O is the
molar mass of H2O, the solubility law x = sPn, and mass conservation
of liquid and volatiles to obtain:
g
H2 O
n
MH2 O
mc
Mli
P−Pi αΔT
Pi
=
−
−
−1
M
M
ρl μ
P
ρl
sRTP n−1
MH2 O
Mci
1
1
P−Pi αΔT
−
− +
+
1
+
M
ρc μ
ρc ρl
ρl
sRTP n−1
−1
MH2 O
1
1
−
+
1
× ð1 + αΔTÞ
ρc ρl sRTP n−1
ð33Þ
where we have neglected density changes due to pressure, and used
mc = Vc(Pi, Ti)ρc(Pi, Ti).
References
Annen, C., Blundy, J., Sparks, R.S.J., 2006. The genesis of intermediate and silicic magmas
in deep crustal hot zones. J. Petrol. 47, 505–539.
Annen, C., Pichavant, M., Bachmann, O., Burgisser, A., 2008. Conditions for the growth of
a long-lived shallow crustal magma chamber below Mount Pelée volcano
(Martinique, lesser Antilles arc). J. Geophys. Res. 113. doi:10.1029/2007JB005049.
Annen, C., Sparks, R.S.J., 2002. Effects of repetitive emplacement of basaltic intrusions
on thermal evolution and melt generation in the crust. Earth Planet. Sci. Lett. 203,
937–955.
Bachmann, O., Bergantz, G.W., 2003. Rejuvenation of the Fish Canyon magma body: a
window into the evolution of large-volume silicic magma systems. Geology 31,
789–792.
Bachmann, O., Bergantz, G.W., 2004. On the origin of crystal-poor rhyolites: extracted
from batholithic crystal mushes. J. Petrol. 45, 1565–1582.
Bachmann, O., Bergantz, G.W., 2006. Gas percolation in upper-crustal silicic crystal mushes
as a mechanism for upward heat advection and rejuvenation of near-solidus magma
bodies. J. Volcanol. Geotherm. Res. 149. doi:10.1016/j.jvolgeores.2005.06.002.
Bachmann, O., Dungan, M.A., Lipman, P.W., 2002. The Fish Canyon magma body, San Juan
volcanic field, Colorado: rejuvenation and eruption of an upper-crustal batholith.
J. Petrol. 43, 1469–1503.
Bandyopadhyay, A.K., Labarbe, P., Zarzycki, J., 1983. Nucleation and crystallization
studies of a basalt glass-ceramic by small-angle neutron scattering. J. Mater. Sci. 18,
709–716.
Brandeis, G., Jaupart, C., 1987a. The kinetics of nucleation and crystal growth and
scaling laws for magmatic crystallization. Contrib. Mineral. Petrol. 96, 24–34.
Brandeis, G., Jaupart, C., 1987b. Crystal sizes in intrusions of different dimensions:
constraints on the cooling regime and the crystallization kinetics. In: Mysen, B.O.
(Ed.), Magmatic Processes: Physicochemical Principles: Geochem. Soc., Spec. Pub.,
No. 1, pp. 307–318.
Brandeis, G., Jaupart, C., Allègre, C.J., 1984. Nucleation, crystal growth and the thermal
regime of cooling magmas. J. Geophys. Res. 89, 187–232.
Brown, M., 2000. Pluton emplacement by sheeting and vertical ballooning in part of the
Southeast Coast plutonic complex, British Columbia. GSA Bull. 112, 708–719.
Burchardt, S., 2008. New insights into the mechanics of sill emplacement provided by
field observations of the Njardvik sill, Northeast Iceland. J. Volcanol. Geotherm. Res.
173, 280–288. doi:10.1016/j.jvolgeores.2008.02.009.
Burckard, D., 2002. Kinetics of crystallization: example of microcrystallization in basalt
lava. Contrib. Mineral. Petrol. 142, 724–737.
Cashman, K.V., Marsh, B., 1988. Crystal size distribution in rocks and the kinetics and
dynamics of crystallization II. Makaopuhi lava lake. Contrib. Mineral. Petrol. 99,
401–405.
Chertkoff, D.G., Gardner, J.E., 2004. Nature and timing of magma interactions before,
during, and after the caldera-forming eruption of Volcán Ceboruco, Mexico.
Contrib. Mineral. Petrol. 146, 715–735.
Coleman, D.S., Gray, W., Glazner, A.F., 2004. Rethinking the emplacement and evolution
of zoned plutons: geochronologic evidence for incremental assembly of the
Tuolumne Suite, California. Geology 32, 433–436.
Corry, C.E., 1988. Laccoliths — mechanics of emplacement and growths. Geol. Soc. Am.
Spec. Pap. 220, 110.
Cottrell, E., Gardner, J.E., Rutherford, M.J., 1999. Petrologic and experimental evidence
for the movement and heating of the pre-eruptive Minoan rhyodacite (Santorini,
Greece). Contrib. Mineral. Petrol. 135, 315–331.
Couch, S., Sparks, R., Carroll, M.R., 2003. The kinetics of degassing-induced crystallization at Soufrière Hills volcano Montserrat. J. Petrol. 44, 1477–1502.
Crisp, J.A., 1984. Rates of magma emplacement and volcanic output. J. Volcanol.
Geotherm. Res. 20, 177–211.
Davaille, A., Jaupart, C., 1993. Thermal convection in lava lakes. Geophys. Res. Lett. 20,
1827–1830.
Davies, G.R., Halliday, A.N., Mahood, G.A., Hall, C.M., 1994. Isotopic constraints on the
production rates, crystallization histories and residence times of pre-caldera silicic
magmas, Long Valley, California. Earth Planet. Sci. Lett. 125, 17–37.
Einarsson, T., 1994. Geology of Iceland. Mal og menning, Reykjavik.
Elliott, R.B., 1956. The Eskdalemuir tholeiite and its contribution to an understanding of
tholeiite genesis. Min. Mag. 31, 245–254.
Fialko, Y., Khazan, Y., Simons, M., 2001a. Deformation due to a pressurized horizontal
circular crack in an elastic half-space, with applications to volcano geodesy.
Geophys. J. Int. 146, 181–190.
Fialko, Y., Simons, M., 2000. Deformation and seismicity in the Coso geothermal area,
Inyo County, California: observations and modeling using satellite radar interferometry. J. Geophys. Res. 105, 21,781–21,793.
Fialko, Y., Simons, M., Khazan, Y., 2001b. Finite source modeling of magmatic unrest in
Socorro, New Mexico, and Long Valley, California. Geophys. J. Int. 146, 191–200.
Fowler, A.D., Berger, B., Shore, M., Jones, M.I., Ropchan, J., 2002. Supercooled rocks:
development and significance of varioles, spherulites, dendrites and spinifex in
Archaean volcanic rocks, Abitibi Greenstone Belt, Canada. Precambrian Res. 115,
311–328.
Geist, D., Larson, K.A.H.P., 1995. The generation of oceanic rhyolites by crystal fractionation : the basalt-rhyolite association at volcán Alcedo, Galápagos archipelago.
J. Petrol. 36, 965–982.
Gibb, F.G.F., 1974. Supercooling and crystallization of plagioclase from a basaltic
magma. Mineral. Mag. 39, 641–653.
Glazner, A., Bartley, J., Coleman, D., Gray, W., Taylor, R., 2004. Are plutons assembled over
millions of years by amalgamation from small magma chambers? GSA Today 14, 4–11.
Gudmundsson, A., 1995. Infrastructure and mechanics of volcanic systems in Iceland.
J. Volcanol. Geotherm. Res. 64, 1–22.
Gudmundsson, A., Brenner, S.L., 2005. On the conditions of sheet injections and
eruptions in stratovolcanoes. Bull. Volcanol. 67, 768–782.
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 Planet. Sci.
Lett. 94, 274–290.
Hildreth, W., Wilson, C.J.N., 2007. Compositional zoning of the Bishop tuff. J. Petrol. 48,
951–999.
Holness, M.B., Humphreys, M.C.S., 2003. The Traigh Bhan Na Sgurra sill, isle of Mull:
flow localization in a major magma conduit. J. Petrol. 44, 1961–1976.
Author's personal copy
C. Michaut, C. Jaupart / Tectonophysics 500 (2011) 34–49
Jackson, M.D., Pollard, D.D., 1988. The laccolith-stock controversy: new results from the
Southern Henry Mountains, Utah. Bull. Geol. Soc. Am. 100, 117–135.
Johnson, A., Pollard, D., 1973. Mechanics of growth of some laccolithic intrusions in the
Henry mountains, Utah, I. Field observations, Gilbert's model, physical properties
and flow of the magma. Tectonophysics 18, 261–309.
Kirkpatrick, R.J., 1977. Nucleation and growth of plagioclase, Makaopuhi lava lakes,
Kilauea volcano, Hawaii. Geol. Soc. Am. Bull. 81, 799–800.
Kirkpatrick, R.J., Klein, L., Uhlmann, D., Hays, J., 1979. Rates and processes of crystal
growth in the system anorthite–albite. J. Geophys. Res. 84, 3671–3676.
Klausen, M.B., 2004. Geometry and mode of emplacement of the Thverartindur cone
sheet swarm, SE Iceland. J. Volcanol. Geotherm. Res. 138, 185–204.
Lipman, P., 2007. Incremental assembly and prolonged consolidation of cordilleran
magma chambers: evidence from the Southern Rocky Mountain volcanic field.
Geosphere 1, 42–70.
Lipman, P., Dungan, M., Bachmann, O., 1997. Comagmatic granophyric granite in the
Fish Canyon tuff, Colorado: implications for magma-chamber processes during a
large ash-flow eruption. Geology 25, 915–918.
Mahan, K., Bartley, J., Coleman, D., Glazner, A., Carl, B.S., 2003. Sheeted intrusion of
the synkinematic McDoogle pluton, Sierra Nevada, California. GSA Bull. 115,
1570–1582.
Mahood, G.A., 1990. Evidence for long residence times of rhyolitic magma in the Long
Valley magmatic system : the isotopic record in pre-caldera lavas of Glass Mountain :
reply. Earth Planet. Sci. Lett. 99, 395–399.
Mangan, M., 1990. Crystal size distribution systematics and the determination of
magma storage times: the 1959 eruption of Kilauea volcano, Hawaii. J. Volcanol.
Geotherm. Res. 44, 295–302.
Marsh, B.D., 1981. On the crystallinity, probability of occurence, rheology of lava and
magma. Contrib. Mineral. Petrol. 78, 85–98.
McNulty, B.A., Tong, W., Tobish, O.T., 1996. Assembly of a dike-fed magma chamber: the
Jackass Lakes pluton, Central Sierra Nevada, California. GSA Bull. 108, 926–940.
Michael, P.J., 1983. Chemical differentiation of the Bishop tuff and other high-silica
magmas through crystallization processes. Geology 11, 31–34.
Michaut, C., Jaupart, C., 2006. Ultra-rapid formation of large volumes of evolved magma.
Earth Planet. Sci. Lett. 250, 38–52.
Mohr, P., 1990. Late caledonian dolerite sills from SW Connacht, Ireland. J. Geol. Soc.
Lond. 147, 1061–1069.
Monecke, T., Renno, A.D., Herzig, P.M., 2003. Primary clinopyroxene spherulites in
basaltic lavas from the Pacific-Antarctic ridge. J. Volcanol. Geotherm. Res. 130,
51–59.
Moore, J.G., Lockwood, J.P., 1973. Origin of comb-layering and orbicular structure, Sierra
Nevada batholith, California. Geol. Soc. Am. Bull. 84, 1–20.
Morgan, S., Stanik, A., Horsman, E., Tikoff, B., de Saint Blanquat, M., Habert, G., 2008.
Emplacement of multiple magma sheets and wall rock deformation: Trachyte mesa
intrusion, Henry Mountains, Utah. J. Struct. Geol. 30, 491–512.
Mourtada-Bonnefoi, C.C., Provost, A., Albarède, F., 1999. Thermochemical dynamics of
magma chambers: a simple model. J. Geophys. Res. 104, 7103–7115.
Muncill, G.E., Lasaga, A.C., 1987. Crystal growth kinetics of plagioclase in igneous
systems: one-atmosphere experiments and application of a simplified growth
model. Am. Mineral. 72, 299–311.
Murphy, M.D., Sparks, R.S.J., Barclay, J., Carroll, M.R., Brewer, T.S., 2000. Remobilization
of andesite magma by intrusion of mafic magma at the Soufrière Hills volcano,
Montserrat, West Indies. J. Petrol. 41, 21–42.
Nakamura, M., 1999. Continuous mixing of crystal mush and replenished in the ongoing
Unzen eruption. Geology 23, 807–810.
Oze, C., Winter, J., 2005. The occurrence, vesiculation and solidification of dense blue
glassy pahoehoe. J. Volcanol. Geotherm. Res. 142, 285–301.
Pappalardo, L., Ottolini, L., Mastrolorenzo, G., 2008. The Campanian ignimbrite
(Southern Italy) geochemical zoning: insights on the generation of a supereruption from catastrophic differentiation and fast withdrawal. Contrib. Mineral.
Petrol. 156, 1–26.
Pedersen, R., Sigmundsson, F., 2006. Temporal development of the 1999 intrusive
episode in the Eyjafjallajškull volcano, Iceland, derived from INSAR images. Bull.
Volcanol. 68, 377–393. doi:10.1007/s00445-005-0020-y.
49
Perring, C., Pollard, P., Dong, G., Nunn, A., Blake, K., 2000. The Lightning Creek sill
complex, Cloncurry district, Northwest Queensland: a source of fluids for Fe oxide
Cu–Au mineralization and sodic-calcic alteration. Econ. Geol. 95, 1067–1089.
Petford, N., Gallagher, K., 2001. Partial melting of mafic (amphibolitic) lower crust by
periodic influx of basaltic magma. Earth Planet. Sci. Lett. 193, 483–499.
Preston, R.J., Bell, B.R., Rogers, G., 1998. The Loch Scridain xenolithic sill complex, Isle of
Mull, Scotland: fractional crystallization, assimilation, magma-mixing and crustal
anatexis in subvolcanic conduits. J. Petrol. 39, 519–550.
Pupier, E., Duchene, S., Toplis, M.J., 2008. Experimental quantification of plagioclase
crystal size distribution during cooling of a basaltic liquid. Contrib. Mineral. Petrol.
155. doi:10.1007/s00410-007-0258-9.
Richet, P., Bottinga, Y., 1986. Thermochemical properties of silicate glasses and liquids:
a review. Rev. Geophys. 24, 1–25.
Rubin, A.M., 1995. Propagation of magma-filled cracks. Ann. Rev. Earth Planet. Sci. 23,
287–336.
Sen, G., 1980. Mineralogical variations in the Delakhari sill, Deccan trapp intrusion,
Central India. Contrib. Mineral. Petrol. 75, 71–78.
Simon, J.I., Reid, M.R., 2005. The pace of rhyolite differentiation and storage in an
archetypal silicic magma system, Long Valley. Earth Planet. Sci. Lett. 235, 123–140.
Sisson, T.W., Grove, T.L., Coleman, D.S., 1996. Hornblende gabbro sill complex at Onion
Valley, California, and a mixing origin for the Sierra Nevada batholith. Contrib.
Mineral. Petrol. 126, 81–108.
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’. Earth Planet. Sci. Lett. 99,
387–389.
Spohn, T., Hort, M., Fisher, H., 1988. Numerical simulation of the crystallization of
multicomponent melts in thin dikes or sills, I, The liquidus phase. J. Geophys. Res.
93, 4880–4894.
Sturkell, E., Sigmundsson, F., 1995. The equivalence of enthalpy and shear stress
relaxation in rhyolitic obsidians and quantification of the liquid-glass transition in
volcanic processes. J. Volcanol. Geotherm. Res. 68, 297–306.
Swanson, S., 1977. Relation of nucleation and crystal-growth rate to the development of
granitic textures. Am. Mineral. 62, 966–978.
Tait, S.R., Jaupart, C., Vergniolle, S., 1989. Pressure, gas content and eruption periodicity
of a shallow, crystallising magma chamber. Earth Planet. Sci. Lett. 92, 107–123.
Tegner, C., Wilson, J.R., Brooks, C.K., 1993. Intraplutonic quench zones in the Kap Edvard
Holm layered gabbro complex, East Greenland. J. Petrol. 34, 681–710.
Turcotte, D.L., Schubert, G., 1982. Geodynamics: applications of continuum physics to
geological problems. J. Wiley and Sons.
Venezky, D.Y., Rutherford, M.J., 1999. Petrology and Fe–Ti oxide reequilibration of the
1991 Mount Unzen mixed magma. J. Volcanol. Geotherm. Res. 89, 213–230.
Viskupic, K.V., Bowring, S.A., 2005. Timescales of melt generation and the thermal
evolution of the himalayan metamorphic core, Everest region, Eastern Nepal.
Contrib. Mineral. Petrol. 149, 1–21.
Walker, F.A., 1930. A tholeiitic phase of the quartz-dolerite magma of Central Scotland.
Mineral. Mag. 22, 368–376.
Walker, F.A., 1935. The late palaeozoic quartz-dolerites and tholeiites of Scotland.
Mineral. Mag. 24, 131–159.
Wark, D., Hildreth, W., Spear, F.S., Cherniak, D., Watson, E., 2007. Pre-eruption recharge
of the Bishop magma system. Geology 35, 235–238.
White, S.M., Crisp, J.A., Spera, F.J., 2006. Long-term volumetric eruption rates and magma
budgets. Geochem. Geophys. Geosyst. 7, Q03010. doi:10.1029/2005GC001002.
Wiebe, R., Wark, D., Hawkins, D., 2007. Insights from quartz cathodoluminescence
zoning into crystallization of the Vinalhaven granite, Coastal Maine. Contrib.
Mineral. Petrol. 154, 439–453.
Worster, M., Huppert, H., Sparks, R., 1990. Convection and crystallization in magma
cooled from above. Earth Planet. Sci. Lett. 101, 78–89.
Yilmaz, S., Ozkan, O., Gunay, V., 1996. Crystallization kinetics of basalt glass. Ceram. Int.
22, 477–481.
Znidarsic-Pongrac, V., Kolar, D., 1991. The crystallization of diabase glass. J. Mater. Sci.
26, 2490–2494.