Earth and Planetary Science Letters 250 (2006) 38 – 52
www.elsevier.com/locate/epsl
Ultra-rapid formation of large volumes of evolved magma
C. Michaut ⁎, C. Jaupart
Institut de Physique du Globe de Paris, France
Received 25 January 2006; received in revised form 12 July 2006; accepted 12 July 2006
Available online 1 September 2006
Editor: R.D. van der Hilst
Abstract
We discuss evidence for, and evaluate the consequences of, the growth of magma reservoirs by small increments of thin
(⋍ 1–2 m) sills. For such thin units, cooling proceeds faster than the nucleation and growth of crystals, which only allows a
small amount of crystallization and leads to the formation of large quantities of glass. The heat balance equation for kineticcontrolled crystallization is solved numerically for a range of sill thicknesses, magma injection rates and crustal emplacement
depths. Successive injections lead to the accumulation of poorly crystallized chilled magma with the properties of a solid.
Temperatures increase gradually with each injection until they become large enough to allow a late phase of crystal nucleation
and growth. Crystallization and latent heat release work in a positive feedback loop, leading to catastrophic heating of the
magma pile, typically by 200 °C in a few decades. Large volumes of evolved melt are made available in a short time. The time
for the catastrophic heating event varies as Q− 2, where Q is the average magma injection rate, and takes values in a range of
105–106 yr for typical geological magma production rates. With this mechanism, storage of large quantities of magma beneath
an active volcanic center may escape detection by seismic methods.
© 2006 Elsevier B.V. All rights reserved.
Keywords: magma reservoirs; crystallization kinetics; thermal evolution; catastrophic melting; magma input rate
1. Introduction
Eruption of large volumes of highly evolved melts
during caldera formation requires the presence of sizable
magma reservoirs beneath active volcanoes. Yet, these
reservoirs often elude the most detailed geophysical
surveys, [1]. According to petrological and geochronological data, they develop over large lengths of time [2–4],
which is not consistent with physical models for the cooling
of large magma bodies, [5–7]. Magma ascent through the
Earth's crust proceeds mostly through dyke propagation,
⁎ Corresponding author.
E-mail address: [email protected] (C. Michaut).
0012-821X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2006.07.019
which can transport large quantities of magma in short
pulses of activity. Field evidence demonstrates that some
plutons develop incrementally as sheeted sill complexes,
[8–11], see Table 1. Where field exposures and late-stage
modifications have completely obliterated traces of internal
intrusive contacts, geochronological studies suggest that
such bodies may get assembled over more than 1 Ma [12].
Thermal models for the progressive growth of magma
reservoirs have been developed by several authors [13–15].
Such models have dealt with either discrete injections of
thick magma sheets (⋍ 100 m) or continuous infilling of a
reservoir, and rely on equilibrium crystallization calculations. They predict many features that are consistent with
the observations, including peaks of activity associated
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
39
Table 1
Sheeted sill complexes
Location
Sierra Nevada Batholith, Onion Valley, California
Upper sill complex
Lower sill complex
Everest region, eastern Nepal
Southeast coast plutonic complex, British Columbia
Scuzzy pluton margins
Icelandic cone-sheet swarms
Geitafell Volcano, SE
Thverartindur Central Volcano
Composition
Sill thickness
Refs
Hornblende gabbro
Hornblende gabbro
Leucogranite
0.1–1.5 (m)
2–4 (m)
From cm to several m
[10]
[49]
Tonalite
cm to 100 (m)
[23]
Gabbro
Gabbro
0.5 to 1 (m)
0.9 (m) on average
[50]
[25]
with each injection event, partial melting of country rock
and progressive changes of magma composition over long
time-intervals. They do not predict the sharp changes of
erupted lava composition that may occur and intriguing
features of igneous complexes such as internal quench
textures and episodic phases of crystallization [16]. By
construction, they imply that melt reservoirs remain present
throughout the lifetime of active volcanic centers, which is
difficult to reconcile with seismic evidence.
These paradoxical observations have motivated us
to develop a new model for the growth of a magma
reservoir by thin sills where crystallization is controlled
by the sluggish kinetics of crystal nucleation and growth,
[17,18]. The model relies on the quenching of magma
such that a large proportion of glass remains and is
applicable mostly to magma intrusions in the upper crust,
in contrast to Annen and Sparks [13]. Successive thin
injections lead to the accumulation of quenched magma
with little crystallization. Kinetic controls on crystallization are such that the rates of nucleation and growth
start from zero at the liquidus temperature, reach a maximum at some finite undercooling and tend to negligible
values at large undercoolings. Rapid cooling to low temperatures thus impedes crystallization and leads to glass
formation. In natural magmatic systems, the temperature
evolution depends on heat loss to the surroundings and
latent heat release, [19]. Latent heat release works in
two different ways depending on the thermal path. For a
cooling path following contact with cold surroundings,
i.e. starting from high temperatures, latent heat release
acts to slow down cooling and to decrease the crystallization rate, [19]. Here, we are interested in another
thermal path starting from low temperatures. Beginning
with chilled magma, punctuated emplacement of small
magma batches acts to increase temperatures slowly until
crystallization sets in. In this case, crystallization occurs
whilst temperatures are rising and latent heat release
enhances heating and crystallization in a positive feedback loop. We develop a quantitative thermal model
which illustrates this process.
In this paper, we first discuss field evidence for the
formation of magma reservoirs out of sheeted sill
complexes, focussing on sill thicknesses. We then review
available data on the kinetics of nucleation and growth in
natural magmas and show that the cooling and crystallization of many sill complexes is kinetically controlled. We
focus on basalts for the purposes of this study, but show
that the same physical mechanism is likely to operate in
andesites and other evolved magmas. The study is set up to
describe the new physical mechanism in the clearest
manner and to evaluate the influence of each control
parameter. We develop a quantitative thermal model and
discuss solutions for a large range of parameters. We
discuss how the vagaries of magma emplacement in a
natural setting may lead to different thermal evolutions.
Requirements for the applicability of the model to natural
situations are established. The paper closes with a short
section on peculiar observations in magmatic and volcanic
systems that can be explained by the model.
2. Cooling and crystallization of magma
In this paper, we consider a new crystallization mechanism due to emplacement in colder continental crust.
Evolved melts with large water contents may also crystallize in response to decompression once they reach
water saturation [20]. We discuss briefly the behaviour of
such melts in a separate section.
2.1. Sheeted sill complexes
At the Onion Valley complex, Sierra Nevada Batholith,
California, thin basaltic sills of 0.1- to 1.5-m thick form the
margins of a large intrusion and are chilled against inter-sill
septa or against one another with finer-grained to aphanitic
margins [10]. In most of the sills, textural evidence
indicates that the cores of plagioclase crystals initially
grew in strongly undercooled melt. In the middle part of
the intrusion, sills tend to be thicker (2–4 m) and include
pillow-shaped masses which suggest emplacement of thin
40
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
flow units. In the pluton center, no clear intrusive contacts
can be traced, but comb-layering and quench textures
suggest that part of the crystallization sequence occurred at
large degrees of undercooling. Such features are best
interpreted as due to individual injection events with large
temperature contrasts between the incoming and resident
magmas. Similar observations in other intrusive bodies
have been interpreted in the same way, for example
elsewhere in the Sierra Nevada Batholith, [21], and in the
Kap Edvard Holm Layered Gabbro Complex, East
Greenland, [22]. In the Scuzzy pluton, Southeast Coast
Plutonic Complex, British Columbia, the outer marginal
zone is made of a spectacular sill complex with individual
units that are typically 1–2-m thick, [23]. Away from the
margins, the interior is made of massive igneous rocks, but
one should not deduce from this that the bulk of the pluton
was injected in one piece. Upon closer scrutiny, the
massive interior contains a large number of thin countryrock lenses which once separated different magma
batches. Thus, it may very well be that evidence for a
sheeted sill structure got obliterated and, in fact, the present
physical mechanism explains how this may occur.
Shallow intrusives provide complementary information on the typical thickness of individual magma pulses.
Preservation of individual contacts is enhanced by the
shorter-lived activity, smaller background temperatures
and possibly by enhanced cooling due to hydrothermal
circulation. The typical thickness of individual intrusions
occurring as dykes, sills or sheets, is about 1 m, [24]. In
the root of the Thverartindur Central Volcano, Iceland,
for example, a cone-sheet swarm is made of 1128
individual sills with an average thickness of 0.9 m, [25].
Table 1 recapitulates sill thicknesses that have been
determined in a variety of igneous complexes. One
should note that thin intrusions are documented in
different environments as well as in magmas of different
compositions. Another way to evaluate the thickness of
individual intrusions relies on the bulk magma production rate, which has been determined at a number of
continental volcanoes by Crisp [26]. Assuming that the
caldera size is equal to the area of the magma chamber,
the rate of growth of reservoirs in the vertical direction
lies between 10− 2 and 10− 3 m yr− 1 in the vast majority
of cases. Over the lifetime of a volcanic system, which is
typically a few times 100,000 yr, such rates lead to pluton
thicknesses of a few kilometers that are consistent with
the geological record. One may further assume that
injection events are associated with volcanic unrest or
eruptions. Using a repetition time between 100 and
1000 yr, the average intrusion thickness is constrained to
be between 10 cm and 10 m, which is consistent with the
data in Table 1.
2.2. The rates of crystal nucleation and growth in
silicate melts
Nucleation and growth rates are zero at the liquidus
because crystallization requires a finite energy. At small
undercoolings, close to the liquidus, the free energy of the
solid phase or the energy difference between crystal and
liquid is the main control on both nucleation and growth.
At low temperature, the limiting process is chemical
diffusion, [27]. Both the nucleation and growth rates
increase with the increasing undercooling until they reach a
maximum and then decrease to zero at large undercoolings.
Direct determinations of the rates of crystal nucleation
and growth in silicate melts are scarce (Table 2). Very few
measurements of the maximum growth rate have been
attempted in the laboratory and we are aware of none for
the maximum nucleation rate. A careful and systematic
study was carried out for the Ab–An system [28], a
synthetic system for which liquidus temperatures are
much higher than those of natural magmas over a large
composition range. For plagioclase crystals that have the
same composition as those found in natural melts, the
maximum growth rate is about 5 × 10− 5 cm s− 1, [28]. One
well-known fact is that, for a given mineral, the peak
growth rate is smaller in a complex multicomponent melt
than in its own melt [27,29]. One should therefore treat the
Ab–An growth rates as upper bounds for plagioclase
crystals in natural magmas.
Available data and laboratory experiments on basaltic
melts indicate that olivine, pyroxene and plagioclase are
ranked in decreasing order with regard to their ease of
nucleation [30]. The growth and nucleation rates of olivine,
pyroxene and plagioclase crystals have been measured in
basaltic lava lakes and flows, either in situ or through the
analysis of crystal size distributions [31–33]. For olivine,
values are in the ranges of 10− 11 to 10− 10 cm s− 1 and 10− 6
to 10− 5 cm− 3 s− 1 respectively. These measurements have
been made at small amounts of undercooling in magmas
that had already started to crystallize. As can be seen in
Table 2, independent estimates are consistent with one
another but do not allow reconstruction of the full
nucleation and growth functions.
Another method has been used to determine the peak
rates of nucleation and growth in natural magmas.
Following emplacement in colder country rock, magma
cooling and crystallization proceed in a highly transient
thermal regime with cooling rates that decrease away from
the contact. One consequence is that crystal sizes increase
with increasing distance from the margin, which provides a
powerful constraint on crystallization kinetics in natural
conditions. By comparing crystal size measurements in
mafic intrusions and numerical crystallization calculations,
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
41
Table 2
Nucleation and growth rates in silicate melts
Y (cm s− 1)
I (cm− 3 s− 1)
Refs
. aboratory experiments on synthetic systems
L
Ab–An
Plagioclase An30
Plagioclase An40
Ab–An
Plagioclase An50
Ab–An
Ym = 1 × 10− 5
Ym = 5 × 10− 5
Ym = 2 × 10− 6
/
/
/
[28]
[28]
[51]
. atural systems–natural conditions
N
Plagioclase
Basaltic lava lake in situ
Plagioclase
Basaltic lava lake
Olivine
Basaltic lava lake
Plagioclase
Basaltic lava lake
Pyroxene
Basaltic lava lake
Plagioclase
Basaltic lava flow
Pyroxene
Basaltic lava flow
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
[31]
[32]
[33]
[52]
[52]
[53]
[53]
. aboratory experiments on natural systems
L
Plagioclase
Basalt
Pyroxene
Basalt
10− 11–10− 10
10− 9–10− 8
10− 6–10− 4
10− 6–10− 4
[52]
[52]
. heoretical calculations of crystal size variations
T
Opx and Plag.
Diabase dykes
Ym ≈ 10− 7
Im ≈ 1
[17]
Mineral
System
I and Y are average values measured in small samples at small undercoolings, Im and Ym are maximum rates over the whole crystallization interval.
Brandeis and Jaupart [17] were able to constrain the peak
rates of nucleation and growth to be about 1 cm− 3 s− 1 and
10− 7 cm s− 1 respectively for both pyroxene and plagioclase
crystals. These values are indeed much larger than local
ones determined at small undercoolings (Table 2).
So far, we have focussed on basaltic melt compositions
because of the rather large number of studies devoted to
them. Table 3 lists values of peak crystal growth rates in
evolved magmas. The data suggest that peak growth rates
are similar to those of more mafic melts and decrease with
increasing water content. Values for a water-rich andesitic
melt are unambiguously smaller than those for more
primitive basaltic compositions.
2.3. Kinetic controls on magma crystallization
Here, we explain the basic physical principle by
comparing the time-scales for cooling and for crystal-
lization. This principle is later put to test with a full
numerical solution of the governing equations. Denoting
the rates of nucleation and growth by I and Y
respectively, the crystallization time-scale is [17]:
s ¼ ðIY 3 Þ−1=4
ð1Þ
Cooling and crystallization proceed in transient
thermal conditions involving two different kinetic timescales. For large undercoolings, the relevant characteristic
time, τm, relies on the peak rates of nucleation and growth,
noted Im and Ym respectively. From the data in Table 2,
τm is ≈ 2 × 105 s in basalts, or about 2 d. This corresponds
to the most rapid crystallization obtained by maintaining
both rates at their peak values. This cannot achieved in
practice, however, because the two peak values are not
reached at the same degree of undercooling [27]. Thus, τm
provides a lower bound to the true crystallization time. At
Table 3
Laboratory determinations of peak rates of nucleation and growth in evolved melts
Mineral
Plagioclase
Plagioclase
Plagioclase
Plagioclase
Alkali Fs
Alkali Fs
Alkali Fs
Ym (cm s− 1)
System
a
Andesite + 6.4% H20
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
‡Not measured.
a
Crystallization is induced by decompression.
−9
1.7 × 10
10− 6
5 × 10− 7
≈ 10− 8
2 × 10− 7
10− 7
≈ 10− 8
Im (cm− 3 s− 1)
−2
3.2 × 10
‡
‡
‡
‡
‡
‡
Refs
[42]
[54]
[54]
[54]
[54]
[54]
[54]
42
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
small undercoolings, which prevails some time after
emplacement in thick magma bodies, the rates of
nucleation and growth are necessarily smaller than their
peak values. Thus, the characteristic kinetic time τ must
be much larger than τm. From Table 2, this characteristic
crystallization time is τ ≈ 4 × 108 s for olivine in natural
basalts. For plagioclase crystals, it is in a range of 107 to
108 s, i.e. from 0.3 to 3 yr. As expected, these values are
larger than τm. The true crystallization time in transient
thermal conditions is bracketed by τm and τ and will be
noted τk. According to [19,34], this crystallization time is
about 106 s (i.e. 11 d) or more for basaltic magmas.
For quenching to occur, cooling must occur on time
scales that are too short to allow a significant amount of
crystallization. For thin sills of thickness d, conduction
is the dominant heat transport mechanism and the
characteristic cooling time is:
sd c
1 d2
4j
ð2Þ
One should note that a sill loses heat from both margins
and that this cooling time must be regarded as an upper
bound. The true cooling time may be much smaller than
this in shallow environments due to hydrothermal circulation. The ratio of the cooling time to the kinetic time,
called the Avrami number, is the relevant dimensionless
number to assess kinetic controls on crystallization [17,18]:
4 2 4
sd
d
Aυ ¼
¼
IY 3
sk
4j
ð3Þ
Analysis shows that quenching occurs if Aυ b 1
[17,18].
For a typical sill thickness of 1 m, τd = 3 × 105 s. With
τk = 106 s, Aυ b 1, implying that little crystallization
occurs and that a large fraction of the injected magma
gets quenched. More precise documentation of how
crystallization proceeds in transient conditions is made
below with numerical calculations. The effects of
changing the kinetic time-scale and the characteristic
cooling time are also discussed then.
Some verification of these ideas in natural conditions
can be obtained by looking at the systematic variation of
crystal size away from the margins of an intrusion, as was
done by [19,34] for example. One relevant study was
made by Dunbar et al. [35] who studied the cooling of an
artificial Ca-rich mafic body with a 1.5-m thickness and a
3-m diameter, i.e. an average width of about 2 m. In this
study, cooling depended on thermal boundary conditions
and the peculiar shape of the melt body, parameters that
were specific to the experimental setup. The important
result is that rapid cooling over about 6 d led to the
formation of 12% glass in the interior of the body. We shall
illustrate below how thin sills achieve more rapid cooling
leading to a larger proportion of glass. Another example of
interest is provided by the ∼30- to 70-m thick Ginkgo
flow of the Columbia River Basalts [36]. Crystallinity is
only about 11% at the margin of the feeder dyke and is
slightly higher in distal flow samples. The data demonstrate that a large fraction of this lava was quenched to
glass upon emplacement and further that crystallization
progressed slowly during the flow. Basaltic pillow lavas
also document glass formation. Such samples correspond
to the lava that drains from the thin tip of long flows,
however, and their complex history of degassing and
cooling prevents simple inferences.
3. Thermal model
3.1. Governing equations
We do not consider the mechanics of emplacement and
focus on a horizontal sill geometry. We consider a
succession of thin magma injections. At t = 0, the first sill
is emplaced instantaneously at depth ZS. Another injection
follows at t =τi and so on. At each emplacement event,
rocks located below the sill complex are instantaneously
displaced downwards. Such motion has a small effect on
the temperature distribution, as will be demonstrated later.
Between two emplacement episodes, heat transport occurs
by conduction alone. For thin sills that are laterally
extensive, one may neglect horizontal heat transport and
temperature obeys the following equation:
qCp
AT
A2 T
AU
¼ k 2 −qL
At
Az
At
ð4Þ
where T is temperature, z the vertical coordinate, t time,
Cp heat capacity, ρ magma density, k thermal conductivity,
U the melt fraction and L latent heat. Density changes
between crystals, magma and glass are neglected. The
crystallization rate, AU
At , drops to zero when T =TL, where
TL is the liquidus temperature, and can be written as a
function of the rates of nucleation and growth, which
themselves depend on the undercooling T* =T /TL [37].
This equation must be solved over very thin units and
extremely long time intervals, implying very large
calculation times (typically several months). As discussed
in the Appendix, we use a 10-cm grid spacing for sill
complexes that grow to be thicker than 1 km. For
simplicity, we lump together nucleation and growth into a
single kinetic crystallization function f (T*), (Fig. 1):
AU
f ðT *Þ
¼ −U
At
sk
ð5Þ
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
43
where TL is in °C, C is composition in wt.% silica and Ce
is the eutectic composition (Ce = 75 wt.% SiO2). Crystals
have an average silica content of 40 wt.% SiO2 and mass
conservation allows calculation of the residual melt
composition, C:
U C ¼ C0 −Cs ð1−U Þ
ð7Þ
with C0, the initial composition, and Cs, crystal
composition, in wt.% SiO2. Melting of country rock is
not kinetically controlled and, following a common
simplification, we assume that the melt fraction is a
linear function of temperature over the melting interval:
AT
A2 T
A T −TSr
qCP
¼ k 2 −qL
ð8Þ
At
Az
At TLr −TSr
with T Lr and T Sr the solidus and liquidus temperatures of
country rock. Values for the various parameters and
physical properties are given in Table 4.
3.2. Time scales and parameters
Fig. 1. a) Effective function for both nucleation and growth rates as a
T
*
*
functionh of undercooling
i TL with T and TL in K. f ðT Þ ¼
i hT ¼
K
K
3
2
CT *exp − T ðT −1Þ2 exp − T * ; with K2 = 10− 3, K3 = 30, and C such
* *
that max( f ) =1. Note the nucleation delay (arrow). b) Simplified phase
diagram. The dashed line corresponds to the maximum crystallization rate.
where τk is a characteristic time for crystal nucleation and
growth. Dimensionless function f (T*) is normalized, such
that the maximum crystallization rate is s1j .
Complexities in the phase diagram are important for
the detailed crystallization sequence but cannot affect
significantly the rather straightforward thermal behaviour. Thus, for simplicity purposes, we take a binary
eutectic solution with a starting magma composition far
from the eutectic, such that crystallization occurs over a
large temperature range. The liquidus temperature is
given by (Fig. 1):
TL ¼ 1000 −
20
ðC−Ce Þ
3
ð6Þ
The primitive melt has a basaltic composition and is
initially at the liquidus, at TL = 1187 °C (Fig. 1). Each
intrusion of thickness d gets emplaced at the same depth,
on top of the preceding ones, (Fig. 2). Other emplacement
configurations, at the bottom of the pile for example,
induce little changes in the results. We return to the
emplacement sequence in the Discussion section. Before
the first injection, the crust is in thermal equilibrium with a
prescribed basal heat flow. We do not consider the effects
of radiogenic heat production, which induce small
temperature changes over the range of times and crustal
depths of relevance to this problem. The pre-existing
geothermal gradient g0 is set at a value of 15 °C km− 1,
which corresponds to surface heat flow Φ ≈ 40 mW m− 2
which typifies stable continental regions before magmatic
and tectonic activity. As long as this background thermal
gradient is small compared to TL /ZS, which corresponds
to geological reality, its exact value has no significant
effect on the numerical results. Boundary conditions are a
fixed temperature at the surface (z = 0) and a fixed heat
flux Φ at the base.
This problem has three time-scales, the characteristic
time for crystallization, τk, the diffusion time-scale τd,
and the time between two injections, τi. A fourth timescale corresponds to diffusion over the thickness of roof
rocks ZS, which is of secondary importance. The
problem is therefore characterized by two dimensionless numbers. One is the Avrami number, which
has already been defined. The second dimensionless
number, σ = τi / τd, compares the time between two
44
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
Table 4
Values for the physical properties
Physical property
Thermal conductivity
Heat capacity
Latent heat of crystallization
Magma density
Solidus temperature of
country rock
Liquidus temperature of
country rock
3.3. Numerical method
Symbol
Value
−1
−1
k
Cp
L
ρ
T Sr
2.5 (W K m )
1.3 × 103 (J kg− 1 K− 1)
4.18 × 105 (J kg− 1)
2500 (kg m− 3)
800 (°C)
T Lr
1100 (°C)
injections and the cooling time. For σ ≪ 1, magma
injection can be treated as continuous, which does not
allow chilling. For σ ≫ 1, magma has been cooled by
the time the next injection comes, which is relevant to
volcanic systems.
The rate of magma injection per unit area is Q = d / τi.
In the calculations, the intrusion thickness d and the
injection rate Q were varied and the other parameters
were fixed. τk was set at a representative value of 106 s,
as discussed above, and ZS at 5 km. In order to facilitate
the physical analysis, the sill thickness and the time
between two injections were kept constant in each
numerical run. In nature, however, these parameters may
both change with time and consequences are evaluated in
the Discussion section.
Governing Eqs. (4) and (8) are solved numerically
with a Crank–Nicholson implicit finite-difference
scheme (Appendix A). Because the computational
domain is much thicker than the size of the thermal
anomaly that develops, the results are not sensitive to
the bottom boundary condition.
Accounting for crystallization kinetics within thin
units requires small grid sizes and time-steps, and hence
extremely lengthy calculations. To reduce the computation time, we use an analytical solution to the governing
equations when the crystallization rate is negligible and
there is no latent heat release. In this case, the timeevolution of an arbitrary vertical temperature profile may
be derived analytically:
Z l
ðz−yÞ2
−ðzþyÞ2
1
−
T ðz; t−t1 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T ð y; t1 Þ e 4jðt−t1 Þ −e4jðt−t1 Þ dy
2 kjðt−t1 Þ 0
ð9Þ
where t1 is some arbitrary time. In this equation, the integral
is calculated numerically using the trapezoid method.
4. A sample calculation
Here, we present results for one particular set of
parameters, d = 1 m and τi = 40 yr, and discuss the
Fig. 2. Schematic representing the emplacement configuration and boundary conditions. The initial thermal gradient is 15 °C km− 1. Temperature is
kept to zero at the surface and a fixed heat flux Φ is maintained at the base of the computational domain. Below the sill complex, country rock gets
displaced downwards by each intrusion event.
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
45
effects of changing these parameters in the following
section. The time-averaged rate of heat supply per unit
area is given by:
/¼
qCp DTd
si
ð10Þ
where ΔT is the temperature drop following cooling.
Taking a representative value of 600 K for this
temperature drop and using the parameter values
specified above and in Table 4, ϕ ≈ 1 W m− 2. The
background heat flux is much smaller than this and
hence plays a minor role in the thermal evolution of the
system.
4.1. Temperature evolution
The first few sills all follow the same thermal and
crystallization evolution illustrated in Fig. 3. Cooling
rates are very large and lead to quenching at the margins.
Magma is rapidly carried to very large undercoolings
and few crystals grow in the interior of each sill.
Crystallinity for this particular case is initially about 3%.
Fig. 4 shows vertical temperature profiles through the
crust after 150, 600, 1000 and 1566 injections. Both the
amplitude and the depth-extent of the thermal anomaly
increase with time. At the end of this sequence, which is
≈ 63,000 yr long, the thickness of the magma pile is
1566 m and the largest temperature is 694 °C, for which
the crystallization rate is still very small. Thus, the
whole magma pile is made of glass with a small crystal
content and has the properties of a solid. At the base of
the computational domain, temperatures decrease with
time at a small rate reflecting the downward motion of
country rock due to sill emplacement (see Fig. 2). The
magnitude of this far-field temperature change is only
22 °C after 1566 injections, which is negligible
compared to the magmatic thermal anomaly.
Fig. 5 shows the residual crystal fraction after 150,
1000 and 1674 sill injections. By construction, each sill
is hotter than the surroundings when it gets emplaced.
The cooling rate is related to the thermal gradient and
decreases away from the contact as well as with time,
which allows a small amount of crystallization.
Temperatures in the magma pile increase with each
new injection, which acts to decrease the cooling rate.
Thus, the time available for crystal nucleation and
growth before quenching increases within each successive injection and the amount of crystals generated in
each new sill gradually increases. Sills no. 150 and 1000
both have chilled margins and weakly crystallized interiors. After emplacement of sill no. 1674, however,
Fig. 3. Cooling (top) and crystallization (bottom) of a single 1-m thick
intrusion emplaced at a depth of 5 km at different times (indicated in
hours) after emplacement.
temperatures are such that crystal nucleation and growth
occur even at the margins.
As the magma pile grows, temperatures rise at a
decreasing rate, as befits diffusive heat transport (Fig. 6).
After a large number of injections, temperatures are
sufficiently large for nucleation and growth to kick in and
the system enters the positive feedback loop described
earlier. For the present set of parameters, this catastrophic
heating event occurs after 1675 injections, corresponding
to a time of 67,000 yr, and the temperature rise is dramatic
indeed: 200 °C in 40 yr (Fig. 6). Because of the positive
thermal feedback loop, crystallization proceeds rapidly
over a large thickness, leading to a sizable volume of
evolved melt. In a few decades, therefore, one goes from a
thick layer of chilled basalt to a large partially molten
region. After this heating event, subsequent magma inputs
occur in surroundings that are at temperatures close to the
solidus, and the system enters a new phase.
46
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
Fig. 4. Vertical temperature profiles for d = 1 m and τi = 40 years after
150, 600, 1000 and 1566 injections, i.e. at t = 6 × 103, 24 × 103, 40 × 103
and 6.26 × 104 yr. At the end of this injection sequence, the pile of
uncrystallized chilled magma is 1566-m thick (shaded area).
Fig. 6. Maximum temperature in the magma pile as a function of time.
From t = 0 to t ≈ 60 × 103 yr, the temperature increase is gradual. At
t ≈ 60 × 103 yr, temperatures are large enough for crystal nucleation and
growth. Latent heat release and crystallization kinetics work in a
positive feedback loop leading to a catastrophic temperature increase
of more than 200 °C in 40 yr.
4.2. Amount of melt generated
The catastrophic heating event also affects the country
rock, which melts. Thus, two different magmas are
generated simultaneously from two different sources, and
hence with different compositions and isotopic characteristics. The amount of melt generated increases with τi, the
time between intrusions (Fig. 7). For d = 1 m and scaling
by a characteristic cross-sectional area, magma input rates
at a large number of continental volcanoes imply that τi is
between 100 and 1000 yr, [26]. In such conditions, large
amounts of evolved magma are generated: 140 m of pure
melt for τi = 100 yr. In comparison, the Crater Lake
caldera collapse event led to the eruption of 40 km3 of
magma, [38]. Scaled to the cross-sectional area of the
reservoir and allowing for the phenocryst content, this
represents a thickness of 400 m, which would be achieved
for τi ≈ 300 yr (keeping the same d value of 1 m).
4.3. Time to the catastrophic event
Fig. 5. Amount of residual liquid or glass remaining in sills no. 150,
1000 and 1674. Prior to sill no. 1674, the margins of each sill gets
completely chilled and few crystals grow in the interior. At the time of
sill no. 1674, temperatures are large enough for the nucleation and
growth of crystals and the margins also crystallize a little.
We have run a number of calculations for different
values of τi and d. Times between two injections and sill
thicknessses ranged from 1 to 100 yr and 1 to 2 m
respectively. We found that the time of catastrophic
melting varies as Q− 2 and is weakly sensitive to the sill
thickness (Fig. 8). This result can be explained by a very
simple physical argument.
Denoting by ΔT the temperature drop, the amount of
heat lost to the surroundings per unit area is ρCpdΔT.
This heat is evacuated by diffusion between two
successive intrusions, over time τi. The largest heat
flux is obtained at the roof and is carried through a
thermal boundary layer of thickness δ which develops in
the country rock. This
pffiffiffiffiffiboundary layer grows by
diffusion, such that df jt . The dominant heat balance
is thus:
qCp dDT
TL
fk pffiffiffiffi
ffi
si
jt
ð11Þ
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
47
where k is thermal conductivity. This simple relation
implies that ΔT decreases with time (i.e. temperatures
rise in the surroundings) because of the increasingly
inefficient conductive heat flux. The catastrophic
melting event occurs at time t = τc such that ΔT = ΔTc,
the critical value for the kinetic heat kick. Rearranging
and using the fact that d / τi = Q, we obtain:
TL 2 −2
Q
ð12Þ
sc f j
DTc
which implies that τc ∝ Q− 2, as shown by the calculations (Fig. 8). Note that this equation predicts that the
critical time depends on d and τi only through the
injection rate Q, as in the numerical results.
For geological rates of injection calculated from [26],
(≈ 10− 3 to 2 × 10− 2 m yr− 1), the time to the catastrophic
melting event is in a range of 105 to more than 106 yr. For
instance, a critical time of 1 Ma is reached for an injection
rate of 5.10− 3m yr− 1, or 1 m every 200 yr. Such large
values are consistent with time scales obtained from
geochronological studies of volcanic and magmatic
systems, and will be discussed further below.
4.4. Key characteristics of runaway devitrification
The thermal mechanism described here, which has
been called Runaway Devitrification by one reviewer of
this paper, has many interesting features. Large volumes
of evolved liquid magma are generated in a short time
after a very long build-up phase (typically 105 yr or more).
Melting of country rock proceeds at an accelerated rate.
One key aspect is that heating of the chilled magma
Fig. 8. Critical time (in years) for the catastrophic heating and melting
event as a function of rate of injection Q in m yr− 1. Results have been
obtained for two different sill thicknesses (1 or 2 m) and for different
values of τi, the time between injections. Geological values of magma
production rates, from [26], are indicated by the horizontal range.
proceeds to almost the same temperature everywhere,
leading to a nearly homogeneous batch of evolved
magma. In contrast, crystallization of a large magma
body due to cooling against cold surroundings leads to
sharp temperature and compositional gradients, [39,40].
Over a long-term perspective, a magmatic system
may go through two different phases, before and after
the catastrophic melting. Before this event, only minor
differentiation can occur. During the melting event,
magma compositions change rapidly due to fractional
crystallization and contamination with country rock.
After this event, a large thermal halo has developed
around the injection site and temperatures remain
consistently large (Fig. 6). Thus, new magma injections
are not cooled efficiently and a liquid reservoir can be
sustained by small magma inputs.
5. Discussion
We have used a simple physical model in order to
identify key controls on runaway devitrification. Application to specific geological cases requires a host of input
parameters including the water content of encasing rocks
and the starting magma composition, which is outside the
scope of this paper. Here, we discuss the applicability of
the model to geological conditions.
Fig. 7. Maximum volume of pure liquid per unit area formed by
fractionation only (solid line), or fractionation + wall-rock melting
(dashed line), as a function of τi, the time between two injections.
Calculations are for a sill thickness d of 1 m.
5.1. Prerequisites
For the present mechanism to work, the sill thickness
must not exceed a critical value. For the same kinetic
48
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
time, i.e. τk = 106 s, calculations with d = 2 m exhibit the
same behaviour, but not those for d = 4 m; in that case,
all sills crystallize following emplacement.
Another condition is that the thermal contrast between
magma and host rock must be large enough for chilling. For
d= 1 m, we find that the initial country rock temperature
must be less than about 500 °C, corresponding to large
parts of the continental crust. The best way to test the
model, of course, is to compare predictions and observations, which will be done in an independent section below.
The model essentially depends on sill thickness and the
kinetic rates of crystal nucleation and growth, as well as
on thermal diffusivity, which may vary slightly among
natural magmas and may change as crystallization
proceeds. Evaluation of the effects of changing one or
all these parameters is best done with the Aυ number. For
example, the same solution is obtained if one simultaneously decreases the kinetic time-scale by a factor of 4
and the sill thickness by a factor of 2 because the Aυ
number stays the same. In the following, we shall use the
sill thickness as control variable for the crystallization
behaviour but the results can be easily recast as a function
of the other parameters through the Aυ number.
5.2. Field evidence
Lava samples emplaced at the Earth's surface or very
shallow intrusions may not be appropriate because they
are likely to have undergone decompression-induced
crystallization at small pressures [20]. By its very nature,
the mechanism explains why evidence may be hard to
find. Firstly, the catastrophic melting event is likely to
eradicate most traces of the initial sill complex. Secondly,
the magmatic system is likely to remain active past this
event with further disruption to the intrusive record. For
example, the Onion Valley mafic complex, Sierra Nevada
Batholith, California, formed in at least five episodes,
with the first one preserved only as dismembered blocks
within later units [10]. Thicker units that are found in the
middle part of this intrusion preserve internal chilled
contacts in a few places. We also note that the margins of
the complex are most likely to preserve the initial sheeted
sill structure. For example, the Urquhart and Scuzzy
plutons, British Columbia, have sheeted margins and
more homogeneous plutonic interiors [23].
5.3. The characteristics of emplacement: Sill thickness
and time
In each run, all parameters were kept constant for
clarity purposes. The assumption which may seem the
most contrived is perhaps the constant sill thickness.
Variations of sill thickness have consequences that are
easy to understand. Indeed, it may very well be that some
differences in the behaviour of natural magmatic systems
can be ascribed to vagaries of emplacement characteristics. The main issue is the emplacement of sills that are
much thicker than those studied here. As stated above, the
present mechanism does not work if all sills are thick and
what is at stake, therefore, is the number of such sills.
Accounting for the large lifetime of magmatic systems,
which typically exceed 100,000 yr, and for pluton
thicknesses, which are typically in a range of 3–5 km,
there can only be a small number of thick injections widely
separated in time. No calculations are needed to find out
what happens in two limit cases. Suppose for example that
the first sill is 100-m thick, say, and that it is followed by
thin ones. The first sill cools rapidly (≈ 200 yr, [6]). If it
has completely crystallized when the next thin sill comes
in, thermal evolution can be understood using the above
results: the problem boils down to that of temperature in
the surroundings, which has already been discussed. The
other limit-case corresponds to a series of thin sills
followed by a large one. In this case, the large sill may
trigger catastrophic melting of the chilled magma pile.
Determining the time between successive injections in
a fossil magmatic system is likely to prove difficult. We
know, however, that continental volcanoes have a large
number of eruptions and that many of them seem to be
triggered by a new injection of magma, as shown by mafic
enclaves and their chilled contacts. Young volcanic
systems, which are most relevant to the present analysis,
are usually characterized by a long initial phase of basaltic
volcanism followed by the emission of differentiated lavas,
as at Mount Adams, for example [41]. Such behaviour
raises the problem of how magma storage zones develop,
which may be rationalized with the present model.
5.4. Complicating factors
Here, we discuss a few complications that may occur
in nature, with emphasis on the mechanism itself and not
on quantitative details, which can only be addressed
with other numerical calculations.
We have verified that results are weakly sensitive to the
details of the emplacement sequence within the magma
pile. For example, forcing injection at the base of the
magma pile instead of at the roof does not modify
significantly the temperature evolution. One may also
entertain emplacement at a random location within the
magma pile, with consequences that are easy to predict. In
this case, each sill would cool faster because the adjacent
sills would be colder than for a fixed emplacement
location. The end result would be a wider thermal
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
anomaly and a smaller heating rate than in the examples of
Fig. 4. In other words, randomization of the emplacement
depth smoothes out temperature differences within the
growing magma pile.
We have assumed that each sill is in contact with
previous sills, with no country rock sandwiched in
between. Interlayering of magma and country rock slows
down heating of the magma pile and hence increases the
time for catastrophic melting. One consequence is that
magma gets contaminated in situ by country rock, to which
we return below. If successive sills get emplaced at
distances that are large compared to the individual sill
thickness, catastrophic heating would be prevented by the
large heat capacity of the magma-country rock assemblage.
Magma may already contain a few phenocrysts when
it gets emplaced in the upper crust, due to prior storage in
a deeper reservoir or to a complex ascent history. In this
case, of course, the amount of glass formed would be
smaller and the heating pulse would be more subdued.
One other simplification is that we have lumped the
crystallization kinetics of all minerals in a single
function, but this may be oversimplified. For example,
one could envisage easy nucleation of the liquidus
phase, such as olivine in basalts, with further crystallization impeded or prevented by very sluggish kinetics.
Such intricacies may be important for petrological
interpretations but require kinetic data for a number of
different crystal phases that is lacking (Tables 2 and 3).
Evidence from the variations of crystal size away from
intrusion margins, [19], does suggest that pyroxene and
plagioclase crystals do not differ markedly from one
another from the standpoint of crystallization kinetics.
5.5. Other magma compositions
The present mechanism can also operate in other
magmas. As shown in Table 1, incremental growth of
plutons is documented in tonalites and leucogranites.
Using the peak rates from [42], τm ≈ 107 s for andesites,
which is larger than the basaltic value by more than one
order of magnitude. As explained above, the true
crystallization time must be larger than this and we
expect the formation of significant amounts of glass in
andesitic sills that are up to 10 m in thickness.
6. Comparison with observations and geological
implications
6.1. Sharp heating events in active magmatic systems
One symptom of the present mechanism is a sharp
heating pulse before an eruption of differentiated lava.
49
Evidence for such a pulse has been found at a number of
volcanoes, [43–45]. The Fish Canyon Tuff provides an
illuminating example, [46]. This large volume deposit
(5000 km3) is made of a surprisingly homogeneous
crystal-rich dacite with near solidus mineral assemblages. Evidence for a late-stage heating pulse include
resorbed quartz crystals as well as reverse zoning in
plagioclase and hornblende phenocrysts, [47].
6.2. Crystallization bursts
The most specific symptom of the present mechanism is a short episode of crystallization occurring a
long time after the beginning of magma emplacement.
In this respect, the most striking observation comes
perhaps from the 13 ka-old caldera-forming eruption of
Laacher See Volcano, Germany. The probability function for the ages of a large number of sanidine
phenocrysts shows a sharp crystallization burst prior
to eruption, in marked contrast with earlier crystallization episodes, (Fig. 9), [48]. We note that this crystal
explosion occurs after about 105 yr of magmatic history,
which is consistent with the time-scales of our model.
6.3. Contamination by wall-rocks
Contamination by wall-rock material is a ubiquitous
feature of natural magmatic systems. The present
mechanism enhances this phenomenon through two
independent effects: a sharp heating event and the very
nature of the emplacement mechanism.
Sudden heating favors wall-rock melting, as shown
by the following argument. In a diffusive regime,
heating material from an interface at a constant heat flux
ϕ leads to a temperature anomaly ΔTb at the interface
that increases with time as:
/ pffiffiffiffiffi
jt
ð13Þ
DTb f
k
Over time t, the amount of heat supplied is ΔH = ϕt
and the expression for the temperature anomaly may be
recast as follows:
DTb f
DH 1
pffiffiffiffiffi
qCp jt
ð14Þ
This shows that, for a fixed amount of heat, ΔTb
increases as the heating time t decreases.
In addition, it is clear that contamination would be
enhanced if successive sills do not abut against one
another, but leave thin lenses of country rock between
them. Those layers would melt with the rest of the magma
50
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
Acknowledgements
This paper benefited greatly from comments and
criticisms by Jon Blundy, Tim Druitt, Steve Sparks and
one anonymous reviewer.
Appendix A. A Numerical method
Fig. 9. Probability function for the apparent age of crystallization of
sanidine phenocrysts from the Laacher See Tephra, Eifel volcanic
field, Germany, determined with 14C, from [48]. A clear peak at an age
of 13 ka coincides with the caldera-forming eruption which produced
the deposits. Before this event, crystallization ages exhibit a few minor
peaks and are spread over a very large time interval (as much as
110 ka).
Diffusion Eqs. (4) and (8) are solved with a finitedifference Crank–Nicholson method, with both explicit
and implicit “Forward Time Centered Space” schemes
for time-stepping. To integrate forward in time, from
step n to the next (n + 1), all terms in the equations are
written at step n + (1/2). The method has second-order
accuracy in time, with a truncation error of O[(Δz)2 +
(κΔt)2], and is stable for large time-steps Δt.
The difficulty comes from the kinetic terms in Eq.
(4), which are written at time t = n + (1/2) as follows:
i
LDt AU nþ1 AU n
LDt h nþ1
þ
Uj f ðTjnþ1 Þ þ Ujn f ðTjn Þ
¼−
2Cp At j
At j
2Cp sk
ð15Þ
during the catastrophic heating event. For a given reservoir
(pluton) thickness, it is obvious that the larger the number
of individual sills is, the larger the amount of country rock
that can get trapped within the magma pile. In that sense,
the extent of contamination may well provide much
information on emplacement characteristics.
7. Conclusions
We have described a new mechanism for the
formation of molten magma reservoirs, which relies
on the injection of thin sills or dykes. In such conditions,
crystallization kinetics act in an interesting way, first by
preventing crystallization in each thin unit and then by
enhancing it over a large number of units after a lengthy
time interval. With this mechanism, magmatic heat gets
released in two distinct events: sensible heat serves to
warm up country rock and latent heat provides the final
push for melting. The requirements for this mechanism
seem to have been met in several well-studied plutons.
Evidence for thin sills may be eradicated by the sharp
burst of heating and crystallization that is predicted by
this model. Changes in the injection sequence, including
variations of sill thicknesses and emplacement location,
lead to a range of thermal behaviours for nascent
magmatic systems. The model has many implications
that deserve separate investigations, such as the sudden
change from solid to liquid in a thick magma pile and
the rapid transition from basaltic to very evolved melts.
Ujn + 1 and f (Tjn + 1) are first calculated using an
explicit scheme:
Dt
nþ1
n
n
Uj ¼ exp logðUi Þ− f ðTj Þ
sk
f ðTjnþ1 Þ ¼ f ðTjn Þ þ
df
ðT n Þ½Tjnþ1 −Tjn dT j
At the end of the calculation, values of Tjn + 1 obtained
are used to correct for the values of f(Tjn + 1) and Ujn + 1
which is calculated as follows:
logðUjnþ1 Þ ¼ logðUjn Þ−
Dt
½ f ðTjnþ1 Þ þ f ðTjn Þ
2sk
ð16Þ
These new values are then introduced in a second
calculation loop for Tjn + 1, and Ujn + 1. The kinetic term
is thus calculated with an explicit scheme with a
truncation error of the order of O(κΔt). The Crank–
Nicholson method is stable for all values of ratio k ¼
jDt
and, in particular, for λ ≈ 0.5, when (Δz)2 is of the
ðDxÞ2
same order as κΔt. Therefore, handling the kinetic term
as explained above is accurate enough.
Calculations must yield accurate values of temperature
and crystal content within each sill, including the first ones
that are emplaced with very large thermal contrasts, and
within the surrounding wall rock. With time, a thermal
halo develops around the injection site and affects an
C. Michaut, C. Jaupart / Earth and Planetary Science Letters 250 (2006) 38–52
increasingly larger thickness. We set the computational
domain so that its lower boundary is deeper than the
thermal halo and so that it always contains a zone of a few
km thick unaffected by the thermal anomaly (i.e. where
the thermal gradient is equal to g0). To restrict the total
computation time, we start with a small domain (typically
10 km in thickness) and increase the domain size as time
progresses. As τi, the time between two injections, is made
larger, the number of intrusions needed for catastrophic
melting increases and hence the thermal halo around the
intrusion develops over a larger thickness, implying that
the computational domain must also be enlarged.
Δz = 10 cm allows sufficient precision at the intrusion
scale for a reasonable number of grid points. Calculations
with Δz = 5 cm yield results that are almost identical.
Calculations are stable for λ = 1.1. Results for λ = 1.1 and
0.5, differ by less than 0.02% after 6570 time-steps.
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