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Earth and Planetary Science Letters 317-318 (2012) 157–164
Contents lists available at SciVerse ScienceDirect
Earth and Planetary Science Letters
journal homepage: www.elsevier.com/locate/epsl
The influence of temperature-dependent thermal diffusivity on the conductive cooling rates of plutons and temperature-time paths in contact aureoles
Peter I. Nabelek a,⁎, Anne M. Hofmeister b, Alan G. Whittington a
a
b
Department of Geological Sciences, University of Missouri, Columbia, MO 65211, USA
Department of Earth and Planetary Sciences, Washington University, St. Louis, MO 63130, USA
a r t i c l e
i n f o
Article history:
Received 2 June 2011
Received in revised form 11 November 2011
Accepted 14 November 2011
Available online xxxx
Editor: R.W. Carlson
Keywords:
Thermal diffusivity
Heat flow
Pluton cooling
Contact aureoles
Numerical simulation
Latent heat
a b s t r a c t
We explore the conductive cooling rates of plutons and temperature-time paths of their wall rocks using numerical methods that explicitly account for the temperature dependence of thermal diffusivity (α) and heat
capacity (CP). We focus on α because it has the strongest influence on the temperature-dependence of thermal conductivity (k = ρ·CP·α) at high temperatures. Latent heats of crystallization are incorporated into the
models as apparent CP's. Two sets of models are presented, one for a 50 m thick basalt sill emplaced into rocks
with diffusivity of the average crust, and one for a 5 km wide and 2 km thick granite pluton emplaced into
dolostones. The sill's liquidus is 1230 °C, the solidus 980 °C, and the sill is emplaced into wall rocks that are
at 150 °C. The pluton's liquidus is 900 °C, the solidus 680 °C, and the pluton is emplaced into wall rocks
with initial geothermal gradient of 30 °C/km. The top of the pluton is at 3 km depth. Incorporating appropriate α = f(T) into calculations can more than double the solidification times of intrusions in comparison with
incorporating constant α of 1 mm2·s − 1 that is used in most petrologic heat transport models. The instantaneously emplaced basalt sill takes ~ 51 a to completely solidify, whereas the granite pluton takes ~ 52 ka to
completely solidify. The solidification time is longer because of low thermal diffusivity of magma and because
wall rocks become more insulating as temperature rises.
In contact aureoles, maximum temperatures reached with α = f(T) are somewhat lower in comparison with
α = 1 mm 2·s − 1; however, temperatures in inner aureoles stay elevated significantly longer with α = f(T),
and consequently promote approach to mineralogical and textural equilibrium. The elevated temperature regime in inner aureoles enhances temperature gradients that may promote greater fluid fluxes if fluid pore
pressures are controlled by temperatures. The results demonstrate the need to incorporate temperaturedependent transport properties of magmas and wall rocks into models of metamorphism and fluid flow in
contact aureoles.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
The cooling rate of a pluton from its liquidus depends on the latent
heat of crystallization (∆Hf) and the thermal conductivity (k) of the
pluton and its wall rocks. Calculating cooling rates of plutons dates
back to the early parts of the twentieth century when Ingersoll and
Zobel (1913) and then Lovering (1934, 1936) used analytical solutions to calculate the solidification times of basalt dikes, flows, and
even 3-D stocks. Lovering (1936) also explored the effects of different
wall rock conductivities as were known then. Jaeger (1957, 1959,
1964) recognized the importance of the latent heat of crystallization
on moderating the cooling rate of a crystallizing melt, and explored
the evolution of temperature gradients inside and outside of cooling
sheet plutons. Using constant values of ∆Hf and k these authors provide analytical solutions for the evolving thermal profiles. However,
⁎ Corresponding author at: 101 Geological Sciences Bldg. University of Missouri Columbia, MO 65211, USA. Tel.: +1 573 884 6463; fax: +1 573 882 5458.
E-mail address: [email protected] (P.I. Nabelek).
0012-821X/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2011.11.009
the salient properties of rocks and magmas, including heat capacity
(Cp), thermal diffusivity (α), and ∆Hf are temperature dependent,
making analytical solutions to the problem of pluton cooling inaccessible. Wohletz (2008) provides a numerical method for calculating
cooling of plutons that incorporate temperature-dependent thermal
conductivity based on parameters of Chapman and Furlong (1992).
This paper uses recent measurements of thermal diffusivities of
various crustal lithologies (e.g., Nabelek et al., 2010; Whittington et
al., 2009) in numerical models to evaluate the sensitivity of conductive cooling of plutons and heat flow in their wall rocks to the
known temperature dependence of CP and α. Thermal conductivity
is related to thermal diffusivity by
k ¼ α " CP " ρ;
ð1Þ
where ρ is density. The temperature dependences of CP and ρ are generally well known for geologic materials (e.g., Robie and Hemingway,
1995), but older laboratory measurements of α are suspect due to
contact losses and unwanted ballistic radiative transfer as shown by
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P.I. Nabelek et al. / Earth and Planetary Science Letters 317-318 (2012) 157–164
Hofmeister (2007) and Hofmeister et al. (2007). However, laser-flash
analyses (LFA) of thermal diffusivity of lithospheric materials avoid
these problems and demonstrate that α is highly dependent on temperature, and is lower for melts and glasses than minerals of identical
compositions (Hofmeister et al., 2009; Pertermann et al., 2008;
Whittington et al., 2009). It is clear then that the thermal diffusivity
in a magma must be lower than in rocks. Moreover, because thermal
diffusivity in rocks decreases with increasing temperature, the rate of
heat transfer through wall rocks of a pluton must decrease as the contact aureole becomes hotter, and therefore the cooling rate of the pluton must decrease beyond the rate related to the diminishing
temperature gradient. Although CP of rocks generally increases with
temperature, the increase does not entirely compensate for the decrease in α. For magma-wall rock systems the temperaturedependence of ρ is so small that it can be neglected, except when
fluids are involved. Therefore, k decreases with temperature also
(Whittington et al., 2009) and for the lithosphere the dependence is
significantly different from that given by Chapman and Furlong
(1992). Consequently, the temperature dependence of k has implications for durations of magmatic systems and related contact metamorphism and hydrothermal fluid convection. This paper examines
the influence of variable thermal diffusivity on the conductive cooling
time scales of crystallizing basalt sills and large granite plutons, and
on the temperature-time paths of their respective wall rocks.
2. Thermal diffusivity of magmas and crustal rocks
Fig. 1 shows fitted thermal diffusivity functions for several relevant crustal rocks and magmas. An exponential form:
α ¼ a⋅ expðTK=bÞ þ c;
ð2Þ
where TK is temperature in Kelvin and a, b, and c are experimentally
determined constants (Table 1), is used because it both reasonably
represents experimental data and does not induce singularities in numerical models. The “average crust” curve is a fit to schist, granite,
and rhyolite data in Whittington et al. (2009) and the “metamorphic”
curve is a fit to granulite and mafic gneiss data in Nabelek et al.
(2010). Coefficients for the various rocks used are listed in Table 1.
The actual data may be as much as 0.3 mm 2·s − 1 above and below
each curve, depending on the foliation and lineation orientation and
mineralogy. Quartz has the highest thermal diffusivity among major
rock-forming minerals in the crust, hence the “average crust” curve
gives more elevated α. A fit to unpublished data for a dolomite
Table 1
Coefficients for α, CP, and ∆Hf functions.
Coefficients for α function (Eq. (2); mm2·s− 1)
a
b
0.534
5.017
Average lithosphere
(granite)1
Metamorphic2
0.319
3.162
0.365
6.953
Dolomite marble3
4
0.534
0
Rhyolite melt
5
0.300
0
Basalt melt
2.5
300
500
a
Coefficients for ∆Hf function (∆Hf = a + b·T; J·kg
a
b
2.465 × 105
0
Granite9
Basalt7
−1.118 × 105 363.1
2.197 × 106
0
2.202 × 107
0
− 2.017 × 104
− 1.847 × 104
− 2.976 × 104
0
0
0
)
marble gives relatively high α at very low temperatures but is nearly
coincident with the “metamorphic” curve at elevated temperatures.
All three “rock” curves underscore that α may be >1.5 mm 2·s − 1 in
the shallow crust but b0.6 mm 2·s − 1 in the temperature regime
where melts can be stable.
Thermal diffusivity of alkali feldspar glasses is significantly lower
at low temperatures than α's of crystalline rocks (Hofmeister et al.,
2009; Pertermann et al., 2008), but it essentially coincides with the
“average crust” curve at high temperatures. α of a rhyolite melt
(Romine, 2008) coincides with the high temperature end of the “average crust” curve. Thus, above 700 °C, high-Si rocks, glasses, and
melts have similar α's of ~0.55 mm 2·s − 1. MORB glasses have similar
α variation with temperature as rhyolite glass, but the α drops to
~0.25 mm 2·s − 1 between the ~ 900 °C solidus and the ~1200 °C liquidus of the measured sample (Galenas, 2008). This value closely corresponds to that of diopside melt (α = 0.29 mm 2·s − 1) and anorthite
melt (α = 0.36 mm 2·s − 1; Hofmeister et al., 2009).
temperature (°C)
900
1100
1300
1600
1.0
100
300
500
900
1100
1300
basalt melt
granite melt
1200
1000
avg. crust & metamorphic
rhyolite melt
rhyolite glass
700
b
1400
Cp (J.kg–1K–1)
1.5
0.5
−1
c
d
2.197 × 106 − 2.017 × 104
Sources:
1
Whittington et al. (2009); 2Nabelek et al. (2010); 3unpublished; 4Romine (2009);
5
Galenas (2008); 6Holland and Powell (1998); 7Bouhifd et al. (2007);
8
Lange (2003); 9Tenner et al. (2007).
metamorphic & basalt
average crust
dolomite marble
2.0
α (mm2s–1)
700
285.4
225.2
0
0
Coefficients for CP function (Eq. (4); J·kg− 1·K− 1)
a
b
−3.911 × 10− 2
Average lithosphere 1916.2
(granite)1
Metamorphic1
1916.2
−3.911 × 10− 2
Dolomite marble6
1946.3
−2.660 × 10− 2
Basalt7
2337.0
−2.773 × 10− 1
Rhyolite melt (liquid 1392.0
0
albite)8
Basalt melt7
1442.6
5.940 × 10− 2
temperature (°C)
100
c
222.1
dolomite marble
800
basalt
MORB
0.0
300
500
700
900
1100
temperature (K)
1300
1500
600
300
500
700
900
1100
temperature (K)
1300
1500
Fig. 1. (a) Exponential thermal diffusivity functions used in models and additional data for silicate melts and rhyolite glass. (b) Heat capacity functions used in models. Coefficients
and data sources for the thermal diffusivity and heat capacity functions from Table 1.
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3. Numerical methods
Calculations of evolving temperature profiles across cooling intrusions and their wall rocks was performed numerically using the inhouse Fortran program SILLS. The program implements a finitedifference algorithm that accounts for temperature-dependent thermal
diffusivity and heat capacity, as well as different densities of rocks and
magmas. The simulations were done in two dimensions (2-D). In each
dimension, SILLS solves the conductive heat-flow equation:
!
"
∂T
1
∂
∂T
þ Γ;
⋅
¼
ρCρ α
∂t ρCρ ∂x
∂x
ð3Þ
where Γ represents the contribution of local heat sources or sinks to
temperature. The chosen time step, dt, was always smaller than the stability criterion dx2/4α by a factor of 10 to reduce numerical errors, using
α appropriate for 25 °C where it is the highest in the lithosphere. Physical properties were computed for each node at each time step. The dependence of α on temperature was discussed above. The polynomial
Maier-Kelly function (Richet, 2001) was used for CP:
CP ¼ a þ b⋅TK þ c⋅TK
−2
−0:5
þ d⋅TK
ð4Þ
:
Constants for the modeled materials are given in Table 1 and the
functions are shown in Fig. 1. For Γ, we considered only the addition
of latent heat of crystallization (∆Hf), which can also be
temperature-dependent (Bouhifd et al., 2007). We did not consider
addition of heat from radioactive decay or consumption of heat by
metamorphic reactions in wall rocks as these can be highly variable
depending on local geologic situations. While a melt is crystallizing
between its liquidus and solidus, Γ during a time step can be
expressed as:
Γ¼
Hf df
⋅
;
Cp dt
In our simulations we used the apparent heat capacity approach,
in which CP of a node that is within the crystallization interval is adjusted for ∆Hf (Hu and Argyropoulos, 1996):
Capp ¼ Cmag þ Hf =ðTL –TS Þ;
ð7Þ
where
Cmag ¼
T−TS
T −T
C
þ L
C
:
TL −TS liq TL −TS xtals
ð8Þ
It is noted that only the CP in the first material property term of
Eq. (3) was adjusted, not the CP in the second diffusive term. Adjustment of the second CP would lead to anomalously high thermal conduction of magma nodes. We tested the apparent heat capacity
approach by modeling the cooling of a near-eutectic pluton with a
1 °C crystallization interval using the above CP and ∆Hf values
(Fig. 2). The shrinking center of the pluton remains liquid as its
outer portions cool below the solidus, as expected. Therefore, the apparent heat capacity approach is a reasonable means of incorporating
∆Hf into the problem of cooling a crystallizing pluton.
4. Model domains
Two intrusion models are shown. In both, temperatures at all
boundaries were allowed to vary. The first model is for an instantaneously emplaced, 50 m thick basalt sill with TL = 1230 °C and
TS = 980 °C (Fig. 3a). These temperatures are appropriate for a midocean ridge basalt as calculated with the computer program MELTS
(Asimow and Ghiorso, 1998; Ghiorso and Sack, 1995; http://melts.
ofm-research.org/). The wall rock was assumed to have temperature
dependences of α and CP of an average crustal metamorphic rock
Eutectic Granite
ð5Þ
0
where df is the fraction of minerals crystallized over a time step dt. df/
dt can be expressed as:
df
T−Tn
;
¼
dt TL −TS
ð6Þ
2
where, at a node, T is temperature at previous time step, T is temperature at current time step, TL is the liquidus temperature and TS is the
solidus temperature. It was assumed that the amount of minerals increases linearly with drop in temperature between the liquidus and
the solidus of the magma.
Simply adding Γ at a node of a crystallizing melt can be problematic.
For example, in the Stefan problem of a moving crystallization boundary
of a near-eutectic melt, the amount of heat added to a node from ∆Hf
may exceed the amount of heat lost from the node by conduction during the chosen time step (Hu and Argyropoulos, 1996). The problem
can be readily seen for eutectic or near-eutectic systems. For example,
assuming a near-eutectic melt that crystallizes over a 1 °C interval and
assuming thermal properties of albite, CP = 1392 J·kg− 1·K− 1 (Lange,
2003) and ∆Hf = 246 kJ·kg− 1 (Tenner et al., 2007), if the temperature
drop due to conduction at a node exceeds 1 °C during the time step, Γ
for the time step is 177 °C. Therefore, depending on the temperature
of the wall rocks and the length of the time step, the apparent temperature in the node may increase rather than decrease. The result is incorrect and the numerical solution rapidly fails. One potential solution to
the problem would be to make the numerical time step so small that
temperature in the magma would be allowed to drop only a fraction
of a degree between the solidus and the liquidus. However, this approach is computationally demanding for big plutons that crystallize
over a large temperature interval, and therefore it is impractical.
depth (km)
n
4
6
8
100
300
500
T (°C)
700
Fig. 2. Isotherms at 5 ka intervals across a 3 km thick near-eutectic granite pluton and
its wall rocks. Dashed line is the initial temperature profile for instantaneously
emplaced melt. The liquidus temperature was 681 °C, the solidus 680 °C. Heat of crystallization (∆Hf) = 246 kJ·kg− 1 (Tenner et al., 2007) and thermal diffusivity (α)
= 1 mm2·s− 1 was assumed throughout the model domain.
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a
150
Basalt Sill
500 m
150°C
depth (m)
1230°C
150°C
1000 m
b
150
200
200
250
250
300
300
b
Granite Pluton
3
10 a intervals
900
350
700
5
10 a intervals
350
0
150
200 400 600 800 1000 1200
c
500
0
150
200 400 600 800 1000 1200
d
300
7
200
200
250
250
300
300
100
T°C
depth (m)
depth (km)
a
9
12
0
2
4
6
width (km)
8
10
12
Fig. 3. Initial temperature conditions in model domains. (a) 50 m basalt sill with liquidus T of 1230 °C and wall rock T of 150 °C. (b) Granite pluton, 5 km horizontally and
2 km vertically and with liquidus T of 900 °C. The wall rock has a 30 °C/km temperature
gradient starting at 90 °C at the top.
(Nabelek et al., 2010; Table 1). The initial temperature of the wall
rock was assumed to be 150 °C. Because the shape of the domain
does not vary in the horizontal direction, the problem is effectively
1-D in the vertical direction. The thickness of the domain was
500 m with grid-spacing of 1 m.
The second model is for a large rectangular granite pluton instantaneously emplaced into dolostones. The pluton has dimensions
x = 5 km and z = 2 km within a model domain with x = 12 km and
z = 8 km (Fig. 3b). Grid-spacing of 10 × 10 m was used. TL and TS
were assumed to be 900 °C and 680 °C, respectively. These temperatures are appropriate for H2O-bearing anorogenic granites. For the
wall rock, a geothermal gradient of 30 °C/km was assumed with top
of the domain at 90 °C. The implied depth of the top of the domain
is 3 km. Although in a strict sense sills are also plutons, for simplicity
we refer to models with the basalt sill as “sill” models and to models
with the granite pluton as “pluton” models.
5. Results
5.1. Intrusions
For each model domain, vertical temperature profiles across the
center of each domain are shown in Figs. 4 and 5 for four cases with
different combinations of CP, ∆Hf, and α. Temperature-time paths
for centers and margins of the intrusions are shown in Fig. 6. The
four cases are: 1) ∆Hf is 0, CP is constant (1300 J·kg − 1 K − 1 for sill,
1200 J·kg − 1 K − 1 for pluton), and α = 1 mm 2·s − 1; 2) ∆Hf for sill is
f(T) and for pluton 246 kJ·kg − 1, CP is constant as above, and
α = 1 mm 2·s − 1; 3) ∆Hf is as above, CP is f(T) and α = 1 mm 2·s − 1;
4) ∆Hf is as above and both CP and α are f(T). The various temperature
350
10 a intervals
0
200 400 600 800 1000 1200
T (°C)
350
10 a intervals
0
200 400 600 800 1000 1200
T (°C)
Fig. 4. Evolution of temperatures across a 50 m basalt sill and its wall rocks. Top and bottom
150 m of the model domain are not shown. The dashed line shows the initial temperature
profile. Vertical line within the domain of the sill marks the solidus temperature of 980 °C.
Temperature profiles are shown in 10 a intervals, with thick lines showing profiles when
the sill is not yet completely solidified. (a) Profiles when ∆Hf = 0, CP = 1300 J·kg− 1 K− 1,
and α =1 mm2·s− 1; (b) Profiles when ∆Hf = f(T), CP = 1300 J·kg− 1 K− 1, and
α = 1 mm2·s− 1; (c) Profiles when ∆Hf =f(T), CP = f(T), and α=1 mm2·s− 1; (d) Profiles
when ∆Hf, CP, and α are all functions of T.
dependencies are listed in Table 1. The constant CP and α values used
in some of these cases are often employed in magmatic and lithospheric heat flow models (e.g., Annen, 2011; Michaut and Jaupart,
2011; Turcotte and Schubert, 1982). A commonly used constant
value of thermal conductivity is 2.5 W·m –1 K − 1 (e.g., Turcotte and
Schubert, 1982). The difference between cases 3 and 4 is meant to
emphasize the importance of α = f(T). Although temperature dependencies of both α and CP are related to material lattice and disorder,
the two properties are entirely independent based on physics and
therefore can be treated separately. Thermal diffusivity is an anharmonic property that is related to damped oscillations (expressed as
peak widths of IR modes) and is connected with thermal expansivity
(Hofmeister, 2010), whereas heat capacity is related to mode frequencies and is 97% harmonic (Kieffer, 1980).
Temperature profiles are shown in 10 a intervals for the basalt sill
and in 10 ka intervals for the granite pluton. Incorporating ∆Hf confirms what has been known for decades, i.e., it moderates the rate
of cooling (Jaeger, 1957). The model sill completely solidifies in 8 a
without ∆Hf and in 18 a with ∆Hf (Figs. 4, 6), assuming constant Cp
and α. The model pluton solidifies in 15 ka without ∆Hf and in 32 ka
with ∆Hf (Figs. 5, 6). Thus, incorporation of ∆Hf at least doubles the
duration of crystallization. The margin of the sill reaches the solidus
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P.I. Nabelek et al. / Earth and Planetary Science Letters 317-318 (2012) 157–164
depth (km)
3
a
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
b
300
500
700
300
700
900
d
c
4
4
5
5
6
6
7
7
8
8
9
9
10
10
300
500
700
T (°C)
900
10 ka intervals
11
100
Rates of temperature increase and durations of elevated temperature regimes in contact aureoles are key factors that drive metamorphic reactions and hydrothermal systems. They depend on the rates
of heat transport that are controlled by thermal conductivity. Fig. 7
shows a comparison of the temperature evolution at four distances
from each intrusion. Temperature-time curves are shown at distances
5 m, 20 m, 50 m, and 100 m from the basalt sill and at distances 20 m,
250 m, 500 m, and 1 km from the granite pluton. Evidently, there are
important differences between models with α = 1 mm 2·s − 1 and
α = f(T), with both models incorporating CP = f(T). With constant α,
peak temperatures at all distances are higher and are reached earlier.
The difference when peak temperatures are reached in the two sets of
models increases with distance from intrusions because diminishing
α in inner aureoles slows down heat transport into outer aureoles.
However, because α diminishes with increasing temperature, once
rocks become hot they stay longer at elevated temperatures, in comparison with when α is constant. The difference in temperatures between the two sets of models diminishes with increasing distance
from intrusions.
6. Discussion
10 ka intervals
11
100
500
3
3
depth (km)
900
10 ka intervals
11
100
because CP, and hence conductivity, in the aureole is lower than the
constant 1200 J·kg − 1 K − 1 used above, the crystallization rate is
slightly lower with variable CP.
Introduction of temperature-dependent α into simulations can
more than double the duration of crystallization, particularly in centers of intrusions (Fig. 6). In the case of the sill, it takes 51 a for the
center to reach the solidus and in the case of the pluton it takes
52 ka. The longer solidification times stem from small α in hot
magma and progressively smaller α in the wall rocks as they are heated. With α = f(T), the centers of intrusions stay at or near their liquidi
for some time, after which the cooling rate becomes nearly linear
until solidi are reached (Fig. 6). In contrast, T-t profiles for margins
are curved with progressively decreasing cooling rate as α's in inner
aureole rocks progressively decrease with rising temperatures. In effect, the aureole becomes progressively more insulating.
5.2. Contact aureoles
10 ka intervals
11
100
161
300
500
700
T (°C)
900
Fig. 5. Evolution of temperatures across a granite pluton and its wall rocks. The dashed
line shows the initial temperature profile. Vertical line within the domain of the pluton
marks the solidus temperature of 680 °C. Temperature profiles are shown in 10 ka intervals, with thick lines showing profiles when the pluton is not yet completely solidified. (a) Profiles when ∆Hf = 0, CP = 1200 J·kg− 1 K− 1, and α = 1 mm2 s− 1; (b)
Profiles when ∆Hf = 246 kJ·kg− 1, CP = 1200 J·kg− 1 K− 1, and α = 1 mm2·s− 1; (c)
Profiles when ∆Hf = 246 kJ·kg− 1, CP = f(T), and α = 1 mm2·s− 1; (d) Profiles when
∆Hf = 246 kJ·kg− 1, and CP and α are f(T).
very rapidly, in 0.013 a without ∆Hf and in 0.039 a with ∆Hf. The margin of the pluton reaches the solidus in 2 a without ∆Hf and in 6 a
with ∆Hf. Incorporation of ∆Hf causes inflections in cooling rates at
the solidi because the moderating effect of ∆Hf disappears.
Incorporating temperature-dependent Cp into simulations has little effect on the crystallization rate. The reason is that the CP functions
above 300 °C begin to flatten out, so the average for basalt and basalt
melt is ~ 1300 J·kg − 1 K − 1, which is the constant value used for the
sill model above. Therefore, while the sill is in magmatic state, the
conductivity is affected only little. Similarly, CP does not vary appreciably within the crystallization range of the granite pluton, although
Heat transport drives many processes in the Earth's lithosphere,
including partial melting, metamorphism, hydrothermal systems,
and cooling of plutons. The temperature dependence of thermal diffusivity influences the shape of the steady-state lithospheric geotherm,
the rate of heat transfer upon thermal perturbation, and heat retention in partially molten crust (Nabelek et al., 2010; Whittington et
al., 2009). Results presented here demonstrate the need to also incorporate temperature-dependent thermal diffusivity into modeling
thermal transport in pluton-wall rock systems. Overall, our calculations show that longevities of model magmatic and metamorphic systems are significantly prolonged when α = f(T) is incorporated. Small
α's of magmatic systems allow liquid or mush states to remain for
quite extended periods of time. Our results agree with observations,
as follows:
Retention of heat in basalt sills keeps viscosity low, thereby allowing
intrusion over large distances as exemplified by the Ferrar sills in Antarctica or Franklin sills on Victoria Island, Canada (Bédard et al., 2007,
in press; Elliot et al., 1999; Leat, 2008). Retention of heat permits extended duration of magma flow through conduits, reintrusion of conduits, and allows sills to differentiate by accumulation and sorting of
crystals. Approximately 50–60% crystallinity is needed for magma to
become stiff due to close-packing of crystals (Marsh, 1981). For the
model basalt sill, 55% crystallinity would occur at ~1090 °C, assuming
a linear increase in crystallinity with decreasing temperature between
the liquidus and the solidus and no crystal settling. Propagation of this
isotherm from margin to center of the 50 m sill is shown for the α = f
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a
b
Sill center
Sill margin
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1200
temperature (°C)
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solidus
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1100
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solidus
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c
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time (a)
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Pluton center
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temperature (°C)
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solidus
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solidus
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0
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time (ka)
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time (a)
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Fig. 6. Temperature-time paths for centers and margins of intrusions. Note different temperature and time scales. (a) Sill center; (b) sill margin; (c) pluton center; (d) pluton margin. Dotted lines show results for ∆Hf = 0 with constant Cp and α , see text; short-dashed lines depict including ∆Hf with constant Cp and α; long-dashed lines depict including ∆Hf,
with Cp = f(T) and constant α; solid lines represent including ∆Hf with T-dependent Cp and α.
(T) in Fig. 8, in which case part of the sill could be mobile as a mush for
~33 a after emplacement of a single batch of melt. If instead
α = 1 mm2s − 1 was applied, the mobility would cease after 15 a.
Our calculations demonstrate that accounting for α = f(T) can more
than double longevities of magmatic states of plutons, which helps explain recent findings that plutons can grow and remain active for extended periods of time as evidenced by improved dating techniques,
in particular U-Pb dating of zircon, and by detailed studies of pluton
construction. These studies (Coleman et al., 2004; Glazner et al., 2004;
Memeti et al., 2010; Michel et al., 2008; Miller and Miller, 2002) suggest
construction time scales ranging from 10's of ka for km-scale plutons to
10's of Ma for very large plutonic centers, such as the Tuolumne intrusive series in the Sierra Nevada. Interpreting the spatial distribution of
radiometric data from intrusive series depends to an extent on constructing thermal models of pluton growth that rely on the timing of
magma replenishment and crystallization through blocking temperatures of relevant isotopic systems in successive pulses of magma
(Coleman et al., 2004; Memeti et al., 2010). The extent of crystallization
of melt pulses affects the ability of successive pulses to mix, to assimilate
previously crystallized parts of plutons, and to potentially cause large ignimbrite eruptions (Lipman, 2007). These processes are largely controlled by how rapidly magma is crystallizing by cooling against wall
rocks and previously injected parts of magma chambers. It is now generally accepted that plutons grow by incremental injection of magma, a
process that allows magma chambers to exist for durations often many
times beyond that suggested by our simple, instantaneously emplaced
pluton model, even when temperature-dependent thermal diffusivity
is accounted for (e.g., Annen, 2009, 2011; Michaut and Jaupart, 2011).
Nevertheless, our models suggest rates at which small melt batches
need to invade a growing magma chamber and keep it at least partially
molten. For example, a 50 m wide pulses of mafic melt injected at least
every few tens of years keep part of a magma chamber at least partly
molten, as suggested by Fig. 6a. However, the frequency at which
magma would need to be injected will progressively decrease because
small α in hot, previously emplaced parts of the chamber and its heated
wall rocks would retard heat loss. Thus, progressively decreasing α in a
growing pluton will prolong the longevity of its magmatic state.
Convection in magma chambers increases the rate of cooling because when hot material from bottom of a chamber is brought toward
the top margin where the temperature gradient is larger, the heat flux
out of the chamber becomes greater. The ratio of the total heat transfer
in the presence of convection to that which would occur with conduction alone is given by the Nusselt number (Nu). When there is only
conductive heat transfer, Nu= 1. For convection to occur, a magma
chamber must be heated from below in order to sustain a temperature
gradient with temperature decreasing upward. In our models with instantaneous melt emplacement, convection could occur only in the
upper part of the magma chamber while the magma had less crystals
than required to reach Bingham body behavior. Nevertheless, within
the convective regime, a low α of the magma will increase Nu due to
the fact that the Rayleigh number (Ra) increases inversely with α.
Given the Nusselt–Rayleigh relationship, Nu = c·(Ra) 1/3 (e.g., Turner,
1973; c is a constant), Nu is related to α by (1/α )1/3. Thus, for example,
a decrease of α from 1 mm2·s− 1 to 0.5 mm2·s− 1 results in 26% increase in Nu. This illustrates that while temperature-dependent thermal
diffusivity should increase the relative rate of convective heat loss, the
increase will be smaller than the reduction in the rate of conductive
heat loss due to a decrease in α .
Author's personal copy
P.I. Nabelek et al. / Earth and Planetary Science Letters 317-318 (2012) 157–164
a
Metamorphism by Basalt Sill
900
800
5m
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600
500
20 m
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50 m
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100 m
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Metamorphism by Granite Pluton
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temperature (°C)
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20 m
500
250 m
500 m
400
1000 m
300
200
100
0
10
20
30
40
50 60
time (ka)
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80
Fig. 7. Temperature-time paths of wall rocks at four distances from basalt sill and granite pluton. Dashed lines are when ∆Hf is included and α = 1 mm2·s− 1; solid lines are
when ∆Hf is included and α = f(T).
1090°C isotherms
Most previous thermal models of pluton growth and crystallization and related metamorphism of wall rocks have used constant
values for material properties, including α and CP. Although constant
values permit analytical heat flow solutions, they do not accurately
describe the thermal evolution of magmatic and metamorphic systems. Such systems are best modeled numerically by incorporating
temperature-dependent material properties. Numerical simulations
presented in this paper demonstrate that model longevities of magmatic systems are significantly extended by accounting for thermal
dependencies for material properties.
α = 1 mm2s-1
α = f(T)
distance (m)
In contact aureoles, decreasing α with increasing T extends the
duration of contact metamorphism and related hydrothermal systems (Fig. 7). Extended periods of elevated temperatures promote
completion of metamorphic reactions and coarsening of minerals
that is evident in most contact aureoles in close proximity to plutons
(e.g., Joesten, 1991; Kerrick et al., 1991). Long retention of heat in wall
rocks allows for extended periods of hydrothermal activity and the
consequent generation of alteration assemblages in plutons and
their contact aureoles (e.g., Cathles, 1981; Criss and Taylor, 1986;
Dilles and Einaudi, 1992; Nabelek, 2009; Nabelek and Morgan, in
press). Particularly, the consequence of α = f(T) is that except for
early stages of metamorphism, the spread in temperatures between
inner and outer aureoles is greater than when α = 1 mm 2·s − 1. A larger temperature gradient will enhance fluid flow velocities due to larger gradients in fluid pressures. As shown for the granite pluton, the
temperature difference between the inner aureole and 1 km outward
is still >50 °C at 10 ka after magma emplacement (Fig. 7).
We did not incorporate enthalpies of metamorphic reactions (∆Hr)
in the presented models because metamorphic reactions depend on
rock and fluid compositions and can be discontinuous or continuous.
Thus, the effect of ∆Hr on heat flow is specific to each contact aureole.
However, prograde metamorphic reactions, being endothermic, will reduce the rate of temperature increase in aureoles. For example, the
effectively discontinuous reaction of dolomite+ quartz to diopside +
CO2 consumes ~448 kJ/kg, which is approximately twice the amount released by crystallization due to ∆Hf, assuming a stoichiometric proportion of the reactants in a rock. Hence, metamorphic reactions in contact
aureoles will affect heat flow in a pluton-wall rock system.
Fluid flow can also affect heat flow in pluton-wall rocks systems,
particularly in the shallow crust where permeabilities can be large.
Advective heat transport becomes significant when permeability
reaches ~10− 16 m 2 (Cui et al., 2001; Manning and Ingebritsen, 1999).
Thus, seepage of a cooler fluid, for example meteoric water, into the
thermal aureole of a shallow pluton will enhance the cooling rate of
the system. However, at depths >5 km where many plutons are
emplaced and permeability is 10− 18 m2 or less (Manning and
Ingebritsen, 1999), advective cooling of a pluton-wall rock system will
be negligible. Nevertheless, even in the shallow crust, within the context of hydrothermal systems, the influence of variable thermal diffusivity on prolonging the longevity of magmatic systems will remain,
particularly while plutons are partially molten and impermeable.
7. Conclusions
center 25
20
163
15
10
Acknowledgments
5
This work was supported by NSF grants EAR-0911116 and EAR0911428. We thank Catherine Annen and an anonymous reviewer
for constructive comments.
margin 0
0
5
10
15
20
time (a)
25
30
35
Fig. 8. Propagation of the 1090 °C isotherm with time from margin to center of sill. The
isotherm corresponds to ~55% crystallinity of the basalt magma. One isotherm is with
α = 1 mm2·s− 1, the second with α = f(T).
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