Foreword
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 191-192
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Special Section: «Keynote lectures of the 16th Conference
on Deformation mechanisms, Rheology and Tectonics (DRT)»
Foreword
This Thematic Section collects original contributions (mainly short papers)
offered by the invited keynote lecturers of the 16th issue of the Conference on
Deformation mechanisms, Rheology and Tectonics (DRT), held for the first time
in Italy at Dipartimento di Scienze della Terra «A. Desio», Facoltà di Scienze
MFN, Università di Milano, from Sept. 24 to Oct. 2, 2007. As a background it is
significant to recall that the DRT Conference series represents one of the most
relevant biennial event for the international community that operates in the
fields of structural geology, tectonics, geodynamics, modeling, experimental
deformation and rheology. The DRT Conference series was promoted in 1976 in
Leiden by Henk Zwart, Richard Lysle, Gordon Lister and Paul Williams (a
history of the DRT meetings until 2001 may be found in the Preface of London
Geological Society Spec. Publication series n° 200, 2002). More recent DRT
Conferences, organized after year 2000, were held in Neustadt, Utrecht, St.
Malo-Rennes, Zuerich and then Milano. Since the beginning, the series of DRT
meetings managed to bring together structural geologists, material scientists
and geophysicists devoted to theoretical and experimental work; successively
field investigators joined the group, that continued to debate periodically
updated advances in tectonics, gained at any scale.
In the Milano 2007 Conference, discussions on 10 topics were based on
presentation of 14 keynote lectures, 38 oral communications and 135 posters.
Subjects encompassed advances in the investigation of lithosphere-scale
tectonic mechanisms, in connection with geological and geophysical results
from various analytical scales. Besides these leading themes of oral and poster
presentations, discussions during two field excursions and at the Workshop,
spread over tectonic mechanisms of subduction-exhumation of ophiolites (Valle
Po-Valle Varaita, excursion to Monviso) and of the continental crust (OropaBiella, excursion to Mucrone-Monte Mars metagranitoids) and on fitting of
modelling predictions with structural and petrologic natural data on orogenic
metamorphites. Workshop discussion helped to individuate non-consistent
results and improvements, introduced by new approaches in gathering and
processing field or laboratory data (P-T estimates, age data, tectonic units size,
and others...) or by different modelling approaches.
With the aim of leaving a published trace in the Bollettino della Società
Geologica Italiana, most Keynote Lecturers accepted to summarize the main
points of their invited contributions in the following short papers, that represent
the summary of the main stream of the scientific contents of the conference
sessions. The scientific sessions were grouped by topics and themes therein
covered: 1) crust and mantle rheology from micro- to mega- scale (strength
contrast between crust and upper mantle, structure and rheology of lithosphere
scale shear/fault zones); 2) numerical and analogical modeling of deformation
processes (role of rheology in mechanical models, identification of rheological
‘knowledge gaps’, (up)scaling and scale dependence of rheological relations,
studies on consistency of field observations and laboratory-derived flow laws);
3) absolute dating vs deformation: the rate of tectonics (advances in
geochronology necessary to discern the micro-scale separation of isotopic
imprints and the relationships between fabrics and mineral assemblages that
reflect step-like evolutions related to displacement of tectonic units in active
lithosphere zones); 4) deformation-metamorphism interaction: what does
condition the memory of a rock? Insigths from natural data, experiment and
modeling (metamorphic reaction progress and deformation history confronted
192
SPECIAL SECTION
with the activation in adjacent rock volumes of contrasted deformation
mechanisms and/or strain rates); 5) interaction between magmatism and
deformation: field studies, numerical models and analogue experiments
(deformation and melting processes, crystallization, segregation, transport and
emplacement of melts or magmas, and the flow of two-phase materials with
very contrasted rheology); 6) palaeorheology (estimates of rock rheology based
on natural rocks and modeling); 7) the geophysical signature of deformation
processes in crust and mantle (global dynamics, and methodologies for inferring
the viscosity profile of the mantle, from GPS data constrains upon predictive
geophysical modeling); 8) quantitative microstructure (microstructures and
textures in rocks: image analysis, electron diffraction, X-ray diffraction, neutron
diffraction; focus on microstructure and texture development, including
polyphase rocks, experimental microstructures and predictions); 9) brittle and
ductile reactivation of compositional and structural heterogeneities (multi-scale
reactivation of structures and strain localization; strain-stress patterns and
rheological contrasts); 10) interaction between climate, erosion and tectonics
(climate, surface processes and tectonics: search for testable predictions of
models). In the post-conference Workshop in Oropa-Biella, conducted over one
of the most intriguing subduction-exhumed Alpine rock associations, containing
remnants of pre-Alpine continental crust, attention was driven to refinement of
analytical strategies and techniques in Geology, that may improve realistic
investigation of geophysical processes through numerical modeling.
Guido Gosso, Anna Maria Marotta, Roberto Sabadini and Maria Iole Spalla
formed the Organising Committee; the Scientific Committee of the 16th DRT
Conference was formed by Ulf Bayer, Jean Pierre Brun, Stephane Bonnet, JeanPierre Burg, Luigi Burlini, Daniel Chateigner, Martyn Drury, Terry Engelder,
Marnie Forster, Taras Gerya, Rob Govers, Harry W. Green II, Djordje Grujic,
Gordon Lister, Giorgio Pennacchioni, Giorgio Ranalli, Claudio Rosenberg,
Bernard Stoeckhert, Holger Stuenitz, Janos Urai, Jean Louis Vigneresse, Igor
Villa, Paul F. Williams, and Michele Zucali; the post-Conference Workshop has
been shaped by Daniele Castelli, Taras Gerya, Rob Govers, Anna Maria Marotta,
and M. Iole Spalla; the field leaders of pre-Conference escursion were Daniele
Castelli and Roberto Compagnoni and of post-Conference excursion were
Daniele Castelli, Guido Gosso, Piergiorgio Rossetti, Maria Iole Spalla, Davide
Zanoni and Michele Zucali. Field guides of pre- and post-conference excursions
were published on volume 9 (2007) of Quaderni di Geodinamica Alpina and
Quaternaria.
The Organising Committee of the 16th Conference of the DRT series thanks
the keynote speakers for their contributions; support is here gratefully
acknowledged from Società Geologica Italiana, Gruppo Italiano di Geologia
Strutturale and the Section of Milano of CNR-IDPA (Istituto per la Dinamica dei
Processi Ambientali), that respectively sponsored and financially sustained the
present publication and the other editorial activities.
The editors of this thematic section are deeply grateful to the reviewers
(Daniele Castelli, Martyn Drury, Taras Gerya, Stefano Poli, Claudio Rosenberg,
Roberto Sabadini and other anonymous, belonging to the Conference Scientific
Committee), for their support in the revision of the key-note contributions.
The editors of the thematic section on 16th DRT Conference
Guido GOSSO (*), (**)
Anna Maria MAROTTA (**)
Maria Iole SPALLA (*), (**)
(Valle Po-Valle Varaita, Milano, Oropa-Biella, Sept. 24-Oct. 2, 2007)
(*) CNR - Istituto per la Dinamica dei Processi Ambientali, Sezione di Milano.
(**) Dipartimento di Scienze della Terra «A. Desio», Università di Milano.
Buiter
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 193-198, 3 figs.
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Piazzale Aldo Moro, 5 – 00185 ROMA
Rheology in numerical models of lithosphere deformation
SUSANNE J.H. BUITER (*)
ABSTRACT
Deformation of Earth’s crust and lithosphere is characterised by
elastic, viscous and brittle material behaviour. The implementation
of this complex rheology is a challenge for numerical models owing
to variations in laboratory and natural data, the choice of equations
to describe the deformation processes and the different flavours of
their numerical representation. I discuss some of the current issues
associated with the role of elasticity in crust- to lithosphere-scale
processes, the implementation of brittle behaviour in continuum
models and the use of laboratory viscous flow laws. Results of
numerical models show that the style of lithosphere deformation is
strongly influenced by strength contrasts between materials. Numerical models can be used to evaluate the role of such strength contrasts and of the individual rheological components by testing model
sensitivity to variations in parameter values.
KEY WORDS: Rheology, brittle failure, elasticity, flow law,
numerical model.
tions) and material behaviour. Deformation of the crust
and lithosphere is characterised by a complex rheology
with elastic, viscous and plastic (brittle) components
(fig. 1). Laboratory measurements of the properties of
rocks can be used to constrain the values of several of
the material properties in models. In addition, models
may not only use laboratory data in this ‘passive’ sense,
but could also constrain rheological values by inversion
of natural observations (e.g. KENIS et alii, 2005). The
aim of this short paper is to discuss some of the choices
and challenges associated with the implementation of an
elasto-visco-plastic rheology in numerical models.
A DISCUSSION OF NUMERICAL RHEOLOGY
ELASTIC BEHAVIOUR
RIASSUNTO
Reologia nei modelli numerici della deformazione litosferica.
La deformazione della crosta e della litosfera terrestre è caratterizzata da un comportamento elastico, viscoso e fragile. L’implementazione di tale reologia complessa è una sfida per i modelli numerici
a causa delle variazioni nei dati naturali e di laboratorio, della scelta
delle equazioni che descrivono i processi di deformazione e dei loro
differenti modi di rappresentazione numerica. Vengono qui discussi
alcuni problemi associati al ruolo dell’elasticità nei processi alla
scala che va dalla crosta alla litosfera, all’implementazione del
comportamento fragile nei modelli continui e all’uso delle leggi di
flusso viscoso dedotte in laboratorio. I risultati dei modelli numerici
mostrano che lo stile della deformazione litosferica è fortemente
influenzato dai contrasti di resistenza fra i materiali. I modelli numerici possono essere utilizzati per valutare il ruolo di tali contrasti
nella resistenza e delle componenti reologiche individuali, testando
la sensibilità del modello alle variazioni nei valori dei parametri.
TERMINI CHIAVE: Reologia, rottura fragile, elasticità, legge
di flusso, modelli numerici.
INTRODUCTION
Numerical models are a useful tool to predict lithosphere deformation styles as a function of sensitivity to
mechanical and thermal properties and driving forces.
Since nature is more complex than can probably ever be
captured in a numerical model, simplifications need to
be made of geometry, driving forces (boundary condi-
(*) Centre for Geodynamics, Geological Survey of Norway, Leiv
Eirikssons vei, 39 - 7491 Trondheim, Norway; [email protected]
Elastic behaviour is characterised by a linear relationship between stress and strain (fig. 1):
σ ij = 2Gε ij
(1)
Here σij is the stress tensor, G the shear modulus and
εij the strain tensor. Elastic stresses can grow with strain
in an unlimited manner and will be released once strain
is removed. This thus introduces a memory of deformation in materials. Elasticity is clearly important for
processes on relatively short timescales (from seismic
wave propagation to post-glacial rebound), but purely
elastic models have also successfully been used to simulate longer-term processes such as the deflection of the
lithosphere at a trench (TURCOTTE & SCHUBERT, 2002)
and under the load of volcanic islands like Hawaii
(WATTS, 2001). Elastic behaviour could be important for
processes that have shorter duration than the Maxwell
relaxation time, which is defined as the ratio of viscosity
over shear modulus. For mantle processes the relaxation
time is so small (on the order of 1000 yrs) that elasticity
can safely be ignored. For processes on the scale of the
lithosphere it may, however, be on the order of a million
years, implying that elasticity may not always be
neglected. A measure of the importance of elasticity is
given by the Deborah number (REINER, 1965), which is
defined by the ratio of Maxwell relaxation time to the
characteristic deformation time. Small Deborah numbers
imply viscous behaviour (though see its limitation in viscoelastic folding (SCHMALHOLZ & PODLADCHIKOV, 1995)
and further discussion in MÜHLHAUS & REGENAUER-LIEB
(2005)).
The numerical implementation of elastic material
behaviour has partly been hampered by technical challenges related to combining large deformations of materi-
194
S.J.H. BUITER
Fig. 1 - Example of the mechanical and thermal setup of a numerical model and schematic representation of its rheology components (at top)
consisting of elastic (spring), viscous (dashpot) and plastic (block sliding on a surface) material behaviour.
– Esempio di un setup meccanico e termico di un modello numerico e rappresentazione schematica delle sue componenti reologiche (sopra) che
consistono di comportamento elastico (molla), viscoso (condensatore) e plastico (blocco che scorre su una superficie).
als (requiring remeshing or an Eulerian approach) with a
stress history (requiring a Lagrangian approach or tracking of stresses with particles). Developments in Earth
Science numerical codes now allow explicit investigation
of the role of elasticity in processes on the lithosphere to
upper mantle scale (MORESI et alii, 2003; MÜHLHAUS &
REGENAUER-LIEB, 2005). KAUS & BECKER (2007) show
that elasticity has negligible effects on the dynamics of
density-driven (Rayleigh-Taylor) lithospheric instabilities, but that viscoelastic models may locally result in different stresses than purely viscous models. Elasticity will
thus not significantly affect the dynamics of mantle convection or lithospheric instabilities. However, it could
play a role on local scales in processes associated with
folding or subduction. Elasticity may also play a role in
shear band formation where stresses build up to their
maximum at which failure occurs and elastic unloading
takes place outside the shear band (VERMEER, 1990). In
these cases, the importance of elasticity should preferably be tested.
PLASTIC BEHAVIOUR
Brittle behaviour (plasticity) limits stresses in regions
in the upper and lower crust and upper mantle. Brittle
stresses depend on the nature of the material, its water
content, the stress regime (extension or compression) and
whether material is newly fractured (failure) or sliding
occurs on pre-existing failure planes (friction). The
empirical Amonton’s law shows a linear relation between
frictional shear stress (σt) and normal stress (σn) and can
be written as:
σ t = µσ n (1 − λ ) + C = tan(ϕ )σ n (1 − λ ) + C
(2)
Here µ is the friction coefficient, ϕ the angle of internal friction, λ pore fluid factor (ratio of pore fluid pressure over lithostatic pressure) and C cohesion. This equation can be written in terms of the principal stresses (σ1
and σ3):
(
) (
)
1
1
σ 1 − σ 3 = σ 1 + σ 3 (1 − λ )sin(ϕ ) + C cos(ϕ )
2
2
(3)
The compilation by BYERLEE (1978) for dry materials
(λ = 0) shows that µ~0.85 for σn < 200 MPa and µ~0.6 for
200 < σn < 2000 MPa. Rock cohesion in this compilation
varies between 0 and 50 MPa. Using these values, extrapolation to depths below the Moho can lead to very high
stresses (on the order of 1000 MPa for a compressional
stress regime). These stresses are likely lower in nature
owing to a different deformation mechanism on the transition between brittle and viscous behaviour (KOHLSTEDT
et alii, 1995). Many numerical models need much lower
RHEOLOGY IN NUMERICAL MODELS OF LITHOSPHERE DEFORMATION
195
Fig. 2 - Two examples of simple two-layer extension models with a brittle upper layer and a linear viscous lower layer. The models show that
the number of shear zones in the upper layer depends on the strength contrast between the two layers. A) Results of 3 models after 30%
extension. The viscosity of the lower layer is 1 and cohesion C of the upper layer varies from 10 to 7 (all values are scaled in these
calculations). After MORESI & MÜHLHAUS (2006). B) Results of 3 models after 2.5% extension (top and middle figure) and 10% extension
(bottom figure). The model is originally 400 km wide and 35 km high. The angle of internal friction is 30° softening to 4°, cohesion is 20 MPa
softening to 2 MPa. The viscosity of the lower layer varies from 1019 to 1021 Pas. After BUITER et alii (2008).
– Due esempi di semplici modelli di estensione a due strati con uno strato superiore fragile e uno strato inferiore viscoso lineare. I modelli
mostrano che il numero delle zone di scorrimento nello strato superiore dipende dal contrasto della resistenza fra i due strati. A) Risultati di
3 modelli dopo il 30% di estensione. La viscosità dello strato inferiore è 1 e la coesione C dello strato inferiore varia da 10 a 7 (tutti i valori sono
scalati). Da MORESI & MÜHLHAUS (2006). B) Risultati di 3 modelli dopo il 2.5% di estensione (figura superiore e centrale) e 10% di estensione
(figura inferiore). Il modello è originariamente largo 400 km e alto 35 km. L’angolo di frizione interna varia da 30° a 4°, la coesione varia da
20 MPa a 2 MPa. La viscosità dello strato inferiore varia da 1019 a 1021 Pas. Da BUITER et alii (2008).
values for the friction coefficient than reported by BYERLEE (1978) to reproduce lithosphere deformation as seen
in nature, especially along subduction faults (e.g., GERYA
et alii, 2007). It is an open question whether these low
numerical coefficients point to weak faults in nature (e.g.,
by foliation development, mineral transformations or
high pore fluid pressures) or to a special feature of the
models.
Some numerical (Lagrangian finite-element) models
can implement a pre-existing failure plane along which
the displacements are limited by the friction coefficient
(e.g., MELOSH & WILLIAMS, 1989). The challenge of this
approach is the need for remeshing as fault offsets
become large. Alternatively, the implementation of plasticity in (finite-element or finite-difference) continuum
models results in the formation of shear bands with a
finite width. Mohr-Coulomb failure follows the above
equations, but the values for ϕ and C may differ from their
values in friction. Construction of the Mohr circle at yield
results in a prediction for the angle of shear zones with
the direction of maximum compressive stress of 45° – ϕ/2.
Shear zones in compression are therefore expected to
have shallow dip angles (30° for ϕ = 30°), while extensional shear zones are steep (60° for ϕ = 30°). Measurements on shear bands in sand show, however, also the
Roscoe angle 45° – ψ/2 (ROSCOE, 1970; see also VERMEER,
1990) or an intermediate angle 45° – (ψ + ϕ)/2 (VARDOULAKIS, 1980). Here ψ is the dilation angle (the ratio of
the rate of volumetric strain and the rate of shear strain).
GERYA & YUEN (2007) obtain the intermediate (45° – (ψ +
ϕ)/2) shear zone angle in their dilatant finite-difference
experiments. In these types of models, the dilation angle
keeps the same value throughout the deformation history.
However, sand shows changes in dilation during loading.
Shear zone formation is associated with dilation which
reaches its maximum rate at peak failure and thereafter
tends to zero when stable sliding is achieved (LOHRMANN
et alii, 2003). This behaviour can be captured with dis-
tinct element methods (EGHOLM, 2007) or sophisticated
plasticity models (CROOK et alii, 2006). Many numerical
models that do well in simulating large deformations are
incompressible (ψ = 0°) and shear zone dip angles in
these models could in theory range between 45° and
the Mohr-Coulomb angle (45° – ϕ/2). Incompressible
viscoplastic models, however, often seem to result in 45°
shear zone dip angles. For this reason, MORESI &
MÜHLHAUS (2006) developed an anisotropic viscosity
method that gives Mohr-Coulomb dip angles. It remains,
however, to be established whether incompressible viscoplastic models may not intrinsically be able to result in
Mohr-Coulomb angles and whether the obtained 45°
shear zone angles point to something overlooked in these
types of models.
The width of shear bands depends on the numerical
resolution. This implies that shear bands can become
extremely narrow for very fine grids and for this reason
some models have introduced intrinsic minimum length
scales (de BORST & SLUYS, 1991). The use of mean stress
(or dynamic pressure) instead of lithostatic pressure in
the numerical implementation of Mohr-Coulomb plasticity (see equation 3) improves localisation behaviour (e.g.,
BUITER et alii, 2006). In numerical models stresses are
followed while they build up until the yield surface is
reached. This stress build-up phase will be different for
viscoelastic and viscous models. In nature, stresses will
never exceed the yield stress, but in numerical models a
stress overshoot can occur. Stresses then need to be
brought back to yield and different techniques exist to do
this (e.g., viscosity iteration or return mapping). It has
not yet been clearly established if these differences in
techniques could have a significant effect on numerical
shear zones (see also BUITER et alii, 2005). After yielding,
associated (ϕ = ψ) or non-associated (ϕ ≠ ψ) plastic flow
occurs. This continued deformation after failure distinguishes many Earth Sciences problems from engineeringtype applications.
196
S.J.H. BUITER
Fig. 3 - Shortening of a continental lithosphere with a 3 km high (200 km wide) mountain which is in isostatic equilibrium with a crustal root.
A) The model is still stable after 10 Myr of shortening (at 1 cm/yr) if the mantle part of the lithosphere is strong. B) A weak mantle lithosphere
leads to an unstable model. Modified from BUROV & WATTS (2006).
– Raccorciamento di una litosfera continentale con una catena montuosa di altezza 3 km (ampia 200 km), che è in equilibrio isostatico con una
radice crostale. A) Il modello è ancora stabile dopo 10 Ma di raccorciamento (a 1 cm/anno) se il mantello litosferico è resistente. B) un mantello
litosferico debole porta ad un modello instabile. Modificato da BUROV & WATTS (2006).
VISCOUS BEHAVIOUR
Viscous behaviour of crustal and upper mantle rocks
is described by an empirical flow law, which relates
strain-rate to stress:
 Q + PV 
ε˙ = Aσ n d − p exp  −

RT 

(4)
Here A is the pre-exponent, n the stress exponent, d
grain size, p grain size exponent, Q activation energy, P
pressure, V activation volume, R the gas constant and T
temperature. The pre-exponent A may include melt fraction, oxygen fugacity and water content. Usually the
.
strain-rate (ε ) is uniaxial and stress (σ) is the differential
stress. Measured flow laws need, therefore, to be converted to effective stress and strain-rate for a general
implementation in numerical models (e.g., RANALLI,
1987). At low-stress conditions, grain boundary diffusion
creep (e.g. Coble creep) is in general important, whereas
deformation by movement of dislocations (dislocation
creep) is more characteristic for high-stress conditions.
Diffusion creep is characterised by n = 1 and is grain size
dependent (p = 3 for olivine (HIRTH & KOHLSTEDT,
2003)). Dislocation creep is grain size independent (p = 0),
while n ≥ 3. Diffusion and dislocation creep may occur
simultaneously (they act in parallel), in which case the
contributions from both mechanisms need to be taken
into account.
Flow laws are determined by measuring stresses for
varying strain-rate and at different temperatures. Ideally,
the conditions should be such that deformation occurs by
one deformation mechanism and at steady state. The laboratory conditions imply that measurements are made on
small rock samples (mm), at low strains and high strainrates (10-3 – 10-6 s-1) and need to be extrapolated to geological conditions. Many of the issues associated with this
extrapolation and the implementation of laboratory flow
laws in models are discussed in, among others, PATERSON
(1987, 2001), RUTTER & BRODIE (1991) and HANDY et alii
(2001). The uncertainty involved in the extrapolation to
low strain rates (10-14 – 10-16 s-1), potentially different
grain sizes and high strains is essentially unknown. However, support to the large extrapolations is given by similarities in microstructures between naturally and experimentally deformed materials (e.g., as discussed for
quartzite by HIRTH et alii (2001)). Current challenges are
to formulate flow laws for polymineral rocks and flow
laws that quantify the effects of water (KORENAGA &
KARATO, 2008) and melt content.
Choosing the flow law to use in a geodynamic model
is not straightforward (RANALLI, 2003; BUROV, 2003).
Extrapolated published flow laws for crustal and mantle
rocks show a large variation in strength, thus giving a
choice of weak or strong materials in models. The best
approach to dealing with this uncertainty in flow law
data in numerical modelling is to treat flow laws as a
variable and to examine model sensitivity to viscous
strength variations. Simple two-layer models of a brittle
RHEOLOGY IN NUMERICAL MODELS OF LITHOSPHERE DEFORMATION
upper layer bonded to a linear viscous lower layer show,
for example, that the number of shear zones in the brittle
layer decreases as the strength contrast between the two
layers increases (fig. 2) (MORESI & MÜHLHAUS, 2006;
BUITER et alii, 2008). The rheology of subducted material
and the surrounding mantle has been shown to influence
the dynamics of subduction zones (e.g., BILLEN &
HIRTH, 2005; 2007). BUROV & WATTS (2006) show that in
their models a weak olivine mantle (which I infer to be
wet Åheim dunite by CHOPRA & PATERSON (1981))
results in deformation styles which are incompatible
with observed lithosphere stability and deformation at
subduction zones, whereas a strong upper mantle
(inferred to be dry olivine by Hirth & KOHLSTEDT (1996))
explains the persistence of mountains and the integrity
of subducting slabs (fig. 3).
CONCLUDING REMARKS
Models of deformation processes in the crust and
lithosphere require a sophisticated rheology description
with elastic, viscous and plastic components. I have
touched upon some of the open questions associated with
especially numerical plasticity and the use of laboratory
flow laws in models. Plastic behaviour in continuum
models results in the formation of grid-size dependent
shear bands. Their dip angle has been shown to vary
between the Roscoe, Mohr-Coulomb or an intermediate
angle and it has until now not been clearly established
whether one of these representations should be preferred.
Elasticity may play a role in deformation processes on the
scale of the lithosphere (e.g. subduction) and its importance should be tested for crust- to lithosphere-scale models. Extrapolated published viscous flow laws for crustal
and mantle rocks show a large variation in strength, giving modellers a choice between weak and strong numerical materials. Results of numerical models show that in
many cases deformation depends on the strength contrasts between layers and less on the absolute values of
the material strengths (e.g., BUROV & WATTS, 2006;
MORESI & MÜHLHAUS, 2006). A useful approach is to
treat rheology in numerical models as a variable and to
test model sensitivity to reasonable variations in the values of the rheological components.
ACKNOWLEDGEMENTS
I would like to thank Janos Urai for his invitation to discuss
numerical rheology at the 16th DRT conference, Florian Heidelbach for
enlightening discussions of laboratory measurements of flow laws and
Susan Ellis, Boris Kaus and Yuri Podladchikov for our many discussions of numerical plasticity. Florian Heidelbach, Susan Ellis and journal reviewer Roberto Sabadini provided helpful feedback on this text.
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Received 5 November 2007; revised version accepted 3 March 2008
Burg&Gerya
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 199-203, 4 figs.
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Modelling intrusion of mafic and ultramafic magma into the continental crust:
numerical methodology and results
JEAN-PIERRE BURG (*) & TARAS V. GERYA (*)
ABSTRACT
INTRODUCTION
Field studies and geophysical imaging indicate that granitic and
non-granitic plutons have both very variable and comparable shapes
and sizes. We simulated numerically intrusion of partially molten mantle rocks from a sub-lithospheric magmatic source region (SMSR). Our
systematic numerical modelling results show that intrusion typically
spans a few hundred kyr spanning three stages: (1) magmatic channel
spreading, (2) emplacement and (3) post-intrusive subsidence and cooling. The duration of each of these stages strongly depends on the viscosity of ascending magma. Upward magma transport from sublithospheric depth is driven by the positive buoyancy of the partially-molten
rocks with respect to the overriding colder mantle lithosphere. By systematically varying the model parameters we document variations in
intrusion dynamics and geometry that range from funnel- and fingershaped bodies (pipes, dikes) to deep seated balloon-shaped intrusions
and flattened shallow magmatic sills. Relatively cold elasto-plastic
crust (TMoho = 400oC) promotes a strong upward propagation of
magma due to the significant decrease of plastic strength of the crust
with decreasing confining pressure. Warmer crust (TMoho = 600oC)
triggers lateral spreading of magma above the Moho.
Field studies and geophysical imaging indicate that
granitic and non-granitic plutons have both very variable
and comparable shapes and sizes (e.g. BEST & CHRISTIANSEN, 2001; PETFORD, 1996; CRUDEN & MCCAFFREY,
2001; BOLLE et alii, 2002). The dimensions of plutons are
therefore rock-type independent. It is generally accepted
that plutons grow by collecting and transferring melt
from a deep source to higher emplacement levels. Melting
(anatexis and dehydration melting of hydrous minerals)
of rocks generates these melts, with felsic to intermediate
compositions when the source is the continental crust, or
mafic and ultramafic compositions when partial melting
affects the upper mantle (e.g. RUDNICK & GAO, 2004).
However, evidence for sublithospheric magma sources
indicates that transfer of non-kimberlitic magma also
occurs on a larger scale, through the lithosphere (e.g.
ANDERSON, 1994; SCHMIDT & POLI, 1998; ERNST et alii,
2005; WRIGHT & KLEIN, 2006). This is especially true for
intrusions of mafic and ultramafic bodies into the lower
density (by 100-500 kg/m3) continental crust that are documented for a large variety of tectonic settings spanning
continental shields, rift systems, collision orogens and
magmatic arcs.
While accepting the conventional ideas concerning
plutonism, we are confronting three intriguing questions:
(1) How can magma move from sub-lithospheric molten
regions to shallower storage chambers? (2) How highdensity, ultramafic and mafic magma can ascend into the
lower density crust, at odds with the common acceptance
that mafic and ultramafic magma stays deep and forms
the lower crust (e.g. RUDNICK & GAO, 2004) and (3) how
temperature-sensitive rheologies of both magma and
country rocks together influence the emplacement of
such ultramafic/mafic magmas?
KEY WORDS: intrusion emplacement, numerical modelling,
magmatic bodies.
RIASSUNTO
Modello di intrusione di magma femico e ultra-femico nella
crosta continentale: metodologia numerica e risultati.
Studi di terreno e immagini geofisiche indicano che plutoni granitici e non-granitici hanno forme e grandezze sia confrontabili che
variabili. Abbiamo simulato numericamente l’intrusione di rocce di
mantello parzialmente fuse da una regione di sorgente magmatica
sub-litosferica (SMRS). I risultati della nostra modellizzazione numerica sistematica mostrano che l’intrusione tipicamente copre tre
stadi su poche centinaia di migliaia di anni: (1) espansione del canale magmatico, (2) messa in posto e (3) subsidenza e raffreddamento
post-intrusivi. La durata di ognuno di questi tre stadi dipende fortemente dalla viscosità del magma ascendente. Il trasporto di magma
verso l’alto da profondità sub-litosferiche è guidato dalla galleggiabilità positiva delle rocce parzialmente fuse rispetto al mantello litosferico circostante più freddo. Variando sistematicamente i parametri
del modello documentiamo variazioni nella dinamica e nella geometria dell’intrusione che varia da corpi a forma di imbuto e colonna
(camini vulcanici, filoni eruttivi) a intrusioni profonde a forma di
pallone (baloon) e filoni strato appiattiti superficiali. Una crosta continentale elasto-plastica relativamente fredda (TMoho = 400oC) favorisce una forte propagazione di magma verso l’alto a causa della significativa diminuzione della resistenza plastica della crosta con la
diminuzione della pressione confinante. Una crosta più calda (TMoho
= 600oC) stimola l’espansione laterale di magma sopra la Moho.
TERMINI CHIAVE: messa in posto di corpi intrusivi, modellizzazione numerica, corpi magmatici.
(*) Department of Geosciences – ETH and University Zürich,
CH-8092 Zürich, Switzerland.
MODELLING TECHNIQUES
We decided to take advantage of recent progress in
hardware and software capabilities to generate twodimensional visco-elasto-plastic numerical models of
mafic-ulramafic intrusion emplacement incorporating in
particular the temperature-sensitive properties of both
magma and country rocks. Thermomechanical modelling
of magma intrusion is numerically challenging because it
involves simultaneous and intense deformation of materials with very contrasting rheological properties: the
country, crustal rocks are visco-elasto-plastic while the
intruding magma is a low viscosity, complex fluid (e.g.
200
J.-P. BURG
& T.V. GERYA
Fig. 1 - Enlarged 20-50 × 215 km areas of the original 1100 km × 300 km reference model. Distribution of rock layers in the intrusion
area during emplacement of the ultramafic body into the crust from below the lithosphere via the magmatic channel. LEGEND: 1) weak
layer (air, water); 2) sediments; 3, 4) upper crust (3 - solid, 4 - molten); 5, 6) lower crust (5 - solid, 6 - molten); 7, 8) mantle (7 - lithospheric,
8 - asthenospheric); 9, 10) peridotite (9 - molten, 10 - crystallized); 11, 12) gabbro (11 - molten, 12 - crystallized). Time (kyr) is given in the
figures. White numbered lines are isotherms in °C. Vertical scale: depth below the upper boundary of the model. Initial numerical setting of
this study is shown on the leftmost section of the model (0 Myr). The lithospheric and asthenospheric mantles have the same physical
properties, different grey tones are used for a better visualization of deformation and structural development. This is also true for the passive
colour-layering in the upper and the lower crust. Initial and boundary conditions are detailed in (GERYA & BURG, 2007).
– Porzione estesa 20-50 × 215 km del modello di riferimento originale di dimensioni 1100 km × 300 km. Distribuzione dei livelli di roccia nell’area
d’intrusione durante la messa in posto del corpo ultra-femico nella crosta da livelli sub-litosferici attraverso il canale magmatico. LEGENDA:
1) strato debole (aria, acqua); 2) sedimenti; 3, 4) crosta superiore (3 - solida, 4 - fusa); 5, 6) crosta inferiore (5 - solida, 6 - fusa); 7, 8) mantello
(7 - litosferico, 8 - astenosferico); 9, 10) peridotite (9 - fusa, 10 - cristallizzata); 11, 12) gabbro (11 - fuso, 12 - cristallizzato). Il tempo (in migliaia
di anni) kyr) è indicato nelle figure. Le linee bianche numerate sono isoterme in °C. Scala verticale: profondità sotto il bordo superiore del
modello. La configurazione numerica iniziale di questo studio è mostrata nella sezione a sinistra del modello (0 Ma). I mantelli litosferico e
astenosferico hanno le stesse proprietà fisiche; differenti toni di grigio sono utilizzati per visualizzare meglio lo sviluppo della deformazione e delle
strutture. Ciò è vero anche per la stratificazione passiva della crosta superiore e inferiore. Le condizioni iniziali e al contorno sono dettagliate in
GERYA & BURG (2007).
PINKERTON & STEVENSON, 1992). We employ the 2-D
code I2ELVIS (GERYA & YUEN, 2003a, 2007), which is
based on finite-differences with a marker-in-cell technique. The code allows for the accurate conservative solution of the governing equations on a rectangular fully
staggered Eulerian grid. New developments allow for
both large viscosity contrasts and strong deformation of
visco-elasto-plastic multiphase flow. The code was tested
for a variety of problems by comparing results with both
analytical solutions and analogue sandbox experiments
(GERYA & YUEN, 2003, 2007).
We simulated numerically intrusion of partially
molten mantle rocks from a sub-lithospheric magmatic
source region (SMSR, fig. 1, 0 Kyr). Developments introduced for intrusion simulation allow for both large viscosity contrasts and strong deformation of visco-elasto-plastic
multiphase flow, incorporating temperature-dependent
rheologies of both intrusive molten rocks and host rocks
(GERYA & BURG, 2007). A magmatic channel is a vertical,
1.5 km wide zone characterised by a wet olivine rheology
and a low 1 MPa plastic strength throughout the lithospheric mantle. The initial thermal structure of the lithosphere is as usually assumed, with a 35 km thick crust
(fig. 1, 0 Kyr) corresponding to a sectioned linear temperature profile limited by 0°C at the surface, 400 oC at
the bottom of the crust and 1300oC at 195 km depth.
The temperature gradient in the asthenospheric mantle is
0.6oC/km below 195 km depth.
The code grey (code colour in the coloured version)
identifying rock types is given in figure 1. The discrimination between «peridotite» and «molten peridotite» is thermal, separating material points (pixels) above/below the
wet solidus temperature of peridotite at a given pressure.
Since the melt fraction is strongly changing with water
content, variations within few % of melt fraction at given
pressure-temperature condition are possible. Therefore, it
MODELLING INTRUSION OF MAFIC AND ULTRAMAFIC MAGMA INTO THE CONTINENTAL CRUST
is illusory to predict the exact melt fraction at any point
of the models, in particular because the simplified linear
melting model implemented here (GERYA & BURG, 2007)
does not allow a very high precision on this question.
DISCUSSION
Modelling results (cf. GERYA & BURG, 2007, for details
of experiments) show that intrusion typically lasts a few
hundred kyr spanning three stages: (1) magmatic channel
spreading (fig. 1, 0-16 Kyr), (2) emplacement (fig. 1, 22-41
Kyr, fig. 2) and (3) post-intrusive subsidence and cooling
(fig. 1, 71-1171 Kyr, fig. 3). The duration of each of these
stages strongly depends on the viscosity of ascending
magma.
Upward magma transport from sublithospheric depth
is driven by the positive buoyancy of the partially-molten
rocks with respect to the overriding colder mantle lithosphere. The gravitational balance controls the height of the
Fig. 2 - Details of temperature distribution (numbered white lines =
isotherms in °C) around intrusive body during the active stage of
emplacement for the reference model. Rock types are the same as in
fig. 1.
– Dettagli della distribuzione della temperatura (linee bianche numerate = isoterme in °C) attorno al corpo intrusivo durante lo stadio di
messa in posto attiva per il modello di riferimento. Le rocce sono le
stesse di fig. 1.
Fig. 3 - Details of culminate intrusive body shape for the reference model. Rock types are the same as in fig. 1.
– Dettagli della forma finale del corpo intrusivo per il modello di riferimento. Le rocce sono le stesse di fig. 1.
201
202
J.-P. BURG
& T.V. GERYA
Fig. 4 - Stability of major intrusion shapes
as a function of lower crust rheology and
magma viscosity. Different color fields correspond to three different types of intrusive bodies (balloons, pipes/fingers and nappes/sills) obtained numerically (GERYA &
BURG, 2007).
– Stabilità dell’intrusione principale in funzione della reologia della crosta inferiore e
della viscosità del magma. Differenti toni di
grigio corrispondono ai tre diversi tipi di
corpi intrusivi (palloni, filoni eruttivi e filoni strato) ottenuti numericamente (GERYA
& BURG, 2007).
column of molten rock but not the volume of magmatic
rocks below and above the Moho. The molten rocks are
pooling along the crust/mantle boundary only if the lower
crust is ductile and very weak (fig. 4, deep grey field or
red field in the colour version), which may be expected at
the base of island arcs. It seems natural that otherwise,
basic – ultrabasic magma is injected into the crust, most
commonly as a finger/pipe-shaped body (fig. 4, intermediate grey field or pink field in the coloured version).
Emplacement within the crust exploits the space
opened by the displacement of tectonic crustal blocks
bounded by localized zones of intense plastic deformation. Temperature is the important player in controlling
crustal viscosities, hence either viscous or elasto-plastic
mechanisms of crustal deformation, which defines modes
and rates of emplacement. Early normal faults (fig. 1, 16
Kyr) produce early surface subsidence in grabens but
rapidly become inverted into thrusts (fig. 1, 22 Kyr)
responsible for surface uplift while the within-crust pluton inflates and rises in the crust (fig. 2).
Late emplacement phases are responsible for cooling
and subsiding of the magmatic body and partial return
magma flow back into the magmatic channel below the
Moho (fig. 3). This event is linked to subsidence of the
surface.
By systematically varying the model parameters we
document variations in intrusion dynamics and geometry
that range from funnel- and finger-shaped bodies (pipes,
dikes) to deep seated balloon-shaped intrusions and flattened shallow magmatic sills (fig. 4). Relatively cold
elasto-plastic crust (TMoho = 400oC) promotes a strong
upward propagation of magma due to the significant
decrease of plastic strength of the crust with decreasing
confining pressure (fig. 4, intermediate and light grey
fields or pink and blue fields in the coloured version).
Emplacement in this case is controlled by crustal faulting
and subsequent block displacements. Warmer crust
(TMoho = 600oC) triggers lateral spreading of magma
above the Moho, with emplacement being accommodated
by coeval viscous deformation of the lower crust and fault
tectonics in the upper crust (fig. 4, deep grey field or red
field in the coloured version).
CONCLUSION
Emplacement of high density, mafic and ultramafic
magma into low-density rocks is a stable mechanism for a
wide range of model parameters that match geological
settings in which partially molten mafic-ultramafic rocks
are generated below the lithosphere. We expect this
process to be particularly active beneath subductionrelated magmatic arcs where huge volumes of partially
molten rocks produced from hydrous cold plume activity
accumulate below the overriding lithosphere (GERYA &
YUEN, 2003b).
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MODELLING INTRUSION OF MAFIC AND ULTRAMAFIC MAGMA INTO THE CONTINENTAL CRUST
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Received 21 November 2007; revised version accepted 10 March 2008
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 205-208, 3 figs.
Queste bozze, corrette e accompagnate dall’allegato preventivo firmato e dal buono d’ordine,
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Magma-controlled tectonics in compressional settings:
insights from geological examples and experimental modelling
OLIVIER GALLAND (*), (**), PETER R. COBBOLD (*), ERWAN HALLOT (*) & JEAN DE BREMOND D’ARS (*)
ABSTRACT
INTRODUCTION
Magmatic activity tends to concentrate at tectonic plate boundaries. At rapidly convergent margins, such as the Andes, intense magmatic activity is coeval with strong tectonic shortening, and some volcanoes and magmatic intrusions have been emplaced near active
compressional structures, usually major thrust faults. In order to
understand the links between magmatic systems and compressional
deformation structures in the upper crust, we describe the structure
of an active volcano (Tromen, Argentina) and an exhumed intrusion
(Boulder Batholith, U.S.A.) emplaced during compressional deformation. In those examples, magmatic systems and thrust faults exhibit
geometrical and chronological relationships. We also present results
of experimental modelling of magma emplacement during compression. The comparison between geological examples and experiments
show close similarities. That suggests that the presence of magma
influences the deformation pattern in the brittle crust. The influence
of deep magma bodies is also to be explored at the scale of the whole
crust during the development of active margins.
Magmatic activity mostly occurs at plate boundaries,
where tectonic deformation also concentrates. Because
magmatic bodies and their country rock have very contrasting rheological properties, one might expect deformation to be influenced by the presence of magmatic bodies
at depth (e.g. BUROV et alii, 2003). Although this problem
has been addressed in the lower crust (e.g. HOLLISTER &
CRAWFORD, 1986; DAVIDSON et alii, 1992; BROWN & SOLAR,
1998; BROWN & SOLAR, 1999; ROSENBERG & HANDY, 2000;
BARRAUD et alii, 2001), very little is known about the
processes of such an interaction between magmatism and
country rock deformation in the brittle upper crust. Which
one comes first? Which one controls the other? Most of
previous research has focused on deformation controlling
magmatism (e.g. HUBBERT & WILLIS, 1957; MARRETT &
EMERMAN, 1992). Here we also attempt to consider the
opposite mechanism, i.e. magma-controlled deformation.
At rapidly convergent margins, such as the Andes, one
might expect that horizontal compression prevents the
rise of magma through the brittle upper crust (HUBBERT
& WILLIS, 1957; HAMILTON, 1995). Nevertheless, volcanic
activity is also common in compressional environments.
Such a contradiction highlights the lack of understanding
of the mechanical interplay between magmatism and
deformation. We therefore address the processes of
magma-controlled deformation in compressional settings.
We first describe two geological examples of magmatic
complexes emplaced in such settings, Tromen volcano,
Neuquén basin, Argentina, and the Boulder batholith,
Montana, USA. Subsequently, we present results of experimental modelling of magma emplacement during shortening. Thus, we show that magma can transport in a
shortening crust, and that magma-controlled deformation
processes can play an important role in the structural
development of the upper crust.
KEY WORDS: magma-controlled tectonics, compressional
tectonics, Tromen volcano, experimental modelling.
RIASSUNTO
Tettonica controllata dai magmi in contesti collisionali:
approfondimenti da esempi geologici e modelli sperimentali.
L’attività magmatica tende a concentrarsi ai margini delle placche litosferiche. Lungo i margini soggetti a rapida convergenza,
come le Ande, l’intensa attività magmatica è coeva con elevato raccorciamento tettonico, ed alcuni vulcani e intrusioni magmatiche si
sono messi in posto in corrispondenza di strutture compressionali
attive, solitamente i sovrascorrimenti principali. Per comprendere i
legami tra i sistemi magmatici e le strutture di deformazione compressiva nella crosta superiore, descriviamo qui le strutture di un
vulcano attivo (Tromen, Argentina) e di un’intrusione esumata
(Boulder Batholith, U.S.A.) messi in posto durante deformazione
compressiva. In questi esempi, sistemi magmatici e sovrascorrimenti
mostrano relazioni geometriche e cronologiche. Presentiamo anche i
risultati di modelli sperimentali di messa in posto di magmi in regime compressivo. Il confronto tra esempi geologici ed esperimenti
analogici dimostra strette similitudini. Ciò suggerisce che la presenza di magma influenzi la configurazione della deformazione nella
crosta fragile. L’infuenza di corpi magmatici profondi deve quindi
essere esplorata alla scala dell’intera crosta durante l’evoluzione dei
margini attivi.
TERMINI CHIAVE: tettonica controllata dai magmi, tettonica
compressiva, vulcano Tromen, modellazione sperimentale.
(*) Géosciences-Rennes (UMR 6118), CNRS et Université de
Rennes 1, Campus de Beaulieu - 35042 Rennes Cedex, France.
(**) Physics of Geological Processes (PGP), Universitet i Oslo,
Physics building, third floor, Sem Selands vei, 24 - NO-0316 Oslo,
Norway (Fax: +47 22 85 51 01; E-mail: [email protected]).
GEOLOGICAL OBSERVATIONS
It is well known that magmatic activity is common at
convergent margins. However, only a few studies have
addressed the association between magmatic complexes
and thrust faults (e.g. HOLLISTER & CRAWFORD, 1986;
PARRY et alii, 1997; TIBALDI, 2005). Noticeable examples
are Tromen volcano, Neuquén province, Argentina (GALLAND et alii, 2007b), and the Boulder batholith, Montana,
USA (KALAKAY et alii, 2001). Tromen is a PleistoceneHolocene back-arc volcano, located in the northern segment of the Southern Andes (fig. 1). It lies in a thick-
206
O. GALLAND ET ALII
Fig. 1 - Two geological examples of magmatic complexes emplaced in compressional tectonic setting: a) Simplified geological map of Tromen
volcano, Neuquén basin, Argentina. Tromen is Andean alkaline back-arc Quaternary volcano, located on top of arcuate east-verging thrust. It
built up during thrusting deformation. Modified after GALLAND et alii (2007b); b) Simplified geological map of Boulder batholith, Montana,
USA. Boulder batholith was emplaced into Sevier fold-and-thrust belt, during thrusting deformation. Locally around Boulder batholith,
thrust front exhibits strongly arcuate trace (Helena salient). Modified after KALAKAY et alii (2001). Structures of both Tromen volcano and
Boulder batholith suggest control of magmatic complexes on deformation.
– Due esempi geologici di complessi magmatici messi in posto in contesto tettonico collisionale: a) Carta geologica semplificata del vulcano
Tromen, bacino di Neuquén, Argentina. Tromen è un vulcano quaternario andino alcalino di retro-arco, collocato sulla sommità di un
sovrascorrimento arcuato vergente a Est. Si è sviluppato durante la deformazione che ha prodotto il sovrascorrimento. Modificato da GALLAND et
alii (2007b); b) Carta geologica semplificata del batolite di Boulder, Montana, USA. Il batolite di Boulder si è messo in posto nella catena a pieghe
e sovrascorrimenti di Sevier, durante la deformazione che ha prodotto i sovrascorrimenti. Localmente attorno a questo batolite il fronte di
sovrascorrimento mostra un contorno molto arcuato (Helena salient). Modificato da KALAKAY et alii (2001). Le strutture del vulcano Tromen e
del batolite di Boulder suggeriscono un controllo dei complessi magmatici sulla deformazione.
Fig. 3 - a) Photograph of map view of typical model without injection. Piston (left) deformed model made of compacted silica powder.
Straight thrusts accommodated shortening. Straight line locates cross section; b) Photograph and corresponding schematic drawing of cross
section. Offset of horizontal markers locates faults (thrusts). Straight thrusts form at base of piston; c) Photograph of map view of typical
model with injection. Straight and arcuate thrusts accommodated shortening. Poorly deformed plateau laid between straight and arcuate
thrusts. Injected molten oil erupted along trace of arcuate thrust; d) Photograph and corresponding schematic drawing of cross section.
Intruding oil (gray) forms basal sill. Straight thrusts form at base of piston. Arcuate thrust nucleate at leading edge of sill. Plateau lies above
sill.
– a) Immagine fotografica dall’alto di un tipico modello senza iniezione. A sinistra un modello deformato a pistone, composto di silice in polvere.
I sovrascorrimenti rettilinei hanno accomodato il raccorciamento. La linea retta individua la sezione verticale; b) Fotografia e corrispondente
disegno schematico della sezione verticale. La dislocazione dei riferimenti orizzontali individua le faglie (sovrascorrimenti). Alla base del pistone
si formano sovrascorrimenti rettilinei; c) Immagine fotografica dall’alto di un tipico modello con iniezione. I sovrascorrimenti rettilieni e arcuati
hanno accomodato il raccorciamento. Tra i sovrascorrimenti rettiliei e arcuati si trovano plateau poco deformati. Il liquido iniettato è eruttato
lungo le tracce del sovrascorrimento arcuato; d) Fotografia e disegno schematico della sezione verticale. L’olio che s’intrude (grigio) forma sill
basali. Alla base del pistone si formano sovrascorrimenti rettilinei. Alla terminazione frontale del sill nucleano sovrascorrimenti arcuati. Il plateau
si trova al di sopra del sill.
MAGMA-CONTROLLED TECTONICS IN COMPRESSIONAL SETTINGS
207
skinned fold-and-thrust belt in the western margin of the
Neuquén basin (COBBOLD & ROSSELLO, 2003). Its volcanic products are unconformable upon Mesozoic strata
of the basin. It built up above the hanging-wall of a major
eastward-verging thrust fault (fig. 1). The Boulder
batholith is a Cretaceous intrusive complex, east of the
major Idaho-Bitterroot batholith (KALAKAY et alii, 2001).
It was emplaced in the upper crust, within the Sevier
fold-and-thrust belt (fig. 1). It has a flat-lying tabular
shape, and an estimated thickness between 5 and 12 km.
Both Tromen volcano and the Boulder batholith have
close chronological and structural relationships with their
substrata (fig. 1):
1) They lie close to major thrust faults.
2) Their emplacement was coeval with thrusting.
3) The thrust fronts have strongly arcuate shapes
around the volcano or batholith.
Geological observations on Tromen volcano and the
Boulder batholith show close relationships between
thrusting and magmatism (KALAKAY et alii, 2001; LAGESON et alii, 2001; GALLAND et alii, 2007b). They show that
magma can ascend and be emplaced in compression and
that thrust faults are likely to control magma transport.
In addition, the arcuate thrusts around the volcano or
batholith may result from the influence of magma upon
Fig. 2 - Schematic drawing of experimental setup (see text for explanations).
– Disegno schematico dell’impianto sperimentale (vedi il testo per le
spiegazioni).
the deformation pattern. The following experimental
results illustrate how magmatic activity may control compressional deformation.
Fig. 3.
208
O. GALLAND ET ALII
EXPERIMENTAL MODELLING
In order to study the mechanical interactions between
compressional deformation and magmatic intrusion, we
resorted to laboratory experiments, in which an analogue
of the brittle crust shortened, while melt was intruding
(fig. 2). We used (1) a cohesive fine-grained silica powder
to represent the brittle crust, and (2) a molten low-viscosity vegetable oil to represent magma (GALLAND et alii,
2006). In the experiments, horizontal shortening and
injection were coeval but independent. Shortening
resulted in thrust faults, while overpressured oil formed
tabular intrusions.
In those experiments where there was no injection,
shortening resulted in a classical thrust wedge, in which
thrusts had straight traces and were 5-6 cm apart (fig. 3;
GALLAND et alii, 2003; GALLAND et alii, 2007a); the apical
angle of the wedge was about 15º. In the other experiments,
where there was injection, oil formed a basal sill, and the
structure of the wedge was very different. Once in place, the
sill lubricated the base of the model, so that arcuate thrusts
formed at the leading edge of the sill (fig. 3). The distance
between thrusts increased, defining a non-deformed
plateau. The apical angle of the wedge was smaller than 10º.
Uplift of the plateau promoted further intrusion of oil at
depth. In general, the pattern of deformation and intrusion
depended on the kinematic ratio R between rates of shortening and injection (GALLAND et alii, 2007a). The lengths
of the basal sill and plateau increased with decreasing R.
Thus, from our experiments we infer that a small amount of
magma in a deforming brittle crust strongly modifies
the deformation pattern. Intrusions control the formation
of arcuate thrusts and slightly deformed plateaus by lubricating their bases.
DISCUSSION AND CONCLUSIONS
There are close similarities between Tromen or the
Boulder batholith and our experimental results. According to the geological observations, melts rose and was
emplaced during thrusting. In addition, thrusts have similar arcuate shapes around the magmatic complexes,
which are in the hanging walls of the arcuate thrusts.
Thus we infer that arcuate structures around Tromen volcano and the Boulder batholith have resulted from the
interaction between compressional deformation and nonsolidified magma. Similar relationships between thrusts
and active volcanoes exist at Guagua Pichincha, Ecuador
(LEGRAND et alii, 2002), Socompa, Chile (VAN WYK DE
VRIES et alii, 2001), and Taapaca, Chile (CLAVERO et alii,
2004). We therefore suspect that similar processes were at
work in those volcanoes.
Our experimental results show that magmatic systems submitted to compression can control the formation
and the shape of thrust faults in a upper brittle crust.
Such magma-controlled processes are likely to be of firstorder importance in the development of compressional
active margins, such as the Andes, and possibly beyond
the scale of the upper crust. At large scale, the potential
mechanical impact of deep magmatic intrusions should
be explored in models of active margins.
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Received 30 October 2007; revised version accepted 28 February 2008
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 209-211, 2 figs.
Queste bozze, corrette e accompagnate dall’allegato preventivo firmato e dal buono d’ordine,
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Segreteria della Società Geologica Italiana
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Piazzale Aldo Moro, 5 – 00185 ROMA
Subduction zone earthquake mechanisms
and the H2O content of subducting lithosphere
H.W. GREEN, II (*)
ABSTRACT
Brittle fracture and frictional sliding are impossible below a few
tens of km yet earthquakes occur in subducting lithosphere to ~700
km. Experimental work shows that at high pressure a small amount
of low-viscosity «fluid» must be generated to enable shear failure
and comparison with the earthquake distribution in the upper 300
km of subducting slabs strongly indicates that the method of earthquake initiation is dehydration of hydrous phases. In contrast, the
earthquake distribution below 400 km shows no correlation with
potential H2O-liberating reactions; indeed assuming that hydrous
phases are present predicts earthquakes in places where they are not
observed. Thus, the earthquake distribution suggests that H2Oreleasing reactions do not take place in the mantle transition zone.
In addition, metastable olivine has now been detected by seismic
means in 2 subduction zones. Such metastable olivine can explain
the earthquake distribution and also is incompatible with the presence of H2O, even in very small amounts. I conclude that subduction
zones are essentially dry below 400 km.
KEY WORDS: antigorite serpentine, dehydration embrittlement, metastable olivine.
RIASSUNTO
I meccanismi sismici in zona di subduzione e il contento di
H2O nella litosfera subdotta.
Fratturazione fragile e scivolamento frizionale sono impossibili
al di sotto di poche decine di chilometri, sebbene i terremoti avvengano nelle litosfere in subduzione sino a ~700 km. I risultati sperimentali dimostrano che ad alta pressione dev’essere generata una
piccola quantità di «fluido» a bassa viscosità per permettere il
cedimento per taglio e il confronto con la distribuzione dei terremoti
nei 300 km superiori delle placche in subduzione indica fortemente
che il modo d’inizio di un terremoto sia la deidratazione di fasi idrate. Al contrario, la distribuzione dei terremoti al di sotto di 400 km
non mostra correlazione con potenziali reazioni che comportano il
rilascio di H2O; inoltre l’assunzione che siano presenti fasi idrate
predice terremoti dove questi non sono osservati. Quindi la distribuzione dei terremoti suggerisce che le reazioni che liberano H2O non
avvengano nella zona di transizione del mantello. In aggiunta, olivina metastabile è stata di recente individuata con mezzi sismici in
due zone di subduzione. Questa olivina metastabile può spiegare la
distribuzione dei terremoti ed è anche incompatibile con la presenza
d’acqua, pure in quantità ridotte. Io concludo che le zone di subduzione siano sostanzialmente anidre al di sotto dei 400 km.
TERMINI CHIAVE: serpentino antigoritico, infragilimento da
deidratazione, olivina metastabile.
Unassisted brittle shear failure and/or frictional sliding on pre-existing faults, the mechanisms by which
(*) Department of Earth Sciences and Institute of Geophysics &
Planetary Physics, University of California, Riverside, CA, USA.
[email protected]
materials fail by shearing at low pressure are impossible
at depths in excess of a few tens of km in Earth because
brittle failure and friction are strongly inhibited by
increasing pressure and plastic flow is enhanced exponentially by increasing temperature (see reviews by
GREEN & HOUSTON, 1995; GREEN, 2007). Laboratory
experiments show that shearing instabilities at pressures
above ~3 GPa only occur in the presence of a small
amount of a phase that has an effective viscosity very
much lower than the dominant material. Generation of
such «fluid» can be a natural consequence of the rising
temperature and/or pressure in subducting material.
Examples are: (a) dehydration embrittlement (RALEIGH &
PATERSON, 1965); (b) transformation-induced faulting
(GREEN & BURNLEY, 1989; GREEN et alii, 1990); (c) thermal runaway leading to melting (KARATO et alii, 2001;
GREEN & MARONE, 2002). The requirement of presence
of a small amount of «fluid» combined with the observed
distribution of earthquakes at high pressure (restriction
to subduction zones) indicates that such «fluid»-producing mineral reactions must be occurring at sites of earthquake generation.
Dehydration embrittlement has been demonstrated in
a variety of hydrous phases (e.g. RALEIGH & PATERSON,
1965; JUNG et alii, 2004) and is strongly implicated as the
trigger mechanism of intermediate-depth earthquakes
(70-300 km) (PEACOCK, 2001; HACKER et alii, 2003).
Antigorite serpentine is capable of initiating such a shearing instability during dehydration under stress at pressures from 0.1 to 6 GPa in the laboratory (JUNG et alii,
2004), a range over which the volume change accompanying dehydration changes from positive to negative, yet the
shearing instability occurs under all conditions. The
instability does not disappear when the net volume
change of reaction becomes negative (∆Vreaction < 0)
because the instability is not dependent upon the net volume change but rather upon the ∆V of fluid and solid
components independently; under all conditions the fluid
remains less dense than the solid matrix (∆Vfluid > 0) and
the nanocrystalline solid reaction products remain more
dense (∆Vsolid < 0); rather than canceling each other out,
they both participate in the instability via formation of
microcracks and microanticracks, respectively (JUNG et
alii, 2004). Exsolution of very small quantities of H2O
from nominally anhydrous phases can also trigger instability in the laboratory (ZHANG et alii, 2004). It is thus
highly likely that dehydration under stress of any reasonably abundant phase in subducting lithosphere can trigger earthquakes.
Here I use this logic to argue that subducting lithosphere is progressively «wrung dry» over the depth interval
210
H.W. GREEN, II
Fig. 1 - Cartoon of an oceanic subduction zone showing 3 populations
of earthquakes: (1) grey dots (red in the colour version) symbolize
earthquakes generated by dehydration embrittlement of hydrous
crustal and mantle phases in the upper 10-12 km of subducting lithosphere; (2) white diamonds symbolize earthquakes generated by
dehydration of antigorite and/or chlorite if hydration of mantle lithosphere at trenches extends sufficiently deep; some of these at depths
< 100 km could also be generated by plastic instability of fine-grained
material in recrystallized subducted outer-rise faults; (3) transition
zone earthquakes (black dots) that could be initiated by dehydration
of DHMS or wadsleyite/ringwoodite (collectively referred to in the fig.
as «spinel»), or breakdown of metastable olivine-evidence presented
in the text-favors the latter. Modified after GREEN (2005).
– Schema di una zona di subduzione oceanica che mostra 3 popolazioni di terremoti: (1) i punti grigi rappresentano terremoti generati per
infragilimento da deidratazione di fasi idrate nella crosta e nel mantello
dei 10-12 km superiori di una litosfera che subduce; (2) i rombi bianchi rappresentano i terremoti generati per deidratazione di antigorite
e/o di clorite se l’idratazione del mantello litosferico si estende sufficientemente in profondità sotto le fosse; alcuni di questi a profondità
inferiore a 100 km possono anche essere generati da instabilità plastica
di materiale a grana fine lungo faglie subdotte e ricristallizate; (3) terremoti della zona di transizione (punti neri) che potrebbero essere iniziati dalla deidratazione di DHMS o wadsleyite/ringwoodite (genericamente indicate in fig. come «spinel»), o destabilizzazione di olivina
metastabile – le evidenze presentate nel testo favoriscono le ultime.
Modificato da GREEN (2005).
50-400 km and that only very small amounts of H2O exist
in such lithosphere below that depth (fig. 1). The evidence
is the following: (1) Earthquake frequency declines exponentially between 70 and 300 km (suggesting that the
cause of the instability is being exhausted) (fig. 2); (2) the
resurgence of earthquakes in the transition zone could, in
principle, be triggered by dehydration of dense hydrous
magnesium silicates (DHMS, the «alphabet phases»;
ANGEL et alii, 2001; POLI & SCHMIDT, 2002) but the continuous production of earthquakes with a maximum at
~600 km and the sudden termination before 700 km (correlating with the breakdown reaction of ringwoodite
spinel to magnesium silicate perovskite + magnesiowüstite
that, because slabs are colder than surrounding mantle,
takes place somewhat deeper in subduction zones than
elsewhere where it occurs at ~660 km; fig. 1) are inconsistent with the conditions under which the DHMS exhibit
Fig. 2 - Semilog plot of global earthquake frequency expressed as a
histogram of the number of earthquakes in 10-km thick concentric
layers. At depths more shallow than ~30 km, earthquakes are distributed widely in the crust but at greater depths essentially all earthquakes are concentrated in subduction zones as shown in fig. 1.
Although there is evidence for subduction to depths much greater
than 700 km, no earthquake has ever been reliably located in the
lower mantle (deeper than 700 km). Of principal note are the bimodal nature of the earthquake distribution, the exponential decrease
in frequency between 30 and 300 km, and the abrupt termination
below 600 km. Modified after FROHLICH (1989).
– Diagramma semilogaritmico della frequenza globale espressa come
un istogramma del numero di terremoti in livelli concentrici spessi
10-km. A profondità minori di ~30 km, i terremoti sono diffusi nella
crosta ma a maggiore profondità sostanzialmente tutti i terremoti sono
concentrati in zone di subduzione come appare in fig. 1. Nonostante
l’evidenza di subduzione a profondità maggiori di 700 km, nessun
terremoto è stato mai collocato attendibilmente nel mantello inferiore
(a profondità maggiori di 700 km). Sono da notare soprattutto la natura bimodale della distribuzione dei terremoti, la decrescita esponenziale
della frequenza fra 30 e 300 km e il drastico esaurimento al di sotto di
600 km. Modificato da FROHLICH (1989).
mineral reactions (indicating that dehydration of these
phases is unlikely to be involved in deep earthquakes); (3)
significant amounts of DHMS can be stable only if the
highly abundant phases wadsleyite and/or ringwoodite
(collectively referred to as spinel in fig. 1) are fully saturated with H2O, which would require more water at
depths of 200-300 km than could be consistent with the
known mineral possibilities in the lithosphere and their
seismic properties (CHEN & BRUDZINSKI, 2001; BRUDZINSKI & CHEN, 2003) (hence saturation is extremely
unlikely); (4) tectonic stresses are not necessary for earthquakes to occur (BRUDZINSKI & CHEN, 2005) – local
stresses generated by the ∆V between adjacent regions of
unreacted and reacted mineral assemblages are sufficient
(hence the ad hoc argument that earthquakes disappear
because there are no stresses is unlikely to be valid); (5) if
significant H2O is present in ringwoodite, wherever lithosphere passes through into the lower mantle there should
be an abundance of earthquakes where that H2O is
released during ringwoodite breakdown (the lack of such
earthquakes implies that even small amounts of H2O in
ringwoodite are unlikely); (6) if H2O gets passed from
hydrous ringwoodite to phase D at the top of the lower
mantle (hence avoiding constraint #5), there should be a
SUBDUCTION ZONE EARTHQUAKE MECHANISMS
flurry of earthquakes in the lower mantle during the dehydration of phase D, the last of the DHMS phases (such
earthquakes are absent, strongly suggesting that phase D
is also absent); (7) there is now strong seismic evidence for
the presence of metastable olivine in two deep slabs
(Tonga (e.g. CHEN & BRUDZINSKI, 2001) and Mariana
(KANESHIMA et alii, 2007)), requiring that slabs in those
subduction zones be essentially dry (because if significant
H2O is present, the kinetics of the ol→spinel reactions
would be enhanced sufficiently that metastable olivine
would not be preserved (DU FRANE & SHARP, 2007).
It could be argued that dehydration of antigorite at
depths of 200-250 km should lead to enhanced dissolution of H2O into olivine and pyroxenes, with that H2O
then carried into the transition zone. However, the empirical evidence cited above strongly suggests that doesn’t
happen. One possible explanation is that the recent evidence for likely low oxygen fugacity at depths in excess of
~250 km (ROHRBACH et alii, 2007) drastically reduces the
H2O fugacity, subverting its solubility in silicates and/or
its catalytic effect on mineral reactions.
In summary, the exponential decline in earthquake
frequency between 70 and 300 km suggests exhaustion of
a critical factor in their generation. The only factor that
seems a logical possibility is exhaustion of the availability
of hydrous phases to initiate the earthquakes. Despite the
experimentally-demonstrated water-carrying capacity of
the nominally anhydrous upper mantle phases and the
DHMS, the abundance and distribution of earthquakes
bears no recognizable relationship to their experimentally-determined properties and phase boundaries; there
is a lack of earthquakes in locations where they would be
expected if H2O is significantly present and being liberated, and an abundance of earthquakes in locations
where H2O, if present, would not be expected to be liberated. In contrast, there are earthquakes where independent seismic evidence strongly suggests the presence of
metastable olivine which is incompatible with the presence of significant H2O. These observations singly and in
concert imply that (i) dehydration embrittlement is at
best a minor trigger of earthquakes in the mantle transition zone and (ii) subduction zones deeper than ~400 km
are essentially dry. A corollary is that subduction does not
significantly recycle H2O into the deep mantle, at least
not at this time.
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subhorizontal slabs. J. Geophys. Res., 110, B08303, doi:10.1029/
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Imbricate Remnant of Subducted Lithosphere. Science, 292,
2475-2479.
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Earth Planet Sci., 17, 227-54.
GREEN II, H.W. (2005) - New light on deep earthquakes. In: Our Everchanging Earth, Scientific American, special edition (electronic
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Received 3 November 2007; revised version accepted 7 March 2008.
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 213-216, 2 figs.
Queste bozze, corrette e accompagnate dall’allegato preventivo firmato e dal buono d’ordine,
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Segreteria della Società Geologica Italiana
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Piazzale Aldo Moro, 5 – 00185 ROMA
Non-equilibrium thermodynamics
and the coupling between deformation and metamorphism
BRUCE HOBBS (*) & ALISON ORD
ABSTRACT
The concept of non-equilibrium phase diagrams is explored and
an example presented for the quartz-coesite reaction. The equilibrium phase boundary is relevant for systems undergoing mineral
reactions only for very high temperatures and very low strain rates.
The influence of strain rate is to modify the equilibrium ClausiusClapeyron slope by a factor that is similar in magnitude to that
slope. Increases in strain rate remove the non-equilibrium phase
boundary further from the equilibrium phase boundary. This
explains the common observation that mineral reactions proceed to
completion in shear zones rather than in adjacent undeformed material.
KEY WORDS: non-equilibrium phase diagrams, entropy production, deformation enhanced reactions.
RIASSUNTO
Termodinamica di disequilibrio ed interazione tra deformazione e metamorfismo.
In questo contributo si analizza il concetto di diagrammi di fase
in non-equilibrio a cui è associato un esempio sulla transizione
quarzo-coesite. Il limite di fase in equilibrio è rilevante per sistemi
che subiscono reazioni mineralogiche solo in condizioni di temperatura molto alta e di bassa velocità di deformazione. L’influenza della
velocità di deformazione è di modificare la pendenza dell’equilibrio
di un fattore dello stesso ordine di grandezza della pendenza. Gli aumenti di velocità di reazione allontanano il limite di fase in nonequilibrio dal limite di fase in equilibrio. Questo spiega l’osservazione comune che le reazioni minerali procedono a compimento nelle
zone di taglio piuttosto che nel materiale adiacente indeformato.
TERMINI CHIAVE: diagrammi di fase di non-equilibrio, produzione d’entropia, reazioni amplificate dalla deformazione.
INTRODUCTION
Non-equilibrium thermodynamics increasingly has
very wide application in many fields of science and engineering (COUSSY, 2004; OTTINGER, 2005) but apart from a
flurry of activity in the 1960’s to 1980’s (KAMB, 1959;
KORZHINSKII, 1959; PATERSON, 1973; FISHER, 1973) there
has been relatively little application within geology.
Recently there has been a resurgence of interest in nonequilibrium thermodynamics with respect to damage
mechanics in seismology (LYAKHOVSKY & BEN ZION,
(*) CSIRO Exploration and Mining, Perth, Australia; University
of Western Australia, Perth, Australia. Corresponding author - Telephone: +61 418 395 545. E-mail: [email protected]
1997) and in structural geology/geodynamics (REGENAUER-LIEB & YUEN, 2003; HOBBS et alii, 2007a; HOBBS
et alii, in press). In this abstract we set out to discuss
some important applications of non-equilibrium thermodynamics to deforming, reacting metamorphic systems.
We make a distinction between classical chemical thermodynamics (CEM) where minimisation of the Gibbs
Free Energy defines the stable states and non-equilibrium
thermodynamics where either minimisation or maximisation of the entropy production rate defines the stable
phases.
The problem in applying non-equilibrium thermodynamics to geological problems derives from the apparent
lack of a set of guiding principles that would allow
progress. In any system, whether at equilibrium or not,
one can define a function, The Gibbs Free Energy. This
function is minimised at equilibrium and so one can proceed to define equilibrium assemblages of minerals.
Another function, the entropy, is maximised at equilibrium. For non-equilibrium systems, it has never been
clear, until recently, that a similar principle was available.
In fact, two apparently opposing views seemed to emerge
in the literature. One is due to PRIGOGINE (1955) who
claimed that in non-equilibrium systems, the rate of
entropy production is minimised. The other view is due to
ZEIGLER (1980) who claimed that the rate of entropy production is maximised in non-equilibrium systems. This
apparent paradox is resolved when one understands that
the Prigogine principle holds for linear steady state systems whereas the Zeigler principle holds for systems that
are not constrained to be at steady state. This now opens
the way to describe the evolution of geological systems
that are forced out of equilibrium by continued deformation, fluid flow, heat flow and chemical reactions. We
employ the Prigogine principle below to understand the
construction of non-equilibrium phase diagrams and why
deformation enhances the progress of metamorphic
reactions.
To focus the discussion we concentrate on metamorphic rocks undergoing only deformation and chemical
reactions and exclude the effects of fluid transport; an
introduction to this area is given by COUSSY (2004). As
such, the energy dissipated during deformation and metamorphism consists of four parts: (i) That due to mechanical processes; this comprises dissipation arising from
deformation and from introducing chemical components
into a deforming system by chemical reaction. (ii) That
arising from the flux of chemical components across gradients in chemical potential and temperature. (iii) That
arising from chemical reactions and (iv) That arising
from thermal conduction.
214
B. HOBBS
In many metamorphic rocks there is evidence of nonequilibrium in the form of partially reacted mineral
assemblages and/or melting. An example is the Monte
Mucrone in the Italian Alps (ZUCALI et alii, 2002). It is
also commonly observed in such areas that in immediately adjacent rocks, the metamorphic reactions reach
completion only in the highly deformed shear zones. The
two important questions are: (i) In regions where the
metamorphic reactions have not proceeded to completion, are estimates of P, T conditions derived from equilibrium theory relevant and/or accurate? and (ii) What
role does deformation play in promoting metamorphic
reactions?
THE NON-EQUILIBRIUM CHEMICAL POTENTIAL
(1)
where σij is the deviatoric stress, P is the pressure, in this
case, the mean stress, T is the absolute temperature and
ξK is the extent of the chemical reaction that produces the
Kth component. Explicit forms of this state equation are
given by PATERSON (1973) and SHIMIZU (1997). To be
explicit here the pressure is
–
1
(σ 11 + σ 22 + σ 33 ).
3
Even in a system not at equilibrium, at a phase
boundary the difference in the sum of the chemical
potentials of the phases on either side of the boundary is
zero, as is also the difference in the affinities of the reactions involved. An important difference between the nonequilibrium and the classical chemical potential is that
the pressure for the non-equilibrium situation is measured by the mean stress. As the stress relaxes to hydrostatic and the chemical reactions proceed to completion,
the non-equilibrium chemical potential evolves to become
the classical chemical potential. From equation (1),
dµ K =
or,
∂µ K
∂µ K
∂µ K
∂µ K
dσ ij +
dP +
dT +
dξ K
K
∂σ ij
∂P
∂T
∂ξ
dµ K = ε ijK dσ ij + V K dP − S K dT + A K dξ K
(2)
(3)
K
where εΚ
ij is the elastic strain of component K, V is the
K
volume of component K, S is the entropy of component
K and AK is the affinity of the reaction that produces K.
At this stage we focus in on a particular simple kind
of chemical reaction, namely,
A⇔B
Then, for example, equation (3) becomes
dµ coesite = ε ijcoesitedσ ij + V coesitedP − S coesitedT + A coesitedξ coesite (5)
–
We define A = Acoesite – Aquarz. Then arguments presented by KONDUPEDI & PRIGOGINE (1998) mean that for
the entropy production rate to be a minimum,
A coesite =
Lquartz A
( Lquartz + Lcoesite )
and A quartz =
Lcoesite A
( Lquartz + Lcoesite )
(6)
where LK is the coefficient that links the extent of reaction K to the affinity of that reaction (KONDEPUDI & PRIGOGINE, 1998).
NON EQUILIBRIUM PHASE DIAGRAMS
The chemical potential of a component is a quantity
that measures the energy required to insert 1 mole of that
component into a system under adiabatic conditions.
This definition is relevant whether the system is or is not
at equilibrium. Consider a system with K chemical components. We define the non-equilibrium chemical potential, µK, of the Kth component inserted into a deforming,
chemically reactive system as (KONDEPUDI & PRIGOGINE,
1998; COUSSY, 2004):
µK = µK(σij, P, T, ξK)
& A. ORD
(4)
where A is, for example, quartz or graphite and B is
coesite or diamond respectively.
From equation (3) we obtain:
d( ∆µ ) = ∆ε ij dσ ij + ∆VdP − ∆SdT + ∆Adξ
(7)
where ∆(.) is the change in (.) during the chemical reaction
and deformation. That is ∆S = Scoesite – Squarz and so on.
At a phase boundary, d(∆µ) = 0 and ∆A = 0 hence,
dP ∆S ∆ε ij dσ ij
=
−
dT ∆V ∆V dT
(8)
Thus, at a phase boundary the classical ClausiusClapeyron slope at equilibrium, ∆S/∆V is modified in the
non-equilibrium case by a term involving the elastic
strain contrast between the two phases and the temperature derivative of the deviatoric stress tensor. If the
strains are elastic then this latter term involves only the
temperature dependence of the elastic moduli. If the total
strains arise from steady state power-law creep of the
.
form σij = L–1/N ε ij (J2)1/(N–1) exp(Q/RT) then,
dP ∆S ∆ε ij Q
=
+
L−1/ N ε˙ ij ( J2 )1/( N −1) exp(Q / RT ) (9)
dT ∆V ∆V RT 2
at a phase boundary. Hence, for a fixed strain rate, dP/dT
approaches the non-equilibrium Clausius-Clapeyron slope
at high temperatures but at low temperatures, the second
term on the right hand side of (9) can be of similar magnitude to the classical Clausius-Clapeyron slope, ∆S/∆V.
As the strain rate decreases at constant temperature, dP/dT
approaches the classical equilibrium Clausius-Clapyron
dP/dT slope. This behaviour is illustrated in fig. 1.
The non-equilibrium phase boundary between two
phases, A and B, represents the boundary between two
regions where A is stable on one side and B on the other
so long as the stress is maintained on the system.
Although these states are stable they are not stable equilibrium states; nor are they unstable equilibrium states so
that terms such as «overstepping» should not be used to
describe these states.
INFLUENCE OF DEFORMATION ON THE EXTENT
OF A CHEMICAL REACTION
Following COUSSY (2004) we can write
ξK =
∂Ψ(ε ij , P, T , A K )
∂A K
NON-EQUILIBRIUM THERMODYNAMICS
Fig. 1 - Non-equilibrium pressure-temperature phase diagram with
the equilibrium phase boundary shown dashed. The full line is the
non-equilibrium phase boundary asymptotic to the equilibrium phase boundary at high temperatures. The non-equilibrium phase boundary moves to the left with decreasing strain rate. The assemblage
at point A is not stable under equilibrium conditions but is stable
under non-equilibrium conditions.
– Diagramma di fase pressione-temperatura di non-equilibrio, con il limite di fase d’equilibrio a tratteggio. La linea a tratto continuo è il limite di fase di non-equilibrio asintotico al limite di fase d’equilibrio ad
alta temperatura. Il limite di fase di non-equilibrio si sposta a sinistra
con velocità di deformazione decrescente. La paragenesi al punto A
non è stabile alle condizioni d’equilibrio ma è stabile alle condizioni di
non-equilibrio.
215
Fig. 2. - Non-equilibrium
phase boundaries at two different strain ra.
tes, εijI (slow) and ε ijII (faster). The assemblage at point A is unstable
at the slow strain rate but stable at the higher strain rate. In this way
an assemblage may grow only in shear zones undergoing higher
strain rates than in the adjacent relatively undeformed rocks.
– Limiti. di fase di non-equilibrio
a due differenti velocità di deforma.
zione, εijI (bassa) ed ε ijII (alta). La paragenesi al punto A è instabile a
bassa velocità di deformazione ma stabile a velocità di deformazione
più elevata. In tal modo una paragenesi si può sviluppare solo in zone
di taglio soggette ad alta velocità di deformazione piuttosto che nelle
rocce adiacenti relativamente poco deformate.
CONCLUSIONS
where Ψ is the Helmholtz Free Energy. We then obtain:
dξ K =
∂2 Ψ
∂A K ∂ε ij
dε ij +
∂2 Ψ
∂A K ∂P
dP +
∂2 Ψ
∂A K ∂T
dT +
∂2 Ψ
∂A K 2
dA K (10)
At constant pressure and temperature this reduces to
dξK = αdεij + βdAK
(11)
where α is the parameter that measures how the extent
of a reaction changes with strain and β is a parameter
that measures how the extent of a reaction depends on
the affinity of that reaction. For the reaction described
by (4) KONDEPUDI & PRIGOGINE (1998) discuss the form
of β for a situation corresponding to minimum entropy
rate production. The relation between the extent of a
reaction and the strain is discussed by COUSSY (2004).
Without proceeding to detail here, equation (11) says
that the extent of a reaction is increased by increases in
strain and by increases in the affinity of the reaction.
Thus increased strain enhances the progress of a chemical reaction although we need to point out that this is
true so long as α ≠ 0. This means that the reaction must
contribute to the strain either through a volume change
or through the preferred diffusion of chemical components.
However there is another important factor involved in
the enhancement of chemical reaction by deformation.
Fig. 2 shows the influence of strain rate on the position of
the non-equilibrium phase boundary. At low strain rates a
particular P, T environment may be below a phase
boundary for low strain rates but be above the phase
boundary for higher strain rates. Thus a region may be
such that a phase such as coesite or diamond is not stable
at the ambient strain rate but is stable within shear zones
in that same environment.
(i) In regions where the metamorphic reactions have
not proceeded to completion, estimates of P, T conditions
derived from equilibrium theory are relevant only at high
temperatures and low strain rates. If reactions have not
proceeded to completion, the relevant measure of pressure is the mean stress and not the lithostatic pressure.
(ii) Deformation plays an important role in promoting
metamorphic reactions through a direct influence on the
extent of the reaction. High strain rates displace the nonequilibrium stability field for a particular reaction from
the equilibrium field so that reactions commonly are
observed to proceed to completion within shear zones
and not in adjacent relatively undeformed rocks.
ACKNOWLEDGEMENTS
BEH thanks the organisers of the 16th DRT meeting for the
opportunity to sit in the Italian Alps and think about non-equilibrium phase diagrams.
REFERENCES
COUSSY O. (2004) - Poromechanics. Wiley, Chichester, 298 pp.
FISHER G.W. (1973) - Non-equilibrium thermodynamics as a model
for diffusion-controlled metamorphic processes. Amer. J. Sci.,
273, 897-924.
HOBBS B.E., REGENAUER-LIEB K. & ORD A. (2007a) - Thermodynamics of folding in the middle to lower crust. Geology, 35, 175178.
HOBBS B.E., REGENAUER-LIEB K. & ORD A. (in press) - Folding with
thermal-mechanical feedback. Jour. Struct. Geol.
KONDEPUDI D. & PRIGOGINE I. (1998) - Modern Thermodynamics.
Wiley, Chichester, 486 pp.
KORZHINSKII D.S. (1959) - Physiochemical basis of the analysis of the
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and friction. Jour. Geophys. Res., 102, 27635-277649.
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B. HOBBS
O TTINGER H.C. (2005) - Beyond Equilibrium Thermodynamics.
Wiley, Hoboken, 617 pp.
PATERSON M.S. (1973) - Nonhydrostatic thermodynamics and its
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PRIGOGINE I. (1955) - Introduction to the Thermodynamics of Irreversible Processes. Charles C. Thomas, Springfield, Ill. 115 pp.
REGENAUER-LIEB K. & YUEN D.A. (2003) - Modeling Shear Zones in
Geological and Planetary Sciences: Solid- and Fluid- ThermalMechanical Approaches. Earth Science Reviews, 63, 295-349.
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SHIMIZU I. (1997) - The non-equilibrium thermodynamics of intracrystalline diffusion under non-hydrostatic stress. Phil. Mag., 75,
1221-1235.
ZIEGLER H. (1983) - An Introduction to Thermomechanics. NorthHolland Publishing Company, Amsterdam, 2nd edition, 356 pp.
ZUCALI M., SPALLA M.I. & GOSSO G. (2002) - Strain partitioning and
fabric evolution as a correlation tool: the example of the Eclogitic
Micaschists Complex in the Sesia-Lanzo Zone (Monte MucroneMonte Mars, Western Alps, Italy). Schweiz. Mineral. Petrogr.
Mitt. 82, 429-454.
Received 14 November 2007; revised version accepted 5 March 2008.
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 217-220, 4 figs.
Queste bozze, corrette e accompagnate dall’allegato preventivo firmato e dal buono d’ordine,
debbono essere restituite immediatamente alla
Segreteria della Società Geologica Italiana
c/o Dipartimento di Scienze della Terra
Piazzale Aldo Moro, 5 – 00185 ROMA
Interaction between brittle fracture
and ductile flow during crustal deformation
NEIL S. MANCKTELOW (*)
La maggior parte dei modelli teorici di deformazione crostale
assume che le rocce si fratturino secondo qualche criterio di cedimento (solitamente considerato come una semplice dipendenza lineare Mohr-Coulomb dalla pressione) o fluiscano in modo viscoso
grazie alla plasticità cristallina o alla diffusione. Le osservazioni di
terreno mostrano che esiste un’interrelazione più stretta tra flusso e
frattura, con fratture precoci che localizzano zone di taglio duttile
successive che possono, a loro volta, sovrapporsi a fratture discrete,
con la conseguente implicazione di cicli multipli di comportamento
fragile-duttile. Frattura e flusso possono essere coevi su piccole distanze, con perturbazioni attorno alle fratture attive che producono
le caratteristiche strutture collaterali (flanking structures). I modelli
reologici devono includere questo comportamento di interazione fragile-duttile se sono rivolti a una modellazione realistica della deformazione delle rocce.
rocks with increasing depth implies distinct «brittle» and
«ductile» rheological layers, corresponding to MohrCoulomb failure or viscous flow respectively (fig. 1)
although, depending on the assumed geotherm, a compositionally layered lithosphere could have several such brittle-ductile transitions (e.g. RANALLI & MURPHY, 1987). A
«Christmas-tree» envelope as shown in fig. 1 predicts very
high differential stress at the depth of the brittle ductile
transition (at least for relatively dry rocks and high strain
rates), which is an artefact of the simplifying assumptions
of constant strain rate and constant linear dependence of
Mohr-Coulomb yield on pressure. Models involving constant force (CF) or strain-rate-dependent force (SRDF) as
boundary conditions overcome the problem of unrealistically high stress levels (e.g. PORTH, 2000) as does recent
evidence for high-pressure brittle fracture with a weaker
dependence on the confining pressure (SHIMADA, 1993;
ZANG et alii, 2007). Nevertheless, these models still imply
that large regions of the crust deform exclusively by
either brittle fracture or viscous crystal-plastic flow.
Increasing the strain rate (fig. 1a) will shift the brittleductile transition to greater depth and this effect may
be enhanced if the pore fluid pressure is also increased
(fig. 1b). Such a shift in the depth of the transition could
occur at the tip of a downward propagating seismic fault
localized in the upper crust (e.g. ELLIS & STÖCKHERT,
2004). Cycles of seismic reactivation and intervening
aseismic creep could thereby lead to periodic brittle and
ductile behaviour in the middle to lower crust. However,
in this simple conceptual model, the major part of the
lithosphere away from the relatively narrow brittle-ductile transition zone is still considered to be either brittle
or ductile at any particular depth and time.
TERMINI CHIAVE: deformazione, reologia, crosta, litosfera,
fragile, duttile, faglie, zone di taglio.
FIELD OBSERVATIONS
ABSTRACT
Most theoretical models of crustal deformation assume that
rocks either fracture according to some yield criterion (usually taken
as a simple linear Mohr-Coulomb dependence on pressure) or flow
in a viscous manner due to crystal plasticity or diffusion. However,
direct field observation shows a much more intimate interplay
between fracture and flow, with precursor fractures localizing subsequent ductile shear zones that may in turn be overprinted by discrete fractures, implying multiple cycles of brittle-ductile behaviour.
Fracture and flow may be coeval over small distances, with the perturbation flow surrounding active fractures producing characteristic
flanking structures. Rheological models must include this linked
brittle-ductile behaviour if they are to realistically model rock deformation.
KEY WORDS: deformation, rheology, crust, lithosphere, brittle, ductile, faults, shear zones.
RIASSUNTO
Interazione tra fratturazione fragile e flusso duttile durante
la deformazione crostale.
INTRODUCTION
The «yield-strength envelope» (GOETZE & EVANS,
1979) is a simple 1D, constant strain rate (CSR) concept
that has been very commonly applied in numerical, analogue and conceptual models of crustal, and on a larger
scale, lithospheric deformation (e.g. RANALLI & MURPHY,
1987). This representation of the mechanical behaviour of
(*) Geological Institute, ETH Zurich, CH-8092 Zurich, Switzerland, E-mail: [email protected]
Nevertheless, it is becoming increasingly clear from
field observation that in reality there is an intimate interplay in space and time between precursor heterogeneities
(either structural or compositional), brittle fracture, fluidrock interaction and more distributed «ductile flow» (e.g.
SEGALL & SIMPSON, 1986; GUERMANI & PENNACCHIONI,
1998; MANCKTELOW & PENNACCHIONI, 2005). In particular, there are now many well-documented examples of
brittle precursors localizing subsequent ductile deformation under metamorphic conditions ranging from upper
greenschist to granulite facies, conditions that are typical
of the middle to lower crust. Fig. 2 is an example of a
«paired shear zone» from the Neves area of the Tauern
window in the eastern Alps (MANCKTELOW & PENNAC-
218
N.S. MANCKTELOW
Fig. 1 - Simple 1D, constant strain rate «yield-strength envelope» for a 30 km thick crustal section approximated by brittle Mohr-Coulomb
fracture with internal friction angle of 30º and power-law viscous flow of «average wet quartzite» according to PATERSON & LUAN (1990). The
effect of increased strain rate (a) and a combination of increased strain rate and pore fluid pressure (b) on the depth of the brittle-ductile
transition in a compressive tectonic regime is shown in cartoon form.
– Semplice profilo di resistenza monodimensionale per una sezione di crosta spessa 30 km, approssimato da frattura fragile Mohr-Coulomb con
angolo di frizione interna di 30° e da una legge esponenziale di flusso viscoso per una «quarzite idrata media» secondo PATERSON & LUAN (1990).
L’effetto dell’aumento della velocità di deformazione (a) e della velocità di deformazione combinata alla pressione dei fluidi nei pori (b) sulla
profondità della transizione fragile-duttile in un regime tettonico compressivo è mostrata in forma schematica.
INTERACTION BETWEEN BRITTLE FRACTURE
Fig. 2 - Precursor fracture localizing fluid infiltration, to produce a
central epidote-rich vein and adjacent bleached zone of fluid-rock interaction, along the boundaries of which subsequent ductile shear
has been localized to form a «paired shear zone» (see MANCKTELOW
& P ENNACCHIONI , 2005). Sense of shear is dextral. Neves area,
Eastern Alps.
– Frattura precoce che localizza l’infiltrazione di fluidi, che produce
una vena centrale ricca in epidoto e una zona di alterazione da interazione fluido-roccia, lungo i cui margini si localizza successivamente
taglio duttile che genera una «zona di taglio appaiata» (vedi MANCKTELOW & PENNACCHIONI, 2005). Il senso di taglio è destro. Area di
Neves, Alpi Orientali.
CHIONI, 2005; PENNACCHIONI & MANCKTELOW, 2007). An
initial precursor fracture has allowed fluid infiltration,
with the development of a thin epidote-rich vein flanked
by a bleached zone to either side as the result of fluidrock interaction. Most of these precursor structures are
sealed joints and show no discernible shear offset prior to
reactivation, even at the microscopic scale. Such joints,
with lengths on the order of tens of metres and widths
less than a millimetre, can only develop by extensional
failure and not by localization of crystal plastic deformation. During dextral reactivation under amphibolite facies
conditions, heterogeneous ductile shearing was localized
on the boundaries of this bleached zone to develop the
characteristic paired geometry. In rare cases, subsequent
straight discrete fractures offset such ductile shear zones,
and these fractures may themselves in turn be loci for
localized shearing. These field relationships provide evidence for cycles of discrete brittle fracture and more distributed ductile shearing within a small area, although
the time scale involved cannot be determined.
Distributed ductile deformation and localized slip on
discrete fractures can occur synchronously. A good example of this is seen in fig. 3, from the same general area as
fig. 2. Precursor discrete fractures were sealed with newly
grown quartz, plagioclase, biotite, and garnet indicating
metamorphic temperatures consistent with regional peak
metamorphic temperatures of around 550-600ºC. These
fractures show a left-stepping geometry typical of Riedel
fault development in an overall dextral shear. The sealed
fractures have subsequently been reactivated, again in
dextral shear, with the formation of a compressive bridge
in the left-stepping zone. The compressive bridge develops
a distributed foliation – a typical «ductile» structure –
whereas slip on the precursor fractures remains localized
(at least initially) on the fracture itself. Completely analo-
219
Fig. 3 - Ductile compressional bridge developed under amphibolite
facies conditions at a step-over between two discrete fractures slipping with a dextral sense (see MANCKTELOW & PENNACCHIONI,
2005). Neves area, Eastern Alps.
– Ponte duttile compressionale sviluppato in condizioni di facies anfibolitica in corrispondenza di una ripresa laterale (step-over) tra due
fratture discrete che scorrono con un senso destro (vedi MANCKTELOW
& PENNACCHIONI, 2005). Area di Neves, Alpi Orientali.
Fig. 4 - Flanking fold structure developed around a discrete brittle
fracture developed under amphibolite facies metamorphic conditions in calcite marble, Naxos, Greece. Sense of shear for this view is
dextral; width of photograph ca. 15 cm.
– Piega collaterale (flanking fold structure), intorno ad una frattura
fragile discreta, sviluppatasi in un marmo in condizioni metamorfiche
di facies anfibolitica, Naxos, Grecia. Il senso di taglio per questa vista è
destro; ampiezza dell’immagine circa 15 cm.
gous structures were described by PENNACCHIONI (2005)
from the Adamello tonalite, with the interplay between
slip on discrete fractures and distributed ductile strain in
the stepovers also occurring under amphibolite facies conditions during cooling of the pluton. It is not necessarily
the case that there are distinct periods of brittle and ductile behaviour, as would be implied by the models of fig. 1.
Flanking structures developed around brittle faults
(e.g. PASSCHIER, 2001; GRASEMANN & STÜWE, 2001;
EXNER et alii, 2004; KOCHER & MANCKTELOW, 2005)
are particularly clear examples of interacting brittleductile deformation, because their geometry can only
be explained if discrete slip occurred synchronously
220
N.S. MANCKTELOW
with the more distributed surrounding ductile flow. In
fact, models assuming perfectly free slip on an isolated
fracture within a viscous surrounding matrix best
explain the observed flanking geometry and can be used
to estimate both the amount of general shear and the
kinematic vorticity number (KOCHER & MANCKTELOW,
2005). Examples of flanking structures developed in calcite marbles under amphibolite facies conditions (e.g.
fig. 4) demonstrate that brittle fracturing can play an
important role even in weak rocks at high temperature
conditions generally taken to imply exclusively ductile
or viscous behaviour.
The interplay between fracture and flow can still
occur at great depth, even under (ultra-) high pressure
conditions. Initial seismic faulting under eclogite facies
conditions in the Bergen Arcs of western Norway allowed
water infiltration into otherwise dry rocks, localizing the
transformation to eclogites and also localizing ductile
shear zones on these precursor fractures (e.g. BOUNDY et
alii, 1992). Both brittle fracture and ductile shearing
occurred under the same metamorphic conditions, with
the crucial factor being the influence of water and fluidrock interaction. A recent study by FUSSEIS et alii (2006)
has also shown that shear zone localization in strongly
anisotropic schists can also be controlled by brittle precursors, that shear zones lengthened by a combination of
fracturing and mylonitic shearing, and that the overall
geometry strongly reflects the interplay between brittle
fracture and ductile flow.
CONCLUSIONS
Field-based studies in granitoids, schists, and even
eclogite facies gneisses have established that brittle precursors and fluid-rock interaction may be critical for the initiation and localization of «ductile» shear zones in otherwise
relatively homogeneous rocks. It follows that natural deformation structures and the bulk rheology and dynamics of
the crust (and lithosphere) cannot be understood in terms
of simple, non-interacting brittle and ductile models. More
realistic models of lithospheric deformation must involve a
combined elasto-visco-plastic rheology, also considering the
importance influence of fluid-rock interaction and compositional heterogeneity on strain localization.
ACKNOWLEDGEMENTS
The ideas presented here are the result of many years of collaboration with Giorgio Pennacchioni, Bernhard Grasemann, Ulrike
Exner, Thomas Kocher and Cees Passchier, whose contributions are
gratefully acknowledged.
REFERENCES
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development and petrofabrics of eclogite facies shear zones, Bergen Arcs, western Norway: implications for deep crustal deformational processes. J. Metam. Geol., 10, 127-146.
ELLIS S. & STÖCKHERT B. (2004) - Elevated stresses and creep rates
beneath the brittle-ductile transition caused by seismic faulting in
the upper crust. J. Geophys. Res., 109, B05407.
EXNER U., MANCKTELOW N.S. & GRASEMANN B. (2004) - Progressive
development of s-type flanking folds in simple shear. J. Struct.
Geol., 26, 2191-2201.
FUSSEIS F., HANDY M.R. & SCHRANK C. (2006) - Networking of shear
zones at the brittle-to-viscous transition (Cap de Creus, NE
Spain). J. Struct. Geol., 28, 1228-1243.
GOETZE C. & EVANS B. (1979) - Stress and temperature in the bending
lithosphere as constrained by experimental rock mechanics.
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GRASEMANN B. & STÜWE K. (2001) - The development of flanking
folds during simple shear and their use as kinematic indicators.
J. Struct. Geol., 23, 715-724.
GUERMANI A. & PENNACCHIONI G. (1998) - Brittle precursors of plastic deformation in a granite: an example from the Mont Blanc
massif (Helvetic, western Alps). J. Struct. Geol., 20, 135-148.
KOCHER T. & MANCKTELOW N.S. (2005) - Dynamic reverse modelling
of flanking structures: a source of quantitative kinematic information. J. Struct. Geol., 27, 1346-1354.
MANCKTELOW N.S. & PENNACCHIONI G. (2005) - The control of precursor brittle fracture and fluid-rock interaction on the development of single and paired ductile shear zones. J. Struct. Geol., 27,
645-661.
PASSCHIER C.W. (2001) - Flanking structures. J. Struct. Geol., 23,
951-962.
PATERSON M.S. & LUAN F.C. (1990) - Quartzite rheology under geological conditions. In: Knipe R.J. & Rutter E.H. Eds., Deformation
Mechanisms, Rheology and Tectonics. Geol. Soc. Lond. Spec.
Publ., 54, 299-307.
PENNACCHIONI G. (2005) - Control of the geometry of precursor brittle
structures on the type of ductile shear zone in the Adamello tonalites, Southern Alps (Italy). J. Struct. Geol., 27, 627-644.
PENNACCHIONI G. & MANCKTELOW N.S. (2007) - Nucleation and initial growth of a shear zone network within compositionally and
structurally heterogeneous granitoids under amphibolite facies
conditions. J. Struct. Geol., 29, 1757-1780.
PORTH R. (2000) - A strain-rate-dependent force model of lithospheric
strength. Geophys. J. Int., 141, 647-660.
RANALLI G. & MURPHY D.C. (1987) - Rheological stratification of the
lithosphere. Tectonophysics, 132, 281-295.
SEGALL P. & SIMPSON C. (1986) - Nucleation of ductile shear zones on
dilatant fractures. Geology, 14, 56-59.
SHIMADA M. (1993) - Lithosphere strength inferred from fracture
strength of rocks at high confining pressures and temperatures.
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Received 1 November 2007; revised version accepted 3 March 2008.
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 221-225, 6 figs.
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Relict meso and micro-structures in orogenic garnet peridotites as tracers
of mantle dynamics and metasomatism at convergent plate margins
M. SCAMBELLURI (*), H.L.M. VAN ROERMUND (**) & T. PETTKE (***)
ABSTRACT
At subduction zones, a number of geologic processes are caused
by influx in the supra-subduction mantle wedge of fluid phases
released by the subducting plates. The distribution of fluids in such
settings affects the mineralogical, chemical and structural transformation of rocks and, hence, the survival of relict minerals and structures of previous events. These features can be investigated by means
of field-based studies of high and ultrahigh-pressure (HP-UHP) orogenic terrains that contain mantle wedge materials tectonically sampled by the subducting plates. Here we review two examples of garnet peridotites hosted in HP-UHP continental crust, which record
different P-T stories: (i) shallow spinel-facies lithospheric mantle
wedge down-dragged to depth during subduction and recrystallized
to garnet + amphibole assemblages due to the infiltration of crustderived fluids (Ulten Zone garnet peridotites, Eastern Alps, Italy); (ii)
transition-zone mantle upwelled and accreted to cratonic roots, and
involved in subduction-zone recrystallization at 200 km depth
enhanced by crustal fluids (UHP garnet peridotites, Western Gneiss
Region, Norway). Our textural and petrologic study shows that the
water distribution controls development of the new assemblages and
the metasomatic imprints of these rocks, independently on the depth
and degree of metamorphism. We conclude that mantle re-fertilization by crust-derived subduction fluids is an effective mechanism
working on a 100-200 km depth range.
KEY WORDS: Convergent margins, mantle wedge, fluid,
UHP metamorphism, mantle metasomatism.
RIASSUNTO
Meso e microstrutture relitte in peridotiti a granato orogeniche, traccianti della dinamica del mantello e del metasomatismo ai margini di placca convergenti.
Una parte dei processi geologici che avvengono nelle zone di
subduzione è causata dall’influsso nel cuneo (wedge) di mantello sopra-subduzione delle fasi fluide rilasciate dalle placche subdotte. La
distribuzione delle fasi fluide in questi ambienti condiziona le trasformazioni mineralogiche, chimiche e strutturali delle rocce, determinando la preservazione dei minerali e delle strutture relitte di
eventi geologici precedenti. Questi processi possono essere investigati
mediante lo studio dei terreni orogenici di alta e altissima pressione
(HP-UHP), che contengono scaglie di materiale proveniente dal wedge di mantello campionate tettonicamente dalle placche subdotte. In
questo articolo vengono rivisti due esempi di peridotiti a granato
ospitate da unità tettoniche crostali con impronta metamorfica di
HP-UHP. Queste peridotiti registrano evoluzioni pressione-temperatura differenti: (i) porzioni superficiali del wedge di mantello (facies a
spinello) trasportato in profondità durante la subduzione e cristallizzato in facies a granato + anfibolo a causa dell’infiltrazione di fluidi
(*) Dipartimento per lo Studio del Territorio e delle sue
Risorse, Università di Genova, Italy.
(**) Faculty of Earth Sciences, University of Utrecht, Netherlands.
(***) Institute of Geological Sciences University of Bern,
Switzerland.
rilasciati dalla crosta subdotta (peridotiti della Zona di Ulten, Alpi
Orientali Italiane); (ii) mantello derivante dalla zona di transizione,
risalito ed accresciuto alla litosfera cratonica ed infine interessato da
metamorfismo di subduzione a profondità di 200 km, innescato
dall’infiltrazione di fluidi crostali (Western Gneiss Region, Norvegia).
Lo studio petrologico e strutturale di queste rocce indica che la distribuzione dell’acqua controlla lo sviluppo delle nuove paragenesi e
le impronte metasomatiche delle rocce, indipendentemente dalla
profondità e dal grado metamorfico a cui avvengono le trasformazioni. La rifertilizzazione del mantello guidata dall’infiltrazione di fluidi
originati dalla crosta continentale è un processo operativo nell’intervallo di profondità comprese tra 100 e 200 km.
TERMINI CHIAVE: Margini convergenti, cuneo di mantello,
fluidi, metamorfismo di Ultra Alta Pressione, metasomatismo del mantello.
INTRODUCTION
Convergent plate margins are highly evolving environments, where significant physical and chemical changes
affect the subducting plates and the overlying mantle
wedges. At subduction zones, mass transfer, hydration
and melting of the supra-subduction mantle domains is
caused by the influx of fluid phases released by the subducting plates. The distribution of fluids and of deformation in such settings affects the extent of mineralogical
and structural transformation of rocks and, hence, the
survival of relict minerals and structures of previous
events.
The interplay of deformation, metamorphism and
fluid infiltration at convergent margins can be investigated through field-based studies of the high- (HP) and
ultrahigh-pressure (UHP) rocks exposed in mountain
buildings, which represent excellent natural observatories
on the Earth interiors and on subduction dynamics
(CHOPIN, 2003; with references). An increasing number of
studies has recently shown that, besides the oceanic and
continental lithosphere recording prograde subduction
metamorphism, the HP-UHP terrains contain mantle
wedge materials tectonically sampled by the subducting
crust (BRUECKNER, 1998; NIMIS & MORTEN, 2000; ZHANG
et alii, 2000). Such mantle rocks may preserve phase transitions attained at exceptional depths, i.e. in the range of
200 to 350 km, representing the deepest transformations
discovered in mantle rocks tectonically exposed at the
surface (DOBRZHINETSKAYA et alii, 1996; VAN ROERMUND
& DRURY, 1998; SPENGLER et alii, 2006; SONG et alii,
2004; SCAMBELLURI et alii, 2008).
So far, much research has been focussed on the slabs
and still few are the observations of mantle wedge peridotites, which are the least known pieces of the subduction
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M. SCAMBELLURI ET ALII
Fig. 1 - Cartoon reporting a subducting
continental slab (grey) and the overlying
mantle wedge. Thin solid lines refer to
isotherms; dashed lines indicate the corner flow motion in the mantle wedge.
The thick solid line shows the possible
path of the Ulten Zone peridotites from
spinel facies conditions (stage 1) to garnet + amphibole facies (stage 2). The
deep seated Norway peridotites derive
from upwelling of transition zone mantle, its accretion to the lithosphere much
before the development of subduction.
– Schema che mostra una placca
continentale in subduzione (grigio) ed il
cuneo di mantello soprastante. Le linee
continue rappresentano le isoterme; le
linee tratteggiate indicano il flusso nel
cuneo di mantello. Le linee continue
spesse rappresentano il possibile percorso delle peridotiti della Zona d’Ultimo dalla facies a spinello (stadio 1) alla facies granato + anfibolo (stadio 2). Le peridotiti profonde della
Norvegia derivano dalla risalita di mantello dalla zona di transizione, la sua accrezione alla litosfera predata lo sviluppo della subduzione.
factory. Information can be achieved from orogenic garnet
peridotites of mantle wedge origin, which may be viewed as
metre to kilometre-scale tectonic ‘xenoliths’ sampled at different depths by the subducted continental plates (fig. 1).
These peridotites can preserve old events, pre-dating their
engagement in the subducted crust and enabling to design
the long-term mantle dynamics at convergent settings.
Here we review two case-sudies of garnet peridotites
hosted in subducted continental basements, which record
different P-T, i.e. physical, trajectories prior to their uptake
in the crust. The two examples correspond to (fig. 1): (i)
shallow lithospheric mantle down-dragged to depth by
corner-flow motion in the mantle wedge (Ulten Zone garnet peridotites, Eastern Alps, Italy); (ii) transition-zone
mantle upwelled and accreted to cratonic roots (UHP garnet peridotites, Western Gneiss Region, Norway). In both
cases the mantle rocks were flushed and metasomatized
by incompatible element-rich fluids sourced from the
continental crust, prior to or during their uptake in the
subducting slabs. Such fluids are crucial to rock recrystallization and to the preservation of former structures.
FIELD-BASED CASE STUDIES
THE HP GARNET PERIDOTITES
ITALIAN EASTERN ALPS
FROM THE
ULTEN ZONE,
The Ulten Zone peridotite bodies are hosted by
Variscan high-pressure migmatites (GODARD et alii, 1996).
They are porphyroclastic spinel peridotites (T=1200°C;
P=1.5 GPa) recrystallized into fine-grained garnet +
Fig. 2 - Textures of the garnet peridotites from the Ulten Zone: A) coronitic garnet peridotites. The coarse porphyroclastic texture of this rock
is inherited from the shallow spinel-facies crystallization: spinel grains (black) are contoured by light grey garnet coronas formed during
low-strain re-crystallization of the mantle peridotite; B) mylonitic garnet + amphibole peridotite. The main foliation dips from left to right
side of the photograph. It consists of amphibole and finegrained garnet associated with olivine, clino and orthopyroxene.
– Tessiture nelle peridotiti a granato della Zona d’Ultimo: A) peridotiti a granato coroniche. La tessitura porfiroclastica della roccia è ereditata
dalla cristallizazione superficiale nella facies a spinello: i granuli di spinello (nero) sono circondati da corone di granato grigio chiaro durante la
ricristallizzazione delle peridotiti di mantello in condizioni di basso strain; B) peridotiti a granato + anfibolo milonitiche. La foliazione principale
immerge da sinistra verso destra della fotografia. È marcata da anfibolo e granato a grana fine associato a olivina, clinopirosseno e ortopirosseno.
RELICT MESO AND MICRO-STRUCTURES IN OROGENIC GARNET PERIDOTITES
223
Fig. 3 - Pb and Sr versus modal amphibole contents in the spinel- and garnet + amphibole-facies peridotites from the Ulten Zone. The
incompatible element contents increase with increasing amounts of amphibole, i.e. with increasing bulk-rock H2O contents. This shows that
the metasomatic imprint affecting the Ulten garnet peridotites is due to fluid infiltration.
– Pb e Sr vs il contenuto di anfibolo modale nelle peridotiti della facies a spinello ed a granato + anfibolo, della Zona d’Ultimo. Il contenuto di
elementi incompatibili aumenta con la quantità crescente di anfibolo, cioè con il crescente contenuto di H2O nella roccia totale. Ciò mostra che
l’impronta metasomatica che registrano le peridotiti a granato della Val d’Ultimo è dovuta a infiltrazione di fluidi.
amphibole peridotites (T = 850°C; P max = 3 GPa) in
response to corner-flow inside a mantle wedge and slicing
into a subducted continental slab (OBATA & MORTEN,
1987; NIMIS & MORTEN, 2000; TUMIATI et alii, 2003). The
rock textures change from coarse porphyroclastic in the
spinel-facies high-temperature domain of the mantle
wedge, to coronitic and mylonitic in the lower-temperature (garnet-facies) hydrated region over the slab (fig. 2).
The coronitic garnet peridotites display relict porphyroclastic textures where spinel (the black mineral spots in
fig. 2A) is contoured by coronitic garnet associated with
minor amounts of amphibole. The highly sheared garnet
peridotite mylonites (fig. 2B) are significantly enriched in
amphibole (up to 20% modal amphibole), suggesting an
open-system fluid influx in the highly deformed zones.
These features indicate that localized aqueous fluid infiltration in such wedge domains highly enhanced deformation development and chemical changes in rocks.
The incompatible element-enriched signature of the
garnet + amphibole peridotites clearly indicate that the
incoming aqueous fluids determined a new, metasomatic,
geochemical imprint of the garnet peridotites (RAMPONE
& MORTEN, 2001; SCAMBELLURI et alii, 2006). In particular, the significant enrichment in Sr, Pb and H2O of the
garnet + amphibole peridotites (fig. 3) indicate that the
fluid phase carried cust-derived components.
In the Ulten Zone peridotites, the heterogeneity in the
fluid flow patterns enabled survival of the precursor
Fig. 4 - Two different generations of majoritic garnets in UHP ultramafic rocks from the Western Gneiss Region of Norway: A) Exsolved px
lamellae in Archean garnet from Ugelvik (Otrøy Island): the lamellae are about 50 µm– thick and several hundred µm– long; B) Scandian
subduction zone garnet, showing the fine-grained exsolved px needles, maximum 5 µm– thick and up to 100 µm long. Same magnification as
in fig. 4A, to show the different size of px lamellae in the Archean and in the subduction zone majoritic garnets.
– Due differenti generazioni di granato majoritico nelle rocce ultrafemiche di Ultra Alta Pressione provenienti dalla Western Gneiss Region della
Norvegia: A) Lamelle di pirosseno smistate nel granato archeano (Ugelvik, Isola di Otrøy): le lamelle sono spesse circa 50 µm e lunghe diverse
centinaia di µm; B) Granato della zona di subduzione Scandinava che contiene finissimi smistamenti aciculari di pirosseno, spessi fino a 5 µm e
lunghi fino a 100 µm. Stesso ingrandimento che in fig. 4A per permettere di confrontare la taglia delle lamelle di smistamento.
224
M. SCAMBELLURI ET ALII
Fig. 5 - Chondrite-normalized REE patterns (ANDERS & GREVESSE,
1989) of the reconstructed bulk REE compositions of the early (Archean) and of the later subduction-zone majoritic garnets. The REE
concentrations in the majoritic garnets have been calculated adding
to the Archean and to the subduction garnet compositions respectively 20 and 1.5 volume % of the coexisting clino and orthporyoxene
compositions (after SCAMBELLURI et alii, 2008).
– Concentrazioni delle REE normalizzate alla chondrite (ANDERS &
GREVESSE, 1989) nei granati majoritici della Norvegia Occidentale. Le
concentrazioni delle REE originarie in questi granati sono state calcolate
aggiungendo alle composizioni dei granati archeani e di subduzione
rispettivamente le quantità di REE corrispondenti al 20 e 1,5% in volume di clino- e ortopirosseno coesistenti (da SCAMBELLURI et alii, 2008).
anhydrous spinel-facies domains unaffected by fluid
influx and by garnet-facies recrystallization aside of
highly sheared mylonitic hydrated garnet peridotites.
Once engaged in the crust, the peridotite lenses behaved
as rigid bodies which escaped the exhumation tectonics
that mostly involved the surrounding softer gneisses.
THE UHP GARNET PERIDOTITES
WESTERN NORWAY
AND WEBSTERITES FROM
The UHP gneisses of the Western Gneiss Region (Norway) record subduction to the coesite and to the diamond
stability fields (SMITH, 1984; DOBRZINETSKAYA et alii, 1995;
VAN ROERMUND et alii, 2002). The diamond-facies gneisses
host garnet peridotites and websterites recording uprise
from extraordinary depths prior to uptake in the continental slab. These ultramafic rocks (exposed in the islands of
Otrøy and Bardane) derive from depleted Archean transition-zone mantle upwelled and accreted to a cratonic
lithosphere (VAN ROERMUND & DRURY, 1998; SPENGLER et
alii, 2006). Evidence for this Archean story are decimetric
garnets preserved in Otrøy, hosting orthopyroxene and
clinopyroxene exsolved from precursor ultradeep majoritic
garnet (up to 20 volume % pyroxene component). Majoritic
garnets form above 5 GPa through the progressive, pressure-dependent, incorporation of pyroxene into garnet,
leading to formation of supersilicic garnets with Si exceeding 3 atoms per formula unit (RINGWOOD & MAJOR, 1971;
AKAOGI & AKIMOTO, 1977; see GRIFFIN, 2008 for a short
review). The Archean garnets from Otrøy contained up to
20 volume % pyroxene, now exsolved as intercrystalline
grains and as coarse exsolution lamellae inside garnet.
Fig. 4A reports such pyroxene lamellae (20-30 µm thick,
hundreds µm long lamellae), exsolved under high-temperatures, as shown by the garnet/cpx REE distribution
(SPENGLER et alii, 2006). The high amounts of pyroxene
exsolutions in this garnet indicates provenance from the
transitions-zone, 350 km deep, mantle. These pyroxenes
and garnets display REE-depleted compositions, indicating that the original majorite crystallized in extremely
refractory peridotite after high degrees of partial melting
during the Archean upwelling history.
This ultradeep mantle was involved in the 430 Ma-old
Scandian subduction cycle, forming new clinopyroxene +
orthopyroxene + phlogopite + garnet + spinel + carbonate,
which host microdiamond-bearing inclusions precipitated
by circulating COH silicate fluids (VAN ROERMUND et alii,
2002; CARSWELL & VAN ROERMUND, 2005). This stage is
mostly recorded in the island of Bardane. The circulating
subduction fluids also crystallized new majoritic garnet at
grain boundaries and in microveins. This new majoritic
garnet hosts maximum 1.5 volume % thin px needles (5 mm
thick, 100 mm long; fig. 4B) exsolved under low-temperatures, as indicated by the garnet/cpx REE distribution.
The amounts of pyroxene needles exsolved indicate that
the new majoritic garnet formed at 7 Gpa and 900-100°C
(SCAMBELLURI et alii, 2008). Pictures in fig. 4 are taken at
the same magnification and refer to the Archean high temperature majorite (fig. 4A) and to the Scandian subduction
majorite (fig. 4B) from Western Norway. They emphasize
Fig. 6 - Trace element compositions of clinopyroxene
from the HP assemblage of the Ulten Zone garnet peridotites (open dots) and from the UHP assemblage of
the Norwegian ultramafic rocks (black dots).
– Composizione degli elementi in traccia del clinopirosseno
dalle paragenesi di Alta pressione delle peridotiti a granato della Zona d’Ultimo (cerchi vuoti) e dalle paragenesi
di Ultra Alta Pressione delle rocce ultrafemiche norvegesi
(cerchi pieni).
RELICT MESO AND MICRO-STRUCTURES IN OROGENIC GARNET PERIDOTITES
the different size of pyroxene exsolutions in these garnets,
which may be taken as textural evidence for distinct exsolution temperatures and geologic environments.
The majorites of fig. 4 also display significantly different trace element compositions. The subduction majorite
has flat REE patterns (fig. 5): this contrasts with the REE
depleted composition of the Archean majorite and indicates re-fertilization of the starting depleted peridotite by
crust-derived fluid at 200 km depth. Distinct generations
of majoritic garnet thus survive in the same terrain, displaying distinct textures, compositions, and exsolution
temperatures.
The majorite microstructures and compositions
enable to discriminate between different crystallization
environments: hot sub-cratonic lithosphere vs. colder
subduction-zones. Crystallization of the new majoritic
assemblage in Bardane was fluid induced, the archean
transition zone majorites in Otrøy likely escaped fluid
infiltration and survived the subduction event.
CONCLUSIVE REMARKS
Our study shows that continental crustal slabs subducted to variable depths entrain mantle wedge peridotites,
the relict structures of which emphasize the stories and
fate of the subcontinental mantle through time. The water
distribution controls development of the new assemblages
and the preservation of relics, independently on the depth
and degree of metamorphism. Comparison of the trace element compositions of clinopyroxenes pertaining to the
metasomatic HP and UHP subduction assemblages in
Ulten and Bardane emphasizes a strong similarity in the
Light Rare Earth Elements and in the Large Ion Litophile
Elements of such phases (fig. 6; data from SCAMBELLURI et
alii, 2006; 2008). Since the clinopyroxenes exchanged components with the incoming metasomatic fluids, this similarity indicates that the fluid phase compositions did not
change dramatically with depth and implies that mantle refertilization by crust-derived subduction fluids is an effective mechanism working on a 100-200 km depth range.
ACKNOWLEDGEMENTS
MS acknowledges Iole Spalla, Guido Gosso and Anna Maria
Marotta for the invitation at the DRT Conference in Milano, a great
oportunity to discuss the structure and petrology of deep mantle
rocks. We thank Stefano Poli and an anonymous reviewer for their
comments. This research has been financially supported by the Italian MIUR, the University of Genova, the Utrecht Institute of Geodynamic Research and the Swiss National Science Foundation.
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CARSWELL D.A. & VAN ROERMUND H.L.M. (2005) - On multi-phase
mineral inclusions associated with microdiamond formation in
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Norway. Eur. J. Mineral., 17, 31-42.
CHOPIN C. (2003) - Ultrahigh-pressure metamorphism: tracing continental crust into the mantle. Earth Planet. Sci. Lett., 212, 1-14.
DOBRZHINETSKAYA L.F., EIDE E.A., LARSEN R.B., STURT B.A., TRONNES R.G., SMITH D.C., TAYLOR W.R. & POSUKHOVA T.V. (1995) Microdiamond in high-grade metamorphic rocks of the Western
Gneiss region, Norway. Geology, 23, 597-600.
DOBRZHINETSKAYA L.F., GREEN H.W., II & WANG, S. (1996) - Alpe
Arami: a peridotite massif from depths of more than 300 kilometers. Science, 271, 1841-1845.
GODARD G., MARTIN S., PROSSER G., KIENAST J.R. & MORTEN L.
(1996) - Variscan migmatites, eclogites and garnet-peridotites of
the Ulten zone, Eastern Austroalpine system. Tectonophysics,
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GRIFFIN W.L. (2008) - Major transformations reveal Earth’s deep secrets. Geology, 36, 95-96.
NIMIS P. & MORTEN L. (2000) - P-T evolution of ‘crustal’ garnet peridotites and included pyroxenites from Nonsberg area (upper Austroalpine), NE Italy: from the wedge to the slab. J. Geodyn., 30, 93-115.
OBATA M. & MORTEN L. (1987) - Transformation of spinel lherzolite
to garnet lherzolite in ultramac lenses of the austridic crystalline
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RAMPONE E. & MORTEN L. (2001) - Records of crustal metasomatism
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RINGWOOD A.E. & MAJOR A. (1971) - Synthesis of majorite and other
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SCAMBELLURI M., HERMANN J., MORTEN L. & RAMPONE E. (2006) Melt versus fluid induced metasomatism in spinel to garnet wedge
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SCAMBELLURI M., PETTKE T. & VAN ROERMUND H.L.M. (2008) Majoritic garnets monitor deep subduction fluid flow and mantle
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SMITH D.C. (1984) - Coesite in clinopyroxene in the Caledonides and
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MASON P.D. & DAVIES G.R. (2006) - Deep origin and hot melting
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Received 9 November 2007; revised version accepted 14 February 2008.
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 227-230, 4 figs.
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Segreteria della Società Geologica Italiana
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Estimation of palaeorheology from buckle-fold geometries
STEFAN M. SCHMALHOLZ (*) & NEIL S. MANCKTELOW (*)
ABSTRACT
The geometry of a natural single-layer fold train is investigated
to estimate the effective viscosity ratio between the folded layer and
the surrounding medium. Different methods based on analytical
solutions for folding yield consistent estimates for the range of
viscosity ratios of between 20 and 70 and for values of the power-law
exponent of the layer of between 1.8 and 5. The error range for such
viscosity ratio estimates is roughly on the order of a factor of 2.
KEY WORDS: buckling, folding, power-law rheology, strain
estimation, palaeorheology.
RIASSUNTO
Stima della paleoreologia dalle geometrie delle pieghe per
buckling.
In questo contributo viene studiata la geometria di un treno di
pieghe naturali su un singolo strato per stimare il rapporto della viscosità effettiva tra il livello piegato ed il mezzo circostante. Differenti
metodi, basati su soluzioni analitiche del piegamento, forniscono
stime consistenti per i rapporti di viscosità nell’intervallo tra 20 e 70,
per valori tra 1.8 e 5 dell’esponente della legge di flusso per il livello.
L’intervallo d’errore per tali stime del rapporto di viscosità è
grossolanamente dell’ordine di un fattore 2.
TERMINI CHIAVE: buckling, piegamento, reologia esponenziale, stima della distorsione, paleoreologia.
INTRODUCTION
Geological structures as observed in the field can be
regarded as results of natural rock deformation experiments. However, in contrast to laboratory or numerical
rock deformation experiments, in nature only the final
geometry is directly measurable and the initial and
boundary conditions, as well as the rock rheology at the
time of deformation, are unknowns that cannot be immediately established. Indeed, one of the main aims of structural geologists is to reconstruct both the kinematics and
dynamics of the natural deformation history from the
observed structures.
This study focuses on buckle-fold structures and aims
to assess the rheology and effective strength ratio (or
«competence» contrast) between the folded layer and the
embedding matrix at the time of folding. In the framework of continuum mechanics, as applied in this study,
rheology means the constitutive equations relating stress
(*) Geological Institute, ETH Zurich, 8092 Zurich, Switzerland.
Tel.: +41 44 632 8167. Fax: +41 44 632 1030. E-mail: schmalholz@
erdw.ethz.ch
tensor components to strain or strain rate tensor components (e.g. JOHNSON & FLETCHER, 1994). In this sense,
this study aims to estimate the effective rheology on the
scale of observation (i.e. the folded layer and the embedding medium). This effective rheology may differ from
the rheology on the micro-scale (i.e. on the scale of individual small crystals building the rock layer).
There are several rheologies, such as elastic or viscous, that can potentially describe the deformation
behaviour of folded layers (fig. 1). However, a purely elastic rheology is unlikely to be appropriate for buckle-fold
formation, because elastic strains are very small (<<1%).
An elastoplastic rheology is also unlikely to be dominant
during buckle-fold formation, because elastoplastic
deformation causes localized shear bands (i.e. bifurcation, e.g. VERMEER & DE BORST, 1990) and not distributed deformation as observed in natural buckle-folds.
The remaining and most likely candidates for the rheology generating buckle-folds are viscoelastic (SCHMALHOLZ & PODLADCHIKOV, 1999; MANCKTELOW, 1999),
linear viscous (BIOT, 1961) and power-law (FLETCHER,
1974) rheologies. This study focuses on folding of powerlaw layers (e.g. FLETCHER, 1974; JOHNSON & FLETCHER,
1994) embedded in a viscous (Newtonian) matrix. The
power-law rheology includes the Newtonian case when
the power-law exponent is 1.
The aims of this study are (i) to present methods for
estimating the effective viscosity ratio and power-law
exponent of a folded layer and (ii) to discuss the accuracy
and reliability of such palaeorheology estimates.
METHOD AND RESULTS
The two interfaces of the natural buckle-fold train
were digitized using MATLAB (fig. 1). The slope of the
two interfaces (i.e. the derivative of the Y-coordinate with
respect to the X-coordinate, fig. 2) was analyzed with an
algorithm that determines the sign of the slope. Every
position on the interface at which the slope of the interface changes its sign was identified and marked with a
diamond-shaped symbol (fig. 2). These locations represent potential fold hinges. Due to natural irregularities,
sometimes more than one hinge position was identified
(fig. 2). Representative fold hinge positions were determined by personal interpretation and five individual
buckle-folds were identified within the fold train. The
ratio of fold-span to layer thickness of the individual folds
varies between 4.6 and 10.6. The average ratio is about 8.
In addition, the Fourier spectrum was calculated (using
the MATLAB fast fourier transform, fft, function) for the
top, bottom and averaged layer interface (fig. 2). The
228
S.M. SCHMALHOLZ
& N.S. MANCKTELOW
Fig. 1 - The photograph shows a natural folded quartz vein embedded in shale from
Vale Figueiras, SW Portugal, with the digitized top and bottom interface of the folded
quartz vein given below.
– La fotografia mostra una vena naturale di
quarzo piegata, incassata in argillite, a Vale
Figueiras, Portogallo SO; nell’immagine sottostante sono rappresentate le superfici digitalizzate di tetto e di letto della vena di quarzo
piegata.
Fig. 2.
ESTIMATION OF PALAEORHEOLOGY FROM BUCKLE-FOLD GEOMETRIES
Fig. 3 - Contours of the dominant wavelength (dashed lines) and the
maximal growth rate (solid lines) versus the viscosity ratio and the
power-law exponent of the layer (matrix is Newtonian).
– Tracce della lunghezza d’onda dominante (linee a tratteggio) e della
massima velocità di crescita (linee continue) confrontate al rapporto di
viscosità e all’esponente della legge di flusso esponenziale per il livello
(la matrice è Newtoniana).
three Fourier spectra are significantly different. The best
Fourier representation of the layer shape is presumably
the Fourier spectrum of the averaged interface, because
the averaged interface is least sensitive to natural interface irregularities around the fold hinges (because such
irregularities are smoothed in the process of averaging).
The Fourier spectrum of the averaged layer interface
yields the largest amplitude at a ratio of wavelength to
layer thickness of about 9. This value is close to the value
of 8 obtained by using the distances between fold hinges,
i.e. the fold-span.
The ratio of fold-span to layer thickness and the ratio
of wavelength to layer thickness can be used to estimate
the viscosity ratio and the power-law exponent of the
layer. In the current analysis, the analytical solution of
FLETCHER (1974) is used. This provides values for the
ratio of dominant wavelength to thickness and for the
maximal growth rate (normalized against the background
shortening rate) as a function of the viscosity ratio and
the power-law exponent of the layer (the embedding
matrix is assumed to be Newtonian, fig. 3). The measured
ratios of fold-span to thickness (about 8) and of wavelength to thickness (about 9) are used as lower and upper
bounds for the ratio of dominant wavelength to thickness.
No correction for the shortening of the dominant wavelength, as described in SHERWIN & CHAPPLE (1968), has
been applied. Furthermore, analytical folding solutions
(e.g. JOHNSON & FLETCHER, 1994) and numerical simula-
229
Fig. 4 - The strain map with A, H and λ being the amplitude, thickness and wavelength, respectively. Solid lines with numbers indicate
strain estimates in per cent and dashed lines with numbers indicate
estimates of effective viscosity ratio. The circles represent the values
corresponding to the five individual folds identified in fig. 2. The
average of the five values is marked with a plus symbol.
– Mappa della distorsione con A, H e λ che corrisondono ripettivamente
ad ampiezza, spessore e lunghezza d’onda. Le linee continue con i numeri indicano le percentuali stimate di distorsione e le linee tratteggiate e
numerate indicano le stime del rapporto di viscosità effettiva. I cerchi
rappresentano i valori che corrispondono alle cinque singole pieghe individuate in fig. 2. La media dei cinque valori è rapprentata da una croce.
tions of folding show that values of the maximal growth
rate should be greater than about 10, in order to generate
observable buckle-folds with a more or less constant layer
thickness. The analytical solution shows that values of the
power-law exponent between 1.8 and 5 and viscosity
ratios between 20 and 70 yield values for the ratio of
dominant wavelength to thickness between 8 and 9 and
values of the maximal growth rate larger than 10 (gray
patch in fig. 3).
Additionally, the values of the amplitude, A, wavelength, λ (i.e. horizontal hinge distance), and thickness,
H, of the five individual folds shown in fig. 2 have been
measured and the corresponding ratios H/λ and A/λ plotted on the strain map developed by SCHMALHOLZ & PODLADCHIKOV (2001). This strain map can be used to estimate the amount of shortening and the viscosity ratio
from buckle-fold geometries and includes a correction for
the shortening of λ and thickening of H during folding.
The individual values are distributed and the average
value for H/λ and A/λ is close to the dashed line for a viscosity ratio of 50 (plus symbol in fig. 4). This is in agreement with the viscosity ratio estimates using the analytical solution of FLETCHER (1974), with values between 20
and 70 (fig. 3).
Fig. 2 - The upper figure shows the top, bottom and averaged fold interfaces. X and Y are the horizontal and vertical coordinates and Tav is
the average layer thickness. Diamond symbols indicate the location of potential hinge points at which the slope of the interface changes its
sign. Different grey levels (colors in the colored version) separate five individual folds. The lower figure shows the Fourier spectra of the top,
bottom and averaged interface of the fold train.
– L’immagine superiore mostra tetto, letto e la superficie media piegate. X e Y sono rispettivamente la coordinata orizzontale e verticale e Tav è
lo spessore medio del livello. I rombi indicano la posizione dei potenziali punti di cerniera ai quali la pendenza dell’interfaccia cambia di segno.
I differenti toni di grigio (colori nella versione a colori) separano cinque singole pieghe. L’immagine inferiore mostra gli spettri di Fourier di tetto,
letto e superficie media del treno di pieghe.
230
S.M. SCHMALHOLZ
DISCUSSION AND CONCLUSIONS
The analytical solution for folding of a power-law
layer embedded in a Newtonian matrix (fig. 3, FLETCHER,
1974) shows that there is no unique solution for the dominant wavelength because it depends on both the viscosity
ratio and the power-law exponent. The range of possible
solutions can be reduced by using the fact that fold
growth rates should be at least ten times larger than the
shortening rate in order to generate observable folds with
more or less constant layer thickness. Smaller growth
rates produce fold shapes with strongly varying layer
thickness due to deformation that is significantly affected
by layer thickening (e.g. JOHNSON & FLETCHER, 1994).
Potentially the best method to quantify a fold shape
is by calculating the Fourier spectrum of the averaged
interface of the folded layer, because no interpretations
and decisions concerning the positions of fold hinges
have to be made. However, several individual folds
should be present within a fold train to yield a representative Fourier spectrum. Also, if the folds have overturned limbs, i.e. more than one vertical coordinate corresponds to one horizontal coordinate, then it is not
possible to calculate a Fourier spectrum. In such cases,
the more interpretative method of defining fold hinge
positions has to be used.
The analysis presented here shows that effective viscosity ratios and power-law exponents can be estimated
from buckle-fold geometries, but the error range is
roughly on the order of a factor of 2 (i.e. between 25 and
100 for an estimate of 50). Better accuracy is difficult to
obtain due to (i) the natural irregularities of fold shapes
that are considerably affected by the initial perturbation
geometry of the layer (e.g. MANCKTELOW, 1999, 2001) and
(ii) by the non-uniqueness of the buckling process, which
means that different combinations of material parameters can generate the same fold shape with a particular
dominant wavelength.
& N.S. MANCKTELOW
The estimates for the effective viscosity ratio between
20 and 70 and for the power-law exponent of the layer
between 1.8 and 5 are well within the range of experimentally confirmed values for mechanically strong quartz
within mechanically weak shale (e.g. CARTER & TSENN,
1984). Additional constraints on the rheology may be
obtained from microstructural observations of the folded
vein, but such observations were not available for the
investigated fold.
REFERENCES
BIOT M.A. (1961) - Theory of folding of stratified viscoelastic media
and its implications in tectonics and orogenesis. Geological
Society of America Bulletin, 72, 1595-1620.
CARTER N.L. & TSENN M.C. (1987) - Flow properties of continental
lithosphere. Tectonophysics, 136, 27-63.
FLETCHER R.C. (1974) - Wavelength selection in the folding of a single
layer with power-law rheology. Am. Jour. Sci., 274, 1029-1043.
JOHNSON A.M. & FLETCHER R.C. (1994) - Folding of viscous layers.
Columbia University Press, New York.
MANCKTELOW N.S. (1999) - Finite-element modelling of single-layer
folding in elasto-viscous materials: the effect of initial perturbation geometry. Journal of Structural Geology, 21, 161-177.
MANCKTELOW N.S. (2001) - Single layer folds developed from initial
random perturbations: the effects of probability distribution, fractal dimension, phase and amplitude. In: H.A. Koyi & N.S.
Mancktelow (Eds.), Tectonic Modeling: A Volume in Honor of
Hans Ramberg. Geol. Soc. of Am., Boulder, pp. 69-87.
SCHMALHOLZ S.M. & PODLADCHIKOV Y. (1999) - Buckling versus
folding: Importance of viscoelasticity. Geophysical Research
Letters, 26, 2641-2644.
SCHMALHOLZ S.M. & PODLADCHIKOV Y.Y. (2001) - Strain and competence contrast estimation from fold shape. Tectonophysics, 340,
195-213.
SHERWIN J. & CHAPPLE W.M. (1968) - Wavelengths of single layer
folds: a comparison between theory and observation. Am. Jour.
Sci., 266, 167-179.
VERMEER P.A. & DE BORST R. (1984) - Non-associated plasticity for
soils, concrete and rock. Heron, 29, 1-64.
Received 8 November 2007; revised version accepted 28 February 2008.
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 231-234, 2 figs.
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Shallow earth rheology from glacial isostatic
adjustment constrained by GOCE
L.L.A. VERMEERSEN (*) & H.H.A. SCHOTMAN (**), (*)
ABSTRACT
The Earth’s asthenosphere and lower continental crust can
regionally have viscosities that are one to several orders of magnitude smaller than typical mantle viscosities. As a consequence,
such shallow low-viscosity layers could induce high-harmonic
(spherical harmonics 50-200) gravity and geoid anomalies due to
remaining isostasy deviations following Late-Pleistocene glacial
isostatic adjustment (GIA). Such high-harmonic geoid and gravity
signatures would depend also on the detailed ice and meltwater
loading distribution and history. ESA’s GOCE satellite mission,
scheduled for launch summer 2008, is designed to map the quasistatic geoid with centimeter accuracy and gravity anomalies with
milligal accuracy at a resolution of 100 kilometers or better. This
might offer the possibility of detecting gravity and geoid effects of
low-viscosity shallow earth layers and differences of the effects of
various Pleistocene ice decay scenarios. For example, our predictions show that for a typical low-viscosity crustal zone GOCE
should be able to discern differences between ice-load histories
down to length scales of about 150 km. One of the major challenges in interpreting such high-harmonic, regional-scale, geoid
signatures in GOCE solutions will be to discriminate GIA-signatures from various other solid-earth contributions. It might be of
help here that the high-harmonic geoid and gravity signatures
form quite characteristic 2-D patterns, depending on both ice load
and low-viscosity zone model patterns.
KEY WORDS: Glacial Isostatic Adjustment, Crust and Mantle
Rheology, GOCE, Gravity Anomalies and Geoid.
RIASSUNTO
Reologia superficiale della terra dall’aggiustamento isostatico glaciale sulla base di GOCE.
L’astenosfera terrestre e la crosta continentale inferiore possono
avere regionalmente una viscosità da uno ad alcuni ordini di grandezza inferiore alla viscosità tipica del mantello. Conseguentemente,
tali strati superficiali a bassa viscosità possono indurre anomalie di
gravità e del geoide alle armoniche elevate (armoniche sferiche da 20
a 200) a causa della deviazione isostatica residua, conseguente all’aggiustamento isostatico glaciale tardo-pleistocenico (GIA). Tali segnali ad elevate armoniche nella gravità e nel geoide dipenderebbero anche dalla distribuzione e storia del carico di ghiaccio e dell’acqua di
fusione. La missione satellitare ESA GOCE, programmata per essere
lanciata nell’estate 2008, è designata a costruire mappe del geoide
quasi statico con un’accuratezza del centimetro e le anomalie di gravità con un’accuratezza del milligal, alla risoluzione di 100 km o migliore. Ciò potrebbe offrire la possibilità di rivelare effetti sulla gravità e sul geoide dovuti agli strati superficiali a bassa viscosità e
differenze degli effetti di vari scenari di riduzione del ghiaccio pleistocenico. Per esempio, la nostra previsione mostra che per una tipi-
(*) DEOS, Faculty of Aerospace Engineering, Delft University
of Technology, Kluyverweg, 1 - 2629 HS Delft, The Netherlands.
(**) SRON, Sorbonnelaan, 2 - 3584 CA Utrecht, The Netherlands.
ca zona crostale a bassa viscosità GOCE sarebbe in grado di distinguere differenze fra storie di carico di ghiacciai fino a scale di lunghezza di 150 km circa. Una delle sfide principali nell’interpretazione di tale segnale del geoide ad armoniche elevate e a scala regionale
sarà di distinguere il contributo del GIA da vari altri contributi della
Terra Solida. Sarebbe d’aiuto che i segnali di gravità e del geoide ad
armoniche elevate avessero delle configurazioni caratteristiche bi-dimensionali, che dipendono sia dal carico del ghiaccio che dal modello della zona a bassa viscosità.
TERMINI CHIAVE: aggiustamento isostatico glaciale, reologia
della crosta e del mantello, GOCE, anomalie di gravità e
geoide.
INTRODUCTION
In Summer 2008 ESA’s Gravity and steady-state
Ocean Circulation Explorer (GOCE) satellite will be
launched. GOCE will observe the Earth’s gravity field
with unprecedented resolution down to 100 km and
accuracies down to 1-2 cm in geoid height and down to
1 mgal in gravity anomaly (e.g., VISSER et alii, 2002).
Such high resolution and accuracies, with almost uniform coverage, have some interesting prospects for the
solid-earth sciences, notably for the shallow parts of the
Earth. One example of this is glacial isostatic adjustment (GIA). In earlier studies (VERMEERSEN 2003; VAN
DER WAL et alii, 2004; SCHOTMAN & VERMEERSEN, 2005;
SCHOTMAN et alii, 2007a) we have shown that crustal
and asthenospheric low-viscosity zones can induce geoid
and gravity signatures that are above accuracy and resolution thresholds of expected GOCE performance. Here
we will concentrate on the question whether it might be
possible to discern the effects of lateral variations in
earth structure, including regional crustal and asthenospheric low-viscosity zones. Continental crust can have
zones of low viscosity for regions with a larger than
average heat flow. Generally such areas can be expected
to occur in regions that are under extension. In order to
give an impression of the perturbations that such lowviscosity zones can give on present-day GIA-induced
geoid anomalies we model resulting GIA geoid anomalies over the northern part of Europe for a specific
laterally varying earth model and a Late-Pleistocene ice
mass decay scenario.
EARTH AND ICE MODEL
In order to compare the effects of a laterally varying
crustal low-viscosity zone with those obtained from a lat-
232
L.L.A. VERMEERSEN
& H.H.A. SCHOTMAN
Fig. 1 - Spherical earth model.
– Modello di Terra sferica.
Fig. 2 - a) Difference in geoid anomalies triggered by the standard earth model of fig. 1, that has no low-viscous lower crust, and the earth
model that has the low-viscous lower crust for Northern Europe; b) Difference in geoid anomalies with respect to fig. 2a, assuming that the
Baltic Shield is not underlain by the low-viscosity zone of the inset in fig. 1.
– a) Differenza nelle anomalie del geoide indotte dal modello di Terra standard mostrato in fig. 1, senza crosta inferiore a bassa viscosità, e il
modello di Terra che ha una crosta inferiore a bassa viscosità nel Nord Europa; b) Differenza nelle anomalie del geoide rispetto alla fig. 2a,
assunto che sotto lo Scudo Baltico non ci sia la zona a bassa viscosità indicata in fig. 1.
SHALLOW EARTH RHEOLOGY FROM GLACIAL ISOSTATIC ADJUSTMENT CONSTRAINED BY GOCE
erally homogeneous one, we use the earth model as
depicted in fig. 1 as standard model. This model consists
of an inviscid core, viscoelastic lower and upper mantle
and elastic upper part. In this elastic upper part a low-viscosity lower crust is sandwiched between the elastic
upper crust and the elastic lithosphere below. Here we
will not model the indicated asthenosphere explicitly, but
consider it to be part of the upper mantle with the same
viscosity as the upper mantle. Results from asthenospheric low-viscosity zones can be found in SCHOTMAN et
alii (2008). Elastic parameters and radial density profile
are based on PREM (DZIEWONSKI & ANDERSON, 1981).
For laterally homogeneous, self-gravitating, spherical
earth models we use the normal mode technique as
described in SABADINI & VERMEERSEN (2004). The rheology is a simple linear Maxwell viscoelastic one. Lower
mantle viscosity is five times the Haskell value, while
upper mantle viscosity is half the Haskell value. Values
for other parameters and variables are indicated in the
figure. Computations with the laterally varying earth
model are performed by means of the finite-element
package ABAQUS (e.g., WU et alii, 2005). The earth model
we use is a viscoelastic halfspace model with the same
layering as the laterally homogeneous spherical earth
model, but it is not self-gravitating. However, it has been
shown in, e.g., SCHOTMAN et alii (2008) that the lack of
self-gravitation is partly compensated by the lack of
sphericity. Furthermore, long-wavelength differences
largely cancel out for the small-scale perturbation signatures related to the shallow crustal low-viscosity zone that
we are interested in here. A validation of using finite elements for computing geoid height perturbations can be
found in SCHOTMAN et alii (2008). It is assumed that the
complete region has the same homogeneous lithospheric
thickness as in the standard earth model in these finiteelement computations. For modeling results with varying
thicknesses we refer again to SCHOTMAN et alii (2008).
The Late-Pleistocene ice mass decay model is based on
ICE-5G of PELTIER (2004), although we have also considered other ice models like RSES of LAMBECK et alii
(1998). It turned out, however, that the background ice
decay model has a negligible influence on the spectral
characteristics associated with the crustal low-viscosity
zone contributions to geoid and gravity anomalies
(SCHOTMAN et alii, 2008). Spatial patterns of these geoid
and gravity anomalies can differ considerably, of course,
as individual ice sheets from various ice models can differ
in position.
MODELLING RESULTS
Fig. 2a shows the difference in geoid anomalies triggered by the standard earth model of fig. 1 that has no
low-viscous lower crust and the earth model that has the
low-viscous lower crust for Northern Europe. Here the
low-viscosity zone is taken as a laterally homogeneous
layer, so also the Baltic Shield is (unrealistically) presumed to have this low-viscosity crustal layer. The geoid
height perturbations are clearly above the expected
accuracy level of 1 cm of maps that GOCE will deliver,
at many places the differential signal will even be more
than an order of magnitude larger than this 1 cm level.
The contours with number «2» or «–2» signify differences between modelling results obtained with the
233
(spherical earth, self-gravitating) normal mode method
and the (halfspace, non-self-gravitating) finite element
model. The numbers are the difference in cm between
the two modelling results, showing that the modeling
results differences between those obtained by the normal mode approach and the finite element model are
only slightly larger than the expected uncertainties in
the GOCE data. This result illustrates what has already
been mentioned in the former section: effects of selfgravitation and sphericity partly annihilate one another,
specifically for small-scale signatures in geoid anomaly.
Fig. 2b shows the effects of lateral heterogeneities.
Depicted is the difference in geoid anomalies with
respect to fig. 2a assuming that the Baltic Shield is now
not underlain by the low-viscosity zone of the inset in
fig. 1. It is obvious from fig. 2b that differences between
the lateral and non-lateral results are generally small
outside the Baltic Shield area and become most prominent prominent underneath those regions (i.e., the
Baltic Shield) that do not have the crustal low-viscosity
zone any longer. The differences are large enough compared to the expected performance of GOCE, even up to
one order of magnitude, that the effects of lateral variations on the geoid might become discernable in GOCE
data. Also the resolution should not be a problem: most
of the patches in figs. 2a and 2b extend over more than
100 km up to even hundreds of km for the more elongated structures. These spatial geoid anomaly patterns,
apart from spectral signatures, might help in identifying
GIA-induced contributions coming from the effects of
lateral, low-viscous, shallow crustal zones and discern
them from other contributions like internal mass anomalies and topographic features from tectonic or geomorphological origins. Finally, it should be emphasized that
the modelling results of figs. 2a and 2b are only meant
to give an indication about what might possibly be
deduced from GOCE data concerning (crustal) lateral
variations and shallow low viscosity zones. More
detailed earth models, based on detailed structural,
compositional and rheological (also non-linear) data,
seismic tomography, electrical conductivity studies, etc.,
are necessary before realistic comparisons can be made
with data from GOCE. For further details we refer to
SCHOTMAN et alii (2008).
CONCLUSIONS
GIA model simulations indicate that information on
shallow low viscosity zones might be deduced from
GOCE geoid solutions, although uncertainties in both
ice load history and earth structure could hamper
unique interpretations to some extent. Combining spectral information with spatial patterns could reduce these
uncertainties, whereby the range of possible ice and
earth models is already constrained through other geodetic and geophysical data (e.g., GPS, ice load dynamics,
tide gauge records). Lateral variations in earth structure, specifically with respect to occurrence of low-viscosity zones as a function of tectonic province, do have
discernable effects on geoid anomalies, although they
appear to be constrained to those regions that do not
have a low-viscosity zone (compared to the laterally
homogeneous low-viscosity zone case) and to their
immediate surroundings.
234
L.L.A. VERMEERSEN
REFERENCES
DZIEWONSKI A.M. & ANDERSON D.L. (1981) - Preliminary reference
Earth model (PREM). Phys. Earth Planet. Inter., 25, 297-356.
LAMBECK K., SMITHER C. & JOHNSTON P. (1998) - Sea-level change,
glacial rebound and mantle viscosity of northern Europe.
Geophys. J. Int., 134, 102-144.
PELTIER W.R. (2004) - Global glacial isostasy and the surface of the iceage Earth: The ICE-5G (VM2) Model and GRACE. Annu. Rev. Earth
Planet. Sci., 32, doi:10.1146/annurev.earth.32.082503.144359.
SABADINI R. & VERMEERSEN L.L.A. (2004) - Global Dynamics of the
Earth: Applications of Normal Mode Relaxation Theory to SolidEarth Geophysics. Modern Approaches in Geophysics Series,
Volume 20, Kluwer Academic Publishers, Dordrecht, The
Netherlands.
SCHOTMAN H.H.A. & VERMEERSEN L.L.A. (2005) - Sensitivity of glacial isostatic adjustment models with shallow low-viscosity earth
layers to the ice-load history in relation to the performance of
GOCE and GRACE. Earth Planet. Sci. Lett., 236, 828-844.
SCHOTMAN H.H.A, VERMEERSEN L.L.A. & VISSER P.N.A.M. (2007a) High-harmonic gravity signatures related to postglacial rebound. In: Dynamic Planet, P. Tregoning and C. Rizos (Editors),
& H.H.A. SCHOTMAN
Springer, International Association of Geodesy Symposia,
130, 103-111.
SCHOTMAN H.H.A., WU P. & VERMEERSEN L.L.A. (2008) - Regional perturbations in a global background model of glacial isostasy. submitted to Phys. Earth Planet. Inter., doi: 10.1016/j.pepi.2008.02.010.
VAN DER WAL W., SCHOTMAN H.H.A. & VERMEERSEN L.L.A. (2004) Geoid heights due to a crustal low viscosity zone in glacial isostatic adjustment modeling: a sensitivity analysis for GOCE.
Geophys. Res. Lett., 31, L05608, doi:10.1029/2003GL019139.
VERMEERSEN L.L.A. (2003) - The potential of GOCE in constraining
the structure of the crust and lithosphere from post-glacial rebound. Space Sci. Rev., 108 (1-2), 105-113.
VISSER P.N.A.M., RUMMEL R., BALMINO G., SUENKL H., JOHANNESSEN J., AGUIRRE M., WOODWORTH P.L., LE PROVOST C.,
TSCHERNING C.C. & SABADINI R. (2002) - The European Earth
explorer mission GOCE: Impact for the geosciences. In: Ice
Sheets, Sea Level and the Dynamic Earth, Mitrovica J.X. & Vermeersen L.L.A. (Editors), Am. Geophys. Union, AGU Geodynamics Series, 29, 95-107.
WU P., WANG H. & SCHOTMAN H.H.A. (2005) - Postglacial induced
surface motions, sea-levels and geoid rates on a spherical, self-gravitating, laterally heterogeneous Earth. J. Geodyn., 39, 127-142.
Received 27 November 2007; revised version accepted 28 January 2008.
Boll.Soc.Geol.It. (Ital.J.Geosci.), Vol. 127, No. 2 (2008), pp. 235-242, 3 figs.
Queste bozze, corrette e accompagnate dall’allegato preventivo firmato e dal buono d’ordine,
debbono essere restituite immediatamente alla
Segreteria della Società Geologica Italiana
c/o Dipartimento di Scienze della Terra
Piazzale Aldo Moro, 5 – 00185 ROMA
Instabilities development in partially molten rocks
JEAN-LOUIS VIGNERESSE (*), JEAN-PIERRE BURG (**) & JEAN-FRANÇOIS MOYEN (***)
ABSTRACT
Partially molten rocks (PMR) are characterized by specific and
contrasting behaviours. For instance, large-scale and smaller scale
structures are consistently oriented in a migmatitic body with those
of the surroundings, indicating that the migmatites were deformed
as a whole. By contrast, ubiquitous strain partitioning and melt distribution are widely present in the same migmatitic body, reflecting
highly heterogeneous strain and intrinsic rheological instabilities. A
continuous transition from a liquid-like to a solid-like rheology, as
many averaging processes implicitly assume, cannot explain this
two-fold information. We develop a full analysis, considering the
stress and strain rate, and the relative proportion of melt and solid
phases. Temperature varies from Tsolidus to Tliquidus in a PMR. We
also assume that the transition to melting is not dual to crystallization. However, we prefer using the viscosity rather than the stress,
since the former is better constrained from experiments. The viscosity of the matrix, which deforms according to a power law, shows
shear thinning, whereas that of the melt remains constant. The viscosity contrast between the two phases thus varies with strain rate.
The lower the strain rate, the higher is the viscosity contrast, hence
instabilities development is controlled by the rheology. The path followed during a transition also controls the intermediate state, and
may lead to instabilities, resulting from mechanical reasons or from
the respective amount in each phase. In the last case, the concentration in one phase induces instabilities. A surface describing viscosity
in a 3D diagram (strain rate-amount of phase-viscosity) is constructed, that presents a cusp shape for low strain rates. The diagram depicts two types of behaviour and a critical state. At high
strain, the viscosity contrast between melt and matrix is lowest. The
rock behaves as a near-homogeneous body and a continuous
description of its rheology may be estimated. Instabilities lead to
fabric development resulting from crystals alignment. At low strain
rate, three domains are separated by a critical state. When the proportion of one phase is very small, the material behaves as the other
end-member. For intermediate proportions, the cusp indicates three
possible viscosity values. Two are metastable, whereas the third is
virtual. Hence, the viscosity of the mixture jumps back and forth
from the viscosity of one phase to that of the other. A similar process
occurs for temperature, since the cusp in the viscosity profile has
also implications in a diagram linking temperature and stress. Different behaviours result, depending on whether the deformation
takes place under a fixed content in each phase, a common stress, a
common strain rate or common temperature. We list several implications for partially molten rocks that may explain fabric development, contact melting between crystals, strain localisation, mineral
banding, shear heating, welding, stick-slip-like melt extraction,
magma fragmentation or formation of strong or fragile glass. A
phase diagram that incorporates temperature, stress and concentration is constructed for PMR that bears much similitude with those
issued for other soft materials.
KEY WORDS: rheology, two-phase material, migmatites.
(*) Nancy-Université, G2R, BP 23, F-54501 Vandoeuvre Cedex,
France, [email protected]
(**) Geologisches Institut, ETH-Zentrum, Leonhardstrasse 19
LEB, CH-8006 Zurich, Switzerland, [email protected]
(***) Department of Geology, Stellenbosch University, Private
Bag X1, Stellenbosch 7802, South Africa, [email protected]
RIASSUNTO
Sviluppo di instabilità in rocce parzialmente fuse.
Le rocce parzialmente fuse (PMR) sono caratterizzate da
comportamenti specifici e contrastati. Ad esempio in un corpo di
migmatiti le strutture a grande e piccola scala sono orientate coerentemente con quelle delle rocce circostanti e ciò indica che le
migmatiti sono state deformate come un unico insieme. Al contrario, le ubiquitarie ripartizione della distorsione e distribuzione del
fuso sono diffuse nello stesso corpo migmatitico e riflettono l’elevata eterogeneità della distorsione e delle instabilità reologiche intrinseche. Una transizione continua da una reologia di tipo-liquido
a una reologia di tipo-solido, così come implicitamente si assume
per molti processi mediati, non può spiegare questa duplice informazione. Sviluppiamo qui un’analisi completa considerando lo
sforzo e la velocità di deformazione e le relative proporzioni di
fuso e fasi solide. In una PMR la temperatura varia da Tsolidus a
Tliquidus. Noi assumiamo anche che la transizione verso la fusione
non riproduce quella alla cristallizzazione. Comunque noi preferiamo usare la viscosità anziché lo sforzo, poiché la prima è definita meglio dagli esperimenti. La viscosità della matrice che si
deforma secondo una legge esponenziale manifesta un assottigliamento per taglio, mentre quella del fuso rimane costante. Il contrasto di viscosità tra le due fasi varia quindi con la velocità della
distorsione. Più bassa è la velocità di distorsione, più si eleva il
contrasto di viscosità, quindi lo sviluppo delle instabilità è controllato dalla reologia. Il percorso seguito durante una transizione
controlla pure lo stato intermedio e può portare all’instabilità,
come risultato di cause meccaniche oppure di diversa quantità relativa delle fasi. Nell’ultimo caso, la concentrazione di una delle
fasi induce l’instabilità. Viene qui costruita una superficie che descrive la viscosità in un diagramma tridimensionale (velocità di
distorsione-quantità della fase-viscosità) e che presenta una forma
a cuspide a basse velocità di distorsione. Sul diagramma sono rappresentati due tipi di comportamento e uno stato critico. Ad alta
distorsione, il contrasto di viscosità tra fuso e matrice è più basso.
La roccia si comporta come un corpo quasi omogeneo e può essere approssimata una descrizione continua della sua reologia. Le
instabilità portano allo sviluppo di un fabric che risulta dall’allineamento dei cristalli. A bassa velocità di distorsione, tre domini
sono separati da uno stato critico. Quando la proporzione di una
fase è molto piccola il materiale si comporta come l’altra fase. Per
proporzioni intermedie, la cuspide indica tre possibili valori della
viscosità. Due sono metastabili mentre il terzo è virtuale. Quindi,
la viscosità della miscela retrocede o avanza dalla viscosità di una
fase a quella dell’altra. Un simile processo si verifica per la temperatura, poiché la cuspide nel profilo di viscosità manifesta anche
implicazioni in un diagramma che collega la temperatura e gli
sforzi. Ne risultano differenti comportamenti, a seconda che la
deformazione si sviluppi a proporzione delle fasi fissa, stesso stato
di sforzi, stessa velocità di distorsione o stessa temperatura. Si
propone la lista delle numerose implicazioni per le rocce parzialmente fuse che possono spiegare lo sviluppo del fabric, la fusione
ai margini dei cristalli, la localizzazione della distorsione, l’alternanza di composizione mineralogica, il riscaldamento per shear,
la risaldatura dei granuli, estrazione del fuso per scivolamento e
bloccaggio (stick-slip), la frammentazione del magma o la formazione di vetro resistente o fragile. Si presenta qui un diagramma
di fase per PMR che incorpora temperatura, sforzo e concentrazione che possiede un grande somiglianza con quelli noti per altri
materiali deboli.
TERMINI CHIAVE: reologia, materiali bifasici, migmatiti.
236
J.-L. VIGNERESSE ET ALII
Most materials that constitute our direct environment
are composed of several phases that all behave differently
when submitted to stress. Rheology and continuum
mechanics are usually the field of investigation for such
behaviour. However, the basics hypotheses assume that
the material presents continuous, or not too contrasted,
properties between phases. Thus can be the case in solid
rocks, where the minerals react similarly to a bulk stress.
It is no more the case when one phase is solid, or highly
viscous, and when the other phase is a liquid or a gas. For
instance, sand usually flows under the wind, resulting in
booming dunes whereas those keep a bulk pile shape.
Conversely, mud saturated with water flows and spreads.
Lavas adopt a similar behaviour. When a high temperature, they flow over kilometres, without any real structural control, except those imposed by the surrounding
topography. Conversely, internal structures develop that
can be used to infer flow directions and internal stress
pattern. Such situations are hard to describe with usual
methods, and any kind of averaging from the laws governing the end-members usually fail to describe instabilities that soon develop. Those are specifically observed
into partially molten rocks, here referred to as PMR.
The present paper started from field observations on
migmatites and crystallizing magmas. Migmatites are
rocks that were partially molten rocks before they crystallized in their actual state (MENHERT, 1968; ASHWORTH,
1985). In contrast, fabrics in magma record the shear
flow of the melt during emplacement (PATERSON et alii,
1998). PMRs can accordingly be envisioned as two-phase
materials. One phase is solid (the rock that melts or crystals in magma); it is hereafter referred to as matrix. Melt
is the other phase, here, essentially referring to felsic
melts, though general term of granitic melt should not be
restricted to any specific composition.
We develop a description of the PMR rheology that
takes into consideration.
RICHET, 2005), granite rheology (PETFORD, 2003),
pastes (COUSSOT, 2007), polymers (DE GENNES, 1979),
foam (KRAYNIK, 1979), dense suspensions (STICKEL &
POWEL, 2005), analogue deformation (ROSENBERG,
2001), friction (PERSSON, 2000), granular flow (JAEGER
et alii, 1996) and wet granular flow (MITARAI & NORI,
2006). Previously, we focused on identifying:
1) The amount of the solid phase (Φ), ranging from 0
to 1. It is similar to a volume.
2) The intrinsic viscosity h of each phase, intimately
linked with the strain rate (γ°).
3) The applied stress (σ).
4) The temperature (T) interval between solidus and
liquidus.
RHEOLOGY OF THE TWO END-MEMBERS OF A PMR
The choice of the viscosity is for convenience, because
it is better constrained by experiments than stress or
strain rate. In consequence, after selection of stress as the
intrinsic variable, a full description could be represented
into a 3D diagram with coordinates stress, temperature
and volume.
The constraints taken into account relate to field
observations. They consist in:
1) The changing viscosity contrast with strain rate.
2) The non-linear aspect of melting rate.
3) The different evolution of viscosity with temperature for melt and matrix
4) The difference between melting and crystallization.
5) The bulk motion «en bloc» at the scale of a magmatic body and the small-scale heterogeneous motion
with instabilities.
The present paper combines information about
parameters identified in previous studies with important review papers about silicate melts (MYSEN &
1) The evidence of two thresholds during melting and
crystallization (VIGNERESSE et alii, 1996).
2) The non-duality between melting and crystallization (VIGNERESSE et alii, 1996).
3) The importance of strain partitioning between
phases (VIGNERESSE & TIKOFF, 1999).
4) The non-linear behaviour of the melting rate and
melt distribution (BURG & VIGNERESSE, 2002).
5) The rheological contrast between melt and matrix
(BURG & VIGNERESSE, 2002).
6) The presentation and solution of a double system
of equations for melt extraction (RABINOWICZ & VIGNERESSE, 2005).
7) The necessity of including pure and simple shear
for melt extraction (RABINOWICZ & VIGNERESSE, 2004;
VIGNERESSE & BURG, 2005).
8) The discontinuous melt extraction rate (VIGNERESSE & BURG, 2005, RABINOWICZ & VIGNERESSE, 2004).
9) The cusped shape of the viscosity as a function of
strain rate (VIGNERESSE & BURG, 2004).
10) The discontinuities the cusp shape induces on a
stress-phase diagram (VIGNERESSE et alii, 2007).
11) The role of nonlinear melting in the melt production,
i.e. on the phase proportion (VIGNERESSE et alii, 2007).
12) The importance of mapping those parameters for
identifying instabilities development (VIGNERESSE et alii,
2007).
Rheology commonly describes the relation between
shear stress (σ) and shear strain (γ), whereas time dependent effects imply a strain rate (γ°) response to stress. We
use a shear strain rather than a plane strain (ε) since most
magmatic flows develop under shear.
Melt and its matrix are the two end-members of the
system. The melt behaves as a Newtonian body for moderate to low strain rates. A constant viscosity relates linearly strain rate to stress. Within the temperature range
of melting (650-900°C), calc-alkaline granitic melts present viscosity value around 106 Pa.s (CLEMENS & PETFORD, 1999). It exponentially decreases with temperature,
in function of the activation energy E, with a typical value
about 300 kJ/mole (MAALØE, 1985). Around 800°C, viscosity decreases by 2.5-3.0 orders of magnitude for an
increase of 100°C.
In contrast, crustal rocks brought at the same temperature range (650-900°C) deform in a ductile manner. We
adopt the case of dislocation creep of a single crystal,
with a power law exponent of 3 (NICOLAS & POIRIER,
1976). Experimentally obtained values for amphibolites,
with values log A = –4.9 and Q = 243 kJ/mole (KIRBY &
KRONENBERG, 1987), are used as a proxy for the restitic
matrix of PMR, yielding a melt of granitic composition.
The effective viscosity is estimated from the local tangent
to the stress-strain rate curve.
INSTABILITIES DEVELOPMENT IN PARTIALLY MOLTEN ROCKS
237
Under those assumptions, the preceding numerical
values provides the equations for the melt
log η = 6
(1)
log η = 10.66 – 2/3 log γ°
(2)
and for the matrix
The rheology of mixed melt and matrix (PMR) cannot
be simply defined as the combination of those two endmembers, depending on their relative proportion (fig. 1).
During crystallisation, the solid particles interact which
each other, leading to the Einstein-Roscoe law (EINSTEIN,
1906; ROSCOE, 1952; ARZI, 1978):
η = η0 (1 – Φ/Φmax)-ne
(3)
in which η0 is the initial melt viscosity, Φmax is the maximum packing assemblage, and ne an experimentally
determined coefficient (LEJEUNE & RICHET, 1995). It has
been experimentally validated up to 0.40 of solid phase,
less than maximum packing, about 0.75 (ROGERS et alii,
1994). Particle interactions become important at higher
concentrations, changing the exponent into –ne.Φmax.
This reduces the exponent value from 2.5 to about 1.8
(KRIEGER & DOUGHERTY, 1959). However, the viscosity
increases by 4 to 5 orders of magnitude near maximum
packing. Indeed, the mixture becomes thixotropic
(BARNES, 1997) with departures from non-linearity in
case of crystallization and pseudo-plastic in case of melting. Nevertheless, the viscosity contrast between melt and
matrix ranges from 10 to 14 orders of magnitude (BURG
& VIGNERESSE, 2002) when restricting the stress values
in between 0.1 and 100 MPa.
PMR SPECIFICITIES
A PMR combines three possibilities to develop instabilities. One is mechanical or rheological, owing to the
large viscosity contrast between melt and matrix. The second is driven by the respective amount of each phase. The
third is chemical and relates to temperature, especially
during the interval between melting and crystallization. A
3D diagram combining stress, temperature and the volume of one phase is suggested that would provide a complete mapping of the complex PMR rheology.
However, before constructing this diagram, one should
take into account the specific points that characterize PMR
rheology. Those are the existence of two thresholds during
the transition between the end-members (VIGNERESSE et
alii, 1996), strain partitioning (VIGNERESSE & TIKOFF,
1999) and feedback loops that develop due to nonlinear
processes (BURG & VIGNERESSE, 2002). The link between
the rheology of a strong matrix and that of a concentrated
suspension, drawn from Einstein-Roscoe equation (RENNER et alii, 2000; ROSENBERG, 2001) is seriously questioned since it does not allow any instability to develop
(BURG & VIGNERESSE, 2002).
The range of threshold values for melting and crystallization overlaps. Thus, a definite rheology cannot be
ascertained in that domain, that sees overlapping of two
behaviours, each being related to one end-member.
In addition, this domain, with two metastable states
varies in size depending on the strain rate or stress acting
on the system. Instabilities develop during melting or
crystallization, when the slope of the flow curve relating
Fig. 1 - Log-log stress-strain rate diagram showing the behaviour of
the melt and its matrix. Viscosity values are indicated in grey.
– Diagramma bilogaritmico sforzi-velocità di ditorsione, che mostra il
comportamento del fuso e della sua matrice. I valori di viscosità sono
indicati in grigio.
the transition from one phase to the other has becomes
negative (SPENLEY et alii, 1993). In case of a system under
common stress, fluid decomposes into a layered structure, with alternate layers of high and low strain rate.
Conversely, in case of deformation under common stress,
shear localisation develops (fig. 2).
The bulk rheology of a PMR should be examined in a
3D (σ – γ° – Φ) diagram. However, the pair σ-γ° is poorly
determined from experiments, that often develop under
constant and fast strain rate. Hence, they are limited by
the total duration of the experiments. We prefer adopting
a 3D (η – γ° – Φ) diagram because the pair η-γ° is experimentally constrained.
We start with the state equations for the melt and its
matrix (Eqs. 1 and 2). Owing to large variations in viscosity, the strain rate response to stress plots in a log-log diagram. A line with constant slope represents the melt,
whereas another line represents the matrix. In between,
the Einstein-Roscoe curve is not strain rate dependent.
The two surfaces constructed from the two end-members
overlap over a wide range of Φ (0.50 to 0.75). The connection between the two end-members takes the form of a
cusp surface in the (η – γ° – Φ) diagram.
Temperature has a differential effect on the viscosity
of melt and matrix, resulting from the activation energy
values for those phases. They respectively plot as two
lines with different slope on a semi-log diagram as a function of temperature. The viscosity for the transitional
state must be computed for fixed values of strain rate
from the 3D diagram (η – γ° – Φ).
238
J.-L. VIGNERESSE ET ALII
cusp shape within this range of temperature. Instabilities
may develop depending on the followed path, i.e. constant stress or constant temperature, identically to the
instabilities with strain rate.
Fig. 2 - Instability occurrence depending on whether the path occurs
under a common strain rate (a) and (b), leading to banding, or under
a common stress (c) and (d), leading to strain partitioning.
– Dipendenza dell’instabilità dall’instaurarsi del percorso in condizioni
di velocità di distorsione comune (a) e (b), che genera un’alternanza di
composizione, oppure in condizioni di stress comune (c) e (d) che
genera ripartizione della distorsione.
All parameters are now settled to build a phase diagram that would determine the limits of PMR rheology,
in function of the phase amount, viscosity and temperature. The strain rate should be introduced to determine
the respective occurrence of instabilities. The basic ingredients to construct a 3D diagram (Φ – η – T) are the preceding diagrams (fig. 3). For a better readability, we use
φ = 1 – Φ, the amount of liquid phase, and because it is
better constrained, we use the viscosity instead of stress.
The three axes (φ – η – T) determine the range of
occurrence of PMR. Whereas the amount of melt ranges
from 0 to 1, the temperature ranges from Tsolidus to Tliquidus, and the viscosity, which is plotted in a log scale
ranges from the viscosity of the melt a Tliquidus to the value
for the matrix at Tsolidus. The resulting diagram adopts the
shape of a quarter of quasi-cylindrical body, the concavity
of which faces the origin. This shape results from the two
limiting values in temperature and phase amount, whilst
the concave pattern results from the melting curve. In case
of cusp development, the quasi-cylindrical body also presents a cusped surface, toward the origin.
Such mapping is useful as much as it can prompt the
design for new experiments through predicting the behaviour of a studied system. A first attempt has been to classify the instabilities in a two-phase material according to
the shape of the flow curve. It corresponds to considering
the concentration, spinodal decomposition, or strain rate,
i.e. essentially adopting a mechanical, point of view
(OLMSTED & LU, 1999).
GEOLOGICAL IMPLICATIONS
MAGMA EMPLACEMENT AND STRUCTURES
Fig. 3 - 3D diagram showing the occurrence of a cusp in the PMR
rheology surface, as a function of decreasing viscosity and temperature.
– Diagramma tridimensionale che mostra la comparsa di una cuspide
sulla superficie della reologia PMR in funzione della viscosità decrescente e della temperatura.
Under high strain situation, the transition from the
solid to the weak phase is monotonous, giving place to a
smooth viscosity variation. In contrast, the low strain rate
case has to take into account the cusp that develops in the
(η – Φ) diagram. Cusp occurs in between 30 and 60% of
the solid phase. The transition in viscosity also adopts a
During felsic body emplacement, the strain rate is
commonly higher than 10-12 s-1, implying a stress level
over 10 MPa (HARRIS et alii, 2000; VIGNERESSE, 2005;
HAWKESWORTH et alii, 2004). The viscosity contrast
between melt and matrix is the lowest, thus relaxation
times for both phases have similar amplitude. The bulk
material responds as a single-phase body with a bulk viscosity. Two situations can be observed that relate the
strain rate and the ability of PMR to flow. Migmatitic
bodies present the same structural trends as surrounding
rocks (NZENTI et alii, 1988) documenting «en masse»
deformation of the PMR massif.
Decreasing the strain rate implies increasing the viscosity contrast between melt and matrix. In a PMR, the
rotation of the first formed crystals results in a fabric
(BOUCHEZ, 1997; ARBARET et alii, 2000). When crystals
interactions develop, it can lead to particle segregation,
controlled by the concentration, as it has been described
as Bagnold segregation (BAGNOLD, 1954). Conversely,
when the strain rate locally exceeds the ability of a PMR
to deform viscously, then it breaks into fragments like
during brittle deformation (PAPALE, 1999), as observed
during volcanic eruptions. Experiments on brittle fragmentation of magmatic melts suggest strain rates ranging
from 50 to 150 s-1 (BÜTTNER et alii, 2006).
INSTABILITIES DEVELOPMENT IN PARTIALLY MOLTEN ROCKS
Mineral banding is one way to accommodate velocity
continuity between phases, as described in flowing liquid
crystals (BONN et alii, 1998). It manifests in PMR through
schlieren and melt-rich segregation (CLARKE & CLARKE,
1998; WEINBERG et alii, 2001; CLARKE et alii, 2002). In
this case, it manifests through crystal sorting by size or
by composition. In obsidian, it also takes the form of
alternating bands of different colour some tens of
microns to decimeters in width (SWANSON et alii, 1989;
SMITH, 2002). The occurrence of shear bands due to
strain localisation in plastic material results from deformation concentration on planes. In PMR, the different
viscosity between the two phases leads to strain partitioning (VIGNERESSE & TIKOFF, 1999). The discrete distribution of localised shear zones with only a few cm in width
profoundly differs from the usual observation that ductile
rocks should present diffuse deformation. In crystallizing
magma, strain localisation develops within a non-yet consolidated framework of touching crystals, leading to formation of dilatant proto-faults (GUINEBERTEAU et alii,
1989; PONS et alii, 1995; SMITH, 2000).
MELT SEGREGATION
Melt segregation at incipient melting results when
both pure and shear stress apply on a PMR (RABINOWICZ
& VIGNERESSE, 2004). A compaction length describes the
resulting space and time discontinuities. Melt-rich bands
form at low angles (within 20°C) when observed on analogue material (ROSENBERG & HANDY, 2000; BARRAUD et
alii, 2004) and natural samples (KATZ et alii, 2006). They
occur both during partial melting (MARCHILDON &
BROWN, 2002) and crystallization (GOURLAY & DAHLE,
2007). Instabilities in time result in cyclic periods of segregation, driven by the amount of melt (RABINOWICZ &
VIGNERESSE, 2004; VIGNERESSE & BURG, 2005).
Grain boundaries wetting by incipient melt is due to
progressive depinning of the melt along the boundary surface, bearing relation to stick-slip motion observed during
friction. Sliding motion is discontinuous and depends on
the differential velocity between the two surfaces in contact leading to stick-slip motion (SCHOLZ, 1990; THOMPSON & ROBBINS, 1990). It results from a competition
between nucleation and growth rate of the pinning zones
on one hand and the sliding velocity on the other hand.
Indeed, stick-slip vanishes as the velocity overcomes a
critical value, just because pinning has no more chance to
develop.
At the end of crystallisation, the high proportion of
the solid phase drastically reduces the melt mobility, isolating small-scale closed systems. The strain rate variation within the solid phase is analogue to pressure dissolution, resulting in important stress gradient between
touching crystals. The gradient relaxes by dissolving one
crystal to the benefit of another one (GRINFELD, 1993),
leading to crystal impingement (MEANS & PARK, 1994;
PARK & MEANS, 1996) in analogue experiments or in natural examples described in a crystallizing gabbro (NICOLAS & ILDEFONSE, 1996; ROSENBERG, 2001). At a larger
scale, similar observations have been realised in metamorphic aureoles induced by granitic intrusions (MARCHILDON & BROWN, 2002).
Sintering and high-pressure aggregation of particles
into a solid bloc is observed in tuff welding (GRUNDER &
RUSSELL, 2005). Competition between compaction and
239
viscous flow results in sintering, adhesion of molten fragments and deformation of glassy clasts (SMITH, 1960).
Superplasticity is been widely observed as related either
to micrograin or microstructural behaviour. It is interpreted as a transition between creep at low stresses and
plastic flow near the yield stress. Viscous heating may
lead to tachylites or pseudo-tachylites formation (SPRAY,
1995).
Dilatancy is a volume expansion in response to an
applied stress, also synonymous with shear thickening,
induced by the increasing viscosity of the crystallising
magma. Nevertheless some dilatant veins also show internal brecciation (SMITH, 1996) indicating that still present
melt overcame the brittle/ductile transition. Dilatant
regions are a sink for the residual melt in a flowing
magma has been widely recognised by a more abundant
glassy material (SMITH, 2000).
MEMORY EFFECTS
Most of the instabilities above described present hysteresis, i.e. memory effect. It means that the transition
from one state to the other is not dual to the reverse transition in terms of energy balance. Hysteresis is commonly
described for induced magnetization (BERTOTTI, 1998),
but also for plastic deformation (PRANDTL, 1928). Indeed,
a plastic body retains some strain (BRIDGMAN, 1950)
when stress returns to its initial state. It profoundly contrasts with elastic deformation during which the strained
body returns to its initial state when the stress is no more
applied.
Hysteresis is the manifestation of stored energy. The
return to initial conditions requires additional forces.
This is the case for plastic deformation, or magnetism
through the magnetic coercive field. In the transition to
melting, the additional energy takes the form of the latent
heat. During crystallisation of viscous material, there is a
continuous reduction in the mobility of elements, manifested by the viscosity increase. Energy is thus continuously released between the liquidus and the solidus, corresponding to the entropy step due to latent heat when
considering the temperature. The correlation between
latent heat and viscosity is linear (GARAI, 2004) for materials that show a good Arrhenius behaviour. This would
correspond to a well-defined heat capacity gap between
the liquid and the solid state, that is, to contrasted values
of entropy of structural configuration (BOTTINGA, 1994).
When this is not the case, as for instance in fragile glass
material, the number of intermediate structural configurations is large allowing intermediate metastable states,
hence departure to Arrhenian behaviour, and nonArrhenian viscosity (ANGELL, 1995) and consequently
larger hysteretic loop. Indeed, hysteretic flow curves have
been observed for non-Newtonian flows (BONN et alii,
1998).
The memory effect or hysteresis in PMR is observed
during successive phases of heating and cooling silicate
melts above their liquidus temperature (YUE, 2004). The
repeated heating and cooling phase lead to a gradual
transition from non-equilibrium to equilibrium states. An
ordered structure is observed up to 70°C above Tliquidus.
The conversion from multi-crystalline phases to a single
phase indicates that the liquid remembers the structures
previously formed (YUE, 2004). Indeed, glassmakers use
240
J.-L. VIGNERESSE ET ALII
cycles of rapid heating and cooling to transform a fragile
crystalline phase into a stronger one (CONRADT, 2004).
In our suggested model, hysteresis should be understood as a dissipation mechanism unable to return to its
initial state without the addition of extra energy. However, repeated cycles of straining could lead to unexpected large strain, especially when the material has not
the time to completely relax and return toward a state
near its initial conditions. This is obviously the case when
seismic waves, which are successive cycles of compression and extension, interact with a two-phase material.
Nonlinear effects develop that indicate no return to initial
conditions before the material is strained again. It usually
leads to soil liquefaction when seismic waves propagate
through saturated sediments (ISHIHARA, 1993).
The preferential reusing of a vein by new magma is
also a sign of hysteresis. Tubes offer a pathway for lava
to flow over large distances (PETERSON et alii, 1994;
CALVARI & PINKERTON, 1994).
Finally, the reusing of already formed plastic shear
zones is also a consequence of the memory effect. Grain
reduction in a shear band or weakened material due to a
former heating are potential sites to localise strain for a
future deformation cycle. In that sense shear heating
(SCHOLZ, 1980) could provide natural conditions for
rapidly deforming magma-present material.
IMPLICATIONS FOR EXPERIMENTAL DEFORMATION
The present paper offers an explanation for the development of much instability observed in natural conditions. However, it should be better regarded as a short
review on the conditions under which those instabilities
are produced. It is a former guide for designing experimental or numerical studies in order to address such
instabilities.
One problem with experiments performed on natural
rocks of analogue materials is the duration of the experiment. It directly points to the effect of strain rate. Adopting the time scale for a one year experiment, a long time
indeed when considering the stability of one experiment,
implies a strain rate of at least 3.2 *10-8 s-1. Each additional order of magnitude implies a factor of 10, that
would results in a maximum strain rate of 10-10 s-1
obtained during a single experimentalist’s life.
One possibility to overpass this difficulty would be to
change the material for some analogue material, resulting
in the application of more reasonable strain rate. Adopting a common value of 10-5 s-1, which is in use in many
experimental press systems, limits in turn the viscosity
contrast between a two-phases material.
The idea of the paper started from a different point of
view. Provided experiments are not able to address the
development of instabilities in terms of viscosity, strain
rate and stress, it should possible to design some specific
experiment that would be designed to address only one
type of instability, depending on the temperature, viscosity and relative percentage of each phase (fig. 3).
CONCLUSIONS
Partially molten rocks (PMR) are commonly described as inhomogeneous, with a locally variable and unpre-
dictable amount of melt. They also show local heterogeneities in strain distribution, with a neat predominance
of non-coaxial deformation and shear. In contrast, at a
large scale, their internal structures are concordant with
those of the surrounding. PMR are by evidence a place
where instabilities develop. Examining the rheology of
PMR, we suggest three types of instabilities, one related
to mechanical reasons, as shear zone localization or stickslip motion, one linked with the concentration of solid
phase, as banding or dilatant zones and a third one linked
to temperature occurs when melting rate overcomes the
rate of melt extraction.
Our model of two-phase rheology presented through a
3D diagram (γ° – Φ – η) shows a transition between a low
strain rate regime during which the transition from one
phase to the other is continuous in terms of rheology. It
corresponds to a bulk motion of magma as a solid body,
as exemplified during magma crystallisation or when
migmatitic bodies are tectonically deformed. In contrast,
at low strain rate, a cusp develops within the surface that
represents the effective viscosity. It is the place of successive jumps between the rheology of each phase. It is naturally also the place where instabilities develop, depending
on whether they develop under common shear rate or
common stress. Crystal impingement during crystallisation reflects the progressive jamming. Melt extraction is
also unstable, leading to competition between discontinuous melt production and melt extraction. All those instabilities strongly depend on the path adopted to go from
one rheology to the other, resulting in strain localisation
or phase banding. At moderate strain rate, crystals orientate toward an equilibrium position, giving place to a fabric in the magma.
The construction of a phase diagram allows designing
specific experiments for better understanding the onset of
those instabilities. It is essentially controlled by the
amount of phase, the available stress and the temperature.
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