Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling

Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling

The Geological Society of London Books Editorial Committee Chief Editor

BOB PANKHURST (UK) Society Books Editors

JOHN GREGORY (UK) JIM GRIFFITHS (UK) JOHN HOWE (UK) PHIL LEAT (UK) NICK ROBINS (UK) JONATHAN TURNER (UK) Society Books Advisors

MIKE BROWN (USA) ERIC BUFFETAUT (FRANCE ) JONATHAN CRAIG (ITALY ) RETO GIERE´ (GERMANY ) TOM MC CANN (GERMANY ) DOUG STEAD (CANADA ) RANDELL STEPHENSON (UK)

Geological Society books refereeing procedures The Society makes every effort to ensure that the scientific and production quality of its books matches that of its journals. Since 1997, all book proposals have been refereed by specialist reviewers as well as by the Society’s Books Editorial Committee. If the referees identify weaknesses in the proposal, these must be addressed before the proposal is accepted. Once the book is accepted, the Society Book Editors ensure that the volume editors follow strict guidelines on refereeing and quality control. We insist that individual papers can only be accepted after satisfactory review by two independent referees. The questions on the review forms are similar to those for Journal of the Geological Society. The referees’ forms and comments must be available to the Society’s Book Editors on request. Although many of the books result from meetings, the editors are expected to commission papers that were not presented at the meeting to ensure that the book provides a balanced coverage of the subject. Being accepted for presentation at the meeting does not guarantee inclusion in the book. More information about submitting a proposal and producing a book for the Society can be found on its web site: www.geolsoc.org.uk. It is recommended that reference to all or part of this book should be made in one of the following ways: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. 2010. Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332. GARDI , A., BAIZE , S. & SCOTTI , O. 2010. Present-day vertical isostatic readjustment of the Western Alps revealed by numerical modelling and geodetic and seismotectonic data. In: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 115– 128.

GEOLOGICAL SOCIETY SPECIAL PUBLICATION NO. 332

Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling

EDITED BY

M. I. SPALLA Universita di Milano, Italy

A. M. MAROTTA Universita di Milano, Italy

and G. GOSSO Universita di Milano, Italy

2010 Published by The Geological Society London

THE GEOLOGICAL SOCIETY The Geological Society of London (GSL) was founded in 1807. It is the oldest national geological society in the world and the largest in Europe. It was incorporated under Royal Charter in 1825 and is Registered Charity 210161. The Society is the UK national learned and professional society for geology with a worldwide Fellowship (FGS) of over 9000. The Society has the power to confer Chartered status on suitably qualified Fellows, and about 2000 of the Fellowship carry the title (CGeol). Chartered Geologists may also obtain the equivalent European title, European Geologist (EurGeol). One fifth of the Society’s fellowship resides outside the UK. To find out more about the Society, log on to www.geolsoc.org.uk. The Geological Society Publishing House (Bath, UK) produces the Society’s international journals and books, and acts as European distributor for selected publications of the American Association of Petroleum Geologists (AAPG), the Indonesian Petroleum Association (IPA), the Geological Society of America (GSA), the Society for Sedimentary Geology (SEPM) and the Geologists’ Association (GA). Joint marketing agreements ensure that GSL Fellows may purchase these societies’ publications at a discount. The Society’s online bookshop (accessible from www.geolsoc.org.uk) offers secure book purchasing with your credit or debit card. To find out about joining the Society and benefiting from substantial discounts on publications of GSL and other societies worldwide, consult www.geolsoc.org.uk, or contact the Fellowship Department at: The Geological Society, Burlington House, Piccadilly, London W1J 0BG: Tel. þ44 (0)20 7434 9944; Fax þ44 (0)20 7439 8975; E-mail: [email protected]. For information about the Society’s meetings, consult Events on www.geolsoc.org.uk. To find out more about the Society’s Corporate Affiliates Scheme, write to [email protected]. Published by The Geological Society from: The Geological Society Publishing House, Unit 7, Brassmill Enterprise Centre, Brassmill Lane, Bath BA1 3JN, UK (Orders: Tel. þ44 (0)1225 445046, Fax þ44 (0)1225 442836) Online bookshop: www.geolsoc.org.uk/bookshop The publishers make no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility for any errors or omissions that may be made. # The Geological Society of London 2010. All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with the provisions of The Copyright Licensing Agency Ltd, Saffron House, 6 –10 Kirby Street, London EC1N 8TS, UK. Users registered with the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, USA: the item-fee code for this publication is 0305-8719/10/$15.00. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 978-1-86239-295-3 Typeset by Techset Composition Ltd, Salisbury, UK Printed by MPG Books Ltd, Bodmin, UK Distributors North America For trade and institutional orders: The Geological Society, c/o AIDC, 82 Winter Sport Lane, Williston, VT 05495, USA Orders: Tel. þ1 800-972-9892 Fax þ1 802-864-7626 E-mail: [email protected] For individual and corporate orders: AAPG Bookstore, PO Box 979, Tulsa, OK 74101-0979, USA Orders: Tel. þ1 918-584-2555 Fax þ1 918-560-2652 E-mail: [email protected] Website: http://bookstore.aapg.org India Affiliated East-West Press Private Ltd, Marketing Division, G-1/16 Ansari Road, Darya Ganj, New Delhi 110 002, India Orders: Tel. þ91 11 2327-9113/2326-4180 Fax þ91 11 2326-0538 E-mail: [email protected]

Preface This Special Publication deals with some of the themes treated during the XVI Deformation Rheology and Tectonics Conference, held in Milano on 27–29 September 2007, and analysed in depth during the workshop, held in OropaBiella on 29 September– 2 October 2007. A preconference excursion was held on the Monviso metaophiolites and Dora –Maira UHP continental crust (23 –26 September 2007). The conference and workshop were mostly devoted to compare results from different field, laboratory or modelling strategies, and to envisage mismatches between outcomes from different analytical or methodological approaches. They help to identify non-consistent results introduced by: (i) new approaches in gathering and processing field or laboratory data (e.g. deformation mechanisms in various thermal environments, P –T estimates during metamorphic and deformation cycles, age data in metamorphic terrains and tectonic units size); (ii) different modelling approaches, combining aspects from surface to the deep interior of the Earth; and (iii) numerical techniques (finite elements or finite differences) applied at different scales. The integration of modelling and natural and experimental data enlighten whether the huge amount of empirical data available on the P–T –t evolution of crust and mantle rocks is at present adequately used to check and interactively optimize numerical models of geodynamic processes. This volume contains 12 contributions dealing with natural and experimental data, model predictions, and integration of natural data and modelling results. Villa examines whether microtextural and microchemical disequilibrium in minerals is compatible with thermal diffusion as the main factor controlling isotope transport in geochronometers. He reviews recent results of laboratory experiments and of integrated petrology –microchemistry – geochronology studies, and argues that dissolution– reprecipitation is both more widespread in natural systems and a more rapid transport process than diffusive reequilibration. In his perspective, the true significance of isotope chronometry is mostly a record of geohygrometry. Forster & Lister propose an example of the refinement of multi-scale data and its integration into numerical modelling of Argon diffusion in K-feldspar. The standard assumption is that the range of diffusion domain sizes is limited. They use fractal geometries instead, in one example based on the assumption that the ‘holes’ in a Menger Sponge define intact diffusion domains

surrounded by a fast-diffusion matrix. Numerical modelling based on this assumption shows that ‘feathered’ Arrhenius plots, as observed in data from the South Cyclades Shear Zone (SCSZ), Ios, Aegean Sea, Greece, can thereby be explained. Based on the UCLA database, diffusion parameters for K-feldspar have been assumed to be relatively unretentive, with closure temperatures for typical cooling rates in the range 250–350 8C. This paper shows that the analysis of Arrhenius data must be carried out so that the results are consistent with a Fundamental Asymmetry Principle: otherwise the values obtained for the diffusion parameters may considerably underestimate the actual retentivity. The new analyses of data from the SCSZ imply that closure temperatures as high as 450– 500 8C may have applied for the larger, most retentive, diffusion domains. Mamtani & Greiling test the application of fractal analysis of quartz grains in a syntectonic granite (Godhra Granite, India) to evaluate temperature and strain rate. They present fractal (ruler) dimension (Dr) data from quartz grain-boundary sutures in 12 thin sections, and demonstrate application of Dr as a geothermometer and its usefulness in investigating superimposition of low-T over high-T fabrics in different parts of the granite. Mamtani and Greiling also calculate the area – perimeter fractal dimension (Da) of quartz grains, which has earlier been proposed as a strain-rate gauge based on experimental studies. Their data yield geologically reasonable strain rate of the order of 10211.4 s21 for low T (300 8C), but extremely high strain rates (c. 1027 –1028 s21) for the high-T calculations (.600 8C). Based on these results they propose that Da can be used as a strain-rate gauge only for low-T calculations and further research is needed to apply the method to higher T. Cirrincione et al. investigate progressively deformed rocks, belonging to a crustal-scale mylonitic shear zone, and quantify, by means of an integrated microstructural and petrophysical study, the relationships between textural and elastic properties. Their results highlight how several parameters, operating as counterbalancing factors, control the seismic anisotropy. Watanabe based on the electrical resistivity change in deforming wet halite rocks (Watanabe & Peach 2002), demonstrates that brine must exist on grain boundaries as a thin fluid film. Although such grain-boundary brine is contradictory to dihedral angle studies (e.g. Holness & Lewis 1997), it is also required to explain the observed rapid grainboundary migration.

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PREFACE

Zucali et al. compare the evolution of microstructures of a polycrystalline aggregate of gypsum observed by optical microscope with those obtained from combined analysis of diffraction data (i.e. X-ray and neutron texture). The comparison shows that during experimental deformation the gypsum accommodated strain by both brittle and plastic deformation mechanisms, developing Riedel-like micro-fault systems together with plastic foliations and no brittle to plastic transition occurs; however, both plastic and brittle structures contemporaneously accommodate and localize strain during the deformation. Using a 2D fully dynamic coupled petrological– thermomechanical model, Baumann et al. have investigated numerically viscous slab breakoff during continental collision. Five phases of model development are distinguished: (a) oceanic slab subduction and bending; (b) continental collision initiation followed by the spontaneous slab blocking, thermal relaxation and unbending; (c) slab stretching and necking; (d) slab breakoff and accelerated sinking; and (e) post-breakoff relaxation. The results also demonstrate: (i) a non-linear dependence of the duration of the breakoff event on slab age; and (ii) localization of viscous slab breakoff at depths of 410 –510 km owing to the buoyancy effects of the olivine/wadsleyite transition. By means of 2D viscoelastic numerical models, Gardi et al. investigate the geodynamic processes proposed in the literature for the Western Alps, comparing the results with available geodetic, geological and seismotectonic data. The main achievements of this study are that an important vertical isostatic readjustment is most probably active in this portion of the Alpine chain and that a slight horizontal compression is also necessary, most likely due to the Africa– Eurasia convergence. Splendore et al. perform a 3D thermorheological analysis to analyse the rheological structure of the lithosphere in the Mediterranean. Their results indicate that a strong lithosphere paves the Tyrrhenian, in particular below the Provenc¸al Basin and the Calabrian Arc, while a soft lithosphere characterizes the Pannonian Area. In addition, at shorter wavelengths, the northern block of the Adria microplate is rheologically differentiated (half an order of magnitude stiffer) than the southern one. Once the predicted lithosphere stiffness is accounted for within a tectonic model, Splendore et al. succeed in predicting the extensive deformation style in the surrounding of Sicily, southern Calabria and part of the southern Tyrrhenian, and the compressive deformation style in the Algerian region, as evidenced by the Word Stress Map 2008 compilation. A x2 statistic test confirms that a significant part of the Africa –Eurasia convergence is absorbed through the Calabrian Subduction.

Meda et al. simulate the oceanic subduction beneath a continent, using a 2D thermomechanical model, to investigate the role played by the mantle hydration in the recycling of continental crust in the wedge region, and compare the results with the natural data from the Sesia– Lanzo Zone (SLZ), Western Alps. The predicted mixing of crust and mantle slices, characterizing the central part of the wedge, can justify the alternance of tectonic units coming from shallow crustal levels with units from deeper crustal levels, similar to the mixing occurring in some portions of the SLZ. The simulated geodynamic scenario generates P– T conditions coherent with those recorded in the subducted continental crust of the SLZ, indicating that the prograde and retrograde P– T evolution of this Alpine nappe can be totally accomplished under an active subduction regime. Salvi et al. evaluate structural and metamorphic memory of polydeformed and polymetamorphic rocks of Central Alps by means of 3D modelling of geological bodies. Structural analysis integrated with petrology allows construction of a map of dominant fabric domains, constituting the base to estimate the per cent volume of textural reworking during polycyclic (pre-Alpine and Alpine) deformations, and shows that fabric evolution and metamorphic transformation degree increase proportionally above the threshold of 60% volume affected by fabric rejuvenation. Structural and metamorphic overprint during the last deformation stage involved less than 50% of rock volume. These estimates of volumes preserving textural and mineral relicts after phase transitions can help to evaluate the potential influence that relict domains exert on the choice of physical parameters for thermomechanical modelling, such as density or viscosity. Hobbs et al. derive feedback relations between deformation and metamorphic mineral reactions using the principles of non-equilibrium thermodynamics. Such relations lead to strain-rate softening, which produces shear zones, folds and boudins by non-Biot mechanisms. These processes are intimately related to the observation that mineral reactions progress to completion only in high-strain areas, driven by energy dissipated from inelastic deformation. Readers interested in additional information on other topics treated during the 16th DRT Conference are referred to the abstract volume (Rendiconti della Societa` Geologica Italiana, 2007, Volume 5-I), to the field guides of pre-Conference Excursion and post-Conference Workshop (Quaderni di Geodinamica Alpina e Quaternaria, 2007, Volume 9) and to the keynote lecture reports (Thematic Section of Italian Journal of Geosciences, 2008, Volume 127-2).

PREFACE M. I. Spalla, A. M. Marotta and G. Gosso are grateful to the Geological Society of London for publishing this volume, and to Angharad Hills, Staff Editor of the Geological Society Publishing House, for her ability to solve editorial problems with kindness and patience. The accurate work of the reviewers U. Bayer, P. Bons, M. Brown, S. Buiter, L. Burlini, E. Burov, A. Camacho, P. Dahl, M. Drury, G. Desbois, T. Gerya, D. Gibson, R. Govers, W. Jens, I. Jimenez-Munt, D. Koehn, O. Lovera, D. Prior, U. Ring, C. Rosenberg, E. Rutter, R. Sabadini, H. Stuenitz, R. Trouw, R. Twiss, J.-L. Vigneresse, J. White and M. L. Williams was fundamental to improving the quality of this Special Publication. During the editing of this volume, our friend and colleague Luigi Burlini passed away after a battle with illness. We honour his scientific contribution to microstructural science at ETH laboratories of Zu¨rich.

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References HOLNESS , M. B. & LEWIS , S. 1997. The structure of the halite– brine interface from pressure and temperature variations of equilibrium dihedral angles in the halite– H2O–CO2 system. Geochimica et Cosmochimica Acta, 61, 795 –804. WATANBE , T. & PEACH , C. J. 2002. Electrical impedance measurement of plastically deforming halite rocks at 125 8C and 50 Mpa. Journal of Geophysical Research, 107(B1), 2004; doi: 10.1029/2001 JB000204.

Maria Iole Spalla, Anna Maria Marotta & Guido Gosso

Contents Preface VILLA , I. M. Disequilibrium textures versus equilibrium modelling: geochronology at the crossroads

vii 1

FORSTER , M. A. & LISTER , G. S. Argon enters the retentive zone: reassessment of diffusion parameters for K-feldspar in the South Cyclades Shear Zone, Ios, Greece

17

MAMTANI , M. A. & GREILING , R. O. Serrated quartz grain boundaries, temperature and strain rate: testing fractal techniques in a syntectonic granite

35

CIRRINCIONE , R., FAZIO , E., HEILBRONNER , R., KERN , H., MENGEL , K., ORTOLANO , G., PEZZINO , A. & PUNTURO , R. Microstructure and elastic anisotropy of naturally deformed leucogneiss from a shear zone in Montalto (southern Calabria, Italy)

49

WATANABE , T. Geometry of intercrystalline brine in plastically deforming halite rocks: inference from electrical resistivity

69

ZUCALI , M., BARBERINI , V., CHATEIGNER , D., OULADDIAF , B. & LUTTEROTTI , L. Brittle plus plastic deformation of gypsum aggregates experimentally deformed in torsion to high strains: quantitative microstructural and textural analysis from optical and diffraction data

79

BAUMANN , C., GERYA , T. V. & CONNOLLY , J. A. D. Numerical modelling of spontaneous slab breakoff dynamics during continental collision

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GARDI , A., BAIZE , S. & SCOTTI , O. Present-day vertical isostatic readjustment of the Western Alps revealed by numerical modelling and geodetic and seismotectonic data

115

SPLENDORE , R., MAROTTA , A. M., BARZAGHI , R., BORGHI , A. & CANNIZZARO , L. Block model versus thermomechanical model: new insights on the present-day regional deformation in the surroundings of the Calabrian Arc

129

MEDA , M., MAROTTA , A. M. & SPALLA , M. I. The role of mantle hydration in continental crust recycling in the wedge region

149

SALVI , F., SPALLA , M. I., ZUCALI , M. & GOSSO , G. Three-dimensional evaluation of fabric evolution and metamorphic reaction progress in polycyclic and polymetamorphic terrains: a case from the Central Italian Alps

173

HOBBS , B. E., ORD , A., SPALLA , M. I., GOSSO , G. & ZUCALI , M. The interaction of deformation and metamorphic reactions

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Index

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Disequilibrium textures versus equilibrium modelling: geochronology at the crossroads IGOR M. VILLA Dipartimento di Scienze Geologiche e Geotecnologie, Universita` di Milano Bicocca, 20126 Milano, Italy; and Institut fu¨r Geologie, Universita¨t Bern, Baltzerstrasse 3, 3012 Bern, Switzerland (e-mail: [email protected]) Abstract: Submicroscopic microchemical disequilibrium in minerals is extremely widespread. Disequilibrium recrystallization is promoted by water in metamorphic terranes and near granites, contact aureoles, and faults. Recrystallization is energetically less costly at almost any temperature than diffusive re-equilibration. Radiogenic isotopes (except 4He) never diffusively re-equilibrate faster than major elements forming the mineral structure. Isotopic inheritance tied to relicts was demonstrated for zircon, monazite, amphibole, K-feldspar, biotite and muscovite. The mechanism for resetting the isotope record in nature depends more on the availability of recrystallization-enhancing water than on reaching a preset temperature. Laboratory diffusion experiments on hydrous minerals were plagued, to a variable but always large extent, by dissolution– reprecipitation. Mineral geochronometers should be viewed as ‘geohygrometers’ that essentially date the fluid circulation episodes. Thanks to submicroscopic petrology, isotopic disequilibria can be put into context with petrogenetic disequilibria. Analytical advances allow the successful dating of each mineral generation. This has opened up a much richer wealth of data on the P– T– A– X– d history of rocks, which in the long run will also improve our ability to develop credible numeric models.

In the 1960s, geochronologists followed the working hypothesis of ‘one mineral, one age’ inspired by the work of Ja¨ger (e.g. Ja¨ger 1967). According to this hypothesis, later called ‘thermochronology’, every mineral was characterized isotopically by diffusive equilibration from the time of its formation to the time of its ‘closure’. Further work by Dodson (1986) extended this working hypothesis by pointing out that, precisely because of diffusive equilibration, every mineral grain was expected to have frozen in a diffusive gradient, that is, to record a variety of ages decreasing in a smoothly monotonic way from core to rim. This concentric spatial age pattern follows a mathematical equation, Fick’s law. In the following discussion it will be essential to distinguish between such a predictable, bell-shaped variation (for clarity, it will be referred to as one single age) and irregular/patchy, nonconcentric, mathematically unpredictable withingrain age variations. The latter are incompatible with simple diffusive re-equilibration as the dominant process. ‘Thermochronological’ modelling makes the explicit assumption that diffusive re-equilibration is ubiquitous, that it is predominant and that its rate can be constrained by laboratory experiments. The question that will be addressed here is to what extent observations confirm these assumptions, and what geological information can be extracted from a series of isotopic analyses.

During the 1980s petrology underwent a burst of progress. Thermobarometric calculations incorporated the activity of end-member phases and the chemical composition of the rock, in addition to pressure and temperature (De Capitani & Brown 1987; Berman 1991; Holland & Powell 1998), and are therefore referred to as P–T –A–X (pressure – temperature –activity–molar fraction). The role of deformation in promoting recrystallization is expressed by the d in P– T–A –X–d. In those years the first scattered observations of disequilibrium within single garnet and other mineral grains were made. Regarding K– Ar mineral chronometers, Wijbrans & McDougall (1986) observed that phengitic micas were not rejuvenated during a subsequent metamorphic event that resulted in the overgrowth of young muscovite. The implication of such an observation slowly became accepted over the following decade as observations of isotopic relicts, despite comparatively high-grade metamorphic and overgrowth events, became more and more common (Onstott et al. 1990; Hammerschmidt & Frank 1991; Hames & Cheney 1997; Cocherie et al. 1998; Villa 1998; Crowley & Ghent 1999). Once the existence of petrological disequilibria was perceived as important, their observation shifted from serendipitous to dedicated, and it became clear that disequilibrium is much more widespread than heretofore assumed. Thus, during

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 1– 15. DOI: 10.1144/SP332.1 0305-8719/10/$15.00 # The Geological Society of London 2010.

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the last decade, unravelling the age records of such heterochemical mineral grains has become a frequently used technique. In this respect it is essential to monitor the chemical indicators of the analysed phase in order to link it to the microstructural and petrological context (Vance et al. 2003 and references therein). The U/Th or the La/Lu ratios in a zircon carry no chronological information per se, but they are vital in understanding the petrogenesis of the analysed mineral (Whitehouse et al. 1999). Similarly, indicators such as Ca/K (Belluso et al. 2000; Villa et al. 2000) or Al/Ti (Kreissig et al. 2001) in amphiboles do not ‘give’ an age, but they do help to clarify the interpretation of geochronological data. The role of electron microprobes in geochronology cannot be overestimated. While backscattered electron (BSE) images and element maps, and cathoduluminescence (CL) images, might sometimes be referred to as ‘just pretty pictures’ (T. Harrison, pers. comm., to S. Samson, 2006) they are, in actual fact, extremely powerful diagnostic tools, without which the petrological understanding is made impossible.

Constraining the underconstrained? Geochronological data are commonly needed for the quantification of rates of tectonic processes. Sometimes these process rates have very scant independent geological constraints and mainly rely on the modelling of isotopic data. This is a very delicate situation, as there is almost no independent (nonisotopic) constraint on the shape of the long segment of the P –T– A–X –d evolution of a rock between the formation of a metamorphic paragenesis and near-surface exhumation. The T –t (temperature–time) curves in Figure 1a depict three possible trajectories between two unambiguously determined points, A and B. Such two points typically are the peak of metamorphism and the outcrop at the Earth’s surface. It is extremely obvious that the rock ‘cooled’ and ‘depressurized’ en route from A to B. What is required by many tectonic models, however, is a more detailed, unique constraint on when, and how, the intermediate P– T stages were reached. The following discussion will focus on a simple question: is there a credible way to identify the T– t trajectory? It is frequently forgotten that it is epistemiologically very different to state ‘these data are compatible with a number of evolutionary scenarios’ or ‘these data precisely constrain a single scenario’. To explain the difference between these two statements, imagine that one needs to model the growth rate of cows. In a first mathematical model, one can assume that cows and humans share the same

growth rate. By fixing two points for humans (e.g. 75 kg at 20 years of age, 150 kg at 50 years) and fitting age by a second-order polynomial of weight, one can then unambiguously determine that a 300 kg cow has an age of 170.2 years (Fig. 1b). Figure 1c uses a different model, the exponential growth rate typical of bacteria. While mathematically flawless, it has a shortcoming conflicting with external constraints, however, in that it predicts that cows spend most of their life being small; which contrasts, for example, with satellite images of pastures showing large numbers of large cows. Unless the total world population of cows is extrapolated to be tens of billions to account for the daily meat production, it is necessary to modify the growth rate curve by assuming that the size of cows does not grow monotonically but shows instead sudden decreases followed by renewed increases (Fig. 1d). Even perfectly correct modelling, where validity in other instances is certain (e.g. the growth of bacteria), cannot per se produce meaningful predictions if the boundary conditions and the mechanisms are not accurately constrained. It is the role of observational scientists to provide mathematicians with the needed external constraints that transform a mathematical mirage into a credible theory.

Understanding isotope exchange The crossroads in the title of this paper refer to the alternatives of an observation-based science, in which facts constrain and limit the models, and model-based studies, in which the underdetermination of all natural systems allows users to freely choose one among multiple answers. Two conceptually opposed approaches have been proposed in the literature to describe the factor(s) governing isotope transport in real-world minerals. The first one, dating back to the explanation of sylvite age discrepancies as an effect of diagenetic heating (Gentner et al. 1954), states that thermally activated Fick’s law diffusion is the one and only cause of isotope mobility. This uniqueness is the requirement of ‘cooling rate’ modelling, which inverts age measurements to derive the corresponding temperature history. The other approach states that in a given system, the fastest operating process will overtake the slower concurrent ones. Such faster processes usually involve the presence of water as an enhancer of recrystallization (e.g. Putnis 2002, 2009). Therefore, even if isotope transport has a broad dependence on temperature, uniqueness is no longer given. Any inversion of isotopic ages as a function of temperature is not legitimate whenever temperature was not the rate-controlling parameter of isotope transport.

GEOCHRONOLOGY AT THE CROSSROADS

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Fig. 1. (a) Possible temperature–time evolutions of a rock between two reliably fixed points, A and B. Any trajectory is a priori plausible; the criteria that may allow one of the infinite possibilities to be selected are the subject of this paper. (b) – (d) Mathematical models for the growth rate of cows. (b) By analogy with the age –weight relationship in humans, a 300 kg cow is calculated to be approximately 170 years old. (c) By analogy with bacteria, cows grow exponentially by doubling their weight. (d) To account for the apparently too low abundance of large cows, a periodic mass loss is required, followed by renewed exponential growth.

The discussion that follows will review the evidence available more than half a century after the original working hypothesis by Gentner et al. (1954), and assess to what extent their hypothesis has predictive potential in real-world geological systems.

Field observations: constraints and ambiguities The age data obtained and reviewed by Ja¨ger (1967) have long been considered the definitive proof that her model was both methodologically correct and regionally accurate, effectively preventing a serious re-examination for about 30 years. One of her key observations was that her calculated ages almost always followed the sequence: t(musc, Rb– Sr) . t(musc, K –Ar) . t(biot, K –Ar and Rb –Sr). Many subsequent studies stressed that they observed the same, reproducible sequence for mineral ages as Ja¨ger (1967), considered this as a validation of Ja¨ger’s (1967) ‘thermochronological’ explanation and added many other mineral

geochronometers (monazite, titanite, etc.) to the sequence. However, can the identification of a retentivity sequence be uniquely identified as being due to temperature? It had already been argued by Dahl (1997, p. 278), and then by Villa (2006 and references therein), that such an age progression is a primary property of the average bond length/strength of a mineral structure, and holds true regardless of the rejuvenation mechanism. Minerals with long, weak bonds, such as micas and feldspars, require less energy to break a bond than minerals such as zircon, formed by high-field-strength elements. This manifests itself in every process that requires bond breaking: diffusion by vacancies, weathering, dissolution, annealing of fission tracks, etc. In this respect, field evidence of reproducible mineral chronometer age sequences is by no means an unambiguous proof that Fick’s law diffusion was the only mechanism causing isotope transport. It merely constrains the reproducibility of the average bond length/strength of the minerals under consideration. One analytical aspect that escaped the attention of many subsequent workers was the way in which

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the Rb –Sr ages had actually been calculated. Up to about 1970 the initial Sr for all gneisses was assumed to be 0.709, and the radiogenic Sr content of a muscovite was calculated by subtracting common Sr with that blanket isotopic composition. In later years muscovite ages were calculated using two-point whole-rock plus muscovite isochrons; no overdetermined internal mineral isochron was ever published by Ja¨ger. A fundamental problem with isochrons is finding suitable phases that were in isotopic equilibrium with the mica at time t ¼ 0. As isotopic equilibrium is a central issue of this paper, which will be discussed in the following paragraphs, at this point I will only refer to Tho¨ni & Jagoutz’s (1992) conclusions on the near-impossibility to produce credible, overdetermined internal isochrons in garnet-bearing highgrade rocks using the Sm–Nd method. This extends to any metamorphic rock with any dating method (U– Pb, Lu –Hf, Rb –Sr, etc.), the more so, the more significant the proportion of common Pb (Hf, Sr, etc.) isotopes is with respect to the radiogenic ingrowth. The conclusion is that the original muscovite Rb–Sr ages by Ja¨ger (1967) should be repeated using a correct internal isochron approach. The systematic inaccuracy of the original data prevents their use as part of a retentivity sequence. The issue of the white mica Rb –Sr v. K –Ar age comparison has been examined by comparatively few field studies. In a study of a Himalayan eclogite, De Sigoyer (1998) documented that some white mica samples, deriving from eclogitic metagranite, showed 39Ar – 40Ar ages in excess of the true eclogitization age owing to 40Ar inheritance. For the only sample analysed both by Rb –Sr and by 39Ar– 40Ar, the Sr isotopic inheritance was almost equal to the Ar inheritance. Similarly, a correlated inheritance of 40Ar and 87Sr was proposed by Villa et al. (2006) for the relict-rich feldspars of the anatectic Montieri granite. The correlation is not ideal, but it indicates that the mobility of Ar and Sr in a feldspar relict entrained by a melt is low for both elements. Both these field observations can be explained by the atomic-scale argument that Sr and Ar should have similar diffusivities, as discussed later.

The relevance of diffusion experiments for natural geochronometers The laboratory diffusion of divalent cations in garnet has been conclusively established by Chakraborty & Ganguly (1992, and references therein). Diffusivities of a variety of cations in many minerals were obtained over more than a decade by Cherniak and co-workers (e.g. Cherniak et al. 1997; Cherniak 2006). In all of these investigations, bona fide Fick’s law diffusion is demonstrated by the observation of concentration profiles that follow the

mathematical prediction of a half-bell-shaped erf(x) curve. Such ‘ideal’ concentration profiles have also been observed in many natural garnets, which is evidence that diffusion of major elements is a very frequent phenomenon in natural garnets. However, very few garnet ages (if any at all) have been used by tectonic modellers, who instead often rely on the K– Ar system in micas. Obtaining accurate values for the Ar diffusivity in micas has been an important area of research over the past 35 years. When Dodson (1973) formalized mathematically the Ja¨ger (1967) approach, which linked a mineral ‘cooling age’ to a unique ‘closure temperature’, he had no mica diffusion parameters to insert into his equations in order to deduce a mica ‘closure temperature’, so he used unpublished data on Sr diffusivity in biotite. These data suggested an activation energy for Sr diffusion of 88 kJ mol21, a value that all subsequent published investigations show to be excessively low. The difficulties in obtaining Ar and/or Sr diffusivities in micas are owing to the limited stability conditions of both biotite and muscovite in P–T – A– X–d space. An excessively low water activity in the sealed experimental capsule causes dehydroxylation of the mica structure. An excessively high one causes dissolution (Wood & Walther 1983) and probably an underestimation of the true activation energy. Accordingly, diffusion experiments on micas are extremely rare in the literature; over almost four decades, only a few have been performed. Villa & Puxeddu (1994) imaged their biotite sample after the hydrothermal experiment by scanning electron microscopy (SEM). Their figure 3 demonstrates that the usual addition of traces of water in the reaction capsule, necessary to prevent dehydroxylation of the mica, was sufficient to dissolve the biotite (which later reprecipitated). The implication is that without water in the capsule the mineral is not stable and reacts away, but with water added the mineral is dissolved – the bulk Ar loss follows the bulk dissolution rate given by Wood & Walther (1983), as discussed by Villa & Puxeddu (1994). Neither option leads to reliable isolation of the Fick’s law diffusion coefficient from the interfering (and often dominating) aqueous dissolution rate constant. Villa & Puxeddu (1994) observed a similar diffusivity, D(T), as other literature experiments that they cited. While these experiments were originally believed to quantify diffusion, similar D(T) values imply that they also were affected by Wood–Walther dissolution to a similar degree to that documented by Villa & Puxeddu (1994). Grove & Harrison (1994) reported 39Ar – 40Ar stepwise heating results from one of their hydrothermally treated mica samples, obtaining a staircaseshaped spectrum. As this is not the manifestation of a diffusive gradient (Hodges et al. 1994), but a telltale sign of degradation/alteration (Hess et al. 1987),

GEOCHRONOLOGY AT THE CROSSROADS

it can be safely assumed that the water in their reaction capsule destabilized the biotite. In the presence of a fast degradation reaction, the total Ar loss determined by Harrison & Grove (1994) is not just due to volume diffusion, but represents the sum of at least two processes and must therefore not be used to infer the rate of the slowest of them. In fact, this artefact had already been abundantly documented by Hess et al. (1987) who had attempted very complex experiments to determine the Ar diffusivity in biotite and muscovite, and concluded that they had not succeeded with either mineral because of within-capsule destabilization.

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The earliest published experiments were performed by Giletti (1974) on phlogopite. Giletti’s original data (1974, table 3) are re-evaluated here. For maximum consistency, I only take into account the 200 mesh data points. In isothermal diffusion experiments, the total diffusive loss, F, is required pffi by Fick’s laws to be linearly proportional to t for F , 0.25; for intermediate losses (F , 0.7) there is a second-order correction that does not exceed a few per cent. Figure 2a shows that the phlogopite data do not follow this prediction. In none of the four plotted temperatures do data points lie on a line through the origin.

Fig. 2. Re-evaluation of Giletti’s (1974) original phlogopite Ar loss experiments. (a) Fractional Ar loss, F, as a function of (time)1/2. The dependence predicted by Fick’s law is shown by curves through the origin. The observed downward deviation of all data points obtained in longer experiments is evidence of systematic violation of Fick’s law. (b) Ideal behaviour of a system undergoing two-path diffusion. Fast path Ar loss of approximately 6%, independently of temperature, is followed by a temperature-dependent Fick’s law diffusion; the slopes of the dashed curves are flatter than those for single-path diffusivity, that is, the corrected diffusivity is smaller than in (a). The fit can be improved by making the fast pathway loss fraction increase with temperature, but this is a second-order detail compared to the demonstration that more than one Ar loss mechanism was operating. (c) Arrhenius diagram of the corrected diffusivities. The dashed line has a slope corresponding to an activation energy of E ¼ 299.4 kJ mol21 and D0 ¼ 205 cm2 s21, and the dotted one to E ¼ 359.2 kJ mol21 and D0 ¼ 93 611 cm2 s21. The diffusivity at the geologically relevant temperatures around 500 8C is very significantly smaller than for the solid line (Giletti 1974).

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Undetected to most users over the last 36 years, the systematic downward deviations of the longduration runs from the curve constrained by the short-duration runs actually show strong evidence for multipath diffusion. Figure 2b shows the expected behaviour of a sample undergoing Ar loss by two separate mechanisms: one by fast pathway transport (which accounts for the temperatureindependent Ar loss of c. 6%) and one by some slower mechanism, which, for the time being, will be called ‘volume diffusion’. The mechanism of the fast pathway loss needs to be clarified before a purely empirical correction is credible. Because the experimental charges contained more water than mica (B. J. Giletti pers. comm. 1990), a very tempting explanation is aqueous dissolution following Wood & Walther (1983). In their formalism, mass has the dimension of (moles of oxygen)/(area) and thus could be termed ‘surface O density’. Giletti’s (1974) phlogopite grains had a thickness of 5 mm, which translates to a ‘surface O density’ of 30 mmol O cm22. In order to completely dissolve them, the required durations range between 63 ks at 900 8C and 294 ks at 650 8C. These durations are lower than the actual hydrothermal run times and the question arises as to why the samples did not undergo total dissolution. Saturation of the solution in the sealed capsule is likely to have occurred. In a similar experiment, reprecipitation was observed by Villa & Puxeddu (1994). One may speculate that saturation could have been reached when approximately 6% of the mica sample had been dissolved. In any case, the exact solubility of Giletti’s (1974) phlogopites exceeds the purpose of the present discussion. What needs to be pointed out here is that the order of magnitude of aqueous dissolution is more than sufficient to account for a significant part of the Ar loss in hydrothermal experiments on silicates and to bias the apparent activation energy towards an excessively low value. Thus, having identified Wood–Walther dissolution as the probable cause, it is legitimate to perform a mathematical correction for the fast pathway Ar loss. To first order, data are attributed to the sum of just two processes: aqueous dissolution and Fick’s law diffusion. This simple correction is straightforward (see also Villa 1994, fig. 2b): the slope of each line gives the corrected diffusivity, Dc(T ). The fit to the Giletti (1974) dataset can be improved by making the fast pathway loss fraction increase with temperature, but this is a second-order detail compared to the demonstration that more than one Ar loss mechanism was operating. Improving the fit would also mean further decreasing the Fick’s law diffusion term in favour of the fast pathway term(s) at higher temperatures, so that the

conclusions of the next paragraph would further be strengthened. Figure 2c shows an Arrhenius diagram of the corrected diffusivities. While the correction on the 900 8C point is almost invisible on a logarithmic scale, the most important effect is a drastic decrease of the low-temperature diffusivities that manifests itself as an increase of the activation energy. The corrected 650 8C results are fairly scattered, a probable effect of the coarseness of the blanket 6.0% fast pathway correction. Because of this scatter a number of possible regressions are plausible: the most conservative one, shown as a dashed line, passes through the uppermost 650 8C point; the dotted line passes through the lowermost 800 8C point. The implications for geological systems are that the calculated ‘Dodsonian closure temperature’ for 100 mm-radius phlogopite grains cooling at 30 8C Ma21 increases from 411 8C to 474 or 523 8C, according to the chosen regression. These corrected values are especially remarkable, as they have two important implications for regional studies. First, a decrease in diffusivity is in better agreement with the observations on biotites from the Lepontine Alps (Allaz 2008, see next paragraph) and those from the Larderello geothermal field (Villa & Puxeddu 1994) than the uncorrected one. Secondly, the corrected biotite retentivity can be tied to Dahl’s (1996) theoretical calculations to give order-of-magnitude estimates of the Ar retentivity in other micas that have a smaller ionic porosity than phlogopite. For instance, white mica (phengite/muscovite) is predicted to have a smaller diffusivity than phlogopite. If Dodsonian ‘closure temperature’ were a universally relevant parameter, its value for Ar retention in white micas would be in excess of 500 8C. In those exceptional cases where the absence of retrogression was documented, the observations indeed require such a high retentivity (Di Vincenzo et al. 2004; Allaz 2008). To sum up the observations on experimental attempts to constrain diffusivities in micas, one should bear in mind that while such attempts have never been successful, they do allow strict upper limits to be placed on diffusivities. When a system undergoes multipath diffusion, the rate of the slowest process will always be smaller than the total bulk transport rate. For Ar loss in biotite, the multipath laboratory data limit the purely thermal Fick’s law diffusion of Ar over geological times to occur only above about 500 8C. The only other geochronometer that was subjected to Ar loss experiments in the laboratory was K-feldspars. While it would seem reasonable that the anhydrous K-feldspar structure could release its Ar following Fick’s law (e.g. Lovera et al. 2002), it has been known for a long time that it does not (Villa 1994). Further evidence that the

GEOCHRONOLOGY AT THE CROSSROADS

process governing Ar loss from K-feldspar is not Fick’s law diffusion was provided by Hetherington & Villa (2007). These authors compared the kinetics of Ar and Xe release from a variety of barian minerals and observed that a hydrous mineral, ganterite, known with certainty to undergo in vacuo destabilization by dehydroxylation showed the same Arrhenian trajectories (quantifying apparent Ar diffusivity) as anhydrous feldspars. This strongly indicates that it is a structure breakdown that effects in vacuo Ar release from feldspars, not Fick’s law diffusion. This makes extrapolations of laboratory experiments to geological temperatures a futile exercise, because they violate the rules of correct experimental design.

Diffusion in natural geochronometers Assessing the presence of bona fide Fick’s law diffusion in natural minerals requires, as a necessary condition, a bell-shaped profile (either in element concentration or in isotope ratios) in the investigated mineral. This has produced excellent, invertible results for Fe –Mg exchange in natural garnets (e.g. Lasaga & Jiang 1995). For most mineral geochronometers, including zircons, monazites, all micas and feldspars, there is no such convincing observational database of diffusion profiles. Zircon is well known for its extreme isotopic retentivity and for the absence of any diffusional re-equilibration of either Pb or the structure-forming cations (that give rise to the prominent growth zonation visible in CL images) under any known geological condition up to 1000 8C (see, e.g. the review by Kramers et al. 2009). It is equally well known that zircons can lose Pb at 180 8C (Hansen & Friderichsen 1989). The subtle difference is that diffusional re-equilibration of growth zoning is energetically very costly, as it would require Fick’s law diffusion of structureforming cations, while hydrothermal recrystallization can occur down to very low temperature, provided that the activity of water, a(H2O), is sufficiently high. The atomic-scale processes that create nanostructures in zircons are only now beginning to be understood (Geisler et al. 2007). For micas, there are clear counter-examples to the hypothesis of Fick’s law diffusion under natural conditions. Onstott et al. (1990) showed that Ar isotopic zonations in their phlogopite were due to retrograde reactions and did not fulfill the necessary requirement of a bell-shaped profile. The unambiguous observations of diffusional re-equilibration of Fe –Mg zoning in garnets beg the question: what about diffusional re-equilibration in the very widely used mineral geochronometers, micas? Three answers can be conceived. (1) Because all garnets always only show diffusional

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re-equilibration, so too do muscovites and biotites. (2) Garnet, itself sometimes in diffusive disequilibrium, is not a good analogy for micas, which never record diffusion. (3) Similarly to garnet, which very often displays diffusive gradients superimposed on disequilibrium textures, there may exist instances in which muscovites preserve diffusioncontrolled core-to-rim K –Ar age profiles. Only answer (1) is compatible with the basis of thermochronology, that an inversion of age data via a ‘closure temperature’ to infer a ‘cooling history’ is legitimized by the uniqueness of the resetting mechanism. It is clear that answer (1) is not universally valid, as counter-examples are abundant in the literature of the last decade (see earlier). Does this imply a wholesale negation of diffusion, such as in answer (2)? From a mathematical point of view, a single counter-example is sufficient to reject a theory. In geology, an incorrect theory that is ‘not too wrong’ in an acceptable number of cases may still yield sufficiently valid indications. This translates into the key question: do dissolution–reprecipitation– recrystallization processes dominate over Fick’s law diffusion in natural geochronometers? At present, with few dozen studies directly addressing the issue with sufficient detail, the petrological evidence appears to favour a marginal role for diffusion compared to dissolution– reprecipitation–recrystallization. However, only when thousands of experimental studies will have produced high-resolution microchemical homogeneity tests (of the sort that now only zircons and monazites undergo) on all analysed micas, and assessed the internal petrological equilibrium of each studied rock sample, will the controversy be settled. However, the validity of answer (3) needs to be assessed for each individual sample. Cases of genuine diffusion profile of one isotope in an unreacted, homogeneous mineral structure must be quantitatively demonstrated beyond qualitative, anecdotal descriptions. Quantitative examples are still comparatively rare; one particularly instructive one, which will be discussed in greater detail in the following section, is the muscovite megacryst studied by Hames & Cheney (1997) using the 39Ar – 40Ar laser microprobe. While there is a very broad pattern of ages becoming younger from centre to rim, a derived plot of age v. distance shows a wide variety of age–distance gradients. Only the steepest gradient can be identified as a diffusion profile, the shallower ones pertain to one or more faster processes. An illustration of the problematics of identifying fast and slow processes may come from an analogy. Suppose a thin-walled rubber balloon is filled with helium. If someone bursts the balloon, then all the gas will escape (fast process). If nobody bursts the balloon, the helium will still escape through the walls of the balloon (slow process). Now

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imagine someone coming across an empty balloon. A thermochronologist would always assume helium leakage through the walls. A petrologist would look to see whether the balloon had a puncture. For such a ‘balloon autopsy’ on mineral geochronometers, however, it is no longer sufficient to rule out fast processes by examining a thin section by optical microscopy and asserting ‘my samples are the freshest’. Petrology has come a long way in the last half century, and an assessment of petrological equilibrium is nowadays both possible and necessary. A direct assessment is possible by (admittedly time-consuming) transmission electron microscopy (TEM). When viewed at high resolution, even standard minerals used to monitor 39Ar – 40Ar

irradiations can contain retrogression minerals, such as the halloysite documented to occur in the McClure Mountain hornblende (Villa et al. 1996). A less labour-intensive, indirect way to test whether a mica has undergone retrograde reactions (in which case it must never be used to study Ar diffusion) or not (in which case it might) is the preservation of thermobarometric equilibrium. P– T–A –X calculations using thermodynamic databases (e.g. De Capitani & Brown 1987) can be performed for all mineral multiplets of a paragenesis. The reaction equilibria in the four-dimensional P– T–A –X space define curves when projected onto the two-dimensional P–T plane, and the intersection of two curves allows the estimation of the

Fig. 3. Example of a thermobarometric calculation (Allaz 2008) showing a rock in near-ideal equilibrium and the absence of retrogression reactions. In rocks such as the one depicted here, biotites completely retain Ar around 500 8C. However, in rocks from the same outcrop, whose thermobarometry shows disequilibrium owing to retrogression, biotite ages are younger. Ar diffusion is detectable against a background of faster transport only when recrystallization was absent and PTAX gives an asterisk as proof of retrogression-free rocks.

GEOCHRONOLOGY AT THE CROSSROADS

P and T recorded by the investigated minerals. If all P–T intersections of a rock agree, all mineral reactions were frozen in at the same point in P–T –A–X space and no subsequent retrogression reactions perturbed the equilibrium compositions. Therefore, a necessary criterion in recognizing a retrogression-free mica geochronometer is the requirement that its thermobarometry agrees with the P–T estimate provided by all other minerals of its paragenesis. Following such a requirement, it is possible to find retrogression-free micas in order to assess their Ar retentivity as a function of temperature. In a recent petrological– microchemical –geochronological study on the Lepontine Alps, Allaz (2008) observed that the rare unretrogressed biotites and muscovites retain practically all of their Ar at temperatures below 500 and 550 8C, respectively. It must also be pointed out, however, that this high retentivity only applies to rocks whose thermobarometry is totally retrogression-free (Fig. 3). In the same field area, Allaz (2008) documented that the majority of micas did show thermobarometric disequilibrium within their paragenesis, and that the significantly younger ages of such micas could be directly tied to the extent of retrogression reactions affecting each rock. This can be understood in terms of the prediction by Villa (1998) that ‘dry closure temperatures’ can only be applied to rocks that did not suffer secondary recrystallization at lower temperatures. It is beneficial to recall that retrogression does not just consist of a mineral cooling, but that it involves reaction –recrystallization. Do the more reliable estimates of ‘closure temperatures’, such as those in the present section, restore the credibility of a neo-Dodsonian approach in which a mineral’s age is determined, and immediately and uniquely associated with a (higher) temperature? From what has been reviewed above, there is a necessary condition for such an approach to be legitimate, namely the complete absence of any aqueous retrogression, of any deformation and of any recrystallization. In theory, the careful documentation of P–T –A–X –d equilibrium by overdetermined thermobarometry on all minerals forming a rock can provide an indication that this necessary condition is, indeed, met. In practice, very few (if any at all) published papers have gone to such a time-consuming effort. The resulting age data are less amenable to a ‘thermochronological’ interpretation when the petrological equilibrium is less constrained. ‘There is no free lunch’ means, in this case, that devoting insufficient attention to the petrological context fatally detracts from the quality of otherwise acceptable isotopic age data. This is an especially serious problem for detrital minerals, whose petrological relations to the other minerals of the paragenesis are disrupted. An

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assessment of the importance of retrogression reactions on the mm-scale is impossible, while a correct interpretation as ‘ideal cooling ages’ would require precisely that.

Disequilibrium: lack of diffusion In contrast to ideal petrological equilibrium, the coexistence of minerals having a different petrogenesis in one and the same rock leads to discordant thermobarometric estimates. The observation is that the isotopic record of rocks in petrological disequilibrium deviates from that of equilibrated rocks from the same location. The persistence of disequilibrium in a rock is a tautology for absence of diffusive re-equilibration, that is, for the irrelevance of diffusion in that particular rock both during and after the establishment of the disequilibrium paragenesis. The discontinuous occurrence of equilibrium and disequilibrium in a field area requires that processes faster than diffusion predominate whenever possible. How frequent is this ‘whenever possible’? A quick look at worldwide occurrences of eclogites and granulites gives, as a first-order answer, ‘very frequent indeed’. The eclogitic and granulitic parageneses very seldom escape greenschist-facies retrogression (the precise reason why greenschistfacies retrogression is so widespread will be addressed later). According to the observational database of today, it is certain that low-temperature retrogression did not erase 100% of the high-grade (pre)histories. It is also certain that fluid ingress during exhumation of a high-grade rock through the ‘greenschist window’ has more often than not caused at least partial recrystallization of the most weak-bonded (sensu Dahl 1997) minerals. Linking the isotope record with petrology can be achieved by combining imaging with spatially resolved ages, or with stepwise release of isotopes under the stringent requirement that chemical information can be extracted from the same data that are used to calculate a ‘step age’. This problem has been discussed extensively in the literature (e.g. Vance et al. 2003 and references therein) and needs not be reiterated here. It is beneficial, however, to point out a few implications that apply to any isotope analysis. The major elements form the mineral structure at the nanometer scale, and their distribution produces the micrometre-scale textures used to reconstruct a rock’s petrogenesis. The elements whose radiogenic isotopes are relevant for geochronology are all trace elements. The relative diffusivities of major and trace elements are the decisive parameters that legitimize or disqualify the search for a link between geochronology and petrology. In order to respect the

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assumptions that are the basis of Fick’s laws, trace elements must freely move in an inert host. In that case, geochronology will be unrelated to microtextures and only record the end of diffusive re-equilibration below some ‘closure temperature’. Instead, it had been proposed (Villa 1998) that the diffusion of trace elements relevant for geochronology is no faster than that of the major elements. Support for this hypothesis has come from both of the diffusivity experiments mentioned earlier. Cherniak et al. (1997) compared diffusion rates of different kinds of ions in zircon, and observed that the diffusivity is lowered by higher ionic charge and larger ionic radius. This means that all geochronologically relevant non-neutral daughter isotopes (87Sr, 143Nd, 176Hf, 187Os, 206 – 208Pb, 230Th, 234U), being larger and/or having a higher charge than all major elements in most minerals, normally will not diffuse out of a crystal unless the major cations also do so – in which case the mineral either recrystallizes or at least readjusts to new P–T –A– X conditions, effectively establishing a new petrological equilibrium, following the discussion above. As for neutral species, Ar was found to be limited in its movement by Si,Al interchange, because its large atomic radius hinders free movement (Nyfeler et al. 1998; see also Villa 2006). This leaves He, with its small atomic radius, as the only possible candidate element for a genuinely Fick’s law diffusion. The diffusivity of H and O is probably connected, as the mobility of H does not occur as protons but as OH ions (e.g. Zhao et al. 2004). Experiments that study OH diffusion require the presence of water. One definitive experiment that settles conclusively the issue of OH diffusion in feldspars is that reported by Labotka et al. (2004). These authors demonstrated that the enrichment of an 18O tracer is observable if, and only if, the sodium–potassium exchange reaction occurred as well (Labotka et al. 2004, fig. 3). K had to be available, and Na had to be removed, in order for the new feldspar generation to crystallize and for OH to be exchanged. Because Ar is a slower diffusant than OH, it follows a fortiori that Ar loss from a feldspar is both associated with, and rate-limited by, diffusion of structure-forming elements. The relevance of water both as a promoter of recrystallization and as an enhancer of trace-element transport can be appreciated by other observations on anhydrous silicates. For major elements such as Fe and Mg in olivine it was documented that diffusion rates depend on the availability of water (Hier-Majumder et al. 2005). Once it has been established that petrographical relicts necessarily imply isotopic inheritance, it is possible to make sense of illustrative examples whose context would otherwise be lost. Pb inheritance in zircons is unambiguously tied to the relict

cores visible by CL (e.g. Gebauer et al. 1988). Similarly, relict/neoformation-related microchemical zoning in monazites is strictly paralleled by age maps (Williams et al. 2007). The muscovite grains imaged by Hames & Cheney (1997) are entirely similar to the other minerals, in that diffusive equilibration did not occur in them at the millimetre scale. At the very most, the length scale over which diffusive equilibration could have occurred is given by the steepest age–distance gradient. In the case of the Acadian muscovite with Grenvillian inheritance (Hames & Cheney 1997, fig. 6), the distance between the two points with the greatest age difference is at most 200 mm (the limit being the IR laser spot size). This (upper limit for the) diffusion length is proportional to D  t. Assuming that t, the duration of the amphibolite-facies event, was between 1 and 10 Ma, one obtains an upper limit for the diffusion constant, D, of between 4  10220 and 4  10221 cm2 s21. This value for D pertains to the metamorphic peak temperatures estimated by Hames & Cheney (1997), viz. 425–450 8C. If diffusion really has been so sluggish, why does one observe rejuvenated ages at all, and what are the flatter spatial age gradients shown by other locations in the same crystal and by other crystals of the same rock? The steepness of the age gradient reflects the inverse of the rate constant for Ar transport. Thus, a shallow gradient represents a fast process, a steep gradient a slow process. The coexistence of shallow and steep gradients is certain proof of multiple Ar transport processes. Diffusion, being slower than recrystallization, is constrained by the steepest gradient, while recrystallization [which Hames & Cheney (1997) identify in a millimetresized shear zone in the lower half of their fig. 5] is the cause of the rejuvenation that occurs patchily and with a shallow gradient. The superposition of slow and fast processes, and the ensuing coexistence of the isotopic signature of both processes in the same mineral grain, requires a change of perspective to take into account the observations of the last two decades. In Ja¨ger’s (1967) hypothesis, the isotope retentivity is tied exclusively to temperature (Fig. 4a). This means that each mineral only has one age (as defined earlier), which allows the unique inversion of a mineral’s age to infer the the temperature of its ‘closure’. Ja¨ger (1967), however, had great difficulties explaining her own observations (Arnold & Ja¨ger 1965) of reaction-limited Alpine ages and preAlpine relict ages in an area supposedly above the biotite ‘closure temperature’. The true implication of the pioneering observations by Arnold & Ja¨ger (1965) is that retention actually starts at much higher temperature, but then [mysteriously, for Ja¨ger’s (1967) understanding] collapses for some,

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Fig. 4. Isotopic retentivity and its relation to ambient conditions. (a) ‘Thermochronological’ working hypothesis: the inversion of a measured age to infer the temperature of its ‘closure’ is always unique. (b) Arnold & Ja¨ger’s (1965) observations require retention at higher T followed by zero retention for some, but not all, samples from one outcrop. An inversion of the measured age to its ‘closure temperature’ (dashed line with question mark) is no longer unique. (c) The actual control on isotope retention is water activity, a(H2O): if aqueous fluid is present, retrogression reactions and recrystallization take place, and the isotope record is reset. What becomes possible is the inversion of a measured age to date the episodes of high water activity. (d) The passage of every high-grade metamorphic rock through greenschist-facies conditions is often accompanied by a sudden fluid ingress (e.g. Proyer 2003). The fluid ingress could have happened at different positions in T –a(H2O) space (dashed lines) and its temperature is not uniquely fixed. (e) The combination of (c) and (d) into a three-dimensional diagram represents the retrogression that affects some (but not necessarily all) rocks from one outcrop. Its projection onto the T –R plane is shown in (b), and is only a partial view; Ja¨ger (1967) was not able to explain her observations by relying only on (b), because the third axis was missing from her interpretive scheme. The full three-dimensional view in (e) establishes mineral geochronometers as geohygrometers.

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but not all, samples collected within a few metres of each other (Fig. 4b). The inverse calculation that assigns a unique temperature to every mineral age is no longer feasible both because the temperature is no longer unique and because partly retrogressed minerals do not have a single age. In actual fact, because the diffusion rate is several orders of magnitude smaller than aqueous dissolution rates (Wood & Walther 1983), the predominant control on isotopic retentivity is the water activity whenever water is present at all (Fig. 4c). To be semantically correct, Earth scientists should refer to isotopic clocks as ‘geohygrometers’. When one takes into account that the transition to brittle deformation opens up exhuming rocks to a sudden fluid ingress (see, e.g. Proyer 2003), it becomes intuitive that there is a broad proportionality between pressure (the depth corresponding to the ductile –brittle transition) and recrystallization ages. Also note that the temperature of the fluid ingress is not fixed by any a priori constraint. Figure 4d is a graphic visualization of three such discrete hygrometric events, each taking place at a slightly different temperature. Combining the hygrometric history of a rock with the isotopic effects (namely, reduced retentivity owing to the formation of discrete greenschist retrogression parageneses) one obtains the threedimensional sketch shown in Figure 4e. Note that the two-dimensional projection of Figure 4e onto the T –R (temperature –rententivity) plane gives exactly Figure 4b; the explanation of the observed patchy rejuvenation pattern that had not been possible in a limited T–R diagram in 1967 becomes obvious if one takes into account the third dimension, water activity. It is also essential to point out that Ja¨ger’s (1967) hypothesis, and its internally consistent continuation by Steck & Hunziker (1994), have inferred a post-orogenic ‘cooling history’ of the Central Alps that was demonstrated to be totally incorrect (Scherrer et al. 2002; Allaz 2008; see also Villa 1998, fig. 1). Any model that produces incorrect results despite its mathematical correctness is likely to be based on unrealistic assumptions, much like the correct but incorrect curve of Figure 1c.

Disequilibrium: unravelling mixtures It has been argued by observational petrologists (e.g. Allen 1992) that minerals more frequently than not are mixtures of diachronically formed, heterochemical generations. There is also compelling evidence that coexisting mineral generations carry an unequilibrated isotopic record. The following discussion will briefly review a decade of geochronological literature on how to extract meaningful

geological information out of heterochemical mixtures. The archetype of polygenic rocks are arguably pseudotachylytes. These are flash-melted friction rocks formed in the brittle regime. The melt is quenched to glass, which devitrifies relatively rapidly. Because of their association with brittle faulting and associated abundant fluid flow, and because of the loose structure and weak bond strength of the chaotic matrix, pseudotachylytes contain high modal proportions of alteration phases. In addition to the devitrified matrix and its alteration products, pseudotachylytes also contain clast relicts of the premelting source rocks. The Ar isotope systematics of pseudotachylytes are discussed in great detail by Mu¨ller et al. (2002). These authors used two 39Ar– 40Ar techniques, spatially resolved dating by UV laser and furnace stepwise heating, in a complementary way. The isotopic and chemical signature of clasts and matrix analysed by laser was identified in the stepwise heating data and subsequently refined with better precision. It will not be necessary to repeat the discussion by Mu¨ller et al. (2002), to which the reader is explicitly referred. What is relevant here is to point out that the commondenominator three-isotope (or three-element) correlation diagrams are a crucial tool in unravelling the isotopic signatures of the coexisting mineral generations. In their study of polygenic amphiboles, Villa et al. (2000) validated the identification of the endmembers in Ca/K– Cl/K mixing diagrams by direct electron-microprobe analyses of texturally identified petrologic end members. Such diagrams are not limited to the Ar isotopic system, and have been used successfully for the U –Th –Pb system as well (Kreissig et al. 2001). The case of white micas is slightly less favourable than that of amphiboles or of pseudotachylytes, as stoichiometric muscovites do not contain Ca, which reduces the abovementioned chemical correlation diagrams by one dimension. Nevertheless, the Cl/K ratio was a sufficiently clear indicator that allowed Villa et al. (1997) to discriminate among, and assign individual ages to, multiple white mica generations from the same hand specimen after petrography and microchemistry had demonstrated their existence and microstructural identity. Similarly, Federico et al. (2005) were able to compare ages of various stages of retrogression from eclogite to greenschist; the scale at which the examined retrogression reactions were documented to occur escapes optical microscope detection (see fig. 3 of Villa 2006). The microchemical observations of sharp compositional boundaries demonstrate negligible diffusive re-equilibration of major structure-forming cations. Negligible diffusion of Ar is shown by mixed ages, intermediate between

GEOCHRONOLOGY AT THE CROSSROADS

eclogite and greenschist ages. As already discussed above, the parallelism between petrological relicts and isotopic inheritance is a consequence of the fact that Ar cannot diffuse faster than the mobility of structure-forming major elements. The 40 year-old hypothesis of ‘one mineral, one age’ is thus neither true nor necessary. It is now routinely possible to diagnose the existence of heterochemical mixtures and to assign each diachronic mineral generation an age.

Conclusions An observation that is important not only for biology but also for petrology is that planet Earth contains liquid water in its uppermost layer. The role of water is vastly dominant in terrestrial geology, especially in comparison with other celestial bodies in our solar system, such as the moon and most meteorite parent bodies. The lack of water on the moon ensures that neither surface alteration nor retrogression reactions have been observed in lunar rocks. A similar lack of water –rock interaction is very rare in terrestrial rocks. Even rocks that are considered ‘dry’, such as eclogites, owe their recrystallization at the metamorphic peak to the availability of water (Bjørnerud et al. 2002; Schneider et al. 2007). The isotope systematics of eclogites are accordingly controlled by precisely those rare fluid circulation events (Glodny et al. 2008). The paradox first pointed out by Tho¨ni & Jagoutz (1992), that especially eclogites (but also many other metamorphic rocks) very rarely give satisfactory internal isochrons, can be understood by examining the presence or absence of complete petrological equilibrium. Absence of relicts and of retrogression is a necessary condition for isotopic equilibrium. It may not be sufficient, however: in the case documented by Tho¨ni & Jagoutz (1992), the diffusivity of the small-radius, divalent cations forming the garnet structure was much higher than that of the large-radius, trivalent radiogenic Nd, preserving isotopic disequilibrium despite achieving petrological equilibrium. In the absence of water, equilibrium is seldom attained; the exceptional instances of true petrological equilibrium need to be documented by concordant thermobarometry before any discussion on the significance of mineral ages can even be started. In the majority of rocks, departures from thermobarometric equilibrium are due to retrogression reactions, which in turn are due to the availability of water. Therefore, the concept of ‘geohygrometry’ should permeate Earth sciences, especially isotope geology. To date, there are very few geochronological investigations in which both the petrological equilibrium and the required absence of water

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have been taken into account. The conclusions of studies in which these fundamental parameters are uncontrolled are very likely to be just as uncontrolled. Their results may or may not have the significance that the model would like them to have, and no reliable geological inference should be based on them. On the contrary, studies that document the presence of petrological disequilibria invariably also document isotopic disequilibria by spatially resolved and/or stepwise release techniques. Isotopic disequilibria are easily understood if one realizes that pure Fick’s law diffusion is an extremely slow process, which plays a subordinate, or even sporadic, role in the mobility of isotopes. This opens up a new paradigm for the contextualization of geochronological data. At the base of any understanding there is the petrology at the submicroscopic scale. A straightforward explanation in terms of equilibria is possible, but only in relatively rare cases. More abundant are petrogenetic disequilibria, which can be tied to microtextural observations that constrain a rock’s deformation history. Isotopic disequilibrium is no longer a nuisance because it can be linked to the petrological – microtextural framework. The spatial distribution of radiogenic isotopes is controlled by the same atomic-scale process as the mobility of the structure-forming major elements, on which the petrological and textural understanding is based. Constructive reviews by P. S. Dahl, who suggested a different version of the balloon analogy, and M. L. Williams are gratefully acknowledged. A. R. Heri is thanked for help with the graphics.

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GEOCHRONOLOGY AT THE CROSSROADS K REISSIG , K., H OLZER , L. ET AL . 2001. Age of early deformation of the Hout River Shear Zone and the metamorphism in the Southern Marginal Zone of the Limpopo Belt, Southern Africa. Precambrian Research, 109, 145 –173. L ABOTKA , T. C., C OLE , D. R., F AYEK , M., R ICIPUTI , L. R. & S TADERMANN , F. J. 2004. Coupled cation and oxygen-isotope exchange between alkali feldspar and aqueous chloride solution. American Mineralogist, 89, 1822– 1825. L ASAGA , A. C. & J IANG , J. X. 1995. Thermal history of rocks – P–T –t paths from geospeedometry, petrological data, and inverse-theory techniques. American Journal of Science, 295, 697– 741. L OVERA , O., G ROVE , M. & H ARRISON , T. 2002. Systematic analysis of K-feldspar 40Ar/39Ar step heating results II: relevance of laboratory argon diffusion properties to nature. Geochimica et Cosmochimica Acta, 66, 1237– 1255. M U¨ LLER , W., K ELLEY , S. P. & V ILLA , I. M. 2002. Dating fault-generated pseudotachylytes: comparison of 40 Ar/39Ar stepwise-heating, laser-ablation and Rb/Sr microsampling analyses. Contributions to Mineralogy and Petrology, 144, 57–77. N YFELER , D., A RMBRUSTER , T. & V ILLA , I. M. 1998. Si, Al, Fe order–disorder in Fe-bearing K-feldspar from Madagascar and its implication to Ar diffusion. Schweizerische Mineralogische und Petrographische Mitteilungen, 78, 11–21. O NSTOTT , T. C., P HILLIPS , D. & P RINGLE -G OODELL , L. 1990. Laser microprobe measurement of chlorine and argon zonation in biotite. Chemical Geology, 90, 145–168. P ROYER , A. 2003. The preservation of high-pressure rocks during exhumation: metagranites and metapelites. Lithos, 70, 183– 194. P UTNIS , A. 2002. Mineral replacement reactions: from macroscopic observations to microscopic mechanisms. Mineralogical Magazine, 66, 689–708. P UTNIS , A. 2009. Mineral Replacement Reactions. Reviews in Mineralogy and Geochemistry, 70, 87– 124. S CHERRER , N. C., E NGI , M., B ERGER , A., P ARRISH , R. R. & C HEBURKIN , A. K. 2002. Nondestructive chemical dating of young monazite using XRF 2. Context sensitive microanalysis and comparison with Th–Pb laser-ablation mass spectrometric data. Chemical Geology, 191, 243–255. S CHNEIDER , J., B OSCH , D., M ONIE´ , P. & B RUGUIER , O. 2007. Micro-scale element migration during eclogitisation in the Bergen arcs (Norway): a case study on the role of fluids and deformation. Lithos, 96, 325–352. S TECK , A. & H UNZIKER , J. 1994. The tertiary structural and thermal evolution of the Central Alps – compressional and extensional structures in an orogenic belt. Tectonophysics, 238, 229–254. T HO¨ NI , M. & J AGOUTZ , E. 1992. Some new aspects of dating eclogites in orogenic belts: Sm– Nd, Rb– Sr, and Pb– Pb isotopic results from the Austroalpine Saualpe and Koralpe type-locality (Carinthia/Styria,

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southeastern Austria). Geochimica et Cosmochimica Acta, 56, 347 –368. V ANCE , D., M U¨ LLER , W. & V ILLA , I. M. 2003. Geochronology: linking the isotopic record with petrology and textures – an introduction. In: V ANCE , D., M U¨ LLER , W. & V ILLA , I. M. (eds) Geochronology: Linking the Isotopic Record with Petrology and Textures. Geological Society, London, Special Publications, 220, 1–24. V ILLA , I. M. 1994. Multipath Ar transport in K-feldspar deduced from isothermal heating experiments. Earth and Planetary Science Letters, 122, 393– 401. V ILLA , I. M. 1998. Isotopic closure. Terra Nova, 10, 42–47. V ILLA , I. M. 2006. From the nm to the Mm: isotopes, atomic-scale processes, and continent-scale tectonic models. Lithos, 87, 155–173. V ILLA , I. M. & P UXEDDU , M. 1994. Geochronology of the Larderello geothermal field: new data and the “closure temperature” issue. Contributions to Mineralogy and Petrology, 115, 415– 426. V ILLA , I. M., G ROBE´ TY , B., K ELLEY , S. P., T RIGILA , R. & W IELER , R. 1996. Assessing Ar transport paths and mechanisms for McClure Mountains Hornblende. Contributions to Mineralogy and Petrology, 126, 67–80. V ILLA , I. M., R UGGIERI , G. & P UXEDDU , M. 1997. Petrological and geochronological discrimination of two white-mica generations in a granite cored from the Larderello– Travale geothermal field (Italy). European Journal of Mineralogy, 9, 563–568. V ILLA , I. M., H ERMANN , J., M U¨ NTENER , O. & T ROMMS39 DORFF , V. 2000. Ar 2 40Ar dating of multiply zoned amphibole generations (Malenco, Italian Alps). Contributions to Mineralogy and Petrology, 140, 363– 381. V ILLA , I. M., R UGGIERI , G., P UXEDDU , M. & B ERTINI , G. 2006. Geochronology and isotope transport systematics in a subsurface granite from the Larderello– Travale geothermal system (Italy). Journal of Volcanology and Geothermal Research, 152, 20– 50. W HITEHOUSE , M. J., K AMBER , B. S. & M OORBATH , S. 1999. Age significance of U– Th–Pb zircon data from early Archaean rocks of west Greenland – a reassessment based on combined ion-microprobe and imaging studies. Chemical Geology, 160, 201– 224. W IJBRANS , J. R. & M C D OUGALL , I. 1986. Ar-40/Ar-39 dating of white micas from an Alpine high-pressure metamorphic belt on Naxos (Greece) – the resetting of the argon isotopic system. Contributions to Mineralogy and Petrology, 93, 187–194. W ILLIAMS , M. L., J ERCINOVIC , M. J. & H ETHERINGTON , C. J. 2007. Microprobe monazite geochronology: Understanding geologic processes by integrating composition and chronology. Annual Reviews in Earth and Planetary Science, 35, 137–175. W OOD , B. J. & W ALTHER , J. V. 1983. Rates of hydrothermal reactions. Science, 222, 413– 415. Z HAO , Y. H., G INSBERG , S. B. & K OHLSTEDT , D. L. 2004. Solubility of hydrogen in olivine: dependence on temperature and iron content. Contributions to Mineralogy and Petrology, 147, 155– 161.

Argon enters the retentive zone: reassessment of diffusion parameters for K-feldspar in the South Cyclades Shear Zone, Ios, Greece M. A. FORSTER* & G. S. LISTER Research School of Earth Sciences, The Australian National University, Mills Road, Acton, Canberra, ACT 0200, Australia *Corresponding author (e-mail: [email protected]) Abstract: 40Ar/39Ar apparent age spectra have been measured for unusually retentive potassium feldspars (K-feldspar) from the South Cyclades Shear Zone, Ios, Greece. Our results imply that the Argon Partial Retention Zone (Ar PRZ) for the most retentive domains in potassium K-feldspar can expand into the ductile regime, even when temperatures in excess of about 400–450 8C apply. In such cases K-feldspar could be used as a geochronometer to estimate the timing and duration of deformation and metamorphism events. Therefore, we have reassessed traditional methods used to analyse Arrhenius plots by simulating the effect of step-heating experiments on argon loss. Fractal multidomain diffusion models were used, with theoretical distributions of diffusion domain size and volume. We discovered a Fundamental Asymmetry Principle that offers objective constraints on slope fitting to allow an analysis to be consistent with the multidomain diffusion hypothesis, and which consistently leads to the estimation of higher activation energies. Reanalysis of existing datasets is encouraged to allow reassessment of the significance of the average values reported. Retentive diffusion parameters for K-feldspar might prove to be commonplace.

The South Cyclades Shear Zone (SCSZ) is an approximately 1 km-thick bowed-up extensional ductile shear zone that defines the carapace of a deeply eroded gneiss dome that outcrops on the island of Ios in the Cycladic archipelago, Aegean Sea, Greece. The SCSZ is of geological importance as it is a key structural element in the tectonic evolution of the Cycladic Massif. It defines the carapace of the dome defining the first-recognized Aegean metamorphic core complex (Lister et al. 1984, 2007; Lister & Forster 1996). More significantly, Forster & Lister (2009) showed that the SCSZ operated from about 35 Ma, for about 5 Ma, and it therefore links the Eocene high-pressure stage of the evolution of the Cycladic eclogite–blueschist belt with the later lithosphere-scale stretching that took place during rollback of the Hellenic slab. There have been several studies using 40Ar/39Ar geochronology on Ios (Lister & Baldwin 1996; Baldwin 1996; Baldwin & Lister 1998; Forster & Lister 2009). The results are often difficult to interpret because deformation occurred at relatively low temperatures, and was partitioned in discrete, often anastomosing zones (from the metre-scale to the ,mm-scale). This geometry allowed preservation of lenses of relict material, on all scales, while the relatively low temperatures involved allowed the preservation of relatively old apparent ages. This is also true in zones where younger deformation has produced intense overprints of earlier fabrics, with minimum recrystallization. Baldwin & Lister

(1998) suggested that the observed heterogeneity of apparent ages obtained using 40Ar/39Ar geochronology was made possible because the SCSZ operated in the Argon Partial Retention Zone (Ar PRZ), and under such conditions the argon system is only partially reset during overprinting events. By analysing the degree of resetting, or more particularly the degree of preservation of older gas reservoirs, they provided data that allowed modelling of the duration of specific events (using the ‘MacArgon’ software: Lister & Baldwin 1996). Baldwin & Lister (1998) suggested that ambient temperatures prior to deformation pulses were lower than about 280 8C, and that during its operation the SCSZ had been subject to short thermal pulses. It was decided to investigate the question of thermal pulses during shear zone operation in more detail, and therefore argon geochronology was conducted along a traverse through the SCSZ (Forster & Lister 2009). The duration of a thermal pulse can be estimated if the following conditions apply: (1) relatively old apparent ages are preserved through a younger thermal event; (2) the peak temperature during the younger thermal event can be estimated using independent means, for example using metamorphic petrology; (3) loss of argon from relict fabrics is related to diffusion; and (4) accurate estimates are available as to the diffusion parameters that applied. At first sight it appeared that these conditions could be met. Relatively old apparent ages

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 17– 34. DOI: 10.1144/SP332.2 0305-8719/10/$15.00 # The Geological Society of London 2010.

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M. A. FORSTER & G. S. LISTER

Fig. 1. There is some latitude in the way that we can analyse Arrhenius data: (a) ignoring outliers defined by the Arrhenius points for the first temperature steps, here for a line fitted to Arrhenius data from a step-heating experiment conducted on irradiated K-feldspar from the South Cyclades Shear Zone by Baldwin & Lister (1998); (b) a line fitted to

ARGON ENTERS THE RETENTIVE ZONE

are preserved in K-feldspar. The SCSZ operated at temperatures sufficiently high as to allow growth of biotite in microdilation sites (i.e. .400 8C). Locally this growth appears to have taken place immediately adjacent to what appears to be new grown garnet, allowing estimates of the temperatures that applied locally in the SCSZ when it was operating to be in excess of about 450 8C (Spear 1993). Forster & Lister (2009) ascertained the timing of operation of the SCSZ by analysing grains that grew during specific deformation events during the life of the shear zone. By inference, the cores of relict Hercynian K-feldspar phenocrysts lost argon only by diffusion. The remaining issue is to provide accurate estimates of the diffusion parameters that applied, and this is the subject of this paper. It is evident that the SCSZ operated at temperatures above what are loosely referred to as diffusional closure temperatures for the minerals in question. The operation of the shear zone may thus have been associated with short-lived thermal pulses, as suggested by Baldwin & Lister (1998). Since it is possible to eliminate potential heat sources such as advection, either due to the emplacement of magma or due to fluid percolation, the origin of these temperature excursions could be associated with the conversion of mechanical work to heat during plastic deformation. Baldwin & Lister (1998) inferred Arrhenius parameters close to the UCLA average reported by Lovera et al. (1997). If these parameters apply, the duration of the inferred thermal pulses would have to have been extremely short, as reported by Baldwin & Lister (1998). This conclusion is valid, however, if and only if K-feldspar is as unretentive as implied by these diffusion parameters. Our data potentially force us to conclude that this may not be the case. Forster & Lister (2009) were able to use cataclased and/or recrystallized K-feldspar to directly date the timing of different movements in the shear zone, and to show that partial resetting of the argon system only occurred when renewed motion in the shear zone led to deformation and/or recrystallization. These circumstances are possible only if K-feldspar in the SCSZ is far more retentive than these earlier estimates would allow. K-feldspar can only be used as a geospeedometer to constrain the duration of these different events if

19

we are able to accurately estimate the diffusion parameters. Traditionally, such estimates are provided by the analysis of Arrhenius plots. There is considerable latitude in respect to how such Arrhenius data can be analysed, however, as illustrated in Figure 1. Different strategies can be applied, either to reduce the activation energy obtained (Fig. 1a) or to maximize the result (Fig. 1b, c). Significant variation in the inferred activation results, depending on the strategy adopted. Without objective guidelines as to how to proceed, one might be tempted to err in the direction that provides the closest answer to the values reported for argon@ UCLA. We can minimize the inferred activation energy if we do not take into account the results obtained from the first Arrhenius points (e.g. Fig. 1a: Baldwin & Lister 1998). This may be a valid approach as these steps are obtained from the first points in the heating sequence and, supposedly, pertain to degassing of the minute, least retentive, diffusion domains. They may be more susceptible to experimental error. Conversely, we can maximize the inferred activation energy if we ignore the overall trend, and select two or three points from the start of the sequence (Fig. 1c: Mahon et al. 1998).

Fractals and multidomain diffusion models In an effort to provide objective guidelines in respect to the analysis of Arrhenius plots we began to explore ramifications of multidomain diffusion (MDD) models based on fractal distributions of size and volume, using a modelling and simulation approach. As we know the values of the diffusion parameters we actually input, we are able to investigate why variation is obtained in estimates obtained by numerical analysis of the results. To achieve this result ‘Program eAr’ (available for download from http://rses.anu.edu.au/tectonics/programs/) was modified to allow simulation of argon release from arbitrary MDD models, and thereafter to allow the resultant Arrhenius plot to be analysed in the traditional fashion. The data required to be input to generate an arbitrary MDD model includes the diffusion parameters: including E, the activation energy, and D0, the frequency factor; and a list of diffusion domains, specifying for each, its overall

Fig. 1. (Continued) the same data, but now constrained by the first few points; (c) a step-heating experiment conducted on irradiated K-feldspar from a sedimentary basin by Mahon et al. (1998). Baldwin & Lister (1998) reported a value for the activation energy, E ¼ 46.5 kcal mol21, which is close to the UCLA average, but their line of best fit (a) was obtained by rejecting outliers defined by the first two steps. The line places Arrhenius points obtained earlier in the sequence to the right of the line, and thus violates the Fundamental Asymmetry Principle (FAP) explicitly enunciated in this paper. In (b) this same data are fitted by a straight line that obeys the FAP, and this fit yields retentivity estimates that are considerably higher.

20

M. A. FORSTER & G. S. LISTER

volume and a characteristic diffusion radius (in this case the half-thickness of a slab). The temperature history used in the simulated step-heating experiment must also be specified. The temperature histories that we use below consist of sequences of one or more isothermal steps, with the temperature incremented by (for example) 50 8C between each step in the isothermal sequence. The temperature reached during an individual step can be the same as that achieved during the preceding heating step, or it can increase. The sequence of temperatures applied is thus monotonically upward. The step-heating schedules for the modelling below have a starting temperature of 400 8C, with the first step in an isothermal sequence set at 5 min. If the simulation uses several isothermal steps, subsequent steps double in their duration (e.g. heating takes place over 5, 10 or 20 min if three isothermal steps are applied in sequence). To simplify matters, if we utilize isothermal steps, the same number of isothermal steps will be applied for each temperature in the sequence. In all simulations reported in this paper a constant value for the activation energy (E ¼ 75.0 kcal mol21) has been applied to all diffusion domains. Based on the trend line describing the best fit to the UCLA data reported by Lovera et al.

(1997) the frequency factor is determined by: log frequency factor ¼ [(activation energy  29:4)=3:33]  3:0: We were drawn to self-similarity in terms of an explanation for the peculiar distributions of volume and diffusion domain radius in published data (Lovera et al. 1989, 1997). Therefore we used two fractal distributions of volume and diffusion domain size for comparative purposes, the fractal cube (Fig. 2a) and the Menger sponge (Fig. 2b). The distribution of volume and diffusion domain radius is shown in Table 1. It should be noted that the choice of fractal is not of particular significance, however; although in many ways the fractal cube allows simulation of the effect of a cubic diffusion domain with a rough surface. In contrast, the Menger Sponge can simulate some of the effects of recrystallization and grain growth or, conversely, the effect of increasing breakdown of the lattice structure in K-feldspar grains in a weathering environment, or when stewing slowly in a sedimentary basin (e.g. Mahon et al. 1998; cf. Lee 1995 and Parsons et al. 1999). It should also be noted that use of fractal MDD models offers a numerical approximation to quantify the effects of roughness, for as temperatures

Fig. 2. Inferred volume–size distributions used in multidomain diffusion (MDD) models are reminiscent of those obtained in fractal distributions of size and volume (Table 1). In this paper such fractals are used to generate artificial MDD models, based on: (a) a fractal cube, shown here in 2D cross-section, reproduced from http://txspace.tamu.edu/ handle/1969.1/3776?show=full; and (b) a Menger Sponge, reproduced from http://en.wikipedia.org/wiki/ Menger_sponge. See the text for further details. ‘Program eAr’ (http://rses.anu.edu.au/tectonics/programs/) was used to calculate the predicted fraction of would-be-released argon if fractal MDD models based on these distributions were subject to step-heating experiments.

ARGON ENTERS THE RETENTIVE ZONE

21

Table 1. The volume –size distribution obtained: (a) from several iterations of a fractal cube; and (b) from several iterations of a Menger Sponge Iteration

Edge

Cumulative volume

Percentage of total

1 6 30 150 750 3750 18 750 93 750 468 750 2 343 750 11 718 750 58 593 750 292 968 750 1 464 843 750 7 324 218 750 36 621 093 750

1.0000000000 0.5000000000 0.2500000000 0.1250000000 0.0625000000 0.0312500000 0.0156250000 0.0078125000 0.0039062500 0.0019531250 0.0009765625 0.0004882813 0.0002441406 0.0001220703 0.0000610352 0.0000305176

1.0000 1.7500 2.2188 2.5117 2.6948 2.8093 2.8808 2.9255 2.9534 2.9709 2.9818 2.9886 2.9929 2.9956 2.9972 2.9983

100 42.9 21.1 11.7 6.79 4.07 2.48 1.53 0.95 0.59 0.37 0.23 0.14 0.09 0.06 0.03

(b) Menger Sponge 0 1 7 2 140 3 2800 4 56 000 5 1 120 000 6 22 400 000 7 448 000 000 8 8 960 000 000 9 179 200 000 000 10 3 584 000 000 000 11 71 680 000 000 000 12 1 433 600 000 000 000 13 28 672 000 000 000 000 14 573 440 000 000 000 000 15 11 468 800 000 000 000 000 16 229 376 000 000 000 000 000 17 4 587 520 000 000 000 000 000 18 91 750 400 000 000 000 000 000 19 1 835 008 000 000 000 000 000 000

0.3333333333 0.1111111111 0.0370370370 0.0123456790 0.0041152263 0.0013717421 0.0004572474 0.0001524158 0.0000508053 0.0000169351 0.0000056450 0.0000018817 0.0000006272 0.0000002091 0.0000000697 0.0000000232 0.0000000077 0.0000000026 0.0000000009

0.2593 0.4513 0.5936 0.6989 0.7770 0.8348 0.8776 0.9094 0.9329 0.9503 0.9632 0.9727 0.9798 0.9850 0.9889 0.9918 0.9939 0.9955 0.9967

100 42.6 24.0 15.08 10.05 6.93 4.88 3.49 2.52 1.83 1.34 0.98 0.72 0.53 0.39 0.29 0.21 0.16 0.12

(a) Fractal cube 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Number

increase so does the characteristic distance associated with diffusion. In effect the domains become smoother. This means that it is potentially futile to search for a one to one correspondence between elements in the microstructure and individual diffusion domain sizes. Diffusion from lamellae surrounded by what is effectively a network of pipe-like fast escape pathways for argon expelled from the surrounding lattice (Fitz Gerald et al. 2006) may be approximated by a MDD model, for example. The choice of fractal is not particularly significant, although interesting analogies can be drawn in consequence of any particular choice (Ben-Avraham & Havlin 2000; Landrenau 2000). In the fractal cube

(http://txspace.tamu.edu/handle/1969.1/3776?show= full) the largest diffusion domain is a cube, with six cubic diffusion domains of dimension half of that cube edge set upon each face. Upon each of the five free faces of those cubes we then attach five cubes of dimension half of that cube edge, and so on. Eventually an octahedron with holes will be outlined, defining a fractal with dimension [(log 5)/(log 2) ¼ 2.32]. The first iteration generates six cubes with the edge of each cube half that of the original cube. The second iteration generates five cubes for each of the cubes generated in the first iteration, with the edge of each cube half that of the first iteration cubes (a total of 30 cubes in total). Subsequent iterations continue in the same fashion,

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M. A. FORSTER & G. S. LISTER

Fig. 3. The fraction of would-be-released argon is computed for each heating step, and the derivative Arrhenius data is examined graphically. Here we show plots for gas released from (a) a single diffusion domain and then (b, c) the variation that would be obtained when several more iterations of a fractal cube are utilized to determine the volume–size distribution. In (b) eight iterations cause the Arrhenius points to feather away from the unique line that can be drawn

ARGON ENTERS THE RETENTIVE ZONE

halving the cube edge each time. For the Menger Sponge (http://en.wikipedia.org/wiki/Menger_ sponge) the holes are assumed to define ‘intact’ diffusion domains, with the sponge considered to be a mineral lattice that is highly diffusive because of the presence of defects or artifacts (defining a fractal with dimension 2.73). It should be noted that other geometries could also be considered if one attempts to more exactly simulate the effect of surface roughness for a 3D diffusion domain geometry.

The Fundamental Asymmetry Principle An important result emerged, which we have termed the Fundamental Asymmetry Principle (or FAP), since it is implicit in (and required by) any MDD model. The FAP applies to any line fitted to data from a sequence of step-heating experiments in which the temperature applied from step to step may be the same, or higher, but may not decrease. Lovera et al. (1997) set out the basis of the MDD method, and shows how a straight line can be fitted to a set of Arrhenius data points, under specific circumstances. The fitted line will faithfully replicate the essential Arrhenius parameters only as long as the data points are derived from a set of partially degassed diffusion domains. In part, this is because the estimated diffusivity from any MDD model will progressively decrease as the less retentive domains are degassed. Mathematically, it follows that, to be consistent with a MDD model, a line fitted to any sequence of Arrhenius points must divide the population by rank order. Points from data obtained earlier in the sequence of step-heating experiments must lie on the fitted line, or to the right of it. Points from data obtained later in the sequence must lie on the fitted line, or to the left of it. Violations of the FAP will lead to consistently underestimated values for the activation energy (E) and for the frequency factor (D0).

Variation owing to domain size range Figure 3 shows modelling for an Arrhenius plot derived from a single diffusion domain, and then the variation that would be obtained if eight then 16 iterations of a fractal cube were utilized to

23

determine the volume– size distribution. This volume –size distribution is shown in Table 1. It is evident that smaller domain sizes define an increasing percentage of the total volume as the depth of the fractal size –volume relationship increases. The simulations show that these smaller grains have a significant impact on the Arrhenius plots that will be obtained. As described by Lovera et al. (1997) for a single diffusion domain (Fig. 3a), we obtain a single straight line and we are able to accurately determine the activation energy used in the simulation (E ¼ 75 kcal mol21). This is also true for a volume –size distribution defined by eight iterations of the fractal cube. However, at higher temperatures the simulation shows a deviation from this straightline trend. As noted by Lovera et al. (1997), this deviation begins to occur as degassing of smaller domains is completed. Nevertheless, applying the Fundamental Asymmetry Principle we were still able to obtain an accurate estimate of the activation energy (Fig. 3b). In the third simulation of this type we utilize a volume –size distribution defined by 16 iterations of the fractal cube (Fig. 3c). The result is that the initial slope is no longer discernible: a direct consequence of this expanded volume–size distribution. Again, as noted by Lovera et al. (1997), the deviation is what is to be expected once smaller grains have completely degassed. The initial slope is no longer visible because, even at the lowest temperatures used in the step-heating experiment, there are smaller grains that have completely degassed. In these circumstances, any attempt at line-fitting will result in a significant underestimation of the activation energy. Using a standard statistical approach to determine the line of best fit (e.g. least squares, allowing a few outliers to be rejected), a lesser value for the activation energy is obtained. Coincidentally, the value obtained (E ¼ 46.5 kcal mol21) is close to the UCLA average (E ¼ 46 kcal mol21). We discover that application of the Fundamental Asymmetry Principle allows by far the most accurate estimates, although only two points could be utilized. Although two points are all that is required to define a maximum slope (thereby here predicting E ¼ 72.0 kcal mol21) this approach would be questionable on numerical grounds if real data were

Fig. 3. (Continued) using the Fundamental Asymmetry Principle (FAP) that stems from the MDD assumption. In (c) 16 iterations of the fractal generate a MDD model that is dominated by this fractal feathering. The single diffusion domain model (a) gives a single straight line with the activation energy used in the simulation (E ¼ 75 kcal mol21). The same slope can be recognized in (b). In (c) the dotted line is compliant with the FAP and produces the closest estimate (72.6 kcal mol21), whereas the solid line might appear more appropriate, statistically speaking, but it is not compliant with the FAP. The effect of fractal feathering produced a fit that yields a value (E  46.5 kcal mol21) equivalent to the UCLA average.

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M. A. FORSTER & G. S. LISTER

being utilized, since errors may be introduced in many ways. Errors that originate from the actual measurements need to be taken into account, which may require some FAP violations to be tolerated. Application of standard statistical methods (e.g. least squares, allowing a few outliers to be rejected) will produce results that consistently underestimate the Arrhenius parameters. Although strict adherence to the Fundamental Asymmetry Principle sometimes requires a choice of only two points to define the fitted line, perhaps this practice should be avoided. Use of too few points may lead measurement inaccuracies to cause the use of the FAP to somewhat overestimate the retentivity. Even so, leaving the topic of error in real measurements for later discussion, theoretical simulation of the effect of step-heating experiments on MDD models show that application of the Fundamental Asymmetry Principle provides the minimum underestimate of the actual value of the activation energy used in these simulations, whereas allowing FAP violations significantly increases the error in any particular determination.

The effect of isothermal duplicates Lovera et al. (1997) pointed out that the use of sequences of isothermal steps in a step-heating experiment provides a strategy that allows discrimination of Arrhenius data that has been affected by degassing of less retentive domains. The experiments shown in Figure 3 were, therefore, repeated using sequences of isothermal steps (see Fig. 4). There are effects that relate to the number of isothermal steps, but in this paper we report only on the consequence of two steps at each temperature. It is evident that once degassing of less retentive domains begins to occur, the estimates of E and D0/r 2 in these two steps begin to differ. The effect is evident only on the last step of the simulation for a single diffusion domain (Fig. 4a). Simulations using isothermal steps on the volume– size distribution for eight iterations of the fractal cube show that the effect is evident immediately the change of slope begins to take place (Fig. 4b). Simulations using isothermal steps on the volume– size distribution for 16 iterations of the fractal cube show that the feathering effect can be evident from the outset (e.g. Fig. 4c). Interestingly, the variation obtained by including two isothermal steps allows recognition that the feathering is taking place. A fitted line obeying the Fundamental Asymmetry Principle allows a far more accurate estimate of the activation energy, E ¼ 69.5 kcal mol21 (i.e. with an error of c. 7%, Fig. 4c), than the line fitted ignoring the initial steps, and which yielded

E  46.5 kcal mol21 (i.e. with an error of c. 40%, Fig. 3c).

Variation owing to discontinuous domain size ranges The final type of simulation explored in this paper considers the effect of variation in fractal dimension, and the effects of non-overlapping but limited domain size ranges. Figure 5a shows an Arrhenius plot derived for a MDD model based on four iterations of a Menger Sponge. This should be compared with Figure 3b, which shows an Arrhenius plot derived for a MDD model based on eight iterations of a fractal cube. A fractal cube volume– size relation has a fractal dimension of 2.32, whereas, with a fractal dimension of 2.73, the Menger Sponge will always have a larger volume fraction of small grains (Table 1). An increase in fractal dimension leads to higher values of D0/r 2 being estimated. The effect of non-overlapping but limited domain size ranges similarly causes larger offsets between the value of D0/r 2 estimated for the largest domains and the average values determined by the fitted line. Figure 5b shows the effect of a population of large domains co-existing with a population of much smaller domains, with a volume– size distribution defined by a Menger Sponge. Four iterations of the Menger Sponge were allowed to take place before new diffusion domains were added to the volume –size distribution. The smaller diffusion domains are approximately 0.05 –1% of the size of the large domain, but they account for around 22% of the total volume. The result is a marked offset in the Arrhenius plot. A feathering effect is observed when the smaller diffusion domains are very much smaller than the larger domains. In Figure 5c, 16 iterations of the Menger Sponge are allowed to take place before new diffusion domains are added to the volume – size distribution. The smaller diffusion domains are now 4–5 orders of magnitude smaller than the smaller domains in the volume –size distribution used in the previous simulation, and the smaller domains now account for a mere approximately 3% of the total volume. The result is a marked deviation in slope, and a trend away from the maximum slope that represents an accurate estimation of the actual activation energy (E ¼ 75 kcal mol21). The inflection in the gradient inferred from the Arrhenius plot results from the population of minute diffusion domains defined by the Menger Sponge. This is interesting because, as we will show later in this paper, actual data from real K-feldspar often reflects similar inflections, and such effects have not been

ARGON ENTERS THE RETENTIVE ZONE

25

Fig. 4. Theoretical Arrhenius plots derived for gas released as a consequence of two heating steps at each temperature (i.e. isothermal duplicates). Once degassing of less retentive domains occurs, the estimates of E and D0/r 2 in these two steps begin to differ. In (a) the effect is evident only on the last step of the simulation for a single diffusion domain; in (b) simulations using isothermal steps on the volume– size distribution for eight iterations of the fractal cube show the effect is immediate once the change of slope takes place; and (c) simulations using isothermal steps on the volume– size distribution for 16 iterations of the fractal cube show the effect is evident from the outset.

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M. A. FORSTER & G. S. LISTER

Fig. 5. Simulations showing effects related to the relative size of large v. small domains. In (a) the simulation uses the Menger Sponge (Table 1) for four iterations. In (b) there is a single large domain, and four iterations of the Menger Sponge take place before new diffusion domains are added to the volume– size distribution. The smaller diffusion domains are approximately 0.01– 1% of the size of the large domain. r/r0 methods can be simply applied where r0 is the half-edge of the reference domain, and r the half-edge under consideration. In (c) 16 iterations of the Menger Sponge are allowed to take place before new diffusion domains are added to the volume–size distribution. Fractal feathering takes place when the smaller diffusion domains are much smaller than the larger domains.

ARGON ENTERS THE RETENTIVE ZONE

taken into account previously in descriptions of variations expected as the result of multidomain diffusion theory (Lovera et al. 1997). Irrespective of the above considerations, application of the Fundamental Asymmetry Principle to the Arrhenius data obtained in these simulations allows more accurate estimates of the activation energy. The effect of ‘iteration level’ may be important since the effect of diffusion in a deformed crystal might be simulated by switching the fractal dimension early in the iteration process, and allowing a greater number of iterations to define the population of smaller grains in the Menger Sponge.

Application of the Fundamental Asymmetry Principle to the analysis of real data The Fundamental Asymmetry Principle can be readily applied to the analysis of real data. Here we compare and contrast the results of Arrhenius data analysed so that the fitted line does not violate the FAP, with the results of the application of statistical methods requiring a minimum number of three or more data points. From the theoretical simulations it is evident that effects related to fractal feathering lead to consistent underestimates of the actual activation energy. The same might be expected in real data. The data from the step-heating experiments performed on K-feldspar from the South Cyclades Shear Zone were therefore re-evaluated, using ‘Program eAr’ to implement methods as described in this paper. Figure 6a, c, e show three examples for which the data are least scattered, and the result obtained using a line fitted using leastsquares regression, but allowing the rejection of outliers. Figure 6b, d, f illustrate the reanalysis of these data in a way that is compliant with the Fundamental Asymmetry Principle (i.e. no FAP violations allowed). There is a systematic increase in the magnitude of the activation energy estimated. Similarly, Figure 7 shows more scattered Arrhenius data, with lines fitted using least-squares regression, allowing the rejection of outliers. Figure 8 shows the effect of reanalysis of these data with no FAP violations allowed. There are quite significant increases in the magnitude of the activation energy estimated. It is also evident that these data provide examples of inflections of the type illustrated in Figure 5c. By analogy, this feathering might be taken to imply a fractal MDD model simulating a high degree of roughness and/or heterogeneity in the domain size distribution. A comparison of the apparent retentivity of the samples determined utilizing these two contrasting strategies is provided in Table 2. This shows that

27

application of the Fundamental Asymmetry Principle causes the average of the activation energies determined to increase from c. 60 kcal mol21 to a total of c. 73 kcal mol21. This is c. 1.6 times the UCLA average reported by Lovera et al. (1997). Figure 9a, b shows the inferred diffusion parameters plotted on a graph modified from Lovera et al. (1997). Figure 9a shows data derived by fitting a line through the Arrhenius points based on the methods developed at UCLA, as described by Lovera et al. (1997). The values obtained are significantly more retentive than the UCLA average (activation energy, E ¼ 46 kcal mol21), although they fall approximately on the same trend line and within the limits of the spread of data reported by Lovera et al. (1997). Figure 9b shows data derived by fitting a line through the Arrhenius points based on the methods developed in this paper, requiring compliance with the Fundamental Asymmetry Principle.

Discussion This study was conducted in order to help us analyse Arrhenius data from a sequence of step-heating experiments applied to K-feldspar from the South Cyclades Shear Zone. We obtained ‘suspiciously high’ values for the activation energy, and therefore put some effort into ensuring that there were no systematic measurement errors (e.g. as would result if temperature was consistently overestimated as a result of calibration errors). Finally, examining the results of Baldwin & Lister (1998) from the same rocks (Fig. 1a), we realized that there were numerical issues in respect to the way different authors analysed their data. Baldwin & Lister (1998) reported a value for activation energy (E ¼ 46.5 kcal mol21) and this is a value that is close to the UCLA average (Lovera et al. 1997). Figure 1a shows, however, that the line of best fit was obtained for five points, rejecting outliers defined by the first two steps. If we used the initial steps, however, we obtained a higher estimate for the activation energy (Fig. 1b). Is one estimate better than the other? The work conducted in this paper was an attempt to resolve this question. In one aspect we obtained a clear answer. Mathematically, there is a fundamental asymmetry required by the set of diffusion equations in an MDD experiment. Arrhenius data points for domains less retentive than those used to determine the line of best fit must fall on, or to the right of, the line, while data points for domains more retentive than those used to determine the line of best fit must fall on, or to the left of, that same line. Baldwin & Lister (1998) were able to obtain a value close to the 46 kcal mol21 UCLA

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M. A. FORSTER & G. S. LISTER

Fig. 6. Arrhenius plots for data obtained from some of the step-heating experiments conducted on irradiated K-feldspar from the South Cyclades Shear Zone, using ‘Program eAr’ (http://rses.anu.edu.au/tectonics/programs/). In (a), (c) and (e) lines have been statistically fitted to the initial data points using a method of least squares, allowing outlying points to be rejected. The FAP has been violated, so the activation energy has been underestimated. In (b), (d) and (f ) the same data is reanalysed without FAP violations being allowed. The activation energies estimated have risen, although within these data there is little scatter.

average, but only by (inadvertent) violation of the Fundamental Asymmetry Principle (FAP). Otherwise they would have obtained higher values, but, again, not as high as reported in this study. The activation energy estimated in Figure 1b is E ¼ 60.2 kcal mol21. This is lower than the values

obtained in our analysis of K-feldspar from the SCSZ, but it is nevertheless close to the minimum values we obtained. Note, however, that Baldwin & Lister (1998) did not use a step-heating schedule that involved isothermal duplicates, whereas our simulations show that this is a key aspect

ARGON ENTERS THE RETENTIVE ZONE

29

Fig. 7. Arrhenius plots for more scattered data obtained from the step-heating experiments. Lines have been statistically fitted to the initial data points using a method of least squares, allowing outlying points to be rejected. The FAP has been violated, so the activation energy has been underestimated.

if one wants to avoid underestimates of the actual activation energy. In any analysis of Arrhenius data, systematic application of the FAP leads consistently to higher estimates of activation energy. The Arrhenius data obtained by applying the FAP (Fig. 9b) can be seen, in part, to plot outside of the range of values recorded by Lovera et al.

(1997) from the extensive UCLA dataset. Are these values realistic? We cannot say. Step-heating experiments with real K-feldspar are as well behaved as theoretical considerations allow. We have shown that there are issues with the different ways that numerical methods can be applied to the Arrhenius data. Described simply, the problem is how to select which points to

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Fig. 8. The same data reanalysed, again using a method of least squares, but now complying with the FAP. It is evident that these data provide several examples of inflections that can be simulated by fractal feathering in the MDD model. Estimates of activation energy are consistently greater than when the FAP is not taken into account.

include in a numerical analysis of a line of best fit when attempting to determine values for activation energy and frequency factor on an Arrhenius plot. We illustrated this point in Figure 1, first using data from Baldwin & Lister (1998). These authors chose Arrhenius points (Fig. 1a) that took no account of outliers defined by the data from the least retentive domains, inadvertently violating the FAP, but nevertheless thereby producing a lower

estimate for the activation energy. Coincidentally, this value was close to the UCLA average. In Figure 1c we illustrate another dataset, this time from Mahon et al. (1998) who appear to have used as few as two or three data points to determine the slope. This strategy will always maximize the estimate obtained for the activation energy, which otherwise in this case would have been far less than the UCLA average. Interestingly, this analysis appears

ARGON ENTERS THE RETENTIVE ZONE

31

Table 2. The values of the Arrhenius parameters obtained during analysis of K-feldspar from the South Cyclades Shear Zone (SCSZ), first using minimum least-squares line-fitting methods allowing the rejection of statistical outliers, and then using the Fundamental Asymmetry Principle to constrain the selection of the line to be fitted. The FAP cannot be significantly violated if MDD theory is to be applied. In consequence, the average of the activation energies (E) determined increased from approximately 60 kcal mol21 to a total of c. 73 kcal mol21 E (kcal mol21)

E using FAP (kcal mol21)

Increase (kcal mol21)

AG03-07 AG01-01 AG04-01 AG03-05 AG03-06 AG03-01 AG02-01 IOS94-5 AG03-03

57.5353 69.8196 60.9885 58.5170 60.3764 64.2191 54.9270 61.9623 57.1245

74.3178 81.7792 75.7240 65.1198 62.7799 74.4001 80.3872 76.7443 64.4695

16.7825 11.9596 14.7355 6.6028 2.4035 10.1810 25.4602 14.7820 7.3450

Mean value Standard deviation

60.6077 4.4462

72.8580 7.0349

12.2502 6.7538

Sample

Fig. 9. Arrhenius plots allow the inference of diffusion parameters by line fitting, with the results here superimposed on an image of a graph of data from the UCLA repository published by Lovera et al. (1997). The values obtained at ANU are significantly more retentive than the UCLA average, although they fall approximately on the same trend line. In (a) results are plotted based on the fitting of a straight line through the initial points, allowing rejection of a few outliers. Such statistical methods consistently underestimate the retentivity of K-feldspar, as shown in (b) where the same data have been reanalysed taking account of the Fundamental Asymmetry Principle (FAP), the discovery of which is reported in this paper. The FAP is a mathematical consequence of the assumption that argon is simultaneously released by diffusion from multiple diffusion domains; i.e. it is a consequence of MDD theory. Hence, line fitting must be compliant with the FAP if MDD theory is to be applied to the subsequent analysis of this data.

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M. A. FORSTER & G. S. LISTER

consistent with the FAP, and is therefore consistent with the principles we advocate in this paper. It is beyond the scope of this paper to reanalyse data from the entire UCLA dataset. We would have to begin by eliminating all experiments that did not perform isothermal duplicates, for example, as such experiments lose vital information and invariably allow lower estimates of the activation energy to be obtained. Perhaps, indeed, systematic application of the FAP would lead to a different distribution of Arrhenius data. The SCSZ K-feldspar does appear to be unusually retentive. The ‘MacArgon’ program (adapted for use with the MacOSX operating systems) was used to compute closure temperatures for cooling through the Argon Partial Retention Zone (Table 3), for a range of cooling rates, and for zero pressure as well as for a pressure of 10 kbar (100 MPa). Calculations for a single sample (AG03-07) are tabulated. A fit of the Arrhenius data that produces values close to the UCLA average can be obtained if the first isothermal duplicate is discounted by discarding the first two Arrhenius points as outliers, and using a mean square regression on the next four isothermal duplicates (i.e. allowing FAP violations). The K-feldspar is relatively retentive compared to the Lovera et al. (1997) analysis of the UCLA archive [E ¼ 46.9 kcal mol21, log(D0/r 2) ¼ 5.37] because there is a 4 orders of magnitude difference

in D0/r 2 values for the most retentive domains compared with data for the fitted line. A fit of the Arrhenius data produces values close to the ANU average (E  73 kcal mol21, see Table 2) if no significant FAP violations are allowed, and a minimum of three Arrhenius points included. There is now a 7 orders of magnitude difference in D0/r 2 values for the most retentive domains compared with data for the fitted line, and this leads to a considerable increase in their estimated closure temperatures. We thus suggest that it is not reasonable to expect that all K-feldspar should display Arrhenius parameters that compare with the UCLA average (Lovera et al., 1997) or with the gem-quality Madagascar K-feldspar studied by Arnaud & Kelley (1997). We note that there are no particular issues with the computation of lines of best fit in many experiments, but for some experiments there can be significant outliers. These same experiments commonly show inflection points in the variation of gradient. We have shown that such inflection points can be readily explained by effects related to fractal feathering and limits to the size ranges encompassed by coexisting fractal populations. In terms of numerical analysis, the MDD method allows the effect of the roughness of a diffusion domain to be simulated, and/or the effect of variations in the spacing of fast-diffusion escape pathways for the argon released from a diffusion domain. Effects as

Table 3. The ‘MacArgon’ program (adapted for use with the MacOSX operating systems) is used to compute closure temperatures that would apply for cooling through the Argon Partial Retention Zone, for a range of cooling rates, for zero pressure, as well as for the case when a pressure of 10 kbar (100 MPa) was applied. A single sample was analysed (AG03-07) with a value close to the UCLA average obtained if the first two Arrhenius points were discounted, and the ANU average if these points are taken into account, and no violations of the FAP allowed. Only data for the most retentive domains are shown Closure temperatures using: ‘MacArgon’ for MacOSX Activation energy Activation volume Most retentive domains D0/r 2 100 8C Ma21, 0 kbar 10 8C Ma21, 0 kbar 1 8C Ma21, 0 kbar 100 8C Ma21, 10 kbar 10 8C Ma21, 10 kbar 1 8C Ma21, 10 kbar D0/r 2 for fitted line 100 8C Ma21, 0 kbar 10 8C Ma21, 0 kbar 1 8C Ma21, 0 kbar 100 8C Ma21, 10 kbar 10 8C Ma21, 10 kbar 1 8C Ma21, 10 kbar

Fitted using statistical method

Fitted using FAP

46.9 kcal mol21 10 cm3

73.5 kcal mol21 10 cm3

33.5 s21 399 8C 360 8C 325 8C 432 8C 391 8C 354 8C

5.65  105 s21 553 8C 514 8C 479 8C 579 8C 539 8C 503 8C

2.32  105 s21 269 8C 243 8C 219 8C 296 8C 269 8C 244 8C

1.34  1012 s21 354 8C 332 8C 310 8C 375 8C 351 8C 329 8C

ARGON ENTERS THE RETENTIVE ZONE

reported in this paper could be obtained by allowing fractal variation in the geometry of fast-diffusion pathways that allow escape of argon; for example, a chicken-wire mesh of irregularly spaced pipes acting as fast escape pathways (cf. Fitz Gerald et al. 2006). We suggest that the MDD model merely approximates the effect of such non-ideal geometries; for example, the effect of a rough-edged diffusion domain. We suggest fractal feathering in Arrhenius plots is commonplace, especially in samples altered by weathering or samples that have been stewed at low temperature in a sedimentary basin before they have been subject to step-heating experiments (e.g. Mahon et al. 1998). Such effects introduce uncertainty that so far has not been taken into account in attempts to extract cooling histories from K-feldspar. The time dependence of microstructural changes during deformation and metamorphism (or while stewing in a sedimentary basin) might be approximated by increasing depth in the fractal distribution of size and volume. Extraction of useful geospeedometry data (e.g. with respect to the cooling path) from a microstructure undergoing such a progressive time-dependent evolution would become increasingly problematic.

Conclusion The Argon Partial Retention Zone using the UCLA average value of E of 46 kcal mol21 would restrict the resting state of the SCSZ so that it could not lie within the ductile field (T . 350 8C) if such values applied. If the ANU average is applied then argon enters the retentive zone: (1) the resting state of the SCSZ can lie within the ductile field; (2) the duration of the thermal pulse associated with the operation of the SCSZ can be as long as the 5 Ma we estimate that this shear zone was active; (3) a far smaller difference in temperature may apply between the shear zone resting state and its operating state. In other words, the more retentive values for the Arrhenius parameters allow a far more realistic geodynamic scenario to be modelled. Therefore, we conclude that the retentive values we have estimated for the Arrhenius parameters are not unreasonable given the constraints offered by the geological environment in which they are found. As noted by Baldwin & Lister (1998), thermal pulses associated with the operation of the SCSZ have to be extremely (and unrealistically) short if Arrhenius values as defined by the UCLA average apply. There are issues to be resolved in respect to the analysis of errors, but these are minimized if step-heating experiments are utilized that involve sequences of isothermal duplicates. Fractal feathering does appear to be commonplace, and this, in part, could explain the differences in the average

33

values obtained for the Arrhenius parameters for argon@ANU compared with argon@UCLA. Nevertheless, any analysis of Arrhenius data from step-heating experiments using K-feldspar should take account of the Fundamental Asymmetry Principle as this is an inherent part of any multidomain diffusion model. If the Fundamental Asymmetry Principle is not applied then numerical analysis will invariably underestimate the actual value of activation energy used in simulating the effect of step-heating experiments on fractal volume – size distributions. M. A. Forster acknowledges the support of an Australian Research Fellowship provided by the Australian Research Council (ARC). Research support provided by ARC Discovery Grants DP0449975 ‘Revisiting the Alpine Paradigm: The Role of Inversion Cycles in the Evolution of the European Alps’ and DP0343646 ‘Tectonic Reconstruction of the Evolution of the Alpine–Himalayan Orogenic Chain’. Irradiations were funded by the Australian Institute of Nuclear Science and Engineering (AINSE Award Grants) and facilitated by the Australian Nuclear Science and Technology Organization (ANSTO) at Lucas Heights, New South Wales, Australia. Argon analyses and microprobe analyses were carried out at the Research School of Earth Sciences Laboratories at the Australian National University. Spectra analysis was performed with ‘Program eAr’ written by G. S. Lister. The Institute of Geological and Mining Exploration (IGME) provided permission for fieldwork and sample collection in Greece. O. Lovera and an anonymous reviewer are thanked for their contribution to the final paper.

References A RNAUD , N. & K ELLEY , S. 1997. Argon behaviour in gem-quality orthoclase from Madagascar: experiments and some consequences for 40Ar/39Ar geochronology. Geochimica et Cosmochimica Acta, 61, 3227– 3255. B ALDWIN , S. L. 1996. Contrasting P– T– t histories for blueschists from the western Baja terrane and the Aegean: Effects of synsubduction exhumation and backarc extension. In: B EBOUT , G. E., S CHOLL , D. W., K IRBY , S. H. & P LATT , J. P. (eds) Subduction Top to Bottom. American Geophysical Union, Geophysical Monograph, 96, 135–141. B ALDWIN , S. L. & L ISTER , G. S. 1998. Thermochronology of the South Cyclades shear zone, Ios, Greece: effects of ductile shear in the argon partial retention zone. Journal of Geophysical Research, 103, 7315– 7336. B EN -A VRAHAM , D. & H AVLIN , S. 2000. Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge. F ITZ G ERALD , J. D., P ARSONS , I. & C AYZER , N. 2006. Nanotunnels and pull-aparts: defects of exsolution lamellae in alkali feldspars. American Mineralogist, 91, 772 –783. F ORSTER , M. A. & L ISTER , G. S. 2009. Core-complexrelated extension of the Aegean lithosphere initiated

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at the Eocene–Oligocene transition. Journal Geophysical Research (Solid Earth), 114, B02401; doi: 10.1029/2007JB 005382. L ANDRENAU , E. 2000. Crystal-like Geometric Modeling. MSc thesis, Texas A&M University (http://txspace. tamu.edu/handle/1969.1/3776?show=full). L EE , J. K. W. 1995. Multipath diffusion in geochronology. Contributions to Mineralogy and Petrology, 120, 60–82. L ISTER , G. S., B ANGA , G. & F EENSTRA , A. 1984. Metamorphic core complexes of Cordilleran type in the Cyclades, Aegean Sea, Greece. Geology, 12, 221–225. L ISTER , G. S. & B ALDWIN , S. L. 1996. Modelling the effect of arbitrary P–T –t histories on argon diffusion in minerals using the MacArgon program for the Apple Macintosh. Tectonophysics, 253, 83– 109. L ISTER , G. S. & F ORSTER , M. A. 1996. Inside the Aegean Metamorphic Core Complexes: A Field Trip Guide Illustrating the Geology of the Aegean Metamorphic Core Complexes. Australian Crustal Research Centre, Technical Publication, 45. Australian Crustal Research Centre, Department of Earth Sciences, Monash University, Melbourne, Australia. L ISTER , G. S., F ORSTER , M. & R ING , U. 2007. Inside the Aegean Metamorphic Core Complexes. Journal of the

Virtual Explorer. Electronic Edition, 27, Paper 1; doi: 10.3809/jvirtex.2007.00167. L OVERA , O. M., G ROVE , M., H ARRISON , T. M. & M AHON , K. I. 1997. Systematic analysis of K-feldspar 40 Ar/39Ar step heating results: I. Significance of activation energy determinations. Geochimica et Cosmochimica Acta, 61, 3171–3192. L OVERA , O. M., R ICHTER , F. M. & H ARRISON , T. M. 1989. The 40Ar/39Ar thermochronometry for slowly cooled samples having a distribution of diffusion domain sizes. Journal of Geophysical Research, 94, 17,917–17,935. M AHON , K. I., H ARRISON , T. M. & G ROVE , M. 1998. The thermal and cementation histories of a sandstone petroleum reservoir, Elk Hills, California. Part 1: 40 Ar/39Ar thermal history results. Chemical Geology, 152, 227 –256. P ARSONS , I., B ROWN , W. L. & S MITH , J. V. 1999. 40 Ar/39Ar thermochronology using alkali feldspars: real thermal history or mathematical mirage of microtexture? Contributions to Mineralogy and Petrology, 136, 92– 110. S PEAR , F. S. 1993. Metamorphic Phase Equilibria and Pressure –Temperature–Time Paths. Mineralogical Society of America, Monograph, 799 pp.

Serrated quartz grain boundaries, temperature and strain rate: testing fractal techniques in a syntectonic granite MANISH A. MAMTANI1* & R. O. GREILING2 1

Department of Geology & Geophysics, Indian Institute of Technology, Kharagpur – 721302, India

2

Lehrstuhl fu¨r Strukturgeologie und Tektonophysik, Geologisches Institut, Universita¨t Karlsruhe (TH), Hertzstrasse 16, 76187 Karlsruhe, Germany *Corresponding author (e-mail: [email protected])

Abstract: In the past fractal (ruler) dimension (Dr) of quartz grain-boundary sutures and area – perimeter fractal dimension (Da) of quartz grains, respectively, have been shown to depend on temperature (T ) and strain rate. However, the application of these methods to gauge temperature and strain rate in naturally deformed intrusive rocks has not yet been tested. In the present study Dr and Da are calculated in 12 thin sections from different parts of a syntectonic granite (Godhra Granite, India). Of these, six belong to the northern part, two to the central part and four to the southern part of the granite. Earlier work on the Godhra Granite showed both a strain and a temperature gradient, with high temperature in the north and high strain in the south. Microstructural studies reveal that the quartz grain-boundary sutures are less serrated in the northern samples compared to those from the remaining part of the granite. The northern samples contain abundant high-temperature solid-state deformation fabrics that formed between 675 and 725 8C (quartz chessboard pattern thermobarometry). Using a Dr v. T plot given by earlier workers, a Dr value of 1.05– 1.14 is expected for the above T range. Dr calculations of quartz sutures from the northern samples give a median of 1.11 and most of the sutures have Dr , 1.14. These data fit well with the expected temperature range in which the quartz chessboard pattern formed in the Godhra Granite. The central and southern parts of the granite are dominated by myrmekites (500– 670 8C), recrystallized feldspars (450–600 8C), deformation twins in feldspar (400–500 8C) and kinked biotite (,300 8C). The expected Dr of quartz sutures under the above medium–low temperature ranges are 1.07–1.23, 1.11–1.25, 1.16–1.28 and ,1.27, respectively. Dr calculations reveal that most of the quartz sutures from the central þ southern part have Dr .1.14, and the median values are 1.18 (centre) and 1.17 (south). Using the Dr v. T plot, these Dr values indicate that most of the textures in the central þ southern part of the Godhra Granite formed in the temperature range of 450–600 8C, which fits well with the temperature range required for the development of medium– low temperature fabrics that dominate this part of the granite. Thus, it is concluded that Dr of quartz sutures can be used as a geothermometer in syntectonic granites. Da for northern and southern samples is 1.10 and 1.14, respectively. Strain rates of the order of approximately 1027 and 10211 s21, respectively, are obtained for high (675 8C) and low temperature (300 8C) using area-perimeter fractal dimension (Da) values. Although these are higher than geological strain rates that are known in nature (10212 – 10215 s21), the calculated values for the lower-temperature range are similar to strain rates estimated for intrusions (10210 –10212 s21). The calculations indicate that the method to calculate strain rate using Da of quartz grains fails to give geologically reasonable strain rates for high temperature in a syntectonic granite. However, the method maybe useful in obtaining reasonable strain rate estimates for lower temperatures.

Over the past few decades it has been generally accepted that many geological phenomena such as frequency –size distributions of rock fragments, faults, earthquakes, volcanic eruptions, mineral deposits and oil fields are non-linear and scale invariant (Turcotte 1997). Such phenomena have therefore been analysed and quantified using fractal geometry techniques. The shapes of mineral grain boundaries are generally irregular, and fractal

geometry is a useful tool to quantify complex irregular patterns (Mandelbrot 1983). Various methods can be applied for fractal analysis, viz.: ruler (variously referred to as compass, divider, perimeter), box counting, Cantor’s Dust and area –perimeter methods. In structural geology, different methods have been applied to analyse different structures or patterns. For example, the ruler and area – perimeter methods are used for quartz grain

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 35– 48. DOI: 10.1144/SP332.3 0305-8719/10/$15.00 # The Geological Society of London 2010.

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M. A. MAMTANI & R. O. GREILING

boundaries (Kruhl & Nega 1996; Takahashi et al. 1998), box counting and Cantor’s Dust for analysing fracture patterns (e.g. Hirata 1989; Velde et al. 1990; Volland & Kruhl 2004). Kruhl & Nega (1996) demonstrated that quartz grain-boundary sutures in rocks show fractal behaviour. They investigated the fractal (ruler) dimensions (denoted here as Dr) of quartz grain boundaries in different deformed metamorphic and igneous rocks, and found that the fractal (ruler) dimension decreases with increasing temperature. Thus, Kruhl & Nega (1996) advocated Dr of quartz grain-boundary sutures as a geothermometer. Apart from temperature, quartz textures also depend on strain rates. Takahashi et al. (1998) carried out fractal analysis of experimentally recrystallized quartz grain shapes by calculating the area –perimeter fractal dimension (referred here as Da). Based on this research they proposed that the strain rate (1˙ ) of a rock can be estimated from the Da and temperature (T ) of dynamically recrystallized quartz grains. However, these techniques have not yet been applied to determine T/1˙ in naturally deformed rocks. Thus, the application of the above techniques to decipher deformation conditions in natural rocks remains to be tested. The present study tries to fill this lacuna by measuring the Dr and Da of quartz grains in the Godhra Granite. This granite is chosen because syntectonic fabrics and aspects related to temperature, depth of emplacement (i.e. pressure, P) and strain-intensity variations in the granite are known from previous investigations (Mamtani & Greiling 2005; Sen et al. 2005; Sen & Mamtani 2006).

The Godhra Granite – fabrics and P – T data The Godhra Granite is located in the southern parts of the Aravalli Mountain Belt (AMB), NW India (Fig. 1a). It is intrusive into the surrounding country rocks and is 955 + 20 Ma in age (Gopalan et al. 1979). It is coarse– prophyritic, with fine-grained varieties cutting through the coarser varieties at a few places. Quartz, K-feldspar, plagioclase and biotite are the major minerals, and sphene, apatite, magnetite and ilmenite are subsidiary. The northern part of the granite (around Godhra and to its north) has magmatic fabrics defined by preferentially oriented feldspar laths, magmatic shear zones (Fig. 2a) and magmatic folds (e.g. Vigneresse et al. 1996). Oscillatory zoning in plagioclase, perthites in K-feldspar and fractures in K-feldspar healed by sodic rims joining adjacent plagioclase grains (Fig. 2b) can be interpreted as magmatic microstructures (Bouchez et al. 1992; Blenkinsop 2000; Pawley & Collins 2002; Sen &

Mamtani 2006). A chessboard pattern in quartz is also prominent in this part (Fig. 2c). The latter is a high-temperature solid-state deformation fabric that forms in the b-quartz stability field and is thus dependent on P –T conditions (Kruhl 1996). During intrusion, garnet in schists of the country rocks developed rims, which re-equilibrated with biotite, muscovite, plagioclase, quartz and ilmenite (Mamtani & Karanth 1997; Mamtani et al. 1999, 2001). Using barometric formulations, pressures were constrained between 4–6 kbar for the corresponding equilibration T (c. 500 8C) in the schist (Mamtani 1998; Bakker & Mamtani 2000). This calculated P in the schists corresponds to the emplacement depth of the Godhra Granite and therefore constrains the pressure in the granite. Plotting the obtained P range (4–6 kbar) in the P–T diagram given by Kruhl (1996), a temperature of 675– 725 8C is needed for the formation of chessboard patterns in quartz in the Godhra Granite within the b-quartz stability field (Fig. 2d). In the central part recrystallized feldspar grains become abundant, indicating solid-state deformation fabric with a temperature of 450– 600 8C (Passchier & Trouw 2005). In these parts, myrmekites are also abundant, which are reported to develop at between 450– 670 8C (Vernon 2004). In the southern part of the Godhra Granite deformation twins in plagioclase (Fig. 2e) are common. These develop at lower temperature and higher strain rate (Vernon 2004). Solid-state deformation at a temperature of 400–500 8C favours their formation (Passchier & Trouw 2005). Kinked biotites (Fig. 2f), which develop at T  300 8C, are also common. Importantly, textures such as zoned feldspars and other magmatic– submagmatic microstructures are not preserved in the southern part, which implies a considerable superposition of lower-temperature fabrics over high-temperature fabrics (Sen & Mamtani 2006). Figure 3a is a map showing the distribution of dominant fabrics in different parts of the granite, which indicates a general decrease in the temperature of fabric development from north to south. In addition, anisotropy of magnetic susceptibility (AMS), as well as intensity of shape-preferred orientation, in the Godhra Granite is higher in the southern part compared to the northern (Sen et al. 2005; Sen & Mamtani 2006) (Fig. 3b). The Central Indian Tectonic Zone (CITZ), which formed due to the suturing of the northern and southern Indian shields during the Palaeoproterozoic period (c. 2200–2100 Ma: Yedekar et al. 1990), lies to the south of the Godhra Granite (Fig. 1a). It has been inferred that the granite emplaced and developed syntectonic fabrics synchronously with the rejuvenation of the CITZ (Mamtani & Greiling 2005; Sen et al. 2005;

FRACTAL TECHNIQUE IN SYNTECTONIC GRANITE

37

Fig. 1. Location map of the study area. (a) Position of the Aravalli Mountain Belt (AMB) located in the NW part of India. The Godhra Granite is located in the southernmost part of the AMB. The Central Indian Tectonic Zone (CITZ) that formed as a result of the accretion of the northern and southern Indian shields lies to the south of the Godhra Granite. (b) Geological map of the study area in the southern parts of the AMB. Locations of the 12 Godhra Granite samples investigated in the present study are shown with open triangles (northern part; six samples), filled circles (central part; two samples) and filled squares (southern part; four samples), respectively.

Sen & Mamtani 2006). Thus, it has been concluded that the higher fabric intensity in the southern part of the Godhra Granite is because of the proximity of this southern part to the CITZ. Based on the above description, it is concluded that fabric development in the Godhra Granite from syntectonic emplacement to syntectonic cooling was a continuum. The entire Godhra Granite developed high-temperature solid-state deformation fabrics during the initial stages of syntectonic emplacement. There was a superimposition of lower-temperature fabric over high-temperature fabrics as the granite cooled with continuing deformation, which became more intense from the north towards the south. With regard to quartz, the samples from the north appear to have less serrated quartz grains compared to those from the south (Fig. 4).

Fractal analysis of quartz in the Godhra Granite – Methodology In the present study two techniques of fractal analysis have been applied to quartz grains of the Godhra Granite, viz. the ruler method and the area –perimeter method. Quartz grains were traced from thin sections of 12 samples from different locations in the Godhra Granite (Fig. 1b). Of these, six samples lie to the NW of Devgadh Bariya (the northern part of the granite; open triangles in Fig. 1b), two samples are from the vicinity of Devgadh Bariya (central part of the granite; filled circles in Fig. 1b) and four samples are from the south of Devgadh Bariya (southern part; filled squares in Fig. 1b). Several photomicrographs of each granite thin section were taken with a Leica DFC-480 digital camera attached to a Leica microscope.

38 M. A. MAMTANI & R. O. GREILING Fig. 2. Textures in the Godhra Granite developed at the high temperatures that are prominent in the northern part (a– d), and at lower temperatures, which dominate the southern part (e, f). (a) Field photograph of a magmatic shear zone from the vicinity of Godhra (see Fig. 1b for location). (b) Photomicrograph of fractured K-feldspar (Kfs) healed by sodic rims joining adjacent plagioclase (Plg). (c) Photomicrograph of chessboard pattern in quartz. (d) T –P estimation in the Godhra Granite based on chessboard pattern geothermobarometry. The P– T space is after figure 3 of Kruhl (1996). Accordingly, the chessboard pattern in quartz is estimated to have formed between 675 8C at 4 kbar and 725 8C at 6 kbar. (e) and (f) are photomicrographs of deformation twins in plagioclase and kinked biotite, respectively. See the text for details.

FRACTAL TECHNIQUE IN SYNTECTONIC GRANITE

39

Fig. 3. (a) Microstructural fabric distribution map of the Godhra Granite highlighting the dominant fabric noted in different parts of the granite. Note the dominance of magmatic and high-temperature solid-state deformation fabrics in the northern part (around Godhra and to its north) and the dominance of medium- to low-temperature solid-state fabrics in the central and southern parts. (The figure is modified after Sen 2006.) (b) Contoured magnetic lineation strength (L) map of the Godhra Granite. Note the increase in L towards the south (after Sen & Mamtani 2006).

Ruler method

Fig. 4. Representative diagram showing differences in the degree of serration of quartz grains in (a) a northern sample and (b) a southern sample of the Godhra Granite. Note the greater serration of grains in (b). (a) and (b) are from samples 442 and 376, respectively.

For the ruler method, grain boundaries between two adjacent quartz grains were magnified 500 times, as suggested in Kruhl & Nega (1996), and traced digitally. Figure 5a is a representative photomicrograph, where the grain perimeter of one of the quartz grains has been traced in white. The dashed box encloses a grain boundary between two adjacent quartz grains, which is reproduced in Figure 5b. The fractal dimensions of such grain-boundary contacts between two adjacent quartz grains were analysed using the ruler method (Kruhl & Nega 1996). For a statistically reliable dataset, at least 40 grain boundaries, and generally more than 60, were traced from every thin section. Each traced grain boundary was saved as a white line in a bitmap file and using the ruler method with the software ‘Benoit’ (version 1.3; Trusoft International Inc.; http://www.trusoft-international.com) its fractal dimension (designated here as Dr) was calculated automatically. Dr is calculated as the slope of the log Nd v. log d plot, where Nd is the number of steps taken (i.e. total length measured) by moving a divider (ruler) of length d on the traced grain boundary. This calculation is based on the formula Nd  dDr . For a fractal, this plot follows a straight line with a negative slope (2Dr). Thus, a less complex quartz suture has a lower Dr value than a more serrated suture (Fig. 6). According to

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M. A. MAMTANI & R. O. GREILING

obtained. The slope of the latter is the Da for that sample. In the present study, the least-square linear fit for every sample was obtained using data (d and P) from at least 30, and generally more than 40, quartz grains.

Strain-rate calculation Takahashi et al. (1998) analysed the fractal dimension of quartz grain boundaries in experimentally deformed samples using the area– perimeter method and suggested its application as a strain-rate gauge. The relation between area –perimeter fractal dimensions (Da in the present study) and strain rate (1˙ ) is Da ¼ f log 1_ þ r=T þ 1:08

(1)

where f ¼ 9.34  1022 f[log(s21)]21g and r ¼ 6.44  102 (K) (Takahashi et al. 1998). The equation shows that Da increases with 1˙ and decreases with increase in T. Thus, if Da and T are known, 1˙ can be calculated using equation (1). Fig. 5. Figure highlighting the two methods of fractal analysis applied to quartz grains in the present study. (a) Photomicrograph showing quartz crystals in a sample of the Godhra Granite (sample 532). The box with dashed margins in (a) demarcates a grain-boundary suture between two quartz crystals, which is reproduced in (b). In the present study, the fractal (ruler) dimension (Dr) was calculated for such sutures using the software ‘Benoit’. The arrow in (a) points to a digitally mapped quartz grain perimeter that is reproduced in (c). In this study, such perimeters (P) were used to calculate the area–perimeter fractal dimension (Da).

Kruhl & Nega (1996), Dr varies between 1.05 and 1.14 for T ¼ 650– 750 8C, 1.14– 1.23 for T ¼ 490 –540 8C and 1.23 –1.31 for T ¼ 300 – 400 8C.

Area – perimeter method Fractal dimensions were also calculated for quartz grains from the same images as above using the area –perimeter method (designated here as Da). For this method, the area (A) and perimeter (P) of every quartz grain in a thin section (Fig. 5a, c) was determined using the digital image analysis software ‘DIAna’ (Version 3.10, Johannes Duyster #). From the measured area (A), the diameter (d ) of a circle having the same area (A) as that of the quartz grain was calculated. Takahashi et al. (1998) demonstrated that d and P of quartz have the relation: P / dDa . P and d for the quartz grains from a particular sample (thin section) are plotted on a log–log plot and a least-square linear fit is

Results Fractal dimension from the ruler method (Dr) Dr was analysed for 377 quartz grain boundaries in the thin sections from the northern part, 111 grain boundaries from the central part and 251 grain boundaries belonging to the southern part of the Godhra Granite (Fig. 1b). The data are tablulated in Table 1. Figure 7a (sample 468), b (sample 217) and c (sample 376) are representative frequency distribution histograms of Dr for a northern, central and southern sample, respectively. Figure 7d –f are the respective histograms of data from all the samples from the north, centre and south. The results indicate that the mean, median and mode of Dr is lower in the northern samples compared to the central and southern samples, which have identical mean and median Dr (Fig. 7e, f ). The northern samples show a maximum Dr of around 1.10, while the central and southern samples do not have this maximum. Instead they have two Dr maxima at around 1.15 and 1.25.

Fractal dimension from the area – perimeter method (Da) A total of 701 quartz grains were investigated using the area –perimeter method of fractal analysis. Of these, 359 quartz grains are from the six northern samples, and 342 are from the central and southern samples. The data are given in Table 2. Dr analysis above showed that the serration of quartz sutures in

FRACTAL TECHNIQUE IN SYNTECTONIC GRANITE

41

Fig. 6. Two examples of quartz grain-boundary sutures in the Godhra Granite and their ruler dimension (Dr) analysis. Digitally traced sutures such as the ones shown in (a) and (c) were saved as white lines in bitmap files. Each bitmap was opened in the software ‘Benoit’ (version 1.3; Trusoft International Inc.), and Dr of the traced suture was calculated automatically using the ruler method. (b) and (d) are the log Nd v. log d plots obtained for the sutures in (a) and (c), respectively. Dr is the slope of the log Nd v. log d plot. Note that the suture in (a) is less serrated than the one in (c), because of which Dr of (a) is less than that of (c).

the northern part differs from the remaining part of the granite, whilst Dr values of the central and southern parts are almost the same. Therefore, the authors treated Da data from the centre and the south of the granite together, and compared this with Da data from the north. Figure 8a and b show representative log P v. log d plots for a northern sample (number 468) and southern sample (number 376), respectively. Whilst the former sample has a Da value of 1.04, the latter has a higher value of 1.16. Whilst Da for the northern samples is generally 1.10 (except sample 471, see Table 2), Da is 1.12 for the central and southern samples (except sample 387, Table 2). Figure 8c shows that Da for all of the northern samples is 1.10, which is lower than Da (1.14) for the central and southern samples (Fig. 8d).

Calculated strain rates As the northern samples are dominantly representative of high-temperature fabrics with limited modification at lower temperature, the Dr as well as Da values of this part are lower. Thus, 1˙ for the northern part can be calculated using T ¼ 675 8C (lowest temperature for quartz chessboard pattern in the Godhra Granite) and Da of 1.10 (determined from Fig. 8c). This yields 1˙ values of 1027.05 s21. As discussed earlier, the central and southern parts of the granite developed medium-temperature fabrics dominantly between 600–450 8C to as low as 300 8C. Deformation twins are abundant in the southern samples and these require a low temperature (400 –500 8C: Passchier & Trouw 2005) and high strain rate (Vernon 2004). Therefore, the

42

M. A. MAMTANI & R. O. GREILING

Table 1. Fractal dimension (Dr) data for quartz grain boundaries from thin sections of the Godhra Granite using the ruler method Sample No.

489 471 468 442 452 532 217 219 376 325 401 387

n

78 61 59 61 46 72 61 50 73 44 80 54

Ruler dimension (Dr) Range

Mean

SD

SE

Median

1.03 –1.44 1.03 –1.31 1.02 –1.27 1.04 –1.54 1.05 –1.46 1.06 –1.30 1.04 –1.40 1.03 –1.36 1.04 –1.74 1.04 –1.45 1.04 –1.34 1.04 –1.28

1.14 1.13 1.11 1.14 1.16 1.14 1.20 1.17 1.26 1.16 1.17 1.14

0.09 0.07 0.05 0.09 0.10 0.08 0.09 0.08 0.13 0.10 0.07 0.06

0.010 0.009 0.006 0.011 0.014 0.009 0.011 0.011 0.015 0.014 0.007 0.008

1.11 1.12 1.10 1.10 1.14 1.14 1.18 1.18 1.24 1.13 1.17 1.12

n, number of quartz grain-boundary sutures studied in each sample (thin section); SD, standard devition; SE, standard error. The sample numbers and data are serially from north to south (see Fig. 1b for locations).

authors used Da ¼ 1.14 (Fig. 8d) and different T (600, 500, 450, 400 and 300 8C) to calculate 1˙ using equation (1). The calculated values of 1˙ are 1027.26 s21 at 600 8C, 1028.28 s21 at 500 8C, 1028.89 s21 at 450 8C, 1029.6 s21 at 400 8C and 10211.40 s21 at 300 8C.

Discussion Dr for quartz grain-boundary sutures has been proposed as a geothermometer (Kruhl & Nega 1996), while Da of quartz grains may be applied as a strainrate gauge (Takahashi et al. 1998). It is important to note that whilst the application of Dr as a geothermometer by Kruhl & Nega (1996) was based on a study of naturally deformed rocks, which developed their fabric under different P –T conditions, the application of Da as a strain-rate gauge (Takahashi et al. 1998) is based on experimental work carried out on quartz aggregates. Indeed, the latter has never been tested on rocks from natural geological settings.

Ruler dimension of quartz sutures, geothermometry and microstructures Kruhl & Nega (1996) reported different ranges of Dr values, viz. Dr ¼ 1.05 –1.14 for T ¼ 650 –750 8C, 1.14–1.23 for T ¼ 490 –540 8C, 1.23– 1.31 for T ¼ 300 –400 8C, and presented a Dr v. T plot, which is shown here as Figure 9. The northern part of the Godhra Granite is dominated by magmatic and high-temperature solid-state deformation fabrics (chessboard pattern in quartz) that developed between 675 and 725 8C. Using this temperature

range, a Dr value varying between 1.05–1.14 should be expected to dominate the northern samples (Fig. 9). The central part of the Godhra Granite has abundant recrystallized feldspars and myrmekites. Whilst myrmekites form between 500 and 670 8C (Vernon, 2004), recrystallization of feldspar indicates 450–600 8C (Passchier & Trouw 2005). Using Figure 9, the expected Dr for 500– 670 8C (myrmekite formation) is 1.07 –1.23, and for 450–600 8C (recrystallized feldspar) the expected Dr is 1.11 –1.25. The southern part of the granite has abundant deformation twins in feldspar, which develop between 400– 500 8C (Passchier & Trouw 2005). This temperature range yields a Dr value of between 1.16–1.28 (Fig. 9). Further, kinked biotite grains are also common in the southern part, which are indicative of temperatures below 300 8C. This would yield Dr values above 1.27. The above theoretical Dr ranges can now be compared with the actually recorded Dr values from different parts of the Godhra Granite. The northern samples show Dr , 1.14, with a major peak of Dr around 1.10 and a median Dr of 1.11 (Fig. 7d). All of these fit well with the expected Dr value for the northern part of the granite. The central (Fig. 7e) and southern samples (Fig. 7f) show most Dr values .1.14 with two peaks, at Dr ¼ 1.15 and 1.25, and a median Dr .1.16. These data fit well with the theoretical Dr values gauged from textures abundant in the central and southern part of the Godhra Granite that formed between 670 8C (highest temperature for myrmekite: Vernon 2004) and 500 8C (highest temperature for deformation twins in feldspar: Passchier & Trouw 2005) down to below 300 8C (temperature for kinks in biotite). Further, for a Dr range of

FRACTAL TECHNIQUE IN SYNTECTONIC GRANITE 43

Fig. 7. Histograms documenting the frequency distribution of ruler-fractal dimension (Dr) of quartz grain boundaries in a sample from (a) the north, (b) the centre and (c) the south, respectively. The statistical details for each sample are mentioned on each histogram. (d)–(f) are the respective histograms for all the data from the northern, central and southern samples. Note the dominance of quartz sutures having Dr ,1.14 and the presence of a maximum of Dr around 1.10 in the northern samples (d). In contrast, sutures with Dr . 1.14 dominate the central and southern samples, and do not have a maximum around 1.10. Instead they have maxima around 1.15 and 1.25 (e, f). SD is standard deviation. In all of the figures n refers to number of quartz grain-boundary sutures analysed.

44

M. A. MAMTANI & R. O. GREILING

Table 2. Area –perimeter fractal dimension (Da) data for quartz grains from thin sections of the Godhra Granite Sample No.

n

Da

r

SD

489 471 468 442 452 532 217 219 376 325 401 387

65 39 30 56 89 80 87 48 48 60 44 55

1.09 + 0.02 1.12 + 0.02 1.05 + 0.04 1.02 + 0.03 1.10 + 0.02 1.09 + 0.02 1.12 + 0.02 1.16 + 0.03 1.17 + 0.02 1.13 + 0.02 1.16 + 0.04 1.08 + 0.02

0.99 0.99 0.98 0.97 0.99 0.99 0.99 0.98 0.99 0.99 0.98 0.99

0.03 0.05 0.06 0.07 0.04 0.03 0.05 0.06 0.05 0.06 0.07 0.04

n, number of quartz grain perimeters analysed in each sample; r, linear regression; SD, standard deviation. The sample numbers and data are serially from north to south (see Fig. 1b for locations).

1.17 –1.18 (median Dr for central and southern samples), a temperature range of 450–600 8C is estimated to be dominantly responsible for the development of medium-temperature fabrics in the Godhra Granite. Progressive deformation at still lower temperature (300 8C) was responsible for the formation of sutures with yet higher Dr values. Thus, the present study indicates that geothermometric studies can be carried out in syntectonic granites using fractal (ruler) dimension analysis of quartz grain-boundary sutures. Further, it is also inferred that superimposition of low-temperature fabrics over high-temperature ones is manifested in the Dr data. It is envisaged that high-temperature fabrics and sutures with low Dr developed in the entire granite during the initial stages of syntectonic emplacement and hightemperature solid-state deformation. As the granite cooled to lower temperature, there was superimposition of low-temperature fabrics in all parts of the

Fig. 8. Log P v. log d plots for the quartz grain perimeters in sample 468 (a, northern part) and 376 (b, southern part), respectively. P is the perimeter of quartz grain, d is the diameter of a circle having the same area as the quartz grain having a particular perimeter P. The slope of the least-square linear fit gives area–perimeter fractal dimension (Da), which is mentioned on each plot along with the number of quartz grain boundaries measured (n), sample number and statistical details, such as standard deviation (SD) and linear regression (r). Following the same procedure, Da was determined for several quartz perimeters in all of the 12 Godhra Granite samples. (c) and (d) are the respective log P v. log d plots obtained for data from all of the northern and central þ southern samples of the Godhra Granite. Note the higher Da value in (d) compared to (c).

FRACTAL TECHNIQUE IN SYNTECTONIC GRANITE

45

Fig. 9. Plot showing the relationship between fractal (ruler) dimension (Dr) of quartz sutures and T (modified after fig. 7 of Kruhl & Nega 1996). The sloping grey rectangle is based on fractal data from Kruhl & Nega (1996) and takes into account the confidence intervals of their data. T ranges gauged from different textures recorded in different parts of the Godhra Granite are shown on the plot, which help gauge the expected Dr values for the various T ranges. In the text, these expected Dr values are compared with the ones actually obtained from different parts of the Godhra Granite. The white rectangle at Dr ¼ 1.10 represents the peak Dr value noted in the northern samples (Fig. 7d). The shaded rectangles at Dr ¼ 1.15 and 1.25 represent the two peaks noted in both the central as well as the southern samples (Fig. 7e, f).

granite. However, this superimposition was less prominent in the northern part, which is more distant from the CITZ. Therefore, 60% quartz sutures from the northern samples have Dr , 1.14 and their frequency distribution shows a maximum of around 1.10 (Fig. 7d). In contrast, the superimposition of low-temperature over high-temperature fabrics was prominent towards the south, which led to increased serration of the quartz sutures. Therefore, 59% of sutures from the central and southern samples have Dr . 1.14, and their frequency distribution shows two maxima, around 1.15 and 1.25 (Fig. 7e, f ), and the absence of a maxima around 1.10 or lower Dr values. It may be noted that peaks at Dr values of 1.15 and 1.25 are also noted in the northern samples (Fig. 7d), which implies that quartz sutures in the north also responded to low-temperature (re)crystallization. However, this response was limited because the majority of the sutures continue to have a lower Dr, thus resulting in a main peak at 1.10. However, this low-temperature superimposition was more complete in the centre and south, due to

which the majority of the quartz sutures recrystallized and developed greater serration. As a result, Dr of quartz sutures in these samples shifted to higher values and most at lower Dr values got obliterated.

Strain-rate (1˙ ) estimation from fractal dimensions of quartz grains Natural and tectonic strain rates are estimated to lie between 10212 and 10215 s21 (Twiss & Moores 1992; Passchier & Trouw 2005). Therefore, a 1˙ value of 10211.40 at T ¼ 300 8C for the southern part of the granite is close to the highest expected natural geological strain rate. According to Vernon (2004), deformation twins in feldspar require low temperature (400 –500 8C; see earlier) and high strain rate. With a Da value of 1.14 and the above temperature range, the value of 1˙ for the formation of deformation twins in the southern part of the Godhra Granite is estimated to be between 1028.28 and 1029.6 s21. According to Vigneresse (2004),

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M. A. MAMTANI & R. O. GREILING

strain rates in partially molten rocks can be of the order of 1029 –10211 s21 or even higher. Recently, Vigneresse (2006) calculated strain rates as high as 10210 s21 for emplacement of granitic batholiths. Harris et al. (2000) argued that the overall timescales of melt segregation and emplacement for many orogenic granites are less than 10 ka. Petford et al. (2000) also argued that granite magmatism is a dynamic process that operates on timescales of 105 years. Miyazaki & Santosh (2005) cited examples of granites in Japan for which cooling rates had been calculated to be 80–90 and 19 8C Ma21. Recent studies in the eastern Pyrenees (France) have also yielded high cooling rates of granitoids of between 3 and 30 8C Ma21 during different stages of their thermotectonic evolution (Maurel et al. 2008). McKenzie (1985) stated that large igneous intrusions cool at a timescale, that is shorter than 105 years after emplacement. As it is known that the southern part of the Godhra Granite experienced high strain due to its proximity to the CITZ, it is concluded that a high 1˙ (10210 –10212 s21) calculated at lower temperatures using a Da value of 1.14 is not totally unlikely. However, strain rates calculated for higher temperatures for the northern as well as the central and southern samples are extremely high (1027 –1028 s21). The latter is closer to experimental strain rate (Hirth & Tullis 1992) and it is difficult to believe that this could be possible in nature. It is thus noted that a realistic strain rate of the order of 10211.40 s21 at 300 8C is obtained only for the southern samples, which yield a Da value of 1.14. The same strain rates can be generated for the northern part using equation (1) under two conditions: (a) T is fixed at 675 8C, which is the minimum temperature for quartz chessboard pattern development in the granite, and Da ¼ 0.69; and (b) Da is fixed at 1.10, which is the value obtained for the northern samples, and T ¼ 322 8C. Both of these possibilities seem unlikely. Possibility (a) is unlikely because 1  Da  2 (Takahashi et al. 1998). Possibility (b) is unlikely because it implies that the obtained Da of 1.10 for the northern samples would have to be indicative of a temperature as low as 322 8C. If this were the case, then the northern part would have been dominated by lowtemperature solid-state deformation fabrics, as well as quartz sutures with high Dr values, which is not the case. Recent calculations by Vigneresse (2006) reveal that 1˙ values of the order of 10210 s21 are possible during emplacement. Using equation (1) this would require a Da of 0.825 (T ¼ 675 8C) or 0.809 (T ¼ 700 8C). All of these are not possible as Da must lie between 1 and 2. Therefore, the technique to calculate 1˙ using area –perimeter fractal dimension of quartz grains fails in providing a reasonable estimate of 1˙ for

high temperatures in syntectonic granites such as the Godhra Granite. Further research is needed to refine the equation for applications such as to partially molten rocks. 1˙ estimations at low temperatures are closer to natural tectonic strain rates and the method, although not very robust, may provide some useful information that can be applied to better understand the tectonic evolution of an area.

Conclusions This research is the first of its kind that involves application of two different fractal analysis techniques to quartz grains in the same rock, viz.: fractal (ruler) dimension (Dr) and area– perimeter fractal dimension (Da). Whilst the former has been advocated earlier as a geothermometer (Kruhl & Nega 1996), the latter has been suggested as a strainrate gauge (Takahashi et al. 1998). The determination of Dr and Da for quartz in the syntectonic Godhra Granite has provided a first database, which enables the extent to which fractal analyses can be applied to gauge deformation conditions in natural rocks to be tested, particularly syntectonic granites. Dr values of quartz grain-boundary sutures provide meaningful results for the calculation of temperature during deformation in the Godhra Granite. The northern samples are dominated by quartz sutures that are less serrated and have a low Dr compared to those from the remaining central and southern parts of the granite. Temperature conditions needed for the development of high- (chessboard pattern in quartz), medium- (recrystallized feldspars and deformation twins in feldspar) and low-temperature fabrics (kinked biotite) are expected to yield Dr values of ,1.14, 1.11–1.28 and .1.27, respectively (Fig. 9). The calculated Dr values fit well with the expected values (above) and it is concluded that fractal (ruler) dimension analysis of quartz sutures in syntectonic granites can be used to estimate temperature. Moreover, superimposition of low-temperature over high-temperature fabrics in different parts of a syntectonic granite can also be investigated using this method. In contrast, the application of Da as a strain-rate gauge in the Godhra Granite indicates that the method does not provide unequivocal data over all temperature ranges. Whilst a high, but geologically reasonable, strain-rate value of the order of 10211.40 s21 is estimated for low temperatures (300 8C), extremely high 1˙ values (c. 1027 – 1028 s21) are obtained for the high-temperature calculations (.600 8C). Thus, the method fails to give reasonable estimates of 1˙ for high temperature, and it is suggested that more research is needed to

FRACTAL TECHNIQUE IN SYNTECTONIC GRANITE

further develop the equation given by Takahashi et al. (1998) for determination of 1˙ over a wider range of temperatures. M. A. Mamtani thanks the Alexander von Humboldt Foundation (Germany) for the award of a Humboldt Research Fellowship to carry out the present research at the University of Heidelberg (Germany). Reviews by J. L. Vigneresse and D. Koehn helped to improve various aspects of this paper. J. Kruhl and P. Bons are thanked for comments on earlier versions of this work. Discussions with J. Kruhl, P. Bons, M. Stipp, S. Cox and B. Hobbs during the DRT-2007 meeting in Milano (Italy) were found useful. M. A. Mamtani also acknowledges interactions with J. Kruhl, M. Peternell and A. Gerik at the Fabric Quantification Workshop 2006 held at Technische Universita¨t Mu¨nchen. However, the authors take full responsibility for the interpretations made in this paper.

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Microstructure and elastic anisotropy of naturally deformed leucogneiss from a shear zone in Montalto (southern Calabria, Italy) ROSOLINO CIRRINCIONE1, EUGENIO FAZIO1, RENE´E HEILBRONNER2, HARTMUT KERN3, KURT MENGEL4, GAETANO ORTOLANO1, ANTONINO PEZZINO1 & ROSALDA PUNTURO1* 1

Department of Geological Sciences, Catania University, Corso Italia 57, 95129 Catania, Italy 2

Department of Geosciences, Basel University, Bernoullistrasse 32, CH-4056 Basel, Switzerland

3

Institute of Geosciences, Kiel University, Olshausenstrasse 40, 24098 Kiel, Germany 4

Institut fu¨r Endlagerforschung, Clausthal University, Adolph-Roemer-Strasse 2A, 38678 Clausthal-Zellerfeld, Germany *Corresponding author (e-mail address: [email protected]) Abstract: A strain gradient was mesoscopically recognized in sheared leucogneisses cropping out near Mount Montalto (Calabria, southern Italy) in the Aspromonte –Peloritani Unit on the basis of field observations. In order to investigate the relationship between textural and physical anisotropy, a microstructural and petrophysical study was carried out on selected mylonites exhibiting different stages of deformation. The main mineral assemblage is Qtz þ Pl þ Kfs þ Wm, displaying S–C and shear-band textures; mica-fish and ribbon-like quartz are widespread. As strain increases K-feldspar, biotite and premylonitic low phengite white mica transformed to synmylonitic high phengite white mica and quartz, accompanied by an increasing albitization. Different quartz c-axis patterns are ascribable to non-coaxial progressive deformation; we suggest that deformation proceeded under greenschist- up to amphibolite-facies conditions owing to a local increase in shearing temperature. Laboratory seismic measurements were carried out on sample cubes (43 mm edged) cut according to the structural frame (foliation, lineation) of the rock. At 400 MPa and room temperature the averages of compressional (Vp) and shear-wave velocities (Vs) are very similar: 5.70–5.91 and 3.36– 3.55 km s21, respectively. Seismic anisotropy and shear-wave splitting are related to the modal amounts of constituent minerals (in particular mica) and their crystallographic preferred orientation. Importantly, anisotropy is lowest in the most strained rock.

Crustal seismic anisotropy may have the potential to be a powerful tool to map tectonic structures and patterns of deformation within the crust (e.g. Mainprice & Nicolas 1989; Christensen & Mooney 1995). Therefore, knowledge of the seismic properties of deformed crustal rocks is of great importance for the interpretation of seismic reflection and refraction data. In most collisional mountain belts naturally deformed rocks provide ‘natural laboratories’, where it is possible to observe and concentrate the investigations on the relationships between microstructural and textural characteristics and physical (elastic) properties of the highly deformed rocks. One interesting site showing these features is located near Montalto (Aspromonte Massif, southern Calabria), where an Alpine crustal-scale shear zone involved Hercynian orthogneiss. Here we present an integrated microstructural and petrophysical

investigation on a selection of variously deformed mylonitic orthogneisses from this area in order to determine the directional dependence of wave propagation, and to understand the nature of seismic anisotropy that may be produced in the crystalline rocks by shearing during deformation. A basic objective is the correlation between textural parameters (i.e. grain size, shape and orientation of grains, and quartz and mica c-axis orientation patterns) and physical properties (i.e. P- and S-wave velocity, seismic anisotropy, Poisson’s ratio and density) on deformed mylonites sampled from a known and well-exposed representative portion of the investigated crustal-scale shear zone. The ultimate goal is to evaluate what kind of relationship exists between fabric and seismic anisotropy within the set of progressively deformed specimens, and to estimate how the mineralogical changes, due to induced shearing reactions,

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 49– 68. DOI: 10.1144/SP332.4 0305-8719/10/$15.00 # The Geological Society of London 2010.

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influence the bulk seismic anisotropy of the sheared rocks.

Geological setting The shear zone of Montalto is located in the southern sector of the Calabria –Peloritani orogen, which represents a segment of the western Mediterranean internal Alpine chain (Fig. 1a). This chain is mostly composed of Hercynian basement rocks, locally reworked during early (Late Cretaceous – Early Palaeocene) to late Alpine (Oligocene – Miocene) metamorphism (Rossetti et al. 2001; Pezzino et al. 2008), and it records part of the tectono-metamorphic history of the Africa– Eurasia composite plate boundary (Gueguen et al. 1998) by its Oligocene –present-day geodynamics. The Montalto Shear Zone (MSZ) (Aspromonte Massif, southern Italy: Fig. 1b) provides the opportunity to study textural, compositional and microstructural modifications within progressively sheared rocks. The outcrop of mylonitic rocks chosen for study is located approximately 2 km ESE of Mt Montalto (geographic co-ordinates: 38890 1.1300 N, 158560 42.1600 E) (Fig. 1c); it is characterized by a very regular deformation fabric marked by a pervasive foliation with a consistent attitude that, on the whole, is only weakly affected by deformation (e.g. Fazio et al. 2007). Leucogneiss samples were selected in order to investigate the different stages of deformation in a monotone shear plane within the major shear zone. Selected rocks are, indeed, characterized by pervasive mylonitic foliation oriented about 1408/208SW (azimuth/ dip) and a marked stretching lineation at about 1858/188 (dip direction/dip) (Fig. 2). A gneissic fabric outlined by feldspar augen up to 1 cm in size is widely observed in the outcrop. Hand samples show a well-defined compositional layering characterized by alternating feldspar- and quartzrich planar domains, representing the mylonitic foliation. Tourmaline-rich layers parallel to the main foliation, as well as pegmatite lenses characterized by centimetre-sized flakes of mica, are frequently observed. Sporadically occurring large tourmaline grains (up to 3 cm) form sigma-type objects. Other interbedded dark layers consist of small garnets (diameter less than 500 mm) arranged within the foliation plane.

Rock samples and methodology Selected leucogneiss gives evidence for a welldeveloped strain gradient. Four rock samples (M13, M8, M3 and M4) were collected from a domain characterized by an increasing strain gradient inferred from different grain sizes (Fig. 3).

Chemical analysis For analyses of bulk samples, glass disks of lithium-tetra-borate and sample powder (,125 mm) were prepared, mixed to a ratio of 6:1. A wavelengthdispersive strument (Panalytical AXIOS) was used for major element determination; hydrogen and carbon were determined as loss on ignition (LOI) at 1150 8C (detection limit 0.6 wt%). Accuracy is maintained by repetitive analyses of international and in-house reference rocks, and is better than 2% for elements Ca –Fe, better than 4% for elements from Al to K and better than 6% for Na and Mg. Precision in the form of standard deviation is in the range of+2% (rel.) for all major elements, except for loss on ignition, which is in the range of+5% (rel.). Individual mineral analyses were performed on a Cameca SX 100 electron microprobe (EMPA) with 15 kV accelerating voltage and 20 nA beam current calibrated with natural mineral standards.

Microstructural investigation by image analysis Grain size and grain-shape analysis, as well as crystallographic preferred orientation (CPO), of quartz grains were determined by image analysis techniques. In particular, we adopted two methods of quantitative image analysis: (1) the Lazy grain boundaries macros (Heilbronner 2000a) for derivation of grain-boundary maps, and to quantify grain sizes and shapes of quartz domains; (2) the Computer Integrated Polarization microscopy method (CIP), introduced by Panozzo Heilbronner & Pauli (1993) and described by Heilbronner (2000b) for deriving crystallographic orientations of quartz c-axis. Moreover, histograms of the equivalent radii of the cross-sectional areas were derived, from which the three-dimensional (3D) volumeweighted histograms were calculated using the StripStar program (as described by Heilbronner & Bruhn 1998). The [001] orientation patterns of mica were established by universal stage measurements. All images were acquired using a Zeiss microscope with magnifications of 2.5 equipped with a Zeiss AxioCam camera. The matrix size of the digital version of the images was typically of the order of 1200  800 pixels. All of the photomicrographs are oriented with the top and bottom borders parallel to the mylonitic foliation. All sections were cut parallel to the stretching lineation, averagely oriented NE–SW and perpendicular to the mylonitic foliation.

Petrophysical investigation The P- and S-wave velocity and density measurements were performed on rock cubes (43 mm edge

ANISOTROPY OF NATURALLY DEFORMED ROCKS 51

Fig. 1. (a) Location of the study area in southern Italy. (b) Geological sketch map of the Aspromonte Massif (after Pezzino et al. 1990; Ortolano et al. 2005). (c) Detailed geological map and outcrop location.

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Fig. 2. (a) Outcrop features and (b) contour plots (Schmidt diagram – lower hemisphere) of mylonitic foliation and stretching lineation.

length) in a multi-anvil pressure apparatus using the ultrasound transmission technique (cf. Kern 1982; Kern et al. 1997) with transducers operating at 2 MHz (Vp) and 1 MHz (Vs), respectively. Measurements of compressional and orthogonally polarized shear-wave velocities were made on sample cubes. Edges of cubes were cut parallel to macroscopic fabric elements: X is parallel to lineation, Y is normal to lineation within the foliation plane and Z is normal to foliation. Measurements of compressional and orthogonally polarized shear-wave velocities were made on sample cubes. The shearwave transducers were oriented such that the predominant S1 and S2 particle motions were either parallel to or perpendicular to foliation and lineation. A detailed description of the device, as well as the transducer–piston– sample arrangement, is given in Kern et al. (1997). Measurements along

the three orthogonal directions, X, Y and Z, were made as a function of pressure up to 400 MPa at room temperature conditions. Each set of results is composed of nine velocity values: three P-wave velocities and six S-wave velocities. Length and volume (density) variations of the sample cubes, resulting from changes in the main stress, are obtained from the piston displacement. The cumulative error in both Vp and Vs was estimated to be ,1%.

Results Compositional features The XRF bulk chemical data of the sheared leucogneiss samples along with the EMPA mineral chemistry results are listed in Table 1. In addition,

Fig. 3. Pictures of investigated mylonites disposed in order of decreasing strain from left to right.

ANISOTROPY OF NATURALLY DEFORMED ROCKS

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Table 1. XRF whole-rock chemical data (LOI values all ,0.6 wt%) and average mineral compositions Wt %

K-feldspar

SD

Albite

SD

Epidote

SD

Muscovite

SD

Biotite

SD

37.34 0.85 18.27 21.19 0.28 6.92 0.05 0.05 9.14 0.00 94.09

(14) 2.35 0.37 1.28 2.11 0.06 0.53 0.03 0.04 0.43 0.00 0.59

M13 SiO2 TiO2 Al2O3 FeOT MnO MgO CaO Na2O K2O P2O5 Total

72.25 0.19 15.05 1.29 0.03 0.41 1.62 3.83 3.80 0.16 98.63

64.79 0.02 18.67 0.01 0.01 0.00 0.02 0.64 15.94 0.00 100.11

(17) 0.40 0.02 0.17 0.01 0.02 0.01 0.03 0.12 0.16 0.00 0.43

66.50 0.02 21.17 0.02 0.01 0.01 1.60 10.67 0.10 0.00 100.10

(10) 0.90 0.02 0.64 0.01 0.01 0.01 0.50 0.22 0.02 0.00 0.29

38.46 0.09 27.84 7.56 0.15 0.16 23.48 0.01 0.18 0.00 97.94

(11) 0.23 0.06 0.38 0.53 0.03 0.28 1.19 0.01 0.27 0.00 0.42

45.65 0.69 34.97 1.73 0.02 0.79 0.00 0.44 10.86 0.00 95.15

(18) 0.36 0.12 0.30 0.14 0.02 0.06 0.00 0.06 0.10 0.00 0.50

M8 SiO2 TiO2 Al2O3 FeOT MnO MgO CaO Na2O K2O P2O5 Total

73.12 0.20 15.06 1.19 0.03 0.43 1.69 4.23 2.50 0.15 98.58

64.89 0.01 18.66 0.01 0.01 0.01 0.01 0.72 15.87 0.00 100.19

(12) 0.68 0.01 0.19 0.01 0.01 0.01 0.02 0.26 0.45 0.00 0.46

68.03 0.02 19.84 0.01 0.01 0.01 0.70 9.97 1.98 0.00 100.57

(9) 1.12 0.02 0.61 0.01 0.01 0.02 0.40 3.19 4.85 0.00 0.28

39.27 0.30 26.71 7.97 0.12 0.07 23.52 0.02 0.05 0.00 98.05

(10) 1.09 0.72 1.08 1.44 0.07 0.06 0.49 0.05 0.11 0.00 0.50

47.38 0.37 31.88 2.94 0.03 1.50 0.01 0.26 10.91 0.00 95.29

(14) 1.19 0.21 2.51 0.89 0.02 0.67 0.01 0.13 0.21 0.00 0.25

M3 SiO2 TiO2 Al2O3 FeOT MnO MgO CaO Na2O K2O P2O5 Total

72.78 0.18 15.30 1.19 0.03 0.38 1.19 4.05 3.59 0.14 98.83

64.76 0.02 18.82 0.02 0.01 0.01 0.00 0.58 16.06 0.00 100.28

(11) 0.17 0.03 0.14 0.01 0.01 0.01 0.00 0.16 0.25 0.00 0.17

67.68 0.02 20.28 0.04 0.00 0.01 0.98 11.15 0.09 0.00 100.25

(10) 0.46 0.03 0.19 0.04 0.01 0.01 0.35 0.16 0.02 0.00 0.31

38.55 0.14 28.15 6.66 0.10 0.02 24.56 0.03 0.02 0.00 98.23

(11) 0.28 0.05 0.54 0.63 0.06 0.02 0.16 0.04 0.03 0.00 0.22

48.74 0.19 28.72 4.25 0.04 1.95 0.01 0.19 10.79 0.00 94.87

(13) 1.29 0.09 0.94 0.27 0.03 0.14 0.01 0.15 0.34 0.00 0.35

M4 SiO2 TiO2 Al2O3 FeOT MnO MgO CaO Na2O K2O P2O5 Total

78.79 0.10 12.94 0.12 0.01 0.06 0.83 6.96 0.09 0.15 100.05

35.88 0.02 19.83 15.35 0.17 15.07 0.34 1.29 0.04 0.00 87.99

(2) 7.12 0.02 1.39 2.54 0.00 3.16 0.18 1.28 0.01 0.00 1.47

67.98 0.03 19.86 0.01 0.00 0.02 0.82 11.57 0.07 0.00 100.35

(11) 0.53 0.02 0.33 0.02 0.01 0.02 0.28 0.26 0.01 0.00 0.30

39.43 0.02 31.81 1.55 0.04 0.03 24.93 0.01 0.03 0.00 97.85

(5) 0.13 0.02 0.21 0.13 0.03 0.02 0.12 0.01 0.02 0.00 0.25

48.88 0.20 31.85 1.50 0.01 2.11 0.01 0.50 10.21 0.00 95.28

(8) 0.61 0.06 0.69 0.22 0.02 0.23 0.03 0.25 0.37 0.00 0.35

SD, Standard deviation; (), the number of measurements are given in brackets.

densities of the optically identified minerals (Deer et al. 1998) are set out in Table 2. Both datasets were used to calculate the modal composition of the sheared rocks (Fig. 4). The calculation of the mineral modes is based on the generalized

petrological mixing model reported by Le Maitre (1979) using the computer program PETMIX (http://www.geologynet.com/programs/ petmix410.xls), which is based on least-square fit. In contrast to point-counting and image analysis,

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Table 2. Mineral densities used for calculation of mineral abundances (after Deer et al. 1998) Density (g cm23)

Mineral Quartz Plagioclase (albite) K-feldspar Biotite Muscovite Epidote (zoisite) Chlorite

2.65 2.62 2.63 3.05 2.83 3.43 3.00

It is clear from Table 3 and Figure 4 that quartz (34– 38 vol%) and albite (34–59 vol%) are the dominant mineral phases in all four samples. In samples M13 and M3, K-feldspar (about 15 and 8 vol%, respectively) is also an important constituent. However, most obvious are the marked differences in the content of phyllosilicate minerals (biotite and muscovite). In samples M13, M8 and M3, phyllosilicates contribute to the bulk volume by about 13 –19 vol%, whereas their content in sample M4 is minimal (about 2 vol%).

Mineralogical and microstructural features

Fig. 4. Mineral abundances (vol %) inferred by whole-rock and mineral chemistry.

this approach includes large rock volumes, thus providing more macroscopic relevant results. Importantly, in strongly foliated rocks with marked shape-preferred orientation (SPO) of major minerals, accurate determination of volume percentages by means of point-counting or image analysis is difficult, even in the case where three perpendicular oriented thin sections are used. This particularly holds for rocks exhibiting, in addition to SPO, pronounced small-scale (mm-size) heterogeneities. In our approach the effects of SPO and small-scale heterogeneities are therefore taken into account. It should be noted, however, that modal composition data based on point-counting and image analysis are accurate as long as the thin-section analysis can represent the bulk sample.

The collected leucocratic gneisses are mostly composed of Qtz þ Ab þ Wm + Kfs accompanied by subordinate amounts of Bt þ Ep þ Chl þ Tur + Grt. Mineral abbreviations are after Kretz (1983), reviewed by Siivola & Schmid (2007). Representative photomicrographs of the investigated samples are presented in Figure 5, and the compositions of feldspar and white mica are plotted on diagrams in Figure 6a, b. Porphyroclastic feldspar and mica-fish (Fig. 5), inducing S–C textures, are typical for samples deformed at a lesser extent (M13 and M8), whereas they are weakly or poorly developed in the most deformed mylonites (M3 and above all M4). Non-pervasive discontinuous shear bands cross-cutting both S and C surfaces are ubiquitous. Ribbon-like quartz levels are very common in all rock samples, and are characterized by different subgrain recrystallization mechanisms such as bulging (rare), subgrain rotation (dominant) and grainboundary migration (rare), typical of greenschistup to low-amphibolite-facies shearing conditions. The occurrence of synshearing tourmaline blastesis further testifies to a significant localized fluid circulation during the shear-zone evolution. Progressive mylonitization is documented by decreasing grain size of the prekinematic porphyroclasts (in the sequence M13, M8, M3 and M4, see Fig. 3), accompanied by progressive increase in SPO, as well as by increasing the reaction-softening process documented by the synmylonitic growth of microcrystalline high phengite white mica, quartz, albite and chlorite.

Table 3. Mineral abundances (vol%) calculated with PetMix spreadsheet, based on whole-rock and mineral compositions Vol%

Quartz

K-feldspar

Plagioclase

Epidote

Muscovite

Biotite

M13 M8 M3 M4

34.5 36.7 34.8 37.9

15.5 – 8.1 –

34.2 42.7 35.3 59.2

3.3 5.0 2.4 1.1

9.2 15.6 19.4 1.2

3.3 – – 0.5

ANISOTROPY OF NATURALLY DEFORMED ROCKS

55

Fig. 5. Representative photomicrographs of mylonitic leucogneisses. M13: K-feldspar porphyroclast wrapped by quartz and phengitic white mica aggregate. M8: pre-mylonitic low phengite white mica wrapped by ribbon-like quartz levels. M3: relicts of premylonitic garnet and white mica within the fine-grained matrix (quartz plus phengite and albite). M4: synmylonitic microcrystalline aggregate of ribbon-like quartz and albite.

The least strained sample M13 (Fig. 3) is characterized by a dominant S–C texture defined by the alignment of mica and quartz-rich layers. Porphyroclasts, up to 7 mm in diameter, are mostly composed of K-feldspar (Or92 – 94) with variably developed cross-hatched twinning. White mica (Phg3 – 9), plagioclase (An4 – 11), and rare garnet, epidote and biotite sometimes form porphyroclasts. Reaction-softening processes are less developed and are limited to an incipient metasomatic albitization, as well as by an initial retrograde muscovitization of K-feldspar porphyroclasts (Fig. 5). The matrix is represented by a microcrystalline aggregate of phengitic white mica and quartz. Ribbon-like quartz layers are well developed, where larger relict grains with diffuse undulatory extinction are surrounded by smaller new ones forming an incipient oblique foliation parallel to the S foliation. Analysis of quartz grain-size distribution evidenced three different clusters of radii (Fig. 7), with a preferential volume concentration of 40% with radii ranging from 8 to 12 mm. The shape analysis of quartz grains’ aspect ratio

(AR ¼ longest/shorter axis of equivalent ellipse), mostly ranging from 1 to 4 (ARmax ¼ 6.9), testifies to a relatively low value of elongation that is consistent with a relative low strain. This is also confirmed by a random distribution of the angle between the longer axis value with respect to the mylonitic foliation. Sample M8 (Fig. 3) is characterized by S–C fabric cross-cut by shear bands. Augen boundaries often look more rounded and smoothed. Moreover, white mica (Phg7 – 12), feldspar (Or90 – 94) and subordinate garnet constitute large porphyroclasts. Mica is often fish-shaped (Fig. 5) and sometimes a bookshelf sliding structure is present. K-feldspar augens (up to 5 mm) are widespread and mantled by fine-grained white mica and ribbonlike quartz. Relicts of prekinematic K-feldspar commonly exhibit at their edges retrograde crystallization of phengitic white mica (Phg25 – 30: Fig. 6) and quartz. Plagioclase (An0 – 4), rare tourmaline and epidote growth are probably linked to substantial fluid circulation operating during the shearing evolution.

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Fig. 6. Mineral chemistry of (a) feldspar and (b) white mica in the studied mylonites with distinction between the pre- and synmylonitic population.

Grain-size distribution analysis of quartz domains showed that approximately 65% of volume is occupied by grains with an equivalent radius of less than 12 mm, which we interpret as resulting from syntectonic recrystallization. The isolated peak found at 40 mm corresponds to the maximum equivalent radii. The shape analysis of quartz grains (ARmean ¼ 3.4) indicates weak relative elongation, consistent with the observed relative low strain gradient. Compared with sample M13, more grains show a significant increase in elongation (i.e. ARmax ¼ 12). Nevertheless, a relatively low increase in strain value is also confirmed by the relatively low increase in grains oriented with angles between the longest axis and the mylonitic foliation in the range of 08 –408 and also 1408–1808. Sample M3 is characterized by relicts of S– C texture within the porphyroclastic domains; fine-grained white mica –quartz aggregates are pervasively developed, and are characterized by subordinate ribbon-like quartz layers alternating with synmylonitic phengitic white mica (Fig. 6) and

chlorite. Augens (with the major axis up to 5 mm) are essentially all K-feldspar grains (Or92 – 95: Figs 5 & 6) with common albite inclusions; they are almost untwinned and characterized by a sigmashaped microstructure. Rare garnet occurs as fragments forming flattened layers parallel to the mylonitic foliation. Prekinematic white mica and subordinately biotite and garnet are less frequent than in the M8 sample, whereas syntectonic recrystallization of a fine mixture of phengitic white mica (Phg24 – 40), quartz and epidote, as well as myrmekite intergrowths, are more evident. Analysis of quartz grain-size distribution shows that the radii of 70 vol% of the grains cluster around 4–8 mm. They are preferentially elongated along the C surface of the mylonitic foliation, and form ribbonlike structures and locally exhibit syntectonic grainboundary migration recrystallization. Fine-grained quartz-rich matrix (ARmean ¼ 3.2; ARmax ¼ 14) developed by dynamic recrystallization, with the major mineral axes elongated parallel to the local foliation, is characteristic of this sample (Fig. 7).

ANISOTROPY OF NATURALLY DEFORMED ROCKS

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Fig. 7. Microstructural analysis of quartz domains. Left: grain-size distribution. Volume% of 3D grains as a function of a sphere of the same size. Right: for each grain, aspect ratio (longest/shorter axis) v. angle formed between the major axis and foliation plane. N is the number of grains.

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The most deformed investigated rock sample M4 is characterized by a dominant porphyroclastic texture given by smoothed microaugens of poikylitic albite that have grain size usually below 2 mm. Relatively subparallel S– C fabric (with intersecting angle c. 128) with later-formed shear bands characterizes this specimen, which is interpreted as an ultramylonite. Quartz ribbons entirely surround the porphyroclastic microdomains, or alternatively constitute independent regularly spaced (c. 1 cm) layers of syntectonically recrystallized white mica (Phg20 – 39) and quartz, the latter comprising 65 vol% of grains, with average radii ranging from 4 to 12 mm (Fig. 7). Epidote and chlorite are very common matrix-forming minerals. Quartz grains have aspect ratios in the range 1– 18, documenting the highest values of elongation (ARmean ¼ 3.6), consistent with the highest observed strain value. This is also confirmed by the bimodal cluster of the angles between the longest grain axis and the mylonitic foliation. Their range is 08–388 and 1408 –1808, respectively.

CPO of quartz and mica Quartz c-axis orientations were determined by the CIP method (Panozzo Heilbronner & Pauli 1993; Heilbronner 2000b) on thin sections cut perpendicular to the mylonitic foliation (XZ-plane) and parallel to the stretching lineation (X ). For each sample we selected quartz domains following two fundamental criteria: (1) the grain size of quartz within the domain is representative of the entire thin section; and (2) regions in the vicinity of large porphyroclasts that have a strong influence on the strain distribution in their surroundings were avoided (in particular fine-grained recrystallized quartz). Quartz c-axis patterns are shown in Figure 8 together with a colour-code legend. For each pixel of the image the orientation of the c-axis is established. Dark areas represent masked regions occupied by other minerals (mainly mica and feldspar). At least three different quartz CPO patterns are evident in rock samples. Samples M8, M13, on the one hand, and sample M3, on the other, suggest dominant slip systems activated during deformation to be rhombic ,a. and basal ,a., respectively (Schmid & Casey 1986; Heilbronner & Tullis 2006). The strong maximum subparallel to Y in sample M13, and partially developed in M8, is typical for amphibolite-facies deformation (Jessell & Lister 1990). Contrary to the latter, the c-axis pattern of sample M3 suggests greenschist-facies deformation conditions (Schmid & Casey 1986). Slip systems inferred for samples M8, M3 and M13 are indicative of deformation temperatures in the range 400 –550 8C. These results are in agreement with geothermobarometric

estimates of the shear event in rocks from the same Aspromonte Peloritani Unit (Cirrincione et al. 2008). For the nominally highest strain sample M4 prismatic ,c. slip prevails, which corresponds to deformation at higher temperatures in that the c-axis distribution is similar to the upperamphibolite-facies patterns reported by Lister & Dornsiepen (1982); however, the activation of prismatic ,c. slip system under greenschist-facies conditions could be enhanced by water weakening (e.g. Blacic 1975; Mainprice et al. 1986). An alternative hypothesis that could explain this hightemperature-related pattern is the localization of shear strain within a narrow layer of rock mass, which in turn causes a local rise in temperature. The tiny grain sizes of the mica minerals and the low volume percentage of mica in sample M4 (1.7 vol%) did not allow representative CPO measurements of mica in all samples. Therefore, U-stage measurements of the orientation fabrics of mica could only be performed on mica-fish in samples M13 and M8. CPO stereoplots of the normal to the (001)-planes of mica are presented in Figure 9. The (001) pole figure of muscovite (biotite)-fish show a strong single maximum normal to foliation, with large angular spread towards Y in sample M13. Consequently, most of the poles to the (010) þ (100)-planes are concentrated on a great circle parallel to the foliation plane.

P- and S-wave velocities P- and S-wave velocities measured in the three fabric directions, X, Y and Z, at various confining pressures are compiled in Tables 4 and 5. The pressure dependence of the three P-wave velocities and the six S-wave velocities measured in the three orthogonal directions is presented in Figures 10–13. The velocity –pressure relations display the well-known initial steep velocity increase with increasing confining pressure. The non-linear rise is due to the progressive closure of microcracks, typically illustrating the pressure sensitivity of P- and S-waves. Quasi-linear behaviour is approached above about 200 MPa, indicating near-intrinsic behaviour of the compacted aggregate. Vp is higher parallel to the macroscopic foliation (XY-plane) and lower normal to it (parallel Z). P-wave anisotropy (A-Vp), as defined by the per cent differences between maximum and minimum velocity with respect to mean velocity (Birch 1961), is presented in Figures 10 –13. A-Vp is higher at low pressures; it decreases with increasing confining pressure due to the progressive closure of microcracks. Near-intrinsic properties are approached at about 400 MPa. Importantly, marked A-Vp is observed in samples M13, M8 and

ANISOTROPY OF NATURALLY DEFORMED ROCKS 59

Fig. 8. Quartz c-axis pole figures of studied samples (contour plots are contoured at eight intervals from 1.0 to 8.0 times uniform; upper-hemisphere projection). The mylonitic foliation plane is east– west oriented, dipping 908. The maximum density distribution is indicated for each stereoplot.

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Fig. 9. Crystallographic preferred orientation of poles to (001) of mica in samples M8 and M13. N is the number of measurements. Upper-hemisphere, equal-area projections. Solid bar, foliation plane.

M3 (7.08– 9.01%), whereas A-Vp is very low (2.26%) in sample M4. The compressional wave velocities measured at various confining pressures in the three structural directions, X, Y and Z, and corresponding anisotropy, as well as the densities, are compiled in Table 4. Another important property of anisotropy is shear-wave splitting (S1–S2), which occurs when

shear waves propagate through anisotropic materials (Crampin 1987). A single shear wave is split into two approximately orthogonal polarizations that travel at different velocities in the same direction. Marked shear-wave splitting is observed in samples M13, M8 and M3, samples that exhibit relatively high P-wave anisotropy. Pronounced shear-wave splitting is apparent parallel to foliation (XY-plane), with the fast-split shear wave being

Table 4. Compressional wave velocities (km s21), velocity anisotropy and densities of mylonites at various confining pressure conditions Sample

M13 mylonite

M8 mylonite

M3 Mylonite

M4 Mylonite

Pressure (MPa) (T ¼ 20 8C)

Propagation direction

X Y Z Mean Anisotropy (%) Density (g cm23) X Y Z Mean Anisotropy (%) Density (g cm23) X Y Z Mean Anisotropy (%) Density (g cm23) X Y Z Mean Anisotropy (%) Density (g cm23)

25

50

100

150

200

300

400

4.89 4.81 3.55 4.42 30.26 2.63 4.61 4.82 3.42 4.28 32.63 2.64 4.87 4.97 3.73 4.53 27.30 2.64 4.78 4.75 3.87 4.47 20.33 2.61

5.15 5.08 3.97 3.97 24.97 2.63 4.99 5.12 3.88 4.67 26.35 2.65 5.24 5.27 4.21 4.91 21.61 2.65 5.14 5.16 4.40 4.90 15.32 2.62

5.407 5.35 4.45 5.07 18.77 2.64 5.35 5.4 4.398 5.05 19.94 2.65 5.518 5.53 4.68 5.24 16.20 2.66 5.48 5.48 4.94 5.30 10.17 2.62

5.54 5.5 4.75 5.27 14.93 2.65 5.52 5.56 4.71 5.27 16.18 2.66 5.64 5.67 4.947 5.42 13.42 2.66 5.65 5.65 5.24 5.51 7.43 2.63

5.64 5.61 4.97 5.41 12.36 2.65 5.62 5.67 4.92 5.40 13.88 2.67 5.722 5.759 5.12 5.53 11.44 2.67 5.75 5.75 5.43 5.64 5.67 2.63

5.77 5.76 5.29 5.61 8.64 2.66 5.759 5.8 5.19 5.58 10.97 2.68 5.84 5.877 5.364 5.694 9.01 2.68 5.87 5.88 5.68 5.81 3.45 2.642

5.85 5.84 5.45 5.72 7.08 2.67 5.84 5.89 5.37 5.70 9.01 2.68 5.92 5.95 5.52 5.79 7.49 2.69 5.96 5.95 5.83 5.91 2.26 2.65

ANISOTROPY OF NATURALLY DEFORMED ROCKS

61

Table 5. Shear-wave velocities and maximum shear-wave splitting (km s21) at various confining pressure conditions Sample

M13 Mylonite

M8 Mylonite

M3 Mylonite

M4 Mylonite

Pressure (MPa) (T ¼ 20 8C)

Propagation direction

X Y Z Mean DVs parallel to foliation X Y Z Mean DVs parallel to foliation X Y Z Mean DVs parallel to foliation X Y Z Mean DVs parallel to foliation

25

50

100

150

200

300

400

2.75 2.68 2.45 2.63 0.42 2.71 2.68 2.36 2.58 0.48 2.79 2.73 2.50 2.67 0.41 2.88 2.87 2.60 2.78 0.110

2.92 2.90 2.65 2.82 0.37 2.91 2.86 2.60 2.79 0.45 2.98 2.94 2.71 2.88 0.37 3.15 3.15 2.85 3.05 0.070

3.08 3.05 2.88 3.00 0.33 3.08 3.03 2.85 2.99 0.38 3.16 3.13 2.94 3.07 0.34 3.41 3.37 3.11 3.29 0.013

3.17 3.16 3.01 3.11 0.30 3.18 3.16 2.99 3.11 0.33 3.26 3.23 3.06 3.18 0.32 3.5 3.458 3.237 3.40 0.015

3.24 3.24 3.11 3.20 0.28 3.25 3.24 3.08 3.19 0.31 3.32 3.30 3.14 3.25 0.30 3.54 3.50 3.31 3.45 0.008

3.33 3.34 3.23 3.30 0.25 3.34 3.34 3.20 3.29 0.28 3.40 3.38 3.24 3.34 0.27 3.59 3.55 3.41 3.52 0.014

3.39 3.96 3.30 3.55 0.22 3.40 3.40 3.27 3.36 0.25 3.45 3.43 3.30 3.39 0.25 3.61 3.58 3.46 3.55 0.014

Fig. 10. Sample M13. Directional dependence of P- and S- wave velocities in mylonites as a function of pressure. X, Y and Z represent the structural frame of the rock: X is parallel to lineation, Y is perpendicular to lineation within the foliation plane and Z is normal to foliation.

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Fig. 11. Sample M8. Directional dependence of P- and S-wave velocities in mylonites as a function of pressure. See the caption to Figure 10.

Fig. 12. Sample M3. Directional dependence of P- and S-wave velocities in mylonites as a function of pressure. See the caption to Figure 10.

ANISOTROPY OF NATURALLY DEFORMED ROCKS

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Fig. 13. Sample M4. Directional dependence of P- and S-wave velocities in mylonites as a function of pressure. See the caption to Figure 10.

Table 6. Comparison between the near-intrinsic physical properties measured at 400 MPa with those calculated by the Hashin–Shtrikman approximation (Hacker & Abers 2004) M3

M4

M8

M13

Calculated physical properties Density (g cm23) Vp (km s21) Vs (km s21) Poisson Vp/Vs

2.68 5.88 3.48 0.23 1.69

2.64 5.92 3.5 0.23 1.69

2.70 5.93 3.51 0.23 1.69

2.67 5.91 3.47 0.24 1.70

Measured physical properties (400 MPa) Density (g cm23) Vp (km s21) Vs (km s21) Poisson Vp/Vs

2.69 5.79 3.39 0.24 1.70

2.26 5.91 3.55 0.21 1.65

2.68 5.70 3.36 0.23 1.69

2.67 5.72 3.55 0.23 1.69

Table 7. Parameters for shear velocity–pressure relationship and derivatives (calculated with MATLAB Program VPPLOT), together with Poisson’s and Vp/Vs ratios Vp ¼ V0 þ DP

Sample

M13 M8 M3 M4

2

Vs0 (km s21)

D ¼ dVp/dP (1024 km s21 MPa21)

R

3.096 3.092 3.171 3.410

6.689 6.576 5.574 3.556

0.991 0.992 0.991 0.987

Poisson

Vp/Vs

(400 MPa)

(400 MPa)

0.231 0.230 0.235 0.213

1.690 1.689 1.699 1.649

R. CIRRINCIONE ET AL.

R

3.417 2.491 2.433 2.342 0.2354 0.5932 0.6895 0.7553 0.0263 20.0085 20.0198 20.0254

MPa )

0.996 0.994 0.992 0.994

b (km s21 MPa21) a (km s21 MPa22)

Vp ¼ a(ln P)2 þ b ln P þ c

2 21

10.353 11.981 10.769 10.477

D ¼ dVp/dP (10

0.300 0.333 0.266 0.281 M13 M8 M3 M4

290 270 247 261

5.595 5.544 5.629 5.758

162.2 141.8 142.0 146.3

V0 (km s21) Vc – V0

5.295 5.221 5.363 5.477

km s

21 24

Vp ¼ V0 þ DP

polarized parallel to foliation. Normal to foliation (parallel to Z) practically no shear-wave splitting is observed. In this direction, the sample behaves quasi-isotropically for shear waves. By contrast, the quasi-isotropic sample M4 exhibits no shearwave splitting (Fig. 13). Shear-wave velocities for the three structural directions, together with shearwave splitting data (parallel to foliation), are listed in Table 5. In Table 6 we compare the arithmetic mean of the near-intrinsic (400 MPa) P- and S-wave velocities measured in the three structural directions, X, Y and Z, with those calculated by the Hashin – Shtrikman approximation, based on the of the volume percentages of the constituent minerals and their single crystal elastic moduli. P-wave velocities are in the range 5.70–5.96 and 5.88 – 5.91 km s21, respectively. The corresponding S-wave velocities vary from 3.36 to 3.55 and from 3.47 to 3.51 km s21, respectively. Measured (400 MPa) densities are in the range 2.64 – 2.68 g cm23 and the corresponding calculated density data vary between 2.64 –2.70 g cm23. The relatively small differences between the measured and calculated velocities and densities give clear evidence that calculation of modal composition based on bulk rock and mineral chemistry using the best-fit solutions is a powerful alternative to point-counting. Finally, based on the regression of the linear segments of the pressure curves, we obtained the intrinsic pressure (D) derivatives of velocities and the intrinsic zero pressure intercept velocity V0. In the linear regime, the velocity –pressure relationship can be described with the equation: V(P) ¼ V0 þ DP(P  Pc )

P0 (MPa) Vc (km s21) Pc (MPa) Sample

Table 8. Parameters for compressional velocity –pressure relationship and derivatives (calculated with MATLAB Program VPPLOT)

c (km s21)

R2

0.999 0.998 0.981 0.997

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where Pc is the critical confining pressure above which a linear velocity increase has been observed. Results of S-wave velocity data in the linear regime are listed in Table 7. In addition, we also calculated the least-square solutions of the Vp –pressure relationship in the non-linear regime (Table 8), according to the expression by Wang et al. (2005): V(P) ¼ a( ln P)2 þ b ln P þ c(P  Pc ) where a and b are constants, c is the velocity when P ¼ 1 MPa. Computations were carried out using the MATLAB program VPPLOT. These calculations provide important constraints for the calibration of seismic data, as they permit the seismic velocities to be extrapolated at any P condition within the stability fields of the mineral assemblage of the rock.

ANISOTROPY OF NATURALLY DEFORMED ROCKS

Discussions The mylonitic leucogneisses collected from a wellpreserved domain examined within the Montalto crustal-scale shear zone are characterized by progressively increasing strain, documented by the development from coarser-grained to fine-grained quartz-rich mylonite (Fig. 3). The relative asymmetry of fabric parameters such as SPO (Fig. 7) and CPO patterns (Fig. 8) of constituent quartz minerals indicates that deformation kinematics are consistent with a non-coaxial flow, showing a top-to-the-NE displacement sense, in the present-day geographical co-ordinates (Fig. 2). The observed directional dependence (seismic anisotropy) of P- and S-wave velocities is mainly controlled by the deformation fabrics: Vp and Vs are faster parallel to foliation and slower normal to it (parallel to the Z direction). The shear-wave polarization is also strongly related to foliation (Figs 10–13). The fast-split shear wave is polarized parallel to foliation (XY-plane) and the slow shear wave is polarized normal to foliation. Parallel to Z (normal to foliation) no shear-wave splitting is observed; that is, in this direction S-waves propagate as if in a quasi-isotropic medium (Figs 10– 13). Furthermore, the marked non-linear increase in wave velocities and decrease in velocity anisotropy with increasing pressure suggests that oriented lowaspect ratio cracks contribute largely to the high Vp anisotropy measured at low pressure. Importantly, intercrystalline and intracrystalline cracks are closely linked to the morphological sheet plane (001) of biotite and muscovite. Increasing pressure and coeval closure of oriented cracks increases Vp and Vs, and decreases A-Vp. Compared to the velocities, anisotropy continues to decrease markedly over the whole pressure range, obviously due to the high compressibility of layered minerals along the c-axes (see Mainprice et al. 2008). The residual Vp anisotropy and shear-wave splitting is related to both the volume per cent of mica and its CPO: this is in agreement with earlier findings of Kern & Wenk (1990) and Rey et al. (1994). In addition, a change of fabric has also been observed as texture develops, as testified by microstructures, grain-size distribution, increased rounded shape of porphyroclasts and mica CPO in samples M13 and M8, and by quartz CPO patterns in all samples. The latter can be associated, on the whole, to greenschistfacies conditions (samples M13, M8 and M3), except for sample M4, which is characterized by prism ,c. slip-system activation, typically occurring under amphibolite-facies conditions (Fig. 8). As deformation increases the mineral assemblages in the mylonitic rocks M13, M8 and M3 change from K-feldspar, biotite and premylonitic low phengite white mica to synmylonitic high

65

phengite white mica and quartz. This retrograde evolution is also accompanied by the albitization process that is controlled by cationic exchange between K-feldspar and Naþ-rich fluid, preferentially channelled during the shear evolution (Hippertt 1998). Indeed, white mica and quartz become the main constituents of the fine-grained matrix, and albite porphyroblasts gradually substitute premylonitic K-feldspar (Figs 4 & 6a, b; Table 3). However, the remarkable grain-size reduction in sample M4, together with the complete albitization of prekinematic K-feldspar, is accompanied by the smallest amount of white mica (Figs 4 & 6). This peculiar feature may be explained by the presence of local mineralogical heterogeneities that favoured a new induced shearing reaction, giving rise to the formation of albite, chlorite and K-rich fluid at the expense of phengite, silica and Na-rich fluid. This dehydration reaction, developed under retrograde metamorphic conditions, can be interpreted as the combined result of local centimetre-scale heterogeneities within the outcrop and of confined temperature increase due to intense shear localization (Burlini & Bruhn 2005). Moreover, the preferential channelling of fluids, as well as the huge increase in net reaction surface due to intense grain-size reduction, probably had a catalytic effect that intensified the mechanical weakening and the rheological evolution towards rock types (i.e. ultramylonite) characterized by enhanced soft behaviour (Williams & Dixon 1982; O’Hara 2007). This interpretation is also supported by microstructural evidence in sample M4, given that the early activation of the quartz prismatic ,c. slip system under greenschist-facies conditions accompanies the complete albitization of the synmylonitic matrix. Observed seismic and microstructural properties can be integrated by plotting the Vp values calculated at 400 MPa for the selected samples at increasing strain (Fig. 14a). This emphasizes a smooth increase in compressional velocities in the XY-plane (parallel to the foliation), whereas normal to foliation (parallel to Z) a steep decrease in Vp is observed in sample M8 compared to an increase in samples M3 and M4. This behaviour can be explained by the fact that M8 is influenced by the large premylonitic mica porphyroclasts with (001)-planes oriented parallel to the foliation, while M3 is characterized by an enrichment of mica content related to the synmylonitic crystallization of small crystalline aggregates of white mica. Finally, compared to M13, M8 and M3, sample M4 shows a marked decrease in mica content counterbalanced by an increase in albite content, which explains the observed steep increase in Vp velocity along the Z direction (Fig. 14a). Figure 14b clearly shows that strain-related changes observed for synmylonitic quartz grain

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Fig. 14. (a) For the studied mylonitic rocks, the relationship between modal amounts of the main constituents and Vp patterns according to the structural frame of the rock (X, lineation, Y, perpendicular to lineation within the foliation plane and Z, pole to foliation). (b) Vp-anisotropy pattern at increasing strain and mean aspect ratio, together with mean equivalent radii for the quartz population. Seismic properties determined at 400 MPa.

ANISOTROPY OF NATURALLY DEFORMED ROCKS

sizes and aspect ratios do not markedly control the bulk anisotropy of P- and S-wave velocities. This also holds for the strain-related changes to the CPO of the quartz c-axes (Fig. 8), where it is evident that the contribution of this mineral to the bulk seismic anisotropy for the studied rocks is negligible. It is important to note that the highest strained sample M4, as testified by the quartz c-axis pattern with a strong maximum around X (Fig. 8), presents the lowest value of seismic anisotropy. Finally, the modal content of the strongly anisotropic phyllosilicate minerals (biotite, muscovite) and their combined CPO and SPO are the dominant factors that control the bulk seismic anisotropy observed in the deformed rocks. Mica has the strongest anisotropy of single-crystal P-velocity (44%) compared to quartz (24%) and feldspar minerals (about 31%). The concentration of [001]-axes parallel to the Z direction (Fig. 9) also explains the directional dependence of experimentally determined velocities, as [001] is the slowest direction for the propagation of compressional waves in a mica single crystal (Vaughan & Guggenheim 1986). The highest single crystal velocities along [100] þ [010] explain the highest velocity values measured along the XY foliation plane.

Final remarks The seismic anisotropy of polycrystalline aggregates is controlled by several parameters operating as counterbalancing factors. Physical properties of constituent minerals, their volume fraction, their spatial orientation (SPO) and their crystallographic preferred orientation (CPO), that on the whole define the mesoscopic fabric of the rock, can interact in a very complex manner. They can give rise either to a positive or a negative contribution to the bulk seismic anisotropy. Even if it is difficult to ascertain whether bulk seismic anisotropy is grain-size sensitive or not, and what mineralogical change could represent a threshold for changing the elastic properties in increasingly deformed rocks, our data suggest that a close relationship exists between bulk seismic anisotropy and fabric-related features that ultimately define the bulk textural anisotropy. Indeed, a positive correlation between seismic- and fabric-related anisotropy has been observed from sample M13 to M8, followed by a smooth negative correlation towards sample M3. In sample M4, which is the most deformed (highest mean aspect ratio of quartz grains), there is no correlation from observations and, importantly, it exhibits the lowest value of seismic anisotropy (Fig. 14b). The observed decoupling between textural development and seismic anisotropy, particularly evident in sample M4, can be explained as the result of an initial local centimetre-scale compositional

67

heterogeneity. This, together with localized temperature increase and preferential channelling of fluids, drove the changes in physical properties during the shearing evolution, such as the pervasive shearinginduced white mica breakdown observed in the most deformed sample. Such evidence is interpreted as the primary factor responsible for the abrupt decrease in bulk seismic anisotropy compared to the other samples because of the strong planar shape of phyllosilicates, which makes them easily and promptly oriented during deformation, being characterized by a very high elastic single crystal anisotropy. Finally, it is worth noting that as a consequence of localized strain into tabular regions within a kilometre-thick shear zone, such as in the area of Montalto, the bulk seismic anisotropy value of sheared rocks could locally be influenced, and even sensitively lowered. This should be taken into account, even in a regional geophysical study of an area where a crustal-scale shear zone is exposed. This work was supported by P.R.A. 2006 (research projects grants of the University of Catania); fund 20104001016 and by P.R.I.N. 2007 (research project grants by the Italian MIUR); fund 20104005015; project title: ‘Strain rate in mylonitic rocks and induced changes in petrophysical properties across the shear zones’. We are grateful for constructive reviews by L. Burlini* and J. White; editorial handling by G. Gosso and I. Spalla is great appreciated. Last but not least, we thank P. Fiannacca for helpful discussion. *L. Burlini died on December 18, 2009. He is deeply missed by his family, friends, and all of those who were touched and inspired by his energy and creativity.

References B LACIC , J. D. 1975. Plastic-deformation mechanisms in quartz: The effect of water. Tectonophysics, 27, 271–294. B URLINI , L. & B RUHN , D. 2005. High-strain zones: laboratory perspectives on strain softening during ductile deformation. In: B RUHN , D. & B URLINI , L. (eds) High-strain Zones: Structure and Physical Properties. Geological Society, London, Special Publications, 245, 1 –24. C IRRINCIONE , R., O RTOLANO , G., P EZZINO , A. & P UNTURO , R. 2008. Poly-orogenic multi-stage metamorphic evolution inferred via P–T pseudosections: an example from Aspromonte Massif basement rocks (Southern Calabria, Italy). Lithos, 103, 466–502. C HRISTENSEN , N. I. & M OONEY , W. D. 1995. Seismic velocity structure and composition of the continental crust: a global review. Journal of Geophysical Research, 100, 9761– 9788. C RAMPIN , S. 1987. Geological and industrial implications of extensive-dilatancy anisotropy. Nature, 328, 491–496. D EER , W. A., H OWIE , R. A. & Z USSMAN , J. 1998. An Introduction to the Rock-forming Minerals. Longman, Essex. F AZIO , E., H EILBRONNER , R. & P UNTURO , R. 2007. Microstructural study and petrophysical characterization of naturally deformed leuco-gneisses from the Montalto shear zone (Aspromonte Massif, southern

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M AINPRICE , D., L E P AGE , Y., R ODGERS , J. & J OUANNA , P. 2008. Ab initio elastic properties of talc from 0 to 12 GPa: interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure. Earth and Planetary Science Letters, 274, 327–338. O’H ARA , K. 2007. Coupled deformation and reaction softening Processes: retrograde shear zones in the Rosslare Complex, south-east Iireland. Irish Journal of Earth Sciences, 25, 63–80. O RTOLANO , G., C IRRINCIONE , R. & P EZZINO , A. 2005. P–T evolution of Alpine metamorphism in the southern Aspromonte Massif (Calabria– Italy). Schweizerische Mineralogische und Petrographische Mitteilungen, 85, 31–56. P ANOZZO H EILBRONNER , R. & P AULI , C. 1993. Integrated spatial and orientation analysis of quartz c-axes by computer-aided microscopy. Journal of Structural Geology, 15, 369 –382. P EZZINO , A., A NGI` , G. ET AL . 2008. Alpine metamorphism in the Aspromonte Massif: implications for a new framework for the southern sector of the Calabria–Peloritani Orogen, Italy. International Geology Review, 50, 423–441. P EZZINO , A., P ANNUCCI , S., P UGLISI , G., A TZORI , P., I OPPOLO , S. & L O G IUDICE , A. 1990. Geometry and metamorphic environment of the contact between the Aspromonte-Peloritani Unit (Upper Unit) and Madonna di Polsi (Lower Unit) in the central Aspromonte area (Calabria). Bollettino della Societa` Geologica Italiana, 109, 455–469. R EY , P., F OUTAIN , D. & C LEMENT , W. 1994. P-wave velocity across a non-coaxial ductile shear zone: consequences for the upper crustal reflectivity. Journal of Geophysical Research, 99, 4533– 4548. R OSSETTI , F., F ACCENNA , C., G OFFE` , B., M ONIE` , P., A RGENTIERI , A., F UNICIELLO , R. & M ATTEI , M. 2001. Alpine structural and metamorphic signature of the Sila Piccola Massif nappe stack (Calabria, Italy): insights for a tectonic evolution of the Calabrian Arc. Tectonics, 20, 112 –133. S CHMID , S. & C ASEY , M. 1986. Complete fabric analyses of some commonly observed quartz c-axis patterns. In: H OBBS , B. E. & H EARD , H. C. (eds) Mineral and Rock Deformation: Laboratory Studies. The Paterson Volume. Geophysical Monograph, American Geophysical Union, 36, 263– 286. S IIVOLA , J. & S CHMID , R. A. 2007. Systematic nomenclature for metamorphic rocks: list of mineral abbreviations. Recommendations by the IUGS Subcommission on the Systematics of Metamorphic Rocks. Recommendations, web version of 01.02.2007. V AUGHAN , M. T. & G UGGENHEIM , S. 1986. Elasticity of muscovite and its relationship to crystal structure. Journal of Geophysical Research, 91, 4657– 4664. W ANG , Q., J I , S., S ALISBURY , M. H., X IA , B., P AN , M. & X U , Z. 2005. Pressure dependence and anisotropy of P-wave velocities in ultra high pressure metamorphic rocks from the Dabie– Sulu orogenic belt (China): implications for seismic properties of subducted slabs and origin of mantle reflections. Tectonophysics, 398, 67–99. W ILLIAMS , G. & D IXON , J. 1982. Reaction and geometrical softening in granitoid mylonites. Textures and Microstructures, 4, 223– 239.

Geometry of intercrystalline brine in plastically deforming halite rocks: inference from electrical resistivity TOHRU WATANABE Department of Earth Sciences, Faculty of Science, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan (e-mail: [email protected]) Abstract: Electrical impedance measurements were performed on deforming fine-grained (c. 300 mm) synthetic halite rocks containing small quantities of water in order to study the distribution of intercrystalline brine. The experimental conditions were 125 8C and 50 MPa confining pressure. The resistivity at the predeformational, heated and hydrostatically pressurized state suggests that brine is interconnected in halite. The resistivity progressively increases with deformation, reflecting the change in distribution. In this paper we applied a simple tube model to the resistivity change, and found that the change must be caused by deformation of a thin fluid path with an initial aspect ratio of less than 2  1024. Brine must, therefore, exist on grain boundaries as a thin fluid film. Previous studies on dihedral angles, however, showed that brine cannot be interconnected under our experimental conditions. The variation in grain-boundary energy cannot explain the coexistence of grain-boundary brine with a positive dihedral angle. The observed resistivity change requires grain-boundary brine to be very thin (,100 nm). Such a thin fluid film might have properties distinct from the bulk fluid, and coexist with brine pores at grain corners and grain faces.

Fluid-bearing rocks should play an important role in the dynamics of Earth’s crust and mantle. Intercrystalline fluid can strongly affect rheological and transport properties of rocks. The influences of fluid are strongly dependent on its distribution. For a good understanding of the behaviour of fluidbearing rocks, it is essential to understand the fluid distribution in rocks. The halite –water system is a simple solid– liquid system. Physical properties and solid– liquid textures have been studied to understand natural salt rocks, and to use this system as a good analogue for crustal and mantle rocks (e.g. Spiers et al. 1988). The dihedral angle between NaCl and brine has been measured systematically at under various pressure and temperature conditions (Lewis & Holness 1996; Holness & Lewis 1997). The dihedral angle has been widely accepted as a key parameter that controls solid– liquid textures (e.g. Watson & Brenan 1987). The liquid phase is interconnected if the dihedral angle is less than 608. Only when the dihedral angle is 08 does the liquid phase completely wet the grain boundaries. The dihedral angle in halite –water system is about 708 at room temperature and pressure (Holness & Lewis 1997). Brine exists as liquid-filled pores at grain corners and grain faces (e.g. Watson & Brenan 1987). The pores are isolated until the liquid volume fraction is increased to a certain amount (1 vol.% or more). This threshold increases with dihedral angle (von Bargen & Waff 1986). The dihedral

angle decreases to less than 608 at higher pressure and temperature conditions (Holness & Lewis 1997). However, observations contradictory to these dihedral angle values have been reported. Watanabe & Peach (2002) showed transmitted light images of thin brine tubes along grain edges at room temperature and pressure. Schenk & Urai (2004), based on SEM observations at room temperature and pressure, argued that brine is at least partly interconnected under these conditions. They suggested that the large variation in interfacial energy might cause significant deviations from textures expected from the reported dihedral angle. Recently, Schenk et al. (2006) and Desbois et al. (2008), based on cryo-SEM observations, showed that brine exists along grain boundaries as thin films at room temperature and pressure. Watanabe & Peach (2002) measured the electrical impedance of deforming wet halite rocks to investigate brine distribution. The experimental conditions were 125 8C and 50 MPa confining pressure. The previously reported dihedral angle is larger than 608 under these condition (Holness & Lewis 1997). Interconnection of brine cannot be expected. The resistivity at predeformational hydrostatic condition, however, implies that brine is interconnected at this condition. The resistivity changed along with deformation, reflecting the change in fluid distribution. Although we have already reported experimental results, the brine distribution in halite rocks has not been fully

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 69– 78. DOI: 10.1144/SP332.5 0305-8719/10/$15.00 # The Geological Society of London 2010.

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understood. How is brine interconnected at hydrostatic conditions? What change in brine distribution caused the resistivity change? In order to solve these problems we applied a simple tube model to the observed resistivity change, and estimated the tube geometry. In this paper we will at first summarize experimental techniques and the results of Watanabe & Peach (2002), and infer the geometry of brine from the observed resistivity change. Causes of grain-boundary wetting will also be discussed.

Experimental techniques Wet halite rock samples were prepared by cold pressing and annealing analytical grade NaCl powder. Annealed samples with a grain size of 200 – 450 mm were machined to right cylinders (120 mm long and 50 mm diameter) with a porosity of 0.1– 0.5% (Table 1). Details of sample preparation were given in Watanabe & Peach (2002). The water content of the samples was measured using a Fourier transform infrared (FTIR) spectrometer (Magna 860, Nicolet). In ‘wet’ experiments, water content before deformation was around 30 ppm. The water content of the ‘dry’ sample measured shortly after deformation was 5 + 2 ppm. Deformation experiments were performed using triaxial Heard deformation apparatus at the HPT Laboratory, Utrecht University. Sample assembly was as described in Watanabe & Peach (2002). The electrical impedance of the sample was measured using the two-electrode method with an impedance analyser (Solartron 1260). Platinum electrodes (45 mm diameter), which were coated with graphite paste to reduce the end friction, were placed at both ends of the sample. Impedance showed a Debye-type frequency dependence to which a parallel array of capacitor and resistor could be applied as an equivalent circuit. The observed resistance was dominated by the sample resistance, while the observed capacitance mainly came from the measurement system, which is

usually called a parasitic capacitance. Our analysis is focused on the resistive part. All deformation tests were conducted at 125 8C and 50 MPa confining pressure, and at a strain rate of approximately 5  1027 s21. Dilatancy is suppressed under these conditions (Peach & Spiers 1996). The confining pressure of 50 MPa was first applied to a sample at room temperature and maintained for about a half day. Temperature was then increased to 125 8C at a rate of approximately 10 8C h21 and maintained for about 3 days before deformation. Resistivity gradually decreased to approach a stationary value during this stage. After stabilization, a constant displacement rate compression test was carried out to a total strain of 7–25% (Table 1). After terminating a deformation test, the sample was cooled to a temperature of about 30 8C over approximately 2 h and then taken out for microstructural examination.

Mechanical data and resistivity change Stress– strain curves and resistivity changes are shown in Figure 1. Wet experiments reproducibly show work hardening up to c. 9% strain followed by a transient stress drop. After the stress drop, wet experiments (p40t111 and p40t114) show work hardening and then a slight stress drop at about 22% strain. Dry experiment p40t109 shows work hardening up to 25% strain without any stress drops. Microstructures of recovered samples were observed using light microscopy in order to study the nature of stress drops. No recrystallized grains were observed in dry experiment p40t109, while large, recrystallized grains with few substructures were observed in wet samples recovered after stress drops (p40t111, p40t112 and p40t114). No recrystallized grains were observed in sample p40t115 recovered before the stress drop (Watanabe & Peach 2002). Recent microstructral studies on recovered sample p40t115, however, showed that migration

Table 1. List of experiments Specimen p40t109 p40t111 p40t112 p40t114 p40t115

Initial porosity (%)

Water content (ppm)

Averaged strain rate (1027 s21)

Final strain

0.5 + 0.1 0.1 + 0.1 0.1 + 0.1 0.2* + 0.1 0.1 + 0.1

5+2 36 + 3 28 + 3 36 + 4 32 + 3

4.862 + 0.004 4.882 + 0.004 4.588 + 0.002 4.905 + 0.005 4.463 + 0.003

0.253 0.252 0.119 0.254 0.070

*This value was calculated from the dimensions and weight of the sample before machining. Machining removes both ends of an annealed sample, which are usually more porous than the middle part. The porosity after machining should have been ,0.2.

GEOMETRY OF INTERCRYSTALLINE BRINE

71

Fig. 1. (a) Stress– strain curves of deformation experiments (125 8C, 50 MPa confining pressure, strain rate c. 5  1027 s21). Run p40t109 containing 5 ppm water is referred to as a ‘dry’ experiment. (b) Resistivity change along with deformation. The electrical impedance of the dry sample (p40t109) was close to the measurement upper limit, causing the measurement to be unstable.

recrystallization was already active before the stress drop. Ter Heege et al. (2005), based on microstructural analysis using light microscopy, reported that minor dynamic recrystallization was evidenced by occasional grain-boundary bulges, and that grain boundaries preferentially aligned at 458 to the compression direction. Both the median and arithmetic mean of grain size, measured in terms of equivalent circular diameters, increased with respect to the starting material by around 50%. An EBSD study on sample p40t115 by Pennock et al. (2006a)

showed that grains in a hard orientation for slip increased in grain size and developed cube shapes. Based on these microstructural observations, we think that the stress drop is related to fluid-assisted migration recrystallization. Migration recrystallization weakens a material by replacing strainhardened grains with annealed dislocation-free grains. The flow stress is controlled by the competition between hardening and weakening mainly due to recrystallization (Watanabe & Peach 2002). When weakening equals hardening in magnitude,

72

T. WATANABE

the flow stress is maximized. Weakening exceeds hardening, leading to the stress drop. All wet experiments show similar resistivity change, although there is some difference in magnitude. Resistivity increases progressively with deformation and shows a small drop slightly ahead of the stress drop, then increases again with further deformation. Migration recrystallization might cause the thickening of fluid paths to lead to the resistivity drop (Watanabe & Peach 2002). In dry experiment p40t109, the initial resistivity was higher than that in wet experiments by about 1 order of magnitude. Resistivity shows no significant change up to 2% strain and then gradually increases. Although the measured value is unstable and unreliable, it seems to reach a stationary value (c. 5  107 V m).

Brine geometry inferred from resistivity change Electrical conduction through intercrystalline brine dominates conduction in bulk. Resistivity of dry halite rocks is estimated to be more than 108 V m at 125 8C by extrapolating the conductivity data of polycrystalline NaCl (Mott & Gurney 1964). Resistivity of saturated NaCl solution is estimated to be 2.8  1023 V m at 125 8C (Watanabe & Peach 2002). Wet samples show a resistivity of 104 – 106 Vm, and even the dry sample shows a resistivity ,108 V m. Measured resistivity implies that intercrystalline brine forms an interconnected network

for electrical conduction. Post-deformational rapid resistivity changes also support the conduction through connected brine (Watanabe & Peach 2002). Resistivity increases with deformation in wet experiments. Watanabe & Peach (2002) ascribed the increase in resistivity to fluid squirt accompanying deformation. Under an axial compression, some fluid paths will expel fluid to shrink, and other paths draw it to dilate. Shrinkage of paths dominates resistivity change, resulting in an increase in resistivity. We will here discuss the geometry of shrinking paths based on observed resistivity change. We focus on the resistivity change during elastic deformation (axial stress ,5 MPa) as elastic deformation of a cavity can be analytically evaluated. Resistivity change during elastic deformation is shown in Figure 2 as a function of the axial stress. The resistivity is normalized by the initial value. We will evaluate the resistance change of shrinking paths using the effective medium theory (Kirkpatrick 1973). The network of fluid paths in a sample is modelled using a network of resistors (Fig. 3). The effective resistance of this network, Reff, is obtained from: ð

(R1 R1 )f (R) eff  z  dR ¼ 0 1 R  2  1 R1 eff

(1)

where R is the resistance of a resistor, z is the number of resistors connected to a resistor at one end and f(R) the distribution function of resistance.

Fig. 2. Resistivity change during elastic deformation. The resistivity is normalized by the initial value, and is shown as a function of the axial stress.

GEOMETRY OF INTERCRYSTALLINE BRINE

73

(1), the effective resistance is obtained from: 1 R1 eff  (bR)   x (bR)1  2z  1 R1 eff

þ

Fig. 3. A network of resistors. A fluid path in a halite is modelled using a resistor. A resistor connects to z resistors at one end. We assume that all resistors have an initial resistance of R0. As the deformation proceeds, a part of fluid path shrinks. A fraction x of resistors increase their resistance by a factor b(b . 1).

For simplicity, we assume that all resistors have an initial resistance, R0, and that a fraction x of resistors increase their resistance by a factor b(b . 1) due to shrinking, along with deformation. From equation

1 R1 eff  R0  (1  x) ¼ 0: z 1 R0  2  1 R1 eff

(2)

The effective resistance is shown in Figure 4 as a function of the fraction of high resistance resistors, x. The effective resistance is normalized by the initial value. The factor of increase in resistance, b, and the number, z, are treated as parameters. As strain due to elastic deformation is very small, the change in sample geometry can be neglected. The normalized effective resistance can be, thus, directly compared with the normalized resistivity. The effective resistance is more sensitive to resistance change in network resistors for smaller number of z. When half of the resistors increase their resistance by a factor of 10, the normalized effective resistance increases to 2.7 (z ¼ 5) or 4 (z ¼ 3). Although the numbers x and z are difficult to constrain, we think that 0 , x , 0.5 and 3 , z , 5 are reasonable ranges. The factor of resistance increase should be larger than 10 in shrinking paths. We will next consider the shape of fluid tube based on this factor. We here consider elastic deformation of fluid paths under axial compression. We first consider a triple-junction fluid tube for a dihedral angle of

Fig. 4. Effective resistance as a function of high resistance resistors, x. The effective resistance is normalized by the initial value. The factor b and the number z are treated as parameters.

74

T. WATANABE

less than 608. The triple-junction tube can be modelled using a tube, the cross-sectional shape of which is given in the x –y plane by the parametric equations:  x ¼ r cos u þ

 1 cos 2u 2þ1   1 y ¼ r  sin u þ sin 2u 2þ1

(3a)

We next consider deformation of a tube with an elliptical cross-section. The initial length of major and minor axes is denoted by a0 and b0, respectively. Compressive stress is applied parallel to the minor axis, which maximizes the deformation of a tube (Fig. 6, inset). Elastic deformation of a tube obeys the differential equation:

(3b)

db 1n a0 ¼ ds G

where r and 1 are constants, and the parameter u varies from 0 to 2p to trace out the entire contour (Mavko 1980). The cross-sectional shape for r ¼ 1 and 1 ¼ 0.2 is shown in Figure 5. Elastic deformation due to compression along the x-axis can be evaluated using the complex variable method of Muskhelishvili (1953). The calculation method is briefly summarized in the Appendix. Deformed cross-sectional shapes are also shown in Figure 5. A triangular tube is so stiff that it cannot show a significant decrease in the cross-sectional area, even at an axial compression of 10 MPa. This should also be the case for other values of 0 , 1 , 1. The triangular tube is sufficiently stiff that it cannot cause the observed increase in resistivity.

(4)

where b, s, n and G are the minor axis, the axial stress, Poisson’s ratio and the rigidity, respectively (Bernabe et al. 1982). The major axis a will not change. The change in the minor axis is expressed as: b ¼ b0  (1  n)

h s si a0 ¼ a0  (1  n) a0 (5) G G

where

a0 ¼

b0 a0

(6)

is the initial aspect ratio of the tube cross-section.

Fig. 5. The cross-sectional shape of a model for triple-junction tubes (r ¼ 1 and 1 ¼ 0.2 in equation 3). Elastically deformed shapes are also shown for the compressional stress in the x-axis direction. The deformation for a compressional stress of 10 MPa is so small that significant change in the cross-sectional shape can hardly be seen.

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75

Fig. 6. Resistance of an elliptical tube (inset) as a function of the axial stress. Resistance is normalized using the predeformation value, R0. The compressional stress is applied parallel to the minor axis. The aspect ratio of the cross-section is varied from 1024 to 1022.

Resistance of the tube is given by: R ¼ rf

L pa 0 b

(7)

where rf and L are the resistivity of brine and the tube length, respectively. The resistance increases with increasing axial stress (Fig. 6). The resistance is normalized using the predeformation value, R0, which is given by: R0 ¼ rf

L : pa0 b0

(8)

Combining equations (5) –(8), the normalized resistance, R/R0, is calculated as a function of the axial stress. Rigidity and Poisson’s ratio are set to be 30 GPa and 0.15, respectively (Turcotte & Schubert 1982). The initial aspect ratio is varied from 1024 to 1022. The normalized resistance increases more steeply for smaller aspect ratios. A tube with an initial aspect ratio of 1024 increases its resistance by an order of magnitude at the axial stress of 3 MPa. In contrast, a tube with an initial aspect ratio of 1023 increases its resistance only by 50%, even at an axial stress of 10 MPa. We thus think that the observed resistivity change must be caused by deformation of thin fluid paths with an initial aspect ratio of less than 2  1024. There might be brine pores with a dihedral angle larger than 608. They are too stiff to cause the observed resistivity change along with deformation. We

think these fluid pores and thin fluid paths form an interconnected network. Thin, compliant fluid paths dominate the resistivity change, along with deformation. The estimated small aspect ratio suggests that brine exists on grain boundaries as a thin fluid film in wet samples. If the major axis is of the order of 100 mm (grain size), the minor axis will be of the order of 20 nm or less. However, it remains unsolved whether grain-boundary brine is a semicontinuous fluid film or it has an island-channel structure (Spiers & Schutjens 1990). van Noort et al. (2007) conducted the halite–glass and halite– halite contact experiments in the presence of brine, and showed that grain boundaries have crystallographically controlled irregularities such as asperities or channels. It is reasonable to think that grain boundaries in our samples have similar irregularities. Grain-boundary brine is required to explain the rapid grain-boundary migration observed in the wet samples (Watanabe & Peach 2002). Ter Heege et al. (2005) showed theoretically that a significant amount of strain in our wet samples was accommodated by pressure-solution creep. This also requires some form of brine layer along the grain boundaries. Resistivity change of the dry sample suggests the lack of grain-boundary brine. No sign of dynamic recrystallization was observed in the dry sample with a water content of 5 ppm. The initial resistivity of the dry sample (c. 5  106 V m) requires the interconnection of brine. Watanabe & Peach (2002) thus thought that grain-boundary brine was

76

T. WATANABE

too thin in the dry sample to promote grain-boundary migration. However, no significant increase in resistivity was observed during elastic deformation. This suggests that the interconnection of brine is maintained through fluid paths with larger aspect ratios. Stiff triple-junction tubes might govern the interconnection of brine. When the dry sample was prepared by heating it in flowing Ar gas, many grain boundaries opened. Grain-boundary water might have been preferentially lost, while a trace amount of water was left in triple-junction tubes.

Causes of grain-boundary wetting Resistivity change and dynamic recrystallization suggest that brine exits on grain boundaries and forms an interconnected network in halite rocks. This seems contradictory to the brine distribution inferred from dihedral angle measurements (Holness & Lewis 1997). As the dihedral angle was larger than 608 at 125 8C and 50 MPa, intercrystalline brine is not expected to interconnect in halite rocks under these conditions (fig. 6 in Holness & Lewis 1997). Three ideas have been proposed to explain the coexistence of grain-boundary fluid with a positive dihedral angle: (i) dynamic wetting; (ii) variation in grain-boundary energy; and (iii) thin fluid films. Dynamic wetting was at first observed in salt– brine systems (e.g. Urai 1983). Pockets of brine spread out into a thin film along grain boundaries during deformation. Later, similar dynamic wetting was reported in a partially molten peridotite system (Jin et al. 1994). Our experiments on wet halite rocks strongly suggest that grain-boundary brine exists even under hydrostatic conditions. Grain-boundary brine before deformation cannot be explained by dynamic wetting. The importance of the variation in grainboundary energy was pointed out by textural studies on a partially molten olivine –basalt system, which has been well studied as a model of partially molten upper mantle. Early studies assumed that grain-boundary energy is uniform in the system (e.g. Bulau et al. 1979). If the grainboundary energy were uniform, mechanical and chemical equilibria would require a constant dihedral angle and constant mean solid– liquid interface curvature. However, significant deviations from equilibrium textures expected from uniform grainboundary energy have been reported. The most notable deviation is the coexistence of flat and faceted solid –liquid interfaces with smoothly curved interfaces (Faul 2000). Deep penetration of melt between grain boundaries was also reported (Waff & Faul 1992), although previous works had reported dihedral angles of 308 –508 (e.g. Waff &

Bulau 1979). These observations were explained in terms of the variation in grain-boundary energy (e.g. Faul 2000). The solid–liquid texture equilibrates to minimize the total interfacial energy. There are two kinds of interface: solid–solid (grain boundary) and solid–liquid interfaces. The energy at the grain boundary comes from unsatisfied bonds existing across the interface. The grain-boundary energy depends on the misorientation of neighbouring grains (Faul 2000). A highly disordered grain boundary might have high interfacial energy and be replaced by a solid–liquid interface to lower the total interfacial energy (Waff & Faul 1992). This would be the case in wet halite rocks. Highenergy grain boundaries might be wet, but lowenergy boundaries fluid free. Using EBSD, Pennock et al. (2006b) studied the nature of grain boundaries in deformed halite rocks, including our samples p40t112 and p40t115. They showed that both high- and low-energy grain boundaries are wet. Therefore, the variation in grain-boundary energy cannot explain the coexistence of grainboundary brine with a positive dihedral angle. When fluids are confined within narrow spaces (,100 nm), their phase equilibria, and structural and dynamical properties become distinct from their bulk values (Israelachvili 1992). Because of the electrostatic forces from solid surfaces, a fluid within such a narrow space cannot behave as a structureless continuum. Such a fluid can exist along grain boundaries, even for systems characterized by non-zero dihedral angles (Hess 1994). Using infrared micro-spectroscopy, de Meer et al. (2005) studied the structure of the water confined between NaCl and CaF2 plates (thickness of ,185 nm), and showed that the hydrogen bonding of the water is modified according to the surface charge distribution of solid surfaces. Our resistivity change during elastic deformation suggests that the thickness of grain-boundary brine is of the order of 20 nm or less. Although the nature of brine confined between NaCl crystals has not been understood, we think that grain-boundary brine in our halite samples might have properties distinct from the bulk fluid and coexist with brine pores at grain corners and grain faces. For greater understanding, properties of thin fluid film should be investigated. Grain-boundary water might exist in crustal materials and play important roles in rheological and transport properties. Textural studies on waterbearing crustal materials have usually discussed the connectivity of water based on a mean dihedral angle (e.g. Yoshino et al. 2002). Estimated dihedral angles are larger than 608 under crustal conditions, and water cannot form interconnected networks in these materials. However, there might be a significant variation in grain-boundary energy. It is, then,

GEOMETRY OF INTERCRYSTALLINE BRINE

77

difficult to evaluate the connectivity of water through a mean dihedral angle. Moreover, there might be a thin film water between grain boundaries, which has properties distinct from the bulk water. The solid– fluid texture should be re-examined taking into account the variation in grain-boundary energy and the existence of a thin film of water. A good understanding of fluid distribution is essential to interpret the geophysical observations. NaCl is a well studied, crystallographically simple ionic solid. Study of the halite –water system provides a good framework for understanding of Earth’s materials.

for the case of plane strain. Poisson’s ratio is denoted by n. The problem is to find stress functions satisfying the boundary conditions. It is convenient to treat irregularly shaped cavities like those in Figure 5 by conformally mapping the outside of the complex z plane into the inside of a unit circle in the complex z plane by means of a function z ¼ v(z), which we choose here as:

Conclusions

where r and 1 are real constants. The values of z ¼ re iu on the unit circle (r ¼ 1) generate the parametric equations for the tube (equation 3). If a uniaxial tensile stress, p, is applied along an axis forming an angle, a with the x-axis, the stress functions are given by Mavko (1980) as:

We applied a simple tube model to the resistivity change during elastic deformation, and found that the change must be caused by deformation of the thin fluid path with an initial aspect ratio of less than 2  1024. Brine must, therefore, exist on grain boundaries as a thin fluid film. Such grainboundary brine is also required to explain rapid grain-boundary migration. The variation in grain-boundary energy cannot explain the coexistence of grain-boundary brine with a positive dihedral angle. The observed resistivity change requires grain-boundary brine to be very thin (,100 nm). Such a thin fluid film might have properties distinct from the bulk fluid and coexist with brine pores at grain corners and grain faces. To understand further the properties of thin fluid film should be investigated.

 z¼r

f (z ) ¼

1 1 2 þ z z 2þ1

  pr 1 1 2 þ 2eia z  z 4 z 2þ1

 (A4)

(A5)

c(z ) ¼ pr  i2a e þ  2z  (2 þ 1)ei2a  [2 þ (2 þ 1)2 ]z þ (2 þ 1)2 ei2a z3 : 2(2 þ 1)2  4(2 þ 1)z3 (A6) Using these equations, we can calculate displacements at the tube surface (r ¼ 1).

Appendix The deformation of a tube under remotely applied stress is conveniently posed using the complex variable notation of Muskhelishvili (1953). Stress solutions of two-dimensional problems in the theory of linear isotropic elasticity can be expressed in terms of two analytic functions f(z) and c(z) of the complex variable z ¼ x þ iy:

sxx þ syy ¼ 2[f0 (z) þ f0 (z)] syy  sxx þ 2isxy ¼ 2[zf00 (z) þ c0 (z)]

(A1) (A2)

where the overbar refers to the complex conjugate. The two functions are usually called stress functions. Displacements u and v can be found by integrating stresses as: 2G(u þ iv) ¼ kf(z)  zf0 (z)  c(z) where G is rigidity and k is given by

k ¼ 3  4n

(A3)

I am grateful to C. J. Peach and C. J. Spiers for numerous discussions on the resistivity change during deformation. G. J. Kasterlein, E. de Graaff, P. van Krieken, and J. L. Liezenberg are thanked for their assistance in experiments. Visits to the HPT Laboratory at Utrecht University (The Netherlands) has been supported by NWO in the form of Visitor’s Award (B 75-358) and JSPS-NWO Bilateral Program (2-178, 2-134). This study was partly supported by MEXT Grant-in-Aid for Scientific Research (19540444). G. Desbois and an anonymous reviewer are thanked for their helpful comments.

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statically recrystallizing halite – evidence from cryo-SEM observations. Geofluids, 6, 93–104. S PIERS , C. J. & S CHUTJENS , P. M. T. M. 1990. Densification of crystalline aggregates by fluid phase diffusional creep. In: B ARBER , D. J. & M EREDITH , P. G. (eds) Deformation Processes in Minerals, Ceramics and Rocks. The Mineralogical Series, 1. Unwin Hyman, London, 334–354. S PIERS , C. J., U RAI , J. L. & L ISTER , G. S. 1988. The effect of brine (inherent or added) on rheology and deformation mechanisms in salt rock. In: H ARDY , H. R. & L ANGER , M. (eds) The Mechanical Behaviour of Salt II. Trans Tech, Clausthal-Zellerfeld, Germany, 89–102. T ER H EEGE , J. H., D E B RESSER , J. H. P. & S PIERS , C. J. 2005. Rheological behaviour of synthetic rocksalt: the interplay between water, dynamic recrystallization and deformation mechanisms. Journal of Structural Geology, 27, 948 –963. T URCOTTE , D. L. & S CHUBERT , G. 1982. Geodynamics, Application of Continuum Physics to Geological Problems. Wiley, Chichester. U RAI , J. L. 1983. Water assisted dynamic recrystallization and weakening in polycrystalline bischofite. Tectonophysics, 96, 125 –157. V AN N OORT , R., S PIERS , C. J. & P EACH , C. J. 2007. Effects of orientation on the diffusive properties of fluid-filled grain boundaries during pressure solution. Physics and Chemistry of Minerals, 34, 95–112. V ON B ARGEN , N. & W AFF , H. S. 1986. Permeabilities, interfacial areas and curvature of partially molten systems: Results of numerical computations of equilibrium microstructures. Journal of Geophysical Research, 91, 9261–9276. W AFF , H. S. & B ULAU , J. R. 1979. Equilibrium fluid distribution in an ultramafic partial melt under hydrostatic conditions. Journal of Geophysical Research, 84, 6109– 6114. W AFF , H. S. & F AUL , U. H. 1992. Effects of crystalline anisotropy on fluid distribution in ultramafic partial melts. Journal of Geophysical Research, 97, 9003– 9014. W ATANABE , T. & P EACH , C. J. 2002. Electrical impedance measurement of plastically deforming halite rocks at 125 8C and 50 MPa. Journal of Geophysical Research, 107, (B1) 2004; doi: 10.1029/ 2001JB000204. W ATSON , E. B. & B RENAN , J. M. 1987. Fluids in the lithosphere, 1. Experimentally-determined wetting characteristics of CO2 –H2O fluids and their implications for fluid transport, host-rock physical properties, and fluid inclusion formation. Earth and Planetary Science Letters, 85, 497– 515. Y OSHINO , T., M IBE , K., Y ASUDA , A. & F UJII , T. 2002. Wetting properties of anorthite aggregates: Implications for fluid connectivity in continental lower crust. Journal of Geophysical Research, 107, (B1), 2007; doi: 10.1029/2001JB000440.

Brittle plus plastic deformation of gypsum aggregates experimentally deformed in torsion to high strains: quantitative microstructural and texture analysis from optical and diffraction data M. ZUCALI1*, V. BARBERINI2, D. CHATEIGNER3, B. OULADDIAF4 & L. LUTTEROTTI5 1

Dipartimento di Scienze della Terra ‘Ardito Desio’, Universita` degli Studi di Milano, Via Mangiagalli 34, 20133 Milano, Italy

2

Dipartimento di Scienze Geologiche e Geotecnologie, Universita` degli Studi di Milano-Bicocca, Piazza della Scienza 4, 20126 Milano, Italy 3

Laboratoire de Cristallographie et Sciences des Mate´riaux (CRISMAT), Ecole Nationale Supe´rieure d’Inge´nieurs de Caen (ENSICAEN), Universite´ de Caen Basse-Normandie, 6 Bd M. Juin, 14050 Caen, France

4

Institut Laue-Langevin, BP 156, 6 rue Jules Horowitz, 38042 Gre´noble Cedex 9, France

5

Department of Materials Engineering, Engineering Faculty, University of Trento, Via Mesiano, 77, I-38100 Povo-Trento, Italy *Corresponding author (e-mail: [email protected]) Dedicated to the memory of Luigi Burlini Abstract: This contribution presents a quantitative microstructural analysis of a polycrystalline aggregate of gypsum, deformed in torsion (T ¼ 70–90 8C) at g (shear strain) ranging from 0 to 4.82. Quantitative microstructural analysis is used to compare the evolution of microstructures observed by optical microscope with those obtained from analysis of X-ray and neutron diffraction data. This analysis shows that during experimental deformation, gypsum accommodated strain by brittle and plastic deformation mechanisms, developing Riedel-like microfaults with plastic foliations and crystallographic preferred orientation (CPO). The relations of microstructures show that with increasing strain, the Riedel systems start from R planes with an angle of 308 to the Imposed Shear Plane. This angle decreases (58–158) when strain increases, and Y planes develop. Quantitative texture analysis (QTA) shows that S-foliations start developing at low g and maintain their orientation up to high g, and that the most active slip system is the (010) along normal to (100) and the [001]-axis. Shape preferred orientation (SPO) of gypsum does not coincide with the theoretical orientation as it does not decrease with increasing strain. This discrepancy is explained by the role of the brittle shear planes that impose a back rotation to gypsum. No brittle to plastic transition occurs. But both plastic and brittle structures contemporaneously accommodate and localize strain.

Many rocks (metamorphic, igneous and sedimentary) show non-random orientation distributions of their crystallites that results in anisotropies of macroscopic physical properties (Turner & Weiss 1963; Wenk 1985; Randle & Engler 2000; Karato 2008). The crystallographic/lattice or shape preferred orientation (SPO) of crystals with respect to macroscopic fabric axes may be attained in response to deformation processes. The interpretation of textures (i.e. crystallographic preferred orientation) in materials (e.g. rocks or rock analogues) relies on a quantitative description of the orientation features. Together with several other texture analysis methods, quantitative texture analysis (QTA) using neutron and X-ray diffraction has

been successfully used in recent years to completely describe the crystallographic preferred orientation (CPO) of naturally deformed rocks (e.g. Kocks et al. 1998; Leiss et al. 2000; Zucali et al. 2002; Zucali & Chateigner 2006; Wenk 2006 and references therein). In this work we present, together with a detailed and complete microstructural analysis, the results of QTA measurements from neutron and X-ray experiments carried out on samples of natural gypsum that were experimentally deformed in torsion at high shear strains and confining pressure, and at various temperatures and strain rates (Barberini et al. 2005). The QTA measurements were performed at the ILL neutron facility (Gre´noble,

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 79– 98. DOI: 10.1144/SP332.6 0305-8719/10/$15.00 # The Geological Society of London 2010.

G BþG G GþB GþB GþB G GþB G G G G G G – D U U U D D U D D D D D D *OM, optical microscopy; NT, neutron texture; XT, X-ray texture.

– 127 70 90 90 90 70 90 70 90 90 90 70 70 – 0.36 0.49 1.12 1.19 1.78 1.80 2.49 2.59 2.61 2.89 4.07 4.11 4.82 VGO V4 V7 V6 V8 V13 V15 V9 V18 V14 V11 V10 V16 V20

– 5  1025 5  1025 up to 1  1024 1  1024 1  1024 1  1024 1  1024 up to 6  1024 2  1024 6  1024 2  1024 1  1024 1  1024

Gypsum/bassanite Drained/undrained Temperature (8C) Shear strain rate (s21) Shear strain ( g)

The studied gypsum samples are cores about 9 mm thick and 7–15 mm long obtained from natural gypsum (gypsum .99%). The specimens (Table 1 and Fig. 1a) were deformed in torsion experiments (Paterson & Olgaard 2000) up to high shear strain values (up to g ¼ 5) at a confining pressure of 300 MPa and at various temperatures (from 70 to 90 8C) and strain rates (between 1023 and 1025 s21) (Barberini et al. 2005). Within cylindrical samples deformed in torsion the deformation varies radially from no deformation at the centre of the core to maximum deformation at the outer surface. Therefore, thin sections for microstructural observations were cut as close as possible to the outer cylinder surface (Fig. 1b).

Table 1. Experimental details of the studied samples

Sample description

Analytical techniques*

France) and at the Laboratoire de Cristallographie et Sciences des Mate´riaux (CRISMAT), Ecole Nationale Supe´rieure d’Inge´nieurs de Caen (ENSICAEN, France). The study of the deformation behaviour of minerals that form evaporitic rocks is of great importance because these rocks can easily localize deformation. Several studies have described the structures and behaviour of gypsum deformed under various conditions (Craker & Schiller 1962; Baumann 1984; Panozzo Heilbronner & Olgaard 1987; Harland et al. 1988; Ko et al. 1995; Stretton 1996; Barberini et al. 2005) and the tectonic implications of gypsum dehydration (Heard & Rubey 1966), but only a few have described the microstructural and textural evolution (Levykin & Parfenov 1983; Kern & Richter 1985; Panozzo Heilbronner & Dell’Angelo 1990; Panozzo Heilbronner 1993). Such studies describe the evolution of a preferred orientation of poles to planes (010), perfect cleavage planes, which is parallel to s1 (main axial stress). (010) , 001. is considered the most common slip plane but others have also been recognized (Muegge 1898; De Meer 1995). The deformation of gypsum has also been studied under various conditions in order to investigate the behaviour under transient drained conditions (Olgaard et al. 1995). Moreover, the instantaneous and long-term behaviour of natural gypsum has been studied to understand the traces of dissolution observed in pillars of underground gypsum quarries (Hoxha et al. 2006; Castellanza et al. 2008) and to model its long-term behaviour (Hoxha et al. 2005). Finally, some authors have studied the creep of wet gypsum aggregates, and the relationships with the mechanics of thrust faults and other large-scale structures, associated with oil and gas accumulations (De Meer & Spiers 1995).

OM– NT – XT OM OM– NT – XT OM OM– NT – XT OM OM OM OM– NT – XT OM– NT – XT OM OM– NT – XT OM– XT OM– NT

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Sample

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GYPSUM QUANTITATIVE MICROSTRUCTURES 81

Fig. 1. (a) Deformation experiments in Paterson gas-medium HPT apparatus equipped for torsion testing (ETH, Zu¨rich, Switzerland). Details about these experiments are in Barberini et al. (2005). (b) Top: comparison between deformation geometry in natural samples, and torsion samples and effect of torsion deformation on pole figures (from neutron and X-ray). Bottom: relationships between fabric reference and sample reference within pole figures. (c) Geometry of the performed texture measurements (neutron and X-ray) using the Curved Position Sensitive detector (CPS). ISP, Imposed Shear Plane; X, Y, Z, principal fabric axes; XY (S) plane, foliation plane.

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Fig. 2. Description of (a) brittle and (b) plastic deformational systems affecting the deformed samples. (c)–(f) Microphotographs (left) and microstructural analysis (right) of gypsum samples deformed in torsion up to g ¼ 2.59. The starting material (c) shows an heterogeneous grain size distribution (20– 250 mm) and a polygonal fabric with only occasional SPO. At g ¼ 1 –2, (d) and (e), undulose extinction and kink bands affect an increasing number of grains of

GYPSUM QUANTITATIVE MICROSTRUCTURES

Microstructural analysis The sample reference frame is defined by the Imposed Shear Plane (ISP) (Fig. 1), which is the plane perpendicular to the optical images and parallel to the long edge of the images, with the rotation axis of the torsion experiments being the vertical axis of the images (Figs 2 & 3). The three fabric axes X–Y –Z define the fabric reference frame (Figs 1b & 2c), with the SPO (S foliation) being close to the XY plane (Passchier & Trouw 1996). Grain-size analysis (Fig. 4a) has been performed using digital microphotographs analysed with the software ImageJ (Rasband 1997–2008; Abramoff et al. 2004). The deformed samples show different microstructures related to their different shear strain states. Here the microstructural features of main strain steps are reported (Figs 2 & 3). † The starting material (sample VGO, Fig. 2c) is characterized by a heterogeneous grain-size distribution from 20 to 250 mm, with mean values at about 80 mm (Fig. 4a). It generally shows a polygonal fabric with few exceptions. Undulose extinction occurs within a few small grains, while larger grains display homogeneous extinction or subgrains with straight boundaries. Large grains are also characterized by twins, and a slight SPO sometimes occurs within small volumes. † At low levels of shear strain (g , 1.00), undulose extinction is visible within the largest grains where kink bands also formed; within small grains, undulose extinction starts and several grain boundaries show lobate shapes. Grains size does not change, ranging from 20 to 250 mm, with mean values at about 70–80 mm (Fig. 4a). † At g ¼ 1.00 –2.00 (Fig. 2d, e), the development of undulose extinction and kink bands affects an increasing number of grains of different sizes. The grain size still ranges between 20 and 200 mm, but mean values decrease at 60 mm (Fig. 4a). Kink bands are mainly developed within larger grains, associated with the SPO, mechanical twinning and undulose extinction. The SPO of flattened grains starts to develop at an angle of about 308–458 (bottom left – top right) with respect to the ISP; it becomes stronger with increasing strain and is also associated with the lattice preferred orientation (LPO). Small grains are characterized by undulose

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extinction and subgrains’ preferred orientation. Narrow disjunctive dark planes occur at a small (,108) angle to the ISP; these planes are generally discontinuous and enclose single grains or aggregates of grains that show undulose extinction and subgrains, and may show an SPO marked by elongated gypsum grains (bottom left –top right). † At g ¼ 2.00 –3.00 (Figs 2f & 3a, b) shear bands become more penetrative and continuous, forming two distinct systems of brittle shear planes at angles of about 108–208 to the ISP. Gypsum aggregates display a strong SPO (S foliation) at a high angle (.458) to the shear bands parallel to the ISP (i.e. bottom left –top right) and the grain-size ranges decrease, ranging from 10 to 150 mm with mean values of less than 50 mm (Fig. 4a). Gypsum aggregates are characterized by strong undulose extinction, subgrain development, and a LPO of old elongated grains and newly formed crystals. Kink bands and mechanical twinning occur within large grains (.100 mm). Grain-size reduction processes are active as the grain size ranges from 10 to 50 mm. † At g . 4.00 (Fig. 3c–e) the grain-size reduction increases and the mean grain size is generally ,50 mm (Fig. 4a). The LPO and SPO of the grain aggregates become more defined and marks a well-defined S foliation, which has an angle of about 458 with respect to the ISP. The shear planes define two sets of planes mostly parallel to the ISP that produce shear lenses; the S foliation, marked by the SPO and LPO of gypsum, is cut by these shear planes. Single gypsum crystals occur as elongated strips parallel to the S foliation and appear slightly curved, showing ‘S’-type symmetry and giving rise to two main SPO orientations: one at a low angle with the ISP, occurring near the shear planes; and the second at a higher angle, far from the shear planes. Very few large grains (30 –120 mm) may preserve kink bands and twinning.

Deformational systems All samples show two sets of microstructures that correspond to two different deformational regimes: (I) brittle (Fig. 2a) and (II) plastic (Fig. 2b) regimes (Passchier & Trouw 1996).

Fig. 2. (Continued) different size. Grain size ranges between 20 and 100 mm. SPO (with associated LPO) of flattened grains develops at an angle of about 308–458 (bottom left– top right) with respect to the ISP. At g ¼ 2– 3 (f) gypsum aggregates, characterized by strong undulose extinction and subgrains development, show strong SPO at high angles (.458) with respect to ISP. Old elongated grains and newly formed crystals both show LPO. Grain size is now reduced to 10– 50 mm. Shear bands become more penetrative and form two systems of brittle shear planes at an angle of 108– 208 with respect to ISP.

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Fig. 3. Microphotographs (left) and microstructural analysis (right) of gypsum samples deformed in torsion up to g ¼ 4.82. At g ¼ 2 –3 (a) and (b) gypsum aggregates, characterized by strong undulose extinction and subgrains development, show strong SPO at high angles (.458)with respect to ISP. Old elongated grains and newly formed

GYPSUM QUANTITATIVE MICROSTRUCTURES

The first system of microstructures is defined by narrow and localized shear planes marked by disjunctive planes (dark planes in Figs 2 & 3). These shear planes evolve from low to high shear strain values constituting an evolving Riedel-like shear system (Fig. 2a) similar to those described by Tchalenko (1970) at various scales. The second system is defined by a set of two foliations marked by an SPO and LPO of gypsum grains that follow the scheme of development within mylonitic shear zones (Fig. 2b). The schematic drawings combined with microphotographs in Figures 2 and 3 summarize the evolution of both the brittle (Fig. 2a) and the plastic (Fig. 2b) shear systems with increasing shear strain.

Brittle (Riedel-like) and plastic (mylonitic-like) structures In this section we will compare the classical models of structure development within localized shear zones (Snoke et al. 1998), either brittle (Riedel – Tchalenko-like, Fig. 2a) or plastic (Fig. 2b) systems, with the observed microstructures corresponding to increasing shear strain (g). The shear strain is measured, for an isotropic homogeneous cylindrical specimen of diameter, d, length, l, and angular displacement (u) (Fig. 1a), as g ¼ du=2l, (Paterson & Olgaard 2000). The angular relationship between planar structures and the ISP are shown [a (angle between ISP and R plane) and x (angle between ISP and S foliation) angles] in Figure 2a, b. In the Riedel scheme (Fig. 2a) the different planes form different angles with the ISP and are also characterized by a specific shear sense (kinematic). In Figure 2b, the angular relations between the foliations and the ISP are shown. Figure 4 shows the evolution of a and x with respect to shear strain. † At g ¼ 0–1.00 (Fig. 2c) no distinctive structure develops. † At g ¼ 1.00–1.50 (Fig. 2d) both R and S planes (Fig. 2a, b) develop. S planes are characterized by an SPO and LPO, and occur at a x angle of about 458 (Fig. 4a). The R planes cut the S planes with a dextral shear sense as shown by small displacements along the R planes and a occurring at high values with the ISP. † At g ¼ 1.50–2.00 (Fig. 2e) S planes develop at x angles of less than 458 (Figs 2e & 4a), and are marked by an SPO and LPO of strained gypsum grains. The R planes show dextral shear sense

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outlined by the displacement of gypsum grains marking the S planes, and dissolution occur along the R planes, as shown by sharp truncations of such gypsum grains. A few R splays also occur, and the a angles are generally lower than at lower g values (Fig. 4a). † At g ¼ 2.00– 2.50 (Fig. 2f) the S planes are marked by well-developed SPO and LPO of gypsum grains, and x angles ranging from 208 to 408. The R planes are more penetrative and occur at smaller a angles. Towards the R planes, the S planes are gently deflected, showing a more plastic behaviour and allowing the interpretation of a dextral shear sense. Several smaller R planes occur, organized in an en-echelon-type geometry. a angles become smaller and few fractures occur parallel to both the R and S planes. † At g ¼ 2.50–3.00 (Fig. 3a, b) the S planes are still well developed and are at high angles (458) to the ISP, but they are also highly deflected by small C-type (Fig. 2b) shear planes that occur at very low angles with the ISP (Fig. 3a, b). These features correspond to the S–C geometry of mylonitic shear zones. Along the C planes small undeformed (no undulose extinction) and equigranular gypsum grains occur. R planes develop at very low angles with the ISP as the a angles approach 0 (Fig. 4b), becoming Y-type planes of the Riedel scheme (Fig. 2b). Along these sets of R– Y planes, diffuse dissolution occurs. A few R–Y splays and P planes also occur. † At g . 4.00 (Fig. 3c –e) S planes marked by the SPO and LPO of gypsum are penetrative, and describe an angular relation with the ISP of about 308 –458 (Figs 3c–e & 4). C planes also develop with a dextral shear sense, and seem to be associated with the development of layers of small strain-free grains of gypsum probably due to dynamic recrystallization. The R– Y planes are characterized by very low a values and the development of shear lenses. Along a few small planes oriented within a C0 -type geometry, a strong LPO of gypsum occurs; several R planes also occur with the same orientation, suggesting a plastic precursor to this second generation of R planes. Figure 4b shows the evolution of a (left) and x (right) with increasing g with respect to the ISP. The theoretical evolution of the R planes with respect to ISP should correspond to a decreasing of a as g

Fig. 3. (Continued) crystals both show LPO. Grain size is now reduced to 10– 50 mm. Shear bands become more penetrative and form two systems of brittle shear planes at an angle of 108– 208 with respect to ISP. At g . 4, (c) – (e), the LPO and SPO of the grain aggregates is more defined and marks a well-defined S foliation, forming an angle of about 458 to the ISP. Shear planes define two sets of planes mostly parallel to the ISP.

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Fig. 4. (a) Box and whiskers representations (left) of the grain-size distribution with respect to g values; mean bars and corresponding standard deviations of the grain-size distribution; both diagrams clearly show the decrease in grain size with the increase in g values. (b) Diagrams show the evolution of a (left) and x (right) angles v. g intervals. a values increase at first (up to g around 2) and then show a decreasing trend. x values slightly decrease at first for increasing g values, then, due to the counterclockwise rotation along Y –R shear bands they oscillate at around 40–458 values, never reaching the theoretical curve of the finite strain long axis (dashed line).

GYPSUM QUANTITATIVE MICROSTRUCTURES

increases, as supported by analogue modelling and field observations (Snoke et al. 1998). Such a decrease in a corresponds to the transition from R to Y planes that lie parallel to the ISP. This theoretical evolution is in agreement with the development and evolution of R planes within gypsum aggregates deformed in torsion from low to high g (Fig. 4b, left). The evolution of the x angles, which correspond to the angle between the S foliation or SPO of grains and the ISP, should follow the theoretical curve of the finite strain long axis as expected for a shear zone (Fig. 4b right). Whereas the theoretical curve shows a continuing decrease of x with increasing g, the actual x values measured for the studied samples from g values from 0 to 4.82 do not follow this function, and stabilize at values of 408–458. Such discrepancies from the theoretical curve may be related to the concurrence of the deformation mechanisms that were accommodating strain. Microstructural analysis showed that the shear bands were active from the beginning of deformation (g , 0.5) and controlled the orientation of grains by imposing shear along the bands. In fact, grains close to the shear bands (R or Y ) are deflected towards the shear band line, and the corresponding x values are generally lower than those measured on grains far from shear bands. The grains that occupy the regions between two shear bands record a back rotation (i.e. counterclockwise) as imposed by sinistral shear, leading to grains characterized by high x values. Meanwhile, simple shear deformation involves various deformation mechanisms acting during the evolution from low to high g; kinking, twinning and passive rotation dominate at low strains, while at higher strains dynamic recrystallization and grain-size reduction (Fig. 4a) become more important and overprint pre-existing microstructures (Barberini et al. 2005). Such progressive change to plastic flow does not dominate the entire deformed volume, as shown by the discrepancies between the theoretical and practical finite strain curves (Fig. 4b), with frictional flow along shear bands making up the majority of the overall deformation. Kinking occurs in large grains oriented at about 308–608 to the ISP.

Quantitative neutron and X-ray diffraction texture analyses In order to use QTA to compare different samples independently of the grain size, phase ratio and porosity, normalization of the pole figures is needed (Bunge & Esling 1982). This normalization can be obtained via the refinement of the orientation distribution function (ODF), which holds information about the texture intensity and all components of

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the texture. The measurement of a single pole figure allows only a limited quantitative analysis as it only shows the orientation distribution and the intensity of the preferred orientation from the corresponding crystallographic planes. The crystallographic recalculation of several pole figures from different crystallographic planes according to their crystal structure in the ODF allows a complete quantitative texture analysis of the specific mineral phase. Measuring single pole figures does not allow a quantitative texture analysis because a single pole figure can be obtained by many different ODFs, meaning that the interpretation of the texture from a single pole figure is doubtful. In this study we performed QTA using neutron and X-ray diffraction data. Neutron diffraction data and refined texture hold information about the whole sample (i.e. volumes of about 1 cm3), while X-rays, due to their low penetration capabilities, are suitable to investigate the marginal domains of the samples. QTA was used to reconstruct the LPO of preferred planes of slip within gypsum grains involved in torsion experiments. Direct numerical integration is not suitable for a reliable ODF determination because the relatively low 2u resolution gives rise to peak overlaps, which we resolved using a combination of Rietveld and texture analyses. The Rietveld texture analysis (Matthies et al. 1997) has the capacity to separate exact and/or strong overlaps. A Rietveld texture analysis (Lutterotti et al. 1997) was performed for all patterns using the software package MAUD (Lutterotti et al. 1999). MAUD uses a Rietveld core routine to compute spectra and a so-called Le Bail algorithm (Matthies et al. 1997) to extract the differences between random and textured intensities for each computed peak. These spectra are the basis of computing the ODF using the eWIMV algorithm (Morales et al. 2002; Lutterotti et al. 2004). The obtained ODF was then introduced into the cyclic Rietveld refinement. The new refined parameters were used for a new eWIMV cycle to correct the ODF. Several cycles of refinement were performed to converge towards a final characterization of the material. The refinement quality was assessed by comparison of the experimental and recalculated diagrams, and by the reliability factors as exhaustively described by Chateigner (2005): RP for the ODF refinement (Lutterotti et al. 1997), and RB and Rw for the Rietveld refinement (Young & Wiles 1982). As a starting model for the gypsum structure, we used the A2/a space group and cell parameters a ¼ 6.28, b ¼ 15.20, c ¼ 6.52 and b ¼ 127.414 (Pedersen & Semmingsen 1982; Schofield & Knight 1996; Boeyens & Ichharam 2002; Grazˇulis et al. 2009). This model allowed complete indexing of the diffracted lines, implying that no extra phase was present in the specimens.

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Table 2. Samples and corresponding g values with reference to reliability factors, goodness of fit, x2, texture strength (F 2 in mrd2) and texture calculation type Rpb (%)

Rexp

x2

F 2 (mrd2)

76

49.6

21.78

4.62

1.45

y

16.16 16.55

23.77 21.94

18.55 17.24

10.64 15.6

3.72 ?

1.42 1.4

y y

20.36 20.53

15.96 15.87

21.61 22.41

17.03 16.51

15.55 14.82

1.9

1.482 1.63

y y

11.78 13.23 9.36 14.83 13.04 10.11

8.95 10.37 7.39 11.68 10.21 8.08

27.49 26.11 15.39 29.35 26.41 15.09

15.05 15.8 10.16 18.12 15.93 11.15

6.41 9.36 6.09 9.51 8.86 5.39

3.09 1.96 3.38 2.25 2.16 3.49

0.54 1.37 1.29 1.83 1.39 1.62

y y y y y y

7.98

6.37

13.3

9.5

3.04

6.86

1.46

y

Sample

g

RW (%)

RB (%)

X-ray VGO V7 V8 V18 V14 V10 V16 V20

0 0.49 1.19 2.59 2.61 4.07 4.11 4.82

46.88

36.61

20.55 21.13

Neutron VGO V7 V8 V18 V14 V10 V16 V20

0 0.49 1.19 2.59 2.61 4.07 4.11 4.82

Rwpb (%)

ewimv

ODF refinement estimators: RW, Intensity weighted factor; RB, R-Bragg factor. Profile refinement estimators: Rwpb, R-weighted pattern factor; Rpb, R-pattern factor; Rexp, R expected (¼1 optimal). ewimv, entropy-WIMV texture model.

The cell parameters were refined during combined analysis (Table 2), but the atomic positions were not, and the crystallite sizes were kept fixed at ˚ . No peak shifts were observed during x 1000 A rotation of the samples, indicating that only very low levels of residual stresses, if any, were incorporated in the material.

Neutron diffraction Neutron diffraction experiments were carried out at the Institut Laue Langevin (ILL, Gre´noble, France) high flux reactor using the Position Sensitive Detector of the D20 beamline. The detector spans a 2u range of 153.68 with a resolution of 0.18, and the neutron wavelength is monochromatized to ˚ . An Eulerian cradle allows the x and f l ¼ 2.41 A angles’ rotations (Fig. 1c). Scans were operated from x ¼ 08 to 908 at steps of 58 using a fixed incidence angle, v, of 108 [corresponding to the (020) Bragg position] and from f ¼ 08 to 3558 (steps of 58) (Fig. 5). Diffraction data were collected for 4 s. The samples used were the cylindrical specimens deformed in torsion.

X-ray diffraction X-ray diffraction texture measurements were carried out at the Laboratoire de Cristallographie et Sciences des Mate´riaux (CRISMAT), Ecole Nationale Supe´rieure d’Inge´nieurs de Caen

(ENSICAEN, France), using a Huber four-circle goniometer (closed eulerian cradle þ u –2u movements) mounted on an INEL X-ray generator, and Cu Ka wavelength monochromatized by an incident flat graphite monochromator. A curved position sensitive detector with a 2u resolution of 0.038 (INEL CPS-120) was used to acquire complete diffraction patterns at different sample orientations in the 2u range of 08 –1208 (Fig. 5). The flat samples were measured in reflection geometry. The incident angle on the sample, v, and the CPS position were chosen in order to get the maximum coverage of the orientation space. The spectra were measured using w values from 08 to 3558, and x values from 08 to 608, at incremental steps of 58 for both angles. Each pattern was acquired for 120 s. The irradiated surface, which was the surface of the cylindrical specimens deformed in torsion from where the thin section was cut (Fig. 1a), was increased by oscillations of the sample perpendicular to the lineation direction (+3 mm of amplitude). Assessment of refinement reliability. Table 2 shows the reliability factors for the refinement of the studied samples. Using X-ray diffraction data, all R-factors (reliability factors) of Rietveld and ODF refinement show tendencies toward lower values for g values above 1.00, while neutron analyses do not show this evolution. This can be attributed to the largest grain sizes for lower g values, which give rise to grain statistic effects when using

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Fig. 5. Theoretical diffraction pattern for gypsum structure (Boeyens & Ichharam 2002) for (a) X-ray and (b) neutron. In (c) the structure of gypsum is represented also showing the main slip plane and direction (Muegge 1898).

X-rays, while neutron measurements still probe enough grains (Zucali et al. 2001). Overall, the reliability factors indicate a reasonably good refinement both of diagrams and ODFs for comparable texture strength levels (Chateigner 2005) ranging up to F 2 ¼ 1.9 mrd2 (multiple of random distribution), which represent moderate textures. The goodness-of-fit values range from around 1.9 for most of the samples to 4.6 for the worst case, with average values also in favour of neutron analysis. The average values obtained in this work are comparable to values reported in the literature. Before discussing the QTA data it is important to emphasize that neutron data reflect the LPO of the whole sample, meaning that any PF represents orientations related to the entire sample that corresponds to the complete g range from the core (minimum) to the edges (maximum). Considering the distribution of shear deformation within a core sample deformed in torsion (Fig. 1b), which is at a maximum at the outer surface and almost zero at the cylinder axis, the obtained textures are the average texture (Fig. 6) and the maximum texture in the external deformed area is probably underestimated. Neither a one-to-one nor a linear relationship between the deformation ratio and the deformed volume fraction from the core to the edge was revealed in the diffracted intensities. Moreover, the surface of the shear zone is not planar, as it

would be in a 1 cm edge cube of a naturally deformed rock, but rather corresponds to the cylinder’s outer surface. Some of the artefacts (small circles) observed in the pole figures might be due to this rather complex geometry (Fig. 6). Even with such strong limitations the pole figures obtained from neutron data (Fig. 6, left) reproduce well the LPO of gypsum aggregates if compared with pole figures from X-ray data (Fig. 6, right). In addition, at extreme conditions (e.g. very low or zero strain values; high strain values) the pole figures obtained from neutron data due to larger irradiated volumes better characterize the sample textures.

QTA results: X-ray diffraction data The starting material shows an unexpected LPO with its maximum parallel to the Y-axis. At g , 1.50, X-ray pole figures show a preferred distribution of the (010) close to the Z-axis, and the corresponding (001) and (100) poles roughly aligned along the XY plane. At higher g (.2.5) values, the (010) poles become stronger and are close to the Z-axis, with progressive localization of the (100) and (001) poles on a circle perpendicular to (010) (Fig. 7). Starting from these g values, recalculated pole figures show a stronger (001) pole density maximum with (001) poles aligning at 908 from the Z fabric axis and dispersed within about 308

90 M. ZUCALI ET AL.

Fig. 6. (a) and (b) Comparison of the recalculated experimental pole figures for X-ray (041), (221), (200) diffracted planes (left column) and for neutron (020), (021), (240) diffracted planes (right column).

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Fig. 6. (Continued).

GYPSUM QUANTITATIVE MICROSTRUCTURES

92 M. ZUCALI ET AL. Fig. 7. (a) and (b) Recalculated pole figures of the main crystallographic planes for X-ray and neutron (left column). Right column shows the inverse pole figures shown in the three main sample directions (Y, N, ISP). Y, Y sample direction; N, normal to ISP direction; and ISP, Imposed shear plane direction.

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Fig. 7. (Continued).

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from Y. Such preferred orientations increase, attaining higher values of g. The well-defined (010) and (001) maxima correspond to an incipient girdle distribution of (100) poles at an angle of about 808 with the ISP and mainly parallel to the X fabric axis (Fig. 7). Meanwhile the (001) poles show a weak preferred orientation that corresponds to a maximum density within the XY plane, close to the Y fabric axis. Such a preferred orientation is better developed only at g . 4.00, where it becomes closer to a girdle distribution mostly parallel to the XY(S) plane but still preserving a maximum close to the Y-axis. The angular relations given by the pole figures with respect to the ISP and the XY(S) foliation plane do not change much above g ¼ 2.50. These show (010) pole maxima to the ISP angles mainly around 308 in the XZ plane, and the corresponding (001) and (100) girdles are perpendicular to the average (010) poles. Consequently, the two latter girdles roughly contain the foliation plane XY(S). Without deformation, (010) poles are located in the ISP plane parallel to the Y-axis, and start deviating from the ISP at g . 1.00, reaching its final orientation at 308 from the ISP at larger g values.

QTA results: neutron diffraction data A strong maximum of the poles to the (010) planes, which were perfect cleavage planes for gypsum, can be observed at low g (Fig. 7). This orientation is almost absent within the starting material, whereas it becomes stronger for shear strains of g . 2.00. At g . 1.00, the (010) poles start moving towards the Z-axis at about 308 from the ISP, and correspondingly the (100) and (001) poles distribute progressively on a girdle characterized by maxima that are parallel to the Y fabric direction for the (00l) poles or close to the XY(S) planes for the (h00) poles. The (010) pole figure shows two distribution components characterized by two maxima. Such distributions also occur as asymmetrical poles that are stronger for the (001) and (100) pole figures, in which a characteristic small-circle distribution develops from low g values (e.g. 1.19). At larger g (e.g. V10 g ¼ 4.11) the (010) maximum becomes more defined, together with the corresponding girdle distribution of (001) and (100) poles, even though they still preserve their small-circle distribution. For these neutron pole figures it becomes more complicated to interpret the angles between the pole distribution maxima or girdle axes orientation and the ISP plane or XY(S) fabric planes. Indeed, as the whole volume of the material is probed by neutrons, the total deformation regimes are seen in the pole figures, with weights that are not determined a priori. As the torsion deformation increases, a more and more important volume

fraction of deformed rock develops, resulting, for most deformed specimens, in pole figures that resemble the X-ray figures. However, in these figures undeformed or slightly deformed contributions can be observed making the small-circle orientation distribution of the pole figures by integration over the deformation regimes. Inverse pole figures in Figure 7 (right) confirm the relations between ISP and the crystallographic axes observed in direct pole figures. The orientation of the ISP direction is distributed at an angle of about 308 from the [010]-axis (i.e. normal to the pole figure plane) from g values above 2.00.

Discussion Evolutionary model If we consider the experiments and related microstructures, from g ¼ 0.49 to g ¼ 4.82, as possible steps of a progressive deformational event, we can ‘observe’ some interesting features during progressive deformation of gypsum within a shear zone at shear strain rates up to 1025 s21 and T ¼ 70 – 90 8C at P ¼ 300 MPa (Fig. 8). This microstructural evolution shows that brittle and plastic deformation mechanisms are both active from the beginning of the deformation of gypsum under dextral shear strain (Fig. 8). The progressive evolution of brittle and plastic microstructures, which are well described by geometrical relationships (Figs 2 & 3), describes a coherent evolution from low to high strain. The brittle, Riedel– Tchalenko-like, evolution is characterized by the progressive decrease of a, which describes the relation of R planes (Fig. 2a) with the shear plane (i.e. the ISP). At g . 4.00 the R planes approach the ISP plane and coincide with the Y planes of the Riedel-scheme (R– Y planes in Fig. 2a). Such a negative correlation is broadly shown by a v. g correlation diagrams (Fig. 4b). The R –Y planes result more from the progressive evolution of R towards the ISP plane than from the development of a second group of planes different from R. This is also corroborated by the disappearance of R planes at high strains. However, plastic microstructural features do not change their geometry during the deformation, as shown by the evolution of x with respect to g (Fig. 4b). Rather, after an initial decrease, they keep more or less the same orientation or do not show any tendency to reduce the angle between the foliation and the ISP plane, as is generally expected in a monoclinic shear zone (Passchier & Trouw 1996), where the range of x (from 208 to 458, Fig. 4) is not related to g intervals. At g . 2.60 and T ¼ 90 8C (stages 5 and 6 in Fig. 8) plastic planes occur at low angles with the

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Fig. 8. Schematic representation of the evolution of the modelled shear zone. U, undrained; D, drained; N, neutron; X, X-ray; G, gypsum; B, bassanite; T70, 70 8C; T90, 90 8C. The microstructural evolution shows that brittle and plastic deformation mechanisms are both active from the beginning of the deformation of gypsum under dextral shear strain. However, although plastic microstructural features do not change their geometry during the deformation, brittle features do change their appearance and orientation. Texture analysis shows a well-developed crystallographic preferred orientation of (010) planes starting at the initial deformational stages (e.g. stage 2). This texture becomes stronger during the evolution of the shear zones, but the orientations of the (010) poles does not change substantially above g ¼ 2, confirming the similar observations at the microscale.

ISP (i.e. C planes), developing the classical S –C features of mylonitic systems. Similar planes do not develop during stages 4, 7 and 8 at similar or even higher g values at T ¼ 70 8C. These differences can only imply that 70 –90 8C is an important temperature interval for the plastic behaviour of gypsum, which most probably starts deforming following the mylonitic scheme (e.g. S–C system), probably reflecting the activation of new deformation mechanisms and/or slip systems. In contrast, this temperature interval does not seem to be of particular interest for brittle behaviour, as we do not notice any difference in microstructures at this scale of observation. This also implies that across these g intervals of deforming gypsum, we observe microstructures related to a brittle plus plastic deformational system rather than brittle v. plastic deformational system or a brittle –plastic transition. Texture analysis (i.e. direct and inverse pole figures) shows a well-developed CPO of (010) planes (Fig. 7) starting at the initial deformational

stages (e.g. stage 2 in Fig. 8). This texture becomes stronger during the evolution of the shear zones, but the orientations of the (010) poles does not change substantially above g ¼ 2, confirming the similar observations at the microscale. The texture analysis also confirms (Figs 7 & 8) that the preferred slip system is parallel to the (010) planes, although a principal direction of slip (e.g. classical [001] slip direction) cannot be clearly recognized, as shown by the tendency towards girdle distributions of the (100) and (001) poles to planes (Fig. 7).

Conclusions † We have conducted a quantitative microstructural study on natural gypsum samples that were experimentally deformed in torsion. In this study we compared information derived from optical microstructural analysis and texture analysis obtained by means of diffraction (neutron and X-ray) techniques. With these

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techniques we studied samples deformed at temperatures of 70 –90 8C and strains (g) from 0 to 4.82. In the microstructural study we could observe, with increasing g values, the development of an SPO of gypsum aggregates (the S foliation) associated with the development and progressive evolution of a Riedel-like brittle deformational scheme. Such brittle microstructures evolve from a relatively early scheme to a substantially completed stage as suggested by the evolution of the angular relationships between the fault planes. The brittle evolution is only slightly sensitive to temperature; brittle systems are present in samples deformed at temperatures of 70 and 90 8C, with the difference that in the samples deformed at 90 8C few C-like planes occur together with Y-like brittle planes. However, the main brittle structure is preserved, suggesting an incipient transition to a plastic deformational regime. A plastic deformational regime is also suggested by deformation mechanisms that start to operate at g . 4.00 as dynamic recrystallization and grain-size reduction; at greater g values these deformation mechanisms become more efficient over kinking, passive rotation and mechanical twinning, but still have to compete/collaborate with frictional flow along shear bands. This is also supported by the large difference between the theoretical and actual curves of the finite strain long axis, which implies the counterclockwise rotation due to the sense of shear along the R planes. From the textural study performed with X-ray and neutron diffraction, we observed that plastic deformation is recorded by gypsum from the initial g values to highest ones. Optical microstructural analysis does not show important differences in the microstructural behaviour of gypsum marking the S foliation, while QTA by means of either X-ray and neutron diffraction clearly shows a tendency to reinforce these textures as can be appreciated either from the distribution of maxima and girdles in pole figures or from texture factors. There is a general agreement between the X-ray and neutron data, as both show an increasingly strong (010) preferred orientation for increasing g values. Nevertheless, the results from the X-ray data are much more coherent with the observations from the optical microstructural analysis. The neutron data are controlled by the volume fraction of deformed gypsum. As torsion increases, the deformed volume fraction increases; giving rise, for most deformed specimens, to pole figures that resemble the X-ray figures and that produce the small-circle

orientation distribution geometry of the pole figures due to the integration over the deformational regimes. † Microstructural and textural analyses showed that at the investigated conditions the behaviour of gypsum is completely characterized by a brittle plus plastic deformation, while a brittle to plastic transition may be induced, keeping other constraints constant, by increasing the temperature above 90 8C. † The QTA shows a monoclinic symmetry of the pole figures of gypsum with respect to the ISP (i.e. direct and inverse pole figures). In contrast, the (010) maximum is generally close to the Z-axis of the fabric. These symmetries may be used, in the first case, to evaluate the kinematics of the deformation (i.e. sense of shear indicators), while little may be said with respect to the shear sense with respect to the S foliation. Where brittle and plastic microstructures coexist, a general sense of shear may be assessed, as shown in this contribution. However, if only plastic microstructures occur, as generally occurs in natural rocks, it is not possible to use texture pole figures (LPO) to assess the sense of shears. We thank all the technical staff at ILL and CRISMAT. We thank G. Gosso for helpful and extensive discussion. We are truly grateful to E. Rutter and J. Walter, whose reviews helped us to improve the manuscript.

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Numerical modelling of spontaneous slab breakoff dynamics during continental collision CYRILL BAUMANN1*, TARAS V. GERYA1,2 & JAMES A. D. CONNOLLY1 1

Department of Geosciences, Swiss Federal Institute of Technology (ETH-Zurich), CH-8092 Zurich, Switzerland 2

Geology Department, Moscow State University, 119899 Moscow, Russia *Corresponding author (e-mail: [email protected])

Abstract: Slab detachment or breakoff is directly associated with phenomena like morphological orogenesis, occurrence of earthquakes and magmatism. At depth the detachment process is slow and characterized by viscous rheolgy, whereas closer to the surface the process is relatively fast and plastic. Using a 2D mantle model 1500 km deep and 4000 km wide we investigated, with finitedifference and marker-in-cell numerical techniques, the impact of slab age, convergence rate and phase transitions on the viscous mode of slab detachment. In contrast to previous studies exploring simplified breakoff models in which the blockage responsible for inducing breakoff is kinematically prescribed, we constructed a fully dynamic coupled petrological–thermomechanical model of viscous slab breakoff. In this model, forced subduction of a 700 km-long oceanic plate was followed by collision of two continental plates and spontaneous slab blocking resulting from the buoyancy of the continental crust once it had been subducted to a depth of 100– 124 km. Typically, five phases of model development can be distinguished: (a) oceanic slab subduction and bending; (b) continental collision initiation followed by the spontaneous slab blocking, thermal relaxation and unbending – in experiments with old oceanic plates in this phase slab roll-back occurs; (c) slab stretching and necking; (d) slab breakoff and accelerated sinking; and (e) post-breakoff relaxation. Our experiments confirm a correlation between slab age and the time of spontaneous viscous breakoff as previously identified in simplified breakoff models. The results also demonstrate a non-linear dependence of the duration of the breakoff event on slab age: a positive correlation being characteristic of young (,50 Ma) slabs while for older slabs the correlation is negative. The increasing duration of the breakoff with slab age in young slabs is attributed to the slab thermal thickness, which increases both the slab thermal relaxation time and duration of the necking process. In older slabs this tendency is counteracted by negative slab buoyancy, which generate higher stresses that facilitate slab necking and breakoff. A prediction from our breakoff models is that the olivine–wadsleyite transition plays an important role in localizing viscous slab breakoff at depths of 410 –510 km due to the buoyancy effects of the transition.

Slab detachment or breakoff is directly associated with phenomena like morphological orogenesis, occurrence of earthquakes and magmatism. This process was hypothesized on the basis of gaps in hypocentral distributions and within tomographic images of subducted slabs (Isacks & Molnar 1969; Barazangi et al. 1973; Pascal et al. 1973; Chung & Kanamori 1976; Fuchs et al. 1979), and is supported both by theoretical considerations (Sacks & Secor 1990; Davies & von Blanckenburg 1995; von Blanckenburg & Davies 1995) and detailed seismic tomography (Spakman et al. 1988; Wortel & Spakman 1992, 2000; Xu et al. 2000; Levin et al. 2002). Slab detachment is often attributed to a decrease in subduction rate subsequent to continental collision (e.g. Davies & von Blanckenburg 1995; Wong A Ton & Wortel 1997), an effect caused by the buoyancy of the continental lithosphere that is introduced into the subduction zone.

In addition to multiple geophysical, geological and geochemical investigations acknowledging slab detachment as a possible explanation for observations in various regions (Andrews & Billen 2009 and references therein), analytical models and laboratory and numerical experiments have been undertaken to characterize the breakoff process (e.g. Davies & von Blanckenburg 1995; Larsen et al. 1995; Yoshioka et al. 1995; Yoshioka & Wortel 1995; Wong A Ton & Wortel 1997; Pysklywec et al. 2000; Chemenda et al. 2000; Buiter et al. 2002; Gerya et al. 2004b; Faccenda et al. 2006, 2008; Mishin et al. 2008; Ueda et al. 2008; Zlotnik et al. 2008; Andrews & Billen 2009). Recent thermomechanical numerical models indicate two modes of detachment (Andrews & Billen 2009): (1) deep viscous detachment that is characteristic of strong slabs (maximum yield strength 500 MPa), and is controlled by thermal

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 99– 114. DOI: 10.1144/SP332.7 0305-8719/10/$15.00 # The Geological Society of London 2010.

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relaxation (heating) of the slab and subsequent thermomechanical necking in dislocation creep regime (Gerya et al. 2004b; Faccenda et al. 2008; Zlotnik et al. 2008); and (2) relatively fast, shallow plastic detachment characteristic of weak slabs (maximum yield strength 300 MPa) and controlled by plastic necking of the slab (Mishin et al. 2008; Ueda et al. 2008; Andrews & Billen 2009). Andrews & Billen (2009) demonstrated that the time before the onset of viscous detachment increases with slab age, indicating that detachment time is controlled by the thickness and integrated stiffness of the thermally relaxing slabs (Gerya et al. 2004b). In contrast, the plastic detachment time is independent of age and is controlled by the rate of flow of the surrounding mantle (Andrews & Billen 2009). Two-dimensional dynamic models of viscous detachment have shown that, following the cessation of subduction, necking of the slab and detachment occur after 8–30 Ma at a depth of 150–400 km (Gerya et al. 2004b; Faccenda et al. 2008; Andrews & Billen 2009). In numerical experiments with viscous dissipation included (Gerya et al. 2004b) shear heating in the necking area causes slab breakoff to occur 8% faster than in models that neglect viscous dissipation (Gerya et al. 2004b). Therefore, the slab necking process is not purely mechanical. Zlotnik et al. (2008) modelled the influence of mantle phase transitions on the viscous slab detachment and demonstrated that the olivine –spinel transition facilitates the detachment by increasing the negative buoyancy of the hanging slab. Thermomechanical models of plastic detachment indicate shallower (Andrews & Billen 2009) and even near-surface (Mishin et al. 2008; Ueda et al. 2008) depths of breakoff. The timing and location of plastic failure is controlled by the dynamics of the loading of the subducting plate by negatively buoyant slab during or shortly after the active phase of convergence. Such models have also shown that, in the case of multiple plates, shallow slab breakoff can affect plate dynamics causing processes such as subduction flip and re-initiation (Mishin et al. 2008). Thermomechanical numerical studies of breakoff (Gerya et al. 2004b; Zlotnik et al. 2008; Andrews & Billen 2009) explored a simplified static breakoff model with two fixed oceanic plates and a kinematically prescribed interruption of subduction. Thus, no dynamic thermomechanical model of this process accounting for the presence of the continental plates and cessation of subduction (e.g. Chemenda et al. 2000; Faccenda et al. 2008) has been studied. The absence of such an effort leaves a gap in our understanding of dynamics and controls of breakoff. In particular, prescribed slab blocking does not allow for the proper balance between the slab

strength, geometry (length, inclination) and negative buoyancy, and may create unrealistic geodynamic situations (e.g. heavy slabs that are too weak to remain coherent during subduction). The simplified models also fail to reproduce the thermal and lithological structure of the collisional orogen that forms in response to continental subduction (e.g. Chemenda et al. 2000; Burov et al. 2001; Faccenda et al. 2008). Therefore, in this paper we investigate the dynamics, geometry and duration of viscous detachment of oceanic slabs after cessation of active oceanic –continental subduction due to collision between two continental plates by means of a high-resolution 2D numerical study. We will account for the effects of phase transformations as well as pressure-, temperature- and strain-ratedependent rheology of the mantle and the subducted oceanic crust. Compared to previous studies dealing with thermomechanical detachment experiments (e.g. Gerya et al. 2004b; Zlotnik et al. 2008; Andrews & Billen 2009), our petrological –thermomechanical model is fully dynamic and considers the presence of the colliding continental plates as causing subduction blocking and breakoff.

Model set-up and governing equations Initial configuration To investigate slab breakoff we developed a 2D, coupled petrological –thermomechanical numerical model of oceanic–continental subduction followed by collision (Fig. 1) using the I2VIS code (Gerya & Yuen 2003a). The spatial co-ordinate frame of the model is 4000  1500 km. The non-uniform 774  106 rectangular grid is designed with a resolution varying from 2  2 km in the studied subduction –collision– breakoff zone to 30  50 km far from it (Fig. 1b). Lithological structure of the model is represented by a grid of around 3 million randomly distributed active Lagrangian markers used for advecting various material properties and temperature. The density of the marker distribution varies from 1 marker km22 in the high-resolution zone to 0.25 marker km22 far from it. The oceanic crust consists of a 3 km-thick basaltic layer underlain by a 5 km-thick gabbroic layer. The continental crust consists of a 35 km-thick granodioritic layer. The velocity boundary conditions are free slip at all boundaries except the lower boundary of the model domain, which is permeable in the vertical direction (Gerya et al. 2008b). An infiniteike external free slip condition (@vx /@z ¼ 0, @vz /@z ¼ vz /Dzexternal, where vx and vz are, respectively, horizontal and vertical velocity components) along the bottom implies that free slip condition (@vx /@z ¼ 0, vz ¼ 0) is satisfied at an external boundary located at a certain depth Dzexternal

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Fig. 1. Initial model set-up (a) and numerical grid (b) for conducted experiments. The flow law and petrological model of lithospheric and asthenospheric mantle are the same (Table 1); different colours are used for better visualization of slab dynamics. White lines are isotherms taken from 100 8C with 200 8C steps.

below the actual lower boundary of the model (Dzexternal ¼ 1500 km for the present study). Similar to the usual free slip condition, external free slip allows global conservation of mass in the computational domain. The top surface of the crust is calculated dynamically as an internal free surface by using initially a 10– 12 km-thick deformable top layer (10 km over the continents and 12 km over the oceanic plate) with low viscosity (1019 Pa s) and density (1 kg m23 for the atmosphere, above z ¼ 11 km sea-water level, 1000 kg m23 for the hydrosphere, below z ¼ 11 km sea-water level, where z is the vertical co-ordinate taken downwards from the top of the model). The validity of the weak layer approach for approximating the free surface has recently been tested and proven (Schmelling et al. 2008) with the use of a large variety of numerical techniques (including our I2VIS code) and comparison with analogue modelling. The interface between this weak layer and the top of the oceanic crust deforms spontaneously and is treated as an internal erosion– sedimentation surface that evolves according to the transport equation solved at each time step

(Gerya & Yuen 2003b): @zes @zes ¼ vz  vx  vs þ ve @t @x where zes is a vertical position of the surface as a function of the horizontal distance x; vz and vx are the vertical and horizontal components of material velocity vector at the surface; vs and ve are, respectively, sedimentation and erosion rates. As demonstrated by Pysklywec (2006) and Gerya et al. (2008b) in the case of subduction and collision variations in erosion –sedimentation rates may significantly affect crustal mass flux and consequently alter the behaviour of the crust –mantle interface. In the present paper we used moderate constant gross-scale erosion and sedimentation rates applicable for continental collision zones (Vance et al. 2003; Gerya et al. 2008b), which correspond to the relation: vs ¼ 0 mm year21, ve ¼ 1 mm year21 when z , 9 km (i.e. for 2 km elevation above the seawater level)

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vs ¼ 1 mm year21, ve ¼ 0 mm year21 when z . 11 km (i.e. below the sea-water level). The influences of erosion–sedimentation rates on the geometry of collisional orogens in the postsubduction collision model similar to the one explored in our study are discussed by Gerya et al. (2008a). Subduction is prescribed by the total convergence rate RT ¼ RR þ RL, where RR and RL are locally imposed constant velocities for the right and the left plates, respectively. Moving continental plates have finite width and are detached from the model boundaries (Fig. 2). This convergence condition was applied for 700 km of forced subduction of the oceanic plate. After this time the model was allowed to evolve spontaneously. The subduction area is initiated by a 5–50 km-wide weak zone of hydrated mantle that cuts across the mantle lithosphere from the bottom of the crust down to 190 km depth. Taking into account the critical role of water for subduction initiation (Regenauer-Lieb et al. 2001), this zone is characterized by wet olivine rheology (Ranalli 1995) and a low plastic strength limit of 1 MPa (Table 1). With the advance subduction, this zone is replaced by weak upper oceanic crust, which is also characterized by low plastic strength (Table 1). This device implies that high-pressure fluids are present along the slab interface during subduction (e.g. Sobolev & Babeyko 2005; Gerya et al. 2008a). The initial geotherm for the oceanic lithosphere is defined by a half-space cooling model (e.g. Turcotte & Schubert 2002; Fowler 2005) for the prescribed lithospheric age (Table 2). The initial thermal structure of the continental lithosphere corresponds to a typical continental geotherm (e.g. Turcotte & Schubert 2002): 0 8C at the surface and 1345 8C at 140 km depth. A gradual transition from the oceanic to the continental geotherm is prescribed over a 200 km-wide section on either side of the ocean–continent boundary (Fig. 1a). The geotherm for the mantle below the lithosphere is defined by a near-adiabatic temperature gradient of 0.46 K km21. Thermal boundary conditions are 0 8C at the upper boundary of the model; no-flux conditions exist at the vertical boundaries. An infinite-like external constant temperature condition along the lower boundary is implemented using the following limitation: @T/@z ¼ (Texternal 2 T )/ Dzexternal, where Texternal ¼ 2140 8C is the temperature prescribed at the external boundary.

Petrological model The stable mineralogy and physical properties for the various lithologies were computed using Perple_X (Connolly 2005) with free energy

minimization as a function of pressure and temperature. For this purpose we adopted the Mie–Grueneisen formulation of Stixrude & Bukowinski (1990), with the parameterization of Stixrude & Lithgow-Bertelloni (2005) augmented for lowermantle phases as described by Khan et al. (2006). This parameterization limits the chemical model to the CaO– FeO–MgO –Al2O3 –SiO2. The mantle is assumed to have a pyrolitic composition, for which the thermodynamic parameterization is adequate to reproduce the expected lower-mantle phase relations (Mishin et al. 2008). Application of the thermodynamic model to the basaltic and gabbroic composition of the oceanic crust is more problematic because phase equilibrium experiments (Irifune & Ringwood 1993; Irifune et al. 1994; Hirose & Fei 2002; Ono et al. 2005) suggest the existence of several high-pressure phases that are not included in our parameterization. In addition, the CaO– FeO–MgO –Al2O3 –SiO2 model excludes volatile oxides, notably K2O and Na2O, that are more significant in the subducted oceanic crust; as a consequence our model is likely to overestimate the basalt–pyrolite density contrast. To calibrate this effect we found that experimentally derived density estimates for K2O– Na2O –CaO –FeO – MgO –Al2O3 –SiO2 (Irifune & Ringwood 1993; Ono et al. 2005) were 1.7 –2.3% below those calculated here. Accordingly, neutral buoyancy in the Earth’s interior most probably corresponds to conditions at which our basalt–pyrolite density contrast is 1.02 + 0.03. Our models therefore overestimate the role of density-induced slab-pull forces in subduction.

Thermomechanical model The momentum, continuity and temperature equations for the 2D creeping flow, accounting for both thermal and chemical buoyancy, are solved using the I2VIS code based on conservative finite differences and non-diffusive marker-in-cell techniques (Gerya & Yuen 2003a). Conservation of mass is prescribed by the incompressible continuity equation: @vx @vz þ ¼ 0: @x @z The 2D Stokes’ equations for creeping flow take the form: @sxx @sxz @P þ ¼ @x @z @x @szz @sxz @P þ ¼  gr(T, P, C, M): @z @x @z

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Fig. 2. Reference model development (model cydd in Table 2). (a) Oceanic slab subduction and bending of the diving plate, high strength of the cold deeply subducted mantle lithosphere prevents its unbending and causes strong slab curvature (Gerya et al. 2008b); (b) thermal relaxation and unbending of the slab; (c) slab stretching and necking; (d) and slab breakoff and sinking. White lines are isotherms taken from 100 8C with 200 8C steps.

104

Table 1. Material properties* used in 2D numerical experiments Material

Hr (mW m23)

Standard density (r0) (kg m23)

Thermal conductivity (W m21 K21)

Rheology

1.0

2700 (solid) 2400 (molten)

0:64 þ

807 T þ 77

Wet quartzite flow law, c ¼ 1 MPa, sin(w) ¼ 0.15

Lower continental crust

0.25

2800 (solid) 2400 (molten)

0:64 þ

807 T þ 77

Plagioclase (An75) flow law, c ¼ 1 MPa, sin(w) ¼ 0.15

Upper oceanic crust (altered basalt)

0.25

3000 (solid) 2900 (molten)

1:18 þ

474 T þ 77

Wet quartzite flow law, c ¼ 1 MPa, sin(w) ¼ 0

Lower oceanic crust (gabbro)

0.25

3000 (solid) 2900 (molten)

1:18 þ

474 T þ 77

Plagioclase (An75) flow law, c ¼ 1 MPa, sin(w) ¼ 0.6

Dry mantle (lithospheric and asthenospheric)

0.022

3300

0:73 þ

1293 T þ 77

Dry olivine flow law, c ¼ 1 MPa, sin(w) ¼ 0.6

Mantle in the subduction initiation zone References

0.022

3200

0:73 þ

1293 T þ 77

Wet olivine flow law, c ¼ 1 MPa, sin(w) ¼ 0

Turcotte & Schubert 2002

Turcotte & Schubert 2002

Clauser & Huenges 1995

Chopra & Paterson 1981; Brace & Kohlstedt 1980; Ranalli 1995

17900 20200 at P , 1200 MPa, þ P þ 54 (P þ 54)2 831 þ 0.06P at P . 1200 MPa 17900 20200 þ at P , 1200 MPa 889 þ P þ 54 (P þ 54)2 831 þ 0.06P at P . 1200 MPa 70400 77800000 þ at P , 1600 MPa, 973  P þ 354 (P þ 354)2 935 þ 0.0035P þ 0.0000062P 2 at P . 1600 MPa 70400 77800000 973  at P , 1600 MPa, þ P þ 354 (P þ 354)2 935 þ 0.0035P þ 0.0000062P 2 at P . 1600 MPa – 889 þ

– Schmidt & Poli 1998; Poli & Schmidt 2002

*Other material properties (density, heat capacity, latent heating, thermal expansion) are computed from the petrological model by Gibbs energy minimization; for models without phase transitions and for the continental crust in all models the following properties are used: Cp ¼ 1000 J kg21 K21, r ¼ r0[1 2 a(T 2 T0)]  [1þ b (P 2 P0)], where a ¼ 3  1025 K21 is thermal expansion, b ¼ 1  1025 MPa21 is compressibility, r0 is standard density at T0 ¼ 298 K and P0 ¼ 0.1 MPa.

C. BAUMANN ET AL.

Upper continental crust

P– T conditions of wet solidus

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Table 2. Description of the numerical experiments used in this work Run cyda cydb cydc (reference model) cydd (reference model) cyde cyaq cyar cyas cydh cydi

Slab age (Ma)

Left plate rate (cm year21)

Right plate rate (cm year21)

Phase transitions

20 40 60 80 100 80 80 80 40 60

þ05 þ05 þ05 þ05 þ05 þ10 0 þ03 þ05 þ05

205 205 205 205 205 0 210 207 205 205

yes yes yes yes yes yes yes yes no no

The density r (T, P, C, M) depends explicitly on the temperature (T ), the pressure (P), the rock composition (C) and the mineralogy (M). The Lagrangian temperature equation includes latent heat effects of phase transformations in the crust and mantle, and is formulated as (Gerya & Yuen 2003a): 

rCp

DT Dt

 ¼

all local rock properties, including effective density, isobaric heat capacity, thermal expansion, adiabatic and latent heating, are calculated at every time step based on Gibbs energy minimization. Viscosity dependent on strain rate, pressure and temperature is defined in terms of deformation invariants (Ranalli 1995) as:

hcreep ¼ (_1II )(1n)=2n F(AD )1=n   Ea þ Va P  exp nRT

@qx @qz  þ H r þ HT þ Hs @x @z

@T , @x @T qz ¼ k(T, c) @z   @H Cp ¼ @T P     1 @H DP HT ¼ r  r @P T Dt qx ¼ k(T, C)

Hs ¼ sxx 1_ xx þ szz 1_ zz þ sxz 1_ xz

where 1_ II ¼ 1=2_1ij 1_ ij is the second invariant of the strain rate tensor, and AD, Ea, Va and n are experimentally determined flow law parameters (Table 1). F is a dimensionless coefficient depending on the type of experiments on which the flow law is based. For example: F¼

2(1n)=n 3(1þn)=2n

F ¼ 2(12n)=n where D/Dt is the substantive time derivative; x and z denote the horizontal and vertical co-ordinates, respectively; sxx, szz, sxz are the components of the deviatoric stress tensor; 1˙ xx, 1˙ zz, 1˙ xz are the components of the strain rate tensor; P is the pressure; T is the temperature; qx and qz are the heat fluxes; r is the density; k(T, c) is the thermal conductivity; Cp is the effective isobaric heat capacity, that is, incorporating latent heat; H is rock enthalpy; Hr, HT and Hs denote radioactive heat production, the energetic effect of isothermal (de)compression and shear heating, respectively. To account for physical effects of phase transitions on the dynamics of breakoff (Zlotnik et al. 2008) we used the coupled petrological–thermomechanical numerical modelling approach described in detail by Gerya et al. (2004a, 2006). In this approach

for triaxial compression for simple shear:

The ductile rheology is combined with a brittle – plastic rheology to yield an effective visco-plastic rheology. For this purpose the Mohr– Coulomb yield criterion (e.g. Ranalli 1995) is implemented by limiting creep viscosity, hcreep, as follows:

hcreep 

c þ P sin(w) (4_1II )1=2

where P is the complete (non-lithostatic) pressure (i.e. the mean stress), c is the cohesion (residual strength at P ¼ 0) and w is effective internal friction angle (Table 1). Assuming high pore fluid pressure in hydrated rocks (e.g. Gerya et al. 2008a), the upper oceanic crust (basalts, sediments) was

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characterized by c ¼ 1 MPa, sin w ¼ 0, a choice that effectively decouples the slabs. As the focus of our paper is viscous breakoff we assigned a high plastic strength to the mantle (sin w ¼ 0.6: Brace & Kohlstedt 1980). Consequently, the plastic strength of subducted slabs in our experiments is always greater than the critical value (500 MPa) reported by Andrews & Billen (2009), and slab detachment occurs in the purely viscous mode by temperatureand stress-activated dislocation creep. The effective viscosity, h, of molten crustal rocks at P– T conditions above their wet solidus (Table 1) was lowered to 1019 Pa s. 1019 and 1026 Pa s are the lower and upper cut values for viscosity of all types of rocks in our numerical experiments.

Modelling results Ten experiments (Table 2) were performed to study the influence of: (1) slab age; (2) plate rate; and (3) phase transitions on the dynamics of subduction. Numerical experiments were performed using the ETH-Zurich Brutus cluster.

Reference models development In our two reference models (models cydc and cydd in Table 2) we distinguish five characteristic phases (Figs 2–4). (a) Oceanic slab subduction and bending; this stage initiates with prescribed convergence rate but thereafter evolves self-consistently (Fig. 2a). (b) Continental collision followed by slab blocking (due to the positive buoyancy of continental crust once it has been subducted to 100 –125 km depth), thermal relaxation and unbending; this stage consists of intense heat exchange between the subducted slab and the surrounding asthenospheric mantle Figs 2b, 3a & 4a); this stage is often manifest by slab decoupling from the overriding plate and asthenospheric upwelling due to the slab roll-back under the orogen (Fig. 4b, c). These latter processes are driven by the slab pull and are most pronounced in experiments with old (dense) oceanic plates (80 Ma, cf. Figs 2c & 3b). The bent slab is unbending under its own weight (Fig. 2b), which is also facilitated by the slab roll-back; (c) Slab stretching and necking (Fig. 2c). Stretching of the slab is directly related to its thermal softening during the previous stage. The former ocean –continent boundary is subducted together with the stretching slab toward depths of more then 200 km; continental crust, however, continuously detaches from the slab and remains at shallower (100 km) levels beneath the orogen (Fig 2c).

(d) Viscous slab breakoff and rapid sinking (Figs 2d, 3b & 4c). This stage is similar to previous models of breakoff with prescribed slab blocking (Gerya et al. 2004b; Andrews & Billen 2009), except that the geometry of the collisional orogen formed at the plate contact evolves naturally with time allowing, in particular, for notable postbreakoff extension (cf. Fig. 4c, d) driven by the positive buoyancy of previously subducted continental crust. (e) Post-breakoff relaxation; this stage is manifest by thermal and topographical relaxation of the orogen associated with exhumation and melting of previously subducted continental and oceanic crust located below the extending plate interface (cf. pink partially molten crustal rocks in Fig. 4d). Within 10–20 Ma from the breakoff event 1 – 4 km of surface uplift occurs in a 200–300 kmwide area above the plate contact (cf. Fig. 4c, d), this uplift is comparable with that observed in previous numerical models (Buiter et al. 2002; Gerya et al. 2004b; Andrews & Billen 2009).

Influence of slab age Experiments on the function of the age of the subducting oceanic plate (Table 3) show that the necking process occurs later in older slabs (Fig. 5). This behaviour is mainly caused by the variations in temperature structure of the subducting oceanic plate (Gerya et al. 2004b) and is characteristic of the viscous mode of breakoff (Andrews & Billen 2009) investigated in our study. In older (i.e. colder) slabs more time is necessary for thermal relaxation to increase temperature and reduce the slab viscosity. Since breakoff process is a direct consequence of the slab necking, the breakoff also correlates with age (Fig. 6). The correlation between slab age and breakoff duration [i.e. the time elapsed between the beginning of slab necking (Fig. 2b) and separation (Fig. 2d)] is more complex (Fig. 7). Two domains are visible in Figure 7: a domain characteristic of young slabs (,50 Ma) and an area for mature slabs (.50 Ma). For young slabs breakoff duration increases with slab age and is related to the influence of the growing slab thermal thickness and strength, which slow the necking process. This trend inverts in old slabs owing to the increasing slab negative buoyancy, higher stresses and stronger thermomechanical feedback, respectively; effects that accelerate necking and breakoff (Gerya et al. 2004b).

Influence of relative plate rates There is no significant effect on the time required for the onset of necking and breakoff when one plate

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Fig. 3. Evolution of geometry and temperature of the detaching slab (Model cydc in Table 2). White lines are isotherms taken from 100 8C with 200 8C steps. (a) Slab stretching and beginning of necking process. (b) Slab breakoff and accelerated sinking.

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Fig. 4. Development of collision zone geometry during spontaneous slab breakoff process (Model cydc in Table 2). (a) Transition from subduction to collision; (b) beginning of necking process, slab roll-back associated with the opening of the asthenospheric window below the collision zone occurs; (c) time of breakoff; (d) post-breakoff time associated with exhumation and melting of deeply subducted crust. The black colour at the bottom of the subducted crust corresponds to partially molten crustal rocks. White lines are isotherms taken from 100 8C with 200 8C steps.

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Table 3. Influences of slab age Name of the numerical experiment cyda cydb cydc cydd cyde

Slab age (Ma)

Beginning of the necking process1 (Ma)

Breakoff time2 (Ma)

Breakoff duration3 (Ma)

Breakoff depth4 (km)

20 40 60 80 100

2.9708 12.6545 22.1207 33.2673 32.8012

6.1320 19.3235 28.5153 38.5978 35.4913

3.1612 6.6690 6.3946 5.3305 2.6901

406.780 455.085 444.915 508.475 432.203

1

With beginning of the necking process meaning an observable necking of the subduted plate (Fig. 2c). Breakoff time means the moment of complete separation of the detached slab from the rest of the plate (Fig. 2d). The breakoff duration is the difference between the beginning of observable necking and complete slab separation. 4 The depth where the separation and breakoff takes place (the deepest point of the separated plate isotherm). 2 3

velocity is adjusted in order to keep the total rate of convergence at 10 cm per year21 (Table 4). This behaviour is to be expected because thermal structure of the slab is mainly affected by the total rate of convergence and not by relative plate velocities. In contrast, breakoff duration and depth (Table 4) are affected by relative plate rates. In particular, in our numerical experiments the duration of breakoff (Table 4) is shortest (5.3– 5.7 Ma) when the velocities of both plates are similar in magnitude. The duration of breakoff is maximized (11.3 Ma) when the magnitude of the velocity of the overriding plate (10 cm year21) is much greater than that of the subducting plate (0 cm year21). Breakoff depth uniformly decreases with decreasing relative velocity of the subducting oceanic plate (Table 4). These variations in breakoff depth and duration are most probably caused by changing slab angle and curvature, which depend on the relative plate

Fig. 5. Correlation between slab age and time (t) at the beginning necking process. Symbols correspond to numerical experiments. The line represents the best fit of numerical results with a second-order polynomial.

Fig. 6. Correlation between slab age and effective breakoff time. Symbols corresponds to numerical experiments. The line represents the best fit of numerical results with a second-order polynomial.

Fig. 7. Correlation between slab age and breakoff duration. Symbols correspond to numerical experiments. The line represents the best fit of numerical results with a second-order polynomial.

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Table 4. Influences of plate rate Name of the numerical experiment cydd cyaq cyar cyas

Slab age (Ma) 80 80 80 80

Plate rates (cm year21) Left plate Right plate 205 00 210 207

þ05 þ10 00 þ03

velocities; for exmaple, if the overriding plate is fast, the slab angle is shallower (van Hunen et al. 2000). In the model with the fastest overriding plate velocity (10 cm year21, model cyar in Tables 2 & 4) the slab is flattened at a depth of 660 km (e.g. Mishin et al. 2008), which is not observed in our reference model with a similar slab age (model cydd in Tables 2 & 4; Fig. 2). In contrast, in the model with the fastest subducting plate velocity (10 cm year21, model cyaq in Tables 2 & 4) the subducting slab is steeper and has stronger curvature than in the reference model (Fig. 2a).

Beginning of the necking process (Ma)

Breakoff time (Ma)

Breakoff duration (Ma)

Breakoff depth (km)

33.2673 29.0926 31.8691 33.1033

38.5978 35.7112 43.2067 38.4074

5.3305 6.61860 11.3376 5.30410

508.475 562.500 460.169 432.203

cydh and cydi in Table 2). Results for pairs of models with and without phase transformations are compared in Table 5 and Figure 8. As follows from our comparison, phase transitions notably accelerate the breakoff process and increase the depth of slab breakoff (Fig. 8). These influences are obviously caused by the stronger negative buoyancy of the subducted slab affected by phase transitions. Consequently, in models with phase transitions the characteristic density contrast of the slab with the surrounding asthenospheric mantle is 100–200 kg m3, which is two–three times larger than the density contrast in models without phase transitions (50– 70 kg m23).

Influence of phase transitions In our experiments a fully coupled petrological– thermomechanical model based on Gibbs free energy minimization is used that accounts for both continuous and discontinuous phase transformations in the mantle and subducted oceanic crust (cf. Mishin et al. 2008 for details). The lowered temperatures in the descending oceanic plate cause it to have a different mineralogy and higher density than the surrounding mantle. Most significant, additional downwards body force on the descending slab is caused by the phase transition of olivine into spinel at a depth of about 410 km, which accelerates viscous detachment dynamics (Zlotnik et al. 2008). Further negative buoyancy effects may come from eclogitization of the subducted oceanic crust. We ran two additional experiments without phase transitions for various slab ages (models

Discussion and conclusions In previous numerical studies (Gerya et al. 2004b; Zlotnik et al. 2008; Andrews & Billen 2009) the cessation of active subduction that is ultimately responsible for breakoff has been kinematically prescribed. The asset of our model is that it simulates both subduction and collision as dynamic processes. Specifically, our experiments model slab detachment induced by slab pull that develops owing to a continental collision after 700 km of oceanic plate subduction. During this latter stage the plate convergence rate evolves realistically (e.g. Chemenda et al. 2000; Faccenda et al. 2008). The chief limitation of our model is that the rheology includes neither elasticity (e.g. Buiter et al. 2002) nor Peierls plasticity (e.g. Kameyama et al. 1999). These deformation mechanisms may influence the transition from viscous to plastic detachment

Table 5. Influences of phase transitions Numerical experiment cydc cydi (no phase transitions) cydb cydh (no phase transitions)

Beginning of the necking (Ma)

Time of breakoff (Ma)

Breakoff duration (Ma)

Breakoff depth (km)

22.1207 29.4116 12.6545 16.5099

28.5153 34.7640 19.3235 36.2608

6.3946 5.3524 06.6690 19.7509

444.915 330.508 457.627 317.790

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Fig. 8. Influence of phase transitions onto the depth of slab breakoff. (a) Model cydc compared with Model cydi. (b) Model cydb compared with Model cydh. The snapshots for different models are taken for different moments in time when breakoff occurs in each model (Table 5). White lines are isotherms taken from 100 8C with 200 8C steps.

(Andrews & Billen 2009) and the development of topography (Buiter et al. 2002). Additional limitations are the simplified rock chemistry and that the petrological model does not account for metastable phase assemblages which might occur in the

subducted gabbroic crust (e.g. van Hunen et al. 2000) and lithospheric mantle (e.g. Schmelling et al. 1998). Thus, negative buoyancy of slabs in our experiments may be overestimated. In natural subduction settings significant along-strike

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variations in the dynamics of breakoff processes are inferred from seismic and topographical data (e.g. Wortel & Spakman 1992, 2000). Such 3D effects cannot be accounted for by the present 2D model. Five evolutionary phases occur in most of our experiments: (a) oceanic slab subduction and bending; (b) continental collision initiation followed by a cessation of subduction, thermal relaxation and unbending; (c) slab stretching and necking; (d) slab breakoff and sinking; and (e) post-breakoff relaxation. The development of our dynamic thermomechanical model of continental collision has a number of similarities with previous analogue and numerical collision models that take into account significant buoyancy effects arising from continental crust subduction (e.g. Chemenda et al. 2000; Burov et al. 2001; Faccenda et al. 2008). In particular, our results confirm a positive correlation between the oceanic slab age and the time of necking and breakoff found in previous thermomechanical studies of the viscous mode of detachment, exploring models without phase transformations (Gerya et al. 2003; Andrews & Billen 2009). However, previous work did not report the non-linear dependence of breakoff duration on the slab age: a positive correlation being characteristic of relatively young (,50 Ma) slabs, while for older slabs the correlation is negative. We also found that at a constant convergence rate, a relatively high velocity of subducting oceanic plate increases the depth of breakoff, most probably because of the geometric consequences of the subducting plate (van Hunen et al. 2000) that influences the dynamics of phase transformations affecting slab buoyancy. A prediction from our breakoff models is that the olivine–wadsleyite transition plays a role in localizing viscous slab breakoff at depths of 410– 510 km owing to the buoyancy effects of the transition (e.g. Schmelling et al. 1998; Zlotnik et al. 2008). Without phase transitions, the slab is significantly less dense. The breakoff in this case is shallower, occurs later and lasts longer. This work was supported by ETH research grants TH-12/ 05-3 and TH-0807-3; SNF research grants 200021113672/1, 200021-116381/1 and 200021-107889; and the RF President Programme ‘Leading Scientific School of Russia’ (Grant #1949.2008.5) to T. V. Gerya. Constructive reviews of S. Buiter and R. Govers are appreciated.

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Present-day vertical isostatic readjustment of the Western Alps revealed by numerical modelling and geodetic and seismotectonic data ANNALISA GARDI1, STE´PHANE BAIZE* & OONA SCOTTI Institut de Radioprotection et Suˆrete´ Nucle´aire, IRSN/DEI/SARG/BERSSIn, BP 17, 92262 Fontenay-aux-Roses, France 1

Present address: Geosciences Consultants, 157 rue des Blains, 92220 Bagneux, France *Corresponding author (e-mail: [email protected]) Abstract: The active tectonics of the Western Alps reveals contrasting regimes: ongoing extension at the heart of the chain and transpression–compression at its external sectors. The active processes currently affecting this region are still a matter of debate. The classical models proposed in the literature invoke: Eurasia–Adria plate collision, counterclockwise motion of the Adria microplate, slab retreat of the subducted continental lithosphere and slab-detachment. More recently, several authors prefer the hypothesis of tectonics driven by isostasy–buoyancy forces. To better understand the influence of these processes on the velocity, strain and stress fields at the surface and in the crust, we developed 2D viscoelastic numerical models along a vertical cross-section perpendicular to the Western Alps. We run our models with different driving forces in order to investigate, one by one, the geodynamic processes proposed in the literature. Results are compared with available geodetic, geological and seismotectonic data. In order to bring into coincidence model predictions and observations, an important vertical isostatic readjustment must be included in the modelling, together with a slight horizontal compression (0.5 mm year21), probably due to Africa– Eurasia convergence. We show that the subduction process in this Alpine region is likely to be dead and that buoyancy forces may be dominating the present-day tectonics.

The Alps are the result of the collision between the Adria microplate (a block of continental crust, now comprising much of Italy and the Adriatic sea floor) and the southern margin of the European platform (Dewey et al. 1973) during Cenozoic times (Fig. 1). Prior to this collision, oceanic lithosphere comprising part of the Eurasian lithosphere was underthrust beneath Adria, starting at about 135 Ma (Tru¨mpy 1980). It seems evident that the convergence between the African and Eurasian plates is still ongoing at a rate of 3 –8 mm year21 (Argus et al. 1989; Calais et al. 2003) in the NW– SE to NNW–SSE direction, while GPS studies (e.g. Nocquet & Calais 2004; D’Agostino 2007) reveal that the Adriatic microplate has a distinct movement pattern relative to Eurasia. Several structures typical of a compressive regime (thrusts, folds and metamorphic zones: e.g. Fig. 2a) have been well documented by a large number of authors (e.g. Choukroune et al. 1986; Fry 1989; Pognante 1991; Spalla et al. 1996; Becker 2000). Seismotectonic and structural analysis highlighted the presence of an extensional regime at the heart of Western Alps (Mancktelow 1992; Eva et al. 1998; Sue et al. 1999; Bistacchi et al. 2000; Lardeaux & Schwartz 2001; Delacou et al. 2004) (e.g. Fig. 2c). This horizontal extension in the internal sector of the chain

while external sectors are submitted to horizontal compression represents a geodynamical problem. The classical geodynamical models for the western Alpine arc involve the push exerted by Adria on the European plate and the progressive migration of the orogen towards the NW (Pavoni 1961; Tapponnier 1977). At first analysis, the extensional regime of the internal sector of the chain, which is radially oriented with respect to the Alpine arc, is not compatible with the simple mechanism of Adria –Europe convergence. Nevertheless, the Adria– Europe convergence could be still active on the Western Alps because of the radial compression in the external regions of the orogen. Another class of kinematic models involve not only the NW translation of the Adria plate but also its counterclockwise rotation first recognized by Vialon et al. (1989). This mechanism would explain the right-lateral strike-slip faults in the whole Western Alps (Me´nard 1988) and along the internal border of the chain (the so-called Periadriatic Line: Schmid et al. 1989; Sue 1998; Nocquet & Calais 2004). The exact position of the Eulerian pole of rotation of the Adria plate has not yet been definitly fixed. A synthesis of the proposed Eulerian poles can be found in Nocquet & Calais (2004) and in D’Agostino (2007).

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 115– 128. DOI: 10.1144/SP332.8 0305-8719/10/$15.00 # The Geological Society of London 2010.

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Fig. 1. Tectonic setting of the Western Alps. The black square highlights the region of study.

A last potentially active mechanism is the detachment of the subducted plate, first proposed for this region by Lyon-Caen & Molnar (1989) in order to explain the uplift of the Alps and the molassic basin after the Miocene. Numerous works exist in the literature addressing the different aspects of a slab detachment (e.g. von Blackenburg & Davies 1995; Marotta et al. 1998, 1999; Wortel & Spakman 2000; Buiter et al. 2002; Gerya et al. 2004; Faccenda et al. 2008). This mechanism implies that hot asthenospheric material replaces the detached cold lithosphere underneath the thickened crust forming the orogen. A first consequence would be a quite rapid uplift of the chain due to isostatic adjustment. Secondly, a thermal adjustment would progressively weaken the crust,

diminishing its strength to body forces. Moreover, the heating of the overriding lithospheric mantle by the upwelling asthenosphere would cause a melting of its enriched layers and thus a bimodal magmatism (Davies & von Blackenburg 1995). In support of the slab detachment thesis, a vertical uplift has been observed in the internal sectors of the Alpine chain (Kahle et al. 1997; Lenoˆtre 1990). In the framework of the NRP20 project (Pfiffner et al. 1997), a tomographic cross-section nearly perpendicular to the chain under the Swiss Alps highlights the presence of positive velocity anomalies at 200–300 km and 600 km depth (Spakman et al. 1993). These are interpreted as the subducted and detached Valais and Piemont ocean lithospheres (Marchant 1993). A similar and

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Fig. 2. (a) Regionalization of the deformation field of the Western Alps, projected along the ECORS profile. Colours give the deformation style: red for shortening; blue for extension; and white for strike-slip (modified after Delacou et al. 2004). (b) Horizontal velocity (mm year21) deduced by permanent GPS measurements along a NW–SE transect, perpendicular to the Alpine chain. SJDV, St Jean De Vigne; FCLZ, La Feclaz; CHTL, Le Chaˆtel; MODA, Modane; TORI, Turin; (1), reverse; (2), normal; (3), strike-slip faulting (modified after Walpersdorf et al. 2006). (c) Vertical movements (mm year21) deduced by levelling data (modified after Lenoˆtre 1990).

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recent work is missing for the Western Alps. Nevertheless, Poupinet (1976) found positive P-wave residuals, indicative of a low upper-mantle velocity, at stations in the French Alps. Babushka et al. (1984) studied P-wave residuals and upper-mantle structures in Europe, finding high velocities beneath the maritime Alps and the Eastern Alps, but not beneath the Western Alps. This could agree with the presence of hot asthenosphere under the Western Alps resulting from a slab detachment. Concerning the bimodal magmatism typical of a slab breakoff, both basaltic and granitoid magmatism occurred in the Alps between 42 and 25 Ma (von Blackenburg & Davies 1995). These magmatic intrusions are located along a strike-slip fault, the Periadric Lineament, corroborating the idea that, once initiated, the rapid lateral migration of slab breakoff would result in a linear trace of magmatism in locally thermally weakened crust. Other mechanisms have been mentioned in the literature, such as subduction roll-back and slab retreat (Royden & Burchfiel 1989; Royden 1993) or collapse and spreading driven by gravity (Molnar & Lyon-Caen 1988), but for several reasons (see Sue & Tricart 2002) they are hardly applicable to the Western Alps. The goal of this work is to investigate the different geodynamic processes proposed in the literature to explain the present-day Alpine stress pattern. We use a finite-element approach to model the deformation along a 2D vertical cross-section nearly perpendicular to the chain. Several different hypothesis have been tested, each one characterized by different driving forces in order to reproduce, one by one, the effects of the Africa–Eurasia convergence, those of the rotation of the Adriatic microplate and those of a detachment of the subducted European lithosphere. Finally, we also investigate the degree of activity of the subduction plane. The results are presented in terms of stress, displacement and velocity fields, and they are compared with seismotectonic data and geodetic measurements (GPS and levelling).

Data As far as seismotectonics is concerned, we relied on the work by Delacou et al. (2004). These authors compiled 389 focal mechanisms of Western– Central Alpine seismicity, providing a mapping of the present-day strain–stress regime thanks to powerful methods of regionalization. Figure 2(a) summarizes the results of the Alpine strain regionalization by Delacou et al. (2004), projected onto a vertical cross-section parallel to the ECORS seismic profile (ECORS-CROP 1989). The most important feature is an alternating stress regime

between external (compression) and internal (extension) zones, which represents the geodynamic problem we are trying to solve in the present work. The region around the Penninic Front (PF), near Le Chaˆtel and Modane, undergoes horizontal extension (green-blue in Fig. 2b), whereas the external regions, near Turin towards the SE and between La Feclaz and Le Chaˆtel towards the NW, are characterized by horizontal compression (pink-red in Fig. 2a). These data are quite consistent with those reported by Jouanne et al. (1998). Continuous GPS data (Walpersdorf et al. 2006) agree partially with seismotectonics showing extension in the internal regions (around the Penninic Front, see Fig. 2b) and a slight shortening between Modane and Turin, that is around the Periadriatic Line. Even if caution must be taken in the interpretation of these data considering the large uncertainty of the geodetic velocity vectors, important information comes from GPS measurements: the values of horizontal movements at the surface are less than 1 mm year21. We used this value as an upper bound to constrain our models. Concerning the vertical movements in the Western Alps, there is a lack of data. The only existing works to our knowledge are: (1) the technical report by Lenoˆtre (1990) about levelling measures in the Southern and Western Alps (Fig. 2c); and (2) the work by Jouanne et al. (1998) dealing with levelling data compared to geomorphological markers. Levelling data of Lenoˆtre (1990) and Jouanne et al. (1998) show, respectively, a slight subsidence (nearly 20.2 mm year21) or a slight uplift (up to 0.8 mm year21) in the westernmost part of our transect, and significant uplift in the inner portion of the chain, with a maximum velocity of 2 mm year21. This is consistent with the vertical velocities (between 0.2 and 1 mm year21) observed by Kahle et al. (1997) in the Swiss Alps, and also by Foeken (2004) in the southern Western Alps (0.421 mm year21 for the last 10 Ma). The French– Italian project ‘Etude Continentale et Oce´anique par Re´flexion et Re´fraction Sismique – Progetto Strategico Crosta Profonda (ECORSCROP 1989)’ constitutes the main source of data concerning the crustal structures of the Western Alps. Figure 3a reproduces the interpretation of this seismic profile by Marchant (1993), but several others have been published (e.g. Tardy et al. 1990; Roure et al. 1996; Schmid & Kissling 2000). We chose to simplify as much as possible the complex structure of the orogenic chain by identifying the principal structures active at the regional scale. For this, we compared the interpretations of the seismic profile to gravity studies (ECORSCROP Gravity and Magnetic Group 1996), geological studies (Mugnier et al. 1993), tomography (Solarino et al. 1997; Paul et al. 2001), tectonic

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Fig. 3. (a) Interpretation of the seismic profile ECORS-CROP. Simplified after Marchant (1993). (b) Geometry and grid of the central part of the models. Thick black lines give the positions of the Penninic Front (PF) and the Periadriatic Line (PL). Thick white lines indicate the contact surfaces with free-slip condition. (c) Geometry, materials and boundary conditions of the whole model.

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reconstructions (Marchant & Stampfli 1997), tectonic models (Beaumont et al. 1996) and seismicity distribution (Sue et al. 1999). The resulting simplified structure chosen for our simulations is represented in Figure 3b and described in details in the next section.

Models description The deformation of the lithospheric structure along a 2D vertical section across the Alpine chain is modelled by means of the finite-element package ABAQUS. The simulations were performed under a quasi-static approximation and the models are purely mechanical, as the coupling between the momentum and the temperature equations is not taken into account. Several models have been carried out, each one representing a 2D vertical cross-section nearly perpendicular to the Western Alps, as shown in Figure 1 by the trace A –A0 . The bidimensional domain is subdivided into quadrilateral plane-strain elements, creating an irregular mesh. The element density is increased around the discontinuities in order to ensure a good approximation of the solution. The modelled cross-section is 1400 km wide and 600 km deep (Fig. 3c). The simplified geometry of the plates and the thickness of the different layers have been defined mainly following the interpretations of the deep seismic sounding of the ECORS-CROP transect (Fig. 3a). The resulting simplified structure chosen for our simulations is represented in Figure 3b. In the present-day structure of the Alps, we can distinguish three great zones from west to east: the Eurasian palaeomargin; the internal metamorphic zone resulting from the subduction of the ancient Liguro– Piemont ocean; and the Adriatic palaeomargin. These three regions are divided by two major discontinuities, the Penninic Front and the Periadriatic (or Insubric) Line. We included these two major structures in our models (Fig. 3b) just as a geometrical reference in order to set the limits between internal and external units of the chain. The PF (Penninic Front) and PL (Periadriatic Line) contacts have not been activated during our simulations. On the contrary, the subduction plane

has been included in the computations as an active contact, whose behaviour has been fixed to ‘locked’ in some tests and to ‘free-slip’ in some others, setting the friction coefficient to 1 or 0, respectively. Another key feature of the modelled vertical structure is the presence of the Ivrea body. It represents the most documented and most studied geophysical body of the Western Alps. It is interpreted as an Adriatic mantle slice included in the superficial crust of the Alps (Closs & Labrouste 1963; Berckhemer 1968; Lanza 1975; Solarino et al. 1997). The model is based on the viscoelastic Maxwell rheology, which accounts for the elastic behaviour of the lithosphere on short timescales and the long-term viscous behaviour. Contrary to more sophisticated models, based for example on viscoelasto-plastic thermomechanical approaches (e.g. Yamato et al. 2007; Burov & Yamato 2008), the goal of this work is not to simulate the initiation of subduction and the progressive deepening and evolution of the down-going plate, but rather to study the actual stress regime by exploring different geodynamic mechanisms. Our approach implies computations with a relativey small deformation time window and in which the large-scale flow is avoided. The rheological properties of the considered materials have been defined based mainly on the work by Burov et al. (1999). These authors, by means of a thermomechanical model, verified the mechanical stability at depth of different properties (density, rheology, temperature) deduced by geophysical and geological data along the ECORS-CROP profile. The viscosity of the upper crust, lower crust, lithospheric mantle and asthenosphere, as well as their relaxation time and elastic properties, are summarized in Table 1. The viscosity of the asthenosphere has been set to 1021 Pa s, as commonly accepted (e.g. Funiciello et al. 2003; Piana Agostinetti et al. 2004). The parameters varying in our experiments are the mechanical behaviour of the subduction plane, the density applied to the subducted lower crust and the kinematic condition applied to the eastern border of the grids; the two latter representing different tectonic components driving the simulations, as we will see in more detail. This allowed

Table 1. Mechanical properties of the materials

Upper crust Lower crust Lithospheric mantle Asthenospheric mantle

Young’s modulus, E (Pa)

Poisson’s ratio, n

Viscosity, h (Pa s)

Relaxation time, Trel

9  1010 9  1010 1.75  1011 1.75  1011

0.25 0.25 0.27 0.27

1024 1020 5  1022 1021

900 ka 90 years 24 ka 300 years

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Table 2. Characteristics of the different models

Subduction plane Density contrast (kg m23) Applied convergence velocity (mm year21) Horizontal velocity obtained in the region of interest (mm year21)

DETA0

DETA1

DETA2

DETA3

SLAB1

locked 2600 –

locked 2600 1

locked 2600 0.5

locked 21200 0.5

free 2600 0.5

c. 0.65

c. 0.3

c. 0.3

c. 0.3

a large number of models to be created, of which only the most significant are presented here. Table 2 summarizes the principal characteristics of the models discussed in the next sections. In order to simulate the effects of a slab detachment, we applied a negative density contrast to the subducted lower crust. As this approach is an artificial way to simulate the buoyancy of the remaining slab after the detachment of its deepest portion, we tested different values of density contrast (Dr). In the following we will show the results obtained with Dr ¼ 2600 kg m23 (DETA1, DETA2 and SLAB1 models) and Dr ¼ 21200 kg m23 (DETA3 model). With the aim of reproducing the Africa–Eurasia convergence, a horizontal velocity of 1 mm year21 (DETA1 model) or 0.5 mm year21 (DETA2, DETA3 and SLAB1 models) is applied to the eastern border of the grids from the surface to a depth of 60 km. These values have been chosen in order to obtain, in the region of interest, an horizontal velocity coherent with the GPS data reported in Figure 2a. This point will be discussed in more detail in the next section. With the aim of separately visualizing the effects of the density contrast and the horizontal velocity, we performed a model driven only by the density contrast (DETA0). Other mechanical boundary conditions are applied as follows. The bottom border of the grid is fixed in the vertical direction, while the western border corresponding to the lithospheric level is fixed in the horizontal direction to account for the presence of stable Eurasia. Gravity is applied to all the elements of the meshes, with a value of 9.8 m s22. In order to avoid an excessive lateral flux of the asthenosphere across the vertical boundaries of the model, and, at the same time, to account for the isostatic response of the medium, we employed a new technique. The part of the grid corresponding to the mantle is wrapped into a series of infinite elements. These are elastic quadrilateral elements with a side at an infinite distance, suitable for problems in which the region of interest is small in size compared to the surrounding medium. To the infinite elements we attributed the same elastic properties as to the asthenospheric mantle. The

final structure of our models is represented in Figure 3c. The infinite elements exert a sort of lateral constraint, thus reproducing, in a very simplified way, the presence of the Earth mantle around the modelled section. In this way it is not necessary to introduce the Winkler foundation (Desai 1979) that used in other similar tectonic models (e.g. Carminati et al. 1999; Negredo et al. 1999; Gardi et al. 2003) to account for the isostatic restoring forces acting at the boundaries between density contrasts. Each simulation is subdivided into two major steps, which are preceeded by a geostatic step for the instantaneous application of the gravity load to all the elements of the meshes. The first major step after the geostatic one has a duration of 200 ka, which we verified as being the time interval necessary to this kind of model to reach a dynamic equilibrium between the tectonic and isostatic forces, and for stress and strain rates to reach steady-state values. This duration is in agreement with the one (250 ka) found by Giunchi et al. (1994, 1996) with a similar approach. During this first step the density contrast of the subducted lower crust, which simulates the effects of slab detachment, is active. During the second major step, which has the same duration of the first one, the kinematic condition that reproduces the Africa– Eurasia convergence is activated.

Modelling results The results of the simulations have been analysed focusing on horizontal velocities, stresses and vertical velocities. Horizontal velocities will be discussed first. Then we will comment on the results in terms of stresses computed along the surface of the different models performed (Fig. 4) and of vertical velocities obtained along the surface of the models (Figs 5 & 6).

Horizontal velocities Horizontal velocities obtained in the crust are the direct consequence of the kinematic condition imposed at the SE border (right-hand side) of our

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Fig. 4. Stresses [(smax þ smin)/2] obtained for the elements along the upper surface. Regions with positive stresses are characterized by horizontal extension, whereas negative stresses indicate horizontal compression.

models for reproducing the Africa–Eurasia convergence. Thus, the value of horizontal velocity at the surface of the models decreases from the maximum value at the SE border (corresponding to the value of the applied convergence velocity, 1 or 0.5 mm year21 depending on the model) to zero at the NW border, this latter being fixed in the horizontal direction by a boundary condition. In the region of interest, that is the central part of the models, we obtain horizontal velocities of between 0.65 and 0.3 mm year21 (Table 2), in agreement with GPS measurements (Fig. 2).

Stresses Figure 4 shows the mean stresses obtained along the surface of different models. The stress value depicted has been computed as

sm ¼ (sMax þ sMin )=2 where sMax and sMin are the maximum and minimum in-plane principal stress components. Negative values of sm indicate a compressional regime, while positive values stand for an extensional regime. It is worth noting that the amount of stress computed in our different tests and depicted in Figure 4 is not really significant owing to the uncertainty of input parameters, especially the values of density contrasts imposed on the subducted lower crust of our models. In the rest of this section we will then focus on the stress pattern

obtained in order to analyse our results from a qualitative point of view. The stress pattern resulting from the first model (DETA0, Dr ¼ 2600 kg m23, Vconv ¼ 0 mm year21) is represented by the black short-dashed line. In this model no convergence rate is applied. It is assumed that only buoyancy forces due to slab detachment are active. DETA0 fits the data only partially, failing in reproducing the compressional regime observed by seismotectonics (Delacou et al. 2004) in the area surrounding the Periadriatic Line. Adding a convergence rate of 1 mm year21 (DETA1 curve of Fig. 4, Dr ¼ 2600 kg m23, Vconv ¼ 1 mm year21) we obtain a compressional regime in the external sectors of the chain, that is NW of the Penninic Front (PF) and SE of the Periadriatic Line (PL), as observed by the surface and seismotectonic data. However, as almost all of the crust turns to a marked compressional regime, model DETA1 is in disagreement with the data in the internal sector of the chain: the extension observed is retrieved only in a very narrow part of the region between the two tectonic contacts. In light of this result, a covergence velocity of 1 mm year21 seems to be too high. Reducing the convergence velocity to 0.5 mm year21 (DETA2 curve of Fig. 4), the stress distribution reflects quite well the alternating pattern of compression– extension, as shown in the observations. We then tested the influence of the density contrast value applied to the subducted plate in order to

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Fig. 5. (a) Vertical velocities along the upper surface. (b) Topography changes obtained at different times after the detachment using a thermomechanical model. Modified after Gerya et al. (2004).

reproduce the post-detachment rebound. Comparing the DETA2 curve (Dr ¼ 2600, Vconv ¼ 0.5 mm year21) with the DETA3 curve (Dr ¼ 21200, Vconv ¼ 0.5 mm year21), we see that the width of the zone undergoing extension increases by increasing the value of Dr. Both the values of 2600 and 21200 kg m23 are completely arbitrary, and using the simple analysis of the stress distribution at the surface presented here we cannot discriminate between these values. Finally, the SLAB1 curve of Fig. 4 (Dr ¼ 2600, Vconv ¼ 0.5 mm year21) is similar to DETA2, the two models differing in just the mechanical behaviour of the subduction plane; locked in model DETA2 and free (zero friction) in model SLAB1.

The stress distribution is affected only in the amount of stress but not in the pattern, thus model SLAB1 also fits quite well with the seismotectonic data. The models that best fit the seismotectonic observations are DETA2, DETA3 and SLAB1, all characterized by a convergence rate of 0.5 mm year21. The analysis of the stress distribution at the surface lead us to conclude that in order to reproduce the observed compressional –extensional– compressional pattern we need to apply to our mechanical models a relatively high value of Dr with respect to the value of convergence velocity. The validity of our results is corroborated by the comparison with the work by Marotta et al. (1998,

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Fig. 6. Comparison between vertical velocities of model DETA2 (Dr ¼ 2600 kg m23, Vconv ¼ 0.5 mm year21, subduction plane: locked) and model SLAB1 (Dr ¼ 2600 kg m23, Vconv ¼ 0.5 mm year21, subduction plane: free).

1999), who simulated the evolution of the lithosphere during its shortening and unrooting by means of a 2D thermomechanical model in order to analyse the relation between the surface stress regime and the processes of plate convergence and lithospheric failure. They found that after unrooting, extension prevails in the centre of their model embedded within a compression stress style and that the occurrence of local extension, its intensity and spatial distribution depend mainly on the decrease/cessation of convergence and on when it occurs.

Vertical velocities The vertical velocities are obviously mainly controlled by the chosen value of the density contrast imposed to the subducted lower crust. Increasing the Dr, the computed vertical velocities increase (Fig. 5a). We obtained values comprised between 0.02 and 0.1 mm year21, which are one order of magnitude smaller than the observations. Conversely, the pattern fits quite well the levelling data (presented in Fig. 2b): the central part of the models uplifts, whereas the regions corresponding to the external zones of the chain are affected by subsidence, especially towards the NW. In order to validate our simulations, we compare our results with those of Gerya et al. (2004). These authors performed a realistic thermomechanical study reproducing the entire evolution of a detachment process caused by thermal diffusion of subducted slab after cesssation of active subduction, using a 2D upper-mantle model 660 km deep and 2000 km wide. They did not model the geodynamics

of the Western Alps. These authors focused on the mechanism of slab detachment in general, without performing an application in a definite region. Notwithstanding, we believe that the comparison between their work and ours is suitable and very interesting. Figure 5b shows the topography changes obtained by Gerya et al. (2004) at different times after the detachment. This corresponds to a vertical velocity of 0.07 mm year21. Our results are of the same order of magnitude as those of Gerya et al. (2004). Moreover, the similarity between their results and ours concerning the shape of the topography changes is also noteworthy: in addition, Gerya and co-authors obtained uplift above the slab and lateral subsidence. If the quantitative agreement between our results and those by Gerya et al. (2004) must be taken with a degree of caution owing to the very different methods of modelling, the qualitative similarities support us in our choice of simulating the effects of a slab detachment by imposing a density contrast to the subducted lower crust of our models. A last crucial remark comes from the comparison between the vertical velocities obtained with models DETA2 and SLAB1. In spite of the fact that these models produced similar results in terms of stress pattern, Figure 6 demonstrates the differences between them in terms of vertical velocities produced. SLAB1, owing to the free-slip condition of the subduction plane, produces a much larger uplift with respect to DETA2. This result highlights that uplift is very sensitive to the friction along the subduction fault, in agreement with the work by Buiter et al. (2002). SLAB1 shows a vertical velocity pattern quite different from the one of

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DETA2 and fails in reproducing the subsidence observed in the westernmost part of our transect, which is also predicted by the accurate simulation of Gerya et al. (2004). In light of these results, DETA2 can be considered as our best-fitting model.

Discussion and conclusions The above analysis illustrates the variety of stress – strain and velocity patterns produced by the different tectonic mechanisms that we investigated with our bidimensional simulations. In our models the slab detachment mechanism has to be necessarily active in order to explain the coeval existence of extensional and compressional regimes in the Western Alps, as observed by seismotectonic and geodetic studies. Actually, the phenomenon of slab detachment produces a vertical instability followed by a relatively rapid uplift of the chain due to isostatic adjustment. This effect has been modelled by introducing a negative density contrast for the subducted lower crust. This artefact allowed us to reproduce the uplift of the central portion of the chain, which implies a horizontal extension at the surface of the uplifted region. Moreover, owing to the elastic response of the upper crust, we could reproduce some subsidence in the external region of the chain towards the NW, which generates a horizontal compressional stress field NW of the Penninic Front, in agreement with seismotectonic and geodetic observations. The activation of a horizontal velocity of 1 or 0.5 mm year21, in order to reproduce the Africa– Eurasia convergence, causes an increase in the compressional stresses. This is necessary for reproducing the compressional –transtensional regime SE of the Periadriatic Line. Our best-fitting model is characterized by a density contrast of 2600 kg m23 and a convergence velocity of 0.5 mm year21 at the border of the grid, which corresponds to a horizontal velocity of about 0.3 mm year21 in the region of interest. Such a slow horizontal velocity probably means that most of the convergence rate between Africa and Eurasia is absorbed in other regions, and that in this part of the Alps it is very weak with respect to the uplift component. The values employed in order to reproduce the post-detachment rebound and the convergence velocity (Dr ¼ 2600 kg m23 and Vconv ¼ 0.5 mm year21 in the best-fitting model) have to be considered with respect to each other: our results clearly show that the contribution of both mechanisms is necessary in order to have a good agreement with the observations, and that the vertical component (post-detachment rebound) has

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to be more important than the horizontal one (convergence velocity). The computed uplift rates, of between 0.02 and 0.1 mm year21, are of an order of magnitude lower than those observed at the surface, whose maximum values range between 1.1 (Kahle et al. 1997) and 2 mm year21 (Lenoˆtre 1990; Jouanne et al. 1998). Partially, this can be due to the uncertainty of several input parameters and to the way of simulating the post-detachment rebound by means of an artefact (the negative density contrast selected for the subducted lower crust). Nevertheless, other mechanisms are invoked in the literature for explaining the uplift of the central part of the Alps, such as post-glacial rebound (Kahle et al. 1997; Stocchi et al. 2005; Barletta et al. 2006) or isostatic adjustment following a strong erosional rate (Schlunegger & Hinderer 2001; Champagnac et al. 2007a, b). As far as the post-glacial rebound is concerned, Stocchi et al. (2005) estimated the vertical uplift due to the Alpine deglaciation. Their computations show that melting of the Alpine glacier may account for uplift rates as large as 0.2 mm year21 at the sites of the permanent GPS REGAL network (which includes the Modane and Turin stations). Barletta et al. (2006) computed the viscoelastic response to present-day glacier shrinkage in the Western Alps, demonstrating that it contributes a substantial fraction of the observed uplift rate. Their models predict uplifts of between 0.1 mm year21 in the periphery of the uplifting region and 0.4–0.5 mm year21 in the centre of the belt; thus, about a factor 2 lower than the measured ones. Champagnac et al. (2007a, b) estimated the average erosion rate for the Alps and calculated the isostatic response of the Alpine lithosphere to erosional unloading. Assuming a steady erosion rate since 1 Ma, these authors obtained vertical movements of approximately 0.5 mm year21, which account for less than half of the measured vertical motion. The mentioned processes (post-detachment rebound, post-glacial rebound and post-erosion rebound) are not only likely to be active at the same time, but they all induce similar signatures on the topography which together may account for the relatively high uplift rate (more than 1 mm year21) observed in the Alps today. The quantitative estimate of the different contributions, performed independently by different authors, seems to confirm this hypothesis. We also tested the activity of the subduction plane. When this discontinuity undergoes a free-slip condition, a part of the crustal deformation is dissipated through the slipping movement. Therefore, the crust does not show any of the elastic bending

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that, in the models without active subduction, helps in reproducing the compressional stresses observed NW of the Penninic Front. Then with an active subduction we are unable to fit the data. It is worth mentioning that this result should be taken with a degree of caution owing to the fact that we tested two endmember cases: a free-slip (zero friction coefficient) and a completely glued (friction ¼ 1) subduction plane. We can speculate that an intermediate condition (e.g. friction ¼ 0.6) would allow us to obtain an uplift of the chain greater than found in the models with a free-slip subduction plane and also to have a partially bending lithosphere, as in the models with a glued subduction plane. However, the concept of a subduction that is no longer active in this region of the Alps is in agreement with the low convergence velocity and with the theory of slab detachment, which implies the absence of slab pull. Regarding the counterclockwise motion of the Adria microplate, we cannot definitely model its rotation component. We are only able to reproduce its residual, parallel to our section, that may itself be due to a non-axial location in our section with respect to the Adria rotation pole. We assume that this parallel residual is included in the value of 0.5 mm year21 (our ‘convergence rate’) necessary for the simulations to retrieve the observations. This research was funded by a IRSN post-doctoral project. A. Gardi is grateful to F. Bonilla for fruitful discussions and suggestions. We thank R. Sabadini and an anonymous reviewer for their constructive comments. Figures are drawn using GMT (Wessel & Smith 1991).

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S UE , C. 1998. Dynamique actuelle et re´cente des Alpes occidentales internes – Approche structurale et sismologique. PhD thesis, Joseph Fourier University, Grenoble, France. S UE , C. & T RICART , P. 2002. Widespread post-nappe normal faulting in the Internal Western Alps: a new constraint on arc dynamics. Journal of the Geological Society, London, 159, 61–70. S UE , C., T HOUVENOT , F., F RECHET , J. & T RICART , P. 1999. Widespread extension in the core of the western Alps revealed by earthquake analysis. Journal of Geophysical Research, 104, 25,611–25,622. T APPONNIER , P. 1977. Evolution tectonique du syste`me alpin en Me´dite´rrane´e: poinc¸onnement et e´crasement rigide-plastique. Bulletin de la Socie´te´ Ge´ologique de France, 7, 437–460. T ARDY , M., D EVILLE , E., F UDRAL , S., G UELLEC , S., M E´ NARD , G., T HOUVENOT , F. & V IALON , P. 1990. Inte´rpretation structurale des donne´es du profil sismique a` re´flexion profonde ECORS-CROP Alpes entre le front Pennique et la ligne du Canavese (Alpes occidentales). In: R OURE , F., H EITZMANN , P. & P OLINO , R. (eds) Deep Structure of the Alps. Me´moires de la Socie´te´ Geologiques de France, Paris, 156; Me´moires de la Socie´te´ Geologiques de Suisse, Zurich, 1; Societa Geologica, Italiana, Roma Special volume, 1, 217–236. T RU¨ MPY , R. 1980. An outline of the geology of Switzerland. In: T RU¨ MPY , R. (ed.) Geology of Switzerland, a Guide-book. Wepf & Co., Basel. V IALON , P., R OCHETTE , P. & M E´ NARD , G. 1989. Indentation and rotation in the Alpine arc. In: C OWARD , M., D IETRICH , D. & P ARK , R. (eds) Alpine tectonics. Geological Society, London, Special Publications, 45, 329– 338. VON B LACKENBURG , F. & D AVIES , J. H. 1995. Slab breakoff: A model for syncollisional magmatism and tectonics in the Alps. Tectonics, 14, 120– 131. W ALPERSDORF , A., B AIZE , S., C ALAIS , E., T REGONING , P. & N OCQUET , J. M. 2006. Deformation in the Jura mountains (France): first results from semi-permanent GPS measurements. Earth and Planetary Science Letters, 245, 365– 372. W ESSEL , P. & S MITH , W. F. 1991. Free software helps map and display data. Eos, Transactions of the American Geophysical Union, 72, 441. W ORTEL , M. J. R. & S PAKMAN , W. 2000. Subduction and slab detachment in the Mediterranean –Carpathian region. Science, 290(5498), 1910– 1917. Y AMATO , P., A GARD , P., B UROV , E., L E P OURHIET , L., J OLIVET , L. & T IBERI , C. 2007. Burial and exhumation in a subduction wedge: mutual constraints from thermomechanical modeling and natural P– T– t data (Schistes Lustre´s, Western Alps). Journal of Geophysical Research, 112, B07410; doi: 10.1029/ 2006JB004441.

Block model versus thermomechanical model: new insights on the present-day regional deformation in the surroundings of the Calabrian Arc RAFFAELE SPLENDORE1*, ANNA MARIA MAROTTA1, RICCARDO BARZAGHI2, ALESSANDRA BORGHI3 & LETIZIA CANNIZZARO2 1

Universita` degli Studi di Milano, Department of Earth Sciences ‘Ardito Desio’, Section of Geophysics, L. Cicognara 7, 20129 Milan, Italy

2

DIIAR, Department of Environmental, Hydraulic, Infrastructures and Surveying, Engineering Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy 3

OGS-Trieste, c/o DIIAR, Department of Environmental, Hydraulic, Infrastructures and Surveying, Engineering Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy *Corresponding author (e-mail: [email protected]) Abstract: A finite-element thermomechanical model is used to analyse present-day crustal deformation in the surroundings of the Calabrian Arc. The major structural complexities of the Tyrrhenian area are taken into account, along with the rheological properties of the rocks resulting from a thermal analysis. A comparison between the results obtained from a model composed of three wide rheologically uniform blocks and those obtained from the thermomechanical model allows us to better constrain the geophysical assumptions and shed light on the roles of the different active mechanisms acting in the Tyrrhenian. Our comparative analysis enlightens the crucial role played by lateral rheological heterogeneities when deformation is analysed at short wavelengths of a few hundred kilometres of the Tyrrhenian, driving the observed diffuse SW–NE extension within the regional context of active Africa–Eurasia convergence. Furthermore, a x 2 analysis based on comparisons with GPS data confirms the hypothesis that a significant part of the Africa– Eurasia convergence is absorbed through the Calabrian subduction.

Present-day crustal deformation in the Tyrrhenian is the result of a complex interplay of various dynamic processes acting at both local (e.g. faults processes) and regional scales (e.g. Africa–Eurasia convergence and Calabrian subduction). A great variability in thermal regime, surface heat flow (Pollack et al. 1993; Artemieva & Mooney 2001; Artemieva 2006) and crustal thickness (Bassin et al. 2000; Tesauro et al. 2008) also characterizes the central Mediterranean and the Italian Peninsula, inducing horizontal heterogeneities in the rheology through the Tyrrhenian area. The combination of the different tectonic mechanisms and the rheological heterogeneities generates an enigmatic regional deformation pattern in which areas subjected to extension, such as the Tyrrhenian and Provencal basins, are embedded in areas subjected to compression and strong crustal thickening, such as the Alps and the Appennine. New advanced GPS (Global Positioning System) techniques and long geodetic time series make it possible to monitor this deformation with more

accuracy at different length and timescales. In addition, geodetic data better constrain the geophysical forward modelling (e.g. Jime´nez-Munt et al. 2003; Marotta et al. 2004; Negredo et al. 2004; Wang & Zheng-Ren 2006) used to test different geodynamic hypotheses in the area that were originally constrained using the geological global models NUVEL-1 or NUVEL-1A (DeMets et al. 1994; DeMets & Dixon 1999). The inadequacy of global models NUVEL-1 and NUVEL-1A in reproducing present-day relative plate motion has been widely demonstrated by several recent geodetic and numerical studies (among others, Sella et al. 2002; Fernandes et al. 2003; Nocquet & Calais 2003; Kremer et al. 2003; Marotta & Sabadini 2008). In Marotta & Sabadini’s (2008) comparative analysis, the authors show that the geodetically constrained tectonic models in the Tyrrhenian area predict a regional deformation that is in better agreement with the observed deformation compared to models constrained with geological data. However, one major limitation of Marotta and

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 129–147. DOI: 10.1144/SP332.9 0305-8719/10/$15.00 # The Geological Society of London 2010.

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Sabadini’s model is the lack of an appropriate coupling between the rheological properties of the lithosphere and the regional thermal field. One main advancement characterizes the present tectonic analysis with respect to Marotta & Sabadini (2008). This approach accounts for three wide rheological uniform units: the stiff East Europe platform (light grey in Fig. 1a); the European plate (intermediate grey in Fig. 1a); and the soft Mediterranean domain (dark grey in Fig. 1a). Marotta et al. (2004) show that a rheological averaged block model can reproduce the observed deformation at the long wavelengths ranging from several hundred to thousands of kilometres. However, when deformation is analysed at the shorter wavelengths of a few hundred kilometres, a deeper knowledge of lithosphere rheology is needed. Within a more localized analysis focused on the Tyrrhenian, Marotta & Sabadini (2008) showed, for example, that a further rheological differentiation between the Tyrrhenian and the Adria microplate (either as an unique block or subdivided into two blocks) is needed to predict at least part of the extensional component of the strain-rate tensor observed within the Tyrrhenian. To improve this analysis a detailed thermal analysis is performed in the present study with the aim of evaluating a rheological structure of the lithosphere, relative to the Mediterranean, that is more realistic than that in Marotta & Sabadini (2008). The results of the present analysis will be discussed in comparison to the models of Marotta et al. (2004) and Marotta & Sabadini (2008), referred to as the ‘block model’ for the remainder of this paper.

Numerical model Tectonic analysis Tectonic deformation in the central Mediterranean is obtained using a finite-element model based on the spherical thin-sheet approach developed by Marotta et al. (2004). The vertically integrated momentum equations    @ @ 2m uu þ ur @u @u    1 @ 1 @ @ m uu þ uf  uf cot u þ sin u @f sin u @f @u    @ 1 @ uu  uf  uu cot u cot u þ 2m @u sin u @f   gr R r @ 2 1 c ¼ c S (1) rm @u 2L

   @ 1 @ @ uu þ uF  uF cot u m @u sin u @F @u    1 @ 1 @ @ 2m uu þ uF  uF cot u þ sin u @F sin u @F @u   gr R r 1 @ 2 1 c S (2)  cot u ¼ c rm sin u @F 2L are solved within a 2D grid composed of linear triangular elements extending through Western Europe (Fig. 1a). uu, uf and ur are the velocity components along the colatitudes, longitude and the radius, u is the colatitude, f is the longitude, m¯ the effective viscosity of the lithosphere, S is the crustal thickness, L is the lithosphere thickness, rc and rm are the density of the crust and the mantle, g is the acceleration of gravity and R is the terrestrial radius. The crustal thickness variation used in the present analysis is obtained from a linear interpolation onto the adopted grid of model CRUST 2.0 (Bassin et al. 2000). Boundary conditions are expressed in terms of velocities. With the exception of the southern boundary of the model, boundary conditions are the same as for the best-fit model of Marotta et al. (2004): (1) at the western boundary of the model, located along the Mid-Atlantic Ridge, zero velocities are assumed (dashed thick line in Fig. 1a), in agreement with the concept that the effects of ridge push forces are negligible at the long wavelengths representative of Europe; (2) along the Aegean trench, trench suction forces (white arrows in Fig. 1a) are based on the geodetic velocities determined by McClusky et al. (2000) in the geodetic sites LOGO, LEON, OMAL, ROML and KAPT; (3) along the Arabian boundary zero velocities are assumed (white triangles in Fig. 1a); and (4) along the eastern boundary of the model, shear-stress-free conditions are assumed (white circles in Fig. 1a). For further details about the motivations for these boundary conditions, we refer the reader to the original paper, Marotta et al. (2004). The boundary conditions along the southern boundary of the model account for Africa–Eurasia relative motion based on ITRF2005 (Altamimi et al. 2007) (black triangles in Fig. 1a) and are calculated following the procedure in Nocquet et al. (2001) to estimate the Eulerian Pole. The geodetic stations PENC, BOR1, BRUS, VILL, ZIMM, POTS, HERS, TOUL, MADR and YEBE (black circles in Fig. 2) define the stable Europe (Devoti et al. 2002), and GOUG, SUTH, MAS1 (white circles in Fig. 2) define the stable Africa (McClusky et al. 2003). Table 1 lists the models considered in the analysis.

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Fig. 1. (a) Geometry and boundary conditions of the domain where the 2D numerical analysis has been performed. The study area is composed of three major blocks: the East European platform (light grey); the European domain (intermediate grey); and the Mediterranean (dark grey). The black triangles reflect the counterclockwise rotation of the African plate with respect to the European plate that is calculated in this study on the base of the ITRF2005 velocity solutions. Along all of the other boundaries the assumed boundary conditions are the same as the best-fit model of Marotta et al. (2004): zero Mid-Atlantic spreading (dashed thick black line); shear-stress-free conditions along the eastern boundary of the study domain (white circles); no-slip conditions along the southern boundary of the Anatolian region (white triangles); velocity along the Aegean trench based on McClusky et al. (2000) (white arrows). (b) Details of the 2D numerical grid used for the analysis. The arrows indicate the Africa– Eurasia convergence velocities as calculated in the present study along the SW portion of the study domain. See the text for further details.

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Fig. 2. Position of the IGS stations used in the geodetic analysis (white stars) and of the geodetic stations used to define stable Europe (black circles) and stable Africa (white circles) in the calculation of the Africa–Eurasia relative motion.

Table 1. List of the models considered in the present analysis Model Block_100% Block_75% Block_50% Block_25% Block_0% TM16_100% TM16_75% TM16_50% TM16_25% TM16_0% TM19_100% TM19_75% TM19_50% TM19_25% TM19_0%

Reference strain rate (s21) – – – – – 10216 10216 10216 10216 10216 10219 10219 10219 10219 10219

TM

x x x x x x x x x x

Block

Convergence percentage (%)

x x x x x

100 75 50 25 0 100 75 50 25 0 100 75 50 25 0

BLOCK MODEL V. THERMOMECHANICAL MODEL

Lithosphere temperature The thermal analysis is based on a 3D numerical model performed in the central Mediterranean (dark grey in Fig. 1a). The steady-state 3D thermal field is calculated by integrating the energy equation: r  (krT) þ rH ¼ 0

qr ¼ 0:6qs :

depending on the type of faulting and assumed to be equal to 3 for thrust faulting, 1.2 for strike-slip faulting and 0.75 for normal faulting (Ranalli & Murphy 1987). Ductile behaviour is controlled by the power law (Weertman & Weertman 1975):

(3)

on a 3D grid composed of prismatic elements obtained by projecting along the depth the 2D numerical grid used in the tectonic model (Fig. 3a). K is the thermal conductivity, T is the temperature, r is the density and H the rate of radiogenic heat production per unit mass. The model accounts for crust (dry felsic granulite) and mantle (dry dunite) up to a depth of 200 km (Fig. 3b). Parameter values are listed in Table 2. Dirichlet boundary conditions are prescribed at the upper boundary of the model, with the temperature fixed to 300 K. Neumann boundary conditions are prescribed at the lower boundary of the model, where the assumed heat flow coincides with the residual heat flow, qr, derived from the observed surface heat flow, qs [Pollack et al. 1993, augmented by Artemieva’s (2006) data] through the Pollack & Chapman (1977) formula:

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sD ¼ (sH  sV )D ¼

 1n   1_ Ea exp nRT 1_ 0

(6)

where 1˙ is the strain rate, and ranges between 10219 and 10216 s21; R is the universal constant of gas, T is the absolute temperature, and 1˙ 0, n and Ea are constant characteristics of the rocks (Table 2). The strength envelope for each type of faulting is determined as:

sy ¼ min{sB , sD }

(7)

and the effective viscosity at each element of the 2D numerical grid as:

meff

11 ¼ 1_ L

ðL

sy dy

(8)

0

where L is the thickness of the thermal model.

(4)

Zero heat flow is prescribed at the vertical sides of the model. The upper surface and the base of the crust are defined by interpolating the topography and the Moho depth from Bassin et al. (2000) at the nodes of the 3D numerical grid. While the base of the crust is physically defined, the base of the lithosphere is defined thermally by the isotherm 1600 K resulting from the integration of equation (3) with the above specified boundary conditions.

Data Crustal thickness Figure 4 shows the crustal thickness variation used in the thermal and tectonic analysis for the Mediterranean and central Europe, based on the CRUST 2.0 model (Bassin et al. 2000). The area is characterized by strong lateral variation with a thin crust in the Provencal Basin, Tyrrhenian Basin and north of the Pannonian Basin. A thick crust characterizes the Alps, the north of the Adriatic Sea and the Ellenic arc.

Lithosphere strength The lithosphere strength is calculated by assuming that rocks behave like a brittle or ductile material according to their composition and thermal state. For the brittle behaviour, a linear failure criterion is assumed (Ranalli & Murphy 1987), as expressed for the dry rheology assumed in the present analysis in the form:

sB ¼ (sH  sV )B ¼ br rg

(5)

where r is the depth along the terrestrial radius, r is the density of the material (2800 kg m23), g is the acceleration of gravity, and b a parameter

Heat flow Figure 5 shows the surface heat flow used in the thermal analysis, obtained by interpolating onto the 2D numerical grid the discrete database of Pollack et al. (1993), augmented by Artemieva (2006) in the areas with no data. Black circles indicate the data of Pollack et al. (1993). The thin southern Tyrrhenian is characterized by rather high values of surface heat flow, higher than 100 mW m22, as occurs also in the north of the Pannonian Basin. Rather low values of heat flow characterize the north Adriatic and the Ionian Sea. Owing to the regional character of the present study, the

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Fig. 3. (a) Scheme of the 3D numerical grid used for the thermal analysis. The assumed linear prismatic elements are obtained by projecting the 2D grid along the depth. The vertical axis is not to scale. (b) Structural sketch of the 3D model.

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Table 2. List of the parameters used in the thermorheological analysis Lithology

1˙ 0 (Pa2n s21)

n

Ea ( j mol21)

Crust Dry felsic granulite 20.095  10222* 3.1* 243  103* Mantle Dry dunite 6.310  10217† 3.41† 444  103†

r (kg m23)

H (W kg21)

K (W m21 K21)

2800 3200

7.5  10210 6.3  10213

2.55 4.15

*Afonso & Ranalli (2004); †Chopra & Peterson (1981). 1˙ 0, exponent of the power-law creep equation; n, pre-exponential factor of the power-law creep equation; Ea, activation energy.

extremely high values of heat flow observed in the Larderello (450–700 mW m22) and Stromboli (250 –822 mW m22) areas have been filtered before the interpolation.

area. Figure 6 shows the maximum principal horizontal stresses in the study area, highlighting areas subjected to extension (white bars) within the regional context of active Africa– Eurasia convergence.

Stress data Numerical results will be discussed in terms of predicted velocities and tectonic deformation. In this last aspect, the most recent World Stress Map 2008 compilation (Heidbach et al. 2008) offers new insights on the present-day enigmatic regional deformation pattern characterizing the Tyrrhenian

The geodetic dataset Continuous GPS (CGPS) data of the most long-lived permanent stations in the surroundings of the Calabrian Arc have been taken into account to compare the tectonic regional deformation model with geodetic observations. The geodetic network consists

Fig. 4. Crustal thickness variation used in both the thermal and mechanical analyses, and obtained by linear interpolation onto the adopted grid of model CRUST 2.0 (Bassin et al. 2000).

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Fig. 5. Surface heat flow variation used in the analysis and obtained by linear interpolation onto the adopted grid of Pollack et al. (1993), augmented by the database of Artemieva (2006). Black circles represent the original data of Pollack et al. (1993).

of 15 permanent GPS stations, which are plotted in Figure 7. For these stations, daily co-ordinate time series have been estimated; they span a variable year number, ranging from almost 10 years to a minimum of 3 years. The GPS data analysis has been carried out with BERNESE Software v.5.0 (Dach et al. 2007), using a standard processing strategy: troposphere parameters have been estimated on an hourly basis and wet delays have been modelled as stochastic parameters using the Dry-Niell mapping function; the ionosphere disturbance has been treated using global ionosphere models by CODE (Hugenobler et al. 2000) in the L1&L2 ambiguity estimation step and using the iono-free observations (L3) in the co-ordinate computation. The Quasi Iono Free (QIF) strategy was selected for ambiguity fixing. According to the IGS (International GNSS Service) standard, absolute PCV (Phase Centre Variation) parameters have been used both for receiver and satellite antennas. The reference frame of the daily co-ordinates is consistent with IGS precise orbits and IGS Earth

rotation parameters. The IGS stations used to fix the frame are WTZR, ZIMM, GRAS, VILL, NICO, EBRE and ANKR (white stars in Fig. 2) depending on the current reference frame realization. Station co-ordinates have all been framed in IGS05 using the IGS transformation parameters at the end of the processing analysis. The velocity estimation (empty arrows in Fig. 7) has been determined by least-square adjustment of individual daily co-ordinate components; the applied functional model consists of linear trends and periodic components (Blewitt & Lavalle´e 2002; Dong et al. 2002; Nikolaidis 2002; Ray et al. 2008). Possible discontinuities owing to reference frame or antenna– receiver changes have also been estimated and reduced. The noise characteristics of the GPS time series have been studied following the Empirical Covariance Function method, as described in Cannizzaro (2008). Black arrows in Figure 7 indicate the same velocity solutions once they have been transformed into the reference frame of the geophysical models.

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Fig. 6. Maximum principal horizontal stresses in the Italian Peninsula, based on the World Stress Map 2008 compilation (Heidbach et al. 2008). NF, normal faulting; TF, thrust faulting.

Results and discussion Lithosphere temperature The lithosphere thermal field predicted in the study area is represented in Figure 8, where the temperature at the base of the crust (Fig. 8a) and the depth of the 1600 K isotherm, assumed to be the base of the thermal lithosphere (Fig. 8b), are shown. It is worth noting the different pattern of the thermal field at the crustal and lithosphere levels. Concerning the temperature at the base of the crust (Fig. 8a), steep gradients are observed throughout the study area, with low temperatures (less than 900 K) predicted at the base of the thin Tyrrhenian crust, and high temperatures (greater than 1000 K)

predicted east of the Apennine. Conversely, smooth variations characterize the depth of the base of the lithosphere (Fig. 8b), with an average lithosphere thickness (neglecting the topography) ranging between 70 and 90 km. The strongest gradients occur south of the Calabrian Arc, in proximity to the trench associated with the Calabrian subduction, where a thermal doubling occurs within 500 km.

Lithosphere strength The predicted 3D lithosphere thermal field is used to determine the strength of the lithosphere in the Mediterranean region, according to (5)–(7). For

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Fig. 7. Geodetic velocity solutions (white arrows) at the permanent stations MATE, NOT1, AQUI, MILO, MSRU, USIX, SVIN, SERS, CUCC, CDRU, ENAV, LAMP, MALT, VAGA and TOLF, obtained by analysing their daily position solutions. Black arrows indicate the same solutions after their transformation into the reference frame of the geophysical models.

ductile behaviour, dry felsic granulite is assumed for the crust and dry dunite is assumed for the lithospheric and sublithospheric mantle (Table 2). Figure 9 shows the vertical strength envelopes at six reference sites distributed in the study area (A– F in Fig. 8a) obtained for two reference values of the strain rate (10216 s21, right-hand side, and 10219 s21, left-hand side, of each panel), and for thrust (dark grey), normal (light grey) and strike-slip (intermediate grey) faults. The black dashed lines represent the corresponding vertical temperature profiles. In agreement with the thermal field, a strong lithosphere paves the Tyrrhenian, with a strong coupling between crust and mantle at the Provencal Basin (site A, Fig. 9a) and south of the Calabrian Arc (site C, Fig. 9c). A relatively softer lithosphere characterizes the Marsili Basin (site B, Fig. 9b) north of the Calabrian Arc, where a local thermal rise occurs. Lithosphere strength profiles predicted at sites D– F (Fig. 9d –f, respectively) are almost coincident. The concurrence of hot lithosphere and thick crust in the eastern portion of the model drives an average soft lithosphere, with a negligible mantle contribution to the total strength as a peculiar feature.

The effective lithosphere viscosity results from the integration of the vertical strength envelopes. For a given lithosphere thermal field, three elemental distributions of effective viscosity are obtained, corresponding to the three types of faulting regime. Thus, Figure 10 shows the variation of lithosphere effective viscosity for thrust (panels ai), strike-slip (panels bi) and normal (panels ci) regimes, and for the two values of reference strain rate, 10219 s21 (panels a1, b1 and c1) and 10216 s21 (panels (a2, b2 and c2). Changing the reference strain rate from 10219 to 10216 s21 induces a global lithospheric softening of almost 3 orders of magnitude. Each tectonic model initially assumes the effective viscosity distribution to be that corresponding to strike-slip faulting; then, it iterates and progressively changes the effective viscosity of each element according to its new strain regime. Figure 11 shows the effective viscosity predicted in the central Mediterranean by a model accounting for reference strain rates of 1˙ ¼ 10216 s21 (Fig. 11a) and 1˙ ¼ 10219 s21 (Fig. 11b). In both cases the Apennine defines a transition from a relatively soft lithosphere, extending from the eastern Alps through to the Pannonian Basin, to a stiffer

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Fig. 8. (a) Predicted temperature at the Moho depth. (b) Predicted depth of the 1600 K isotherm (assumed base of the thermal lithosphere). Capital letters in panel (a) indicate the six sites where the strength envelopes shown in Figure 9 are calculated.

lithosphere, below the Tyrrhenian Sea, where changes of up to 1.5 orders of magnitude occur in the effective viscosity. At shorter wavelengths our rheological analysis supports the rheological

differentiation between north and south Adria (e.g. Oldow et al. 2002), with, in our case, the northern block stiffer by about half an order of magnitude than the southern one (Fig. 11a).

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Fig. 9. Predicted lithosphere strength envelopes at the six sites shown in Figure 8a obtained assuming a reference strain rate of 10216 s21 (right-hand side of each panels) or 10219 s21 (left-hand side of each panel), and for thrust (dark grey), normal (light grey) and strike-slip (intermediate grey) faults. Black dashed lines represent the corresponding vertical temperature profiles.

Tectonic deformation Figure 12 compares the strain rate predicted by tectonic model TM16 using the effective viscosity of Figure 11a to the tectonic strain rate predicted by an equivalent tectonic block model in which a rheologically uniform lithosphere is assumed below the Mediterranean, with an effective viscosity of 1025.5 Pa s (Fig. 11b). The block model predicts compression through the study area, with the strongest gradients localized along a longitudinal stripe

between 378 and 408 of latitude and dominant SE –NW eigen-directions (Fig. 12b), according to the Africa–Eurasia convergence direction. Rather small and uniform deformation is predicted at the higher latitudes. Similar to the block model, the TM16 model predicts compression at intermediate latitudes; however, the lateral rheological heterogeneities concur with the variation in lateral crustal and lithosphere thicknesses to induce strong lateral gradients in the deformation field that also shows strong compression at the high

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Fig. 10. Variation of lithosphere effective viscosity obtained for the thrust (panels ai), strike slip (panels bi) and normal (panels ci) regime, and for a reference strain rate of 10219 s21 (panels a 1, b 1 and c 1) and 10216 s21 (panels (a 2, b 2 and c 2).

latitudes. Furthermore, a slight counterclockwise rotation of the compressive component of the strainrate eigenvectors occurs at the intermediate latitudes. Another peculiar feature that differentiates the deformation patterns predicted by the two models is the diffuse SW –NE extension, as intense as the compression, predicted by the TM16 model in the region below 388 of latitude (Fig. 11a). This last result is in agreement with the evidence of extension within the regional context of active Africa–Eurasia convergence, as evidenced by the World Stress Map 2008 compilation (Fig. 6). Despite these promising results, the TM16 model has the tendency to underestimate the extensional component of the strain-rate tensor, in particular at the intermediate latitudes. Trying to overcome this limitation and to improve, in particular, the extension in the southern Tyrrhenian, we embrace Marotta & Sabadini’s (2008) hypothesis that partially ascribes the underestimation of the extension to the absence of subduction in the

thin-sheet tectonic model. We therefore implement new models in which only 75, 50, 25 and 0% of the estimated Africa–Eurasia convergence is transmitted to the Eurasian plate through the Calabrian subduction zone. Numerically, this corresponds to a progressive decrease in the magnitude of the prescribed velocities at the nodes of the numerical grid delimiting the Calabrian trench, represented by the black arrows in Figure 1b. Figure 13 shows the deformation pattern predicted by the model TM16_50, using 50% of Africa –Eurasia convergence. This model moves the extension–compression interface further to the north, and enhances extension throughout Sicily and south of Calabria. Furthermore, a SE–NW compression appears in the Algerian region, as indicated by the World Stress Map 2008 (Heidbach et al. 2008). TM16_50 is the model that best reproduces the regional tectonic deformation in the Mediterranean; this role is also demonstrated by a direct comparison

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Fig. 11. Effective viscosity predicted for a reference strain rate of (a) 10216 s21 and (b) 10219 s21.

between observed and predicted velocities at 15 permanent geodetic stations distributed around the Calabrian Arc. Predicted velocities (coloured arrows) for all the analysed models of Table 1 are shown in Figure 14 compared to the velocities from the geodetic measurements (black arrows).

One major comment is that all of the block-type models predict velocities apparently in great disagreement with the geodetic velocities, both in magnitude and azimuth (Fig. 14a). The fit from the block model gets worse with the progressive decreasing of the per cent of Africa– Eurasia convergence allowed

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Fig. 12. Strain-rate regime (colour map, with red indicating compression and blue extension) predicted by (a) model TM16 and the (b) block model, including 100% of the Africa–Eurasia convergence transmitted to the Eurasian plate through the Calabrian subduction zone.

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Fig. 13. Strain-rate regime (colour map, with red indicating compression and blue extension) predicted by model TM16_50, including 50% of the Africa–Eurasia convergence transmitted to the Eurasian plate through the Calabrian subduction zone.

to be transmitted through the Calabrian subduction; this induces a progressively larger disagreement, in both magnitude and azimuth, of the predicted velocities and the GPS velocities. Models that take into account the temperature dependence of viscosity show a better agreement in magnitude (e.g. at AQUI and MATE) and a greater variation in the azimuth with the variation of the per cent of Africa–Eurasia convergence transmitted through the Calabrian subduction, which makes model TM16_50% the best-fitting model. This conclusion is further strengthened by a x 2 analysis. As the CGPS velocities have been

estimated for each component of each station independently, the quadratic form of the n normalized horizontal velocity residuals is chi-square distributed, with n degrees of freedom, Uþ C1 U ¼ xn2 where U is an n-dimensional vector formed by the difference values between the horizontal GPS velocities (vN, vE) after a datum shift to ITRF2005 and geophysical model velocities, and C is the diagonal matrix containing the velocity variances.

BLOCK MODEL V. THERMOMECHANICAL MODEL

Fig. 14. Predicted (coloured arrows) v. GPS velocities (black arrows). The per cent represents the amount of Africa– Eurasia convergence transmitted through the Calabrian subduction.

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Final remarks

Fig. 15. Classification of the geophysical models obtained with the x 2 analysis. The per cent indicates the percentage of the Africa– Eurasia convergence transmitted to the Eurasian plate through the Calabrian subduction zone.

The geophysical models are qualitatively classed by using this statistic; the lower the x 2 value, the better the agreement between the tested model and the geodetic data. Figure 15 shows the final classification of all the geophysical models using the statistic x 2 test. A first general observation can be made by focusing on the effects of a local reduction of the convergence due to Calabrian subduction; the lower the percentage, the better the agreement between model and observation. The block-type model shows the best agreement with the data only when 100% of Africa –Eurasia convergence is included. A decrease in the per cent of Africa– Eurasia convergence transmitted through the Calabrian subduction does not significantly affect the block model, as shown in Figure 14a and Figure 15 (dotted line). The model that exhibits the best agreement with observation is the TM16_50 model, in which the assumed effective viscosity is obtained from a reference strain rate of 10216 s21 and 50% of Africa–Eurasia convergence is transmitted though the Calabrian Arc. From about 95% on the thermomechanical model TM16 is the best-fit model, within any configuration. Note the poor quality of thermomechanical model TM19; it is always the worst in comparison with the block and TM16 models. It is worth noting that for all types of models, the x 2 minimum value is reached for a reduction of 50% of Africa –Eurasia convergence, confirming the hypothesis that a significant part of the Africa –Eurasia convergence is absorbed through the Calabrian subduction, as proposed by Marotta & Sabadini (2008).

A thermorheological analysis is performed to study the rheological structure of the lithosphere in the Mediterranean. Our results show that a strong lithosphere paves the Tyrrhenian, with a crust strongly coupled with mantle below the Provencal Basin and the Calabrian Arc surroundings, and the concurrence of hot lithosphere and thick crust in the Pannonian Area drives an average soft lithosphere. Furthermore, the southern portion of the Adria microplate can be rheologically differentiated from the northern portion, with the northern block stiffer than the southern one by about half an order of magnitude. Once the predicted lithosphere stiffness is included within a tectonic model, the results confirm the crucial role played by the lateral rheological heterogeneities when deformation is analysed at the short wavelengths of a few hundred of kilometres. In fact, strong rheological gradients concur with crustal and lithosphere thickness variations to drive a diffuse SW– NE extension within the regional context of active Africa–Eurasia convergence. In particular, tectonic model TM16_50, accounting for 50% of Africa–Eurasia convergence transmitted through the Calabrian subduction zone, predicts extension in Sicily, southern Calabria and part of the southern Tyrrhenian, as well as compression in the Algerian region, as shown by the World Stress Map 2008 compilation. A x 2 analysis confirms this hypothesis that a significant part of the Africa – Eurasia convergence is absorbed through the Calabrian subduction, as already proposed by Marotta & Sabadini (2008). This work was supported by the Italian Space Agency project SISMA. The authors thank the editor and the reviewers I. Jime´nez-Munt and U. Bayer for their constructive criticisms. All figures have been created by GMT plotting software (Wessel & Smith 2001).

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The role of mantle hydration in continental crust recycling in the wedge region MARCO MEDA1,2, ANNA MARIA MAROTTA3 & MARIA IOLE SPALLA4,5* 1

Dipartimento di Scienze della Terra ‘A. Desio’, Universita` degli Studi di Milano, Milano, Italy 2

Present address: GEBA Basin Geology Department, ENI E&P, San Donato Milanese, Italy

3

Dipartimento di Scienze della Terra ‘A. Desio’, Universita` degli Studi di Milano, Sezione di Geofisica, Via Cicognara 7, 20129 Milano, Italy

4

Dipartimento di Scienze della Terra ‘A. Desio’, Universita` degli Studi di Milano, Sezione di Geologia, Via Mangiagalli 34, 20133 Milano, Italy 5

C.N.R. – I.D.P.A., Sezione di Milano, Via Mangiagalli 34, 20133 Milano, Italy *Corresponding author (e-mail: [email protected])

Abstract: In orogenic belts high pressure– low temperature (HP–LT) metamorphism can widely affect units derived from both the oceanic and the continental lithosphere. In order to verify whether high P/T (pressure/temperature) ratios recorded in the continental lithosphere can result from tectonic erosion, ablative subduction and recycling in the mantle wedge, we implemented a 2D numerical model to simulate oceanic subduction beneath a continent. Particular attention is paid to the role played by mantle hydration within the continental crust recycled in the wedge region. A comparison between hydrated and non-hydrated models highlights that hydration is fundamental in allowing the recycling of crustal material at shallow depths (150 km for a convergence rate of 1 cm year21), making the uprising and exhumation of buried crustal material during active subduction possible. The recycled crustal material can originate from any crustal level. The Tmax and Pmax distributions within the final marker configuration show that crustal recycling induces the coupling of volumes that reached different depths during their paths in the corner flow. To verify the reliability of this model we compare predictions with natural geological data from the Austroalpine Sesia– Lanzo Zone (SLZ), the largest eclogite-facies crustal fragment of early Alpine age and whose Alpine tectonic evolution has been interpreted as compatible with a cycle of burial at depth and exhumation during active subduction of the oceanic lithosphere. The relationships between natural P– T estimates and predicted P– T values show that the simulated geodynamic scenario generates a thermal regime coherent with that affecting the subducted continental crust of the SLZ, which may have been stable for a long time during Alpine subduction, allowing the SLZ rocks to accomplish their burial and exhumation path under an active subduction regime.

Subduction reorganizes the initial structural and metamorphic configuration of the lithosphere along the active margins, and where the tectonic system involves an oceanic subduction underneath a continental plate edge, continental crustal units, together with lithospheric mantle, may constitute the orogenic belt (e.g. Cowan & Silling 1978; Cloos 1982, 1984, 1985; Platt 1986; Polino et al. 1990; Ernst 2001; Rondenay et al. 2008). This crust– mantle mixing is allowed by tectonic erosion and ablative subduction, which, during the precollision phase, drag crustal material of continental type to remarkable depths (over 100 km), where they undergo high pressure –low temperature (HP –LT) or ultra-high pressure– low temperature (UHP– LT) metamorphic transformations. The dynamics

developing in the wedge area drives the exhumation of buried continental units, which will be successively involved in the continental collision, eventually preserving signatures of their deep evolution. The subducting oceanic lithosphere and the convection in the surrounding mantle should induce the vigorous circulation characterizing corner flow in the mantle wedge, confined above the subducted plate (Turcotte & Schubert 1982); the occurrence of such a flow has been inferred by seismic imaging after earthquakes in the Japan subduction zone (Nakajima & Hasegawa 2004). Mantle partial melting, feeding the magmatic activity of the volcanic arc, and diffuse mantle hydration, by means of fluids evolved during dehydration reactions in the subducted oceanic lithosphere, characterize this

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 149–172. DOI: 10.1144/SP332.10 0305-8719/10/$15.00 # The Geological Society of London 2010.

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mantle volume. In particular, mantle hydration might promote a dramatic viscosity reduction (Billen & Gurnis 2001). Hydration also causes a density and consequent pressure decrease that, in turns, sucks in more fluids. The existence of a low viscosity wedge (LVW) has been supported by the uppermantle low viscosity (1019 Pa s) inferred in NE Japan, underneath an elastic 30 km-thick continental crust (Honda & Saito 2003). Water acquired by the oceanic slab, during hydrothermal alteration in the ridge –transform system or by hydration of normal faults near the trench, may represent 2.5–6.5% by weight of the crust and the lithospheric mantle, respectively (Schmidt & Poli 1998). A further amount is added to the water budget of the subducting oceanic lithosphere by the pelagic sediments, rapidly accreted to the sedimentary prism and dragged into the subduction zone. Slab dehydration during subduction starts with the breakdown of serpentine and chlorite in mantle rocks, and of amphibole and lawsonite in the oceanic crust (e.g. Tatsumi & Eggins 1995; Schmidt & Poli 1998; Maruyama & Okamoto 2007). Their effects on the mantle wedge dynamics and the consequent hydration of the overlying lithospheric plate have been modelled by Arcay et al. (2005), demonstrating that the wedge geometry and amount of water conveyed in the deep mantle, with related viscosity changes, depend on the subduction rate. Mantle wedge serpentinization during subduction has been considered to play a dominant role in the exhumation of eclogites (Guillot et al. 2001). In active subduction systems the seismic tomography and gravimetric anomalies point out localized mantle upwelling, compatible with small-scale convective cells, supporting the existence of an active corner flow able to move large lithospheric volumes (e.g. NE Japan: Honda & Saito 2003). The movement of lithospheric slices and the role of interactions between hydration –dehydration and viscosity variations have recently been simulated numerically to explore exhumation paths of highpressure rocks in a subduction channel (Gerya et al. 2002; Stoeckhert & Gerya 2005; Gerya & Stoeckhert 2006; Yamato et al. 2007; Gorczyk et al. 2007; Warren et al. 2008). In the Western Alps HP–LT metamorphism widely affects not only units derived from the Thethyan oceanic lithosphere (i.e. ophiolites and related sedimentary sequences), but also large volumes of pre-Alpine continental crust, among which the Austroalpine Sesia–Lanzo Zone (SLZ) is the largest fragment with the Late Cretaceous (early Alpine) eclogite-facies imprint; its Alpine evolution has been interpreted as compatible with a cycle of deep burial and exhumation during the active subduction of oceanic lithosphere (e.g. Spalla et al. 1996 and references therein). This

tectonic interpretation and the peculiar high P/T ratio of this slice of continental crust inspired us to verify whether the SLZ evolution could be due to tectonic erosion, ablative subduction and recycling in the mantle wedge by means of a 2D numerical model simulating an oceanic subduction beneath a continent (Marotta et al. 2006; Marotta & Spalla 2007; Spalla & Marotta 2007). We analysed the effects induced by the change in rheological parameters, boundary conditions and geometries, paying particular attention to the ‘corner-flow’ area, where the continental crust of the upper plate can be involved in deep recycling.

Modelling The physics of the crust –mantle system during subduction is described by the coupled equations for continuity, conservation of momentum and energy, expressed in the form: rv¼0 @tij @p ¼  rg @xj @xi   @T þ v  rT ¼ r  (KrT) þ rH rc @t

(1) (2) (3)

respectively, where r is the density, v the velocity, p the pressure, g the gravitational acceleration, tij the deviatoric stress tensor, c the thermal capacity at constant pressure, T the temperature, K the thermal conductivity and H is the heat production rate per unit mass. Equations (1)–(3) are solved by means of the 2D finite-element code SubMar (Marotta et al. 2006). This numerical code uses the penalty function formulation to integrate the equation for the conservation of momentum and the streamline upwind/ Petrov –Galerkin method to integrate the equation for the conservation of energy. This code has been modified to account for the mantle hydration mechanism in the wedge area, as described in the next paragraph. For further details about the code the reader is referred to the original paper (Marotta et al. 2006). The crust is compositionally differentiated from the mantle via the Lagrangian particle technique (e.g. Christensen 1992), as in Marotta & Spalla (2007) and Spalla & Marotta (2007). In each time step, the non-dimensional function C describing the elemental composition is calculated on the basis of the density of the oceanic or continental crust particles in the element, such that C ¼ 1 2 C o 2 C c, with C o and C c being the densities of the oceanic and continental crust particles in the element,

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respectively. C, C o and C c vary from 0 for pure mantle to 1 for pure crust (either oceanic or continental). An incompressible viscous fluid is assumed with a temperature and composition-dominated viscosity and density, and is calculated for each element as follows:

to a fixed convergence velocity, imposed along the top of the subducting oceanic plate, and to no shear conditions along the other sides of the model. The positions of the individual particles (markers) during the dynamic evolution of the system are calculated by solving the equation:

m ¼ mm [1  Co  C c ] þ mo C o þ mc C c

dx=dt ¼ v

with: Eai (C) 1 1 T  T0

mi ¼ mi0 (C)e ni R




and

r ¼ r0 [1  a(T  T0 )] þ Do rC o þ Dc rC c where m is the elemental viscosity, the index i stands for mantle (i ¼ m), continental crust (i ¼ c) and oceanic crust (i ¼ o), mi0 is the reference viscosity at the reference temperature T0, Eai the activation energy, T the temperature, a the thermal expansion factor and n i is the exponent of the power-law creep equation. We assume a viscous behaviour for the whole system, and do not account for elasticity and brittle behaviour at the shallow levels of the lithosphere. As only a viscous rheology is considered, the crust is expected to remain strong at much higher stresses than the brittle limit and we introduce a viscosity cut-off value of 1025 Pa s to minimize this effect (Marotta & Spalla 2007). Complexities such as phase transitions, viscous dissipation and adiabatic heating are not taken into account. The 2D domain where the numerical solution was performed is 1700 km wide and 600 km deep (Fig. 1a). An irregular numerical grid was used, composed of 6300 six-node quadratic triangular elements, with a denser nodal distribution in the wedge area. The size of the elements varies, both horizontally and vertically, from 5 km in the wedge area to 30 km in the lower left border of the domain. While the base of the crust is defined compositionally, the base of the lithosphere is defined thermally by the isotherm 1600 K. For the set of parameters used in our model (Table 1), the temperature of 1600 K corresponds to the maximum horizontal average temperature of the upper thermal boundary layer of the steady-state convective cell. Boundary conditions are specified in Figure 1b. The thermal boundary conditions correspond to fixed temperatures at the top and bottom of the model domain, at 300 and 1600 K, respectively, and to zero thermal flux imposed at the vertical side walls. The velocity boundary conditions correspond

using a first-order Runge –Kutta scheme, with x and v indicating the position and velocity of each particle; the last one is evaluated by interpolating the velocities of the six nodes delimiting the element containing the marker through the same shape functions used in the numerical approximation. Numerical simulations last 70 Ma, with the beginning of subduction corresponding to 95 Ma absolute age, to facilitate the comparison between model predictions and natural data from the SLZ. A convergence velocity of 1 cm year21 is assumed to make the simulation compatible with the subduction rates involving the SLZ rocks, estimated as 0.7–1 cm year21 on the basis of radiometric and petrological data (Rubatto 1998; Rubatto et al. 1999).

Model results Table 2 lists all of the numerical experiments performed. In a system where hydration of the wedge area is not considered, during the first 10 Ma of active convergence, ablative subduction is observed, favoured by a large-scale convective cell below the subduction plane (Fig. 2a). Isotherms deflect downwards, as occurs in a typical subduction zone. After 15 Ma (Fig. 2b) a small-scale convective cell develops in the wedge area. This pattern persists until late simulation times, when a significant horizontalization of the subducted slab also occurs (Fig. 2c, d). For all of the analysed models no crustal markers are involved in the small-scale mantle circulation, allowing exhumation of the crust at the surface. We tried to overcome this model limitation by accounting for the possibility of mantle hydration in the wedge area. A rigorous implementation of the hydration process would require numerous mechanisms to be taken into account, such as initial fluid diffusion from the subducting oceanic crust, compaction and diagenesis of the sediments, collapse of the primary and secondary porosity of basaltic rocks, and dehydration due to destabilization of hydrated phases, all at different depths and in different proportions. It would be difficult to calibrate all of these mechanisms because they are poorly supported experimentally. Thus, we implemented

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Fig. 1. (a) 2D geometry and numerical set-up of the model. (b) Thermal and velocity boundary conditions used for active oceanic subduction. Modified after Spalla & Marotta (2007).

numerical models to account for the only consequence of the presence of a hydrated portion in the wedge area, in terms of variations of viscosity of a few orders of magnitude with respect to the surrounding mantle (Arcay et al. 2005). We tested different geometries for the area where hydration occurs, with a closed, right and open configuration (Fig. 3a –c, respectively). Previous studies (e.g. Gerya et al. 2002) indicate that the closed configuration is preferred when the hydration mechanism is not implemented directly. When the hydration mechanism is implemented rigorously (e.g. Arcay et al. 2005) the geometry of the hydrated area is

very similar to that associated with the right configuration. Our numerical experiments indicate that the open geometry allows a huge amount of crustal markers to be involved within the small-scale circulation of the wedge area and, thus, to reach shallow depths. Several tests have been performed to define the dimensions of the hydration wedge area. Few data are available and many of them are coarse. Regarding the vertical extent, many authors (e.g. Maruyama & Okamoto 2007) agree that some minerals can issue fluids up to great depths (e.g. 200 km for phlogopite and 300–400 km for lawsonite). For the horizontal

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Table 1. Properties and rheological parameters used in the numerical analysis

Composition

r H K

Mean densitya (kg m23) Heat generationb Thermal conductivityb

Ea m0 n

Rheology Activation energy Pre-exponential factor Exponent of the powerlaw creep equation

Oceanic crust

Continental crust

Mantle

7% basalt þ 16% dolerite þ 77% gabbros

66% gneiss þ 34% granite

100% dunite

2961 (W m23) 0.4  1026 (W m21 kg21) 2.10 (kJ mol21) (Pa2n s21)

dry diabased 260 8.04  10225 3.04

2640 2.5  1026 3.03

3200 0.2  1028 4.15

dry granitec 123 7.92  10229 3.02

dry dunitee 444 6.31  10217 3.41

a

Dubois & Diament (1997) and Best & Christiansen (2001). Rybach (1988). c Ranalli & Murphy (1987). d Kirby (1983). e Chopra & Paterson (1981). b

extent of the hydration zone, many authors (e.g. Gerya et al. 2002) define the maximum distance from the trench at about 150 km, on the basis of estimations of the amount of fluids released by the oceanic crust. In the present analysis the dimensions of the hydration area are dynamically growing with the progress of oceanic subduction. The most horizontally advanced crustal marker is considered at each iteration to define the instantaneous horizontal dimension of the hydration area, with a limit of 150 km. The lower limit of the hydration area is defined by the surface of the subducting slab, approximated by a cubic function. Using this approach the geometry of the hydration area resembles a right triangle, with a curved hypotenuse. Our approach implies that the growth of the hydration area is coeval to the advancement of the subducting oceanic crust. Previous studies (e.g. Arcay et al. 2005) have shown that the time span between the advancement of the subducted oceanic crust and the onset of hydration is rapid, making our approximation reasonable. Regarding the effects of hydration, in our analysis we consider only the consequent variation of the rock viscosity. Gerya et al. (2002) demonstrated that peridotite viscosity decreases when hydration occurs. In our analysis, the hydrated mantle viscosity ranges between 1018 and 1020 Pa s; the viscosity of the hydrated oceanic crust is fixed to 1019 Pa s, an index of a strength decrease due to fracturing of the surface basaltic material, while the viscosity of the continental crust within the wedge area, from 30 km upwards, is allowed to vary between 1020 and 1022 Pa s. Within the hydrated area, viscosity becomes independent of temperature.

Figure 4 shows predictions of the hydrated model described above. During the first 10 Ma of simulation, non-hydrated and hydrated models evolve in a similar way (Figs 2a & 4a, respectively), with ablative subduction and large-scale convection below the subducting plate. After 15 Ma the hydrated model shows that the small-scale convection in the wedge area involves previously subducted continental markers in upwelling trajectories (Fig. 4b). In addition, with respect to the nonhydrated model, over longer periods the hydrated model shows a significantly lower temperature in the wedge area and a more vertical subducted slab (Fig. 4c, d). In order to estimate the amount of continental crust that is exhumed after subduction, we evaluated the recycled percentage in terms of the ratio between the total number of markers buried below 30 km depth (crustal base at the beginning of the evolution) and the number located or ascended at different depths at the end of the evolution (Fig. 5). The ascended markers are identified on the basis of a difference 0.05 GPa between the maximum pressure and the pressure at the end of the simulation. As expected, for the non-hydrated models the percentage of recycling is zero as the continental crustal material, once buried, is not exhumed again. Hydrated models, on the other hand, favour crustal exhumation, with a maximum recycling percentage of about 40% after 70 Ma of evolution. Figure 6 shows the variation in time of the distribution of crustal markers, with particular attention paid to the wedge area. For a better understanding of the viscous flow dynamics, different colours have been used to distinguish continental markers originating from different crustal levels: red for initial

Table 2. List of the numerical experiments performed with their peculiar characters. In each section, the complete set of boundary conditions is listed only for a reference model (1, 15 and 1  1 idrataz); for the other models only the variations with respect to the reference are specified Dimension length (km)  depth (km)

Name

1

1400 3 700

Subduction angle

45

Subduction velocity (cm/y)

Thermal boundary conditions

5

Upper side: T ¼ 300 K Bottom side: T ¼ 1600 K Lateral sides: zero flux

Oceanic subducting plate

Bottom side

Left side

Right side

Vx ¼ 0 Vy ¼ 0

Vx ¼ 5 Vy ¼ 0

txy ¼ 0 Vy ¼ 0

Vx ¼ 0 txy ¼ 0

Vx ¼ 0 txy ¼ 0

6.4

60.0

3

40.0

txy ¼ 0 Vy ¼ 0

4.4

dynamic

1

Vx ¼ 1 Vy ¼ 0

10.0

6

dynamic

3

Vx ¼ 3 Vy ¼ 0

10.0

7

dynamic

10.0

8

2

Vx ¼ 2 Vy ¼ 0

9.7

9

1

Vx ¼ 1 Vy ¼ 0

5.5

txy ¼ 0 Vy ¼ 0

6.9

11

1

txy ¼ 0 Vy ¼ 0

Vx ¼ 1 Vy ¼ 0

12

1

txy ¼ 0 Vy ¼ 0

Vx ¼ 1 Vy ¼ 0

13

1

txy ¼ 0 Vy ¼ 0

Vx ¼ 1 Vy ¼ 0

14

1

txy ¼ 0 Vy ¼ 0

Vx ¼ 1 Vy ¼ 0

9.7 Vx ¼ 0 Vy ¼ 0

Vx ¼ 0 Vy ¼ 0

9.7 Vx ¼ 0 Vy ¼ 0

9.7

Vx ¼ 0 Vy ¼ 0

9.7

M. MEDA ET AL.

5

10

400 3 200

Continental overriding plate

Minimum distance between nodes (km)

2

4

NO Hydration

Velocity boundary conditions (cm/y)

154

A

Table 2. Continued B

Dimension Name length (km)  depth (km)

15

16

Subduction angle

45

Subduction velocity (cm/y)

Thermal boundary conditions

1

prescribed*

Velocity boundary conditions (cm/y) Continental overriding plate

txy ¼ 0 Vy ¼ 0

Oceanic subducting plate No plate

Bottom side

Left side

dVx/dx ¼ 0 dVx/dx ¼ 0 dVy/dy ¼ 0 dVy/dy ¼ 0

Right side**

Vx ¼ 0 txy ¼ 0

Hydration wedge geometry

Viscosity of hydrated mantle (Pa . s)

Viscosity of hydrated crust (Pa . s)

4.00

acute

1019

1021

5

non-hydrated crust

17 2

Vx ¼ 0 Vy ¼ 0

obtuse

non-hydrated crust

obtuse

non-hydrated crust 1020

19

21

Hydration

250 3 250

1020

dVx/dx ¼ 0 dVy/dy ¼ 0

20 Vx ¼ 20.3 Vy ¼ 0

1020

Vx ¼ 20.3 txy ¼ 0

22

1020

23

1020

24

non-hydrated crust 1019

Vx ¼ 0 Vy ¼ 0

25

non-hydrated crust

26

non-hydrated crust

27

1019

28

1020

29

1020

30

1020

1022 1019

31 32

MANTLE HYDRATION IN CRUST RECYCLING

18

Minimum distance between nodes (km)

Vx ¼ 20.3 txy ¼ 0

*Top side: T ¼ 300 K; Bottom side: zero flux; Lateral sides: conductive gradient from T ¼ 300 K at surface to 1600 K at 80 km depth; constant downward. ** Specified boundary conditions are applied to the portion of the right side extending throughout the overriding continent. For the rest of the right side, conditions of dVx/dx ¼ 0 and Vy ¼ 0 are assumed.

155

Vx ¼ 20.3 Vy ¼ 0

156

Table 2. Continued C

Hydration

Dimension length (km)  depth (km)

Name

1400  700

11 idrataz

11 no_idr

Hydration

22 idrataz

NO hydration

22 no_idr

45

Subduction velocity (cm/y)

Thermal boundary conditions

1

Upper side: T ¼ 300 K Bottom side: T ¼ 1600 K Lateral sides: zero flux

Velocity boundary conditions (cm/y) Continental overriding plate

Oceanic subducting plate

Bottom side

Left side***

Right side

txy ¼ 0 Vy ¼ 0

Vx ¼ 1 Vy ¼ 0

Vx ¼ 0 Vy ¼ 0

Vx ¼ 0 Vy ¼ 0

Vx ¼ 0 Vy ¼ 0

Minimum distance between nodes (km)

Hydration wedge geometry

Viscosity of hydrated mantle (Pa . s)

Viscosity of hydrated crust (Pa . s)

Density of markers

5.0

dynamic rect

1020

1021

1 marker/ km2

no

no

no

4 marker/ km2 no

no

*** Specified boundary conditions are applied though the portion of the left side below the oceanic plate. For the rest of the left side, conditions of dVx/dx ¼ 0 and Vy ¼ 0 are assumed.

no

4 marker/ km2

M. MEDA ET AL.

NO hydration

Subduction angle

MANTLE HYDRATION IN CRUST RECYCLING

157

Fig. 2. Thermomechanical evolution of the non-hydrated model, at (a) 10 Ma, (b) 15 Ma, (c) 30 Ma and (d) 40 Ma from the beginning of subduction. Continental crustal markers are in brown and oceanic markers in black; the background, grading from blue to violet (see the legend), represents the temperature field, and white arrows the velocity field. Axial co-ordinates are in km.

depths of between 0 and 10 km; yellow for between 10 and 20 km; and green for between 20 and 30 km. Note that this stratification is not related to any physical or rheological differentiation. However, it allows us to recognize the provenance of each of the markers involved in the wedge area circulation, even at very late evolution stages. By the first 10 Ma, ablative subduction dominates and the descent of the crustal markers occurs

along the subduction plane (Fig. 6a). The small counterclockwise flow, which started after 20 Ma from the beginning of subduction and localized at 30 km depth near the active margin, drives crustal markers towards the surface. A significant thinning of the crust also occurs in the wedge owing to the local rising of hydrated mantle material during the initial stages of subduction (Fig. 6b). Underplating of exhumed crust also begins below the continental

158

M. MEDA ET AL.

Fig. 3. Assumed geometric configurations of the hydrated mantle wedge: (a) closed, (b) right (b) and (c) open; the grey straight line represents the subduction plane, inclined at 458.

crust, producing a remarkable crustal thickening (Fig. 6c). After 40 Ma continental crust begins to uplift between 0 and 50 km and, after about 55 Ma of subduction, the deep continental crust reaches 10 km depth (Fig. 6d). In the last phases of the evolution, the crustal markers that have not been exhumed develop a crustal root (Fig. 6e, f). At the end of the evolution, two different regions can be distinguished: an area where non-significant deformation dominates, far from the wedge; and an area characterized by strong deformation and mixing of crustal markers, originally belonging to different levels, particularly at depths of between 25 and 100 km (Fig. 6f). Mixing of continental crust rocks has recently been described by Faccenda et al. (2008), who show that, without hydration in the subduction wedge, a substantial part of the continental crust from the overriding plate may sink to great depth to be recycled into the mantle. In their models shear heating plays a key role for the occurrence of continental crust recycling, and induces local thermal softening in the wedge area. The predicted thermal regimes are coherent with the development of HP–MT (medium temperature) or LP –LT metamorphic units, but are too high to allow HP– LT metamorphism similar to that affecting the SLZ. For comparing with natural P–T data it is useful to identify, for each crustal marker involved in the wedge circulation, the maximum values of pressure, temperature and depth reached during the stages of progressive evolution. These values are represented in Figure 7 by superposing a coloured scale on the final configuration of the markers, after a 70 Ma evolution. Regarding the temperature, Figure 7a shows the maximum temperature (Tmax) assumed by each marker, whereas Figure 7b shows the difference DTmax between Tmax and the temperature reached at the end of the evolution; in both cases the markers are represented in their final position. Continental markers outside the hydration area show a Tmax increasing linearly with depth (Fig. 7a) throughout the crust. Tmax variations occur mainly along the vertical direction even at shallower crustal levels within the wedge area, indicating that crustal particle movements occur basically along the horizontal. At intermediate and deep structural levels within the wedge area, on the other hand, strong vertical and horizontal Tmax gradients occur, indicating a significant mixing of crustal portions that run along different paths during their involvement in the corner flow, before reaching the same final position (Fig. 7a). In any case, Tmax values are less than 700 8C, which is the initial temperature at the base of the crust. Positive values of DTmax (Fig. 7b) indicate that, when hydration is allowed, the subduction process induces a general cooling in the continental crust.

MANTLE HYDRATION IN CRUST RECYCLING

159

Fig. 4. Thermomechanical evolution of the hydrated model at (a) 10 Ma, (b) 15 Ma, (c) 30 Ma and (d) 40 Ma from the beginning of subduction. Colours and co-ordinates are as in Figure 2.

In particular, outside the wedge cooling of up to 100 8C occurs, showing that subduction can perturb the thermal field even far from the wedge area. Much closer to the active margin DTmax increases up to 300 –400 8C for the crustal markers coming from the original deep crust and recycled only at shallow depths in the wedge (compare Fig. 6f with Fig. 7b). Where the remixing of material from all different crustal levels is more effective DTmax is minimal (150 8C) and quite homogeneous inside the wedge. The reasons for this low value are different for markers coming

from different original crustal levels: crustal particles originally from the deep continental crust move in a quite stable thermal field produced because of the peculiar configuration of the depressed isotherms (Fig. 6f), while crustal particles originally from shallow and intermediate continental crust reach their Tmax during their involvement in the corner flow. A similar analysis was conducted for pressure (Fig. 7c for Pmax and Fig. 7d for DPmax). With respect to temperature, continental crust located far from the wedge area does not undergo significant

160

M. MEDA ET AL.

within the final marker configuration, after 70 Ma of simulated subduction. Regarding the transition zone, between 0 and 50 km from the trench, and within 30 km depth, the markers reach their baric maximum during the first few million years as they start the decompression phase from their initial position. In the central part of the wedge area baric maxima are reached after 40– 50 Ma of subduction; a comparison with the DPmax distributions (Fig. 7d) indicates that these crustal volumes have risen from deep levels of the wedge. The youngest ages of Pmax achieved at the boundary with the subduction plane, together with the corresponding negligible DPmax values (Fig. 7d), indicate that in this region crust particles are still sinking.

Case study

Fig. 5. Histogram of continental crustal markers located (top of the figure) or uplifted (bottom of the figure) at different depths within the hydrated mantle wedge, with respect to the total number of markers buried below 30 km depth. 1  1 and 2  2 stand for densities of 1 and 4 markers per km2, respectively.

changes in pressure, with differences of less than 0.05 GPa (Fig. 7d). For markers involved in the wedge circulation, on the other hand, Pmax experienced by recycled crustal material can be of the order of 2 GPa (Fig. 7c). If we look at DPmax we can identify markers risen after reaching great depths. It is then evident that between 250 and 0 km a large portion of continental crust undergoes a decrease in pressure of the order of 1 GPa, rising up to less than 30 km depth (Fig. 7d). Furthermore, between 0 and 50 km from the trench, and within 30 km depth, a regular gradient in the pressure changes can be distinguished. This can be related to the local uplift of the lower crustal levels, up to 15 km depth, as already observed for the thermal field. Finally, Figure 8 shows the time at which the baric maximum is reached, always represented

The Austroalpine is the uppermost tectonic domain of the Western Alps, and consists of a thin sheet of continental rocks tectonically overlying the Penninic pile of oceanic and continental slices (ocean suture); it does not comprise Mesozoic ophiolites, but is infolded with them and the related Mesozoic sediments. In the Western Alps the metamorphic evolutions of the Austroalpine continental crust indicate its involvement in subduction processes. Lithological affinities between the protoliths of the subducted continental crust and the rocks of the Southern Alps (Adria Alpine hinterland) suggest a provenance of these metamorphic rocks from the upper plate of the Alpine subduction system, constituted by continental lithosphere. The Western Austroalpine system consists of the internal Sesia–Lanzo Zone (SLZ) and the external Dent Blanche nappe, and was deformed and metamorphosed from the Cretaceous –Palaeogene to the Eocene–Lower Oligocene with different dominant metamorphic imprints (quartz-eclogite- to greenschist-facies conditions: e.g. Dal Piaz et al. 1972, 1983; Hunziker 1974; Compagnoni et al. 1977; Gosso 1977; Lardeaux et al. 1982, 1983a; Spalla et al. 1983; Oberha¨nsli et al. 1985; Balle`vre et al. 1986; Pognante et al. 1987; Vuichard 1989; Pognante 1991; Hunziker et al. 1992; Venturini et al. 1994). The SLZ (Fig. 9) is the widest slice of eclogitized pre-Alpine continental crust of the Western Alps, and its Alpine tectonic evolution is compatible with uplift during active oceanic lithosphere subduction (e.g. Spalla et al. 1996; Zucali et al. 2002). During the Alpine subduction this portion of the Adria microplate was tectonically detached (ablative subduction) from the continental margin (Polino et al. 1990) and subducted to depths of at least 60 km. After a pervasive Cretaceous eclogite-facies

MANTLE HYDRATION IN CRUST RECYCLING

161

Fig. 6. Thermomechanical evolution of markers from the continental crust in the hydrated mantle wedge after (a) 10, (b) 30, (c) 40, (d) 50, (e) 60 and (f) 70 Ma from the beginning of subduction; the colours of the markers indicate their initial structural levels: red, upper crust (0– 10 km); yellow, middle crust (10–20 km); green, lower crust (20–30 km); and black lines are isotherms every 200 K, until 1500 K.

162 M. MEDA ET AL. Fig. 7. (a) Tmax (8C) recorded by each continental crustal marker; (b) DTmax (8C) calculated for each continental crustal marker; (c) Pmax (GPa) recorded by each continental crustal marker; and (d) DPmax (GPa) calculated for each continental crustal marker. Each marker is represented in its final configuration, after 70 Ma from the beginning of subduction.

MANTLE HYDRATION IN CRUST RECYCLING

163

Fig. 8. Time interval (Ma) between the beginning of subduction and the achievement of Pmax for each continental crustal marker, here represented in its final configuration; only markers undergoing a difference Pmax 2 P70 Ma  0.05 GPa are shown.

recrystallization, multistage exhumation emplaced the SLZ into the uppermost part of the Tertiary nappe pile and produced partial greenschist-facies re-equilibrations (e.g. Compagnoni et al. 1977; Gosso 1977; Passchier et al. 1981; Inger et al. 1996; Ducheˆne et al. 1997; Rubatto et al. 1999; Zucali et al. 2002, 2004). The SLZ is bounded in its footwall by ophiolitic relicts of the Ligurian– Piemontese Tethys and in its hanging wall by lowercrustal rocks of the Southern Alpine Ivrea Zone, from which it is separated by the Periadriatic Lineament (Fig. 9a). It is traditionally subdivided into two elements (e.g. Dal Piaz et al. 1972; Compagnoni et al. 1977): the upper element, or ‘II Zona Diorito–Kinzigitica’ (IIDK), comprises metapelites and metabasics of the lower crust with a dominant pre-Alpine metamorphic imprint under amphibolite-/granulite-facies conditions; the lower element consists of polymetamorphic metapelites, metagranitoids, metabasics and marbles, from the lower and upper crust, intruded by Permian igneous bodies (Oberha¨nsli et al. 1985; Bussy et al. 1998; Rebay & Spalla 2001) and further divided into three metamorphic complexes (Fig. 9b). These are the ‘Gneiss Minuti Complex’ (GMC), which shows a dominant Alpine metamorphic imprint under greenschist-facies conditions, the ‘Eclogitic Micaschists Complex’ (EMC), which is characterized by a dominant Alpine imprint under eclogitefacies conditions, and the ‘Rocca Canavese Thrust Sheet’ (Pognante 1989a, b; Spalla & Zulbati 2004), in which a lawsonite blueschist-facies

metamorphic imprint develops during the retrograde exhumation path. Cover sequences of possible Mesozoic age (Venturini et al. 1991, 1994) and Piemonte Zone serpentinized peridotites (Ferraris & Compagnoni 2003) have been reported from the central SLZ. The tectonic evolution of the SLZ is polycyclic and was accomplished during pre-Alpine times under granulite- to amphibolite- and then to greenschist-facies conditions (e.g. Lardeaux 1981; Lardeaux & Spalla 1991; Spalla et al. 2005), and during Alpine times under eclogite-facies peak conditions, with a final retrogression under greenschistfacies conditions. In the lower element, the EMC and GMC, both pervasively eclogitized, strongly differ in the volume percentage of greenschist-facies re-equilibration (Stuenitz 1989; Spalla et al. 1991). The EMC, which constitutes the innermost part of the SLZ (Fig. 9b), shows a greenschist-facies overprint confined to discrete shear zones, more pervasively developed toward the Insubric Line, its inner boundary with the Southern Alps. Pre-eclogitic fabrics are here locally marked by P –T prograde blueschist-facies minerals (e.g. Pognante et al. 1980; Rebay & Messiga 2007). Recently, different structural domains have been contoured in the GMC and EMC complexes of the central SLZ (Babist et al. 2006), and different P– T evolutions are described in the two complexes in Valchiusella (Konrad-Schmolke et al. 2006), perfectly adherent with the tectonometamorphic outline inferred at the regional scale (e.g. Pognante 1991).

164

M. MEDA ET AL.

Fig. 9. (a) Tectonic outline of the Alps; the star shows the Sesia–Lanzo Zone; P. L., Periadriatic Lineament. (b) Simplified geological map of the Sesia–Lanzo Zone. Legend: 1, II Dioritic-Kinzigitic Zone; 2A, Gneiss Minuti Complex (GMC); 2B, Eclogitic Micaschists Complex (EMC); 2C, Rocca Canavese Thrust Sheets; 3A, post-metamorphic Oligocene intrusive bodies of Biella and Traversella; 3B, contact metamorphic aureole; symbols locate: (4) the main metagabbro bodies; (5) metagranitoids; (6) calc-silicate marble lenses; and (7) eclogite relics in the GMC (7).

The Alpine evolution is characterized by polyphase deformation under blueschist- to eclogitefacies conditions followed by retrogression under blueschist- to successive greenschist-facies conditions (e.g. Compagnoni et al. 1977; Gosso 1977; Pognante et al. 1980; Williams & Compagnoni 1983; Tropper & Essene 2002; Zucali et al. 2002, 2004). Absolute age estimates and field relationships allow us to attribute an age of 270 Ma to the granulite-facies stage, 240 Ma to the amphibolitefacies and 170 Ma to the greenschist-facies events;

mineral ages ranging between 60 and 70 Ma (Table 3) have been related to the Alpine eclogitefacies peak (e.g. Lardeaux & Spalla 1991; Inger et al. 1996; Rubatto 1998; Rubatto et al. 1999; Rebay & Spalla 2001). Different P –T and P–T– dt (P–T-deformation time) evolutions have been inferred on the grounds of mineral phase equilibria, geothermometry and thermodynamic analysis, which are stable throughout the complex (e.g. Andreoli et al. 1976; Compagnoni et al. 1977; Desmons & Ghent 1977;

Table 3. Review of the metamorphic assemblages of P–Tmax for the Sesia –Lanzo Zone HP rocks, with radiometric ages. Labels correspond to those of Figure 10 Label

P– T climax assemblage

P– T climax age (Ma)

References

60– 75

2 3 4

marbles Cal, Dol, Qtz, Na-Cpx, GrtI, PhI, ZoI, Al-Ttn metapelites Cpx, Grt, Ky, Ph, Qtz + Gln + Cld; metagranitoids Grt, Cpx, Kfs, Ph, Zo, Qtz; metagranitoids Jd, Qtz, Grt, Zo, Qtz, Ph

– 71.2 + 3.2 129 + 15; 114

5

90– 67

6

metapelites Grt + Omp, Ph + Pa + Gln + Ep, Qtz; metabasics Grt, Omp, Rt + Ph + Gln + Ep; marbles Cal (Dol, Ank) + Grt + Ph + Zo + Qtz Grt, Omp or Jd, Zo, Rt, Cld

7 8 9 10 11 12 13 14

Grt, Omp, Zo, Ph, Rt, Qtz Omp, Grt, Zo + Gln + Bar Jd, Qtz, Grt; Omp, Grt Qtz, Ph, Omp/Jd, Gln, Rt + Pa + Ctd + Zo Jd, Ph, Grt + Kfs + Zo Gl, Lws, Ep, Sph, Qtz Grt, Omp, Pa Grt, Omp, Gln

65 + 3 – – – – – – –

15 16

eclogitised granulites Omp, Grt, Qtz, Amp; eclogitic gneisses Jd, Grt, Ph, Ky; metabasics Omp, Bar + Gln metagranitoids Grt, Jd, Ph, Zo, Qtz; metapelites Omp, Grt, Amp, Ph, Qtz; metabasics Grt, Omp, Amp, Qtz

17 18

Jd, Zo, Qtz + Ky + Kfs Omp, Grt, Zo + Gln + Bar þ Qtz

– 65+5 (Mucrone); 65+3 (Aosta Valley); 68+7 (Bonze) 65 66 + 1; 55–48

19

Ph + Omp, Grt, Ttn

c. 65

20 21 22 23

Omp, Czo + Zo, Ky, Qtz, Rt Par, Gln, GrtII Grt, Jd, Zo, Ph Omp, Czo, Grt, Ky, Rt, Qtz + Zo

– 69,4 + 0,7; 140,5 + 0,6 – –

24

metapelites Ph, Qtz, Grt, Omp, Amp, Rt; metabasics Ph, Amp, Grt, Rt + Omp + Zo + Cc; metaintrusives Ph, Amp, Grt, Rt + Omp + Zo/Czo + Cc; quartzites Ph, Grt, Rt, Ky, Cld + Cc metapelites Jd, Wm, Qtz, Kfs; metabasics Gl, Grt, Rt, Omp + Zo + Tc + Wm metapelites Wm + Zo + Cld + Gl + Ky; metabasics Gl/Act, Zo Jd, Qtz, Wm, Kfs Fe-Omp, Ab, Ph, Grt Omp, Ky, Ep, Grt

65 + 5

Compagnoni (2003) Castelli & Rubatto (2002) Castelli & Rubatto (2002) Giorgietti et al. (2000) Ruffet et al. (1995) Tropper et al. (1999) Tropper & Essene (2002) Zucali et al. (2002)

– – – – 60– 70

Pognante et al. (1987) Pognante et al. (1987) Pognante et al. (1987) Lardeaux et al. (1983) Zucali et al. (2004)

25 26 27 28 29

90– 70

Rebay & Messiga (2007) Castelli (1991) Hy (1984) Oberhansli et al. (1985) Compagnoni et al. (1977) Williams & Compagnoni (1983) Spalla et al. (1997) Lardeaux et al. (1982) Andreoli et al. (1976) Pognante (1989a, b) Pognante (1989a, b) Pognante (1989a, b) Reinsch (1979) Desmons & Ghent (1977) Gosso et al. (1982) Rubatto et al. (1999)

165

metagabbros Gln, Cld, Ep, Grt + Ph + Pa

MANTLE HYDRATION IN CRUST RECYCLING

1

166

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Reinsch 1979; Gosso et al. 1982; Lardeaux et al. 1982, 1983b; Williams & Compagnoni 1983; Hy 1984; Oberha¨nsli et al. 1985; Pognante et al. 1987; Pognante 1989a, b; Castelli 1991; Ruffet et al. 1995; Spalla et al. 1997; Rubatto et al. 1999; Tropper et al. 1999; Giorgetti et al. 2000; Castelli & Rubatto 2002; Tropper & Essene 2002; Zucali et al. 2002, 2004; Compagnoni 2003; Rebay & Messiga 2007), and experienced a quartz eclogitefacies imprint peaking at about 600 + 50 8C and 2.1 GPa (Spalla et al. 1997; Tropper & Essene 2002; Zucali et al. 2004: on the basis of the occurrence of bimineralic eclogites and Ky-bearing assemblages in metabasics). During blueschist- to greenschist-facies retrogression, kilometre-scale folding of the eclogitic foliation occurred in places associated with a new penetrative foliation. Large-scale shear zones developed during the final stages of greenschist-facies re-equilibration in the central SLZ (Handy et al. 2005), and a later brittle–ductile faulting post-dates the Oligocenic andesitic dykes and igneous stocks of Biella and Traversella emplacement (Zanoni 2007; Zanoni et al. 2007, 2008). Both plutons are intruded in the innermost part of the SLZ (Eclogitic Micaschist Complex) close to the internal boundary of the Alpine metamorphic units (Insubric or Periadriatic tectonic line). Natural data exclusively referred to the metamorphic peak conditions estimated for different portions of the SLZ lower element are reported, with the error bars, in Figure 10, with the aim of comparing them with the P –T predictions of the numerical simulation. Mineral assemblages characterizing different chemical systems are listed in Table 3. The P –T-climax conditions were obtained from different bulk-rock chemistries and using different approaches such as geothermobarometry and phase petrology over a large span of time (1977–2007; see the references in Table 3). The wide range of bulk compositions may be the cause of apparent differences in the estimated baric and thermal peak values. Despite the different equilibria and solution models used, and the large span of time over which the data were produced, there is surprisingly good consistency in the P –T estimates of the metamorphic climax derived from a large number of SLZ HP rocks with contrasting bulk compositions cropping out in the same location (Table 3). This consistency makes the P– T data shown in Figure 10 reliable. The natural data are scattered on a Ppeak interval of 1.8 GPa, and an average P–T ratio, corresponding to 0.003 GPa 8C21, is inferred by the regression line traced from the origin (P ¼ 0 GPa, T ¼ 0 8C), which is equivalent to a T/depth ratio of about 9 8C km21 for a density of 2670 kg m23. This density corresponds to the average for the

continental crust and also approximates well the serpentinite (e.g. Deer et al. 1993; Dubois & Diament 1997). This value indicates a thermally depressed environment, compatible with active oceanic subduction (cold subduction of Cloos 1993) and already envisaged for the early evolution of the Alps (e.g. Polino et al. 1990), with a low T/depth ratio maintained during the exhumation, testified by the blueschist-facies re-equilibration following the eclogite-facies imprint. This peculiar metamorphic evolution led to the interpretation of a burial –exhumation cycle, accomplished in a regime of active oceanic subduction (e.g. Spalla et al. 1996; Zucali et al. 2004). Similar P–T evolutions, characterizing continental units, have been interpreted in the Aegean as the effect of continuous deep underthrusting and slab retreat (Ring & Layer 2003 and references therein). However, with respect to the SLZ, the Aegean units occupy a different position in the nappe pile at the core of the suture zone, interlayered with several oceanic slices. In addition, the Alps are commonly considered an advancing plate boundary (Royden 1993; Ring & Layer 2003). In order to verify the geodynamic interpretation adopted in this study for the SLZ, in Figure 11 we plot the predicted P–T conditions recorded by subducted continental crustal markers, partly recycled in the mantle wedge, at different times during the numerical simulation. The different colours represent successive time intervals of 5 Ma from the beginning of the simulation (0 Ma in Fig. 11), and the two grey lines are the minimal and maximal P– T ratios that can be recorded by crustal slices involved in the modelled subduction zone; in other words, they represent the boundary P–T conditions in which the subducted and partially exhumed P–T paths of continental markers are accomplished. The maximal and minimal P –T ratios are 0.007 and 0.001 GPa 8C21, corresponding to 4 and 23 8C km21 T/depth ratios, respectively, calculated for the same reference density of 2670 kg m23. We note that these two limits bound all possible P–T values achieved by the continental markers, at different times, during the beginning of the numerical simulation. During the late stages of the evolution, while the lower T/depth ratio remains fixed, the maximum T/depth ratio decreases down to 16 8C km21 (black line in Fig. 11). This global thermal evolution may derive from an initial setting of a normal continental geotherm (average geotherm c. 25 8C km21: e.g. Thompson 1981; Peacock 1989; Spear 1993) followed by a subsequent cooling, induced by the involvement of the crust in a thermally depressed subduction zone. Natural data (black squares in Fig. 11) plot in a P –T space occupied by the P–T conditions predicted for the markers (with the exception of only two of them– namely data 2 and 3 of Fig. 10)

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Fig. 10. Review of the P –Tmax natural data for the HP rocks of the Sesia– Lanzo Zone (black squares) with error bars (grey). Legend: EMC: 1, Corio and Monastero gabbros (Rebay & Messiga 2007); 2, metacarbonates (Castelli 1991); 3, metapelites, metabasics and metagranitoids in the Monte Mucrone area (Hy 1984); 4, metagranitoids at Monte Mucrone (Oberha¨nsli et al. 1985); 5, metapelites, metabasics and metagranitoids in the lower Aosta Valley and Monte Mucrone area (Compagnoni et al. 1977); 6, metapelites, metabasics and metagranitoids in the lower Aosta Valley (Williams & Compagnoni 1983); 7, metabasics in the lower Aosta Valley (Spalla et al. 1997); 8, metapelites, metabasics and metagranitoids in Val Vogna (Lardeaux et al. 1982); 9, metapelites, metabasics and metagranitoids in Valchiusella (Andreoli et al. 1976); 10, metapelites, metabasics and metagranitoids in Val Malone (Pognante 1989a, b); 13, glaucophanites in Valchiusella (Reinsch 1979); 14, metabasics NE SLZ (Desmons & Ghent 1977); 15, metapelites, metabasics and metagranitoids in Val Vogna (Gosso et al. 1982); 16, metapelites, metabasics and metagranitoids (Rubatto et al. 1999); 17, metagranitoids at Monte Mucrone (Compagnoni 2003); 18–19, metapelites, metabasics, metagranitoids and metacarbonates (Castelli & Rubatto 2002); 20, metabasics in Valle Ianca (Giorgetti et al. 2000); 21, metapelites and metabasics in the Monte Mucrone area (Ruffet et al. 1995); 22, metapelites in Val Savenca (Tropper et al. 1999); 23, metabasics in Valle Ianca (Tropper & Essene 2002); 24, metapelites, metabasics, metagranitoids and quartzites in the Monte Mars–Mombarone area (Zucali et al. 2002); 25, metapelites and metagabbros in the Orco Valley (Pognante et al. 1987); 28, metasienite at Boccioleto (Lardeaux et al. 1983b); 29, metagabbros at Ivozio (Zucali et al. 2004). GM: 11, metagranitoids, metabasics and Ca-bearing metapelites in the Orco Valley (Pognante 1989a, b); 27, gneiss and carbonatic schists in the Orco Valley (Pognante et al. 1987). RCTS: 12, metagranitoids, metabasics and ultramafics at Rocca Canavese (Pognante 1989a b). II DK: 26, gneiss, metabasics and impure marbles in the Orco Valley (Pognante et al. 1987). The inclined line is the regression line computed from these P –T data.

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Fig. 11. Comparison between P– T conditions reached by the subducted continental crustal markers at different times from the beginning (0 Ma) of the numerical simulation (see the legend for the reference time). The two grey lines represent the minimal and maximal P/T ratios; the black line represents the maximal T/depth ratio during the late stage of evolution; black squares plot the natural data represented in Figure 10, with their error bars.

and cover a large time interval (from 5 to 70 Ma), suggesting that the numerically simulated geodynamic scenario reproduces, for the whole simulation time, P –T conditions coherent with those recorded by the SLZ subducted continental crust, although radiometric ages obtained on HP –LT climax assemblages are restricted to a shorter time interval (90–60 Ma absolute age; Table 3).

Conclusions We used a 2D thermomechanical model to simulate the oceanic subduction beneath a continental margin and to investigate the role played by mantle hydration in the continental crust recycling in the wedge region. This numerical analysis allows us

to compare the predicted geodynamic scenario with the tectonic interpretation proposed by geologists to justify the peculiar high P –T ratio characterizing the Alpine tectonometamorphic evolution of the SLZ. A comparison between hydrated and nonhydrated models highlights that, while in both configurations ablative subduction facilitates the sinking into the mantle of a huge amount of crustal material scraped off the overriding continental plate, hydration is fundamental to the recycling of crustal material at shallow depths before continental collision (150 km depth for a convergence rate of 1 cm year21). This recycling process makes the uprising and exhumation of buried crustal material possible during a still active subduction; after 70 Ma of active subduction about 40% of the

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subducted crustal material rises at depths shallower than the base of the unperturbed continental crust. The recycled crustal material originates from all crustal levels and its rising generates crustal thickening by underplating, making the wedge area the maximal thickening region that, at the end of the simulation, holds a crustal root of up to 100 km thick. In the core of the wedge area small-scale convection, responsible for the crustal recycling, induces a complete mixing of the subducted crustal units: thin crustal slices, coming from distant structural levels (upper crust of depth 10 km and lower crust of depth 20 km), may come into contact. In the upper part of the wedge area, a horizontal laminar flow dominates until the rising of the recycled material produces a folding, with a wavelength of about 75 km, with consequent uplift. The deformation dampens beyond 100 km from the trench at all crustal levels. Hydration produces a thermal depression in the wedge area, causing crustal particles originating from the deep continental crust of the upper plate to move in a stable thermal field. It also favours, at low convergence rates, the steepening of the subducted plate, inducing a widening and opening of the wedge area. In addition, in the hydrated model both mechanical and thermal thickening occur in the wedge area; in contrast to the mechanical and thermal thinning characterizing the wedge area in the non-hydrated model. The Tmax and Pmax distributions within the final marker configuration show that crustal recycling induces the coupling of volumes that reached different depths during their paths in the corner flow. The P and T climaxes are achieved at different times: in the upper part of the wedge, where crustal material undergoes exclusively uplift, the climax is reached early (Pmax coincident with Pinitial). In the central part, the climax occurs during the intermediate phases of the evolution, before the involvement of crustal particles in the recycling. The predicted mixing characterizing the central part of the wedge can justify the alternation of tectonic units coming from shallow crustal levels (metagranitois rich) with units from deeper crustal levels (kinzigites rich) that naturally occurs in some areas of the SLZ. The relationships between natural P–T estimates and predicted P–T values show that the simulated geodynamic scenario generates P–T conditions coherent with those recorded in the subducted continental crust of the SLZ and that these conditions may have been stable for a long time during Alpine subduction, allowing the accomplishment of prograde and retrograde P– T evolutions of the SLZ totally under an active subduction regime. Finally, our geodynamic model makes the ablative subduction and recycling of crustal particles able to produce HP–LT metamorphism in the

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continental crust for subduction systems characterized by an advancing plate boundary such as the Alps. This is an alternative to underthrusting, which is envisaged as the effective mechanism in subduction characterized by a retreating plate boundary. Careful reviews by T. Gerya and U. Ring are greatly appreciated. G. Gosso is acknowledged for stimulating discussions. A. M. Marotta was funded by FIRST07, and M. I. Spalla was funded by FIRST07 and CNR-IDPA. American Journal Experts Association provided the English revision of the text.

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T ROPPER , P., E SSENE , E. J., S HARP , Z. D. & H UNZIKER , J. 1999. Application of K-feldspar–jadeite– quartz barometry to eclogite facies metagranites and metapelites in the Sesia–Lanzo Zone (Western Alps, Italy). Journal of Metamorphic Geology, 17, 195– 209. T URCOTTE , D. L. & S CHUBERT , G. 1982. Geodynamics: Applications of Continuum Physics to Geological Problems. Wiley, Chichester. V ENTURINI , C., M ARTINOTTI , G., A RMANDO , G., B ARBERO , M. & H UNZIKER , J. 1994. The central Sesia– Lanzo Zone (western Italian Alps): new field observations and lithostratigraphic subdivisions. Schweizerische Mineralogische Petrographische Mitteilungen, 74, 115– 125. V ENTURINI , G., M ARTINOTTI , G. & H UNZIKER , J. C. 1991. The protoliths of the ‘Eclogitic Micaschists’ in the lower Aosta Valley (Sesia– Lanzo zone, Western Alps). Memorie di Scienze Geologiche – Padova, 43, 347–359. V UICHARD , J. P. 1989. La marge Austroalpine durant la collision alpine: evolution tectonometamorphique de la Zone Sesia– Lanzo. PhD thesis, Universite´ de Rennes. W ARREN , C. J., B EAUMONT , C. & J AMEISON , R. A. 2008. Modelling tectonic styles and ultra-high pressure (UHP) rock exhumation during the transition from oceanic subduction to continental collision. Earth and Planetary Science Letters, 267, 129–145. W ILLIAMS , P. F. & C OMPAGNONI , R. 1983. Deformation and metamorphism in the Bard area of the Sesia– Lanzo zone, Western Alps, during subductionand uplift. Journal of Metamorphic Geology, 1, 117–140. Y AMATO , P., A GARD , P., B UROV , E., L E P OURHIET , L., J OLIVET , L. & T IBERI , C. 2007. Burial and exhumation in subduction wedge: mutual constraints from thermomechanical modeling and natural P– T– t data (Schistes Lustre´s, western Alps). Journal of Geophysical Research, 112, B07410; doi: 10.1029/ 2006JB004441. Z ANONI , D. 2007. Messa in posto di plutoni tardo collisionali in unita` continentali profonde esumate. L’esempio della Zona Sesia Lanzo (Alpi Occidentali interne). PhD thesis, Universita` di Milano. Z ANONI , D., B ADO , L., S PALLA , M. I., Z UCALI , M. & G OSSO , G. 2008. Structural analysis of the Northeastern margin of the Tertiary intrusive stock of Biella (Western Alps, Italy). Bollettino della Societa` Geolica Italiana, 127, 125 –140. Z ANONI , D., S PALLA , M. I., G OSSO , G. & Z UCALI , M. 2007. Emplacement of late-collisional plutons in exhumed deep crustal slices: the case of the Sesia Lanzo Zone (Internal Western Alps). Rendiconti della Societa` Geologica Italiana, 5, 228 –230. Z UCALI , M., S PALLA , M. I. & G OSSO , G. 2002. Fabric evolution and reaction rate as correlation tool: the example of the Eclogitic Micaschists complex in the Sesia– Lanzo Zone (Monte Mucrone–Monte Mars, Western Alps Italy). Schweizerische Mineralogische Petrographische Mitteilungen, 82, 429–454. Z UCALI , M., S PALLA , M. I., G OSSO , G., R ACCHETTI , S. & Z ULBATI , F. 2004. Prograde LWS-KY transition during the Alpine continental crust subduction of the Sesia– Lanzo Zone: the Ivozio Complex. Journal of Virtual Explorer, 16, 4-1– 4-21.

Three-dimensional evaluation of fabric evolution and metamorphic reaction progress in polycyclic and polymetamorphic terrains: a case from the Central Italian Alps FRANCESCA SALVI1,2*, MARIA IOLE SPALLA3,4, MICHELE ZUCALI3 & GUIDO GOSSO3,4 1

Dipartimento di Scienze della Terra ‘A. Desio’, Universita` degli Studi di Milano, Milano, Italy

2

Present address: Eni E&P Division, Via Maritano 26, 20097 S. Donato Milanese, Italy 3

Dipartimento di Scienze della Terra ‘A. Desio’, Universita` degli Studi di Milano, Sezione di Geologia, Via Mangiagalli 34, 20133 Milano, Italy 4

C.N.R. – I.D.P.A., Sezione di Milano, Via Mangiagalli 34, 20133 Milano, Italy *Corresponding author (e-mail: [email protected])

Abstract: The 3D reconstruction of geological bodies is an excellent tool for the representation of crustal structures and is applied here to understand related heterogeneities in the grain-scale fabrics; the western portion of the Languard–Tonale Alpine tectono-metamorphic unit (Austroalpine domain, Central Alps) allows evaluation of the per cent volume of textural reworking during polyphase pre-Alpine and Alpine deformations. The structural and metamorphic overprinting during the last deformation imprint involved less than 50% of rock volume; this estimate is obtained by discriminating domains that homogeneously recorded structural and metamorphic re-equilibration during crenulation–decrenulation cycles. These domains are reconstructed using a geograhpical information system (GIS) to manipulate field data and interpretative crosssections as a means to constrain their 3D volumes. The degree of fabric evolution is integrated at the microscale with the estimate of the reactants/products ratio to infer the progress of metamorphic transformation related to advancing degree of mechanical reactivation. The correlation between degree of fabric evolution and progress of synkinematic metamorphic reactions shows that differences between pristine mineral assemblages v. pre-existing fabrics influence the rate of reaction accomplishment. Fabric evolution and degree of metamorphic transformation increase proportionally once above the threshold value of 60% of volume affected by fabric rejuvenation; metamorphic degree also influences the progress of metamorphic reactions.

The terrains affected by polyphase deformation and metamorphism during polycyclic tectonic evolution are characterized by different crustal units that show distinct structural and metamorphic imprints. The definition of the shape and size of these units is a crucial step in unravelling deep-seated mechanisms active during tectonic processes such as crustal accretion or consumption (Spalla et al. 2005). The complete reconstruction of the tectonometamorphic evolution of these terrains is possible through a multidisciplinary approach based on detailed correlation of superposed fabric elements, microstructural analysis and recognition of fabric gradients (Turner & Weiss 1963; Park 1969; Hobbs et al. 1976; Williams 1985; Passchier et al. 1990; Johnson & Vernon 1995; Spalla et al. 2000; Zucali et al. 2002). The extent, degree and timing of metamorphic re-equilibrations, and associated

fabric changes, can be used to define the size and shape of rock volumes that underwent the same crustal path during a defined time interval and constitute a tectono-metamorphic unit (TMU: Spalla et al. 2005). Rocks belonging to a single TMU record a heterogeneous partitioning of the total deformation, resulting in heterogeneous distribution of metamorphic re-equilibration due to the catalysing effect of deformation on the metamorphic reaction progress. The resulting patchy distribution, at the end of each deformation episode, of the dominant fabrics and metamorphic assemblages provides an evaluation of the percentage volume of mechanically and chemically reacting rock portions during successive stages of the tectonic evolution, in which different crustal slices, which correspond to different

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 173–187. DOI: 10.1144/SP332.11 0305-8719/10/$15.00 # The Geological Society of London 2010.

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TMUs, were engaged. A quantitative 3D estimate of the final distribution of differently re-equilibrated volumes may give fundamental insights into the accomplishment of structural and/or metamorphic re-equilibrations at different structural levels along active plate margins. This is fundamental to estimating, for example, the influence of different deformation mechanisms, phase transitions or density and viscosity variations in the thermomechanical setting of geophysical models. In this paper we estimate the reacting rock volumes within a single TMU, starting from a map reporting in detail the heterogeneous distribution of textural and metamorphic transformations (map of dominant fabric domains). Taking into account the degree of mechanical and mineral –chemical transformation of different rocks at micro- to megascales, every TMU includes a mosaic of domains with an homogenous fabric evolution and degree of metamorphic transformation, each of them showing a dominant fabric and metamorphic imprint. In addition, showing the low- and high-strain domains for each deformation phase, the map of dominant fabric domains corresponds to a map of deformation partitioning at all scales and identifies the distribution of geometric patterns (homothetic micro- to mega-structures). A quantitative volume estimate of the dominant fabric domains is performed by means of a 3D model, for which subsurface constraints are provided by carefully rendered geological cross-sections. The case of the Languard–Tonale TMU (Austroalpine domain in the Central Alps: Zucali 2001; Spalla et al. 2005) is used as a case study to compare the relationships between the structural and metamorphic memories recorded at contrasting (pressure/temperature) P/T ratios [from high pressure–intermediate temperature (HP –IT), intermediate pressure– high temperature (IP–HT) to low pressure –low temperature (LP –LT)] during the polyphase tectonic evolution of this portion of continental crust involved in Permian–Triassic lithospheric thinning and subsequent Alpine subduction. The highly contrasting thermal regimes under which successive tectonic imprints have been recorded in the Languard –Tonale TMU make the conclusions drawn from this case study applicable to different geological contexts in which temperature can serve as catalyst to reaction kinetics or deformation mechanism activation.

Geological background The investigated portion of the Languard– Tonale TMU (Fig. 1) is located between the upper Val Camonica and Valtellina of the Central Alps. This polydeformed and polymetamorphosed unit

belongs to the Austroalpine domain, which is considered to have originated from the Adria plate based mainly on its lithological affinities, and occupies the uppermost structural level in the Alpine nappe pile. This TMU includes two lithostratigraphical units, the Languard–Campo Nappe (LCN) and the Tonale Series (TS) (Bigi et al. 1990; Schmid et al. 1996), and is bounded southwards by the eastern segment of the Tonale dextral strike-slip fault (Stipp et al. 2004). The mapped portion of the Languard –Tonale TMU mainly consists of low- to medium-grade muscovite, biotite- and minor staurolite-bearing gneisses, and micaschists with interlayered amphibolites, marbles, quartzites and pegmatites occupying the northern and central sector of the map. The southernmost portion of the TMU is mainly composed of high-grade sillimanite-bearing gneisses and micaschists, garnet- and biotite-bearing amphibolites, marbles and pegmatites. Post-Variscan intrusives (granitoids, diorites and minor gabbros) occur throughout the area (Ragni & Bonsignore 1968; Bonsignore et al. 1971; Del Moro et al. 1981; Tribuzio et al. 1999 and references therein). The Alpine metamorphic evolution of this portion of the Austroalpine domain consists of a high-pressure imprint followed by retrogradation to greenschist-facies conditions (Spalla et al. 1995, 2003, 2005; Tomaschek & Blu¨me 1998; Gazzola et al. 2000; Zucali 2001; Gosso et al. 2004); large-scale Alpine mylonitic belts, such as the Mortirolo or Insubric Line, are often associated with greenschist-facies metamorphism (Werling 1992; Viola et al. 2003; Stipp et al. 2004). In the mapped region (Fig. 1), the rocks recorded the same tectono-metamorphic evolution (P–T –t– d path, where t and d are ‘relative time’ and ‘deformation’ respectively) during Alpine convergence (e.g. Gazzola et al. 2000; Zucali 2001; Spalla et al. 2003), even where the distribution of Alpine polyphase deformation and metamorphic transformations is highly heterogeneous, and responsible for the localization of different dominant and diachronous structural and metamorphic imprints in adjacent domains (Fig. 1). The tectono-metamorphic history of the TMU Languard–Tonale consists of six synmetamorphic deformation phases (Fig. 2): three are pre-Alpine (D1a, D1b, D2) and three Alpine (D3, D4, D5: Gazzola et al. 2000; Zucali 2001; Spalla et al. 2003, 2005; Gosso et al. 2004); the widespread Permian diorites and granodiorites (Del Moro & Notarpietro 1987) were used as markers to distinguish Alpine from pre-Alpine structures and metamorphic imprints because they are post-Variscan and pre-Alpine. The pre-Alpine evolution, recorded only in the country rocks, developed under medium- to high-grade conditions: D1 is poorly preserved in

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Fig. 1. Simplified foliation trajectory and petrographical map of the Languard–Tonale TMU. Colour intensity gradient of the Permian meta-intrusives (metadiorites and metagranitoides) qualitatively reproduces the increase in planar fabric intensity from coronitic (pale) to mylonitic (deep) textures (modified after Spalla & Zucali 2004). The square includes the selected area for Figure 3 and the 3D modelling. Location and simplified tectonic map of the Alpine belt are included in the inset.

176 F. SALVI ET AL. Fig. 2. Sequence of the geological events recorded by the Languard– Tonale TMU. Deformation phase, structures and their associated mineralogical support, inferred metamorphic imprints and available radiometric data are reported. HT, high temperature; IT, intermediate temperature; LT, low temperature; HP, high pressure; IP, intermediate pressure; LP, low pressure; Ab, albite; And, andalusite; Bt, biotite; Chl, chlorite; Cld, chloritoid; Czo, clinozoisite; Ep, epidote, Grt, garnet; Ilm, ilmenite; Kfs, K-feldspar; Ky, kyanite; Pl, plagioclase; Qtz, quartz; Sil, sillimanite; St, staurolite; Ts, tschermakite; Ttn, titanite; Wm, white mica). *Tho¨ni 1981. **Del Moro et al. 1981; Del Moro & Notapietro 1987.

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low-strain D2 domains, while D2 produced folds and a pervasive composite axial-planar foliation (S2). Pre-Alpine structures and related metamorphic assemblages are preserved in significant volumes mainly in the southern and northernmost parts of the area, whereas in the central part, dominated by Alpine structures, they occur as small relict domains. D1 structures are marked by contrasting mineral assemblages (Fig. 2), indicating that in different volumes these early structures developed under different thermal regimes (e.g. Spalla et al. 2003). The common tectonic history started with D2, marked by HT– LP mineral assemblages, indicating T ¼ 650– 750 8C and P  0.6 GPa (Gosso et al. 2004 and references therein). D3 occurred under HP –IT conditions, and mineral assemblages marking these structures provide an estimate of 500 –600 8C and 1.1 + 0.2 GPa; D4 and D5 took place under lower-greenschist-facies conditions (T  350 8C and P  0.5 GPa: Gazzola et al. 2000; Gosso et al. 2004). The Alpine deformation (D3 –D5) is responsible for the development of km-scale shear zones and isoclinal folds (Fig. 1). The Alpine uplift to shallow structural levels after subduction, compatible with greenschist-facies conditions, was accomplished early during Alpine convergence (.78 Ma), as suggested by radiometric data on syn-D4 micas (Del Moro & Notarpietro 1987; Gazzola et al. 2000). A synthesis of preAlpine and Alpine mineral assemblages marking successive fabric elements in metapelites and meta-intrusives is shown in Figure 2. During the complete structural evolution, strain partitioning produced fabric gradients (from coronitic to mylonitic) at different scales during each deformation phase. In the Languard–Tonale TMU

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it is common to observe that the more mylonitic fabrics correspond to more complete metamorphic re-equilibration (Spalla et al. 2005).

Data analysis, 3D volumetric reconstruction and discussion of model results The correlation of the structural and metamorphic re-equilibration stages performed for the Languard –Tonale TMU represents the starting point for the recognition of the dominant fabric domains. Low- and high-strain domains for each deformation phase have been identified by a semiquantitative estimate of the degree of fabric and associated metamorphic transformation, both in the host rocks and in the Permian meta-intrusives. A wide range of lithologies is generated locally, providing a complex mosaic of rocks volumes with variable mineral composition and planar fabrics. Recognition of gradients in fabric intensity and abundance of the associated metamorphic mineral products was the focus of field analysis and sampling strategy. The evaluation of the dominant fabric domains, degree of metamorphic reaction progress and their volumetric estimate has been performed in a selected area (about 54 km2) in the southern part of the Languard –Tonale TMU (Fig. 3), in which the pre-Alpine (D1 – D2) structures are preserved in extensive domains and the Alpine (D3 –D4 – D5) sequence of superposed structures are detectable with good continuity at map scale. The 3D model is constructed using five main steps, shown in the flowchart of Figure 4.

Fig. 3. Tectono-metamorphic map of the southern part of the Languard– Tonale TMU selected for the 3D volumetric modelling of the dominant fabric domains. The number of dots and colours used for the foliation symbols represent a given deformation phase and the P– T conditions associated with their formation.

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Fig. 4. Flowchart for the 3D reconstruction of the dominant fabric domains.

Hereinafter, the analytical input data and the model results are described and discussed for each step of the procedure.

Meso- and microstructural data input into a GIS database As a starting point, the numerous meso- and microstructural data of the Languard–Tonale TMU (Gazzola et al. 2000; Spalla et al. 2003, 2005; Gosso et al. 2004) were georeferenced and stored in a GIS (Geographic Information System) database. The tectono-metamorphic map of the Languard–Tonale unit was drawn at 1:10 000 scale within the GIS environment. The archive of the data in GIS allows rapid management, representation, querying and manipulation of the structural data. Finally, the estimates of the degree of fabric evolution and of the metamorphic reaction progress, using combined meso- and microstructural descriptions, were organized in a georeferenced database in order to represent the distribution of fabric domains.

Analysis of the degree of fabric evolution and metamorphic transformation The interpretative map of dominant fabric domains derives from the foliation trajectory map of the Languard–Tonale TMU (Fig. 2) (Spalla et al. 2003, 2005; Gosso et al. 2004), where the foliations

(S1 –S5) are chronologically distinguished on the basis of overprinting criteria and compatibility of metamorphic assemblages. Contouring of the homogeneous fabric domains is facilitated by integrating mesostructural information with microstructural analysis to estimate the degree of fabric evolution and metamorphic transformation in volume percentage; microstructural analysis was performed on 154 thin sections strategically distributed throughout the modelled area. The mineralogical assemblages and microstructures were described using a semi-quantitative estimate of the degree of fabric evolution (volume percentage of planar fabric distribution) and metamorphic transformation (modal amount of mineral assemblages expressed in %) corresponding to each of the successive fabrics; an example of the microstructural relationships between the degree of fabric evolution and metamorphic reaction for originally foliated and isotropic rocks is shown in Figure 5. The estimate of the fabric evolution is based on the degree of grain-scale reorganization of the dominant fabric (Fig. 6); the successive stages of crenulation cleavage development proposed by Bell & Rubenach (1983; see also Passchier & Trouw 2005), up to the complete transposition, were used as a guide. Two different evolutionary schemes of the planar fabric were considered in accordance with the character of the original rock (Fig. 6): the first for the fabric evolution of the pre-Permian host rocks that evolved from an originally foliated

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Fig. 5. Example of microstructural relationships between the degree of fabric evolution and metamorphic transformation in originally foliated (left) and massive samples (right): (a) S2 foliations in garnet-, biotite- and sillimanite-bearing micaschist; (b) porphyroblast of staurolite (pre-Alpine fabric) in syn-D3 tectonitic fabric; (c) white mica, garnet- and chlorite-bearing micaschist with mylonitic fabric (S3 Alpine foliation); (d) syn-D3 coronitic fabric in a metadiorite; (e) syn-D3 differentiated crenulation cleavage in a metadiorite; (f) syn-D3 mylonitic fabric in a metadiorite showing the degree of fabric evolution and a metamorphic transformation of 100%.

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Fig. 6. Reference ranges for the comparative fabric v. mineral growth analysis of the tectono-metamorphic histories in the Languard– Tonale TMU. Schematic fabric evolution from originally foliated country rocks and initially isotropic Permian intrusives. For each stage the range of the degree of fabric evolution is indicated. Only when the degree of fabric evolution (F.E.) is 60% of the volume of the rock (thick dashed line) does the degree of metamorphic transformation increase proportionately (modified after Bell & Rubenach 1983). Asterisk indicates mean values.

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fabric; the second for the Permian meta-intrusives that exclusively recorded the Alpine history and are assumed to have started from an isotropic igneous fabric. Three main evolution stages are proposed (Fig. 6) both for originally foliated and isotropic fabrics. The ranges chosen to discriminate the different stages of deformation derive from the microstructural observation. Two thresholds were observed: the first is approximately 20%, under which the new planar fabric is not persistent; and the second is approximately 60%, under which the new planar fabric is developed but it is still possible to recognize the enveloped previous one; over about 60% the new fabric is pervasive and only relicts of the previous fabrics are preserved. The first stage (LD, low degree of deformation) corresponds to the early development of the new fabric (degree of fabric evolution 0–20%); this includes an incipient crenulation or the appearance of a non-persistent new foliation for the country metapelites and for the meta-intrusives, respectively. The second stage (MD, medium degree of deformation) corresponds to a successive evolution up to the differentiation of a new foliation, which can reach the stage of a differentiated crenulation cleavage or a pervasive foliation (20 –60% fabric evolution degree), in originally foliated or isotropic rocks, respectively. The last stage (HD, high degree of deformation) coincides with the progressive obliteration of the relicts of earlier fabrics in metapelites and meta-intrusives (crenulated foliation remnants in microlithons or the presence of magmatic porphyroclasts, respectively) by complete grain-scale reworking and development of new continuous foliations (c. 60 –100% overprinting of the previous fabric). Ranges of volume percentage occupied by Alpine mineral assemblages synkinematic with D3 and D4 þ D5 are also indicated in Figure 6 (metamorphic transformation, M.T.), at each stage, for both originally foliated and isotropic rocks. The first stage is characterized by an average of 20%, with a maximum value of 50%, of synkinematic metamorphic transformations in originally foliated rocks, and by an average of 40%, with a maximum of 60%, of metamorphic transformation in originally isotropic rocks; the second stage shows an average metamorphic transformation of 45 and 55% in originally foliated and isotropic rocks, respectively, with a maximum of 75% for both; in the last stage, the mineral– chemical re-equilibration can reach up to 100% of the volume in both cases (M.T. ranging between 70 and 100%). In addition to the catalysing effect of deformation, a significant role is played by the thermal regime under which deformation and associated metamorphic transformations occur, with the

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condition that the fabric evolution remains below the HD stage (60%). For the same degree of fabric evolution, the metamorphic reaction progress is more evolved during D3 (IT– HP) than during D4 or D5 (LT –LP). At the LD stage (fabric evolution (F.E.) of 0 –20%, Fig. 6) the mean value of M.T. for originally foliated and isotropic rocks is about 31% during the IT–HP D3 phase and 22% during the LT –LP D4 þ D5 phases; correspondingly, at the MD stage (F.E. of 20–60%) the mean value of the M.T. is 60% during D3 and 37% during D4 þ D5. Generally, the degree of metamorphic transformation is dependent on the original composition of the rocks: for example, the metadiorites show M.T. values higher than the metagranitoid rocks at the same degree of F.E. and P –T conditions. The resulting microstructural semi-quantitative scheme synthesized in Figure 6 highlights the relationship between fabric evolution and degree of metamorphic transformation: the two processes do not necessarily develop at the same rate, but an F.E. threshold of approximately 60% appears to coincide with the transition to the HD stage, above which the mechanical and mineral–chemical transformations of the rocks increase proportionally; only above the 60% threshold is the related synkinematic mineral assemblage capable of reaching 100% of the total volume.

2D dominant fabric domain map The rock volumes that reached an F.E. of at least 60% are shown in a map (Fig. 7a, b) that is divided into different HDi domains (high degree of grain-scale deformation during the i-phase of deformation), each corresponding to the most intensely reacting rock volumes during each deformation phase. The size of the HDi domains is related to the interpreted geological map scale (1:10 000). According to the previous microstructural observations of HDi domains, the metamorphic reactions coeval with the i-phase of fabric development were nearly totally accomplished and related mineral assemblages occupy a mean volume 75% (Fig. 6). The representation of the dominant fabric domains has been simplified by grouping the pre-Alpine structures (D1 þ D2) and the two greenschist-facies Alpine deformation phases (D4 þ D5). D1 structures are, in fact, poorly preserved mainly as metre-scale relict fold hinges in D2; similarly, D5 structures are rare and extremely localized. Boundaries between different domains that homogeneously recorded deformation were traced based on estimates of the fabric evolution and of the degree of metamorphic transformation at the meso- and micro-scale for each deformation

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Fig. 7. 3D prospective of the map of dominant fabric domains. (a) Downwards view and (b) from the south. Three main domains were recognized: the pre-Alpine HD(1þ2) domain, corresponding to the high concentration of pre-Alpine D1 þ D2 deformation at the grain-scale; the HD3 domain corresponding to the high concentration of D3 (Alpine) at the grain-scale; and the HD(4þ5) domain corresponding to the high concentration of D4 þ D5, greenschist-facies, Alpine-related deformation at the grain-scale. In the HD3 and HD(4þ5) domains a few lenticular relicts are preserved in which syn-D1 þ D2 and syn-D3 tectonitic–mylonitic fabrics are dominant, respectively. (c) Subsurface construction of the surfaces bounding HDi domains based on their intersection with the topographic surface (boundary) and the projected traces of the buried surface on each cross-section. (d) Triangulated buried surface obtained by interpolating the constraints shown in (c). (e) and (f) Structural model of the map of the dominant fabric domains showing the bounding buried surfaces: 3D prospective view from (e) the west and (f) the SW.

phase, according to the subdivision of Figure 6. Bounding surfaces of each domain are enveloping surfaces, compatible with the structural pattern represented on the foliation trajectory map. The precision of the boundaries ranges from 5 m, where the structures are widely exposed, to 100 m, where the outcrops are scanty and the degree of interpretation is higher. Three dominant fabric domains were recognized in the southernmost portion of the Languard– Tonale TMU (Fig. 7a, b): southwards a pre-Alpine dominant fabric prevails (D1 þ D2); the central part is characterized by a D3 Alpine dominant fabric; and the northern part by the greenschist D4 þ D5 fabrics. A few 10 m-scale HD(1þ2) and HD3 relicts are preserved in the HD3 and in the HD(4þ5) domains, respectively (Fig. 7a, b). The HD(1þ2) domain is characterized by a pervasive differentiated S2 foliation, in places continuous,

developed under HT –LP metamorphic conditions. This domain includes the low-strain D3 domains, where S2 is locally folded without the development of an axial-planar S3 foliation. In the HD(1þ2) domain the Alpine mineralogical assemblages represent a low volume percentage (25% mean value), showing incomplete metamorphic transformations such as pseudomorphic replacement of the pre-Alpine minerals or fine-grained reaction rims. In HD3 domains the earlier HP Alpine foliation (S3) is dominant and associated with a foliated to mylonitic fabric. The HD(4þ5) domain is characterized by the prevalence of D4 with minor D5 Alpine structures developed at LT –LP conditions: the dominant S4 differentiated foliation with mylonitic fabric was locally folded during D5. In the Alpine domains the relict pre-Alpine fabrics are scarce or absent. Moreover, in the northern HD(4þ5) and in the HD3 domains some lenticular

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Fig. 7. Continued.

low-strain domains ranging from 1 to 100 m-scale were recognized in which syn-D3 and syn-D2 structures are dominant, respectively (Fig. 7a, b). Comparison of the tectono-metamorphic map (Fig. 3) with the map of the dominant fabric domains (Fig. 7) clearly shows that the distribution of the high- and low-strain domains is not controlled by lithology as HDi domains cross-cut lithological boundaries. This may be related to strain softening that resulted from strain localization during successive deformation stages, making the originally competent rocks, such as diorites or granulites, softer by grain-size reduction along deformation zones that may develop independently of preexisting lithological boundaries.

3D dominant fabric domain model The map of dominant fabric domains (Fig. 7a, b) makes the 3D reconstruction of the different HDi domains feasible, considering only the surface data, when this is made possible by using a DTM of the topographical surface (Digital Elevation Model, from www.cartografia.regione.lombardia.it),

and the lithological and structural field data stored in the georeferenced database. The main constraints used for the subsurface reconstruction are the boundaries between the HDi domains projected from the topographical surface onto a set of parallel north– south cross-sections, chosen perpendicular to the regional trend of Alpine fold axes. For each cross-section the topographical profile, its intersection with the dominant fabric domain boundaries and the projection of structural measurements onto the plane of section were automatically obtained (Zanchi et al. 2009). As much as possible, the geological cross-sections honour the mesostructural data, the fold geometry and the deformation style of each of the superimposed deformation phases. A test of the geometric consistency of the geological structures in the cross-sections is performed with an iterative process through 2D and 3D. This procedure consists of: (i) interpretation of the 2D parallel geological sections; (ii) import of the 2D sections in the 3D modeller software; (iii) a check of the geometrical coherence between the adjacent sections (e.g. axes or axial plane surfaces continuity); (iv) export from 3D to 2D

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environment for editing the possible mistakes; and (v) return to the 3D software. Sometimes, this procedure induces a check of the interpretation of the geological boundaries on the map. Buried surfaces are reconstructed by projecting the boundaries between HDi domains on the topographic surface onto each cross-section, as well as important structural elements such as the fold hinges; successively, the geometry of the buried surfaces fitting these constraints are interpolated (Fig. 7b, c). Reconstruction of buried surfaces is an excellent tool for the 2D geological interpretation test: the 3D representation of the geological structure highlights the geometric inconsistencies of the interpretation. This may require the rigorous revision of the 2D interpretative tectono-metamorphic map, the map of the dominant fabric domain and the cross-sections. In detail, the contact surface striking NE–SW between the northern HD(4þ5) and the HD3 domains dips southwards at approximately 608 (Fig. 7e, f ). The boundary shows a wide, gentle anticline northwards, coherent with that described by Viola et al. (2003). The HD(1þ2) domain is bounded to the north by a surface that strikes

east –west and has a steep dip southwards that decreases with depth, and to the south by a surface that strikes NE–SW and dips 708 towards the south; this surface also bounds the southern HD(4þ5) domain (Fig. 7e, f). In summary, the modelled portion of the Languard –Tonale TMU is bounded northwards and southwards by two HD(4þ5) domains, while in the central zone D3 structures are dominant and a lenticular pre-Alpine HD(1þ2) domain is preserved (Fig. 7e, f ).

Volumetric estimate of the dominant fabric domain The reconstructed buried surfaces, corresponding to the boundaries between HDi domains, constitute the 3D topological model. As already pointed out in the third phase, we assume that the degree of F.E. for any particular deformation phase is homogeneously distributed at the map-scale, and is therefore strongly influenced by the chosen representation scale. From this, the reconstruction of the volumetric model is possible in which the whole volume is represented through a discrete grid, made up of 3D equal volume cells (Fig. 8).

Fig. 8. (a) 3D volumetric model of the map of the dominant fabric domains of the southern part of the Languard– Tonale TMU (about 53 km3). (b) HD(4þ5) dominant domains corresponding to about 52% of the whole volume; (c) the HD3 domain corresponding to about 37%; and (d) the pre-Alpine HD(1þ2) domain corresponding to about 12%.

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The grid is intersected by the reconstructed buried surfaces, which allows computation of the volumes, corresponding to volumes of the different dominant fabric domains. The total volume of the Languard– Tonale TMU considered in this study equates to approximately 53 km3. The volumes estimated for the three domains are: 12% for the HD(1þ2) domain, of which 0.2% is represented by the lenticular relics preserved within the HD3 domain; 37% for the HD3; and 51% for the HD(4þ5) domains. Using the fabric evolution v. metamorphic transformation relationships synthesized in Figure 6, it is possible to estimate the minimal volumes occupied by the mineral assemblages synkinematic with each group of structures characterizing the three HD domains: pre-Alpine assemblages (synD1 þ D2) occupy a minimal volume ranging from 7 to 12%; early Alpine assemblages (syn-D3) occupy a minimal volume of 22–37%; and the late Alpine assemblage (syn-D4 þ D5) represents a minimal volume of 30– 51%.

Conclusions From the micro- and mesostructural analysis performed in this volume of Alpine continental crust, which was repeatedly tectonized in an active margin during Permian extension and Alpine subduction, it appears that the dominant metamorphic imprint corresponds to that associated with the most pervasive strain fabric. This is in agreement with other results from the Alpine belt (Spalla et al. 2005) where the recognition of tectonic units that record a common structural and metamorphic evolution (TMU) has been similarly based on detailed structural mapping and microstructural analysis. The construction of the map of dominant fabric domains, based on the degree of fabric evolution, has demonstrated that the localization of the highstrain domains is not controlled by the lithological setting, because high-strain domains of successive deformation stages appear to be discordant with lithological boundaries. The correlation between the degree of fabric evolution and the progress of synkinematic metamorphic reactions in different lithologies has shown that differences in original mineral assemblages and fabrics (i.e. originally foliated or isotropic) exerts more of an influence on the degree of reaction accomplishment. For instance, originally isotropic meta-intrusives achieved a greater degree of mineral transformation, at low and medium degree of deformation (Fig. 6), than the originally foliated country rocks at the same degree of deformation. At a high degree of deformation (HD) the

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related metamorphic transformations are similar in spite of their different original texture or mineral composition. Fabric evolution and degree of metamorphic transformation do not increase proportionally at LD and MD stages. There is a threshold at the transition to the HD stage (c. 60% evolution) above which mechanical and chemical transformations increase proportionally and the synkinematic minerals can achieve total replacement of pre-existing mineral phases. These results highlight the influence of strain energy as a catalyst on metamorphic transformation, as discussed by Hobbs et al. (2010). Temperature can also catalyse the progress of metamorphic reactions: D3 took place under epidote– amphibolite-facies conditions (IT –HP) that yielded a greater degree of synkinematic metamorphic transformation than during D4 and D5, corresponding to lower greenschist-facies conditions (LT –LP). The 3D modelling permits us to obtain the volumetric estimate of the degree of fabric evolution and associated metamorphic recrystallization of rock volumes within a TMU. Only half (51%) of the total rock volume was mechanically and chemically re-equilibrated during the late stages of the tectono-metamorphic evolution (D4 and D5 under LT –LP metamorphic conditions); one-tenth of the total volume preserves the structural and metamorphic imprints related to the earlier stages (D1 and D2 pre-Alpine deformation phases), presumably due to a very poor mechanical reactivation during the whole Alpine orogeny. The diffuse heterogeneity of textural and metamorphic imprints requires that the characterization of TMUs be based on detailed field mapping over an area reaching a critical size, which depends, in part, on the scale of strain partitioning in the region. Results from this type of detailed field and laboratory procedure are useful to refine, constrain and verify geophysical modelling that simulates the mechanical behaviour at active plate margins, assuming the changes of continental or oceanic rheology on the basis of a full accomplishment of the predicted phase transitions that drive changes of the dominant active deformation mechanisms. Our 3D modelling allows an estimation of the volumes preserving textural and mineral relicts after phase transitions, and may help to evaluate the potential influence that relict domains have on the choice of the physical parameters for thermomechanical modelling, such as density or viscosity. The authors thank D. Gibson and R. Trouw for their constructive criticisms and helpful suggestions. This work was developed within the gOcad Consortium; A. Zanchi is also thanked. Funding by FIRST07-08 of the Universita` degli Studi di Milano and CNR-IDPA. America Journal Experts Association provided the English revision of the text.

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The interaction of deformation and metamorphic reactions BRUCE E. HOBBS1,2*, ALISON ORD1,2, MARIA IOLE SPALLA3, GUIDO GOSSO3 & MICHELE ZUCALI3 1

CSIRO Exploration and Mining, PO Box, 1130 Bentley, Western Australia 6120, Australia 2

School of Earth and Geographical Sciences, The University of Western Australia, Perth, Western Australia, Australia

3

Dipartimento di Scienze della Terra, Universita` degli Studi di Milano, Via Mangiagalli 34, 20133 Milano, Italy *Corresponding author (e-mail: [email protected]) Abstract: Feedback relations between deformation and metamorphic mineral reactions, derived using the principles of non-equilibrium thermodynamics, indicate that mineral reactions progress to completion in high-strain areas, driven by energy dissipated from inelastic deformation. These processes, in common with other time-dependent geological processes, lead to both strain, and strain-rate, hardening/softening in rate-dependent materials. In particular, strain-rate softening leads to the formation of shear zones, folds and boudins by non-Biot mechanisms. Strain-softening alone does not produce folding or boudinage and results in low-strain shear zones; strain-rate softening is necessary to produce realistic strains and structures. Reaction– mechanical feedback relations operating at the scale of 10–100 m produce structures similar to those that arise from thermal– mechanical feedback relations at coarser (kilometre) scales and reaction–diffusion– mechanical feedback relations at finer (millimetre) scales. The dominance of specific processes at various length scales but the development of similar structures by all coupled processes leads to scale invariance. The concept of non-equilibrium mineral stability diagrams is introduced. In principle, deformation influences the position of mineral stability fields relative to equilibrium stability fields; the effect is negligible for the quartz ! coesite reaction but may be important for others. Application of these results to the development of structures and mineral reactions in the Italian Alps is discussed.

Metamorphism is commonly associated with deformation and regional thermal gradients, together with chemical reactions, all of which are non-equilibrium situations. In many instances, another non-equilibrium process, namely fluid flow (either of melt or fluids comprised of H2O, CO2 and so on), accompanies the metamorphic process and is intimately coupled to deformation, infiltration, heat advection and chemical reactions. This paper explores the influence of energy dissipated by such processes upon: (i) the development of meso-scale structures commonly observed in deformed metamorphic rocks; (ii) the kinetics of metamorphic reactions; and (iii) the thermodynamic stability of metamorphic assemblages. To focus the discussion we concentrate on metamorphic rocks undergoing deformation and chemical reactions with no large-scale mass infiltration (i.e. metasomatism) so that fluid flow is considered insignificant and also any regional thermal gradients are neglected. As such, the energy dissipated during deformation and metamorphism consists of four parts: (i) that due to mechanical processes; this

comprises dissipation arising from inelastic deformation and from the work done in forming chemical components in the deforming –reacting system by mineral reactions; (ii) that arising from the diffusive flux of chemical components across gradients in both chemical potentials and local gradients in temperature; (iii) that arising from chemical reactions; and (iv) that arising from thermal conduction. In the absence of metasomatism at the outcrop scale, many effects arising from mass flux and thermal transport turn out to be unimportant. An important ingredient of the interaction between these dissipative processes is that of spatial scale, so that various processes dominate at different length scales. Thermal feedback effects dominate at scales larger than the outcrop scale, and mass transfer feedback effects dominate at the micro-scale. The dominance of these different processes at specific length scales and the development of similar geological structures at all scales by different coupled processes are the two principles at the heart of the scale invariance of structures observed in geology. This separation of spatial scales also

From: SPALLA , M. I., MAROTTA , A. M. & GOSSO , G. (eds) Advances in Interpretation of Geological Processes: Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, Special Publications, 332, 189–223. DOI: 10.1144/SP332.12 0305-8719/10/$15.00 # The Geological Society of London 2010.

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greatly simplifies any study of the coupling effects. We include a brief consideration of the influence of metasomatism in the Discussion section. In many metamorphic rocks there is evidence of non-equilibrium in the form of partially reacted mineral assemblages. An example is the Monte Mucrone–Monte Mars region in the Italian Alps (Sesia –Lanzo Zone: Zucali et al. 2002) where igneous relicts are preserved in a subduction complex; here in rocks immediately adjacent to each other, the metamorphic reactions reach completion only in highly deformed shear zones which are ubiquitous. In such rocks the progress of mineral reactions correlates with the amount of strain and not necessarily with the maximum pressure and temperature (P, T ) conditions reached. The important questions are: (i) What roles do mineral reactions play in promoting deformation and influencing the style of deformation, particularly the development of shear zones, folds and boudinage? (ii) What role does deformation play in promoting metamorphic reactions? (iii) In regions where the metamorphic reactions have not proceeded to completion and deformation accompanies the reactions, are estimates of pressure and temperature conditions derived from equilibrium theory relevant and/or accurate, and what influence does deformation have on the position of the mineral phase boundary in pressure –temperature space established by using equilibrium thermodynamics? The questions are summarized in Figure 1. Although there is a wealth of literature published on these issues, there is no discussion that integrates them within one framework. We attempt to do this within the concepts of modern non-equilibrium thermodynamics. The structure of the paper is as follows. We first outline the observations of partially completed metamorphic reactions and the relationship to deformation observed in the Italian Alps. We then briefly review the development of non-equilibrium thermodynamics as it is relevant to the coupling of geological deformation and metamorphic reactions. Next we consider the influence of metamorphic reactions on constitutive behaviour, together with

Fig. 1. The interplay between deformation and metamorphic mineral reactions.

the influence of mineral reactions on deformation with particular emphasis on localization, folding and boudinage. The concept of non-equilibrium mineral phase diagrams is then introduced, and we discuss the influence of deformation on mineral stability. Finally, we consider the implications of this non-equilibrium framework for metamorphic rocks in general, and the Italian Alps in particular. Terms are defined as they are introduced in the text and in Table 1.

Examples of extent of reactions as a function of degree of deformation from the Italian Alps Western Alps Within the Monte Mucrone –Monte Mars area of the Sesia –Lanzo Zone (internal part of Western Italian Alps, location 1 in Fig. 2) six periods of deformation and related metamorphism (commonly associated with incomplete mineral reactions) are recorded (Zucali et al. 2002). The first and oldest (pre-D1) of these is a pre-Alpine high temperature –low pressure (HT–LP) event, and five subsequent events are recorded during the Alpine evolution in which D1 corresponds to the P– T prograde path. D2 and D3 structures developed under Tmax –PTmax conditions and are dated at approximately 65 Ma, whereas D5 is earlier than 30 Ma. Events D2 –D5 represent a progressive evolution in P– T conditions from approximately 2 GPa and 550 8C for D2 to 0.4 GPa and 200 8C for D5, associated with the exhumation after subduction of this portion of Austroalpine continental crust (Fig. 3a). The Monte Mucrone–Monte Mars is a single tectono-metamorphic unit (Spalla et al. 2005) in which the heterogeneous distribution of deformation and metamorphic imprints can be represented in a map of deformation imprints (Fig. 3b) emphasizing the metamorphic assemblages that predominate in each region. The important point is that the dominant metamorphic imprint in a particular region is not coincident with the P –T peak of the tectono-metamorphic unit but rather is the one associated with the most pervasive deformation of each region (Spalla et al. 2005). The dominant metamorphic imprint is also better developed as the intensity of the most pervasive deformation event increases with a direct correlation at the micro-scale with the progressive development of schistosity, as proposed by Bell & Rubenach (1983). The progress towards completion of mineral reactions is distributed in a patchy manner with lozenge-shaped, low-strain regions of very little mineral reaction embedded in an anatomosing network of highstrain shear zones that are apparently unrelated in

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Table 1. Symbols used in this paper, units and typical values Quantity A A B c cp Da E Gr g h h h chemical J2 JK k L l thermal l chemical M mK N Pe thermal P p Q Q mechanical Q chemical q R s s˙ T Tc t u u xi g 1ij 1elastic ij 1viscous ij 1_ ij _ H@

h hchemical kthermal kchemical kij l@ mK v vK v0

Description

Typical values, Units

Affinity of mineral reaction Affinity vector in stress space Constant Concentration of chemical species Specific heat at constant pressure Damkohler Number Young’s modulus Gruntfest Number Acceleration due to gravity Depth below surface of Earth Mechanical hardening coefficient Chemical hardening coefficient Second invariant of the deviatoric stress Mass flux of K-th chemical component Shear zone thickness Pre-exponential constant Thermal diffusion length scale Chemical diffusion length scale Power-law dependence of chemical reaction rate on strain rate Concentration of K-th chemical component Stress power-law exponent Thermal Peclet Number Pressure Mean stress Power-law dependence of strain hardening Activation enthalpy Activation enthalpy for mechanical deformation Activation enthalpy for mineral reaction Power-law dependence of strain-rate hardening Gas constant Specific entropy Rate of specific entropy production Absolute temperature Critical temperature where thermal–mechanical feedback becomes important Time Specific internal energy Local material velocity vector Spatial co-ordinates Integrated shear strain Strain tensor Elastic strain tensor Viscous strain tensor Strain-rate tensor Rate of heat production from latent heat production from the @-th mineral reaction Mechanical viscosity Chemical viscosity Thermal diffusivity Chemical diffusivity Diffusivity tensor Dimensionless group that is an expression of dissipation of the @-th process Specific chemical potential of K-th, chemical component Specific volume Specific volume of K-th chemical component Specific volume under hydrostatic stress

J kg21 J kg21 Pa s mol kg21 1450 J kg21 K21 Dimensionless 4.5  109 Pa Dimensionless 9.8 m s22 m Pa Pa Pa2 kg m22 s21 m Pa2N s21 m m Dimensionless kg m23 Dimensionless Dimensionless Pa Dimensionless J mol21 J mol21 J mol21 Dimensionless 8.3143 J K21 mol21 J kg21K21 J kg21 K21 s21 K K s J kg21 m s21 m Dimensionless Dimensionless Dimensionless Dimensionless s21 J kg21 s21 Pa s Pa s 1026 m2 s21 m2 s21 m2 s21 Dimensionless J kg21 m3 kg21 m3 kg21 m3 kg21 (Continued)

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Table 1. Continued Quantity

j j˙ r sij s ij0 s F Fmechanical Fchemical Fdiffusive Fthermal w x C v vchemical @

Description

Typical values, Units

Extent of chemical reaction or other process Rate of chemical reaction Mass density Cauchy Stress tensor Deviatoric stress tensor Stress vector in stress space Specific dissipation function Specific mechanical dissipation Specific chemical dissipation Specific diffusive dissipation Specific thermal dissipation Thermal-hardening parameter Thermal efficiency Specific free energy of the system Strain-rate-hardening parameter Chemical strain-rate-hardening parameter Number of mineral reactions or processes

orientation to the axial planes of folds, and within which the coeval mineral reactions have progressed to completion.

Central Alps A belt of steep, thin slices of continental crust bounds the Periadriatic Lineament (internal part of Central Italian Alps, location 2 in Fig. 2) and contains portions of Austroalpine crust as the Languard–Tonale tectono-metamorphic unit where Alpine and pre-Alpine structures have been distinguished by their relationships with Permian intrusives (Gazzola et al. 2000). The pre-Alpine deformations (D1 and D2) took place under amphibolite–granulite-facies conditions, and the most pervasive S2 pre-Alpine fabric is marked by mineral assemblages peculiar to a HT –LP imprint of Permian age. The Alpine structures developed under epidote –amphibolite-facies (D3) and greenschist-facies conditions (D4) and are recorded during the progressive P –T evolution from approximately 1.2 GPa and 500 –600 8C, representing the Alpine metamorphic climax, to 0.5 GPa and 350 8C, characterizing the retrograde exhumation path (Fig. 4a). Structural mapping assisted by microstructural and petrological analysis indicates that since the Permian the deformation history and associated metamorphic evolution has been spatially and temporally consistent throughout the Languard– Tonale package, which therefore represents a single tectono-metamorphic unit throughout its postPermian evolution (Gosso et al. 2004; Spalla et al. 2005; Salvi et al. 2010). Also in this case, as in

Dimensionless s21 2750 kg m23 Pa Pa Pa J kg21 s21 J kg21 s21 J kg21 s21 J kg21 s21 J kg21 s21 Pa K21 Dimensionless J kg21 Pa s Pa s Dimensionless

the Western Alps, the distribution of successive deformation and metamorphic imprints is highly heterogeneous. At the micro-scale (Fig. 4b), progress to completion of mineral reactions is achieved in high-strain zones where the fabric is mylonitic; this is true for each rock type of the lithostratigraphical association comprising metapelites, metaintrusives, marbles and metabasics (Zucali 2001). Figure 4b displays the progressive decrease of the modal amount of pre-Alpine minerals from the poorly deformed (coronitic, Fig. 4b1, b2) to the highly deformed (mylonitic, Fig. 4b4–b6) metapelites, where pre-Alpine relicts are less than 10% in volume. In this case again, different dominant metamorphic imprints coexist in spatially adjacent volumes of a single tectono-metamorphic unit and they do not reflect the P–T peak reached within the tectono-metamorphic unit, but correspond to the metamorphic conditions associated with the most pervasive fabric of each region. An important observation is that the microstructures of less deformed domains (Fig. 4b1–b3) in many cases suggest that grain-size reduction resulting from the nucleation of new phases is not sufficient to localize granular-scale deformation.

Examples of natural synmetamorphic shear zones, folds and boudins in contrasted thermal environments from the Italian Alps Shear zones Many Alpine shear zones display V and Y shapes in addition to the expected X shapes where conjugate shear zones intersect (Fig. 5).

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Fig. 2. Tectonic map of the Alps with the location of the natural examples described in the text (numbered black stars). Legend: 1, Southalpine basement; 2, Austroalpine basement; 3, Penninic basement; 4, Helvetic– Dauphinois–Provenc¸al basement; 5, Tertiary intrusive stocks.

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Fig. 3. (a) P –T– d–t path inferred from rocks of Monte Mucrone– Monte Mars area (simplified after Zucali et al. 2002). Pre-D1, D1, D2 þ D3, D4 and D5 represent P– T conditions estimated with respect to the successive deformation periods (d). Pre-Alpine and Alpine P –T– t– d evolutions are compared with Vi (stable geotherm after Thompson & England 1984). (b) Map of deformation imprints, simplified after Zucali et al. (2002) with domains recording the same relative timing of superposed structures. The areal extent of every new planar synmetamorphic fabric (dominant fabric) has been estimated and reported in the legend. Close-up displays the petrographic–structural map between Monte Mucrone and Monte Mars (Zucali et al. 2002). Legend: 1, metagranitoids with coronitic fabrics; 2, micaschists and paragneisses; 3, metagranitoids showing mainly mylonitic fabrics, with minor tectonites; 4, eclogites.

In the Monte Mucrone area (location 1 in Fig. 2) Y-shaped shear zones localize (Fig. 5a1) in the eclogitized metagranitoids during the early deformation period under eclogite-facies conditions. Starting from the Permian igneous assemblage (plagioclase, quartz, biotite, K-feldspar and accessory minerals)

the new eclogite-facies minerals ( jadeite-rich clinopyroxene, zoisite, phengite, garnet, K-feldspar and quartz) developed both in poorly deformed granitoids as coronas (Fig. 5a2) or as fine-grained aggregates or trails in the mylonitic bands (Fig. 5a3). In the weakly deformed rocks the modal amount of

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Fig. 4. (a) P– T– t– d path inferred for the Alpine-subducted metaintrusives and surrounding metapelites of the Languard–Tonale tectono-metamorphic unit (redrawn after Gazzola et al. 2000; Gosso et al. 2004). D2, D3 and D4 represent P– T conditions estimated with respect to the successive deformation periods; pre-Alpine and Alpine P– T–t –d evolution compared with Vi (stable geotherm after Thompson & England 1984). (b) Alpine microstructures characterizing the IT –HP metamorphic re-equilibration in metapelites from the Languard–Tonale tectono-metamorphic unit. Weakly deformed domains. (b1) Coronas of Alpine garnet (Grt II) around pre-Alpine relict garnet (Grt I); Alpine phengitic white mica (Wm) and ilmenite (Ilm) partly replace pre-Alpine biotite (Bt), and the plagioclase site is fully pseudomorphed by a thin aggregate of epidote and white mica. (b2) Pre-Alpine staurolite (St) totally replaced by a very thin aggregate of new Alpine minerals, easily detected with the SEM (photomicrograph b3); sillimanite, marking with biotite the pre-Alpine S2 foliation, is overgrown by small kyanite Alpine grains. (b3) BSE image of Alpine chloritoid (Cld) and kyanite (Ky) micro-aggregate occurring in microsites such as that of pre-Alpine staurolite in photomicrograph b2. Highly deformed domains. (b4) The S3 pervasive foliation is marked by an Alpine phengitic white mica (Wm) and chloritoid (Cld) shape-preferred orientation; garnet porphyroblasts (Grt) preserve the pre-Alpine compositions at the cores, which represent the only pre-Alpine relicts in these rocks. (b5) The Alpine mylonitic foliation is marked by ribbon-quartz, phengitic white mica and trails of new garnet (Grt II); pre-Alpine relicts exclusively consist of garnet porphyroclasts (Grt I). (b6) New bluish amphibole (Amp) underlines the Alpine mylonitic foliation, together with garnet (Grt), white mica and ribbon-quartz; pre-Alpine red biotite (Bt) is rarely preserved as small relict grains.

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Fig. 5. Examples of shear zones in the continental crust of the northern Sesia– Lanzo Zone of the Western Alps, eclogitized during Alpine subduction: (a) metagranitoids from Monte Mucrone and (b) eclogitized amphibolites from Valsesia. (a1) Convergent early Alpine shear zones developed under eclogite-facies conditions in a metagranitoid (base of the ENE slope of Monte Mucrone). In the shear zone (a3) the mylonitic fabric is associated with a widespread development of Alpine eclogitic minerals, such as garnet (Grt), phengitic white mica (Wm), jadeitic clinopyroxene (Cpx) and zoisite/clinozoisite (Ep); rare K-feldspar porphyroclasts (Kfs) represent the igneous pre-Alpine relicts (here c. 3% in volume). In the weakly deformed domains (a2) the igneous texture, as well as the igneous minerals (Kfs, Qtz and Bt), are well preserved (c. 60% in volume) and Alpine eclogitic minerals (Grt, Wm, Cpx, Ep) are confined to thin coronas around igneous biotite (Bt) and to the plagioclase (ex-Pl) microsite. (b1) Relicts of pre-Alpine amphibolites from middle Val Sesia (redrawn after Lardeaux & Spalla 1991). In the shear zone (b2) pre-Alpine relicts are preserved (small grains of brown-green hornblende) only as inclusions in the omphacite porphyroblasts; the fine-grained foliated matrix consists of omphacite, phengite, zoisite, blue-green amphibole and minor garnet that occur also within porphyroblasts. Conversely, outside the shear zone (b3) the pre-Alpine granoblastic fabric and mineral assemblages (brown hornblende) are still preserved; the plagioclase microsite is fully pseudomorphosed by a thin aggregate of zoisite, jadeite and white mica.

the preserved igneous relict minerals can reach 70% in volume, whereas in the high-strain domains, where the fabric is mylonitic, the HP Alpine minerals can occupy more than 90% in volume. The paragenesis observed in shear zones, or as coronas in the

weakly deformed rocks, and the related mineral compositions indicate that phase transitions occurred under consistent P –T values in eclogite-facies conditions (Fig. 3a) (Fru¨h-Green 1994; Rubbo et al. 1999; Tropper et al. 1999; Zucali et al. 2002).

DEFORMATION AND METAMORPHIC REACTIONS

In the northern part of the Sesia–Lanzo Zone (middle Val Sesia, location 3 in Fig. 2) Y- and X-shaped shear zones developed in the pre-Alpine granulites and amphibolites (Fig. 5b1) during the early Alpine deformation periods under eclogitefacies conditions (Lardeaux & Spalla 1991). As in the previous case, the modal amount of eclogitic Alpine minerals increases strongly from nearly undeformed eclogitized amphibolites to mylonitic eclogites (Fig. 5b2, b3). Also in this case, thermobarometric estimates performed on the weakly (coronitic) and highly (mylonitic) deformed rocks yield coherent P–T estimates. In the Languard– Tonale unit of the Central Alps (location 2 in Fig. 2) Alpine deformation, occurring under epidote–amphibolite-facies conditions, localizes in metre- to 10 m-scale Y-shaped shear zones in Permian intrusives (Fig. 6a). Here, as pointed out in the two previous cases, a direct relationship between deformation progress and reaction accomplishment exists, and is indicated also by the matching of fabric gradients with metamorphic reequilibration as evidenced by comparisons of quantitative fabric data with compositional variations in amphiboles of metadiorites (Spalla & Zucali 2004). In the low-strain domains of metadiorites (Fig. 6b) the Alpine IT–HP minerals (blue-green hornblende, albite-rich plagioclase, phengitic white mica, garnet, Mg-rich chlorite, quartz, epidote and ilmenite) occupy no more than 20% in volume with respect to the igneous mineral relicts (brown

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hornblende, plagioclase, biotite, quartz and accessory minerals). In the mylonitic metadiorites (Fig. 6c) the modal amount of Alpine IT–HP minerals increases up to more than 80% in volume (Salvi et al. 2010). P– T estimates performed both on weakly deformed rocks and on mylonites give very close values (P ¼ 1.1 + 0.2 GPa and T ¼ 500– 600 8C) for the syn-D3 mineral assemblages (Gazzola et al. 2000). In the Oetztal nappe (southern Venosta Valley, location 4 in Fig. 2) conjugate shear zones (Fig. 7a) are widespread in metagranitoids deformed under amphibolite-facies conditions during Alpine times (Spalla & Zucali 2004 and references therein). As already described for the previous examples, new Alpine minerals replace the pre-Alpine relicts that in D2 shear zones mainly consist of partly recrystallized K-feldspar porphyroclasts (Fig. 7b, c). Mineral assemblages developing in D2 shear zones are compatible with parageneses observed in normally foliated L– S tectonites and coronites. P–T estimates derived from metagranitoids and from the metapelitic country rocks indicate that D2 took place at 550–600 8C and 0.6–0.8 GPa.

Folds Natural cases of extreme thinning of fold limbs are frequent at the 10 m to km scales in a wide range of lithological associations and metamorphic

Fig. 6. (a) Form surface map of a Permian diorite deformed and metamorphosed during Alpine subduction (Gazzola et al. 2000; Gosso et al. 2004) at Monte Pagano, Central Alps. The metamorphic reactions developing during the early Alpine deformation (D3) reach completion in the shear zones (c) where the sole igneous relics are brown hornblende (Amp I) porphyroclasts representing approximately 30% of the mineral modal amount and wrapped by a mylonitic foliation contemporaneous with the growth of new amphibole (Amp II), white mica and garnet (Grt). In the weakly deformed domains the magmatic fabric is preserved and the Alpine transformations are only incipient (b).

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environments (greenschist, blueschist, eclogite to amphibolite –granulite) in the Alpine orogen. An example at the scale of 100–1000 m comes from the southern termination of the Schneeberg Complex, in the core of the Oetztal nappe (location 4 in Fig. 2), which consists of marbles, micaschists, gneisses and amphibolites, giving rise to the largest megastructure of the Schlingen Zone of the Eastern Alps (Fig. 8a). This rock association forms a multilayer unit several kilometres thick, which is folded under amphibolite- and successively greenschistfacies conditions during D2 and D3 deformation periods of Alpine exhumation, respectively (Helbig & Schmidt 1978; Hoinkes et al. 1987; van Gool et al. 1987; Spalla 1990). Extreme thinning of limbs in isoclinal fold systems, alternatively marked by micaschist layers within thick marbles (Fig. 8a close-up), or vice versa, is ubiquitous; amphibolite metamorphic conditions dominated the entire D2 deformation period. In the Languard –Tonale unit of the Central Alps (location 2 in Fig. 2) coupled scales of shortening (from kilometres to centimetres) are manifest in natural fold systems traced by schistosity (S3) that are overprinted by localized zones of intense development of a new foliation (S4) positioned as a conjugate axial-planar set (Fig. 8b). Strain accommodation within grain-scale layering during S4 is assisted by the replacement of IT –HP to LT – LP assemblages in schists (garnet, white mica, +chloritoid, +kyanite, +tschermakitic amphibole, plagioclase and quartz to chlorite, albite, epidote, white mica, +biotite), metadiorites (tschermakitic amphibole, plagioclase, phengitic white mica, zoisite/clinozoisite, garnet, quartz, +Mg-rich chlorite, +ilmenite to actinolitic amphibole, albite, white mica, epidote, chlorite, quartz, +biotite and titanite) and metagranitoids (plagioclase, phengitic white mica, zoisite/clinozoisite, garnet, minor bluegreen amphibole, quartz, ilmenite/titanite to chlorite, white mica, albite, quartz, epidote +biotite). In the high-strain D4 zones the metamorphic replacement of earlier mineral assemblages exceeds 65% in volume (Salvi et al. 2010).

Boudins Fig. 7. (a) Conjugate shear zones at the map scale in the Tschigat metagranitoids, east of Milchsee Scharte (Texel Gruppe, Oetztal nappe) above Merano, South Tyrol (Spalla 1990, 1993), developed under amphibolite-facies conditions during Alpine times, as testified at the micro-scale by mineral assemblages marking the mylonitic foliation within the shear zones (b, plane polarized light; c, crossed polars, with a slight birefringence compensation). Map legend: orange, metagranitoids; light brown, gneisses; violet, mylonitic bands; green, amphibolites; dashed bands, shear zones; light yellow, red dots and blue and red fans, Quaternary deposits.

In the axial part of the Alpine belt, synmetamorphic boudinage always displays coupling of the deformation gradient with progress of metamorphic transformations at boudin margins. Layered basic granulites of the Valpelline Series of the Dent Blanche nappe (location 5 in Fig. 2) are generally boudinaged in the high-grade Sill–Grt – Bt-bearing paragneisses during HT–LP pre-Alpine deformation. The Valpelline Series is mainly composed of metapelites, mafics and carbonate rocks, with a dominant metamorphic and structural

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Fig. 8. Natural examples of folding and boudinage. (a) Extreme thinning of fold limbs in a marble and schist sequence of the Schneeberg Complex SW termination (Schlingen zone, Oetztal nappe of the Eastern Alps, on SW face of Cima Fiammante, Schnalstal). Upper image: map view (in Spalla 1990); legend: turquoise, marbles; yellow, micaschists. Lower image: close-up of a cross-section of the fold system (arrow) of a natural slope dominated by light-coloured marbles (photograph width is 600 m long); slope orientation differs from that of the average profile plane of the fold system of +158. (b) Lozenge-shaped kilometre-scale LT– IP relict rocks (1 in the legend) carrying an internal fold system traced by a S3 foliation; it is dissected by conjugate sets of localized zones of newly generated S4 schistosity within the same rock types (2 in the legend). Sequence of deformation imprints (3 in the legend) in the Languard– Tonale unit (Austroalpine domain of the Central Alps) showed the existence of two deformation periods during pre-Alpine evolution and two others during Alpine convergence (D3 and D4; from Gosso et al. 2004). (c) Granulite- to amphibolite-facies transition synchronous with boudinage in metabasics from Valpelline Series (Dent-Blanche nappe, Western Austroalpine domain; Lac Mort locality in Valpelline). The boudin core consists of orthopyroxene, clinopyroxene, plagioclase, +amphibole + biotite, and displays granoblastic texture. Rims and boudin necks are constituted by an extremely foliated garnet-bearing amphibolite characterized by large plagioclase and dark amphibole grains that define the newly developing foliation (c2, close up of detail 2 in c1).

imprint at amphibolite- to granulite-facies conditions of pre-Alpine age (Nicot 1977; Gardien et al. 1994). Three deformation events are related to the pre-Alpine evolution and are characterized by a complex (Roda & Zucali 2008 and references therein) evolution from granulite to amphibolite conditions, predating the Alpine tectonometamorphic evolution, which is heterogeneously recorded. In the example of Figure 8c1, a lowpressure granulite-facies basic boudin, composed of Cpx–Opx þ Pl + Amp + Bt, formed during a higher-pressure –lower-temperature amphibolitefacies stage that produced an Amp þ Grt þ Cpx þ Pl + Bt-bearing metamorphic assemblage. The mineral reaction zone migrates from the rim to the core of the basic boudin (Fig. 8c2) and mimics

the fabric gradient: plagioclase and garnet porphyroblast growth is the most prominent result of this metamorphic transformation. Similar features characterize also the boudinage of eclogites (Fig. 9a) within the Eclogitic Micaschists Complex of the southern Sesia– Lanzo Zone (location 6 in Fig. 2). Here the early S1 foliation is preserved together with the associated eclogitic assemblage (omphacite, epidote, garnet, quartz, rutile) at the core of the boudins, which are from centimetre to decimetre in size. At the boudin margins a new planar fabric appears and intensifies towards the boudin margins, marked by the syn-D2 blueschist mineral assemblage (glaucophane, clinozoisite, quartz, garnet, +white mica and titanite) developed during the P-retrograde path of this

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Fig. 9. Natural examples of boudinage. (a1) Field structural setting within the Eclogitic Micaschists Complex at Alpe Mecio– Rocca Lunga, southern Sesia– Lanzo Zone (Western Austroalpine: Spalla & Zulbati 2003). Legend: light orange, quartz-rich eclogitic micaschists; orange, glaucophane-rich micaschists; brown, metagranitoids; violet, albite– epidote–chlorite schists; light blue, glaucophanites; red, pre-Alpine relict amphibolites. Foliation trajectories: red, S1 eclogite-facies foliation; blue, S2 blueschist-facies foliation; green, S3 greenschists-facies foliation; double line trajectories, mylonitic foliations; inferred trajectories are dashed. S1 syneclogitic foliation is preserved in the boudins and is variably reoriented with respect to S2 of glaucophanites (a2). Amphibole-free eclogite boudins (a3) progressively isolated within intensely foliated (S2) glaucophanites representing their transformation products. (b) Variscan amphibolitized eclogite boudin in the metatexites of the Argentera–Mercantour massif (Helvetic–Dauphinois– Provenc¸al domain of the external SW Alps). Strain gradient localized at the boudin margins (b1) corresponds to a modal mineral amount gradient with increase of amphibolite-facies minerals with respect to the eclogite-facies minerals, more abundant at the boudin core (b2).

portion of the Sesia –Lanzo Zone (Pognante 1989; Spalla & Zulbati 2003). This assemblage of the structural and mineralogical features seems to be related to a mechanism for the formation of conjugate shear zones, initially affecting an eclogite layer (Fig. 9a3) and localizing the initiation of its transformation into glaucophanites; successively, a boudin-like structure is intensified within a newly formed eclogite– glaucophanite multilayer (Fig. 9a2). In a HT–HP-dominated metamorphic environment, as for that of the Argentera –Mercantour massif (Helvetic –Provenc¸al domain of the external

SW Alps, location 7 in Fig. 2), a correlation may similarly be established between fabric intensification and metamorphic reaction progress. Variscan eclogites boudinaged within biotite-bearing metatexite layers (Fig. 9b1) display marked textural and grain-size variations from boudin core to boudin rims. The mineral assemblage of the poorly amphibolitized eclogite in the boudin core (garnet, omphacite – totally replaced by diopside þ plagioclase symplectite – hornblende, rutile, quartz and opaque minerals), where the texture is granoblastic and weakly foliated with a centimetre grain size, is gradually replaced by the new HT

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amphibolite-facies assemblage (hornblende II, plagioclase, quartz, opaque minerals) associated with a progressively stronger mineral layering and grain-size reduction, down to mm size (Fig. 9b2). Thus, there is a 1:1 correlation between both the relative and absolute intensity of a particular deformation and the tendency for a metamorphic reaction corresponding to the P–T conditions of that deformation to proceed to completion. This observation is not unique and has been discussed by many authors including Beach (1973, 1976, 1980), Rubie (1983, 1990), Austrheim & Griffin (1985), Mork (1985), Koons et al. (1987), Austrheim (1990) and Brodie & Rutter (1990). Examples from experiments on the influence of deformation on phase transitions and chemical reactions in mineral systems are reported by Coe & Paterson (1969), Brodie & Rutter (1985) and Delle Piane et al. (2007), and in metal systems by Levitas et al. (1998a, b). The influence of mineral reactions on the constitutive behaviour of materials has been explored by Shigematsu (1999), Burlini & Bruhn (2005) and de Ronde et al. (2005). A common response to the observation that deformation influences the progress of metamorphic reactions is that the deformation influences the kinetics of the reaction perhaps by decreasing the grain size or enhancing diffusion, particularly (but clearly not exclusively) that of water (Rubie 1983, 1990; Vernon 2004). These processes are undoubtedly important and are considered later, where we show that not only is there a direct influence of deformation on the kinetics of a reaction, there is also a small influence on the stability field of that reaction (at least at geological strain rates). More importantly, the feedback between the kinetics of mineral reactions on the deformation is fundamental in influencing the strain hardening/ softening, and particularly the strain-rate-hardening/ softening behaviour, and hence controls the development of structures such as shear zones, folds and boudinage that otherwise would not develop. In order to progress the discussion we proceed to consider some aspects of non-equilibrium thermodynamics.

Non-equilibrium thermodynamics and metamorphic petrology Non-equilibrium thermodynamics increasingly has a very wide application in many fields of science and engineering, and has proved invaluable in explaining aspects of the Earth’s climate (Paltridge 1975, 1978, 2001), the oceanic general circulation (Shimokawa & Ozawa 2002), the fundamental mechanical and chemical behaviour of materials (Ziegler 1983a, b; Kondepudi & Prigogine 1998;

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Houlsby & Puzrin 2006), and fluid flow in deforming reactive porous media (Coussy 1995, 2004). Apart from applications in metamorphic petrology in the 1970s and 1980s (Fisher 1970, 1973; Fisher & Lasaga 1981; Foster 1981; Joesten 1977) there has been relatively little application within metamorphic geology. There has been a growing interest in non-equilibrium thermodynamics with respect to the application of damage mechanics to seismology (Lyakhovsky & Ben Zion 1997; Main & Naylor 2008) and in structural geology/ geodynamics (Lehner & Bataille 1984; Shimizu 1992, 1995, 1997, 2001; Regenauer-Lieb & Yuen 2003; Hobbs et al. 2007, 2008; Ricard & Bercovici 2009; Hobbs & Ord 2010; Regenauer-Lieb et al. 2009). In this paper we set out to discuss some applications of non-equilibrium thermodynamics to deforming, reacting metamorphic systems at the outcrop scale. We make a distinction between classical equilibrium chemical thermodynamics, where minimization of the Gibbs Free Energy defines the stable states (Gibbs 1906), and non-equilibrium thermodynamics, where either minimization or maximization of the entropy production rate defines the evolution of the system (Zeigler 1983a; Kondepudi & Prigogine 1998). It is also important to make a distinction between the area of study concerned with the thermodynamics of elastic solids under the influence of non-hydrostatic stress (with and without fluids) and that concerned with nonequilibrium thermodynamics. The isothermal deformation of elastic solids by non-hydrostatic stress is reversible. Thus, much of the varied literature that is concerned with the chemical potential of a stressed elastic solid (e.g. Kamb 1959, 1961; Green 1970, 1980; Paterson 1973) is a part of equilibrium chemical thermodynamics, as is emphasized by McLellan (1980). We are concerned in this paper with dissipative deformations and mineral reactions where the subject is a part of non-equilibrium thermodynamics. The hesitation 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; care being taken to define meaningful thermodynamic state variables (Callen 1960; Kestin & Rice 1970; Shimizu 2001). This function is minimized at equilibrium, and so one can proceed to define equilibrium assemblages of minerals as discussed by many authors such as Kern & Weisbrod (1967). Another function, the entropy, is maximized at equilibrium. For non-equilibrium systems, it has never been clear, until recently, that a similar guiding principle was available. In fact, two apparently opposing views seemed to emerge in the

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literature. One was emphasized by Prigogine (1955), who claimed that in non-equilibrium systems the rate of entropy production is minimized. This view was also adopted by Biot (1965a, 1984). The other view is due to Zeigler (1983a), who claimed that the rate of entropy production is maximized in non-equilibrium systems. This apparent paradox is resolved when one understands that the Prigogine principle holds for thermodynamically linear systems at steady state, whereas the Zeigler principle is more general and holds for systems that are not constrained to be at steady state. The issue is discussed by Martyushev & Selezvev (2006). Both principles have a foundation in statistical mechanics (Dewar 2005), where it has been shown that a state of maximum entropy production rate is the most probable state for large systems not at equilibrium and not constrained to be at steady state. This opens the way to describe the evolution of geological systems that are maintained out of equilibrium by the continued supply of energy in the form of deformation, fluid flow, heat flow and chemical reactions (Hobbs & Ord 2010). The incorporation of non-hydrostatic stress into classical equilibrium thermodynamics was discussed by Gibbs (1906) for the situation where a fluid is in contact with a stressed solid. This approach was elaborated upon by Kamb (1959, 1961), Green (1970, 1980), Fletcher (1973) and Paterson (1973) in particular; the interface between the solid and fluid is particularly important in such circumstances, and the emphasis is on gradients in the chemical potential of the solid dissolved in the fluid induced by elastic deformation of the solid by variations in the normal stress across the interface. A number of other aspects of the thermodynamics of non-hydrostatically stressed solids (sometimes in the absence of fluids) were considered by Ramberg (1959), Bowen (1967), Bowen & Wiese (1969), Coe & Paterson (1969), Nye (1957), McLellan (1980), Truskinovskiy (1984), Bayly (1983, 1987, 1992) and Paterson (1995). Most of these discussions, although concerned with nonhydrostatic stresses, are in the realm of equilibrium thermodynamics where the entropy production rate is zero, and some are concerned with deriving the equilibrium thermodynamic properties (such as thermal expansion coefficient, specific heat, elastic moduli) for crystals. The deformation of the solid is elastic so that the deformation is reversible and there is no dissipation of energy. Although mass diffusion (a non-equilibrium process) is considered in many of these publications, the approach is always to consider what the configuration of the system is at equilibrium and the dissipation of energy associated with the diffusion process is not considered.

Authors who concentrated on gradients in chemical potential to drive diffusion in deformed and metamorphosed rocks include Korzhinskii (1959), Kamb (1959, 1961), Thompson (1959), Green (1970, 1980), Fisher (1973), Fletcher (1973), Paterson (1973), Foster (1981) and Lasaga (1998), but of these only Fisher, Lasaga and Foster developed approaches based on non-equilibrium thermodynamics. For instance, Fisher uses the principle of minimum entropy production rate to constrain the phenomenological coefficients describing the diffusion process. All of the other workers adopt what is essentially an approach based on equilibrium thermodynamics, and the concepts of ‘local equilibrium’ (Thompson 1959) or ‘mosaic equilibrium’ (Korzhinskii 1959) are introduced to handle the need for gradients in chemical potential to exist. Dissipation of energy or entropy production arising from the diffusion process itself is not considered. The same is true of workers in the metals literature of the time (Larche & Cahn 1985), although a notable exception is Kocks et al. (1975). Shimizu (1992, 1995, 1997, 2001) is one of the few who have applied the principles of non-equilibrium thermodynamics in the geosciences to the behaviour of stressed solids coupled to diffusion driven by chemical potential gradients generated by non-hydrostatic stress. Other authors include Lehner & Bataille (1984) and Ghoussoub & Leroy (2001). An approach sometimes considered in metamorphic petrology when non-hydrostatic stresses are developed (Verhoogen 1951) is to accept that the ‘pressure’ relevant to phase equilibria studies is actually the mean stress and so can differ from the lithostatic pressure, which is calculated as rrgh, where rr is the mean rock density, g is the acceleration due to gravity and h is the depth of the rock unit of interest below the surface of the Earth. The implications of this are explored by Mancktelow (1993) and Stuwe & Sandiford (1994) amongst others. These considerations have no influence on the mineral stability field derived using equilibrium thermodynamics. Essentially the identification of the mean stress as the thermodynamic pressure in a deforming metamorphic rock is equivalent to an overestimation of the depth of burial corresponding to the metamorphic conditions if the thermodynamic pressure is equated with rrgh. Such an overestimate can be significant, especially in dry deforming brittle materials where the mean stress is 3rrgh (Petrini & Podladchikov 2000). Other aspects of the influence of non-hydrostatic stress on the stability of mineral assemblages, particularly modifications of the phase rule, are discussed by Kumazawa (1961, 1963) and Shimizu (2001). The early historical development of nonequilibrium thermodynamics is given by Truesdell

DEFORMATION AND METAMORPHIC REACTIONS

(1969) and more recently by Maugin (1999). An important early contribution came from Duhem (1911), although many people worked in this area prior to Duhem including well-known names such as Kelvin, Thomson and Rayleigh. The chemical approach to non-equilibrium thermodynamics was developed by Prigogine (1955) and de Groot & Mazur (1969) based essentially on the approach of Gibbs (1906). The mechanics end of the spectrum was developed by Coleman & Gurtin (1967), Truesdell (1969), Ziegler (1983a) and Biot (1984), and more recently by Coussy (1995, 2004), Collins & Houlsby (1997), Collins & Hilder (2002), Rajagopal & Srinivasa (2004) and Houlsby & Puzrin (2006). There have been attempts to integrate chemical and mechanical non-equilibrium approaches by Coussy (1995, 2004), Levitas et al. (1998a, b) and Rambert et al. (2007). Biot in particular published many papers that brought chemistry and mechanics together (Biot 1984). This array of publications sets a well-defined process for studying the evolution and behaviour of non-equilibrium systems. We outline this process in the next section.

Coupling of deformation, chemical reactions, diffusion and thermal transport The general case We present below a discussion of the coupling between deformation and mineral reactions. The aim is to arrive at a general expression that describes constitutive behaviour, particularly strain and strain-rate softening. We consider general dissipative processes so the discussion does not involve purely elastic materials, as is common in many discussions of related problems (e.g. Kamb 1961; McLellan 1980; Larche & Cahn 1985). Our discussion is restricted to elasto-plastic –viscous solids. In discussing chemically reacting systems where the reaction has not proceeded to completion it is convenient to use the state variable, j (called the extent of the reaction, with 0  j  1), and the conjugate variable, A, the affinity of, or driving force for, the chemical reaction (Kondepudi & Prigogine 1998). Some authors (e.g. Lasaga 1998) refer to j as the progress of the reaction. j˙ is the rate of the reaction where the overdot indicates the material derivative with respect to time, t. A is a linear function of the difference in the sum of the chemical potentials of the reactants and products in the reaction (Kondepudi & Prigogine 1998). Non-equilibrium thermodynamics attempts to describe the evolution of systems not at equilibrium in terms of two potentials, the Helmholtz Free Energy (or, if pertinent to the problem, the Gibbs

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Free Energy) and the dissipation function (Houlsby & Puzrin 2006). The Helmholtz Free Energy is useful if we want to describe the system in terms of a thermodynamic state variable such as the elastic strain, whereas the Gibbs Free Energy is useful if we want to use the conjugate state variable, stress (Callen 1960; Houlsby & Puzrin 2006). For instance, if we consider the specific Helmholtz Free Energy, C, as the relevant function then we can choose the thermodynamic state variables , the absolute temperature, as the elastic strain, 1elastic ij T, and the extent, j, of a diffusive process or of a mineral reaction (Kondepudi & Prigogine 1998). We assume that the system is closed, so that there is no exchange of mass with the outside of the system and write: , T, j ) C ¼ u  Ts ¼ C(1elastic ij

(1)

where u is the specific internal energy and s is the specific entropy. From equation (1) and following standard arguments in continuum thermodynamics (Callen 1960; Coussy 2004) we obtain: vo sij ¼

@C @C @C ; A¼ ; s¼ @T @j @1elastic ij

(2)

where sij is the Cauchy stress. The kinetics of the reaction are given by (Coussy 1995):

j_ ¼

  Qchemical exp  RT vo hchemical A

(3)

where hchemical has the dimensions of viscosity, Q chemical is the activation enthalpy for the chemical reaction and R is the universal gas constant. One can see from equation (3) that since A is a function of chemical potential, it is in turn a function of nonhydrostatic stress (Shimizu 2001); the kinetics of a chemical reaction are also influenced by a non-hydrostatic stress. In order to proceed we need to couple temperature to these processes and derive an expression (the Energy Equation) that describes the ways in which the various dissipative processes contribute to temperature changes. In order to achieve this (equation 9 later) we rewrite equation (1) with mK, the concentration of the K-th chemical species: , T, mK ): C ¼ C(1elastic ij

(4)

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A combination of the First and Second Law of Thermodynamics may be written (Coussy 2004):

_ K  ðFdiffusive þ vo mK m cp T_ ¼ xvo sij 1_ viscous ij

T s_ ¼ F

þ Fchemical þ Fthermal Þ

¼ Fmechanical þ Fdiffusive þ Fchemical þ Fthermal  0

(5)

where F is the dissipation function, and Fmechanical, Fdiffusive, Fchemical and Fthermal are the contributions to the total dissipation from purely mechanical processes, chemical diffusion, chemical reactions and thermal diffusion, respectively. The mechanical dissipation includes the introduction of the K-th chemical species into the deforming material through a chemical reaction, whilst the diffusive term includes the mass flux, JK of the K-th chemical species across gradients in both the chemical potentials, mK, and the temperature. These dissipation functions are given by (Coussy 2004): @C elastic @C _ @C _ K (61 ) Tþ m 1_ ij þ @T @m @1elastic K ij   @m (62 ) ¼ vo JK gradmK  K gradT @T

Fmechanical ¼ Fdiffusive

processes and, hence, does not include contributions from elastic deformations:



Fchemical ¼ AK j_ K þ H_ @

(63 )

and Fthermal ¼ kthermal cp r2 T:

(64 )



In equation (62) represents the scalar product of vectors; in equation (63) H_ @ is the volume rate supply of heat of reaction from the chemical reaction, @. kthermal is the thermal diffusivity and cp is the specific heat at constant pressure given by: @2 C cp ¼ T 2 (Nye 1957). @T @C and Using the expressions s ¼  @T @C vo mK ¼ , equation (61) becomes @mK _ K: Fmechanical ¼ vo sij 1_ elastic  sT_ þ vo mK m ij

(7)

If we assume that ¼ 1_ elastic þ 1_ viscous 1_ total ij ij ij

(8)

then, using equation (5), we can eventually arrive at the Energy Equation that expresses the change in temperature arising from all of the dissipative

(9)

where x is the Taylor–Quinney coefficient, and represents the proportion of mechanical work arising from dissipative deformation that is available to increase the temperature or to drive diffusion, chemical reactions and structural adjustments such as fracturing or grain-size reduction. At high strains, where the energy arising from deformation is stored in crystal defects, x is generally in the range 0.85  x  1 (Taylor & Quinney 1934). We assume in what follows that x ¼ 1. Equation (9) is a critical equation and will be used when we come to examine the coupling between deformation and mineral reactions at the scale of 10 –100 m.

Scale effects In this paper we are concerned with the coupling between deformation and mineral reactions at the field-outcrop scale (measured in tens to hundreds of metres), and we draw a distinction between this scale and the regional scale measured in kilometres and the hand specimen –thin section scale measured in fractions of a metre. The length scale, l, associated with a particular diffusion process is defined by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lthermal,chemical  2 (kthermal,chemical )=1_

(10)

where the superscripts thermal and chemical refer to the diffusion process with the diffusivity k. For thermal diffusion a geologically realistic strain rate of 10212 s21 leads to a length scale for the related process of approximately 2 km for a diffusivity of kthermal ¼ 1026 m2 s21. At the same time a chemical feedback process with a diffusivity of, say, kchemical ¼ 10218 m2 s21 has a typical millimetre length scale. This vast separation of scales, as well as providing a guide for the dominant scale of any localization, folding or boudinage phenomena, also provides an opportunity to simplify the numerical treatment. The models we consider in this paper are at the outcrop scale and measured in tens to hundreds of metres. This means that with a thermal diffusivity of 1026 m2 s21, any heat produced by deformation, diffusion or chemically induced dissipation diffuses out of the system on timescales of 108 –1010 s. These timescales are short compared to timescales of, say, 1011 s (the time taken to reach 10%

DEFORMATION AND METAMORPHIC REACTIONS

shortening at a 10212 s21 strain rate), and so we consider that the deformation is isothermal. We also suppose that the surrounding rock does not heat up appreciably, so that the entire system remains isothermal. We also neglect the latent heat of any chemical reactions as expressed by equation (63), so that the chemical dissipation arises only from the term AKj˙ K in equation (63). We consider the effect of including the latent heat term in the Discussion section. This means that in the Energy Equation, equation (9), T˙ ¼ 0, rT ¼ 0 and Fthermal ¼ 0. Thus, the Energy Equation during the chemical reactions reduces to: _ K Þ ¼ Fdiffusive þ Fchemical : (11) þ mm vo ðxsij 1_ viscous ij Similarly, for timescales as long as 1011 s and chemical diffusivities of 10218 m2 s21, material does not diffuse more than about 6  1024 m. Thus, we propose that a negligible part of the deformation-induced dissipation results in a diffusive flux at the scale we are considering, and ˙ K ¼ Fdiffusive ¼ 0. Such processes operate hence mm at a finer scale and are expressed in the form of the constitutive equation used to describe the deformation. Thus, equation (11), at the scale of tens to hundreds of metres, reduces to:

x vo sij 1_ viscous ¼ Fchemical : ij

(12)

Note that this chemical –mechanical formulation has an interpretation in hyperplasticity (Houlsby & Puzrin 2006). We can think of equation (12) as expressing a chemical ‘yield’ condition where the chemical reactions do not begin until the mechanical dissipation matches the chemical dissipation. If processes such as chemical reactions occur at the same time as other deformation processes and contribute to strain, then one way of considering the coupled problem is to represent both the deformation and the chemical reactions as different yield (or dissipation) surfaces. The coupled deforming– reacting system is then represented in stress space as a set of nested yield surfaces that change size and translate together according to the rules set out by Puzrin & Houlsby (2001) and Houlsby & Puzrin (2006). If we represent the chemical reaction as a von Mises yield surface (the chemical yield surface), then the radius of the yield surface is proportional to the affinity of the chemical reaction. In this paper we follow this formalism and consider that the chemical yield surface is reached when the mechanical dissipation is large enough to be equal to the chemical dissipation, as indicated in equation (12). This means that mechanical dissipation drives the chemical reaction once the

205

products of the reactions are in their stability field. Chemical hardening or softening then depends both on the value of (s –A) and of A, where s and A are the stress and affinity vectors in stress space. Once the chemical reaction is completed, the mechanical dissipation is available at a fine scale to drive diffusion away from the reaction site (Regenauer-Lieb et al. 2009) and to generate increases in temperature at a coarser scale (Hobbs et al. 2008). Compared to equation (9) we have now decoupled the thermal and chemical diffusion equations, and assumed that the thermal and chemical diffusion processes operate on different length/ time scales and there is no feedback between the scales. We have also assumed that the coarse length scale process loses heat such that the net temperature change in the large-scale system is zero. We have also ignored the heat of the chemical reaction and assumed that all of the mechanical dissipation at the grain-scale is taken up in creating new phases or in driving chemical diffusion. This simplification means that the problem reduces to one identical in form to shear heating feedback in mechanical systems; the latter is intrinsically simpler and is much better understood (RegenauerLieb & Yuen 2004). We now have the following relationships expressing the coupling of deformation and chemical reactions at the outcrop scale: vo sij 1_ viscous ¼ Fchemical ¼ Aj_ ¼ vo h

 chemical  Q : (13) j exp RT

chemical _ 2

Coupling of deformation and mineral reactions First consider the situation where the viscous part of the strain is achieved (in a one-dimensional deformation) through power-law flow (specifically not at steady state), such that for a given deviatoric stress, s 0 , and in the absence of chemical reactions the steady viscous strain rate, 1˙ viscous, is given by:   Qmechanical 1_ viscous ¼ Ls 0N exp  RT

(14)

where L is a material constant, N is the stress exponent and Q mechanical is the activation energy for steady deformation. The deviatoric stress is given by s 0 ¼ s 2 P, where s is the total stress and P is the mean stress. The mineral reaction can influence L by the generation of new weaker or stronger phases and/or by reducing the grain size (Rubie 1983, 1990; Vernon

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B. E. HOBBS ET AL.

2004). The reaction can also influence Q mechanical through the diffusion of species into the crystal structures, as occurs in the hydrolytic weakening effect or in the case of the influence of water on the kinetics of the coesite to quartz transition (Mosenfelder & Bohlen 1997). Q mechanical can also be substantially changed as mineral reactions change the proportions of populations of different minerals in the rock. Chemical species diffused into the crystal structure can also influence the mobility of dislocations, and individual fine phases can influence the migration of grain boundaries so that the dominant mechanism of deformation changes; thus influencing the value of N. Thus, chemical processes are capable of influencing L, N and Q mechanical and so influence the strain rate for a given stress. Conversely, the deformation can influence the chemical kinetics through a direct influence of stress on Q chemical, as has been documented by authors such as Hillig & Charles (1964), Lawn (1993), Schmalzried (1995) and Gilman (2003); an example of this effect with respect to the influence of stress on the incorporation of water into quartz is given by Zhu & Yip (2005). Deformation also influences hchemical through structural changes that result in the production of more reaction sites arising from increased dislocation density or smaller grain size. The affinity of the reaction also depends on stress through an influence on the chemical potentials of the products and reactants. While the chemical reaction is in progress there is, therefore, a dependence of some or all of L, N and Q mechanical on the amount of strain, thus resulting in reaction-induced strain hardening or strain softening. Thus, equation (14) can be rewritten for nonsteady deformations as: 1

s ¼ L1 _ N0 exp 0 1



 Qmechanical 0 1p RT

(15)

where p is the strain-hardening exponent so that p . 0 represents strain hardening and p , 0 represents strain softening. The null subscripts refer to values of the respective quantities at some reference strain. This means that all of the strain hardening/softening is expressed in the term 1 p. We admit this may be simplistic but, failing systematic experimental data, it is a start. Experimental data on reaction-induced strain softening are rare but the limited data (Brodie & Rutter 1985, 1990; Burlini & Bruhn 2005; de Ronde et al. 2005; Delle Piane et al. 2007) suggest that, over a significant range of strain, the softening can be linear with strain; however, we assume that, overall, the dependence can be represented by a power-law relation such as equation (15).

In many materials (Estrin & Kubin 1991), processes that evolve at the same time as deformation and that are coupled to the deformation result in strain-rate hardening (or softening), as well as strain hardening (or softening). Thus, a material deforming solely by dislocation creep with diffusion playing a negligible role needs to increase the density of mobile dislocations for an increase in strain rate, thus resulting in strain-rate hardening. Notice that this effect is greatest for Newtonian viscous materials (N ¼ 1 in eq. 14) and decreases as N increases. The use of the terms strain-rate hardening and softening   in this paper is meant to refer to @s . From equation (15) this is the sign of @_1 1,T  1   mechanical  L Q p 1N N , which is always exp 1 1_ N RT positive for N  1. In particular, a strictly Newtonian material where the viscosity is a constant is a strain-rate hardening material, since an increase in strain rate always produces an increase in stress. It is important to note that Smith (1977) referred to the strain-rate response of power-law materials with N . 1 as strain-rate softening because the 1N is negative for N . 1, which means term N that the tangent viscosity (Smith 1977) decreases with an increase in strain rate. This usage has been followed by some workers in structural geology. We emphasize that for any power-law material with N  1 the response to an increase in strain L1 is always positive. rate is a hardening, since N Thus, in our usage of the term, which coincides with that of Poirier (1980) and of the general mechanics community, the case N  1 in equations (14) and (15) corresponds to strain-rate hardening material. However, if time-dependent processes, such as thermal or mass diffusion, chemical reaction, melting, grain-size reduction or the development of a crystallographic preferred orientation, play a dominant role and are coupled with the deformation, then an increase in strain rate can lead to hardening or softening depending on the relative values of the timescales associated with a strain-rate change and the relaxation of the stress due to diffusion or the extent of the relevant process. Examples are softening due to thermal diffusion coupled to a decrease in viscosity with temperature increase (RegenauerLieb & Yuen 2003) and various diffusive processes related to dislocation motion in some alloys (Estrin & Kubin 1991; Zaiser & Hahner 1997). The strainrate hardening (softening) effect depends on the ratio of the two timescales mentioned above and resembles the velocity weakening effect observed in frictional sliding (Ruina 1983). The formal

DEFORMATION AND METAMORPHIC REACTIONS

analogy between strain-rate softening in viscous materials and velocity weakening in frictional sliding has been examined by Mesarovic (1995) and Kameyama (2003). Thus, there is always a transient hardening effect associated with a rapid increase in strain rate for a power-law material with N  1, but with the elapse of time and the progress of a chemical reaction or some other timedependent process an ultimate weakening may result (Estrin & Kubin 1991). Strain-rate stepping experiments in reacting– deforming materials were conducted by Delle Piane et al. (2007), but a strainrate softening effect arising from a mineral reaction will be difficult to demonstrate at laboratory accessible strain rates. The behaviour of strain-hardening (softening) and strain-rate-hardening (softening) materials is discussed by a number of workers including Estrin & Kubin (1991), Hahner (1995), Wang et al. (1997) and Zaiser & Hahner (1997). These authors define three hardening (softening) parameters, namely:   @s (161 ) h¼ @1 1_ ,T     @s @s v¼ or v ¼ (162 ) @_1 1,T @(ln 1_ ) 1,T   @s w¼ : (163 ) @T 1,_1 If the timescale for the diffusive relaxation process associated with strain-rate softening is short compared to the timescale for a strain-rate change then a criterion for the onset of localization in materials with an exponential dependence of stress on strain rate is: hs , 0: v

(17)

Thus, localization can occur in these materials even for strain-hardening situations for h , s and v . 0, and for (h 2 s) . 0 and v , 0. For the constitutive law assumed by Estrin & Kubin (1991), strain and strain-rate effects are coupled so that the competition between the evolution of forest (dislocations that intersect the glide plane) and mobile dislocation densities with continuing strain results in a situation where strain softening (h , 0) coupled with strainrate softening (v , 0) does not arise. An important outcome from these studies is that for materials that are both strain-softening and strain-rate-hardening, deformation becomes localized into individual zones that remained fixed in space during the deformation. Localization phenomena similar to this have been investigated by Mancktelow (2002). For strain-hardening, strain-rate-softening materials the competition

207

between strain hardening and strain-rate softening results in propagating (i.e. migrating) shear bands (Estrin & Kubin 1991; Wang et al. 1997). The softening induced by thermal effects is identical to that considered by Regenauer-Lieb & Yuen (2003) and Hobbs et al. (2007, 2008). The development of shear bands in strain-ratesensitive materials appears to be different from localization in strain-rate-insensitive materials (Needleman 1988). Localization in strain-rateinsensitive materials has been summarized in Hobbs et al. (1990), where the localization process appears as a bifurcation in material behaviour once some critical parameter such as amount of strain or strain-softening magnitude is exceeded. In strain-rate-sensitive materials the localization is an instability phenomenon that is quite sensitive to initial imperfections such as variations in viscosity and does not involve bifurcation behaviour (Needleman 1988; Wang et al. 1997). To this extent the theory for localization in strain-ratesensitive materials has many features in common with the theory of folding proposed by Biot (1965a), where instability in the deformation is related to the growth of initial imperfections from the moment visco-plastic deformation begins. Needleman (1988) and Wang et al. (1997) show that for quasi-static deformations the thickness of shear zones in strain-rate-sensitive materials is related to the size of the initial imperfection. Viscous materials with N  1 can never localize unless at least strain softening exists (Anand et al. 1987; Hobbs et al. 1990). Anand et al. (1987) analysed the onset of localization in viscoplastic strain-rate hardening materials and showed that localization arising from strain softening grows fastest just after initiation for weaker strain-rate hardening sensitivity. Another aspect arises from the intrinsic anisotropy of power-law viscous materials. Smith (1977) emphasized that viscous materials with N . 1 are, in effect, anisotropic because the viscosity measured in simple shearing is always less than the viscosity measured in pure shearing. Anisotropic viscous materials can become unstable (Biot 1965b; Hobbs et al. 2000) and form shear zones but the instability does not develop unless N is large (N . 10) which is unrealistic for rocks. In the models of Mancktelow (2002) strain-rate hardening is always present as the response of a power-law material (with no coupled diffusive processes) to an increase in strain rate but the effect is greater for materials with N ¼ 1 than for materials with N . 1. The strain softening invoked by Mancktelow (2002) competes with this strain-rate hardening so that localization is better developed for N ¼ 3 rather than for N ¼ 1, in agreement with Anand et al. (1987). However, as pointed out by Mancktelow

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(2002), the intensity of strain in these shear zones always remains small. This contrasts with the high intensity of strain developed in natural shear zones (Ramsay & Graham 1970) and in strain-rate softening materials as demonstrated later. The thermal–mechanical feedback process is worth further investigation. Fleitout & Froidevaux (1980) show that the effective viscosity within a shear zone that arises from thermal–mechanical feedback in a developing shear zone is given by: R Tc2  8krcp : Q_12 k2

heff

1_ 20 1_ 2

(19)

where h0 is the viscosity when the strain rate is 1_ 0 . There is an additional feature that arises during the thermal–mechanical localization process. The shear zone initially nucleates with a width k0 but collapses in width as deformation proceeds until the conduction of heat away from the shear zone balances that being generated by the shearing deformation. This results in a transient dependence of the viscosity on time, or equivalently, on strain. Thus, the evolution of the effective viscosity during deformation with thermal –mechanical feedback consists of a transient strain hardening arising from the collapse in shear-zone thickness and a strain-rate softening that is inversely proportional to the square of the strain rate. The general form of the constitutive equation considered by Estrin & Kubin (1991) can be written in incremental form as: ds ¼ hd1 þ vd(ln 1_ ) þ wdT:

(20)

It would appear that to consider the coupling between chemical reactions and deformation we need an incremental constitutive relation of the form: ds ¼ hd1 þ vd(ln 1_ ) þ wdT þ hchemical dj þv

chemical

dj_

2 hmechanical (_1viscous )2 ¼ Bj_

(21)

where h chemical and vchemical are the chemical strain and strain-rate hardening/softening parameters:   @s chemical ¼ (221 ) h @j 1,_1,T,j_   @s vchemical ¼ : (222 ) @j_ 1,_1,T,j

(23)

or

(18)

The effective viscosity is inversely proportional to both the square of the strain rate and the square of the shear-zone thickness, k; equation (18) can be written more generally for constant k as:

heff  h0

Unfortunately, we have little knowledge of the precise forms of these parameters and so we proceed in a somewhat simplistic manner below in order to explore some scenarios of possible behaviour. For a Newtonian viscous material, equation (13) can be rewritten as:

hmechanical ¼ B

2 j_

(24)

viscous 2

(_1

)

 chemical  Q . Equation (24) RT is of similar form to equation (18). The difference between thermal–mechanical and reaction – mechanical coupling is that the strain rate has only a small effect on the rate of heat flow and can be neglected. However, the strain rate can potentially have large effects both on B and j_ . We need more theoretical and experimental work to define the precise form of equation (24), in particular the dependence of reaction rate on strain and on strain rate. Neglecting thermal effects, we for the moment write the general relation between the mechanical viscosity and the strain and strain-rate softening as: where B ¼ hchemical exp

hmechanical ¼ h0 (1viscous ) p



1_ 0 1_ viscous

q (25)

where h0 is the mechanical viscosity at zero strain and a reference strain rate of 1_ 0 , p is the strainhardening exponent, and q is a parameter that measures the dependence of viscosity on strainrate; there is viscosity strain-rate hardening for q , 0 and viscosity strain-rate softening for q . 0.

The development of localized shear zones, folding and boudinage In order to gain some insight into the influence of mineral reactions on the development of localized zones of shear, and on folding and boudinage at the outcrop scale, we explore a simple generic model that incorporates the issues discussed earlier. For simplicity, we assume that the reacting deforming material can be represented at each instant as an elastic –viscous material with Newtonian viscosity but with reaction-induced strain and strain-rate hardening/softening. This behaviour is expressed as a dependence of the instantaneous viscosity on the strain and on the strain rate as expressed by

DEFORMATION AND METAMORPHIC REACTIONS

equation (25). This follows the approach of authors such as Smith (1977) and Needleman (1988), where the instantaneous viscosity is expressed via a dependence on strain and strain rate. The constitutive law at each instant is the Maxwell constitutive relation: 1_ ij ¼

sij s_ ij þ hmechanical E

(26)

where E is the elastic modulus. The governing equations are, first, the Continuity Equation: @r þ rr u ¼ 0 @t







band during progressive deformation is arrested due to strain hardening, and the site of localization moves to a new area where strain-rate softening dominates. This results in shear zones that localize as the deformation proceeds up to a stage where the localization switches from one to another during the deformation. In both cases shear zones are strongly developed with localized integrated strains, as measured by g, the integral of the square root of the second invariant (I2) of the viscous strain rate 0 1 ðt rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 viscous viscous @g ¼ )dtA. Values of g reach 5 1_ ij (_1 2 ij 0

(27)

where u is the local material velocity vector. Secondly, the momentum equation describes equilibrium of forces: r sij ¼ 0

209

(28)

where r sij is the divergence of the Cauchy stress tensor. These equations are solved together with equations (25) and (26) using the finite-difference code FLAC2D version 6.0 (ITASCA 2008). We consider first a body of rock with no layering but with a random distribution of heterogeneities in the viscosity (corresponding to initial variations in mineral composition). The viscosity is initially 1021 Pa s and the standard deviation of the viscosity distribution is 1020 Pa s. The model is shortened isothermally at a constant velocity parallel to x2 in Figure 10 corresponding to an initial strain rate of 10212 s21. The velocity parallel to x1 is varied with time to maintain an isochoric deformation. In Figure 10a, b, p from equation (25) is 20.8 and q ¼ þ2, so that both viscosity strain softening and viscosity strain-rate softening occur. This corresponds to a situation where the mineral reaction rate is independent of the strain rate, but the reaction results in strain softening. Shear zones are strongly developed, and intensify and decrease in number as the deformation proceeds; with increasing shortening the strain progressively becomes focused in wider spaced shear zones. In Figure 10c, d, p ¼ 0.5 and q ¼ þ2, corresponding to a situation where the mineral reaction results in strain hardening and the reaction rate is independent of the strain rate. In this case, the shear zones become more localized with increasing shortening. This resembles the development of so-called propagating shear bands in rate-sensitive materials (Estrin & Kubin 1991; Wang et al. 1997) where the growth of an individual shear

in shear zones after 50% total shortening and 6 after 70% shortening. In both strain-hardening and strainsoftening situations the shear zones develop, in addition to the expected X-shaped ‘conjugate’ configuration, a characteristic Y- or V-shaped geometry so that quite intense shear zones suddenly stop and are replaced by a diffuse pattern of deformation. This resembles shear-zone geometries seen in the field (Figs 5 & 6). Next, in Figure 11a –e we add a single layer of slightly greater viscosity initially parallel to x1 and shorten the model parallel to x1. In all models the initial viscosity contrast between the layer and the embedding material is 10. This viscosity contrast is too small to produce folding according to a Biottype of mechanism (Biot 1965a; Hobbs et al. 2008). The model is shortened parallel to the initial orientation of the layer at 10212 s21 with p ¼ 20.8 and q ¼ þ2. No folds develop if only strain softening is present, as the resultant shear zones develop total accumulated strains too small to result in significant inhomogeneous deformation. However, once strain-rate softening is incorporated, shear zones are strongly developed and any slight deflection in the layer is amplified because of the influence of strain-rate softening. The strain and, hence, the strain rate is larger in the inner hinge of the incipient fold than in the outer hinge. Thus, the viscosity decreases in the inner hinge more than in the outer hinge (Fig. 11b). This generates a buckle that is further amplified by the viscosity strain-rate softening. Thus, folds ultimately develop with wavelengths controlled by the distribution of imperfections in the system such as the initial random variation of viscosity and the boundaries of the model. The folds that ultimately form at high total shortening have thinned limbs where shear zones cross the layer and thickened hinges in between. This is a non-Biot-type of folding mechanism that is identical in principle to folds developed from thermal –mechanical coupling at the kilometre scale (Hobbs et al. 2007, 2008) and folds developed by chemical reaction diffusion–mechanical coupling

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Fig. 10. Localization in unlayered Newtonian viscous strain-rate-softening materials (q ¼ þ2.0). Plots of the pffiffiffiffi logarithm of I2 . (a) p ¼ 20.8; shortening 40%. (b) p ¼ 20.8; shortening 50%. (c) p ¼ 0.5; shortening 40%. (d) p ¼ 0.5; shortening 50%.

at the centimetre scale (Regenauer-Lieb et al. 2009). We emphasize that in a Newtonian material, strain softening alone is not sufficient to generate folds; strain-rate softening is the important control. The intensity of strain increases in the inner hinges. For reaction rates independent of strain rate the extent of the reaction is directly proportional to the amount of viscous strain, so that Figure 11b –d are proxies for the extent of mineral reaction. Figure 11f, g shows the same situation but with p ¼ 20.1 and q ¼ þ2. The shortening strains are 50 and 70%, respectively. Note the thinning on the limbs and boudinage in the hinge position. Figure 12 shows the development of shear zones and boudinage. In Figure 12a, p ¼ 20.8 and q ¼ þ2, and the deformation history comprises pure shortening parallel to x2. In Figure 12b, c, p ¼ 0.5 and q ¼ þ2, and the deformation history comprises simultaneous pure shearing parallel to x2 and simple shearing parallel to x1.

Figure 12a (lower panel) shows the distribution of the total integrated strain (g) for the situations shown where the reaction rate is independent of the strain rate. Again, these diagrams are equivalent to maps of j, the extent of the mineral reaction. Thus, one sees that the mineral reactions are more advanced in shear zones that form an anastomosing network around the boudins, with reactions tending to completion in the necks of boudins or on the margins of boudins particularly near necks. Again, the characteristic V- or Y-shaped geometry of shear zones is developed resembling examples seen in the field (Figs 5 & 6). Notice that for a material with N ¼ 1, with no coupled mineral reactions and with the initial viscosity ratio between layers of 10, no boudins would develop (Smith 1977; Schmalholz et al. 2008). Strain-rate softening is essential. Figure 12d shows part of a mutilayer sequence shortened 80% parallel to x2. The extreme thinning

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Fig. 11. Folding and localization in Newtonian strain-softening, strain-rate-softening materials ( p ¼ 20.8, q ¼ þ2.0). Layer viscosity initially 1021 Pa s. Embedding material initial viscosity 1020 Pa s. (a) Shortening 20%, passive markers show deformation in embedding material. (b) Same as (a) but showing logarithm of current viscosity. Dark green is 1021 Pa s, light green is 1019 Pa s, purple is 1017 Pa s. Viscosity is higher in the outer arc of the folded layer and low in the inner arc. (c) Shortening 30%; logarithm of g. (d) Shortening 40%; logarithm of g. In both diagrams blue is g ¼ 3.5, green is g ¼ 0.2. (e) Deformed layer at 40% shortening. (f ) Single-layer fold with p ¼ 20.1 and q ¼ þ2; 50% shortening. (g) Same as (f ) but with 70% shortening.

of fold limbs is notable and resembles the extreme thinning observed in the field (Fig. 8a). Although this technically is a Class IC fold in the terminology of Ramsay (1967), the intense thinning on the limbs is due to the development of a shear zone rather than flattening as in the Ramsay model.

Influence of deformation on phase stability: simple chemical reactions The chemical potential of a chemical component is a quantity that measures the energy required to insert a unit quantity (usually expressed as 1 mole but here expressed as unit mass) of that component into a system isothermally. This definition is relevant whether the system is or is not at equilibrium and

whether the stress is hydrostatic or non-hydrostatic. Consider a system with K chemical components. We define the non-equilibrium specific chemical potential, mK, of the K-th component inserted into a deforming, chemically reactive system as (Kondepudi & Prigogine 1998; Coussy 2004):

mK ¼ mK (sij0 , P, T, j K )

(29)

where sij0 is the deviatoric stress, P is the mean stress and j K is the extent of the chemical reaction that produces the K-th component. To be explicit, here 1 the mean stress is P ¼  skk where the Cauchy 3 stress, sij, is related to the deviatoric stress through sij0 ¼ sij  skk =3:

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Fig. 12. Deformation in multilayer materials. (a)–(c) boudinage in a three-layer sequence (d) Very strongly appressed fold. (a) Deformation is pure shearing; p ¼ 20.8, q ¼ þ2. Shortening is 70%. Plots of geometry and logarithms of viscosity, strain rate and g, which is taken as a proxy for the extent of reaction. In the viscosity plot: dark green is 1022 Pa s, yellow is 1020 Pa s, purple is 1018 Pa s. In the strain-rate plot, red is 10214 s21, blue is 10210 s21. In the strain plot: blue is g ¼ 8, red is g ¼ ,1. (b) Boudinage developed with combined pure shearing (shortening 70%) parallel to x2 and simple shearing through 708 parallel to x1; p ¼ þ0.5, q ¼ þ2. (c) Map of log g for the geometry of (b): blue is g ¼ 8, red is g ¼ ,1. (d) p ¼ 20.8. q ¼ þ2; shortening 80%.

DEFORMATION AND METAMORPHIC REACTIONS

Even in a system not at equilibrium, at a phase boundary where the reactants and the products of a mineral reaction coexist, 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 non-equilibrium 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 (29): dmK ¼

@m K 0 @m K @m K dP þ dT dsij þ 0 @sij @P @T þ

@m K K dj @j K

(30)

or dmK ¼ v0 1K(elastic) dsij0 þ vK dP  sK dT ij þ AK dj K

(31)

1K(elastic) ij

is the elastic strain of component K, where vK is the specific volume of component K, s K is the specific entropy of component K, AK is the affinity of the reaction that produces K and v0 is the specific volume of the reactant at hydrostatic stress. For a discussion of the term v0 one should consult Houlsby & Puzrin (2006). From equation (31) we obtain: ) dsij0 þ Dv dP d(Dm) ¼ D(v0 1elastic ij  Ds dT þ DA dj

(32)

where D(.) is the change in (.) during the chemical reaction and deformation. That is, for the reaction quartz ! coesite, Ds ¼ scoesite  squartz and so on. At a phase boundary, d(Dm) ¼ 0 and DA ¼ 0 hence: ) dsij0 dP Ds D(v0 1elastic ij : ¼ þ dT dT Dv Dv

(33)

Thus, at a phase boundary the classical Clausius– Ds Clapeyron slope at equilibrium, , is modified in Dv the non-equilibrium case by a term that involves the elastic strain and specific volume contrasts between the two phases and the temperature derivative of the deviatoric stress tensor. If the total strains arise from steady-state power-law creep of the

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form involving sij0 ¼ L1=N 1_ ij (J2 )1=(N1) exp(Q=RT) then: ) dP Ds D(v0 1elastic ij ¼  dT Dv Dv Q 1=N L 1_ ij (J2 )1=(N1) exp(Q=RT) RT 2

(34)

at a phase boundary. Hence, for a fixed strain rate, dP approaches the equilibrium Clausius –Clapeyron dT slope at high temperatures, but at low temperatures and/or fast strain rates the second term on the righthand side of equation (34) can be of similar magnitude to the classical Clausius –Clapeyron slope, Ds/Dv. As the strain rate decreases at constant dP approaches the classical equilitemperature, dT brium Clausius –Clapeyron slope. As the non-equilibrium phase boundary is asymptotic to the equilibrium phase boundary as T ! 1, the equation for the non-equilibrium phase boundary becomes through the integration of equation (34): Ds T Dv   elastic D(v0 1 ) 2=N 2=N Q L 1_ exp þ Dv NRT     Q 1  exp NRT

P ¼ P0 þ

(35)

where P0 is the intercept of the equilibrium phase boundary on the pressure axis for T ¼ 0 and taken as D(v0 1elastic ) ¼ D(v01elastic) has been  quartz coesite v v and E is the Young’s s 0 0quartz  0coesite E E modulus. Note that for the quartz ! coesite reaction, D(v01elastic) is positive whereas Dv is negative. The general behaviour of this boundary is illustrated in Figure 13. For the reaction quartz ! coesite P0 ¼ 13 000 Ds ¼ 14:8  106 Pa K21, vquartz ¼ 3.76  Pa, 0 Dv 24 3 24 3 coesite 10 m , v0 ¼ 3.41  10 m and Dv ¼ 22.7  1026 m3 mol21 (Kern & Weisbrod 1967). At atmospheric pressure the elastic moduli C11 and C33 of a-quartz are approximately 90 and 80 GPa, respectively, whereas the equivalent elastic stiffnesses for coesite are about 150 and 200 GPa, respectively (Carpenter et al. 1998; Kimizuka et al. 2008). We take the elastic stiffness of a-quartz to be about half of the equivalent moduli of coesite and so the elastic strain of quartz is about twice that of coesite for the same stress. We assume

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Fig. 13. Non-equilibrium phase diagram. (a) Theoretical non-equilibrium pressure– temperature phase diagram with the equilibrium phase boundary shown as a dashed line. The full line is the non-equilibrium phase boundary asymptotic

DEFORMATION AND METAMORPHIC REACTIONS

D1 ¼ 0.5s 0 /E quartz, where E quartz is taken to be 1011 Pa. If we also assume that the constitutive parameters for coesite are the same as for quartz we can write Q ¼ 135 KJ mol21, N ¼ 4 and L ¼ 1.33  10234 Pa2N s21. The resulting phase boundary for the quartz ! coesite reaction is shown in Figure 13c for strain rates ranging from geological realistic values of 10214 s21 to laboratory accessible strain rates of 1024 s21 and assuming E quartz ¼ 0.5E coesite. At geological strain rates the nonequilibrium phase boundary is within 0.1 MPa of the equilibrium boundary at temperatures above 400 K and so the difference would not be detectable for geological systems. However, examples for coesite forming outside of its equilibrium stability field during laboratory deformation experiments are well known (Hobbs 1968; Green 1972), where the strain rates are 1026 – 1024 s21. This analysis seems to confirm that this is a metastable development of coesite, as discussed by Green (1972). As the difference between the equilibrium and non-equilibrium curves is strongly dependent on D1elastic there may be examples where the effect is larger than indicated for the quartz ! coesite reaction, especially in reactions involving ‘hard’ phases reacting to form ‘soft’ phases. Moreover, we have assumed that the constitutive behaviour and parameters for the reactants are the same as for the products, and certainly this is not the case for more complicated reactions. Hence, the magnitude of the difference between the equilibrium and non-equilibrium behaviour in more complicated reactions remains open, and would benefit from detailed, high-precision P–T investigations on assemblages in suitable structural environments.

Discussion We have presented a model whereby mineral reactions occurring during metamorphism are coupled to the accompanying deformation. A two-way feedback process operates whereby mineral reactions influence deformation and deformation influences mineral reactions. If the deformation enhances the reaction rate then the deformation is enhanced. If the reaction softens the material either by strainsoftening or strain-rate-softening mechanisms then deformation is enhanced, which in turn further enhances the mineral reaction. An important outcome of employing non-equilibrium thermodynamics at the scale of tens to hundreds of metres is

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that the thermal and chemical diffusion effects can be decoupled from the mineral reaction effects so that the energy dissipated by the inelastic deformation is coupled only to the mineral reaction rate. Thermal diffusion effects dominate at larger scales (c. 1 km), whereas coupled mineral reactions – chemical diffusion effects dominate at smaller scales (c. 1 mm). This results in a strain-ratesoftening effect at the scale of tens to hundreds of metres, in addition to any strain hardening or softening. If the reaction rate is independent of the strain rate then the effective viscosity is inversely proportional to the square of the strain rate as it is in thermal –mechanical feedback situations (Fleitout & Froidevaux 1980). Unfortunately, we have no experimental data on the dependence of reaction rates on strain rate and so we have explored (but not presented) several simple situations where the reaction rate, j˙, is proportional to 1˙ M. If M ¼ 0 then the reaction rate is independent of the strain rate. For M ¼ 0.5 the stress becomes independent of the strain rate for a material that otherwise would behave as a Newtonian material with no reaction coupling. For M ¼ 1 the viscosity is independent of strain rate and the material always behaves as a Newtonian material. These relations are illustrated in Figure 14 where the structural behaviour of the material for various values of M and q are also indicated. At the scale of tens to hundreds of metres, although reaction-induced strain softening is commonly quoted as a mechanism for generating shear zones, it appears from the modelling reported here and by Mancktelow (2002) that strain softening, although capable of inducing localization in a viscous material, does not lead to the strain intensities observed in natural examples such as those studied by Ramsay (1967) and Ramsay & Graham (1970) at least for values of N  3. Part of the reason for this is that viscous materials with N  1 and with no coupling to time-dependent processes inherently exhibit strain-rate hardening, which largely offsets the strain softening (Anand et al. 1987). The hardening is less for non-Newtonian materials with N . 1, so that localization is better developed in these power-law materials than in Newtonian materials (Mancktelow 2002). Even so, strain softening alone does not reproduce the strain intensities observed in natural shear zones (Mancktelow 2002). In order to generate high-strain shear zones, nonlinear strain-rate softening is necessary. Then three

Fig. 13. (Continued) 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. (b) Theoretical non-equilibrium phase boundaries at two different strain rates, 1_ Iij (slow) and 1_ IIij (faster). The assemblage at point A is unstable at the slow strain rate but stable at the higher strain rate. (c) Non-equilibrium phase boundary constructed for the quartz–coesite transition.

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Fig. 14. Behaviour of strain-rate hardening/softening viscous materials. M is the exponent that links reaction rate to strain rate: j_ ¼ 1_ M . q is the exponent that describes strain-rate softening. q ¼ þ2 means that the reaction rate is independent of the strain rate. (a) Changes in deformation behaviour with changes in M and q. (b) Changes in constitutive behaviour with changes in M and q.

possibilities arise. For strain hardening together with non-linear strain-rate softening shear zones that nucleate at a given site grow until the strain hardening arrests the growth. The localization then shifts to an adjacent site. Such a shear-zone network is to be expected in metamorphic terrains where the mineral reactions result in new phases stronger than the original assemblage, such as in temperature-prograde metamorphic environments. Alternatively, non-linear strain softening accompanied by strain-rate hardening leads to localized shear zones that remain at the same site throughout the deformation but never progress to high-strain shear zones. Lastly, strain-rate softening accompanied by strain softening leads to the formation of zones of intense localization that tend to evolve so that the strain is taken up in just a few wide shear zones. The result is relatively widely spaced shear zones with relatively high intensity of strain. Such a shear-zone pattern is to be expected in metamorphic terrains where the mineral reactions result in new phases weaker than the original assemblage. This is typical of areas undergoing temperatureretrograde metamorphism. The development of localization in strain-ratedependent materials is strongly sensitive to initial perturbations in mechanical properties or geometry so that shear-zone thickness is influenced by the size of initial departures from homogeneity (Needleman 1988; Wang et al. 1997). This is particularly true if layers of slightly different material are present. Slight departures from plane interfaces are amplified because of differences in strain rate between the outer and inner arcs of a slight buckle. The strain rate is marginally higher in the inner arc and

strain-rate softening induces a lower effective viscosity than in the outer arc. This amplifies the incipient buckle, and folds grow even if the initial viscosity contrast between the layers and the embedding material is small (10). This folding mechanism is different from the classical Biot mechanism (Biot 1965a) but is identical to that which develops from thermal–mechanical and from chemical reaction –diffusion–mechanical feedback processes at the kilometre and millimetre scales, respectively (Hobbs et al. 2007, 2008; Regenauer-Lieb et al. 2009). The folds that develop from reaction – mechanical feedback are remarkably similar to natural folds with thickened hinges and attenuated limbs, an irregular wavelength distribution and wavelength to thickness ratios of 4–7. Folds do not develop by this mechanism if strain softening exists without strain-rate softening. However, both strain softening and strain hardening can lead to fold development when coupled with strain-rate softening. Similarly, in strain-rate softening materials, boudinage is readily developed even for very small initial viscosity ratios (10) between the layer and the embedding material, and for materials that are Newtonian in the absence of coupled mineral reactions. This, again, is quite different from the development of boudinage in power-law materials with no strain-rate softening (even for such materials with strain softening) as discussed by Smith (1977) and Schmalholz et al. (2008). Deformation not only influences the kinetics of mineral reactions, it also influences the stability of mineral phases. Thus, while a mineral reaction is in progress and viscous deformation is proceeding

DEFORMATION AND METAMORPHIC REACTIONS

the field of stability of the minerals is shifted relative to the equilibrium stability field. This nonequilibrium stability field relaxes to the equilibrium stability field as the mineral reaction proceeds to completion and as the deformation wanes. The effect is negligible at geological strain rates and temperatures for the quartz ! coesite reaction. The P–T estimates performed in natural cases presented earlier also indicate no difference in P –T estimates from deformed and undeformed assemblages. Nevertheless, the effect is present and may influence nucleation to be preferentially favoured or inhibited in high-strain zones as is well documented for martensitic transformations in deformed metals (Ericksen 1998). Moreover, in the analysis presented here we have treated a very simple reaction, and we have assumed that the constitutive behaviour is identical for the reactants and products. This is not the case in general and so the issue needs further attention. Thus, in regions where mineral reactions have not gone to completion the estimates of pressure and temperature obtained by assuming equilibrium are within the error of the physical conditions operating in the non-equilibrium system except that the relevant measure of pressure defining equilibrium is the mean stress rather than the lithostatic pressure. This is true even with only elastic deformation if the stresses remain non-hydrostatic. It must be noted that the difference, although small, corresponds to a value that may be in the medium error margin in barometric estimates (e.g. Worley & Powell 2000) and so is amenable to high-precision P –T studies. In the Italian Alps the shear zones that host mineral reactions that have proceeded to completion are widely spaced with little evidence that localized deformation has migrated from one site to another. This would be interpreted as arising from strain softening coupled to strain-rate softening. The lack of obvious geometrical relation between the orientation of shear zones and coeval folds is to be expected from the models presented here. The outcrop pattern of shear zones, the widespread occurrence of Y- and V-shaped shear zones, the widespread occurrence of boudins, and the association of mineral reactions with the necks and margins of boudins are all hallmarks of a terrain where deformation and chemical reactions are strongly coupled to produce strain-rate-softening behaviour. An important point to emphasize is that the results depicted in Figures 10–12 are, in principle, scale invariant. Although we have developed the theory here specifically for the outcrop-scale, the same principles hold at all scales except that different processes lead to strain-rate softening at different scales. Thus, thermal feedback processes dominate at the regional scale, chemical kinetic

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feedback processes dominate at the outcrop scale and chemical kinetic– diffusion processes dominate at the micro-scale. All of these processes produce similar structures (shear zones, folds, boudinage) at the relevant scale and this leads to scale invariance of structures in deformed metamorphic rocks. The reader will have noted that we have specifically not included large-scale fluid and mass influx (metasomatism) into the deforming rock mass nor the heat generated (or absorbed) during mineral reactions. The reason for this neglect is that the inclusion of such processes adds a significant additional level of complexity that is best left for other papers. Our intent in this paper has been to emphasize that coupling between mechanical dissipation and that part of the chemical dissipation represented by the term AKj˙ K in equation (63) is sufficient to result in strain-rate softening and, hence, the development of instabilities in the deformation. In order to understand the complexity introduced by the inclusion of metasomatism and the heat of mineral reactions, it is convenient to recast the Energy Equation, equation (9), in the form:   X DT @2 T E@ ¼ kij þ l@ exp  RT Dt @xi @xj @

(36)

where DT/Dt is the time material derivative of the absolute temperature, T, kij is the thermal diffusivity tensor and xi is a spatial co-ordinate. l@ is a parameter that is a measure of the dissipation for the @-th process; for thermal–mechanical, thermal– chemical and thermal–fluid-flow processes it is the Gruntfest Number, the Damkoehler Number and the Peclet Number, respectively (Law 2006; Veveakis et al. 2010). These dimensionless groups are measures of the heat generated by deformation, chemical reactions and heat advection relative to that generated by conduction. E @ is the activation energy for the @-th process. Equation (36) is a special form of the reaction –diffusion equation and its form derives from an Arrhenius dependence of the relevant geological processes upon temperature. Coupled processes that obey equation (36) have been widely studied and the solutions of many of the equations involved are well known, especially in combustion physics (Law 2006). In equation (36), if T is replaced by c, the concentration of a mineral species, and the exponential dependence is replaced or augmented by non-linear terms involving c, then the result is the reaction –diffusion behaviour discussed by Ortoleva (1994) for a diverse range of processes including metamorphic differentiation. All of these relations are special forms of the Energy Equation, equation (9). One outcome of this approach is that ratedependent mechanical instabilities are inhibited by

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endothermic reactions such as most anhydrous or dehydrating silicate reactions (including melting), and enhanced by exothermic reactions such as hydrating retrograde reactions and melt crystallization. The feedback relations also depend on the timescale for the deformation with different effects at seismic strain rates as opposed to orogenic strain rates. Veveakis et al. (2010) show that for seismic strain rates endothermic reactions (which include most prograde reactions as well as grainsize reduction and melting) stabilize the deformation and produce a narrowing of the zone of localized deformation within a developing wide shear zone. Thus, the development of ultramylonites and pseudotachylites within a wide mylonite zone is explained by a coupling between these processes. The widening of the study of coupled processes in metamorphic geology to include these other processes offers the opportunity to integrate many of the features that we see in deformed metamorphic rocks under the single umbrella of continuum thermodynamics.

Conclusions We have examined the coupling between deformation and mineral reactions at the scale of tens to hundreds of metres, and developed a model whereby the coupling results in both strain hardening (or softening) and strain-rate hardening (or softening). The precise forms of these relationships await further theoretical and experimental developments, but we have explored several realistic scenarios in answer to the specific questions raised in the beginning of the paper. † The application of non-equilibrium thermodynamics at the scale of tens to hundreds of metres shows that mineral reactions induce both strain and strain-rate hardening (or softening) in ratedependent materials. Strain softening alone in strain-rate hardening materials such as Newtonian and power-law viscous materials with no coupling produces shear zones, but the accumulated strain remains much smaller than is observed in natural shear zones. Folds and boudinage do not form. Coupling to mineral reactions leads to strain-rate softening, and highintensity strain localization now develops with strain intensities similar to natural examples. This occurs no matter whether the mineral reaction results in strain hardening or strain softening. Moreover, folds and boudinage also develop in layered materials with small (c. 10) viscosity ratios between the layers and the embedding material. This is a non-Biot folding mechanism and is driven by strain-rate (and, hence, viscosity) differences that develop between the inner and

outer arcs of incipient buckles or between planar and curved parts of incipient boudins. The folds that develop through this process resemble natural folds with strongly attenuated limbs. The ratios of wavelength to thickness are also in the range of naturally observed ratios but depend on the initial distribution of geometrical or material heterogeneities. † Deformation influences mineral reactions at the scale of tens to hundreds of metres where the energy dissipated by inelastic deformation enhances the mineral reaction rate. As indicated by many authors (see Vernon 2004 for a discussion), this is equivalent to influencing the kinetics. More theoretical and experimental work, coupled with detailed field and laboratory analysis, is required to develop better quantitative models for this influence, particularly the influence (if any) of strain rate on mineral reaction rate. The result is a feedback relation resulting in both strain and strain-rate hardening (or softening). † Although the concept of a non-equilibrium mineral phase diagram can be developed, at geological strain rates the non-equilibrium phase boundary for the quartz ! coesite reaction is within error of the equilibrium phase boundary. The departure depends strongly on the differences in elastic strain and specific volumes between the reactants and the differences in constitutive behaviour between the reactants and the products. At least for the quartz ! coesite reaction the differences between the nonequilibrium and equilibrium phase boundaries do not exceed approximately 0.1 Pa in the temperature range 600–1200 K at strain rates of 10214 –10212 s21. High-precision P and T estimates are needed, simultaneously on weakly and highly deformed rocks (from coronites to mylonites), to explore the effect for more complicated assemblages. The non-equilibrium phase stability depends on the mean stress (P) instead of the hydrostatic pressure and so this effect may be important in some regions (Burg & Schmalholz 2008). We thank K. Regenauer-Lieb for long and inspirational discussions. C. Detournay is thanked for help with the remeshing routines in FLAC2D. R. Twiss and M. Brown are thanked for critical reviews that greatly improved the paper.

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Index Note: Page numbers in italics denote figures. Page numbers in bold denote tables. activation energy see Arrhenius data analysis Adria microplate collision with European plate 115 modelling 120 –125 rheology 146 rotation 115, 126 Sesia–Lanzo Zone 160 Aegean trench 130, 131 Africa–Europe convergence Calabrian Arc 130, 131, 141, 143, 144, 145, 146 Western Alps 115, 121, 122, 125 albitization 65 Alps see Austroalpine domain; Central Alps; Western Alps amphibolite-facies 65 anisotropy, seismic, deformed Montalto leucogneiss 49, 58, 60 –65, 66, 67 Aravalli Mountain Belt 36, 37 Argentera –Mercantour massif, synmetamorphic boudinage 200 –201 argon isotope dating 4 K-feldspar, South Cyclades Shear Zone 17– 33 loss experiments 4– 7 pseudotachylytes 12 transport 10 Argon Partial Retention Zone, K-feldspar, South Cyclades Shear Zone 17, 32, 33 Arrhenius data analysis K-feldspar 7 South Cyclades Shear Zone 18, 19, 22, 23, 24, 25, 27– 33 Aspromonte Massif 50, 51 asthenosphere, upwelling, Western Alps 116, 118 Austroalpine domain Languard–Tonale TMU 174 –185, 192, 193 Sesia–Lanzo Zone 150, 193 tectonic evolution 160 –168, 164 Biella pluton 160, 164 Biot folding mechanism 209, 216 biotite Ar loss 6 destabilization 4 –5 kinked, Godhra Granite 36, 38, 42

blueschist-facies 160 bond strength 3 boudinage development 210, 212, 216, 217 synmetamorphic 198–201 breakoff see slab detachment brine intercrystalline 69 conductivity 72 electrical resistivity 69 –70 geometry 73–76 grain boundary 75– 77 Calabrian Arc, crustal deformation 129–146 numerical modelling 130–146 crustal thickness 133, 135 GPS data 135–136, 137, 145 heat flow 133, 135, 136 lithospheric strength 133, 137–139, 140 stress-strain data 135, 137, 142, 143 tectonic deformation 140–146 thermal analysis 133, 134, 137, 139 Cauchy stress 203, 211 Central Alps Languard –Tonale TMU 174–185, 193 deformation and metamorphism 192, 198 Central Indian Tectonic Zone 36 –37 chessboard pattern, quartz, Godhra Granite 36, 38, 41, 42, 46 Clausius –Clapeyron slope 213 closure temperature 10 Dodsonian 4, 6, 9 collision see continental collision conductivity, intercrystalline brine 72 constraints 2 continental collision slab detachment 100, 102, 106 collision zone geometry 107, 108 continental crust exhumation 150, 151, 153, 157–160 Sesia–Lanzo Zone 160 corner flow 149–150 coupling deformation and mineral reactions 203–208 scale effects 204–205 crust see continental crust

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crystallographic preferred orientation (CPO) 79 Montalto Shear Zone leucogneiss 50, 58, 59, 60, 65, 67 Cycladic Massif 17 deformation brittle, gypsum 82, 83, 85, 94–95, 96 crustal, Calabrian Arc 129 –146 gypsum 79 –96 microstructural analysis 83–87 quantitative texture analysis 87 –94 shape-preferred orientation (SPO) 79, 83 –87 influence on phase stability 211, 213 –215, 216– 217 interaction with metamorphism 189 –218 leucogneiss, Montalto Shear Zone 50 mineral reactions 203 –208, 211, 213 –215, 216– 217 scale effects 204– 205, 215, 217 plastic gypsum 82, 85, 94–95, 96 halite, electrical resistivity 69 –77 polyphase, Languard– Tonale TMU 173 –185 twins 36, 38, 41, 42, 45, 83 deglaciation, Alpine 125 Dent Blanche Nappe, synmetamorphic boudinage 198 –199 detachment see slab detachment diffusion chemical potential gradients 202 Fick’s law 1–7, 10 multidomain models 19 –33 natural geochronometers 7–9 OH 10 diffusivity H and O 10 mica 4–6 dihedral angle 69, 76 –77 Dioritic–Kinzigitic Zone 163, 164 disequilibrium 1–2, 9–13 heterochemical mixes 12 –13 dissipation coupling 203– 204, 217 energy 189 viscous 100 dissolution, Wood–Walther 6 East European Platform 131 eclogite, exhumation 150 Sesia –Lanzo Zone 160, 163 –168 eclogite-facies, Monte Mucrone 194, 196

Eclogitic Micaschists Complex 163, 164 synmetamorphic boudinage 199–200 ECORS-CROP seismic profile 117, 118, 119, 120 energy dissipation 189 Energy Equation 203, 204, 205, 217 entropy 201–202 equilibration, diffusive 1–13 European plate 130, 131 Fe –Mg zoning, garnet 7 feldspar, recrystallization 42 Fick’s law diffusion 1 –7, 10 fluid, intercrystalline, deformed halite 69 –77 fluid flow 189, 202 folds 197–198 Biot mechanism 209, 216 development 209–211, 212, 216 fractal cube, multidomain diffusion models 20 –23, 24, 25 fractal feathering 24, 27, 30, 33 fractal geometry, quartz grains area-perimeter method 37, 40–41, 44 Godhra granite 37–47 ruler method 35 –36, 37, 39–40, 41, 42, 43, 44 Fundamental Asymmetry Principle 23 –24, 27–33 garnet cation diffusion 4 diffusional re-equilibration 7 Montalto Shear Zone 55, 56 geochronometers, diffusion 7–9 experiments 4–7 geohygrometry 11, 12, 13 Gibbs Free Energy 201, 203 gneiss, Languard– Tonale TMU 174 Gneiss Minuti Complex 163, 164 Godhra Granite 37, 38 fabric development 36–37 fabric distribution 39 kinked biotite 36, 38, 42 quartz grains chessboard pattern 38, 41, 42, 46 fractal analysis 37– 47 serration 39, 45 strain rates 40, 41– 42, 45 –46 grain boundaries, brine 75 –77 granite cooling rates 46 syntectonic see Godhra Granite greenschist-facies 65, 160, 163–166 Languard– Tonale TMU 174, 177 retrogression 9, 160

INDEX

gypsum, deformation grain size analysis 86 microstructural analysis 83 –87 quantitative texture analysis 87–94 shape-preferred orientation (SPO) 79, 83–87 torsion experiments 79 –96, 81 halite, deformation electrical resistivity 69 –77 fluid paths 72–76 grain boundary wetting 75 –77 stress-strain curves 70–72 halite-water systems 69 dihedral angle 69, 76 –77 grain boundary fluid 75 –77 Hashin–Shtrikman approximation 63, 64 Helmholtz Free Energy 203 hydration, mantle 149 –169 hydrogen, diffusivity 10 Imposed Shear Plane (ISP), gypsum 81, 83, 84, 85 –87, 94 Insubric Line see Periadriatic Lineament Ios, South Cyclades Shear Zone 17 –33 isostatic uplift, Western Alps 115 –126 isothermal duplicates 24, 25, 32 isotope retentivity 10 –12, 11 isotope transport 2–13 Ivrea body 120 K-feldspar albitization 65 Ar loss 6–7 Ar retention, South Cyclades Shear Zone 17–33 augens, Montalto Shear Zone 55, 56 kink bands biotite 36, 38, 42 gypsum 83 Languard– Campo Nappe 174 Languard– Tonale TMU 174– 185, 175 deformation 174, 176, 177, 180, 181 –183 and metamorphism 192, 195 folding 198 shear zones 197 fabric evolution 177 3D modelling 177 –185 metamorphic evolution 174, 176, 177 modelling 177 –185 tectono-metamorphic history 174, 177 –185

227

Lattice Preferred Orientation (LPO), gypsum 83, 85, 89 leucogneiss, mylonitic Montalto shear zone 49 –67 albitization 65 chemical analysis 50, 53 image analysis 50 mineralogy 54–58, 65 petrophysical analysis 50, 52 P-wave velocity 58, 60, 61, 62, 63, 65 S-wave velocity 58, 60– 64, 65 MacArgon program 32 magmatism, bimodal, Alps 116, 118 mantle hydration 149–169 modelling 150–160 P–T data 158–160 wedge convective cell 151, 157–160, 169 corner flow 149, 169 P–T conditions 169 serpentinization 150 MATLAB program VPPLOT 63, 64 MAUD software 87 Mediterranean, central crustal deformation 129–146 see also Calabrian Arc Menger Sponge 20, 21, 23, 24, 26 metamorphism HP– LT, Western Alps 150, 169 interaction with deformation 189–218 non-equilibrium mineral reactions 190 polyphase 173–185 retrograde, Montalto Shear Zone 65 metasomatism 189, 217 mica Ar isotope dating 4, 6 diffusion experiments 4–6 Montalto Shear Zone, CPO 55, 58, 59, 65, 67 retrogression 8–9 mica, white, Montalto Shear Zone 55, 56, 58, 65 micaschist, Languard –Tonale TMU 174, 179 Mid-Atlantic Ridge 130, 131 mineral reactions and deformation 203–208, 211, 213–215, 216–217 scale effects 204–205, 215, 217 non-equilibrium 190 modelling crustal deformation, Calabrian Arc 130–146

228

INDEX

modelling (Continued) fabric evolution and metamorphic transformation, Languard–Tonale TMU 177 –185 isostatic readjustment, Western Alps 115 –126 multidomain diffusion models 19 –33 oceanic-continental subduction, mantle hydration 150 –160 slab detachment 100 –112 reference model 103, 106 thermomechanical model 102, 105– 106, 107, 108, 110, 111, 112 Montalto Shear Zone 51 deformed leucogneiss 49 –67 Monte Mucrone–Monte Mars TMU 193 deformation and metamorphism 190, 194 shear zones 194, 196 non-equilibrium 190 multidomain diffusion models 19–33 fractals 20 –23, 24, 25 muscovite destabilization 4–5 diffusive equilibration 10 mylonite Montalto Shear Zone leucogneiss 49–67 P- and S-wave velocities 60, 61, 62, 63 myrmekite 36, 42, 55 necking, slab 100, 103, 106, 108 neutron diffraction, deformed gypsum 87, 88–89, 90– 93, 94, 96 oceanic plate see subduction, oceanic-continental Oetztal Nappe folds 198, 199 shear zones 197, 198 olivine–spinel transition, slab detachment 100, 112 orientation distribution function (ODF), deformed gypsum 87, 88 orthogneiss, mylonitic, Montalto Shear Zone 49– 67 oxygen, diffusivity 10 P–T –A –X equilibrium 1, 2, 8–9, 10 P-wave velocity Alps 118 leucogneiss, Montalto Shear Zone 58, 60, 61, 62, 63, 65 Penninic Front 118, 119, 120, 125 Periadriatic Lineament 118, 119, 120, 125, 163, 164, 166, 193

petrology, metamorphic, and non-equilibrium thermodynamics 201–203 phase stability, influence of deformation 211, 213–215, 216– 217 phlogopite, Ar loss experiments 5– 6, 7 plate velocity, relative 106, 109–110 polycyclic tectonic evolution, Languard– Tonale TMU 173–185 pseudotachylytes 12 quantitative texture analysis (QTA), gypsum 79, 87 –94, 95, 96 quartz grains fractal geometry methods 35–36 Godhra Granite fractal analysis 37– 47 serration 39, 45 Montalto Shear Zone 55, 56, 57, 59 CPO 58, 59, 65, 67 quartz ! coesite reaction 213, 214, 215, 217 Rb –Sr isotope dating 3–4 re-equilibration diffusional 7 Languard– Tonale TMU 173–174, 177 rebound, Western Alps 125 recrystallization deforming halite 70–72, 76 eclogite-facies 160 hydrothermal 7 resistivity, electrical, deforming halite 69 –77 retrogression 8– 9, 12 –13 greenschist-facies 9 rheology, lithospheric, central Mediterranean 130–146 Riedel-like deformation, gypsum 85, 94, 96 Rietveld texture analysis, deformed gypsum 87, 88 Rocca Canavese Thrust Sheet 163, 164 roll back, slab 106, 108 S-wave velocity leucogneiss, Montalto Shear Zone 58, 60, 61, 62, 63, 64, 65 shear-wave splitting 60, 61, 62, 63, 64, 65 Schlingen Zone, folds 198, 199 Schneeberg Complex, folds 198, 199 seismotectonics, Western Alps 118 serpentinization, mantle wedge 150 Sesia– Lanzo Zone 150, 164 deformation and metamorphism 190 shear zones 196, 197 non-equilibrium 190

INDEX

synmetamorphic boudinage 199 –200 tectonic evolution 160 –168 P –T conditions 164, 165, 166, 167, 168 –169 shape-preferred orientation (SPO) deformed gypsum 79, 83–87, 96 Montalto Shear Zone leucogneiss 54, 65, 67 shear strain, gypsum 83, 85–87, 89, 90– 93, 94 –96 shear zones development 208 –209, 211, 215 –216, 217 strain-rate-sensitive materials 207 –211, 216 synmetamorphic, Italian Alps 192, 194, 196 –197, 198, 217 viscosity 208 –209, 215 see also Montalto Shear Zone; South Cyclades Shear Zone slab detachment Alpine chain 116, 118, 124, 125– 126 bending 103, 106 collision zone geometry 107, 108 necking 100, 103, 106, 108 numerical modelling 99–112 petrological model 102 reference model 103, 106 thermomechanical model 102, 105 –106, 107, 108, 110, 111, 112 olivine–spinel transition 100 phase transitions 110, 111 plastic 100 relative plate velocity 106, 109 –110 relaxation 103, 106 roll back 106, 108 slab age 106, 109, 109 stretching 103, 106 viscous 99– 100, 106 South Cyclades Shear Zone, K-feldspar 17– 33 Sr isotope dating 4 step-heating experiments 20, 22, 23 –24, 25, 27, 28 strain deformed gypsum 83 leucogneiss, Montalto Shear Zone 50, 52, 65, 66, 67 Western Alps 117, 118 strain hardening/softening 206– 211, 215, 216 strain rate 206 –207, 215 –216 Godhra Granite 36, 40, 41– 42, 45 –46 stress Cauchy 203, 211 mean 202, 211 non-hydrostatic 202

subduction, oceanic-continental ablative subduction 151, 153, 157, 168 mantle hydration 149–169 modelling 150–160 slab detachment 99–112, 103, 106 collision zone geometry 107, 108 relative plate velocity 106, 109–110 Western Alps 125– 126 subduction plane, Western Alps 120, 125–126 tectonomorphic units (TMU) fabric evolution and metamorphic transformation 173–185 3D modelling 177–185 temperature –time evolution 2, 3 texture analysis, quantitative (QTA), gypsum 79, 87–94, 95, 96 thermobarometry 8 –9, 13 thermochronology 1, 11 thermodynamics First and Second Laws 204 non-equilibrium 190 and metamorphic petrology 201–203 Tonale Series 174 see also Languard– Tonale TMU torsion experiments, gypsum deformation 79–96, 81 trace elements, diffusion 10 Traversella pluton 164, 166 twinning 36, 38, 41, 42, 45 mechanical, gypsum 83 Tyrrhenian Sea, crustal deformation 129–146 numerical modelling 130–146 uplift, Western Alps 116, 125–126 Valpelline Series, synmetamorphic boudinage 198–199 viscosity, in shear zones 208–209, 215 water importance for equilibrium 10, 11, 12, 13 oceanic-continental subduction 150 wedge see mantle wedge Western Alps HP– LT metamorphism 150 isostatic readjustment 115–126 numerical modelling 120–126 horizontal velocities 121–122, 125

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230

Western Alps (Continued) stresses 122– 124, 125 vertical velocities 123, 124 –125 seismotectonics 118 Sesia –Lanzo Zone 150, 193 deformation and metamorphism 190 tectonic evolution 160 –168

INDEX

stress-strain 117, 118, 125 uplift 125–126 X-ray diffraction, deformed gypsum 87, 88–89, 90 –93, 96 zircon, diffusion 7

Iterative comparison of analytical results and natural observations with predictions of numerical models improves interpretation of geological processes. Further refinements derive from wide-angle comparison of results from various scales of study. In this volume, advances from field, laboratory and modelling approaches to tectonic evolution – from the lithosphere to the rock scale – are compared. Constructive use is made of apparently discrepant or non-consistent results from analytical or methodological approaches in processing field or laboratory data, P–T estimates, absolute or relative age determinations of tectonic events, tectonic unit size in crustal scale deformation, grain-scale deformation processes, various modelling approaches, and numerical techniques. Advances in geodynamic modelling critically depend on new insights into grain- and subgrain-scale deformation processes. Conversely, quantitative models help to identify which rheological laws and parameters exert the strongest control on multi-scale deformation up to lithosphere and upper mantle scale.