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_Journal of Structural Geology 32 (2010) 1114e1124_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg_ _Scale-invariance and self-organized criticality in migmatites of the southern Hualapai Mountains, Arizona_ _Chlo? E. Bonamici*, Ernest M. Duebendorfer_ _School of Earth Sciences and Environmental Sustainability, Geology Program, Northern Arizona University, P.O. Box 4099, Flagstaff, AZ 86011-4099, USA_ _article info_ _Article history: Received 16 February 2010 Received in revised form 25 June 2010 Accepted 28 June 2010 Available online 19 August 2010_ _Keywords: Migmatite Melt-?ow network Scale-invariance Self-organization_ _abstract_ _Whether or not melt-?ow networks are scale-invariant may have broad implications for the understanding of the development of magmatic systems. Leucosome distributions in natural migmatite samples from the Hualapai Mountains, Arizona, are used to investigate the scaling characteristics of an inferred syndeformational melt-?ow network. One-dimensional line and two-dimensional box-counting analyses yield distributions that are consistent with a scale-invariant relationship between the leucosome size and leucosome frequency. We infer from these results that the interactions of anatexis and deformation gave rise to a self-organized critical system in which melt distribution and melt movement were linked over a range of scales (at least millimeterto meter-scale). We conclude that, even in metatexite migmatites volumetrically dominated by solid phases, melts can play an active role in determining the architecture of the melt-?ow network. One implication of the envisioned self-organized critical system is the potential for rapid, large-scale melt coalescence and escape without long-term maintenance of a fully interconnected melt-?ow network._ _1. Introduction_ _Although most workers agree that migmatite terranes were sources for larger granitoid intrusions, they continue to debate the driving forces and mechanisms of melt coalescence and transfer between these anatectic zones and plutons (e.g., Mehnert, 1968; Petford et al., 1993; Brown et al., 1995a, b; Weinberg, 1999; Sawyer, 2001; Simakin and Talbot, 2001; Vanderhaeghe, 2009). The ultimate driving force of large-scale melt migration is buoyancy gravitational instability (e.g., Clemens, 1998; Petford et al., 2000), but it remains unclear how ef?cient buoyancy alone is at driving melt coalescence locally within the anatectic zone, where the viscous properties and deformational states of solid-dominated migmatite (metatexite) exert a strong control on the location and interaction of melt segregations (e.g., Sawyer, 1994; Brown et al., 1995a, b; Holyoke and Rushmer, 2002; Vanderhaeghe, 2009). While plutons and batholiths testify to the ef?cacy of long-range, large-volume melt migration, predominantly local control of melt migration in the anatectic source region begs the question of how_ _* Corresponding author. Present address: Department Geoscience, University of Wisconsin-Madison, 1215W. Dayton Street, Madison, WI 53706, USA._ _E-mail addresses: bonamici@wisc.edu (C.E. Bonamici), ernie.d@nau.edu (E.M. Duebendorfer)._ _0191-8141 $ e see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2010.06.019_ _to affect the focused and organized extraction of melts from the source to the ?nal site of emplacement._ _Several studies document the rock record of melt-?ow networks in migmatite terranes, which typically comprise leucocratic segregations (leucosomes) of variable shapes and sizes within a host of high-grade metamorphic rock (e.g., Brown and Solar, 1998; Brown et al., 1999; Vanderhaeghe, 1999; Ledru et al., 2001; Sawyer, 2001). Most workers infer that leucosomes represent crystallized partial melt segregations that interacted periodically through mass transfer (e.g., Brown et al., 1999; Marchildon and Brown, 2001). Various melt migration processes have been suggested from these natural examples, including pervasive grain-scale percolation (e.g., Sawyer, 1994; Laporte and Watson, 1995; Brown et al., 1999; Rabinowicz and Vigneresse, 2004), mesoscale diapirism (e.g., Weinberg and Podladchikov, 1994), dyking (e.g., Petford et al., 1993; Clemens, 1998), pervasive ?ow (Weinberg and Searle, 1998; Weinberg, 1999), and ductile fracturing (e.g., Eichhubl and Aydin, 2003; Brown, 2004), most of which are actuated and or controlled by local stress and strain gradients (e.g., van der Molen, 1985; Brown et al., 1995a, b; Marchildon and Brown, 2001; Sawyer, 2001; Simakin and Talbot, 2001; Rabinowicz and Vigneresse, 2004)._ _Migmatite melt-?ow networks are potentially useful analogues for magma transfer networks throughout the crust, provided that their spatial features can be scaled (Brown, 2001). Only a few studies have attempted to quantify the spatial distribution of_ _C.E. Bonamici, E.M. Duebendorfer Journal of Structural Geology 32 (2010) 1114e1124_ _1115_ _natural migmatites and investigate the scaling properties of the melt-?ow networks they represent. Different authors have applied subtly different techniques that assess different physical aspects of migmatites. Soesoo et al. (2004) found a power-law (scale-invariant) relationship between leucosome size and leucosome frequency over 2e3 orders of magnitude in Estonian drill core samples. In the most comprehensive study of its kind, Marchildon and Brown (2003) determined that migmatites of southern Brittany do not preserve evidence for a power-law distribution, either between the size and frequency of leucosomes (from 1-D analyses) or between the size and shape of leucosomes (from 2-D analyses). Tanner (1999) documents a power-law relationship in Bavarian migmatites over >1.5 orders of magnitude; however, as Marchildon and Brown (2003) point out, his analyses do not distinguish the signal associated with leucosome shape from that associated with leucosome frequency, making the resultant power-law relationship somewhat harder to interpret._ _In this study, we ?nd that stromatic (layered) migmatites of the southern Hualapai Mountains consistently show a scale-invariant relationship between the size and frequency of leucosomes both in one-dimensional line traverse analyses and two-dimensional surface analyses. The southern Hualapai migmatites differ from those previously investigated in that they generally contain a signi?cantly higher percentage of leucosome (>65_) and yield signi?cantly higher fractal dimensions (D ? 2.68e3.87 in twodimensional analyses). Because a single melting event is unlikely to produce >65_ melt, we argue that the observed abundance of leucosome probably arose from temporal superposition of multiple melt-?ow networks, each with a scale-invariant sizeefrequency relationship. The inferred tectonic history of the area (Bonamici and Duebendorfer, 2009) and outcrop-scale structures are consistent with diachronous melt-?ow network creation and abandonment, and we argue that the preserved scale-invariant leucosome distribution signals an important, perhaps fundamental, mode of melt?ow network development in this system. We further discuss the possibility that the scale-invariant distribution indicates achievement of self-organized criticality (SOC) in this system and some associated implications for initiation and maintenance of throughgoing melt conduits in the crust._ _2. Geologic background_ _The southern Hualapai Mountains study area (Fig. 1) is located within the proposed N-S-trending boundary zone between the isotopically distinct Mojave and Yavapai crustal provinces (Bennett and DePaolo, 1987; Wooden and DeWitt, 1991; Duebendorfer et al., 2006). The provinces were conjoined and then accreted to southwestern Laurentia during the Paleoproterozoic (Duebendorfer et al., 2001). Regionally, two phases of Paleoproterozoic deformation are distinguished: an early phase produced NNW-striking, moderately to gently dipping foliations and recumbent folds, and a later phase reoriented early fabrics to produce a NE-striking, steeply dipping foliation and upright folds (Karlstrom and Bowring, 1991; Albin and Karlstrom, 1991; Ilg et al., 1996; Duebendorfer et al., 2001). Early deformation occurred, at least in part, at 5e6 kbar and 650e750 x14C (Ilg et al., 1996; Duebendorfer et al., 2001). Later deformation took place at similar temperatures but at reduced pressures of 3e4 kbar (Williams, 1991; Ilg et al., 1996; Hawkins et al., 1996; Duebendorfer et al., 2001)._ _The southern Hualapai Mountains study area comprises Paleoproterozoic basement exposed through a combination of Laramide uplift and high-angle normal faulting at the southwestern margin of the Colorado Plateau during Tertiary Basin-and-Range extension (e.g., Anderson, 1989). The area contains predominantly subhorizontally foliated granitoids and paragneisses with abundant textural and compositional evidence for pervasive partial melting. The following is a summary of structural and metamorphic data for the area, which are described in detail in Bonamici and Duebendorfer (2009)._ _2.1. Structures and deformation_ _The study area contains four types of deformational structures. The most prominent is a penetrative, gently dipping, variably striking foliation, de?ned in migmatitic units by well-developed compositional layering of leucosomes._ Ключевые слова: weinberg, model curve, central range, thickest leucosomes, evidence, leucosome thickness, nature, marchildon, law, size category, melt distribution, righthand plot, journal structural geology, spatial, bonamici, melt extraction, parallel, mountains, migmatite package, melanosome schlieren, power-law distribution, society, frequency, thickness, k-type distribution, loriga, slope, anatectic zone, segment, tightly clustered, greater thickness, proterozoic geology, power-law relationship, bak, type, binary image, surface, greater, model, subvertical shortening, self-organized critical, active role, temporal superposition, vigneresse, migmatite terranes, traverse length, outcrop, geology, melt segregation, brown, leucosome geometry, traverse, solid earth, migration, structural, field, field photograph, curve, cumulative, self-organized, traverse insets, sciences, tectonic, leucosomes, self-organized criticality, journal structural, leucosome frequency, sawyer brown, power law, duebendorfer, tectonophysics, power, structural geology, stromatic leucosomes, williams, stroma thickness, university, analysis, study, anatectic, bonamici duebendorfer, hualapai, total, leucosome, correlation coefcient, geophysical, matrix inversion, image, foliation, cumulative frequency, stromatic migmatites, scale, layering, migmatites, subvertical, central segment, physical, organized, gillespie, crosscutting dike, melt-?ow, soc, number, continental crust, geological, earth, segregation, angle, hand, box-counting curve, shorter traverse, transfer, melting, limited range, leucosome distribution, distinct peak, long-term maintenance, recumbent fold, relationship, melanosome, area, magma, network, journal, correlation, plot, turcotte, karlstrom, petford, address, migmatitic, traverse analysis, study area, wa, melanosomes, scale-invariant, cumulative thickness, mm, measurement, tanner, length, deformation, jensen, marchildon brown, box-counting, scale-invariant distribution, dike, geological society, vanderhaeghe, journal geophysical, critical, fractal, bons, southern, crust, stroma, elsevier, melt-ow network, power-law, blue traverse, shear, range, zone, strain gradient, spatial distribution, layer-parallel leucosomes, underlying distribution, melt migration, migmatitic gneiss, scale-invariant relationship, geometry, dimension, cambridge, ore deposits, total number, mass transfer, box, counting, fractal dimension, logelog plot, larger leucosomes, granite, one-dimensional analysis, distribution, usa, adjacent horizontal, data set, size, crustal, melt-?ow network, image analysis, burg bons, proterozoic, sornette, sawyer, common, rock, leucosome volume, melt, southern hualapai, migmatite, represent, van milligen, leucosome size, data, arizona, thickness greater, traverse data