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_Journal of Structural Geology 32 (2010) 2072-2092_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com/locate/jsg Review Article Vorticity analysis in shear zones: A review of methods and applications P. Xypolias* Department of Geology, University of Patras, GR-26500, Patras, Greece Article info Article history: Received 11 April 2010 Received in revised form 30 July 2010 Accepted 15 August 2010 Available online 24 August 2010 Keywords: Vorticity number Vorticity methods Ductile flow Monoclinic flow Flow path Abstract Quantitative vorticity analyses in naturally deformed rocks are essential for studying the kinematics of flow in shear zones and can be performed using a range of methods, which have been developed over the last two decades. The purpose of this review is to act as a starting point for the reader who needs a current overview of the existing methods and to indicate in what circumstances these methods can be most suitably applied. The review begins by providing an overview of deformation theory, followed by description of the most promising methods—in terms of assumptions, analytical procedures, and possible sources of uncertainty. Finally, the methods are compared on the basis of their uncertainties and strain memory, and discussed in terms of how they can be used to retrieve information about temporal and spatial variation of flow vorticity in shear zones. This review confirms that although the existing methods are valuable, they are at an immature stage of development and suffer from limitations and uncertainties leading to interpretational problems, which, at present, can be alleviated by applying as many methods as possible to a given sample. Additional studies are recommended to advance the development of existing and new methods. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Natural deformation is commonly concentrated into shear zones that range from the centimetre to the kilometre scale in width. Understanding the kinematics of flow in these zones is a prerequisite for elucidating critical aspects of the tectonic evolution of the Earth’s crust, as well as for deciphering the kinematic significance of fabrics in deformed rocks. The ideal model of simple shear (Ramsay and Graham, 1970; Ramsay, 1980) has strongly influenced the thinking of structural geologists about the formation of high-strain zones and, for many years, it was a standard of reference for interpreting geological structures. Semi-qualitative studies during the decade of the 80s inferred from crystallographic fabrics that the deformation path within some naturally occurring shear zones was not strictly progressive simple shear but included a pure shear component, emphasizing that it is more appropriate to interpret structures in terms of the degree of non-coaxiality rather than in terms of either perfectly non-coaxial or coaxial flow (Law et al., 1984, 1986; Platt and Behrmann, 1986). Such observations generated the necessity to find practical ways for determining the degree of non-coaxiality, or in other words, for evaluating the relation between the vortical and the stretching components of flow using numerical quantities such as the kinematic vorticity number (e.g., Truesdell, 1953; Means et al., 1980). The challenge was, and continues to be, the effective use of structural fabric data to quantify kinematic flow, commonly referred to as vorticity analyses. Since the first vorticity analysis in naturally deformed mylonites by Passchier (1987a), significant progress in developing practical methods for vorticity analysis has been made by many geologists (Passchier and Urai, 1988; Wallis, 1992; Simpson and De Paor, 1993; Tikoff and Fossen, 1995; Grasemann et al., 1999; Holcombe and Little, 2001; Jessup et al., 2007; Gomez-Rivas et al., 2007; Johnson et al., 2009a; Xypolias, 2009), accompanied by numerous theoretical works about the kinematics of rock flow (e.g., Ghosh and Ramberg, 1976; Lister and Williams, 1983; Passchier, 1987b, 1997; Weijermars, 1991; Fossen and Tikoff, 1993; Jiang, 1999; Iacopini et al., 2010). Especially in the last ten years, different vorticity analysis methods have been applied, either in isolation or in combination, to study shear zones from various tectonic settings (Xypolias and Doutsos, 2000; Bailey and Eyster, 2003; Law et al., 2004; Marques et al., 2007; Iacopini et al., 2008; Sullivan, 2008; Frassi et al., 2009; Xypolias et al., 2010; Law, 2010; Thigpen et al., 2010a). Such vorticity studies have confirmed that simple shear is the exception rather than the rule in natural deformation and have presented preliminary data about the temporal and spatial variation of the vorticity of flow in natural shear zones. The identification of a pure shear component of deformation in a shear zone is critically important since during a persistent flow governed by a simultaneous combination of pure and simple shear, it is possible P. Xypolias Journal of Structural Geology 32 (2010) 2072-2092 2073 to cause elongation of deforming material parallel to the walls of the zone (Wallis et al., 1993). It has been also shown that even for a relatively small pure shear component, this transport-parallel elongation can be significant if strain magnitude is enough (Xypolias and Kokkalas, 2006). Such observations, especially when they are combined with data about the spatio-temporal variation of flow vorticity, may shed light on the tectonic setting of the shear zone in question (e.g., Wallis et al., 1993; Northrup, 1996; Grujic et al., 1996; Grasemann et al., 1999; Law et al., 2004). From the above, it becomes apparent that vorticity analysis is a relatively new and valuable tool for solving problems in structural geology and tectonics. However, due to the complexity of natural deformation, we have a fragile sense of confidence about numbers extracted from rocks using methods of vorticity analysis, which are still in a relatively immature stage of development. Therefore, apart from the analytical procedure, one must be aware of possible sources of error linked to the application of vorticity methods. In many application studies, however, the error is rarely discussed in detail. The purpose of this review is to present a state-of-the-art summary of current knowledge about vorticity analysis methods in terms of theoretical background, analytical procedure, limitations and possible sources of errors and uncertainties. Due to a limited database, a comprehensive comparison of existing methods is currently impossible but an attempt is made to discuss the consistency or discrepancies between various methods in the context of uncertainties and the length of strain memory associated with the different methods. This review begins with an overview of the basics of deformation theory. 2. Overview of theory At any instant of time, the velocity field around a point in a deforming continuum can be described, with respect to a Cartesian coordinate system, by the flow or velocity gradient tensor, L (Lij = vvi vxj; i, j = 1, 2, 3) and the related velocity gradient equation: vi = Lijxj (1) where vi is the velocity at spatial coordinate xj (Malvern, 1969; Ramberg, 1975). If L is space-independent and remains unvarying throughout the deforming volume of material, the flow is considered to be homogeneous, otherwise it is heterogeneous. Also, the flow is considered to be steady if L is time-independent, otherwise it is non-steady. The vast majority of analytical works in the geological literature are limited to homogeneous and steady-state flows mainly because their mathematical description is simple. Notice, for example, that homogeneous flows have linear velocity gradients and can be described in a straightforward manner while heterogeneous flows have non-linear velocity gradients, which require application of numerical integration methods of governing equations for their description. However, it is widely accepted that deformation in nature is generally heterogeneous and non-steady. The problem of heterogeneity can be partly overcome by subdividing the deforming continuum into smaller domains where the flow can be approximately viewed as homogeneous. The assumption of steadiness of flow, however, remains a fundamental problem since little is known about flow paths in progressive deformation. Thus, for the present state of knowledge, homogeneous steady-state flows represent a valuable standard of reference for investigating more complex natural systems. 2.1. Decomposition of the flow tensor The flow tensor can be decomposed into the symmetric tensor D and the anti-symmetric tensor W, which are related to the stretching and the vortical (or rotational) components of velocity field, respectively (Malvern, 1969; Ramberg, 1975; McKenzie, 1979; Lister and Williams, 1983): L = D - W (2) Note that the flow tensor does not contain the translating component of the velocity field. This component vanishes by fixing the coordinate system to the particle in question. The symmetric quantity D is the stretching tensor and its three orthogonal eigenvectors are known as the Instantaneous Stretching Axes, ISAi (i = 1, 2, 3), of flow. The eigenvalues of D describe the stretching rates, si (i = a, b, c), of material lines instantaneously parallel to these axes and can have any magnitude. For planar deformation zones in isotropic material, the ISAs are thought to be parallel to the stress axes (Weijermars, 1991). 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