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_Journal of Structural Geology 32 (2010) 127–130_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg _Comment on ‘Folding with thermal-mechanical feedback’_ _Daniel W. Schmid a,*, Stefan M. Schmalholz b, Neil S. Mancktelow b, Raymond C. Fletcher a,c_ _a Physics of Geological Processes, University of Oslo, Oslo, Norway_ _b Geological Institute, ETH Zurich, Switzerland_ _c Department of Geosciences, Pennsylvania State University, University Park, PA 16803, USA_ _article info_ _Article history: Received 16 April 2009 Accepted 5 October 2009 Available online 13 October 2009_ _‘Deeper physical insight combined with theoretical simplicity provides the short-cuts leading immediately to the core of extremely complex problems and to straightforward solutions. This cannot be achieved by methods which are sophisticated and ponderous even in simple cases. The process of thought which is involved here may be described as ‘cutting through the scientific red tape’ and bypassing the slow grinding mills of formal scientific knowledge. Of course, formal knowledge is essential but, as for everything in life, the truth involves a matter of balance.’_ _M. A. Biot Acceptance speech, 1962 Timoshenko Medal_ _1. Introduction_ _In their recent paper, Hobbs et al. (2008) claim that (1) effective viscosity ratios and power-law stress exponents for natural rocks, as extrapolated from experimental conditions to those appropriate for the middle to lower crust, are typically too low for folds of finite amplitude to form according to the ‘traditional’ buckle fold theory, referred to by them as the ‘Biot process’; (2) the geometry of natural folds, in particular the rather irregular fold form and the limited range in arc-length layer thickness ratios, cannot be predicted from this theory; and (3) a coupled model of folding with thermal-mechanical feedback is more appropriate for explaining natural folds. We strongly disagree with all these statements and show that correct application of the analytical buckling theory and its extension to finite amplitude using numerical models based on the same mechanical principles can explain the range of natural fold geometries. This approach provides relatively simple and clear insights into the fundamental processes governing such folds._ _* Corresponding author. E-mail address: schmid@fys.uio.no (D.W. Schmid)._ _0191-8141 $ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2009.10.004_ _development. Our primary aim in this comment is to demonstrate that the ‘traditional’ theory is appropriate for explaining natural examples. We therefore limit our discussion of the ‘Biot process’ to interfacial viscous buckling (IVB) in the absence of elastic effects, as in the original work of Biot (1957, 1961, 1964, 1965), Sherwin and Chapple (1968), Fletcher (1974, 1977) and Smith (1975, 1977). However, for us the ‘Biot process’ encompasses all aspects of the mechanical folding instability, such as multilayer systems, nonlinear rheology, and finite strain effects._ _2. Model of single-layer folding with thermal-mechanical feedback_ _Hobbs et al. (2008) discredit the ‘Biot process’ as a viable folding mechanism without providing any sound counterarguments. They show no natural fold or any comparison between natural examples and numerical models that would illustrate any apparent problem with this approach. It is not sufficient to argue that low effective viscosity ratio between pairs of rocks may occur in nature – of course they do, and in this case buckle folds of finite amplitude simply do not develop (although passive amplification of irregularities can still occur in high strain zones). The necessary argument is to give examples of natural folds that have formed even when it can be reliably established that the viscosity ratio was too low for viscous buckling to have occurred. Such examples are not presented in the paper of Hobbs et al. (2008)._ _Hobbs et al. (2008) cite the absence of parasitic folds in the ‘Biot process’ as an argument. However, there are at least five ways in which purely mechanical folding can produce parasitic folds: 1) at large strains a fold train starts to behave as an effectively thicker layer and larger wavelength folds develop, 2) the effective layer-to-matrix viscosity ratio (R) changes due to ambient temperature changes, 3) thickness (Frehner and Schmalholz, 2006) or 4) viscosity (Ramberg, 1964) variations in multilayer systems or 5) matrix anisotropy (Kocher et al., 2008) cause the simultaneous development of several wavelengths._ _Hobbs et al. (2008) also claim that the type of boundary condition has a crucial influence on fold development in the ‘Biot process’. Clearly, for the linear viscous case this is incorrect as the development of fold amplitude with respect to bulk strain will always be the same, irrespective of the type or magnitude of the boundary conditions. They go as far as stating that ‘‘for constant velocity and strain rate conditions amplification rates are relatively small and realistic folds do not develop unless viscosity ratios are of the order of 3000’’. This contradicts basically all previous analogue and numerical folding research, where folds were readily developed for much lower viscosity ratios for the given boundary conditions. It is even more puzzling because their own previous (Zhang et al., 2000) and current work also shows this, as can be seen from their Fig. 3b for R ≥ 20 folds. Note though that the ‘sensitivity analysis’ presented in their Fig. 3 aims at illustrating viscous folding, but a quick analysis (cf. Schmalholz and Podladchikov, 1999) reveals that elastic effects are strong for these runs (especially e to f). Also, what they call a ‘perturbation velocity field’ is misleading, as it is usually termed the ‘total velocity field’ (e.g. Cobbold, 1976; Passchier et al., 2005)._ _The alternative model presented by Hobbs et al. (2008) and the specific numerical model of single-layer folding are not appropriate either for the claimed middle to lower crustal conditions or for explaining the micro-to-mesoscale folds directly observed by field geologists. They present a single-layer example for the case of thermal-mechanical feedback, in which a 400 m layer of ‘feldspathic rock’ embedded in quartzite is shortened at temperatures of 510 K to 570 K (240 x14C – 300 x14C). The resulting large scale structures would usually be termed ‘pop-up structures’ and the ‘fine-scale crenulations’ referred to as a mesh sensitive phenomenon. In fact, the ‘feldspathic rock’ is the aplite whose creep parameters were reported in an abstract by Shelton and Tullis (1981) and the quartzite flow law is that of Hirth et al. (2001). From the range of feldspar-rich rock and quartzite flow laws available, this aplite is the weakest and this quartzite one of the strongest. The aplite – quartzite R varies between 2 and 1.2 over the model temperature range, and IVB is negligible, as the authors intend, to highlight the effects of thermal-mechanical instability. The model scale is w10^2–10^5 times larger than that at which folds are commonly observed at outcrop, hand specimen or thin section scale (Fig. 1). The temperature range of 510 K to 570 K (240 x14C – 300 x14C) is comparable to that of the brittle-ductile transition. It is not what is expected under middle to lower crustal conditions and is also rather low for crystal-plastic flow of quartzite and aplite. The effects of thermal-mechanical feedback are indeed significant at the model scale for the chosen rock properties and rate of shortening, at which the Peclet Number, written here as Pe ? W^2jDxxj_k ? 0:9, where initial model thickness is W ? 3000 m, rate of shortening is the large value jDxxj ? 10?13s?1 and thermal conductivity is k ? 10?6 m2 s?1. For the single-layer folds shown in their Fig. 6, they report jDxxj ? 10?15s?1, but this yields Pe w 0:009, contrary to the reported value. The use of W rather than layer thickness H is problematic when considering the case of a layer effectively embedded in an infinite medium. Scaling accordingly for a layer of thickness H ? 4 cm, which is greater than that of any of the layers shown in Fig. 1, yields Pe w 10?8 and the effect of thermal-mechanical feedback is negligible, as the authors themselves point out. Thus, unless a thermal-mechanical feedback associated with ‘small-scale compositional – fabric heterogeneities’ produces regular folding in single layers in which mean fold arc-length scales with layer thickness, a result neither adequately examined in their paper nor intuitively plausible, the authors, discounting IVB, leave us with no viable mechanism for the folds commonly observed in nature._ _We now show that, with proper implementation of IVB for power-law layer and matrix pairs and with plausible creep parameters, folds will develop with a scale and geometry directly comparable to those formed in the middle and lower crust._ _3. Arc-length thickness ratios for natural folds_ _Hobbs et al. (2008) assert that the mean fold arc-length to thickness ratio (FR) of natural single-layer folds is so small as to imply that R is too low for the folds to have initiated via IVB. They estimate R inappropriately, by equating FR with the dominant wavelength to thickness ratio Ld/H, rather than the most-amplified value Lp/H, which takes into account the basic-state layer-parallel shortening. For Ld/H, they use also inappropriately the thin-plate approximation (e.g. Biot, 1961). In the limit of R ? 1, Ld/H ? 2π for linear viscous media, and use of the thin-plate equation to estimate Fig. 1. Single-layer folds: (a) calc-silicate layer in coarse-grained calcite marble, Adamello, Italy; (b) pegmatitic quartz-feldspar layer in coarse-grained calcite marble, Adamello, Italy; and (c) pegmatitic quartz-feldspar layer in quartz-feldspar-biotite gneiss, Roveredo, Switzerland._ _D.W._ Ключевые слова: tectonophysics, nite amplitude, folding, buckling, velocity, model scale, claim, layer thickness, author, ramberg, amplitude, crust, ivb, chapple, royal society, problem, rate, absence, hudleston, strain, biot, effective, quartzite, crustal condition, buckle, geological, yield, reported, limit, layer-parallel shortening, single layer, elastic, code, demonstrate, misleading, geophysical, middle crust, develop, wavelength selection, matter, ?nite, single-layer fold, type, matrix, rock, pair, earth, tullis, power-law, physics, structural, marble pair, simulation, schmalholz, rutter, developed, theory, geological society, kohlstedt, large, rheology, ratio, smith, geology, single, schmid, thermalmechanical feedback, process, aplite, natural fold, range, viscosity ratio, mechanical, numerical model, occur, numerical, brittle-ductile transition, hobbs, stress, wavelength, thermal-mechanical, hirth, layer-matrix pair, statement, thermal-mechanical feedback, exponent, presented, development, fold, viscous buckling, negligible, viscous, thickness, result, thick-plate result, feedback, typical, dry diabase, boundary condition, comparison, brodie, limestone, approach, single-layer folding, single-layer, layer, law, cobbold, scale, paper, discussion, numerical modelling, journal structural, viscosity, nature, shortening, sherwin, order, temperature, marble, america bulletin, fr, multilayer, boundary, middle, argument, mancktelow, model, natural, case, small, rock pair, fletcher, formed, arclength, structural geology, quartz vein, initial, buckle fold, ?ow, society, biot process, journal, condition, estimate, observed, fold arclength, geometry, american, ?eld