Discussion

2D sections through 3D spirals

Millipede microstructures
Previous studies have described an unusual inclusion pattern that consists of outwardly opening pairs of concave microfolds, termed ‘clamshells’ by Rosenfeld (1970), ‘millipede microstructure’ by Bell & Rubenach (1980) and ‘oppositely concave microfolds’ by Johnson & Moore (1996). Bell & Johnson (1989) interpreted these microstructures to indicate conditions of progressive inhomogeneous shortening, but subsequent studies have shown that the microstructures can form by any deformation path between bulk coaxial shortening and bulk simple shearing (Gray & Busa, 1994; Johnson & Moore, 1996; Johnson & Bell, 1996). Examples of millipede microstructures can be seen in sections through both the non-rotation and rotation spirals presented in this study (e.g. Fig. 4c,e,f; see also fig. 13 from Gray & Busa, 1994). This further supports the proposal that millipede microstructures represent a peculiar 2D slice through a 3D spiral, and are not an indication of the deformation path or mode of spiral formation.

Closed loop microstructures
Another inclusion pattern commonly observed in 2D sections through spirals is one that consists of closed loops of inclusion trails (e.g. Fig. 6b). These have been previously described in both real rocks (e.g. fig. 4 of Johnson, 1993a) and numerical simulations (Gray & Busa, 1994). The shapes of the loops vary from sub-circular (e.g. Fig. 6c,e) to elliptical (Fig. 6b) and crescentic (Fig. 6e), and the loops may be located entirely within the porphyroblast, or straddle the porphyroblast margin. The closed loops result from the non-cylindrical geometry of the spirals, and represent a section through either the ‘sheath fold’ part of the spiral (see above), or alternatively, a section through that part of the matrix foliation that had wrapped around the sphere prior to being overgrown. These closed loops are observed in any section oriented away from the XZ plane, but are best observed in sections cut parallel to the spiral axis.

2D spiral geometry in off-centre sections
The changing 2D inclusion trail geometries encountered with sections cut at progressively greater distances from the spiral centre can be seen in Fig. 6. In the XZ sections (Fig. 6a,d), the amount of apparent relative rotation between sphere and matrix decreases steadily in sections cut further from the centre of the sphere. This is of significance for studies that seek to measure the orientation of inclusion trails within a population of porphyroblasts, or correlate sections of inclusion trail between different porphyroblasts on the basis of orientation. In XY sections (Fig. 6c,f), the central section contains elliptical and crescent-shaped closed loop geometries, and these loops change in size as sections are cut at progressively greater distances from the sphere centre. Close to the sphere margin, the closed loops form circular shapes, representing a section through matrix foliation (and equivalent foliation included within the sphere) that has wrapped around the growing sphere. In YZ sections (Fig. 6g,e), the central sections contain near-symmetrical closed loops. With increasing distance from the sphere centre (toward positive X values), the upper loops change shape from elliptical to circular, while the lower loops remain elliptical. With increasing distance toward negative X values, the lower loops change shape from elliptical to circular, while the upper loops remain elliptical. The circular loops represent intersection of the narrow ‘sheath fold’ parts of the spiral, while the elliptical loops represent intersection of the more open parts of the spiral, further from the leading tip of the sheath fold.

Simulation conditions and range of possible spiral geometries

As noted by Johnson (1999), both the rotational and non-rotational models can be modified in various ways to account for specific geometries that are not predicted by the models in their simplest forms. This may involve varying the amount of flattening in the matrix (e.g. Williams & Jiang, 1999), the ratio of pure to simple shear (Mandal et al., 2001), the timing of porphyroblast growth relative to deformation, rate of porphyroblast rotation (e.g. Biermeier et al., 2001), or the geometry of the pre-deformation foliation relative to the shear plane (e.g. Masuda & Mochizuki, 1989). In this study, the sphere volume increases by a constant amount at each time step in the simulation. Accordingly, the rate of sphere growth has no effect on the 3D geometry of the spiral, although spiral size increases with faster rates of growth (see above). Alternative growth laws, which approximate diffusion-limited growth or linear growth, affect the shape of the spiral while having only minimal effect on the bulk 3D geometry (e.g. fig. 5 of Gray & Busa, 1994). The spiral geometry is also affected by the degree of coupling between sphere and porphyroblast, and variation in the proportions of simple and pure shear matrix deformation. Deformation by simple shear, when paired with strong coupling between sphere and matrix, creates favourable conditions for producing spirals, although the modelling of multiple foliations enables spiral formation despite the non-rotation of the sphere. Additional complexity results if the angle between the matrix foliation and shear plane is altered, although the unusual geometries observed at high theta values (e.g. Fig. 7g,h) may not actually occur in rocks.
Summary and conclusions
In this study, we presented 22 Quicktime movies, and related figures, that illustrate the progressive development and final 3D geometry of simulated spirals. We also documented the effect on spiral geometry of varying simulation conditions. Both the rotation and non-rotation simulations, in their simplest forms, produce similar 3D spiral geometries, although the mode of spiral formation is different in the two models. Movies of serial slices through the simulated spirals can be used as an aid to interpret the wide range of inclusion trail geometries commonly observed in thin sections in terms of the orientation of the thin section with respect to the spiral axis. While growth rate of the sphere doesn’t influence spiral geometry, factors such as angle between initial foliation and shear plane, and simple shear/pure shear ratio have a strong effect on the resulting pattern of inclusion trails.

Interpretation of spirals in thin section

The simulation results presented in this study have a number of implications for our interpretation of spirals in real rocks. The similar geometry of the rotation and non-rotation simulations suggests that 3D spiral geometry is not diagnostic of the mode of spiral formation (cf. Williams and Jiang, 1999), and as such, a reliable test of the two models remains to be found. The competing models of spiral development also record opposite senses of spiral asymmetry in relation to the matrix shear sense. Accordingly, interpretation of shear sense from spiral asymmetry remains speculative in the absence of reliable evidence concerning the mode of spiral formation.
The simulations presented in Fig. 7 also illustrate the potential uncertainty of interpreting the significance of reversals in spiral asymmetry. Reversals may indicate either a change in the sense of matrix shear, or alternatively a change in the relative rate of rotation of sphere and matrix foliation.

Thin sections cut through spirals often contain complex inclusion trail geometries, and these can be better understood once the orientation of the thin section has been determined with respect to the spiral axis (e.g. Fig. 6). This study emphasises the need for an understanding and description of 3D spiral geometry when conducting any kind of research that measures or interprets spiral geometry, for example determining FIAs or measuring the orientation of subplanar sections of inclusion trails.

Summary and conclusions

In this study, we presented 22 Quicktime movies, and related figures, that illustrate the progressive development and final 3D geometry of simulated spirals. We also documented the effect on spiral geometry of varying simulation conditions. Both the rotation and non-rotation simulations, in their simplest forms, produce similar 3D spiral geometries, although the mode of spiral formation is different in the two models. Movies of serial slices through the simulated spirals can be used as an aid to interpret the wide range of inclusion trail geometries commonly observed in thin sections in terms of the orientation of the thin section with respect to the spiral axis. While growth rate of the sphere doesn’t influence spiral geometry, factors such as degree of coupling between sphere and matrix, and simple shear/pure shear ratio have a strong effect on the resulting pattern of inclusion trails.

Acknowledgements
A. Stallard acknowledges the aid of a JSPS fellowship at Shizuoka University, and a ‘Grant-in-Aid for Scientific Research’ from the Japanese Society for the Promotion of Science. The authors acknowledge assistance with MATLAB programming by Bert Jagers of Delft Hydraulics, The Netherlands, and Michael Robbins of the Canadian Imperial Bank of Commerce.

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