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Results
Results
of the simulations are presented in Figs. 3-6. Figure 3 shows the 3D development
of the central inclusion surface during spiral formation, while Fig. 4
shows 2D movies of spiral development in the XZ, XY and YZ planes. The
XZ plane is perpendicular to both the shear plane and axis of relative
rotation between sphere and matrix, while the XY plane is parallel to
the shear plane, and the YZ plane is perpendicular to the shear plane
and parallel to the axis of relative rotation between sphere and matrix.
Figure 5 compares the final rotation and non-rotation simulation geometries,
and Fig. 6 shows 2D movies of serial slices through the spirals parallel
to the XZ, XY and YZ planes. Results of both the rotation and non-rotation
simulations are presented in these figures.
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| Figure 3.
3D movies showing the progressive development of spiral inclusion
trails according to both the non-rotation (a) and rotation (b) models
of spiral formation. For maximum clarity, the movies show the development
of the central inclusion surface only. A green marker has been added
to the sphere margin to provide a visual trace of sphere rotation
during the simulation. Note, this marker has been positioned on the
sphere margin in all frames of the movie, and does not represent equivalent
marker points within the simulation at different stages of spiral
development. In the non-rotation movie, arrows indicate the sense
of matrix shear. The long axes of the arrows are parallel to the orientation
of the shear plane, and parallel to the orientation of the theoretically
developing matrix foliation. |
3D
spiral development, non-rotation simulation
In the
non-rotation simulation, the matrix wraps around the irrotational growing
sphere, and it is this shape, in combination with the rotation of the
matrix foliation about the sphere, that defines the 3D geometry of the
spiral (Figs. 3a, 4d-f). Further from the sphere, the matrix is unaffected
by the flow perturbation about the sphere and maintains a sub-planar geometry
during the course of the simulation. The maximum perturbation of the matrix
occurs in the plane through the centre of the sphere, perpendicular to
the axis of relative rotation between the sphere and matrix. The perturbation
decreases to zero along the axis of relative rotation as we move away
from the centre of the sphere, resulting in a strongly non-cylindrical
spiral geometry.
From
the centre to the rim of the sphere, the central inclusion plane develops
with intervals of relatively gentle curvature separated by intervals of
relatively tight curvature. The reason for this is related to the repeated
cycles of foliation development in the matrix. At the initiation of each
new foliation, the matrix foliation is oriented at approximately 90°
to the new shear plane, and is thus quickly rotated toward the shear plane.
The rate of rotation slows once the matrix foliation is oriented close
to the shear plane. Thus the spiral development is marked by periods of
relatively rapid rotation of the foliation about the sphere, coinciding
with the development of a new foliation, and periods of relatively slower
rotation as the new foliation matures and is reoriented closer to the
shear plane.
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| Figure 4.
Movies showing the progressive development of spiral inclusion trails
according to both the rotation and non-rotation models of spiral formation.
Spiral development is shown in sections oriented parallel to the XZ,
XY, and YZ planes. Equivalent colours represent equivalent surfaces
in different sections of each simulation. Subplots show the orientation
of the 2D sections with respect to the spiral. In all figures, the
sphere is represented by a dashed black line. |
3D
spiral development, rotation simulation
As with
the non-rotation simulation, the non-cylindrical spiral geometry reflects
the wrapping of the matrix foliation around the growing porphyroblast,
although in the rotation simulation, it is the rotation of the sphere,
rather than the matrix, that is the driving force behind spiral formation.
Once captured within the sphere, the foliation is progressively rotated
away from the equivalent foliation in the matrix, thus forming the spiral
geometry. The central inclusion surface forms a doubly curving non-cylindrical
geometry, whose development can be visualised as a symmetrical pair of
sheath folds. Once formed, the sheath fold is progressively rotated and
stretched about the axis of relative rotation between the porphyroblast
and matrix (Fig. 3b). The central surface develops relatively smoothly
compared to the non-rotation simulation. The development of foliation
surfaces that were originally oriented off-centre with respect to the
sphere can be visualised as the progressive rotation and elongation of
a single sheath fold (Fig. 4a). The part of the spiral further from the
‘sheath fold’ tip has a more open geometry due to the wrapping
of the matrix foliation about the growing sphere.
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