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Development of snowball structure: numerical simulation of inclusion trails during synkinematic porphyroblast growth in metamorphic rocks
Tectonophysics, Elsevier
170 (1989) 141-150
Science Publishers
141
B.V., Amsterdam
- Printed
in The Netherlands
Development of snowball structure: numerical simulation of inclusion trails during synkinematic porphyroblast growth in metamorphic rocks TOSHIAKI MASUDA and SACHIYO MOCHIZUKI Institute of Geosciences, Shizuoka University, Shizuoka 422 (Japan) (Received
May 30, 1988; revised version
accepted
March
3,1989)
Abstract Masuda,
T. and
during
Mochizuki,
synkinematic
This paper
S., 1989. Development
porphyroblast
presents
a hydrodynamic
lasts. The porphyroblast
is represented
represented
by a Newtonian
track the motion move along velocity.
of matrix
trails of the marker Single rotation
particles
marker
that simulates
are arranged the sphere,
in the sphere
of curvature
simulate
rotation
types
line and the shear plane was initially
between
calculated
results are found
the porphyroblast
by two reversals
to be very similar
are produced
of snowball
in inclusion
structures
volumetric
line before
and incorporated
the inclusion
center
which are characterized
in a straight
of curvature
from the porphyroblast
types appear
the development
simulation
within
to its margin.
rotate
the growing
pattern
particles
sphere.
angular
The resultant Three
and triple rotation
the marker
main types.
line and the shear
with the marker
This type is formed
is
used to
Those in the matrix at a constant
porphyroblasts.
single, double
when the initial angle between
show a complicated
in porphyrob-
rate, while the matrix
deformation.
trails of synkinematic
in the sphere:
trails
170: 141-150.
while those in the sphere passively
captured
by the appearance
O” and 45O. Double
numerical
Tectonophysics,
that grows at a constant
types show very simple spiral trails formed
is between
reversal
around
structure:
rocks.
by simple shear well away from the sphere. The marker
are progressively
types of trails are distinguished plane
analysis
and porphyroblast
in the matrix
of snowball
in metamorphic
by a rigid sphere
fluid that deforms
flow lines that deflect
Those
growth
line showing
when the angle between
a the
45 o and 135 O. When this angle exceeds 135 O, then triple rotation of curvature
to some natural
in the similar way to what happens
Introduction
of the marker
examples,
line from center
implying
in the computer
that such natural
to the margin. spiral patterns
The in
simulation.
Spiral inclusion trails sometimes occur in spherical porphyroblasts of metamorphic minerals such as garnet and albite. These trails have long been considered as evidence of paracrystalline ro-
sense indicator (e.g., Ramsay and Huber, 1987). So far there are many descriptions on snowball porphyroblasts from several erogenic belts around the world (e.g., Spry, 1963; Rosenfeld, 1968; Cox, 1969; Schoneveld, 1977; Powell and Vernon, 1979). Among them, “doubly rotated” garnets from
tation since the early work of Schmidt (1918) and Miigge (1930). They, known as rotational structure if the apparent rotation of the crystal is less than 90” and snowball structure if it is more (Spry, 1969; p. 253), have been taken as one of the useful tools for structural analysis, and their pattern has been regarded as an unequivocal shear-
rotated” garnets from Hunza Valley, northern Pakistan (Powell and Vernon, 1979) seem to be the convincing evidences for the rotation of porphyroblasts as they grew (e.g., Nicolas, 1984). Recently, however, Bell and Johnson (1989) proposed a radical re-interpretation for spiral-
0040-1951/90/$03.50
0 1990 Blsevier Science Publishers
B.V.
southern Vermont (Rosenfeld, 1968) and “540°-
T. MASUDA
142
shaped trails by examining “da&c” snowball garnets from Vermont, central Nepal and Hunza Valley. The main points of their interpretation are that (1) such porphyroblasts did not form by rotation of the growing po~h~oblasts relative to geographic coordinates, (2) they formed instead by progressive growth by ~~hyroblasts over several sets of Ned-orthogon~ foliations that successively overprint one another, and (3) the ductile deformation environment in which a porphyroblast can rotate relative to geographic coordinates during orogenesis is spatially restricted in ~n~~~~tal crust to vertical, ductile tear/tran~u~ent faults across which there is no component of bulk shorte~g or transpression. Bell and Johnson (1989) emphasized common occurrence of “truncation” of inclusion trails in the porphyroblasts, and divided each porphyroblast into several domains which are separated by the “ truncation zones”. On the other hand, according to the conventional inte~~tation (e.g., Ramsay and Huber, 1987), the inclusion trails are continuously traced throughout porphyroblasts with no large “truncation”. The trace of trails is usually made up by appropriately connecting each inclusion material into a line. Inclusion materials are sometimes discretely and scarcely dist~but~ in the po~hyroblasts. Thus it will be sometimes difficult to trace the trails into a line. The difference between the conventional and the “radical” interpretations may depend on how one traces the discrete inclusion trails, continuously or discontinuously. This paper, assuming the ~nventional interpretation, presents a new model that simulates how inclusion t&s distribute in growing porpbyroblasts. The aim of this paper is, however, not to insist that the conventional interpretation should be better than the ““radical” one, but to give a physical basis to help the decision on which interpretation is better for each “spiral-shaped” inclusion trail which we encounter in the field. Previous models based on the ~nven~onal inte~retation qualitatively explained the development of the spiral trails (Spry, 1963; Cox, 1969; Rosenfeld, 1970; Dixon, 1976; Schoneveld, 1977). However, their models are presumably unsatisfactory, because their models depend onlyon the geometrical
AND
S. MOCHIZUKI
treatment of the spiral trails in po~hyroblasts and paid less or no attention to the constraints of the plastic deformation around the porphyroblasts. As the deformation around porphyroblasts turned out to be complex through experimental analysis (e.g. Ghosh, 1975; Fasschier and Simpson, 1986) and h~~ody~a~~al analysis (Masuda and Ando, 1988), a new model should be needed which correctly incorporates the deformation around the porphyroblasts. This paper, intending to meet the need, presents a numerical analysis based on the hydrodynamical equations, dealing with simple shear as for the far-field matrix defo~ation by which the growing sphere is forced to rotate. HydrodyxwmicaI model
A model presented here assumes that a matrix is composed of a Newtonian viscous material and a po~hyroblast is represented by a rigid sphere, Although actual spiral inclusion patterns in po~h~oblasts are three Dimensions, a two dimensional model is considered mainly due to the simplicity of the analysis. Cartesian co-ordinates (x, z> for the simulation were taken. The velocity vector field around the sphere is given by Masuda and Ando (1988) which is shown in Fig. 1: deformation of the matrix is dextral simple shear far from the sphere, while a deflecting movement of the matrix around a sphere is prominent near the sphere. The sphere is forced to rotate at a rate of ti, which is expressed as & =: q/2 (Jeffery, 1922)”
z -.-.---C-L-~-“.-...---__
-cc-/-c----__-_---
I+-.------.._-----
-x.-“--e,_---~.~----
.
c5
Fig. 1. Velocity vector field after Masuda and Ando (1988).
DEVELOPMENT
OF SNOWBALL
STRUCTURJZ
where + is the far field simple shear strain rate. The radius of the sphere increases at a constant rate of volume, which simulates growth of the porphyroblast. Marker particles set in the matrix passively move as the matrix deforms, while those included in the sphere rotate as the sphere rotates. If a particle outside the sphere of radius a is placed at (xi, zr) at which the velocity vector is given as (u,, wr), then the particle moves after a very short time (d t ) to (x2, t2) which is expressed as:
and the radius of the sphere becomes a + dn, where da is an increase in the radius. After the next dt the position of the particle and the radius of the sphere are also calculated in the same way. Such repeated calculation enables the tracing of a smooth path of the particle if dt is sufficiently short. Some particles will be captured by the growing sphere during defo~ation. The captured particles will rotate as the sphere rotates. Marker particles are arranged into straight lines (marker lines) before deformation. The initial angle between the shear plane and the marker line is defined as 8,. How the marker lines deflect around the growing sphere and how the marker particles are captured by the sphere during deformation are given by the calculation using a personal computer. For the growth rate of the sphere (da), a constant increase in volume is assumed. Thus the increase rate of the radius of the sphere gradually decreases with the growth of the sphere. In the actual calculation the initial radius of the sphere is set at 0.1, the increase in volume in a unit time (dt) at 0.001, and the far field strain rate to be 0.002 per unit time. 8i is set from 0” to 170° with a 10” step or a 5 o step when needed. The initial marker particles are set with a constant distance of 0.05 between neighbouring particles, or 0.025 and 0.0125 when needed for the clear presentation of the results. RWlltS Casel:t9,=0
The results of the calculation are shown in Fig. 2, in which the rotational pattern and snowball
Fig. 2. Spiral and snowball patterns when Bi = 0. Simple shear strain (y) far from the sphere is shown. Initial radius of the sphere is 0.1. Marker lines composed of marker particles of initial distance of 0.05 are arranged parallel to the shear plane with a distance of 0.2. Eight large marker dots are shown in figures of y = 0 to 4. Two of them are out of bounds in those of y=6and&
structure are clearly demonstrated by the marker lines inside the sphere (Si). Such marker lines can be smoothly traced beyond the sphere-matrix boundary to the external foliation (S,) throughout whole growth stage of the sphere. In the initial stage (y c 0.2), the inclusion trails of the marker lines appear linear due to rapid growth of the radius of the sphere. At about y = 0.6, the marker lines begin to show observable deflection in the outer part of the growing sphere due to gradual decrease in growth rate of the radius during a constant rotation of the sphere. After that stage the deflection becomes more pro~nent with growth of the sphere, and a sigmoidal, pattern clearly appears at y = 1. The distance of the marker particles seems to be uniform when y K 1. However, it becomes quite heterogeneous when y > 2, neighbouring particles in the outer part of
144
Fig. 3. Variation
T. MASUDA
of spiral patterns
at y = 8 as a function
of ffi. These patterns
double (45 o < Bi 4 135 * ) and triple rotation
the sphere becoming more distant with each other. When y > 6, the area where the particles can not be properly traced appears in the outer part of the sphere. Outside the sphere the distribution of marker particles and lines is uniform only when y < 0.2 (Fig. 2). Up to y = 1, it is disturbed only close to the sphere. As y increases, the disturbed area becomes wider both horizontally (parallel to the shear plane) and vertically (perpendicular to it). The most characteristic feature is the asymmetric development of the “pressure shadow” areas, where no marker particles exist. The areas are clearly visible when y exceeds 4 (Fig. 2). The distances of neighbouting particles are highly heterogeneous around the areas; they are much denser above the upper right area and below the lower left area.
are divided
into three types:
AND S. MOCHIZUKI
single (0 o < Bi 5 45 “)
types (135 o < 8; < 180 o f.
and triple rotation types, which appear when 0” 45”<8ij135”, 135”<6i<1800, <8,245”, respectively. The critical angles (45 * and 135” )
I
A--30”
/o/
/cY r=O
7=0.2
.’ . ..’
.. .
. Q
,...
..’
:
_..
Case2:Bi#0 When Bi is not equal to zero, the pattern obtained (Fig. 3) becomes strikingly complex with changing &. The marker line which passes the center of the sphere is only drawn in Fig. 3. Three types of patterns are distinguished on the basis of the appearance of the trails (Fig. 3); single, double
Fig. 4. Development
of single-rotation patterns
7=4
type spiral and snowball
when 8, = 30.
DEVELOPMENT
OF SNOWBALL
145
STRUCXURE
which divide the pattern types are related to the relative velocities between particles on the margin of the sphere and those outside the sphere as explained later. Single rotation type (0 o < 8,s
45 o )
This type is characterized
by a simple sigmoidal trail or snowball structure of inclusions (Fig. 4). The shape of the trail is just as inferred in the dextral shear sense because marker particles on the margin of the sphere rotate faster than those outside the sphere. Here these trails are called “normal”. Marker particles, which are set homogeneously before deformation, become less dense in the outer part of the sphere. The pattern of the trail hardly varies with varying 19~(see Fig. 3, for 6,=20”, 30” and 400). The pattern at Bi=O shown in Fig. 2 is essentially assigned to an end member of this type. Double rofation type (45 o < 6, s 135 * ) This type is characterized by the reverse sense curvature in the inner and the normal sense curva-
\ \
y=O.
6
;. ‘s .. *cl r=l
..-’
.....
..-
*..*
..f’ ,..-
8
7=4
( i
Fig. 6. Development
of triple-rotation
7=8
type spiral and snowball
patterns (L$ subtype) when Bi = 150 O.
Fig. 5. Development of double-rotation type spiral and snowball patterns when ei =llO”.
ture in the outer part of the sphere (Fig. 5). When traversing from the center to the margin along the marker line, marker particles first become denser at the reverse-sense curve and then rapidly become infrequent when entering into the normalsense curve. The reverse-sense curve develops at the margin of the growing sphere when the orientations of the marker particles at matrix-sphere boundary from the x-axis lie between 45“ and 135O, where the marker particles just outside the sphere rotate faster than those at the margin of the sphere (Masuda and Ando, 1988; and see, e.g., Fig. 5 at y = 1 and 2). As the marker particles subsequently move and the orientations become less than 45 o (e.g., Fig. 5 at y = 3 and 4), a normal-sense curve appears in the overgrowing sphere (e.g. Fig. 5 at y 2 4). The change of the pattern with changing 8i is well depicted by measuring the position of the reversal point where the sense of curvature changes from reverse to normal (see Fig. 9). The reversal points at ‘8, = 50 o and 60” can not be measured because their change of curvature is not prominent (Fig. 3).
146
T. MASUDA
Triple rotation type (135 o < tii < 180 o ) This type is characterized by the composite
AND
S. MOCHIZUKI
ei=170*
inclusion pattern, although the pattern varies with changing
ei (Fig.
3). The
sense of curvature
changes from normal through reverse and then
r=O
y-1
back to normal when traversing the marker line from the center to the margin of the sphere. Thus patterns of this type have two reversal points; one represents a point from normal to reverse sense (inner reversal point) and the other is from reverse to normal sense (outer reversal point). The density of
marker
reverse-sense
particles
becomes
denser
on
the
curve, then rapidly becomes infre-
quent on the outer normal-sense curve. The inner normal pattern is produced when the orientations from the x-axis of the marker particles just outside the sphere stay in the range between 145” and 180” (e.g., see Fig. 7 at y = l), and the reverse sense curve forms when they stay in its range of 45O < ei 6 135O (e.g., see Fig. 7 at y = 4). At the later stage of the deformation the marker particles move into the area where the orientations range
QTZ7
Qz9
..
z
-...., ...i .:‘
.i /.*
/ ..... C3
Fig. 8. Development
/
..:.:
r=ll of triple-rotation
i 0
\
\ : ,.y-.. \. ‘I,. ....-’ j i-=14
type spiral and snowball
patterns (T3 subtype) when Bi = 170 “. Note that folding of S, (0 < y -C7) and unfolding of S, ( y = 9) are clearly visible. &=160”
between -45” and 45” (e.g., see Fig. 7 at y = 6), and there the outer normal pattern ultimately develops in the overgrowing sphere. The outer reversal point is formed in the identical way as the reversal point of the double type described above. The triple-rotation pattern is further subdivided into three subtypes (T,, T2 and T3) based on the
Fig. 7. Development
of triple-rotation
type spiral and snowball
patterns ( T2 subtype) when Bi = 160 O.
appearance of the trails and the relation of the two reversal points (Fig. 9). In the TI subtype the orientation angle of the outer reversal point is negative while that of the inner one is positive (Fig. 6). In the T2 subtype both angles are positive, and the inner one is larger than the outer one (Fig. 7). In the T3 subtype both angles are also positive, but the inner one is smaller than the outer one (Fig. 8). The shear strains at which the reversal points appear at the margin of the sphere are measured on the resultant patterns as shown in Fig. 9. As t$ increases, both the inner and outer reversal points appear at the larger shear strains.
DEVELOPMENT
OF SNOWBALL
147
STRUCTURE
Single l.Odr
I
Triple
Double I
#8 : A"
Oute~g.O; -0.fl i
o.8
0.6.
0” -0-o
___o_.‘O---
0 ,d’Inner
o.._o_..o---o _..__...
0.4.
,oo$
,A---,
,/-G+,
k$(ft-
$8~:
.”
@
20.
7
-1
1 TI
JT2: d
’
y/y; %;or
{loo0
20
4$
Inner
0
14 -2
: a'/ 0 ---._o._ -+--o-..o_ --o-...*...O_-~-".~~ter 4 -20
-20. I
I
30
0
1
I
60
90
I
I
120
150
I
I
180'
ei Fig. 9. Measured fZ, and d, as functions of Bi. 0, is defined as an angle between the line which passes the reversal point and the center and the initial marker line which is preserved in the center of the sphere, while d, is defined as a distance between the reversal point and the center of the sphere. The inlet figures schematically show 0, of the double rotation type (left) and the position of the outer and inner reversal points of the triple rotation type (right). Note that both the 0, and d, of the double rotation type are smoothly traced to those of the outer reversal points of the triple rotation type. Reversal points at Bi = SO0 and 60°,
and the inner
reversal point at 140 o cannot be measured because the change of curvature for these are not prominent. Far field simple shear strain (y) in the upper diagram shows at what strain the reversal points appear in the sphere. 0, of the inner and outer reversal points at Bi of 165 o are indicated by Z and 0, respectively. For TI, T, and T3 subtypes, see text.
Comment on the resultant curvature of trails The most important
factor
that controls
the
sense of curvature of trails is relative velocity of the particles just outside the sphere and those on the margin of the sphere (Fig. 3). Thus when the sphere rotates several times as a result of large shear strains, the marker lines should alternately display reverse and normal sense curves. However, the resultant patterns do not show such complicated curves at large shear strains but actually show at most only two reversal points (e.g., Fig. 8 at y = 14). This discrepancy is mainly due to the paucity of marker particles in the outer part of the sphere and the slow increase in radius of the sphere which apparently gives rise to a lack of observable reversal points on the trail at larger strains. Thus the above three-type division is sufficient. Discussion The simulated continuous Si and S, patterns in various growth stages (Figs. 4-8) differ from the results of earlier workers in their unsmooth con-
nection of Si and S, (e.g., Spry, 1963; Cox, 1969; Dixon, 1976). These are unsatisfactory because the complex deformation around the porphyroblast was not incorporated in their models. The simulated patterns are, however, roughly similar to the results of Schoneveld’s (1977) string model in the case of f3i= 0. His model explains well the continuous relationship of Si and S, as well as the deflection of S, in and around the “pressure shadow” areas. The most conspicuous difference of the new hydrodynamical model from his string model lies in the constitution of the marker lines. In the new model marker lines are defined by the arrangement of marker particles which can move separately with the velocity vector field, consequently the distance between neighbouring marker particles can vary during deformation. On the other hand in the string model the marker lines made of strings seem to be too strong to change in length and width even when they are disturbed, and thus heterogeneous elongation of the marker lines cannot be depicted by the model. In the case of 13~ # 0, Dixon (1976) performed
148
7‘. MASUDA
kinematic curved tion. Dixon,
calculations inclusion
The
and
trails
inclusion
produced
during
patterns
deforma-
3 and
1976) are very useful and are similar
current
results,
porphyroblasts Figures
although
growth
was not specified
10 and
garnet porphyroblasts
S. MOCHIZUKI
composite
a single (figs.
AND
rate
1
4 of
@ S18e
to the of
1
the
in his model.
11 show natural
examples
of
which were selected to dem-
Fig. 11. Natural examples of complex spiral patterns selected after Rosenfeld (1968,
fig. 14-S). The sense of rotation
is
deduced from the fold-drag pattern shown in fig. 14-5 of Rosenfeld (1968). Judging from the pattern and sense of rotation, S18e and C31a is considered to belong to the double rotation type pattern, while S32f, S34d and R38a are considered to belong to the triple rotation type pattern, although these have only one reversal point. Note that C31a is similar to a simulated pattern of Bi = 130 o (see Fig. 3) and R38a is very
a
similar to Fig. 7 at y = 4 in which 19,=160°.
It is rather
difficult to find out the simulated pattern which resembles S18e, S32f and S34d.
onstrate tween rences. shown blance b
the strong morphological resemblance bethe simulated results and actual occurThe typical examples of the single type are in Fig. 10. Such a morphological resemmay suggest that rotational inclusion trails
in natural porphyroblasts manner to what happens
develop in a very similar in the computer simula-
tion. Rosenfeld (1968) presented various complexly curved inclusion trails in Appalachian garnets, some of which are shown in Fig. 11 with a sense of rotation deduced from the drag pattern of folds shown in fig. 14-5 of Rosenfeld (1968). For such
Fig. 10. Natural examples of single rotation type snowball structure. a. Sketch after plate 4 (a) of Cox (1969). This pattern resembles simulated patterns of y = 1 or y = 2. b. Reproduced after fig. 3A of Wilson (1971). This resembles simulated pattern of y = 4 of Fig. 2. c. Sketch after fig. 3 of Schoneveld (1977). This resembles simulated pattern of y = 8 of Fig. 2. Exact research shows that simulated patterns at y = 1.4, 3.8 and 7.4 are the most similar pattern to a, b and c, respectively.
trails at least three different interpretations have been proposed. Rosenfeld (1968) simply ascribed the reverse of the trails to the inversion of shear direction during porphyroblast growth. Dixon (1976) insists on the possibility that these are “doubly rotated” garnets which were produced during a single deformation in which the trails are arranged oblique to the shear plane before deformation. Bell (1985) and Bell and Johnson (1989) proposed the idea of the foliation reactivation around the porphyroblast by which such patterns may be produced. Here it is pointed out that
DEVELOPMENT
OF SNOWBALL
149
STRUCTURE
Dixon’s (1976) proposal is incomplete, and can be
each po~hyroblast
supplemented by adding another interpretation
structural and petrological data.
as
when considered
with other
follows. Consider the compositely
curved trails of Fig.
11. These trails are “doubly rotated”, and Dixon (1976) proposed the possibility that these belong to the III type; his III type is qualitatively identical to the double rotation type. However, it is also possible to interpret the trail as being a member of a triple rotation point.
This
type without the outer reversal
interpretation
insists
Acknowledgements We thank Dr. K. Kano for his critical discussion, Dr. K. Kosaka and Dr. T.H. Bell for critical reading of a part of the manuscript,
and Mr. T.
Shibutani for help in the preparation of figures.
that the con-
tained reversal point in the porphyroblast is the inner one, and the porphyroblast ceased to grow
References
before it would contain the outer reversal point. If the shear sense of defo~ation can be deduced
Bell, T.H., 1985. Deformation partitioning and ~~h~oblast
from other evidence as examples of Fig. 11, it can be established to which type (double or incomplete triple) the doubly rotated pattern belongs, because the central spiral of double rotation type
rotation in methodic
rocks: a radical ~~te~retation.
J.
Me&morph. Geol. J., 3: 109-118. Bell, T.H. and Johnson, S.E., 1989. Porphyroblast trails: the key to orogenesis. J. Metamorph.
inclusion Geol.,
7:
299-310. Cox, F.C., 1969. Inclusions in garnet: discussion and suggested
shows reverse sense whereas that of triple rotation type shows normal sense. For example, in Fig. 11,
mechanism of growth for syntectonic garnets. Geol. Mag.,
S32f, S34d and R38a can possibly be interpreted as belonging to the triple rotation type (T, or T,), while C31a and Sl8e possibly belong to the double rotation type. If Rosenfeld (1968), in his garnets, had given not only appropriately traced spiral inclusion trails but also the distribution of inclusions, the above discussion might be more reliable by comparing the density ~s~butions of inclusions in the natural with the simulated porphyroblasts.
Dixon, J.M., 1976. Apparent “double rotation” of porphyrob-
In this paper we present hydrodynamically possible rotational and snowball structures when the porphyroblasts rotate in a deforming matrix with the assumption of simple shear far from the po~hyrobl~ts.
It is also assumed that the volu-
metric growth of a porphyroblast is constant. Such assumptions may be ideal, as for example growth rate and/or rotation rate of porphyroblasts presumably vary under the changing temperature during progressing metamorphism. Thus it may be difficult to quantitatively apply the simulated results to natural samples. However, it is suggested that these results may be very useful when considering the natural spiral-shaped inclusion trails because they may provide a guide of the possible inclusion trails at variable conditions. They may also be helpful to prove or disprove the rotation of
106: 57-62. lasts during a single progressive deformation.
Tectono-
physics, 34: 101-115. Ghosh, S.K., 1975. Distortion of planar structures around rigid spherical bodies. Tectonophysics, 28: 185-208. Jeffery, G.B.,
1922. The motion of ellipsoidal particles im-
mersed in a viscous fluid. Proc. R. Sot. London, Ser. A., 102: 161-179. Masuda, T. and Ando, S., 1988. Viscous flow around a rigid spherical body: a hydr~~~c~
approach. Tectonophys-
its, 148: 337-346. M&age, O., 1930. Bewegungen von Porphyroblasten in Phylliten und ihre Messung. Neues Jahrb. Mineral. Geol. Palaontol., Beilage Bd., 61A: 469-510. Nicolas, A., 1984. Principes de Tectonique. Masson, Paris, 196 PP. Pass&x,
C.W. and Simpson, C., 1986. Porphyroclast systems
as kinematic indicators. J. Struct. Geol., 8: 831-843. Powell, C.McA. and Vernon, R.H., 1979. Growth and rotation history of garnet porphyroblasts with inclusion spirals in a Karakoram schist. Tectonophysics, 54: 25-43. Ramsay, J.G. and Huber, MI., 1987. The Techniques of Modem Structural Geology. 2. Folds and Fractures. Academic Press, London, 700 pp. Rosenfeld, J.L.,
1968.
Garnet
Paleozoic defo~ations
rotations
due to the major
in southeast Vermont.
In: E-an
Zen, W.S. White, J.B. Hadley and J.B. Thompson Jr. (Editors),
Studies of Appalachian
Maritime. Wiley-Interscience,
Geology:
Northern
New York, N.Y.,
and
pp. 18S-
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150 Schmidt, W., 1918. ~we~n~pu~ in Po~h~oblasten Kristrdliner Schiefer. Sitzungsber. Akad. Wiss. Wien, Math.Naturwiss. KJ., Abt. 1, 127: 239-310. Schoneveld, C., 1977. A study of typical inclusion patterns in strongly pamcrystalline-rotated garnets. Tectonophysics, 39: 453-471.
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AND S. MOCHIZUKI
Spry, A., 1963. The origin and significance of snowball stmcture in garnet. J. Petrol., 4: 211-222. Spry, A., 1969. Metamorphic Textures. Pergarnon, Oxford, 352 PP. Wilson, M.R., 1971. On syntectonic Tectonophysics, 11: 239-260.
porphyroblast
growth.
170 (1989) 141-150
Science Publishers
141
B.V., Amsterdam
- Printed
in The Netherlands
Development of snowball structure: numerical simulation of inclusion trails during synkinematic porphyroblast growth in metamorphic rocks TOSHIAKI MASUDA and SACHIYO MOCHIZUKI Institute of Geosciences, Shizuoka University, Shizuoka 422 (Japan) (Received
May 30, 1988; revised version
accepted
March
3,1989)
Abstract Masuda,
T. and
during
Mochizuki,
synkinematic
This paper
S., 1989. Development
porphyroblast
presents
a hydrodynamic
lasts. The porphyroblast
is represented
represented
by a Newtonian
track the motion move along velocity.
of matrix
trails of the marker Single rotation
particles
marker
that simulates
are arranged the sphere,
in the sphere
of curvature
simulate
rotation
types
line and the shear plane was initially
between
calculated
results are found
the porphyroblast
by two reversals
to be very similar
are produced
of snowball
in inclusion
structures
volumetric
line before
and incorporated
the inclusion
center
which are characterized
in a straight
of curvature
from the porphyroblast
types appear
the development
simulation
within
to its margin.
rotate
the growing
pattern
particles
sphere.
angular
The resultant Three
and triple rotation
the marker
main types.
line and the shear
with the marker
This type is formed
is
used to
Those in the matrix at a constant
porphyroblasts.
single, double
when the initial angle between
show a complicated
in porphyrob-
rate, while the matrix
deformation.
trails of synkinematic
in the sphere:
trails
170: 141-150.
while those in the sphere passively
captured
by the appearance
O” and 45O. Double
numerical
Tectonophysics,
that grows at a constant
types show very simple spiral trails formed
is between
reversal
around
structure:
rocks.
by simple shear well away from the sphere. The marker
are progressively
types of trails are distinguished plane
analysis
and porphyroblast
in the matrix
of snowball
in metamorphic
by a rigid sphere
fluid that deforms
flow lines that deflect
Those
growth
line showing
when the angle between
a the
45 o and 135 O. When this angle exceeds 135 O, then triple rotation of curvature
to some natural
in the similar way to what happens
Introduction
of the marker
examples,
line from center
implying
in the computer
that such natural
to the margin. spiral patterns
The in
simulation.
Spiral inclusion trails sometimes occur in spherical porphyroblasts of metamorphic minerals such as garnet and albite. These trails have long been considered as evidence of paracrystalline ro-
sense indicator (e.g., Ramsay and Huber, 1987). So far there are many descriptions on snowball porphyroblasts from several erogenic belts around the world (e.g., Spry, 1963; Rosenfeld, 1968; Cox, 1969; Schoneveld, 1977; Powell and Vernon, 1979). Among them, “doubly rotated” garnets from
tation since the early work of Schmidt (1918) and Miigge (1930). They, known as rotational structure if the apparent rotation of the crystal is less than 90” and snowball structure if it is more (Spry, 1969; p. 253), have been taken as one of the useful tools for structural analysis, and their pattern has been regarded as an unequivocal shear-
rotated” garnets from Hunza Valley, northern Pakistan (Powell and Vernon, 1979) seem to be the convincing evidences for the rotation of porphyroblasts as they grew (e.g., Nicolas, 1984). Recently, however, Bell and Johnson (1989) proposed a radical re-interpretation for spiral-
0040-1951/90/$03.50
0 1990 Blsevier Science Publishers
B.V.
southern Vermont (Rosenfeld, 1968) and “540°-
T. MASUDA
142
shaped trails by examining “da&c” snowball garnets from Vermont, central Nepal and Hunza Valley. The main points of their interpretation are that (1) such porphyroblasts did not form by rotation of the growing po~h~oblasts relative to geographic coordinates, (2) they formed instead by progressive growth by ~~hyroblasts over several sets of Ned-orthogon~ foliations that successively overprint one another, and (3) the ductile deformation environment in which a porphyroblast can rotate relative to geographic coordinates during orogenesis is spatially restricted in ~n~~~~tal crust to vertical, ductile tear/tran~u~ent faults across which there is no component of bulk shorte~g or transpression. Bell and Johnson (1989) emphasized common occurrence of “truncation” of inclusion trails in the porphyroblasts, and divided each porphyroblast into several domains which are separated by the “ truncation zones”. On the other hand, according to the conventional inte~~tation (e.g., Ramsay and Huber, 1987), the inclusion trails are continuously traced throughout porphyroblasts with no large “truncation”. The trace of trails is usually made up by appropriately connecting each inclusion material into a line. Inclusion materials are sometimes discretely and scarcely dist~but~ in the po~hyroblasts. Thus it will be sometimes difficult to trace the trails into a line. The difference between the conventional and the “radical” interpretations may depend on how one traces the discrete inclusion trails, continuously or discontinuously. This paper, assuming the ~nventional interpretation, presents a new model that simulates how inclusion t&s distribute in growing porpbyroblasts. The aim of this paper is, however, not to insist that the conventional interpretation should be better than the ““radical” one, but to give a physical basis to help the decision on which interpretation is better for each “spiral-shaped” inclusion trail which we encounter in the field. Previous models based on the ~nven~onal inte~retation qualitatively explained the development of the spiral trails (Spry, 1963; Cox, 1969; Rosenfeld, 1970; Dixon, 1976; Schoneveld, 1977). However, their models are presumably unsatisfactory, because their models depend onlyon the geometrical
AND
S. MOCHIZUKI
treatment of the spiral trails in po~hyroblasts and paid less or no attention to the constraints of the plastic deformation around the porphyroblasts. As the deformation around porphyroblasts turned out to be complex through experimental analysis (e.g. Ghosh, 1975; Fasschier and Simpson, 1986) and h~~ody~a~~al analysis (Masuda and Ando, 1988), a new model should be needed which correctly incorporates the deformation around the porphyroblasts. This paper, intending to meet the need, presents a numerical analysis based on the hydrodynamical equations, dealing with simple shear as for the far-field matrix defo~ation by which the growing sphere is forced to rotate. HydrodyxwmicaI model
A model presented here assumes that a matrix is composed of a Newtonian viscous material and a po~hyroblast is represented by a rigid sphere, Although actual spiral inclusion patterns in po~h~oblasts are three Dimensions, a two dimensional model is considered mainly due to the simplicity of the analysis. Cartesian co-ordinates (x, z> for the simulation were taken. The velocity vector field around the sphere is given by Masuda and Ando (1988) which is shown in Fig. 1: deformation of the matrix is dextral simple shear far from the sphere, while a deflecting movement of the matrix around a sphere is prominent near the sphere. The sphere is forced to rotate at a rate of ti, which is expressed as & =: q/2 (Jeffery, 1922)”
z -.-.---C-L-~-“.-...---__
-cc-/-c----__-_---
I+-.------.._-----
-x.-“--e,_---~.~----
.
c5
Fig. 1. Velocity vector field after Masuda and Ando (1988).
DEVELOPMENT
OF SNOWBALL
STRUCTURJZ
where + is the far field simple shear strain rate. The radius of the sphere increases at a constant rate of volume, which simulates growth of the porphyroblast. Marker particles set in the matrix passively move as the matrix deforms, while those included in the sphere rotate as the sphere rotates. If a particle outside the sphere of radius a is placed at (xi, zr) at which the velocity vector is given as (u,, wr), then the particle moves after a very short time (d t ) to (x2, t2) which is expressed as:
and the radius of the sphere becomes a + dn, where da is an increase in the radius. After the next dt the position of the particle and the radius of the sphere are also calculated in the same way. Such repeated calculation enables the tracing of a smooth path of the particle if dt is sufficiently short. Some particles will be captured by the growing sphere during defo~ation. The captured particles will rotate as the sphere rotates. Marker particles are arranged into straight lines (marker lines) before deformation. The initial angle between the shear plane and the marker line is defined as 8,. How the marker lines deflect around the growing sphere and how the marker particles are captured by the sphere during deformation are given by the calculation using a personal computer. For the growth rate of the sphere (da), a constant increase in volume is assumed. Thus the increase rate of the radius of the sphere gradually decreases with the growth of the sphere. In the actual calculation the initial radius of the sphere is set at 0.1, the increase in volume in a unit time (dt) at 0.001, and the far field strain rate to be 0.002 per unit time. 8i is set from 0” to 170° with a 10” step or a 5 o step when needed. The initial marker particles are set with a constant distance of 0.05 between neighbouring particles, or 0.025 and 0.0125 when needed for the clear presentation of the results. RWlltS Casel:t9,=0
The results of the calculation are shown in Fig. 2, in which the rotational pattern and snowball
Fig. 2. Spiral and snowball patterns when Bi = 0. Simple shear strain (y) far from the sphere is shown. Initial radius of the sphere is 0.1. Marker lines composed of marker particles of initial distance of 0.05 are arranged parallel to the shear plane with a distance of 0.2. Eight large marker dots are shown in figures of y = 0 to 4. Two of them are out of bounds in those of y=6and&
structure are clearly demonstrated by the marker lines inside the sphere (Si). Such marker lines can be smoothly traced beyond the sphere-matrix boundary to the external foliation (S,) throughout whole growth stage of the sphere. In the initial stage (y c 0.2), the inclusion trails of the marker lines appear linear due to rapid growth of the radius of the sphere. At about y = 0.6, the marker lines begin to show observable deflection in the outer part of the growing sphere due to gradual decrease in growth rate of the radius during a constant rotation of the sphere. After that stage the deflection becomes more pro~nent with growth of the sphere, and a sigmoidal, pattern clearly appears at y = 1. The distance of the marker particles seems to be uniform when y K 1. However, it becomes quite heterogeneous when y > 2, neighbouring particles in the outer part of
144
Fig. 3. Variation
T. MASUDA
of spiral patterns
at y = 8 as a function
of ffi. These patterns
double (45 o < Bi 4 135 * ) and triple rotation
the sphere becoming more distant with each other. When y > 6, the area where the particles can not be properly traced appears in the outer part of the sphere. Outside the sphere the distribution of marker particles and lines is uniform only when y < 0.2 (Fig. 2). Up to y = 1, it is disturbed only close to the sphere. As y increases, the disturbed area becomes wider both horizontally (parallel to the shear plane) and vertically (perpendicular to it). The most characteristic feature is the asymmetric development of the “pressure shadow” areas, where no marker particles exist. The areas are clearly visible when y exceeds 4 (Fig. 2). The distances of neighbouting particles are highly heterogeneous around the areas; they are much denser above the upper right area and below the lower left area.
are divided
into three types:
AND S. MOCHIZUKI
single (0 o < Bi 5 45 “)
types (135 o < 8; < 180 o f.
and triple rotation types, which appear when 0” 45”<8ij135”, 135”<6i<1800, <8,245”, respectively. The critical angles (45 * and 135” )
I
A--30”
/o/
/cY r=O
7=0.2
.’ . ..’
.. .
. Q
,...
..’
:
_..
Case2:Bi#0 When Bi is not equal to zero, the pattern obtained (Fig. 3) becomes strikingly complex with changing &. The marker line which passes the center of the sphere is only drawn in Fig. 3. Three types of patterns are distinguished on the basis of the appearance of the trails (Fig. 3); single, double
Fig. 4. Development
of single-rotation patterns
7=4
type spiral and snowball
when 8, = 30.
DEVELOPMENT
OF SNOWBALL
145
STRUCXURE
which divide the pattern types are related to the relative velocities between particles on the margin of the sphere and those outside the sphere as explained later. Single rotation type (0 o < 8,s
45 o )
This type is characterized
by a simple sigmoidal trail or snowball structure of inclusions (Fig. 4). The shape of the trail is just as inferred in the dextral shear sense because marker particles on the margin of the sphere rotate faster than those outside the sphere. Here these trails are called “normal”. Marker particles, which are set homogeneously before deformation, become less dense in the outer part of the sphere. The pattern of the trail hardly varies with varying 19~(see Fig. 3, for 6,=20”, 30” and 400). The pattern at Bi=O shown in Fig. 2 is essentially assigned to an end member of this type. Double rofation type (45 o < 6, s 135 * ) This type is characterized by the reverse sense curvature in the inner and the normal sense curva-
\ \
y=O.
6
;. ‘s .. *cl r=l
..-’
.....
..-
*..*
..f’ ,..-
8
7=4
( i
Fig. 6. Development
of triple-rotation
7=8
type spiral and snowball
patterns (L$ subtype) when Bi = 150 O.
Fig. 5. Development of double-rotation type spiral and snowball patterns when ei =llO”.
ture in the outer part of the sphere (Fig. 5). When traversing from the center to the margin along the marker line, marker particles first become denser at the reverse-sense curve and then rapidly become infrequent when entering into the normalsense curve. The reverse-sense curve develops at the margin of the growing sphere when the orientations of the marker particles at matrix-sphere boundary from the x-axis lie between 45“ and 135O, where the marker particles just outside the sphere rotate faster than those at the margin of the sphere (Masuda and Ando, 1988; and see, e.g., Fig. 5 at y = 1 and 2). As the marker particles subsequently move and the orientations become less than 45 o (e.g., Fig. 5 at y = 3 and 4), a normal-sense curve appears in the overgrowing sphere (e.g. Fig. 5 at y 2 4). The change of the pattern with changing 8i is well depicted by measuring the position of the reversal point where the sense of curvature changes from reverse to normal (see Fig. 9). The reversal points at ‘8, = 50 o and 60” can not be measured because their change of curvature is not prominent (Fig. 3).
146
T. MASUDA
Triple rotation type (135 o < tii < 180 o ) This type is characterized by the composite
AND
S. MOCHIZUKI
ei=170*
inclusion pattern, although the pattern varies with changing
ei (Fig.
3). The
sense of curvature
changes from normal through reverse and then
r=O
y-1
back to normal when traversing the marker line from the center to the margin of the sphere. Thus patterns of this type have two reversal points; one represents a point from normal to reverse sense (inner reversal point) and the other is from reverse to normal sense (outer reversal point). The density of
marker
reverse-sense
particles
becomes
denser
on
the
curve, then rapidly becomes infre-
quent on the outer normal-sense curve. The inner normal pattern is produced when the orientations from the x-axis of the marker particles just outside the sphere stay in the range between 145” and 180” (e.g., see Fig. 7 at y = l), and the reverse sense curve forms when they stay in its range of 45O < ei 6 135O (e.g., see Fig. 7 at y = 4). At the later stage of the deformation the marker particles move into the area where the orientations range
QTZ7
Qz9
..
z
-...., ...i .:‘
.i /.*
/ ..... C3
Fig. 8. Development
/
..:.:
r=ll of triple-rotation
i 0
\
\ : ,.y-.. \. ‘I,. ....-’ j i-=14
type spiral and snowball
patterns (T3 subtype) when Bi = 170 “. Note that folding of S, (0 < y -C7) and unfolding of S, ( y = 9) are clearly visible. &=160”
between -45” and 45” (e.g., see Fig. 7 at y = 6), and there the outer normal pattern ultimately develops in the overgrowing sphere. The outer reversal point is formed in the identical way as the reversal point of the double type described above. The triple-rotation pattern is further subdivided into three subtypes (T,, T2 and T3) based on the
Fig. 7. Development
of triple-rotation
type spiral and snowball
patterns ( T2 subtype) when Bi = 160 O.
appearance of the trails and the relation of the two reversal points (Fig. 9). In the TI subtype the orientation angle of the outer reversal point is negative while that of the inner one is positive (Fig. 6). In the T2 subtype both angles are positive, and the inner one is larger than the outer one (Fig. 7). In the T3 subtype both angles are also positive, but the inner one is smaller than the outer one (Fig. 8). The shear strains at which the reversal points appear at the margin of the sphere are measured on the resultant patterns as shown in Fig. 9. As t$ increases, both the inner and outer reversal points appear at the larger shear strains.
DEVELOPMENT
OF SNOWBALL
147
STRUCTURE
Single l.Odr
I
Triple
Double I
#8 : A"
Oute~g.O; -0.fl i
o.8
0.6.
0” -0-o
___o_.‘O---
0 ,d’Inner
o.._o_..o---o _..__...
0.4.
,oo$
,A---,
,/-G+,
k$(ft-
$8~:
.”
@
20.
7
-1
1 TI
JT2: d
’
y/y; %;or
{loo0
20
4$
Inner
0
14 -2
: a'/ 0 ---._o._ -+--o-..o_ --o-...*...O_-~-".~~ter 4 -20
-20. I
I
30
0
1
I
60
90
I
I
120
150
I
I
180'
ei Fig. 9. Measured fZ, and d, as functions of Bi. 0, is defined as an angle between the line which passes the reversal point and the center and the initial marker line which is preserved in the center of the sphere, while d, is defined as a distance between the reversal point and the center of the sphere. The inlet figures schematically show 0, of the double rotation type (left) and the position of the outer and inner reversal points of the triple rotation type (right). Note that both the 0, and d, of the double rotation type are smoothly traced to those of the outer reversal points of the triple rotation type. Reversal points at Bi = SO0 and 60°,
and the inner
reversal point at 140 o cannot be measured because the change of curvature for these are not prominent. Far field simple shear strain (y) in the upper diagram shows at what strain the reversal points appear in the sphere. 0, of the inner and outer reversal points at Bi of 165 o are indicated by Z and 0, respectively. For TI, T, and T3 subtypes, see text.
Comment on the resultant curvature of trails The most important
factor
that controls
the
sense of curvature of trails is relative velocity of the particles just outside the sphere and those on the margin of the sphere (Fig. 3). Thus when the sphere rotates several times as a result of large shear strains, the marker lines should alternately display reverse and normal sense curves. However, the resultant patterns do not show such complicated curves at large shear strains but actually show at most only two reversal points (e.g., Fig. 8 at y = 14). This discrepancy is mainly due to the paucity of marker particles in the outer part of the sphere and the slow increase in radius of the sphere which apparently gives rise to a lack of observable reversal points on the trail at larger strains. Thus the above three-type division is sufficient. Discussion The simulated continuous Si and S, patterns in various growth stages (Figs. 4-8) differ from the results of earlier workers in their unsmooth con-
nection of Si and S, (e.g., Spry, 1963; Cox, 1969; Dixon, 1976). These are unsatisfactory because the complex deformation around the porphyroblast was not incorporated in their models. The simulated patterns are, however, roughly similar to the results of Schoneveld’s (1977) string model in the case of f3i= 0. His model explains well the continuous relationship of Si and S, as well as the deflection of S, in and around the “pressure shadow” areas. The most conspicuous difference of the new hydrodynamical model from his string model lies in the constitution of the marker lines. In the new model marker lines are defined by the arrangement of marker particles which can move separately with the velocity vector field, consequently the distance between neighbouring marker particles can vary during deformation. On the other hand in the string model the marker lines made of strings seem to be too strong to change in length and width even when they are disturbed, and thus heterogeneous elongation of the marker lines cannot be depicted by the model. In the case of 13~ # 0, Dixon (1976) performed
148
7‘. MASUDA
kinematic curved tion. Dixon,
calculations inclusion
The
and
trails
inclusion
produced
during
patterns
deforma-
3 and
1976) are very useful and are similar
current
results,
porphyroblasts Figures
although
growth
was not specified
10 and
garnet porphyroblasts
S. MOCHIZUKI
composite
a single (figs.
AND
rate
1
4 of
@ S18e
to the of
1
the
in his model.
11 show natural
examples
of
which were selected to dem-
Fig. 11. Natural examples of complex spiral patterns selected after Rosenfeld (1968,
fig. 14-S). The sense of rotation
is
deduced from the fold-drag pattern shown in fig. 14-5 of Rosenfeld (1968). Judging from the pattern and sense of rotation, S18e and C31a is considered to belong to the double rotation type pattern, while S32f, S34d and R38a are considered to belong to the triple rotation type pattern, although these have only one reversal point. Note that C31a is similar to a simulated pattern of Bi = 130 o (see Fig. 3) and R38a is very
a
similar to Fig. 7 at y = 4 in which 19,=160°.
It is rather
difficult to find out the simulated pattern which resembles S18e, S32f and S34d.
onstrate tween rences. shown blance b
the strong morphological resemblance bethe simulated results and actual occurThe typical examples of the single type are in Fig. 10. Such a morphological resemmay suggest that rotational inclusion trails
in natural porphyroblasts manner to what happens
develop in a very similar in the computer simula-
tion. Rosenfeld (1968) presented various complexly curved inclusion trails in Appalachian garnets, some of which are shown in Fig. 11 with a sense of rotation deduced from the drag pattern of folds shown in fig. 14-5 of Rosenfeld (1968). For such
Fig. 10. Natural examples of single rotation type snowball structure. a. Sketch after plate 4 (a) of Cox (1969). This pattern resembles simulated patterns of y = 1 or y = 2. b. Reproduced after fig. 3A of Wilson (1971). This resembles simulated pattern of y = 4 of Fig. 2. c. Sketch after fig. 3 of Schoneveld (1977). This resembles simulated pattern of y = 8 of Fig. 2. Exact research shows that simulated patterns at y = 1.4, 3.8 and 7.4 are the most similar pattern to a, b and c, respectively.
trails at least three different interpretations have been proposed. Rosenfeld (1968) simply ascribed the reverse of the trails to the inversion of shear direction during porphyroblast growth. Dixon (1976) insists on the possibility that these are “doubly rotated” garnets which were produced during a single deformation in which the trails are arranged oblique to the shear plane before deformation. Bell (1985) and Bell and Johnson (1989) proposed the idea of the foliation reactivation around the porphyroblast by which such patterns may be produced. Here it is pointed out that
DEVELOPMENT
OF SNOWBALL
149
STRUCTURE
Dixon’s (1976) proposal is incomplete, and can be
each po~hyroblast
supplemented by adding another interpretation
structural and petrological data.
as
when considered
with other
follows. Consider the compositely
curved trails of Fig.
11. These trails are “doubly rotated”, and Dixon (1976) proposed the possibility that these belong to the III type; his III type is qualitatively identical to the double rotation type. However, it is also possible to interpret the trail as being a member of a triple rotation point.
This
type without the outer reversal
interpretation
insists
Acknowledgements We thank Dr. K. Kano for his critical discussion, Dr. K. Kosaka and Dr. T.H. Bell for critical reading of a part of the manuscript,
and Mr. T.
Shibutani for help in the preparation of figures.
that the con-
tained reversal point in the porphyroblast is the inner one, and the porphyroblast ceased to grow
References
before it would contain the outer reversal point. If the shear sense of defo~ation can be deduced
Bell, T.H., 1985. Deformation partitioning and ~~h~oblast
from other evidence as examples of Fig. 11, it can be established to which type (double or incomplete triple) the doubly rotated pattern belongs, because the central spiral of double rotation type
rotation in methodic
rocks: a radical ~~te~retation.
J.
Me&morph. Geol. J., 3: 109-118. Bell, T.H. and Johnson, S.E., 1989. Porphyroblast trails: the key to orogenesis. J. Metamorph.
inclusion Geol.,
7:
299-310. Cox, F.C., 1969. Inclusions in garnet: discussion and suggested
shows reverse sense whereas that of triple rotation type shows normal sense. For example, in Fig. 11,
mechanism of growth for syntectonic garnets. Geol. Mag.,
S32f, S34d and R38a can possibly be interpreted as belonging to the triple rotation type (T, or T,), while C31a and Sl8e possibly belong to the double rotation type. If Rosenfeld (1968), in his garnets, had given not only appropriately traced spiral inclusion trails but also the distribution of inclusions, the above discussion might be more reliable by comparing the density ~s~butions of inclusions in the natural with the simulated porphyroblasts.
Dixon, J.M., 1976. Apparent “double rotation” of porphyrob-
In this paper we present hydrodynamically possible rotational and snowball structures when the porphyroblasts rotate in a deforming matrix with the assumption of simple shear far from the po~hyrobl~ts.
It is also assumed that the volu-
metric growth of a porphyroblast is constant. Such assumptions may be ideal, as for example growth rate and/or rotation rate of porphyroblasts presumably vary under the changing temperature during progressing metamorphism. Thus it may be difficult to quantitatively apply the simulated results to natural samples. However, it is suggested that these results may be very useful when considering the natural spiral-shaped inclusion trails because they may provide a guide of the possible inclusion trails at variable conditions. They may also be helpful to prove or disprove the rotation of
106: 57-62. lasts during a single progressive deformation.
Tectono-
physics, 34: 101-115. Ghosh, S.K., 1975. Distortion of planar structures around rigid spherical bodies. Tectonophysics, 28: 185-208. Jeffery, G.B.,
1922. The motion of ellipsoidal particles im-
mersed in a viscous fluid. Proc. R. Sot. London, Ser. A., 102: 161-179. Masuda, T. and Ando, S., 1988. Viscous flow around a rigid spherical body: a hydr~~~c~
approach. Tectonophys-
its, 148: 337-346. M&age, O., 1930. Bewegungen von Porphyroblasten in Phylliten und ihre Messung. Neues Jahrb. Mineral. Geol. Palaontol., Beilage Bd., 61A: 469-510. Nicolas, A., 1984. Principes de Tectonique. Masson, Paris, 196 PP. Pass&x,
C.W. and Simpson, C., 1986. Porphyroclast systems
as kinematic indicators. J. Struct. Geol., 8: 831-843. Powell, C.McA. and Vernon, R.H., 1979. Growth and rotation history of garnet porphyroblasts with inclusion spirals in a Karakoram schist. Tectonophysics, 54: 25-43. Ramsay, J.G. and Huber, MI., 1987. The Techniques of Modem Structural Geology. 2. Folds and Fractures. Academic Press, London, 700 pp. Rosenfeld, J.L.,
1968.
Garnet
Paleozoic defo~ations
rotations
due to the major
in southeast Vermont.
In: E-an
Zen, W.S. White, J.B. Hadley and J.B. Thompson Jr. (Editors),
Studies of Appalachian
Maritime. Wiley-Interscience,
Geology:
Northern
New York, N.Y.,
and
pp. 18S-
202. Rosenfeld, J.L., 1970. Rotated garnets in metamorphic rocks. Geol. Sot. Am., Spec. Pap., 129: 105 pp.
150 Schmidt, W., 1918. ~we~n~pu~ in Po~h~oblasten Kristrdliner Schiefer. Sitzungsber. Akad. Wiss. Wien, Math.Naturwiss. KJ., Abt. 1, 127: 239-310. Schoneveld, C., 1977. A study of typical inclusion patterns in strongly pamcrystalline-rotated garnets. Tectonophysics, 39: 453-471.
T. MASUDA
AND S. MOCHIZUKI
Spry, A., 1963. The origin and significance of snowball stmcture in garnet. J. Petrol., 4: 211-222. Spry, A., 1969. Metamorphic Textures. Pergarnon, Oxford, 352 PP. Wilson, M.R., 1971. On syntectonic Tectonophysics, 11: 239-260.
porphyroblast
growth.