THE STABILITY AND EVOLUTION OF TRIPLE JUNCTIONS
Theodore G. Apotria
B.S., The University of Connecticut, 1982
A Thesis
Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science
at The University of Connecticut 1985 APPROVAL PAGE
Master of Science Thesis
THE STABILITY AND EVOLUTION OP TRIPLE JUNCTIONS
Presented by- Theodore G. Apotria, B.S.
Major Adviser
Associate Adviser |
Peter Dehlrnger
Associate Adviser 7 Peter Geiser
Associate Adviser Thomas Moran
The University of Connecticut
1985
ii DEDICATION
This thesis is dedicated to my parents, George and Cleo
Apotria, and to my "brother and best friend, John. Their support was essential in completing this work.
iii ACKNOWLEDGEMENTS
There are several individuals whose assistance and support contributed to this thesis, and to whom I am indebted:
I am especially thankful to Dr. Norman Gray, who, as my major adviser, contributed many hours of stimulating discussion and attention. As the thesis topic evolved, his assistance in the theoretical development, computer programming, and critical commentary on the manuscript, all proved invaluable.
Dr. Peter Dehlinger, Dr. Peter Geiser, and Dr. Thomas
Moran, contributed their time reviewing the text and provided useful perspectives. Dr. Randolph Steinen contributed useful comments on the preliminary research proposal.
I am also grateful to the Dept, of Geology and Geophysics for providing financial support in the form of a Teaching Assistantship, and an Amoco sponsored Master's Fellowship in
Geophysics. I also thank Dr. Anthony Philpotts for providing a Research Assistantship from K.S.F. grant if EAR8017059*
- iv ABSTRACT
Triple junction stability, as introduced by McKenzie and
Morgan (1969)> assumes constant relative velocities and can only be defined instantaneously. Factors that affect the stability of triple junctions as a function of time include changes in the boundary orientations and relative velocities. Changes in the velocity triangle can be induced by the effects of motion on a sphere, and the motion of the lithospheric plates with respect to the mantle.
Each type of triple junction has been assigned a number of degrees of freedom, where a degree of freedom is an imparted change in boundary configurations or relative velocities without upsetting the stability condition. The number of degrees of freedom correlates positively with the junction's ability to maintain stability under imposed changes.
The exact stability condition of any triple junction can be calculated by least squares analysis. An equation for the location of the triple junction with respect to each boundary is introduced with respect to any reference frame.
- v - The equations are linear and of the form J = a + hh. A unique simultaneous solution results if the junction is stable. If the junction is unstable, the sum of the squares
of the residuals is proposed as a measure of the degree of
geometric instability. This analysis is particularly useful when several boundary types are consistent with a single
velocity triangle. The type least affected by imposed changes can be regarded as the most geometrically stable,
and can be considered in evolutionary schemes.
The absolute motion of the Bouvet triple junction with respect to the mantle (mesosphere) plays a significant role
in its evolution. Absolute poles of rotation from the AM1-2 model (Minster and Jordan, 1978) for the South American,
African, and Antarctic plates were utilized to calculate an "isosceles line" in the South Atlantic. Along this locus (trending 91°)* the relative velocity triangle for the
three plates is isosceles. The actual Bouvet triple
junction lies about 5 degrees north of this line, presumably due to uncertainties in the AM1-2 poles. Assuming the Bouvet
junction is actually isosceles, the absolute triple junction velocity vectors for the motion of the Ridge-Ridge-Ridge
(RRR) and Ridge-Fault-Fault (RFF) configurations are respectively 14*7 km/yr, with an azimuth of 34j5> and 5*8 km/my with an azimuth of 157* A least squares analysis
- vi - suggests the RFF configuration is geometrically more stable in the vicinity of the isosceles line and is the preferred mode.
Consequently, a RRR triple junction located south of the present position of the Bouvet junction vould, relative to the mantle, migrate northwest until it intersects the isosceles line and spontaneously changes to a stable RFF configuration. As a RFF junction, its new velocity would take it southeast away from the isosceles line, rendering it increasingly unstable. As it migrates, the rotations of the relative velocities are such that both growing transform fault develop a small component of extension normal to the transform. Hence, the junction may survive for an appreciable time in a geometrically unstable state. At the point of mechanical instability, the junction would change back to a stable RRR configuration and migrate northwest until it intersects the isosceles line once again. Each time the junction switched to RFF mode, paired transform faults would develop. The RFF configuration of the Bouvet junction was apparently able to migrate approximately 110 km
(about 1 degree latitude) from the isosceles line before RRR mode intervened.
vii A rigorous examination of the unstable Kendocino triple junction in velocity space reveals possible modes of evolution. If the requirement is relaxed that the trench is fixed with respect to the North American plate, trench rotation and subsequent back-arc spreading are possible. At specific orientations of the trench boundary, spreading at the North American-Pacific boundary allows the junction to regain stability. A third alternative is the migration of the San Andreas shear zone eastward, transferring part of the North American plate to the Pacific plate.
- viii - CONTENTS
APPROVAL PAGE ii DEDICATION iii ACKKOVLEDGEKENTS iv ABSTRACT v LIST OF FIGURES xi LIST OF TABLES xiii
Chapter page I. INTRODUCTION 1 II. PREVIOUS WORK 5 Triple Junction Geometry 5 III. THE FRAMEWORK OF PLATE GEOMETRY 10 The Angular Displacement of a Plate 10 The Tectonic Character of Plate Boundaries 13 The Stability of Triple Junctions 14 Nomenclature 14 Defining Stability 16 IV. THE QUANTIFICATION OF GEOMETRIC INSTABILITY 22
Triple Junction Equations 23 The Best Fit Solution 26 least Squares Estimation 27 Relative Stability 29 V. THE PERTURBATION STABILITY OF TRIPLE JUNCTIONS 32
Analysis of Stability 33 The Problem on a Sphere Absolute Plate Motion 41
- ix - VI. GEOLOGIC APPLICATIONS 44
The Bouvet Triple Junction 44 Introduction 44 The Absolute Notion of the Eouvet Triple Junction 46 Eouvet Triple Junction Evolution 48 The Question of Relative Stability 51
Triple Junction Evolution in the Western United States 55 Introduction 55 Previous Work 55 Velocity Space Analysis 58
VII. CONCLUSIONS 61 BIBLIOGRAPHY 65
FIGURES 68
TABLES 132
Appendix page
A. SINGULAR VALUE DECOMPOSITION 139
B. SUMMARY OF CONSTRAINT AND PERTURBATION
STABILITY ANALYSIS 144
X -T LIST OP FIGURES
Figure page 1. A stable configuration for the RRF triple junction. 68
2. An unstable RRR triple junction...... 70 3* A stable TTT(b) triple junction...... 72
4. A geometrically stable quadruple junction...... 74
5* Instantaneous displacement and angular velocity. . . 76 6. The vector circuit around a triple junction...... 78
7. The possible types of plate boundaries...... 80
8. Nomenclature and labeling scheme...... 82 9* An FRT triple junction defining the stable condition...... 84 10. Possible velocity diagrams for the RRR junction. . . 86 11. Direct space representation of Figure 10...... 88
12. Four possible rff direct space configurations. ... 90 13» Development of vectors describing the location of (J)...... 92
14. Sum of square of the residuals...... 94
15* The effects of boundary rotation on stability. ... 96
16. The effects of triangle distortion on stability. . . 98 17. Constraints on the relative velocity...... 101
18. A description of perturbation stability...... 103
- xi 19* Stability analysis of the RPP Junction...... 105 20. Stability analysis of the RRF and RRR junctions . 107 21. The location of the Bouvet triple junction. . . . 109
22. The position and orientation of the isosceles line. 111
23. Velocity space representation of the Bouvet junction...... 113
24* A model of evolution for the Eouvet triple junction...... 115
25* The evolving velocity triangle for the RFF junction...... 117
26. Relative geometric stability...... 119 27* Location and tectonic map of the western United States...... 121 28. The geometric evolution of the NE Pacific...... 123
29. Stable and unstable FFT configurations...... 125 30. Cumulative deformation at an unstable FFT junction. 127
31. Tectonic consequences of the unstable Mendocino junction...... 129
- xii - LIST OP TABLES
Table page
1. The stability of triple junctions...... 132 2. Stability involving oblique symmetrical spreading. 133 3« Topologically distinct varieties of triple junctions...... 134 4. Topologically distinct varieties of stable T- junctions...... 136
5. Perturbation stability of some existing triple junctions...... 137
6. AMI-2 absolute poles of rotation ...... 138
- xiii Chapter I INTRODUCTION
The success of plate tectonics as a framework for our understanding of the surface motions of the earth has unfortunately not been accompanied by similar progress in determining the driving mechanisms.^ Although the driving mechanisms remain elusive, most interactions of plates at the surface have been effectively described by the geometry of rigid plate motions.
In a benchmark paper, Mckenzie and Morgan (1969) first established the geometric rules that govern the existence of triple junctions, points where three lithospheric plates meet. By combining information about the relative velocities and boundary orientations, McKenzie and Morgan defined what they termed the 'stability' of a triple junction. As defined by Mckenzie and Morgan, a stable junction is one which retains its relative velocities and boundary configurations through time. By considering the possible combinations of the three types of boundaries (ie. ridges,
1 For a discussion of plate driving mechanisms, see Turcotte (1983).
1 2 trenches, and transform faults), they determined that 16 junction types were possible, fourteen of which were potentially stable. Since then, a number of authors (York,
1973, 1975; Eguchi, 1979) have pointed out inconsistencies in Mckenzie and Morgan's initial results, and it appeared that the treatment of triple junction stability was incomplete. This was recognized by Sclater (1978) regarding the stability of the Bouvet triple junction:
"The reasons for the abrupt change in geometry are not known. ...The answers to these questions awaits more work in the area and a more extensive consideration of the stability of triple junctions.”
In an effort to clarify the geometric evolution of plates and to thoroughly define the two-dimensional stability of triple junctions. Gray and Apotria (1983) analyzed the sensitivity of triple junctions to changes in velocity and boundary orientations. Although the number of of potentially stable junctions is 16, their study suggested that some types included more than one distinct variety, and that 21 varieties exist. Each variety possesses differing degrees of stability with respect to changes in the relative velocities and boundary orientations.
A least squares algorithm is developed here to quantify the exact stability or degree of instability of any triple 3 junction. Three linear equations of the form J = a + bh involving only two unknowns describe the location of the triple junction with respect to each boundary in velocity
space. For a stable junction, a unique solution is obtained.
Otherwise, the sum of the square of the residuals of a least squares regression can be considered a measure of the degree of geometric instability. In the event that a number of different junctions are compatible with the same velocity triangle, the junction type least affected by small changes
in the relative velocities can be considered the preferred mode.
Consideration of the motion of triple junctions with respect to the mantle is essential in evaluating their evolution. In an ’absolute’ reference frame, the velocity triangle of a particular triple junction may change drastically, rendering it unstable. Thus, it is important to recognize that stability can be defined only instantaneously. The evolution of triple junctions with respect to the mantle, introduces complexities not considered in McKenzie and Morgan’s (1969) treatment.
Stable triple junctions may become unstable through changes in their boundary orientations or in the relative velocities. An attempt is made in this thesis to model the 4 tectonic effects of instability and to assess the possible routes of reorganization of unstable triple junctions.
This thesis presents a treatment of the stability and evolution of triple junctions by considering the following:
1 . the stability criteria established by McKenzie and Morgan (1969) and the interpretation of velocity
space diagrams. 2. a discussion of unstable triple junctions,
introducing quantitatively, the degree of geometric instability using a least squares analysis. This
precedes the evaluation of alternate modes of triple
junction evolution.
3. the role of triple junction motion with respect to
the mantle causing changes in the velocity triangle during evolution. Such changes may spur instabilities that can be evaluated as in (2).
4. an analysis of geologic examples of triple junctions
in the light of (1), (2), and (3)* The Bouvet
junction in the South Atlantic and the Mendocino
junction in northern California will be considered. Chapter II PREVIOUS WORK
TRIPLE JUNCTION GEOMETRY Mckenzie and Parker (1967) realized that the movement of
plates on a sphere is best understood in terms of rotations, and hence introduced Euler's theorem (see Chapter 3) to explain such motion. As a result, the linear velocity
relations at triple junctions on a plane, and the concept to angular velocities on a sphere, could be described.
Mckenzie and Morgan (1969) extended this concept by
including a description of triple junction evolution based
on geometric relationships between relative plate velocities
and boundary configurations. In this classic treatment, the
notion of 'stability' was introduced. A stable junction was one in which the relative velocities and boundary
configurations remained unchanged with time. More
specifically, a stable junction is one for which a reference frame can be located in velocity space that is stationary with respect to a point of uniform velocity on each boundary. Unstable junctions should not survive for long
- 5 - 6 periods of time (Mckenzie and Morgan, 1969) since they would probably undergo a change in relative velocities, boundary orientations, or the nature of the boundary itself, such that it regained a stable state. Mckenzie and Morgan enumerated all possible types of triple junctions based on combinations of three types: trenches, ridges, and transform faults, and concluded that of sixteen possible types of junctions, fourteen were potentially stable. They applied this concept to the San Andreas transform and triple junctions.
Since Mckenzie and Morgan's treatment of triple junctions, several papers have been published which point out confusing aspects in the original paper. York (1973) found that one of the triple junctions that Mckenzie and
Morgan had claimed as unstable, was in fact stable. The RRF (ridge-ridge-fault) junction (Figure 1) of York (1973) made apparent the need to consider all possible configurations and relative velocities for any given junction. York (1975) realized that certain configurations of RRR were unstable. This instability existed whenever all ridge boundaries encompassed 180 degrees (i.e., the angle between the first and third ridge, including the second, is less than 180 degrees) (Figure 2). York supported his argument by postulating that an extinct ridge within the Nazca plate 7 reported 1)7 Herron (1972) was caused by such an instability.
The junction was initially forned by a mantle-induced rifting event, generating a third ridge segment, resulting in a RRR triple junction. As a result of the instability, the Nazca-Pacific boundary was rendered inactive and a new boundary formed (Figure 11b of York, 1975)*
Eguchi et al, (1979) described another case where the supposedly unstable TTT(b) (trench-trench-trench) type triple junction of McKenzie and Morgan (1969) is geometrically stable (Figure 3)»
Hey and Milholland (1979) described several situations where quadruple junctions could exist, contrary to Mckenzie and Morgan's claim of unconditional instability (Figure 4)*
If all 4 boundaries are orthogonal, five possible quadruple junctions may exist (RRRR, RFRF, RTRT, TTTF, TTTT). However, in such cases, plate velocities are constrained by strict geometric requirements. Hey and Milholland cite two examples where 'near' quadruple junctions exist: one where the Pacific, Phillippine, China, and Australian plates meet near the tip of the Phillippine plate; the second where the
Pacific, Cocos, Riviera, and North American plaxes meet.
Recent work by Patriat and Courtillot (1984) illustrate the fact that discussions of stability show an increasing 8 degree of sophistication over earlier studies. Three additional aspects of evolution are now considered for junctions that involve only ridges and faults: a) the possibility of oblique or asymmetric spreading,
b) rift propagation at triple junctions, where at least one of the ridges meeting at an RRR triple junction
may recede c) spontaneous jumps from one configuration to another.
Point c demonstrates that in general, there are several types of triple junctions that are compatible with a given velocity triangle. Patriat and Courtillot contend that the main parameters that influence triple junction evolution involving ridges and transform faults are the lengths of the transform faults, the availability of magma and related connectivity of magma chambers, and the spreading velocities. They believe that the activity at such junctions occurs in two preferred modes: the effusive and tectonic modes, corresponding to RRR and RRF-RFF configurations respectively. They contend that these modes alternate in episodes of 1 my duration.
Similar complications were recognized by Thede (1983) for the case of an RRR junction with an obtuse velocity triangle. The obtuse configuration requires the fastest spreading ridge to shrink, shortened by the advancing triple 9
junction. If the ridge is offset by a transform fault, the
path of the triple junction is abruptly truncated. This instability may be resolved only if one of the following occurs: a) an instantaneous jump of the triple junction to
another ridge segment,
b) gradual translation of the triple junction along the offsetting transform,
c) in the case of an isosceles velocity triangle, a
change from RRR to RFF mode. As a result of case (c), two paired transforms are created
on the two slow spreading (and lengthening) ridges. Such
paired transforms have been observed near the Indian Ocean and Bouvet triple junctions (Thede, 1983).
In summary, the previous work suggests that although the
stability criteria described by McKenzie and Morgan (1969) are geometrically sound and correct, their analysis is incomplete as several authors have shown. Stability conditions for each type of junction are defined only instantaneously. The stability and evolution of triple junctions must take into account changes in relative velocities and boundary orientations as a function of time. Chapter III THE FRAMEWORK OF PLATE GEOMETRY
THE ANGULAR DISPLACEMENT OF A PLATE
A plate in motion relative to the mesosphere occupies different positions at different times, its change in position is referred to as an angular displacement, and the rate of change its angular velocity. Such a displacement only specifies a change in position, and not the actual path of the displacement. For a spherical earth, an angular displacement may be described as a rotation about a line through the center of the earth. That such a line exists for any two positions of a plate follows from Euler's theorem (Beer and Johnston, 1973)5
"The motion during a time interval dt of a rigid body with a fixed point 0 is considered as a rotation through d© about a certain axis. Given along that axis a vector of magnitude d0/dt, and letting dt approach zero, we obtain at the limit the instantaneous axis of rotation and the angular velocity w = d0/dt of the body at the instant considered" (Figure 5)*
This rotation axis intersects the earth at two antipodal points, either of which define the pole of rotation for a plate. These poles are described with respect to a reference frame assumed fixed, usually another plate or the mantle.
10 - 11
A finite angular displacement differs from the instantaneous angular displacement just described. The finite displacement has a direction and magnitude associated with it, but is not a vector quantity. Vector quantities obey the laws of vector addition, one of which states that the sum of several vectors is independent of the order in which they are combined. For example, consider a rectangular object rotated 180 degrees about two different axes. If the same object is rotated about the same axes but in a different order, the final configuration of the object will be different.
The angular velocity of a plate specifies the instantaneous angular speed at which the plate is rotating and its orientation about its instantaneous axis of rotation. The velocity of a particle P on the surface of the earth is obtained by forming the vector product of omega
(w) and the radius vector (r) of the particle:
(1) v = dr/dt = w X r = (w)(r sin 6) where dr/dt is the change in the position vector with time and 0 is the angle between the radius vector and omega. The fact that particle velocities such as v are directed tangentially to small circles about the instantaneous pole of rotation gives rise to a method used by Morgan (1968) and 12
LePichon (1968) to locate poles of rotation for plates separated by a ridge-transform boundary. The method assumes that transform faults are instantaneously parallel to small circles to the relative pole of rotation for the two plates.
At triple junctions, if any two relative velocity vectors are known, the third can be determined (Mckenzie and Parker, 1967). This concept is most easily understood for a plane rather than for a sphere (Figure 6). Starting from a point
X on plate A and moving clockwise, the relative velocity of B with respect to A (denoted herein as aVb) is represented by the vector AB in the velocity vector diagram. Similarly bVc and cVa are represented by vector BC and GA respectively. The vector diagram closes as the circuit returns to point X. Thus:
(2) aVb Vc cVa = 0 or
V.-Vb a + V c -V,b + V a -Vc = 0
To construct a vector circuit, 3 parameters must be known. One must be the length of a side and the other two can be lengths or included angles.
Transform faults and magnetic lineations have been useful in determining the directions and magnitudes of portions of 13 the vector circuit in vhich case, the other parameters can he calculated.
Equation (2) is applicable to the corresponding problem on a sphere because angular velocities behave like vectors
(Mckenzie and Parker, 1967):
(3) a W,b + be, k’ + c Va 0
The sign convention obeys the right hand rule and is such that a positive vector points outward along the rotation axis for a clockwise rotation when viewed from the center of the sphere.
THE TECTONIC CHARACTER OF PLATE BOUNDARIES
One of the basic consequences of the assumption of rigid plates is that either creation, destruction, or preservation of lithosphere occurs at plate boundaries. Intraplate stresses and boundary deformation are closely linked to geometric plate evolution but are not fundamental to the basic postulates. Trenches are defined as boundaries which consume lithosphere from one side. Ridges are boundaries which generate lithosphere from both sides. It is usually assumed that spreading at ridges occurs symmetrically with the boundary at right angles to the relative velocity 14 vector; however, this constraint is relaxed in many situations. Transform faults conserve lithosphere and, instantaneously, trend parallel to the relative velocity vector.
The instantaneous character of the boundary between two plates depends on the relative orientations of the boundary and the relative velocity vector (bVa). For example, if bVa is directed west to east and is parallel to the boundary between plates A and B (Figure 7), that boundary must be a transform fault with dextral strike-slip. All possible relations between bVa and the boundary orientation are summarized in Figure 7, where bVa is the relative velocity of B with respect to A, and i is a unit vector normal to the plate boundary, tangent to the surface of the earth, and pointing from the reference plate A toward B (Hobbs et al, 1976).
THE STABILITY OF TRIPLE JUNCTIONS Nomenclature
The analysis of geometric stability is simplified if a slightly different notation from that of McKenzie and Morgan
(1969) is used. 15 The following four varieties of plate boundaries are considered:
f,F: a transform fault. r,R: a symmetrical spreading ridge oriented per pendicular to the spreading direction. t+,T+: a trench or subduction zone attached to the plate plotting clockwise ’ahead' of the subducting plate. t”,T“: a trench or subduction zone attached to the plate which, in a clockwise sense, plots 'behind' the subducting plate. For example, assuming boundary AB is a trench; if A is subducted beneath B, the trench is T+, if B is subducted
beneath A, the trench is T”.
The triple junction classification proceeds according to
the cyclical sequence of boundaries in either 'direct' or
'velocity vector' space. The lowercase symbols f, r, t+, t” are used to label boundaries in direct space; the uppercase symbols F, R, T+, T“ are assigned to the velocity space labeling. Labels in either space, by
convention, are assigned in a clockwise manner.
In Figure 8, for example, two junctions are considered in direct space as frt+ and ft+r (Figure 8a). It should be noted that the labeling schemes in direct and velocity space are not entirely equivalent. For example, the velocity space configuration FT+R (Figure 8b) may 16 correspond to either frt“ or ft+r in direct space.
Therefore, several direct space configurations may be consistent with one velocity space configuration.
The variety of triple junctions involving combinations of
the four simple boundary types should number 24 (ie. the number of distinct ways of filling 3 boxes with 4 different
objects without regard to cyclical order). However, because T+ and T” (t+ and t”) are mirror images, the
number of topologically distinct cyclical arrangements is only the 16 listed in Table 1. Of these 16, only one (FFF) is unconditionally unstable. All others may be stable or unstable depending on the relative velocities and the
boundary orientations. A stable example of each type is given in Appendix B.
Defining Stability
The simple constraint that the relative velocities form a
triangle at triple junctions was described in the previous section (Equation 1). This constraint does not impose any restrictions on relative velocities or on the orientation of the plate boundaries. However, if the junction is required to be stable, i.e., maintain the sane velocities and orientations, certain additional constraints must be 17 satisfied. Unless these constraints are satisfied, the triple junction is unstable. While unstable junctions do exist, they must be transient, changing their boundary orientations or relative velocities so that they become stable. Otherwise, continuous and predictable plate evolution is prevented.
To clarify this definition of stability, which was first introduced by McKenzie and Morgan (1969)» the triple junction between a ridge, a trench, and a transform fault in both direct space and in velocity space is examined in some detail (Figure 9a and 9b). This junction type was selected because the stability criteria involved are established for each type of boundary. In this and in all other examples, the relative velocity vectors at the junction are required to satisfy Equation (2). This is automatically satisfied if the plates are rigid.
In defining stability, the following points in velocity space are of particular signifigance: a) the relative velocity triangle. b) the locations of the plate boundaries.
c) the locus of points which move with a uniform velocity
along a plate boundary. d) the location of the triple junction. 18 The velocity triangle is defined in Figure 9b where the
lengths AB, BC, and CA are proportional and parallel to the relative velocities aVb, bVc, and cVa respectively. This triangle represents the conditions imposed by Equation (2).
Assuming that ridge boundary AC spreads symmetrically and at right angles to its strike, a point on its axis will move with a velocity cVa/2 relative to plate A. This velocity corresponds to the midpoint of AC in Figure 9b. Also, any point moving with uniform velocity along ridge AC would have its motion represented by a point on the perpendicular bisector ac of side AC of the velocity triangle. Similar constraints apply to the two other boundaries. The boundary between plates B and C is a trench (T”). Assuming plate
C is not consumed, the trench does not move relative to C.
Therefore, the location of the trench in velocity space is at point C. Any point moving with a uniform velocity along the trench boundary is represented by a position on the line be, parallel to the trench boundary, through point C. For the case of transform fault AB, a point along the boundary is fixed with respect to A and B. Hence any point motion along the transform would be represented by a point on the line ab, which passes through both A and B in velocity space. If the three lines ab, be, and ca intersect at one point in velocity space, the intersection J is stationary 19 with respect to three points of uniform velocity along each boundary. J then represents the velocity of a reference frame in which the triple junction is stationary and is thus stable in the sense of Mckenzie and Morgan (1969)*
To summarize, a triple junction is defined as stable if two criteria are met according to Mckenzie and Morgan
(1969): a) the three relative velocity vectors of the rigid plates must form a closed vector triangle (Equation 1). b) a reference frame must exist which is stationary with respect to a point of uniform velocity on each of the
boundaries.
Various directions and velocities that are determined once the triple junction is located. Figure 10a is the p acute velocity triangle for a stable RRR triple junction. Figure 11 represents the direct space analogs of the velocity triangles in Figure 10. The rate of growth for ridge AC (Figure 10) is given by vector zJ from the midpoint of aVc to the triple junction (J). The vectors xj, yj, and zJ are referred to as boundary growth vectors. The ridge is lengthening if J is located in the direction of the interior
2 An acute triangle has all angles less than 90 degrees. An obtuse triangle has one angle greater than 90 degrees. 20 of triangle ABC from the midpoint of aVc. For an acute triangle, the intersection of the perpendicular bisectors are always in the triangle’s interior. If the triangle is equilateral, the junction does not move along any boundary
(Figure 10a and 11a), new ridge segments are generated from the triple junction.
An obtuse velocity triangle behaves differently. The fastest spreading ridge of an obtuse velocity triangle is shortening while the two other slower spreading ridges lengthen (Figure 10b and 11b). The boundary growth vector for the fastest spreading ridge is directed toward the exterior of the velocity triangle. An obtuse velocity triangle will always have one perpendicular bisector located in a direction away from the triangle's interior.
In the case of a RRR right triangle, the triple junction is located on the hypotenuse. Only the two slower segments propagate and the third one (BC) remains constant in length
(Figure 10c and 11c).
For the triangles in Figure 10, vectors AJ, BJ, and CJ parallel the traces of the triple junction on the respective plates (i.e., the rough-smooth boundary separating crust formed by different ridges). The area of the triangle bounded by such traces (ABJ, ECJ, and ACJ) are proportional 21
to the rate at which material is created on the new parts of the respective rises.
For a given velocity triangle, several direct space
configurations, including different triple junction types,
are possible. For velocity triangles with several consistent direct space configurations, boundary growth vectors vary considerably. For example, the velocity triangle in Figure 12a has four possible RFF configurations.
In b, d, and e, the ridge segments lengthen by a rate xC In
12d, the ridge shortens by the same rate xC. In all cases, the transforms lengthen by yC and zC (both are equal if the
triangle is isosceles). A person standing at y along BC sees
plates B and C moving in opposite directions at the same rate, y is therefore a point in velocity space on the
transform. This point moves at a rate yC with respect to the triple junction C and is therefore the rate of transform lengthening. This rate may be easier to understand by visualizing a ridge segment on the transform BC in figure
12b. The rate of transform lengthening is half the spreading rate of that ridge segment corresonding to a vector yC. Chapter IV THE QUANTIFICATION OF GEOMETRIC INSTABILITY
McKenzie and Morgan's (1969) notion of triple junction
stability is largely a geometric concept. The calculation
of the stability condition and the varying degrees of
stability has been done graphically.
In this chapter, triple junction stability will be
introduced from a more quantitative viewpoint. Vectors representing velocities and orientations can be represented algebraically by linear equations. Hence, a set of equations representing the location of a triple junction can be written including the position and orientation of each of the boundaries. These three equations, between two unknowns, may be solved simultaneously. If a unique solution exists, the junction is stable and the velocity of the triple junction is determined. There will be no unique solution if the junction is unstable. In this case, a best estimate of the velocity space location of the triple junction is given by a least squares fit performed to minimize the sum of the squares of the residuals between the best fit and each boundary. The magnitude of the sum of the
22 23
squares of the residuals can be considered a measure of the
degree of geometric instability (Figure 14) • For a given
type of triple junction, zones of stability and instability
can be outlined. This type of procedure may be performed on
a computer to increase the speed of calculation and to
enhance the graphic representation.
As a triple junction moves on a spherical earth, the
relative velocity triangle continually changes. As a consequence, triple junctions are susceptible to periods of stability and instability. The ability to calculate
stability may provide more insight to the evolution of unstable triple junctions.
TRIPLE JUNCTION EQUATIONS
This section develops an algebraic expression for the location of a triple junction (J) with respect to each of the four different types of boundaries. The vector equations describing the velocity space location of the triple
junction are linear and of the form:
(1) J = a + bh where J is the location of the triple junction in a 2-dimensional coordinate system, a and b are 2-dimensional vectors describing the relative velocities and boundary 24 configurations respectively, and h is a scalar quantity representing a point of uniform velocity along a given plate boundary.
To illustrate the derivation, consider a FT+R triple junction in both direct (Figure 13a) and velocity vector space (Figure 13t>)» A coordinate system can be chosen arbitrarily. A simple reference frame is one fixed to one of the plates. An external reference frame, such as the mantle, may in some circumstances be more appropriate.
The triple junction equations involve several known quantities, namely velocities and boundary orientations
(i.e., AB and ab in Figure 13)» and one unknown vector (J), the location of the triple junction. The point J is common to a point of uniform velocity on each of the boundaries at all times if the junction is stable. Each equation must equate J to a series of vectors connecting the external reference frame (0) to the location of the triple junction (J). This is performed with respect to each of the three boundaries. For ridge AC in Figure 13h, the vector 1/2(0A + OC) describes a line from 0 to the intersection of AC and ac
(P). Then, the product of a scalar quantity h^ and vector ac is a vector from P to J. Hence, the triple junction equation corresponding to the ridge boundary is:
(2) J = 1/2(0A + OC) + h1(ac) 25
Similarly, for the trench boundary BC, the location of the triple junction is described by the equation:
(3) J = OC - h2 (be)
If the trench has the same relative velocity vector but had an opposite polarity (i.e., T“), the equation would be:
(4) J = OB - h2 (be)
Notice in this case, a unique solution could not exist because the relative velocity vectors in Figure 13a and 13b
For the fault boundary AB, the location of J is described by:
(5) J = 1/2(OA + OB) + h5(ab)
This equation fixes the transform fault to the midpoint of
AB so that transforms not purely strike-slip can be evaluated.
Thus, for a FT+R triple junction, the three equations:
J = 1/2(0A + OC) + h1 (ac)
J = OC - h2 (be)
J = 1/2(0A + OB) + h5(ab) 26 describe the location of the triple junction (J).
THE BEST FIT SOLUTION
The next step is to find the location of the triple junction in velocity space by solving three equations sinultaneously. Three situations are possible. If there are an equal number of unknowns as there are independent equations, a unique solution exists. If there are more unknowns than equations (underdetermined), there is no analytic solution. If there are more independent equations than unknowns (overdetermined), a best"fit value can be calculated for the unknown parameter.
The solution of the location of the triple junction is overdetermined. In the vector equation:
(6) J=a+bh a and b are two dimensional vectors representing velocities, and h is a scalar quantity representing a length along a velocity vector. Each vector equation contributes component values in the x and y direction:
(7) + bxh
(8) +
These two eqautions can be solved in terms of h: 27
(9) h = Jx - ax/bx , Jy - By/by
00) h - Jx/bx - Jy/by . ai/,x . ay/l)y
The coordinates of the triple junction can be solved by eliminating the parameter h. The equation then can be written as a matrix:
(11) AX = B which, including all three triple junction equations, expands to: N ’ > 1/bx1 - T/hyl Xx axl/bxl “ ayi./byl
(12) 1/bx2 " 1^by2 Xy ax2^bx2 ” ay2^by2 L J J 1/bx3 “ 1/by3 ax3/bx3 " ay3/by3 . - Three equations with two unknowns are thus obtained, and the system is overdetermined.
Least Squares Estimation
The best“fit solution is calculated by least squares estimation. The general form of the equations in matrix form can be written in a linear form expressing both x and y components as:
(13) A1x + A2y - B = r 28
where and A2 are the constant values in the first and second columns of matrix A, B are the constant values in matrix B, x and y are the unknown quantities Jx and Jy
respectively, r is the value of the residual and is equal to
zero if a unique solution is found (ie., a stable triple junction). To arrive at a least squares solution of x and
y, the sum of the squares of the residuals for each of the three equations is minimized. We proceed by squaring both sides of Equation (13) and finding the minimum value by
taking the derivative with respect to both x and y:
(14) (A1)2x2 + 2(A1A2)xy - 2(A1B)x - 2(A2B)y + (A2)2y2 + B2 = r2
Taking the derivative with respect to x and equating to zero
produces:
(15) 2(A1)2x + 2(A1A2)y - 2(A1B) = 0
and with respect to y:
(16) 2(A1A2)x - 2(A2B) + 2(A2)2y = 0 then substituting for A and B the sum of these constants for each of the three equations, the derivative with respect to x reduces to: 29
(17) 2(An2 + A122 + A132)x + 2(A11A21
— 2B^ A^ + 2B2A^ 2 2B^A^ ^ = 0 and with respect to y:
2( A^ ^ A2^ + 2^22 ^2^1^21 2 2 2 + 2B2A22 + 2B^A2^) + 2(A2^ + A22 + A2^) (18)
We thus obtain 2 equations with 2 unknowns (x,y)f which can be solved analytically to arrive at the location of the triple junction.
In practice, when many numbers are involved in the least squares calculation, numerical problems arise in finding the solution. Such problems are avoided by solving the equations utilizing a singular value decomposition (see, e.g., Nash, 1979; Appendix A)
RELATIVE STABILITY
The applications of the least squares fit triple junction and the residuals include the quantification of geometric instability. The calculation of the exact stability condition of any triple junction can be made by precise 30 drafting. However, using the above algorithm, the effects on geometric stability of changing boundary orientations and relative velocities can be determined easily.
For example, the effects of rotating the trench boundary in an FFT+ triple junction (similar to the Mendocino junction) are shown in Figure 15* The sum of the squares of the residuals are plotted versus the azimuth of the trench boundary (0). The junction has zero residuals and is stable when the trench and the San Andreas transform are co-linear
(Figure 15» inset).
Instead of rotating a boundary, the velocity triangle can be distorted. Making predictable changes in the velocity triangle is particularly useful when one velocity triangle is consistent with several configurations. The analysis then provides a means of determining which configuration is least affected by these changes.
Figure 16 plots the analysis of an isosceles right triangle in which RRR, RFF and RRF are all initially geometrically stable. In Figure 16a, the relative velocity of plate B is varied in the x-direction. In this case RRF remains stable while RRR and RFF modes show the resulting geometric instabilities (the triangle remains a right triangle, but deviates from isosceles). For large values of 31
Bx, the RRF junction also is unstable. In order for a right triangle to exist, velocity point B must lie along an arc whose center is the midpoint of line AC. Interestingly, the two unstable junction have exactly the same residuals, even though the location of the triple junction is different for both triangles.
In Figure 16b, the relative of plate B varies in the y-direction. In this case, RRF mode becomes unstable
(because the triangle is no longer a right triangle), and both RRR and RFF modes remain stable (the triangle remains isosceles). Chapter V THE PERTURBATION STABILITY OP TRIPLE JUNCTIONS
In their classic treatment of triple junctions, McKenzie
and Morgan (1969) contended that a stable triple junction
would maintain the same relative velocities and boundary orientations with time. This condition in turn, would allow
continuous plate evolution.
However, it is now known that continuous plate evolution involving triple junctions may lead to potential
instabilities (Dickinson and Snyder, 1978; Thede, 1983)*
For example, as the Mendocino triple junction migrated
northwestward in the late Tertiary, it encountered changes
in boundary orientations caused by the irregular shape of
the western North American plate. Such changes in orientation arise independently of the triple junction, making it possible for the stability of a junction to be
threatened by evolution. The question must then be asked "How stable is stable?" What might be the effects of changes in relative velocities or boundary orientations on
the stability of triple junctions?
- 32 - 33
Using the methods described in the previous chapter, a portion of this thesis work was completed to extend and clarify the concept of stability. The following is taken from the geometric analysis by Gray and Apotria (1983) in which degrees of stability were prescribed to each topological variety of triple junction. The analysis originated in an effort to evaluate the geometric constraints on the evolution of unstable triple junctions by exhaustively examining every possible triple junction configuration.
ANALYSIS OP STABILITY Two different aspects of the geometric stability of triple junctions were examined:
]) Constraints - The geometric stability of a triple junction generally implies certain relationships between relative velocities and boundary orientations. The number of applicable constraints is a function of the junction variety. Herein, 'variety' is distinguished from 'type' in that there are 16 different types of triple junctions as introduced by McKenzie and Morgan (1969)* yet some of these types can be distinguished further by a special case. A complete list of junction types, including varieties, is contained in Table 1 . Two types of constraints are considered: 34
i) constraints on relative velocities. ii) constraints on the triple junction velocity.
For each variety of triple junction, the constraints on the value of one plate velocity (arbitrarily taken as B) when the other two (A and C) are known, are illustrated in velocity space in Figure 17* Because only clockwise labeling of velocity points are considered around the triple junction, the location of B must always lie above the line AC. Velocity point B has zero constraints if it can exist anywhere in the region (2 degrees of freedom^1) one constraint if it is confined to a line (B* , 1 degree of freedom), and two constraints if it is confined to a point
(B*’, 0 degrees of freedom). In certain situations, the location of B is specified by the location of the triple junction, and is noted.
2) Perturbation Stability -The stability of some triple junctions may be very sensitive to small changes in the relative velocities or boundary configuration. The degrees of freedom for perturbations (i.e., imparted changes) which do not upset the stability of a junction can be deduced from the velocity vector space diagram. For example,T+T~R has two degrees of freedom in relative
3 Degree of freedom - a change in relative velocities or boundary configuration that does not upset the stability. 35
velocities given fixed boundary orientations (Figure 18a). The position of B along ac may vary and the size (area) of the velocity triangle may change. T+T**R also has
two degrees of freedom in its boundary orientations, given fixed velocities (Figure 18b). ab and be may independently
change their orientations.
The overall perturbation stability of triple junctions is widely variable. Of the 21 potential stable varieties (of
the 16 types):
4 have 1 degree of freedom (DOF)
5 have 2 DOF
3 have 3 DOF 6 have 4 DOF 2 have 5 DOF 1 has 6 DOF
Determining the sensitivity of triple junctions in terms
of degrees of freedom can only be accomplished by careful examination of the velocity space diagram. In the following discussion, the analysis of constraints and perturbation stability is explained in more detail for all types of triple junctions that involve only ridges and faults.
Figure 19 is constraint analysis for the RFF configuration. In Figure 19&» an example of a stable configuration of 36 boundary orientations and relative velocities is illustrated. Figure 19b portrays the constraints placed on the choice of one plate velocity (here arbitrarily taken as
B) when the other two (A and C) are known. As in this case, where the choice of B has only one degree of freedom, the feasibility curve is shown as a double line. For the RFF junction this line must be the perpendicular bisector of line AC (i.e., the junction is stable only if isosceles).
Figure 19c illustrates the constraints on the triple junction velocity. Again, a double line is used to outline its possible locations when the junction velocity has one degree of freedom. For the RFF junction, the triple junction velocity must lie on the perpendicular bisector of AC.
A second variety of the RFF junction, one in which the configuration has two boundaries co-linear (T“junction), exhibits somewhat different constraints. Figure 19d is a stable example of an RFF T” junction. Note that all velocity points are co-linear. This is expressed in Figure 19e, where the velocity of plate B has one degree of freedom and is constrained to the line AC. Because boundary AC is a ridge, the velocity of the triple junction must lie at the midpoint of AC. The triple junction velocity then has zero degrees of freedom as shown by the open triangle (Figure
19f). 37
Figure 20a illustrates a stable example of an RRF junction, where boundaries AB and BC are ridges and AC is a transform fault. Note in Figure 20b, velocity point B is constrained with one degree of freedom to lie on a semi-circle as shown. The junction is only stable if the velocity triangle is a right^triangle. The triple junction velocity has zero degrees of freedom and is constrained to lie at the midpoint of the fault boundary AC
(Figure 20c). Anywhere else along AC, either the triangle is not a right“triangle or the ridges must deviate from orthogonal spreading to maintain stability.
The RRR triple junction is the least constrained among those combining only ridges and faults and is almost always stable (see Figure 2). The velocities of plate B and of the triple junction can lie anywhere in the region above AC and the location of each in velocity space is not independent of the other. The triple junction location in velocity space is defined only after the velocity of plate B is specified.
For the junctions previously mentioned, each of the boundaries is constrained to fit the given velocity triangle to maintain stability. Boundary orientations and relative velocities are not mutually independent, although this 38
property may be relaxed for certain junctions involving
trenches (see Chapter 6).
Perturbation degrees of freedom, or changes in the velocity triangle or boundary orientations that do not
disturb the stability criteria, can also be deduced for the
RPF, RRP, and RRR junctions. These junctions are very
sensitive to such perturbations and have few degrees of freedom. With fixed relative velocities, each of the three boundaries has zero degrees of freedom (i.e. the triple
junction becomes unstable if the boundary orientations are
changed). With fixed boundaries, each junction has one perturbation degree of freedom (i.e. the area of the velocity triangle may change. However, the Injunction case of an RPF junction has two perturbation degrees of freedom with fixed boundary orientations. The line ABC may change in length, and the location of B along that line may change without upsetting the stability.
Tables 1 through 3 summarize the constraints and perturbation stabilities for all varieties of triple junctions.
T-type junctions are a special class of triple junctions in which two of the three boundaries are colinear. They may arise as a result of triple junction collisions and 39
more importantly, as a result of the breakdown of unstable
junctions. It appears that T”type junctions are likely
to form from unstable junctions because co-linear boundaries
most often meet the criteria for stability. Table 4
cataloqs the potentially stable types of T“type
junctions along with their perturbation stability.
Table 5 portrays the overall perturbation stability of
several existing triple junctions assuming that oblique
spreading at ridges is allowed. The triple junctions listed
have a high perturbation stability, suggesting that
junctions exist in, or evolve to, the most stable state
possible.
THE PROBLEM ON A SPHERE
The previous treatment of triple junction stability was
described on a plane, where all displacements are
translations. This is justified because the behavior of
triple junctions depends only on the motion of the three
plates at the point where they intersect. McKenzie and
Morgan's (1969) analysis applies only to the instantaneous
domain and to the infinitesimal plate tangent to the triple
junction. Their discussion of triple junction evolution applies rigorously only on a plane. 40
The stability and evolution of triple junctions is more complicated on a spherical earth. It has been shown
(McKenzie and Morgan, 1969; LePichon, 1973; Cox, 1973) that it is not possible for three plates to rotate simultaneously through finite angles about their instantaneous relative rotation axes. Hence, through some finite time interval, the instantaneous relative pole of rotation of at least one plate must change its location. Concurrently, a boundary must change its orientation as it tracks its corresponding pole^. For example, assume that the lithosphere is divided into three plates, A, B, and C with different rotation poles
Pab, Pbc, and Pac. If the transforms between A and B are parallel to small circles about Pab, and similarly with transforms between B and C about Pbc, then it is not possible for the transforms between plates A and C to be parallel to small circles. This is equivalent to saying that there is no pole Pac that remains fixed with respect to both plates A and C (Cox, 1973)*
In summary, these authors have pointed out that the motion of plates on a sphere requires that the pole of relative motion, and subsequently, the corresponding plate boundaries, must change orientation with time. Furthermore,
4 This point was used to explain why transform faults may not trend small circles through time (LePichon, 1973)* 41 motion on a sphere requires that triple junctions move with respect to the relative poles of rotation, possibly disturbing stability.
ABSOLUTE PLATE MOTION
A velocity space diagram that describes triple junction motion with respect to the three plate system contains a limited amount of instantaneous information about plate and triple junction velocities and rates of boundary growth and consumption.
With time, plate boundaries and triple junctions move with respect to external reference frames as well. Morgan
(1971) described 'hotspots' or convection plumes rooted in the mantle which are manifest at the surface as seamounts and volcanoes. Morgan (1971) proposed the use of hotspots as stationary bodies providing a reference frame from which lithospheric plate motion could be measured.
' An absolute reference frame is one that is external and independent of the three plate system and in which the poles of rotation are stationary. A hotspot reference frame, one that describes the motion of plates with respect to the mantle, is herein considered an absolute frame of reference. 42 Minster and Jordan (1978) developed an absolute^ plate motion model (their AMI-2) based on hotspot data younger than 10 my. Hotspot data older than 10 my BP may be erroneous due to the possibility of hotspot motion with respect to the mantle. The AM1-2 model of Minster and Jordan (1978) provides a means of measuring the velocity of any point on a plate network with respect to the deep mantle.
The absolute motion of triple junction has important consequences with regard to the associated velocity triangle. If a triple junction is not fixed with respect to the mantle, its relative velocity triangle will deform with time. The velocity of a point on a sphere is a function of the angle between the angular velocity vector of the plate and the radius vector to that point. Hence, as a triple junction changes latitude or longitude, the relative velocities change.
The notion of absolute triple junction velocities and accelerations was recognized by Hey (1977). Like Minster and
Jordan (1978), Hey et al, (1977) developed a relative and absolute plate motion model (PRM1 and PAM1 respectively) derived from hotspot data in the Pacific basin. Hey (1977) plotted a vector velocity triangle for the Galapagos triple junction (Pacific, Cocos, and Nazca plates), including its 43 velocity with respect to the hotspot reference frame (Hey
1977, Figure 2). The velocity of the Galapagos triple junction with respect to the mantle was important in this case because the triple junction was presumably created by the Galapagos hotspot. Hence, the evolution of the triple junction could be determined by noting its location with respect to the hotspot with time. Chapter VI GEOLOGIC APPLICATIONS
THE BOUVET TRIPLE JUNCTION Introduction
Having discussed the implications of the absolute motions of triple junctions on their stability, the notion will be applied to the Bouvet triple junction. As will become apparent, the absolute motion of the Bouvet junction has significant implications on its evolution.
The boundaries between the African, Antarctic, and South American plates intersect near latitude 55°S, longitude
0°E (Figure 21), at what has been termed the Bouvet triple junction (Forsyth, 1975)* Johnson et al (1975) on the basis of topographic data, postulated the existence of the Conrad fracture zone at 56°S, 4°W. Furthermore, they suggested that the triple junction was to the northeast of the Conrad fracture zone and that it was an unstable ridge-fault^fault (RFF) triple junction.
- 44 45
By matching synthetic and observed magnetic anomalies, Sclater et al, (1976) deduced that the relative velocity triangle around the Bouvet junction was, within observational error, isosceles (i.e., the relative velocities of the two transforms are equal in magnitude). If exactly isosceles, the junction is stable, in the sense of
McKenzie and Morgan (1969)> in the RFF configuration.
Sclater et al, (1976) also showed that the velocity triangle for the Bouvet triple junction is kinematically compatible with both RRR and RFF configurations, and that the junction has probably alternated between the two modes in the past 20 my.
In an attempt to understand the evolutionary history of the
Bouvet triple junction, Thede (1983) modeled the development of an obtuse RRR junction truncated by a transform fault in which paired transforms subsequently develop in RFF mode.
However, the cause for changes in the configuration of the junction is unclear.
This section discusses the implications of different absolute velocities of the Bouvet triple junction in RRR and RFF configurations. A model is presented here which demonstrates: 46
a) the observed transitions from RRR to RFF are a result
of the unique orientations of their absolute velocity vectors with respect to an "isosceles line".
b) the RFF configuration of the Bouvet, is considered to be geometrically more stable than RRR with respect to
the small changes in their velocity triangles which accompany their migration away from the isosceles
line.
The Absolute Motion of the Bouvet Triple Junction
The absolute velocity of each plate at a triple junction can be calculated in terms of a hotspot reference frame.
Given the angular velocity vector for each plate at a triple junction and the radius vector to the junction, the absolute velocity of each plate at the junction is determined by forming the cross product of these two vectors (w X r) (see Figure 5)* The direction of the velocity vector can be found by forming the dot product with vectors representing true north and due east. This yields the cosines of the angle between the two vectors and hence, the azimuth. 47
The evolution of the Bouvet triple junction is best understood in terns of its notion with respect to the mantle. Minster and Jordan's (1978) AM1-2 absolute poles of rotation for the African, South American, and Antarctic plates (Table 6) were used to calculate the velocity triangle for several locations in the vicinity of the present Bouvet triple junction. An "isosceles line", along which the velocity triangle is always isosceles, was calculated in the South Atlantic (Figure 22). Along the isosceles line, the velocities of the slower spreading ridges in RRR mode (transforms if in RFF mode) change direction but remain equal in magnitude.
The present Bouvet triple junction does not fall exactly on the calculated isosceles line. This discrepancy presumably reflects uncertainties in the absolute poles of rotation in the AK1-2 model since it is known that the relative velocities of the Bouvet junction form an isosceles triangle within observational error (Minster and Jordan, 1978). To maintain consistency with the AMI-2 velocities, all subsequent calculations assume the Bouvet triple junction is at 61 .8°S, 0°E, 6.8° south of its actual location. 48
Vith the absolute velocity triangle known, the absolute velocities of the triple junction in RRR and RPF modes are
readily calculated. Triple junction velocity vectors along
the isosceles line are shown in Figure 22. With respect to
the mantle, the model Bouvet triple junction in RRR mode has a velocity of 14.7 km/my with an azimuth of 343°• The
RFF configuration has a velocity of 5*83 km/yr with an azimuth of 157*1°« The orientation of these vectors
changes systematically along the isosceles line. These
absolute trajectories are very close to those calculated by
Sclater et al, (1976) relative to the mid-Atlantic ridge
(MAR). This reflects the fact that the MAR is nearly stationary with respect to the mantle in this vicinity. It
is clear from Figure 22 that the RRR and RFF mode triple
junction trajectories are nearly oppositely directed with respect to the isosceles line, an observation that has significant implications for the junction's evolution.
Bouvet Triple Junction Evolution
Figure 23 is a velocity space diagram for the Bouvet triple junction illustrating both RRR and RFF configurations. The vectors a-J2 and b-J2 are the rates of 49 growth relative to the triple junction, of the paired transforms that form in RFF mode. A person standing on the South American-Antarctic ridge sees the triple junction moving at half the spreading rate with an azimuth of 095°• These two vectors are equal in magnitude if the triangle is isosceles. In RRR mode, vectors a-JI and b-J1 represent the growth rates of the SOAM-ANTA, AFRC-ANTA ridges. Vector c-J1 is the rate that the SOAM-AFRC ridge is consumed.
Topographic data from the area of the triple junction suggest both the Conrad and the Bouvet fracture zones are about 150 km in length (Sclater et al, 1976). Assuming that the absolute velocities of the three plates has not changed significantly during the last 30 my, the rate of lengthening of both transforms would have been 8.1 km/my (vector a-J2, Figure 23) This would have required that the Bouvet junction was capable of remaining in the RFF mode for approximately
18.5 my.
Using only the lengths of the Bouvet and Conrad fracture zones, information derived from the AMI-2 model, and the present configuration of the plate boundaries in the vicinity of the Bouvet junction, the following history for the Bouvet junction from the time of the inception of the
Bouvet and Conrad fracture zones is deduced. 50
Some time "before 31 ny BP, the Bouvet junction existed south of its present location. Its absolute velocity carried it 14 km/my at 343° (Figure 24a). At 31 my BP the
junction intersected the isosceles line and changed to RFF
mode (Figure 24 b). The junction migrated in RFF mode with a velocity of 5*8 km/my at 157° away from the isosceles
line until about 12.5 my BP (18.5 my) when its increasingly unstable state required that it change back to RRR (Figure 24c). From 31 to 12.5 my BP, the ridge lengthened 120 km. and both the Conrad and Bouvet transforms grew to their present length of 150 km. The junction then migrated once again northwest (343°)» consuming the SOAK-AFRC ridge and lengthening the SOAM-ANTA, AFRC-ANTA ridges. At 5 my BP, the junction once again intersected the isosceles line (Figure 24d) and switched to RFF mode. For the past 5 my the
junction has evolved in RFF mode moving southeastward away from the isosceles line (Figure 24e). According to this interpretation, the present mode will continue another 13»5 my, barring any major plate reorganizations. Such reorganizations are proposed to have occurred in the vicinity of the Bouvet triple junction during the late
Cretaceous (Lawver, 1984)*
The above described sequence of events is similar to the history deduced by Sclater et al, (1976) from magnetic 51 isochron data. They suggest that the triple junction evolved from 20 to 10 my BP in RFF mode by increasing the lengths of the paired Bouvet and Conrad fracture zones. The plate geometry changed 10 my BP by either a triple junction jump or a change to RRR mode. At 5 my BP, the junction regained its RFF configuration and generated new paired transforms.
Minor discrepancies between our model and these observations probably reflect inaccuracies in the absolute motion model or indicate that the plate motions have changed slightly in the past 30 my.
The Question of Relative Stability
The RFF configuration of the Bouvet triple junction is stable, in the sense of McKenzie and Morgan (1969)» on the isosceles line. The RFF junction migrates away from the isosceles line and becomes immediately unstable. Yet the
Bouvet seems to have persisted for at least 18.5 my (31 "to 12.5 my BP) without reverting to RRR mode. Why? Furthermore, why does the RRR triple junction switch to RFF mode at the isosceles line? 52
The persistence of the RFF configuration for 18.5 my in an unstable state seems mechanically reasonable in view of
Figure 26. Figure 26 portrays the evolving velocity triangle, with respect to the mantle, as the RFF triple
junction migrates southeast away from the isosceles line. Assuming the fault boundary orientations remain the same,
the transforms become slightly "leaky" (i.e., there is a
small component of extension normal to the transform). Actual rotation of the fault boundaries amounts to less than
1 degree over a period of 20 my. resulting in about 1.0 to
1.5 km of normal displacement for a 150 km transform. The leakiness of the transforms alleviates some of the mechanical problems associated with an unstable RFF junction. On the other hand, if the sense of rotation generated "sticky" transforms, (i.e., a small component of
compression normal to the transform), it is unlikely the RFF mode could persist. When the triple junction finally changes to the RRR configuration, the newly generated slow spreading ridge segments must undergo rotation as the relative motion vector changes during northwest migration. The MAR must also undergo a change in the spreading direction (Figure 26). This rotation is possibly accommodated by forming secondary transform faults along the ridge segments. 53 A partial answer to the later question can he derived by by employing the least squares analysis developed in Chapter
IV. For the Bouvet triple junction, the stability for both RRR and RFF modes with respect to the change in the velocity triangle which accompanies motion away from the isosceles line are plotted in Figure 25* Constraining the boundary orientations to remain fixed and that spreading remain symmetric, the residuals increase in both RRR and RFF modes as the junction migrates. Interestingly, the data in Figure
25 demonstrate that the geometric instability of the RFF configuration builds up much more slowly than the RRR configuration. In the vicinity of the isosceles line (the origin in Figure 25)» the RFF configuration is more geometrically stable and may be mechanically the preferred mode. Thede (1983), suggested that an obtuse RRR junction would develop paired transforms in RFF mode as a result of intersection with a transform along the MAR. However, the extent of northwesterly triple junction migration has not been clearly documented.
In summary, the Bouvet triple junction is presently "trapped" along an "isosceles line" in the South Atlantic.
On this line, the junction can exist in the RRR or RFF configuration, although RFF is more stable. Since the absolute motion of the RFF junction moves it southeastward 54 away from the isosceles line, it eventually becomes mechanically unstable and switches to the RRR configuration. The RRR junction has an absolute motion which carries it back to the isosceles line where it reverts to the RFF configuration. Because of the orientations of the absolute triple junction trajectories relative to the isosceles line, the Bouvet junction will continue to alternate between RRR and RFF modes, until a major plate reorganization occurs. 55
TRIPLE JUNCTION EVOLUTION THE WESTERN UNITED STATES
Introduction In this section, the literature will be reviewed concerning triple junction stability in the western United States, primarily the Mendocino triple junction (Figure 27).
Recent publications qualitatively discuss the geometric and tectonic consequences of an unstable junction. In an effort to strengthen these proposals, an examination of the
Mendocino velocity triangle reveals the geometric possibilities of transient unstable triple junction evolution.
Previous Work
Modeling the development of western North America in terms of triple junction evolution began when McKenzie and
Morgan (1969) postulated that the Pacific ridge had collided with the North American trench, resulting in an incipient transform fault bounded by two triple junctions^. In Figure
28, note that as the character of the southernmost triple junction changed as a result of the interaction of the
Murray fracture zone and the San Andreas transform, the resultant plate motion and triple junction velocity changed
^ This general concept was initiated by Vine and Matthews (1963) from the ccymmetrical pattern of marine magnetic anomalies. 56 drastically (Figure 28c).
In a landmark paper, Atwater (1970) extended McKenzie and
Morgan’s idea by utilizing marine magnetic anomaly data to conclude that the inception of the San Andreas transform must have occurred not earlier than- 30 my BP. Furthermore, the age of the anomalies along the coast allowed her to date the extinction of the North American trench as it was replaced by the transform boundary. This dating method confirmed McKenzie and Morgan's hypothesis.
In a detailed study of the Mendocino triple junction,
Silver (1971) determined that the Gorda basin and escarpment had undergone extensive internal deformation and underthrusting. This deformation suggested that a local departure from the assumption of plate rigidity must have occurred. It also suggested that the Mendocino triple junction was geometrically unstable. Dickinson and Snyder
(1979) showed that this instability was caused when the Farallon plate broke up into the Juan de Fuca and Cocos plates. The accompanying change in the velocity triangle caused the relative motion vector between the Juan de Fuca and Pacific plates to deviate from that along the existing transform fault, resulting in underthrusting along the Gorda escarpment. 57
The concept of triple junction instability was developed by Dickinson and Snyder (1979) to explain the existence of
specific structural provinces. The Neogene migration of the Mendocino and Rivera triple junctions along the continental margin of North America had the tectonic affect of terminating subduction and initiating transform motion (Atwater, 1970). Providing the triple junctions could maintain stable configurations, no other tectonic effects would be expected as a result of their evolution. Dickinson and Snyder (1979) claimed that the two junctions could maintain stability only if the trench and transform remain co-linear (Figure 29a). Due to the irregular trend of the
North American plate boundary, the two junctions incurred transient unstable configurations (Figure 29b). Such instabilities were thought to induce extensional tectonics within a nearby region. The northward migration of the
Mendocino triple junction coincided with pulses of subsidence in Neogene sedimentary basins with local volcanic eruptions in the Coast Ranges of northern California. Passage of the Rivera triple junction southward was associated with the rifting events that formed the
California continental borderland and the Gulf of California (Dickinson and Snyder, 1979)* 58
An extension of the concept of junction instability was used to explain Cenozoic extension and fragmentation of the western United States (Ingersoll, 1983)* In the event that the trench-transform boundary becomes non co-linear, the triple junction becomes unstable (Figure 29U). Ingersoll (1983) claimed several processes allowed the unstable
junction to persist: 1 . clockwise rotation of the continental margin of the North American plate (Figure 30b).
2. If truly rigid plates exist, a "hole" would form at the triple junction to be filled with oceanic crust
by spreading (Figure 30a).
3* eastward transfer of the San Andreas shear zone,
transferring a portion of the Pacific plate to the
North American plate.
Velocity Space Analysis
The formation of a "hole" and subsequent back-arc extension proposed by Ingersoll (1983) can be tested by showing that they are a possible means of achieving geometric stability.
In the following analysis, the Pacific, Juan de Fuca, and the North American plates are referred to as A, B, and C 59 respectively. As the BC (trench) boundary in Figure 31 rotates clockwise fron its co-linear position with respect to AC (San Andreas transform), several readjustments are possible. They are denoted as a,b, anc c in response to points 1,2, and 3 above: a) (Point 1) The trench boundary EC no longer remains fixed to plate C, but moves with the triple junction fixed with respect to plate A. This requires the trench to rotate clockwise in the vicinity of the
triple junction (Figure 31c). Back-arc extension is possible as long as the trench is fixed to plate A.
Alternatively, accretion of plate B (Farallon plate)
to the trench would accommodate its rotation. Fixing the trench to plate A also causes the crustal rotation of plate C, as proposed by Ingersoll (1982).
b) (Point 2) At a special orientation of the trench BC, an FT+R triple junction may resume. In this case,
spreading occurs between plates A and C. This spreading causes the triple junction to migrate along the trench boundary (Figure 31d). Note that a small change in orientation of the trench boundary will render the FT+R triple junction unstable, perhaps
switching back to the FFT+ junction in (a). 60
c) (Point 3) Readjustment of the transform boundary AC will also regain a stable velocity triangle. In this
case the trench can still remain fixed to plate C.
The AC transform must migrate inland parallel to itself until the stable configuration is reached
(Figure 31 f) Slivers of plate C are transferred to plate A. Readjustment of the AC transform alone will regain junction stability.
Figure 31 e schematically depicts an episode of back-arc
spreading followed by spreading. An evolution combining both back-arc spreading and inward stepping of the AC
transform is illustrated in Figure 31g*
The inferences made by Ingersoll (1982) and Dickinson and
Snyder (1979) are feasible, as shown by velocity space
analysis. Back-arc spreading, accretion, the development of spreading centers, and eastward stepping of the active San
Andreas shear zone may have occurred in the Keogene as a
result of triple junction instability.
By examining velocity space diagrams for unstable triple
junctions, several geometric possibilities can be
enumerated. Knowledge of the actual geologic history in the wake of such junctions may shed light on which alternative configurations are the most mechanically feasible. Chapter VII CONCLUSIONS
McKenzie and Morgan's (1969) concept of triple junction stability and evolution is a landmark paper in plate geometry and kinematics. Nonetheless, several authors have noted inconsistencies in McKenzie and Morgan's treatment of triple junction geometry (York, 1973; York, 1975; Eguchi,
1979)* In addition, stability lias been shown to be a time dependent concept; a triple junction that is stable at one moment is not guaranteed continuous stability. Plate motion on a spherical earth, and changes in boundary orientations may render a junction unstable (McKenzie and Parker, 1973;
Dewey, 1975; LePichon, 1973; Dickinson and Snyder, 1978;
Ingersoll, 1983)» Furthermore, many triple junctions may be kinematically compatible with a single velocity triangle, and switch configurations (i.e., the Bouvet triple junction)(Sclater et al, 1978).
The aforementioned problems in triple junction kinematics have been addressed in three parts: 1. quantification of stability algebraically,
- 61 62
2. the stability of triple functions as a function of
time, 3. applications to the Bouvet and Mendocino triple junctions.
In Part 1 (Chapter IV), the degree of instability of a
triple junction has been determined algebraically by solving
three equations simultaneously. Each equation, of the form J
= a + bh, represents the location of the triple junction in velocity space with respect to each of the three boundaries.
If the triple junction is stable, an analytic solution is possible. If unstable, a least squares estimate determines the residuals to a best fit triple junction, which are a measure of the degree of instability. If several configurations are consistent with one velocity triangle,
the configuration which is least susceptible to instability
can be determined.
In Part 2 (Chapter V), the notion of constraints is
introduced to measure a triple junction's requirements on relative velocities and the location of the junction in velocity space. Perturbation degrees of freedom are determined to measure of the sensitivity of a junction to imparted changes in relative velocities and boundary orientations. The number of constraints and degrees of 63
freedom has been enumerated for each type of triple junction. The nomenclature has been simplified by considering four types of plate boundaries: ridges (R),
transform faults (P) and trenches of opposite polarity (T+, T”). Of the 16 types of junctions introduced by
McKenzie and Morgan (1969)> six have special cases which have different constraints and degrees of freedom, such that
21 varieties exist.
In Part 3 (Chapter VI), the affects of motion with
respect to the mantle has been recognized as important in
the evolution of the Bouvet triple junction. If the velocity triangle is isosceles, both RRR and RPF modes are feasible. However, when the junction is in RPF mode, its velocity causes it to migrate southeastward, causing the velocity triangle to deform. At the point of mechanical
instability, the junction switches to RRR mode because the velocity triangle is no longer isosceles. The RPF junction can evolve 18.5 niy in an increasingly unstable state because the transforms develop a component of extension normal to fault surface at the triple junction.
The unstable Mendocino triple junction in northern California has been evaluated to determine the tectonic effects of instability. The idea that a 'hole' forms, at 64 the triple junction, filled with new seafloor, and that rotation of the North American plate has occurred (Dickinson and Snyder 1978, Ingersoll 1983), is tested by a velocity space analysis. It has been determined that back-arc spreading, trench accretion, the development of spreading centers, and eastward stepping of the active San Andreas shear zone are all geometrically feasible.
Future work in triple junction stability and evolution might focus on applying the tools presented in this thesis to other triple junctions, particularly using a mantle reference frame. Secondly, the connection between triple junction geometry and mechanics is still unclear. The kinematic observations of the Bouvet triple junction with respect to mantle alludes to its mechanical behavior. These quantities include: the changes in the relative motion vectors with time, the terminal lengths of the transforms in RFF mode, and the distance traveled in each mode. These parameters may allow the quantification of the mechanical stability of the triple junction. In any event, the forces at plate boundaries must play a significant role in junction stability and evolution. BIBLIOGRAPHY
Apotria T.G., and Gray N.H., 1985» Evolution of the Bouvet triple junction; the implications of its absolute motion, EOS trans. Amer.Geophys. Union, in press. Apotria T.G., and Gray, N.H., 1985f The affects of absolute motion on the evolution of the Bouvet triple junction, Nature, in press.
Beer,F.P., and Johnston,E.R., 1977, Vector Kechanics for Engineers: Statics and Dynamics, 957 PP*» McGraw Hill Cox,A., 1973 Plate Tectonics and Geomagnetic Reversals. 700pp., W.H. Freeman and Co., San Francisco.
Dewey,J.F., 1975 Finite plate implications: some implications for the' evolution of rock masses at plate margins American J. Sci., 275-A, 260-284•
Dickinson,V.R., Snyder ,V.T.S., 1979 Geometry of triple junctions related to the San Andreas transform JGR, 84, B2, 561-572. Eguchi,T., Katsubara,Y., Seno,T., 1979 Stable condition for the TTT(b) type triple junction Japanese J. Seismology, 32, 191-195.' Grant,N.K.,1972, South Atlantic, Benue Trough, and Gulf of Guinea Cretaceous triple junction, G.S.A7 Bull, v 82, 2295-2298.
Gray,N.H., and Apotria,T.G., 1983> The stability and evolution of triple junctions, G.S.A7 Abstr. with programs, 15> no. 6. Hayes,D.E., and Ewing,M., 1970, North Brazillian ridge and adjacent continental margin, A.A.P.G. Bull., 54, 2120-2150. Hey,R., Milholland,P., 1979 Stability of quadruple junctions Nature, 277, 201-202.
- 65 - 66
Hey, R., 1977 Tectonic evolution of the Cocos-Nazca spreading center, G.S.A. Bull., v 88, p. 1404-1420 Hey,R.N., Deffeyes,K.S.-, Johnson,G.L., Lowrie,A., 1972 The Galapagos triple junction and plate notions in the east RacificTTature, 237, 20-22.
Hobbs,B., Means,V., and Williams,P., 1976 An Outline of Structural Geology 571 pp., John Wiley. Ingersoll,R.V., 1982 Triple junction instability as a cause for late Cenozoic extension and fragmentation oT The western United States Geology, 1 621 -624 •
Johnson,G.L., Hey,R.N., Lowrie,A., 1973 Marine geology in the environs of Bouvet island and the south Atlantic triple junction Marine Geophysical Res., 2, 23-36.
Lawson, C.L., and Hanson, R.J., 1974 Solving Least Squares Problems, 340 pp., Prentice Hall
Lawver, L., 1984, The Cenozoic evolution of the Bouvet triple junction, E.O.S., v. 65, no. 45, p•11OTT LePichon,X., Francheteau,J., Bonnin,J., 1973 Plate Tectonics 300pp., Elsevier, New York.
Patriat, P., and Courtillot, V., 1984, On the stability of triple junctions and its relation to episodicity in spreading, Tectonics, v. 3» no. 3>—517-332. McKenzie,D.P., Parker,R.L., 1967 The North Atlantic: an example of tectonics on a sphere Nature, 216, T276-1280.
------, Morgan,W.J., 1969 Evolution of triple junctions Nature, 224, 124-133* Morgan, W.J., 1971, Convection plumes in the lower mantle Nature, 230, 41-4T^
Nash, J.C., 1979» Compact Numerical Methods for Computers: Linear Algebra and Function Minimization John Wiley, 224 pp.
Sclater,J.G., Fisher,R.L., Patriat,P., Tapscott,C., Parsons,B., 1981 Eocene to recent development of the South-west Indian ridge, a consequence of the evolution of the Indian Ocean triple junction GeopHys. J.K. astr. TJoc., 64, 587-604* 67
Sclater,J.G., Bowin,C., Hey,R.t Hoskins,H., Pierce,J., Phillips,J., and Tapscott,C., 1976 The Bouvet triple junction J.G.R., 81, 1857-1869* Searle,R., 1980 Tectonic pattern of the Azores spreading center and triple junction Ear. Plan. Sci. Letters, *>1 , 415-434"------
Tapscott,C.R., Patriat,P., Fisher,R.L, Sclater,J.G., Hoskins,H., Parsons,B., 1980 The Indian Ocean triple junction J.G.R., 85* B9, 4723-4739»
Thede,J.C., 1983* Paired Transforms, M.S. Thesis, Stanford University.
Turcotte, D.L., 1983 Mechanisms of crustal deformation, J. geol Soc. London, 1707 701-724"------Vine,F.J., Matthews,D.H., 1963 Magnetic anomalies over ocean ridges Nature, 199* 917-949*
York,D., 1973 Evolution of triple junctions Nature, 224, 341-342. ------, 1975 Evolution of triple junctions Can. J. Earth Sci., 12, 516-519* 68
Figure 1: A stable configuration for the RRF triple junction.
(A) Example of a stable RRF triple junction with the corresponding velocity triangle (B). J is the location of the junction in velocity space (After York, 1973) 69
a B
b
'••B Figure 2: An unstable RRR triple junction.
(A) An RRR triple junction where all ridge boundaries encompass 180 degrees. (B) An attempt to construct the velocity triangle is futile because no unique point C can be found to express the velocity of plate C with respect to both A and B (After York, 1975). 71
b
A B 72
Figure 3: A stable TTT(b) triple junction.
(A) A stable case for the TTT(b) type triple junction and (B) the corresponding velocity diagram (After Eguchi et al, 1979). 73 Figure 4: A geometrically stable quadruple junction.
(A) Example of a geometrically stable quadruple junction and (B) its corresponding velocity diagram. J is the location of the triple junction (After Hey and Mulhollandf 1979). 75
A B
C D
b
J ac A C D B
ad, cd 76
Figure 5: Instantaneous displacement and angular velocity.
The application of Euler's theorem to the instantaneous displacement (w) of a plate and the instantaneous tangential velocity (v). c- r- Figure 6: The vector circuit around a triple junction.
The vector circuit around a triple junction from which the relation: aVb + bVc + cVa = 0 follows. 79 Figure 7: The possible types of plate boundaries.
Diagram representing all possible types of plate boundaries. aVb is a particle velocity on plate B with respect to A, i is a unit vector tangent to the earth's surface in the direction of a fixed plate B. Ridgt I I
Plate B
Sinistra! __ Dn»»l •ansform >tstorm
I O • *
Trench Figure 8: Nomenclature and labeling scheme.
The importance of a labeling scheme and correct nomenclature in describing triple junction topology. Note that a label in velocity space corresponds to more than one direct space labeling. ac
FT+R 84
Figure 9; An FRT triple junction defining the stable condition.
An FRT triple junction in (A) direct and (B) velocity vector space. The junction is stable as denoted by the intersection (J) of ab,bcf and ca. 85 86
Figure 10: Possible velocity diagrams for the RRR junction.
Parameters determined vhen an acute (A), an obtuse (E), and a right triangle (C) velocity vector diagram is constructed for the RRR junction. xJ, yJ, and zJ represent the lengthening or shortening of a particular boundary (see text). 87
b
bb c Figure 11: Direct space representation of Figure 10.
The acute, obtuse, and right angle RRR configurations in direct space showing ridge lengthening and consumption. Arrows show the direction of ridge growth with respect to the triple junction. 89
a
b
c 90
Figure 12: Four possible rff direct space configurations.
Four direct space configurations are consistent with a single velocity space diagram for the RFiF triple junction (see text). 91 92
Figure 13: Development of vectors describing the location of (J).
Direct space and velocity space diagram for the FT+R triple junction. Vectors representing known quantities can be constructed from a reference frame to delineate the location of the triple junction (J). The linear equations are of the form J = a + bh (see text). 93
Jac=2 + h, *ac ^5^;= OC ~ h2 *bc Jatj=2 ^A+OB) +h3*ab 94
Figure 14 Sum of square of the residuals.
The "best fit solution of the location of the triple junction minimizes . the sum of the square of the residuals (r1+r2+r3) • 95 96
Figure 15; The effects of boundary rotation on stability.
The residuals measure the degree of geometric instability upon rotation of the trench boundary for a FFT+ junction. The inset illustrates the angle theta between the trench boundary and 0°. 97
Mendocino (FFT+) Triple Junction Effects on Stability of Trench Rotation
Theta BC Figure 16: The effects of triangle distortion on stability.
An isosceles right triangle was plotted in which RRR, RFF, and RRF configurations were all initially stable. In 16a, the relative velocity of plate B varies along the x-direction (as shown), forcing RFF to go unstable. In 16b, the relative velocity of plate B varies along the y-direction (as shown), rendering RRR and RFF unstable (see text). 99
Relative Geometric Stability Effects of a Distorted Velocity Triangle
RFF & RRR
o .003*
.002*
.001- 100
b Relative Geometric Stability Effects of a Distorted Velocity Triangle
-B- RRF
By 101
Figure 17: Constraints on the relative velocity.
Constraints on the relative velocity of B if A and C are fixed. The velocity is: i) nowhere confined (B), ii) confined to a line (B'), and iii) confined to a point (B") corresponding to 2, 1, and 0 degrees of freedom, respectively. 102 103
Figure 18: A description of perturbation stability.
Perturbation stability for the T+T-R triple junction. (A) B has two degrees of freedom in relative velocities, it may change along line ac or the size of the triangle may change. (B) B has two degrees of freedom with respect to boundary orientations, ab and be may independently change their orientations (see text). 104 Figure 19: Stability analysis of the RFF junction
A stable example of the RFF junction (a); the velocity of plate B (b) and the velocity of the triple junction (c) are constrained to lie on the perpendicular bisector of AC. The triangle must be isosceles. For the T-type variety of the RFF junction (d), the velocity of plate B must lie along line AC (e), and the triple junction must lie at the midpoint of AC (f).
107
Figure 20: Stability analysis of the RRF and RRR junctions
A stable example of the RRF junction (a); the velocity of plate B is constrained to lie on the semi-circle (b); the triple junction must lie on the midpoint of the fault boundary AC (c). The triangle must be a right triangle. For the RRR triple junction (d), the velocity of plate B (e) and of the triple junction are not constrained and can lie anywhere in the region above AC. RRF RRR o
A ' E
A c
F
A c 108 109
Figure 21: The location of the Bouvet triple junction.
The location and orientation of the Bouvet triple junction in the South Atlantic (from Sclater et al, 1976). Map includes the interpretation of magnetic anomalies. o Figure 22: The position and orientation of the isosceles line.
The isosceles line in the South Atlantic calculated from Minster and Jordan's (1978) AM1-2 absolute motion model. Several velocity triangles and triple junction trajectories have been plotted in the vicinity. The velocity of the triple junction in RRR and RFF modes is 14.7 km/my at 343° and 5»8 km/my at 157°* respectively. In RFF mode, the junction migrates southeast away from the isosceles line becoming increasingly unstable as the velocity triangle is distorted. 5.8 (157 Model Bouvet TJ Isosceles Line
Vector Scale
20 knymy 112 113
Figure 23s Velocity space representation of the Bouvet junction.
A velocity space diagram for the Bouvet triple junction with both RRR and RFF modes superimposed. In RRR mode vector aJI and bJ1 are the growth rates of the SOAM-ANTA, AFRC-ANTA ridges and vector cJI is the rate that the SOAM-AFRC ridge is consumed. In RFF mode, vectors aJ2 and bJ2 are the rates of growth for the two respective transforms, and vector cJ2 is the rate of ridge growth, m is the location of the mantle reference frame. Vectors mJI and nJ2 are the velocities of the triple junctions in RRR and RFF modes, respectively. The two insets illustrate the direct space configurations. 114
J| RRR
ANTA 115
Figure 24: A model of evolution for the Bouvet triple function.
A model of evolution for the Bouvet triple junction "based on the AMI-2 model of Minster and Jordan (1978) and the length of the Bouvet and Conrad fracture zones. At the isosceles line (a) the junction switches from RRR to a more geometrically stable RFF mode. As it migrates southeast (b) the junction generated the Conrad and Bouvet transforms and extended the length of the MAR. In (c), the transforms reached their terminal length of 150 km and the junction switched to RRR mode and began propagating northwest. The cycle begins again in (d) generating new paired transforms. 31 my BP
a 20
12.5
150 km 117
Figure 25: The evolving velocity triangle for the RFF junction.
A diagram of the velocity triangle for the Bouvet triple junction at the isosceles line (0 my) and after 20 my of southeast migration. The transforms become leaky as the junction migrates away from the isosceles line. This allows the junction to migrate 110 km in a geometrically unstable state from 31 to 12.5 my BP. Note that the KAR relative velocity vector also rotates with time. 118
Bouvet Velocity Triangle Distortion due to Southeast Migration
AFRC
SOAM 274.5(16.2) ANTA
0 my 20 my Figure 26: Relative geometric stability.
The degree of geometric instability (residuals) as a function of time as the RRR and RFF configuration of the Bouvet triple junction drifts away from the isosceles line. The orientations of the boundaries are assumed fixed, and spreading must remain symmetrical. Instability arises because the relative velocities change with time. The stability of the RFF configuration is least affected by small changes in the velocity triangle and is the preferred configuration in the vicinity of the isosceles line. 120
RRR-RFF GEOMETRIC STABILITY Bouvet Triple Junction
RRR RF
.0004-
Time (my) 121
Figure 27: Location and tectonic map of the western United States.
Diagrammatic map shoving the present plate configurations related to the current San Andreas transform fault in the western U.S. (from Dickinson, 1979» Figure 1).
Figure 28: The geometric evolution of the NE Pacific.
(A) The geometry of the NE Pacific at about the time of anomaly 13* (B) A anomaly 9> the meeting of the East Pacific Rise and the North American trench initiate two stable triple junctions. Double headed arrows show velocity with respect to the North American plate (note corresponding velocity diagrams). (C) Later, when the Murray fracture zone collides with the transform, the nature of the triple junction (2) changes and becomes fixed relative to plate C (After McKenzie and Morgan, 1969)* a b c
Juan de Fuca Plate Farallon Plate
MENDOCINOFZ. N. American I Plate Racilic Plate
U MURRAY F.Z. 124 125
Figure 29s Stable and unstable FFT configurations.
(A) A stable FFT triple Junction with co-linear BC-AC boundaries. (B) An unstable configuration occurs when BC-AC non co-linearity exists. a 127
Figure 30: Cumulative deformation at an unstable FFT junction.
(A) As a result of the unstable configuration in Fig.6b, regional spreading or extension may be induced. (B) Postulated cumulative deformation as the San Andreas transform steps inland (eastward relative to C) and the continental margin of C is rotated clockwise with extension of plate C as shown in (A) (After Ingersoll, 1982). 128 129
Figure 31: Tectonic consequences of the unstable Mendocino junction.
Non co-linearity of the trench and San Andreas transform induce periods of instability. In an effort to regain stability, back-arc spreading, accretion, ridge spreading, and eastward stepping of the San Andreas are all geometrically feasible (see text). 130 131
d 132
TABLE 1
SUbi) ity of Tr pi* Junction*
Jwcli#*. Tm Ittbiiitr Cwitniili ffMprbllld* - ItBffU *f Frrrdoi fcfrill Wflocitr Sfict Ufetlliig) Vilocitr iMUdjr/ Milk turd •iU Find PtrUrbltin CwiigoritiM CafipritiM r»liti*« Hltciliii bcaidir; trintiliwn Itkbilitr Otai ■ 2) Olu ■ 3> tb be ct i> rni i) 1 3 1 1 1 2 2 7. IF ii) 1 7 1 1 1 1 2
I) F 7.1 1 3 1 1 1 2 2 F7.I
J) 747_l 1 1 1 1 1 2 4
t) 7_7+l 1 2 ( I 3 t 3 4
5) 74J_F i) 2 ( 1 ) 1 3 4 ii) 1 1 1 1 1 2 4
4) 7.7+F 1 2 1 1 1 ) 4
7) 7+7,F i) 1 2 1 1 1 2 3 7_ 7_ F ii) 1 2 1 1 1 3 4
1) 7,7,7. i) 1 1 1 1 1 ) 3 7. 7. 7. ii) 1 1 1 ( 1 ) 4 4
f) 7,7,1 1 2 t 1 ) 1 2 3 7.7.1
II) F F 7, i) 1 3 1 1 1 2 2 FF7l ii) 1 2 1 1 1 2 1
II) FFI 1) 1 3 1 1 1 1 1 ii) 1 3 1 1 1 2 7
17) 7.11 1/2 3 1 1 • 1 1 7.11
13) 1 t F 1 3 1 1 1 1 I H) 7 7 7 1 1 ( 2 ) 1 s L \ \ 13) III 1 3 1 1 1 1 1
W FFF ■KCNditiMilly Htltbli 133
TABLE 2
Stability ii Perturbation* Involving Oblique Symetrical Spreading at Ridge* Pottible
finctiw Tn» tUtilitr CwitreiMt lerterbetiw - frreti * frerell (Vilecitx S*ict Ubtllitg) Velocitr touter? ■itl tiled •itb filed fertr-betiw Ceaiigerttiea Ceeiigvritien relttiee eelecitiei bovadary orieilitiMl Stability Olai ■ 2) Olu ■ 3) ab be (i 1) 7 *F i) 1 2 1 1 ) 1 3 4 T4f » ii) 1 2 1 1 1 2 3
l) 1 2 1 < I ) 2 3
1) T+l. 1 1 1 1 I 1 3 3
«> T.I,* 1 1 ( 2 ) 3 3
f) T. T. 1 1 1 { 2 1 3 3
111 FFI i) 1 3 1 1 1 1 1 ii) 1 2 1 t 1 2 3
Ii) 7.11 1 1 ( 2 ) 3 3 7_ 11
13) IIF 1 2 ( I ) 1 2 3
13) III 1 1 ( 2 ) 3 3 134
TABLE 3 Topologically Distinct V*ri»ti»* ot TripU Junctions
Junction Trot Junction Topology (Dirsct Spice Labelling) (Ignoring hirror Images) Y f I-
t 4 t ** - « - - r r r (0,1) t • s t _ t (2,3) (2,3)
t 4 4 • • • 4 • t 4 4 (1,2) (0,2)
4 r r - 4 (0,1)
*♦ r r - « _ (0,1)
r 4 4 4 - 4 4 (0,1) (0,1) (0,2) • - ft « (1,2) (1,2) (1,3) r t^ t+ • * • r t_ t_ (1,2) (1,2) (1,2) • « • ft U »- <- (2,3) (2,3) (2,3) (2,3) «- U or (2,4) TABLE 3
Junction Type Junction Topolooy > \ / \ ••••Iv \ I K | t r 1 1 1 ' 1 1 1 f <- Va - - - - < f 4 4 4 « < (2,2) (2,2) (2,2) (2,2) (1,3) r V 1 4 4 4 4 « r f t_ <1,1) d,i) 0,1) (i.i) (0,2) r U 4 4 4 4 4 ( (2,2) (2,2) (2,2) (2,2) (2,2) r t_ f • t • * r 1 *♦ (0,2) (0,2) (0,2) (0,2) r «- u « • • * • < (1,3) (1,3) (1,3) (1,3) (1,3) Onconditiontlly Untttbl* “ - Pottntially Etablr with « - 1 constraint on ralativa valocitias. • - 0 constraints on ralativa valocitias. • Nota - Tha dtgraas of fraadon of T-junction typas ara givan with raspact to tha junction ranaining stabla not nacassarily rmaining T. 136 TABLE 4 Topological!)' Distinct Vtrittin ol 6t»bl» T-Junction* Jandiw Tm I Cytttl-'tlt Dumtet ot frettfea wiU »t*Mcl (birtcl Space L*btlli*g) B" IP PtrUrbUton* of the Tnt WlocitlM . - U O It ii) - a o - « t4) It i> I - <1 t > - (I t) II i) "T" t t t II ii> t t t - U o 7 ii) I ♦ - «♦ <> 8 i) / I 4 - V I I 1 - <»_ t4> 8 ii) « | 1 I 8 «■« i»»*^-- - I - «4 l_) S i) 8 5 ii) 2 - Jill ill■•••"* uu * <<4 «_> 2 137 Table 5 Existing Junctions Junction Type Ovsrall (Dirsct Bpacs) Psrturbatlon Stability (obllqus sprsadlng) Kuls - Farsi11on - Pacific rrr s North ftssrlca - Kula - Faralllon rt*t- s African - Arabian - Boaallan rrr s JDF - Pacific - North Assrlca rft-(?) 4 Pacific - JDF - North Assrlca t.ff 3 South Assrlca - Africa - Antarctic rff 3 Africa - India - Antarctic rrr s Pacific - Cocos - Nazes rrr s North Assrlca - Eurasian - Africa rfr 3 Pacific ~ Nazea - Antarctic rff 3 Naica - South Assrlca - Antarctic rt*t- s Cocos -South Assrlca - Nazea ft.t- 4 Asia - Pacific - Phllllplnss t.t.t- S Indian — Pacific - Antarctic rff 3 TABLE 6. Model AMI-2 Absolute Rotation Vector Error EllIp se 9 (1) 0 ^max o e °e u max °ain Plate •N deg *E deg deg/m.y. deg/m.y. deg deg deg AFKC 18.76 33.93 -21.76 42.20 0.139 0.055 S73*E 40.40 33.24 ANTA 21.85 91.81 75.55 63.20 0.054 0.091 N12*E 93.01 56.12 AKAB 27.29 12.40 -3.94 18.22 0.388 0.067 S76*E 16.38 12.11 CASB -42.80 39.20 66.75 40.98 0.129 0.104 N30*E 43.21 23.90 COCO 21.89 3.08 -115.71 2.81 1.422 0.119 S32*E 3.35 2.25 EOXA 0.70 124.35 -23.19 146.67 0.038 0.057 S67*E 151.10 118.90 XRDI 19.23 6.96 35.64 6.57 0.716 0.076 S25*E 7.16 5.97 RAZC 47.99 9.36 -93.81- 8.14 0.585 0.097 S02*E 9.37 5.43 ROAM -58.31 16.21 -40.67 39.62 0.247 0.080 S57*E 23.12 12.14 rcrc -61.66 5.11 97.19 7.71 0.967 0.085 SWB 5.23 3.50 SOAM -82.28 19.27 75.67 85.88 0.285 0.084 «03*E 19.28 11.38 from Minster end Jordan (1970? Appendix A SINGULAR VALUE DECOMPOSITION The following discussion relies heavily on the treatment by Nash (1979, pg. 16-40). Singular value decomposition is a type of orthogonal transformation that allows the expression of the problem of minimizing the sum of r in the matrix equation: (1 ) Ax - b = r to be solved in the following manner. Find an orthogonal matrix V, n by n, which transforms the real matrix A into another real m by n matrix B whose columns are orthogonal. That is, find a matrix V such that (2) B = AV = (bl, b2, --- bn) where B.T b. = S.2 d. . i 3 i 1*3 and V^V = 1n - 139 - 140 The Kronecker delta takes on the values 0 for i * j d. . = 1,3 1 for i = j The quantities nay be positive or negative since only their square is defined. They are the singular values of the matrix A. The vectors u.J = b.J / S.(S.J J > 0) which are computed if none of the singular values are zero, are the unit orthogonal vectors. Collecting these vectors into a real m by n matrix and the singular values into a diagonal n by n matrix, it is possible to write (3) B = US where uTu = in A combination of (2) and (3) gives (4) AV = US or using the orthogonality of V (5) A = USVT which expresses the singular value decomposition of A. 141 The matrix V sought to accomplish the orthogonalization (2) is built up as the product of simpler matrices (6) V = V1 V2...Vk The matrices used in this product will be plane rotations, v If V is a rotation angle 0 in the ij plane, then all V elements of V * will be the same as those in a unit matrix of order n except for (7) cos 0 k k sin 6 v Thus V affects only two vectors of any matrix it multiplies from the right. These columns are labeled x and y* The effect of a single rotation involving these two columns is: * cos e -sin 0 (8) (x.y) (X.Y) k sin G cos © ✓ If the resulting vectors X and Y are to be orthogonal then (9) XT Y = 0 = -(x^x - yTy) sin 8 cos G + x^y (cos2 0 - sin2 0) 142 The specific formulae for the sine and cosine of the angle of rotation are given in terms of the quantities: P = x Ty q =xTx - yTy v = (4p2 + q2)0*5 They are cos 9 = ((v + q) / (2v))°**5 sin 0 = p / (v cos 0) for q 0 and sin 0 = sgn (p) ((v - q) / I2v))0*5 cos 8 = p / (v sin 0) for q < 0 where 1 for p ^ 0 sgn (p) = -1 for p < 0 Hence the rotation matrix has been specified which will orthogonalize V such that B = AV and B = US and condition (5) is true. 143 Nov that the singular value decomposition has been performed on matrix A, the problem of least squares is solved via (10) x = V S+ UT b where S+ is the generalised inverse of S where ’I / s,i. i for S.^ f 1. > q 1.1 for S. . 4: q 1.1 H where q is a user set tolerance. The sum of squares is defined as r^r = (b - Ax)'*' (b - Ax) rTr = bTb - bT USS+UTb Appendix B SUMMARY OP CONSTRAINT AND PERTURBATION STABILITY ANALYSIS The following diagrams illustrate the number of constraints and perturbation degrees of freedom of each of the 21 varieties of the 16 types of triple junctions. These diagrams follow the same analysis as described in chapter IV. 144 - A given B .. . 145 146 147 V < £ 148 T+T_F /* / A c "r A c A C A c A -r*> vo TTT A given 8 HTT A given B XRR FFT^ n 1/ . hS A C z7 X.\ t ti 4^1 A ic A C Ba A C A C VJl ro $ FFR d FFR «.) 1 B , ^.bt A C A c A c RRF RRR 8 A ' A c A c 155 xT e Li. Li.