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STRESS MODELING OF THE NAZCA PLATE:
ADVANCES IN MODELING
RIDGE-PUSH AND SLAB-PULL FORCES
by
Billie Lea Cox
A Thesis Submitted to the Faculty of the
DEPARTMENT OF GEOSCIENCES
In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE
In the Graduate College
THE UNIVERSITY OF ARIZONA
STATEMENT BY AUTHOR
This thesis has been
fillment of requirements for
University of Arizona and is
Library to be made available
of the Library„
submitted in partial ful­
an advanced degree at The
deposited in the University
to borrowers under rules .
Brief quotations from this thesis are allowable
without special permission, provided that accurate
acknowledgment of source is made. Requests for permis­
sion for extended quotation from or reproduction of this
manuscript in whole or in part may be granted by the
head of the major department or the Dean of the Graduate
College when in his judgment the proposed use of the mate­
rial is in the interests of scholarship. In all other
instances, however, permission must be obtained from the
author.
SIGNED:
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
R. M. RICHARDSON
Assistant Professor of Geosciences
'
Date
ACKNOWLEDGMENTS
I am very grateful to my thesis advisor, Professor
Randall M. Richardson, for suggesting this thesis topic,
convincing me that finite elements could be understandable
given time and effort, and for patiently guiding my pro­
gress, supporting my work with a research assistantship
and computer time.
I want to thank Professors Spencer R. Titley and
William R. Dickinson for reading my thesis, and for stimu­
lating my interest in plate tectonics through course
lectures.
The Department of Geosciences supported me with
teaching assistantships in mineralogy, and I also received
a research assistantship from Dr. Dennis K. Bird. Further
Support for my graduate studies came from Amoco and Conoco
scholarships awarded by the Geophysics Committee, and from
earnings from summer employment with GeothermEx, Inc., and
Sohio Petroleum Company. I appreciate all of this support.
I want to express a special thanks to Murray C.
Gardner for encouraging me to pursue graduate studies at
the University of Arizona, and for sending flowers at the
right times.
And finally, I want to thank my mother, Marcilie
G. Cox, for providing a coastal haven from desert heat
and allergies, and for the moral support I needed to see
my way through to the completion of this thesis.
iii
TABLE OF CONTENTS
Page
TABLE OF CONTENTS .........................
iv
LIST OF ILLUSTRATIONS ................. . .
vi
LIST OF TABLES
ix
.........................
.
x
ABSTRACT
I.
INTRODUCTION
. . ; ............
II.
DESCRIPTION OF THE NAZCA PLATE
.1
6
East Pacific Rise Ridge Boundary . . . . . .
Galapagos Spreading Center Ridge Boundary .
Chile Ridge Spreading Center and
Transforms . . . .
. . . . .
South American Trench Boundary ........ .
Intraplate Features . . . . . . . . . . . .
11
13
STRESS DATA . . . .
IV.
DESCRIPTION OF MODELING METHOD
Elastic Modeling
Viscous Modeling
. . ....
...
. .
III.
V.
....
» . . . . . . . . ...
. .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
MODELING PLATE-DRIVING FORCES
Ridge-Push Forces .........................
Distributed Ridge Forces Applied to
the Nazca Plate . . . . . . . . . .
Line Ridge Forces Applied to the
Nazca Plate ........ . . . . . . .
Slab-Pull Forces . . . . . . . . . . . . .
Drag Forces . . . . . . . . . . . . . . . . .
Transform Fault Shear Forces . . . . . . .
Hot Spot Forces . . . . . . . . . . . . . .
Distant Forces From Other Plates . . . . .
iv
15
16
20
24
30
31
38
40
40
48
48
51
57
63
63
64
V
TABLE OF CONTENTS— Continued
Page
VI.
STRESS MODELS . .
. . . ...
. . . . . . . .
Distributed Ridge Models . .. .. . . . .
DR Models . . . . . . . . . . . . .
. .
DRD Models . . . . . . .
. .. . . . .
Refined Grid Models . . . . . . . . . .
Line Ridge Models . . . . . . . . . . . . .
LR Models . .. . . . . . . .. . . . . ■
LRD Models . . . . . . . . . . . . . .
Slab-Pull Models
. . . . . . . .. . . . .
SD Models . . . . . . . . . . . . . . .
Angle Dependent Models
.. .. . . . .
Age Dependent Models . .. .. . . . .
Combined Ridge-Push and Slab-Pull Models .
DRSD Models
.. . . . . . . .. . .
. .
LRSD Models . . . . . . . . . . . . . .
Viscous Models . . . . . . . . . . . . . .
VII.
CONCLUSIONS . . . .
REFERENCES
.. . . . . . . .. .
65
65
65
68
71
74
74
78
78
78
80
83
87
87
90
90
. . . v 97
101
LIST OF ILLUSTRATIONS
Figure
Page
1.
Location map of the Nazca plate . . . . . „ .
2.
Nazca plate boundaries and intraplate
f e a t U r e S
o
e
e
e
o
e
o
e
o
o
e
e
p
o
o
o
7
o
I Q
Topography along the East Pacific Rise Ridge
Boundary . .■ „ . • . . . , . . . . . . .
12
Segmentation of the South American Subduction
Zone
17
Distribution of intraplate earthquakes in the
Nazca plate recorded between 1901 and
1981 . .
25
Location of the 1965 and 1944 intraplate
earthquakes
.............. ..
26
7.
Triangular elements on a half-sphere
32
8.
Finite element grid of the Nazca plate
(coarse)
o o p e p p o p p p o p e o o a o
35
Finite element grid of the Nazca plate
(re fined)
9 * * * 9 9 9 9 * 9 0
37
4„
5.
6.
9.
10.
. . , . .
* 9*9
Plate-driving forces acting on the Nazca
plate * .............. ..
41
Test case for comparing distributed ridge
and line ridge forces
46
12.
Distributed ridge forces on the Nazca plate . .
49
13.
Line ridge forces applied to the Nazca plate
.
52
14.
Constant slab-pull forces on the Nazca plate
.
56
15.
a.
b.
11.
Angle dependent slab-pull forces
Age dependent slab-pull forces ...........
vi
58
vii
LIST OF ILLUSTRATIONS— Continued
Figure
16.
17.
Page
Drag forces balancing distributed ridge
forces e o <
» o -<
<> •- • • « o o
62
a.
b.
67
Stress state for Model DR-a
Stress state for Model DR-b .
18.
Stress state for Model DRD--a
• » • v ° o
69
19.
a.
be
Stress state for Model DRD-b
Stress state for Model DRD-C
o • • •
70
a»
be
ae
be
Distributed ridge forces for refined grid
Stress state for Model *DR-a »
• - • o o
-'
Stress state for Model *DRD-a
«, o
Stress state for Model *DRD-b - • » •
75
a.
be
Stress state for Model LR-a
Stress state for Model LR-b » •
76
20.
21.
22.
* -
° °
o
« < • e
73
23.
Stress state for Model LR-(: » • > • •
24.
a.
b.
Stress state for Model LRD-a
Stress state for Model LRD-b
• - e
a o
79
a.
b.
Stress state for Model SD—a
Stress state for Model SD—b a « • •• °
o o
81
a.
b.
Stress state for Model SD-c
Stress state for Model SD—d e • • • «
o
82
a.
b.
Stress state for Model SD-ang
Stress state for Model SD-ang2
o o
84
28.
Stress state forModel
Model SD-age
SD-age .... . .... .... ....
85
29.
Composite of stress states for Models SD-c,
SD-ang, SD-ang2, and SD-age . . .. .. ..
86
25.
26.
27.
30.
31.
32.
. . a
a.
b.
Stress state for Model DRSD-1
Stress state for Model DRSD-2 . . . . . .
a.
b.
Stress state
Stress state
for Model DRSD-3
for Model DRSD-4 . ,.
a.
b.
Stress state
Stress state
for Model LRSD-1
for Model LRSD-2 , .. . . .
..
77
. 8 8
..
89
. 9 1
viii
LIST OF ILLUSTRATIONS— Continued
Figure
Stress state for Model LRSD-3
Stress state for Model LRSD—4 . . . « . . .
92
fdA
Stress state for Model V-DR
Stress state for Model V-DRD
. . . . . . .
94
ftirQ
Velocities for Model V--DR
Velocities for Model V--DRD
o o o e o o e o
96
33.
34.
35.
Page
LIST OF TABLES
Table
1.
2.
Page
Plate Motion Data Used in Nazca Plate
Stress Modeling . . . . . . . . . . . . . .
Description of Nazca Plate Models . . . . . . . 6 0
ix
9
ABSTRACT
The forces which drive the plates are evaluated by
finite element stress modeling of the Nazca plate, where
stresses resulting from various combinations of forces are
compared with the record of intraplate earthquakes and
other stress data.
are modeled.
Both elastic and viscous rheologies
Advances in modeling both ridge-push and slab-
pull forces are presented.
Ridge forces are distributed
over the entire plate, and then compared with line ridge
force models.
Angle and age-dependent slab-pull models
are compared with constant slab-pull models.
The distri­
buted ridge force models produce more realistic stresses
and represent a significant improvement over line ridge
forces.
Age-dependent slab-pull models fit the data better
than other slab-pull models.
Angle-dependent slab-pull
models do not fit the data in regions of shallow dipping
slab segments.
Both elastic and viscous models with drag
forces are less correlatable with the stress data than
models without drag forces.
x
I.
INTRODUCTION
The earth's lithosphere is made up of about a
dozen plates, defined by seismicity, and although we can
directly observe and measure the relative motion of the
plates, we have no direct method for measuring the forces
which drive the plates and must rely on theoretical
modeling.
The object of this thesis is to utilize one
of these methods, stress modeling, to investigate the
role and relative importance of possible forces in.driv­
ing and stressing a single plate, the Nazca plate.
The
forces considered are ridge-push, slab-pull and resistive
drag at the base of the plate.
The Nazca plate is modeled by the finite element
method as in Richardson (1978) where combinations of the
various plate-driving forces are applied and the resulting
stress states compared with the stress data.
Several
advances to this earlier work are presented here.
One
improvement is a new, more realistic way of modeling
ridge-push forces, by distributing the ridge-push over
the entire plate.
Distributed ridge forces are also
modeled with a refined grid.
Advances in modeling slab-
pull forces include both age-dependent and angle-dependent
slab-pull models.
A fourth innovation is the use of an
incompressible viscous rheology for two of the eight
1
distributed ridge models„
The viscous solution results
in velocities and stresses rather than displacements and
stresses obtained from elastic modeling.
The relative velocities of the plates were worked
out by Minster et al. (1974) and then improved by Minster
and Jordan (1978) by inverting data from transform fault
azimuths, spreading rates and earthquake slip vectors.
The relative strength of the various plate-driving forces
were then studied by inverting the data on velocities and
plate geometries, minimizing the net torque action on each
plate (Forsyth and Uyeda, 1974; Chappie and Tullis, 1977).
Intraplate stresses for various driving force
models were calculated by the finite difference technique
(Solomon et al., 1975, 1977; Richardson et al., 1976)
utilizing elastic rheology and comparing the stress states
of the models with the earthquake data.
The ridge-push
forces were found to be at least comparable to any of the
other forces acting on the lithosphere.
The results of
stress modeling were improved by the finite element
modeling technique and a finer grid (Richardson, 1978;
Richardson and Solomon, 1979) showing that ridge-push
was required for all models which matched the stress data.
Drag forces that were resistive could fit the data, while
driving drag force
models did not correlate with the
stress data. , Viscous stress modeling of the global set
Of plates by Richardson (1982, 1983) indicated that
ridge-push may be the most important plate-driving force
on the basis of predicting the direction and rate of rela­
tive plate motion, while the effective force of slab-pull
may be less than half the ridge-push.
In this thesis,
new ways of modeling ridge-push and slab-pull forces are
applied to elastic models of the Nazca plate, and the
effects of a viscous rheology are also tested.
The Nazca plate is ideal for stress modeling be­
cause it is a small oceanic plate with nearly comparable
lengths of both ridge and trench boundaries.
The size
is important because a small sized plate economically
allows detailed modeling with a fine grid*
Oceanic plates
are easier for modeling stresses than continental plates
because oceanic lithosphere is young and relatively homo­
geneous , and is not so complicated by the preservation of
past stress states.
The section which follows this introduction in­
cludes a detailed description of the Nazca plate bound­
aries and intraplate features.
In detail the Nazca plate
is surprisingly complex with, asymmetrical spreading along
the ridges, and a segmented subduction zone.
A fossil
spreading ridge in the middle of the plate, and the
presence of several aseismic ridges and island seamount
chains are potential sources of stress.
This stress is
modeled by distributing the ridge forces.
The stress data for the Nazca plate is the sub­
ject of the third section.
This data includes the set
of intraplate earthquakes, extensional features along
the plate boundaries, and a new kind of stress data from
VLB! (Very Long Wavelength Interferometry) and lunar
laser ranging techniques.
The next section describes the use of the finite
element method to model stress for both elastic and
viscous rheologies, and is followed by a section which
explains how the driving forces are calculated.
These
driving forces include ridge-push, slab-pull and resis­
tive drag, and other forces which are not modeled but
are described such as transform fault resistance, radial
push from hot spots and distant forces from other plates.
Thirty different models represent combinations
of the plate-driving forces and variations of the initial
and boundary conditions.
These models are described in
section six, followed by the conclusions.
Distributed
ridge forces are found to be a more realistic method of
modeling ridge-push than simple line ridge forces.
The
refined grid was found to result in nearly identical
stresses to those of the coarser grid, indicating that
the coarse grid, with approximately 5° grid spacing in
the plate interior, is sufficient for stress modeling of
the Nazca plate.
Slab-pull models with age-dependence
fit the stress data better than other slab-pull models.
Ridge-push models without drag forces fit the data better
than ridge models which were balanced by drag forces.
The viscous models with distributed ridge forces alone
qualitatively matched the relative velocity data for the
Nazea-South American plate boundary.
However, if drag
forces were added to the viscous distributed ridge model,
the resulting velocities indicated non-plate-like
behavior„
II.
DESCRIPTION OF THE NAZCA PLATE
The Nazca plate is a small oceanic plate in the
Southeast Pacific Ocean, extending southward from lati­
tude 5° N to 40° S and westward from longitude 75° W to
115° W.
It is bounded by the Cocos plate on the north,
the Pacific plate on the west, and Antarctic plate on
the south, and the South American plate on the east
(Fig. 1).
The plate is squarish in shape, pinched at
the north end and elongated to the southeast, with a
total area of approximately 1.6 X 1 0 ^ cm'2.
The thickness of the plate is not known in detail,
but estimates of oceanic plate thickness based on thermal
models (Parsons and Sclater, 1977) indicate a range from
essentially zero at the ridge crest to over 125 km for
80 my old lithosphere.
The maximum thickness of the Nazca
plate, based on a maximum age of 50 my (Herron, 1972),
would be 92 km, using the following equation from Turcotte
and Schubert (1982) :
Yl
=
2.32. (kt)1/2
(2.1)
where YL is the thickness of the lithosphere, k is the
thermal diffusivity, and t is the age.
A constant thick­
ness of 50 km is used for the Nazca plate modeling.
6
The
Eurasian Plate
Juan de Fuca Plate
r'N o rth American Plate
Caribbean
Plate
Philippine
Plate
Arabian Plate
Pacific Plate
Cocos Plate
African
Plate \
/ South
American
Plate
Nazca Plate
Indian Plate
Transform F a u lt----------Ridge Axis
Subduction Zone M i LI
Unknown
'
—
Antarctic Plate
75° 90° 105° 120° 135° 150° 165° 180° 165° 150° 135° 120° 105° 90°
Figure 1.
75°
60° 45°
30°
15°
0°
15° 30°
Location map of the Nazca plate
(From Turcotte and Schubert, 1982)
45°
60°
75° 90'
effect of doubling the plate thickness is to halve the
stress magnitudes.
The relative motion poles for the Nazca plate and
adjacent plates, as well as the absolute motion pole in
the hot spot reference frame are taken from Minster et al.
(1974) and Minster and Jordan (1978) who inverted a data
set of spreading rates, transform fault azimuths, and
earthquake slip vectors.
The 1974 plate motion models
were used in previous stress modeling of the. Nazca plate
by Richardson (1978) to determine the direction of plate­
driving forces.
In the present stress modeling, the more
recent plate motion data from Minster and Jordan (1978)
were used to determine the direction of some of the forces
These poles and rotation rates appear on Table 1.
The Nazca plate includes significant lengths of
all three kinds of plate boundary:
a subduction zone or
convergent boundary adj acent to South America, and spread­
ing centers and transform faults along the Other three
boundaries.
The length of spreading ridge along the East
Pacific Rise boundary is nearly the same as the length of
the South American trench, making it feasible to test the
roles of both slab-pull and ridge-push forces.
The plate
also contains intraplate ridges, seamounts, hot spots, and
two possible microplates.
The Nazca plate boundaries and
intraplate features are shown in Figure 2 and described
below.
TABLE 1„
Plate Motion Data Used in Nazca
Plate Stress Modeling
(from RM-2 and AMl-2 models of
Minster and Jordan, 1978)
RELATIVE MOTION
Latitude,
°N
Longitude,
°e
Rate, 10 ^
degrees/year
Nazca-Pacific
56.64
-87.88
15.39
Nazca-South America
59.08
-94.75
8.35
Nazca-Antarctic
43.21
-95.02
6.05
5.63
-124.40
9.72
47.99
-93.81
5.85
Plate Pair
____
Cocos-Nazea
ABSOLUTE MOTION
Nazca Plate
When viewed from above the pole, the first named plate
moves counterclockwise with respect to the second.
10
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PLATE
Figure 2.
Nazca plate boundaries and intraplate features
(Numbers indicate whole rates in cm/yr. Arrows
indicate direction of relative plate motion.
Double lines are ridges. Dashed lines are
transform faults. Solid line is trench.)
11
East Pacific Rise Ridge Boundary
The East Pacific Rise portion of the Nazca plate
boundary separates the Nazca plate from the Pacific plate
to the west, and extends 4,000 km south from the Galapagos
triple junction at 2° N, 2° W to a poorly defined triple
junction with the Chile Ridge at about 35° S, 110° W (Rea,
1981).
The East Pacific Rise is spreading more rapidly
than any other ridge, with a whole rate of approximately
16 cm/yr (Rea, 1981).
Rea and Scheidegger (1979) observed
that the East Pacific Rise spreading rates have fluctuated
during the past.
Also, spreading is asymmetrical south of
the Garret Fracture Zone
at 13.5° S, spreading faster to
the east than to the west.
Although variable spreading
rates and asymmetrical spreading may be important to the
state of stress of the Nazca plate, they have not been
included in the present modeling.
The East Pacific Rise has a complex topography,
ascending to the south in three broad steps.
The regional
depth of the rise is around 2,850 m between 8° N and 4° S,
ascending to 2,650 m between 5° S and 22° S.
South of
the Easter plate the axial depth is around 2,350 m (Rea,
1981).
The axial ridge tends to deepen at intersections
with offsetting fracture zones, with the exception of the
Garret Fracture Zone (Fig. 3).
This complex ridge topog­
raphy is an indication that the constant force per unit
EAST
PACIFIC
RISE
2200
2200
Easter
2400
2400
Plate
2600
■
2800
•
2600
2800
3000
3000
Figure 3
Topography along the East Pacific Rise Ridge Boundary
(after Rea, et al, 1979)
13
length for ridge forces assumed by Richardson (1978) was
an oversimplification.
The present East Pacific Rise portion of the
Nazca-Pacific plate boundary is young, and began forming
in Late Oligocene by three large westward jumps from the
now fossil Galapagos Rise.
The jumps covered a distance
of 900 km, spanning a time period of 2.5 my, and coincid­
ing with similar adjustments involving the Mathematician's
Ridge at 5° N to 20° N, and the opening of the Gulf of
California (Anderson and Sclater, 1972).
This realignment
may have arisen because of the breakup of the Farallon
plate and/or complex interactions of the East Pacific Rise
with North America (Anderson and Sclater, 1972).
Since we
are dealing with a single plate in this modeling, stresses
from outside the Nazca plate boundaries are ignored.
How­
ever , stress resulting from the elevation of the fossil
Galapagos Rise above the surrounding sea floor, as well
as variations in stress resulting from the complex topog­
raphy of the East Pacific Rise will be included in this
new modeling.
Galapagos Spreading Center Ridge Boundary
The Galapagos Spreading Center, also called the
Cocos-Nazca Spreading Center, is an asymmetrical, moderatevelocity spreading center which extends from the Galapagos
triple junction on the west, to the Panama,Fracture Zone
14
at 83° W,along one large and three smaller ridge segments
offset by transforms (Fig. 2).
The spreading rate along
the Galapagos Ridge was very rapid in the past, with a
whole rate of 14 cm/yr, but now is spreading at a moderate
6 to 7 cm/yr (Hey et al., 1977).
Scheidegger et al. (1981)
claim that most of the older crust of the Nazca plate was
formed at the Cocos-Nazca Spreading Center.
Recent accre­
tion on the center is asymmetrical, with more material
added to the Cocos plate than to the Nazca plate (Hey
et al., 1977)
The Galapagos Spreading Center is not a simple
spreading ridge, but can be divided into four different
regions (Johnson et ad., 1976),
The western segment,
located between the Galapagos triple junction and 98° W,
is very rugged and has a rift valley characteristic of
slow spreading ridges.
The central portion, between
98° W and the Galapagos Plateau, has moderate relief and
no well developed rift valley.
The Galapagos Plateau is
an elevated region which may be entirely of Pleistocene
age (Johnson and Lowrie, 1972) and contains the Galapagos
hot spot, and several box-shaped depressions which may
be submarine canyons or grabens.
The fourth segment is
the eastern Galapagos Spreading Center, which has a rela­
tively subdued relief, no rift valley, and may represent
a recent axial shift to the south (Johnson et al., 1976).
The stresses resulting from these complexities cannot be
15
modeled with simple line ridge forces, but are modeled
with distributed ridge forces.
Chile Ridge Spreading Center and Transforms
The Chile Ridge boundary extends from approxi­
mately 35° S, 110° W, to the Chile Margin triple junction
at 46° S, 76° W (Herron et aT., 198]).
The Chile Ridge
was not recognized as a major plate boundary in early
papers because the area was remote and the data were
sparse (Herron et al., 1981).
Most of the west side of
this boundary consists of transform faults, with rugged
relief.
The east side is almost a single ridge segment
with low bathymetric relief superimposed on a gradual
increase in depth away from the ridge crest (Klitgord
et al., 1973).
The Chile Ridge may be a remnant of the
former Galapagos Rise (Klitgord et al., 1973).
The whole
spreading rate along the Chile Ridge has averaged 5.6
cm/yr over the past 700,000 yr, slowing from a previous
rate of 11.2 cm/yr before 5.12 my B.P.
(Herron et al.,
1981).
Herron et. al. (1981) studied the collision of
Chile Ridge with the subduction zone at the Chile Margin
triple junction.
Continental seismic activity stops
60 km north of this triple junction and there is a gap
in volcanism landward from the triple junction north to
the intersection with the fracture zone at 49
S (Herron
et al., 1981).
This is also within the region which
slipped during the 1960 Chilean earthquake, the largest
earthquake ever recorded (Ms =8.3, Mw = 9.5; Kanamori,
and Cipar, 1974; Kanamori, 1978).
The Chile Ridge col­
lided with the subduction zone 300,000 years ago, but
the spreading process does not appear to have been modi­
fied by proximity to the subduction zone (Herron et al.,
1981).
However, this may have affected the stresses
within the adjacent South American plate.
South American Trench Boundary
The subduction zone between the Nazca plate and
the South American plate is divided into two sections,
separated by the Grijalva Scarp (Fig. 4).
The northern
trench extends southward from the Panama Isthmus to the
Grijalva Scarp, near 4° S, and is much shallower than
the Peru-Chile Trench to the south.
The entry of the
Carnegie Ridge into the subduction zone during the past
3 my may have shoaled the trench off Ecuador, which has
a minimum axial depth of only 2,920 m ( L o n s d a l e 1978) .
The Ecuador Trench is subducting younger (less than 27
my old) lithosphere from the Cocos-Nazca Spreading
Center, while the Peru-Chile Trench is subducting litho­
sphere up to 50 my old originating from the fossil
Galapagos Rise (Lonsdale, 1978).
17
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p£^u
AMERICAN)
SEGMENT
PLATE
N orth
CHILE
Se g m e n t
30*
N AZ_CA
P late .
Figure 4.
Segmentation of the South American Subduction
Zone
(After Barazangi and Isacks, 1976)
O "
'■ 18
The northern subduetion zone is very complex
because of the proximity of active and extinct ridges,
transforms, and possible hot spot traces to the trench.
Pennington (1981), using seismicity data, divided this
subduction zone into three segments:
the Bucaramanga
segment in the north, and the Cauca and Columbia-Ecuador
segments to the south.
There may be a tear between the
Bucaramanga segment, which is dipping at a 20 to 25 degree
angle, and the other two segments, which are dipping at
35 degrees (Pennington, 1981).
Because of the complexity
of this region, and its small area relative to the rest
of the Nazca plate, only the Ecuadorean segment was in­
cluded in the stress modeling.
This segment is moving
obliquely with a rate of 9 cm/yr (Lonsdale, 1978).
The Peru-Chile Trench extends from 4° S to 45° S,
and subduction between the Nazca plate and the South
American plate is occurring at around 9 cm/yr (Minster
and Jordan, 1978).
Barazangi and Isacks (1976, 1979)
attributed the transition from the flat Peru segment to
the steeper North Chile segment to a major tear in the
descending Nazca plate, basing their argument on first
motion studies of P waves.
Hasegawa and Sacks (1981)
argue on the basis of ScSp observations that instead of
a tear, there is contortion in the shallow Peru segment.
They argue further that the Peru segment dips initially
at an angle of 30 degrees, then bends below 100 km, where
19
it then subducts nearly horizontally.
Whether a tear or
a contortion, the lengths of the slab segments seem to
be different, and the tectonics of the Andes Mountains
show the same segmentation as the Nazca plate slab
(Jordan et al., 1983).
Schweller et al. (1981) observed five morphotectonic provinces along the Peru-Chile Trench, from a data
set of over 200 bathymetric profiles, showing distinct
changes in tr&nch depth, axial sediment thickness, extensional fault structures, and dip of the seaward trench.
These provinces are similar to the angle-dependent seg­
ments of Barazangi and Isacks (1976).
Grellet and Dubois
(1982) found a correlation between the rate of subduction,
the age of the subducting slab, and the depth of the trench
Increasing subduction rate and older lithosphere both
correlate with deep trenches.
This correlation was con­
sistent with either elastic or viscous rheologies for the
oceanic lithosphere (Grellet and Dubois, 1982).
The age of the subducting slab varies from as
young as 12 my at the north end, to a maximum of 50 my
in the middle, and becoming as young as 3 my at the south
end near the Chile Margin triple junction.
Age affects
the thickness and density of the subducting slab (Wortel
and Vlaar, 1978; Wortel and Clotingh, 1981) and young
lithosphere may be less dense than the mantle into which
it is being driven, causing shallow dip angles (Sacks
et al., 1978).
Other factors which may contribute to the
complexity of the South American subduction zone are the
subduction of buoyant aseismic ridges and seamount chains
(Vogt et al., 1976; Pilger, 1981) and the curvature of
the South American subduction zone (Rodriguez et al.,
1976).
The possible contributions of slab age and slab
dip angle will be considered and evaluated in the slabpull stress modeling.
Intr aplate Feature s
In addition to the plate boundaries, the Nazca
plate has several other important geologic and tectonic
features: possible hot spots, microplates, several
aseismic ridges, a fossil ridge, and island seamount
chains (Fig. 2).
At least two hot spots or mantle plumes have
been proposed for the Nazca plate, one in the Galapagos
Islands and one either west of or beneath Easter Island
(Johnson and Ldwrie, 1972; Pilger and Handschumacher,
1981).
Hot spots are thought to be narrow plumes of
mantle material which rise and spread radially in the
asthenosphere.
As oceanic plates move over the hot
spots, basaltic volcanism occurs, forming linear aseismic
ridges (Johnson and Lowrie, 1972).
The hot spots could
locally affect the state of stress in the Nazca plate by
a variety of mechanisms, including the heating of the
plate.
However, in terms of this paper, only the effect
of the elevation of these regions has been included in
the distributed ridge models discussed below.
The Nazca plate also has two
as possible microplates.
regions described
The Easter plate and the Panama
plate are complex regions along the East Pacific Rise and
Cocos-Nazca Spreading Centers, respectively.
The Easter
plate is a small 3.2 my old plate defined by seismic,
magnetic and bathymetric data.
There are two active
spreading centers along the east (Este) and west (Oeste)
boundaries, and active fault zones along the north and
south borders (HandSchumacher et al., 1981).
The Este
Ridge is more active, spreading at greater than 13 cm/yr,
while the Oeste Ridge is spreading at less than 2 cm/yr
(Engeln and Stein, 1983).
This may lead to the future
"capture" of the Easter plate by the Pacific plate
(Handschumacher et ad., 1981).
In the present modeling,
the Easter plate has been treated as part of the Nazca
plate.
The Panama platelet (Lonsdale and Klitgord, 1978)
is bounded by the Panama Fracture Zone, the South American
and Cocos plates, and some poorly defined boundary between
the Carnegie and Malpelo Ridges.
The platelet encompasses
part of the thick sediment filled Panama Basin, which dis­
plays many extensional features (Van Andel et al., 1971).
22
The now seismically active transform at 83° W, the Panama
Fracture Zone, may have recently jumped westward, detach­
ing the region from the rest of the plate (Van Andel
et alo, 1971? Lonsdale and Klitgord, 1978; Pennington,
1981).
In modeling the Nazca plate, this platelet is
not included because of its complexities, poorly defined
boundaries, and small size relative to the Nazca plate.
The Galapagos Rise, a fossil ridge in the middle
of the Nazca plate, extends from near 15° S to 30° S .
It
is elevated up to a kilometer above the surrounding sea
floor (Hammerickx and Smith, 1978).
The East Pacific
Rise spreading jumped from the Galapagos Rise and became
active 6.5 my B.P.
However, the Galapagos Rise may have
continued spreading after the jump, up to 4 my B.P., so
that the two ridges were both active for a period of
over two million years (Anderson and Sclater, 1972).
Aseismic ridges, which include the Nazca Ridge
and the Cocos Ridge, may have been formed syncronously
with adjacent sea floor at the mid-oceanic spreading
centers (Pilger, 1981).
These are continuous features,
unlike island seamount chains such as the Juan Fernandez
and Easter Island seamount chains, which are discontinu­
ous and may have been formed by thermal anomalies (Pilger,
1981).
The subduetion of these possibly buoyant ridges
and islands may be responsible for shallow dips and the
segmentation of the subduction zone (Pilger, 1981;
23
Pennington, 1981).
In the present modeling, the effects
of the elevation of the ridges and islands are considered
in the distributed ridge models.
III.
STRESS DATA
The stress data for the Nazca plate consists of
the record of intraplate earthquakes (Pig. 5).
One of
these occurred on November 25, 1965 with a magnitude
(m^) of 5.75, located at 17.1° S, 100.2° W.
Mendiguren
(1971) used both body and surface wave data to determine
the source mechanism.
The analysis indicated nearly
east-west maximum horizontal compression.
Two earlier
earthquakes were recorded in 1944 and the first motion
studies of the P waves were consistent with the mechan­
ism solution of the 1965 earthquake.
The location of
these earthquakes is shown on Figure 6.
Bergman and Solomon (1980) studied 159 oceanic
earthquakes to determine their association with large
bathymetric features and pre-existing faults.
They con­
cluded that large earthquakes are often associated with
old zones of weakness which appear to be reactivated by
the present stress field.
One earthquake (m^ of 5.0),
located along the Nazca Ridge at 21.3° S, 82° W, occurred
on December 8, 1964.
Bergman and Solomon (1980) sug­
gested that the Nazca Ridge may be a weak zone, but that
the bathymetry may also be contributing to the stresses.
The November 25, 1965 earthquake occurred on the
24
25
IlFw U 0*«
105*M
IOO'm
9^
MAGNITUDE
o 0.00 M 3.99
X 3.99 It9.99
A 9.99 It9.90
Figure 5.
Distribution of intraplate earthquakes in
the Nazca plate recorded between 1901 and
1981
(Earthquake epicenter map is from NOAA)
26
COCOS
PACIFIC
S. AMERICA
NAZCA
ANTARCTIC
Figure 6.
Location of the 1965 and 1944 intraplate
earthquakes
(Plate boundaries are defined by earthquake
epicenters; boundaries are as described in
Figure 2. Arrows indicate orientation of hor­
izontal compression axis inferred from the
focal mechanism of the 25 November 1965 earth­
quake. The segment of the Peru-Chile trench
boundary that failed during the 22 May 1960
earthquake is hatched. After Richardson,
1978.)
southwest flank of the Galapagos Rise, near the Mendana
Fracture Zone, and Bergman and Soloman (1980) suggest
that this earthquake was responding to the stress field
of the East Pacific Rise.
The epicenter of another Nazca
plate earthquake, recorded on May 28, 1972 (m^ of 4.9)
was located at 32.78° S, 92.2° W, just east of the
southernmost trace of the fossil Galapagos Rise (Fig. 6),
Because of its small size, there has been no attempt to
determine a fault plane solution.
Studies of oceanic intraplate earthquakes world­
wide indicate that oceanic lithosphere older than 20 my
is under deviatoric compression (Sykes and Sbar, 1973;
Wiens and Stein, 1983).
Most of the Nazca plate east of
the 1965 and 1944 earthquakes is older than 20 my, and
so might also be in a compressive state of stress.
The
number of intraplate earthquakes per unit area of oceanic
floor and per unit volume of lithosphere decreases with
lithospheric age (Wiens and Steip, 1983) indicating that
stresses might be greater in younger lithosphere hot too
distant from the spreading ridges.
A recent study by Hilde and Warsi (1983) claims
that the Mendana Fracture Zone near 10° S is undergoing
rifting with the creation of new oceanic lithosphere at
the subduction zone.
They suggest that the Nazca plate
may be undergoing, extension in this region in response
to subduction-induced stresses along a zone of weakness.
28
which may lead to the separation of the entire Nazca plate
in 35 my.
Another region of extension is the Panama Basin,
in the northeast corner of the Nazca plate (Van Andel
et al., 1971) .
A new kind of stress data from VLBI (Very Long
Wavelength Interferometry) and lunar laser ranging is
being accumulated (Bender et al., 1979; Smith et al.,
1979; Allenby, 1979; Clark, 1979; Niell et al., 1979).
This new data has the potential to measure distances
between widely separated stations with an accuracy of
centimeters.
Relative plate motions and internal defor­
mation of plates could be measured on time scales of years
or less.
The Goodard Space Flight Center has developed a
highly mobile satellite laser tracking system called
TLRS2 (Transportable Laser Ranging System) which arrived
on Easter Island in early January, 1983, and began routine
tracking in March of 1983.
The station is going to col­
lect data for six months, be moved to a new site, then
return annually to Easter Island to measure baselines
between the Nazca plate and the Pacific and South American
plates (Nasa, 1983).
Permanent stations are located at
Arequipa, Peru and Maui, Hawaii.
Stress modeling of the
Nazca plate will be valuable for evaluating this new data
and for possibly guiding the location of future data
collection stations.
29
The choice of model rheology and plate thickness
can be constrained by experimental data from rock defor­
mation studies (Kohlstedt and Goetze, 1974; Kirby, 1980).
Kirby (1980) observed that the maximum strength of the
lithosphere is reached at some critical depth, and that
below this depth strength falls off rapidly due to increas­
ing temperature.
The strength maximum increases in both
magnitude and depth with increasing age of lithosphere,
and the strength distribution is strongly influenced by
the state of stress.
Except for lithosphere less than
2.5 my old, the elastic strength of oceanic lithosphere
is predicted to be in the kilobar range (Kirby, 1980).
The effective elastic thickness varies from 2 to
54 km, based on studies of the flexure of oceanic litho­
sphere (Watts et al., 1980).
Further, oceanic lithosphere
appears to be able to support stresses of at least a kilobar for long periods of time (Watts et al., 1980).
The
Nazca plate is here modeled as an elastic plate 50 km
thick.
An elastic lithosphere of 50 km would have stresses
twice the magnitude of the stresses of 100 km thick litho­
sphere, but the distribution of stresses would be the same.
Thus, the choice of model thickness does not affect the
results of attempting to fit the orientation of the Nazca
intraplate earthquake data, or a general sense of compres­
sion or extension in any region of the plate.
IVo
DESCRIPTION OF MODELING METHOD
The Nazca plate is modeled as a linearly elastic
medium, using the displacement method of finite elements
in which all external loads are balanced by internal
strains (Zienkiewicz, 1971; Bathe and Wilson, 1976).
The
plate is divided into a discrete number of regions of
finite area (elements), connected at a finite number of
points (nodes).
The displacements of the internal points
within a given element are approximated by some polynomial
function of the nodal displacements„
The dimension of the
matrix of polynomial functions is determined by the number
of degrees of freedom (coordinate directions of displace­
ment) at any point and the total number of degrees of
freedom for an element.
Therefore the necessary calcula­
tions are simplified by minimizing the number of nodes
per element and the number of degrees of freedom per node.
The Nazca plate model is made up of constant
strain triangular elements with two in-plane degrees of
freedom per node, in the latitudinal and longitudinal
directions.
per element.
Therefore, there are six degrees of freedom
In addition to decreasing the number of
degrees of freedom needed to solve the problem, the con­
stant strain triangles are also a good choice of element
30
31
because they can approximate displacements on a sphere
while still retaining their planar geometry, since three
points on a sphere define a plane„
Any other kind of
element would lead to a less satisfactory approximation
to a sphere.
The elements can be visualized as triangles
situated on a sphere, with springs between the nodes
(Fig. 7).
The triangles are deformed as a result of
forces applied to the nodes.
Elastic Modeling
Strain at any point in the element can be deter­
mined from the displacements of the node points.
Element
stresses can then be found from stress-strain relation­
ships.
For a two-dimensional isotropic elastic medium in
a plane state of stress, Hooke's law leads to the follow­
ing relationship between stress and strain (Zienkiewicz,
1971) :
G11
a22
012
1
v
V 0
1 0
0
0
1—v
2
511
522
512
(4.1)
where o^j is the stress,
is strain, E is Young's
Modulus, v is Poisson's Ratio, and [C] is the elasticity
matrix.
MODES
Figure 7.
Triangular elements on a half-sphere
33
For each constant strain triangular element, the
two displacement degrees of freedom,
and
, for an
interior point in the element, are functions of the six
nodal degrees of freedom.
A strain operator matrix [B]
is computed for each element.
This matrix depends on
the geometry and area of the element, including the rela­
tive difference in nodal locations.
A stiffness matrix
[K] is then calculated for each element as (Zienkiewicz,
1971) :
[K.]
[C\]
=
[B^] h dxdy
(4.2)
area
where h is the element thickness.
The local element
stiffness matrix [K^] can then be transformed into the
global coordinate system and added to a global stiffness
matrix [K].
The equilibrium condition for a finite ele­
ment approximation to an elastic medium is (Zienkiewicz,
1971):
[K]
{U}
=
{F}
(4.3)
The solution of the equation for displacements leads to:
{U}
=
[K]"1
{F}
(4.4)
where {U} is the vector of unknown nodal displacements,
and {F} is the vector of all equivalent nodal loads.
The inputs to any problem are thus the elastic properties,
34
the geometry of the grid, and any forces acting on the
system.
The solution of Equation 4.4 used here incorporates
the Wave Front solution technique of Irons (1970) and
Orringer (1974).
This technique is based on Gauss elim­
ination, and trades external computer storage and core
space off against each other.
A Certain number of degrees
of freedom are assembled, sent to storage, and later re­
trieved for further calculations.
The maximum number of
degrees of freedom required at any one time in in-core
storage is called the front width and depends on the ele­
ment numbering scheme.
The front widths for the coarse
and fine grids for the Nazca plate were 30 and 42 respec­
tively.
The first grid used for modeling the Nazca plate
was modified from one used by Richardson et al, (1979)
for global modeling.
The modifications included a better
fit to the plate boundaries from two bathymetric maps of
the Southeast Pacific Ocean (Mammerickx and Smith, 1978;
A.A.P.G., 1981).
This grid is composed of 116 nodes and
183 elements, for a total of 236 degrees of freedom (Fig.
8).
For the distributed
ridge models, discussed below,
elevations of node points were needed, and the grid points
were moved in some cases to include the bathymetric detail
that might otherwise have been missed by the grid.
17
2
total area of this grid was 1.601 x 10
cm .
The
#o°w
IO°S,
z o ms
30*5
4 0 mS
izo'kj
Figure 8.
no°w
90"*/
70°U/
Finite element grid of the Nazca plate
(coarse)
(There are 183 constant-strain triangular
elements with six degrees of freedom per
element and a total of 116 nodes.)
A finer grid was also constructed to see if it
would give significantly different results for the eleva­
tion-dependent distributed ridge force models„
A grid
with approximately 2-1/2 degree spacing was constructed
(Fig. 9), utilizing 467 elements and 264 nodes. The
17
2
total area of this grid was 1.599 X 10
cm „ The re­
sults of the distributed ridge force models for the two
grids are compared and discussed later.
Elastic constants used in the modeling, appro­
priate for oceanic lithosphere, were 7 X 10"*"1 dyne-cm 2
for Young's Modulus and 0.25 for Poisson's Ratio.
The
finite element method allows the material properties to
vary on an element by element basis.
However, the mate­
rial properties of an oceanic plate are relatively homo­
geneous and there were no data which would indicate the
need for ah inhomogeneous model, except possibly along
the plate boundaries.
An inhomogeneous model with a
viscous rheology along the subduction zone boundary was
included in the Nazca plate stress modeling and is dis­
cussed later.
Calculated stresses represent an average across
the plate thickness, with the vertical component removed.
They are, in a sense, stress differences or a kind of
deviatoric stress, except that the vertical, rather than
hydrostatic stress has been removed.
The calculated
stress represents the actual stress field if one assumes
37
u
o
*
n
Figure 9.
n
o
1
N
Finite element grid of the Nazca plate
(refined)
(There are 467 constrant-strain triangular
elements with six degrees of freedom per
element and a total of 264 nodes.)
38
that the material contains no memory of past stress
states.
In order to constrain rigid body motion, a
minimum of three appropriate degrees of freedom must
be constrained.
One point is pinned from motion in both
the latitudinal and longitudinal directions and another
point is chosen so that there is no rotation about the
pinned point.
-•
Viscous Modeling
The oceanic lithosphere can be modeled with
either viscous or elastic rheologies and still fit the
data on oceanic flexure at subduction zones (DeBremaecker,
1977; Grellet and Dubois, 1982).
The major advantage of
viscous modeling is the ability to include the data set
s
■
on present-day relative plate motion as well as the data
on intraplate stress as constraints on the driving mechan­
ism.
The viscous equations are homologous to the elastic
equations where stress is related to strain rate by the
material property of viscosity.
aij
where
=
^ ^kk
+
The viscous equation:
2 v ^ij
^4,5)
^ is the stress, £ is bulk viscosity, £ is the
shear viscosity, and
is strain rate, and be solved in
the same way as the elastic equations by using appropriate,
values for the shear and bulk viscosities.
Thus, if you
have solved an elastic problem in terms of the Lame con­
stants e and p for unknown displacements, you have also
solved the equivalent viscous problem in terms of the
viscous constant %■ and p for unknown velocities.
The volumetric strain rate 4 ^ must be zero for
an incompressible Newtonian fluid.
This can be accom­
plished if the bulk viscosity £ becomes infinite.
Thus,
for
e
_
“
EV
(1+v) (l-2v)
!A
V °
a Poisson's ratio of 0.50 fulfills this requirement of
incompressible fluid flow.
The viscosity of the litho24
sphere is on the order of 10
P (deBremaecker, 1977),
and appropriate values for Young's Modulus are chosen to
achieve this viscosity.
The subduction zone is modeled
with a viscosity which is a factor of 20 lower than the
rest of the plate.
V.
MODELING PLATE-DRIVING FORCES
The plate-driving forces are not well understood,
but attempts to model the forces, and then match the re­
sulting model stress states or plate motions with actual
observations, have shown that the three most significant
forces seem to be ridge-push forces, slab-pull forces,
and drag forces along the base of the plates (Forsyth
and Uyeda, 1975; Chappie and Tullis, 1977; Richardson,
1978; Richardson et al., 1979; Carlson, 1981).
forces are illustrated in Figure 10.
These
The methods of
modeling these three major forces for the Nazca plate
are discussed below, followed by a brief description of
other potential forces acting on the plate.
Ridge-Push Forces
Ridge-push forces arise from horizontal density
differences resulting from thermally induced elevation
and thickness variations throughout the oceanic litho­
sphere.
The lithosphere thickens as the cooling iso­
therms descend deeper into the mantle (Hager, 1978).
The oceanic lithosphere then sinks to compensate for
the thickening, and the depth of the sea floor increases
with the age of the lithosphere (Sclater and Francheteau,
40
Figure 10.
Plate-driving forces acting on the Nazca plate
(Fr and Ft are forces per unit length of ridge and trench,
respectively, acting in the direction of relative plate
motions; Fp is drag force per unit area, resisting the
motion of the other forces; V is fixed so that torques
due to Fr and FT are balanced by drag forces. From Richard­
son, 1978.)
H
42
1970; Parsons and S.clater, 1977; Hager, 1978) „
The follow­
ing equation describes the variation in lithospheric thick­
ness due to cooling from Parsons and Sclater (1977):
h = 2 (kt)1//2
erf-1
(Tj/1^)
(5.1)
where h is the lithospheric thickness, k is the thermal
diffusivity, t is the age of the lithosphere,
is the
temperature of the lithosphere, and Tm is the temperature
in the mantle.
Parsons and Sclater (1977) also describe
the variations in the depth of the sea floor for young
oceanic lithosphere (less than 70 my old) by the follow­
ing equation:
d
=
2500
+
350 t1//2
where d is the depth of the sea floor.
(5.2)
The variation
in sea floor depth resulting from isostatic compensation
is given by Hager (1978) :
dt
=
db / ((Pi - Pw)/(pl " pm ))
(5.3)
where d^_ is the depth to the top of the lithosphere, d^
is the depth to the bottom of the lithosphere, and p^,
p , and p are the densities of the lithosphere, sea.w
m
water, and mantle respectively.
If isostatic compensation is assumed, the hori­
zontal mass difference from the change in depth of the
base of the lithosphere is equal to the horizontal mass
43
difference from the changes in elevation of the sea floor
(Hager, 1 •fe}.
Numerical modeling of mantle flow by
Hager (1978) and Hager and O'Connell (1981) shows that
these ridge-push forces may be more important than slabpull forces.
This theory is supported by previous stress
modeling by Richardson (1978) where models with predom­
inately ridge-push forces matched the stress data better
than models with predominately slab-pull forces.
The horizontal ridge-push forces, perpendicular
to the ridge crest, are analogous to glacier flow (Frank,
1972) and give rise to compressive stresses in the litho­
sphere which decrease close to the ridge crest, diminish­
ing to zero at the ridge crest.
Lister (1975) describes
this force as:
FR
=
1/2 h 1 g (P]L - pw )
(5.4)
where FR is the horizontal ridge-push force, h is the
drop in height from the ridge crest reference height,
1 is the lithospheric thickness, g is the gravitational
constant, and p, and p are the densities of the litho^1
^w
sphere and seawater, respectively.
The ridge-push force can also be quantified using
thermal and time parameters as (Parsons and Richter,
1980):
FR
=
9 “ pm
Tm
k
t
(5.5)
44
where a is the volume expansion coefficient,
mantle density, k is the thermal diffusivity,
is the
is the
temperature of the mantle, and t is the age of the litho­
sphere.
Parsons and Richter (1980) show that this driving
force is proportional to the geoid anomaly associated with
the ridge.
Estimates of the magnitude of this ridge-push
force per unit length of ridge, utilizing thermal models,
range from 1 to 5 x 10 15 dyne/cm, averaging between 100
and 500 bars across a 100 km thick plate (Forsyth and
Uyeda, 1975; Lister, 1975; Chappel and Tullis, 1977;
Richardson, 1978).
An independent calculation of ridge-
push force from distributed ridge forces on the Nazca
plate will be described below, and the magnitude will be
compared with the estimates from thermal models.
Previous finite element stress modeling of the
Nazca plate by Richardson (1978) utilized a constant force
15
per unit length of 1 x 10
dyne/cm along the ridge bound­
aries in the direction of relative plate motion.
These
directions were taken from the RMl model of Minster et al.
(1974).
This kind of ridge-push modeling will be referred
to as line ridge force, and is adequate as a first approxi­
mation of ridge-push forces.
models have three weaknesses.
However, line ridge force
For the region near the
ridge crest the horizontal compressive stresses due to
the ridge-push do not decrease in the model, while they
45
would decrease in reality.
Secondly, there should be force
contributions from regions of elevated topography within
the Nazca plate, such as the fossil Galapagos Rise and
the aseismic ridges and island seamount chains.
Potential
forces from these elevated regions are not included in the
line ridge force models.
Finally, the ridge force itself
should be distributed over the entire area of the plate,
rather than concentrated at the ridge crest (Hales, 1969?
Lister, 1975? Hager, 1978).
A new way of modeling ridge-push, by distributing
ridge forces due to topographical gradients, was developed
in order to improve the ridge-push stress modeling.
Dis­
tributed ridge forces as well as line ridge forces were
first applied to a test case, and the resultant stresses
of the two types of forces were compared.
In the test
case, a strip of elements 2,000 km long, 100 km wide, and
100 km thick was used to approximate a 100 km segment of
ridge crest pushing across 2,000 km of plate (Fig. 11).
The right side of the strip was pinned to approximate the
behavior of a subduction boundary in between major seismic
slip events. The topographic gradient from the ridge
crest to the abyssal sea floor was calculated from the
depth equation (Eg. 5.2) for young ocean floor described
above (Parsons and Sclater, 1977).
The distributed ridge
forces were then computed for each element from the follow­
ing equation:
*4K0m0-*
Figure 11.
Test case for comparing distributed ridge and line ridge forces
47
FDR
=
Ap
9
a
h
(sin 0)
(5.6)
where FDR is the distributed ridge force for the element,
Ap is the difference between the lithospheric and seawater
densities, a is the area of the element, h is the thick­
ness of the lithosphere, and 0 is the angle of the slope
of the bathymetry.
The total force applied to the line
ridge test case was the same as the total force applied to
23
the distributed ridge test case, 2 x 10
dyne, equivalent
7
2
to a stress of 5 x 10 dyne/cm , or 500 bars over the 100
km thickness.
Stresses resulting from the two cases were differ­
ent.
For the line ridge force case, the stresses were
compressive and equal in magnitude for all of the elements.
For the distributed ridge case, the stresses were also com­
pressive, and were equal in magnitude to the line ridge
stresses on the non-elevated portion of the strip, approxi8
9
mately 6.5 x 10 dyne/cm . However, the magnitude of the
stresses on the ramped portion of the strip decreased
8
2
toward the ridge crest, from 6.5 x 10 dyne/cm to 6.6 x
10 7 dyne/cm 2 , nearly an order of magnitude decrease in
stress.
Therefore, the two cases resulted in identical
stresses for regions with no topographic relief far from
the ridge, but the distributed ridge force resulted in a
better approximation of stress near the ridge crest and
in areas of relief away from the ridge crest.
Distributed Ridge Forces Applied to the Nazca Plate
The nodal elevations for the Nazca plate were
taken from a bathymetric map of the Southeast Pacific
Ocean (Mammerickx and Smith, 1978).
A computer sub­
routine was used to calculate the dip of each element and
to apply a force in the direction of dip to each element
which was proportional to the dip angle and the area of
the element (Eg. 5.6).
In other words, the force is due
to the weight of the element and the sine of its topo­
graphic gradient.
The force was distributed equally
between the three nodes of each element.
If the element
had no slope, there was no force applied to its nodes.
Also, if any nodes were pinned, no forces were applied
to the pinned nodes. These distributed ridge forces are
plotted on Figure 12.
A comparison with a bathymetric
map of the data verified that angles and magnitude of
forces corresponded with topographic gradients.
Line Ridge Forces Applied to the Nazca Plate
In some models, line ridge forces were applied to
nodes along the ridge boundaries, proportional to the
length of ridge associated with the nodes.
The direction
of the force is the direction of relative plate motion of
the corresponding plate pairs from Minster and Jordan
(1978; Table 1).
The force for each node was computed
49
Figure 12.
Distributed ridge forces on the Nazca plate
(The arrows originate at node points and are
proportional to the topographic slope of each
element. The actual forces are also propor­
tional to the area of each element but the
vectors presented here have been normalized
to show the slope direction.)
50
as a product of the sum of the half-distances between
adjacent nodes and 1.00 x 10‘L dyne/cm, making the force
constant per unit length of ridge crest, equivalent to
200 bars across a 50 km thick plate.
The distance between nodes was computed by first
converting the node location to xyz (cartesian) coordi­
nates :
x
= cos (lat) cos (long)
y
= cos (lat) sin (long)
z
= sin (lat)
(5.7)
The arc-cosine of the dot product for the position vectors
of the two node points gives the angular distance between
the nodes. To input the force in the direction of lati­
tude and longitude, the following steps were taken.
First, the relative velocity in the xyz coordinates for
each node point was computed by finding the cross product
of the position vector of the node point and the pole of
relative motion.
These relative velocity vectors, V^.,
V^., and V g , were then converted to latitudinal and longi­
tudinal vectors, u^at and u^on, according to the follow­
ing relationship:
Ulat ^ -Vx cos (lon.g)°sin (lat) - V
sin (lat) + V z cos (lat)
J^on = V
cos (long) -
sin (long)
sin (long)
(5.8)
51
The velocity vector,
(Ulat,
) was first normalized
to unit length, then multiplied by the sum of the half­
distances belonging to each node, and scaled so that the
average stress across the plate was 100 bars„
The force
vectors for the line ridge forces are shown on Figure 13 =
Slab-Pull Forces
Slab-pull forces are primarily derived from
gravitational or negative buoyancy forces resulting from
the entrance of a cold dense oceanic lithosphere into a
hot, less dense mantle.
This negative buoyancy force is
described by McKenzie (1969) and Davies (1980) to be:
FT = g a Tm
p2 h3
Cp
V (sin 0)/(24 K)
(5.8)
where a is the thermal expansion coefficient, p is the
density of the slab, H is the thickness of the slab, C
is the heat capacity, v is the veolocity of subduction,
0 is the dip angle of the slab, and K is the thermal con­
ductivity.
This force is enhanced by the thickening of
the lithosphere near the trench due to the cooling effect
described above under ridge-push forces.
The slab-pull
forces may also receive a contribution from two major
pressure-induced phase transitions in the mantle, one from
olivine to spinel at 350 to 400 km depth, and the other
from spinel to post-spinel at 650 km depth.
The olivine-
spinel transition is elevated in the descending slab,
52
Figure 13.
Line ridge forces applied to the Nazca plate
(Vectors originate at nodes along spreading
ridges and are applied in the direction of
relative plate motion, proportional to the
length of ridge segment belonging to each
respective node.)
53
contributing a significant force to the total slab-pull
forces (Schubert et al., 1975).
If these slab-pull forces are transmitted to the
surface plate, then slab-pull has the potential to be the
dominant plate-driving force. While ridge-push forces
are On the order of 100 bars across a 100 km thick plate,
those predicted for slab-pull are on the order of kilobars
(McKenzie, 1969; Smith and Toksoz, 1972; Chappie and Tullis
1977; Davies, 1980).
Resistive forces due to friction may significantly
reduce the pull of the slab and may be as great in magni­
tude as the gravitational forces pulling the slab (Forsyth
and Uyeda, 1975; Chappie and Tullis, 1977; Davies, 1980).
The record of large magnitude earthquakes along the subduction zone is evidence for the resistance to plate motion
at the trench.
Since the forces acting on the surface
plate are the sum of a slab-pull and slab resistance term,
the resulting net slab-pull may well be of the same magni­
tude as ridge-push forces, between 100 and 500 bars for a
100 km thick plate or up to a kilobar for a 50 km thick
plate.
The South American subduction zone, described
earlier, is very complex, with large variations in both
age and dip angle of the subducting slab.
The dip angle
might have a significant effect on the slab-pull force,
since a smaller dip angle would decrease the component
54
of the negative buoyancy force acting down-dip along the
slab, as well as increasing the friction between the Slab
and overridging plate.
Therefore, the subducting slab
along the South American trench would have a variable pull,
and these variable forces would result in a different
stress state than would result if the pull were constant
along the trench.
The age of the descending slab may also affect the
slab-pull forces since older lithosphere is thicker and
would have a greater pull (Molnar and Atwater, 1978? Wprtel
and Clotingh, 1981).
Age varies greatly along the Nazca
plate subduction boundary, from 11 my at the north end, to
50 my near 30° S , to 3 my near the Chile Margin triple
junction.
This age variation would also cause a variation
in stresses along the boundary and into the plate.
The
effects of both age and dip angle variations have been
included in the present model for the Nazca plate.
Another factor which could conceivably affect the
slab-pull force is the crossover depth, the depth along
the slab where the stress on the slab changes from downdip
tension to downdip compression (Chappie and Tullis, 1977).
This crossover point should represent the depth where the
deviatoric stress is zero, and would be the depth below
which slab-pull forces are not transmitted to the Nazca
plate.
Earthquake fault solutions for the Nazca plate sub­
duction boundary do not give any clear evidence that this
55
crossover depth varies, along the Nazca plate boundary
(Isacks and Molnar, 19.71; Barazangi and Isacks, 1976 and
1979; Stauder, 1975; Chinn and Isacks, 1983)„
Therefore,
the effects of variable crossover depth have not been
included in the Nazca plate modeling.
Slab-pull forces on the Nazca plate are modeled
by applying a line force along the length of the trench,
scaled in the case of constant force per unit length to
l x 10 15 dyne/cm, equivalent to 200 bars deviatoric tension
across a 50 km thick plate.
Slab-pull forces are applied
in the direction of relative plate motion between the Nazca
and South American plates from Minster and Jordan (1978,
Table 1).
For the case of constant line force per unit
length of trench, the procedure follows that of Richardson
(1978) and Richardson et al. (1979) in previous Nazca plate
and global plate modeling.
shown on Figure 14.
The forces for this case are
In addition to the constant slab-pull
model, angle and age-dependent slab-pull models have also
been considered.
The angle dependent model forces were multiplied
by the sine of the dip angle, and then scaled so that the
total torque exerted was the same as that of the constant
line force model.
A second angle-dependent model, where
the force,was multiplied by the sine squared of the dip
angle, was run in order to include the added effect of
56
Figure 14.
Constant slab-pull forces on the Nazca plate
(Vectors originate at nodes along the trench
boundary and point in the direction of rela­
tive plate motion, proportional to the trench
length belonging to each respective node.)
57
friction exerted by a smaller dip angle.
These angle
dependent forces are shown in Figure 15.
The age dependent model was constructed using an
equation from Carlson et al. (1983), which also includes a
velocity dependence:
Fc
=
S
v
t3/2
(5.9)
where S is a slab force constant, v 3. is the absolute
velocity of the plate, and t is the age of the lithosphere.
These age dependent slab-pull forces were also scaled so
that the total torque from the slab-pull force was the
same as that for the other slab-pull models.
dependent forces appear on Figure 15.
The age
The results of
these various slab-pull models are discussed later.
Drag Forces
The movement of the plates over a viscous asthenosphere may lead to drag forces which resist the plate
motion.
The magnitude of these forces is unknown, but
may be less than a few bars (Melosh, 1977) or as great as
10 bars, depending on the velocity and area of the plate
(Richardson, 1978).
These drag forces may not be signifi­
cant for oceanic lithosphere (Chappie and Tullis, 1977) .
However, there will be a large net torque acting on the
Nazca plate if only ridge and slab forces are modeled.
the plate is to be in mechanical equilibrium, there must
If
p
0®
♦
»
id' s
10® 3 -
*
p
p
»' s-
20* S-
3DeS-
30® 3-
40s 3 -
40® 3 -
90®31
U0®M
30® Si
110* M
lOO'H
90k H
Figure 15.
80l H
a.
b.
70k H
120 M 110* M
100® M
90k W
Angle dependent slab-pull forces
Age dependent slab-pull forces
(Force vectors originate at trench nodes and point
in the direction of relative plate motion. The angle
dependent forces shown here are for the sine squared
of the dip angle.)
oo^h
59
be some balancing forces.
As discussed later, forces from
distant plate boundaries and from transform faults prob­
ably do not contribute significantly to the torque balance.
Thus, drag forces are a likely candidate for the source of
resistive forces to balance the net torque acting on the
plate.
Drag forces are thus chosen to balance the total
torque due to boundary forces from ridges and slabs.
The drag forces are assumed to be proportional to
an absolute velocity of the plate where the pole of rota­
tion of the plate is determined by the torque pole of
other boundary forces.
Thus, the rotation pole need not
bear any relationship to an absolute reference frame de­
fined by hot spots.
A linear drag law is assumed where
the force acts in a direction opposite to the absolute
velocity:
FD
C V abs
(5.10)
— Cm abs
where C is a constant, Vahg is the absolute velocity of
a point on the plate with radius r and a)a^g is the ab­
solute rotation pole for the plate determined from the
torque pole of the non-drag forces.
The constant C is
chosen so that a velocity of 1 cm/yr produces shear
stress of one bar.
The absolute motion poles for the
models are shown on Table 2, along with the magnitude of
the non-drag torque.
The drag forces for one of the dis­
tributed ridge force models is shown on Figure 16.
TABLE 2.
Model
Torque x 10
dyne/cm
32
Description of Nazca plate models
Deg.
Latitude
:
Deg.
Longitude
:
Rate, 10
deg/yr
Ht
1
: ' '
Deg.
Latitude
~
Deg.
Longitude
Figure
DR-A
2.324
60.6 N
52.5 W
DR-B
2.394
61.4 N
52.4 W
——“
DRD-A
2.324
60.6 N
52.5 W
2.6
33.9 N
74.4 W
18
DRD-B
2.735
63.1 N
61.8 W
3.1
32.2 N
80.2 W
19a
DRD-C
2.790
63.0 N
61.1 W
3.2
32.9 N
79.7 W
19b
*DR-A
2.295
64.0 N
66.0 W
——“
———
*DRD-A
2.295
64.0 N
66.0 W
2.7
31.6 N
82.4 W
*DRD-B
2.743
65.3 N
70.8 W
3.1
35.7 N
83.8 W
LR-A
2.731
61.6 N
41.1 W
———
———
-- —
22a
LR-B
2.731
61.6 N
41.4 W
-- —
———
—
22b
LR-C
3.629
60.2 N
59.8 W
LRD-A
3.629
60.2 N
59.8 W
LRD-B
3.629
60.2 N
59.8 W
SD—A
3.364
66.2 N
SD—B
3.364
66.2 N
17a
----
———
17b
20b
,
21a
21b
1 ^
1
1 LO
0J
*3
Z
16.6 N
82.9 W
24a
5.4
16.6 N
82.9 W
24b
79.7 W
3.9
32.0 N
95.8 W
25a
79.7 W
3.9
32.0 N
95.8 W
25b
TABLE 2— Continued
-7
Model
Torque x 10 ^
dyne/cm
Deg.
Latitude
Deg.
Longitude
SD-C
3.364
66.2 N
79.7 W
3.9
32.0 N
95.8 W
26a
SD-D
1.273
79.0 N
53.1 W
3.6
39.5 N
90.0 W
26b
SD-ang
3.364
64.6 N
80.9 W
4.9
19.3 N
94.8 W
27 a
SD-ang2
3.364
63.6 N
81.4 W
5.5
14.2 N
94.4 W
27b
SD-AGE
3.364
65.6 N
80.1 W
4.3
26.7 N
95.4 W
28
DRSD-1
11.278
65.0 N
71.5 W
13.0
32.8 N
84.6 W
30a
DRSD-2
14.587
65.8 N
77.8 W
16.9
32.7 N
87.2 W
30b
DRSD-3
17.916
66.1 N
81.9 W
20.8
32.6 N
88.9 W
31a
DRSD-4
21.256
66.3 N
84.7 W
24.7
32.6 N
90.0 W
31b
LRSD-1
6.899
64.5 N
77.1 W
9.2
23.2 N
88.0 W
32a
LRSD-2
10.230
65.5 N
84.3 W
13.1
25.9 N
90.2 W
32b
LRSD-3
13.577
65.8 N
88.2 W
17.0
27.3 N
91.4 W
33a
LRSD-4
16.930
66.0 N
89.5 W
20.9
28.2 N
92.2 W
33b
V-DR
2.324
60.6 N
52.5 W
———
~~ —
V-DRD
2.324
60.6 N
52.5 W
Rate, 10
deg/yr
2.6
Deg.
.Latitude
33.9 N
Deg.
Longitude
Figure
34a
74.4 W
34b
Cl
H
62
Ml
o'
10*
20*
SO*
40*
80*
Figure 16.
Drag forces balancing distributed ridge forces
(The force vectors originate at the nodes.
In this figure, the drag forces are balancing
in the torque due to distributed ridge forces.)
Transform Fault Shear Forces
Shear stress acting along transform faults would
resist plate motion.
This resistance is evidenced by
seismic activity along the faults.
Estimates of shear
stresses along the San Andreas Fault by heat flow measure
ments indicate a few hundred bars maximum (Brune et al.,
1969).
Extrapolation of in situ measurements to depth
along the San Andreas by Zoback et ad. (1980) did not
resolve the issue of the magnitude of shear stresses
along the San Andreas, because the stress measurements
could be explained by both low and high shear stress
fault models.
Schols et al. (1979) interpreted local
thermal metamorphism and argon depletion along New Zea­
land’s Alpine fault to indicate a shear stress of at
least a few kilobars.
Theoretical modeling of plate motion by Forsyth
and Uyeda (1975), however, showed that shear forces along
transforms were hot an important part of the driving
mechanism of plate tectonics.
Transform fault resistance
has not been included in this present modeling of the
Nazca plate.
Hot Spot Forces
Hot spots may be the result of hot mantle plumes,
and the forces produced by these hot spots would be
64
radially outward directed.
Chappie and Tullis (1977)
estimated that these forces would produce little or no
net force if on a single plate.
Hager and O'Connell
(1981) stated that small scale density contrasts, such
as would be produced by the hot spots, would not exert
a significant net force.
Driving forces from hot spots
on the Nazca plate have not been included in the model­
ing, except to the degree that the elevation pushes on
the surrounding plate for the distributed ridge models.
Distant Forces From Other Plates
There are potential forces from outside the Nazca
plate, but these forces are ignored in the one plate
problem.
Richardson (1978) observed that ridges, being
composed of hot material, do not transmit stresses well
from one side to the other.
Transform faults and slab
margins may transmit stresses from outside the Nazca
plate, although the nature of how well they are trans­
mitted is not known. To the extent that a wide variety
of slab pull models are considered, variable forces
transmitted from the South American plate to the Nazca
plate are at least partially considered.
Transform faults
have been discussed above, and since the total length Of
transform fault boundary is relatively small (except the
Nazca-Antarctic boundary) they will be ignored.
VI„
STRESS MODELS
The results of the stress modeling are shown on
Table 2 and on Figures 17 through 35.
The model names
have the following meanings:
DR — ;—
Distributed Ridge
DRD -- Distributed Ridge with Drag
L R -- -
Line Ridge
LRD -- Line Ridge with Drag
SD — -—
Slab Force with Drag
V — ■--- Viscous Rheology
The results of varying the initial and boundary condi­
tions for each of these kinds of models are described
below, followed by the results of combining various
ridge and slab models.
Distributed Ridge Models
DR Models
These models have a pinned subduction boundary
and no drag forces.
The subduction boundary nodes are
pinned in both the latitudinal and longitudinal direc­
tions to approximate the behavior of the boundary in
between major stress release events.
65
In model DR-A the
66
entire subduction zone (16 node points) was pinned.
The
total torque due to the distributed ridge forces was
32
2.32 x 10 dyne-cm. For model DR-B, the three northern­
most subduction zone nodes were unpinned. The total
32
torque increased slightly to 2.40 x 10
dyne-cm. The
poles for this distributed ridge torque were only mar­
ginally affected by this change in boundary conditions
(Table 2).
The stress states of the Nazca plate for these
two models are Shown on Figure 17.
The only difference
in the two models is that there is much less deviatoric
tension in the northeast corner of the plate for model
DR-B than for model DR-A.
The Nazca plate can be divided into five different
stress regions for the DR models.
The region along the
Spreading ridges shows compression oriented parallel to
the ridges .
A .second region is in the vicinity just west
of the 1965 earthquake, where the stress is biaxial com­
pression.
A third region is to the east and south of the
earthquake, covering most of the interior of the plate,
where northwest-southeast to east-west uniaxial compres­
sion occurs.
A fourth stress region is located to the
southeast, where primarily east-west and northwestsoutheast compression occurs, with a smaller secondary
compression axis.
And finally, the northeast and south­
west corners of the plate show deviatoric tension.
•
*
i
<4,i— <
/,
v
XX'
/
J
XX
/:
r
\ \
XX
%
XX
s
"
-
O
'
x x
X
3
X
l—
1
IO C
B«ur^
Figure 17.
a.
b.
Stress state for Model DR-a
Stress state for Model DR-b
(Principal horizontal deviatoric stresses are represented by arrows.
Compressive stresses have no arrowheads. Deviatoric tensional stresses
have outward pointing arrowheads. Tensional stresses smaller than the
depicted arrowheads are not shown. Stresses are centered at element
centroids and are displayed for only some of the elements for ease of
viewing. See text for desription of models.)
68
The 1965 earthquake is located in an area of
stress transition between two regions of different stress:
biaxial compression to the west, and northwest-southeast
uniaxial compression on the east.
This may be a region
of instability because there is a change in stress orienta­
tion and therefore is an area where earthquakes might be
more likely to occur.
D.RD Models
These models have distributed ridge forces and
drag forces which balance the torque due to the ridge
forces.
In model DRD-A, the entire subduetion zone was
pinned.
In model DRD-B only three degrees of freedom
were pinned on the subduction boundary, the latitudinal
and longitudinal directions at the Chile Margin triple
junction, and the longitudinal direction at the northern­
most node point.
In DRD-C, three degrees of freedom were
pinned in the interior of the plate, rather than along
the subduction zone.
The stress states in the interior of the Nazca
plate for the DRD models is significantly different from
the non-drag DR models (Figs. 18 and 19).
The two types
of models are similar to the west of the 1965 earthquake
where there is biaxial compression, and along the East
Pacific Rise, where there is compression parallel to the
69
\ I i
Figure 18.
Stress state for Model DRD-a
(See Fig. 17 for explanation of symbols.
See text for description of models.)
__JL£
&
o
\ <p
■
'
I
Figure 19
Stress state for Model DRD-b
Stress state for Model DRD-c
(See Fig. 17 for explanation Of symbols.
See text for description of models.)
o
71
ridge.
However, in the DRD models most of the plate is
in a state of nearly north-south compression, with a
small region of east-west compression along the subduction
boundary adjacent to Peru.
Deviatoric tension occurs at
the northeast and southwest corners again, and in addi­
tion, there is tension in the center of the plate and in
the southeast corner.
Model DRD-B differs from DRD-A in the following
ways.
There is more east-west tension in the middle of
the plate.
Also, the tension in the northeast corner is
no longer present, and the east-west compression adjacent
to Peru vanishes.
The torque pole shifted slightly to
the north and west for model DRD-B, and the rotation rate
increased.
Model DRD-C is practically identical to the DRD-B
model, with slightly more tension in the northeast corner.
Refined Grid Models
A refined grid was created to test the sensitivity
of the model results to grid size.
fined grid are denoted by *DR.
Models with the re­
The resulting stress
models below are not significantly different from those
using the coarser grid.
This indicates that the coarser
grid is probably adequate for stress modeling of the Nazca
plate and that future modeling does not need to be as
detailed as the fine grid.
In model *DR-A, similar to model DR-A, the subdue
tion zone is totally pinned and there is no drag force.
The total torque is almost the same as for the DR models,
32
2.30 x 10
dyne-cm. The pole for the torque is slightly
north and west of the coarser grid pole for model DR-A.
The distributed ridge forces for the refined grid are
illustrated on Figure 20.
The stress state for model
*DR-A (Fig. 20) is very similar to DR-A with the same
five stress regions described earlier.
However, the
biaxial compressive region to the west of the 1965 earth­
quake in model DR-A extends eastward to longitude 90° W
in the *DR-A model.
Another difference is that there is
less tension in the southwest corner for the refined grid
model.
Model *DRD-A is analogous to DRD-A with the subduction zone pinned, balanced by drag.
The refined grid
absolute velocity pole is almost 10 degrees further west
than that of the DRD-A model.
The rate of rotation is
slightly faster for the refined grid, but the stress
states of the two models are essentially identical.
Model *DRD-B has three points pinned on the subduction zone, analogous to model DRD-B.
The pole for
absolute motion is further north and west than that of
model DRD-B, and the rate of rotation is slightly slower.
The stress states for models DRD-B and *DRD-B are nearly
w ■
T
4
X
;r(l
4,
t
'/ /
•'
*- * -
*
»
>?A
f j ; > 1/|V
! 1f n * **
4 ; ; ‘v r
9 1
A
Jg
W /
9
uo* n
liei"m
ico'*
Figure 20.
a.
b.
m k*
1
I
4
m 1*
H• m
Distributed ridge forces for refined grid
Stress state for Model *DR-a
(See Fig. 1] for explanation of force vectors. See Fig. 17
for explanation of stress symbols. See text for description
of models.)
74
identical.
The stress states of models *DRD-A and *DRD-B
are shown on Figure 21.
Line Ridge Models
LR Models
The LR models have a line ridge force which is
equivalent to 200 bars across a 50 km thick plate, and no
drag.
Model LR-A has a completely pinned subduction zone,
and only the East Pacific Rise ridge forces were applied.
The total torque was 2.73 x 10"^ dyne-cm, about 18 percent
larger than the total torque on the distributed ridge
models. Model LR-B is identical except the subduction
zone is only partially pinned.
For both of the models
the stress state is very similar (Fig. 22) with uniform
compression in the direction of plate motion, and one
area of biaxial compression in the northwest corner of
the plate.
For model LR-B, the northeast corner becomes
north-south instead of east-west compression.
In model LR-C the line forces for the Galapagos
Spreading Center and the Chile Ridge were added.
The
torque pole shifted westward, even further westward than
the distributed ridge model torque pole. The total
32
torque increased to. 3,62 x 10
dyne-cm. The stress
state did not change significantly (Fig. 23).
100 M
UD* M 110* M
a.
b.
Stress state for Model *DRD-a
Stress state for Model *DRD-b
(See Fig. 17 for explanation of symbols.
See text for description of models.)
Figure 22.
a.
b.
Stress state for Model LR-a
Stress state for Model LR-b
(See Fig. 17 for explanation of symbols.
See text for description of models.)
-j
CTi
LATITUDE
77
Figure 23.
Stress state for Model LR-c
(See Fig. 17 for explanation of symbols
See text for description of models.)
LRD Models
The LRD models have line ridge forces at all
three ridges equivalent to 200 bars, and the torque from
this line ridge force is balanced by drag.
Model LRD-A
has a totally pinned subduction zone, and the torque pole
is much further south and west than the pole for the LR
models (Table 2).
The absolute velocity was much slower
than that of the distributed ridge models.
The stress
state for LRD-A (Fig. 24) differs from the LR models
primarily because the east-west compression diminishes
east of the 90° W longitude.
Also, tension in the middle
of the Chile Ridge is present.
Model LRD-B has only three degrees of freedom
constrained from motion on the subduction zone.
This
has no effect on the torque pole, but the stress state
changes.
The LRD-B model stress state (Fig. 24) has
compression parallel to the subduction zone along the
subduction boundary and tension across the Chile Ridge.
Slab-Pull Models
SD Models
The slab-pull SD models have line forces along
the subduction zone which average 200 bars across the
50 km thick Nazca plate, and the torque is balanced by
drag forces.
Model SD-rA has three degrees of freedom
pinned along the subduction zone.
Models SD-B and SD-C
Figure 24.
a.
b.
Stress state for Model LRD-a
Stress state for Model LRD-b
(See Fig. 17 for explanation of symbols.
See text for description of models.)
V£>
have three degrees of freedom pinned within the plate.
32
The torque fof these models is 3.36 x 10
dyne-cm,
similar to the total torque of the LR models (Table 2).
The stress states for these three models are almost
identical (Figs. 25 and 26), and all show deviatoric
tension in the direction of plate motion from about
longitude 95° W to the subduction boundary.
There is
also a secondary stress axis along the subduction zone,
which is north-south compression.
Model SD-D was run with only slab-pull along
the north end of the subduction zone to see what effect
this would have on the north-south compression along the
southern part of the Peru—Chile trench.
The compression
was still present on the south end (Fig. 26), even though
there was no pull on this end.
This may indicate that
the north-south compression is related to the geometry
of the subduction zone.
These models are essentially
stress-free in the vicinity of the 1965 earthquake.
Angle Dependent Models
The first angle dependent model, called SD-Ang,
had three degrees of freedom pinned in the interior of
the plate, and the forces were balanced by drag.
The
slab forces were multiplied by the sine of the angle of
dip of the slab and then scaled so that the total torque
was the same as for the SD models.
The only difference
Figure 25.
a.
b.
Stress state for Model SD-a
Stress state for Model SD-b
(See Fig. 17 for explanation of symbols.
See text for description of models.)
00
H
Figure 26.
a.
b.
Stress state for Model SD-c
Stress state for Model SD-d
(See Fig. 17 for explanation of symbols.
See text for description of models.)
00
K>
83
between this model and SD-Ang2 was that the latter model
forces were multiplied by the sine squared of the dip
angle.
These forces are shown on Figure 15.
The stresses for these two models are shown on
Figures 27 and 29.
The stresses for SD-Ang are tensile
over most of the plate east of longitude 100° W.
The
magnitude of the stresses are greatest in the northeast
corner, in the middle of the trench, and in the southern
border of the plate, responding to the larger dip angles
in these sections.
The two regions of shallow dip show
north-south compression.
The stresses on model SD-Ang2 were similar to
those of model SD-Ang except there was more compression
along the shallow zones because the differences in the
angle-dependent forces were accentuated.
Age Dependent Model .
The age dependent slab-pull model, called SD-Age,
had three degrees of freedom pinned in the interior of
the plate, and was balanced by drag.
The total torque
was scaled to that of the other slab-pull models, 3.364 x
32
10
dyne-cm. The stresses for this model are displayed
on Figures 28 and 29.
The stresses are entirely tensionsi,
in the direction of plate motion, and are greatest
between 20° and 35° S, where the slab is the oldest.
There is a small component of tension perpendicular to
Ml *•
Figure 27
Stress state for Model SD-ang
Stress state for Model SD-ang2
(See Fig. 17 for explanation of symbols.
See text for description of models.)
00
85
Figure 28.
Stress state for Model SD-age
(See Fig. 17 for explanation of symbols.
See text for description of models.)
86
Figure 29.
Composite of stress states for Models SD-c,
SD-ang, SD-ang2, and SD-age
(See Fig. 17 for explanation of symbols.
See text for description of models.)
87
the major tensional stresses.
Tension along the slab
north of 10° S and south of 40° S disappears where the
subducting slab is the youngest.
Combined Ridge-Push and Slab-Pull Models
DRSD Models
These models combined a distributed ridge force,
a slab-pull force, and were balanced by drag.
Three
degrees of freedom were pinned in the interior of the
plate.
The distributed ridge force was scaled differ­
ently from the previous distributed ridge models, a factor
of .348 larger.
The slab-pull forces were increased pro­
gressively from 200 bars to 800 bars in the four DRSD
models.
The total torque resulting from the non-drag
32
forces ranged from 11.28 to 21.26 x 10
dyne-cm.
The stress states for these four models are shown
in Figures 30 and 31.
As expected, the west side of the
plate indicates predominately compression and the east
side indicates mostly tensile stresses.
The region of
the 1965 earthquake fits the data best for the first of
these models, DRSD-1, where the forces from distributed
ridge forces and slab-pull forces are nearly the same.
As the slab-pull increases in models DRSD-2, DRSD-3 and
DRSD-4, the region becomes increasingly tensile, although
the north-south compression remains.
Mf
M» HI
H
O'
10*s
ao's
90* S
40* S
90*9
U0*M
Figure 30.
a.
b.
Stress state for Model DRSD-1
Stress state for Model DRSD-2
(See Fig. 17 for explanation of symbols.
See text for description of models.)
00
CO
J
^
/
J
i
44-”
\
y
.
/j
1
1
'its*'
\
\
t
---4
4— ;
^30
sJ-r--120*M no*M
3aci
100* N
90
M
1
90 M
Figure 31.
i
70
a.
b.
120 M 110” M
100” M
Stress state for Model DRSD-3
Stress state for Model DRSD-4
(See Fig. 17 for explanation of symbols.
See text for description of models.)
90
LRSD Models
The LRSD models combined a line ridge force with
slab-pull forces, balanced by drag forces.
The line ridge
and slab-pull forces were equal on LRSD-1, equivalent to
200 bars compressive and tensile stress respectively
across a 50 km thick plate.
The other three models had
slab-pull forces equivalent to 400, 600 and 800 bars
tensile stress combined with a line ridge force of 200
bars.
The total torque from the non-drag forces for these
32
four models ranged from 6.90 to 16.93 x 10
dyne-cm, and
the pole of absolute motion moved increasingly northwest­
ward, responding to the increasing pull from the slab.
The stress states for these four models are shown
on Figures 32 and 33.
The first of these models, LRSD-1,
is dominated by the line ridge force, except along the
subduction zone where tensile stresses occur.
The com­
pression in the region of the 1965 earthquake is still
present in model LRSD-2, with about half of the plate in
compression, and the other half dominated by tension.
The slab-pull forces dominate the line ridge forces in
the models LRSD-3 and LRSD-4.
Viscous Models
Stresses and velocities for two viscous models
were calculated using distributed ridge forces, and then
compared with the relative velocity data for the
Figure 32.
a.
b.
Stress state for Model LRSD-1
Stress state for Model LRSD-2
(See Fig. 17 for explanation of symbols.
See text for description of models.)
VO
I-1
tv N Y
Figure 33.
a.
b.
Stress state for Model LRSD-3
Stress state for Model LRSD-4
(See Fig. 17 for explanation of symbols.
See text for description of models.)
VO
KJ
93
Nazca-South American plate boundary.
The first model
V-DR had distributed ridge forces, no drag, and a totally
pinned subduetion zone.
The second model, V-DRD, had
distributed ridge forces which were balanced by drag
forces, with a pinned subduction zone.
Elements along
the subduction zone had a viscosity that was smaller
than plate interior viscosity by a factor of 20.
The stresses resulting from the two models are
shown on Figure 34.
Model V-DR is similar to the elastic
DRD-A model, with the following differences.
There is
less tension in the northeast corner for the viscous
model, and there is less compression in the interior of
the plate relative to the Subduction zone.
Large biaxial
compressive stresses occur along the subduction zone, and
the largest stresses are located in the southeast region
of the plate.
A line of tension crosses the plate in an
east-west trend at around 23° S.
The viscous V-DRD model differs from the elastic
DRD-A model in the following ways.
First, there is less
compression along the East Pacific Rise boundary, and the
compressive stress axes change orientation along the trace
of the ridge.
Also, in the viscous model there is a
linear trend of tensile stresses between 20° S and 30° S.
A third difference is that there is more east-west compres­
sion in the northern part of the plate for the viscous
case.
W
»V IV
0* -
x^-
\
IV
o* -
i
'
>K
10* 3 -
<r
+
10*3-
+f \
\
ao's-
\
x \-
\
aoe 8 -
\
/
*
\
30* 3*
'
1\
I
>
\
40*9
l-------------<
80 Sio *M
Figure 34.
a.
b.
110* H
72.0 Baits
100* M
80l M
e 1*
70fcll
Stress state for Model V-DR
Stress state for Model V-DRD
(See Fig. 17 for explanation of symbols.
See text for description of models.)
VO
The velocities for the two models are displayed
on Figure 35.
The data for the displacements were in­
verted to obtain the best fitting pole of rotation
(Richardson, 1982).
The velocity pole for the V-DR
model was located at 17.8° N, 53.5° W, with a rate of
1.6 degrees per million years.
The rate is not signifi­
cant because it depends on a trade-off between the
magnitude of the forces and the viscosity of the plate.
Increasing the viscosity by a factor of two will decrease
the velocity by one-half to the observed rate of 0.8
degrees per million years (Minster and Jordan, 1978).
Such an increase in viscosity is well within the uncer­
tainty of the appropriate viscosity for the lithosphere.
The location of the velocity pole was over 50 degrees
away from that of Minster and Jordan (1978, Table 1),
indicating at best a qualitative agreement between the
two poles.
The velocities for the other viscous model,
V-DRD, were scattered (Fig. 35) and not resolvable into
a single pole of rotation, indicating that this model
did not behave as a single plate.
mi
n
O'
id ' s
5
v
(
v
x
\l
ux
\ ^
3 0 '3
40* 3
ft
v ^v:
\ \
ao's
Ml
N X '
V x X
X .*
X X
5 0 '3 :
jj
:
\ \
to'3
ao's
3 0 '3
4 0 '3
4
\x
X\ \
\ \ v \ \ ' k• ‘
•
~ r i 1
f1
4 ‘
#
♦
i 4
, 4
y
&
4
y
/ /‘ (
• f
& ♦ r# *
» ♦•
5 0 '3
5 0 'N
Figure 35.
i
f
>*
• X* u
\ \ \ >
>» *
\ \ \ ^
N \
4S
/ / /
120'H
a.
b.
110'N
lOO'M
80* M
Velocities for Model V-DR
Velocities for Model V-DRD
(Velocity vectors originate at element centroids.
See text for description of models.)
80*8
VII.
CONCLUSIONS
By comparing the various stress models with each
other and with the stress data, the following conclusions
can be made.
1.
The stress states for the distributed ridge
models are significantly different from those of the line
ridge models.
The distributed ridge models represent an
improvement over the previous models, because they are
probably more realistic, reflecting several different
Stress regions in the Nazca plate.
There is compression
in the vicinity of the 1965 earthquake, and this region
is also an area of stress transition.
Deviatoric tension
in the northeast corner of the plate agrees with the
extensional features observed in the Panama Basin.
2.
The use of a refined grid, with approximately
2.5° grid point spacing, for the distributed ridge force
models does not significantly change the stress state of
the models.
This indicates that the coarse grid, with
approximately 5° grid point spacing, is sufficient for
modeling distributed ridge forces on the Nazca plate.
3.
The inclusion of drag forces results in very
different models for both line ridge and distributed
ridge models.
The stresses are rotated approximately
90 degrees in both cases when drag forces are introduced.
97
98
The drag force models do not fit the stress data as well
as the models without drag.
This may indicate that drag
forces are hot significant for the Nazca plate, or that
the drag forces are more complex than modeled.
4.
Age-dependent and angle-dependent models show
some of the possible variations in stress along the com­
plex subduction zone.
The age-dependent models are in
better agreement with the proposed rifting along the
Mendana Fracture Zone at around 10° S.
The other slab-
pull models show north-south compression in the region
of the Mendana Fracture Zone, which is just the opposite
orientation expected in an area which may be experiencing
rifting.
5.
The angle-dependent slab-pull models are in
poorest agreement with the stress data for the region of
possible extension near the Mendana Fracture Zone.
This
region is one of the shallow dipping slab segments
described by Barazangi and Isacks (1976) and yet the
resulting stresses are compressive parallel to the trench.
Since the shallow angle stress results do not fit the
data, a shallow subduction angle in this region may not
be appropriate.
Hasegawa and Sacks (1981) suggested that
the slab is contorted rather than segmented, and that the
entire length of slab dips initially at a 30 degree angle.
The contortions of the slab occur at a few hundred kilom­
eters depth which is below the cross—over depth for
99
transition from down-dip tension to down-dip compression„
Therefore, a contorted slab model would have a uniform
dip angle in the upper region where slab-pull forces
occur.
The stress results favor this contorted model
over the segmented model.
6.
Combination models which have both slab-pull,
ridge-push, and drag forces, with increasing amounts of
slab-pull fit the data best when the ridge-push and slabpull forces are approximately equal, or when the slabpull is at most twice the ridge-push force.
This result
agrees with previous stress modeling of the Nazca plate
by Richardson (1978).
The absolute velocities become
exceedingly high as the slab-pull is increased (Table 2)
but this is a function of the torque magnitude which could
be scaled down.
7.
The viscous models of distributed ridge forces
with no balancing drag roughly agree with the relative
velocity direction for the Nazca-South American plate
boundary.
The viscous models with drag forces result in
non-plate behavior where it is impossible to obtain a
single pole of rotation for the plate.
This suggests
that the drag forces may not be important, or that perhaps
they should be modeled in a different manner.
The viscous
models allow expansion at the three ridge boundaries.
Since the southern ridge boundary has significant amounts
of transform fault lengths between the ridge segments,
100
the viscous model should somehow be constrained from
expanding along the southern boundary.
This might result
in more realistic velocity directions.
Several areas of future research could signifi­
cantly improve the stress modeling of the Nazea plate.
The addition of transform fault resistance is one obvious
area of improving the driving-force modeling.
Another
area of future research might be the effect of modeling
thickness variations as a function of age throughout the
entire plate.
The inclusion of the two microplates, the
Panama and Easter plates, might lead to a better under­
standing of the dynamics of ridge and transform fault
jumping.
Other improvements would include the thermal
effects of hot spots and for viscous modeling, the con­
straint of expansion along the southern boundary.
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