The gravitational potential energy of the Earth's lithosphere.
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The gravitational potential energy of the Earth's lithosphere
Coblentz, David Dwight, Ph.D.
The University of Arizona, 1994
V·M·I 300 N. Zeeb Rd. Ann Arbor, MI48106
------The Gravitational Potential Energy of the Earth's Lithosphere
by David Dwight Coblentz
A Dissertation Submitted to the Faculty of the DEPARTMENT OF GEOSCIENCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA
1 994 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that we have
read the dissertation prepared by ____~D~a~v~i~d~D~w~i~g~h~t~C~o~b~l~e~n~t~z ______
entitled The Gravitational Potential Energy of the Earth's Lithosphere
and recommend that it be accepted as fulfilling the dissertation
requirement for the Degree of Doctor of Philosophy
Anril 111 1994 Date April 11, 199LI Date
April 11, 1994 Date
Date
Date
Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation r.equirement.
7!c.~tct(~. ,2· /fc<-r~.L_~ Randall M. Richardson April 11, 1994 Dissertation Director Date I.'
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the Uni versity Library to be made available to borrowers under rules of the library. Brief quotations from this dissertation are allowable without special per mission, provided that accurate acknowledgment of SOUl'ce is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole 01' in part may be granted by the head of the major department or the Dean of the Graduate College when in his 01' her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED:L2~ ~ ACKNOWLEDGMENTS
Many people have helped to make this project a reality. To my principal ad visor, Randall Richardson, I extend special thanks for providing motivation, tech nical expertise and guidance during my tenure at the University of Arizona. I should not neglect mentioning the fruitful time we spent discussing geodynamic problems at the ping-pong table. I also thank other members of my committee and faculty of the geophysics group for their advice and criticism, including Susan Beck, Roy Johnson, Clem Chase, and Terry Wallace. I am especially grateful for the help Steve Sorenson provided on computer-related problems and bugs. Many of my fellow students, including Steve Myers, Sterling Cook and Wolfram Schuh, made the path easier with their helpful distractions. My minor advisor Barbara Kosta provided me with inspiration, encouragement and introduced me to many wonderful "points of departure".
Much of the research presented in this dissertation "vas completed during a short visit to the University of Adelaide in 1993. I thank Richard Hillis for the opportunity to work on the Indo-Australian stress project and Shaohua Zhou for all his assistance with LaTex and ANSYS. I am especially indebted to Kurt Stuewe, Karin Ehlers, and Martin a.nd Eva Kennedy who made our visit most enjoyable, both in the department and on the cliffs of Mount Arapiles. I reserve special thanks for Mike Sandiford who taught me the magic of Mathematica and LaTex. Without his motivation, help and guidance this dissertation would have been significantly less complete. I am particularly indebted to him for his help with the mathematical formulations presented in Chapters 2 and 6. Chris Pigram of AGSO is thanked for input into the location and nature of the northern Indo-Australian plate boundary.
I would like to thank my non-departmental friends Mike, Andrea, James and Beverly for helping me keep it all in perspective. Finally, and most importantly, Kristine is especially acknowledged for her help in overall "stress management".
Parts of this study were funded by the Australian Cooperative Research Centre as part of a study of the factors controlling the stress field of the North West Shelf of Australia. Partial funding was also provided by NSF Grant No. EAR-921931 l1. 5 TABLE OF CONTENTS
LIST OF FIGURES. 7
LIST OF TABLES. 9
ABSTRACT .... 10
CHAPTER 1 : INTRODUCTION 12
CHAPTER 2 : THE POTENTIAL ENERGY FORMULATION 18 2.1 Introduction ...... 18 2.2 Theoretical Basis...... 18 2.3 Lithospheric Density Models ...... 20 2.:3.1 Young Oceanic Lithosphere. 22 2.3.2 Oceanic Basin ...... 22 2.3.3 Continental Lithosphere. . . 23 2.3.4 Constraint on the Density of the Continental Crust 25 2.4 Global and Plate Scale Lithospheric Potential Energy Estimates 27 2.5 Discussion ...... 32 2.5.1 Potential-Energy Torques. . . 33 2.5.2 The Tectonic Reference State. 37 2.5.3 The Ambient Stress State. . . 39
CHAPTER 3 : TRENDS IN THE INTRAPLATE STRESS FIELD 114 3.1 Introduction ...... ,14 3.2 Statistical Method ...... '15 3.3 Trends in the World Stress Map Da.tabase . 47 3.4 Discussion ...... 55
CHAPTER 4 : SOUTH AMERICAN INTRAPLATE-STRESS MAG- NITUDE...... 60 '1.1 Introduction ...... 60 ,1.2 Topography and Deformation Style. 63 4.3 Finite-Element Modeling ...... 64 '1.3.1 Modeling Location. . . . 65 L1.3.2 Description of the Finite-Element Grid 69 4.3.3 Modeling Results ...... 71 4.4 Discussion ...... 76
CHAPTER 5 : INTRAPLATE STRESSES DUE TO POTENTIAL- ENERGY VARIATIONS: A FINITE-ELEMENT ANALYSIS 80 5.1 Introduction ...... 80 5.2 Discussion of Tectonic Forces ...... 81 6
TABLE OF CONTENTS - Continued 5.2.1 Buoyancy Forces .... 81 5.2.2 Collisional Boundaries .. 83 5.2.3 Basal-Shear Tractions .. 85 5.3 Discussion of the Modeling Method 86 5A The African Plate ...... 87 5A.1 Description of the Plate. 89 5A.2 The Regional Intraplate Stress Field. 89 5.'1.3 Modeling Results 91 5AA Discussion ...... 95 5.5 The South American Pla.te ..... 97 5.5.1 Description of the Plate. 99 5.5.2 The Regional Intraplate Stress Field . 101 5.5.3 Modeling Results 104 5.5.'1 Discllssion ...... 118 5.6 The Indo-Australian Plate .... . 121 5.6.1 Description of the Plate. 123 5.6.2 The Regional Intraplate Stress Field. 125 5.6.3 Modeling Results ...... 130 5.6,4 The North West Shelf Stress Field . 146 5.6.5 Discussion ...... 1<19
CHAPTER 6 : TIME-EVOLUTION OF PLATE-SCALE POTENTIAL- ENERGY DISTRIBUTIONS...... 158 6.1 Introdudion ...... 158 6.2 Geoid Anomalies and Potential-Energy Variations 159 6.3 Circular Pla.te Model ...... 165 6.'1 The Aging of the African and Antardic Plates 173 6A.l The African Plate .. 173 6.'1.2 The Antarctic Plate . 180 6.5 Discllssion ...... 183
CHAPTER 7 : CONCLUSIONS 190
APPENDIX A: THE CLOSED-FORM SOLUTIONS FOR THE POTENTIAL ENERGY OF THE LITHOSPHERE...... 196
APPENDIX B: THE DEPENDENCE OF THE POTENTIAL EN ERGY CALCULATION ON THE MECHANISM OF ISOSTATIC SUPPORT. . . . 198
REFERENCES. 200 7 LIST OF FIGURES
Figure 2.1 Schematic of the Lithospheric Depth-Density Distributions 24 Figure 2.2 Potential Energy as a Function of pc and Topography. 26 Figure 2.3 Topography and Potential-Energy Distributions . . . 28 Figure 2,4 Calculated Potential Energy Versus Topography . . . 31 Figure 2.5 U/ Torques vs Absolute Plate Velocities, Normalized 35 Figure 2.6 U/ Torques vs Absolute Plate Velocities, Absolute 36
Figure 3.1 Global Stress Indicators . . . 48 Figure 3.2 Average Stress Regime . . . . L19 Figure 3.3 Average SH,l1H1X Orientations. 51 Figure 3A Trends in the SH,mnx Orientations. 52 Figure 3.5 SH,ma,,' Trends Versus Ridge-Push Torques 53 Figure 3.6 SH,l1H1X Trends Versus Absolute Plate Velocities M
Figure 4.1 Stress Indicators for the Cordillera Blanca Region of Peru 66 Figure 4.2 Stress Indicators Plotted Versus Topography Along the Profile. 68 Figure 4.3 Boundary Conditions Applied to the Finite-Element Grid 70 Figure 4,4 Principal Non-Hydrosta.tic Stresses Along the Profile . . . 73 Figure L1.5 Contours of the Calculated Shear Stress ...... 7Ll Figure 4.6 Predicted Compressional and Extensional Stress Regimes. 75
Figure 5.1 Total Torques and Misfits ...... 84 Figure 5.2 Geometry and Stress Indicators ror the African Plate 90 Figure 5.3 The Finite-Element Grid Used for the African Plate 92 Figure 5,4 Predicted Stresses, African Plate ...... 93 Figure 5.5 Geometry of South American Plate ...... 100 Figure 5.6 Stress Indicators for the South American Plate ... 103 Figure 5.7 The Finite-Element Grid Used for the South American Plate. 105 Figure 5.8 Predicted stresses for Modell, SAP 107 Figure 5.9 Predicted Stresses for Model 2, SAP 108 Figure 5.10 Predicted Stresses for Model 3, SAP 110 Figure 5.11 Predicted Stresses for Model Ll, SAP 112 Figure 5.12 Isodepths to the Subducting Nazca Plate. 11L1 Figtll'e 5.13 Predicted Stresses for Model 5, SAP . . . 116 Figtll'e 5.1Ll Predicted Stresses for The Andes Region, Model 5 117 Figure 5.15 The Geometry of the Indo-Australian Plate 124 Figure 5.16 Stress Indicators for the Indo-Australian ...... 127 Figure 5.17 SH,ntax Trends in the Indo-Australian Plate 131 Figure 5.18 The Finite-Element Grid Used for the Indo-Australian Plate 132 Figure 5.19 Predicted Stresses ror Modell, lAP 134 Figure 5.20 Predicted Stresses for Model 2, lAP 136 Figure 5.21 Predicted Stresses for Model 3, lAP 138 8
LIST OF FIGURES - Continued Figure 5.22 Predicted Stresses for Model 4, lAP ...... 140 Figure 5.23 Predicted Stresses for Model 1, Central Australia 142 Figure 5.24 Predicted Stresses for Model 2, Central Australia 1.44 Figure 5.25 Predicted Stresses for Model il, Central Australia 1i15 Figure 5.26 Predicted St.resses for the North West Shelf 147 Figure 5.27 Predicted Stresses for the North West Shelf ... 148
Figure 6.1 Geoid Anomaly over Cooling Oceanic Lithosphere. 161 Figure 6.2 Observed Geoid Anomaly over Continental Margins. 1M Figure 6.3 Percentage of Lithospheric Types, Circular Case. 1.68 Figure 6A Mean Potential Energy vs Age, Circular Model 170 Figure 6.5 L1 U vs Age for a Range of U~, Circular Plate .. 171 Figure 6.6 Dependence of ~ U{ It on 10', Circular Plate ... 17il Figure 6.7 Area of the African Plate Over the Last 163 Ma . 176 Figure 6.S Area, Percentage, and ~ U{ for the African Plate 177 Figure 6.9 Cumulative Area of the Antarctic Plate Over the Last 163 Ma . 181 Figure 6.10 Area, Percentage, and ~ U{ for the Antarctic Plate 182 Figure 6.11 Age of Ocea.nic Lithosphere for UI = UI (Africa) . 187
Figure B.l ~Uh for Three Surface Elevation Values ..... 199 .'
9 LIST OF TABLES
Table 2.1 Values of Parameters Used in Calculations ...... 41 Table 2.2 Geoid Anomaly Across a Typical Continental Margin ...... 41 Table 2.3 Topogra.phic Distribution of the Earth and the Seven Ma.jor Plates ill Table 2.4 uf for the Earth and the Seven Major Plates 42 Table 2.5 Continental Elevation Information ...... ,12 Table 2.6 Total Torque and Angular Misfit Information i13
Table 3.1 Global Stress Indicators . . . . 58 Table 3.2 Absolute Plate Velocity Poles. 59
Table 4.1 Stress Indicators for the Cordillera Blanca Region, Peru 79
Table 5.1 Relative Torque Contributions, African Plate ... 153 Table 5.2 Potential-Energy Means and Stress, African Plate 153 Table 5.3 Summary of Stress Indicators, SAP .. . . 154 Table 5A Description of Force Models, SAP ...... t54 Table 5.5 Torque Pa.rameters for Force Models, SAP 155 Table 5.6 Boundary Forces and Torque Contributions, SAP 155 Table 5.7 Summary of Stress Indicators, lAP...... 156 Table 5.8 Boundary Force Ma.gnitudes and Torque Contributions, lAP 157
Table 6.1 Values of Parameters Used in Calculations .... 188 Table 6.2 Definitions of Abbreviations Used in Calculations 188 Table 6.3 Area of the African plat.e for the Last 163 Ma 189 Table 6.'1 Area of the Antarctic plate for the Last 163 Ma . 189 .'
10
ABSTRACT
This dissertation explores the tectonic implications of lateral potential-energy variations within the Earth's lithosphere. The central hypothesis of this study is that much of the observed intraplate-stress field is explainable in terms of these variations. The study develops the concept of a tectonic TeJe1'encc state (TRS), defined here as the mean potential energy of the lithosphere, U/. A simple, first-order lithospheric density model is used to calculate U/, which was found to correspond to both near-sealevel continental lithosphere and cooling oceanic lithosphere at a depth of ,1.3 km. Although the potential energy of the continental lithosphere with elevated topography is sensitive to the assumed crustal density, pc, both global and plate-scale TRS values were found to be robust. A finite element analysis of the intraplate stress field in the African, South Amer ican and Indo-Australian plates was used to demonstrate that many of the long wavelength features in the intraplate stress fields of these plates can be explained in terms of lateral variations in the lithospheric potent.ial energy. Constraints for the numerical modeling were provided by an analysis of the long-wavelength trends in the intraplate stress field and a study of the intraplate stress magnitude in the South American plate. It was found that the value of U/ changes significantly in response to the aging of oceanic lithosphere such that continental regions become increasingly susceptible to extensional collapse as the plate ages. In the case of the African and Antarctic plates, the aging of the ocean lithosphere since the late Jurassic has contributed a mean stress difference of about 5 MPa and 7 ..5 MPa (averaged over a 125km-thick lithosphere) in the respective continents. This extensional stress may contribute as much as 50% of cumulative force needed to deform continental lithosphere. One important implication of this calculation is that rifting in present-day continents 11 and the breakup of supercontinents may be explained in terms of the time-evolution of the potential-energy distribution. 12
CHAPTER 1 INTRODUCTION
Plate tectonics has provided a kinematic framework that has revolutionized our understanding of the behavior of the lithosphere in the modern Earth. While' it is now widely recognized within both the geological and wider scientific commu nities that many important geological phenomena can be related to the relative motion of the lithospheric "plates" [Minslc/' and Jordan, 1978], fundamental un certainty remains in our understanding of the dynamics of lithospheric motion. In particular, the origin and nature of the forces that maintain plate motion remain highly contentious, with the debate focusing on the roles of va.rious contributing processes such as ridge push (and otlter intra-plate density variations), sla.b pull, and tractions imposed by the convective mantIc at the base of the plates. This study explores the relationship between variations in the gravita.tiona.1 potential energy of the Earth's lithosphere and the intraplate Htress field in order to evalu ate the role lateral density variations play in the dynamics of plate motion. The principal hypothesis of this study is that much of the global intraplate stress field and associated lithospheric deformation, especially on the continents, is the prod uct of lateral density variations in the lithosphere. Thus, the results of this stuely have important implications for our understanding of many fundamental tectonic processes including the evolution of orogenic systems, rift and basin development, and the source of the intraplate stress fielel. Lateral density variations in the lithosphere have long been associated with variations in lithospheric gravitational potential energy, and are recognized as an important source of intraplate stress and associated deformation [e.g., Frank, 1972; Artyushkov, 1973; Listcr, 1975; Molnar and TapjJonicr, 1978; Houscman ct at., 1981; England and McR:cnzic, 1982; Flcilo'ltl and F/,oidevCl'llx, 1982, 1983; England, 13
1987; Zho7t and Sandiford, 1992]. On a global scale, horizontal stresses due to lateral density variations are known to be a major contributor to the intraplate stress field through ridge push, which is thought to be a major component of thc driving mechanism for plate tectonics. This force arises from the cooling and thickening of oceanic lithosphere with age and has been estimated be on the order of 3 x 1012 N per mcter of ridge length [Frank, 1972; Liste!', 1975; Parsons and Richte'/', 1980]. Note that since ridge push arises from lateral density variations within the oceanic lithosphere it is, strictly, an intraplate source of stress rather than a plate boundary force [Fmnk, 1972]. The results of a number of studies have confirmed the relationship between the ridge push force and intraplate stress orientations [e.g., Richardson el ai., 1979; Richardson and CO;!:, 1984; Riclul1'(!son and Reding, 1991; Worlel el ai., 1991; Richardson, 1992]. The slab pull force also arises from density variations, hut in this case between subdllcted lithosphere and the surrounding sub-lithospheric mantle. On the basis of density variations alone, the slab pull force is predicted to be an order of magnitude greater than the ridge push force [e.g., Turcotte and Schubert, 1982]. The fact that the global intraplate stress data set [Zoback, 1992] of over 8000 stress indicators is dominated by compressional and strike-slip indicators implies, however, that the surface plates do not feel the large slab pull. This is consistent with a large, oppositely directed slab resistance acting locally on the slab [Forsyth and Uyeda, 1975]. Thus, while slab pull is an important component of the driving mechanism, other processes, such as lateral density variations within the plates, probably have a more important role in the intraplate stress field. Lateral density variations in the continental regions also contribute to the in traplate stress field. For example, the buoyancy force associated wjth the density structure of high topography in continents can be several times larger than the ridge push force [e.g. Fleilo'llt and Fl'Oideva'll:!:, 1982, 1983; Molnar and Lyon-Caen, 14
1988; Fleito1lt, 1991; Zoback and Magee, 1991; Sandiford and Powell, 1990; Zhou and SandifO'/'{Z, 1992], and are thought to be responsible for the normal stress regime in regions such as the Altiplano and the Tibetan plateaux. Similarly, stresses aris ing from lateral density variations at passive continental margins can perturb the regional stress field and locally alter the tectonic st.yle of deformation [Coblentz and Richardson, 1992J. In attempting to understand stresses associated wi th gravitational potential energy variations, it is useful to first define the tectonic reference state (TRS). In the absence of external forces, a lithospheric column in potential and isostatic balance with the TRS would remain undeformed. The reference lithosphere is assumed to be isostatically compensated at, or beneath, its base. Isostasy does not specify a complete force balance, however, and it often has been assumed that the reference continental lithosphere is in potential energy balance with mid-ocean ridges [e.g., Turcotte et al., 1977; Mc[(enzie, 1978; Le Pichon and Angeliel', 1979; Cochran, 1982; Le Pichon, 1983; Turcotte, 1983; Crough, 1983; Houseman and England, 1986; SondeI' et at., 1987; England and Houseman, 1988, 1989; Zho'll and Sandz{ord, 1992J. The extensional nat nre of mid-ocean ridges, however, implies that they are in an state of excess potential energy and it would be more appropriate
to define the TRS in terms of the mean potential energy of the lithosphere, [1/. The concept of the TRS is useful for understanding the relationship between the tectonic stress state and the lateral density variations discllssed above. The state of tectonic stress (whether compressional, neutral or extensional) can be directly related to the plate-scale potential energy distribution and deviations from t.he TRS. Following Dahlen [1983J, t.he TRS is termed the ambient slale. In the ambient state, parts of the plate with potential energy in excess of the [1/ will experience deviatoric tension, while parts of the plate with potential energy less than [1/ will experience deviatoric compression. 15
The second part of this study demonstrates the utility of the potential energy formulation discussed above through a finite element analysis of the intraplate stress field in three plates: Africa, South America, and Indo-Australia. Each of these plates possesses a unique combination of tectonic features making them ideal locations to demonstrate tectonic implications of the potential energy formulation. The African plate, which is slow moving in a hotspot reference frame (producing negligible drag tractions along the base of the plate) and predominately surrounded by mid-ocean ridges, is expected to best approximate the ambient stress state. It provides a way to test the concept of the TRS and the ambient state of stress as well as place bounds on the magnitude of the intraplate stresses due solely to potential energy variations. The intraplate stress field in the South Ameri can plate is not expected to be in the ambient state given the large amount of collisional boundaries along the western margin. However, the plate itself is not at tached to a significant length of subducting slab and thus the torque balance of the plate will not be dominated by the slab pull force. It is therefore a good setting to investigate the relative contributions of both lateral density variations and collisional boundary forces. Finally, the role of lateral density variations in the Indo-Australian stress field is studied. This plate contains a large number of first order tectonic features including active subduction zones, an extensive mid-ocean ridge system, significant areas of both continent-continent and continent-island arc collision, which have made it difficult to explain the observed intraplate stress field in terms of a single tectonic process. In addition, the magnitude of the intraplate stress field in the Indo-Australian plate remains a subject of considerable debate with estimates differing by a factor of ten. The principal aims of this study are to evaluate what part of the observed stress field in the Indo-Australian plate can be explained in terms of the ridge push force and to constrain the intraplate stress magnitude. Constraint for the modeling is provided by the long wavelength trends 16
111 the maximum horizontal compressive stress orientation, Sll,l1laX' of the stress indicators in the World Stress Map database and by estimates of the intraplate stress magnitude in the South American plate. The final topic addressed in this study is the time-evoilltion of plate-scale po tentiaJ energy distributions in continental plates. Although it is widely recognized that lateral density variations in the ocean lithosphere produce variations in UI and thus provide important torques on the plates [e.g., Forsyth and Uyeda, 1975; Richardson el al., 1979; Riehm'rlson, 1992; Zobaek, 1992], it has not been widely appreciated that continents may be associated with similar variations in UIChanges in UI attendant with the aging of a plates will produce a corresponding change in UI and the ambient stress state. The central focus of this part of the study is how changes in UI accompanying the growth of large plates may enhance the possibility of extensional failure within continentaJ lithosphere. This hypothesis is tested by evaluating the plate-scale evolution of potential energy for a simple circular plate surrounded by an oceanic ridge system. Next, the change in UI in the African and Antarctic Plates since the late Jurassic is evaluated. This study demonstrates that the ambient tectonic stress state within continental Africa and Antarctica has become increasingly more extensional with time, possibly contributing to the stress field responsible for the observed continental rifting in these plates. The organization of this dissertation is as follows: Chapter 2 presents the meth ods and assumptions used in calculating UI. The tectonic implications of UI, TRS and the ambient stress state are discussed, and this chapter lays the theo retical groundwork for the methods used in the finite element analysis presented in Chapter <1. Chapter 3 presents information about statistical trends in the long wavelength intraplate stress field. Chaptet <1 presents constraints on the magnitude of the far-field intraplate stress field in the South American plate. Results of a fi nite element analysis of the intraplate stress fields in the African, South American, 17 and Indo-Australian plates are presented in Chapter 5. The focus of this chapter is to understand what part of the observed intraplate stress field in these plates is explainable in terms of the forces due 1;0 lateral variation in the gravitationaI potential energy. Chapter 6 addresses the tectonic implications of changes in the mean potential energy of the lithosphere in response to growth and aging of the the oceanic lithosphere within the African and Antarctic plates. Finally, concluding remarks about each of theses studies are presented in Chapter 7. 18
CHAPTER 2 THE POTENTIAL ENERGY FORMULATION
2.1 Introduction
The purpose of this chapter is to provide an approximate formulation for the potential-energy distribution of the lithosphere at both global and plate scales in order to constrain the tectonic reference state. The main problem with address ing the potential energy of the Earth's lithosphere arises from uncertainties in our knowledge of the density structure, particularly in the continental crust, and one of the main aims of this research is to assess the sensitivity of the estimates of the mean potential energy to such uncertainties. In addition Lo defining the mean potential energies at the global and plate scales, the magnitude of the spatial varia tions in potential energy is quantified. The calculations presented below are based on a simple, first-oreler model of the lithosphere which reflects density variations associated with topographic features at a 10 x 10 resolution. The initial work on this topic [Richa'l'dson and Coblentz, 1992] has been extended by the use of a more realistic density structure for the lithosphere based on the assumption of a linear geotherm and the use of reference densities for the crust and mantle that sat isfy constraints imposed by observed geoid anomalies across continental margins. Finally, the correlation between the torque poles associated with the potential energy distributions and the observed plate velocity poles for the individual plates is evaluated.
2.2 Theoretical Basis
The gravitational potential energy per unit area, U, of a column of material above a given depth, z, is given by the integral of the vertical stress, O'zz, from z 19 to the Earth's surface II. [e.g. Molnar and Lyon-Cae'll, 1988]:
{It' {It {It I I U = }z O"zz(r)dr =9 }z }T p(r )dr dr (2.1 ) where p(z) is the density at depth z, II. is the surface elevation and 9 is the gravita tional acceleration. The potential energy of the lithospheric column, U" is defined by (2.1) when z corresponds to the equipotential surface at which the lithosphere is compensated, Ziso. For the pmpose of this study it is useful to define the mean potential energy of the lithosphere at both the global and plate scale (Uf and Un. Because the Earth's lithosphere can be considered to be in isostatic equilibrium for wavelengths greater than a few hundred kilometers [[(aula, 1970, 1972; T/trcolle and McAdoo, 1979; Sandwell and Smith, 1992], horizontal stresses can be directly related to the vertical-density distribution [lfaxby and Turcotte, 1978; Dahlen, 1981]:
-0" xa: = -91,£ 6.. p( z) Z dz (2.2) L h where L is the lithospheric thickness and O"xx is the horizontal stress averaged over the thickness of the lithosphere, relative to a reference state against which the 6.. p is measured. Equation (2.2) shows tha.t the mean horizontal stress is related to the local dipole moment of the density distribution, Ai:
- 9 O"xx = L 111 (2.3)
Using the definition of gravitational potential energy in (2.1), the horizontal stress can be expressed in terms of the potential energies:
(2.4) where 6..UI is the difference between the potential energy of the local lithospheric column, U" and the potential energy of some column defining a reference tectonic
state, U1.: 20
To a good approximation, geoid anomalies in isostatically compensated regions also can be related to the density moment [Ilaxby and Turcotte, 1978; Turcotte and Schubert, 1982]. Thus, the geoid-height anomaly provides valuable, indepen dent information that ca.n be used to constrain the density distribution within the lithosphere. The geoid-height anomaly can be expressed as [1kJ'colle and Schuberl, 1982] L t1N = !, t1 p(z) z dz (2.5) 9 .It where G is the gravitational constant. t1 N can be rewritten in terms of the potential energy differences, t1U/:
t1N (2.6)
As will be seen later, the relationship between the geoid-height a.nomaly and litho spheric potential energy gradients provides a.n important constraint on the density structure of the continental lithosphere. These simple planar expressions are good approximations when the density anomalies are in isostatic equilibrium as assumed in the present study.
2.3 Lithospheric Density Models
The calculation of U/ requires information about the density structure of the lithosphere about which there is considerable uncertainty, particularly in the conti nental crust. The purpose here is to devise a first-order view of the potential-energy distribution and thus a very simple density structure was assumed (in defense of the simplistic assumptions it will be shown that the main results are robust to the principal uncertainty, which is the reference density for the continental crust, Pc). For the purpose of the present study, the Earth's lithosphere has been categorized into fotll' principal types (young oceanic, oceanic basin, submerged continent and 21
exposed continent) based on global topographic [ETOPO.5, National Geophysical Data Genter 1988] and sea floor age information [Royel' el al., 1992]. For each of these lithospheric types, a thermally stabilized mantle lithosphere was assumed and a simple density distribution appropriate to a linear geotherm was used, with the density of the crust defined by
(2.7)
and that of the mantle lithosphere by
Pm(Z) = Pm [1 + av (T/ - T(z))] (2.8)
where Pc and Pm are the crustal and mantle densities at T/, the temperature at the
base of the lithosphere, and all if; the thermal expansion coefficient. Becnuse the average surface elevation of the lithosphere at length scales appropriate to local isostatic compensation reflects the local density structure of the lithosphere, the lithospheric density structure for each of the four lithospheric types has been for mulated in terms of topography (or bathymetry), wi th a few sirn pIe assumptions about the nature of the isostatic mechanism (Appendix A). The density distribu tion for the Ii thosphere is based on two assumptions: 1) the lithosphere columns are in local isostatic equilibrium, and 2) the contribution to the potential energy variations from lateral density variations deeper than 125 km below sea level is negligible. Therefore, it is assumed that the equipotential surface appropriate to global isostatic compensation, Ziso, is 125 km below sea level. This is equivalent to assuming that the thickness of the lithosphere is about 125km [Turcotte and McAdoo, 1979; Morgan and Smith, 1992]. Further constraints used for the litho spheric types are described below. 22
2.3.1 Young Oceanic Lithosphere
For the purposed of the present study, young ocea.nic lithosphere was assumed to pre-date the end of the Creataceus normal mega-polarity epoch at 8'1 Ma which approximately corresponds to the age when the subsidence of oceanic lithosphere is not predicted by the half-space cooling model [e.g., Turcotte and Schubert, 1982; J\l{OI:qan and Smith, 1992]. The potential energy for a lithospheric column of young oceanic material was calculated assuming that the thickness of the oceanic crust. is constant. Conse quently, variations in bathymetry are correlated with variations in the thickness of the lithosphere (where the base of the lithosphere is defined by the intercept of the lithospheric conductive geotherm and the mantle adiabatic temperature, 1/). At the ridge, the thickness of the lithosphere is assumed to be that of the oceanic crust. The bathymetry of normal oceanic ridges is assumed t.o be 2.5 km. The oceanic crust is assumed to have a thickness of 7 km [White el al., 1992] and a density of 2960 kg m-3 at the reference temperature, 1/. The density of the
asthenosphere constrained by Pratt isostacy to be 3238 kg m-:3. The effects 011 mantle density 61' melt extraction from the upper part of the mantle during the formation of the oceanic crust have been ignored in this formulation [e.g., OJ.:b'Il'I'f}h and Parmentier, 1977].
2.3.2 Oceanic Basin
The ocean basins were assumed to have a constant lithospheric thickness with a base at a depth of 125 km below sea level (= Ziso), consistent with t.he thermal plate model [Parsons and J\l{cI{enzie, 1979]. To account for potential-energy varia tions associated with anomalously shallow regions of the ocean basins, such as sea 23
mounts, variations were allowed in thickness of the oceanic crust to preserve local isostatic balance with the mid-ocean ridges.
2.3.3 Continental Lithosphere
In this formulation, the lithospheric density structure for the submerged conti nental margins (oceanic regions close to the continents wi th bathymetry less than 2000 m) and exposed continents differs only in the incorporation of the contribu tion of the water column in the former. In hoth cases variation in topography (or bathymetry) is assumed to reflect variations in crustal thickness assuming the
base of the lithosphere, at T" is fixed at Ziso = 125 km depth. This formulation represents a significant approximation, since the elevated topography in some ar eas of the continents is likely compensated by anomalously hot mantle rather than thick crust, as for example in the Basin and Range Province of the western US. In Appendix B it is shown that for regions of continents with elevated topography on the order of 1 - 3 kill above sea level, the assumed mechanism of isostatic :mpport,
whether it be variation in the thickness of the mant.1e lithosphere 01' of the crust, results in only minor variations in the potential energy of the lithospheric column, and therefore this approximation does not significantly affect the results. A significant uncertainty regarding the continental lithospheric density structtll'e pertains to the crustal density, pc, which is known to be heterogeneous across a wide range of scales. As described in the following section the density of the continental crust, pc, relative to oceanic lithosphere, is constrained by modeling the observed geoid anomalies across the ocean-continent margins. Representative lithospheric columns for each of the lithospheric types are shown as Figure 2.1. Each 1°x1 0 region of the Earth's surface was categorized as one of the four lithospheric types (on the basis of age in the oceanic regions and bathymetry in the case of continental margins) and U/ calculated by evaluating (2.1) for the appropriate topography (or bathymetry). The parameters used to evaluate U/ are listed in Table 2.1 [Parsons I
a) b) c) d)
...... -.:.: ~ ..."". "----- Sea Water Continental Crust ------Oceanic Crust------
Asthenosphere z Density Density Density Water I~ d wr Wuter dwr Continelllal I t Continental Crust d Crust d Oceunic 1+ cc cc Crust ~ doc
sthcnosphcrc Mantle Muntle Mantle
...... •••••••••••••••••••••• 1.... , L' Equipotential z z z z
II) Occlln Ridgc b) Occlln nllsin c) Contincntlll Mllrgin d) E1cvlltcd Contincnt
Figure 2.1, Schematic of the lithospheric depth-density distributions for the four lithospheric types: ypung oceanic (1) , oceanic basin (b), submerged continent (c) and exposed continent (d). The density of the continental crust and mantle lithosphere varies as a linear function with depth. See text for details. 25
and Sclate1', 1977; Tlll'Coite and McAdoo, 1979; Tltl'cotte and Schubert, 1982]. The full expressions used for the calculation of the potential energy for each of the lithospheric types is given in Appendix A.
2.3.4 Constraint on the Density of the Continental Crust
The strong dependence of U/ on the near-surface density distribution makes it critical to constrain the value of pc. Figure 2.2 shows that this dependence of U/ on pc is particularly strong for continental lithosphere with high elevation. For lithospheric columns supporting 4 km of elevation, this difference in poten tial energies amounts to 6.7 x 1012 N m-1 for pc equal to 2600 and 2900 kg m-3 but diminishes to only 1 x 10 12 N m-1 for lithospheric columns supporting 1 km of ele vation. Indeed, the recognition of the sensitive dependence of the potential energy on pc for lithospheric columns supporting high elevation led England and /'dolnar [1991] to suggest that it was pointless to at tempt direct calculation of potential energies from topography alone, as is proposed here. While this dependence of U/ on pc does have important consequences for the range in intraplate potential energy which dictates the magnitude of the local lithospheric stress state, it will be shown that it has a relatively small affect on the global- and plate-mean poten tial energies, Uf, due to the relatively small amount of the Earth's surface with elevation greater than 1 km. While admitting there may be considerable regional variation in crustal density values within the continents, one way to constrain typical mean crustal densities is to use independent information from geoid anomalies (e.g., TltTcolte and Sclw bert [1982]). As discussed above (e.g., Equation 2.5), the geoid anomaly can be directly related to the local dipole moment of the density-depth distribution, and thus provides an independent constraint on pc. The geoid anomaly across passive 26
S 2.45 Z L.....J
Crustal Density [kg/m3] -- 2600 -- 2700 --2800 --- 2900 2.30 -t------r----r---.----r---.-----.-~___f_ o 123 4 Elevation [Ian]
Figure 2.2, Potential energy, U" as a function of pc (2600, 2700, 2800 and 2900 3 kg m- ) and topography. U/ is the potential energy for a continental lithosphere column based on the formulation described in the text. 27
Atlantic-style continental margins has been estimated to be about 6 m [Haxby and TU1'cotte, 1978; Tu'/'cotte and McAdoo, 1979; TU'l'cotte and 8chube'/'t, 1982; Coblentz and Richardson, 1992; see also a review in Chase, 1985]. Predicted geoid anomaly across the continental margin calculated from (2.5) for pc in the range of 2600 to 2900 kg m-3 are listed Table 2.2. There is close agreement between the observed
and predicted geoid anomaly for a value of pc in the range of 2700 to 2800 kg m -3. Unless otherwise stated, a value of 2750 kg m-3 for Pc is used in the following calculations.
2.4 Global and Plate Scale Lithospheric Potential Energy Estimates
Global bathymetry and topography information was obtained from the ETOP05
topographic dataset. The average bathymetry 01' topography was calculated for surface elements of dimension 10 x 10 (about 12,300 km2 at the equator). Thus the effect of isolated seamounts, and other anomalous topographic features on the mean elevation can be considered to be negligible. ''''hile the bathemetric informa tion in the ETOP05 dataset has bep.n shown to contain large errors [Smith, 1993], the use of the average bathymetry within lOx 10 windows minimizes the effect of these errors on this analysis. Histograms of the topographic distributions for the entire Earth and the seven major plates (Africa, Eurasia, Indo-Australia, North America, South America, Pacific and Nazca) are shown as Figure 2.3a, while the surface areas, average elevations and percentage of the plates composed of young oceanic, old oceanic, continental margin and exposed continental lithosphere are listed in Table 2.3. The topographic distributions for the Earth and the majority of the continental plates (Africa, Indo-Australia, North and South America) are strongly bimodal, 28
(a) Topography (b) Potential Energy
700t---~-'-'r.-~----r N.lIc., N.l.4,.. 350 S .. l.thlll~
5000 I'.ldflc NIIW,IW 2500 SIl1.f,'I0~ o JI1 111
1500t---~-.r-,-~----r fw.mthAnwric.l SuulhAml'rlc.l NII4,OHl N·t(JHl 500 750 5",).5,,10·\
North Aml'rk" N.HJlS 2000 1500 5.5.6"11' Ol-__.... nw..f1-
1500 t---~-,--,-~----r Indn·Atulr.ll1" N.. S,62t1 750 750
5000 3500t---~-r.,,_~----r [u.ul.1 1 L Eu ... '., n N.H,S(I) 250:~~.-_~..z:11,ll.ll;.~N~."'~~ll-.--l 1750 SIII.H,Ull
'=:11--.-...... ~UJ.l=lj,WJ,-l _~~i'~~I)-.--I 1000 Ol--__--ollW
7000 t--'~~~~~~~"--; 7000r--~---r~----' WI",IL' [.uth Whult·E.trlh N=M,Htltl N=M,NOII N 3500 3500 S,.1.thlttl
~JO -8 -6 -4 -2 0 2 4 6 8 10 2.30 2.35 2.40 2.45 Topography [kill] U XIO I4 [N/Ill]
Figure 2.3, (a) Histograms of the topography and bathymetry for the Earth and the seven major plates. The number of elements for each histogram is also shown. Bin width = 500 meters. Based on 1° x 1° global topography, (ETOP05). (b) Histograms of the potential energy, U/, for the Earth and the seven major plates. 11 I Bin width = 1 x 10- N m- • U/ was calculated as described in the text. Thick~ vlOR dashed line designates the potential energy of the mid-ocean ridges (U/ ), 2.391 14 I 14 x 10 N m- • Thick line designates the global potential mean, Uf, 2.378 x 10 N rn-I. Thin line designates the potential mean of the individual plates. 29
reflecting the relative proportion of continents and oceans (continental topography has a mean elevation of several hundred meters while oceanic bathymetry has a mean depth of about -/1 km). The anomalous amount of topography between 0 and 2 km for the African plate reflects the elevated topography associated with the East African rift [Cazenave et ai., 1989; Anderson, 1989]. The Eurasian, Pacific and Nazca plates show anomalous topographic distributions. The Eurasian plate is dominated by continent (63%) and continental margin (23%) with very little deep oceanic areas (less than 6%). As a result, the mean elevation for the plate is anomalously high (about 200 m above sea level). In addition, the mean elevation of the continental part of the Eurasian plate is nearly 1 km, reflecting, in part, the influence of the Tibetan plateau topography. While the Pacific and Na2ca plates are both oceanic plates, they have significantly different topographic distributions. The Pacific plate is dominated by old, deep basins (50%), while the Nazca plate is principally young, cooling oceanic lithosphere (92%). Histograms of UI for the Earth and the seven major pla.tes are shown as Fig ure 2.3b. The varied topographic distributions of the plates produces significant variation in their potential-energy distributions. Importantly, the potential-energy distributions do not share the distinctive bimodal distribution of topography due to the fact that, for a given surface elevation, ocean lithosphere has much greater potential energy than continental lithosphere. With the exception of the African and Pacific plates, the plate-scale potential-energy distributions are characterized by a sharp peak near the mean potential energy value, UI. The mean potential energies, UI, for three topographic subsets (all topography, topography belmv sea level, topography above sea level) for three values of pc (2700, 2750 and 2800 kg
3 m- ) are listed in Table 2.4. Table 2.4 highlights the fact that the mean potential energy on both global and the plate scales are insensitive to the assumed conti nental crustal density, pc. The greatest sensitivity is shown by the Eurasian plate 30
because of contribution of the eleva.tion related to the Tibetan plateau, with the
3 12 difference in Uf for pc of 2700 and 2800 kg 111- amounting to only 0.6 x 10 N m-1 (Table 2.'1). For the other fOllr major continental plates, the difference in potential energies for this range in pc amounts to 0.2 - 0.4 x 10 12 N m-\ while at
12 1 the global scale the difference is 0.2 x 10 N m- • The principal features of the potential-energy distributions can be understood by considering the potential energies associated with the major tectonic features. The potential energy associated with a represent.ative topographic profile which includes each of the four lithospheric types is shown as Figure 2.<1. Also shown in Figure 2.<1 are the mean elevation of the global topography, the mean global potential energy, Uf, and the potential energy of the mid-ocean ridge,
1 1 U/,o'Ion. Note that the mid ocean ridges (U/,WII = 2.391 X 10 '1 N m- ) and regions of the continents with significant elevated topography have potential energies greater than the global mean, while the oceanic b"sins are potential-energy lows. Table
3 2.5 shows that for the range pc = 2800-2700 kg m- , continental lithosphere with a surface elevation in the range 940 - 1650 m has the same potential energy as the
14 1 mid-ocean ridge. The global-mean potential energy (Uf = 2.379 X 10 N 111- ) is equa.l to the potential energy of cooling oceanic lithosphere at. a depth of about 4.3 km and to continenta.llithosphere with about 70 m elevation (for pc = 2750 3 kg m- , Table 2.5). The unimodal potential-energy distributions of the major plates reflects this similarity in the potential energy of cooling oceanic lithosphere and neal' sea level continental lithosphere. The sharp peak in the potential-energy distribution for the Pacific plate near 2.36 x 1014 N m-1 reflects the large amount of deep ocean basin in the plate. In general, the mean potential energy of the continental plates, Ur, are very
12 1 close to Uf, with the range in Ur for all plates being only 1.2 x 10 N m- • The
14 1 anomalously high mean 1'01' the Eurasian plate (Ur = 2.383 x 10 N m- ) is due to 31
S 2.44 Z 2.42 L-...I 2.40 Ridge Potential ------.------_._._------_.----_ .... _------_._._--
l~:;.:;.;;..;-___-"- ______2.38
2.36
8
4
o ...... S!:!jl-',,!:!X~!. ___ ...... _.. ___ ... __ ...... _.. _... _,...-=..:.;,"_-~ ...... ____ g~o~~1 !~e.o.s~ap~i~ ~!~n______-4 'd e C(lutin\!ntal Eler·ated Rl Basin Margm Con ment _8~~~------~~~------~------L
Figure 2.4, Calculated potential energy, Ul, across a topographic cross section rep resenting the mid-ocean ridge, oceanic basin, passive (Atlantic-type) continental margin, and elevated continental lithosphere. Also shown are the global topo 14 1 graphic mean (-2880 m), ridge potential (2.391 x lO N m- ) and mean global 1 1 3 potential energy (2.378 x 10 "N m- ) calculated with pc = 2750 kg m- • I L'
32
the high topography of the Tibetan plateau (where VI attains values of up to 2.472
14 1 X 10 N m- ) and the mean calculated without this contribution is consistent with the means of other continental plates. The mean for the Indo-Australia plate (V!, = 2.:375 x 1014 N m -1) is slightly lower than the other continental plates due to its relatively large proportion of oceanic surface area (60%). The means of the subsets of the continental plates incorporating only those regions exposed above sea level (see Table 2.'1) is substantially greater than V?, but, importantly, is still less than the potential energy of the mid-ocean ridge. The oceanic plates, represented by the Pi:tcific and the Nazca plates, have sig nificantly difFerent mean potential energies. The mean for the Pacific plate (Vr =
14 1 2.371 x 10 N m- ), which is dominated by old oceanic lithosphere, is much lower
1 1 than Vr, In contrast, the mean for the Nazca plate (V!, = 2.382 X 10 '1 N m- ), which is dominated by young oceanic lithosphere, is much greater than Ur, Thus on the global scale, the mean potential energy of the Pacific and Nazca plates arc anomalously low and high, respectively.
2.5 Discussion
This study has demonstrated the important correlation between potential en ergy variations and major topographic features of the Earth. The oceanic ridges are features with an excess potential energy !J..VI (= VI- vn, of about 1.2 x 1012 N 1 m- . Passive continental margins and the ocean basins are features with potential
12 1 12 energy less than the global mean, with !J..UI of 0 - -1.0 X 10 N m- and -1A x 10
1 N m- , respectively. The magnitude of the potential energy of continental litho sphere with elevated topography is particularly sensitive to the assumed crustal density and thus quantitative estimates of the excess potential energy of regions with high topography and, consequently, the intraplate range in potential energy 33
are significantly less robust than the estimates of the mean potential energy at the
plate and global scale. For a reference continental crustal density of 2750 kg m-3 , the global range in potential energy estimated at the lOx 10 scale is estimated to be about 11 x 1012 N m-1 with the high corresponding to the highest parts of the Himalaya - Tibetan Plateau system (U, = 2.472 x 101-1 N m-1 at elevation 6400
14 1 m) and the low corresponding to old ocean basin (U, = 2.364 X 10 N m- ). In general, the potential-energy means of the continental plates are very close to the global mean.
2.5.1 Potential-Energy Torques
The influence of lateral density variations on the lithospheric stress field is pro pOl'tionaI to the density moment of the mass di pole formed by the mass anomaly, with positive mass anomalies producing tectonic compression [Fleito'ltt, 1991]. The contribution of potential energy variations within the lithosphere to the total torque a.cting on the plates was calculated by using the 'moment law' in a plane stress finite element analysis. This was accomplished by applying a basal shear force to each element proportiona.! to the horizontal gradient of the local dipole mo ment (which is proportional to U" see (2.3) and (2.4)) [Fleilolll, 1991; Richardson and Reding, 1991]. For plane geometry this force can be expressed as
1 oM loAf (2.9) rYx:: = L ox and rYxy = L oy where M is the density moment. The resulting torque is calculated as
T = rxF (2.10) where r is the radius position vector and F is the force acting on the plate. The total torque acting on the plate is found by integrating T over the surface of the plate. Calculating the torque acting on the plates due to potential energy differ ences provides a way of quantifying the importance of potential-energy differences within the plate. The total torque contribution from the potential-energy distribu tions within the plates is considered for three cases: 1) mid-ocean ridges alone; 2) non-mid-oceanic-ridge potential energy; and 3) all potential energy sources. The total torques and a.ngula.r misfit between the torque poles and the absolute plate velocity poles [Minstcr and Jordan, 1978; Gripp and Gordon, 1990] are listed in Table 2.6. Whereas the mean potential energies of the plates are inherently clus tered about the globa.l mean potential energy, there is significant varia.tion in the relative torque contributions from the potential-energy distribut.ions. The angular misfit between torque poles due to the cooling oceanic lithosphere and the absolute
velocity poles varies from less than 20 degrees 1'01' the Pacific, North American and South American plates, to between 20 and 30 degrees for the Indo-Australian and Nazca plates, and to greater than 50 degrees for the remaining plates. These results are in substantial agreement with the observation that a wcak correlation exists between ridge torques and plate velocities [FO'I'syth and Uycda, 1975]. A stronger correlation exists between the velocities and the torques due to a.ll potential en ergy sources. Torques associated with the continental topography are typically about 2.5xl025 Nm and are considerable smaller than the ridge t.orques which ex ceed 'lx1025 Nm for all the plates (with the exception of Nazca). No correlation was found between the topographic torques and the absolute plate velocities. Further more, the angular misfit between the torque and velocity poles is very large for the topographic torques, in excess of 900 for most of the plates. The absolute plate velocities and torque directions associated with the ridge, topographic and total potential energy torque poles given in Table 2.6 are plotted at selected locations on the seven major plates on Figure 2.5 and 2.6. In order to facilitate the comparison between the torque directions and plate motions for plates with small velocities and torques, the vectors have been normal ized to unit length in Figure 2.5. In Figure 2.6, the length of the vectors rcflect I L'
35
_30 0
_60 0
00 45 0 90 0 135 0 180 0 225 0 315 0 360 0
Figure 2.5, Comparison of ridge torque (solid arrows), topographic torque (medium gray shade), total potential energy torque (light grey shade) and absolute velocity directions (open arrows) for the seven major plates.The magnitude of the torques and velocity vectors have been normalized to unit length. Plate velocity informa tion is from Minster and Jordan [1978J for the Indo-Australian plate, and NUVEL-1 [Gripp and GO'l'dan, 1990J for all other plates. The Pacific ridge and total torque arrows nearly coincide because the total torque acting on the plate is dominated by the ridge torque. 36
-30·
-60·
o· 45· 90· 135· 180· 225· 270· 315· 360·
Figure 2.6, Comparison of ridge torque (solid arrows), topographic torque (medium gray shade)' total potential energy torque (light grey shade) and absolute velocity directions (open arrows) for the seven major plates. The length of the vectors reflect the absolute magnitude of the torques and velocities. The Pacific ridge and total torque arrows nearly coincide because the total torque acting on the plate is dominated by the ridge torque. See Figure 2.5 for other details. 37
the absolute magnitudes. As discussed in Richardson [1992]' there is a very strong correlation between the ridge torque directions and the azimuth of the absolute plate velocities. This is particularly true for the North American, South Amer ican, Pacific and Indo-Australian plates. Large topography torques exist in the Eurasian, Indo-Australian, North and South American plates. In contrast to the ridge torque directions, the topographic torques act at a large angle to the plate motion directions. In the case of the total potential energy torques, there is a strong correlation between the torque directions and the azimuths of the absolute plate velocities. The relationship between the torque directions and the plate mo tions demonstrate that while topographic forces are certainly an important sources of stress, mid-ocean ridge torques would seem to playa more important role in the plate-scale dynamics. These results support the Botion that ridge push forces are an important component of the driving mechanism and global intraplate de formation [Richardson, 1992]. The strong correlation between the potential-energy distributions and plate velocities, at least for the fast-moving plates, is evidence that the absolute reference frame is determined by the surface plates themselves, rather than sub lithospheric flow.
2.5.2 The Tectonic Reference State
The concept of a tcctonic 1'cfcrcncc statc has proved useful in evaluating the con tribution of lateral density variations to the intraplate stress field [e.g., ZhO'/l and Sandiford, 1992]. Moreover, it is a concept that has been employed in a number of studies of continental deformation to illustrate the importance of the changes in potential energy (or buoyancy forces) accompanying progressive deformation of the lithosphere [ e.g., Houscman and England, 1986; Sondcr ct al., 1987; England and Houscman, 1988, 1989; Zholl and Sandifm'd, 1992]. As discussed above, the tectonic reference state is best defined in terms of the potential energy distribu tions at either the plate or global scale, which can be characterized by Ur and Uf, 38
respectively. This study has shown, however, that the cont.inental plates (with the exception of the Eurasian plate) have mean potential energies, Ur, close to the global mean, Uf. Moreover, if the contribution of Tibetan Plateau is excluded, this relationship also holds true for the Eurasian plate. Thus, as far as continental tectonics is concerned, it would appear to make little functional difference whether the tectonic reference state is defined by a column in potential energy balance with the global mean, Uf, or individual plate mean, Ur The mean global potential en ergy corresponds to the potential energy of continentalliLhosphere of neal' sea level elevation, as well as cooling oceanic lithosphere at a depth of about '1.3 km below sea level. Therefore, it is self-consistent to use sea level continental lithosphere as the tectonic reference state for evaluating continental deformation. The definition of the continental tectonic reference state proposed here contrasts with that of a number of previous studies which assumed a reference state in potential energy balance with mid ocean ridges [Turcotte, 1977; McKenzie, 1978; Le Pichon and Angclier, 1979; Cochran, 1982; Le Pichon, 1983; Tllrcotte, 1983; Crough, 1983; HO'llseman and England, 1986; England and HO'llseman, 1988, 1989; SondeI' cl al., 1987; Zhou and San dij'o nI, 1992]. The difference in definitions amounts to a.bout
12 1.2 x 10 N m-1• Some uncertainty in these calculations is obviously introduced by our inadequate knowledge of the density structure of the lithosphere. While independent constraints from the geoid anomalies associated with continental mar gins provide a way to estimate the mean density of the continental crust, it remains the least constrained parameter. However, it has been shown that the global and plate mean potential energies are insensitive to the assumed continental crustal density. Thus, this definition of the tectonic reference state can be considered to be robust. 39
2.5.3 The Ambient Stress State
In the absence of forces applied along the base of the plate or along plate boundaries, intraplate variations in potential energy must provide a major control on the intraplate stress field. The intraplate stress field in slow-moving plates that are surrounded by mid-ocean ridges may be expected to approximate the ambient stress state [Crough, 1983J. It has been shown above that the potential energy of the mid-ocean ridges is in excess of the mean plate potential energy with the
difference, flU/, approximately equal to 1.2 x 10 12 N m-1 for plates containing a large amount of continental lithosphere. Consequently, in the ambient state, the ridges and much of the ridge flanks are expected to exhibit an extensionaJ stress
regime with the maximum horizontal compressive stress direction, S'U,mllx, oriented parallel to the ridge axis. Correspondingly, the direction of maximum tension will be aligned with the gradient in potential energy and, hence, with the gradient in bathymetry. The definition of the tectonic reference state in terms of the mean plate potential energy can be used to clarify some fundamental aspects concerning the nature of the ridge push forces that have been the subject of some confusion in the literature. First, while the ridges have an excess potential energy over the old ocean basins
12 1 of about 2.6 x 10 N m- , the compressional force witnessed by old ocean basin lithosphere in the ambient state must be somewhat less since it reflects the potential energy difference relative to the plate mean (that is, flU/ which is about -1,4 x 1012 N m-1 for old ocean basin) anclnot relative to the ridge. Similarly, while the ridges have potential energy in excess of much of the continental mass with low elevations, it is potential energy relati ve to the plate mean (and not the ridges) that dictates the ambient stress state in the continents. The implication of the results presented above is that, for reference continental crustal densities in the range of
2750 kg m-3 , the ambient stress state in the exposed continental mass should be 40 extensional (or strike slip). Because large regions of compression are thought to be the dominate state of stress in most of the continental plates, particularly in eastern North American and western Europe [Zoback and Magee, 1991; Zoback, 1992J, this conclusion suggests that other sources of tectonic stress (i.e., plate boundary forces) play an important role in the nature of intraplate stress field in these non-ambient plates. The goal of the following chapters is to evaluate the relative contribution of ambient and non-ambient sources to the observed intraplate stress field. temperature at base of lithosphere 1280 0 e temperature at surface of lithosphere oDe water depth above mid-ocean ridge 2.5 km oceanic crust thickness above mid-ocean ridge 7km sea water density 1030 kg/m3 oceanic crust density 2960 kg/m3 asthenosphere density at, Tt 3200 kg/m3 coefficient of thermal expansion 3 x 10-5 I(-l depth of equipotential surface 125 km
Table 2.1, Values of Parameters Used in Calculations
N (m) 2600 3.30 2700 4.70 2800 6.70 2900 10.0
Table 2.2, Geoid Anomaly Across A Typical Continental Margin
2 Plale Area, xIO? km zel, (m) %-,6. 1m) if,c • (m) YOel (%1 00· 1%1 CM' (%1 C 9 (%1 Whole Earlh bO.B5 .2436 .3647 836 26.0 31.5 14.4 28.1 Africa 7.78 ·21b2 ·3884 6bO 32.6 21.1 8.1 38.2 Eurasia 6.12 206 ·1041 902 1.3 1.5 26.9 67.3 Indo-Audtralia 6.13 ·2708 .3524 339 42.6 19.5 18.,1 19.6 North America 5.29 ·1000 ·2409 658 12.5 14.5 27.0 45.9 South America. 4.37 ·2059 .3755 608 34.7 15.9 10.6 38.0 Pacific 10.78 .4499 .4520 354 50.2 44.9 4.5 0.4 Nazca 1.66 .3752 .3752 02.1 5.0 2.8 0.0
a Average topography, b Average of topography below sea. level, C Average of topography above sea level, cl Young occ.lnic lilhodphcrc, e Old oceanic lithosphere, J Continental margin, Y EXpOdCd continent.
Table 2.3, Topographic Distribution of the Earth and the Seven Major Plates 42
--{I b b b Plate U~oll U1,_ U~± U1,oll U1,_ U1,± U~,oll U~,_ U~.± Whole Earth (Un 2.378 2.371 2.386 2.378 2.375 2.389 2.380 2.375 2.392 Africa 2.377 2.373 2.383 2.378 2.374 2.386 2.380 2.374 2.390 EUl'asia 2.383 2.375 2.387 2.386 2.377 2.390 2.389 2.379 2.394 Indo-Australia 2.375 2.371 2.381 2.376 2.375 2.383 2.377 2.375 2.386 North America 2.378 2.371 2.384 2.380 2.375 2.386 2.382 2.376 2.390 South America 2.377 2.373 2.384 2.379 2.374 2.386 2.380 2.371 2.390 Pacific 2.371 2.371 2.381 2.371 2.371 2.383 2.371 2.371 2.386 Nazca 2.382 2.382 2.382 2.383 2.383 2.383
(all) = All Topography, (-) = Topography below sea level, (+) = Topography above sea level. o Pc=2700 kg 111-:1; b Pc =2750 kg 111-:1; c Pc =2800 kg 111-:3. All potential energies are given in units of 10 14N 111- 1 .
Table 2.'1, Ur 1'01' the Earth and the Seven Major Plates
Uf (111) 2800 -160 940 2750 70 1100 2700 220 1650
Table 2.5, Continental Elevation Information 43
Plate Torque Magnitude Laillude Longitude Al1gul>< MI'fil (xl025 Nm) (des) (deg) (deg) Ridge Torque PAC 10.0 77.5 S 94.b E 17.01 NAM 7.2 00.2 S 43.9 W 15.0 SAM O,b 08.0 S 109.1 E 12.3 EUA 0.7 '14.5 S 128.3 W 00.5 APR 3.2 57.0 N 34.·1 W 111.0 IND 8.9 40.0 N 30.1 E 21.3 ANT 5.8 38.6 S 158.0 E 83.1 NAZ 8.2 04.2 N 120.4 W 24.9 COC 1.0 04.01 N 149.0 E 75.0 CAR 0.7 52.0 N 170.9 W 100.2 ARD 0.2 31.8 N 101.0 E 110.0 PIlL Topographic Torque PAC 0.5 57.3 N 132.0 W 158.4 NAM 1.1 38.3 N 79.1 E 125.0 SAM 2.1 22.1 N 17.0 E 145.9 EUA 3.5 G.5 N 20.0 W 80.7 APR 0.8 10.8 S 701.2 W 117.0 IND 0.5 39.3 S 100.8 E 00.5 ANT 0.4 3.4S 178.5 W 113.7 NA7, 0.3 03.1 S 011.1 E 147.0 COC CAR 0.2 33.2 N 0.1 E 95.8 ARD 0.9 25.4 S 117.5 E 105.2 PIlL Total Potenh ..11 Energy Torque PAC 11.9 72.0 S 87.7 E 12.5 NAM 0.4 00.1 S 30.0 W 10.9 SAM 7.3 58.2 S 72.0 \V 12.2 EUA 3.3 1.8 S 29.0 \V 80.7 APR 1.8 20.3 N 07.0 W 112.1 IND 5.7 53.2 N 93.4 E 55. j ANT 0.1 30.9 S 100.0 I, 84.8 NAZ 7.9 04.2 N 110.7 \V 2·1.7 COC 1.0 0·1.4 N H8.0 E n.o CAR 0.7 00.0 N 175.4 \V 174.3 AIlD J.I 16.0 S 124.7 E 109.0 PIlL 0.8 50.1 N 04./ W 122.8
Angular misfit between the torque poles and the absolute plate velocities poles are based on plate velocity information from Minster and Jordan [1978] for the Indo-Australian plate, and NUVEL-l [Gripp and Gordan, 1990] for all other plates. Plate abbreviations are EUR=European, AFR=African, NAM=North American, SAM=South American, PAC=Pacific, NAM=North American, SAM=South American, EUR=European, AFR=Afl'ican, IND=Indo Australian, ANT=AntarcLic, NAZ-=NAZCA, COC=Cocos, ARB=Arabian, PlIL=Philippine, respectively.
Table 2.6, Total Torque and Angular Misfit Information 44
CHAPTER 3 TRENDS IN THE INTRAPLATE STRESS FIELD
3.1 Introduction
It has been hypothesized, on the basis of visual inspection of the maximum
horizontal compressive stress (Sll,11IClx) orientations of the stress indicators in the World Stress Map database, that broad regions of the Earth's crust (on the order of 1000's of kilometers) exhibit uniform patterns of stress orientation [Zobad:, 1992J. This observation has profound implications for the origin of the intraplate stress field, namely, the sources of this stress field must be broad scale in nature [Zoback, 1992; Zoback and Magee, 199L]. These long-wavelength trends are an important sout'ce of constraint for numerical models of the intrapla.te strcss field [e.g., Richardson el ai., 1979; Richardson and Reding, 1991; Worlel and Cloelingh, 1985; Stefanick and .fun/y, 1992; MeUer and Worlel, 1992J. Thcre is also evidence
for general agreement between the the SU,ma:v orientations and the azimuths of the ridge-push direction and absolute plate velocities [Richardson, 1992; Zoback, 1992J. Richardson [1992J used histograms of the angular difference between the
observed SU,1I!ClX orientation and ridge-push torque directions to quantify the misfit between the two for a number of plates. Such histograms, while providing a good picture of the overa.ll fit, contain no information about the spatial distribution of the misfit in the plates. Since the global SU,71H1X orientations show considerable scatter throughout many of the plates, it is essential to employ a trend ana.lysis which provides such spatial information. One such analysis was performed by Assumpc;ao [1992J who calculated the regional SU,11Ia,'L' orientation by computing the average direction ahout grid points spaced several degrees apart to demonstrate that the regional stress field in western South America is consistently oriented E-W 45
despite changes in the strike of the Andean chain and the Peru-Chile trench. This study presents results from a quantitative analysis of the long-wavelength trends in the SU,max orientations of the stress indicators in the World Stress Map database. The stress indicators used in this analysis are limited to those of suffi cient quality (A-C, after Zoback [1992]) to provide reliable information about the tectonic stress field (4496 out of a total of 6691 indicators). Trends were eval uated for the stress data lying within bins with a dimension of several degrees, corresponding to a wavelength of several hundred kilometers. The Rayleigh test [Mantia, 1965], which tests the null hypothesis that the orientations within a bin are random at a given confidence level, was used to quantify the existence of a trend within a given bin. A second aspect of the analysis involved the evaluation of the stress regime represented by the stress indicators within the bins. A weighted measure of the stress regime within the bins was determined to assess the state of stress (compressional, strike slip or extensional) exhibited by the stress indicators within the bins. Finally, the angular difference between the the average SU,mnx orientations and ridge-torque directions and the absolute plate-velocity azimuths is calculated for bins which pass the Rayleigh test.
3.2 Statistical Method
Tectonically significant trends in the intraplate stress field have wavelengths of several hundred kilometers. In order to extract information about trends of these wavelengths, the following analysis was carried out for the stress indicators falling within bins having square dimensions of several degrees. To ensure that the interpreted trends are not a function of the bin size, the analysis was conducted with a bin dimension of 3, 4,and 5 degrees. The first step in the analysis was to quantify the state of stress within the bin as reflected by the stress indicators. This was accomplished by assigning an arbitrary value to each indicator based on the stress regime it represents (l00, 200, and 300 have been chosen for compressional, strike slip, and extensional indicators, respec tively), then calculating the mean value of this quantity for each bin. The result is a weighted measure of the stress regime within each bin, where, for example, a value of 100 indicates that all the indicators within in the bin are compressional, 300 indicates all the indicators are extensional, and intermediate values indicate a mix of stress regimes skewed to a higher or lower value depending on the relative number of compressional, strike slip and extensional indicators.
Next, the average SU,1/I(lX orientation within each bin was calculated. This was accomplished by determining the mean resultant length R [see Equation 5.43,
Davis} 1986] of the SU,1II(1X orientations of the indicators within the bin. R is calculated as
(3.1 )
where
_ 1 n 1 n C = - L cosOj, S = - L sinOi, an(Wi = S'u,1/I(lxazimuth. (3.2) 11. i=1 11. i=1
The value of R varies from a to 1 and is a measure of the dispersion of the data within the bin, with values closer to 1 indicating a high cOLTelation of the SU,mo.v orientations within the bins. The Rayleigh test [see Chapter 5, Davis} 1986] provides a way to quantify the existence of a trend in directional data. The test is used to reject the hypothesis that the directional observations are random, based on the assumption that random orientations sample a von Mises distribution [see Mardia, 1965] which is the circular equivalent of the normal distribution. The Rayleigh test is applied by calculating 47
R and comparing the resulting value to a critical cutofI' value for the desired level of confidence [e.g., Appendix A-I, Marida, 1965; Appendix D-1 Games and [{la'l'e, 1967; Table 5.7, Davis, 1986]. If R exceeds the critical value, the null hypothesis that the data come from a random distribution of data can be rejected, and the observations can be presumed to come from a population showing a preferred orientation. For bins which pass the Rayleigh test, the difference between the orientation
of Sfl,Tllctx and the ridge-push torque and absolute velocity azimuths also was com puted. This allowed us to evaluate the correlation bet.ween the long-wavelength trends in the Sfl,lIIrlx orientations and the ridge-push torque and absolute-velocity azimuths throughout the plates. The ridge-push torques at each bin centroid were taken from the information in Table 2.6. The velocity poles taken from the NUVEL-1 model of Gripp and Gordon [1990], are are listed in Table 3.2.
3.3 Trends in the World Stress Map Database
The locat.ion, type and orientation of the maximum horizontal compressive
stress, Sfl,7nax for the indicators of the World Stress Map dataset which have suffi cient quality to provide tectonic stress information [quality A-C Zoback, 1992]' are shown as Figure 3.1. The distribution of the indicators by type is listed in Table 3.1. Of the more than 6000 indicators in the complete dataset, 4496 have A-C quality. The weighted stress regimes for 5x5 degree bins are shown as Figll1'e 3.2. Twenty levels of grey-scale shading were used to produce an image of the relative distri bution of the stress regimes of the indicators within the bins, with the darkest shading corresponding to predominance of compressional stress indicators. The image shows large regions of compression in eastern North America, central South IHO' 270'
0'
,70' r.-___=====- ___-=====- ___====- ___ac:==~ IHO' 225' 270' 0' 45' 90' IHO'
Figure 3,1, The global stress indicators from the World Stress Map database with quality A - C (N=4L196) [Zoback, 1992], Circular symbols represent focal mecha nisms,square symbols represent geologic indicators, and triangular symbols repre sent wellhole data, Solid, open, and grey-shaded stress indicator symbols represent compressional,extension and strike-slip deformational style, respectively, The ori entation of the maximum horizontal compressive stress (SII,max) is also shown, Orientations without symbols designate volcanic vent alignments, The lengths of the vectors representing SH,max have been weighted by the indicator quality (A - C) after Zoback [1992], The distribution of the stress indicators are listed in Table 3,1. Also shown are the plate boundaries as defined by De Mets, et at" [1986], ••••••••••••II.RtDrE]r'~'i CompressIonal StrIke Slip ExtensIonal
35' 35'
0'
-35' -35'
-70' ~ ___====-I"" __c:::===-"""'_-=====-_"'_-====~ -70' 180' 225' 270' 315' 0' 45' 90' 135' 180'
Figure 3.2, Grey-scale image of the average stress regimes within 5x5-degree bins l,
50
America, some areas of central Asia and Australia. A more neutral stress regime (corresponding to either a large number of strike-slip indicators or an equal number of compressional and extension indicators) is present in western Europe, central Africa, and eastern Asia. The continental areas of eastern Africa, the central area of the Tibetan plateau, western North America and the Altiplano of South America are dominated by extensional stress indicators. Oceanic areas near the ridge crest show a mix of stress indicators. Deep ocean basins arc characterized by compression.
The average SH,max orientations for the bins are shown as Figure 3.3. The length
of the resultant vectors reflect the degree of correlation of the SH,ma.v orientations in the bins. While there is a great deal of scatter present in the data, large areas exist where the orientations are consistent between bins. Examples of areas with consistent orientations include eastern North America, western Europe, western South America, central Asia, and the central Indian ocean. In order to quantify the existence of these apparent trends, the Rayleigh test was applied to the bins.
The SH,max orientations for bins which pass the Rayleigh test are shown as Figure 3.4. The solid and open bars designate bins in which the null hypothesis that the
SU,mcLx orientations of the indicators arc randomly distributed has been rejected at the 90 % and 95 % confidence levels, respectively. Note that the bins with open bars are necessarily a subset of the solid bars; that is, bins with an open bar also contain a solid bar. Many of the apparent trends in Figure 3.3 are shown to be statistically significant. Robust trends in the data exist in eastern North America, western South America, western Europe, and central Asia. Less-robust trends exist in eastern Africa and central Australia.
The angular difference between the SH,max orientation in bins which pass the Rayleigh test at a 90 % confidence level anel the ridge-push torque and the absolute velocity azimuths are shown as Figures 3.5 and 3.6. 51
180' 225' 270' 315' 0' 45' 90' 135' 180' 70' ~"-IIiII!.I!~~:;:::r;~."IIIW-==:::::;:::;::::::::.-...IIItIIIIa:;=:;;;;;:;:=---~==:;;;;;;:::;;R 70'
35' 35' /.
0' 0'
_35' _35'
_70' ~ ___~==::::J_"'_-====:1."''''~~==::::J ___-====::fl-70' 180' 225' 270' 315' 0' 45' 90' 135' 180'
Figure 3.3, Average SH,max orientations within 5x5 degree bins. The length of the vectors reflects the relative correlation of the orientations of the indicators within the bins. 52
180' 225' 270' 315' 0' 45' 90' 135' 180' 70' ~~-~"'-;;;;:;::;~~~~~=::;::;;::::::-III!IJI!~..ii:;:=;:~;:::::---~==;;;;~ 70'
35' 35' /.
" 0' I 0'
-35' -35'
-70' ~-_~=:===::~IIIIIII-IIIC=:===:"'''-I1.::=~==~---====~- -70' 180' 225' 270' 315' O· 45' 90' 135' 180'
Figure 3.4, The SH,max orientations for bins which pass the Rayleigh test. The solid and open bars designate bins in which the null hypothesis that the SU,max orientations of the indicators are randomly distributed has been rejected at the 90 % and 95 % confidence level, respectively. Bins with open bars are a subset of the solid bars. 53
180' 225' 270' 315' 0' 45' 90' 135' 180' 70'~""~~~~~~~~C=~~=3--"~~~~~"""~==~~70'
35' 35' ..-:
0' 0'
-35' "' -35'
••• I -70' ~ __-====_""'_-====-IIIIIIIIIiiiI .... _c=:===-_~-=====l! -70' 180' 225' 270' 315' 0' 45' 90' 135' 180'
Figure 3.5, The angular difference between the SU,max orientation in bins which pass the Rayleigh test at a 90 % confidence level and the ridge-push torques. Square symbols designate bins in which the difference is less than 20 degrees; cir cular symbols, difference between 20 and 40 degrees; triangular symbols, difference between LlO and 60 degrees; diamond symbols, difference between 60 and 80 de grees; shaded diamond symbols, difference greater than 80 degrees. Solid arrows indicate the ridge-torque orientations calculated with the torque poles listed in Table 2.6. 180' 225' 270' 315' 0' 45' 90' 135' 180' 70' ~""'-I111!'!;;;:;;;;:~~~""'.-===;::::;::=--IIJIIIIIII!IIIIII:;:=:;;;:::;:rr:=---II!II==~;;::;&l70'
35' 35'
0' 0'
-35' -35'
:-- ' -70' ~ ___====-""""'_-=====-"""'iiiiiII-======___ -=====:!l-70' 180' 225' 270' 315' 0' 45' 90' 135' 180'
Figure 3.6, The angular difference between the SlI,max orientation in bins which pass the Rayleigh test at the 90 % confidence level and the absolute plate velocity azimuths. Solid arrow indicate the absolute plate velocity azimuth calculated from [G1'ipp and Gordon 1990]. For other details see Figure 3.5. 55
The open bars in these Figures designate the average SH,ma,v orientation in the bins. The solid arrows show the ridge-push torque orientation in Figure 3.5 (Table 2.6), and absolute plate velocity azimuths in Figure 3.6 (from Gripp and Gordon [1990], velocity poles listed in Table 3.2). In both figures, geometric symbols have been used to designate an angular difference of 0-20, 20-40, 40-60, 60-80 and greater than 80 degrees. In general there is a good correlation between the trends and both the ridge push torques and absolute plate velocity azimuths throughout the North Amer ican, South American, and western European plates. The correlation is poor in Asia, Africa and parts of Indo-Australia. It is important to note that there exists considerable variation in the misfit values within the individual plates, particularly in Indo-Australia. This suggests the use of histograms to compare the misfit be tween plates can be misleading. For example, while the histogram of the misfits in the Indo-Australian plate shows considerable scatter [e.g., Richardson, 1992], Figures 3.5 and 3.6 provides a way to identify areas where the misfit is small (i.e., northern Australia and the central Indian ocean) and other areas where the misfit is large (i.e., south of Java and southeastern Australia).
3.4 Discussion
The results of the present study are in substantial agreement with the observa tion made by previous investigators that large regions of the intraplate stress field
have consistent SH,max orientations[Zoback and Magee, 1991; Zoback, 1992]. These trends are particularly robust in eastern North America, western Europe and west ern South America. In many regions, there is a tectonically significant trend, the
SH,max orientations are closely aligned with the ridge-push torque directions and the absolute plate-velocit.y azimuths, and thus, the results are in agreement with 56
other studies of possible trends in the intraplate stress field [Richardson, 1992J. Figures 3.5 and 3.6 clearly show that there are regions of large misfit between
SU,max and the ridge-torque orientations, even within plates where histograms of the overall misfit for the plate suggest a good correlation (i.e., the Aegean region of southern Europe, along the Aleutian trench, and the central Indian Ocean). In areas where the tectonics are dominated by second-order tectonic effects, such as eastern Africa, southern Europe, and central Indian ocean along the trenches, a large misfit exists between the observed SU,maJ: orientations and both the torques due to the mid-ocean ridges and the absolute plate-velocity azimuths. Zoback [1992J concluded that the interior regions of the plates are dominated by compressional stress indicators and that extensional indicators are confined to high topographic regions (i.e., the Tibetan plateau, western North America, East Africa, and western South America) and some near-sea level extensional provinces (i.e., the Aegean and the around coastal Gulf of Mexico). The intraplate stress field in slow-moving plates which are surrounded by mid-ocean ridges may be expected to approximate the ambient stress state [Crough, 1983J. Of the Earth's tectonic plates, these conditions may be satisfied by the African and Antarctic plates. The results of the present study show that the interior of the African plate is close to a neutral state of stress with a predominance of strike-slip indicators, with compression limited to the northern boundary. This prediction differs significantly from the assumptions made by many previous investigators who assumed that the state of stress within the continents is dominated by stresses transmitted from the mid-ocean ridges, and is therefore compressional [e.g., Turcotte el al., 1977; McI{enzie, 1978; Le Pichon and Angelier, 1979; Cochran, 1982; Le Pichon, 1983;
Turcotte, 1983; 01'0 ugh, 1983; Houseman and England, 1986; Sonder el al., 1987; England and Houseman, 1988, 1989; Zhou and Sandiford, 1992J. The large regions of compression seen in the other largely continental plates, particularly in eastern 57
North American and western Europe [Zoback and Magee, 1991; Zoback, 1992], suggests that other sources of tectonic stress (i.e., plate boundary forces) may dominate the intraplate stress field in these non-ambient plates. Because the stresses predicted by numerical modeling must be consistent with the broad-scale regional stress field in the regions where the observed stress field is well constrained, the results presented here will provided valuable constraint for the finite element analysis presented in Chapter i1. 58
Indicator Type Number of Indicators Total Total Focal Mechanism 2430 Focal Mechanism Thrust 972 Focal Mechanism Normal 531 Focal Mechanism Strike Slip 894 Focal Mechanism Unknown 33 Total Geologic <167 Geologic Thrust 29 Geologic Normal 297 Geologic Strike Slip 26 Geologic Unknown 115 Total Wellhole 1599 Wellhole Thrust 29 Wellhole Normal 297 Wellhole Strike Slip 43 Well hole Unknown 1399
Table 3.1, Global Stress Indicators .'
59
Platel Latitude Longitude Velocity [degJ [degJ [Deg/MyrJ PAC 60.2S 90.0E 0.98 NAM 67.2S 11.1W 0.28 SAM 70.3S 7l1. 7E 0.32 EUA 4L1.8S 58.1E 0.09 AFR 5.5S 3.6E 0.15 IND 19.2N 35.6E 0.72 ANT 1l1.8S 65.9E 0.11 NAZ l15.7N 90.2W 0.'16 COC 18AN 115.9W 1.29 CAR 62.4S 5.7W 0.17 ARB 16.8N 18AE O.M PHL 49.'1S 19.9W 1.11
Velocity information is from Gdpp and Gordon [1990J, and Seno et al. [1987J for the Philippine Plate. 1 Plate Abbreviations: PAC=Pacific, NAM=North America, SAM=South America, EUA=Eurasia, AFR=Africa, IND=Indo-Australia, ANT=Antarctica, NAZ=Nazca, COC=Cocos, CAR=Caribbean, ARB=Arabia, PHL=Philippines
Table 3.2, Absolute Plate Velocity Poles I
60
CHAPTER 4 SOUTH AMERICAN INTRAPLATE-STRESS MAGNITUDE
4.1 Introduction
Uncertainty about the magnitude of intraplate stresses limits our understand ing of a number of plate-tectonic processes includillg the dynamics of the plate driving mechanism, the mechanics of earthquake faulting, the tectonics of conti nental interiors and the dynamics of mountain building. While the World Stress Project [Zoback, 1992] has increased our knowledge about the orientation of the in traplate stresses in the lithosphere, it has provided only limited information about the magnitude of these stresses. In part, this is due to both technical and economic constraints which limit direct measurement of the stress tensor to the upper few kilometers of the Earth's crust. Predictions about intraplate-stress magnitudes a.re therefore ba.sed primarily on indirect methods such as the analysis of earthquake focal mechanisms, numerical modeling of intraplate-stress fields, scaling of labo ratory experiments, and extrapolation of stress meaSUl'ements made in the near surface. The results of in situ stress measurements in the upper 3 km of the crust worldwide are consistent with the Mohr-Coulomb frictional faulting theory and laboratory-derived friction coellicients [e.g., McGarr and Gay, 1978; Brace and Kohlstedt, 1980; McGarr, 1980; McGa'l'/' et al., 1982; Zoback et al., 1980; Zoback and Hickman, i982; Slock et al., 1986; Vel'nik and Zoback, 1992; Zoback and Healy, 1992]. For the case of hydrostatic pore pressure, Byerlee's law [Bye'l'lee, 1978] pre dicts levels of several hundred MPa for the deviatoric stress at mid-crustal depths, which is in agreement with these observations. The prediction of high tectonic stress magnitudes is supported by a number of studies, including flexUl'al analysis of outer arc bathymetry [e.g., McNutt, 1980; BoeHne et at., 1981; McNutt and 61
k[enal'd, 1982; I{i1'by, 1983; Ida, 1984]' studies of the Sierra Nevada uplift [Chase and Wallace, 1988], constraints from seismological investigations [Stein and Pelayo, 1991; Grovel's et ai., 1992], numerical modeling of the intraplate-stress field of the Indo-Australian plate [Cioetingh and Wodel, 1985, 1986], and investigations of the possible buckling of the Indo-Australian plate [McAdoo and Sandwell, 1985; Stein et ai.} 1989]. In contrast to the predictions of Byerlee's law, however, a number of recent studies suggest that motion along major plate-bounding transform faults such as the San Andreas is resisted by very small shear stresses. Models for the fric tional strength of fractured rock predict that the depth-dependent shear strength
of faults, T, in the brittle continental crust should be about 35, 60 and 150 MPa in normal-, strike-slip- and thrust-faulting environments, respectively [Hicknwn, 1991]. The results of conductive heat-flow measurements along the San Andreas
imply that T is less than 20 MPa averaged over the upper ILL km of the fault [see Hickman, 1991 for a review]. These low shear stresses can either be explained by appealing to such factors as changes in pore pressure, fault-strength anisotropy, or rheological and lithological heterogeneities [e.g., Hickman, 1991; Slcep and Btan pied, 1992]' or by the existence of low (ten's of MPa) far-field intrapla.te-stress magnitudes. In general, results of numerical modeling studies and of the intraplate stress field for a number of plates support the prediction of low tectonic intraplate stress magnitudes, [e.g., Richardson et al., 1979; Richardson and Cox, 198 Ll; Ju/'{iy and Stefanick, 1991; Richardson and Reding, 1991; Wodel et ai., 1991]. These studies also have served to emphasize the important role that the ridge-push force plays in orientation of the maximum horizontal compressive stress, SU,I1WX' A re cent study of the relationship between the ridge-push force, the absolute plate motions and SU,max suggests that the ridge-push force may be the source of the majority of the long wavelength features in the intraplate-stress field [Richardson, 62
1992]. Because the ridge-push force is well constrained to range from 2 - 3 X10 12 Nm-1 [e.g., Frank, 1972; Lister, 1975; Parsons and Richter, 1980], corresponding to a horizontal tectonic stress in the range of 20-30 MPa averaged over a 100 k111 thick lithosphere, this correlation further supports the predidion of low intraplate-stress magnitudes. An alternative approach to prediding the magnitude of the horizontal tectonic stress is to evaluate the interaction between tectonic and buoyancy forces near high plateaus. On a global scale, there is a correlation between high topography
and extensional stress regimes which implies that the buoyancy force, O"z:;, due to lateral density variations associated with the topography locally exceeds the regional horizontal tedonic stress,O"xx, [Zoback and Magee, 1991]. This observation has been used to predict the magnitude of the horizontal tedonic stresses which support regions of high topography [e.g., A'I'i,yushkov, 1973; Dalmayrac and fI![o!nal', 1981; Fleito'ltt and Fl'oideva'lt:r, 1982, 1983; Filkao and Yamaoka, 1983; Fl'oidevau:r and Isacks, 1984; Pl'oidev(t'llx and Ricard, 1987; MoinaI' and Lyon-Caen, 1988; Zhou and Sandiford, 1992]. These studies established an upper bound of about 50 MPa for the horizontal tectonic-stress magnitude (averaged over a 100 km thick lithosphere) which supports a 4 km topographic load. The results of these studies were limited, however, by the need to average the magnitudes of 0":;:; and O";L'X over the thickness of the lithosphere. As a result, the possible variations of 0":;:; and
O".vx with depth were not evaluated. In addition, these studies assumed that the long-wavelength approximation was applicable to the topographic load, implying that the average vertical shear stress across the lithosphere is negligible. In the present study, we use an elastic finite-element ana.lysis to evaluate the lithospheric stress state across the high topography of the Cordillera Blanca region of the Andes. Because the finite-element method provides a way to calculate the principal stresses explicitly throughout the lithosphere we will be able to study the 63 state of stress in detail, including its dependence on depth. The purpose of this paper is thus twofold. First, we seek to evaluate the possible depth dependence of
(J'zz and (J'xx in the vicinity of a high topographic region in order to refine the upper bounds placed on the horizontal tectonic-stress magnitude made in the previous studies. Secondly, we evaluate the depth dependence of the shear stress, (J'xz, in order to verify that the long-wavelength approximation is appropriate for the topographic load of the Andes.
4.2 Topography and Deformation Style
The tectonic style of deformation is governed by the quantity, ((J'xx - (J'::z)
[Fldtout and F/,oidevaux, 1983], where (J'XJ' and (J'::z are defined as the horizontal and vertical stresses averaged over the thickness of the lithosphere. If extensional stresses are defined as positive, this quantity is positive in extensional regimes. The relationship between topography and deformation style can be evaluated (af ter Fleilo1lt and Fl'Oideva1lx, 1983) by considering the two-dimensional mechanica.l equilibrium equations: f)(J' xx f)(J' x:: 0 (4.1) f)x + f)z = and f)(]' ::z + f)(J' xz A -- = upg (4.2) f)z ax where x and z are horizontal and vertical , ~ p is the density difference from a reference state and 9 is the gravitational acceleration. If the density variations in the lithosphere have a wavelength that is long compared to the lithospheric thickness, lateral variations of the shear stress within the lithosphere are small, and the equilibrium equations reduce to
= 0 (4.3) 64
and
aO"Zz A -az = w.pg (4A)
The average value of the vertical stress, O"zz, then reduces to
9 O"zz = L 1£%0 ~ p(z) dz (4.5)
(where L is the depth of compensation and Zo is the surface elevation). Variations in O"xx are then related only to shear stresses acting along the base of the lithosphere and variations in the lithospheric thickness. While the magnitude of these shear stresses is not well constrained, it can be assumed that they are much smaller than the variations in O"zz. Thus, in equilibrium cases where the long-wavelength approximation is applicable, O"xx is independent of the lateral density variations, and the sign of the quantity (O"a,x - O"zz) is dependent only of the magnitude of
0" zz. Because 0" z:: can readily be calculated from (L1.5), this approach provides a. powerful way to place constraints on the horizontal intraplate-stress magnitude.
4.3 Finite-Element Modeling
In the present study, we have used an elastic finite-element analysis to evaluate the state of stress within lithosphere in the vicinity of a high topographic re gion. The finite-element method provides a way to calculate the principal stresses throughout the lithosphere, and therefore will allow us to build upon the results of previous studies which used the reduced equilibrium equations (4.3) and (4.4) to evaluate the stress state. We will be able to evaluate O"xx, O"zz, O".vz and the quantity
(O"xx - O"zz) explicitly throughout the lithosphere and therefore be able to study the depth and lateral dependence of these quantities. 65
4.3.1 Modeling Location
The high topography used for the stress modeling in the present study was the Cordillera Blanca region the Andes. Extension in the Cordillera Blanca region is well-documented with many examples of normal-faults displacing Quaternary glacial deposits [Dalmayrac and Molnar, 1981J. The major normal-fault zone is over 200-km long and shows recent normal motion with right- and left-lateral components striking Nl.SoE and N150oE, respectively, thus indicating roughly NS lengthening [Merciel' cl al., 1992]. Other examples of active normal faulting include
the 19 L16 Ancash earthquake which indicated normal faulting on a plane parallel to the Cordillera [Doser, 1987], and the results of a microearthquake survey [Dcv crchcl'c ct al., 1989J. The NS extension observed in the Cordillera Blanca region is
evidence that 0'1 and 0'2 lie in a plane nearly orthogonal to the orientation of the
Cordillera Blanca. In regions of little topography, 0'1 is oriented in the direction of the east-west regional stress field [Ass'Ilmpfao, 1992]' whereas in high topography,
0'1 is oriented vertically. This orientation of the principal stresses allows the use of' a t.wo dimensional analysis, oriented orthogonal to the trend of the Cordillera, to model the stress field. This provides a significant advantage for conducting the stress modeling in the Cordillera Blanca region rather than other areas with similar topographic relief such as the Altiplano to the south or the Tibetan plateau. The relationship between topography and the distribution of stress indicators in the Cordillera Blanca region is shown as Figure L1.l. Details of the individual stress indicators are listed in Table 4.1. The thirteen stress indicators located in this region define both the compressional stress state of the Andean foreland and the extensional stress state of the high topography regions. The extensional stress regime is restricted to regions where the elevation exceeds 3000 meters, which is consistent with observations in other areas of the Andes and in the Tibetan plateau [England and Molnar, 1991]. As pointed out by Ass1l'Inpfao [1992]' the 66
-8 0 ~-"'-=~===~-"-;::======---======+l South America
Pacific Ocean _10 0
•...
282 0 2840 286 0
Figure 4.1, The relationship between topography and the distribution of stress incli cators for the Cordillera Blanca region of Peru. Light-grey regions have elevations from 3-4 km; mediumgrey, 4-5 km; dark gray, over 5 km. Circular symbols repre sent focal mechanisms and square symbols represent geologic indicators. Contour interval in 1000 meters. .'
67
SU,nwx orientation is this region is nearly uniformly EW in both the foreland and high topography regions, which suggests that the regional horizontal stress field in western South America reflects far-field rather than local sources. The stress modeling performed in the present stuely was conducted along the profile shown in Figure L1.1. The profile is centered on 9.5 0 S and extends from 281 0 to 285 0 E with a lateral extent of about 500 I West 3000m Contour East + o 13 o 4 11 12- o 1 ~ - ~-20 10 OJ - Q -30 - 8 -40+---~----~--~----~--~----~----~---r------+ o 100 200 300 400 500 Distance Along Profile [km] Figure £1.2, The relationship between the topography and stress indicators along the profile shown in Figure 4.1. Topography along the profile was determined by averaging the ETOP05 topography dataset in a 50-Inn swath about the profile latitude. See Figure 4.1 for details of the stress indicators. " 69 transition occurs at an elevation of 3000 meters based on the observation that nearly all crustal extension in the Andes occurs at greater elevations [e.g., Siigado, 1951; Mercier, 1981; Sllarezet al., 1983; Sebrier et ai., 1985, 1988a, 1988b; Cab'rem et al., 1987; Doser, 1987; Mercie'r et al., 1992J. 4.3.2 Description of the Finite-Element Grid A two-dimensional vertical slice through the lithosphere was represented by a finite-element grid composed of an assembly of isotropic, elastic, quadrilateral elements in a state of plane strain with a Poisson's ratio of 0.25. Model stresses are caused by the density contrast which exists between lithosphere beneath the Andes and the far-field, which we have taken as the stable Brazilian Shield. Due to the limited knowledge of the crustal structure under the Andes, we have assumed a crustal model where excess topography is isostatically compensated by a crustal root. The crustal model is based on a density structure which balanced the posi tive 3 density contrast of the topography (~p = +2600 kgm- ) with a negative crustal 3 root (~p = -300 kgm- ), and is consistent with the available crustal information for the Andean region based on seismic and gravi ty data [James, 1971; Ocola et al., 1971; Grow and Bowin, 1975; Kulln et al., 1981; Gotze et al., 1988; [{ono et ai., 1988; Wigger, 1988; Jilukao et ai., 1989J. In order to minimize edge effects, the grid was extended 100 km past both ends of the profile and to a depth of 200 km. The lithosphere was represented as a stiff region (Young's modulus of 70 GPa) overlying a less stiff asthenosphere (Young's modulus of 70 MPa). The boundary between the lithosphere and asthenosphere was fixed at 100 km. In total, the finite-element grid contained 2616 nodes and 2L18 Ll elements, with a uniform nodal spacing of 5 km for the lithosphere directly beneath the high topography region. This nodal spacing was chosen to provide sufficient resolution to study lateral variations of the stress field in detail. The boundary conditions applied to the grid for the stress modeling are shown as I' 70 W E Topogrllphy 0 Lithosphere - 0 - Fx Crus!1l1 nOli! - - Figure L1.3, Boundary conditions applied to the finite-element grid for the stress modeling. The upper surface of the grid is traction-free and represents the Earth's surface. Free-slip boundary conditions were used along the left and bottom edges of the grid such that the left side was fixed in the horizontal and free in the vertical dimensions, and the bottom was fixed in the vertical and free in the horizontal dimensions. Equivalent nodal forces, representing the far-field horizontal intraplate stress in the South American plate averaged over the lithospheric thickness, were applied to the right side of the lithosphere. The right side of the region representing the asthenosphere was free to move in all directions. 71 Figure 4.3. The upper surface of the grid is traction-free and represents the Earth's surface. Equivalent nodal forces, representing the far-field horizontal intraplate stress in the South American plate, were applied to the right. side of the lithospheric layer. The right hand edge of the grid representing the asthenosphere was free to move in all directions. Free slip boundary conditions were used along the left and bottom edges of the grid such that the left side was fixed in the horizontal and free in the vertical dimensions, and the bottom was fixed in the vertical and free in the horizontal dimensions. There is general agreement that the current phase of mountain building in the Aneles has occurred over at least much of the late Cenozoic, with significant uplift during much of Miocene time [see review in Isacks, 1988J. Thus, unlike the Tibetan plateau, where the onset of uplift has been much more recent [e.g., !I;{olna1' and Tapponniel', 1978; England and Houseman, 1989], the Andes can be considered as steady state. This assumption is reflected in the boundary conditions used in the present study which allow us to place constraints on the horizontal stresses in South America without completely specifying the dynamics of the Andean uplift. 4.3.3 Modeling Results The principal non-hydrostatic stresses along the profile with an applied far-field horizontal stress of 35 MPa are shown as Figure 4A. The st.resses are compressional throughout the lithosphere. As predicted by the long-wavelength approximation, the horizontal stress is uniform throughout the lithosphere with slight variations due to non-zero shear stresses. The vertical stress throughout the lithosphere varies as a function of the topography. The maximum vertical stress occurs at l11id crustal depths at about 325 kl11 along the profile and corresponds to the highest topography. Along the eastern flank of the Cordillera, the vertical stress exceeds the horizontal stress at about 525 kl11 along the grid (225 kl11 along the profile) which corresponds to the location of the 3000 meter contour. Also shown in Figure 72 {LA are the compressional and extension stress indicators along the profile. We note that the location of the compressional and extensional stress regimes predicted by the modeled principal stresses are in good agreement with the observed stress indicators. Contoured values of the calculated shear stress, O''''Z' where 0'1 - 0'2 ' O'xz = ( 2 ) 8'/,11.(20) (4.6) where 0 is the angle between the horizontal and 0'1, for a model with an applied far-field horizontal stress of 35 MPa are shown as Figure 4.5. The magnitude of the shear stress approaches zero as 0'1 approaches 0'2 or as 0 0 approaches a or 90 • Regions of non-zero shear stress correspond to areas where the principal stresses are not oriented vertically and horizontally. It can be seen in Figure 4.5 that the largest shear stress values correspond to regions where the surface topography is changing most rapidly, such as near the crest of the Andes, at about. 425 km along the grid (125 km along the profile). The shear stresses are near-zero near the edges of the grid and beneath the 1110ho. The maximum deviation of the shear stress from a zero value is less than 4 MPa. The resulting deviation in the quantity O'xx-O'zz from the value based on assuming that the shear stress O'a'z is zero is less than 5 MPa, or less than 5%, with the vertical average of the shear stress across the lithosphere being negligible throughout the grid. Because the principal stresses are oriented nearly horizontally and vertically throughout the lithosphere, the change in deformation style can be related to the relative magnitudes of O',rx and O'zz. An upper bound for the horizontal stress mag nitude was established by evaluating the quantity O'xx - 0':::: for a range of far-field horizontal stress values and modeling the change in the deformation regimes. The compressional and extensional regimes within the lithosphere, as determined by the sign of the quantity O'xx - 0':;:;, for three far-field horizontal stresses magnitudes (la, 35, and 100 MPa) are shown as Figure 4.6. The magnitude of O'z:: was calculated from the assumed density structure of the I I' 73 West East 3 5 6 9 3000m Contour 100MPa , , '2 ~ 13 o -r-- + f t -b~t t t t + +-.. .. + + - -11 - + + t t hot t t + -I- .. .. + + - 12'! ..-_ -25 - 10 + + t t t t t t + x .... + ·e-8- f t t t + + + '" e7 - + + + '" • -75· -100 +------r--~--_r_----_._--~-__._----_+ 300 400 500 600 700 800 Distance Along Grid [km] Figure ilA, The principal non-hydrostatic stresses along the profile with an applied far field horizontal stress of 35 MPa.. The principal stresses are compressional throughout the lithosphere. West East 3000m Contour 2,3,5,6,9 ~ 13 o G 11 12- _. ·25 10 -7 8 ~ - o:Sg. ·50- Q \) ·75 ~@:11 ·100 300 400 500 600 700 800 Distance Along Grid [kmJ Figure 4.5, Contoured values of the calculated shear stress, O"xz, for a model with an applied far-field horizontal stress of 35 MPa. Contour interval is 1 MPa. 75 3000m Contour West East ·20 . , ·40· ·60· Sxx =10 MPa ·80 -I------,--~-_r_--___,_---_r__--_+ O·h~mc0'Gf~~~----~i ~ . , ·40· • • ·60 Sxx =35 MPa ·80 .I---__,_-~-,.---__,_---,_--__l ~ o~----~~~~~------.------+. , ~.20. ® 0 • o:S ·40 • 0.. ~ ·60 Sxx=100MPa ·80 .I---__,_-~-r----_,_--__,,_--_+ 300 400 500 600 700 800 Distance Along Grid [kml Figure 4.6, The predicted compressional and extensional regimes within the litho sphere beneath the Andes for far-field horizontal stress magnitudes of 10, 35, and 100 MPa, respectively. The shaded region defines the extensional regime where (J" zz exceeds (J" xx' l, 76 Andes discussed above, and is the same in all three cases. The shaded region de fines the extensional regime, where (J'zz exceeds (J'xx' The 10 and 100 MPa models are poor predictors of the deformational state of a majority of the stress indica tors, and represent lower and upper bounds on the far-field horizontal stress (J':L':L" respectively. A range of intermediate far-field stresses between these bounds was applied and the resulting deformational regimes evaluated. As noted earlier, the exact location of the change in deformation style is not well defined by the stress indicators along this particular profile. If we choose the location of the 3000 meter contottL' as indicative of the transition from compressional to extensional tectonics, consistent with other studies in the Andes, we obtain a magnitude of 35 MPa for the far-field horizontal tectonic stress. A value of 35 MPa also predicts the correct deformational state of all the stress indicators along the profile. We therefore con clude that 35 MPa averaged over a 100 km thick lithosphere represents OttL' best estimate of the horizontal intraplate-stress magnitude of the South American plate in the vicinity of the Cordillera Blanca. 4.4 Discussion We interpret the horizontal stress magnitude established by the stress modeling in the present study to be representative of the intraplate-stress magnitude in the South American plate. The orientation of SH,1I1(tx (maximum horizontal compres sive stress) in western South America is quite uniform (see Figures 3.3 and 3A), given the possible variations in plate coupling, angle of subduction, and changes in the strike of the Andes. We conclude that SH,max therefore reflects far-field sources rather than local sources such as the forces acting along the Peru-Chile trench and variations in the buoyancy force due to the high topography [Ass'Ilmp9ao, 1992J. Extrapolating our study, based on the nearly uniform EW orientation for SH,max along the Andes [ASS'llmp9aO, 1992]' we conclude that the stress magnitude Pre- .' 77 dicted in the present study is representative of the far-field intraplate-stress mag nitude throughout the South American plate. A far-field tectonic stress of 35 MPa supported across 100 km is consistent with a ridge-push origin. The magnitude of lithospheric stresses associated with lateral density variations is dependent on t.he distribution of the density variations with depth. The analysis performed in the present study was based on a crustal model which assumed the topography to be entirely compensated by the negative buoyancy of the deflected moho. The topography may be compensated by a lithospheric thermal root, in which case the crustal root is absent and the topography is supported by a de flection of the lithosphere-asthenosphere boundary [Froidcva1lx and [sacks, 1984]. The source of the high elevation in the Andes is probably not simply related to thickening of the crust but to a combined effect of a thickened crust and a thinned mantle [Isacks, 1988]. The presence of a thinned lithosphere beneath the Andes is also supported by the large number of volcanic centers in western Sout.h America [Jordan ct al., 1983]. A thermal root, however, would result in a ductile lower lithosphere and cause a concentration of the horizontal stress in the upper part of the lithosphere. For rea.listic rheologies, it has been estimated that a stress am plification by a factor of 2-3 is possible in the upper lithosphere in regions with heat flows of 60 mWm-2 [I(1lsznir, 1991]. The existence of a thermal root beneath the Cordillera Blanca would therefore locally reduce the thickness over which the far-field tectonic stress is supported. The range (10 to 100 MPa) and preferred (35 MPa) values for the far-field stress from our modeling are based on supporting the stress over a 100 km thick lithosphere. If the thickness over which the far field stress is supported is locally reduced, a smaller far-field stress can produce the same pattern of extensional and compressional stress regimes. Therefore, our estimates of the far-field tectonic-stress magnitude are conservative. The results of the present study serves to emphasize the important role played 78 by lateral density variations in the intraplate-stress field. It is well documented that topography generated on density interfaces during continental deformation affects the potential energy of the lithosphere and therefore has important implications for stresses in the vicinity of orogenic belts [e.g., Al'tyushkov, 1973; Molnm' and Tapponicl', 1978; England and Mc[(cnzic, 1982; England and Houseman, 1988, 1989; Sonde'/' et ai., 1987, MoinaI' and Lyon-Caen, 1988; Sandiford and Powell, 1990, 1991; Zlwu and Sandiford, 1992). Oul' results agree with the suggestion that many of the extensional stress indicators observed globally are the result of hori zontal buoyancy forces due to high topography exceeding the regional compressive stress [Zoback and Magee, 1991; Zoback, 1992). Other models have been proposed which explain the observed pattern of deformation in the Andes in terms of cou pling between flow in the viscous asthenosphere and the lithosphere [Wdowinski and O'Connell, 1991). While recent investigations of the attenuation of seismic waves beneath the Andes suggests a correlation between upper mantle structures and changes in the sUl'face featUl'es [Whitman el ai., 1992)' we feel that the results presented here demonstrate that density variations within the lithosphere play an important role in the dynamics of lithospheric deformation. 79 Map Latitude Longitude Azimutha Typeb Depth Site Elev QualityC Stress Index (OS) (OE) ( degrees) (km) (meters) Regime 1 8.4 282.5 120 FMS 16 2995 C Normal 2 8,4 282.5 120 GFS 0 2995 A Normal 3 9.1 282.3 100 GFS 0 4608 A Normal 4 9.3 282.5 150 FMA 5 4827 A Normal 5 9.4 282.5 94 GFS 0 4525 A Normal 6 9.6 282.5 100 GFS 0 4395 A Normal 7 9,4 284.3 95 FMS 32 2003 C Thrust 8 9.1 284.4 80 FMS 2 897 C Thrust 9 10.0 282.7 79 GFS 0 4570 A Normal 10 10.5 285.2 81 FI'vIS 18 906 D Thrust 11 10.6 285.3 72 FMS 14 1103 C Thrust 12 11.0 285.1 55 FMA 14 864 C Thrust 13 11.2 284.6 109 GFI 0 1726 A Thrust All stress indicators are from the World Stress Map database [Zoback, 1992]. aSIl,max orientation, measured in degrees clockwise from North, 0° to 180°. bStress Indicator Type: FMS: Single focal mechanism; FMA: Focal mechanism average. GFS: Geologic fault slip orientation; GFI: Striae inversion. Frol11 Zoback [1992]. cQuality ranking frolll Zoback [1992]. Table 4.1, Summary of Stress Indicators for the Cordillera. Bla.nca. Region, Peru 80 CHAPTER 5 INTRAPLATE STRESSES DUE TO POTENTIAL-ENERGY VARIATIONS: A FINITE-ELEMENT ANALYSIS 5.1 Introduction This chapter presents the results from a finite-elements analysis which incorpo rates forces due to potential-energy variations. Predicted stresses for three plates (African, South American (SAP) and Indo-Australian (lAP)) are presented. The analysis begins with all evaluation of the predicted stress field for the African plate. The African plate, which is slow moving and is predominately surrounded by mid-ocean ridges, can be expected to approximate the ambient stress state (see discussion, Chapter 2). It provides an ideal location to evaluate the applicability of the concept of the Tectonic Reference State (TRS) anel the relationship between potential-energy variations and the intraplate stress field in a continental plate which is thought not to be dominated by other tectonics forces. The study turns next to the SAP. The tectonic forces acting on the SAP are more complex than ill the African plate, with collision occurring along its entire westcl'll margin. How ever, because the SAP itself in not attached to a grcat length of subducting slab, the slab-pull force docs not dominate the tectonic balance of the SAP [e.g., Mei je'!' and Worlel, 1992; Stefanick and Jurdy,1992]. lVlodeling the intraplate stress field in the SAP therci'ore provides a way to evaluate thc relative contribution of forces due to potential-energy sources to the intraplate stress in a continental plate where boundary forces also play an important role. The final plate studied, the lAP, contains a large number of tectonic features including active subduction zones, an extensive mid-ocean ridge system, significant areas of both continent-continent and continent-island arc collision, and regions of intraplate oceanic-lithosphere de- I 81 formation. The lAP therefore is ideal for studying the relationship between the full spectrum of tectonic processes and intraplate stresses. At the outset it is necessary to emphasize that the main constraints used in this study are the SH,mo:!, orientations of the in sit'll stress field. A study of the statistical trends in the regional intraplate stress field [Chapter 3J provides the bulk of this constraint. A study of the magnitude of the intraplate stress field in the South American plate (Chapter 4) provides additional information about stress magnitudes which is used to constrain the predicted stress in this plate. 5.2 Discussion of Tectonic Forces Three principa.l tectonic processes were used to model the forces acting on the plates. These are, buoyancy forces, compressional stresses transmitted across plate boundaries and shear forces acting along the base of the plates. Buoyancy forces result from lateral density variations within the lithosphere (see Chapter 2) and include the well-known ridge push and slab-pull forces as well as forces associ ated with continental margins and high topography regions. Compressional stress are transmitted across boundaries where plate collision is taking pla.ce. Exam ples include, continent-continent collision, continent-island arc collision and forces transmitted to the overriding plate in a subduction zone. Basal drag forces a.rise from shear tractions acting along the ba.se of the lithosphere due to the motion of the plate over the underlying mantle. Each of these forces is discussed below. 5.2.1 Buoyancy Forces Buoyancy forces arise as a consequence of lateral density variations within the lithosphere and are known to produce significant stresses within the lithosphere (see 82 discussion in Chapter 2). The influence of lateral density variations on the litho spheric stress field is proportional to the density moment of the mass dipole formed by the mass anomaly, with positive mass anomalies producing tectonic compression [F'leitollt, 1991J. The contribution of potential-energy variations within the litho sphere to the total torque acting on the plates was calculated by incorporating the 'moment law' into the finite-element analysis. This was accomplished by applying a basal shear force to each element proportional to the horizontal gradient of the local dipole moment (which is proportional to UI, see equation (2.3) and (2.4) ) [F'leito1l1, 1991; Richardson and Reding, 1991J. For plane geometry this force can be expressed as L (8(7x3.,' 8(73.,'Y) = L _ L8M L (8(71/Y 8(7Xy) L L8M + ~., + = (7 (5.1) 8x 8y w_ 8x 8y 8:v 1/Z 8y or L 8M L 8M (7 xz C\' --;:;- a11 d (71/Z C\' --;:;- (5.2) ux uy where x and yare the two horizonta.l directions, (7~z and (7~z are the resistive shear stresses due to the mantle drag along the base of the plate and Al is the density moment. Information about the torque contribution from the buoyancy forces provides a way of quantifying the relative contribution to the intraplate stress field with respect to other tectonic forces acting on the plates. The torque exerted on the plate from a force F is calculated as T = rxF (5.3) where r is the radius position vector and F is the force acting on the plate. The total torque acting on the plate is found by integrating T over the surface of the plate. The total torque due to forces associated with cooling oceanic lithosphere, 83 topographic forces and all potential-energy sources is plotted versus the absolute plate veloci ty in Figure 5.1. Whereas the mean potential energies of the plates (see Figure 2.3) are inherently clustered about the global mean potential energy, there is significant variation in the relative torque contributions from the potential-energy distributions. The weak correlation between ridge-torques and plate velocities was originally pointed out by Forsyth and Uyeda [1975]. It should be noted that the angular misfit between the torque and velocity poles is considerably less for plates with large potential-energy torques, and is less than 25° for the South American, Nazca, Indo-Australian and Pacific plates. Torques associated with continental topography are typically about 2.5x1025 Nm and are considerable smaller than ridge torques which exceed 4x1025 N m for all the plates (with the exception of Nazca). There is little correlation between the topographic torques and the absolute plate velocities. Furthermore, the angular misfit between the torque and velocity poles is very large for the topographic torques, in excess of 90° for most of the plates. While topographic forces are certainly an important sources of stress, it is clear that mid-ocean-ridge torques playa 1110re important role in the plate-scale dynamics. This observation supports the notion that ridge push forces are an important component of the driving mechanism and global intraplate deformation [Richardson, 1992]. The strong correlation between the potential-energy distributions and plate velocities, at least for the fast-moving plates, is evidence that motion relative to the absolute reference frame is determined by the surface plates themselves, rather than sub Ii thospheric flow. 5.2.2 Collisional Boundaries Substantial forces are expected to be transmitted across the collisional bound aries of the lAP and the SAP. For the purposes of the present study, the force transmitted across the collisional boundaries of the lAP were approximated by 84 S a) Ridge Torques Z 211 -l--'--1-.-,--1-.-,--1-.___ ...L.--,-+ IXII ..... ~ PA l;::: CIl 8 15 APR >< EUR ~ 911 ~ III - NAM SAM NAZ IAI' 1a ~ 5 - AI'I] ~ hAl' .-. EURn 1/ NA~~hAM flAZ I'An f! II -l-.q.J.I...,.-LJ,lL--r-l-I.,-...-...,LL...,--.-JJ!:- ~ ~ (1.11 11.2 II..! II.X 1.11 (1.(1 11.2 II,,) 11.6 II.X 1.11 Velocity [deg/m.y.J Velocity [deg/m.y.J S b) Topographic Torques Z 211 -l--'-....L..--'-...l---'-....I-~--L---'--I- - NAZ PA ~ SAM 8 15' A~&M >< lAP gill _ EUR 8" ~ 5 ~ EUR ] II -t-J.!-'l:;:.:.....u,u...-,-N:..:.;A,:.::Z:....,--..,a.IA_1',..--,....I'A4 - ~ (1.(1 11.2 II..) 11.6 II.X 1.11 11.11 11.2 II,,) IIll II.X 1.11 Velocity [deg/m.y.J Velocity [deg/m.y.J c) All Potential Torques I'A NAZ (1.(1 11.2 11.4 1.11 (1.(1 11.2 11,4 (1.(, II.X 1.11 Velocity [deg/m.y.J Velocity [deg/m.y.J Figure 5.1, Total torque versus absolute plate velocity for potential-energy dif ferences due to mid-ocean ridges, topographic, and all potential-energy sources, respectively. Also shown are the angular misfit between the torque poles (listed in Table 2.6) and the absolute plate velocities poles. Plate velocity information is from Minster and Jordan [1978J for the Indo-Australian plate, and NUVEL-1 [Gripp and Gordan, 1990J for all other plates. Plate abbreviations are EUR=European, AFR=African, NAM=North American, SAM=South American, NAZ=Nazca, IND=Inclo-Australian, PAC=Pacific, respectively. 85 applyjng a constant force per unit length along the boundary. The magnitude of the coupling force applied along the Peru-Chile trench in the SAP is proportional to the sine of the subducting angle [Fl'Oidcvaux and [sacks, 198L1J and thus varies significantly along the strike of the trench. In both cases, the magnitude of these forces is poorly constrained. One constraint on the magnitude of the forces acting along boundaries involving continent-continent and continent-island arc collision is provided by the excess potential energy associated with the elevated continental topography. For example, the horizontal force due to crustal thickening in Tibet 12 l has been estimated to be as great as 6 to 8 xl0 Nm- , depending on the crustal density distribution [Zho'/t and Sandiford, 1992J. However, dynamic support of the high topography may significantly reduce the net contribution of this force to the clastic parts of the plate. Bouguer gravity anomalies along profiles across the western Himalaya and Ganga Basin show large deviations from local isostatic equilibrium [Lyon-Caen and Molnar, 1985], which can be explained in terms of flexural loading of the Indian plate by the distributed load of the high topography. In as much as the exact magnitude of the force acting on the boundary is not well constrained, the sensitivity of the predicted stresses within the plate for a range of boundary forces is tested. The magnitude of the boundary force applied to major 12 1 plate boundaries was limited to values between 2 and 6 xl0 Nm- • 5.2.3 Basal-Shear Tractions The third class of tectonic force which acts on the plates is due to the relative motion of the plates across the underlying asthenosphere. This force is often referred to as the 'drag force' and may be either resistive or driving in nature. The relative contribution of the drag force to the intraplate stress field is difficult to constrain since the nature of the coupling between the lithospheric plates and the underlying asthenosphere is poorly understood. A major uncertainty centers on 86 the fact that the relative motion between the lithosphere and the asthenosphere is not understood. Because there is little correlation between plate area and absolute plate velocity [Forsyth and Uyeda, 1975J, the nature of the coupling probably is quite complicated. Furthermore, there appears to be little evidence for a strong correlat;ion between the ast.henospheric flow pattern due to counter flow and the present-day plate motions [Chase, 1979; Richards and Hage'l', 1984J. In the present study a basal drag is used only as needed to balance the net torque acting on the plate (see discussions in Richardson et at., 1979 and Richardson and Reding, 1991). The drag force per unit area, Fel!,) can be expressed as 1~11' = D Wabs X r (5.'1) where D is a drag constant which converts plate velocity into shear stress, Wa.bs is the absolute rotation vector of the plate (see Minster and Jordon, 1978 (AM1-2), Chase, 1978; Gripp and Gordon, 1990 (I-IS2-NUVEL1)), and r is radial vector at a point on the plate. 5.3 Discussion of the Modeling Method The tectonic stresses in the plates have been modeled using an elastic finite element analysis. While the use of an elastic rheology to model whole-plate de formation is obviously an oversimplification, it is appropriate for the modeling of first-order tectonic stress. Alternative rheologies, such as visco-elastic, are useful for studying how tectonic stresses relax over time. Because the first-order stresses modeled in this study have renewable sources on geologically significant time-scales (e.g., forces due to long-term lateral density variations in the cooling oceanic litho sphere and collisional boundary forces) they can be considered to be in steady state. Thus, the use of purely elastic rheology is appropriate. 87 In order to adequately model stresses due to the potential-energy variations discussed in Chapter 2 (i.e., high topography areas, continental margins and mid ocean ridges), a grid with a resolution of at least 2° is required. Previous finite element modeling efforts [e.g., Richardson ct al., 1979; Richardson and Reding, 1991] used grids which provided a resolution of 5° x 5° a,way from plate boundaries and approximately 3° along plate boundaries. In this study, grids with a. spatial resolution of approximately 2° were used (l degree ror the SAP), which allow the incorporation of forces associated with topographic features with a. wavelength of about 200 kilometers. The sensitivity of the modeled stresses is therefore lim ited to large-scale tectonic features with wavelengths of a few hundred kilometers. Constant-strain triangular clements in a state of plane stress were used because they adequately a.pproxima.te a spherical surfa.ce. The predicted stress magnitudes are calculated for a lithosphere of constant thickness, assumed to be 100 km, and stress concentrations may occur where variations in the lithospheric thickness are present. In this study, bending moment stresses associated with the subducting plates arc ignored and the predicted stresses cannot be rega.rded as significant for locations within the flexural wavelength of such boundaries. All clements were assigned a. Young's modulus of 7 xlO lO Nm-2 and a Poisson's ratio of 0.25. Based on the fact that the plate is not accelerating, static equilibrium is assumed. As discussed above, given the uncertainties associated with quantifying the nature of the coupling rorce between plates and the underlying mantle, it was assumed that the basal shear provides the resistance to other forces acting on the plate. 5.4 The African Plate In the ambient state (see Chapter 2), the intraplate stress field is governed solely by lateral density variations in the lithosphere. These variations produce 88 corresponding gradients in the gravitational potential-energy distribution, which govern the local state of stress. Thus, information about the ambient state has important implications for understanding the source of tectonic stresses responsible for sedimentary basin development, mountain-building processes, and the source of the intraplate stress field. In most plates it is difficult to isolate tectonic stresses due to the ambient stress state from stresses associated with other processes, such as tractions acting along the plate boundaries or along the base of the plate due to viscous drag. The major continental plates, including North America, South America, and Indo-Australia, are fast moving (suggesting a large amount of drag along the base of the plate) and have long segments of convergent or transcurrent plate boundary [Po/'syth and Uyeda, 1975]. The African and Antarctic plate are unique in that both are slow moving (producing negligible drag tractions along the base of the plates) and are predominately surrounded by mid-ocean ridges. Thus, of the Earth's tectonic plates they, can be expected to best approximate the ambient stress state [Orough, 1983]. In this section, the ambient stress state is evaluated with a finite-element analy sis of the African intraplate stress field. The plate is nearly surrounded by oceanic ridges with convergence limited to a small amount of plate boundary along the northern margin. The modeling is constrained by 248 stress indicators extracted from the World Stress Map database [Zoback, 1992]. The goal of this study is twofold. First, constraints will be placed on the magnitude of the ambient tectonic stresses in continental plates. Second, this study will provided a way to evaluate the part of the observed African intraplate stress field explicable in terms of lateral variations in the lithospheric potential energy. This will help clarify the signifi cance of other postulated sources of intraplate stress such as viscous drag at the base of the lithosphere [Pa'Voni, 1992; Weslaway, 1993]. 89 5.4.1 Description of the Plate The African plate is dominated by oceanic lithosphere, with young (post-dating the end of the Creataceus normal mega-polarity epoch at 8Ll Ma) oceanic lit;ho sphere comprising about LLO % of the plate area, and old oceanic lithosphere making up about 20 % of the plate area, respectively. The mean elevations of the whole plate, the oceanic and continental areas are -2768, -3524 and 339 meters, respec tively. 5.4.2 The Regional Intraplate Stress Field The present day intraplate stress field in the African plate is well documented by 2L18 stress indicators compiled as part of the World Stress Map Project [Zoback, 1992]. Most of the stress indicators in the African plate are located on the con tinent, providing an excellent basis for evaluating the resu1t,s of stress modeling studies aimed at elucidating the dynamics of the African continent. In compari son, the stress field in the oceanic portions of the African Plate is much more poorly constrained. The location, type and orientation of SI1,1IlClx, for the indicators are plotted as Figure .5.2. Also shown in Figure 5.2 are boundaries of the African plate as defined in the present study. In North Africa, Sll,max is oriented N-S and the stress regime is strike-slip to compressional. In the mid-plate regions, SIl,l1l<1X is oriented E- Wand the stress regime is strike slip. In the East African rift region, Sll,max is oriented roughly N200E [Bosworth el al., 1992] and the stress regime is dominately extensional. The magnitude of SIl,max - SIl,min in eastern Africa has been estimated from the rotations of the principal stresses in the East African rift at approximately 14 MPa [Zoback, 1992]. 90 Figure 5.2, The African plate with boundaries as defined by De Mets, et al. [1990J. Also shown are the stress indicators from the World Stress Map database [Zoback, 1992]. Circular symbols represent focal mechanisms, square symbols represent ge ologic indicators, and triangular symbols represent borehole data. Solid, open, and gray-shaded stress indicator symbols represent compressional, extensional and strike-slip deformational style, respectively. The orientation of the maximum hor izontal compressive stress (SlI,max) is also shown. Orientations without symbols designate volcanic vent alignments. The lengths of the vectors representing SlI,max have been weighted by the indicator quality (A - D) after Zoback [1992J. 91 5.4.3 Modeling Results The finite-element grid consisted of 3348 constant-strain triangular elements composed of a network of 1762 nodes (Figure 5.3). The spatial resolution of the finite-element grid is about 2° in both latitude and longitude. For the purpose of this study, the plate was assumed to be stationary and no tra.ctions due to drag forces were applied along the base of the plate. A condition of static equilibrium was assumed which requires the sum of the torques acting on the plates to be zero. Because the focus of the present study is on the ambient state, the zero net-torque condition is readily achieved by fixing all plate boundaries. A small amount of torque (1.8 x10 25 Nm, Table 2.6) is being balanced by the fixed boundaries since the total torque acting on the plate in not identically zero. The principal stresses predicted by the finite-element analysis appropriate to the ambient state are shown in Figure 5,4. The solid and open bars with arrowheads in Figure 5,4 designate deviatoric compression and extension, respectively. The main features of the modelled stress field are described below. The mid-ocean ridges and much of the ridge flanks exhibit all extellsional stress regime with SH,max oriented parallel to the ridge axis. Correspondingly, the di rection of maximum tension is aligned with the gradient in potential energy and, hence, with the gradient in bathymetry. The maximum extension along the ridge axis is about 12 MPa. At a depth of about 4 km along the ridge flanks the state of stress is neutral. In the older, deeper parts of the oceanic basins, the predicted ambient stress state is compressional, with a maximum compressive stress of about 12 MPa. Continental regions near sea level are in a near neutral state of stress, with large extensional stresses present in the Ethopian highlands (15 MPa), the East Africa rift (9 MPa), and in southern Africa (8 MPa). With the exception of East and South Africa, the predicted SH,max orientation is E-W throughout the 92 40° 20° _60° ~_-====-_-=== __-=== __ .I!l 300° 320° 340° 0° 20° 40° 60° 80° Figure 5.3, Finite-element grid for the African plate. The grid is composed of 1762 constant-strain plane-stress triangular elements and 3348 nodes. The spatial resolution is about 2° at the equator. 93 - - . Pc = 2650 kg/m3 ._-- Pc = 2750 kg/m~ ...... Pc = 2850 kg/m 0 _60 ~--====---====---====--~0 300 Figure 5.4, Predicted stresses for the African plate at selected element locations. Solid bars indicate deviatoric compression, open arrows indicate deviatoric tension. Maximum extension in the Ethiopian Highlands is 20 MPa; maximum compression in the oceanic basins is 18 MPa. Also shown are the zero contour of the excess potential energy ~ U/ (the difference between U/ and the plate mean, Ui])) for three 3 continental crustal density values 2650, 2750, 2850 kg m- , respectively. The mid ocean ridges and continental regions have a potential energy in excess of the plate mean for each of the continental crustal density values. 94 plate. It is worth noting that in the ambient state, lateral variations in potential energy provide the only force acting on the plates and the net torque resulting from the potential-energy distribution should be zero. For the density model assumed here, the net torque is estimated as 1.8 x 1025 Nm about a pole at 20oN, 67°W (Table 5.1). This estimate is significantly smaller than for other plates containing a large amount of mid-ocean ridge (Chapter 2), which range up to 13 X 1025 Nm for the Pacific Plate. It therefore is justifiable to conclude that the density structure used in the present study predicts tectonic stresses that are consistent with a near ambient stress state. The main contribution to the net potential-energy torque in the African Plate arises from the disposition of the mid-ocean ridges around the plate (Table 5.1). Although the ridges make up about 85% of the plate boundary, they are absent along the northern margin. The small resultant potential-energy torque acts to drive the plate northwards and may be expected to be resisted by collisional processes acting along the northern boundary of the plate. Such resistance would be expected to result in some N-S oriented compression along the northern part of the plate, as observed in the Atlas Mountains (Figure 5.4). The greatest uncertainty in lithospheric potential energy relates to the density structure of the continental lithosphere. In the model described here, the density model assumed was consistent with small positive geoid anomalies across continen tal margins (Chapter 2). Changing the reference density of the continent, pc, can affect the predicted stress regimes throughout the plate. The plate-scale potential energy mean is dependent on pc, increasing to 6 xlO 11 N m- I (from 2.375 to 2.381 14 1 3 X10 N m- ) as pc varies from 2650 to 2850 kg m- . The zero contour of the excess potential energy ~ UI ( UI - Ulp,) for three continental crustal density values 2650, 2750, 2850 kg m-3 respectively, are shown as dashed lines in Figure 5A. The depth along the oceanic ridge corresponding to the plate-scale potential mean is a 95 strong function of pc. It should be noted, however, that continental regions remain in an excess potential state for all values of pc in this range. 5.4.4 Discussion The sources of the stresses that drive continental extension remain controversial. Much current debate exists about the relative contribution of tractions exerted at distant plate boundaries, of viscous forces applied at the base of the plate, and of buoyancy forces arising from lithospheric density distributions [e.g., Houseman and England, 1986; Pa'Voni, 1992; Westaway, 1993]. One obvious constraint on the various contributing sources is provided by the observed intraplate stress-field. The comparison of stresses predicted by numerical modeling studies with the observed stress field provides a means of resolving the current controversies. A comparison between Figures 5.2 and 5,4 shows that the stress regimes pre dicted by the modeling correlate well with the observed distribution of extensional and compression stress regimes in the African continent. Moreover, the agreement between the predicted and observed SJ/,7I1(IX orientations is good in most regions of the plate, particularly in the mid-plate regions where the calculated SJ/,7I1UX orien tation is dominately E-W. This reflects the important influence of the mid-ocean ridges bounding the easteni and western margins of the plate. In the East African rift region, the predicted SH,7I1ux is roughly N-S. This orientation compares well with the observed orientation of roughly N200E [Boswol'th et al., 1992]. These results underscore the important contribution of lithosphere sources to intraplate stresses, with the general agreement between observation and modeling suggesting that many of the long-wavelength features of the African intraplate stress field may be explicable in terms of lateral variations in the lithospheric potential energy. An important corollary of this analysis of the African Plate stress field, in terms of ambient processes, is that there may be no need to appea.l to poorly constrained :' 96 sub-lithospheric processes stich as the viscotls drag imposed by a diverging man tle plume [i.e., Pavoni, 1992; Westaway, 1993] to explain the observed African intraplate stress field. In the ambient stress state, variations in the stress regime are governed by the difference between the local potential energy and the plate mean, Ut (see Chapter 2) with areas that have a potential energy greater than Ut in deviatoric tension and areas with potential energy less than Ut in deviatoric compression. The potential energies, potential differences, and corresponding horizontal stress associated with the various lithospheric density variations within the African plate are listed in Table 5.2. The mid-ocean ridges provide the principal potential-energy source in the African plate, with a potential energy excess over Ut of about 1,4 x 1012 N 1 m- • The principal potential-energy sinks in the plate are the ocean basins with 12 1 !:l.Ut of -1.2 x 10 N m- • The magnitude of the ridge forces is generally accepted to be 2-3 x 1012 N per meter of ridge length [Frank, 1972; Lister, 1975; Parsons and Richter, 1980]. Although the ridges have an excess potential energy over the old 12 1 ocean basins of about 2.6 x 10 N m- , the actual compressive stress experienced by old ocean basins will be less than this value since it is the difFerence in potential energy between Ut and the ocean basin which governs the stress magnitudes. The potential-energy difference of the old ocean basin relative to LIt is -1,4 x 10 12 N m-1 corresponding to a compressive horizontal stress of about 11 MPa, rather than the value of 21 MPa predicted by the potential difference between the ridge crest and the old ocean basin. An important implication of these results is that the predicted ambient st.ress state in exposed continental regions is extensional to strike slip. This predic tion differs significantly from the assumptions made by many previous investiga tors who assumed that the state of stress within the continents is dominated by stresses transmitted from the mid-ocean ridges, and is therefore compressional [e.g., 97 Crough, 1983; Houseman and England, 1986 ]. In the case of Africa, the potential difFerence between U/ and the mean potential energy of the exposed continental regions is about 0.9 x 1012 N m-1 (corresponding to an extensional horizontal stress of about 7 MPa). The prediction of an extensional or strike-slip stress regime for conliinental regions is consistent with the World Stress Map database, which con tains only one thrust focal mechanism stress indicator for the continental areas of the African plate. This is in sharp contrast with the large number of com pressional stress indicators in the other largely continental plates, particularly in eastern North American and western Europe [Zoback and Magee, 1991; Zobacl.:, 1992], suggests that other sources of tectonic stress (i.e., plate boundary forces, flexural stresses, and basal drag) may dominate the intraplate stress field in these non-ambient plates. 5.5 The South American Plate Mid-ocean ridges make up only about half of the South American plate bound ary, with the remainder dominated by collisional tectonics. Thus, the South Amer ican plate is not expected to be in the ambient stress state. However, because the plate itself is not attached to any significant length of subducting slab, the slab-pull force does not dominate i;he tectonic balance in the plate [e.g., Mel)er and Worfel, 1992; Stefanick and J'Il'l'dy,1992j. The plate therefore provides an ideal location to investigate the relative contributions of stresses from other tectonic source such as lateral density variation within the lithosphere or forces transmitted across the collisional boundaries. Significant horizontal stresses are associated with the mid Atlantic ridge, the Brazilian continental margin and the high topography of the Andes and both the intraplate stress field and the torque balance of the plate can be expected to reflect a balance between the stresses associated with each of these 98 features. The study builds on the previous modeling studies through the use of better constrained forces acting along the collisional and subduction zone boundaries and the inclusion of forces due t.o other lateral density variations within the lithosphere, such as those associated with continental margins and high topography. Previous stress modeling studies of the South American plate have employed coarse grids. In the study by Richa1yisan et ai. [1979], they evaluated the intraplate stress field of the South American plate as part of the global modeling. The emphasis of this work was on a first-order global analysis, without much detail on individual plates and tectonic forces involved. The spatial resolution of this work was about 5°, too coarse to accurately represent lithospheric stresses due to local density structures. Stefanick and Jurdy [1992]' modeled the intraplate stress field with a regular finite difference grid with a spatial resolution of 300 km. The large-scale observed stress pa.ttern in the plate was modeled with a balance of ridge push, a. small amount of slab-pull at the Scotia and Caribbean arcs and trench suction near the Peru-Chile trench. The coarse resolution of their grid prevented any analysis of other stresses. MeUer and Wadel [1992J addressed the dynamics of the South American pla.te by studying the balance of forces required to maintain the equilibrium condition of zero net torque. Their results suggest that basal shear cannot be rejected on the basis of a dynamical analysis. They also showed that the ridge push force is by far the largest contributor to the torque balance on the plate. Their study was intended to establish the dynamical framework for a future stress modeling study and as such no predicted stresses were presented. As discussed in Chapter 2, lateral density variations in the lithosphere are an important source of intraplate stress and associated deformation. The results pre seJIted above for the African plate have been used to speculate about the nature of the predicted intraplate stress field in plates which approximate the ambient :' 99 stress state. However, the majority of plates are not expected to be in the am bient stress state due to the existence of large boundary forces and the possible existence of drag forces acting along the base of the plate. This analysis of j;Jw intraplate stress field in the South American plate provides a way to evaluate the non-ambient intraplate stress field in a major continental plate. The central focus of this study will be to understand what part of the observed intraplate stress field can be attributed to forces due to lateral density variations as opposed to other tectonic forces acting on the plate, such as collisional boundary forces and basal drag. 5.5.1 Description of the Plate The South American plate consists of about equal amounts of oceanic (51 %) and continental (49 %) lithosphere. The young oceanic lithosphere (younger than 8fl Ma) makes up about 35 % of the plate area, and old oceanic lithosphere about 16 %. The continental areas are nearly equally divided between submarine passive margins (11 %) and regions above sea level (about 39 %). The mean elevations of the whole plate, oceanic and continental areas are about -2100, -3800 and 600 meters, respectively. The geometry of the South American plate as defined in the present study is shown as Figure 5.5. The boundary geometry was originally defined by the work of Sykes [1969], Isacks et al. [1968], and Dewey and Bird [1970], among others. For the purpose of this study, the geometry used in DeMet.') et al. [1990] has been adopted. The plate is attached to only about 500 km of subduction slab, mostly along the Lesser Antilles and South Sandwich arcs. Convergent boundaries make up over 9000 km of the plate boundary. The nature of the plate boundary along the western and eastern margins are well defined by seismicity. Subduction occurs along the entire western boundary of the plate. North of about f150S, the Nazca 100 20' o· . _20' Nazca Plate _40' ~ NR _60' Antarctic Plate ...... 280' 300' 320' 340' 0' Figure 5.5, South American plate with boundaries as defined by Minster and Jor dan, [1978] and Pelayo and Wiens [1989]. Abbreviations are: !vI AR = Mid-Atlantic ridge, SS=South Sandwich subduction zone, SR=South Scotia ridge, N R=North Scotia ridge, PCT=Peru-Chile trench, N AB=North Andean block, LA=Lesser Antilles arc, N AM=North-South American boundary. 101 plate sub ducts under South America and the axis of the Peru-Chile trench (PCT in Figure 5.5), which defines the western margin. South of 45° S, the Antarctic plate sub ducts under South America and the boundary is well defined by an extension of the trench. In contrast to the northern part of the westel'll margin, this region is characterized by a low level of seismicity activity. The southel'll boundary is defined by the North and South Scotia ridge (NR, SR) [Fo'l'sylh, 1975J. A small amount of convergence occurs along this ridge, suggesting a transgressive nature [Pelayo and JiVicns, 1989J. Active subduction occurs along the South Sandwich arc (SS). The eastern boundary, defined by the mid-Atlantic ridge (MAR), is characterized by both high topography of ocean floor and zero-age magnetic isochron [Royc'/' ct al., 1992J. This ridge system extends more than 7000 km and is the dominate tectonic feature of the South American pla.te. The boundary between North and South America (N AM) is thought to be a diffuse boundary, and therefore is poorly constrained. For the purpose of the present study the plate boundary was chosen to intersect the Lesser Antilles arc (LA) at about 18 ON and the mid-Atlantic ridge at about 20.1°N. West of the Lesser Antilles arc, the Caribbean-South America plate boundary is complex, consisting of a zone of primarily right-lateral strike-slip deformation [Molnar and Sykcs, 1969J. 5.5.2 The Regional Intraplate Stress Field The nature of the regional intraplate stress field in South America is based primarily on earthquake focal mechanisms and Quaternary fault-slip inversions [A ssump~ao, 1992J. The major part of the data used is located in the westel'll part of the plate, anel the orientation of the observed stress field shows a great deal of scatter in the other parts of the plate. The first-order pat tel'll of the regional intraplate stress field of the SAP is relatively well defined by the available World Stress Map stress indicators [Zoback, 1992J. Figure 5.6 shows the 213 indicators 102 by type and orientation of the maximum horizontal compressive stress orientation (SU,max)' The distribution of the indicators by type is listed in Table 5.3. The orientation of SH,max in western South America is consistently E- Wand does not seem to be affected by either changes in the strike of the Andes near the Arica deflection (about 200 S), nor to changes in the angle of the Benioff zone of the subducting Nazca plate (see the discussion in Assump<;ao, 1992). The regional SH,max orientation is more closely aligned with the absolute plate motion than with the direction of convergence with the Nazca plate, suggesting that basal shear forces acting in the central and eastern regions of the plate may be dominating the regional stress field. Meijer and Wodel [1992J have suggested that the concept of a driving basal shear stress cannot be discarded on the basis of their dynamical analysis of the motion of the South American plate. Throughout the Andes, N-S extension is observed at elevations greater than 3000 meters [e.g., Silgado, 1951; Mercier, 1981; Suarez, et al., 1983; Sebrier, ct ai., 1985, 1988a, 1988b; Cabrc'/'a et al., 1987; DoseT, 1987; MeTcieT, et al., 1992J. This extension is thought to be the result of buoyancy forces due to lateral density variations within the lithosphere associated with the crustal root of the Andean topography which is exceeding the regional compressive stress (see Chapter 4). The regional stress field in the central part of the plate is thought to be domi nated by E- W compression, despite the low levels of seismicity. Two focal mech anisms which show N-S compression in the vicinity of the Amazonas Basin have been shown to be the result of lower crustal loading [Richardson and Zoback, 1990J. In northeastern Brazil, strike-slip deformation is thought to be due to the inter action between the regional stress field and stresses due to continental margin [Assumpr-ao, 1992; Coblcntz and Richa'l'dson, 1992J. The stresses predicted by numerical modeling must be consistent with the broad-scale regional stress field in the regions where the observed stress field is I L' 103 20' o· -20' -40' -60' 280' 300' 320' 340' o· Figure 5.6, Stress indicator data for the South American plate from the World Stress Map database [Zoback, 1992]. Circular symbols represent focal mechanisms, square symbols represent geologic indicators, and triangular symbols represent well hole data. Solid, open, and gray-shaded st.ress indicator symbols represent compressional, extension and, strike-slip deformational style, respectively. The orientation of the maximum horizontal compressive stress (SH,max) is also shown. The lengths of the vectors representing SH,max have been weighted by the indicator quality (A - D) after Zoback [1992]. Details of the individual stress indicators are listed in Table 5.3. I :' 10,1 well constrained. Within the SAP, these regions include: (1) The Brazilian Shield region where SH,ma:!: is oriented E - W, and the magnitude of the intraplate stress field has been estimated to be about 35 MPa (Chapter 4) and (2) The Cordillera Blanca region of Peru where N - S extension is observed at elevations greater than 3000 m (see discussion in Chapter 4). There is poor constraint on stress orienta tion in the oceanic areas in the Southern Atlantic and in the southern continental regions of South America. 5.5.3 Modeling Results The finite-element grid, shown as Figure 5.7, consists of 3100 equi-angular, constant-strain triangular elements composed of a network of 5933 nodes, which provides a spatial resolution of about 1° in both latitude and longitude at the equator. The first models presented show the predicted stresses due to forces associated with potential-energy variations wit.hin the cooling oceanic lithosphere (i.e, the ridge push force). The force provides a significant torque (9.5 x 1025 N m) about a pole at 68.°8, 109.1°E which is close to the pole of the plate velocity (70.30 8, 7,1. 7°E; misfit = 12.3 degrees) and t.herefore potentially drives the motion. Because the ridge push force is well-constrained by bathymetry data [e.g., Turcotte and Schubert, 1982]' the first three models have been designed to evaluate the influence of the boundary conditions applied along the western margin and the drag force. A description of the force models used to model the stresses in the SAP are listed in Table 5.'1. Modell: Fi:ved Western Margin, No Basal Drag In Modell mechanical equilibrium was achieved by fixing all the collisional 105 00 _20 0 _40 0 _60 0 280 0 300 0 320 0 340 0 00 Figure 5.7, Finite-element grid for the South American plate. The grid is com posed of 3100 constant-strain plane-stress triangular elements and 5933 nodes. The spatial resolution is about 10 at the equator. .' 106 boundaries. Along the western margin (the primary collision boundary), this is equivalent to assuming that all of the resistance to the ridge force is transmitted from the Nazca plate to the SAP along this boundaries. The principal stresses for Model 1 are shown as Figure 5.S. The predicted stress field for Model 1 is characterized by E-W compression throughout most of the plate with a maximum stress magnitude of about 20 MPa (200 bars). This model demonstrates that the dominant E - W compression in the observed stress field can be explained by a very simple model which combines the effect of the ridge push force and forces acting along the western margin. Mode! 2: Ridge Push and Resistive Basal Drag Model 2 employed a different set of boundary conditions to balance the ridge torque. In this model the ridge torque was balanced with resistive drag applied along the base of the plate. The drag force required to balance the torques acting on the plate are listed in Table 5.5. The predicted stresses for Model 2 are shown as Figure 5.9. The predicted stresses in the oceanic and eastern-continental parts of the plate are very similar to those in Model 1. The stress field is characterized by E - W compression in this part of the plate with a maximum magnitude of about 20 MPa. The predicted stresses decrease in the western part of the plate, becoming zero along the western margin. The apparent rotation of the 8II,max orientation is due to the balancing drag required to maintain mechanical equilibrium (66.3°8, 115.1°E, 5.1 cmjyr). The predicted SII,max orientation in the northern and western parts of the plate differ significantly from the broad-scale patterns in the observed stress field. Thus, while the drag forces necessary to balance the torques will change with the addition of other tectonic forces (such as boundary forces and buoyancy forces), this model suggests that resistive drag may be an inappropriate 107 20' o· X -+- .,. ·20' -I- 4"- -+- -\- .,. -t- -r- -\- I -I- -t- I ·40' ...- ...- - -"" ./ ",..- / /' ,- .yo;> / / ;< ·60' .. , :. 280' 300' 320' 340' O· Figure 5.8, Predicted stresses for Model 1, SAP. Solid bars indicate deviatoric compression, open arrows indicate deviatoric tension. 108 20 0 0 0 X -+- ...\-' -+- _20 0 -- ~ -+- ...... - --,... 0 _40 ...... ,...... -- -t- ..... ~ - - _60 0 280 0 300 0 320 0 340 0 00 Figure 5.9, Predicted stresses for Model 2. Solid bars indicate deviatoric compres sion, open arrows indicate deviatoric tension. See text for other details. 109 way to balance the torques acting on the plate. Model 3: Driving Basal Drag Model 3 was used to evaluate the predicted stresses for a driving drag force. In this model the torque acting on the plate was balanced by pinning the collisional boundaries as in Model 1. A driving basal drag was applied along the base of the plate (the drag coefficient was chosen to produce 0.1 MPa shear stress for a velocity of 1.0 cmfyr) in the direction of absolute plate motion based on a hot-spot reference 'frame [(HS2-NUVEL1), Gripp and Gonlon, 1990j. The predicted stresses for Model 3 are shown as Figure 5.10. Because the plate velocity pole is very close to the ridge-torque pole, the drag force acts very nearly in the same direction as the ridge-push force. As it consequence, the predicted stresses for Model 3 are very similar to Modell. The predicted SH,max orientations everywhere are in good agreement with the broad-scale patterns in the observed stress field. A significant difference between the models exists in the predicted stress magnitudes. While the stress magnitudes predicted with Modell arc uniform throughout continental South America, the magnitude of the stresses predicted by Model 3 increase from east to west across the plate reaching a. maximum value of about 30 MPa along the western margin. Given the non-uniform distribution of stress indicators in the plate and the lack of information about variations in the magnitude of the intraplate stress field, it is difficult to test the existence of such a stress magnitude gradient. It is not possible, therefore, to rule out the existence of driving basal drag force acting on the base of the plate. Model 4, Topographic Forces Given the large amount of elevated topography in the western part of the South American plate, forces due to topographic buoyancy forces can be expected to play 110 00 'f...'><- l( -!- ~ -I- >< _20 0 y.. -+- ok ~ -\- X I A-" ok ~ 0 _40 ,....y' x ,It" / ./ /' ... / / / ~ / I ;' _60 0 ... , 280 0 300 0 320 0 340 0 0 0 Figure 5.10, Predicted stresses for Model 3. Solid bars indicate deviatoric com pression, open arrows indicate deviatoric tension. See text for other details. 111 an important role in the nature of the intraplate stress field. Model 4 was used to evaluate the predicted stresses due to tectonic forces from all potential-energy sources in the plate. In this model, the torques acting on the plate were balanced by pinning the collisional boundaries. No basal drag was applied along the base of the plate. The relative contribution to the total torque acting on the plate from the various potential-energy sources are listed in Table 2.6. The predicted stresses for Model il are shown as Figure 5.11. The predicted stress field in the eastern part of the plate is dominated by E - W compression with a maximum compressive stress magnitude of about 10 MPa. In the oceanic regions the stresses are similar in magnitude and orientation to those predicted by Models 1 and 3. The magnitude of the predicted stresses in the continental regions of the Brazilian shield are about 10 MPa, which is about a factor of two to three less than predicted by Models 1 and 3. Theses magnitudes are also about a factor of two less than the estimate of intraplate stress magnitude made in Chapter L1. This suggests tha.t the forces due to lateral density variations in conjunction with a pinned western margin are unable to explain the observed intraplate stress magnitude in this part of the plate. The Sll,1IItlx orientation of the predicted stresses remains dominantly E - W throughout most of the plate except in the Andean foreland where substantial rotation occurs. The stress field in the western part of the part is dominated by extension associated with the high topography of the Andes. The orientation of the extension is roughly N - S in the northern Andes which is consistent with observed orientations. South of about 200S the extension is oriented E - W. The lack of stress indica.tors in this regions ma.kes it difficult to evaluate the plausibility of this rotation. The maximum magnitude of the extensional stress in the high topography regions is about 30 MPa. 112 0 0 + -t- ..\-' 0 _20 -I- -r- -I- -+- -+- -+- -+- -+- -I- _40 0 ;- ;- ...r -I- .-\- ~ ,If' /' ~ "'" / / _60 0 ... , :. 2800 3000 3200 3400 0 0 Figure 5.11, Predicted stresses for Model 4. Solid bars indicate deviatoric com pression, open arrows indicate deviatoric tension. See text for other details. 113 Model 5, Western Bounda1'Y F01'ces Model 5 evaluated the effect of employing an alternative set of boundary con ditions to represent the forces acting along collisional boundary segments. In this model, a constant force pel' unit length of boundary segment was applied along the boundary segments listed in Table 5.6. Since this does not produce a torque bal ance, a basal shear forces were applied along the base of the plate such that a plate velocity of 1.0 cm/yr produced a basal shear stress of 0.1 MPa, as discussed above. A driving drag force was used to balance the torques acting on the plate. The nondrag forces exert a net torque about a pole located at 77.3 N, 5.9 E, resulting in a net force acting eastward. The use of a resistive drag force would require a net motion of the lithosphere eastward with respect to the asthenosphere to balance this torque. Although it could be argued that a complex flow pattern beneath the lithosphere could product a net force in this direction, the use of driving drag to balance the torques requires plate motion relative to the asthenosphere which is more consistent with the observed absolute plate motion. The magnitude of the forces transmitted across the boundaries are not well constrained. The forces used in this study were chosen to produce torques which are consistent with those estimated by [ivlci;jel' and Worfel, 1992]. Significant variation exists in the dip angle of the subducting Nazca plate beneath western South America (see review in Isacks, 1988). This variation may well produce a corresponding variation in the forces transmi tted across the plate boundary. The angle of subduction is thought to vary as much as 15° along the Peru-Chile trench. The isodepths to the top of the Wadati-Benioff zone in the subducting Nazca plate is shown as Figure 5.12. If it is assumed that the coupling force is proportional to the sine of the sub ducting angle [Froidevau.'V and Isacks, 1984], this change in subduction angle can I 280 0 288 0 296 0 00 ."".'" .... I, ....."" " .... \ .... ' '\ \ " I 1\\ I \ , \ \ I } I ~ I I I I I 0 _10 0 _10 \;\ , \ ,,\, \ '\ " , \ " ,',,,,.... " '~\. ,<~~, '\ , ~, , ," ,\ ,\~ \\, 0 _20 0 _20 \ \, \1\ \ \ \ \ , II'I I , ! ,: \ I \ I II I "\ I I" , ~\ \ 0 I"I \ ,,\, \140 km _30 I \ I I J \ I,' I 100 k~ , I .,-,.,; I 0 280 0 288 0 296 Figure 5.12, Isodepths to the Top of the Wadati-Benioff Zone (average hypocen tral depth) in the subducting Nazca plate along the Peril-Chile Trench. Contour interval is 20 km. [After Bevis and [sacks, 1984 and Assumpfao, 1992] 115 produce a reduction in the coupling force by a factor of 0.25. In this study, the force acting along the margin was allowed to vary by a factor of 0.5 (from 1 to 12 1 2 x10 Nm- ) in response to changes in the angle of subduction. Such a large variation was chosen in order to evaluate the dependence of the predicted stresses on variations in the coupling forces acting along the Peru-Chile trench. The predicted stresses for Model 5 are shown as Figure .5.13. The broad-scale features of the predicted stress field are similar to those predicted by Model tl, demonstrating that a similar stress field is predicted whether the collisional bound aries are pinned or are subjected to a boundary force. The largest difference in the predicted stress field exist ill the northern part of the plate in the vicinity of the Caribbean boundary, where substantial compression is predicted by Model 5. Given the nature of the observed stresses it is difficult to further constrain the magnitudes of the collisional resistance acting on the plate. In order to facilitate comparison of stresses predicted by Model 5 with the results of the modeling discussed in Chapter 3, the predicted stresses for the Cordillera Blanca region of Peru are shown as Figure 5.14. The predicted stress field in this region is characterized by roughly E - W compression to the east and west of the high topography of the Cordillera Blanca region, with a magnitude of about 25 to 35 MPa. In the region above 3000 m the predicted stresses are extensional with N - S to NE - SW oriented extension. Although the precise location of the transition from a compressional to extensiollal stress regime is llot well determined by the stress indicators, it is generally assumed that extension is confined to regions with elevations greater than 3000 m [e.g., Silgado, 1951; Mercier, 1981; S'Ila·/,ez et al., 1983; Sebl'ie·/' et al., 1985, 1988a, 1988b; Cabrera et al., 1987; Doser, 1987; Mercier et ai., 1992]. The stresses predicted by Model 5 for the Cordillera Blanca region are in general agreement with the observed intraplate stress field. The extension predicted immediately to the east of the 2000 m contour in Figure 5.14 is due to " 116 0 0 .;..:".: + ~ -I- x 0 _20 -f- ...-\' -t- -- 'I- " _40 0 --- - >1&' -- ... _60 0 ... , ~ 280 0 300 0 320 0 340 0 0 0 Figure 5.13, Predicted stresses for Model 5. Solid bars indicate deviatoric com pression, open arrows indicate deviatoric tension. See text for other details. 117 - _10 0 - + 25MP(/ - _120 ~_-====-___=====- __ ... 280 0 282 0 2840 286 0 Figure 5.14, Predicted stresses for the Andean region, Model 5. Solid bars incli cate deviatoric compression, open arrows indicate cleviatoric tension. Light gray, gray and clark gray designate regions with elevations from 2-3km, 3-4 km and 4-5 km, respectively. Contour interval is 1000 m. Circular symbols represent focal mechanisms and square symbols represent geologic indicators. See text for other details. 118 limitations of the spatial resolution used for the finite-element grid. The 1 degree resolution, while able to predicted long-wavelength variation in the stress field, is too coarse to precisely model the transition from extension to compression in this region where the transition occurs rapidly. 5.5.4 Discussion The predicted stresses for a number of models have been presented with the aim of demonstrating the relative contribution of the various tectonic forces to the SAP intraplate stress field. The first set of models (Models 1,2 and 3) were used to evalua.te the relationship between ridge-push forces, the boundary conditions applied along the northern margin, and the predicted stress field. These models demonstrated that the principal features of the observed intraplate stress field are readily explainable in terms of the relatively homogeneous boundary configuration of the plate, with a long mid-ocean ridge along the trailing (eastern) margin and long continental arc-related mountain tracts along the leading (western) margins. Model LJ was used to evaluate the predicted stresses from all potential-ellergy source (cooling oceanic lithosphere, continental margins, elevated topography). The pre dicted stresses were found to have a magnitude of about 10 MPa throughout the eastern continental areas of the plate, which is smaller by about a factor of two to three than those predicted by the magnitudes predicted in Chapter 3. Additional tectonic forces acting on the plate are therefore required to explain the existence of stress magnitudes of the order of 25 MPa in the Brazilian shield region. Model 5 was used to evaluate the possibility that forces acting across the Peru-Chile trench supply the needed additional force. The stress field predicted by this model, which included forces due to all potential-energy sources, collisional boundary forces and balancing basal drag, was found to be consistent with most of the large-scale fea tures of the observed intraplate stress fielcl. A comparison between Figures 5.13 119 and 5.6 shows that the stress regimes predicted by the modeling correlate well with the observed compression in the eastern continental regions and oceanic areas as well as the distribution of extensional stresses in the western part of the plate. The features of the predicted stress field in the eastern part of the plate (magnitude and orientation) for this model are not significantly difFerent from those predicted in the simple model cases (Models 1,2 and 3). It is therefore not necessa.ry to appeal to complex tectonic models to explain the observed intra.plate stress field in the South American plate. Without further information about the details of the observed st.ress field in the Brazilia.n shield region, it is difficult to further constrain the ma.gnitudes of the collisionall'esistance acting on the plate. An important result of the models presented above is the prediction of low tectonic stress throughout the plate, on the order of 20-30 MPa averaged over a 100km thick lithosphere. These magnitudes are consistent with the predicted intraplate stress magnitude from Chapter 4, as well as those predicted for most of the other plates including North America, Africa, Nazca, Pacific and Europe. Forces due to lateral density variations within the lithosphere have a significant affect on the intraplate stress field, particularly in the western part of the plate. In the central continental regions the principal effect of the topographic forces is to reduce the magnitude of the regional E - W compression. It is possible that the magnitude of the horizontal stress is reduced enough to allow the vertical stress to become the intermediate principal stress and thus produce strike-slip deformation. This is difficult to evaluate, however, given the two-dimensional nature of the predicted stresses and the lack of information about the vertical stress. Furthermore, there are other factors which may a.lter the local stress field and which were not incorporated into the modeling. Northeast Brazil is an area of high heat flow (greater than 100 m W In~2) [Assllmp~ao, 1992J which may reduce the thickness of the elastic lithosphere and produce local stress concentration. Local r l, 120 variations in the rheology and density variations could also serve to amplify stresses due to local forces. The fact that the observed seismicity along the northeast coast of Brazil is concentrated near the Potiguar Basin and is not observed to extend along the entire coast suggests that local heterogeneities in the crust may play an important role in the local stress field in this region. Modeling studies of the North America,n plate have been used to evaluate the dif ferences in the stress predicted by models incorporating driving and resistive drag [Richardson and Reding, 1991J. The driving drag models yield principal stress ori entations parallel and perpendicular to absolute plate motion and predict a stress gradient across the plate in the direction of plate motion. The existence of driving drag acting along the base of the North American plate was ruled out because such a gradient is not observed [Zoback and Magee, 1991J. In the case of the South American plate a similar gradient in the stress magnitude must exist across the plate if a driving drag force is acting along the base of the plate (Figure 5.9). How ever, the results of the present study indicate that the relative motion of the plate required to balance the total torque is more consistent with the observed absolute plate motion for driving drag than for resistive drag. Because little information exists about the change in stress magnitudes across the South American plate a driving drag force was used to balance the torques acting on the plate. The mod eling results are therefore in agreement with the conclusions of MeUer and Wortel [1992J that it is not possible to rule out the existence of a driving drag force acting on the South American plate. The analysis of the African intraplate stress field suggests that the the ambient stress state in exposed continental regions is extensional to strike slip. In contrast, the predicted stresses for the models presented above are compressional throughout the near-sea level continental regions of the South American plate. The results of this study suggest that forces or resistance associated with the collisional boundary 121 forces acting along the western margin of the South American plate are responsible for the non-ambient nature of the South America intraplate stress field. 5.6 The Indo-Australian Plate The Indo-Australian plate (lAP) is unique among the Earth's plates in its va riety of first-order tectonic features, including active subduction zones, an ex tensive mid-ocean ridge system, significant areas of both continent-continent and continent-island arc collision, and regions of intraplate oceanic-lithosphere defor mation. These features make the lAP an important place to study the relationship between tectonic processes and the intraplate stress field. However, while defor mation within the Indo-Australian plate is well documented, the origin of the tectonic stresses within the plate remains controversial. With the exception of the N-S compression in India, it has proved very difficu1t. to relate specific tec tonic forces to specific features in the observed stress field. From a plate dynamics perspective, one of the unusual features of the Indo-Australian plate is the lack of a direct correlation between the ridge-push force and the observed stress field, despite the extensive mid-ocean ridge system which dominates the entire southern plate boundary. Thus, in sharp contrast to most other plates (i.e., North and South American, European, Pacific and Nazca), it appears that the lAP intraplate stress field is not readily explained in terms of any single principal source, i.e., simple ridge-push, slab-pull, absolute plate motion, 01' collisional resistance force models [Hichan/son, 1992J. The magnitude of the tectonic stresses within the lAP are also controversial. Two previous investigations which modelled the broad-scale features of the lAP intraplate field [Cloclingh and J;Vol'lel, 1985;1986; Richa'rdson, 1987J predicted sig nificantly different intraplate stress magnitudes. Cloetingh and Worlel [1985;1986J I ??..,"" predicted very large intraplate stresses (of the order of several hundred MPa) throughout much of the lAP, including in the central Indian Ocean and the Coral Sea region. Indeed, Richa'/'dson [1987] showed that large stresses are not required to match the observed stress pattern and argued that forces associated with the Himalayan collision and the ridge-push force play an important role in the in traplate stress field. Stress magnitudes of the order of 100's of MPa were argued to be consistent with the amount of deformation of oceanic lithosphere observed in Indian Ocean [e.g., Stein and Okal, 1978; Weissel et al., 1980; Celie'/' ct al., 1983; Wiens and Stein, 1983,1984; Be'/'gman and Solomon, 198 L1, 1985; Stein ct. al., 1989;]. However, recent investigations of this deformation suggest that high angle faults within the crust may have modified the basement topography causing the accentuation of some of the basement highs and, thus, to a misinterpretation of the magni tude of lithospheric buckling [King and Slein, 1984; Bull, 1990; Bull and Scntllon, 1990]. While whole-lithosphere buckling requires the existence of very large stress (l00 to 1000's of MPa) within the lithosphere [MacAdoo and Sand'lVell, 1985; NejJ1'Ochnov el al., 1988], stress magnitudes required for such fault-involved deformation may be less than 100 MPa [Bull, personal communication]. Thus, it is not clear whether the geological evidence supports the stress magnitudes predicted by Cloelingh and Worlel [1985;1986]. While the principal goal of this study is to evaluate the relative contribution of the ridge-push force and other intraplate sources due to lateral density variations to the observed stress field in the lAP, it will also help clarify questions about the magnitude of the intraplate stress field. Namely, whether it is of the order of 100 MPa (kbars), as suggested by Cloetingh and Worlel [1985;1986]' or 10's of MPa (100's of bars), as suggested by Riehm'dson [1987]. 123 5.6.1 Description of the Plate The geometry of the lAP as defined in the present study is shown as Figure 5.15. The plate is dominated by oceanic lithosphere, with young (post-dating the end of the Creataceus normal mega-polarity epoch at 84 Ma) oceanic lithosphere compris ing about ilO % of the plate area, and old oceanic lithosphere making up about 20 % of the plate area, respectively. The continental areas are nearly equally divided between submarine passive margins (18 percent) and regions above sealevel (about 20 percent). The mean elevations of the whole plate, the oceanic and continental areas are -2768, -3524 and 339 meters, respectively. The boundary geometry was originally defined by the works of Sykes [1969J and Heezen and Thal'p [1965J. The nature of lAP boundaries is discussed in detail in Isacks et al. [1968J and Dewey and Bird [1970], among others. Most of the lAP boundary is well defined by seis micity, as shown by Espinosa ct al. [1981J. The southern boundary, defined by the mid-ocean ridge (MOR), is characterized by both high topography of ocean floor and zero-age magnetic isochron [Royer el al., 1992J. The Himalayan boundary (1-1), a continent-continent collision zone between the India subcontinent and Tibet, is clearly defined by a topographic change and strong seismicity zone associated with the Himalayan mountain range. The plate boundary along the major subduction zones including Sumatra (S), Java (J), Solomon (SM) and New Hebrides (NH) arcs, are defined both by contrasting bedrock geology across the boundary, and a relatively deep trench system associated with plate subduction. The plate bound ary along the continent-arc collision zone between the Australian northern margin and Banda Arc (B) and Papua New Guinea (PNG) is less well constrained. The boundary adopted in this study is based on seismological and surface geological studies of Abers and McCaffrey [1988J and Pigmm and Symonds [1991J. Along the Tonga-Kermadec trench (TK), the plate boundary has been modified by Pelletie'l' Pacific Plate , ...... ,.... , ,,_ ... , _40 0 ( ••.•. Alllarctic Plate Figure 5.15, Indo-Australian plate with boundaries as defined by Minster and JO'l'dan, [1978] and Chase, [1978]. Abbreviations are: H=Himalaya, S=Sumatra Trench, J=Java Trench, B= Banda Are, PNG= Papua New Guinea, SNI=Solomon Trench, NH=New Hebrides, Tf(=Tonga-Kermadec Trench, NZ= New Zealand, JVIOR=Mid Ocean Ridge, OR=Owen's Ridge, and N ER= Nine tyeast Ridge. Solid arrows represent the boundary forces acting on the plate. Open arrows represent the forces associated with the Indian and Australian pas sive continental margins. Forces arrows are not draw to scale. 125 and Louat [1989] to correspond to the back-arc spreading center on the basis of new seismological and relative plate motion data. South of New Zealand (NZ), the boundary is characterized by both strike-slip and compressional motion along the plate boundary. The forces considered to be acting along the lAP boundaries are schematically shown in Figure 5.15. The forces due to subduction, and to a lesser extent collision, along the northern boundary are poorly understood with the possibility of large variations in the direction and magnitude occurring over relatively short distances. The open arrows shown in Figure 5.15 represent the forces due to lateral·density variations associated with the continental margins of India and Australia. Early plate motion studies treated the Indo-Australian plate as a single plate [e.g., Minster and Jordan 1978; Chase 1978]. More recent work based on seismicity [Wiens et al., 1985] and plate motion studies [DeMets el al., 1990; Ol'ipp and Gordon, 1990], suggest. that the lAP should be considered as separate Indian and Australian plates with the boundary defined by the Owens Ridge (OR) and the Ninetyeast Ridge (NER), shown as the segmented line in Figure 5.15. For the study of the intraplate stress field, the lAP was treated as a single plate for two reasons. First, the observed stress data do not show any significant change in orientation across the proposed boundary. Secondly, the boundary forces acting across the diffuse plate boundary are difficult to constrain. Furthermore, the modeling results in the present study will be compared to those of the previous modeling efforts [e.g., CloeHngh and Worlel, 1985;1986 and Richa·l'dson, 1987] which treated the lAP used a single tectonic plate. 5.6.2 The Regional Intraplate Stress Field The first-order pattern of the regional intraplate stress field of the lAP is rel atively well defined by the available World Stress Map stress indicators [Zoback, I L' 126 1992]. Figure 5.16 shows the 251 indicators by type and orientation of the max imum horizontal compressive stress orientation (SU,max)' The distribution of the indicators by type is listed in Table 5.7. The orientation of SU,max reflects the large-scale features of the regional stress field, showing a smooth transition from a roughly N-S orientation in the Indian subcontinent to NW-SE in the northeastern Indian Ocean to roughly E-W in west ern Australia [Fitch et at., 1973; Weissel et al., 1980; Bergman and Solomon, 1980. While the stress orientation data within the oceanic regions of the lAP are fairly uniform, there is considerable scatter in the data for continental regions. Details of stress field for the major subregions of the lAP and discussion of some of the possible reasons for the observed scatter are described below. The regional stress field of the Indian subcontinent is characterized by four provinces based on regionally consistent SU,max orientations [Gowd et at., 1992]. These provinces are the midcontinent, southern shield, the Bengal basin and the Assam wedge. SU,maJ: is generally NNE-ENE in central and northern India with a mean orientation of N23°E . This orientation is sub-parallel to the direction of compression expected from the forces present along the Himalayan collision, which resist the northward movement of the lAP. The large amount a scatter in the orientation of SU,mllx suggests that, at least within the Indian continent, some of the stress indicators reflect local sources of stress rather than the stresses associated with plate-scale processes. In the central Indian Ocean the stress field changes smoothly from a N-S ori entation near India to a E- W orientation in western Australia. The stress field in the central Indian Ocean is oriented NW-SE, sub-parallel to the Java and Sumatra trenches. This orientation suggests that the source of the stress field in this region is dominated by far-field sources (such as collisional forces) rather than by the nearby subduction zones. Seismicity in the central Indian Ocean is thought to be 127 Figure 5.16, Stress indicator data for the Indo-Australian plates from the World Stress Map database [Zoback, 1992]. Circular symbols represent focal mechanisms, square symbols represent geologic indicators, and triangular symbols represent wellhole data. Solid, open, and gray-shaded stress indicator symbols represent compressional, extension, and strike-slip deformational style, respectively. The orientation of the maximum horizontal compressive stress (SH,max) is also shown. Orientations without symbols designate volcanic vent alignments. The lengths of the vectors representing SH,max have been weighted by the indicator quality (A - D) after Zoback [1992]. Details of the individual stress indicators are listed in Table 5.7. [, 128 related to intraplate oceanic-lithosphere deformation [e.g., Stein and Okal, 1978; Weisse! et al., 1980; Gellel' et ai., 1983; Wiens and Stein, 1983,19811; Bergman and Solomon, 1984, 1985; Stein et al., 1989J. The nature of the intraplate deforma tion of the central Indian Ocean near the Bay of Bengal is well doctl)nented from geophysical data [e.g., Stdn and Okal, 1978; Weisse! et al., 1980; Geller et al., 1983; Wiens and Stdn, 1983,1984; Be'rgman and Solomon, 1984, 1985; MacAdoo and Sandwell, 1985; Neprochnov et al., 1988; Stein et al., 1989; Bull and Scrlliton, 1990J, and has been interpreted as being associated with lithospheric buckling. For example, gravity highs in this region have been attributed to 200-km wavelength basement undulation associated with lithospheric buckling [McAdoo and Sandwell, 1985J. However, recent seismic reflection studies in the area have shown that high angle faults have modified the basement topography, accentuating some of the basement highs [I(ing and Stein, 1984; Bull, 1990; Bull and Scrlltton, 1990J. While whole-lithosphere buckling requires the existence of very large stress (100 to 1000's of MPa) within the lithosphere [MacAdoo and Sandwell, 1985; Nepl'ochnov et al., 1988], stress magnitudes required for such fault-involved deformation may be less than 100 MPa [Bull, personal communicationJ. The observed stress field of the mid-ocean ridge in the vicinity of the Central and Southeast India ridges is char acterized by ridge-parallel extension. This extension is thought to be related to relaxation of thermo-elastic stresses due to cooling of the oceanic lithosphere [e.g., Bergman et al., 1984; Wiens and Stein, 1984; Bergman and Soloman, 1984; Bratt et al., 1985J. Cloeiingh and Worle! [1985;1986J suggested that the unique near ridge tensional deformation in the lAP is the consequence of subduction along the Java and Sumatra trenches. The stress field in continental Australia is characterized by E- W compression in the western and eastern regions and N-S compression in the north [Denham et at., 1979; Lambeck et al., 198 11; Fredrich et al., 1988; Denham and Win do 1', 1991J. This 129 change in orientation is thought to be related to the resistance of the Indian Ocean Australian plate to subduction a.long its northern margin [Filch el al., 1973], or the position of Australia relative to the subduction zones along the northern lAP boundary [Cloelingh and Worlel, 1985; 1986). The grea.t Sumba earthquake of 1977 [Stewa'l'l, 1978], suggests the presence of a large slab pull force acting along the Sunda arc. Along the North West Shelf, SIl,ma.v is oriented roughly northeast in the Timor Sea region, and nearly east-west in the Barrow-Dampier sub-basin [Hillis, 1991; Hillis and Williams, 1992;1993). The stress data for continental Australia show a large amount of scatter in the SU,11lua,' orientation, possibly reflecting a strong influence from local sources of stress. These local sources have the greatest affect on stress indicators based on shallow in-situ stress measurements (such as hydraulic fracturing), many of which have been ma.de at depths less tha.n 1 km. Furthermore, stress orientations based on earthquake focal mechanisms may reflect the orientation of pre-existing zones of weakness rat.her than t.he contemporary stress field [McKenzie, 19(9). The stresses predicted by numerical modeling must be consistent with the broad-scale regional stress field in the regions where the observed stress field is well constrained. Within the lAP, these regions include: (1) The mid-ocean ridges where SIl,71lC1X is oriented normal to the trend of the ridge axis. (2) Continental India where the stress field is dominately N-S despite the large amount of scatter in the observed stress orientation. (3) The central Indian Ocean where the observed stress field smoothly changes from a N-S to an E- W orientation. ('1) Continental Australia where SU,meLa,' is oriented E- W in the western and eastern regions and N-S in the north. And (5) the North West Shelf region of Australia where SIJ,maa,' is E- W in the Barrow-Dampier region and NE-SW in the Timor Sea. There is poor constraint on stress orientation in the oceanic areas south and east of A us tralia, and along the southeastern plate boundary between the New Hebrides arc 130 and the Tonga-Kermadec trench. Constraint for the modeling is provided by the long-wavelength trends in the intraplate stress field discussed in Chapter 3. Figure 5.17 shows the trends in the SIl,max orientations for the indicators in the lAP. 5.6.3 Modeling Results Four models have been used to study the influence of the possible sources of tectonic stress in the lAP. As discussed above, the ridge torque in the lAP is very significant. Because the ridge-push force is well-constrained by bathemel;ric data [e.g., Turcotte and Schube1'l, 1982]' all of these models have been designed to evaluate the influence of various boundary conditions in conjunction with the ridge-push force. Predictions about the magnitude and orientation of the tectonic stresses in the lAP were made using an elastic finite-element analysis. The finite-element grid, shown in Figure 5.18, consists of 2527 constant-strain triangular elements com posed of a network of 137'1 nodes, which provides a spatial resolution of about 2° in both latitude and longitude. The sensitivity of the modeled stresses is there fore limited to large scale tectonic features with wavelengths of a few hundred kilometers. It should be noted that the lateral variations in the lithospheric potential energy provide substantial torques in a number of plates, including the lAP (Figure 5.1). For plates with the largest gravitational torques (the Pacific, Indo-Australian and South American plates ), there is good correlation between the gravitational torque poles and velocity poles suggesting gravitational torques may be an important driving mechanism for plate motion. In the lAP the potential-energy distribution produces a torque of 5.7 x 1025 N m about a pole at 53.2° N, 93.4° E which closely matches the velocity pole 29.8° N, 31.6° E (the angular misfit is about llO). This torque mostly reflects the potential-energy distribution associated with the 131 60· 80· 100· 120· 140· 160· 20· 20· I o· 1/ I : o· --- __ L~:, , _20· " _20· ,/ .; _40· I _40· _60· _60· 60· 80· 100· 120· 140· 160· 180· Figure 5.17, Long-wavelength trends in the SH,max orientations. The bars are centered in bins in which the null hypothesis that the orientations are random can be rejected at the 90 % confidence level (see Chapter 3). 132 20 0 , . 00 _20 0 _40 0 _60 0 40 0 60 0 80 0 1000 120 0 140 0 160 0 180 0 Figure 5.18, Finite-element grid for the Indo-Australian plate. The grid is com posed of 2527 constant-strain plane-stress triangular elements and 1374 nodes. The spatial resolution is about 2°. I 133 ocean ridge systems along the southern and western margins of the plate (which contribute 8.9 x 1025 N m about a pole at Model 1: Homogeneous N01'thern Boundn1'Y In Model 1 mechanical equilibrium was achieved by homogeneously fixing the entire northern margin. This is equivalent to assuming that all of the resistance to the ridge force is transmitted frOIll the Pacific and Eurasia plates to the lAP plate along these boundaries, balancing any tendency for the boundary to be deflected. No differentiation was made between the collisional and subduction zone bound- aries along the northern margin. In this model it was assumed that the drag force acting on the base of the plate is negligible. The principal stresses for Modell are shown as Figure 5.19. The stress field due to the ridge-push force is characterized by NE-SW compressive stresses throughout most of the plate and therefore is clearly inconsistent with the observed dataset. The maximum stress magnitude is about 17 MPa (170 bars), and located along the northern boundary and in western Australia. In the continental areas of India, the stresses are oriented NE-SW and have a magnitude of about 7 MPa (70 bars), and no significant stress focusing effect occurs along the Himalayan boundary. This moclel clearly demonstrates that a homogeneous northern margin condition cannot match the broad-scale patterns in the observed stress field. 134 40 0 25MPa 20 0 , . , 0 0 -20 0 U -40 0 -60 0 40 0 60 0 80 0 100 0 120 0 140 0 160 0 180 0 Figure 5.19, Predicted stresses for ModelL Solid bars indicate deviatoric com pression, open arrows indicate deviatoric tension. 13.5 Model 2: Fixed Collisional Boundaries Model 2 employs a set of boundary conditions to balance the ridge torque which involved focusing the ridge-push stresses along the continental collisional boundaries (i.e., Himalayan, Papua New Guinea, and New Zealand). In this model the collisional boundaries were held fixed, while the subduction zone segments were left free. While still simplistic, these boundary conditions probably represent a more geologically plausible situation than those of ModelL The principal stresses for Model 2 are shown as Figure 5.20. The fixed segments produce considerable stress focusing for this heterogeneous northern boundary con dition in northern India and northern Australia, where the stress magnitudes ap proach 70 MPa. Importantly, the predicted stresses reflect many of the major features of the observed stress field, including N-S orientation in India, chang ing to E-W compression in western Australia, and NE-SW stress orientations in northern Australia and the North West Shelf region. This model thus highlights the dependence of the predicted stress field on the boundary conditions applied along the northern boundary. One important aspect of stresses predicted by Model 2 is that, if significant stress focusing occurs along the collision boundary segments, major features of the observed stress field can be related to the ridge-push force. Model 3: Subduction Zone Forces Model 3 demonstrates the effect of including subduction zone forces on the pre dicted stress field. The subduction zone forces applied along the northern boundary in this model are listed in Table 5.8 along with the relative torque contributions. As discussed above, the magnitude of the forces are poorly constrained. The mag nitude of the forces used in this model are geologically plausible with a constant force of 2.0 X10 12 Nm-l applied along the Java and Summatra trenches (directed 136 40MPa 20 0 , . , 00 I \ 0 \ \ _20 ... U ;t I! .... / _40 0 40 0 60 0 80 0 120 0 140 0 180 0 Figure 5.20, Predicted stresses for Model 2. For other details see Figure 5.19. 137 outward), 0.5 X10 12 Nm-1 along the Solomon and New Hebrides trenches( directed outward) and 1.0 X10 12 Nm-1 along the Tonga Kermadec Trench (directed inward). As in Model 2, the continental collisional boundaries (i.e., Himalayan, Papua New Guinea, and New Zealand) were held fixed. A net force of 2.0 X10 12 Nm-1 was applied a.long the New Zealand segment of the eastel'll boundary to represent the buoyancy forces due to the high topography. The predicted stresses for Model 3 are shown as Figure 5.21. Significantly, many of the major features of the predicted stress field are similar to those of Model 2. With the exception of the immedi ate area in the vicinity of the trenches, inclusion of subduction zone forces of the above magnitudes does not alter the broad-scale features of the predicted stress field. Specific areas for which there a.re differences in the predicted stress include: (1) A large region of extension in the eastel'll part of the plate due to subduction zone forces acting along the Solomon and New Hebrides arcs and trench suction along the Tonga-Kermadec trench. The magnitude of the extensional stresses in this region approach 25 MPa. (2) Large (50 MPa) extensional stresses along the Java trench associated with the subduction zone force. The extensional stresses are confined to the immediate vicinity of the trenches and do not extend into continen tal Australia or the North West Shelf region. (3) Sma.ll (10 MPa) compressional stresses in eastern Australia in response to the forces applied along the eastern boundary. And (/1) A reduction (from 70 to 110 MPa) in the magnitude of the com pressional stresses due to the boundary conditions along Himalaya segment. The reduction in the compressional stresses due to the Papua New Guinea boundary segment is significantly less, and the stress magnitude predicted in the North West Shelf region is very nearly the same as those predicted by Model 2 (about 60 MPa). This model shows that while the forces acting along the subduction zone bound ary may have an important influence locally, they do not influence the broad-scale features of the lAP stress field which seem to be controlled by the combination of 138 50MPa 200 .- 00 -20· U _40 0 _60 0 40 0 60 0 80 0 tOO° 120 0 140 0 160 0 180 0 Figure 5.21, Predicted stresses for Model 3. For other details see Figure 5.19. 139 ridge-push forces and collision(tl boundaries. It is important to note that Models 2 and 3 both predict the observed NW - SE SU,maxorientation in the central Indian Ocean. Model 4: AlteTnative Bo'ltnda1'Y Conditions Model <1 evaluates the effect of employing an alternative set of boundary con ditions to represent the forces acting along collisional boundary segments. In this model, a constant force per unit length of boundary segment was applied along 12 1 12 1 the Himalayan (4.0 X10 Nm- ), Banda (0.5 Xl0 Nm- ) and Papua New Guinea 12 1 (4.0 Xl0 Nm- ) segments (see Table 5.8). Since this does not produce a torque balance, shear forces were applied to the base of the plate such that a plate ve locity of 1.0 cm/yr produced a basal shear stress of 0.1 MPa. The tectonic forces acting on the plate were otherwise identical to those used in Model 3 The pre dicted stresses for Model 4 are shown as Figure 5.22. The broad-scale features of the predicted stress field are similar to those predicted by Model 3. While slightly different stress magnitudes are predicted for several areas of the plate, the bl'Oad scale features of the predicted stress field are robust to the change in the boundary conditions. Significant differences in the predicted stress field include: (1) Very large (80 MPa) compressional stresses in the central Indian Ocean. (2) Tension parallel to the spreading ridge along the Southeast Indian ridge. (3) Large (50 MPa) compressional stresses in continental India. (4) A smooth transition from N-S to NW-SE SU,max orientation in the central Indian Ocean. And (5) A signif icant reduction in the magnitude of the predicted stresses in northern Australia (35 MPa). Model 4 demonstrates that similar broad-scale features of the predicted stress field are produced whether the collisional boundaries are pinned or they are subjected to a boundary force. Given the lack of uniform spatial distribution of observed stress indicators it is difficult to further constrain the magnitudes of the I· 50MPa 20' .- , . o· -20· U.' \ 'N. '\ ..... _40' .. _60' 40' 60' 80' 100' 120' 140' 160' 180' Figure 5.22, Predicted stresses for Model 4. For other details see Figure 5.19. 141 collisional resistance and subduction zone forces acting on the plate. Continental A 1lstmlia The Indo-Australian plate (lAP), along with the North and South American plates form a group of fast moving "continental" plates [Minste'l' and JO'l'dan, 1978]. In the continental regions of North and South America the orientation of the maxi mum horizontal compression (Sll,max) is well defined and is clearly aligned with the absolute plate velocity and ridge-push directions [Richa'l'dson, 1992; Section 5.4]. In contrast, the intraplate stress field within the continental regions of Australia is complex, and not readily explicable in terms of any single tectonic process. Like North and South America, but unlike the continents on the slower moving plates such as Europe and Africa, the stress field within the interior of the Australian con tinent is largely compressional. In the northern part of the plate SU,max is aligned N to NNE, more or less orthogonal to the collisional boundary in New Guinea. In the southern part of continental Australia SU,max is oriented E - W. Previous studies have been able to model the stress field orientation data in the northern and western part of the continent in terms of plate-scale tectonic processes, [e.g., Cloelingh and W01'lei, 1985, 1986; Richardson, 1987] resulting in differing views of the contribution of subducting slabs to the intraplate stress field. However, the origin of the E - W compression in SE Australia has not been explained and remains enigmatic. The predicted stress field in continental Australia for Models 1, 2, and 3 are evaluated below in an attempt to understand the origin of the observed stress field. The principal stresses predicted by Modell for the continental Australia re gion are shown as Figure 5.23. The predicted orientation of Sll,max is NE - SE throughout continental Australia. Although this orientation agrees with observed stress indicators in the northern pa.rt of the continent, this is clearly inconsistent 142 0 0 lOMPa _10 0 r 0 I; _20 I ;' -30 0 I I 0 ... _40 I \ V \ 100 0 120 0 140 0 160 0 180 0 Figure 5.23, Predicted stresses for Modell, continental Australia. Solid bars incli cate deviatoric compression, open arrows indicate deviatoric tension. In this model the potential-energy torques were balanced by fixing the whole of the northern and eastern boundary of the plate. Open bars designate the trends in the SU,max ori entation (see Figure 5.17). I with the observed dataset in the southern part of Australia. It can be concluded that for continental Australia, as for the plate-scale stress field, a homogeneous northern margin condition cannot match the broad-scale patterns in the observed stress field. The principal stresses predicted by Model 2 for the continental Australia region are shown as Figure 5.2'1. The fixed segments produce considerable stress focusing in northern Australia, resulting in more pronounced N-S oriented SU,71WX in this region. However, a large amount of misfit still exists between the observed and predicted SU,lnClX orientations in western and eastern Australia. The principal stresses predicted by Model tl for the continental Australia region are shown as Figure 5.25. The SU, 11IClxorientation is roughly NE - SW in the north ern part of AustraJia and E - W in the southeastern and southwestern regions. The match between the observed and predicted stress fields for Model A3 is therefore far better for than for either Model 1 or 2. The stresses predicted by this model suggest that much of the compressional stress within the lAP is the result of focus ing the potential-energy torque (mostly arising form ridge push) along collisional boundary segments. It appears that the SII,lIlClx orientations in eastern Australia is strongly influenced by the forces acting along the New Zealand boundary segment. The predicted occurrence of EW compression in SE Australia in Figl\l'e 5.25 is of particular importance in that it is consistent with the evidence of a range of in situ stress indicators [e.g., Zoback, 1992J. Perhaps more importantly, the combined effect of the stress focusing along the collisional boundary segments can produce a near isotropic state of stress in continental Australia. It is difficult to make an accurate estimate of the SU,11Iax orientation if the horizontal principal stresses a.rc isotropic. It is possible that the large amount of scatter in the observed SU,11IC1X orientations in continental Australia is the consequence of this near-isotropic stress state. 144 30MP~ .... -20' , -+- -40' / l "- lOa' 120' 140' 160' Figure 5.24, Predicted stresses for Model 2, continental Australia. In this model the potential-energy torques were balanced by fixing the collisional boundary seg ments along the Himalayan, Papua New Guinea, and New Zealand segments. The orientation of the maximum horizontal compressive stress (SH,max) from Figure 5.16 is also shown. H5 o· 40MP~ Figure 5.25, Predicted stresses for Model 4. Note that the predicted stresses are nearly isotropic in central Australia. The orientation of the maximum horizontal compressive stress (SH,max) from Figure 5.16 is also shown. 146 5.6.4 The North West Shelf Stress Field The orientation of the regional stress field with respect to the Australian conti nental margin in the North West Shelfregion makes it an ideal location to evaluate the effect topographic stresses have on the predicted stress field at a local-scale. Since the horizontal stresses associated with continental margins have a magnitude of about 15 MPa (corresponding to a geoid anomaly of 6 m) [e.g., TIl'l'cotte and 8chube'l't, 1980; Coblentz and Richardson, 1992], they have the greatest effect on the predicted stress field in area where the magnitude of the regional stresses are of the order of several tens of MPa. Because the regional stress magnitudes in continental Australia are small (less than 25 MPa, as predicted in Model i1), the horizontal stresses due to the continental margin along the North West Shelf are expected to play an important role in the predicted stress field. The stresses pre dicted for the North West Shelf region (bounded by 50S, 25°S, 1l00E, and 135°E), without and with topographic forces included in the modeling are shown as Fig ures 5.26 and 5.27, respectively. Figure 5.27 shows that along the continental shelf, the effect of the topographic forces is a function of the orientation of the regional stress field and the continental margin. Where the regional stress field is perpendicular to the coastline (for example, along coastline of westel'Jl Australia at 250S), the topographic stresses reduce the E- W compression and induce a minor amount of N-S extension. Further north along coast, where the regional field is parallel the coastline, the topographic stresses induce a greater amount of tension in a direction perpendicular to the coast. It is possible that the magnitude of the horizontal stress is reduced enough to allow the vertical stress to become the inter mediate principal stress and thus produce strike-slip deformation. This is difficult to evaluate, however, given the two-dimensional nature of the predicted stresses and the lack of information about the vertical stress. Off the northwest coast of 147 _50 FP--~~======?I-!!!IfIPII!--======:J_-""'~ 25MPa _10 0 _15 0 _20 0 0 _25 ~ __"'-======~- ___======::J ___~ 0 - 0 0 0 0 0 110 115 120 125 130 135 Figure 5.26, Details of the predicted stresses for Model 4 for the North West Shelf region of Australia, topographic forces due to the continental margin were not included in the modeling. See Figure 5.19 for details. 148 -5" FP--III!!IJ!I~======::;::JIII-!!IIIPI!--======:::=:J--""'~25MPa -10" -15" -20" ;( .,. I II! I I JIf I~ 'i lit II! II! ;f )f f. I I - ; lit lit #t I I I ~ ~ 18 -25" 110" 115" 120" 125" 130" 135" Figure 5.27, Details of' the predicted stresses for Model 4 for the North West Shelf region of Australia, with topographic forces included in the modeling. See Figure 5.19 for details. lt19 Australia, the predicted SU,max orientations are in good agreement with the ob served orientations. However, while the predicted stress regime is compressional, local seismicity, as defined by two focal mechanisms in the area, is strike-slip. 5.6.5 Discussion The results of the modeling presented above are in substantial agreement with the observation that the nature of the intraplate stress field within the lAP is a consequence of the unique combination of tectonic forces acting on the plate [C/octingh and Worlel, 1985; 1986; Richardson, 1987], with the observed stress field best explained in terms of a combination of ridge push and the forces acting along the northern boundary. The results demonstrate that other tectonic processes such as subduction zone forces produce significant variation in the predicted stress orientation on a local scale. A number of models have been presented which demonstrate the relati ve con tribution of the various tectonic forces to the lAP intrapla.te stress field. The first set of models (Models 1,2 and 3) was used to evaluate the relationship between ridge-push forces, the boundary conditions applied along the northern margin, and the predicted stress field. These models demonstrate that many of the major fea tures of the observed stress field can be explained in terms of the ridge-push forces if significant stress focusing occurs along the collisional boundaries (Himalaya and Papua New Guinea, and to a lesser extent, New Zealand). An important aspect of the modeling results is that, while other tectonic forces such as the subduction zone forces, basal shear and topographic forces may be locally important, a combi nation of ridge push and the collisional boundary establishes the basic broad-scale features of the observed stress field. In other continental plates where ridge-push force is considered to dominate the intraplate stress field (e.g., South America), 150 the nature of the plate boundary opposite the mid-ocean ridge system is relatively simple. In the South American plate, the entire western margin is basically colli sional (i.e., no subduction of the plate). As discussed above, the ridge-push force produces a uniform stress field, which also corresponds well with the absolute plate motion. In the lAP, however, the plate boundary opposite the mid-ocean ridge is not uniform, but rather a mixture of collisional and subduction zone segments. The main consequence of the complex northern boundary configuration seems to be significant stress concentration along the collisional segments, and a complex regional stress field. While there is little agreement between the observed Sl/,max orientation and the ridge-push torque in the lAP [Richardson, 1992]' the results presented here suggest that the ridge-push force is a very important control on the intraplate stress field in the lAP. An important result of the models presented in the present study is the predic tion of low tectonic stress for most of the India and Australia plates, on the order of 20-40 MPa averaged over a 100km thick lithosphere. These magnitudes are con siderably less than those predicted by Cloctingh and Wodel [1985;1986]' and are more consistent with the stress magnitudes predicted for most of the other plates including North and South America, Nazca, Pacific and Europe. In the Central Indian Ocean, the stress magnitudes are greater, approaching 100 MPa (1 kbar) in the Bay of Bengal region of the Indian Ocean, and are in agreement with the suggestion that the oceanic lithosphere in the Central Indian Ocean is deforming in response higher stresses than elsewhere, but to much lower stress than previously thought [i.e., Bull, 1990; Bull and 5'crutton, 1990; Bull, personal communication]. The main features of the Australian continental stress field have been shown to be explainable in terms of the interaction of only two governing processes, namely gravitational potential-energy torques (mainly due to ridge push) and collisional resistance. Model 3 (Figure 5.21) clearly shows that the main com plexi ty in the 151 stress field reflects the heterogeneous disposition of the collisional segments along the northern and eastern convergent boundaries of the lAP. If this interpretation is correct it raises important questions about the role of subduction at conver gent boundaries in the intraplate stress field. Namely, that subduction processes provide, at best, a second-order control on the orientation of the lAP stress field. Of course, it is possible that subduction along the northern margin of the lAP provides a tension to the trailing plate [Cloelingh and Worfel, 1985,1986J. The re sults of further modeling have demonstrated that such an effect results in increased stress focussing in collisional zones with corresponding increases in the magnitude of SlI,max, but with little reorientation of SlI,ma:L" This suggests the complexity in the Australian stress field, in comparison with other continents such as South America, may simply reflect the heterogeneous convergent boundary conditions operating on the northern and eastern boundary. While the role of the northern boundary of the lAP has long been suspected, this new interpretation of the origin of the E-W compression in the SE Australia further emphasizes the importance of stress focussing at collisional boundaries. Forces due to lateral density variations within the lithosphere were found to have a significant affect on the intraplate stress field in the continental regions of India and Australia. The topographic stresses reduce the magnitude of the compressive stresses of the regional stress field, and in some areas, induce a significant rotation in the SlI,max orientations. These results support the suggestion that these forces are an important source of stress even for plates dominated by boundary forces [Fleitout and Froidevau;l:, 1982,1983; Fleitout 1991J. Because the stress magnitude in the continental regions of Australia were predicted to be quite small, 20-30 MPa (200-300 bars) local sources of stress can be expected to have a significant influ ence on the observed stress orientations. This may help explain the considerable variation in the azimuth of P-axes throughout continental Australia, as well as the 152 large number of observed strike-slip events. Furthermore, the stresses predicted by this model suggest that much of the compressional stress within the lAP is the re sult of focusing the potential-energy torque (mostly arising from ridge push) along collisional boundary segments. Perhaps more importantly, the combined effect of the stress focusing along the collisional boundary segments can produce a near isotropic state of stress ill continental Australia. It is difficul t to make an accurate estimate of the SU,mul' orientation if the horizontal principal stresses arc isotropic. It is possible that the large amount of scatter in the observed SU,11IUX orientations in continental Australia is the consequence of this near-isotropic stress state. The analysis of the lAP stress field has bearing on an important geodynamic problem concerning the magnitude and origin of the stresses that drive collisional orogens such as the Himalaya. As stated above, the potential energy changes accompanying the construction of the Himalayan mountains, while poorly con 12 1 strained, are believed to be of the order of 5 - 10 X 10 N m- , requiring shear stresses of the order of 25 - 50 MPa averaged over a 100 km thick lithosphere [e.g., England and MolnaT, 1992J. Moreover, it is likely that the potential en ergy of mountain ranges such as the high Himalaya i::; significantly greater than the potential energy of the mid-ocean ridges. These simple considerations of the potential-energy changes accompanying collisional mountain belt formation have been used to implicate subduction as an important driving mechanism in plate tectonics. In this context, the important aspect of the modeling presented above has been to illustrate how the focussing of potential-energy torques arising from the asymmetric distribution of ridge systems around the lAP may lead to com pressional stresses of the magnitude needed to drive collisional orogens. 153 Source Magnitude Latitude Longitude [x 10 25 Nm] [deg] [deg] Whole Plate 1.81 20.3 N 67.0 W Young Oceanic 3.21 57.9 N 34.4 W Old Oceanic 1.97 63.6 S 159.7 W Submerged Continent 0.32 25.4 S 106.6 E Elevated Continent 0.77 16.8 S 74.2 W Table 5.1, Relative Torque Contributions, African Plate Source Potential Energy Potential Difference! Horizontal Stress2 [x 10 14 Nm- 1] [x 10 12 Nm- 1] [MPa] Plate Mean 2.377 o o Ridge Crest 2.391 1.4 11.2 Young Oceanic Mean 2.378 0.1 0.8 Old Oceanic Mean 2.365 -1.2 -9.6 Submerged Continent Mean 2.376 -0.1 -0.8 Elevated Continent Mean 2.386 0.9 7.2 1 Difference between the absolute potential energy and the plate mean. 2 Calculated for a unit 125km-thick lithospheric column. Table 5.2, Potential-Energy Means and Stress, African Plate 154 Stress Indicator Type Number Total Number 217 Total Focal Mechanism 97 Focal Mechanism Thrust 65 Focal Mechanism Normal 11 Focal Mechanism Stike-Slip 21 Total Geologic 100 Geologic Thrust 17 Geologic Normal 72 Geologic Strike-Slip 4 Geologic Unknown 7 Total WeI/hole 20 Borehole Breakouts 19 Hydraulic Fracturing 1 Stress indicators extracted from the World Stress Map database [Zoback, 1992). Table 5.3, Summary of Stress Indicators for the South American Plate 0 ~ c ,i Model F"iclge Ji/'ounclar'" Ftol!.o9'·Ol!.hl!. FcI"a,q Figure 1 1 P 0 0 5.8 2 1 0 0 R 5.9 3 1 P 0 D 5.10 4 1 P 1 0 5.11 5 1 F 1 D 5.13 a Ridge-push force based on potential-energy gradient. as described in the text. b Plate boundary forces as described in text. An index of P designates the use of pinned boundary segments, F designates the use of force boundary segments, and 0 designates the use of free boundary segments. C Topography forces based on potential-energy gradient as described in the text. cI Drag forces based method described in text, An index of R designates the use of a resistive drag force, D designates the use of driving drag force, an index of 0 indicates no drag force was applied. Table 5A, Description of Force Models 155 NOlldrag Torque Balancing rrorquc U Model MagnHudc Ll1titudc Longitude Ihle Lathude Longitude [x 1025 Nm ] [deg] [deg] [cm/rr] [deg] [deg] I 9.5 07.6 S 10B.7 E 2 0.5 67.6 S 10B.7 E 5.0 66.3 S 115.1 E 3 0.5 67.B S 10B.7 E 3.5 70.3 Sb 71.0 E 4 5.9 50.2 S 64.B E 5 5.5 77.3 N !J,g E 5.4 52.0 S 01.7 W a n.. ed on equaliou (5.3). See Richard.on el al. [1070] and Richard.on and Reding (IDOl] for olher delail •. b Ab.olule rolation pole from Gripp and Gordon [IDOO]. Table 5.5, Torque Parameters for Force Models, SAP Force Total Torque Latitude Longitude [xl025 Nm] [deg] [deg] Collisional Nazca 9.0 68.8 N 104.7 W ANT-CHI 0.5 68.8 N 104.7 W Car 1.3 68.7 N 40.9 E Other Boundaries NAM 0.3 0.5 S 141.1 W SCO 0.2 33.2 N 60.5 W LA 0.6 69.6 S 12.9 W SS 0.7 31.3 S 170.9 E Non-Boundary Forces Ridge Push 9.5 67.8 S 108.7 E Passive Margin 1.76 67.1 N 2.0 E Topography 2.11 22.1 N 17.6 E Positive forces are directed towards the interior of the plate. Note: force magnitude of lxl012Nm-1 is equivalent to a stress of 10 MPa (100 bars) across a plate with a thickness of 100 km. Abbreviations are: Nazca = NAZ- Nazca Subduction Zone; Car = Caribbean; NAM = North America; ANT = Antarctic Transform; SCO = Scotia Trench; LA = Lesser Antilles Arc; SS = Sandwich Island Arc. Table 5.6, Boundary Forces and Torque Contributions, SAP 156 Stress Indicator Type Number Total Number 251 Total Focal Mechanism 141 Focal Mechanism Thrust 54 Focal Mechanism Normal 31 Focal Mechanism Stike Slip 56 Toial Geologic 6 Geologic Normal 1 Geologic Unknown 2 Volcanic Vent Alignment 3 Total Well/tole 104 Overcoring 18 Borehole Breakouts 65 Hydraulic Fracturing 21 Stress indicators extracted from the World Stress Map database [Zoback, 1992]. Table 5.7, Summary of Stress Indicators for the Indo-Australian Plate 157 Force Magnitude Total Torque Latitude Longitude [Xl0 12 Nm-1] [xl025Nm] [deg] [deg] Collisional Himalaya 4.0 11.1 0.3 S 171.0 E PNG 2.0 2.78 30.8 S 122.5 W Banda 0.5 0.70 15.8 S 1'12.3 W Subduction Zone Sumatra 2.0 2.78 10.0 N 0.5 E Java 2.0 3.13 18.8 N 23.4 E Solomon 0.5 0.52 73.0 N 105.8 E New Hebrides 0.5 0.43 60.5 N 132.5 E Other Boundaries Tonga-Kermadec 1.0 2.25 63.2 S 0.0 E New Zealand 2.0 1.74 44.1 S 21.1 E Non-Boundary Forces Ridge Push 8.0 40.0 N 30.1 E Passive Margin 2.2 25,4 S 177.0 W Topography 0.6 30.3 S 100.8 E Positive forces are directed towards the interior of the plate. Note: force magnitude of 1x1012Nm-1 is equivalent to a stress of 10 MPa (100 bars) across a plate with a thickness of 100 km. Table 5.8, Boundary Force Magnitudes and Relative Torque Contributions 158 CHAPTER 6 TIME-EVOLUTION OF PLATE-SCALE POTENTIAL-ENERGY DISTRIBUTIONS 6.1 Introduction This chapter evaluates the time-evolution of plate-scale potential-energy distri butions in continental plates. Although it is widely recognized that lateral density variations in the ocean lithosphere produce variations in lithospheric gravitational potential energy, U/ and thus provide important torques on the plates [e.g., Forsyth and Uyeda, 1975; Richardson et at., 1979; Richa'l'dson, 1992; Zoback, 1992J, it has not been widely appreciated that continents may be associated with similar vari ations in U/ . Understanding the role of potential-energy variations in the origin of continental extension is especially relevant given the current debate concern ing the source of stresses responsible for continental extension [e.g., Cl'ough) 1983; IJ O1lseman and England) 1986; Pavoni) 1992 J. Much of the debate focusses on the relative roles of lithospheric sources associated with distant plate-boundary activ ity (which contributes to so-called passive rifting) and the asthenospheric mantle (so-called active rifting) [e.g., Turcotte, 1983J. As discllssed in Chapter 2, the mean potential energy of a plate, U/, depends on the relative amounts of young and old oceanic lithosphere and continental lithosphere, and will vary as the plat.e ages. The controls on this variation in U/ with plate aging form the main focus of this chapter. As discussed above, in the ambient state, parts of the plate with potential en ergies in excess of U/ will experience deviatoric tension, while parts of the plate with potential energies less than U/ will experience deviatoric compression. The ambient state is particularly relevant to the intraplate stress field in slow-moving 159 plates largely surrounded by mid-ocean ridges, such as the African and Antarctic plates [e.g., Crough, 1983]. Changes in U/ attendant with the growth of old ocean lithosphere in such plates should therefore correspond to changes in the ambicnt intraplate stress field. In this chapter, the change in U/ accompanying the growth of large plates is calculated. The observed geoid anomalies associated with ocean ridge systems and continental margins are used as a first-order constraint 011 the potential-energy distribution within the present-day plates. This chapter seeks to evaluate how the change in the potential-energy distribution of aging plates may engender the possibility of extensional fa.ilure within continenta,llithosphere. This hypothesis is tested by evaluating the plate-scale evolution of potential energy for a simple circular plate surrounded by all oceanic ridge system. Changes in the potential-energy distribution in the African and Antarctic Plates since the late Jurassic are then evaluated using the sea-floor age data of Roycl' cl al. [1992J. It; is shown that the ambient tectonic stress state within continental Africa and Antarc- tica has become increasingly extensional as the plate bas aged, thus potentially contributing to the stress field responsible for the observed continental rifting in these plates. 6.2 Geoid Anomalies and Potential-Energy Variations Because variations in the gravitational potential energy of the lithosphere, .6..U/, correlate with the dipole moment of the near-surface density distribution [e.g., Flcitout and Froidevaux, 1983], they can be directly related to the lithospheric component of the observed geoid anomalies, .6..N/ [e.g., IJaxby and Tlll'coUe, 1978; TIl'l'cotte and Schubcl't, 1982]: (6.1 ) , l, 160 where 9 is the gravitational acceleration, and G is the gravitational constant (note that the potential energy varies with the geoid anomaly as approximately 0.23 x 1012 N m-1 pel' meter). The main problem with using this relationship is resolving the lithospheric contribution of the geoid anomalies from the much larger amplitude anomalies associated with the dynamic processes of the deep mantle [Tu1'cotte and McAdoo, 1979]. Positive geoid-height anomalies of up to 10 - 15 m associated with a number of mid-ocean ridge segments [Haxby and Tll'I'Coite, 1978; Pa1'sons and Richte1', 1980; Sandwell and Sclwbe1't,19S0j Forsyth 1992], as well as age-correlated geoid offsets across fracture zones [Ol'ough, 1979j Wessel and Haxby, 19S9] imply that aging of the ocean lithosphere is accompanied by a decline in potential energy. In a recent summary, Forsyth [1992] concluded that the semi-infinite half-space model for the cooling ocean lithosphere provides an adequate approximation for geoid anomalies in ocean lithosphere younger than about SO - 100 Ma. The geoid anomaly relati"e to the mid-ocean ridge (assuming a semi-infinite half-space) associated with aging of the ocean lithosphere can be expressed as [Haxby and TU1'cotte, 1975]: !:1No = _ 27f' G pm a Tm I~ t (1 + 2pm a '1~n ) (6.2) 9 7f' (Pm - Pw) where Tm is the reference temperature of the mantle, pm and Pw are the density of the mantle (at Tm) and water, respectively, a is the thermal coefficient of expansion, and ~ is the thermal diffusivity. It should be noted that the cooling half-space model the geoid anomaly is lineal' with age. The lithospheric contribution to geoid anomalies in old oceanic lithosphere is poorly constrained. Parsons and Richte1' [19S0] showed that there are significant discrepancies between the predictions of the thermal plate and half-space models for lithosphere older than about 40 - 60 Ma (Figure 6.1). For the purposes of the present study, it was assumed that the old lithosphere 161 0 -2 -4 -6 -E -z -8 -10 -12 0 50 100 150 200 250 Age (Ma) Figure 6.1, The predicted geoid for the modified half-space (Curve 1) and thermal plate (Curve 2) models. The geoid anomaly used for Curve 1 is given by the half-space model for t < 8i1 Ma is assumed to be age independent for t > 84 Ma. Curves l' and 2' show the mean geoid anomaly for the oceanic lithosphere for the geoid anomalies shown as Curves 1 and 2, respectively. Geoid anomalies are shown relative to mid-ocean ridge with corresponding potential-energy anomalies of 0.23 x 1012 N m-1 per meter. 162 For the purposes of the present study, it was assumed that the old lithosphere is thermally stabilized and the geoid anomalies in the oceanic lithosphere can be approximated by a combination of the cooling half-space model for tc < 84 Ma and an age-independent geoid anomaly for tc > 84 Ma (Curve 1 in Figure 6.1) . This behavior approximates the geoid anomalies predicted by the thermal plate model (Curve 2 in Figure 6.1) [Pa'l'sons and Richlcr, 1980], and has the virtue of admitting very simple analytical expressions for the potential-energy evolution for simple plate geometries. The use of Curve 1 to approximate Curve 2 produces only a slight difference in the mean geoid anoma.ly for the oceanic lithosphere, shown as Curves l' and 2' in Figl\l'e 6.1, respectively. The geoid anomaly predicted for the cooling half-space model (as well as the thermal plate model) is strongly dependent on uncertain parameters. The set of thermal parameters used in the calculations below (Table 6.1) are consistent wi th d (!~No) /dt = -0.15 m/Ma, which compares favorably with the observed geoid anomaly over the Mid-Atlantic Ridge at 44.5°N [IJa:l:by and Turcottc, 1978J and elsewhere [Sandwcll and Schubcrt, 1980J as well as with the geoid offsets across fracture 7,Ones [Wcsseland JIa:z:by, 1989J. This parameter set. yields a total geoid anomaly of -12.7 mover 8il Ma with a corresponding decline in [1/ of 2.9 x 1012 N m-1 (which therefore defines all absolute lower limit to the mean plate potential energy relative to the mid-ocean ridges (see Figure 6.1). Following the method outlined in Chapter 2, the potential energy of the mid-ocean ridge ([lAwn) is equated with a column of unit surface area extending to the depth 125 I\m, with the density structure appropriate to the mid-ocean ridge system. The estimate 1 1 of 2.391 x 10 ,1 N 111- for UAwn is significantly greater than ocean lithosphere 1 1 of age 8il Ma (Uo /t=S4Mn = 2.362 x 10 ,1 N m- ), with the ratio of the potential energy of 84- Ma ocean Ii thosphere to that of the mid-ocean ridge estimated at U~/t=84Mn = 0.988. 163 In comparison with the mid-ocean ridges, the geoid anomalies associated with continental margins and the interior of continents are far less clear. On the basis of averages taken over large areas, TIl1'Cotte and Il!lcAdoo [1979] concluded that there was no systematic difference in the geoid height between old ocean basins (older than Cretaceous) and continental masses. Such an interpretation implies that the mean potential energy of the continental lithosphere is equivalent to old ocean basins, that is, U~ = 0.988 (where U~ is the mean potential energy 0[' the continental lithosphere relative to the potential energy of the mid-ocean ridge, U~ = UcIUlIwH). However, the data show very substantial differences between continents, with the mean geoid of the African continent some 40 m higher than the North American continent and 10 m higher than the mean for the Atlantic and Pacific ocean basins older than Cretaceous. This observed intercontinental variation far exceeds the plausible lithospheric contributions to geoid anomalies and therefore must reflect long-wavelength sub-lithospheric contributions. Moreover, a. number of continental margins are characterized by distinct positive anomalies on the order of 6 m across the transition from the ocean basin to sea-level continent [Ha:vby and Tllrcotte, 1979; TIl'I'colle and Schubert, 1982] (see Figure 6.2), and imply that a continental lithospheric column supporting sea level elevation has the potential-energy equivalent to ocean lithosphere of age about 4Ll Ma (that is U~ = 0.9936). Since the lithospheric contribution to the geoid anomaly reflects the dipole mo ment of the near-surface density distribution, the observed geoid anomalies across continental margins can also be used to constrain the continental lithospheric den sity structure. A lithospheric thickness of 125 km and a crustal density of 2750 kg 1m3 is consistent with a continental marginal geoid anomaly of + 6 m. More over, such a density structure are consistent with the interpretation proposed by a number of workers [Crough, 1983; Houseman and England, 1986; Sonde'/' ct al., 16<1 8+------~------+ 6 ~ ~ 4 oS ~ "1j 2 .~o OJ C) o o NEBr.lzll 100km o Western Austr.ll1.1 A Eastern Australia o North America -2+------r Figure 6.2, Observed geoid anomalies, b.N, across seven passive continental mar gins. The North American profile is from lIaxby and Tu1'cotte [1978], the Brazilian profile from Coblentz and Richan/son [1992]' with the other five margins from this study. The geoid anomalies have been normalized such that b.N = 0 corresponds to the oceanic basin with bathymetry of -,!OOO m. The transition from ocean to continent is depicted from left to right. 165 1977; Zh01l and Sandiford, 1992J that an isostatically compensated continental lithospheric column supporting about 1 km of surface elevation above sealevel is in potential-energy balance with the mid-ocean ridges (i.e., U~ls=11(1Il = 1). While the generally poor resolution of the geoid in mountainous regions precludes definitive correlation between topography and potential energy within the continents, some evidence of the correlation is provided by the lithospheric contribution to the geoid anomaly of 24 - 27 m for the Andean Altiplano inferred by Fl'oideva1t:l: and [sacks [1982J. Such inferences are consistent with a geoid that; varies with continental topography as 6 - 7 m/km, corresponding to a potential-energy variation of about 1.3 x 1012 N m-1/km. For a continent with an average elevation of 500 m, this correlation suggests a mean continental potential energy of U~ = 0.997. The main purpose of the calculations presented below is to assess time-dependent variations in the potential-energy distribution accompanying plate growth about a continent with a density structure consistent with a positive geoid anomaly of 6 m across the continental margin. As we have shown above, for a mean elevation 01' 500 m, which is close to the global continental average, the mean potential energy of such a continent is approximately U~ = 0.997. However, because the a.vaila.ble geoid data imply that considerable uncertainty is attached to the potential-energy distribution in the continental lithosphere, the results for the range of continental lithospheric potential-energy distributions, U~ = 0.988 - 1.0, corresponding to the full range of potential energies reflected in the aging ocean lithosphere, is also presented. 6.3 Circular Plate Model To demonstrate the importance of aging of ocean lithosphere on the mean poten tial energy of a plate, this study begins by assuming a very simple plate geometry 166 consisting of a circular continent, of radius r' c , surrounded at time t = 0 by a system of incipient mid-ocean ridges spreading with a constant velocity, V•• ll. The mean potential energy of the continent, Uc , is assumed to be invariant through time. Note tha.t abbreviations used in the calculations below are listed in Table 6.2. As the oceanic ridge spreads and the plate ages and grows, the rela.tive percent ages of the principal lithospheric types (young oceanic, old oceanic, and continental lithosphere) change. The relative proportion of the lithospheric types making up the total plate area at any time t is given by: 1 Ac = (6.3) (w't + 1)2 (w't + 1)2_1 Ayo It the surrounding ocean grows, while the proportion of young oceanic lithosphere increases rapidly until te, and thereafter declines as the proportion of old ocean lithosphere begins to increase. As discussed above, the changing makeup of this simple lithospheric plate with time must result in corresponding changes in V,and below is presented a simple analysis of the time evolution of V, accompanying this plate growth model. Assuming the bilinear functional dependence of geoid anomaly on age, as shown in Figure 6.1, the difference in potential energy between the mid-ocean ridge and aging ocean lithosphere (I~.u!1OR) at any time, t < te , after the onset of spreading is given by: A VAlOR I T ( 2pm aTm ) u 0 t (6.9) Since the geoid anomaly is linear in time for young ocean lithosphere, t < te : _ ~VII1OR I ( 1 ) V~ It U~ Ite ((w't + 1)2 - (w' (t - te) + 1)2) 2 + (6.11 ) (w't + 1) (1 + f:l.VtfoRbrc) ((w' (t - t ) + 1)2 -1) VAlOR e + (6.12) (w'l + 1)2 U' e (6.13) 168 100~~--~----~------~ 80 < 60 '#. 40 20 50 100 150 200 Time (Ma) 100r---~~==~~----~------~ 80 < 60 o~ 40 50 100 150 200 Time (Ma) (c) old ocean 40 10 50 Time (Ma) Figure 6.3, Percentage areas of the principal lithospheric types for the circular continental plate. Contours are for normalized time constant, w' = 0.001, 0.003, 1 0.006, 0.01, 0.1 Ma- • 169 where the first term is the contribution of the young ocean lithosphere, the second is the contribution of the old ocean lithosphere, and the third is the contribution of the continental lithosphere. The term U~ Ite gives the mean potential energy of the ocean lithosphere younger than te and is given by: e U'I = 1+ ~U:fOR It::t e (10' (t - t ) + 1) (6.14) o te 2 Ul'l'fOR 10' l + 1 The analysis presented above shows that the time evolution of Uf for this simple plate geometry depends only on U~, the ratio of the spreading rate to the continental radius, 10', and te. In all calculations, it was assumed that Ie = 84 Ma.. Figure 6.4 shows the time evolution of U/ for a U~ = 0.997 (that is, for a continent with a mean elevation of 500 m and a density stl'llcttll'e appropriate to a geoid anomaly of + 6 m across the continental margin) and for various normalized time constants 10'. It has been assumed in these calculations that the potential energy distribution in the continent is time-invariant. A decline in U/ of -1 x 12 I 10 N m- 1'0), U~ = 0.997 (Figure 6.'1) corresponds to an increase in the mean extensional stress difference (O'zz - O'.,;,v) in the continental lithosphere (over 200 Ma) of up to 8 MPa, averaged over the thickness of the lithosphere (125 km). In view of the uncertaiuties, results are shown 1'0), U~ corresponding to the range from Uflt::te to UIIlOH (i.e., 0.988 - 1.0). The change in mean potential energy, ~ U{ It as a function of time 1'01' the aging circular plate model for U~ = 0.988, 0.9936 and 1.0 is shown as Figure 6.5. Figure 6.5 demonstrates the dependence of ~ U{ It on U~. For U~ < 1 the initial growth of the ocean ridges increases the mean plate potential energy. However, the mean plate potential energy begins to decline once the aging of the plate results in U~ It < Ufo In the ambient state the requirement for extension in the continents is that U~ > Uf. It is interesting to note that for U~ = 1, the growth of ocean lithosphere causes a. decline in U/ for all times. 170 0.5 ,~------, I /' 01. \ \ , , 0.25 ,," \ ,, I \ ,....., I .... ----_0.01 \ , ~------...... \ .§ 0 ---- ..... ::---...... " Z ~::::,..;;- ,, C\I -...::::.. ...~ " T'"' -0.25 "~ " , b ~~ T'"' 0.006' "'::--... ' ~ ~...... " ::> -0.5 ~ ...... " Cl ...... '" "" '- ...... '" ...... ', -0.75 " .... ':::::- ... ~~ ,,~ -....~ ...... : Uc' = 0.997 """ ",::::,,, -1 ''':::''-:::::- o 50 100 150 200 Time (Ma) Figure 6.4, The change in mean potential energy, ~ Ur It as a function of time for the aging circular plate model assuming U~ = 0.997 (that is, for a continent with a mean elevation of 500 m and a density structure appropriate to a geoid anomaly of + 6 m across the continental margin). ~ Ur It is shown for normalized time 1 constants, 10' = 0.001, 0.003, 0.006, 0.01, 0.1 Ma- • 171 2.5 -~ (a) Uc' = 0.988 :[ 2 z ~ C\J 0.Q1 '-. ~ 1.5 ~~~---""- ~~ ;; 0.006 ----_ 1 ,/ ~---~------x / .. ,..".------.. _------:J / //' 0.003 ------Cl 0.5 /,/I ' ...... _------. /I"'- 0.001 ,/ Ow-~______~ ______~ ______~ ______~ o 50 100 150 200 Time (Ma) 1.25 (b) Uc' =0.9936 1 ~ 0.75 ..-C\J 0.5 ;'" ------~ , ..-b ,/ ,------" ~ 0.25 1/1,"""'- _--_ -...... ' ...... :J "".,.,,--- -- ...... ' ...... Cl 0 ~ ...... ,;;:....:::..~ -0.25 --~- ..... ----~ 0 50 100 150 200 Time (Ma) o 50 100 150 200 Time (Ma) Figure 6.5, The change in mean potential energy, ~ U{ It for the circular continental plate for U~ = 0.988, 0.9936, and 1.0. 172 If U~ is greater than about 0.995, the growth of the oceanic lithosphere causes a significant decline in U/ over tirrles of 200 Ma. An alternative way of evaluating the data plotted in Figure 6.5 is to plot .6. Vf It as a function of 10' for given ages. The dependence of .6. Vf It on w' for t = 5, 84, 163 and 200 Ma is shown as Figure 6.6 . The decline in UI after 200 Ma is greatest for small values of the normalized time constant (Figure 6.6el), and increases with U~. The slight minima at values of w' around 0.005 Ma-1 in Figure 6.6c and 6.6d suggests that for high spreading velocities, the rapid production of new lithosphere around the circumference of the plate partly counteracts the effect of aging of older oceanic lithosphere. Figure 6.5 illustrates that the change in UI is dependent on the value of V~assumed. For example, the decline in UI over 200 Ma is -1.7 x 1012 N m-1(about 14 MPa) for a value of U~ = 1.0, nearly double that predicted for U~ = .997 (Figure 6.4). When V~ is substantially less than unity (e.g., Figure 6.5a and solid curves in Figure 6.6), the initial growth of the ocean lithosphere may lead to a tran sient increase in Vf, and, therefore, to the possible development of compressional deviatoric stress regimes in ambient continental lithosphere. Figures 6.5 and 6.6 show that the magnitude of this efrect is strongly favored by low values of V~ and high values for the normalized time constant. For very low V~ (Figut'e 6.5), corresponding to the potential energy of old ocean lithosphere ( V~ = 0.988), plate growth leads to an overall increase in VI for all times, although distinct UI peaks accompany the formation of the young ocean lithosphere. For high normalized time constants (w' = 0.1 Ma-1) and low V~ (= 0.988) the calculated transient 12 12 increase in VI is up to 2.7 X 10 N m-1, with UI greater than 2.5 x 10 N m-1 during the interval 20 - 100 Ma and an increase in VI over 200 Ma of about 1.3 x 10 12 N m-1 (Figure 6.5a). The changes in potential energy accompanying the initial growth of ocean lithosphere for U~ < 1 may provide an explanation for basin inversion processes. However, note that the circular plate geometry used 173 in these calculations assumes uniform sea-floor spreading initiated synchronously around the entire margin of a continent, which somewhat limits the application of this simple model to realistic plate scenarios. 6.4 The Aging of the African and Antarctic Plates The preceding discussion implies that the growth and aging of ocean lithosphere in plates largely surrounded by spreading ridges, such as the African and Antarctic plates, may have important ramifications for the stress regimes in the continents. The magnitude and sign of the changes in ~ U{ (where ~ U{ is the difference, U/ - Uc ) is clearly a function of Uc , about which there is considerable uncertainty. Both the African and Antarctic plates are experiencing (or in the recent geological past have experienced) extensional failure which makes it plausible that the change in the mean potential energy of these plates as they have aged has produced the observed extension. In view of the obvious limitations imposed by the highly idealized circular-plate model described above, it seems sensible to pursue the implication of this formulation using more realistic time-evolutions for the growth of the ocean lithosphere in these plates. In this section the sea-floor age data of Royer et al. [1992J is used to assess the changes in the potential energy of the African and Antarctic plates over the last 163 Ma (that is, since the Late Jurassic). 6.4.1 The African Plate The time evolution of U/ for the African plate is estimated using the method discussed above for the circular plate model.' The mean elevation of continental Africa (including the continental margins) is 492 m. Thus, a continental density structure appropriate to a continental marginal geoid anomaly of + 6 m yields 174 1.5 (a) t =5 Ma 2.5 1.25 0.988 2 E ~ Z 1.5 C\J ...... -C\J ...... ------...... / .... ~ 0.75 0.9936 ...... ~ ..- ...... T- / 0.9936 ~ // ~ / 5 0.5 // ::l ,,-,,/ Cl / /--·--·-O~99;-·-·-·-·-·-·- // ~.~?J-.-.-.-'-'-'- 1 ./ // .. .".. .. ,...... ,,- '// """ .. ."..""," \-~ ---~------.:::: •...... :.1:!t ...... -0.5 \,, __ ------1.0 o 0.02 0.04 0.06 0.08 0.1 o 0.02 0.04 0.06 0.08 0.1 W' W' 1.5r:~=:=~==~ a 0.988 a 0.988 ~ C\J 0.9936 ~ ------~ O' _------~ 0 / ...... -- § I\~ ...... -- 0.9936 ..- .... / ~ I ~ -0.5 I ::l ~ -0.5 ~ 5 1\ ._.-.-.-.-.-.-.-.-. Cl \\ \\ ------1 I '-.-' 0.997 \\'-'-'-'-'-'0:997'-'-'-'-'-'-'- -1 \ \, \, \,. ____ ------J ...O------· -1.5 \ 1.0 -1.5L~_~_~ __~_-..l \~------o 0.02 0.04 0.06 0.08 0.1 o 0.02 0.04 0.06 0.08 0.1 W' W' Figure 6.6, Dependence of ~ Ur It on Wi for given times, (a) 5 Ma, (b) 8tl Ma, (c) 163 Ma, (d) 200 Ma, after the onset of sea-floor spreading for the circular plate model. Contours are for U~ = 0.988, 0.9936, 0.997, 1.0. 175 u~ = 0.997. There have undoubtedly been some changes in the elevation (and hence potential energy) of the continent since the Late Jurassic. However, it has been assumed in the calculations presented below that the potential-energy distribution in the continent has not varied significantly over the age of the plate. This assumption is justified for two reasons. First, changes in the elevation of the continents does not have a significant effect of the mean potential energy of the continental region. Second, changes in the mean potential energy of the continental regions would have the effect of increasing the potential-energy difference between the continental regions and the plate-scale mean, and thus serve to increase the predicted magnitude of the extensional stresses. Because the principal purpose of this study is to establish the first-order bounds on the applicability of this theory, it is desirable to make conservative estimates of the the extensional stresses related to the aging of the plate. The calculation of the changes in the mean potential energy is therefore based only the contribution of the aging oceanic lithosphere which is readily approximated on the basis of age. Changes in the extensional stresses due to changes in the elevation of the continental regions is a second-order efFect which is not taken into account in the present study. The calculations summarized here neglect the changes in plate configuration associated with convergence along the northern boundary with the Eurasian plate, which has undoubtedly resulted in the subduction of some oceanic lithosphere in this time interval. However, plate convergence rates constrained by palaoemagnetic data [e.g., Royel' et al., 1992J imply that the loss of ocean lithosphere through sub duction along the northern margin of the African plate has been small compared to the contribution of new oceanic lithosphere through spreading along the South ern Atlantic and Indian ocean ridge segments (the ratio of ocean loss to growth is estimated at about 1 : 6). The growth of the African Plate, based on the sea-floor age data of Royer et I 176 Figure 6.7, The area of the African plate over the last 163 Ma. Shaded areas correspond to the area added to the plate during the Pliocene (5.3 Ma), Miocene (23.7 Ma), Oligiocene (36.6 Ma), Eocene (57.8 Ma), Paleocene (66A Met), Late Cretaceous (84 Ma), Mid Cretaceous (119 Ma), Early Cretaceous (144 Ma), and Late Jurassic (163 Ma), respectively (see Table 6.3). .' 177 90 (a) ~80 ~ 70 ~ 60 whole plate ~ ~ 50 40 -50 0 Time (Ma) 100r.-~----~----~----.----. 80 60 40 young ocea.!J._-/ .--/' 20 ,----- _.. -" a ------_.-'.-.--/ old ocean _.-'-' ·150 ·100 ·50 o Time (Ma) ---- t\J ~ ...... - ...... E.-O.5 o:J ·1 -150 -100 ·50 o Time (Ma) Figure 6.8, (a) The cumulative area of the African plate over the past 163 Ma, and projected into the future llsing average Cenozoic growth rates. The primary data are listed in Table 6.3 and are from Royel' et al. [1992J. (b) The relative percentages of the principal lithospheric types for the African plate over the past 163 Ma. (c) The change in mean plate potential energy, ~ U{ for the African plate. As discussed in the text, Uc was assumed to be constant during the aging of the plate. Each figure is contoured for the range in U~ = 0.988, 0.9936, 0.997, 1.0. 178 at. [1992] is shown in Figure 6.7. The relative areas of the regions defined by the magnetic lineations are listed in Table 6.3. The cumulative area of the plate is listed in Table 6.3 and is plotted as Figure 6.8a. The plate has grown by L1.31 i 2 X 10 km over the last 163 Ma, increasing smoothly with the exception of an anomalously rapid spreading interval between the Oligiocene (36.6 Ma) and the Miocene (23.7 Ma), followed by an interval between the Miocene (23.7 Ma) and Pliocene (5.3 Ma) when growth slowed by a factor of four (of interest here, is that extension in the East African Rift system was most active in this interval - e.g., Bosworth el at., [1992]). Figure 6.8a suggests that the average growth rate of the African Plate has accelerated slightly throughout the last 163 Ma. Note that Figure 6.8 also shows the projected changes over the next 37 Ma assuming that the Cenozoic growth rate (averaged over the last 84 Ma) is maintained. The relative percentages of the principal lithospheric types in the African plate have varied significantly over the last 163 Ma, as shown in Figure 6.8b. The percentage of the continent has decreased systematically to about 46%. The proportion of young oceanic lithosphere increased rapidly between the Jurassic (163 Met) and the Miocene (23.7 Ma) when it stabilized at about LlO%. Since t.he Late Cretaceous (84 Ma) the percentage of old oceanic lithosphere has grown to about 1Ll%. The projected growth rates show the relative proportion of old ocean lithosphere will increase at the expense of both the continental lithosphere and the young ocean lithosphere over the next 37 Ma. As discussed above, temporal variations in the relative percentages of young and old oceanic lithosphere may have a profound effect on the mean potential energy of the plate. The variation in U/ with time for the African plate is shown as Figure 6.8c, for a range of U~. For U~ = 0.997, U/ has decreased from 2.385 to 2.379 14 1 12 x 10 N m- over the last 163 Ma, which corresponds to Ll U{ = - 0.6 X 10 N I m- , and to an increase in the magnitude of the mean extensional stress difference 179 (CT zz - CTxx ) within the continents of about 5 MPa (for a 125 km thick plate) over this time period. Assuming continuation of the average Cenozoic growth rates, the 12 projected decline in .6 U{ is a further - 0.25 x 10 N m -1 over the next 37 Ma. Figure 6.8 highlights the contrasting effects of plate growth and plate aging on the evolving potential-energy distribution within the African plate. During the rapid growth phase from the Oligiocene to Miocene (36.6 to 23.7 Ma) the excess potential energy due to production of young ocean lithosphere counteracted the effects of aging of the existing ocean lithosphere. Consequently, there was little net change in U/ during this period. In contrast, during the Miocene to Pliocene (23.7 to 5.3 Ma) interval plate growth slowed by a factor of four, and the evolving potential-energy distribution was dominated by plate aging, with a relatively minor contribution through production of new ocean lithosphere. The net effect was a 12 dramatic decline in U/ which, for U~ =0.997, amounted to .6 U/ = - 0.25 X 10 N m-1 over 18.'1 Ma. Figure 6.8c shows that a column of the African continental lithospheric with potential energy equal to the mean potential energy of the African continental lithosphere (that is a column supporting surface elevation of about 500 m) may be expected to witness extension in the ambient state provided U~ > 0.993. The pre dicted mean extensional stress differences are small for all reasonable values of U~, attaining CTzz - CTxx = 5 MPa and 8 MPa for U~ = 0.997 and 1.0, respectively. Such results only apply directly to those parts of the continent with Uc = Un and those parts of the continent wi th greater potential energy will wi tness correspondingly greater extensional-stress differences. A rough estimate of the magnitude of the variations of the potential energy within the African continent is provided by the estimate (admittedly, poorly constrained) that potential energy varies with topog raphy as about 1.3 x 10 12 N m-1/k111 [Fl'Oidevaux and Isacks, 198£1]. In the East African dome, ""here the average elevation approaches 2 k111, the excess potential 180 energy over the plate mean is therefore about 2.5 x 1012 N m-1, corresponding to a mean extensional stress difference of about 20 MPa. The rate growth aver aged over the 163 Ma gives a normalized time constant of about 10' = 0.003 Ma-1 1 (see Figure 6.3). For 10' = 0.003 Ma- and U~ = 0.997, the circular plate model 12 1 predicts ~ U{ = - 0.68 X 10 N m- over 163 Ma (Figure 6.4), which compares very favorably with the more detailed estimate of -0.6 x 10 12 N m-1 derived from analysis of the sea-floor spreading data presented above. 6.4.2 The Antarctic Plate The fact that the Antarctic plate is almost completely surrounded by mid-ocean ridges with only a very small length of convergent margin along the Antarctic Peninsula suggests that, of all the plates, it is most likely to approximate the ambient state. The growth history for the Antarctic plate over the last 163 Ma is shown in Figures 6.9 and 6.10a, and summarized in Table 6.'1. In compa.rison with the Africa.n plate, the somewhat smaller continental area of Antarctica, combined with a growth history which is approximately linear in time over the last 163 Ma (Figure 6.10a) has resulted in a much higher proportion of old ocean to continent in the present-day Antarctic plate. As a result the time-integrated, mean potential-energy changes are significantly larger than those calculated for the African plate. Figure 6.10c shows that for U~ = 0.997 the 12 1 change in mean plate potential energy ~ U{ is estimated at -0.911 x 10 N m- , corresponding to a increase in 0-:;:; - O-xx of about 7.5 MPa for a 125 km thick lithosphere. 181 N ...... :J \0 o o o o Figure 6.9, The area of the Antarctic plate over the last 163 Ma. Shaded areas correspond to the area added to the plate during the Pliocene (5.3 Ma), Miocene (23.7 Ma), Oligiocene (36.6 Ma), Eocene (57.8 Ma), Paleocene (66.4 Ma), Late Cretaceous (8 Ll Ma), Mid Cretaceous (119 Ma), Early Cretaceous (144 Ma), and Late Jurassic (163 Ma), respectively (see Table 6.4). 182 60 ;,!! o .--'..... 40 ...... - ..... ,,/' young ocean r 20 / .~ o '--/ .--... -----old oceaft... ' -150 -100 -50 o Time (Ma) C\I ~ -0.5 -'--"-i E -1 -1.5 -150 -100 -50 o Time (Ma) Figure 6.10, (a) The cumulative area of the Antarctic plate over the past 163 Ma, and projected into the future using average Cenozoic growth rates. The primary data are listed in Table 6.3 and are from Royer et al. (1992). (b) The relative percentages of the principal lithospheric types for the Antarctic plate over the past 163 Ma. The primary data are listed in Table 6.4 and are from Royer et al. (1992). (c) The change in mean plate potential energy, .6. VIc , for the Antarctic plate. As discussed in the text, Vc was assumed to be constant during the aging of the plate. Each figure is contoured for the range in U~ = 0.988, 0.9936, 0.997, 1.0. 183 6.5 Discussion The recognition that the growth and aging of individual plates through sea floor spreading may significantly alter the mean plate potential energy should provide important insights into the evolution of intraplate stress fields. Assuming a continental potential-energy distribution consistent with observed geoid anoma lies across continental margins, it has been shown that the likely net changes in U/ range up to about. -1.0 x lO12 N m- 1 over 200 Ma, and could be significantly greater if the mean potential energy of the continental lithosphere has been under estimated. In the absence of tractions imposed at the plate boundaries or along the base of the lithosphere (i.e, in the ambient state) such changes may be ex pected to create a substantial amount of deviatoric tension within the continental regions of aging plates. While the magnitude of the stresses will depend on the potential energy of the local lithospheric column, in the ambient state, the mean stress difference (O"zz - O"X3') should depend only on .6. U{ and is estimated to be up to 8 MPa averaged over the 125 km thickness of the lithospheric column. These predictions, which are based on a very simple circular plate model, are in close agreement with estimates based on the detailed assessment of the sea-floor growth history in the Antarctic and African plates. In order to understand the role of this process in the mechanics of an aging plate, it is necessary to assess the strength of the continental lithosphere. Relatively little is known about the absolute strength of the lithosphere and, in particular, how such strength is distributed with depth [e.g., Zoback et ai., 1993]. Laboratory based strength estimates predict very heterogeneous strength distributions, with discrete strength maxima associated with compositional and rheological stratifica tion within the lithosphere [Brace and [(hoisted!, 1980]. The strength of such a lithosphere is clearly dependent on both the lithologi cal constitution and the thermal regime, both of which are known to vary widely. 184 On the basis of such laboratory estimates a number of workers [e.g., Houseman and England, 1986; Kus/miT and PaTk, 1987J have suggested that thc extcnsional strength of the lit.hosphere at the limit of geologically significant strain ratcs is of 12 1 the order or 3 - 4 X 10 N m- • More reccntly, KushniT [1992J argucd that exten sional strength may be considerably rcduced, possibly to as little as 2 x 10 12 N m-1, due to the visco-elastic amplification of strcsses in the lithosphcre. Estimates of the upper-crustal strcngth inferred from strcss measurements in the KTB (Contincntal Deepborc Drilling Program) wellhole in Gcrmany show that the cumulativc force necded to deform crustal material is in the range of 2 - 5 X 1012 N m-1(Zoback et al., 1993). While such cstimates are highly unccrtain, they arc of inLcrest here in as much as the predicted changes in ~ U{ represent a significant fraction (up to 25 - 50%) of these strength estimates. The implication is that, providing thc continental stress field fecls the changes in .6. U{ that accompany plate aging (as it should in the ambicnt statc), the cvolving dcnsity structurc of an aging platc may creatc a favorable environment [01' extensional failure and fragmentation. In the modern Earth, the only plates largcly surrounded by mid-ocean ridgc systems, and therefore likely to approximate thc ambient strcss state, arc t.hc African and Antarctic plates, and both show evidence for active, 01' recently active, continental extension. \"'hile other processes sut.:h as thc active involvement of mantle plumes may have contributed to the location and initiation of such extcnsion, it is possible that the conditions for such extension havc been significantly aided by the aging of these plates, as their peripheral ocean ridges spread outwards, and UI declines. An important question concerning the interpretation of thc ambicnt stress state is whether the continental lithosphere will see the cffects of the plate-scale potential-energy distribution as proposed here. A number of previous workers have touched briefly on the question of the potential-encrgy variation across litho spheric plates [e.g., Fleitout and Fl'oideva1tx, 1983; Houseman and England, 1986J. 185 Houseman and England [1986J acknowledged the "potential energy contrast be tween continental lithosphere and old ocean basins" but argued "that in general this does not result in (continental) extension, presumably because the oceanic lithosphere acts rigidly to transmit the stresses arising from the midocean ridges". This argument assumes implicitly that the ocean ridge system defines a tectonic reference slale and in the ambient state must therefore be close to potential-energy balance with the mean plate potential energy (see Chapter 2 for a discussion of the concept of a tectonic reference state). This notion is founded on the belief that the micl-ocean ridges are sufficiently weak that they could not sustain su bstan tial extensional stresses (but see Fleiloul and Fl'oidevaux [1983J for an altel'llative view). Because the mid-ocean ridges must be very weak, the acti ve extension which is manifestly occurring at the ridges requires not only an excess potential energy (i.e., a driving force) but also the kinematic requirement that accommo dates the extension (i .e., a displacement). The extension along the ridges bounding most plates, and particularly the African and Antarctic plates, is not currently be ing accommodated by the internal compressional deformation of these plates but rather by global plate-kinematic constraints. This implies that the parts of the plate with potential energy less than the plate mean (such as the old oceanic litho sphere) are sufficiently strong to withstand the compressional forces imposed by the ridges and any other parts of the plates with excess potential energy. The lack of accommodating displacements internally within these plates allows (but does not require) that ridges are able to withstand significant excess potential energy independently of their inherent strength. One test of the notions developed in this chapter concerns the nature of t.he stress fields in the ridge systems and ocean basins surrounding supposedly near ambient plates. Figure 6.11 shows how the age of the ocean lithosphere with the potential energy corresponding to mean plate potential energy (i.e., U/ = U/) 186 evolves with time for the African plate. In the ambient state, ocean lithosphere of this critical age will experience no deviatoric stress while younger lithosphere should experience deviatoric tension and older lithosphere should experience de viatoric compression. For U~ = 0.977, this study predicts that this cd tical age increases from about 20 Ma to 38 Ma over the 163-Ma history since the late Juras sic. One implication of this is that, in the modern African plate, the compression witnessed by old ocean lithosphere (l > 84 Ma) is only about half the ridge-push force which is traditionally formulated in terms of the difference between the po tential energy of the mid-ocean ridges and old ocean lithosphere. This critical age range is consistent with the global oceanic seismic dataset which suggests the transition from tension to compression in ocean lithosphere typically occurs in the lithospheric age range 20 - 40 Ma [e.g., Stein and Pelayo, 1991J. However, since the global dataset is strongly biased by events 111 the Inclo Australian plate, where the intraplate stress field is far from ambient, a more detailed analysis from the African and Antarctic ocean basins is required to test the ideas presented here. This study predicts low stress magnitudes in the oceanic basins which is in general agreement with the low levels of recorded seismicity within the ocean lithosphere of the African and, particularly, Antarctic plates. Unfortunately, the stress magnitudes predicted for the ocean basins in these plates is too low to be evaluated with passive testing techniques such as earthquake focal mechanisms. I IS7 80 ~-- ...... _----....,...... 0.988 .... ----...... ,.,.,------' ...... 60 -_ .. 0.9936 /. --- ~-~-~-~-----~--.~-~-~-~~ // 20 ...-- ...... 1.0 ...-""""- o .--__e------150 -- -100 -50 o Time [Ma] Figure 6.11, Age of ocean lithosphere with potential energy corresponding to the mean plate potential energy (i.e., U/ = U/) plotted as a function of age for the African plate using the plate growth data summarized in Figure 6.S. Contours show the range in U~ = 0.988, 0.9936, 0.997, 1.0. 188 Parameter Value 6 2 /), 0.87 X 10 m s 1 5 1 0' 3 X 10- °K- pm 3238 kg m-3 pm 1030 kg m-3 Tm 1280°C tc 84 Ma UMOn 2.391 X 101<1 N m-1 9 9.82 m s-2 G 6.673 10-11 m 3 kg-l s-2 The thermal parameter values used here are consistent with geoid anomalies giving d (6.No) /dt = -0.15 m/Ma in the aging ocean lithosphere (see text; for discus sion). Table 6.1, Values of parameters used in calculations Parameter Description Mean PE of the Lithosphere Mean PE of the Continents The difference U/ - Uc Mean PE of the Plate Relative to the Mid-Ocean Ridge Mean PE of the Continental Lithosphere Relative to U/lwn Change in Mean PE as a Function of Time Mean PE of the Ocean Lithosphere Younger than tc Spreading Rate Vs]!/1'c , Normalized Time Constant (Large 'W' = Fast Spreading Rate) Table 6.2, Definitions of Abbreviations Used in Calculations 189 Age Ma Incremental Area Cumulative Area 6 2 7 2 [x 10 km ] [x 10 km ] Present 7.78 Pliocene 5.3 3.72 7A1 Miocene 23.7 3.22 7.09 Oligiocene 36.6 9.65 6.12 Eocene 57.8 6.92 5A3 Paleocene 66,4 3.87 5.04 Late Cretaceous 8L1.0 5.78 4A7 Mid Cretaceous 119.0 4.50 L1.02 Early Cretaceous HL1.0 L1.00 3.62 Late Jurassic 163.0 1.52 3,47 Table 6.3, Area of the African plate for the last 163 Ma Age Ma Incremental Area Cumulative Area 6 2 7 2 [x 10 km ] [x 10 km ] Present 6.26 Pliocene 5.3 0.99 6.16 Miocene 23.7 5.30 5.63 Oligiocene 36.6 L1, 75 5.15 Eocene 57.8 6.02 4.55 Paleocene 66,4 4A9 4.10 Late Cretaceous 84.0 L1.3Ll 3.67 Mid Cretaceous 119.0 5.27 3.14 Early Cretaceous 144.0 5AO 2.60 Late Jurassic 163.0 6.58 1.95 Table 6A, Area of the Antarctic Plate for the last 163 Ma 190 CHAPTER 7 CONCLUSIONS This study has explored the hypothesis that much of the global intraplate-stress field and associated lithospheric deformation, especially on the continents, is the product of lateral density variations in the lithosphere (Chapter 2). Constraint for numerical modeling of the intraplate-stress field was presented in Chapters 3 and L1. It was demonstrated (Chapter 5) that many of the features of the intraplate stress field can be explained in terms of variations about the plate- and global scale mean potential energy. It also was shown that the mean potential energy of the plates changes as the plates age (Chapter 6) and that this change can induce significant extensional stresses in the continental regions of the plates. The principal conclusions of the individual chapters are discussed below. Chapter 2: Theoretical Basis Chapter 2 focussed on evaluating the mean potential energy of the global litho sphere, Uf, of indi vidual plates, Ur, and the intraplate variation about this mean using a simple, first-order lithospheric density model. Constraint for this model was provided by the observed geoid anomaly across continental margins. In formulating the potential energy in terms of topography it was assumed that the continental geotherm is lineal', and density variations below a depth of 125 km have negligible influence on the potential energy of the lithosphere. The estimated global mean 14 1 potential energy (UP = 2.378 x 10 N m- ) is equivalent to the potential energy of both ncar sea level continental lithosphere (-160 to +220 m for pc in the range 2800 - 2700 kg m-3 ) and cooling oceanic lithosphere at a depth of 11.3 km. With the exception of Eurasia, which has anomalously high mean potential energy (Ur = 2.383 x 1014 N m-1), the mean potential energies of the continental plates are nearly identical to the global mean Uf. The mean potential of the oceanic plates 191 was found to be a strong function of the mean age of' the oceanic lithosphere. The 14 Pacific plate has a mean potential energy (U/, = 2.371 x 10 N m-1) significantly lower than Uf, and the Nazca plate, dominated by cooling oceanic lithosphere, has 14 I U/' = 2.382 x 10 N m- , significantly greater than Uf. Although the potential energy of the continental lithosphere with elevated topography was found to be very sensitive to the assumed crustal density, pc, both the global and plate-mean potential energies are relatively insensitive to a wide range in pc. This robustness suggests that the tectonic reference state (TRS) in the continental lithosphere is best defined by a lithospheric column with potential energy corresponding to Uf, that is with elevation close to sea level. III the absence of external forces applied at the base or along plate bounclaries a lithospheric column with the potential energy of the TllS woulcl remain undeformed. Thus, the difference between the potential energy of a lithospheric column and the TllS determines whether the column is in an extensional, neutral, or compressional state of stress. Elevated 3 continental lithosphere with a height of about 70m (for pc = 2750 kg 111- ) has an equivalent potential energy to Uf', suggesting that the in the absence of external forces continental regions will be in a slightly extensional state of stress. Chapter 3: Trends in the Intraplate-Stress Field A statistical analysis of the maximum horizontal compressive stress (Sll,max) orientations in the World Stress Map database was performed to quantify the existence of long-wavelength trends in the data. The existence of trends in the Sll,max orientations within bins having a dimension of several degrees was based on the Rayleigh test, a standard statistical method in the analysis of directional data to test the null hypothesis that the orientations are random. The analysis was performed using the 4L196 indicators in the World Stress Map Dataset which are of sufficient quality (A-C, after Zoback [1992]) to provide reliable information about 192 the tectonic stress field. The analysis demonstrates that long-wavelength trends exist in the data at both the 90 and 95 % confidence levels, and are particularly robust in eastern North America, western South America, and western Europe. Other, less robust, trends exist in central Asia and central Australia. The results of the present study are in substantial agreement with the observation made by previous investigators that large regions of the intraplate-stress field have consistent SH,ma:z; orientations [Zoback and Magee, 1991; Zoback, 1992J. The existence of large areas of strike-slip stress regimes imply that the Earth's tectonic plates are in a near-neutral state of stress. This is particularly important in the African plate which closely approximates the ambient stress state. Chapter 4: Constraints on the South American Intraplate-Stress Magnitude The Cordillera Blanca of the Andes provides a unique tectonic setting to place constraints on the intraplate-stress magnitude in the South American plate. An elastic finite-element analysis was used to evaluate the lithospheric state of stress. The vertical stresses due to the buoyancy force of the high topography were cal culated using a crustal structure which assumed the topography to be entirely compensated by the negative buoyancy of the deflected moho. The modeling was constrained by both extensional stress indicators for high topography and com pressional deformation at lower elevations along the profile. The modeling results estimate the magnitude of horizontal intraplate stresses in the South American plate to be about 35 MPa averaged over a 100-km-thick lithosphere. Chapter 5: Numerical Modeling of the Intraplate-Stress Fields 193 Finite-element analysis of the intraplate-stress fields in the African, South Amer ican and Indo-Australian plates was used to illustrate a number of important rela tionships between the forces due to potential-energy distributions and the observed intraplate-stress field. The African plate provided a setting to evaluate the predicted stresses in a continental plate which closely approximates the ambient stress state. Predicted stresses in the oceanic regions range from deviatoric tension along the mid-ocean ridges to compressional in the ocean basins with a maximum of about 12 MPa (averaged over the lithospheric thickness of 125 km). Continental regions near sea level were found to be near a neutral state of stress, with large extensional stresses present in the Ethiopian highlands (15 MPa), the East Africa rift (9 MPa), and in southern Africa (8 MPa). The genera.! agreement between the predicted and observed stress fields suggests that the principal long-wavelength features of the intraplate-stress field can be explained in terms of stresses arising from potential energy variations within the lithosphere without appeal to poorly constrained sttb lithospheric processes. The intraplate-stress field in the South American plate was found to be readily modeled in terms of simple collisional forces acting across the western margin of the plate in conjunction with the forces due to lateral density variations in the lithosphere. Importantly, complex tectonic models were not required to explain the principal features of the intraplate-stress field. The South American stress field is non-ambient in that predicted stresses are compressional throughout the near-sea level continental regions of the plate. Despite its complex natlll'e, many featlll'es 111 the stress field in the Indo Australia plate were also explained in terms of the interaction between the bound ary forces and the forces due to lateral density variations. Namely, it was demon strated that if substantial focusing of the ridge-push torque occurs along the col- 194 lisional boundaries (i.e., Himalayan and Papua New Guinea), many of the bl'Oad scale features of the observed stress field can be reproduced without appealing to either subduction or basal drag forces. Lateral density variations within the con tinental areas of Australia and India were found to have a significant effect on the regional stress field in these areas. Predicted stresses for most of the plate were found to be robust for a large range in the magnitude of applied boundary forces. In contrast to the previous modeling work of Cloelingh and Worfel [1985;1986]' large stresses of the order of hundreds of MPa (kbars) are not required to match the observed stress field. MallY of the models predict low tectonic stress magnitudes for most of the plate, on the order of 30 MPa (300 bars), averaged over a 100km thick lithosphere, except in the Central Indian Ocean region were they approach 100 MPa (1 kbar). The stress magnitudes predicted in the present study for the Indo-Australian plate are therefore consistent with the magnitudes predicted for most of the other plates including North and South America, Europe, Nazca and the Pacific. Forces due to lateral density variations within the lithosphere were found to have a significant efFect on the intraplate-stress field in the continental regions of India and Australia. The topographic stresses reduce the magnitude of the compressive stresses of the regional stress field, and in some areas, induce a significant rotation in the Sll,7nax orientations. These results support the suggestion that these forces are an important source of stress even for plates dominated by boundary forces [Fleito1lt and Froideva1l3.:, 1982, 1983; Fleito'ltt 1991]. Because the stress magnitude in the continental regions of Australia were predicted to be quite small, 20-30 MPa (200-300 bars) local sources of stress can be expected to have a significant influ ence on the observed stress orientations. This may help explain the considerable variation in the azimuth of P-axes throughout continental Australia, as well as the large number of observed strike-slip events. Furthermore, the stresses predicted by 195 this model suggest that much of the compressional stress within the lAP is the re sult of focusing the potential-energy torque (mostly arising form ridge push) along collisional-boundary segments. Perhaps more importantly, the combined effect of the stress focusing along the collisional bounda.ry segments can produce a. near isotropic state of stress in continental Australia. It is difficult to make an accurate estimate of the SIJ,max orientation if the horizontal principal stresses are isotropic. It is possible that the large amount of scatter in the observed SIJ,max orientations in continental Australia is the consequence of this near-isotropic stress state. Chapter 6: Thne-Evolution of Plate-Scale Potential-Energy Distributions The recognition that the growth and aging of individual plates through sea-floor spreading may significantly alter the mean plate potential energy has provided important insights into the evolution of intraplate-stress fields. It was shown that the likely net changes in U/ range up to about -1.0 x 10 12 N m- 1 over 200 Ma. In the ambient state such changes may be expected to create significant deviatoric tension within continents in aging plates. While the magnitude of the stresses will depend on the potential energy of the local lithospheric column, in the ambient state increments in the mean stress difference ((J" zz - (J" xx) depend only on Ll U{ and are estimated to be as great as 8 MPa averaged over the thickness of the lithospheric column. These predictions, which are based on a very simple circular plate model, are in close agreement with estimates based on the detailed assessment of the sea floor growth history in the Antarctic and African plates. While the predicted stress changes associated with aging of the oceanic lithosphere are smaller (by at least a factor of two) than the required cumulative force required to deform continental lithosphere, they may provide an important contribution to the stress fields that eventually lead to the fragmentation of aging plates. 196 APPENDIX A THE CLOSED-FORM SOLUTIONS FOR THE POTENTIAL ENERGY OF THE LITHOSPHERE This Appendix presents t.he closed-form solutions for the potential energy for each of the lithospheric types, subject to the constraints outlined in the text. For young oceanic lithosphere, the lithospheric potential energy is given by: U, 3+2w (1 ~) (1 )./ (1-8) (1+2w) 1/J2 --2 = -6- + - + u + w~) + 2 + [I Pm ::, (3+2w-31/J+381/J-3w1/J+38w1jJ) 1/J!+ 3 2 (3 + 2w) 1/J1 - (w 1/J3) + 8Wlf,3 6 + 3 (1 +1jJt) For old oceanic lithosphere, the lithospheric potential energy is given by: 3+2w (1 ~)(l ),/ (1-8)(1+2w)1jJ2 --'J = -6-+- +u +w ~)+ 2 + [/ Pm zr (3+2w-31/J+381/J-3w1/J+38w1/J-382) 1/Jl 3 + 2 (3 + 2w - 382 ) 1Pl + - (w1jJ3) + 8w1jJ3 6 3 (1 +r/Jd For submerged continental marginal lithosphere, the lithospheric potential energy is given by: U, --q [/ Pm::, For exposed continental lithosphere, the lithospheric potential energy is given by: U, 1 (3+2w-31jJ+381jJ-3w1/J+38w1jJ-382) 1/'1 --'J !l Pm zr = 6'+ 3 + 2 (3+2w-382 )1/J1 -(w1/J3)+8w1/J3 6 + 3 (1 + 1/Jd where the denoted parameters are defined as: Zc W = aT, 1/) = - Ziso ZI pc 1/J3=- 8=- Ziso Pm 197 61 = .E!.. 62 = Pw Pm Pm In these expressions II is the surface topography (above sea level), Zw is the bathymetry, Zi30 is the depth of iostatic compensation (beneath sea level), Zc is the crustal thickness (see below), ZI is the lithospheric thickness, and Pc, Pm, and Pw are the crustal, mantle, and seawater densities, respectively Topography was used as the basic constraint and the crustal thickness for continents und old ocean basin was calculated assuming the base of the lithosphere base is at depth Zi30' For exposed continental lithosphere the crustal thickness is given by: J Ze ZJ 0 - ---== Q vc:;:; Tl [ zJ2t;.pJ + 2{3zJ (lIoPb -I'Pe -I'OPm -lIw Pm -lIwPw zlot;.PJ) - {32 pezJ ] The crustal thickness for submerged continental marginal lithosphere (continental margin) is calculated as: tem ZJ 0 - --==- Q vc:;:; Tl [/ZJ2 t;.PJ + 2{3zJ ("OPb -IlPe - 110 Pm - "wPm -I,wPw + I,pw zlOLlPJ) - {32 peZJ ] In old oceanic lithosphere, crustal thickness for anomalous topography is calculated as: J zob = ZJ 0 - --==-- u ,;-;:;; Tl [ zJ 2 t;. P2 + 2/3 zJ (110 Pb - II Pb - 110 Pm - I,w Pm - IIw PILI + I, Pw zlO t;. P2) - {32 Pb ZJ] In young oceanic lithosphere the thickness for the cooling oceanic lithosphere is calculated, assuming constant ocean crustal thickness, as follows: 110 Z pIU _ Z .6. P2 2f3 hoPm Zip =-- ( 1) _ 1 + -'---0.'--"'- f3 Pm Pm Z .6. P2 In these expressions: .6. P1 = (Pm - Pc) .6. P2 = (Pm .- Pb) f3=o:1} zl=(h+ z1o) O=(I+~) 198 APPENDIX B THE DEPENDENCE OF THE POTENTIAL ENERGY CALCULATION ON THE MECHANISM OF ISOSTATIC SUPPORT In this study, the assumption was made that regions of elevated topography in the continents are supported mainly by thickened crust calculated on the assump tion that the base of the continental lithosphere is everywhere at a constant depth corresponding to Ziso = 125 km. In many regions of the continents elevated to pography is likely to be supported by thinned mantle lithosphere, with or without thickened crust. In order to evaluate the errors in the potential energy calculations introduced by this approximation the approach of Sandiford and Powell [1990J is adpotecl. In this case, the difference in potential energy of two lithor-pheric column in isostatic equilibrium (subject to the constraints adopted in this paper, i.e., no internal heat production) can be approximated by: 2 fjJJ = 6(1- 6) (.r. 2 _ 1) _ edt (.e _ 1 _ 3 (1 - 6) U~.Ii -1)) _ a T? (.e - 1) 9 Pm z; 2 e 6 '1/)2 / 8 '1/)2 / where fe and f/ are the ratios of the thickness of the crust and whole lithosphere for the two columns (taken relative to the reference column against which the potential energy difference is to be measured), 'IP is the ratio of crustal to whole lithospheric thickness in the reference column, and Ze is the thickness of the crust in the reference column (in this formulation the approximation that ape = apm)) was used. In order to evaluate the effect of the isostatic support mechanism, Figure A1 shows the potential-energy difference, b.Uh, between a lithospheric column with thinned lithosphere and a reference lithosphere with a base at Ziso = 125 km, 199 4 ,...-, 3 S 2 Z ~ 1 ~ ~ 0 0 ~ :>< -1 ~ -2 -4 70 80 90 100 110 120 130 Lithospheric Thickness [km] Figure B.l, ~Uh = Ulz/=z - Ulz/=125km for three values of surface elevation, h= 1,2 and 3 km. with the results for elevations in the range h = 1000 - 3000 m shown in Figure AI. For this topographic range the errors in the calculated potentia.! energy introduced by assuming isostatic support by thickened crust are no greater than about 3 x 1 1011 N m- , thus implying that, in this range of topography, the potential energy of the lithospheric columns is rather insensitive to nature of the isostatic support mechanism. 200 REFERENCES Abel's, G. and R. 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