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Topics in extended dynamical systems

Bhagavatula, Ravi S., Ph.D.

The Ohio State University, 1994

UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 T o pics in E x t e n d e d D y n a m ic a l S y st e m s

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Ravi S. Bhagavatula, M. S

The Ohio State University

1994

Dissertation Committee: Approved by

Prof. C. Jayaprakash

Prof. C. A. Ebner Adviser Prof. C. D. Andereck Department of Physics 0 Copyright by

Ravi S. Bhagavatula

1994 to my Parents

ii A cknowledgements

First, and foremost,, I sincerely thank my adviser Prof. C. Ja.yapra.kash for providing me excellent, opportunities to work on a wide variety of interesting topics. I am grateful for his patience, suggestions, openness to my ideas, and the efforts he made to teach me physics.

1 thank Prof. C. Ebner and Prof. ('. 1). Andereck for serving on my final exam committee. I also wish to express my gratitude to my general exam com m ittee members, Prof. T. L. Ho, Prof. R. J. Perry and Prof. C. D. Andereck.

I would also like to thank Prof. Ebner for his collaboration on the dynamics of Joscphson junction arrays (section 1.5 of chapter 1) and for freely sharing his wisdom with me on t he evolutionary aspects of physical sciences. Sincere thanks go to collaborators Dr. G. Grinstein and Dr. Y. He on the project, “Algebraic

Correlations in Conserving Chaotic Systems” (chapter II) and Dr. K. Chen for the continuing collaboration on earthquake models (chapter III). I also thank

Dr. Chen and Dr. H-J.Xu for the collaboration on the Green’s function method for random fuse networks, which has not explicitly entered into this thesis.

I would like to thank Prof. F. Hayot for his valuable help and support,; I always admired his interests in Sanskrit and Hinduism and enjoyed discussions with him.

I wish to thank my friend, the late Dr. Jayesh, for helpful conversations on the experimental aspects of grid turbulence. Special thanks to Dr. S. Hebboul for many interesting conversations on experimental features of Josephson junction arrays.

1 am indebted to a number of colleagues and oflicemates for their interest in my work and their helpful comments. I thank Lizzette, Luke, Sven and Eric for proof reading parts of this thesis. 1 would like to specially thank Lizzette for inspiring me in numerous ways and sharing with me the culture that I never knew existed before coming to the other side of the globe. 1 will remain grateful to her for the help, kindness and support during many difficult times. I also thank Shastry and

Wen bin for helping me with the iATjrX style file for this thesis.

Finally, I am indebted to my parents, Ananda Murthy and Rama Lakshmi, for their support and for letting me freely choose some unconventional paths in my career. I also thank my sister Renu for giving company to my parents in my absence during the past few years.

iv V it a

August 29, 1967 ...... Born - Eluru, India.

1989 ...... Master of Science in Physics, (5-year Integrated) Indian Institute of Technology. Kanpur, India, 1989-1990 ...... Graduate Teaching Associate, Department of Physics, The Ohio State University. 1990-Presen t ...... Graduate Research Associate, Department of Physics, The Ohio State University.

v Publications

R. Bhagavatula, I\. Chen, C. Jayaprakash, and H. Xu, “Green’s Function Mel,hod for Random Fuse Networks”, Phys. Rev. E. 49, 5001 (1994).

R. Bhagavatula, K. Chen, and C. Jayaprakash, “A Self-Consistent Description of Ruptures in an Elastic Medium : An Application to Earthquakes”, ./. Phys. A, 27, L I55 (1994).

R. Bhagavatula and C. Jayaprakash, “Non-Gaussian Distributions in Extended Dynamical Systems”, Phys. Rev. Lett., 71, 3657 (1993).

R. Bhagavatula, G. Grinstein, Y. He, and C. Jayaprakash, “Algebraic Correlations in Conserving Chaotic Systems”, Phys. Rev. Lett., 69, 3183 (1992).

R. Bhagavatula, C. Ebner, and C. Jayaprakash, “Dynamics of Capacitive Joseph- son Junction Arrays Subjected to Electro-Magnetic Radiation”, Phys. Rev., B45. 4774 (1992).

R. S. Bhagavatula, K. Shahi, and S. Gupta, “A Low Cost, Differential Thermal Analyzer for Research and Teaching”, 7. of lustrum. Soc. India, 20, 177 (1990).

Fields of Study

Major Field: Physics

Studies in: Condensed Matter Theory Statistical Physics T a b l e o f C o n t e n t s

DEDICATION ...... ii

ACKNOWLEDGEMENTS ...... iii

VITA ...... v

LIST OF FIGURES ...... x

CHAPTER PAGE

Non-Gaussian Distributions in Extended Dynamical Systems 1

1.1 Introduction ...... 1 1.2 Experimental O verview ...... 1 1.2.1 Temperature fluctuations ...... 5 1.2.2 Concentration Fluctuations ...... 15 1.2.3 Fluctuations in Velocity and its G radients ...... 1 (> 1.2.1 Fluctuations in Other Quantities ...... Hi 1.2.5 Experimental Issues for Temperature PD F s ...... 17 1.3 Theoretical Developments ...... 19 1.3.1 Moment Balance A pproach ...... 20 1.3.2 Phenomenological Approach ...... 23 1.4 Our Approach ...... 27 1.4.1 Passive Scalar E q u a tio n ...... 29 1.4.2 Models for the Passive S c a la r ...... 31 1.4.3 Probability Distributions of S c a la r ...... 34 1.4.4 Toy M o d els ...... 49 1.4.5 Multi-Variable Fokker-Planck equation ...... 58 1.4.6 Role of Boundary C o n d itio n s ...... 67 1.5 Another ExtendedSystem: Josephson Junction A rra y ...... 71 1.5.1 M o d e l ...... 72 1.5.2 Chaotic S t a t e s ...... 74 1.5.3 Local P ro p e rtie s ...... 75 1.5.4 Experimental Issues ...... 82 1.6 Summary and Conclusions ...... 84

List, of References ...... 88

II Generic Scale Invariance in Conserving Chaotic Systems . . . 91

2.1 Introduction ...... 91 2.2 Generic Scale Invariance and Chaotic Systems ...... 96 2.2.1 An Exactly Solvable Linear M o d e l ...... v ...... 96 2.2.2 Inclusion of Nonlinearities ...... 101 2.2.3 Induced Scale Invariance ...... 106 2.2.4 Lattice M o d e ls ...... 108 2.2.5 Arguments for Chaotic S ta te s ...... 116 2.3 Two-Dimensional M o d e l ...... 118 2.3.1 Phases of the M odel ...... 119 2.3.2 Nature of Chaotic S tates ...... 123 2.3.3 Spatio- temporal C orrelations ...... 125 2.3.4 Domain Dynamics in AE I S ta te s ...... 138 2.3.5 R em arks...... Ill 2.4 One-Dimensional M odel ...... 143 2.5 Summary and Conclusions ...... 116

List of References ...... 148

III Scaling of Earthquakes in Quasi-Static Seismic Zone Models 150

3.1 Introduction ...... 150 3.2 Earthquake Phenom enology ...... 154 3.2.1 Spatial Features ...... 154 3.2.2 Characaterization of Earthquakes ...... 156 3.2.3 Statistical Features...... 157 3.2.4 Some Numbers for San Andreas Fault Z o n e ...... 159 3.2.5 Issues in Earthquake M o d e lin g ...... 159 3.3 Representation of Ruptures in Seismic Z o n e ...... 163

viii 3.3.1 Equilibrium Linear Elasticity ...... 164 3.3.2 Dipole R epresentation ...... 165 3.3.3 Double-Couple Representation ...... 175 3.3.4 Self-Consistent Method ...... 185 3.4 Seismic Zone Model ...... 189 3.4.1 Independent-Rupture Description ...... 191 3.4.2 Self-Consistent Description ...... 193 3.4.3 R e s u lt s ...... 193 3.4.4 Discussion on the fi V a lu e ...... 203 3.5 Model with a Pre-Existing F a u l t ...... 206 3.5.1 Modeling Aspects ...... 206 3.5.2 Features of the M odel ...... 211 3.6 Summary and Conclusions ...... 219

List of References ...... 221

APPENDICES

A Appendix for Chapter I ...... 223

A.l Example Probability Distributions ...... 223 A.2 Derivation of Fokker-Planck Equations ...... 221 A.3 Some Details of the Toy m odel ...... 226 A. l PDFs in Noisy Linear Diffusive Process ...... 229

B Appendix for Chapter I I ...... 231

B.l Single Variable M a p ...... 231

C Appendix for Chapter III ...... 233

C.l Evaluation of Green’s functions ...... 233

Bibliography ...... 235

ix L is t o f F ig u r e s

FIGURE PAGE

1.1 Rayleigh-Benard Convection Experiment: Top figure shows the schematic experimental set up with an externally applied tempera­ ture difference A = 0\ — O2 . Bottom figure shows the experimental temperature PDFs measured in the middle of the cell at different Ra values. This figure is obtained from the Figure 1‘2 of Ref. [ 1]. The logarithm of the PDF is plotted vs. the normalized scalar deviation X ...... 7

1.2 Grid Turbulence Experiment: lo p figure shows the schematic ex­ perimental set up with a temperature gradient applied along z di­ rection. Bottom figure shows the experimental temperature PDFs at different x. 'Phis figure is obtained from the Figure 7a of Ref. [ 6], The logarithm of the PDF is plotted vs. the normalized scalar de­ viation X ...... 10

1.3 Stirred Fluid Experiment: Top figure shows the schematic exper­ imental set up. A temperature difference is applied along x di­ rection. Bottom figure shows the experimental temperature PDFs obtained from the Figure 7b of Ref. [ 8 ]. The logarithm of the PDF is plotted vs. the normalized scalar deviation X. The straight lines correspond to the strict exponential PDF ...... 11

1.4 PDF for the normalized fluctuation A' = 80/ag for the stochastic model: with L = 48, k = 0.1, f3 = 0.1, r„ « 10 and 13 ss 0.35. The white noise terms are uniformly distributed between -1 and 1. The variances are ag « 0.034,v0 ~ 0.21 and the kurtosis is 4.22 ...... 37

x 1.5 Data for deterministic Model: (a) PDF for v normalized by v0 has Gaussian tails, (b) PDFs for the normalized fluctuation A' are shown for two parameters. The dashed line is drawn with the same slope as that of the tail, (i) Upper curve (shifted up by two decades) is for L = 48, k = 0.1,7 = 0.5, /? = 0.1. The variances are: ag fa 0.045, Vo fa 0.2, the kurtosis is 4.62 and B « 0.45. (ii) Lower curve is for L = 48, k = 0.05,7 = 0.25,/? = 0.1. The variances arc ag ~ 0.031, Vo ^ 0.1, the kurtosis is 4.12 and B fa 0.3...... 10

1.6 Correlation functions of the velocity used in Figure 1.5. A semilog plot of C\{n) vs n is shown. The straight lines have slopes fa 1.3 and fa 0.24. The inset shows a semilog plot of CVn) vs. n ; the straight line has a slope 0.69. See text for discussion. (The variance i>o is set to unity.) The error in the value of correlation function C,t is with in the size of the symbol for n < 16...... 13

1.7 PDF for the normalized fluctuation A' = SO/ag for stochastic Model with L = 48, k = 0.1,/? = 0.1, ra fa 10 and B fa 0.45. The noise amplitude has exponential tails. The variances are ag fa 0.045 and n0 ~ 0.28, and the kurtosis is 7.8. The dashed line corresponds to the stretched exponential with an exponent 2/3 ...... 11

1.8 PDFs for the passive scalar fluctuations with B >fa 1 using the con­ tinuum model. The data are obtained with L = 48, k = 0.1, /? = 0.1 using the discretizations St = 0.05 and SI = 0.5, and velocity cor­ relations « SI and tc = 2. The upper three curves (a,b,c) are shifted up from the lowest curve (d) by 6,4 and 2 decades respec­ tively. The curves a,b,c,d correspond to vu = I.I7 where 7 = (0.4,0.35,0.3,0.25) with kurtosis values (15.6,6.1,4.1,3.5). The dashed lines are modified Lorentzian (ML) fits with S = (1.6,2.6, 1.21,6.89) respectively for curves a to d ...... 15 1.9 PDFs obtained using the 3-d version of the model: (a) Small L3 regime: PDF shows exponential tails. The parameters are k — 0.1,/? = 0.1 and B « 0.4. The variances are crg ss 0.04, v0 ~ 0.3 and the kurtosis is 4.1. The velocity field exhibits Gaussian amplitude fluctuations over the time scale r„ « 10. (b) Large B regime: PDF shows Modified Lorentzian Behavior. The parameters are k = 0.1,/? = 0.1, no « 0.5 and the kurtosis is 7.1. The velocity

field is Gaussian with t c ~ 1...... 47

1.10 PDF for .r0 = x / aT in the toy model of Eqns. 1.31 and 1.32 with a = k. The noise variables 7/1 and 7/2 are uniformly distributed with cr, = er2 = 1 and the measured aT = 0.5...... 53

1.11 PDF for X = x/(Tx in the toy model of Eqns. 1.42 and 1.43 with

a — k . The noise variables 7 / 1 , 7/2 and £2 are uniformly distributed between -1 and 1 ...... 55

1.12 The conditional expectations for the 2-d model with the parameters of Figure 1.8b: The conditional expectations f„,h and <7 are shown. 63

1.13 The variance profile for a give velocity distribution. The inset shows log-log plot and the drawn straight line has a slope —3/2. The error in the variance data is within the size of the symbolused ...... 68

1.14 Schematic picture of an 8 x 8 Josephson Junction Array: The crosses with circles in the middle denote superconducting islands. The thick lines indicate Josephson junctions between neighbouring is­ lands. Input/output currents along ,r direction at left/right edges are also shown ...... 73

1.15 The distribution of the voltage difference across a transverse junc­ tion shows non-Gaussian exponential tails. The upper curve cor­ responds to parameter set P\ (shifted up by one decade) and the lower curve to P2. The data are obtained with L = 12 using 500,000 points at intervals of To ...... 77

1.16 PDF for 77(7?.) for the model defined by Eq. 1.65with 7= 0.85 (?.c, tj 0.5), ra ~ 2 and variance of £ rs 0.1 ...... 80 1.17 A snapshot of the phase configuration in a 16 x 16 array for the pa­ rameters P\. Positive/negative vortices are shown by crossed/empty circles. The current flow is along x and the motion of the crossed (empty) vortices is along negative (positive) y...... 81

2.1 Spatio-temporal correlations in the linear lattice model with con­ serving anisotropic noise. The parameters are v = 0.5 and a = 0.05. (a) Upper data are the log-log plot of (7(f) vs. r along :r direction. The straight has slope -2. Lower curve is simply the function given in Eq. 2.42 and is shifted down by one e-cade for clarity, (b) Log- log plot, of the autocorrelation function C(n) vs. n. The straight has slope — 1...... 112

2.2 Largest Lyapunov exponent \ as a function of the size of the lattice L. The error in the value of the exponent is ±0.005 which is of the order of the symbol size used ...... 124

2.3 Scaling of the width of the fluctuations in the structure factor as a function of the size of the system L. The straight line shows the slope of -1 indicating \/L scaling. The error is within the size of the symbol used to denote the data ...... 126

2.4 Log-log plot of G(rx) vs. r for noisy 1-cycle phase of model, with v = 0.8, a = 0.25, p = 0.10, and a = 0.02, on a 100 x 100 lattice. The straight line has slope -2 ...... 128

2.5 Log-log plot of G(rx) vs. r for AF I phase of model, with u = 1.30, tv = 0.25, p = 0.10, and tr = 0.02, on an 80 x 80 lattice. The straight lines have slope -2. Staggering of points for even and odd r shows “induced scale invariance” in Q — (7r, 7r) mode. The error in the correlation function is within the size of the symbol for v < 24. 130

xiii 2.6 The static structure factor G{q) is shown along three directions: (a) at q = 0 along qy = 0 ,qx = qy and qx — 0 (from the top to the bottom curve at q = 0). The symbol q denotes the variable along the direction of approach, (b) at q — Q along qy = 7T,qx = qy and qx = 7r. Symbol q denotes the magnitude of the variable Q — q along the directions of approach ...... 132

2.7 Log-log plot of G(rx) vs. r for chaotic (noisy) checkerboard phase of the model with v = 1.91, o = 0.25, p = 0.10, and a = 0.05, on an 80 x 80 lattice. The largest Lyapunov exponent, for these parameters is 0.415 ± 0.01. The straight line has slope -2. The error in the correlation function is within the size of the symbol for r<24...... 134

2.8 Log-log plot of G(rey) vs. r for chaotic (noiseless) checkerboard phase of 2-d model, with v = 1.91, a = 0.25, p = 0.10, and a = 0.00, on an 100 x 100 lattice. For these parameters the largest, Lyapunov exponent A is approximately 0.42. The straight lines indicate a slope -2. The data along x also have similar scaling. The error in the correlation function is within the size of the symbol for r < 24. 135

2.9 Spatial correlation functions on 80 x 80 lattice: (a) along x in the deterministic case with smaller a. The parameters are p = 0.1, v = 1.91, a = 0.1 and a = 0. The straight lines have slope of -4 and -2. This figure shows the crossover from \/rA to 1 / r 2 behavior. (b) along y in the deterministic case with x — y symmetry. The parameters are p = 0.1, u = 1.91 and a = a = 0. The straight line shows a slope of -4. The error in the correlation function is within the size of the symbol for r < 12...... 136

xiv 2.10 Data, for 1 — d model: (a) Log-linear plot of G(r) vs. r for chaotic (noiseless) phase of 1 — d. model, with v = 1.95, a = 0.25, p = 0.00, and er = 0.00, on a size 1024 lattice. The largest Lyapunov exponent is 0.51 ± 0.01. The data indicate exponential decay with a correlation length of 2 or 3 lattice spacings. The inset shows ordering at some q yk 0 mode. The error in the correlation function for r < 18 is within the size of the symbols, (b) Log-log plot of the autocorrelation function C(n) vs. n for the same parameters. The straight has a slope —1/2, indicating algebraic decay of C(n) ss 7?-1 / 2...... 144

3.1 Schematic diagram of the discretized elastic medium. The shear rupture is shown by a thick line between the blocks centered at r0 + Hj/Cj and f 0...... 107

3.2 Forces acting on a block centered at r*o are shown. Arrows point the direction of the force. Fr are tensile forces given by F\ =

3.3 Double-couple force distribution on the blocks around a single shear rupture is shown. The distribution is drawn for aT = ay...... 179

3.4 Spatial patterns of typical large earthquakes in 2d for the self- consistent dipole model. The triangles represent the ruptured blocks in a 2d earthquake in a 80 x 80 system ...... 195

3.5 Spatial patterns of typical large earthquakes in 3 — d for the self- consistent dipole model. The ruptured blocks in two adjacent layers (.r — c planes) of the 3 — d system are shown (in this particular earthquake, only blocks in these two layers rupture). The triangles and open circles denote the locations of the ruptured blocks in x — z planes at y = 16 and y = 17 respectively in a 20 x 20 x 20 system. 190

3.6 Log-log plot of E vs. crack length /0. The straight line slope is 2. 198

xv 3.7 Self-consistent dipole model: log-log plot of the probability distri­ bution of the energy release P(E) vs. E: (a) for the 2d model (straight line has a slope « — 1-8 ); (b) for the 3 d model (straight line has a slope rs —2.0)...... 200

3.8 Independent-rupture dipole model: log-log plot of the probability distribution of the energy release P(E) vs. E: for the 2d, model

(straight line has a slope r s —1.35). This figure is shown for com­ parison. For more data, see Refs. [23,25] ...... 201

3.9 Self-consistent dipole model: log-log plot of the probability distri­ bution of the number of broken blocks P(N) vs. N: (a) for the 2cl model (straight line has a slope — 2 .1); (b) for the 3 d model (straight line has a slope « —2.3) ...... 202

3.10 2-d seismic zone model along with an embedded strike-slip fault in the middle at y=0. The stresses (along x direction) on a single block are also shown ...... 207

3.11 The displacement field (shown in circles), as a function of // for a fixed .r, due to a slip (displacement discontinuity) at the fault (j) = 0) in a large earthquake with case A ...... 213

3.12 Clustering of epicenters of earthquakes in the vicinity of the fault is shown. One thousand earthquakes are included. An epicenter is taken to be the mean of the x and y coordinates of the ruptures or slips. A 40 x 40 system with Case A is used in the simulation. . . 214

3.13- Log-log plot of distribution vs. the energy released E for both cases A and B of the model. Plot B is shifted down by 5 e-cades for clarity. The slope of the line drawn for A is Rs —1.7; for B the slope is rs —1.3 ...... 216

3.14 Log-linear plot of the distribution P(St) vs. St. The distribution is close to exponential for small St with a minor deviation for large St. This deviation is with in the statistical error ...... 218

xvi C H A PTER I

Non-Gaussian Distributions in Extended Dynamical Systems

1.1 Introduction

Occurrence of fluctuations is ubiquitous in the course of natural evolution.

Particularly, temperature fluctuations caused by abrupt changes in weather are of serious concern in our everyday life. Such fluctuations are typically character­ ized by a Probability Distribution Function (PDF) which assumes a specific shape for the physical quantity of interest. Indeed, the explanation of the form of this distribution function poses an interesting and challenging problem to scientists and engineers in a variety of disciplines. For example, in several experiments on fluid turbulence [1-1-1], PDFs of velocity gradients and passive scalars (( .

Gaussian forms with exponential and stretched exponential tails implying more likelihood of unlikely events in turbulent regimes. Clearly, it requires an exten­ sive theoretical investigation to provide insight into the possible interconnections between the underlying dynamics and the form of the PDF. The aim of this chapter 1 is to present two novel mechanisms by which non-

Gaussian behavior can arise in the PDF of local variables in diffusive extended dynamical systems such as fluid systems mentioned earlier. We elucidate these mechanisms using diffusive models of a passive scalar. We also take the point of view that local fluctuations do not particularly reflect the nature of the long wavelength (collective) properties, and hence may be understood by simple, few variable, toy models that capture the physics at short wavelengths. Supporting this view, we demonstrate that toy models can mimic the form of the PDFs of local variables in extended systems very well in different regimes.

This chapter is organized in the following way:

• In Section 1.2, we describe three recent fluid experiments which show t he

existence of exponential tails in the PDF of local temperature fluctuations.

We also mention a few other experiments on quantities such as concentration

fields, velocity and its gradients and pressure.

• In Section 1.3, we present theoretical approaches proposed by other re­

searchers to explain the form of the temperature PDFs in fluid turbulence.

• We devote section 1.4 to describing our modeling approach and mechanisms

to understanding the origin of nontrivial temperature PDFs in fluid turbu­

lence. This section is organized in the following way:

’Some of the results presented in this chapter are published in Ref. [20]. 1. After motivating our approacli by considering the passive scalar equa­

tion, vve present lattice models that enable a comprehensive numerical

study of the fluctuations of passive scalar.

2 . We then demonstrate the mechanisms by using both stochastic and

deterministic versions of our model.

3. Using few variable toy models, we further elucidate the mechanisms.

4. We (try to) justify our approach for the passive scalar problem by

drawing connections between toy models and the fluctuations in the

extended system using simple mean field type of arguments. We also

provide convincing numerical evidence for these connections.

5. Finally, discuss the possible role of boundary conditions on the scalar

PDFs.

• In Section 1.5, we briefly present a different, dynamical system, a “.Josephson

Junction Array” that exhibits chaotic behavior when harmonically driven

by an external current2. In chaotic states, interestingly, the PDF of the

voltage of a single junction, transverse to the input current direction, ex­

hibits exponential tails. We apply a mechanism presented in Section IV to

understand the origin of these tails.

• We conclude in Section 1.6 with a discussion of our results, possible gener­

alizations, connections with experiments and future research.

2The contents of this section are to be published in Ref. [52]. For an overview of various dynamical states in such capacitive Josephson Arrays, see Ref. [51]. A

1.2 Experimental Overview

There have been several experimental studies that explore detailed character­ istics of fluctuations in fluid turbulence. Some of them focus on long-wavelength properties while the others examine local variations in a particular quantity such as temperature, concentration, velocity and its gradients [1]-[14]. The local fluc­ tuations are characterized by a Probability Distribution Function (PDF) which is obtained by the use of time series .3 Typically, in these experiments, at low values of the control parameter, PDFs exhibit Gaussian behavior. On the other hand, when the control parameter is gradually increased, non-Gaussian behavior starts showing up, i.e., the PDF develops exponential or stretched exponential t ails.'1

We now summarize a few turbulence experiments that explore the PDFs of local fluctuations. We do not intend to provide an extensive list of experiments, instead we present a few representative experiments that focus on certain physical quantities, vi~., temperature, velocity and its gradients. We concentrate mainly on temperature fluctuations, since unlike velocity which is a vector, temperature is a scalar quantity and therefore poses a simplified problem for theoret ical analysis especially in cases where it acts as a passive scalar .5

3See Ref. [15, 1G, 17] for some of the statistical features of fluid turbulence.

Quantitatively, the PDF tails are characterized by the largeness of numbers such as Kurtosis, also known as flatness (ratio of the fourth moment to the square of the second moment). Note that the flatness values of Gaussian and strictly exponential PDFs are 3 and G respectively.

5Teinperature fluctuations that are generated by the velocity field do not in turn affect the velocity. 1.2.1 Temperature fluctuations

Variations in temperature have considerable implicat ions for both geophysical and astrophysical systems. Though the fluct uations in temperature are important,, the detailed behavior of the probability distribution function especially in highly turbulent regimes was first studied only recently by Castaing et al. [1] using a

“Rayleigh-Benard connection" cell. Earlier studies focused on temperature spectra and temperature gradients in light of scaling and long wavelength phenomena [18].

Other recent experiments that characterized the temperature fluctuat ions include

“grid-generated turbulence ” in a wind tunnel [5] and “stirred fluid experiments ” in a plexiglass cell [7]. Here, we describe the above three experiments which differ considerably both in the boundary conditions and in the driving mechanisms by which turbulence is generated.

Rayleigh-Benard Convection Experiment

In 1989, Castaing et al. [1] studied the temperature fluctuations in the middle of a fluid cell filled with gaseous Helium heated from below (see Eig.l ).'1 A con­ trolled temperature gradient is applied along the vertical by means of maintaining the temperature at different values at the bottom and the top of the cell. The advantage of this experiment is that the boundary conditions (temperature) can be accurately realized unlike open flow experiments on fluid turbulence. However,

°This experimental set up is named after Rayleigh and Benard who first examined the temper­ ature driven convection in such a cell. See for example the review article Ref. [21] for more details. complete analysis of the system starting from the basic equations (Navier-Stokes with buoyancy, continuity equation, heat conduction equation), known as Boussi- nesq equations [19], is complex as the temperature difference (A) itself drives the system in to a turbulent state. The so-called Rayleigh number (Ra), which is proportional to A, is the control parameter. Typically, with the increase in Ra, the system undergoes several convective instabilities leading to a turbulent behav­ ior at large values of Ra. The details of the instabilities and the boundary layer phenomena are well documented in the literature. The interested reader is urged to look thorough the review articles [ 20, 21]. In this experiment, by exploiting the physical properties of gaseous Helium, Castaing et al. could achieve large Ra values up to 1012.

The turbulence in the system is generated by hot plumes going from the bottom to the top of the cell which causes mixing in the middle of the cell. In the turbulent state, for moderately low Ra values, i.e., Ra < Rac = 'I x 10', the temperature distribution in the middle of the cell is Gaussian. As the Ra is increased beyond Rac, the distribution abruptly becomes non-Gaussian displaying exponential tails (see Figure 1.1). Since the shape of the PDF is qualitatively different, the terms soft and hard turbulence regimes are used by Castaing et al. to denote the Gaussian and non-Gaussian regimes respectively. The characteristic of hard turbulence, i.e., non-Gaussian behavior, persists over the entire range of the explored large Ra regime: 4 x 10 7 < Ra < 1012. Also, in this regime, the distributions are found to collapse to the same specific shape when the PDFs for z

Fluid Cell

0 x

10 i

10*»

4 0 4

X

Figure 1.1: Rayleigh-Benard Convection Experiment: Top figure shows the schematic experimental set up with an externally applied temperature difference A = #i - 02. Bottom figure shows the experimental temperature PDFs measured in the middle of the cell at different Ra values. This figure is obtained from the Figure 12 of Ref. [1], The logarithm of the PDF is plotted vs. the normalized scalar deviation .V. different Ra, are plotted for the normalized scalar deviation .7 The tails of the

PDFs can be fit well with an exponential over two to two and a half decades (see

Figure 1.1).

One can ask several questions at this point. W hat is the characteristic differ­ ence in the dynamics between the two types of turbulent states? What makes the

PDF of the temperature in hard turbulence state to be non-Gaussian? Is there any specific universal form for the tails, i.e., exponential tails behavior? What is the role of plumes and coherent, structures such as vortices on the temperature

PDF? Note that, in this experiment temperature is driving the system, i.e., ve­ locity fluctuations are generated by a temperature gradient and hence the role of temperature (scalar) is active. Again, we point out that the detailed study of temperature dynamics starting from the dynamical equations is complicated for this problem, even though this experiment is accurate in the sense of precise realization of boundary conditions.

We finally note that several details of the t urbulent state have not yet. been completely characterized. For example, the dynamical behavior of the velocity field is not explored in the system. The time scales, if any, associated with the plumes that go from the bottom to the top of the cell are not understood both experimentally and theoretically. In addition, the mean temperature profile, which would enable a better theoretical understanding of the internal dynamics, is not well studied.

form alized scalar deviation (A') is the deviation of the scalar value from the mean divided by the standard deviation. With this choice, in addition to the usual normalization of the PDF, the standard deviation is also normalized to unity. 9

Grid Turbulence Experiment

This experiment is done by Jayesh and Z. Warhaft [5, 6 ] to study the temper­ ature fluctuations in a grid-generated turbulence as shown in Figure 1.2. Air is injected through the grid with a certain velocity Vxo into a wind tunnel extending along the x direction. The obstructions due to the grid generate turbulence in the wind tunnel. The (turbulent) flow is characterized by the mesh Reynolds number which could be varied from 1600 to 50,000. Such wind tunnels have been exten­ sively used to invest igate a variety of aerodynamic phenomena. The t urbulence generated in the grid has distinct characteristics on its own. First, it has some mean flow along .r. Second, it decays along ;r, i.e., the standard deviation (n0) of t he velocity fluctuations decays as a power law as one goes farther away from the grid [22].

To study the temperature variations in the middle of the wind tunnel, a variety of temperature boundary conditions are imposed along the direction of flow using a toaster and mandolin wires. The toaster is capable of generating both uniform and non-uniform temperature gradients (/?) along c. An important feature is that t he temperature does not drive the velocity field unlike the Rayleigh-Benard convection experiment described earlier. Hence, the role of temperature is passive and one can hope that the heat conduction equation along with an advective term might give a clue to the origin of any non-trivial behavior in the temperature PDF.

The temperature and velocity fluctuations are measured in an applied uniform temperature gradient /? along z. The focus is on measurements far away from 10

Grid

Turbulent Flow

10*

10 1

10'*

to'# a 4 o 4 a

Figure 1.2: Grid Turbulence Experiment: lo p figure shows the schematic exper­ imental set up with a temperature gradient applied along z direction. Bottom figure shows the experimental temperature PDFs at different x. This figure is obtained from the Figure 7a of Ref. [ 6]. The logarithm of the PDF is plot ted vs. the normalized scalar deviation .V. the grid and wind tunnel boundaries, so that edge effects are negligible. Closer to the grid, the distributions of temperature have exponential tails and as one goes farther away from the grid, the temperature fluctuations gradually become

Gaussian. This behavior is quite different from the velocity PDF which is observed to relax very quickly to a Gaussian as one goes little fart her away from the grid .8

Similar to the R13 experiment, the temperature PDFs at. different distances from the grid for the normalized scalar deviation collapse on to a simple curve.

Also, the PDFs show the existence of exponential tails over three decades (see

Figure 1.2). In addition, in contrast to the symmetric scalar PDFs in the middle of the tunnel, skewed distributions are found in the vicinity of the wind tunnel boundaries. Another important feature is that, the mean temperature profile is linear along the c direction with the slope inside the tunnel being close to the applied fi.

In this experiment, the nature of the temperat ure PDF is also explored in t he presence of other choices of non-uniform temperature boundary condit ions. The main conclusion is that the PDFs are observed not to have exponential tails and display roughly Gaussian behavior when the boundary conditions are such that the mean profile along z is almost flat. This behavior suggests that a mean gradient is necessary to see non-Gaussian temperature PDF with exponential tails!

8IJowever, in the vicinity of the grid, considerable deviations from Gaussian are observed for the velocity PDFs; also, the shape of the velocity PDF depends on the location of the m easurem ent, see F ig.‘2a in Ref. [6]. 12

The length scales lv and /#, associated with the velocity and the temperature fluctuations respectively9, are obtained from the corresponding spectra .10 The experimental spectrum clearly shows considerable qualitative changes at these 1 length scales. However, since all the spectra are obtained using the measurement s at a single spatial point, it is not clear how well one can extract the length scales associated with the fluctuations by using such spectra.

Motivated by a dimensional argument in Ref. [35], the PDFs are collapsed by scaling the temperature fluctuations by filv, i.e., bO —> bO/(/jl„). The collapse is not as good as that, for the case with the standard deviation (shown in Figure 1.2).

However, there is evidence for (To /f, indicating a linear dependence of the inverse of the slope of tails on ft.

Stirred Fluid Experiment

'Phis experiment is done by CJollub cl «/. [7, 8 ] to study the statistics of the scalar fluctuations in a turbulent fluid mixed by a vertically oscillating horizont al grid. A plexiglass cell of fluid (water-glycerol mixture) is stirred with an external oscillating grid (see Figure 1.3) with periods ranging from 0.5 to 10 secs. The grid is also made of a thin plexiglass sheet with hexagonal holes of diameter d punched in a regular fashion. The turbulent state is characterized by a Reynolds number R defined using the grid characteristics and the driving parameters. The value of R

9These numbers may be taken as the correlation lengths associated with the velocity and the temperature fluctuations. They do seem to increase along a\ see Table. 1 in ltef. [(!].

10The time series data is converted into a spatial data by using the Taylor frozen flow hypoth­ esis [16] while obtaining the spectrum. could be varied from 100 to 2000 for different, forcing parameters. In addition to this forcing, a temperature gradient is applied along the x (horizontal) direction by fixing the boundaries at different temperatures. Temperature is measured by the use of thermistor probes in this experiment.

The main result of this study is that the temperature PDF at large R values displays exponential tails (see Figure 1.3) unlike the Gaussian behavior observed at, low Reynolds number regime. The data for the flatness as a function of increasing

R suggest that the PDF develops exponential tails gradually with the increase in

R. 'I'lie maximum value of the flatness is around (i corresponding to an exponential

PDF. However, the data are insufficient to preclude the possibility of higher values of the flatness, i.e., it is conceivable that the PDF can develop other types of non-Gaussian tails for higher values of R which could not be explored due to experimental limitations. If is also pointed out that close to the boundaries, similar to grid turbulence experiment, skewed PDFs are observed for the scalar fluctuations. In a recent paper [ 8 ], the authors also report the observed PDF of the velocity fluctuations which appears to have a well defined Gaussian core with some minor deviations at the tails.

In addition to the PDFs, other characteristics such as eddy diffusivity, mean scalar profile and spatial correlation length of the temperature have also been measured in this experiment. The eddy diffusi vity, which measures the enhanced diffusion due to turbulence, is shown to increase as a function of /?. This increase M

Grid

z

Fluid

CDj

0* 6 - 4-2 0 24 6

X

Figure 1.3: Stirred Fluid Experiment: Top figure shows the schematic experi­ mental set up. A temperature difference is applied along x direction. Bottom figure shows the experimental temperature PDFs obtained from the Figure 7b of Ref. [8 ]. The logarithm of the PDF is plotted vs. the normalized scalar deviation A'. The straight lines correspond to the strict exponential PDF. appears to occur at values of R where temperature PDF starts developing ex­ ponential tails, indicating a connection between the dilfusivity enhancement and the appearance of exponential tails in the PDFs. Furthermore, the mean scalar profile over a region in the middle of the cell is observed to be linear along the direction of applied gradient (with some temperature drop at the boundaries ).11

The correlation length of temperature fluctuations is obtained from the exponen­ tial decay of the measured equal time spatial correlation function. This length is observed to be of the order of the diamet er d of the holes which is also expected to be the correlation length of the velocity field. It, is also pointed out that the

PDF tails can be characterized by cxp(—')\0\) with 7 -1 ~ fid, consistent with the result of a dimensional argument in Hef. [35].

Though the experiment is better characterized, t he question of the existence of exponential tails in the temperature PDF in the absence of an externally imposed temperature gradient case, which is raised in the grid turbulence experiment, was not explored here.

1.2.2 Concentration Fluctuations

Concentration fluctuations are of considerable interest as they are very impor­ tant in industrial applications such as transport of chemicals through fluid flows.

Srinivasan cl al. [9, 10] have measured the fluctuations in the dye concentra­ tion injected in to a fluid through a jet. The resultant PDF is observed to be non-Gaussian and suggests exponential nature of the tails. A modified Lorentzian

11 The gradient, values could be obtained from the figures and figure captions of their paper [7]. form (see Appendix A.l) is also strongly suggestive, but the data have not been

fit to this form. Gollub el al. [8 ] also tried looking at concentration fluctuations

in the stirred fluid experiment, however, the results are inconclusive due to ex­

perimental limitations. In recent years, there appears to be more interest, in the

detailed charact erization of concentration fluctuations.

1.2.3 Fluctuations in Velocity and its Gradients

Velocity fluctuations, measured in both grid turbulence and stirred fluid exper­

iments above, reveal close to Gaussian behavior with minor deviations. However,

recent studies indicate that one can obtain non-Gaussian behavior even for the

velocity field.

Velocity gradients, measured in atmospheric flows by Van Atta and Chen in as

early as 1970, are known to be non-Gaussian [11, 12]. This study appears to have

done the first detailed characterization of the PDFs of velocity gradients in highly

turbulent regimes. The observed PDF is shown to exhibit stretched exponential

.tails and this non-Gaussian behavior of the PDF is believed to be the signature

of small scale “ interiniUency ” in fluid turbulence. Since then several experiments

including shear flows and turbulent jets have been conducted, which also revealed

similar phenomenon [18].

1.2.4 Fluctuations in Other Quantities

In addition to the above mentioned local variables, the other important quan­

tities in fluid turbulence are pressure and vorticity. PDFs of Pressure fluctuations, 17 which have been measured recently in a swirling fluid experiment [23], revealed strong asymmetry along with some exponential tail behavior .12 The experiments on vorticity are currently in progress [24].

1.2.5 Experimental Issues for Temperature PDFs

• In highly nonlinear regimes non-Gaussian tails are observed in the PDFs

of temperature. The origin of these tails may be attributed to the change's

in the dynamical behavior of the turbulent state. The detailed connection

between the PDF shapes and the dynamics in highly turbulent states is a

challenging issue. Moreover, one may ask whether it is possible to identify

the underlying dynamical mechanism just, by looking at, the shape of the

PDF of a local variable in an extended dynamical system.

• The PDFs for temperature appear to assume a specific non-Gaussian form

with exponential tails. This raises the issue whether or not there are any

universal features for the scalar PDF in highly turbulent regimes. On con­

trary, it would be interesting to see if t here are any other forms of PDF t hat

can be observed in different regimes. Another interesting observed feature

is that the inverse of the slope of the exponential tails is proportional to the

mean gradient [i. How can we understand this?

• Boundary conditions appear to be crucial in seeing non-Gaussian behavior.

As mentioned in the grid turbulence experiment, one is unable to see st rong

'"Note that, pressure is not a simple diffusive quantity such as velocity, vorticity and passive scalars. non-Gaussian behavior in the presence of (low) zero mean scalar gradient.

One may ask: What are the limitations on seeing exponential tails? What

is the role of the scalar gradient? How does the scalar PDF depend on the

mean scalar profile?

Close to the boundaries, skewed PDFs are observed in most of the exper­

iments. This behavior is probably expected due to boundaries and should

naturally come out of a passive scalar model.

• The role of coherent structures such as vortices and plumes (in the RB

experiment) 011 the temperature PDF is an issue of serious concern. 'Phis

is related to the details of the dynamics in the turbulent state such as hard

vs. soft turbulent states described in RB experiment. The time scales

associated with the coherent structures and the connection between small

scale intermittency and the non-Gaussian PDFs are worthy of investigation.

This chapter addresses some of the above issues and questions. It, also eluci­ dates possible underlying mechanisms for the origin of non-Gaussian behavior for the temperature PDF. The hope is that the mechanisms presented here may also be relevant for other diffusive fields such as velocity (gradients) and vorticity; this remains to be seen in future. 19

1.3 Theoretical Developments

There have been several theoretical studies which provide explanations for the origin of non-Gaussian behavior in the PDFs of local variables in Fluid Turbu­ lence. Since our primary focus is on temperature PDFs, in this section, we present the following two recently developed approaches which, however, complet ely differ from each other while understanding the shape of temperature PDFs in fluid tur­ bulence: (i) the moment balance approach suggested by Sinai and Yakhot [27] and

(ii) a phenomenological approach 1,3 using a model non-linear evolution equation for the probability distribution, suggested by Pumir, Shraiinan and Siggia [35].

In addition to these studies, several others have also investigated the temperature' fluctuations [28, 30, 31, 32]. On the numerical front, Kerstcin [33] has developed a model that includes the eddy diffusive effects and examined the temperature' fluctuations. Direct simulations of the underlying passive scalar expiations also indicate nontrivial behavior in the scalar PDF in highly nonlinear regime's [31].

On the other hand, for the fluctuations of velocity and its derivative's, the situation is more complex. Recently, using a mapping closure method, it has be>e'ii shown that the velocity gradients can attain exponential tails [37]. In addition, a method that relies on an appropriate superposition of Gaussian or log-normal distributions has also be'en adopted to understand the non-Gaussian nature of the velocity gradients [38, 14].

13This approacli has been originally pointed out in a different context by S. B. Pope [29], however, possible general forms for the PDFs were not extracted. We now briefly present the basic ideas involved in the moment balance and the phenomenological approaches, so that we can contrast these with our approach in understanding the shapes of temperature (scalar) PDFs later.

1.3.1 Moment Balance Approach

In this approach, Sinai and Yakhot, [27], consider the passive scalar equation for the temperature field Q(r,t) adverted by a velocity field V:

0/1 — = kV 20 - V -VO. (1.1) di

The coefficient k is the thermal difiusivity and the velocity is assumed to be incompressible: V • V = 0. We do not intend to go through the algebra here, but give a brief sketch along with some remarks which will help connect this approach with our models described later.

At the outset, we point out that the velocity can be allowed to have arbit rary correlations in this approach. Moreover, one can write an equat ion for Q =< 02 > in terms of Q\ =< |V 0|2 >, where < ... > indicates the spatial average', as the following:

-^- = - 2 kQ1. (1.2)

This shows the decay of Q as time goes on, since the thermal difiusivity k > 0.

Thus, asymptotically as /. — > oo, the solution for 0 would be flat unless there are some nontrivial conditions imposed on this decay. These are some of the concerns and confusions that arise especially while extending this approach. 21

The moment balance relations are obtained by using an ensemble average,

which is defined to be the spatial average in the problem. By using the incom­ pressibility of the velocity field and by multiplying the Eq. 1.1 with powers of 0 both sides, one can obtain the following moment balance relation for any positive integer n :

(2n - 1) < x2,,~2y2 > -< x2n >, (1.3) where, x = 0/yjQ and y = \^0\/\/Q\. This equation contains expectation values of the temperature with its gradients. If the temperature is independent of its gradient,, one retrieves the moment relation for Gaussian behavior trivially: < x2" > / < x2n~2 >= 2n — l leading to < x2n > = (2n — 1)!!, a. characterist ic of the

Gaussian PDF with zero mean and unit variance. Now, the key is to propose an appropriate expansion around this Gaussian and thereby extract the nature of the possible PDF.

The moment balance relation can be rewritten in terms of the distribution

P(x,y) in the following way:

(2n — 1) J x2n~2y2P(x,y)dxdy = J x2nP(x, y)dxdy. ( 1. 1)

Now, by using P(x,y) = p(x)q(y/x), where q(y/x) is the conditional probability of y given x and p{x) is the PDF of .r, we get

J ~j^{x2n~l )p(x)dx j y2q{y/x)dy = J x2np(x)dx. (1.5)

To obtain a solution for p(x), all we need is

f/o(j-) = J y2q{y/*)dy, (1.6 ) which is the expectation value 14 for the variance of y given x. By using the normalization properties of the probabilities, we get

Now, to determine the p(x) uniquely, by invoking spatial isotropy one demands p(x) = p{—x) and qo{x) = qa{—x). By appealing to these symmetries, they show that the term in the square bracket should be identically zero. This implies

( 1. 8 )

Note that this equation looks like a stationary condition for a PDF for the single variable Fokker-Planck equation as shown in Appendix A.2. Indeed, we draw a connection with this while discussing our mechanisms using toy models. Speeifi- cally, the function r/0, which is the conditional expectation of the* variance of //, may be viewed as a diffusion coefficient, in the evolution equation for the probability p{x) as will be seen later in our approach.

By assuming a specific form for

(1.9)

When a simple quadratic expansion for #/0 is invoked,

( 1. 10)

14Special case

P('r ) ~ ( /1 1 + i C.TZ)C 2 U+fi» 1+* (]-H ) with 2e£> = 1 + t. In Ref. [27], it is also shown that the above form fits well for the numerically obtained scalar PDF in direct simulations [39].

This approach has been used by others to fit experimental data showing the' strength of this method [31]. Recently, moment balance relations have also been derived in a general context [32]. However, the main difficulty (that 1 have) with this approach is that the spatio-temporal correlations of the velocity field do not play any direct role in determining the scalar PDF.

1.3.2 Phenomenological Approach

This approach is centered around postulating an evolution equation for the probability distribution function using phenomenological considerations.1'’ It is suggested by Pumir, Shraiman and Siggia [35]. In this approach, the evolution of the distribution P(0,l) for the local scalar fluctuation 0 is considered in discrete time steps (t) following the method pointed out by Pope [29].

It is argued that in turbulent state one has to look into time scales while studying the local fluctuations. The focus is on two time scales: (i) One is the diffusive time scale of the scalar fluctuations and (ii) the other is the mixing

15One lias to be careful while choosing = 1. If not, one obtains an incorrect value for the exponent 6. This point is important especially while fitting experimental data to this form. Also, note that the form given in Ref. [27] does not satisfy < x~ >= 1 due to this reason.

is The author is presenting his understanding of this phenomenological approach. time scale set by the velocity field that advects the passive scalar. The relative importance of these two time scales is emphasized in this approach.

Then, it is argued that due to the mixing in the presence of random advection. the value of the scalar at a particular site at. time 1, + r would be an average of the variables one correlation length apart, from that, site at time /,. Using this, t he central theme is to write a phenomenological evolution equation for the probability, which takes the following form in a one dimensional model:

P(0J. + r) = + J P(0u l)P(02J.)6(0 - °^±2l)d0xd02. ( 1. 12)

Mere, r is a characteristic mixing time of the t urbulent advection, and are the values of the scalar at, sites which are one correlation length away from the site of interest,.1' Hy using a complicated argument, in Ref. [35], it is further shown that I) « /f2£2, where £ is correlation length of the velocity field and /f is the mean scalar gradient. However, in the above equation 1.12, one can view the first term on the RI1S as a diffusive term corresponding to the random walk of 0 as pointed out in Appendix A.2, while deriving the Fokker-Planck equation for a simple Langevin equation .18 The second term is the mixing term that, models the averaging caused by the advection.

17By assuming the spatial isotropy and translational invariance they argue that 0(/ + r) s» [q t / k ) which plays a crucial role. It is useful to consider the characteristic function C{k,1) defined as:

C(k,t) = f P(O,t)e-'0kdO =< c~m > . (1.13)

Now, following Eq. 1.12, the evolution of the the above characteristic function

C(k%t) is given by:

C (kJ + r) = — DkzC(k, I) -f Cz(k/2, /,). (1.1.1)

The stationary solution for this equation can be found by removing the time dependence on both sides; this leads to:

r - ’(A') = (i + /^ 2)n^,(i + Dk2/ 22,)2^ (i.i5)

Now, it is arguedthat, since t his characterist ic function has a leading pole at \k\ = i/v/7;. exponential tails arise for the PDF of the scalar (k i.r.. the dist ribution takes the form

P{0) « c"lffl/ ^ . (1. Hi)

'I'his approach has been applied to a one dimensional passive scalar model in an externally imposed gradient boundary condition with both mixing and diffusion.

This model is a simplified version of Kerstein’s model of eddy diffusion [33] for a passive scalar in fluid turbulence. The mixing due to velocity field is modeled by random flips of adjacent sites. It is argued that one would see exponential tails for large mixing rates in this model; the resulting PDFs and their non-Gaussian nature are presented in a later publication on this (and variants of this) model [36].

Note that, since D « /?2£2, the inverse of the slope of the tails is expected to scale as where is the mean gradient and £ is the correlation length of the velocity fluctuations. This expectation is shown to be consistent with the data of stirred fluid experiment [7]. However, the role of time scales in determining the value of D is unclear and the relevance of their phenomenological model to some real systems are questionable. 27

1.4 Our Approach

As has been motivated earlier, our aim in this chapter is to present a new

approach, based 011 stochastic Langevin equations19, to understand the origin of non-Gaussian PDFs for local variables in diffusive systems especially in the

context of passive scalars such as temperature. In our view, t here are two possible

mechanisms by which non-Gaussian behavior can arise in the scalar fluct uations:

(i) One mechanism is due to the “correlations” of velocity field. Specifically, it relies on the nature of the temporal correlations of the velocity fluctuations that

“ndditively" couple to a diffusive field. We show that a variety of non-Gaussian

PDFs can arise; for example, exponential and stretched exponential tails occur in certain regimes when the amplit ude of the random velocity varies 011 a time scale comparable to the intrinsic diffusive time scale of the passive scalar. Note that the amplitude variations may be interlinked with the intermittent'-}) of the velocity field in turbulent states.

(ii) The other mechanism is operative for larger magnitudes of the velocity fluctuations. In this case the scalar fluctuations become non-Gaussian due to the

“multiplicative" coupling between the scalar and the velocity fluctuations via t he convective term. In contrast to the previous mechanism mentioned above, the resultant non-Gaussian distributions develop algebraic tails taking forms such as modified Lorentzians.

19In this chapter, we interpret the (noisy) Langevin equations as Ito stochastic differential equations as opposed to Stratanovich stochastic equations [40, 41]. This choice is made primarily to facilitate a discretization in time for the stochastic equations. Following these mechanisms, in experimental fluid systems, exponent ial tails could arise if the velocity fluctuations exhibit intermittent behavior in their mag­ nitude on relevant time scales. In addition, non-Gaussian algebraic tails can also arise due to the dominance of the multiplicative coupling. In reality, both these mechanisms may be relevant and may as well be co-operative in a variety of dif­ fusive systems including the passive scalar. However, the two mechanisms are distinctly different,. Even though, the presence of exponential t ails in the temper­ ature PDFs is overwhelming in diverse experiments, it is yet to be verified whether the tails of the PDFs are predominantly algebraic in certain turbulent, regimes. 29

1.4.1 Passive Scalar Equation

Wo consider the diffusion of the temperature field 0(rj.) adverted by an incom­ pressible velocity field V described by the passive scalar equation 20

^ = h V 20 - V • VO, (1.17)

where the coefficient k is the thermal diffusivity. For adetailed discussion on this equation, see e.g., Ref. [42]. Since the velocity is assumedto be incompressible, we have:

V-\/ = 0. (1.18)

2()This equation follows from the law of conservation of energy under certain assumptions (see chapter 5 of lief. [17] and lief. [19] for an elaborate discussion). To see how it arises, first consider the heat, conduct ion in a certain volume of stationary, isot ropic, incompressible fluid. The heat flux per unit, area ./ t hrough any surface is related t o the variation of the temperature through the fluid. Specifically, / = - VV0,

where' \ is the thermal conductivity. Now the continuity equation for heat can be written by equating the change in heat content. (//) in that, volume per unit time to the heat flux gone out of the same' volume in that, unit time. This gives us dll/dt — —V ■ Since the heat content, is proportional to pcrQ, where p and r,, are density and specific heat at constant pressure respectively, the resultant, heat, conduction equation is

% -

where k - \/(pc,,) is the thermal diffusivity. In the presence of fluid motion, since the heal is also transported by the fluid motion due to convection, partial time derivative gets replaced by the total time derivative (this can be derived by writing a continuity equation for heat in a control volume); the total derivative gives the advect.ive term: V -VO, leading to Kq. 1.17. Note that the viscous heat generation due to shear in the fluid is neglected here. The velocity is assumed to be much smaller that, the velocity of sound. The temperat ure differences between the boundaries are also assumed to be small so that the assumption that the density ( p ) is constant, is valid. Also note that for a concentration field one obtains the same equation with k replaced by the corresponding diffusivity. Motivated by the passive scalar experiments on grid-generated and stirred fluid turbulence, we apply a gradient ft along the direction z to study the temperature fluctuations .21

Before presenting more on our approach, we consider the following rescaling of the passive scalar equation and identify the relevant parameters that characterize the scalar PDF. From Eq. 1.17, it is clear that the characteristics of velocity are

crucial. Let the velocity be correlated over a time t c with a correlation length and a characteristic magnitude vq (the standard deviation of velocity fluctuations) in a turbulent state of interest. Using these quantities, we can rescale the time, space and velocity by setting I = tc1', f = £vr ' and V' = ?>0V" and obtain the following equation for the scalar:

= k ' V ,20 — VqV' • V '0 . (1.19) dV where k' - ktc/£?, and Vo = t>0tc/£v. The gradient also gets scaled to ft' — ;i£v.

Since the equation is linear, rescaling of 0 simply does not alter the equation including its coefficients. Hence, any changes in ft can be simply absorbed into a rescaling of 0. Then, we can rescale 0 by a factor £„ and maintain a constant ft.

Consequently, if we find the PDFs for a specific ft, the PDFs for other ft values are determined .22 If we absorb the Vo back into the velocity so that the variance of the velocity is Vo instead of unity and dropping the primes we get back the

'-'Here, is the applied temperature difference per unit length along r direction. While referring to experiment, we use ft as the observed local mean gradient where the PDFs have been exam ined.

"This feature provides an explanation for the observed proportionality erg

rescaling does not facilitate any further simplification.

From the rescaled equation, one can see that two important quantities are

relevant in determining the scalar fluctuations: ( 1) the difTusivity k and ( 2 ) the

magnitude of the velocity fluctuations v0 and its correlations. Typically, in ex­ periments, it is evident, that as the driving parameter such as Reynolds number is increased the nature of the velocity fluctuations change. This change can be manifested either in the temporal correlations of the velocity field or in the form of an increase in the magnitude of the velocity fluctuations. As the coupling with the temperature, V -VO term is multiplicative, it is reasonable to expect not only the temporal correlations but also the magnitude of the velocity field can influence the form the scalar fluctuations. Furthermore, the diffusivity k can also change in a turbulent state. This brings in a question about the value of k in turbulent states [12] which is beyond the scope of this thesis. Indeed, as we will see later,

the dimensionless number (t^ t c/ k ), which is the ratio of velocity characteristics to k, plays a major role in determining the shape of the PDF. Now, we first illus­ trate these aspects by devising simple stochastic and deterministic models for the passive scalar.

1.4.2 Models for the Passive Scalar

One can numerically study the behavior of passive scalar in the presence of prescribed turbulent velocity fields directly in the continuum using passive scalar equation (1.17). However, the case of a velocity field arising from a deterministic 32 chaotic dynamics is computationally prohibitive if one uses Navier-Stokes equa­ tions for a given geometry. Hence we simplify the problem by a discretization of the system in space in units of the correlation length of the velocity field; this provides a system of size Lx x Ly x Lz. Then the passive scalar equation takes the following form:

= KV2L0(fi,l) - V ■ V L0(rtJ). (1.20) where V/. is a symmetric, lattice gradient 23 This can be viewed as a coarse grained equation in space. The k here can be taken as a coarse grained or renormalized effective difTusivity over the correlation length of the velocity field. Determinat ion of the effective k in turbulent states still remains an open problem as pointed out in a recent review article 011 the advection of passive scalar [42].

We further assume periodic boundary conditions along r and y directions and apply a gradient boundary condition with 0 = 0 at, z = 0 and 0 = (L: + \)j1 at, z = L: -f- 1 boundaries. Thus the effects due to thermal boundary layers are ignored. Now, this model can be used to study the fluctuations of 0 for variety of stochastic and deterministic velocity fields. Moreover, the system of equations Eq. 1.20 can be considered as a set of coupled Langevin equations

-''Specifically we use: = £[/(*,)-/(r)], U

where t.lie sum on n is over all the 2d nearest neighbours of site r 0 11 a r/-dimensional lattice. The gradient along the direction of the unit vector ca w ith a = (jis:

^Lam = ^[f(r + ea)- f(r -c a)].

Note that the lattice spacing is taken to be unity. when the velocity field is a. white noise. This feature facilitates an analysis of the temperature PDFs using the Fokker-Planck formalism as will be seen later in

Section. 1.4.5.

Note that for computational convenience, one can reduce the dimensionality of the problem by considering a 2-d version of the model by simply dropping the y dimension. Also, the 2-d version of the model provides better numerical results as one can obtain longer time series for the fluctuations using numerical simulations for a given velocity field.

To explore the case of a deterministic chaotic velocity field, we consider a coupled map lattice model in discrete time and space which can be viewed as coarse-grained version of the continuum equation in both space and time. Such models have been extensively used in the past few years to examine spatio- temporal chaos in spatially extended dynamical systems (see the next chapter, i.e., Section 2.2.4)t The scalar 0(i) is defined on each site i of a Lr x l,~ square lattice (in the ,r — c plane) with a mean gradient fi along the c direction. The system evolves synchronously in discrete time n according to

On+i(i) = 0n(i) + KVjOn(i) - VU*) • V/A(*)- (1-21)

First, we illustrate the mechanisms by a numerically study of the scalar PDFs for different velocity fields. For all the cases, the incompressibility of the velocity field, i.e., Eq. 1.18, is enforced by the choice V(i) = V/, x A, where A is a stochastic vector field with given correlations among its components. In the 2- d version of the model, we require only one component of A, namely the stream function Ay = rj> which defines the velocity components in the following way2'1: u =

Vr{i) = 7 ?; = Vz(i.) = —7 V^x0(?‘). We choose the boundary conditions so that the velocity vanishes at both the 2 = 0 and 2 = L. -f 1 boundaries.

1.4.3 Probability Distributions of Scalar

In the absence of V, when a gradient is imposed along 2 direction, the system relaxes to a solution with linear scalar profile, i.e., 0(z) = ftz, along 2. Hence, in the presence of the velocity fluctuations one expects the value of 0 to fluctuate around this mean profile. Indeed, in our numerical simulations, for all t he models we explored corresponding to different dynamics of V(i), the mean scalar profile is linear confirming this expectation.

Now the scalar fluctuations in the presence of a fluctuating velocity field can be ex. 02 Yd around the mean profile by 0 = 0 — 0(z):

0(r, I) = h-VjO - - V(r) ■ V/,0. ( 1.2 2 )

Note that this separation occurs in all our models. When the velocity field is viewed as a stochastic field, we obtain both an additive noise term, fiV., and a convective (multiplicative) noise term. The strengths of these two terms are expected to be /?e 0 and (ToVo/io, where ctq is the variance and £0 the characteristic length scale of 0 fluctuations.

■‘'The parameter •) controls the variance n0 of the velocity field preventing instability of the (discrete time) model at large i)0. Wc find that different regimes of behavior of the scalar PDFs can be charac­

terized empirically by a single parameter

which measures the relative strengths of the two noise terms. From now on wt

drop the tilde on the 0 to denote the fluctuations unless otherwise specified.

Small B Regime

Let us first consider the IJ < 1 regime where the additive noise term is domi­

nant. In the absence of convective noise term, for a large system, one can analyt­ ically obtain the distribution of 0 to be Gaussian for the case when the velocity is obtained from a stream (potential) function which is a simple white noise with delta-function spatial correlations25. The magnitude of the scalar fluctuations is given by: c j n n i v i r , ...... *, = —— = — -— , where C,t is a constant of the order of unity and depends on the spatial dimension.

Since, we expect to be the order of the small B regime occurs for v$tc/ k < 1.'

Mechanism for Exponential tails in this regime:

When the velocity field has Gaussian amplitude fluctuations on a time scale r„ comparable to the diffusive time scale r,t = £2/ k we obtain PDFs with exponential tails for 0.

■5The solut ion of this problem is illust rated in Appendix A A 36

We now illustrate this mechanism in cases when the velocity field is obtained from both stochastic and deterministic dynamics using the 2-d discrete time model and then comment on the distributions in the continuous time model. The size of the system that we use is 48 x 48. We focus on the PDFs of scalar fluctuations in the middle of the system; also we obtain the PDFs by using a time series of

5 Million data points. All the simulations are performed on C'ray-Y MP at the

Ohio Supercomputer Center.

Stochastic Model : An easy way to obtain amplitude fluctuations is to use a stochastic model and choose the stream function (/’„(*) = Vn(i)v!,(*) where ?/'(/’) is a white noise and 7]„(i) a Gaussian noise with a correlation time r„. both hav­ ing delta-function spatial correlations. The velocity thus displays two correlation

times: (i) the mixing time tc governed by //',(?) chosen to be one timestep and (ii) the time scale t„ on which the amplitude varies. This model exhibits exponential tails2'' in the PDF of 0(i) when r„ ss r,/ as shown in Figure 1.4. On the other hand

when t „ << t ,i the distributions are Gaussian.

We have also implemented such stochast ic velocity field in the cont inuous time model with (/’(d, 0 = ?/(F,, l)i]'(rt, t) and checked that the results are unaltered.

For this case the temporal correlations of the velocity field can be analysed in the following way. Since the velocity field is a white noise the autocorrelation function

C2(/) = < Vj(OK-(0) > at site r, will be a delta-function in time. But the higher

■eThe fit to the logarithm of the PDF is linear and logarithmic corrections, i.e., power law corrections to the exponential form could not be extracted. -10 -5 0 5 10 X

Figure 1.4: PDF for the normalized fluctuation .V = SO/ctq for the stochastic model: with L — 48, n = 0.1,/tf = 0 .1, r(1 « 10 and 13 ss 0.35. The white noise terms are uniformly distributed between -1 and 1. The variances are 00 ~ 0.034, e0 « 0.21 and the kurtosis is 4.22. order autocorrelation function given by C4(1) = < V 2(t.)V2(0) > — < K-(O)2 > 2 is not a delta-function and reveals the time scale ra associated with the amplitude fluctuations.

To understand how the time scale r„ (associated with amplitude variations) can be extracted, it is convenient to consider the stream function and compute its autocorrelations:

Instead of being a white noise, more generally, if the //' is correlated over a time rc, we will have

C2(t) =< 0(/)0(O) > - < 7/’(0) >2= C (1.25) where r„, is a new time scale given by (the harmonic mean): r„, = (l/r,.-f 1/r,,)-1.

The higher order autocorrelation function for the stream function is:

C4(l) =< 0 2(/) - < 0 2(O) > 2, (1 -2(>) which, upon s r ^1 / ‘on of 0 = ip/i since ;/ and if are uncorrelated, (.akes the form:

a t(l.) =< 7;2(/)7/2(0) X 7,/2(/)7,/2(0) > - < 7/2(0) > 2< 7/2(0) > 2 . (1.27)

Now, the basic idea is that each part of the first, term in the RHS can be t hought of as a constant plus a term that decays to zero with increasing /. Since, the two terms are decoupled one can extract the time scale associated with the amplitude with the use of the constant part of the < if2(t)if2{0) > term. This is the key

by which the time scale t„, that could not be extracted out through a simple two point, correlation function, gets unraveled. As an extremely simple example, if one considers the case in which both ?] and ?/' are Gaussian, the function C\ exhibits the following three decays:

C4(t.) = G02[4r-2f/T’" + 2(~2t/Tr + 2e-2‘/r"] (1.28) associated with time scales rm,rc and r„ respectively. However, in the limit rr < <

r„ we have rm « t c . We stick to this limit while carrying out further analysis.

Note that the above analysis becomes complicated especially when t here are correlations between 7/ and rj'\ and this is likely to be the case in reality. How­ ever, one can explicitly compute the higher order autocorrelations and examine the amplitude variations as shown in the following deterministic example.

Dcierm.inist.ic Model : We now provide numerical evidence to show that expo­ nential tails also occur when the stream function is derived from a dctcrinini.stic chaotic model provided it leads to amplitude fluctuations 011 the appropriate time scale.

We investigate the following model: define the stream function in terms of an auxiliary variable „(?■) by tf'„(i) — V2^2n(0;

^n+i(0 = FEW ’»(*)), (1-29) 5 * where the sum includes the site i and its nearest neighbours in the 2-d square lattice.27 The function F is chosen to be F{x) = (1 — 2//).r + 2//.r3 and its behavior

J7Such models are commonly referred to as coupled-map Lattice models, which exhibit spatio- temporal chaos for large values of the control parameters when F is non-linear. See chapter II, Section 2. 2.4 for more details. 40

_T

Dato Gaussian fit -2

> CL o cn o

-6

-8 -6 -3 0 3 6

V

2

0

4

-6

-8 -10 -5 0 5 10 X

Figure 1.5: Data for deterministic Model: (a) PDF for v normalized by e 0 has Gaussian tails, (b) PDFs for the normalized fluctuation .V are shown for two parameters. The dashed line is drawn with the same slope as that of the tail, (i) Upper curve (shifted up by two decades) is for L = 18, k = 0.1 ,7 — 0.5, /i = 0.1. The variances are: a g « 0.045, v 0 « 0.2, the kurtosis is 4.62 and B ~ 0.15. (ii) Lower curve is for L — 48, k = 0.05,7 = 0.25, = 0.1. The variances are (To ~ 0.031, vq w 0.1, the kurtosis is 4.12 and B ~ 0.3. 41 as a mapping is described in Appendix B. We consider u = 2 for which the largest

Lyapunov exponent, whose positivity signals the chaotic behavior, is « 0.95 in the extended system. The spatial correlation length, computed using the techniques discussed in chapter II, is one lattice spacing.

The PDFs for 14, Vy display Gaussian tails (See Figure 1.5a). The PDF for the normalized scalar fluctuation is shown in Figure 1.5b which clearly shows exponential tails over five decades. We have checked that similar results persist for a range of values of a. As expected from the ao in Eq. 1.24, the slope of the fails is proportional to y/H/div0).

We now argue that this deterministic model exemplifies the mechanism pro­ posed above. The analysis of the autocorrelations of (/’ in stochastic model sug­ gests considering the autocorrelation function C,|(u) = < > — < "o >2 f°i' this problem, where < .. > indicates discrete time average in the usual way.

Note that, in the stochastic model with intermittent amplitude fluctuations G.i(u) displays more than one time scale while the two-point, autocorrelation function

G'2(n) =< v „ v q > — < v0 > 2 exhibits only one. In Figure 1.6, we show (\t(ii) for the deterministic model; it. shows two regimes of exponential decays associ­

ated with time scales tc/ 2 ^ 1, r„/2 ~ 4. The inset shows G ^u) which displays one decay time of approximately 1.6 consistent with the expected decay time of

(1 / t c -f 1 /t„ )-1 . More generally, such higher-order autocorrelations can be used to distinguish well-separated time scales. Also, the above results suggest that, some 42 of the local characteristics of deterministic systems can be analysed by using the techniques developed for the stochastic models.

Note that in Figure 1.5b, the diffusive time scale t j is comparable to t „ and the kurtosis decreases with increasing tj. Such dependence of the PD F’s charac­ teristics on time scales will be discussed later in the Toy model subsection.

Other non-Gaussian behavior : We have also considered other possibilities for the scalar distributions in the stochastic model. For example, the stream function is obtained by the same rule 1/’ = w ', but with an amplitude 7/ having exponential tails instead of a Gaussian distribution. The resultant scalar distributions display stretched exponential tails with an exponent ss 2/5 as shown in Figure 1.7. 1'his indicates that the distribution of the amplitude fluctuations is important rather than the distribution of the velocity itself in determining the shape of the scalar

PDFs. This feature is elucidated in a later section 011 Toy Models.

Large B Regime

We now discuss the model in the regime B > ~ 1 with a Gaussian random ve­ locity field obtained from = 7 /,,(/), where 7/„(?) is a Gaussian noise correlated over a time t c with a correlation length £v. While the PDFs of 0 are Gaussian when B < 1 in both the discrete and the continuum model, for B > « 1 the PDFs are non-Gaussian.

Our study of the continuum model (Eq. 1.17) leads to the PDF's for the nor­ malized fluctuation X shown in Figure 1.8 for different parameters, (i) For B > 1 n

o -2 c

CJ> - 4 -10

-8

°o

0 6 12 18 n

Figure 1.6: Correlation functions of the velocity used in Figure 1.5. A semilog plot of vs n is shown. The straight lines have slopes ss 1.3 and » 0.23. The inset shows a semilog plot of C'2 {n) vs. n; the straight line has a slope 0.69. See text for discussion. (The variance t >0 is set to unity.) The error in the value of correlation function (\ is with in the size of the symbol for n < 16. Figure 1.7: PDF for the normalized fluctuation A' = bO jag for stochastic Model with L = 48, k = 0.1,/? = 0.1, t„ fa 10 and B fa 0.45. The noise amplitude has exponential tails. The variances are ag fa 0.045 and i>o ^ 0.28, and the kurtosis is 7.8. The dashed line corresponds to the stretched exponential with an exponent 6

Data 4 ML fit

2

X 0

- 4

6

-8 15 10 0 55 10 15 X

Figure 1.8: PDFs for the passive scalar fluctuations with J3 >fa 1 using the continuum model. The data are obtained with L = 48, k — 0.1,/i = 0.1 using the discretizations St = 0.05 and SI = 0.5, and velocity correlations £„ fa SI and t c = 2. The upper three curves (a,b,c) are shifted up from the lowest curve (d) by 6,1 and 2 decades respectively. The curves a,b,e,d correspond to ru = I.I 7 where* 7 = (0.4,0.35,0.3,0.25) with kurtosis values (15.6,6.1,4.1,3.5). The dashed lines are modified Lorentzian (ML) fits with S = (1.6,2.6 , 4.21,6.89) respectively for curves a to d. 46 the distributions (a,b,c correspond to B ~ 1.7,1.3,1.05) clearly do not have expo­ nential tails; however, strikingly, we could fit the PDFs very well with a modified

Lorentzian: p(X) = C/( 1 + eX 2)i+8 where C is fixed by normalization of p(X) and c by that of the standard deviation. This function is a variant of the form proposed by Sinai and Yakhot [27], as pointed out in an earlier footnote (15).

(ii) For B rs 1 the tails of the scalar PDFs can be fit by an exponential form over a narrow range of parameters 28 (this case corresponds to Figure' 1.8(1 with

B Rs 0.85). However, the fit, with a modified Lorentzian agrees over a wide range' of B values including this narrow range; moreover, the fit is superior as me’asured by the rms deviation from the data. Also note that in Figure 1.8, one can fit the core of the PDFs with a Gaussian.

When the velocity field exhibits amplitude variations, the scalar distribut ions have algebraic tails but with some deviations from Gaussian behavior at the core.

This behavior of the PDF is expected and is elucidated later in the Toy model subsection.

PDFs of 3-d Model

We have done the simulations for 3-el model and e?xamined scalar PDFs. The results are similar in spirit, exhibiting non-Gaussian behavior as expected in both small and large B regimes depending on the characteristics of the velocity field.

The system size that we explored is smaller, i.e., Lr = Ly = 16 and Lz = 10. The

28A recent preprint by E. S. C. Ching anel Y. Tu reaches the conclusion that the tails are Gaussian, exponential or stretched exponential in different regime's [43], 41

0

-2

X

4 O J

6

8 10 -8 6 4 2 0 2 4

- 2 Ml M

x rT

-15 - 10 -5 10 1b

Figure 1.9: PDFs obtained using the 3-d version of the model: (a) Small B regime: PDF shows exponential tails. The parameters are k = 0.1,/? = 0.1 and B « 0.4. The variances are cro « 0.04, no Rs 0.3 and the kurtosis is 1.1. The velocity field exhibits Gaussian amplitude fluctuations over the time scale r„ « 10. (b) Large B regime: PDF shows Modified Lorentzian Behavior. The parameters are k = 0.1,/? = 0.1, ~ 0.5 and the kurtosis is 7.1. The velocity field is Gaussian with t c f a 1. PDFs, obtained by using a time series over one million time steps, in both small and large B regimes are shown in Figure 1.9. The velocity is obtained from a vector potential A with components /tr , Ay and Az being uncorrelated from each other. The correlations among the components of A did not affect, the behavior of the tails of t he PDF in different, regimes in our explorations. 49

1.4.4 Toy Models

Now, we elucidate the mechanisms from the point of view that local fluctu­ ations do not particularly reflect the nature of the long wavelength behavior in extended systems and may be modeled by simple toy models that, capture the physics at short wavelengths. Note that we have assumed the existence of a char­ acteristic length £o for t he temperature fluctuat ions .29

Small B Regime

Origin for the tails: In this regime, the mechanism for 11011-Gaussian behavior relies on the amplitude variations in the fluctuations (that couple to the scalar) over a time scale comparable to the diffusive time scale of the scalar. As. has been shown earlier, the* shape of the tails crucially depends on the distribution of the amplitude. This suggests that the PDF of the scalar may be interpreted as

/>(./•) = J Q(x/y)R{y)dy, (1.30) where Q is the conditional probability of x for a given amplitude // and R is the distribution of the amplitude. When both. Q and R are Gaussian10, p gets exponential tails as the resultant distribution is /\ 0(|.r|), i.e., modified Bessel

Function of order zero, which has exponential tails for large x. This is simply

2aEven though, we have a small length scale in the problem, e.g., Eq 1.21 where the lattice spacing sets a length scale, algebraic spatio-temporal correlations can occur for the scalar at larger scales; for example, due to features such as local conservation and spatial anisotropy as described in chapter 2.

J0For more complex amplitude distributions one may resort to saddle point integration methods to examine the form. 50 a superposition of Gaussian distributions and such superposition methods have already been adopted in fitting the experimental PDFs in turbulent states [38].

However, the above formulation ignores the time scales and hence cannot ac­ commodate the effects of the temporal correlations of the velocity fluctuations on t he scalar PDFs. We emphasize that, in our approach time scales are naturally introduced in to the problem. Specifically, what we show here is that not only the distribution of the amplitude but also the time scales are import ant in determining the PDFs. This is illustrated explicitly by using the following toy model.

We consider a mean-field-like toy model with only additive noise, since the multiplicative noise term is small for 13 < 1. The model is described by the

Langevin equations,

■r = — k x - f i/i/t, (1-31)

i t = -n -//+ f/2, (1-32) where the variable .r couples to a noise with an amplitude y that fluctuates in time. The* ?/, with i = (1,2) represent white noise withexpectations < ?/, > = 0 and < > = (T'fSijS(l — /'). The time scales int his problem are r„ = 1/n and t,i — 1/ k .

The Fokker-Planck equation for this problem is the following:

OP 0 , d , a\y2d2P

From now on, it is convenient to use scaled variables xq — x / >/1)\ Di> and

Vo — y/y/D 2 where D\ =

^ r ^ 2&Q-\ ° 1 ® n®Q ■ M + y ^ r -] = — 0 - ti) dxQ ' dx0 k R dy0 fyjo

This equat ion can be solved iteratively for small o/w by using Q = 5Z^=o Qn as shown in Appendix A.3. For large .r 0 we have obtained the first three terms of p{x0) = f .7oW?/o expanded as a. formal power serif's:

PM = i t ( n / K)nPn(xo)- ( 1-:ir>) n =0

We have obtained the terms: />o,pi,/> 2. These are combinations of modified Bessel funct ions that have exponential tails for large values of .r(). In particular, t he zerot h order term po(A>) = [ 7r]— 1 /\o(|;t’o|), where / \ 0 is the modified Bessel function of order zero .'11 Note that this zerot h order ^ 1 / ’on can be viewed as a super, ^ of Gaussian [38], and also for the PDF we have an expansion in terms of the rat io of the time scales t,i/ t„.

Starting form the Langevin equations, we can directly calculate the moments as pointed out in Appendix A.3. By using the second and fourth moments, we find that the kurtosis K of the resultant PDF to be,

r* K = 3 d — . (1.36) o + K

31 Note that the exponential tail results from a linear FP equation due to a branch cut in the characteristic function in contrast to Ref. [35] in which a nonlinear equation leads to simple poles in the characteristic function. 52

This interpolates between 9 for small q / k and 3 for large o / k corresponding to p0 and the Gaussian distribution respectively. The Gaussian limit is expected since, in the limit of very large a / k the noise correlations suggest modeling the the Eq. 1.31 by a simple Langevin equation with a white noise.

For intermediate regimes the PDF exhibits exponential tails with a non-Gaussian core. In Figure 1.10 we show the numerically obtained PDF of ,r 0 for o = /,• for which K = 6 corresponding to that of an exponential distribution. Note the devi­ ations from the strict exponential behavior at the con' even though the tails are exponential for this case.

In addition to exponential tails discussed above other non-Gaussian behavior can occur for different amplitude distributions. For example, tin* introduction of another variable c using

i = -p~ + ./•//;, (1.37) in the toy model (Eq. 1.31 and 1.32) leads to a stretched exponential PDF for c with an exponent 2/3 for appropriate n, k and //. This result is consistent with the PDF obtained in the stochastic model with an amplitude (of velocity) having exponential tails in its distribution (see Figure 1.7). We can further extend this approach hierarchically and eventually obtain log-normal distributions which ap­ pear to be successful in describing observed statistical features of t urbulence [14]. 53

0 T- -

2

- 4 cn

-6

8 12 6 0 XO

Figure 1.10: PDF for :r0 = xja r in the toy model of Fqns. 1.31 and 1.32 will) a = k. The noise variables i]\ and //2 are uniformly distributed with ax ~ a2 = 1 and the measured ar = 0 .5 . Large B Regime

Since the convective noise term is not negligible we model the D >ss 1 regime by a single variable with both an additive (Ci) and a multiplicative (( 2) noise term:

X = -KX + Cl + ( 2 ^ - ( 1 -;18)

When C; arf> white noise with < Q(t) >— 0 and < >= 2D,bKlb{1 — /'). the PDF p{x,t) satisfies the following Fokker-Planck equation:

m = l t {Kxp) + + lhx2]p (1 M)

'Phe stationary solution for this equation should satisfy:

(k-i'P) + + l h * 2]p = 0 ( 1. 10)

The equation we have here is exactly the same as Eq. 1.8 derived from the moment balance relation by Sinai-Yakhot [27]. This has the solution:

]>{X) = (1 + c.t2)i+s ’ (1-11)

Conseciuently, the single variable model with a multiplicative noise yields exactly the modified Lorentzian form 32 for the PDF with c = D2 /D \, 6 = k/'IDi-

32Note that to normalize the standard deviation, we need to rescale x so that r = D-j/(k — />>)• Also, for k < D> the variance is infinite and this normalization is not possible. For such cases, one has to he careful while writing an expansion for

-6

-8 15 10 5 0 5 10 15 X

Figure 1.11: PDF for A' = xfa r in the toy model of Eqns. 1.42 and 1.43 with o = />. The noise variables // 1, 7/2 and (2 ai‘p uniformly distributed between -1 and 1. 56

Other Toy Models

Wo can consider a toy model with both amplitude fluctuations and multiplica­ tive noise terms.

X = -KX + ?/?/] + ^C'2, ( 1 -12)

?/ = -aty + Tj 2, (1.13) where 0 is replaced by //?/] in the Eq. 1.38. The analysis of the Fokker-Planck equation is complicated, but one can examine the PDFs numerically and fit the experimental data, of interest. The main feature of this model is that, even though the tails become algebraic when the multiplicative term is dominant, the core of the PDF remains non-Gaussian (See Figure 1.11), as is observed in several fluid experiments on temperature (see for example the data presented for temperature

PDFs in Kef. [31]). When B ~ 1, as stated earlier, it is conceivable that both the mechanisms could be co-operative in fluid turbulence. In that case this would be an appropriate toy model to study scalar PDFs. For example, note that the estimated B values for the stirred fluid experiment [7] are dose to unity; the above toy model is expected to be relevant for this experiment.

Remarks on Toy Models

One may view our toy models as extensions of the Langevin equation (that describes the well known Ornestein-lJhlenbeck process [44]) which is well studied in the context of Brownian motion (see Appendix A.2). While our toy model ap­ proach seems promising especially in understanding the shape of the PDF, there are some issues that need to be clarified. The toy models can fail in some cases; primarily due to dimensionality or constraints such as local conservation. For example, a 1-d diffusive equation with additive noise does not yield stationary distributions for the local fluctuations of the diffusive field since <7 = 0 mode it self executes random walk. While the toy models can mimic the shape of the distri­ bution, they cannot capture features such as temporal correlations in extended systems. For example, a one dimensional diffusive process with an additive con­ serving white noise exhibits algebraic autocorrelations (r'2(/) « I//'/2) for local variables (as shown in the next chapter), but a simple toy model with one variable with additive white noise does not. 58

1.4.5 Multi-Variable Fokker-Planck equation

The evolution equations for the passive scalar in the model defined by Eq. 1.20 can be treated as coupled Langevin equations when the velocity field is stochast ic.

The analysis becomes simpler for the case when the velocity is derived from a stream (or potential) function that has delta,-function correlations in both space and time. With this simplification, one can make an attem pt to obtain an effective

('volution equation for the PDF of the scalar at a single site in the middle of t he system by adopting a reduct ion of the multi-variable Fokker-Planck equation corresponding to these coupled Langevin equations. One may wish to call this a mean-field-like reduction procedure which gives a clue for t he appropriate form of the terms entering into the toy models.

Following the above procedure, we now obtain a single variable FP equation in the large B regime and a few variable FP equation in the small B regime of the passive scalar model (Eq. 1.20). One can readily see that these FP equations correspond to the Toy models described in the previous subsection.

Large B Regime

Since the mean scalar profile is linear along z in the presence of a gradient, the scalar fluctuations due to a noisy velocity field can be expanded around the mean value (see Eq. 1.22) leading to:

0(v,t) = KVl0 + ii(r), (I-14) 59 where ij is the stochastic term given by:

*l(r) = ~ V(r) ' V L0(f,/,), (1.15) which has two parts (one is a simple additive part associated with the mean solution and the other is a multiplicative part) and has a mean < >= 0

(< .. > indicates the noise average).

To carry out further analysis, it is useful to define the quantity:

< 7/(r, /)/;(r /') > = 2D(r, f ')&{!. — /,'). (1.10)

In particular,for the velocity field used in the model, wherethe derivatives are defined as symmetric lattice gradients, we have,

/;(/-, r) = I),fj(r) = ^ { /i2 + ‘2/tV i/~0(r) + £ V La0(?)VLa0(r)]. ( 1,17) ~ (I

Summation on a is over all

We can write the Fokker-Planck equation for the Probability distribut ion func­ tion for all the variables P{{0(rj)}) as the following:

f = E jfefrEWM - +E ooSo(p)W f'n i‘] (us)

In this equation, the sum on » is over all the nearest neighbours of site r and the sum on f and r ' is over all the sites in the lattice. The values of 0 an* fixed at the boundaries c = 0 and z — Lz + 1 and the above equation should be interpreted appropriately at these boundaries. We are only interested in the PDF of 0 — 0^ at a site f0 in the middle of the lattice which is sufficiently far from the boundaries 60 so that the boundary effects are negligible (this will be justified further in the next subsection). Hence, we integrate the above equation 1.48 over all 0(f) except

0(7ro). While integrating, we demand the probability to be zero for 0(f) = oc, since finite probability at 0(f) — oc is unphysical. Due to this reason, all the terms of the right hand side of the equation with derivatives djd0(f) vanish except the ones with d/dO(fo). Moreover, while carrying out the integrations, since we are interested in the PDF of 0o at, a site r0, it is convenient to write

/'({«(’•))) = i>(»o)g(t»(0}i»„). (i,mi where Q is the conditional probability of all other 0(f) for a given 0a.

After the integration, the evolution (FP) equation for p(0a) would take the form ^ r 1 = ikM{0M + iklFit°',p] (l-r,0) This equation looks similar to the single variable FP equation (see Appendix A.2) with F(0o) and f(0o) being the explicit dependences in the diffusive and drift parts respectively. The functions F and f are given by:

(i) Diffusive pari:

F(0o) = J Dejj(f0)Q, (1.51) which is the conditional expectation of Dtjj and the integration is over all 0(f) except 0Q. Using Eq. 1.17, the function F(0o) can be written as:

F(0o) = ~ [ ^ + 2l3hz(0o) + '£ g a(0o)], (1.52) 61 where, h and g are conditional expectations given by:

ha{0 o) = < ^ L ( i0o \0q > (1.53)

f)a(@o) — < (V;jO0o)2|^o > ( 1 .5-1 )

Here, the < .. |#o > indicates the expectation of a variable for a given value of 0(). i.e., it represents the integration over the conditional probability which is what we performed earlier in Eq. 1.51 for F(0o).

(ii) Drift part: The function f(0Q) is defined as:

71 with the summation on n being over all the 2d nearest neighbours of r{). Each fn(0o) is given by:

f n(0o) = < 0{rn)\0u >, (1.56) which is the conditional expectation of the 0 at the neighbouring site it for a given

0o.

One can seek a variety of symmetric PDFs using the following stationary solution 33 of the above equation 1.50:

C . r°o nf(0')d0', ,i0°) = TW )'r!'ll " w r ] with the following criteria:

33This is exactly the form obtained by Pope and Cliing in their general formulation [32]. 62

We need, for any m > 0, f (0)2m+i p(0)d0 — 0 for the symmetry of the p(0o).

This can be achieved when: (1) F(0o) is a symmetric function of 0Q, i.e., the conditional expectations /?. and g are symmetric in 0o.

(2 ) / is an odd function of 0O as we have a derivative of 0o in the drift part of Eq. 1.50. This condition is satisfied when fn(0) is an odd function of 0o.

Moreover, since we expect the distribution to be the same at all sites in the middle of the system, we must have < 0(rn) >= f fn(Oo)p(0o)d0o = 0.

Now using the lowest order possible dependences of the functions / and F that satisfy the above criteria, i.e., with a / linear in 0Q (with a negative coef­ ficient) and a function F quadratic in fl0, it is straightforward to check that the above symmetric solution (Eq. 1.57) yields exactly the modified Lorentzian for the scalar PDF. Thus, similar to the moment balance approach [32], not only the denominator of the integrand inside the exponential but also the numerator can have corrections in Eq. 1.57. Moreover, the quantity F can be simply viewed as the variance of the effective noise '5'1 that couples to the variable in the effective

Langevin equation for 0u- In other words one can write an effective Langevin equation for the FP equation given in Eq. 1.50 as the following:

0o = Kf(0o) + f], (1.58) with the expectations < g >= 0 and < g(t)g(t') > = F(0o)S(i — /').

34Note that, unlike the function < 7 0 (3 :) (see Eq. 1.6) which is only the conditional expectation of the square of the gradient, other terms such as the expectation of the gradient itself also enter into the function F(0a) here. The other terms could be important, while fitting the experimental data. 63

O o

O o o o

in oI 1 0 - 0.5 0.0 0.5 1.0 1. 0 - 0.5 0.0 0.5 1.0 0 6

o o

o o o d

»n oi 1. 0 - 0.5 0.0 0.5 1 0 .!: 0 0

o

Q} o o

1. 0 - 0.5 0.0 0.5 1

o o

O O'** O

CM (N o o I I 1.0 1.0

Figure 1.12: The conditional expectations for the 2-d model with the parameters of Figure 1.8b: The conditional expectations /,,,/t and g are shown. In large B regime, it is easy to check the validity of this reduction procedure by using numerical simulations. We consider the 2-d model defined by Eq 1.21 with a Gaussian velocity field and explicitly compute the expectation values us­ ing standard techniques. The computed f n(0o) are shown in Figure 1.12. The expectations of /„ transverse to the applied gradient direction are odd functions and have clearly linear parts with a large scatter for large 0o with f +T zz /_ r . On the other hand, along the direction of the gradient the situation is different as

/+- /_ ; (see Figure 1.12). However, /+- -f /_ . is an odd function, making / an odd function. Moreover, the function h~ oc [f+: — /_;], which is the expectation of the local gradient along z, is an even function (see Figure 1.12). The functions

<) are also symmetric in 0o as shown in Figure 1.12. In addition, the functions h and (j can be approximated by a constant, plus a quadratic correction in 0U.

'Phis behavior of the conditional expectations is in agreement with the observed

Modified Lorentzian PDF. In summary, the reduction procedure we have adopted here, describes the shapes of local PDFs of scalar fluctuations in B > 1 regime very well.,!r’

1,5Note that, in experiments typically one finds asymmetric (skewed) scalar PDFs near the boundaries, see e.g., Refs. [5, 7]. Such behavior may be understood by appropriate corrections in the functions f and F in Eq. 1.50. We also note that one finds asymmetric PDF’s in the vicinity of the boundaries in our extended model Eq. 1.20 and thus may be a good starl ing model to explore issues related to boundaries in future. One issue that may be of interest, both experimentally and theoretically, is to examine how the skewness goes to zero as one goes farther away from the boundaries. Small B Regime

In this regime the multiplicative term can he neglected, but the amplitude

fluctuations enter into the effective equation. We can start with a set of Langevin equations for the entire system, including the equations for the amplit ude i/A in

the stream function i /1 = Va7}'- F°r example consider:

VA{ri) = -ni)A(fi) + Ci, (1 -59) where C is a white noise anti is uncorrelated from site to site with zero mean and width (Tq. My using the reduction procedure, since K is a derivative of the stream function, we can obtain an evolution equation for the PDF of the scalar and the amplitudes at, two sites, lJ{0 a,ijAi,i}A2 ), the following:

M t r,n\n\ + 7I42) P 7 ^

This equation is basically the FP equation for the Langevin equations of the toy model with two uncorrelated noise terms with amplitudes i]A\ and i\Ai. Note that, one can increase the spatial correlation and reduce this to the Toy model of

Hqns. 1.31 and 1.32.

In summary, the above analysis, in both small and large B regimes, suggests that one can adopt, a coarse graining procedure along with a stochastic descrip­ tion to study the effects of the underlying couplings and the time scales of the fluctuations on the shape of the PDF of a local diffusive variable in an extended system. Even though this procedure appears simple for the passive scalar coupled to velocity fluctuations (since we could treat the velocity as a stochastic field), for 66 other cases viz velocity and its gradients, however, the situation is complex. We hope that the methods and mechanisms presented here might help understand the shapes of the PDFs in more complex systems. 67

1.4.6 Role of Boundary Conditions

In addition to a simple gradient boundary condition in wliicli the scalar values arc' kept fixed at z = 0 and z = L~ + 1 boundaries, we have also considered other cases in which the scalar boundary values themselves exhibit spatio-temporal fluc­ tuations. This study is motivated in part by the observation that in experiments such as Ra.ylcigh-Bena.rd convection and stirred fluid experiments, large gradient drop occurs in the thermal boundary layer and the effective gradient in the mid­ dle of the system is considerably smaller than the applied gradient. The thermal boundary layer can exhibit both spatial and temporal fluctuations. If we were to model the region between the thermal boundary layers, then it is important that we take the boundary fluctuations into account.

So far, we have focused on the mean scalar profile which characterizes the additive and multiplicative noise 1 terms that enter into the evolution c , 1767 (sex 1

Kq. 1.2 2 ) of the scalar fluctuations. In addition to t he mean profile, we now argue 1 that the variance 1 ag(z ), i.e., as a function of c, would provide useful information regarding the dynamics, especially associated with the boundary fluctuations. VVe illustrate this by considering our 2-el model using a Gaussian velocity field with the following boundary conditions: The scalar value 0^ at each site i at, z = 0, L: -f 1 boundaries is assumed to independently evolve, according to the simple Langevin dynamics,

Qbl = -fiOg, + C,, (1.61) 68

CT*a) 1 O 7 —* „

0 1 2 -2 ? logc z

:>4 48 /

Figure 1.13: The variance profile for a give velocity distribution. The inset shows Iog-Iog plot and the drawn straight line has a slope —3/2. The error in the variance* data is within the size of the symbol used.

where* C is a white noise with zero mean and unit variance. The time* scale* 1/// is the correlation time for the boundary fluctuations. For this evolution, < o,„ >= 0 and the mean scalar profile is flat in our simulations.

In the above two-dimensional model with boundary fluctuations, we find that erg decays to the middle of the system as shown in Figure 1.13. Moreover, as is shown in the inset, ag decays roughly as 1/c3/2, where c is the normal distance

(i.e., along c direction) from the closest boundary36. This behavior tells us that if

JhThe decay of

(correlation lengths) from the boundaries. Thus, in experiments, measurement of this variance profile might give us a clue regarding the boundary eflec.ts on scalar fluctuations.

We now describe a few qualitative features of the PDFs when the boundaries exhibit fluctuations following Kq. 1.61. The PDFs are found to be non-Claussian with some skewness in the vicinity of the boundaries. As one goes slightly farther from the boundaries, PDFs gradually become symmetric, however, exhibit expo­ nential tails or algebraic tails depending on the magnitude of the* velocity field. As one goes towards the middle, the PDFs tend to become closer to Gaussian for the case with a Gaussian velocity field. When the velocity field exhibits amplitude fluctuations, PDFs remain non-Gaussiau in the middle exhibiting appropriate non-Gaussian tails; this behavior can be understood by employing appropriate

Toy models in small 13 regime described earlier in Section 1.1.1.

of the fluctuation at a boundary site, where r is the distance between these sites. This leads to a contribution of \fr~d to

3‘Note that the time series data presented in Figure (i of Ref. [1] on Rayleigh Benard system appear to show this feature. The width of the fluctuations at the boundary is much larger than the width in the middle of the system. We have also considered stationary but non-uniform boundary conditions. For

this case, as expected, the shape of the PDF shows site dependence. When the

mean gradient is reduced to zero, still keeping the spatial variations at the bound­

ary, for the case with a simple Gaussian velocity field, the shape of the local

PDF crucially depends on local gradients of the mean scalar profile. The t ails of

the non-Gaussian PDFs in the middle of the system can be fit with a modified

Lorcntzian in some cases. Also, the < (To{z) >, which is averaged over the other

dimension .r, becomes smaller as one goes to t lie middle of the cell.

We conclude this discussion on boundary conditions by admit ting this work as

preliminary. More understanding along these lines, especially for large magnitudes of velocity fields, remains an open problem. 1.5 Another Extended System: Josephson Junction Array

In addition to the passive scalar models studied in the previous section, non-

Gaussian PDFs can also occur in a variety of extended dynamical systems. As an example, in this section, we consider a model for the square array of capacitive

.Josephson .Junctions'38 and examine t he I’DFs of local junction voltage fluct uations in chaot ic states (which typically occur over a range of parameters when the array is harmonically driven by uniform input/output currents [51]). The main result that we present in this section is that in these arrays, the PDF of the voltage across a junction transverse to the input current, direction exhibits exponential tails in chaotic states. This behavior is in contrast, to the Gaussian beliavior of the IPDI's in noisy periodic states of the arrays.

We present a mechanism by which non-Gaussiau behavior arises: specifically, we argue that the coarse-grained dynamical equations contain an effective noise term characterized by slow random variations of the amplitude leading to expo­ nential tails in chaotic states. We attribute the variations in the a:.'., , 6108J 1 1 to the random generation and motion of vortices in the chaotic states. This mechanism is closely related to the one that we presented in small B regime of the passive

38Note that a single Capacitive Josephson junction (i.e., a junction between two superconduct­ ing islands) is well known to exhibit a variety of non-linear phenomena including chaos when harmonically driven by an external current [45]. In recent years, experimentally, square ar­ rays of such junctions are shown to manifest interesting current-voltage characteristics (e.g., Shapiro steps) and unusual dynamical states [46, 48]. Such dynamical behavior of these arrays stimulated considerable interest, both theoretically and experimentally. Note that, in addi­ tion to external currents, applied external magnetic field can also significantly influence the behavior of the arrays. For a recent review on these arrays, see Ref. [50]. 72 scalar model in the previous section 1.4.4. We emphasize that, this study not only suggests that the phenomena, of exponential tails is widespread, but also motivates a study of coherent structures such as vortices and their role on the local PDFs in turbulent states.

A discussion on the local and global behaviors of chaotic states in capacitive

Josephson junction arrays (JJA) from the point of view of extended dynamical systems is provided in our recent paper [52]; also, occurrence of a variety of dy­ namical states including periodic, quasi-periodic and chaotic states in capacitive

Josephson junction arrays is discussed in our earlier paper [51].

1.5.1 Model

Our model for a JJA consists of superconducting nodes described by the phase

at points r of an L x L square lattice (see Figure 1.14). The HSJ model [49] is used to describe the junctions between the nodes. Each junction has resistance

It, capacitance C, and critical current Jc. We apply an external driving current i-crt = Uc + hf sin(u;o/), normalized in units of /r, to each node along the edge of the array at x = 1 and extract the same current from each node at, the edge

.r = L.

The dynamical evolution of (f)(r) is determined by charge conservation at each node [51]:

X [^(^)-^(n) + «(^(»v)-^ (n )) + si»(^(jV)-0 (n )) +»W,] = (1.02) n£t'(r) 73

i

n u E X n m ^ nrnrrixxG3^ i l i n L i u ^ n ^ Pljnrrixrn^ K P t J n r r i x ^ W n x j i r p ^

m

Figure 1.14: Schematic picture of an 8 x 8 Josephson Junction Array: The crosses with circles in the middle denote superconducting islands. The thick lines indicate Josephson junctions between neighbouring islands. Input/output currents along x direction at left/right edges are also shown. 74

Here, u(f) denotes the neighbours of site r, dots represent time derivatives, and time is measured in units of ' = JhC/2cl c\ a is related to the McCumber parameter (3C = u 2R2C2 by a = fi~xl2. Finite temperature is introduced by a

Langevin noise current across the junction between adjacent nodes r and r\.

The noise has zero mean and correlations given by < i/viv, (/,)?yv^vj(F) > = a2h(l —

where a2 = 2kfjTu)1,/R I2. We restrict our discussion to .J.JA in zero magnetic field. The evolution equations 1.62 are numerically integrated on a Cray Y-MP using a fourth order Runge-Kutta method along with the techniques discussed in Ref. [61].

1.5.2 Chaotic States

We find chaotic states for both i,ic = 0 and idc ^ 0 similar to the case with the single capacitive junction [45]; note that we use the positivity of the largest

Lyapunov exponent A as the signature of the chaotic behavior (see the next chap­ ter). For finite idc chaos typically occurs for parameters that cones, : 64 1 to t he regions between the plateaus39 in currcnt-voltage (idc-V) characteristics [51]. Wo work well within the chaotic regime and present results for two typical sets of parameters Pi and /J2 which are 0.69, irf= 1.154, u;o= 0.1 x 27r. with /3,-=-\ for Pi and M-=(i for many additional sets of parameters have been investi­ gated with qualitatively similar results. For comparison we study the behavior of laminar states, i.e., states with A non-positive that are either time-independent

•19Notc that the voltage fluctuations in a chaotic state are typically few hundred times larger than the noisy voltage fluctuations expected due to thermal effects. This feature may be experimentally interesting since one can study the dynamical behavior of the system in the presence of large fluctuations in chaotic states. or periodic with period To = ‘I tt/ujo on the average. In chaotic states we focus on the dynamics of junction phase differences s„(r) = [{r — c„)], voltages va(r) = [<^(r) — — ea)], and supercurrents Sa(r) = sin[.s„(r)] where the subscript, a denotes x or y and e„ is the unit vector along a.

In chaotic states the power spectrum of a longitudinal junction voltage shows a characteristic broadband with peaks at the harmonics of the driving frequency.

On t he other hand, a transverse junction voltage shows the broadband behavior without any peaks. We have computed equal-time spatial correlation functions of junction voltages and supercurrents in chaotic states. We find that the correlat ion lengt h along x is approximately 1 in units of the lattice spacing and % 3 — 1 for parameter set. P\ and 5 — 6 for parameter set l}2 .m Also, our study of global features suggests that the chaotic state can be considered as fluctuations on top of an underlying periodic state with a period To [52].

1.5.3 Local Properties

We now discuss the distributions of local variables in the chaotic states of a

JJA. Since the chaotic state can be viewed as fluctuations superimposed on top of an underlying state with period To, we define the distribution of a junction voltage, using its values at times I -f j r 0 with j = 1,2 N for 0 < / < To.

In Figure 1.15, we show the PDF for the voltage distribution across a transverse

40Since the underlying state lias a periodicity To the eyual-time spatial correlation function of f(r,t) is defined by Gj(r,t) =<< /(/? + r,t)f(R,t) >/{>r - << f(R + f,i) >r< f(ltj) >r>n where the subscripts 11,T indicate spatial and time averages. The time av­ erage is taken over times t + jt0 with j = 1,2,..., JV. Also, note that the spatial correlat ions for large f could be algebraic due to local current conservation in the problem following the results presented in the next chapter. junction vy(r) in two typical chaotic states. The distributions show a Gaussian core and well-defined exponential tails. This behavior is independent of the choice of / as is to be expected since the mean value of the vy(f) vanishes. We have checked that the exponential behavior persists in experimentally more relevant situations. In particular, it is present in the voltage distribution for arbitrary time series (not necessarily at intervals of r0) and also in the presence of small

(10%) disorder in the critical currents of individual junctions. The distribution of the voltage differences between two nodes along the transverse direction becomes

Gaussian as we increase the distance between the nodes beyond the (transverse) correlation length. Thus one needs to measure a time series for the voltage across at most a few junctions to see exponential tails in the probability distribution.

Note also that the time window for the voltage measurement must be an order of magnitude smaller than r0. In contrast to its behavior in the chaotic state, the distribution of local voltage fluctuations is Gaussian in any laminar state in the presence of noise currents.

The distributions of longitudinal junction voltages in chaotic states are signif­ icantly different from those of transverse junctions. They are broadly asymmetric without long tails, similar to distributions of single junctions; also, the form of the distribution depends on / which is not surprising since the mean value of the t>j(r) itself depends on t. Again one recovers a Gaussian distribution for voltage across nodes separated by distances larger than 77

O

/--N '-->- n_ o o k S CJ> 1 o o lO

'- 2.0 - 1.0 0.0 1.0 2.0 v

Figure 1.15: The distribution of the voltage difference across a transverse junction shows non-Gaussian exponential tails. The upper curve corresponds to parameter set P\ (shifted up by one decade) and the lower curve to l\. The data are obt ained with L = 12 using 500,000 points at intervals of r0. To elucidate the behavior of PDF’s we rewrite Eqs. 1.62 in terms of local junction phase differences .s„(r), voltages v„(r ), and supercurrents Sa(r). For convenience we express the noise current isr,r-ra as iN„(r). Then v„(r) and s„(r) evolve according to:

va(r) + m\,(r) + ^ Clab(r, r, )[,S,(Fi) + f/v/,(rj)] = i0„. (1.6.'!) 6,f|

.s„(r. /,) = v„(r,t), (1-61) where i0r = i^c + irfs\n(u}at), ?oy = 0, and Ctai, are lattice Green’s functions.

In a laminar state Fqs. 1.63 and 1.61 can be linearized and assume the stan­ dard form of Langevin equations. One can solve the corresponding Fokker-Planck equation and obtain a Gaussian PDF for u„(r) with a width ~ yV 2/ln . Our numerical results for laminar states are in good agreement, with this estimate.

Consider the dynamics of a transverse junction. That is relatively simple because (i) there are no phase slips (of 27r) resulting from vortex motion caused by the Magnus force (see Ref. [53]) which is t ransverst' to the direct ion of the net current flow and (ii) the mean value of the t ransverse voltage vanishes. Since the state exhibits coherent oscillations with a period To = 27t/u.’o, we coarse grain the equations in time in units of r0. Then we can write the evolution in discrete t ime n in the following discrete Langevin equation form:

v(n + 1) = (1 - 7 )e(?j) + C(»), (1.65) where £,"(») isan effective noise term. The characteristicdissipation time 77 for this dynamicsis 77 = l/\loge(l —7 )!; 7 is set by choosing 77 ss 2/( 077 ) in units of 79 r0. The detailed behavior of the effective noise £ is complicated because it arises from the sum over the supercurrent fluctuations weighted by the Green’s functions in Eq. 1.63.

We will first show that Eq. 1.65 yields exponential tails for the PDF of ?>(??.) if the noise satisfies certain conditions and then argue how these conditions can arise from the dynamics of vortices. Assume that, £ has an amplitude that, varies on a time scale large compared to the dissipation scale, i.e., £(n) = A(n)j/(n) where ?/ is a Gaussian white noise and A is the amplitude which is assumed to have Gaussian tails with a correlation time t„ > t j . We obtain v(n) numerically and find exponential tails in the PDF as shown in Figure 1.16. The continuous time version of this model is described in the previous section (see Fqns. 1.31 and

1.32) while discussing the origin of exponential tails in the passive scalar PDFs in small B regime.

The dynamics of the vortices can yield a noise with the above-mentioned char­ acteristics. A snapshot of a typical vortex configuration is displayed in Figure 1.17.

From our numerical studies, we find the mean number of free vortices scab's as

/,2/(£r£y). Thus we can, roughly speaking, associate a free vortex with each corre­ lation volume and interpret the dynamics via the motion of these vortices. Since the effective noise £(w) is caused by the supercurrent fluctuations, the motion of vortices causes the amplitude of £ to vary. It is reasonable to estimate the charac ­ teristic time Ta for these fluctuations as the time during which the vortex moves a distance ~ £y. We find ra = (2. ±0.4)ro from numerical simulations for parameter 80

O o

o C \l

qT q o cn I O _i o co

o 00

- 2.0 - 1.0 0.0 1.0 2.0 v

Figure 1.16: PDF for v(n) for the model defined by Eq. 1.65 with 7 = 0.85 (/'.r, Tj « 0.5), r„ w 2 and variance of £ rs 0.1. cSl

y A

> X

Figure 1.17: A snapshot of the phase configuration in a 16 x 16 array for the parameters P\. Positive/negative vortices are shown by crossed/emply circles. 'Fhe current How is along x and the motion of the crossed (empty) vortices is along negative (positive) y. sets P] and P2- For these parameters, = 2/a « 0.5ro yielding t„ > rj. We also

find that the slopes of the tails in Figure 1.15 are consistent with the expectations

based on the model defined by Eq. 1.65.

Finally, we note that the dynamics of a longitudinal junction crucially dif­

fers from that of a. transverse junction in that vortices move directly across it.

This phenomenon along with the non-zero drive term [See Eq. 1.63] render the

assumptions made above invalid and no simple distribution need be expected.

In summary, we have shown that exponential tails occur for the PDFs of the

transverse junction voltage fluctuations in chaotic states of capacitive Josephson

junction arrays; and also proposed a mechanism for the origin of these tails based

on the random motion and generation of vortices in chaotic states.

1.5.4 Experimental Issues

To obtain the PDF of voltage fluctuations experimentally, one has to measure

the time series with sufficient accuracy, i.e., since the underlying state has a period

equal to the driving period (to) in the chaotic states, the window of measurement

should be much smaller than the driving period.

Typically, capacitive arrays have the following experimental characteristics for

each individual junction (see, e.g., Ref. ['17]): (i) Frequencies : order of 1 to 100’s

of GHz; (ii) Resistance : order of few Ohms; (iii) critical current : order of milli

to micro Amperes; (iv) capacitances : order of 10~lu Farads; (v) dc voltages : order of milli to micro Volts. The fluctuations in individual junction voltages

(especially in chaotic states) are expected to be in the order of micro volts, since in our ,simulations we find that the width of the chaotic fluctuations is in the order of dc voltage in certain regimes. Thus experimentally one could characterize the chaotic fluctuations by a power spectrum.41 Moreover, in the near future by taking advantage of the new high frequency measurement technology, one may try measuring the Fourier transform of the voltage signal and convert, it back into time series and thereby examine the local PDFs.

We end this section by noting that it would also be interesting to see the phenomena of exponential tails in the PDFs of local variables in other non-linear experimental systems.

11 Not.p that, power spectrum of the voltage across a non-capacitivc array has already been measured in Ref. [48]. Also, these arrays have smaller individual junct ion dc voltages ranging from nano to micro volts in the frequency range 10fl Hz. 1.6 Summary and Conclusions

In this chapter, we have presented an approach based on Langcvin equations

to study the PDFs of local variables in extended diffusive dynamical systems.

We have proposed two mechanisms by which non-Gaussian behavior arises in

local PDFs and illustrated these mechanisms by using simple models of passive

scalar (temperature) advected by turbulent velocity fields. In addition, in the

passive scalar models (see Eq. 1.20 and 1.21), we have empirically identified a

parameter 13 that measures the relative importance of additive and multiplicative

noise couplings and demonstrated the validity of the two mechanisms (?.r.. one in

small 13 and the other in large 13 regimes).

The first mechanism, which is operative in the “additive” (small 13) regime,

relies on the temporal correlations of the velocity field and provides an explanation

for the origin of non-Gaussian PDFs with exponential and stretched exponential

tails for local diffusive variables in extended systems. We have shown that when

the velocity field exhibits amplitude fluctuations over a time scale comparable

to or larger than the diffusive time scale, this mechanism is operative for the

passive scalar. Moreover, the shape of the PDF tails depends on the distribution of the amplitude of the velocity field. We have also shown that the existence of

amplitude fluctuations in the velocity field can be investigated by computing tin*

higher order autocorrelations.12 In addition to stochastic models, we have also

42Note that the amplitude fluctuations may be related to the motion of the coherent struc­ tures [24, 25] such as vortices in the turbulent state. Also, the time scale ra may be taken as the mean persistence time of such structure at the spatial point of interest. devised a deterministic passive scalar model with a velocity field that exhibits amplitude fluctuations and gives rise to exponential tails for the scalar PDFs. This

study also demonstrates the applicability of stochastic methods while exploring some time scale issues of the deterministic chaotic dynamics at short wavelengths.

The second mechanism, occurs in the large B regime, is based on the '‘mul­ tiplicative” coupling between the diffusive field and the fluctuations. Unlike the first mechanism, this yields algebraic tails for the scalar PDF (in particular, the

PDF assumes a modified Lorcntzian form). In this regime, we have also derived a form for the scalar PDF which agrees with the form derived using moment balance' approach [27].

We have elucidated both the mechanisms using few variable Toy models and presented a reduction of multi-variable Fokker Planck equation that provides some justification for the use of toy models for the description of local PDFs in extended systems. The toy models may be viewed as extensions of the Langevin model used to study fluctuations in Brownian motion [Tl].

('onIndians to experiments: We focus on the grid turbulence [5] and stirred fluid [7] experiments for which B values are available. Recall that these exper­ iments have shown the existence of exponential tails in the PDFs of local tem­ perature fluctuations. In the presence of a mean gradient /i, the B value in grid turbulence experiment appears to lie in the range 0.3 to O.fi, where exponential tails are prominent (see Fig.3b in Ref. [()]). This suggests that the mechanism 86 presented in small B regime, where the amplitude fluctuations on appropriate time scales are crucial, is applicable here. The intermittent amplitude variations in the velocity field may be connected with the structures such as vortices and small scale intermittency of the velocity field. In the stirred fluid experiment,, the value of B is close to unity. Hence, both the mechanisms can be co-operative leading to non-Gaussian PDF tails in this narrow region. However, it is yet, to be experimentally verified whether modified Lorentzian form (i.e., large B mecha­ nism) occurs in highly nonlinear regimes. On the other hand, in Rayleigh-Benard convection experiment [1], where temperature gradient, drives the system into a turbulent state, the situation is not clear, since the behavior of the mean scalar profile is not, experimentally well characterized. Since there appears to be a large temperature drop in the thermal boundary layers, the fluctuations in the thermal boundaries might, play a crucial role on the temperature PDF in this experiment.

Our study on the role of boundary conditions indicat es that, in •addition to mean scalar profile, variance profile also provides insight into the internal dynamics and t he effects of the fluctuating boundaries.

Other extended system: We have also shown that the phenomenon of non-Gaussian

PDFs is more widespread by illustrating its presence in a completely different ex­ tended dynamical system: a capacitive “Josephson Junction Array” that, exhibits spatio-temporal chaos when driven by both external dc arid ac currents. Specifi­ cally, numerical simulations in chaotic states indicate exponential tails in the PDFs 87 of local junction voltages transverse to the external current. This behavior can

be ascribed to intermittent current fluctuations caused by random vortex motion.

In particular, we have shown that the mechanism (in small B regime) that relies on additive coupling is operative in the chaotic states of a, Capacitive .Josephson

.Junction Array while understanding the non-Gaussian behavior of the PDFs of individual junction voltage differences. Thus, this study provides some motivation to explore the role of coherent structures such as vortices on the local PDFs in turbulent states.

Future research: (1) Establishing the occurrence of amplitude fluctuations start­ ing from the underlying equations, e.g., Navier-Stokes equations with appropriate boundary conditions, remains a challenging problem; (2) another is to examine the scalar PDFs in the presence of turbulence (with vortex structures) generated by simple geometries, e.g., a back-step flow problem, using the techniques devel­ oped in computational fluid dynamics; (3) generalization of our Langevin equation approach to other quantities and with more complicated noise correlations; (1) a detailed study of the role of boundary conditions.

In conclusion, in this chapter, we have devised models; proposed mechanisms; analysed toy models; justified the relevance' of t oy model approach to the extended systems; to understand the occurrence of non-Gaussian behavior in the PDFs of local variables in extended systems. Our stud)' indicates that local PDF shapes in extended systems are lion-universal and crucially depend on the physics associated with the couplings at short wavelengths and the time scales. L ist o f R e fe r e n c e s

[1] B. Ca.st.ia.ng, G. Gunaratne, F. Helsot, L. K ad an off, A. Libchaber, S. Thomae. X.-Z. Wu, S. Zalcski and G. Zanetti, J. of Fluid Mech., 204, 1 (1989).

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88 89

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[53] See e.g., M. S. Rzchowski, S. P. Benz, M. Tinkham, and C. J. Lobb. Phys. Rev. B 42,2041(1990). C H A PT E R II

Generic Scale Invariance in Conserving Chaotic System s

2.1 Introduction

During the last decade it has become clear that a broad class of extended (>.(., many-body) non-equilibrium (i.e., driven) systems with external white noise ex­ hibits Generic Scale Invariance [1] “correlations that decay slowly like power laws in both space and time without the tuning of external parameters.'’ The first example of this phenomenon was provided by fluids in a temperature gra­ dient, where the static structure factor1 was argued theoretically to diverge* for arbitrary parameter values [2], Recently, this prediction has also been verified experimentally [3], In addition, scale invariance in nonequilibrium systems is of particular interest as it occurs for arbitrary parameter values in contrast to the situation in classical equilibrium systems - wherein typically the tuning of one or more parameters is required to achieve such behavior [4],

Recently, from a different perspective, a concept along similar lines, referred to in literature as Self-Organized Criticality [5], has also been proposed - which

’Static structure factor is the Fourier transform of the equal time spatial correlation function. See later for definitions.

91 92 states that a certain class of extended dissipative systems evolve themselves into states with no intrinsic length or time scales. This phenomenon has originally been demonstrated in the so called sand-pile models, where one finds power law distributions for sizes and time scales of events such as avalanches. Triggered by this intriguing suggestion, both numerically [6] and analytically [7], several models have also been shown l.o exhibit such behavior. On the current experimental front, phenomena such as earthquakes strongly appear to manifest this behavior as will be discussed in the next chapter.2

At this juncture, it is not dear how the phenomena of Generic Scale Invariance and Self-Organized Criticality tie in together; however, they do seem to have some features in common such as local conservation of a particular quantity*. Inspired by such issues, several analytical studies have shown recently that d-dimensional single-component, field theoretical models with a “locally conserved field" when subjected to external uncorrelated white noise1 will exhibit Generic Scale Invari­ ance [10, 11, 12]. The main exception, however, is the one-dimensional strictly conserving case, where one expects exponential decay for the spatial correlat ions due to lack of sufficient, phase space. In addition, numerical evidence for Generic

“liven though small sand- piles are observed to exhibit scale invariance, large sand piles appear to show relaxational oscillations with a time scale set. by experimental parameters. See Ref. [8] for details.

:illere, the term “local conservation” is used with the meaning that one can write a continuity equation for the conserving field. Note that, a model without any obvious local conservation law has also been shown to exhibit the characteristics of Self-Organized Criticality [9].

'Note that it is easy to show that noisy nonequilibrium systems without a conservation law behave like equilibrium systems and hence, they do not exhibit Generic Scale Invariance. Scale Invariance, consistent with these predictions, in simple stochastic models has also been accumulating [13].

Therefore, it appears that, at least, in one-component systems, “local conser­ vation” is an important ingredient in achieving generic algebraic spatio temporal correlations. In particular, for systems with a single-component strictly con­ serving field, the exponents governing the asymptotic decays of correlations are extremely simple. For example, on hypercubic lattices in <7 > 2 spare dimen­ sions, spatial correlations are predicted [11, M] to decay as 1 /v'1 and 1 /rd+2 in systems that respectively break or preserve the hypercubic symmetry. In deriving the above results from perturbative renormalization group arguments [11], one tacitly assumes that the systems in question have modest, nonlinearities and so are presumably nonchaotic. Thus, it remains to be seen whether Generic Scale

Invariance continues to hold in noisy conserving systems driven strongly enough to produce chaos. One is further led to ask whether in noiseless (deterministic) conserving chaotic systems the “chaotic fluctuations” simulate [15] the effect, of

“external noise” and so produce algebraic correlations with similar exponents as that of noisy models.

In this chapter, we present numerical confirmations along with heuristic ar­ guments which suggest that both in the presence and absence of noise, many

“conserving chaotic systems” do indeed exhibit Generic Scale Invariance for spa­ tial dimension d > 2. Specifically, we construct a coupled map lattice model with

5A short, note on these results is published in Ref. [16]. d — 2 and demonstrate the existence of Generic Scale Invariance in chaotic sys­ tems. In addition, we find that, in agreement with the heuristic arguments, the exponents characterizing the power law decays in chaotic systems are the same as those predicted analytically for noisy non chaotic ones. These results suggest, that, in the presence of local conservation, the intriguing notion of Generic Scale

Invariance with simple predictable exponents may hold equally widely in both noisy and deterministic chaotic systems, whose analytic intractability is notori­ ous. Though, work on multi-component, systems is clearly needed, our results raise the hope of seeking a characterization, through simple arguments, for the large- scale properties of the general class of chaotic syst ems with local conservation laws.

This chapter is organized as follows:

• In Section 2.2, we provide some background on noisy-nonequilibrium sys­

tems that exhibit Generic Scale Invariance with a strictly conserving field.

We then describe the reasons to expect, generic algebraic decay for spatio

temporal correlations in conserving chaotic systems.

• In Section 2.3, we present a coupled map lattice model in two-dimensions

and describe parts of the phase diagram. We elucidate the occurrence of

power law spatio-temporal correlations in different phases including chaotic

phases of the model. We also discuss a few issues related to domain dynamics

in both chaotic and nonchaotic phases. In Section 2.4, we consider a one-dimensional version of the model presented in Section 2.3 and show the absence of Generic Scale Invariance.

We conclude in Section 2.5 with a summary of our results. 96

2.2 Generic Scale Invariance and Chaotic Systems

In this section, we consider a simple example (see Ref. [11]) and illustrate how

Generic Scale Invariance can occur in diffusive systems with a strictly conserving quantity subjected to an external noise.6 Next we provide heuristic arguments for the occurrence of Generic Scale Invariance in the chaotic states of extended dynamical systems. Note that, in this chapter, our focus is only on strictly con­ serving systems as our goal is to examine correlations in a deterministic (chaotic) extended system with a locally conserved quantity.

2.2.1 An Exactly Solvable Linear Model

First,, let us examine a simple model that exhibits Generic Scale Invariance.

Consider the following diffusive evolution for a field

1 ^ = "o'V2^(?\ /) + ?/(/% I). (2.1)

When (j) is strictly conserving, e.g., without, loss of generality, let the Fourier component {q = 0,/) = f t)d[ir = 0 (or a constant,), then we have:

d'(j){(l = 0 , / ) = 0 . ( 2 .2 ) dt

The noise is characterized by:

GNote that Generic Scale Invariance, i.e., power law correlations in both space and time, also occurs in a variety of other systems viz systems that conserve a quantity on the average [11] and systems with special symmetries (e.g., interface models which have implicit translational invariance for the interfacial height variable [17]; in such systems the correlations typically grow in time and space). 97 where < ... > indicates noise (ensemble) average, and

< >= 2D(q) (2ir)d 6(q + q ')8(1 - /'). (2,1)

The strict conservation condition (Eq. 2.2) is satisfied when D(q) vanishes as q —> 0, where q denotes the magnitude of the vector q. Examples of such D{q) include:

D(q) = D0q2 (2.5)

D(q) = Diq2x + D2ql, (2.(>) where the former is spatially isotropic and the latter is anisotropic, since q± lies in the subspace spanned by all other directions excluding qT in d-dimensions. Now. the spatio -temporal correlations for this linear problem can be easily obtained in the Fourier space, where Eq. 2.1 can be written as:

- iw[q, u>) = -l'0q2j){q,u>) + fj(q, u>). (2.7)

From the above expiation, it follows that the structure function C(

C (? ,« ) = < l = w2 ^ ^ 2 , 2 - (2 -S)

Now, using this structure function one can obtain the two-point correlation func­ tions. First let us examine the temporal correlation function, which is also com­ monly denoted as the autocorrelation function in literature. Temporal correlation function:

This function provides information on how a local variable is correlated with itself in later (or previous) times and is given by:

CIO = < « f , w m > = A M

After the integration over u> (with t > 0) we obtain:

C(0 = 7(27 ^zY T j/AJ i/0q2 (2.10) for example, when we consider the isotropic case D(q) = D0

= ( 2. 11) where for one dimension f 'o = A)/(27r1/,2//0^2), for two-dimensions C ’0 = A j/( l7r;/,(2) and for threedimensions C '0 = /T0/(8 it'^U q 2). Moreover, it is easy to check that the 1 /t(ll2scaling forthe autocorrelation function is unaltered whenthe noise correlations are anisotropic, i.e., D(q) = D\q2+ D2qj_ (this choice is possible only in <1 > 1). For this case, the coefficient Ca gets appropriately modified:

Specifically, the Du in the above expressions for C0 gets replaced by ( D\ + D2)/2 and (D\ -\-2D2)/2 in two- and three-dimensions respectively. The autocorrelation function t hen, irrespective of spatial anisotropy, exhibits power law decay with an exponent —d/2 in this problem. Thus, generic algebraic decay of temporal correla­ tions is easy to achieve in conserving systems. We now turn to spatial correlations.

7This behavior is simply due to local conservation and it is the well studied long-time tail phenom ena in conserving equilibrium system s [18]. Spatial Correlation Function:

Starting from the structure function (see Eq. 2.8), it is straightforward to show that the (equal time) spatial correlation function G(r) is given by:

(?(r) =< I)M0, l)>=-^—t jddq (2.12) (27T) J Vq(]

Now, one can readily see that for D(q) = Doq2, the spatial correlation function is a delta-function, i.e., G(r) = Dq/ vo 6 (r), for all d > 1 and thus simply can be interpreted as an exponential decay with a correlation length £ = 0. Note that this case with appropriate nonlinearities corresponds to the well studied time depen­ dent. Ginzburg-Landau equation with a conserved field, where spatial correlation function genetically (i.e., without tuning the parameters) exhibits exponential decay [18].

However, if there is spatial anisotropy in the noise correlations, as is in the case with Eq. 2.(5, the situation is quite different, (with the exception of one-dimension), since this anisotropy can not be removed by a simple rescaling of length scales8.

Then the spatial correlation function G(f) attains the following power law form9:

G(r) = G o— j - (2.1.1) r“ in d > 2 dimensions, where /(fi) denotes the angular dependence, with H being the d-dimensional angular element. Note that in one-dimension, spatial anisotropy does not have any role to play and hence Generic Scale Invariance can not. occur.

8Note that, a rescaling of t/± to achieve D\ — D-> leads immediately to an anisotropy in the diffusion coefficient vq and vice-versa.

9One can expect this behavior by simply counting the powers of q inside the integral in JOq. 2.12. 100

In two-dimensions, G(r) is given by:

and in three-dimensions,

r ,. {P 2 ~ fli) 2 ^.r 2 - .7y2 - -c 2 ,,, 1M G(r ) = —A ITTttvq rZs------’ (2-ir>) which are simple power laws.

More generally, the “static structure fad,or: 6'(

* D(

where /l(

< * « = M ± M . ,2.17) I'a

Note that the limiting value for G, as q —> 0, depends crucially on the direction of approach. For example, it assumes values A //'u, Az/f'o and (D1 + D2)/{'2i/o) respectively along qy — 0, qT = 0 and qx = qy directions of approach to q = 0. This is the characteristic that yields Generic Scale Invariance with “anisotropic power laws” in this problem. This is evident since one can write the spatial correlation 101 function 10 (Eq. 2.14) as Eq. 2.13 with f(0) = cos(20) in two-dimensions; one can seek appropriate generalizations to higher dimensions .11

In summary, we have seen the following ingredients to be responsible for generic algebraic decay of correlations in this simple linear problem: (i) local conservation

(ii) spatial anisotropy, i.e., D\ ^ D2 (iii) absence of detailed balance, i.e., D(q) ^

A{q). Even though (iii) appears to arise from the anisotropy in this problem, it particularly denotes that spatial anisotropy can not be removed by a simple rescaling of length scales.

This linear problem now provides us with a framework to further explore the correlations in strictly conserving extended systems with more complications as will be discussed next..

2.2.2 Inclusion of Nonlinearities

In the linear problem defined by Eq. ( 2 .1-2. 1), one can risk what happens when the evolution equation Eq. 2.1 has non-linear terms; it is of particular in­ terest to see if the non-linearities can change the long wavelength properties, i.e., the spatio temporal correlations. To investigate this one can add non-linear terms

1Hln general, if we have Dxql + lhq* (’(

the spatial correlation function is given by:

G'(r) = D~/a~ ~ Dl X' ~ 2na (x- + y~/a-)-

However, we can rescale 1/ by setting xf — y/a and get back the Eq. '2.13 in two-dim ensions.

"Note that in two-dimensions the angular part has the property: f f(0)d0 = 0. However, in higher dimensions the integrated angular part does not in general vanish, i.e., f f(Q)dQ ^ 0. 102 that conserve (j) in Eq. 2.1 and examine their relevance using standard perturba- tive renormalization group techniques. However, as shown in Ref. [10, 1 1], it t urns out that all the non-linear terms are irrelevant in d > 2 and hence do not alter the spatio-temporal correlations for the strictly conserving single-component prob­ lem. Consequently, the scaling behavior governed by the linear terms persists in d > 2, and therefore, d = 2 is the upper critical dimension for this problem. Here we briefly sketch the analysis and do not go into details (see Refs. [ 11, 10] for more details).

To investigate the influence of nonlinearities, first consider the following mod­ ification of Eq. 2.1:

__ = V„ , 6_ / / + k (2.IS) where t he symbol h denotes the functional differentiation and II is a free energy functional. This equation is the time-dependent. Ginzburg-Landau equation for a conserving field [18]. Note that // = j~(p2 corresponds to the linear problem, i.<

Eq. 2.1. Now one can include a variety of non-linearities to this funct ional 11 and see if they are capable of changing the long wavelength behavior, for the sake of argument, let us add ‘1 with a coefficient /q by taking 11 = va(f>1 + v\4>x. Then, in Fourier space, the above equation (Eq. 2.18) can be written as:

- iuJ(p{(j,u>) = - l'uq2(j>{(J, u>)

~ ‘W / ((7l^i)(

+ 7 /(9 , w). (2.19) 103

In the above equation, the integration subscripts q and uo denote f ddq j{‘l~ )d and f du/(2ir) respectively. Now, following the standard dynamical renormalization group procedure, one can examine the rescaling of the coefficients (/'o./q dc.) when a simple scale transformation is performed [4]. Moreover, power counting can be used in determining the leading terms that dictate the long wavelength behavior. Consider the following scaling transformation 12 with b > 1: q'r —

= bxq±,u>' = tfu) and

iJ\q ', u/) = -[i'fQrq? + i\)Lq'l\(t>'{

~ \ U'\r(ir + U'\L(l'i\ / , , , , {$2 Jq, ,72,u/,,u;2

~

+>/'(

"a,- = /r'_ 2"o, (‘2.2 1 )

,/01 = b'~2xv0, (2.22)

,/lr = (2.23)

u\L = \ ^ - 2-2dx-:vx. (2.24) rI'he rescaled noise term and its correlations are given by:

W ' V ) = bs~crj(q,u)), (2.25)

'“This transformation denotes that when there is scaling in the system, a change in the scale along x (by a factor b) requires rescaling of other lengths, time and the variable itself [4]. 104

< W V )W i >0 >= 2(DWr2+ D'2q'l){2n)d+'8(J + J x)8(q' + q, '), (2.26) with the new rescaled coefficients given by:

(2.27)

d '2 = yte-K+'M'i-vx d 2' (2.28)

First, let us assume the noise normalization, i.e., noise persists for small q, by choosing D\ = Z), for i. = 1,2. For this we need ( = (d — 2 + 3c)/2 and\ = 1.

Then, the fixed point behavior for the diffusion coefficients requires z = 2. Note that these exponents simply correspond to the scaling behavior associated with the linear problem, i.e., Eq. 2.1.

Now, with the above values of the exponents c, \ and the coefficient of the non linearity turns out to be b~a (i.e., eigenvalue is —d) and hence, becomes smaller as one keeps performing the scale transformation. Thus, this nonlinearity is irrelevant and does not change the scaling behavjor. Instead of a cubic lionlin- earity which can be derived from a //, let us consider a lower order nonlinearitv

V 2(V(£)2, which violates detailed balance and preserves spatial isotropy [11]. It is easy to show that this term also becomes irrelevant with an eigenvalue: —[r //2 + 2j.

More generally, the procedure for deciding the relevance of non linearities based on a simple power counting method works in the following way. Any term in the dynamical equation can be written as qm f (f)n in Fourier space (some powers of q can lie with in the integral as well). This term will have n — 1 integrals with the integration being over < 7, and u;, (see, e.g., Eq. 2.19 which has a 11011 linearity with m = 2 and n — 3). Then, for this term, one can check that the eigenvalue is given by: 2 — in — (n — 1 )d/2. Since the system is conserving, at least one q should he there in any nonlinear term13; and thus we need to explore the relevance of all the nonlinear terms with in > 1 and n > 2. This leaves us with the option of having a nonlinearity of the sort q f in two-dimensions. Clearly, in three-dimensions all the nonlinear terms are irrelevant. Since a nonlinear term with a single spatial derivative (i.e., q f 4>2) breaks spatial isotropy, one concludes that all nonlinear terms that, sat isfy spatial isotropy are irrelevant even in two-dimensions.

From the above power counting analysis, it, is clear that the only nonlinear terms with non-negative eigenvalue are of the sort dr 4>2 for d = 2. Note that, in addition to being spatially anisotropic, such terms also break the reflection symmetry (e.g., the term dr 2 breaks ,r to — .?• symmetry). Moreover, since the eigenvalue is 1 — d/2, which is zero in 2 — d, these terms are marginal1'* in 2 — d.

However, such anisotropic terms also turn out to be irrelevantlfi as has been argued in Kefs. [10, 11]; thus, the nonlinearities do not change the scaling behavior even in two-dimensions. 1(1

In summary, when detailed balance is violated, a strictly conserving single­ component system, in d > 2 dimensions, exhibits Generic Scale Invariance with

13Note that a simple linear term with rn = n = 1 can be removed by redefining the time appropriately. Also note that higher order time derivatives on the left hand side of the Eq. ‘2.18 are also irrelevant .

HNote that if there is —■ — invariance in the problem, then all the allowed nonlinearities are irrelevant in *2 — d and marginal terms such as dT

15Note that this marginality can lead to logarithmic corrections to the correlations.

16Note that, for other cases such as when the variable is not strictly but only conserving on the average the nonlinearities play a crucial role as the upper critical dimension gets changed. 106 simple exponents in the presence of spatial anisotropy. On the other hand, for one dimensional strictly conserving systems, even though the temporal correlation function exhibits algebraic decay, since the phase space can not support, spatial anisotropy, one expects exponential decay for the spatial correlation function.

From now on let us focus on two-dimensions, since it is relatively easier to

(numerically) explore two-dimensional chaotic systems. Recall that in the presence of spatial anisotropy in two-dimensions, the exponents are simple as the spatio- temporal correlations are given by: (7(f) = Gof(0)/r2,C(i) — C0/i. for a strictly conserving single-component system.

2.2.3 Induced Scale Invariance

So far, we have considered only single-component strictly conserving systems driven by external conserving noise and examined the spatio-temporal correla­ tions. Now, we envisage a simple situation in which the system has two compo­ nents (or modes) which are linearly coupled to each other with one mode being strictly conserving and the other not. The interesting feature that emerges is that the conserving mode induces algebraic correlations ( i.e., Generic Scale Invariance) for the non-conserving mode. This phenomenon may be relevant for experimental systems that exhibit fluctuations in more than one mode, e.g., nonlinear regimes in surface wave experiments. Here we briefly illustrate a simple case, h'or more elaborate discussion see Ref. [19]. 107

Consider two fields \ and fa linearly coupled to each other in the following way:

= -a(q)fa(q,t) - b(q)2 (qj) + in(qJ), (2.29)

M q,1) = -c(q)\{q,i) - d(q)2(qj) + irtqj), (2.30) where \ is strictly conserving and fa is non-conserving. The coefficients a,b,e,d are assumed to be insensitive to t he sign of q. The noise correlations are given by:

< »/.■(£ /.') > = 2r,(

Since the variable \ is strictly conserving, the coefficients ajt and also the coef­ ficient P] {(f) of the noise-noise correlation function must vanish as q2 (the terms can be anisotropic) in the limit q —► 0.

By solving the above coupled linear equations one obtains the following static structure factors (7, corresponding to each

+ r2r , , r 2 ^'2 (q) — —— 777—)— 7—7 + .• (2.33) (o + d)(od — be) a -f a

Note that the denominator of the first term of RHS contains a factor proportional to q2 for both G\and G2. Also, the numerators contain the conserving noise term. i.e., F]. This resembles the situation we had before in single-component system.

Thus the non-conserving mode can attain algebraic correlations due to a simple linear coupling with a conserving field. As an extremely simple case, consider the following choice of coefficients, a = Vaq2, b — 0, d = q2 -f A, Fj = D \q2., 1'2 = 0 and 108 c a constant. For this choice, \ is independent, of (j)2 and hence, it exactly follows the linear problem considered earlier (cf Eq. ‘2.1), leading to (7] (q) = D^q2TfvGq2.

However, the structure factor for

^ = v0q2(q2 + A )(tw 2 + q* + A )’ {2'™ ] which exhibits algebraic decay, since it goes as q2/q2 for small q. Based on this observation, one may view the coupling in Eq. 2.30 between the non-conserving field ((j>2) and the conserving field as an additive coupling between (f>2 and a noise

(cob) which itself has algebraic spatio-temporal correlations; this provides an

('xpla.na.tion for the induced scale invariance in the non-conserving field. The effects of nonlinear couplings and other related issues are discussed in lief. [19].

The above example clearly shows that in multi-component, systems a variety of nontrivial phenomena can occur. Note that, interestingly, it turns out that the model with a conserving q — 0 mode presented in section 2.3 robustly exhibits induced scale invariance in q = (n, k ) mode in some of its phases (this is illustrated in subsection 2.3.3).

2.2.4 Lattice Models

Numerical confirmation for Generic Scale Invariance in non-equilibrium sys­ tems may be sought by devising models on hypercubic lattices similar to Ising models in equilibrium systems. For example, a two-dimensional stochastic parti­ cle model [13] on a lattice with .r — y spatial anisotropy has been shown to exhibit 10!) algebraic spatial correlations17 decaying as 1/r2. Here we consider a different, set of models, which enable us to study spatio-temporal correlations in chaotic states.

Linear Models

First, we consider a simple linear lattice model in 2 — d which exhibits 1 /r2 scaling behavior for the spatial correlation function similar to the linear problem

Eq. 2.1 in the continuum. This will guide us (later in section 3) in developing a strictly conserving nonlinear model to investigate the possible existence of Generic

Scale Invariance in chaotic states. Next, following Ref. [11], we illustrate that., unlike the continuum models, even in the isotropic case (i.e., ,r — ;/ isotropy is preserved), when detailed balance is violated in the 2 — d lattice, spatial algebraic

correlations can persist but. with a faster power law 1/r'’ (for r/-dimensions o i k * expects \/r,i+2).

The model is defined as follows: A variable S, at each site i of a square lat tice* of size L x L, is assumed to evolve in discrete* time n according te> the rule*:

Sn+i(i) = S„(i) + j 52[S„(j) ~ £„(»)] + ;/„(/), ( 2 . 33) where j is summed over all the four nearest neighbours of site i. When the sum of site variable's is (strictly) conserved ( e.g., S(i) = 0), it inquires that, the* e*xternal noise //„(?) is also conserving. This can be accomplished by deriving ?/„(/) from

1' A general form in two-dimensions G(r) « — Btj~ )/r'1 has be*t*n suggested in Ref. [ 1 ,'j]. Howe>ver, note that, this form dot's not satisfy the condition / f(0)d0 = 0 for arbitrary A and B, i.e., A ^ B, moreover, it corresponds to the divergence of the static structure factor at e/ = 0; this appears to be not the case even in the model considered in Ref. [13]. Recall that A = B case corresponds to anisotropic convergence for the static structure factor at

(gradient.) choice generates a noise with anisotropic correlations:

?/n(0 — d" f’-x) (2.36) where each i]xn(i) is chosen randomly from a Gaussian distribution18 with zero mean and < t]xn(i)i]xni(j ) > = 2f)] Then the noise (?/) correlations in q space are given by:

< Vn(t7)i)n'{(f') >= 2D\ [2 - 2 cos(<7j.)] (27t )2 <$„,„< f>(q + q'). (2.37)

Similarly, the definition,

= [^/j-ti(? d~ ( j-) ?/jn(0] d* [ijji7i(? d" ( y) ?/.V’i(0]i (2.3b) yields noise correlations that preserve .r — y symmetry, when //x and qy are uncor- related and havt* identical noise strengths.

In the absence of noise //„(?), it is easy to show that, in the regime [0 < v < 1], rule (2.35) evolves to a spatially uniform (flat) state with trivial correlations.

However, in the presence of anisotropic noise given by Eq. 2.36, one expects the correlations to decay as 1/r2. One can numerically compute the correlations by defining the equal time correlation function as the following:

Ci(r) = < S(r,n)S(0,n) >, (2.39) where < ... > represents both spatial and time average. As a control, i.e., for fu­ ture comparisons, we show19 the numerically obtained spat ial correlation function

18However, especially in numerical simulations, for convenience qr can also be chosen simply from a distribution function uniform for |r/j-1 < a and 0 otherwise. lyWe plot |(7(F)| since, (!(r) need not always be positive. See for example E<). 2.1*1. along x direction for this model with L = 80 in Figure 2.1a. Note the power law decay of the correlation function G{r) consistent with the expected power -2. We have also computed the discrete time autocorrelation function defined as:

C (n ) = < $(r, n)S(r, 0) > . (2.10)

As shown in Figure 2.1 b, it decays as a power law with a power -1. The coefficients of the power laws are found to be in good agreement, with constants f»{) and (\} in the solutions of the linear problem, i.e., Eq. 2.1 in two dimensions.

A modified version of the above discrete time model in continuous time can be written as a spatial discretization of the continuum model Eq. 2.1 as follows:

*V'l) = jVjS(iJ.) + ;/(/,/.), (2,11) where Vy is the Laplaeian operator. For this model the spatial correlations are given by20:

«W> - 7 ^ 3 - f (' < M », 2-2'-Mlr) ^ (2.42) ( J7T) ^/^o J-K J-n 1 — CO s((/j.) — COS(<7tf) when* v — nic3 with c, being the unit vector along direction 'This funct ion is also plotted (in Figure 2.1a) along .r which shows the 1/r2 behavior. This result shows that an analysis based on continuous time models is applicable for simple laminar states such as uniform states in the discrete time lattice systems. One can also justify such connections in different ways.

-'“Note that this function C!(r) is exactly the same as the Green’s function derived in the next chapter (Eq. 3.27 with ■) — 1) for the stress redistribution due to a shear rupture in the dipole model. This suggests that in the presence of anisotropy, one can view the conserving anisotropic noise as a source of dipoles causing spatial scale invariance in the system. The interesting part is that the nonlinearities are incapable of changing this scaling behavior. 112

o 00

o o

55 o 00 rsi JOr~\ 'j*x

1 0.0 3.0 Loge r o

o

o

d i . . _

1 0.0 2.0 5.0 Loge n

Figure 2.1: Spatio-temporal correlations in the linear lattice model with conserv­ ing anisotropic noise. The parameters are u - 0.5 and a = 0.05. (a) Upper data are the log-log plot of Ci'(f) vs. 7‘ along x direction. The straight has slope -2. Lower curve is simply the function given in Eq. 2.42 and is shifted down by one e-cade for clarity, (b) Log-log plot of the autocorrelation function ('{n ) vs. n. The straight has slope —1. We now turn to the situation in which the lattice model preserves x — y sym­

metry. In the discrete time model, since the noise correlations (i.e., with a noise

defined as in Eq. 2.38) are proportional to the linear part of the dynamics (i.e., detailed balance is satisfied) one expects exponential decay. Indeed, in the above continuous time model we get. delta-function spatial correlations for this case.

However, as shown in Ref. [14], the inclusion of non-linearities can introduce terms that reflect, the broken symmetry due to the hypercubic lattice. Therefore, in 2 — <7, genetically one expects a faster power law decay: 1 jrA behavior for (>(r) in the absence of x-y anisotropy. This requires lack of detailed balance i.t., the noise-noise correlations and the linear coefficient in the dynamical equation should not cancel each other. To see this, consider an alternate definition for the noise

?/„ (which yields correlations that, preserve x — y symmetry but doesnot load to the detailed balance on a lattice) in the following way:

Vn(i-) — [llrn(i 4" er) fjrn (i ^r)] 4" [^/;/ri (^ 4" (y) Vi/n Ev)]' (2.13)

Then the noise correlations go as : 2 — cos(2qx) — cos(2r/w),when yr and //,, are uncorrelatcd with equal strengths. Now, using the continuous time model Kq.2.T2, it is easy to check that this noise, though isotropic, produces 1 //*'* for (!(r), since the structure factor contains terms with the form: ((ft. + (ft)/(ft, for small q. Note also that when the anisotropy is present only in the higher order powers of q in the noise-noise correlation D(q), one obtains faster power laws even in the continuum model, i.e., Eq. 2.1. Chaotic Models

The discrete time lattice model described above, e.g., rule (2.35), is a linear version of the coupled map lattice models [20], which have been extensively studied over the past few years to investigate the spatio-temporal characteristics of chaot ic states in extended systems with short, ranged interactions.21 Such models include a non-linear evolution rule22 for the dynamical variable S defined at each site i on a hypcrcubic lattice with periodic boundary conditions given by:

s„+,(i) = F({S„(j))) + !,„(,) (2, 11)

The function F is non-linear {e.g.. F(S) = S — S 2) and ${j) arc' variable's in some defined local neighbourhood of site i, which can also include the site' i. The* term 7/ is a uoise with certain correlations. Thus only short, ranged interactions between the sites are included. Such models typically exhibit spatio - temporal chaos when the coefficients of nonlinear terms are large and are well suited to explore issue's such as spatio-temporal characteristics of chaotic slate's, onset of chaos and dynamical phase transitions in chaotic regimes of exteneh'd dynamical systems. Note that these issues are relevant for experimental chaotic systems, e.g., vertically forced surface wave phenomena [23], where a transition from order to disorder (spatio-temporal chaos) has been observed.

21 In addition to these discrete time models, continuum model systems such as complex Ginzburg-Landau [21] and Kuramoto-Shivasinsky [22] equations, which exhibit spatio- temporal chaos, have also been studied. In this chapter we consider only discrete time lattice models to illustrate the possible existence of Generic Scale Invariance in chaotic systems.

22The deterministic model considered in the previous chapter Eq. 1.29 provides an example, which exhibits intermittent amplitude fluctuations in the local chaotic fluctuations. Typically, in coupled map lattice models one observes a variety of instabilities

(bifurcations) with a gradual increase in the control parameter; eventually, for large values of the control parameter the model exhibits spatio-temporal chaos.

This behavior is similar to the bifurcations observed in a single variable logistic map [24, 25] with a quadratic nonlinearity: 5„+i = —,Sn). Consequently the phase diagram for the lattice model is very rich and includes interesting phases. In regimes between successive instabilities, the model generally exhibits states with certain spatio-temporal characteristics. However, beyond a particular value of the control parameter the system becomes unstable i.e., runaways occur similar to v > 4 for the logistic map. This is one set back which appears to be unavoidable in these models (especially with polynomial non-linearities) without including further complications. Furthermore, one can not use unbounded noise //„(/) as the system can become unstable when a local variable exceeds some maximal value.

Apart from these difficulties, these models otherwise are efficient for numerical explorat ions of the characteristics of chaotic states in extended systems.

Next we provide heuristic arguments by which one expects algebraic decay for the spatio-temporal correlations in chaotic states of extended systems. 116

2.2.5 Arguments for Chaotic States

Since chaotic states in extended systems arise due to non-linear couplings23, analytic calculations of correlations are extremely difficult. However, one can formulate some expectations based on heuristic arguments and test their validity by using numerical simulations. Such expectations can be hypothesized more easily in the presence of external noise than in the deterministic cases.

Over the past few years, it has been argued and demonstrated that in systems with short-ranged interactions, generically (i.e., in the presence of noise), collec­ tive chaotic states do not occur, e.g., there are no (appropriately normalized)

Fourier modes that exhibit chaotic behavior [27]. Moreover, the onset of chaos in such noisy systems, as a control parameter is varied, is often purely a local phenomenon, rather than a collective one or a phase transition: consequently, out' can regard the chaotic fluctuations as simply superimposed upon the underlying nonchaotic phase. Arguments and supporting numerical evidence for this asser­ tion are provided in Refs. [M, 27]. Since there is no pha.se transition (when one

'-'•'Note that., in systems with a few degrees of freedom, one can characterize the chaotic states hy the posit.ivity of at least one of the exponents commonly referred to as the Lyapunov exponents. The positivity of at least, one exponent denotes the existence of exponential sensitivity to initial conditions (a characteristic of chaotic systems) and provides the informat ion on how the difference in two closely spaced initial conditions in phase space grows in time in the chaotic state. Negativity of all the exponents reflects the nonchaotic nature of the state. For a detailed discussion see, for example, Ref. [25]. However, an extended system such as a coupled map lattice model of size L x L (i.e., Is degrees of freedom), will have L~ exponents. The set. of the exponents is referred to as the Lyapunov spectrum. Typically, when an extended system with short -ranged interactions is in a state of spatio-temporal chaos, the number of positive Lyapunov exponents increases linearly with the size of the system, supporting the notion of the absence of global chaos in spatially extended systems. This statement is equivalent to asserting that, there is a fixed density of positive Lyapunov exponents in the continuum limit, i.e., L —* oo. For numerical techniques to compute Lyapunov exponents, see Ref. [20], roaches from the nonchaotic to the chaotic phase in the presence of noise), the generic algebraic decays of spatial and temporal correlations characterizing the noisy nonchaotic regime would continue to hold in the chaotic states, and with the same exponents.

Now, let us imagine reducing the external noise to zero in the chaotic state by assuming, without loss of generality, that chaos persists in the noiseless limit. The expectations in this deterministic limit are merely speculative, since the fluctua­ tions in chaotic states are typically bounded. However, if we suppose the simplest assumption that the noiseless limit can be taken smoothly, then the chaotic fluc­ tuations essent ially should behave like an external noise and hence the asymptotic correlations are unaltered. Though one cannot argue for this smooth limit with certainty, given that the effect of noise is typically decorrelating, one’s strong ex­ pectation is that correlations in noiseless chaotic systems s' 0^05 1 1 ' cay at. least, as slowly as those in noisy ones. Therefore, one can conclude that conserving, noise­ less. chaotic systems can also display Generic Scale Invariance, with exponents which, if they differ at all from the noisy case, correspond to slower decays.

In the next section we present a 2 — <1 lattice model that has algebraic correla­ tions in its chaotic states with simple exponents, both in the presence and absence of external noise, consistent with the above expectations. 118

2.3 Two-Dimensional Model

In this section, vvc present a 2 — d lattice model, similar to the chaotic models outlined in the previous section 2.2.4, with a locally conserved quantity and show that the model exhibits Generic Scale Invariance in chaotic states.

Consider the following coupled maps evolving in discrete time n on a square lattice of size L x L with periodic boundary conditions: The dynamical variables

{•Si( i defined at each site i at time n, are updated synchronously following the rule given by,

Sn+,(>') = S .(') + j D ^ ( - '„(>))] + ||S „2(i)-.9„J(i + 2i)]+.,„(i), C2.-15) where j is summed over the four nearest neighbors of site i and F(S') = S' —

S'*. The control parameter, //, regulates the non-linear diffusive co , '' 78 g between sites and tin* term with coefficient n introduces x — y anisotropy and also breaks the reflection symmetry in .r. By inspection one can see that the above model conserves the variable locally: p = jjYJi ■(*■) >s independent of n, where

N is the number of sites.

The last term in Eq. 2.15, ?/„(/), denotes the external noise and is generated from a second set of noise variables, i)r„(i), so as to satisfy the conservation law:

( / ) = //j>i(? + f r ) ~ — f^j ). Tlie ;/j-„(/) are chosen independently and randomly from either a Gaussian distribution with < i]rn(i) > = 0 and < i}Xn{i)Vxu'(j) > =

2D\6,j8Uitli, or (as is more typical in the simulations), from a distribution function uniform for \i]2 \ < a and 0 otherwise. 119

Note that in the absence of non-linearities the equation is simply the stochas­

tic diffusion equation on a lattice with local conservation as considered earlier

(see Eq. 2.35). The non-linear coupling between neighbouring sites is crucial in generating the chaotic behavior. Moreover, it is important to note that when the

model exhibits chaotic behavior the term proportional to cv acts as an effective

noise term with correlations that behave as q for small q modes.

2.3.1 Phases of the Model

We now briefly summarize parts of the phase diagram of our model (Eq. 2.-15).

We consider // > 0 regime where chaos occurs for large values of // and include a finite but small p. Here, we do not explore all possible instabilities and the complete phase diagram, since we are interested in spatio -temporal correlations

('specially in chaotic phases. We simply out line certain prominent phases which occur over a wide range of parameter values so that the ideas of (leneric Scale

Invariance can be tested in different regimes in the model.

To start with, consider o = 0 which poses a simplified problem especially while examining the stability of certain phases. For small v and in the presence of noise the system remains in a spatially homogeneous 1-cycle phase (with fluctuations) which we denote as uniform state. In other words a long time average of each individual site assumes the same value (the density p) at all sites. One can perform the linear stability analysis for this state by inserting S„(i) = p + f>S„(i) in to the original model (2.45). This yields the following equation (only the linear terms 120 are included) for SSn(i):

(2,10) j

Now in the Fourier space we can write the linear part, as:

6Sn+i(q) = C(q)6Sn(q), where

r(9) = [!-//(]-V )(1 -

For stability we need |C'(r/)| < 1 for all q. Now, this condition tells us that at, // = //,* = 2 the uniform state first becomes unstable at the wavevector q = Q = (7r, 7r). Thus, in the absence of n. uniform slate is stable in the regime

(0 < v < /q*). However, when the o term is included, the uniform state is stable only above2'1 //„ > 0 and becomes unstable at //,* exactly similar to o = 0 case.

As a result of the instability at the system undergoes a transition into a spatially-ordered checkerboard (“antiferromagnetic'’), temporal 2-cycle state in which the odd and oven sublattices assume values a and b at alternate times. This state can be viewed as a flip-flop state and here we refer to it as the “AF 1" phase, since the background is antiferromagnetic. Note that, in the presence* of small noise, a long-time average of even and odd times at individual sites reveals such behavior; one may also use Fourier methods (e.g., spectral analysis) to characterize this state. The origin of this state can be understood in the following way: Since the system is conserving, it requires that (a -f b)/2 = p. By substituting the even

24 From stability analysis one can show that e(* Rs 8o/>(l + «/>)/( 1 — 3 /r ). 121 and odd time values in the noiseless equation (Eq. 2.45) at a site j i , we get:

a = b+ u[F{a) - F{b)]. (2.19)

After solving this equation, we simply get

a = p + y/l - 3p2 - (1/;/) (2.50)

h = p - \J\ - 3 p2 -(Ifv). (2.51)

Note also that this state is possible only when // > //,* = ] \ 2. Straight forward linear stability analysis shows that, AF 1 state is stable up to //£ = (3/2);/,* when n = 0.However, for finite a, in the vicinity of ;/j, i.e., in the transition region, thesystem exhibits complicated behavior. Mere, we do not gointo details of this as they are irrelevant for the issues discussed in this chapter.

Now as // is increased beyond //j, the system undergoes a transition into a temporal 2-cycle wherein the odd sites assume values a.\ and a-i at odd and even times respectively, while the even sites assume values b\ and 62 consistent with the conserved value of p. Thus the antiferromagnetic order parameter develops a nonzero temporal average. We refer to this as the “AF II” state and this state occurs over a wide region even for o ^ 0.

With further increase of ;/ (around // = 1.7 for p = 0.1) transition to chaos occurs, i.e., the model starts exhibiting chaotic behavior. We use the posit ivity of the largest Lyapunov exponent as the signature of chaos. The transition region is narrow and is typically difficult to characterize, since it requires large numerical accuracy. In the chaotic regime, with further increase in // (e.g., at ;/ = 1.8 for 122 p = 0.1), the model displays a chaotic 2-hand phase with long-ranged antiferro- magnetic order, i.e.., local, chaotic fluctuations superimposed on the AF II state.

When // is gradually increased further (e.g., v = 1.91 for p = 0.1), the model eventually enters a chaotic 2-band phase where it exhibits local chaotic fluctua­ tions superimposed on AF I background. The underlying AF I and AF II periodic

2-cycle states in 2-band chaotic regime are insensitive to the presence of finite n.

Now, this chaotic behavior which is observed over a range of large r values is of currently interest to us.

Though for sufficiently large // (e.g., v > 2), model (2.15) can become unstable, and runaways to arbitrarily large values of .S’, can occur, we consider only //s for which, at least within our numerical limitation of several million time steps per site, the variables remain bounded. One can argue analytically that there exists, for zero noise, a range of v above the onset of chaos for which the variables remain bounded by unity provided their initial values are bounded. We also note that the inclusion of next-nearest neighbor interactions in the model (2.45) allows for the possibility of more intricate spatial ordering such as striped phases. In addit ion, for zero noise other phases (e.g., temporal 4-cycles) can occur. Consequently, the phase diagram can depend on initial conditions.

Note that one can also gain a qualitative and some quantitative understanding of the occurrence of various phases in the model by examining a single variable cubic map as shown in Appendix B. Also note that, in general, the presence of n term does not alter the underlying periodic states in different regimes (uniform and both noisy and chaotic AF states) over a wide range of parameters; it can, however, make the system to exhibit complicated behaviors at the transition regions, e.g., the transition from AF I to AF II state.

2.3.2 Nature of Chaotic States

We now turn to the characteristics of the chaotic states, as our goal is to study correlations in chaotic phases rather than details of phase diagrams. For the purpose of illustration, we work exclusively in the 2-band chaotic AF I phase, as far as possible from transitions to other phases.

The power spectrum for the time series of a single variable in the lat t ice has a broadband background, with sharp features at frequencies u; = ±7r cones, : 74 'ing to the 2-band oscillations. As mentioned in the previous section, the posit.ivitv of the largest Lyapunov exponent, (A) signals the chaotic behavior in the system.

We now discuss the behavior of A as a function of the size of the system L (see

Figure 2.2. Note that A saturates to a constant with a small uncertainty beyond a length L — 8. Also, the exponent is close to zero for small L - 8,1. Such behavior of A was noted first in similar diffusive coupled map lattice models [28], where the crossover length £.\ (which is 4 in Figure 2.2), was shown to provide a measure of the spatial correlation length of the local “chaotic” fluct uations.

Furthermore, the saturation of A for large L also suggests that, the system can 124

in d i— M o r-O d

,< CM o

o o o

d i

0.0 16.0 3 2 .0 4 8 . 0 64.0 L

Figure 2.2: Largest Lyapunov exponent A as a function of the size of the lattice L. The error in the value of the exponent is ±0.005 which is of the order of the symbol size used. he viewed as a collection of correlation volumes executing independent chaotic fluctuations, consistent with the notion of the absence of collective chaos25.

In addition, following Ref. [27], since the underlying state is AF I, we have computed the width of the fluctuations in the normalized Fourier mode $(Q ,n ), where Q = (7r,7r), as a function of the system size L. The width displays \/L scaling as shown in Figure 2.3, suggesting the absence of collective chaos.2(1

From the above observations, it. is reasonable to hypothesize that, AF 1 chaotic states can be described as local chaotic fluctuations superimposed on an underly­ ing AF I background (we have also checked that, chaotic AF II state has similar features). Moreover, since the chaotic states do not exhibit, collective chaos, one expects that, the arguments provided in section 2.5 to go through, resulting in al­ gebraic spatio-temporal correlations in noisy (and perhaps also in deterministic) chaotic states.

2.3.3 Spatio-temporal Correlations

We now discuss the spatio-temporal correlations in both chaotic and non­ chaotic states of our model. All the correlations are computed using one-million discrete time steps in the simulations, after skipping the first 100,000 time steps so that the system reaches the asymptotic state. We first present correlations

-'5We have also computed the number of positive Lyapunov exponents, on modest size systems up to L — 16. This number appears to scale linearly with the size of the system. However, confirmation for this linear scaling behavior on larger systems is awaited.

-6One can also compute the width of the fluctuations in any normalized (i.e., divided by the volume) Fourier mode S(q,n) and find the l/L scaling for the width of the fluctuations in both real and imaginary parts of that mode. 126

lO

o

q 1.0

5.0 3 .5 logeL

Figure 2.3: Scaling of the width of the fluctuations in the structure factor as a function of the size of the system L. The straight line shows the slope of -1 indicating \/L scaling. The error is within the size of the symbol used to denote ( lie data. 127 in noisy nonchaotic states where linear theory predicts Generic Scale Invariance with simple exponents. Next we display correlations in chaotic states which also exhibit. Generic Scale Invariance with the same exponents. The data were taken for p = 0 .10, but the results are independent of the value of p.

Noisy Uniform state

First, consider the uniform state which is stable below Since the density p is nonzero we define the spatial correlation function to be:

- < S„(i) >< ^ U ) >• (2.52) where r = rt- — rr The expectation < ... > denotes both temporal and spatial average. Similarly, the temporal correlation function is defined as:

C(n) =< $„{i)S0{i) > - < .S'0(/) > 2 . (2.53)

Hot.li spatial and temporal correlations in the uniform state simply resemble the previously mentioned linear lattice model in section 2.2.4. However, as a control, we show G(rcr ) in Figure 2.4, for the spatially-uniform (temporal 1-cycle) phase in the presence of anisotropic noise. The straight line in the figure has a slope of - 2 , consistent with the expected value for noisy anisotropic phases in two dimensions.

Noisy AF I State

This state occurs in between and //j. Since the background is AF I, which is periodic (i.e., 2-cycle) with antiferromagnetic background, one has to define an 128

o

o

0.0 1.0 2.0 0.0 4.0 l.oge r

Figure 2.1: Log-log plot of Ct(r.r) vs. r for noisy 1-cycle phase of model, with v — 0.8,o = 0.25,p = 0.10, and a = 0.02, on a 100 x 100 lattice. The straight line has slope - 2. 129 appropriate spatial correlation function that removes the effects due the underly­ ing periodicities and spatial background.

The correlation funct ion in AF states is defined as follows: Let us denote the odd time and even times by T0 and 7r; and let < F > t, denote the average of a function F at odd (for i = o) and even (for i — c) times respectively. Also, let

< F(R) >v denote the spatial average (over R c V the volume, i.e., allI 2 sites) of a function F(R ), for example F(R) = S(R + f)S(R). Let the odd time average of the site variables be p„(r) and even time average of the site variables be pr(f). i.e. p„{r) =< S(f) >-/•„, pr(v) = < S{r) >r,- Then we compute the odd and even time spatial correlat ions funct ions denoted by (!„ and Ci, as follows:

C0(r) = « S( R + r )A( R) >v >Tu - < p„( R + r)pu( R) > v , (2.5.1)

C,(f) - < < S{ R -f f)S(R) >v>r, - < p,{R + r)pr(R) > v . (2.55)

In noisy AF 1 states, we find both (7„ and Cl, to give the same results. 'I bis is expected, since the system has odd-even symmetry in AF 1 states2'. We consid­ ered initial conditions with small Antiferromagnetic (AF) component, otherwise the system can reach states with mult iple domains, since the system has 2-cycle behavior. Some of the characteristics of domains (e.g., islands of the “flip" state in a sea of the ‘“flop” state) are discussed in the next subsection.

Figure 2.5 shows the spatial correlation function for the AF 1 phase in the presence of anisotropic noise. The slope is again consistent with the value of

27In AF II state Cl0 and G< yield different results for the coefficients of the correlation function, since there is no odd-even symmetry. 130

o

o CM

O QC)

1 0.0

I OCJe r

Figure 2.5: Log-log plot of G(i\r) vs. r for AF 1 phase of model, with 11 = 1.30,o = 0.25, p — 0.10, and a = 0 .02, on an 80 x 80 lattice. The straight lines have slope - 2 . Staggering of points for even and odd r shows “induced scale invariance” in Q = (7r, 7r) mode. The error in the correlation function is within the size of the symbol for r < 21. 131

—2 (similar results were obtained for the AF II phase). Note that, although the

—2 result was derived earlier (sec also Ref. [ 11]) for spatially uniform phases,

essentially identical arguments can be made for the ordered checkerboard phase’s

that occur here. Also, note that the data for odd and even values of r r in Figure 2.5

are staggered. Indeed, these data are consistent with algebraic decay both in

the <7 = 0 and q = Q modes. This seems surprising at first glance, since the

antiferromagnetic order parameter is not, conserved in the system. The result

can, however, be understood by considering a simple two-component model with

two coupled variables, one conserved and the other not as described earlier in

section 2.1. Thus our model, in addition to CJeneric Scale Invariance, also exhibits

the phenomenon of Induced Scale Invariance. We have also checked that the

autocorrelation function computed at (odd or even times) displays power law

( 1//.) behavior.

Now, since the spatial anisotropy is crucial for the algebraic decay of spatial

correlations, as a check we have also computed the static structure factor (!(<])

along three different directions of approach to both q — 0 and q = Q = (7r, 7r).

The data shown in Figure 2.6, clearly display the anisotropic convergence' of the

structure factor at both <7 = 0 and q = Q, consistent, with the expectations based on the corresponding linear problems. 0.05

0.04

0.03 CX

0.02

0.01

0.0 0 00 0 1 0.2 04 0.5 q/(2?0

0.05

0.04

0.03 O'

0 .0 ?

0.01

0.00 0.0 0.1 0 ? 0 4 O.b P/(2n)

Figure 2.6: The static structure factor C(q) is shown along three directions: (a) at q = 0 along qy = 0 ,r/r = qy and qr — 0 (from the top to the bottom curve at q = 0). The symbol q denotes the variable along the direction of approach, (b) at q = Q along qy = n , qr = qy and qr = 7r. Symbol q denotes the magnitude of the variable Q — q along the directions of approach. Chaotic AF I State

Wc now turn to the chaotic regime (where a density of Lyapunov exponents is nonzero). The spatial correlations are computed using the same techniques employed for the noisy AF I states.

First, in Figure 2.7 wc present the data for the spatial correlation function along x in the presence of “anisotropic noise” in chaotic AF I state, i.r., chaotic checkerboard 2-cycle phase. Again, the straight line behavior in the log-log plot shown in Figure 2.7 with a slope of -2, indicates the presence of Generic Scale

Invariance. This behavior is consistent with the heuristic prediction that the asymptotic behaviors of chaotic and nonchaotic phases with external noise are identical. The correlation function also shows a tiny antiferromagnet ic component suggesting the possibility of the phenomenon of induced scale invariance similar to noisy nonchaotic AF I state. However, the component appears to be small to draw any firm conclusions regarding this aspect. We have also checked that the autocorrelation function exhibits I/I behavior for this case.

Now, let us consider the deterministic AF 1 chaotic state, i.t., wit hout noise.

Figure 2.8 shows data for G(r) along y in the chaotic checkerboard phase with no noise. Again the results are in good agreement with the heuristic arguments presented earlier, yielding an exponent of -2. It is important to emphasize that no parameters have been tuned to produce the observed power law decays. We have taken data for several values of i/ in the chaotic checkerboard phase, to verify that the results are indeed characteristic of the entire phase. We have* also 134

o

o cn I

1 0.0 1.0 2.0 3.0 4.0 Loge r

Figure 2.7: Log-log plot of G(rx) vs. r for chaotic (noisy) checkerboard phase of the model with v = 1.91, a = 0.25, p — 0.10, and er = 0.05, on an 80 x 80 lattice. The largest Lyapunov exponent for these parameters is 0.415 ±0.01. The straight line has slope -2. The error in the correlation function is within the size of the symbol for r < 24. 135

o u~> .i__

1 0.0 1.0 7.0 3.0 4.0 Logc r

Figure 2.8: Log-log plot of G(rey) vs. r for chaotic (noiseless) checkerboard phase of 2-d model, with u = 1.91, q = 0.25, p = 0.10, and a = 0.00, on an 100 x 100 lattice. For these parameters the largest Lyapunov exponent A is approximately 0.42. The straight lines indicate a slope -2. The data along x also have similar scaling. The error in the correlation function is within the si/e of the symbol for r < 24. 136

cn .o

CP Csl

lO

0 2 3

■OQe r rO

O cn

• •

00

0 2 3 -oge r

Figure 2.9: Spatial correlation functions on 80 x 80 lattice: (a) along x in the deterministic case with smaller o. The parameters are p = 0.1, v — 1.91, a =0.1 and <7 — 0 . The straight lines have slope of -4 and -2. This figure shows the crossover from 1 /r4 to 1/ r 2 behavior, (b) along y in the deterministic case with x — y symmetry. The parameters are p — 0.1, v = 1.91 and a — a = 0 . Tlie straight line shows a slope of -1. The error in the correlation function is 54 ' 1 ' the size of the symbol for r < 12. 137 looked at. the static structure factor along three directions at both q — 0 and q = (7T, 7r). The anisotropic convergence at q = 0 is evident in the static structure factor. At q = (7T,7r), there appears to be a small anisotropy (see Figure 1.8 for large r values) suggesting induced scale invariance; however it lies within the numerical inaccuracy to draw firms conclusions. The autocorrelation function for this deterministic case also shows \/t decay consistent with the expectations.

Finally, we turn to the isotropic (i.e., x — y symmetry in the lattice) deter­ ministic case by gradually decreasing a to zero. Note that for a = 0 the model preserves the square lattice symmetry (i.e., the system does not break the x — y symmetry). Recall that, for this case one expects 1 jrA spatial correlations as dis­ cussed earlier in section 2.1. Numerically computed spatial correlations with small and zero values for o are shown in Figure 2.9. One can see a crossover behavior from 1 /?*'* to 1 jv2 behavior for small o. The coefficient of 1 /r2 is roughly propor­ tional to o 2, consist, (Mil. with the expectation that the a term generates effective' anisotropic conserving noise in the deterministic case. Also note the l/r‘l rapid fall ofr for the case with o = 0 ; even though, the evidence is not as persuasive as that for the anisotropic case which shows 1/ r 2 behavior, the first few data points

(up to v = 12) are consistent with the expected 1 jr‘l decay.

These numerical results strongly support both the simple physical arguments predicting Generic Scale Invariance in conserving chaotic systems with d > 1, and the equivalence of the exponents produced by external noise and deterministic chaotic fluctuations. Note that, the local nature of the chaos which seems to 138 underlie t he similarity of the large-distance properties of chaotic and nonchaotic phases has a simple interpretation in terms of the renormalization group: The fixed point governing asymptotic properties of the system is independent of the presence of chaos, i.e., the system looks less and less chaotic as one examines it on progressively longer length scales (which is the absence of “collective chaos” as discussed earlier).

2.3.4 Domain Dynamics in AF I States

In t his subsection, we present, some cpia.lita.tive features observed in bot h noisy and chaotic AF I state's. Detailed understanding of some of these features and related issue's in other state's such as AF 11 state's are eleferre'd to future research.

Recall t hat AF I state's e'.xhibit not only an antiferromagnetic spatial ordering but also a 2-cycle temporal behavior.

Non-chaotic AF I State

As mentioned earlier, linear stability analysis shows that noisy AF 1 state's are locally stable. However, one' cannot draw any conclusions for global stability using such analysis. By demoting the even and odd time AF I state's as “flip” and “flop” states respectively, let us try to see if the AF 1 state* is asymptotically stable {i.e., globally stable) when domains of “flip” states (with size in x in) are present in the sea {i.e., the size of the system L x L) of a “flop” state.

When a = 0 we find that the flip domain, in a sea of flop domain gets eroded as \/l, where t is the number of time steps (i.e., the number of flip state sites 139 that are changed to the flop state in a time t scales as \/J). This behavior is consistent with the expectations for diffusive systems [29]. However, for n ^ 0 domains are observed to get eroded faster; moreover, the domains are observed to move along x while getting eroded, indicating that there is non-zero current in the model along x direction. Note that the erosion of domains tells us that the

AF I state is globally stable for all o.

To explore the motion of domains further, we have created domain walls be­ tween “flip” and “flop” states with the wall boundary being along either x or y direction. Specifically, we have used an initial condition in which half the lattice is in the “flip” state and the other half in the “flop” state. Note that, since we use' periodic boundary conditions, we have two wall boundaries. The domain walls along x are found to remain stationary for all values of n. This behavior is ex­ pected, since there is y —* —y symmetry in the system. However, when the wall boundaries are along y, for non-zero a, domains can move along x direction, since the o term breaks the reflection (i.e., , x —> — x) symmetry. We have observed that for a given non-zero o, for »/,* < v < uc, the domain walls along y move towards positive (or negative) x direction for positive (or negative) n . However, above i/c the domain wall appears to remain stationary with in the few hundred thousand time steps that we explored in numerical simulations. Such behavior for the domain wall motion is observed over a range of a values and even in the presence of small noise. Such features of domain walls may be of future interest .28

-8Note that the domains can be distinguished on the basis of their temporal characteristics and the density profiles. Also, when there is a domain wall present in the system, the specific 110

Chaotic AF I State

This state resembles the noisy AF 1 non chaotic state (recall that, a long-time average at even and odd times at, sites gives us the AF I background). To check the global stability in the chaotic regime, after evolving the system to a AF 1 chaotic state (this can be accomplished by starting with a large antiferromagnetic component in the initial conditions), we have replaced a part, (size in x in) of the system at, an odd time T -f 1 wit h t he corresponding values at the even t ime

T. Further evolution of this mixed state indicates the erosion of the even time domain in the sea of the odd time domain. This behavior is similar to the noisy non-chaotic AF I state and shows that, AF I chaotic state is globally stable. In addition, wo have also observed that for certain initial conditions domain walls are formed, and they typically appear to remain stationary (all the above conclusions are based on our explorations using some (e.g., 2 0 ), random initial conditions with

// = 1.91 and o = 0 or o = 0.25).

We end this discussion on domains with a speculative remark that, may pro­ vide some motivation for future research. One interesting issue is the connection between the domain dynamics and the observed Generic Scale Invariance. This might provide a connection between the phenomena: Self-Organized Criticality and Generic Scale Invariance. Specifically, if one could construct a model that ex­ hibits algebraic correlations (similar to the model Fq.2.45 that we already have)

site values can differ considerably from the values of AF I state solution (i.e., a and b values provided in Eqns. 2.50,2.51). and find power law size and time scale distributions for domains, understanding of generic scale invariance in extended systems will (perhaps) be advanced. At the moment, the role of domains on the scale invariance in our model is an open problem.

2.3.5 Remarks

In addition to the model (Eq. 2.15) presented in the beginning of this section, we have also considered other locally conserving non-linear models. One class of such models is the following:

Yjl .W O = C(.S,,(j,)-A’,,(i)) + y]C\ /'’(*„(j„) - -S..(')) + '/.(>)• CW6) Jx J y where j T and j y are nearest neighbours of site i along r and y direct ions respectively .2

The above model, similar to the previous model defined by Eq.2.15, is rich in having its own phases, when F is non-linear. We have considered non-linearities of the sort: F(S) = sin(.c>') and F(S) — S — S'*. While in the presence of anisotropic noise, the noil-chaotic states are found to exhibit Generic Scale Invariance, in the chaotic regimes30, however, the fluctuations art' found to decay rat her slowly but without any persuasive evidence for Generic Scale Invariance. Note that in the deterministic limit one expects a faster decay, i.e., I/?-1 in the chaotic phases for these models, since the effective noise-noise correlation is expected to scale as

-9Note that in addition to local conservation, this model has an additional symmetry similar to interface models as it is invariant under uniform translations of the variables (i.e., it has S —- S + C invariance). 30The chaotic regimes have typically antiferromagnetic 1-cycle behavior, unlike the model Eq. 2.45 which exhibits 2-cycle behavior. Aq* + Bq* for small q. Recall that, in the model defined earlier, i.e., Eq. 2 .45, the

noise-noise correlation is expected to scale as q%, since the non-linear o term has

a single derivative. Moreover, it is also plausible that the effective noise generated

by the chaotic fluctuations satisfies detailed balance and thereby produces only

exponential decay for the spatial correlations.

However, we could fit the spatial correlation function in the chaotic regime to a

slow exponential decay (in Q = (7r, i t ) mode). It is interesting to note that unlike

Ising models in equilibrium systems, for these models, spatial correlations decay

slowly with a large correlation length (often 20 to 30 lattice spacings) even when one is farther away from the transition regions. At this point, it is not dear, how

such slow exponential decay arises in chaotic states and whether there is Generic

Scale Invariance in this class of models buried underneath this exponential decay.

Thus, this chaotic model (numerically) does not appear to display Generic Scale

Invariance even though it satisfies the criteria of local conservation and spatial

«j.nisotropy. 2.4 One-Dimensional Model

In one-dimension algebraic decay of the spatial correlation function is not expected, since there is not (enough) phase space to support spatial anisotropy.

To verify this numerically in chaotic states, we have considered the following model which is a 1 — d version of the 2 — d model Eq. 2.45 presented in the previous section:

))] + - [

The nonlinear function F(S) = S — .S’3 and the sum on j is over the two nearest neighbours. The density p = 1/jV $„{i) can be taken to be zero for s' 82 , ’’city as finite p does not change the qualitative features.

The above 1 — d model, similar to the 2 — d model, as the control parameter // is gradually increased, shows nonchaotic phases such as uniform, AF I and Ah' II states with similar stability boundaries, e.g., the uniform state becomes unstable at v = = 1/(1 — 'ip2) and above this value the system exhibits AF I state.

Also, the transition to chaos (i.e., the appearance of positive largest Lyapunov exponent) occurs at v « 1.7 similar to 2 — d model.

However, in contrast to 2 — d, where AF I state is observed for large values of u before the systems becomes unstable (runaways), the 1 — d model exhibits a chaotic state in which the temporal behavior is not a 2-cycle. In this state the system does not exhibit any temporal background order, i.e., the temporal average is zero and the power spectrum shows a broadband. To compare our 141

2

-3

5

C(q) 6

-7 0 10 1b 70

ID '1—

O rsi

0.6 1.2 1.8 2.'I 3.0 3.6 4.2 4.8 5.4 Loge n

Figure 2.10: Data for 1 — d model: (a) Log-linear plot of G(r) vs. r for chaotic (noiseless) phase of 1 — d model, with w = 1.95, a = 0.25,/) = 0.00, and a = 0.00, on a size 1024 lattice. The largest Lyapunov exponent is 0.51 ±0.01. The data indicate exponential decay with a correlation length of 2 or 3 lattice spacings. The inset shows ordering at some q ^ 0 mode. The error in the correlation function for v < 18 is within the size of the symbols, (b) Log-log plot of the autocorrelation function C(n) vs. n for the same parameters. The straight has a slope —1/2, indicating algebraic decay of C(n) ss 145 results with the deterministic case in 2 — d, we have computed spatial correlations

Ci(r) both with and without noise in this chaotic state. The data for C(r) do not fit a straight line 011 a log-log plot, consistent with the theoretical prediction for noisy nonchaotic systems that Generic Scale Invariance does not occur in

1 — d. Figure 2.10a shows the data 011 a semilog plot; the data are consistent with a correlation length of 2 to 3 lattice spacings. Moreover, the structure factor typically shows a broad peak at some <7 near Q indicating short-ranged order as shown in the inset of Figure 2.10a. On the other hand, the autocorrelation function decays as 1 / \fl, as expected for this case both in noisy nonchaotic and deterministic chaotic states (see Figure 2.10b). In summary, this study shows t hat spatial scale invariance does not (generically) occur in 1-d for strictly conserving chaotic systems, consistent with the theoretical expectations. 146

2.5 Summary and Conclusions

In this chapter, we have provided evidence for the existence of Generic Scale In­

variance in conserving chaotic systems in d > 2. Specifically, we have constructed

a simple coupled-map lattice model Eq. 2.45 in two dimensions and elucidated the

generic algebraic decay of spatio-temporal correlations in the chaotic states of the

model. Moreover, in the presence and the absence of external noise, chaotic states

display algebraic correlations with the same exponents in the model. This char­

acteristic reveals the similarity between noise and chaos as far as long-wavelength

properties are concerned. If, as we suspect., this holds rather generally, then it

constitutes a powerful tool for analyzing the macroscopic properties of conserving

chaotic systems.

Furthermore, the model exhibit s Generic Scale Invariance in the chaotic phases

wit h simple exponents that, correspond to the ones predicted by a linear theory.

Recall that,, standard perturbative renormalization group analysis of the con­

tinuum version of our model with external noise shows that, at least for small

coupling constants, the nonlinearities are all irrelevant, i.e., they do not alter the

long-distance algebraic correlations predicted by the linear theory. Though such

analysis need not necessarily provide any information about the strong coupling

problem, where chaos appears, our results demonstrate that the noisy linear the­ ory seems to describe correctly the long-distance behavior of the noisy, and even of

the noiseless, chaotic phase. Therefore, the model presented in this chapter stands 147 as an example, where the exponents of the algebraic decay of spatio-temporal cor­ relations in the chaotic phase are simple integers, consistent with the exponents of a linear theory.

In addition to Generic Scale Invariance, in some of its phases, the model also robustly exhibits induced scale invariance, a phenomenon expected in two- component systems. Since our model also displays a variety of intricate phases, variants of our model may be used for future explorations of spatio-temporal features of conserving chaotic (multi-component) systems.

In conclusion, we have shown that algebraic correlations can occur in con­ serving chaotic systems with predictable exponents and elucidated t he similarity between noise and chaos as far as large-distance and long-time properties are concerned. L ist o f R efer en c es

[1] G. Grinstein, J. Appl. Phys, 69, 5441, (1991).

[2] T.R. Kirkpatrick, E.G.D. Cohen, and J. Dorfman, Phys. Rev. B. 26, 950 (1982); D. Ron is and 1. Procaccia, Phys. Rev. A ., 25, 1812 (1982).

[3] B.M. Law, R.W. Gammon, and J.V. Sengers, Phys. Rev. Lett., 60, 1554

S. K. Ma, “Modern Theory of Criticeil Phenomena" (Benjamin, NY 1976).

P. Bak, C. Lang, and K. Wiesenfeld, Phys. Rev. Lett., 59, 381 (1987); Phys. Rev. /!., 38, 364 (1988).

[6] P. Bak and K. Chen, in Fractals anel their application to Geology, (Geological Society of American, Denver, 1991), edited by C. Barton and P. Lapointe, and references therein. References on Self-Organized Criticality in earthquake models are provided at. the end of the next chapter.

1). Dliar and R. Ramaswamy, Phys. Rev. Lett., 64, 1613 (1990).

[8 ] G. Held, D. 11. Solina, 1). T. Keane. W. .1. Haag, P. M. Horn, and G. Grinstein. Phys. Rev. Lett., 65. 1120 (1990).

[9] Z. Olami, H. S. Feder, and K. Christensen, Phys. Rev. Lett., 6 8 , 1214 (1991).

[10] 'I'. Ilwa and M. Kardar, Phys. Rev. Lett., 62, 1813, 1989; Phys. Rev. A., 45,7002 (1992).

[11] G. Grinstein, D.-Il. Lee, and S. Sachdev, Phys. Rev. Lett., 64. 1927 (1990).

[12] P.L. Garrido, J.L. Lebowitz, C. Maes, and H. Spohn, Phys. Rev. A., 42, 1954 (1990).

[13] Z. Cheng, P.L. Garrido, J.L. Lebowitz, and J.L. Valles, Europhys. Lett., 14, 507 (1991).

[14] G. Grinstein, Y. He, C. Javaprakash, and B. Bolker, Phys. Rev. .4., 44, 4923 (1991).

[15] S. Zaleski, Physica , D 34, 427 (1989); T. Bohr, G. Grinstein, C. Jayaprakash, M.H. Jensen, J. Krug, and D. Mukamel, Physica, D59, 177 (1992).

148 149

[16] R. Bhagavatula, G. Grinstein, Y. He, and C. Jayaprakash, Phys. Rev. Lett.. 69,3483 (1992).

[17] M. Kardar, G. Parisi. and Y.-C. Zhang. Phys. Rev. Lett.. 56. 889 (1986).

[18] P. C. Hohenberg and B. 1. Halperin, Rev. Mod. Phys., 49, 435, (1977).

[19] G. Grinstein, C. Jayaprakash, and J.E S. Socolar, Phys. Rev. E.. 48. 643 (1993).

[20] See, e.g., H. Chate and P. Manneville, Phys. Rev. Lett., 58, 112 (1987); J. Crutchfield and K. Kaneko in Directions in Chaos , edited by B.-L. Hao (World Scientific, Singapore, 1987), Vol. 1.

[21] B. I. Shraiman, A. Pumir, W. van Saarloos, P. C. Hohenberg, II. Chate, and M. Holen, Physica , D57, 241 (1992).

[22] Y. Kuramoto and T. Tsuzuki, Proy. Theor. Phys., 63, 2112 (1980); G. I. Shivasinsky, Acta Astrona.uti.ca., 6 . 569 (1979).

[23] N. B. Tufilla.ro, H. Kamashankar, and .). P. Gollub, Phys. Rev. Lett.. 62, 422 (1989).

[24] M. J. Feigenbaum, Los Alamos Science, 1 , 4 (1980).

[25] P. Berge, Y. Pomeau, and ('. Vidal, “ Order within Chaos'1'1 (John Wiley V Sons, New York, 1986). [26] Y. Pomeau, A. Pumir, and P. Pelce, Stat. Phys., 37, 39 (1981); also set', K. Kaneko, Proy. Thro. Phys., 74, 1033 (1985).

[27] T. Bohr, G. Grinstein, Y. lie*, and (V Jayaprakash, Phys. Rev. IaH.. 58. 2155 (1987).

[28] T. Bohr and O. B. Christiansen, Phys. Rev. Lett., 63, 2161 (1989).

[29] See, e.g., CL 11. Bennett, CL Grinstein, Y. He, CL Jayaprakash, and 1). Mukamel, Phys. Rev. A., 41, 1932 (1990). CHAPTER III

Scaling of Earthquakes in Quasi—Static Seismic Zone Models

3.1 Introduction

The phenomenon of earthquakes is mostly confined to seismic zone's located around the mutual boundaries of tectonic plates [ 1]. The boundaries occur in various regions in the crust of the earth. Typically, earthquakes in seismic zones are caused by “faulting instabilities", which occur suddenly, releasing the stress accumulated due to slow relative motion of the tectonic plates. The instabilities include brittle fracture of rocks and slips along already fractured surfaces (faults) and lead to several interesting phenomena both on short and long time scales. The short time dynamics during an earthquake involve complex phenomena such as rupture propagation, seismic waves, stick-slip friction and energy dissipation. On the other hand, the long-time statistical analysis of seismological data strongly manifests scale invariant features of earthquakes. For example, the well known empirical Gutenberg-Richter law [ 2] (which states that the number of earthquakes with energy greater than E, n(E) scales as a power law: n(E) oc E~^) and the observed fractal features of the fault structures in the seismic zone [3], suggest

150 scale invariance in earthquake dynamics .1 Thus, statistical aspects of earthquakes merit special attention, mainly in understanding qualitative features such as the emergence of scale invariance in earthquakes dynamics.

During the past few years, to investigate the origin of scale invariance espe­ cially in the context of Self-Organized Criticality [5], many models for earthquakes have been proposed and studied. Most, of them are spring-block models, origi­ nally suggested by Burridge and KnopofT [ 6], which mainly focus on the stick slip behavior at the “faults”. Particularly, two types of models have been explored:

(i) quasi-static2 models [7]-[H] which ignore the short time dynamics during an earthquake and only focus on long-time statistical aspects; (ii) dynamical mod­ els, which include frictional features such as velocity weakening [15, Hi] and ra­ diation damping [17], and study both the dynamical and statistical aspects of earthquakes. These models, being conceptually simple, nicely demonstrate' the emergence of some of the scaling behaviors observed in earthquake dynamics. e.g., the Gutenberg-Richter Power law for the frequency size distributions. How­ ever, they do not include seismic wave radiation (for dynamical models) [17] or long-ranged redistribution of stress (for quasi-static models) [18] following a slip

’Scale invariance in extended systems has been a topic of considerable interest, in recent years as many non-equilibrium (i.e., driven) systems in nature exhibit, self-similarity in their spatio- temporal characteristics. Examples include cloud formations, dendritic growth, fractal be­ havior of landscapes and 1 // noise in a resistor (see Ref.[4]). Note that the coupled map lattice model presented in the previous chapter is an example of a chaotic extended system that exhibits scale invariance.

■’Note that, the duration of an earthquake (of the order of a few seconds) is much smaller t han the time scale (of the order of years) over which the stress builds up in the seismic zone. This feature provides a justification for the implementation of quasi static models, which, with appropriate earthquake descriptions, ignore short-time dynamics during an earthquake. 152 event,. In other words, the interaction of the faults with the surrounding medium is not explicitly taken into account. Furthermore, these models cannot capture realistic features such as spatial complexities of faults and formation of new fault, structures.

Thus, a detailed study of the earthquake dynamics has to begin with a model for the entire seismic zone that can include prescribed fault, structures and exhibit scale invariance. Our primary aim in this chapter is to present such a model so that, it may be appropriately extended to study earthquakes in a specific zone' of interest. Towards the end of this chapter, we investigate a simple quasi st atic seis­ mic zone model that “self-consistently” incorporates both the stick slip behavior at the fault surfaces and multiple ruptures in the seismic zone. For this purpose, we devise a “self-consistent method” to compute stress redistribution due to mul­ tiple ruptures and slips when the seismic zone also includes fault surfaces which remain stuck due to stick-slip friction during an earthquake. We also investigate

“spatial” and “scale invariant” features of large earthquakes in our model and compare them with the seismological observations.

This chapter is organized in the following way:

• In Section 3.2, wc describe basic phenomenology of earthquakes and out line

a few important issues that need to be addressed while constructing a model

for earthquakes. • We devote Section 3.3 to describing our formalism to include non -elastic

features and present discretizations of an elastic medium which enable mod­

eling of earthquakes in a seismic zone. Specifically, we consider both dipole

and double-couple formulations for rupt ures and present a self consistent

method to compute the stress redistribution due to seismic events.

• In Section 3.4, we consider a simple quasi-static seismic zone model driven

by an external shear stress and examine statistical and spatial aspects of

earthquakes in the model. We consider two distinct quasi static descrip­

tions, corresponding to the exclusion'* and inclusion '1 of the rupture rupture

interactions during an earthquake respectively, and show that they yield

different scaling behaviors for the earthquakes in the model.

• In Section 3.5, we construct a seismic zone model with an embedded pre

existing fault and include stick slip behavior at the fault with in the quasi

static description. We elucidate the spat ial and statist ical features when the

system is driven by the relative motion of tectonic plates and compare the

results with seismological observations.

• We conclude in Section 3.6 with a summary of our results.

•’This description lias already been considered by earlier researchers in Refs. [23, 25].

4The results of the inclusion of rupture-rupture interactions are published in Ref. [24]. 3.2 Earthquake Phenomenology

In this section, we summarize a few of the observed features in earthquake dynamics. For a detailed discussion, see Refs.[l, 3, 19, 20], from which the in­ formation presented here is extracted. At the end of this section, we also outline some basic issues that need to be addressed while construct ing a model to study earthquake dynamics in a seismic zone.

3.2.1 Spatial Features

Seismic zones, where earthquakes typically occur, are located at the intersec­ tions of tectonic plates; examples include the regions around the margin of the

Pacific ocea.ii and mid oceanic ridges under the surface of oceans. The main fault is typically formed in t he intersecting region between the tectonic plates .5 The fault surfaces are observed to be stuck in between earthquakes and suddenly slip during an earthquake. This behavior is termed as stick-slip behavior and clearly frict ion plays a major role in determining how the stress is redistributed in a slip event.

An important spatial feature of the seismic zone is that the main fault is con­ nected to many other small faults (and so on) typically forming a fractal network.

The small faults, while they cannot support very large slipping events, nevert he­ less can rupture causing the formation of a new large fault. Also, most of the earthquakes are observed to occur in the vicinity of the fault surfaces. Thus, one

5The faults are classified on the basis of the type of displacements that occur at the fault, the direction of tectonic loading and the geometry of the fault, e.g., strike-slip fault, dip-slip fault etc. Here we do not go into details of this classification. may view the main fault and its vicinity as a weak zone subjected to an increasing shear stress due to slow relative motion of the tectonic plates. However, the ac­ curate determination of the loading of seismic zone still remains an open problem in Seismology.

The length of the main fault is typically of the order of few hundreds of kilo­ meters and the fault depth ranges from few lens to a few hundreds of kilometers.

The depth is typically taken to be the width of the crust on top of the upper mantle. In several seismic zones, the main fault itself is observed to have complex shapes often exhibiting fractal behavior.

It is generally observed that in some seismic zones most of the earthquakes are shallow focus, i.e., the location of the rupture, commonly referred to as a hypoci nt( 7\h is close (few fens of kilometers) to the surface* of the earth. The San

Andreas Fault zone in Southern California is an example where most of I lie earth­ quakes are shallow. In some tectonic intersecting regions deep focus earthquakes occur with hypocenter having a depth as large as 2-5 hundred kilometers (e.g., the fault zone between the Australian plate and the Pacific plate).

Moreover, most of the earthquakes are caused by shear instabilities such as slips and ruptures. Typically, the displacement discontinuities (slips) at the fault are observed to range from centi-meters to a few meters depending on the size of an earthquake. By using a long time average of the observed slip along the main fault, the (tectonic) relative plate motion (on the upper mantle) is found to be a

°T he term epicenter is used to denote the vertical projection of this hypocenter on to the surface of the earth. few cm/year. Also, since most of the displacement at the fault, is caused by large earthquakes, one can naively estimate the recurrence time for large events to be few centuries at the main faults.

3.2.2 Characaterization of Earthquakes

An earthquake is characterized by quantities such as “seismic moment” and

“energy released” which are defined below in the next, paragraph. However, it is convenient to specify the strength of an earthquake by a measured' surface wave magnit ude M„ on a logarithmic scale, commonly known as the Richter scale.

The magnitude is directly proportional to the logarithm of the seismic moment (or energy released). Thus, a unit increase in the Richter scale magnitude corresponds to a ten-fold increase in the energy released in an earthquake. Other characteristics of earthquakes include the durat ion, location (hypocenter), rupture geometry and displacement, discontinuities at the ruptures.

An earthquake is a result of a shear rupture in the seismic zone or a slip at the fault, (already fractured) surface8, for large earthquakes the rupture area is found to be as large as 10J to 10 1 kins2. For a linear rupt ure, the seismic moment .

Mo, is defined as the slip or displacement discontinuity, [

M0 = fiJ[u]dA, (3.1)

7In experimental Seismology, one measures the surface wave magnitude M , , which in turn can be related to energy released during an earthquake [32].

8Often, rupture and slip are used synonymously in Seismology literature, since the resultant displacement discontinuities due to these two faulting instabilities appear to have identical features. where //, is the shear modulus. Mo ranges in between 1022 — 1028 Dyne-cm for earthquakes with magnitudes (3 < m < 9). The energy released, E, is defined as the work done by the stress drop (

E = j\n](T0dA. (3.2)

This quantity ranges from 10,b — 10 25 ergs for small to large earthquakes. The two quantities Mo and E are linearly related to each other, i.e., Mo oc E. The Richter scale magnitude (M.,) is related to seismic moment. M 0 (or energy released E) by the following empirical law:

l o ' I w M o = r M, - f (I. (3.3)

For large earthquakes c fa 3/2 and for small earthquakes c fa 1. For a large

(earthquake with M 0 ~ 1028 Dyne-cm, the Richter scale magnitude M., fa 7 8 .

3.2.3 Statistical Features

liven t hough earthquakes are known to occur over m " ’ 15 s of years, t he em­ pirical understanding of the statistical behavior is only based on the available seismological data gathered over the past hundred years. The empirical observa­ tions suggest self similarity in the dynamics of earthquakes which is currently of interest to us.

One well known feature is the “Tlutenberg-Richter law” (already mentioned in the introduction) [2], which states that the number of earthquakes, n(E), with en­ ergy greater than E follows a power law: n(E) oc E~[i. The value of the exponent ft is found to bo close to unity. However, there appears to be a crossover from large to small earthquakes both due to the measurement and the dimensionality.

Large earthquakes typically have rupture lengths much larger than the depth of the seismogenic layer (crust, of the earth) and thus can be treated as two dimen­ sional. In contrast small earthquakes are three dimensional. We will address this crossover issue later while discussing our results in Section 1.

During an ea.rtluiua.ke ruptures occur in a very short, time. After the earth­ quake some of the ruptures are healed over a period of time (months to years) which is much larger than the duration of the earthquake. During t his healing time

“aftershocks’' are observed to occur which often exhibit, scale invariant temporal and spat ial characteristics. For example, they follow the “Omori law” which st ates that the number of aftershocks, n(l). after a time / following a large earthquake falls ofr as a power law \/i. [21]. There is also some evidence for the occurrence of

“foreshocks” before a large earthquake exhibiting a power law behavior similar to aftershocks, however, this phenomenon appears to be limited to only a few fault regions [ 1].

In addition to the above power laws, spatial moments and distributions of hypocenters are also observed to exhibit, scaling behavior. Furthermore, the spatial structure of the main fault is often found to be a fractal exhibiting self similarity.

In some tectonic regions near Japan, three dimensional fault, structures when projected (or mapped) on to a two-dimensional plane (surface of the earth) are found to have fractal dimensions as large as l.G [3]. 159

3.2.4 Some Numbers for San Andreas Fault Zone

Here we summarize some of the features of San Andreas fault (SAP) in South­ ern California. The seismic zone has a width of approximately 100 kms, length

400 kms and depth 50 kms. The main fault (when projected on to the surface of the earth) is almost straight with a fractal dimension slightly larger than unity.

It has complex geometric features such as intersections with other faults; and it often splits in some places also.

The shear stress at. the fault surface is estimated to be a few MPa (few tens of bars) and t he stress drop during an earthquake is observed to be in the range .‘ 1-8

MPa. The elastic constants, k and ft which are bulk and shear moduli, are in the order of few tens of Cl Pa. (30 CIPa for most of the SAP zone). The strains are small

(of t he order of 10-'1) implying that over a distance of a kilometer perpendicular to the direction of the fault, the displacement differences would be in the order of a few tens of cms. The relative plate velocity is estimated to be 35 — 55 mm/year.

3.2.5 Issues in Earthquake Modeling

• An important issue is the prediction of an earthquake. However, a theo­

retical investigation of this issue requires a model that exhibits some ba­

sic features of earthquakes. For example, a model, that captures some

of the geometrical complexities of the ruptures in an earthquake and ro­

bustly reproduces scale invariant features such as Clutenberg-Richter law for

frequency-size distributions, would be a suitable start ing point. 160

• Most of the earthquakes are caused by shear ruptures. Hence, one needs to

incorporate the tensorial feat ures of elast icity in the model.

• The tectonic loading is an important issue as it depends on the geometrical

and frictional features of faults in the seismic zone. It is also conceivable

that, in addition to tectonic plate motion, some of the loading of the seismic

zone also accrues due to mantle convection underneat h the crust.

• Material characteristics and spatial non-uniformities play a major role in

determining the locations of possible ruptures. Thus a model that can in­

corporate spatial disorder appears to be crucial. However, note that there

are many models that exhibit scaling behavior without disorder showing

that disorder is not crucial to achieve scaling (e.g., see [11] and [15]).

• One has to model the rupture criteria and the stress redistribution after the

earthquake. This is crucial especially for quasi-static models that neglect,

short-time dynamics as the boundary conditions need to be specified for the

fractures during an earthquake. Also, note that the flow of ground water can

play a major role in changing the st rength of rocks under largo stress [ 22].

Thus the rupture criteria can depend on ground water flows.

• There are three important time scales in the problem: (i) the duration of

an event, (ii) the healing time scale for the ruptures and (iii) the geological

time scale over which stress builds up in the seismic zone. It is important

that the model can be generalized to include some of the specific feat ures 16]

that occur during these time scales. Note that, in addition to the above time

scales t here are also others such as the time scale of static fatigue associated

with the occurrence of after shocks.

• The statistics of earthquakes manifest scale invariance in earthquake dy­

namics which is characterized by exponents such as /i. An important issue

is to examine the origin of scale invariance in earthquake dynamics. While

the idea of “Self-Organized Criticality” proposed by Bak cl nl [5] “who

pointed out that a class of extended dissipative dynamical systems r volve

themselves in to stales with no intrinsic length scales and time scales other

than the ones set by the system size and microscopic cut offs'" seems appli­

cable to earthquake dynamics, however, it, is not clear what, causes such scale

invariance. Alt hough, symmetries and const raints such as local conservation

laws (or force balance conditions) appear to play a major role in achieving

scale invariance in several models, recently a model, without (any obvious)

conservation laws [14], has been shown to exhibit scale invariance suggest­

ing that scale invariant behavior can arise purely due to many degrees of

freedom or the extended nature of the system.

• Moreover, if one is interested in the statistical features, it may be useful to

adopt, a quasi-static description for earthquakes. One can propose several

quasi-static descriptions, but, then the question is which description would

be appropriate for the study of eart hquakes. 162

• During an earthquake, part of the energy is released in the form of seismic

radiation emitted from the rupture. Moreover, rupture at one site can effect

a rupture at another site during an earthquake. This suggests that a model

that captures the effects of the seismic wave radiation such as a long ranged

stress redistribution following a rupture, would be appropriate for the study

of earthquakes.

• In addition to geometrical features, a model that also incorporates the st.ick-

slip behavior at the fault surfaces woidd enable a bet ter understanding of

the role of friction on the earthquake dynamics.

In the following sections, we address some of the above issues by considering quasi-static seismic zone models that exhibit scale invariance. We also construct a model that, can incorporate' complex boundary conditions associated with stick- slip behavior at the faults and long-ranged stress redistribution due to a rupture or a slip. 163

3.3 Representation of Ruptures in Seismic Zone

In this section, we present discretizations of an elastic medium that facili­ tate the construction of seismic zone models which can include prescribed fault structure. Specifically, the discretizations enable a consistent implementation of the equilibrium conditions of elasticity theory and also provide a general frame­ work to represent the discontinuities (ruptures, slips and friction at the faults) by appropriate non-elastic tractions (forces) which can be viewed as 'e4 (or double-couple9) forces.

We first, derive the stress redistribution due to a single shear rupture in bot h dipole and double-couple representations 10 by assuming the seismic zone as an infinite elastic medium. Next we devise a self-consistent method by which one can compute stress redistribution due to multiple ruptures and slips when the seismic zone also includes fault segments that are stuck. We emphasize t hat the self-consistent method proposed here is general and can be applicable to st udy the spatial and statistical aspects of multiple ruptures in an elastic medium .11

!'The term double-couple is commonly used in seismology to describe the earthquake source.

lnThe discretizations presented here are motivated by the dipole and double-couple models suggested in Ref. [23, 24] and Ref. [25] respectively.

"The self-consistent method developed here can be successfully implemented to study other physical problems that fall into this category, e.g., random fuse network problems, as illus­ trated in our paper [26]. Thus, this method can be used as an alternative to the well known conjugate gradient method which is commonly used while studying such problems. In addition to its implementation, the self-consistent method also allows a simple formulation to study superconductor-insulator-resistor networks. Moreover, the method is efficient especially for low defect (insulator or superconductor) concentrations. 164

3.3.1 Equilibrium Linear Elasticity

Before getting into the details of discretizations, let us recall the equilibrium conditions and the stress-strain relations (Hooke’s law) with in the linear elasticity in an elastic medium [20, 27]. Let the stress tensor,

?7, be defined at every point r in an clastic medium. In a three dimensional medium, the indices i,j denote the directions x,y,z. In the absence of external body forces (e.g. gravity), the stress tensor satisfies the following conditions in static equilibrium:

d,atJ{r) = 0, (3.1)

( ;L 'r>)

In the above equations summation convention is used, i.e., the sum is assumed over every repeated index over all its three' values. The first equation relied,s force* balance and the second is a consequence of torque balance. The strain tensor e,, in the linear elastic medium is dcfineel in terms of (the small) displacements f/,(r) as:

(,j = {fhuj + djui)/2. (.‘LG)

For an isotropic elastic medium, linear elasticity provides us with the follenving relation between the stress tensor rrtJ and the strain tensor c(J:

aij(t) = Kc„(f)S,j + 2//(c,J(r) - ^ uc//(r)), (3.7) where K and // are bulk and shear moduli respectively. 165

One can use the same equation but with a rescaled K for simple cases such as plane stress and plane strain which often reduce the problem to a two-dimensional description. For example, for the plane stress condition, i.e., when the stresses acting on the system are only along x — y plane with < 73, = 0, it requires cn — k'(e 11 + €22) with k' = 2fi/(K -F 4//./3). Then K in Eq.(3.7) gets replaced by

K' = Kk' + 2//(l — k')/3 with the indices (?, j , /) only taking values 1 and 2. For the case of a plane strain Eq.(3.7) does not change and the stress a:i:i ^ 0 even though C 3, = 0. From now 011. we assume plane stress situation while adopting a two-dimensional description and use linear elasticity condition given by Eq.(3.7).

We now present discretizations of the elastic medium in which ruptures can be represented by appropriate non-elastic boundary conditions such as dipole and double-couple forces. First we describe the dipole representation as it is relat ively simple and provides insight into the discretization scheme.

3.3.2 Dipole Representation

This relies 011 a simplification of the tensorial nature of the elasticity equations, which is accomplished by considering only the x displacements, ur, and neglecting the displacements along other directions .12

Consider the two-dimensional case corresponding to plane stress. Since we are only concerned with x displacements, the equilibrium condition for the system is

'-This simplification may bo justified for the study of earthquakes as most of the observed displacements are along the direction of the fault. Indeed, most of the existing models only consider x displacements while studying the dynamics of earthquakes. 166 only the force balance (along x) satisfied by the shear and compressional stresses.

^3 :^X 3 ' T d y & r y — 0* ( 3 ■ S)

The stresses along x are related to the displacements by a Hooke’s law:

^XX ^ 0 ^.7' ^ 7' • (33))

&xy — /I dyUT. (3.10)

We now explore a (physically appealing) discretization by making a particular choice for the spatial representation of the stresses and displacements. Later we will see that this choice corresponds to simple definitions for the discrete (lattice) derivatives.

The infinite elastic medium is discretized in to regular blocks of size ar xa y xa~. with the thickness a,, so that the blocks form a two dimensional lattice as shown in Figure 3.1. The displacements iiT are defined at. the centers of individual blocks and the stresses at, the interfaces between the blocks. Specifically, the shear stress cTj-y is defined at the block surfaces along x and the compressional stress

When there are no displacement, discontinuities, the elastic stresses at the surfaces are given by the Hooke’s law, which can be written as:

(Txy{r) = fi [ux(r + auey) - u T{r)]/ay, (3.12) 167

Figure 3.1: Schematic diagram of the discretized elastic medium. The shear rupture is shown by a thick line between the blocks centered at Fo + «y< y and F0. 168

F iy ------

a

F 2x F ix a v y u x

a X u ------►

p 2y

Figure 3.2: Forces acting on a block centered at ?r0 are shown. Arrows point the direction of the force. Fx are tensile forces given by F\x = (Txx(ru)ayaz and p 2r = Vrr(^) ~ axtr)aya,. Fy are shear forces given by F\y — rrxy(fi))axa. and I'iy — (lyC y)(lxU z. 169

and the force balance 13 condition along x is given by:

[orAr) - V x A r- axex)]/ax + [

where c, denote unit vectors along the direction i. This equation, upon substitu­ tion of stresses in terms of displacements, becomes:

Ko [M r(r + dxex) + ux(f— axex) - 2?iJ,(r)]/a£

+/i [?/,x( 7r + aycy) + ux( r - ayey) - 2ux{r)]la2y = 0. (3.M)

We are now interested in finding the stress distribution in the elastic medium due to a shear rupture between the blocks centered at f 0 and f0 + «vcv. Note t hat a shear rupture is equivalent to having a displacement discontinuity between the blocks. Also, due to a rupture the final shear stress on the ruptured surface is reduced to a lower value <7„ (commonly referred to as arrest, stress).

Let the equilibrium stresses and displacements without the rupture bo and it'’}'1. Note that, the old stress variables should satisfy the force balance condit ion for equilibrium. The new equilibrium stress due to the shear rupture can be writ­ ten as : (j’lfw = a01'1 + a \ where the a' itself satisfies the force balance condition.

The additional stress a' due to the rupture can be related to the additional dis­ placements uel everywhere except, at the rupture by Hooke’s law .1'1 By separating

13Thc forces (tractions) on a block surface are given by F, = ^ (TjjHjA, where it is tin* unit normal and .-1 is the area of the surface. Thus the shear force on an interface along !• is ±(Tryara:\ the tensile force along the y interface is ±crJ.ya;/a! depending on the outward normal. The balance of these forces acting on a block are represented by this equation.

14Here we are considering the Hooke’s law for the deviatoric stresses, i.e., a' [20]. 170 this violation at the rupture, we can write:

CM = + <*•!«

»™(r) =

can be viewed as the elastic response of the system to this non elastic part.

In particular, this non-elastic part can be viewed as being due to a displacement discontinuity between the blocks on either side of the rupture. By using the

Hooke’s law, we can arrive at, the following equation which relates the elastic displacements to the non-elastic st ress:

Ko K ' ttrfr) + n'J{r- r ) ~ {r)\/d],

+// [»;'(r + a„ty) + (iy(y) - '2u'J{r)\/ ay

= “<»/«» “ ?To) -b {r- dyty - Co)]. (3.18)

The right hand side of the equation contributes only at, sites r = r() and r —

H) + ayt >n w’^ 1 opposite signs. By setting a ’"y = /0, this contribution is:

f s ’,o,f(n = - i ^ y - ^ i,] (3.i9) ay which is equivalent to placing “dipole” forces with the magnitude F, = faar(i: on the blocks centered at r*o and -f uyi y in opposite directions.

If we know the non-elastic part, we can determine the elastic displacement s at the centers of all the blocks by using Eq. 3.18 in Fourier space. Thus we define 171

the Fourier transform:

ii.cJ(k) = Ji i rj (r)cxp(-ik.r), (3.20) where thesum/integral is over all the sites r = maxcT + nayey with m, n being integers ranging from —oo to + 00. The components of the vector k take values

: kr c ( — 7r/rtj., 7r/a r] and ky t ( — n/ay. Tt/ay]. Using Eq. 3.18, in Fourier space we obtain,

,yi(Z) = (1 - e-'^y^xpi-tk.ro) ft [2 — 2cos(krar)] + 7(2 — 2cos(kyay)} with 7 = fKij./K0(iy. 'Fhe parameter 7 is a measure of spatial anisotropy and we take it to be unity in later sections for simplicity. By taking inverse Fourier transform of the above equation, we get:

<.'(,■) - 2 * 2 * fj,), ( 3.22) /' whore the Green’s function Gr is given by

] r'l* r lx (] _ ( ~ ’P« + J o rfM ft[2-2«»^H -rl2-2CMff,)- (:U:1) with r = n)(ijCr + nayey.

Now by using the Hooke’s law, Eq. 3.11 and 3.12, we find

= fol [G,(r - r ) + aycy) - Gr( r - r0)]

arAr ) = ' — — »*0 + aA j ) — Gx[r — Co)]. ay

In summaiy, the elastic part of the shear stress can be written as:

e / / -*\ __ * 7 . / o ^ y 1 -* -> \ <7a-y(J') — G ry(v 7‘o), /' 172 where Gxy is the Green’s function given by:

„ 1 [2* f 2n , , [2 — 2 cos(»„)]e'mp;r+'nP!' ry{r) ~ (2tr)2 Jo Jo PrPy [2 - 2 cospT] + 7 [2 - 2cospy]' (r27) with r = maxcx + naycy.

Now the key is to determine axcy = f 0 by using a boundary condition such as a known residual shear stress cra after the rupture. This implies:

< V o) +

= g" ~ a^ {Po) rv 1 + 7A with A = Gxy(0) = [G,. («,/,,) — (7r (0)]. For 7 = 1, we get A = — <7-1 when* d is the spatial dimension. In two dimensions the dipole force Fx = —'2

Discretization based on Lattice Derivatives:

So far, we have implemented the discretization by defining the stresses and dis­ placements at appropriate places. The same results can be obtained by using discrete derivatives, D, along i = (.(',(/), defined on any function g(r) as follows:

Dig(r) = [

However, one needs to be careful while interpreting the derivatives. Since the elastic stresses are derivatives of the displacements, clearly the definition of dis­ placements at r = maTex + nayey is sufficient to determine the stresses 011 the interfaces: r,- = r - f a,c,/‘2. Moreover, the stresses, whose derivatives are defined at r. consistently satisfy the force balance condition for all the blocks, since they are centered at r. We can carry out the same analysis that we have performed earlier by including the non-elastic stress appropriately at ry0 = f0 + «vcv/2 ‘, and obtain the stress redistribution due to a shear rupture. The connect ion bet,ween this and the formulation we adopted earlier is simply relabeling! Note that in this derivative formulation, shear stresses are defined at surfaces ry = f + «wcv/2 and the tensile stresses at rr = r-f arcr/2. Now, if we relabel the stresses at r, as r we get back our original equations Eq. 3.1 1 3.13. Thus, defining appropriate discrete derivatives enables a convenient discretization scheme for the elastic medium and this becomes evident in the next subsection on double-couple representation.

Three Dim fusions: For the sake of completeness, here we provide the (boon's function and stress distribution due to a rupture along r — c plane at r() in 3-d:

= 7 < l av az _ axyar(Tnc(l [G'.r(r - r0 + ~ Cir( r - rn)] <£(?) Cly

u'J(f) l arlaV 174 where the Greens function Gr is now given by

a (f) = — f f r dP*dP«dP* ( \ - e ~ i»y)eim^ +in^ ‘ x (2 tt)3 J J Jo 2 — 2cospT + 7(2 — 2cospy\ + 7'[2 — 2cospz] ’ with r = maTeT -f naycy + lazez and 7 = K0al/fia*; 7 ' = 3.3.3 Double-Couple Representation

We now consider a discretization which preserves the full tensorial nature of the elasticity by including both x and y displacements. Similar to the case with the dipole representation considered earlier, let the elastic medium subjected to a plane stress condition be divided in to regular blocks of size ar x ay x ar.

Displacement vectors { t /( r ) } are defined at the centers (f) of the blocks, while the stresses {

For this spatial representation, it is convenient to adopt the following discrete derivatives /), defined11’ for any function g(r):

Dxg(r) = [g(f + b) + )

Vyfli?) = l>) + g ( r - J) - fj{r + d) - g(r - b)}. (3.37) ~ay where d = (arcr — ayey)/2 and b = {arcr + ayey)/2. With these definitions of lattice derivatives, the discretization of elasticity theory becomes straightforward.

The strain tensor, which characterizes the distortion of the medium, is then given by:

+ Djit,{r))- (3.38)

15From now on r = maTe.T + naycy and r ' — 7V+ arcr/2 + ayey/ ‘2.

H‘One can consider a variety of derivative definitions. But some of them lead either to unphysical stress redistributions due to ruptures (such as sub-lattice decoupling) or to a degeneracy problem for the direction of a rupture. See, for example, the double-couple representation considered in Ref. [25]. The latter can be avoided by using the average stress along a surface in determining the possibility of the fracture direction. Then the rupture involves non-elastic stresses at more than one corner as will be seen next. 176

Also, when all deformations are elastic (no ruptures exist), this tensor issimply related to the stress tensor through the generalized Hooke’s law:

crt,(r) = K eu{r)6ij + 2//,(e,,(r) - ^ft;jeH(r)), (3.39) where the summation convention is assumed and t he indices take values 1,2 . More­ over, in terms of the discrete derivatives the equilibrium conditions for the system are simply given by:

Dr(TTT + Dy

/JxOVy + DyVyy = 0. ( 3 •‘1 1 )

Note that (Try = eryr holds here. Let us expand the first equation3.10 using the definitions (3.36 and 3.37) of the discrete derivatives:

k«(»7 +b) +

+ kx./(T + ft) + Vry{r- d) - (Try{r - ft) - (Txy(v + d)]/‘2ay = 0. (3.12)

We can interpret this equation in tin* following simple way: Consider the stresses

dry = Wry{r + ft) + <7rtf(r - d)\/2. (3.13)

Now it is easy to see that in terms of the average stresses er, the above l\q.(3.12) takes a simple form similar to the one we have already seen for t he dipole case, i.(.,

Eq. 3.13. The force balance along y also takes similar form. Thus the definition of 177

&ij provides a clearer connection with the physical picture of blocks presented in the dipole case. Moreover, physically it also makes sense to consider the average stress along a surface to determine whether the surface will be ruptured or not.

We now derive the stress redistribution in an infinite medium due to a local shear rupture along .r direction between blocks centered at Vo and f 0 + following the procedure described for the dipole case. Recall that we have assumed that after the rupture the original shear stress (crxy) on the fractured surface' is reduced to a known residual stress cra in the dipole case. Also, this stress reduction causes a force imbalance leading to a stress redistribution in the elastic medium.

Similar to the dipole case, we can compute the stress distribution, by denoting the stress after the rupture as a\lJw and the stress before the rupture as a '-j'1 with a\ V =aijd + aij-> where

0 ^ = 0 . (.1 1 1 )

Now, since linear elasticity is valid everywhere except at the two corners (denoted by r*i = f{i + />, f '2 = 7*0 — (I) of the ruptured surface, we can use Hookes law for the deviatoric stress a' except at the rupture. This exception can be isolated as a non-elastic part and the new stress distribution can be written as:

< yw{r') = < ( f ' ) t < ( r " ) + * :;[f,w 4 i,v ,l (3,1.5)

< ”(>■') = <"(>■') + <-',(>■') (3,16)

+ (3,17) The now displacement fields are given by:

- <"(r) +

K;™(f) = u^fr) + <'(?)• (3.-10)

By using the equilibrium condition Eq. 3.44 for the additional stress field due to the rupture, we obtain the following equations for arjy.

+ = 0 (3.50)

Vjayj(r) + + •Wj] = 0 (3.51)

Now, we seek a solution for arl for a given o"y. This solution in turn can be used to obtain a"'y self-consistently by using the boundary condition for the ruptured surface':

< / ’ =

Before proceeding further, note that the equations 3.50 and 3.51 for a'1 can bo rewritten as:

= / * . (3.5:i) where

!rC = (fo/2ay)[26?A+ay'y -26?,?0

T^r,ro+Ujei+ayey — ^r,fo+<>r>‘s

T^r.i'o-aj es +ayey ^r,r0—iixfj ] (3.0-1)

J y ~ (/o/2«J-)[^r,r0+'ix^+iiyty + ^r.fc0+« xtj 179

Figure 3.3: Double-couple force distribution on the blocks around a single shear rupture is shown. The distribution is drawn for ar = ay.

the* symbol / 0 is used to represent for convenience. In the equation 3.53, o ik * can view that a'Jy is generated by a source / ',r, which is nonzero only for the blocks around the rupture as shown in Figure 3.3. This force distribution satisfies the force and torque balance conditions, and hence forms a double-couple distribution.

Thus, the discretization scheme adopted here is consistent with the well known double-couple representation of a shear rupture in elastic medium [ 6].

Next, we turn to the determination of cr-jl for the above double couple force distribution. The solution can be obtained easily in Fourier space. We start with 180 defining the Fourier transform as:

Then, in Fourier space the discrete derivatives Z), are given by

(Dxg)(k) = [(2i/aT)s\n(kxaT/2)cos{kyay/2)]g(k) = d.r(k)g(k), (3.57)

- [{2i/ay) cos(krar/2) x\n(kyay/2)]g{k) = dy(k)g(k ), (3.58) and the double-couple forces are given by:

/f =

rf; = d A h .n h . (3.00) with

(3.01)

Recall that the elastic stresses (rrjj) are related to elastic displacements (a]1) by the Hooke’s law given by Eq. 3.39; thus we simply have:

e x ' = [(A + + iiD'IK' + (A’ + /‘/3)e,e„i/;', (3.62)

= IM' + i/'/:i)eJ + i‘D ‘ju ',1 + (A’ + /i/;))e ,e v»;'. (3.63)

Using the above equations, the Fourier transform of equation (3.53) can be written as:

[(A' +

[(A + .1,1/3)4 + K K /e ) + (A + i,/:t),l,d,,rj(k) = ~J[k)

These are coupled linear equations and have the following solution:

f i ’J ( k ) = 77^11 K + i l ' P K - (/' - (3.66) LA{k}

*?<*> = + 4;*/3)<§ - (/i - 2 /./3 R 4 ) , (3.67) A(fc)

where A (h) = //.(A' + rf )(rf2 + f^ ) 2- Now we can get (it is straightforward hut tedious) the stress tensor (Tcl(k) using the Hooke’s law given by Eq.(3.39):

arJr(h) = ( K - 2//./3)(drvrJ + dyuj) + 2fi(lTujl

\dr

where o = (/\ + ///3)/(A' + -1///3).

By taking the inverse Fourier transform of the above equations, we can obtain the stress redistribution due to a given double-couple force distribution, e.g., cor­ responding to a shear rupture. Below we provide the results for the shear stress a' on x surfaces and the x displacements of the blocks due to a shear rupture.

These results will be used in later chapters. 182

Shear Stress: We start with the acJ defined at, the corners r ' = f-\-b by using

Eq.(3.70) and the ruptured surface corners (r*i, 7^2):

f2n/ax r2n/av (j n rfl* rjb O ' d 2 d 2 c l / - > \ e / I Ur UyUhx Uhye f „,fcT„Tlr 4 u'ruy ,t

) = /oX /„ ------(S )5 l1+c ( J ' 7 I )

Now, since the average stress <7^ , defined at the center of the x surface v " = r T a.yCy/2 is given by the average of the corners, we have:

r2n/n7 r2Tr/ay ,, n rlL' rfL' f'Mr — ro) 4q. (p (p

W'-hL I ]■

This gives us the additional stress &’ (=

Ky(if") = fo'y C-i,ry(f - f 0), (3.73 ) with the following Green's function

r" (~)- ° f f (lP'dPy ( ""Pr + "iVy [1 + cos^.)] sin 2 pr sill2 ])y ’j y 7 Sir2 J J [sin 2(pT/ 2 ) cos2(pv/ 2 ) + 7 cos 2(pT/ 2 ) sin 2(pv/ 2)]2 1 where all integrals are from 0 to 2n with r = marer + naycy and 7 = a2/a2. The

Green’s function in 2d can be simplified for 7 = 1; numerically it can be easily obtained by reducing the 2d integral to Id as shown in Appendix C. Note that the elastic part of the average stress is given by:

< ( ? " ) = ")- - f„). (3.75)

From thenotation of f,f' and f " (for centers, corners and surfaces of a block respectively), it is clear that one can relabel the stresses at the corners r 1 with r and the average stresses on the surfaces r " with f and consider the problem with in a single lattice, similar to the dipole case with appropriate discretizations of the equilibrium conditions and Hooke’s law, see Eq.(3.11-3.13). From now on we refer to the relabeled variables, i.e., represent ary(r ") as

Now the non-elastic part, = f 0 can be determined by the Green's function and the following equation corres, A.A '' g to the boundary condition given by

Eq.(3.r,2):

ary(r o) + /o7^»ry(0) =

This simply gives us the non-elastic part. old er„ - (T•ry

In summary, by using only the final stress value at tin* rupture, we can de­ termine the stress distribution in the entire system. The same idea will be used in determining the stress distribution due to multiple ruptures (or slips) in the seismic zone having arbitrary embedded fault structures in the next subsection.

Next we provide the x displacements of the blocks due to a single shear rupt ure.

Displacements along x: Using Eq. 3.66, the x displacement, can be obtained as the following:

-e, = k ^ L [ CxGxr{k) + c2G'2j.(A0](1 - r-'**“») (3.78) tl with ci = 7 /2 and c 2 = —\dj2. The constant = {K —2p/'i)/(K +1///3) depends only on the elastic constants (011 the material characteristics) and 7 = aj./a2y. We can now write this equation in real space:

ucJ(r) = '^—j^[c\{G \3{r — fo) — G\r(r — r0 — o,ycy)}

+c2{6 '2r(r - r0) - Ci2x( r - r0 - «„£„)}]

= i>).

The Green’s functions are given by:

dpTdpyc'b”Pi-+"Pv) cos 4(;i,/2 ) sin 2(py/2) r l?\ - 1 [ f (,pr dl ~’U (2n)22tt)2 JJ J J fsin [sin 2(pj./(px/ 2 )( ) cos 2(py/ 2 ) + 7 cos 2(/v / 2 ) sin 2(p 1// 2)]2 and

l _ f f dprdprdpyt dpyc’lm’’T+nPl') cos 2(/r, / 2 ) sin 2(pr / 2 ) cos 2(/)y/ 2 )

(2( 2 7t tt)2)2 7J 7J fsin2(wr[sin 2(/j,/2 )cos 2(;>!// 2 ) + 7 cos 2(/ir / 2 )sin' 2(/;y/ 2)]2 with 7r = inaxcr + all(l integrals are from 0 to 27r. 185

3.3.4 Self-Consistent Method

This method is equally applicable to both dipole and double-couple rupture representations. Since, in this chapter we restrict ourselves to only shear insta­ bilities while studying earthquakes, we illustrate the method by considering only the shear stress axy and the x displacement ux. Generalization to oilier stress variables and displacements in other directions is straight forward.

Following the results of the previous subsections, we begin with the shear stress and displacement fields due to a single shear rupture along .r. Recall that the new shear stress is given by:

= <"('■)+«<•>+m-.«, r i - s - ' i where the elastic part is:

Also, the extra x displacement at the center of a block due to the rupture is given by:

u'J(r) = ^ G A f - r0). (.1.81)

The Green’s functions Gx,Gry can be obtained from the results presented in the previous subsections (for either dipole or double couple case, see Eqns. .1.21, .1.27,1.75 and 1.78).

Next we describe the method to self-consistently compute the stress redistri­ bution due to multiple ruptures first and then generalize it to the case where some blocks are stuck (at the fault surfaces) due to non-elastic features such as friction. 186

Multiple Ruptures:

Our aim is to obtain the stress redistribution due to many ruptures simultaneously.

In analogy with the single rupture case, the additional stress caused by many ruptures can be viewed as being due to dipole (or double-couple) forces placed on the blocks around the rupture. Suppose there are ii\ ruptures located at

{r*oii i = 1, . . . , »i }; the elastic part of the extra stress due to these ruptures can be expressed as:

= J 2 n fo i Gxy(r- roi), (8.85) i=i where {/o;} are the dipole (or double-couple) forces at the rupture sites. Now, the magnitude of these forces ({/o,}) can be obtained self-eonsistently by solving a set of linear equations corresponding to the boundary conditions with final stress

{o'a(^oi)} at the ruptured surfaces:

} by using:

< „ w(r) = < V ) + 7 E /o iGry{v - nu). (8.87) !=i

Note that this method can be easily extended to include other types of ruptures by choosing appropriate boundary conditions. 187

Multiple Ruptures and Stuck Surfaces:

Consider the case when the seismic zone has multiple ruptures (as described above) along with some blocks that are stuck at the fault say, in x direction. One may view that the blocks on one side of the fault are stuck with the blocks on the other side due to friction. This means that the additional x displacement due to ruptures should be the same for the blocks on both sides of a stuck surface.

This situation can be implemented by placing non-elastic stresse s {/oj} at the mutual surface of the stuck blocks located at {roj

<'(?•) = — " i f M C A r - rok). (3.88)

Since the blocks are stuck at. the surfaces ?r0;, we need:

ur (r0j + (ly(y) — "t (?’oj) = 0, (3.89) which is equivalent to

Y fok[(jr(rOj + ay*:U ~ ?’UA-) “ j ~ ^’oA-)) 1 0 (3.90) k= 1 for j = ii\ + + 7?2- Now using the n\ + »2 linear equations defined by

Eq. 3.86 and 3.90, we can solve for fob for all k = 1 ... n.j -f n-2 , which immediately determine the stresses everywhere. Note that the elastic parts in Eq. 3.86 are now determined by all the ti\ -f ??2 non elastic parts. Also, the non-elastic partat the stuck surfaces may be interpreted as a frictional force. 188

Remarks:

The self-consistent method can be generalized appropriately to include other fea­ tures such as normal forces at the surfaces and displacement discontinuities in other directions depending on the problem of interest. Moreover, one can easily incorporate arbitrary zone boundaries for the seismic zone by using appropriate boundary conditions for the boundary sites, see e.g., Ref. [26], where a random fuse network1' of size L x L has been considered with both constant input current and constant voltage boundary conditions using this self-consistent method.

11 Also note that, the dipole model is directly related to random fuse network on a square lattice in the following way: (i) aTy —► iy,

3.4 Seismic Zone Model

In this section, wc present a quasi-static seismic zone model subjected to a slowly increasing external shear stress and study the statistical features of earth­ quakes in the model. Such models have been studied previously and have been shown to exhibit Self-Organized criticality, i.e., Gutenberg-Richter power law for the frequency-size distributions [23, 25]. However, these studies have ignored rupture-rupture interactions during an earthquake for simplicity. Here we include this essential feature by employing a self-consistent description for an eart hquake and show that it leads to a new scaling behavior consistent with the seismological data.

By using the discretizations proposed in the previous section, the seismic zone is modeled as a lattice of blocks with size ax x a.y x a- subjected to a slowly in­ creasing external shear stress in the xy plane, where z is chosen along the vertical.

In the two-dimensional model, corresponding to plane stress condition, the width of the crust (along z) is set equal to the thickness of the block (see, for example,

Figure 3.1). The stresses are defined at the interfaces between the blocks so that equilibrium conditions are satisfied for each block. We restrict our attention to the shear mode of fracture and consider only shear ruptures along x in 2d and xz planes in 3d. For this purpose, each interface (along x) is random]}' assigned a normalized threshold stress between 0 and 1 which sets the maximum shear stress 190

(erXy) the interface can sustain and beyond which it will rupture .18 For simplicity, we set the anisotropy parameters to unity, i.e., 7 = 7 ' = 1.

The slow increase of the external shear stress is realized by adding a small stress to all aIy at the interfaces until a rupture occurs at an interface initiating an earthquake. This stress build-up at all r can be represented as:

t +f>1-) = <7ry{r, I) + S8t (3.91 ) where ,S is the slow external shear stress increase rate and 6t is the time interval between successive earthquakes.

Now, since the Green’s function (e.g., Gxy in Eq. 3.27) decays as l/r'1 (where d is the spatial dimension) with the distance r, the occurrence of a rupt ure leads to a long-ranged elastic stress redistribution that decays as a power law from the rupture. The resulting stress configuration in turn could trigger more interfaces to rupture causing an avalanche in the system which we define as an earthquake.

Thus the earthquake proceeds in a sequence of steps after the initiation. In each step .s, ruptures occur causing a long ranged stress redistribution19:

<'(rl = <„(f) + £ -*) + £f>6„. • (3.92)

18One may view these surfaces as potential rupture sites in the seismic zone. We are considering a simple case in which these sites form a square lattice of size LxL . As an alternative, however, one may also consider a specific spatial pattern for these potential rupture sites.

19 We assume that the redistribution of elastic forces occurs over an infinite medium and solve the equations for an infinite system. For our finite system this corresponds to an open boundary condition with the activity outside the system being neglected. 191

Here the index i stands for the non-elastic, i.e., ruptured sites. The non-elastic stresses /* are determined by the choice of the adopted quasi-static earthquake de­ scription. The earthquake ends once the stress distribution reaches below the threshold, i.e., when the stresses at all the interfaces are below their corre­ sponding thresholds .20 The stress redistribution can be considered instantaneous since it occurs with the speed of sound, much faster than the geological time-scales involved in the build-up of stress. Therefore, we also hold the external stress fixed during an earthquake. After the earthquake, all the thresholds of the fractured surfaces are re-set, to a random number between 0 and 1 signifying healing of the rupt ures before the start, of next, earthquake .21 This process is repeated and statistics of earthquake properties are then studied.

3.4.1 Independent-Rupture Description

As mentioned in the beginning of this section, the above evolution for earthquakes in a seismic zone has been considered previously using both the dipole model [23] and the double-couple model [25].22 However, these studies, for simplicity, do not include the rupture-rupture interactions, i.e., treat each rupture independently of

2UOne may view the earthquake as a sequence of unstable stress configurations that obey force balance condition.

21 One can consider fixed thresholds corresponding to quenched disorder, however, the scaling features are found to be insensitive to this choice. Also note that this choice leads to a deterministic evolution for the model.

"In these models, shear ruptures are modeled by dipole forces and double-couple forces respectively. 192 the other ruptures during an earthquake. Specifically, in the formulation consid­ ered above with Eq.(3.92), each // is obtained independently of other ruptures.

For example, in Ref. [23], f- = —d/(d— is used in d-dimensional model.

With this independent-rupture quasi-static description, both the dipole [23] and double-couple [25] versions are shown to display identical scaling behaviors exhibiting Gutenberg-Richter law with a ft « 0.35 in 2-d [23, 25]. While it has already been shown that the 3-d version of the dipole model yields a ft « 0.65 with the independent-rupture description [23], based on the 2-d results, one may also expect, ft « 0.65 for the 3d double-couple model which still awaits the investigation due to its numerical complexities.

However, the exponent, ft for 2-d, obtained with this simple quasi-static earth­ quake description, significantly differs (see at the end of this section, i.e., 3.1.1) from the value suggested by the recent analysis of earthquake data for large 2'5 earthquakes [30].

*M Recall that large earthquakes are essentially two dimensional since the rupture lengths are much larger than the width of the crust. 193

3.4.2 Self-Consistent Description

Here, in contrast to the above simplified description, we consider a different

quasi-static description for an earthquake, which includes rupturc-rupture inter­

actions during the earthquake .24 This leads to a new scaling behavior with an

exponent ft consistent with the seismological data.

Consider the case in which the stress at the rupture within an earthquake is

maintained at zero during the rest of that earthquake. Then we need to solve the

following set of equations “self-consistently”:

<„(*) + £ /*G,tf(f - r,) + ft = 0 (3.93) I

to obtain /* at each rupture site r, during the earthquake. This self-consistency

introduces rupturc-rupture interactions. Once f* is determined, the stress every­

where can be obtained easily using Eq.(3.92) in each step of the earthquake.

3.4.3 Results

Now we illustrate the spatial and statistical features of the model with the self-consistent quasi-static description and contrast them with the results of the

independent-rupture description. For simplicity we first consider dipole Green’s functions.

24The model st udied in Ref. [28] appears to obtain stress redistribution self-consistently within each step of an earthquake, using a conjugate gradient technique, yielding exponents /i similar to independent-rupture description. Our simulations also show that the inclusion of rupture- rupture interactions only in a given step of an earthquake do not change the scaling behavior from that of the independent rupture description. 194

As the stress is slowly increased, earthquakes occur and the ruptures are healed, the system evolves to a critical state exhibiting Self-Organized Critical- ity [5]. Note that in the quasi-static model the stress distribution should always satisfy the equilibrium conditions. This feature is observed to be crucial in achiev­ ing Self-Organized Criticality in the seismic zone model.

We characterize each earthquake by quantities analogous to the energy released

E and seismic moment Mo for real earthquakes. Since, the net dipole force, fi = £ ,.// , is the body force equivalent [ 20] of the slip across the broken surface i, it is a measure of the local slip. The stress drop at a broken surface is simply the stress just before the earthquake a01'1 since anr"’ = 0. The energy released in an earthquake E is taken to be the elast ic work done by the stress drop to create a mean slip integrated over the seismic zone [19], and therefore, we define:

(T91) t where the sum on i is over all the ruptured surfaces. Similarly, since the magnitude of the seismic moment in the xy plane is proportional to the total mean slip along

,r, we define

(d.95) I where the sum is over all the broken surfaces. We also monitor the number N of the ruptured surfaces during each earthquake. 195

80

60

70

0 0 70 40 60 80 X

Figure 3.4: Spatial patterns of typical large earthquakes in 'Id for the self-consistent dipole model. The triangles represent the ruptured blocks in a 2d earthquake in a 80 x 80 system. 196

20

5 O O O

0

5

0 0 5 10 15 20

Figure 3.5: Spatial patterns of typical large earthquakes in 3 — <7 for the self-consistent dipole model. The ruptured blocks in two adjacent layers (.c — z planes) of the 3 — d system are shown (in this particular earthquake, only blocks in these two layers rupture). The triangles and open circles denote the locations of the ruptured blocks in x — z planes at y = 16 and y = 17 respectively in a 20 x 20 x 20 system. 197

Spatial Features

First,, we display in Figures 3.4 and 3.5 typical spatial patterns (broken surface configurations) of large earthquakes that occur in 2 — d and 3 — d respectively.

The pattern of a 2 — d earthquake (Figure 3.4) is that of a linear crack with some scatter due to the rupt ure of weak spots caused by long-ranged interact ions. For

3 — d, the ruptures are confined to adjacent xz planes with some scat t er as shown in Figure 3.5 and have the form of a planar crack.

In particular, for the 2 — d model, one notable feature with the self-consistent description is that, the energy E scales as /„, where /0 is the length of the crack formed in a moderately large earthquake. To understand this, we compute the energy released due to the creation of a linear crack of size /(> when the seismic zone is subjected to a uniform shear stress. Numerical results shown in Figure' 3.(i indicate E oc for Iq > 10. This result, is also consistent with the expectation of the energy released during the formation a crack of size l0 in an elastic medium with appropriate (zero shear stress) boundary conditions. Furthermore, the con­ tinuum limit, of this model for a linear crack of length /« amounts to solving the

Kolosov-Inglis problem of an elliptic hole (with force balance along one direction for dipole case) in an elastic medium subjected to constant, shear stress boundary condition at infinity [29]. This gives 1 /y/r scaling for the stress near the crack tip in the limit of vanishing width of the hole. Thus, our discrete model, despite 198

o o V —

O 00

o LJ d (D Cn .3 °

o C\j

o i o 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 Loge lo

Figure 3.6: Log-log plot of E vs. crack length lQ. The straight line slope is 2. 199 its simplicity, captures the universal features of stress redistribution due t.o frac­ tures in elastic media, further justifying its use in the investigation of earthquake dynamics.

Statistical Properties

Next we discuss the distributions of the energy released ( E ) in both 2d and 3 d for the self-consistent description and contrast with the results of the independent rupture description. The results are obtained using a system of size 80 x 80 at d = 2 and 20 x 20 x 20 at d = 3 for 100,000 earthquakes after skipping the first,

20,000 earthquakes to evolve the system to a critical state. In Figure 3.7 we show a log-log plot of the distribution P(E) vs. E ; the linear portion which extends up to E corresponds to a power law P(E) « E~l~fi with li = 0.8 ± 0.1 for 2 tl and /3 = 1.0 ± 0.1 for 3-d. Note that, the value for (3 in 3-d is in agreement with the mean field exponent derived in Ref. [9] using a model with gap-dynamics.

In contrast, the independent-rupture description [23] yields (3 « 0.35 for 2-d and [3 « 0.6 for 3-d. (In Figure 3.8 the distribution for 2-d is shown with a system of size 80 x 80 for this case.) Moreover, for the self-consistent descrip­ tion, beyond E the distribution has a hump which is a finitc-size effect unlike the independent-rupture case. We find that, E increases with the size of the system

L and consequently the probability in the hump scales to zero with increasing L.

Such humps also occur in the distributions of other quantities.

The distribution for the seismic moment M 0 (not shown here) scales similarly with the same exponents as that of the energy. This is expected since the average ‘200

o

4

-8

2 0 2 6 8 LogeE

o

4

8

2 0 2 4 6 8 LogeE

Figure 3.7: Self-consistent dipole model: log-log plot of the probability distribu­ tion of the energy release P(E) vs. E: (a) for the 2d model (straight line has a slope — 1.8 ); (b) for the 3 d model (straight line has a slope « — 2.0 ). 201

O

t-U Q CD QJ I CD o _J cn

1 3 5 7 Loge E

Figure 3.S: Independent-rupture dipole model: log-log plot of the probability distribution of the energy release P(E) vs. E: for the 2d model (straight line has a slope Rs —1.35). This figure is shown for comparison. For more data, see Refs. [23,25]. 202

o

3

-6

-9

2 0 2 4 LogeN

o

3

6

-9

2 0 2 4 6 8 LogeN

Figure 3.9: Self-consistent dipole model: log-log plot of the probability distribu­ tion of the number of broken blocks P{N) vs. N: (a) for the 2d model (straight line has a slope « — 2 .1); (b) for the 3 d model (straight line has a slope k, —2 .3 ). 203 slip due to rupture-rupture interactions is proportional to the linear size of the macroscopic rupture. However, the distribution P(N) for the number of fractured blocks TV follows a different, power law as shown in Figure 3.9. We find P(N) oc

./V-1-^1, with 0\ = 1.1 ±0.1 in 2-d and 0\ — 1.3 ±0.1 for 3-d models25. Thus the self-consistency of the stress redistribution leads to 0 / 0\ in contrast, to t he independent-rupture description [23] for which E and N have identical scaling behaviors.

3.4.4 Discussion on the f t Value

We now compare our results with seismological data and discuss crossover ef­ fects. 'Phe original form of the Cutenberg-Hichter law [2] states that the rate of earthquakes with (surface wave) magnitude larger than A/,, »(A/.,) scales as l°gio[u(A/.,)] = A — hMs with an exponent, b. In terms of the total energy release

E, t he rate of earthquakes with energy larger than 1C, i>( E). scales as n( E) = a E~lj with an exponent 0. It. is generally accepted that, A/, is proportional to the loga­ rithm of E: log,0(£) = (I ± cMa; this yields b = c0.

There are two possible reasons for a crossover in the scaling behaviors from large to small earthquakes: (i) Ms calculated from seismic-wave amplitudes at a period of 20 seconds underestimates the energy of large earthquake's of longer duration. Recent analysis of data [31] indicates a crossover from c Rs 3/2 for

A/, > G.8 to c « 1 for Ms < 5.3 in agreement with earlier theoretical analysis [32].

25We also find that modifications of our model, such as (i) the inclusion of non-shear mode of ruptures due to the tensile stress air exceeding its threshold in d = 2 and (ii) choosing the threshold distribution to be uniform bet ween po and 1 with po = 0.1, 0.2, leave the exponents ± /3j invariant while changing non-universal feat ures such as the shape of the hump. 201

(ii) The second reason is geometrical. While the extent of small earthquakes is not constrained, the down-dip width of a large earthquake is limited by the thickness of the seismogenic layer. Therefore, large earthquakes correspond to d = 2 and small ones to d, = 3. Recent analysis of the frequency-size distribution [30] reveals a break in the self-similar behavior at a magnitude around 6.5 to 7.5 due to the change in effective dimensionality. This leads to a change in h from roughly 1 for small earthquakes to 3/2 for large ones.

If is reasonable to view earthquakes with Ma between 2.0 and 5.5. as 3 d earthquakes, and hence, use c « 1 and deduce a value of ft ss ]. The situation for determining ft for 2-d earthquakes, on the other hand, is far less clear due to the two crossovers involved. If we assume that earthquakes with M„ > 7.5 are two-dimensional and use c ~ 3/2 , we again obtain /f r; 1.

With the above caveats, the ft values obtained for the self-consistent descrip­ tion in d = 2 and 3 are in rough agreement with the above ft values for large and small earthquakes respectively. Moreover, since the model with the independent rupture description yields smaller ft values both in 2-d and 3-d, it also appears that self-consistent quasi-static description perhaps is more relevant for exploring earthquake dynamics in a seismic zone.

We have also studied the scaling behavior of earthquakes within the self consistent description using the double-couple Green’s functions in 2-d. The 205 results indicate that the scaling features are similar to that of the self-consistent

2-d dipole model.

Also note that similar to the investigations of Ref. [25], one can introduce a healing time scale and study the occurrence of aftershocks following a large earthquake using the self-consistent description.

In the next section we consider a model with a pre-existing fault driven by the tectonic plate motion. 206

3.5 Model with a Pre-Existing Fault

In this section, we present a two-dimensional quasi-static model with a linear vertical strike-slip fault and use dipole stress redistribution for the purposes of illustration. Using the self-consistent methods described in Section 3. the model incorporates both the stick—slip behavior at the fault surfaces and long ranged stress redistribution following a rupture or a slip event.

Driven by a slow tectonic plate (relative) motion, the model evolves to a crit ical state exhibiting Gutenberg- Richter power law for the frequency-size distributions.

We emphasize that the basic ideas and methods presented here are not limited to this specific model and can be easily generalized to study seismic zones with arbitrary fault, geometries and zone boundaries. Furthermore, our formulation allows more realistic representations of slipping and stuck regions of the faults.

3.5.1 Modeling Aspects

We model the seismic zone using a Lr x Ly lattice of blocks (each block is char­ acterized by a few kilometers) with a linear strike-slip fault along the ,r direction embedded in the middle as shown in Figure 3.10. Local stresses are defined on the surfaces between adjacent blocks and displacements at the centers of individual blocks. The displacements are assumed to be small compared to the size of a block: aT x ay x where a, is the thickness of the crust taken to be a constant for simplicity.

Tectonic loading of the seismic zone is represented by the driving of plates on both sides of the fault in opposite directions: the plate on one side of the fault 207

0 ,7*

Ly F a u l t

Lx

Figure 3.10: 2-d seismic zone model along with an embedded strike slip fault in the middle at y=0. The stresses (along x direction) on a single block are also shown. 208

(positive y) moves to the right (positive x) with a speed u 0/2 while the other plate moves to the left with the same speed, leading to a relative velocity v0 for the plate motion. The mechanism for earthquakes in the model is as follows: The blocks on one side of the fault experience frict ional forces with the blocks on the other side, resulting in a zero relative displacement at the fault, i.e., the fault surfaces are stuck while there is slow relative motion of the tectonic plates in progress. This causes stress to build up in the vicinity of the fault, which, in particular can be viewed as an extra elastic deformation everywhere except at the fault where the no slip condition is enforced by an appropriate increase in the frictional force .2(1

Karthquakes are then caused by faulting instabilities such as .s -hear ruptures in the zone or slips at the fault surface's when the stress (or frictional force) exceeds the appropriate (static or dynamic) local threshold value.

In addition to t>0, which sets a time scale for measuring the time interval between successive earthquakes, we introduce three local parameters following

Ref. [,‘M]: we assign a static threshold aj >

26 While implementing these boundary conditions, we have assumed that the plates move (slip) past each other without any friction at all other parts of the plate interface except at t he fault. One may interpret that the fault in the model corresponds only to the interface where frictional features are dominant and the rest of the interface is some kind of oceanic or low frictional region. 209

The stress increase in a time St, between successive earthquakes, can be ex­ pressed as:

0 x y (r ,t + St) = axy(r,t) + %—a(r)St, (3.96) a,y where / is a constant that appropriately normalizes the stress and a(r) are con­ stant coefficients self-consistently determined by the stuck boundary conditions2' at the fault, using the methods outlined in Section 3.3.4

Similar to the seismic zone model in the previous section, an earthquake can proceed in several steps after the initiation. In each step, at the ruptures/slips

(where the shear stress/force exceeds the current threshold) the new shear st ress is set to rrnrw = (Tn and the threshold is reset to for the subsequent steps. Also, the surface of a new rupture is treated as being stuck for the subsequent, steps unless it, slips again within that, earthquake. The stress distribution in the seismic zone after each step is obtained self-consistently using the procedure to be outlined below. 'File eart hquake ends when the stress field reaches a state that is below the threshold. After each earthquake, ruptures in the seismic zone are healed (i.e., no new faults are generated in the zone; the healing time is assumed to be much smaller than the time scale for stress build-up) while the surfaces along the fault are treated with stuck boundary conditions. We also assume that plates are held fixed during an earthquake in accord with our quasi-static description.

-'7Specifically, the stuck boundary conditions can be implemented by superimposing the follow­ ing two displacement fields. First field is simply (say, one unit of) relative displacement for the plates. Then consider the non-elastic stress fields at the fault that generate the same (one unit of) relative displacement at the fault but in the opposite direction. Adding these two give us the coefficients both for stress and displacement fields. 210

We now briefly describe our formalism for calculating stress redistribution during each step of an earthquake .28 By assuming the validity of linear elasticity everywhere except at the fault surfaces and newly ruptured places, we write the new stress and displacements due to ruptures/slips as the following:

< ‘( 0 = ?,) + £/*%-,• (3.97) i t

K +i(r) = <(r) + £/?<7a(r - *), (3.98) I where axy is the shear stress at the mutual x boundaries of the blocks and ur is the x displacement of the blocks. The non-elastic behaviors on the fault surfaces and newly ruptured surfaces ({r,} denote their positions) are reflected in the non elastic stresses { ( r,) = /■ }, which are determined by a self-consistent procedure outlined in Section 3.3.4. The self consistent solution for {/',(r,)} is obtained with the following boundary conditions: (i) we set a"'yw = a„ on the interfaces between slipping blocks at the fault and new ruptured surfaces off the fault; (ii) we maintain the extra displacement, fiux(r) = X), — r,), in each step to be the same for the blocks on both sides of the interfaces of the stuck regions at. the fault, and the previously ruptured surfaces off the fault.

One can consider a variety of choices for the thresholds as^ia. For simplicity, we set <7„ = 0.1 and choose as to be uniformly distributed between eru and 1. After each earthquake the threshold cs can either be reset to a new value signifying a

28Note that, in some special cases with simple fault geometry, one can also use known continuum Green’s functions to compute stress redistribution and study the evolution of the faults, as has been illustrated in Ref. [33]. 211 change in strength at the ruptured/slipped sites, or be fixed for the entire evo­ lution. The latter corresponds to quenched disorder and leads to a deterministic evolution for the system. We consider the former case for the purposes of illus­ tration, and focus 011 two extreme cases, A and B, corresponding to the choice of dynamic thresholds: (A) ad = a„ (B) ad = as.

3.5.2 Features of the Model

Similar to the seismic zone model considered in the previous section, the earth­ quakes are characterized by the energy released (E) and the seismic moment (Mo), defined in terms of the displacement discontinuity ([«*]) measured as the differ­ ence in the elastic displacements of blocks on either side of the rupture/slip. By considering E as the elastic work done by the stress drop to create a mean slip integrated over the seismic zone and M 0 as proportional to the total slip along .r

(see Ref.[20, 19]), we define E « — iCi[Mx,'],(o-c'

Spatial features

We first present the spatial features of earthquakes in our model. One notable feature is that, in moderately large earthquakes, the sequence of ruptured/slipped surfaces roughly assumes the form of a linear crack of length l0 (in units of ar ) along the direction of the fault with some scatter associated with ruptures at 212 weak spots. For case A, the energy B is found to scale as /q, as expected from the self-consistent description considered in the previous section.

The global stick-slip behavior of the model is illustrated in the displacement field urTTtTa(x,y) during an earthquake. Figure 3.11 shows this for a fixed r as a function of y due to a large earthquake occurred at the fault. The slip (displace­ ment discontinuity) at the fault displays a well known feature of large earthquakes observed in surface faulting [35]. Moreover, the average slip rate at the fault agrees with the relative plate velocity29. We also find that most of the earthquakes occur in the vicinity of the fault as shown in Figure 3.12.

Statistical Features

Next, we present statistical features of earthquakes in the evolution of the model for both cases A and B, by using a 80 x 80 system with 100,000 earthquakes after skipping the first 10,000 (so that the system is evolved to a critical state in the sense of “Self Organized Criticality” [5]). In Figure 3.13, we display log-log plots of the distribution P(B) vs. B for both cases A and B; the linear portion which extends up to B corresponds to a power law P[B) ~ with ft = 0.8 ± 0.1 for case A and ft = 0.3 ± 0.1 for case B. Clearly, the scaling behavior (exponent ft) crucially depends on the choice of aj and therefore is a relevant parameter

29Note that, mechanical fault models [15, 13] also display this feature. However, such models appear to exhibit scaling behavior with a cut off set by the parameters related to the plate motion. Thus these models do not exhibit Self-Organized Criticality in the strict sense. As will be seen later, unlike these models, our model exhibits scaling with a cutoff depending on the system size. Thus the interaction with the seismic zone appears to be crucial for scale invariance in earthquake dynamics. Figure 3.11: The displacement field (shown in circles), as a function of y for a fixed .?•, due to a slip (displacement discontinuity) at the fault (y = 0 ) in a large earthquake with case A. 214

20

-20 0 X 40

Figure 3.12: Clustering of epicenters of earthquakes in the vicinity of the fault is shown. One thousand earthquakes are included. An epicenter is taken to be the mean of the x and y coordinates of the ruptures or slips. A 40 x 40 system with Case A is used in the simulation. 215 in determining the universal features of earthquake dynamics in the model. In addition, the scaling of the seismic moment M q is found to be similar to that of

E , as expected for both cases. We also find that the above scaling features are unaltered when a deterministic evolution, i.e., with fixed er,, is considered.

Note that the exponent values for cases A and B are in agreement with the results obtained for the seismic zone model (without pre-existing faults), driven by a uniformly increasing shear stress, described in the previous section with self consistent and independent- rupture descriptions respectively. This suggests that the presence of a simple linear fault structure and the choice of driving do not alter scaling behaviors. However, it is yet to be seen if this remains the case for a complex fault structure such as a fractal net work of faults.

The value of ft obtained for the model with case A is consistent with seismolog- ical data gat hered over the past, one hundred years [30] as discussed in the previous section, suggesting that, case A is more relevant than case B for the description of earthquakes in seismic zones. Also, note that beyond E the distribution for case

A has a hump. This is a fmitc-sizc elTect, as we find that E increases with the size of the system L and the probability in the hump scales to zero with increasing

L. This behavior is in contrast with the parameter dependent cut off (< .

Moreover, for case A, the earthquakes with energy greater than E typically have ruptures extending from one side to the other and hence, can be viewed as

“great earthquakes”. Though these events are less frequent, they release most of fteln rw fr i « —.; o B h lp i s —1.3. ss is slope the B —1.7; for « is A for drawn line the of A and B of the model. Plot B is shifted down by 5 e-cades for clarity. The slope slope The clarity. for e-cades 5 by down shifted is B Plot released model. energy the of the B vs. and A distribution of plot Log-log 3.13: Figure

Loge P(L) I -1 Ooo

1

oe E Loge 3

5

E o bt cases both for 7 216 217 the energy accumulated due to the stress build-up. For real earthquakes, however, the size effects have not yet been well established due to the limited seismologi- cal data on great earthquakes. By using typical numbers (e.g., for San Andreas fault zone: ?;0 ~ 0.35 cm/year, g - 30 GPa, fault length « 400 kms and crust thickness « 50 kms), we find that the mean recurrence time of great earthquakes in our model to be of the order of a few hundred years; this is consistent with the recurrence times estimated using the seismic moments observed in great, earth­ quakes [1],

We have also studied the distribution of time intervals between successive earthquakes. The distribution is found to be closer to an exponential as shown in

Figure 3.14, suggesting a Poisson process for the occurrence of earthquakes; this can be ascribed to the absence of spatial correlations in the thresholds. Note that the experimental data [36] also shows Poisson distribution for the time intervals between successive earthquakes with energy larger than / i , when E is small com­ pared to the energy released in a large event. However, the effects of correlations among the thresholds on the statistical features remains an interesting issue. 218

O O

o CN

o •st-'

>• • • o co • • • • CD • • • • • • CD 1 0.00 0.02 0.03 0.04 0.05 <5t

Figure 8.14: Log-linear plot of the distribution P(St) vs. St. The distribution is close to exponential for small St with a minor deviation for large St. This deviation is with in the statistical error. 219

3.6 Summary and Conclusions

In this chapter, we have constructed a simple quasi-static model for a seismic zone and investigated the spatial and statistical aspects of earthquakes. The model incorporates long-ranged stress redistribution and exhibits generic features of ruptures in elastic media. We have also shown that the model can include pre-existing fault structure. In addition, one can implement both the stick-slip behavior at the fault and the occurrence of ruptures in the seismic zone in the model by using appropriate boundary conditions. We have also developed a self consistent method to compute stress redistribution due to multiple ruptures and slips in the seismic zone. This model can also be easily generalized to include complicated (irregular) zone boundaries using the self-consistent method.

After presenting discretizations of the seismic zone (consistent with elasticity theory), we have considered a simple seismic zone model driven by a uniform external shear stress. It is shown that the scaling behavior (e.g., the exponent governing the power law for the frequency-sizc distribution) crucially depends on the quasi-static description chosen for an earthquake in the model. Specifically, we have considered a self-consistent quasi-static description that includes rupture- rupture interactions during an earthquake and demonstrated that this leads to a different scaling behavior from that obtained in previous studies which excluded rupture-rupture interactions during an earthquake [23, 25]. Moreover, we find that this self-consistent description yields an exponent for the Gutenberg-Righter law consistent with recent analysis of seismological data [30]. 220

We have also studied the evolution of earthquakes for the case of a linear strike-slip fault in a two-dimensional seismic zone driven by a slow relative tec­ tonic plate motion. This model is investigated for two cases corresponding to the choice of dynamic thresholds: (A) (Td = <7„; (B) Od = &s- The scaling behaviors of the frequency-size distributions for these two cases are strikingly different; more­ over, the exponents for cases A and B correspond to the ones for the seismic zone models with self-consistent and independent-rupture earthquake descriptions re­ spectively. Since the exponent for case A agrees with the seismological data [30], it appears that the threshold choice for close to being the arrest strength an is more relevant for the study of earthquakes in seismic zones with faults.

We point out that an appropriate dynamical seismic-zone model which includes short-time dynamics during an earthquake is yet to be investigated and this is differed to future research.

Finally, we conclude by noting that our model can be readily extended to in­ clude other (e.g., geometrical and tensorial) features and may as well be employed to study earthquakes in a specific seismic zone of interest by choosing appropriate boundary conditions, parameters and time scales. L ist o f R efe r en c es

[1] L. Knopoff, Disorder and Fracture , edited by C. Channel et al, (New York: Plenum) 279, (1990).

[2] B. Gutenberg and C. F. Richter, Ann di Geofis. 9, 1 (1956).

[3] See, e.g., Fractals in Geophysics , edited by C. H. Scholz and B. B. Mandel­ brot, (Birkhauser), (1989).

[4] See, e.g., B. B. Mandelbrot “‘Fractal Geometry of Nature ”, (Freeman, San Francisco, 1982).

[5] P. Bak, C. Tang, and 1\. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).

[6] R. Burridgc and L. KnopofT, Bull. Seismol. Soc. Am. 57,341 (1967).

[7] P. Bak and C. Tang, J. Geophys. Res. B94, 15635 (1989).

[8 ] P. Bak and K. Chen, 1991, in Fractals and their application to Geology (Ge­ ological Society of American, Denver), edited by C. Barton, and P. LaPointe and references therein.

[9] A. Sornette and D. Sornette, Europhys. Lett. 9, 192 (1989)

[10] H. Takayasu and M. Matsuzaki, Phys. Lett. A131, 244 (1988).

[11] S. R. Brown, C. 11. Scholz, and J. B. Rundle, Geophys. Res. Lett ., 18, 215 (1991).

[12] K. Ito and M. Matsuzaki, J. Geophys. Res., 95, 6853, (1990).

[13] H. Nakanishi, Phys. Rev. A, 41, 7086-7089 (1990).

[14] Z. Olami, H. S. Feder, and K. Christensen, IJhys. Rev. Lett., 6 8 ,1244, (1991).

[15] J. M. Carlson and J. S. Langer, Phys. Rev. Lett. 62, 2632 (1989); Phys. Rev. A, 40, 2803 (1989).

[16] B. E. Shaw, J. M. Carlson, and J. S. Langer, J. Geophys. Res., 97, 479 (1992).

[17] L. Knopoff, J. A. Landoni, and M. S. Abinante, Phys. Rev. A, 46, 7445 (1992). 222

[18] J. Lomnit,z-Adler, J. Geophys. Res , 98, 17745 (1993).

[19] C. H. Scholz, “ The Mechanics of earthquakes and Faulting ” (Cambridge Univ. Press, 1990).

[20] K. Aki and P. G. Richards, “ Quantitative Seismology, Theory and, Methods'' (Freeman, San Francisco, 1980).

[21] F. Omori, J. Coll. Sci. hnper. Univ. Tokyo , 7, 111 (1894).

[22] A. Nur and J. R. Booker, Science, 175, 885 (1972).

[23] K. Chen, P. Bak, and S. P. Obukhov, Phys. Rev. A , 43, 625 (1991).

[24] R. Bhagavatula, K. Chen, and C. Jayaprakash, J. Phys. A ., 27, LI55-162 (1994).

[25] H-J. Xu, B. Bergersen, and K. Chen, J. Phys. A., 25, L1251 (1992); .7. Phys. I (France), 3, 2029 (1993).

[26] R. Bhagavatula, K. Chen, C. .Jayaprakash and II.-J. Xu, Phys. Rev. E, 49, 5001 (1994).

[27] L. D. Landau and F. M. Lifshitz, Theory of Elasticity" (Pergamon, 1986).

[28] P. Miltenbcrger, D. Sornette, and C. Vanneste, Phys. Rev. Lett., 71, 3604 (1993).

[29] J. R. Rice, in Fracture, a new treatise II, Ed. by H. Liebowitz (Academic Press, New York), 191 (1968).

[30] J.F. Pacheco, C. II. Scholz, and L. R. Sykes, Nature 355, 71 (1992).

[31] G. Ekstrom and A.M. Dziewonski, Nature, 332, 319 (1988).

[32] II. Kanamori and D. Anderson, Bull. Seism. Soc. Am. 65, 1073 (1975).

[33] Y. Ben-Zion and J. R. Rice, J. Geophys. Res., 98, 14109 (1993).

[34] J. B. Rundle and H. Kanamori, J. Geophys. Res., 92, 2606 (1987).

[35] H. F. Reid, Univ. of Calif. Publ. Geol. Sci., 6,413 (1911).

[36] A. C. Jhonston and S. J. Nava, J. Geophys. Res., 90, 6737 (1985). A ppendix A

Appendix for Chapter I

A.l Example Probability Distributions

1. Gaussian Distribution: p(;r) = J— c-r2l2o\ \Z‘2tt(t2

2. Exponential Distribution:

r) =

3. Order Zero Modified Bessel Function (/\0) Distribution:

,,(,•) = — h'0(D\x\). 7T U

Modified Lorentzian Distribution:

C p(x) = (1 +e.r2)1+*’

where the constant C is given by the normalization. A.2 Derivation of Fokker-Planck Equations

Here we briefly outline the derivation of the Fokker-Planck equation for a

Langevin equation that yields Gaussian distributions. For more details see Refs. [40,

41] of chapter I. Consider the following single variable Langevin equation:

x = — k x + i]. (A.5) with the noise correlations given by: < ?/ > = 0 and < ?/(/)?/( 0 ) > = 2D6(t), where

<’ ... > indicates ensemble average. We interpret, the above equation as an Ito stochastic differential equation to enable a discretization in time.

Now, by using the evolution equation for the Probability p(x, / ) one can obtain the stationary PDF for the problem: The evolution in a small time A I for the probability can be written as:

p(. r, t + At) = J ;>(.r\ /) < 6(x — [.»•' — kx 'AI + f/(/.)A/]) > dx'. (A.6)

Using Taylor series expansion in the above equation, we arrive at the following:

p ( x j + 61.) = £ [ p(x\ t)6l%v - ,•') < > dx', (A J) Tl=0 ,K where 6 ^ denotes n th derivative of 6-function with respect to x'. Using the characteristics of the white noise, i.e., f ^ 1 < i](t)i)(t') > dt'dt" = 2D At, we get the following equation for the evolution of the probability up to order A/: 225

In the small A t limit we get the Fokker-Planck equation for the evolution of the probability given by:

! - & - ■ * * ”3 -

In the above equation, the first, term on RHS is commonly referred to as the drift term and the second is the diffusive term. Note also that,the coefficient of the diffusive term D is the coefficient, of noise-noise correlations. Thus the random walk of the variable .r is reflected in the diffusion of the probability.

The stationary solution for t he above equation is given by the condition:

'() , „f)p, -[-K.rp+D— ]=Q. (A.10) Or Ox

It is easy to check that the solution for this is a Gaussian with a width \JI)/ k.

Multi-variable Fokker-Planck Equation

Similar to the above single variable Langevin equation, one can have a set of coupled Langevin equations:

= /.({•?•}) + :ff, ({.r})?/, (A .ll) with noise correlations < > = 0 and < //,■(<)ilj(l') >= '21)j.j6(t — /'). '1'hen the

Fokker-Planck equation for the probability /-'({.r}) is given by:

= ErWWIfl + E j4 ;[WWWWin (A.12) t ‘ 1 i j ‘ 1 J

Note that we followed Ito calculus while obtaining the above equation. A.3 Some Details of the Toy model

The two-variable toy model defined by Eqns. 1.31 and 1.32 can be analysed by using the two-variable Fokker-Planck equation given by:

dP ® m cr2y2d 2P&ld2P — - ~(,:xP) + g -{ c y l ) + — (A.I.'l)

It is convenient to use scaled variables .r 0 = xj\JD \ and 7/0 = y/y/Th

where D i = a 2/ 2 k , D 2 =

<2(.r01?/o)/?(?/(,), where R(yo) = [ln]~'l2c.xp{— ?/q/2 ) is simply a Gaussian with unit width. The (stationary) conditional probability Q(.ro(i/o) satisfies the following equation

[.ToQ + //o t:—] = 7: tj— P ~,— (A. 1 1) f)x0 ' 0.r() k R ()yu f)y0

This equation can be solved iteratively for small ci/ k by using Q = Q„.

The above equation A. 1-1 can be rewritten as:

(J Q x a Q _ A 0 / Jo , . , .) T 2 ~ 2 I *( (A.lo) fh 'o US //o Jo where /1 0 = The homogeneous solution for this equation is given by:

Qo = 7— je - , o/2V (A. 16) ZxiJo

As mentioned above, by denoting Q — 'jP.^LoQn* we can obtain Q„ iteratively by using the following: 227

for all n > 0. The first and second order solutions, Q\ and Q 2 are given by:

Q\ = —Qo[{- + 7- 5 } - { ^ 4 + 3 ( 3 - 7?)}] (A-llS) K 2 4?/q 2j/q 2/5 ?/o 2/0 Q2 ] 1 11 ^ o l_ { 8 + ^ 2 + 3 2 ^ } 1 1 r 2+ ± 1 ( r f 2o _ rfsL)\ 4 2 2,7/q 7/0 Z/o l?/d

f-4 , i_/M _ lifo , 1 8.V0 ?/o2 8?/o2 Ki//

Note that the above conditional expectations normalize the probability distribu­ tion up to order 2. Now using these expressions one can obtain the first three terms in the expansion for the probability p(.rp). In the following, .r() denotes the absolute value of .r0.

Po (■'■()) =— A’o(-i'o) (A. 20) 7r

/ \ 0 ^ rA° , -roAi .CoA.) — - A j+ — 1 (A'2I) ou 2 1 i . f A 0 A'l 11^2, HA'2 , 1 1 .r,.<0/110A'i , •Co • , *'q I<2 m _ M-CpA'a -cdA'-i 1 8 8 16 8 f 1 1 3 A a 39 A',, M.r 0A>, .c2A'(i ,, 2,t0 ~ 16 + ” 2d diT (A>22)

In the above expression A' 7,(.c0) denotes modified Bessel function of order n. It is emphasized that, the above expansion is obtained so that the probability is normalized at every order n. One can readily compute the moments using the above p„(;c0). For example, the fourth moment up to order o//> is given by

9 - 6o / k. One can directly compute the moments by using a variety of methods. The choice of the method depends on the problem. For example, the second moment can be obtained by evaluating correlations < .r(/).r( 0) > and then setting / = 0.

Another way is to write the equations in Fourier space and compute < |.r(u ;)|2 > and then Fourier transform it back. 229

A.4 PDFs in Noisy Linear Diffusive Process

Lei us consider a noisy diffusion equation on an infinite hypercubic latt ice in

(/-dimensions:

, = « 53 ( ~ fa) + ’ (A. 23) 3 where j is summed over the nearest neighbours of site i. In Fourier space, this can be written as:

4k = ~f(k)dk + Vi (A.24) where f(k) = k[2(I — 5Zf=i 2 cos(A:,)]. Let the noise correlations in Fourier space be: < i)h > = 0 and < i)k{t)Vk>(t') >= 2Dk,k'Ht ~ i') with D being symmetric. By solving the multi-variable Fokker-Planck equation, one can obtain the probability distribution for all the k modes as the following:

/J({^}) oc (A.2Ti)

Since we are interested in the local PDF, i.e., p(j = a) at a site j , by using the above with (j>j = f k ke,k’r>, we can obtain ]>{a ) as the following:

P(j =a)= i 6{a - *;)/>({&}), (A.26) where the integration over k represents a functional integration. This can further be rewritten as:

p(k

Further simplification of this yields:

P(*i = «) = ^ T dpe^e-M 2, (A.28) JuTT J —ck> 230 where D = f k k, Dk,k'/f(h) with Dk,k< — e,k'T}Dk,k'e'k > T}- Integration over // yields the PDF, which is simply a Gaussian:

The task is to find the width of the Gaussian using

D = [ ^ .

Note that in the additive regime, when the velocity is obtained from a stream function (in 2 — d case) we have f(k) = /»-.[2d — H; 2 cos(A:,-)] and D(k,k') =

Vjf12[2 — 2cox(2kx )}6(k + k '). Substitution of these in the expression for I) immediately give us the result in Eq. 1.21. A ppendix B

Appendix for Chapter II

B.l Single Variable Map

Let us start with the following initial condition in the extended system defined by Eq. 2.45. Imagine having an anti-fcrromagentic initial condition in the system with «o + P on even sublattice and — a() + p on odd sublattice, where p is the density. Then the future evolution of a0 is simply given by:

rtT1+] = an{ 1 - 2v + iwp 2) + 2va'n . (B.l)

This provides a single variable map for the problem .1 Note that a solution of this mapping is also a solution of the extended system. However, it may not bo a stable solution for the extended system.

By examining the above map we gain some understanding of the ext ended sys­ tem, especially the information regarding possible phases in the extended system.

This map is a variant of the Logistic map which exhibits period doubling route to chaos due to a quadratic maximum. Thus as the control parameter v is varied the map exhibits a sequence of period doubling instabilities and then enters into a chaotic regime.

'Note that we have used this non-linear mapping with p = 0 in the first chapter, see Eq. 1.29. 232

Lot us first explore the fixed point behavior of this map:

o* = a*(1 - 2 1/ + §vp2) + 2 z/a*3. (13.2)

This gives us simply that a' = 0 or «* = ± \/l — *V2- ^ >s easy f° check that

the solution a* = 0 is stable below i/j = 1/(1 — 3p2) and becomes unstable at //,*.

Above ;/,* the map simply gets into a two-cycle in which at even and odd l imes

the variable assumes values a and —a respectively. By examining the stability of

this two-cycle state, one can see that this 2-cycle solution becomes unstable at

"2 = •!/['“( 1 — ’V2)]- Above v2, the single variable map breaks the symmetry and

becomes a 1 at even times and a2 at, odd times. Note that the two-cycle with —a\

and — n2 at even and odd times is also a solution. Thus the map develops two

solutions with equal basins of attraction. In the extended system, this corresponds

to AF II state.

Further increase in the value of the u makes the map undergo a series of bifurcations and eventually for large 1/ the map exhibits chaos with a positive

Lyapunov exponent. One can see that the map becomes unstable for v > ■

2 /(1 — 3p2). This suggests that even in the extended system there exists a solution that evolves indefinitely below this value. However, since the globally chaotic slates are unstable in the extended system, runaways can occur prior to this

Note that at exactly u = /z^, the Lyapunov exponent is log<3 for the map B.l. A ppendix C

Appendix for Chapter III

C.l Evaluation of Green’s functions

In order lo numerically evaluate the 2d Green functions efficiently, one can reduce the 2d integrals to 1 <7 integrals. This can be achieved because we are only inter­ ested in the values of the Green’s functions at locations with integer coordinates.

Let ns consider the double-couple Green’s function for the stress redistribution with 7 = 0 = 1 in 2-d (see Eq. 3.74), i.e., (7J. (r) with r = m arer + nay<-y with

/}i,ii being integers:

r« , i r dk* (* dky sin2 kx s\n2 kucos2{kr /2) _ikrm+ikli„ ,,,,, Grv(r) = (» (;»»,;?.)= / — / ------— — c T » . (( .1) J -n lit J -n 2ir (1 — COS A’j. COS Ky)*

We first integrate over ky by performing a complex integration with the change of the variable c = <

r: , , t’ ^ r . 2 , a \ l dz [£<= - 1/--)] L ” G ,.„(m ,n)= / ——sin A-., cos — e J

'Die integral in the square bracket can be evaluated as follows:

f dz (z2 — 1 ) V 1_1 Jc 2m (2z — cos kxz i — cos kxy

233 Let us consider the case n > 0. The poles of the integrand can be obtained by solving the equation

2z — cos kTz2 — cos kT = 0. (0 , 1)

There are two double poles: z\t2 = (1 ± jsiii kT\)/cos kT; only the root zi =

(1 — |sin kx\)/cos kT - cosL'r /(l + | sin Av|) is inside the unit circle. Using the standard complex integration methods, 7j can be written as:

2_ri-l 2 1 \2 „y-1 dz (~ - 1) ~ d_ (~ — I ) /> = - (C.5) Jr 2?n cos2 kx{z - r,)2(r - z2)2 dz COS2 kr{z - Z2)2 or equally as

'i(-f - 1 + (j - 1)(-? - 1 r=r2 -2(z2 - 1 y2z’r 1 i /. = - (C.6) cos2 k . (~~. - c2)2

Using the values of the poles, we can simplify the above expression and obtain:

‘o-H+2 i .7 1 - 2 “~1 ~\ -1 /i = (n- I) cos 2 kr -l sin 2 kx + | sin kr \ ( 1 + 1 sin krI)2 (G.7)

One can use t he above / 1 and compute the Green’s function by performing a single numerical integration over kr . Special care need to be taken while evaluating the

Id integral at kr close to 0, and ±7T (the values of the integrand can be shown to be finite at these points). Moreover, for n < 0 the Green function can be obtained by using the symmetry:

(C.8)

The procedure can be adopted for all the 2d Green’s functions. However, in

3d, the above procedure helps in only reducing the 3d integrals to 2d. BIBLIOGRAPHY

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