Introduction

Nonlinear dynamics is a rich subject, still only partly understood. Progress is being made in different ways through experiments, computer simulations and theoretical analysis. In this latter category, perturbation theory and asymptotic methods are classical but useful techniques. The aim is to introduce the reader to the nuts and bolts of their use. These notes correspond to an introductory course given at the Beg-Rohu Summer School in the Spring of 1991. It was for graduate students and young post-docs who were interested in theoretical computations in hydrodynamics. The emphasis is put on practical applications rather than on theory. We present, as completely as possible, several calculations on some examples that we find interesting both from a physical and mathematical standpoint. For the expert reader, we should apologize because many subtleties are left aside in these lectures. However, more thorough treatments are referred to as we proceed. Finally, this is an appropriate place to acknowledge my debt to Yves Pomeau from whom I have learned almost everything that I know in this field. I would also like to thank Claude Godrèche for giving me the opportunity to present these lectures in Beg-Rohu and even for convincing me of writing them up, all the “students” for their interest, for catamaran sailing and for sharing unexpected salted baths and crêpes which made teaching in Beg-Rohu an enjoyable experience. Finally, I am grateful to Angélique Manchon for patiently typing these notes, and to Paul Manneville for carefully editing them.

Fluid mechanics, one of the oldest branches of continuum mechanics, is still full of life. However, it is too often taught in a very formal way, with a heavy insistence on the analytical formulation and on peculiar solutions of the equations. But it has been recognized for a long time that many of the “classical results” are of little or no help for understanding real flows. This was to the great disappointment of G.I. Taylor who realized that the niceties of Rayleigh's stability theory of parallel flows were off the mark most of the time, but that made him happier at last to see his theory of the Taylor–Couette instability explain his experiments. Perhaps the best example of the irrelevance of classical stability theory is that it predicts the most simple shear flow –the plane Couette flow – to remain always linearly stable, although it becomes experimentally highly turbulent as soon as the Reynolds number goes beyond a few hundreds.

There is a curious example of what might be called conservatism in the exposition of fluid mechanics. Around the time of World War I, Henri Bénard in Paris did very careful experiments on the wake behind a cylinder. He basically showed that this wake changes around Reynolds 40–50 from stationary to time periodic, something now called a Poincaré– Andronov bifurcation.

Introduction

Instabilities in nonlinear systems driven far from equilibrium often consist of a transition from a motionless state to one varying periodically in space or time. Various examples, widely studied in the past, are Rayleigh – Bénard convection, Couette–Taylor flow, waves in shear flows, instabilities of liquid crystals, oscillatory chemical reactions,…. The appearance of periodic structures in these systems driven externally by a forcing homogeneous in space or constant in time, corresponds to a bifurcation, characterized by one or several modes that become unstable as a control parameter is varied. Linear stability analysis of the basic state gives the critical value of the control parameter for the primary instability onset, the nature of the most unstable modes and their growth rate above criticality. Many examples have been studied for a long time and can be found for instance in the books of Chandrasekhar (1961) or Drazin and Reid (1981). However, linear stability analysis does not describe the saturation mechanism of the primary instability, and thus a nonlinear analysis should be performed to determine the selected pattern, its dynamics and in particular the secondary instabilities that occur as the control parameter is increased above criticality. Before considering these problems, we present some examples of the characteristic phenomena that occur above a pattern-forming instability onset.

Example: the Faraday instability

As a first example, consider a cylindrical vessel containing a liquid and its vapor (or any other gas), vertically vibrated at frequency ωe (see Fig. 1.1).

Introduction

The objective of the following lecture notes is to provide a consistent account of the development of hydrodynamic instabilities in open flows such as mixing layers, wakes and boundary layers. These prototype shear flows are commonly encountered in a variety of technological applications in aerodynamics, mechanical and chemical engineering, and in many geophysical processes in the oceans and atmospheres. A physical understanding of the transition to turbulence in “simple” shear flows gives rise to fundamental issues that constitute the core of this course.

Flows are typically classified according to their open or closed nature and to their laminar or turbulent character. In open flows, fluid particles are not recycled within the physical domain of interest but leave it in a finite time (Huerre and Monkewitz, 1990), as in the mixing layer generated downstream of a splitter plate by the merging of two parallel streams in relative motion. By contrast, in closed flows, fluid particles always remain within the same physical region, as in Taylor–Couette flow between two counter-rotating cylinders (Andereck et al., 1986). The distinction between laminar and turbulent flows is related to the degree of spatial and temporal coherence of a given flow (Tennekes and Lumley, 1972; Monin and Yaglom, 1975; Landau and Lifshitz, 1987a; Lesieur, 1991; Frisch, 1995). Typically, turbulent flows arise at high Reynolds numbers and are characterized by the presence of a wide range of spatial and temporal scales, three-dimensional vorticity fluctuations and a certain degree of unpredictability.

This is a book intended for everyone interested in one of the most exciting and ambitious current developments in the field of physics and complex systems. A little bit of mathematics background may be helpful in certain sections of the book, but large parts can be read without any special prerequisites.

First we describe, in Chapters 1 and 2, what is meant by the notion self-organized criticality (SOC). We also list the characteristics of this behavior. In Chapter 3 we discuss a variety of systems that might exhibit the kind of behavior denoted by SOC. We discuss such systems as sandpiles, superconductors, earthquakes, and biological evolution. In Chapter 4 we describe various computer models of SOC. Most of these models are so simple that anyone with a PC can start right away to do numerical experiments on the models. Chapter 5 contains some mathematics. This chapter is dedicated to a discussion of different mathematical formalisms developed in order to understand the behavior of the computer models and so perhaps supply a mathematical description of real systems. Chapter 6 contains an attempt to assess to what extent the dream has come true – is, in fact, SOC ubiquitous? Several computer codes are included as appendices.

It is hoped that this will help the interested reader to start out on the path of numerical experiments. Some mathematical details are likewise deferred to the appendices. I believe the book can be read in at least three different ways. Those who are interested only in the overall “philosophical” impact of the ideas behind SOC can read Chapters 1, 2, and 6 for a start.

Introduction

In this chapter we examine the extent to which self-organized criticality is of relevance to real physical systems. A large number of experiments have purported to reveal generic SOC behavior. We do not pretend to know the final and definite answer. Some systems are more accessible to experimentation than others. It is easier to settle the question concerning the distribution of avalanche sizes in a pile of a certain granular material than to determine the properties of biological evolution. Self-organized criticality might, in the end, not be the most useful way to describe some of the dynamical systems we discuss in this book. However, I consider it an undeniable achievement the degree to which SOC developments have revived interest in the dynamics of (say) sandpiles. In this chapter we discuss the phenomenological implications of a set of experimental observations.

According to the seminal 1987 paper by Bak, Tang, and Wiesenfeld (BTW), the hallmark of SOC is its lack of any scale, in time as well as in space. As a consequence, we observe spatial fractals and temporal 1/f fluctuations. Thus the strategy for our experimental search for SOC is clear. Measure some of the time-dependent quantities of the system. Construct the power spectrum of the signal. If the spectrum behaves like l/fβ with β ≃ 1, must we then be dealing with SOC? No, not necessarily so (O'Brien and Weissman 1992, 1994). In fact, most power spectra found in connection with the search for SOC extend only over a narrow frequency interval.

Introduction

Self-organized criticality was introduced in 1987 by Bak, Tang, and Wiesenfeld (BTW), who employed some appealing yet heuristic handwaving arguments. To substantiate the hypothesis, computer simulations of a simple algorithm – inspired by the avalanches induced when one plays with a pile of sand – were presented. Substantial analytic understanding was lacking for some time. Soon, however, standard mathematical tools were being applied to the new set of models. Statistical mechanics traditionally makes use of a combination of approaches, one of which consists of defining models that have a structure that allows an exact calculation of specific quantities. The art is to formulate a model of the right degree of complexity. One wants a model with sufficient structure to contain nonobvious behavior, but the model should not be so complicated that analytic approaches cannot be carried through. This last property obviously depends strongly on who is going to perform the analysis. The mathematical power of Deepak Dhar and his co-workers made it possible for them to solve an only slightly altered version of the original BTW cellular automata (Dhar and Ramaswamy 1989; Dhar 1990). After Dhar's work it was clear that, at least in some cases, the observed critical behavior was not merely an artefact of simulations on too-small systems. We shall outline the approach developed by Dhar and co-workers in Section 5.3.

Despite their undeniable beauty, the exact solutions have one drawback: the specific mathematics tends to be tailored to the details of the solved model. This means that generalization to other, similar models is often not possible.

Ever since the term “self-organized criticality” was introduced by Bak, Tang, and Wiesenfeld (BTW) in their 1987 paper for Physical Review Letters, the concept has been surrounded by a hectic air of controversy. There are a number of reasons for this. One reason is the bold and optimistic claims that were made. The attitude was that here finally is a line of thinking that will allow us to bring the statistical physics of Boltzmann and Gibbs in touch with the exciting real world of nonequilibrium physics, and that SOC is powerful enough to explain everything from mountain formation to stock-market variation. Supergeneral theories always meet a certain amount of skepticism from expert scientists working in the specific fields. It is difficult to draw a precise line between the general and the specific. It might not appear likely to the geologist that the many specific details of earthquakes can be understood in terms of a simple numerical cellular automaton. The biologist working on the immensely complicated interconnected web of evolving species might not find it anything but a bad joke to represent evolution in terms of a string of random numbers with nearest neighbor interaction only.

So what, then, is SOC good for? Let us consider some important questions.

1. Can we identify SOC as a well-defined distinct phenomenon different from any other category of behavior?

2. Can we identify a certain construction that can be called a theory of selforganized critical systems?

3. Has SOC taught us anything about the world that we did not know prior to BTW's seminal 1987 paper?

4. […]

Consider a collection of electrons, or a pile of sand grains, a bucket of fluid, an elastic network of springs, an ecosystem, or the community of stock-market dealers. Each of these systems consists of many components that interact through some kind of exchange of forces or information. In addition to these internal interactions, the system may be driven by some external force: an electric or a magnetic field, gravitation (in the case of sand grains), environmental changes, and so forth. The system will now evolve in time under the influence of the external driving forces and the internal interaction forces, assuming we can break the system up into internal and external components in an unproblematic way. What happens? Is there some simplifying mechanism that produces a typical behavior shared by large classes of systems, or will the behavior always depend crucially on the details of each system?

The paper by Bak, Tang, and Wiesenfeld (1987) contained the hypothesis that, indeed, systems consisting of many interacting constituents may exhibit some general characteristic behavior. The seductive claim was that, under very general conditions, dynamical systems organize themselves into a state with a complex but rather general structure. The systems are complex in the sense that no single characteristic event size exists: there is not just one time and one length scale that controls the temporal evolution of these systems. Although the dynamical response of the systems is complex, the simplifying aspect is that the statistical properties are described by simple power laws. Moreover, some of the exponents may be identical for systems that appear to be different from a microscopic perspective.

There are four different sets of configurations (see Figure E. 1). The weight factor Wα can be written as a product of the probability that α sites are occupied (and 4 - α are unoccupied) by a degeneracy factor nα: Wα = nαρα (1 - ρ)4-α. The expressions for nα are given in Figure E.1.

The calculations are performed as follows. Let us first consider the α = 2 case. There are K4,2 = 6 configurations for which two of the four sites are critical. Two of these configurations correspond to the critical sites being placed along the diagonal. Such configurations are ignored because they do not meet the spanning criterion. This leaves us with four configurations. For each of these remaining configurations, we have two possible choices when we turn one of the critical sites into an unstable site. In total there are eight different but dynamically equivalent configurations of α = 2. Next, we turn our attention to α = 3 configurations. We can pick three critical sites in K4,3 = 4 different

ways. The α = 3 sites break up into two classes of different dynamical behavior depending on where we choose to position the unstable site among the three critical sites. There are four configurations with the unstable site positioned between the two critical sites (see Figure E.1) and eight configurations where the critical site is located at either end of the string of three sites.

Introduction

The present chapter is about models. All the models are inspired by some physical system: sandpiles, earthquakes, magnetic vortex motion, forest fires, interface growth, or biological evolution. The models are defined in terms of a dynamical variable – for example, the local slope of the sand heap or the stress in the earthquake fault. The dynamical variable or field is updated in every time step according to some algorithm. The choice of the updating algorithm is, to some degree, arbitrary. The criteria for choosing the relevant definitions are, for the most part, simplicity and intuition. Statistical mechanicians have this overall belief that complexity arises from simplicity: that the intricate and complex behavior found in many systems is due to the large number of degrees of freedom, rather than caused by some very complicated behavior of the individual degrees of freedom. All the models described in this chapter are formulated according to this paradigm. In addition, they are all designed with an eye toward numerical ease and efficiency when simulated on computers.

For ease of presentation, for each model we shall first simply present the definition of the model and the conclusions derived from numerical studies thereof. Since the aim is to understand the world around us, we shall use separate sections to discuss the relevance and shortcomings of each model. As always when building models of Nature, one proceeds through steps of increasing refinement, and so also for models of SOC systems.