Chaos Theory Xubin Zeng- AtmospheriDepartmenc Sciencet, and Its Applications Colorado State University, to the Atmosphere Fort Collins, Colorado Abstract specific subjects in detail, we attempt to give an overall picture of the field. We usually cite the original refer- A brief overview of chaos theory is presented, including bifurca- ences and more recent review papers so that inter- tions, routes to turbulence, and methods for characterizing chaos. The paper divides chaos applications in atmospheric sciences into ested readers can easily find more detail concerning three categories: new ideas and insights inspired by chaos, analysis particular fields of study and can find places to start of observational data, and analysis of output from numerical models. their own research in this field. Chaos theory is re- Based on the review of chaos theory and the classification of chaos viewed in section 2. Applications of chaos theory to the applications, suggestions for future work are given. atmosphere are discussed in section 3. Conclusions and suggestions for future research are given in section 4. 1. Introduction Nonlinear phenomena occur in nature in a wide 2. Chaos theory range of apparently different contexts, such as hydro- dynamic turbulence, chemical kinetics, electronics, a. Background ecology, and biology; yet they often display common Li and Yorke (1975) seem to be the first to introduce features or can be understood using similar concepts, the word chaos into the mathematical literature to permitting a unification of their studies. The similarity denote the apparently random output of certain map- of complicated behaviors is not a superficial similarity pings, although the use of the word chaos in physics at the descriptive level; instead, it concerns experi- dates back to L. Boltzmann in the nineteenth century mental and theoretical details. This similarity results in another context unrelated to its present usage. from the modern theory of nonlinear dynamical sys- However, there is still no universally accepted defini- tems, which describes the emergence of chaos out of tion of the word chaos. Usually, chaos (deterministic order and the presence of order within chaos. This chaos) refers to irregular, unpredictable behavior in includes such features as solitons, coherent struc- deterministic, dissipative, and nonlinear dynamical tures, and pattern formation, as well as chaos theory, systems. It should be emphasized that chaos cannot which makes use of fractal dimensions, Lyapunov be equated simply with disorder, and it is more appro- exponents, the Kolmogorov-Sinai entropy, and other priate to consider chaos as a kind of order without quantities to characterize chaos. In this paper, only periodicity. It was demonstrated in Lorenz (1963) that chaos theory will be reviewed. the sensitive dependence on initial conditions of a Many of the publications in the past few years nonlinear system is related to the aperiodic behavior of concerning chaos applications to the atmosphere the system. have concentrated on the evaluation of fractal dimen- By dynamical system we mean any system, what- sions from observational data. The existence of low- ever its nature, that can be described mathematically dimensional climate and weather attractors is a highly by differential equations or iterative mappings. Some- debated subject. A more difficult task is to find con- times, we also include systems where the exact present crete examples that show the significance of such state only approximately determines a near-future computations. However, it needs to be emphasized state: this extended definition of a dynamical system that these computations are just a small portion of the admits many real physical systems (such as the atmo- applications of chaos theory to the atmosphere. sphere plus its ocean and terrestrial boundaries), The purpose of this paper is to give a brief overview whose behavior commonly involves at least some of chaos theory and to discuss its applications to the randomness or uncertainty (Lorenz 1990). In a dissi- atmosphere. Instead of discussing any concepts or pative dynamical system, a vast number of modes die out due to dissipation, and the asymptotic state of the 'Department of Physics system can be described within a subspace of a much ©1993 American Meteorological Society lower dimension, called the attractor. Chaos can also Bulletin of the American Meteorological Society 631 Unauthenticated | Downloaded 10/11/21 05:07 AM UTC occur in Hamiltonian or conservative systems, but this solutions, limit cycles, which lead to periodic solutions, subject requires special methods due to the absence and tori, which lead to quasiperiodic solutions. The of an attractor. Chaos may also occur in quantum name strange attractor refers to its unusual proper- systems where the number of degrees of freedom is ties, the most significant being sensitivity to initial larger than the number of independent commuting conditions: two initially close trajectories on the attractor operators (Berman 1990). Another behavior, called eventually diverge from one another. Strange attractors intransitivity, which is different from chaos but related are finite-dimensional, and, in some sense, they cor- to it, also occurs in some, but not all, dynamical respond to exciting only a finite number of degrees of systems. An intransitive system is one with a freedom; yet they have an infinite number of basic positive probability of acquiring any one of several frequencies (Ruelle 1990). These two independent, sets of infinite-term (or very long-term) properties pioneering papers triggered an upsurge of interest (Lorenz 1990). In this paper, only deterministic chaos among researchers in different fields in an attempt to in dissipative nonlinear dynamical systems will be gain new insights. Since then, especially since 1975, discussed. publications related to chaos have grown extremely There were classical physicists and mathemati- rapidly, and it is not the purpose of this paper to review cians, even in the previous century, who had thought all of this progress. Many of the historical papers on about nonlinear dynamical systems. Hadamard(1898) chaos were assembled into a single reference volume first observed the sensitivity of solutions to initial by Hao (1984). A comprehensive treatment of chaos conditions at the end of the last century in a rather theory with a readable account of many aspects of the special system called geodesic flow. Subsequently, subject may be found in Berge et al. (1984). A Poincare (1908) discussed sensitivity to initial condi- nontechnical discussion of chaos is given in Gleick tions and unpredictability at the level of scientific (1987). The concepts of chaos, fractal dimensions, philosophy. (Poincare even went on to discuss the and strange attractors, and their implications in me- problem of weather predictability!) However, their teorology, are presented in Tsonis and Eisner (1989) ideas seem to have been forgotten until Lorenz (1963) in a very readable way. Some more recent references rediscovered them independently more than half a can be found in, for example, Campbell (1990) and century later in his elegant paper entitled "Determinis- Marek and Schreiber (1991). However, most of the tic Nonperiodic Flow." The primary reason for this long progress in this field so far may be roughly divided into hiatus is that chaos defies direct analytic treatment, two different categories: one involves bifurcations and making numerical computation essential. Therefore, routes to turbulence, and the other consists of quan- Lorenz is generally regarded as the first to discover the titative means to recognize, characterize, and classify irregular behavior and to analyze it quantitatively in attractors. Here, we give only a brief review: usually completely deterministic, dissipative systems. He used only the original references and a few review papers a set of three equations, drawn from the spectral will be cited. equation set of Saltzman (1962), to model the nonlinear evolution of the Rayleigh-Benard instability (i.e., the b. Bifurcations and routes to turbulence instability that results when a fluid layer subjected to Ruelle and Takens (1971) showed thatthe Landau- gravity is heated sufficiently from below). This model Hopf route to turbulence (Landau 1944; Hopf 1948) is is also equivalent to a low-order quasigeostrophic unlikely to occur in nature, and they instead proposed model derived from shallow-water equations (Lorenz a route based on four consecutive bifurcations: fixed 1980) or derived from a two-level baroclinic model point limit cycle 2-torus -> 3-torus strange (Klein and Pedlosky 1992). By a careful analysis of the attractor (turbulence). A few years later, in collabora- numerical solutions combined with analytical reason- tion with Newhouse, they reduced this scheme to fixed ing, Lorenz was able to deduce that the solution of his point limit cycle ^ 2-torus strange attractor. In equations is eventually trapped in a region of the other words, quasiperiodic motion on a 2-torus (i.e., system's phase space that has a very intricate (strange) with two incommensurate frequencies) may lose sta- geometric structure, and this solution is very sensitive bility and give birth to turbulence directly (Newhouse et to the initial conditions. al. 1978). This result also implies that, usually, there Eight years later, Ruelle and Takens (1971), mak- are only four types of stable attractors (fixed point, limit ing use of then-recent developments