NONLINEAR DYNAMICAL systems that have sensitive dependence on initial conditions may exhibit chaotic behavior. In other words, if initial conditions are available only with some finite precision, two solutions starting from undistinguishable initial conditions (i.e., whose difference is smaller than the precision) can exhibit completely different future evolutions after time. Thus, the system behavior is unpredictable. Sensitive dependence on initial conditions can occur even in deterministic systems whose solutions are not influenced by any stochastic effects. Chaos theory attempts to find an underlying order in such chaotic behavior

In the early 1900s, H. Poincare noticed that simple nonlinear deterministic systems can behave in a chaotic fashion. While studying the three-body problem in celestial mechanics, he found that the evolution of three planets could be complex and sensitive to their relative initial positions. Other early pioneers in chaotic dynamics from a mathematical viewpoint include G. Birkhoff, M.L. Cartright, J.E. Littlewood, S. Smale, and A.N. Kolmogorov, for a wider range of physical systems. Experimentalists also observed chaotic behaviors in turbulent fluid motion and radio circuits.

Weather is, without a doubt, one of the best examples of chaotic behaviors in nature. Future weather can never be predicted correctly beyond a certain period of time. The first identification of chaotic behavior in atmospheric sciences was made accidentally by E. Lorenz in 1960s. To study the problem of predicting atmospheric convection, he developed a simple deterministic model with 12 variables. Simplified models such as these are quite useful in shedding light on the corresponding complex system. He followed a numerical approach by solving the model on a digital computer. On one occasion, he repeated a numerical experiment using an initial condition that was rounded-off to the first three digits (e.g., 0.506) from the original six digits (e.g., 0.506127), expecting that the difference between the two solutions would remain very small. To his surprise, the new solution stayed close to the original one for about a month of the model simulation, and then suddenly transitioned to a completely different behavior. This discovery has come to be known as the "butterfly effect."

By repeating similar experiments, Lorenz also noticed that transitions happened randomly, irrespective of the size of the perturbations in the initial conditions. To conduct further investigations, Lorenz developed a simpler model, with three variables, able to mimic the chaotic behavior of the 12-variable model, in terms of sensitivity to the dependence on

initial conditions. By plotting the solutions, he discovered another astonishing aspect of chaotic behavior: all solutions were attracted to a densely nested curve with a shape of double spirals. Starting from an arbitrary initial condition, a solution eventually reached the double spirals and never left them. It looped around one spiral, occasionally made a transition to the other spiral, then looped around, returning to the first spiral, and so on. The curve clearly moderates the chaotic behaviors of this model and is called the strange attractor. Qualitative and quantitative description of chaotic behavior can be made by understanding the properties of the strange attractor.

Since then, advances in scientific computation brought full appreciation to the significant implications of chaotic behaviors in a variety of fields. Many mathematical concepts and techniques have been developed to study such behaviors and make quantitative statements about them. Collectively, they form the chaos theory. Naturally, much of the chaos theory was developed to study and demonstrate the predictability limit.

Fundamental questions in today's atmospheric sciences include the extent of the predictability of weather and climate, as well as the reliability of forecasts made by numerical models. Not only simple but also many realistic models are known to exhibit chaotic behaviors. Chaos theory implies that a single definitive forecast made by such a model will fail because future evolution is unpredictable. This unpre dictability is due to the impossibility of obtaining the exact initial condition with arbitrarily high precision, even if the model truthfully describes the evolution of the complex system. Instead, chaos theory suggests the need for "probabilistic forecasts" that express the uncertainty associated with a forecast. For example, the probabilistic forecast of tomorrow's precipitation may be expressed as "95 percent to exceed 0.1 inch," while a definitive forecast may be "0.1 inch."

To represent the uncertainty, and establish a probability distribution associated with a forecast, most leading weather forecast centers use the ensemble forecast approach, in which an ensemble of forecasts is made starting from the initial conditions slightly different from the best possible estimate of the current state of the atmosphere. If all forecasts in the ensemble stay close together, then real future evolution is likely to be close to the mean of the forecasts. In contrast, if the forecasts show a wide spread, then the forecast is considered to have high uncertainty. Uncertainties can be quantified if the ensemble consists of a large number of forecasts. Probabilistic forecasts can be used to study and quantify impact of human activities on future climate change by running the ensemble forecasts of multiple climate models under certain forcing scenarios as described in the Intergovernmental Panel on Climate Change (IPCC) reports.

see also: Climate Models; Computer Models; Measurement and Assessment; Simulation and Predictability of Seasonal and Interannual Variations.

bibiography. James Gleick, Chaos: Making a New Science (Penguin Group, 1988); Ed Lorenz, The Essence of Chaos (University of Washington Press, 1996); Ed Lorenz, "Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas," Seminar at American Association for the Advancement of Science, http://en.wikipedia.org/wiki/ American_Association_for_the_Advancement_of_Sci-ence, 1972 (cited January 2008); Intergovernmental Panel on Climate Change (IPCC), "Third Assessment Report," (2001), and "Fourth Assessment Report," (2007), http:// www.ipcc.ch/ipccreports/assessments-reports.htm (cited January 2008); T. Palmer and R. Hagedorn, Predictability (Cambridge University Press, 2006); L.A. Smith, Chaos: A Very Short Introduction (Oxford University Press, 2007); J.C. Sprout, Chaos and Time-Series Analysis (Oxford University Press, 2003); Ian Stewart, Does God Play Dice? The

New Mathematics of Chaos (John Wiley & Sons, 2008); M.M. Waldrop, Complexity: The Emerging Science at the Edge of Order and Chaos (Simon & Schuster, 1993); G.P. Williams, Chaos Theory Tamed (Taylor & Francis, 1997).

Carl Palmer Independent Scholar

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