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Unformatted text preview: Why the Weather is Unpredictable, An Experimental and Theoretical Study of the Lorenz Equations AN HONORS THESIS PRESENTED TO THE FACULTY OF THE DEPARTMENT OF MATHEMATICS AND THE DEPARTMENT OF PHYSICS BATES COLLEGE IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE by Christopher A. Danforth Lewiston, Maine March 16, 2001 1 2 Acknowledgments I would like to thank several people who have made this thesis possible. Thanks to Chip Ross for your guidance, patience, and mastery of all things chaotic. Id be a clueless weatherman if it werent for you! Thanks to Mark Semon for your technical support and magical proofreading skills. Thanks to George Ruff for your enthusiasm towards physics and life. I wouldnt have majored in physics if it werent for your influence. Thanks to Molly and Lank for making life fun even when thesis wasnt. Thanks to Fowler, Josh, Kelley, Lindsay, Bouchard, and all of the other physics majors who gave me a crash course in the world of circuits and soldering. Thanks to Laura for making me laugh. Thanks to Mom for making me appreciate the importance of being positive. Finally, thanks to Dad for being so supportive and showing me how rewarding science can be. Contents Chapter 1. The Nature of Deterministic Systems 4 Chapter 2. The Lorenz Equations 7 Chapter 3. A Simple Model of the Weather 18 Chapter 4. The Lorenz Attractor 27 Bibliography 35 3 CHAPTER 1 The Nature of Deterministic Systems In the early 1960s, a meteorologist and mathematician named Edward Lorenz was work ing on a problem in fluid dynamics. He was trying to create a theoretical model of the atmosphere in order to better understand atmospheric dynamics. As part of his model, Lorenz simplified the usual complicated numerical weather forecasting equations into a set of ordinary differential equations. In this seemingly simple step in the process of mathemati cal modeling, he stumbled upon a new phenomenon, one which subsequently became known as Chaos. Lorenz was attempting to find a model in which the usual fluid equations describing the atmosphere were especially simple. He chose as his simplified model a particular set of ODEs which are now referred to as the Lorenz equations, for two reasons. First, the equations for atmospheric dynamics involve thousands of variables and parameters (BUR). Such systems are so complicated that even modern computers cant always use them to make accurate predictions. In Lorenzs time, any attempts to model the atmosphere were painfully slow and revealed little insight into the behavior of the actual weather. Secondly, as Lorenz became more convinced that unpredictability was an inherent part of these deterministic equations, he wanted to show that even a simple set of equations derived from them will have solutions whose behavior is unpredictable (LOR1, 134)....
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 Spring '11
 Johnopera
 Equations

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