GENERAL ARTICLE Edward Norton Lorenz DiscovererofChaos V Krishnamurthy Edward Lorenz discovered nonperiodic behavior in deter- ministic nonlinear systems and laid the foundation of chaos theory. He showed that chaos exhibits sensitive depen- dence on initial conditions implying that long-range weather prediction is difficult because of errors in the observations used as initial conditions. Lorenz described the intricate structure of chaotic attractors and quantified predictability. His important contributions in meteorology include energy V Krishnamurthy is at cycle, slow manifold and general circulation. the Center for Ocean- Land-Atmosphere Discovery of Chaos Studies, George Mason University, USA, and Weather forecasts routinely issued nowadays by major prediction has worked at MIT, University of Maryland centers in the world are prepared by integrating global-scale and Abdus Salam ICTP. numerical models on supercomputers. Weather prediction models He was a doctoral are based on dynamical equations governing the atmosphere, student of Edward ocean, land and other components. The first dynamical weather Lorenz. His interests include chaos, monsoon forecast was reported by the project led by Jule Charney and John variability and climate von Neumann at the Institute for Advanced Study in Princeton in change . 1950 using the pioneering Electronic Numerical Integrator and Computer (ENIAC). However, during the 1950s, the weather forecasts were usually made by statistical models that were primarily linear methods relying on past observed data. In 1955, the Department of Meteorology at the Massachusetts Institute of Technology (MIT) appointed Edward Lorenz as a new faculty member to lead the on-going statistical forecasting project. Lorenz examined numerous statistical schemes and convinced himself that the statistical predictions were similar to subjective predictions and that even one-day forecasts were mediocre. He Keywords Lorenz, chaos, nonlinear dy- also showed that many statistical forecasters had misinterpreted a namical systems, weather pre- paper by the great MIT mathematician Norbert Wiener to wrongly diction, predictability. RESONANCE March 2015 191 GENERAL ARTICLE The time series of the conclude that linear methods using limited past data were capable solutions was of producing good forecasts. He set out to test this hypothesis in a unmistakably systematic manner [1,2]. He proposed to conduct this test by nonperiodicand generating solutions to a set of deterministic equations that were exhibited a continuous nonlinear. He also recognized that the solutions should not be power spectrum. stationary or vary in a regular manner since linear methods can Lorenzhad discovered easily predict such solutions. Lorenz settled on a two-layer atmo- chaos. spheric model with advection of wind as the nonlinear term, heating as forcing and friction for dissipation, and simplified it to be a 12- variable model represented by 12 ordinary differential equations. At about the same time, Lorenz acquired a Royal–McBee LGP-30 computer solely for usein his own office. Hehad a PC in 1958 itself, but it was about the size of a large desk with an internal memory of 4096 32-bit words! The solutions to his two-layer atmospheric model were generated by numerically integrating it on the Royal– McBee computer. The twelve variables of the model represented large-scale features of the weather such as the speed of the westerly winds. After many experiments, Lorenz obtained solu- tions which varied like observed weather patterns with no evidence of periodicity. The time series of the solutions was unmistakably nonperiodic and exhibited a continuous power spectrum. Lorenz had discovered deterministic chaos. He applied a linear regression method on the solutions generated by the model and showed that it produced mediocre and progressively worsening forecasts. The discovery of chaos was reported by Lorenz at the International Symposium on Numerical Weather Prediction in Tokyo in Novem- ber 1960, and appeared in the proceedings of the symposium published by the Meteorological Society of Japan in March 1962 [3]. Early Life and Education Edward Norton Lorenz was born in West Hartford in the state of Connecticut, USA on 23 May 1917. His mother was a school teacher who became active in civic organizations, and his father was a mechanical engineer educated at MIT. His parents provided him a happy childhood by introducing and teaching different 192 RESONANCE March 2015 GENERAL ARTICLE interests and activities, some of which remained his lifelong passion. His love for mountains and hiking was acquired from his parents while his mother taught him chess, card games and board games. He later became the captain of his high school and college chess teams. Hetook lessons in playing violin and developed a lifelong passion for music. Although he was not good at team sports, he would compete with his friends in swimming, but hiking in themountains became his favorite activity. His father also taught him about science and mathematics, making him especially fascinated with numbers. He was interested in weather as a hobby and used to go through weather records. For a while, Lorenz seriously considered becoming an astronomer1 but his interest in mathematics returned as he grew 1 Lorenz learned about all the older [1]. planets and the Saturn’s rings from his father. When he was When Lorenz entered Dartmouth College, an Ivy League university eight years old, he even saw the shadow of the total solar in New Hampshire, to start his undergraduate studies, he decided to eclipse in Hartford. Much later, major in mathematics. After receiving his bachelor’s degree from he spent many nights observ- Dartmouth in 1938, he joined Harvard University as a graduate ing Jupiter at Lowell Observa- student to continue the study of mathematics. He acquired a diverse tory in Flagstaff. background by taking a wide range of courses in mathematics and chose to work under the supervision of the renowned mathematician George Birkhoff towards a doctorate degree. Birkhoff was well known for many important contributions in mathematics, including a rigorous proof of a special case of Poincaré’s three-body problem. Poincaré, who had laid the foundation of dynamical systems theory, had indicated the possibility of chaotic behavior by dynamical systems. Birkhoff was considered a successor to Poincaré and had written a book on dynamical systems. However, it was not the association with Birkhoff that generated Lorenz’s interest which led to his discovery of chaos. In fact, Lorenz worked in algebra involving 2 Dirac spinors for his doctoral thesis. He even published a paper on 2 Dirac spinors are column the generalization of Dirac equations in the Proceedings of the matrices representing spin in National Academy of Sciences in 1941. the wavefunction solution of the Dirac equation in relativis- About a few months before Lorenz was expected to submit his tic quantum field theory. doctoral thesis, World War II interrupted his studies at Harvard. Because of his long fascination with weather, he responded to the announcement of the Army’s program to train weather officers and RESONANCE March 2015 193 GENERAL ARTICLE joined the first batch of the program at MIT. After completion, he served as a weather forecaster in the US Army Air Corps in the Pacific during 1942–46. After the war, he had to decide whether to return to Harvard to complete his mathematics degree or to study meteorology at MIT. Meanwhile, his advisor Birkhoff had died in 1944. After considerablethought and consultation, he chose to pursue meteorology at MIT. Mainly working on his own, he wrote his doctoral thesis on methods to predict the motion of cyclones from the dynamical equations of the atmosphere, which was accepted in January 1948. A few weeks later, he married Jane Loban, who was working as a research assistant in the same department. In addition to her interest in weather, Jane was also a licensed pilot who had served with the Women’s Army Service Pilots during World War II. Lorenz continued as a post-doctoral scientist at MIT working with Victor Starr whom he considered a mentor and who became his close friend. He joined the MIT faculty as an Assistant Professor in 1955. Further Work on Chaos After discovering the nonperiodic solutions in the 12-variable atmospheric model, Lorenz continued his experiments with the model on the computer. One day, in order to examine a solution in more detail, he restarted the integration of the model from an earlier time step. After returning from a coffee break an hour later, he found that the new solution was different from the original one. A careful comparison between the two solutions revealed that the Sensitivedependence two solutions were almost the same at first but diverged as time on initial conditions is progressed and became unrecognizably different later. He soon thefundamental realized the source of the problem.When he restarted the integra- property of chaos in tion, he had typed the twelve numbers of the initial state by nonlinear systems. truncating them to three decimal places. In other words, the two Lorenz’sdiscoveryof initial states had a small difference which grew with time until it this property had a became as large as the difference between two randomly selected profound implicationon solutions. Lorenz had just demonstrated that such solutions exhibit the limitation of long- sensitivedependenceon initial condition, which is now known to be rangeweather a fundamental property of chaos. He immediately recognized the prediction. impact of this property on weather prediction. Since real weather 194 RESONANCE March 2015 GENERAL ARTICLE observations arenot accurate, it would beimpossible to make long- range weather predictions. In common folklore, this property has come to be known as the ‘butterfly effect’, drawn from the title of a paper (‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?3’) he presented at a meeting in 1972 [4]. 3 See Classics, p.260.