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GENERAL  ARTICLE Lorenz DiscovererofChaos

V Krishnamurthy

Edward Lorenz discovered nonperiodic behavior in deter- ministic nonlinear and laid the foundation of . He showed that chaos exhibits sensitive depen- dence on initial conditions implying that long-range prediction is difficult because of errors in the observations used as initial conditions. Lorenz described the intricate of chaotic and quantified . His important contributions in include energy V Krishnamurthy is at cycle, slow 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 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 von Neumann at the Institute for Advanced Study in Princeton in change . 1950 using the pioneering Electronic Numerical Integrator and (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 Institute of Technology (MIT) appointed Edward Lorenz as a new faculty member to lead the on-going statistical 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 to wrongly diction, predictability.

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The series of the conclude that linear methods using limited past data were capable solutions was of producing good forecasts. He 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 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 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 with no evidence of periodicity. The 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 and Education

Edward Norton Lorenz was born in West Hartford in the state of Connecticut, USA on 23 . His mother was a school teacher who became active in civic , and his father was a mechanical 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 , making him especially fascinated with . 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 , 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 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 , 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 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 in relativis- About a few months before Lorenz was expected to submit his tic quantum 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

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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 , 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 ‘’, 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.

Lorenz provided a quantitative theory of the growth of small initial errors in nonlinear systems in a paper published by Tellus in 1965. He showed that a small sphereof initial errors becomes an ellipsoid during the linear phase of growth. The long-term averages of the magnitude and orientation of the axes of the ellipsoid are now well known as Lyapunov exponents and Lyapunov vectors. The cur- rent practice of the leading weather prediction centers is to provide not just a single forecast but to issue probabilistic forecasts from an ensemble of predictions which start from initial states suitably separated by small differences based on some form of Lyapunov vectors.

Lorenz’s discovery of chaos is known to the scientific morefromhis famous 1963 paper entitled ‘Deterministicnonperiodic ’ which describes chaotic solutions in a 3-variable model4 of 4 The three-variable convection convection [5]. This is one of the greatest papers in science and model is the famous Lorenz model used in many disciplines. provides a fundamental theory of nonlinear forced dissipative It is one of the simplest systems dynamical systems and describes the complete geometric struc- to exhibit chaos. Because of ture of the solutions including the structure. The asymptotic just three variables in the model, set of solutions, known as the , is confined to a particular Lorenz was able to provide the complete geometric structure of region of the . The nonperiodic solutions are also the attractor. known as strange attractors. Lorenz was also a pioneer in studying the logistic as a dynamical . In a 1964 paper, he had described the periodic and chaotic solutions of the , stability of the periodic solutions and the scaling of the periodic windows. Lorenz’s work was unnoticed outside the meteorology community until the early 1970s. The nonperiodic solutions came to be called ‘chaos’. The is a revolution in science and has brought a . Chaos has now become ubiquitous and has reached all branches of science, mathematics, , and even the financial markets.

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Other Important Contributions

Even if Lorenz had not discovered chaos, his contributions to meteorology would have made him quite famous. Beginning with his doctoral thesis work, he was interested in the dynamics and of the weather system. In 1955, he introduced the concept of available potential energy to describe the conversion of solar energy into kinetic energy of the atmospheric motions and replacing the loss of energy due to friction. The detailed process 5 Lorenz’s first significant sci- came to be known as Lorenz’s energy cycle5. Lorenz provided a entific finding was the energy complete description of the general circulation of the atmosphere cycle. He worked out the entire formulation of the available po- in a monograph published by the World Meteorological Organiza- tential energy in a single night, tion in 1967. During the 1960s, he also published papers on the waking up to derive and write mathematical models of the laboratory simulations of atmospheric down all the equations. circulations conducted by Raymond Hide and David Fultz. He explained the possibility of different climate regimes through the ‘almost intransitive’ of multiple attractors. In the 1980s, he worked on the solutions of primitive equations which admit both Rossby modes and high-frequency gravity . He examined 6 The primitive equations are the thepossible existenceof a slowmanifold6 which is devoid of gravity dynamical equations govern- waves and found it to be unstable. ing the atmosphere. The slow manifold is a set consisting of Apart from his pioneering work on chaos during 1960s, Lorenz only the slower Rossby waves made other important contributions in nonlinear dynamics. He in the solutions of the primitive equations. The slow manifold showed that chaotic behavior can arise dueto approximations in the was found to be unstable be- numerical model used before computational sets in and cause gravity waves eventu- referred to it as computational chaos. He examined the detailed ally appear in the solutions. geometrical structure and the variation of the power spectra in the chaotic attractors of different models. He conducted extensive research on predictability and published several papers quantifying predictability in low-order models and applying those ideas on forecasts by large models.

Career and Honors

Lorenz spent his entire career at MIT. He became a full Professor in 1962 and served as the Chair of the Department of Meteorology during 1977–81. He was a visitor at the Lowell Observatory in

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Flagstaff, Arizona, University of California at Los Angeles and at Suggested Reading San Diego, the Norwegian Meteorological Institute in Oslo, the [1] E N Lorenz, A scientist by European Centre for Medium-Range Weather Forecasts in Read- choice, Kyoto Prize Lec- ing, UK. He was a frequent visitor at the National Center for ture, 1991. Atmospheric Research in Boulder, Colorado. He was a dedicated [2] E N Lorenz, The essence teacher and received the award for best graduate teaching at MIT of chaos, University of Washington Press, Se- for a successive number of years. He advised more than twenty attle, 1993. graduate students (including the author) toward their doctoral [3] E N Lorenz, The statis- degree. He was an excellent writer with superb clarity and self- tical prediction of so- sufficiency. His papers are still used in teaching scientific writing lutions of dynamic equations, Proceedings at MIT. Although he officially retired in 1987, he continued to be of the International Sym- active in research as Emeritus Professor at MIT and published posium on Numerical several papers. Lorenz was elected a Fellow or Honorary Fellow Weather Prediction, To- of several academies including the National Academy of Sciences kyo, Meteorological Society of Japan, (USA) in 1975 and the Indian Academy of Sciences in 1981. He pp.629–635, 1962. received several honorary doctorates and more than 20 awards and [4] E N Lorenz, Does the flap prizes. These include the from the Swedish Acad- of a butterfly’s wings in emy of Sciences in 1983, the Kyoto Prize in 1991 and the Buys Brazil set off a tornado in Texas?, American As- Ballot Medal in 2004. There are now honors and awards named sociation for the Ad- after Lorenz. In 2011, MIT created the Lorenz Center devoted to vancement of Science, fundamental understanding of climate as a . 139th Meeting, Wash- ington DC, December Lorenz died on 16 April 2008 at the age of 91 in Cambridge, 29, 1972. Massachusetts. Just before his death, he was working on the final [5] E N Lorenz, Determin- istic nonperiodic flow, version of his paper on periodic windows in the Hénon map which Journal of the Atmo- later appeared in the August 2008 issue of Physica D. Lorenz and spheric Sciences, Vol.20, his wife, Jane, were a loving and dedicated couple who took great pp.130–141, 1963. pleasure in traveling all over the world. They served as host family to several international students in area. Lorenz had a great passion for hiking and had climbed White Mountains, many peaks in the Rockies, the Cascades and the Alps. He climbed one of the Address for Correspondence White Mountains a few months before his death. His other big V Krishnamurthy interests were music, skiing and chess. Above all, Lorenz was Center for Ocean-Land- known as a modest, unassuming and kind person. Atmosphere Studies George Mason University, Fairfax, VA 22030 USA Email: [email protected]

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