# Edward Norton Lorenz

### Born: 23 May 1917 in West Hartford, Connecticut, USA

Died: 16 April 2008 in Cambridge, Massachusetts, USA

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**Edward Lorenz**studied mathematics at Dartmouth College in the town of Hanover in western New Hampshire. He graduated from Dartmouth College with a bachelor's degree in 1938, then went to Harvard where he studied for a Master's Degree in mathematics. He received the A.M. degree in 1940 and submitted his first mathematical paper for publication. The paper

*A generalization of the Dirac equations*appeared in the

*Proceeding of the National Academy of Sciences*in 1941.

After the award of his master's degree, Lorenz undertook war service with the United States Army Air Corps. His work with the Army Air Corps involved applying his mathematical skills to weather forecasting and soon he was looking to undertake further study of meteorology. He did a second master's degree, this time an S.M. in meteorology, at the Massachusetts Institute of Technology graduating in 1943. Then, after World War II ended, he continued to study for his doctorate at the Massachusetts Institute of Technology with James Austin as his thesis advisor. He was awarded his Sc.D. in 1948 after submitting the dissertation

*A Method of Applying the Hydrodynamic and Thermodynamic Equations to Atmospheric Models*.

Lorenz had been employed as an assistant meteorologist at the Massachusetts Institute of Technology from 1946, but when he was awarded his doctorate in 1948 he was promoted to meteorologist. He held this post until 1954 and during these eight years he published some major works. He submitted

*Dynamic Models Illustrating the Energy Balance of the Atmosphere*to the

*Journal of the Atmospheric Sciences*in June 1949 and it was published in February 1950. Lorenz's abstract for this paper reads:-

The paperA generalized vorticity equation for a two-dimensional spherical earth is obtained by eliminating pressure from the equations of horizontal motion including friction. The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields. The first few terms of a particular series solution are obtained explicitly. The series appear to converge near the north pole, and determine a model of a polar air mass. Within the air mass, the coldest winds are northeasterly and the warmest are southwesterly, while the coldest air of all is at the north pole. Heating occurs in the northwesterly winds and cooling in the southeasterlies, while aside from the effect of friction the air mass as a whole is cooled. The energy balance of the air mass is investigated. It is suggested that an analogous distribution of heating and cooling may be instrumental in maintaining the general circulation of the atmosphere.

*Seasonal and Irregular Variations of the Northern Hemisphere Sea-Level Pressure Profile*appeared a year later in the same journal. Lorenz's abstract begins:-

In 1952 he publishedThe variations of five-day mean sea-level pressure, averaged about selected latitude circles in the northern hemisphere, and the variations of differences between five-day mean pressures at selected pairs of latitudes are examined statistically.

*Flow of Angular Momentum as a Predictor for the Zonal Westerlies*. The first sentences of the abstract read:-

Lorenz spent his whole career at the Massachusetts Institute of Technology, being appointed assistant professor in 1954, then promoted to associate professor before becoming Professor of Meteorology in 1962. He served as Head of Department from 1977 to 1981, retiring in 1987 when he was made professor emeritus.An approximate differential equation is presented, relating the change in speed of the zonal westerly winds to the contemporary zonal wind-speed and the meridional flow of absolute angular momentum. This equation is tested statistically by means of values of the momentum flow and the zonal wind-speed, computed with the aid of the geostrophic-wind approximation, from pressure and height data extracted from analyzed northern-hemisphere maps.

It was a simple event in 1961 which led Lorenz to results which brought him worldwide fame. He was using a computer to investigate models of the atmosphere which he had devised involving twelve differential equations. Having obtained results from running his program, he decided that he would like to carry the calculations further. Rather than start the whole program again (for although he was using an up-to-date computer, it was painfully slow by later standards), he started the program in the middle of the calculation inputting the data as calculated by the machine at that middle point. After going away for a cup of coffee, he came back to see how the computer was getting on with the calculations. He was surprised to see that the computer had found significantly different answers over the range that it had calculated before. At first he thought it must be a hardware problem, since the software should come up with the same answer every time when given the same input data. Eventually he discovered that the data he had input to begin the second run had not been printed out to the same number of decimal places as the machine had stored, so the initial data was slightly different for the second run (differing in the fourth decimal place). He then set about trying to understand how a tiny change in the initial data could have such a major effect on the calculations. Lorenz had discovered chaos.

Lorenz was not the first person to discover chaos. Poincaré had discovered chaos in the 1880s when studying the 3-body problem. However, Poincaré's discovery had not led to any significant developments. Now that Lorenz understood the significance of his discovery he wrote it up in the paper

*Deterministic Nonperiodic Flow*and it appeared in the

*Journal of the Atmospheric Sciences*in 1963. The abstract begins:-

The paper, like Poincaré's work 80 years earlier, had relatively little impact in the time immediately after it appeared. However, it has gone on to become one of the most quoted papers of all time. The set of equations and the attractors described by this set of equations are now called the 'Lorenz equations' and 'Lorenz attractors', respectively. It would be fair to say that Lorenz began a scientific revolution with this paper which he and many others developed over the following years. In 1972 he addressed the American Association for the Advancement of Science giving a talk with the titleFinite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

*Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?*By encapsulating the essence of chaos theory in the title of his talk, Lorenz succeeded in capturing the public's imagination and the term "butterfly effect" was soon the popular term for chaos.

Another account of aperiodic behaviour in ordinary differential equations, and difference equations, in which Lorenz describes how he arrived, starting from the description of convection in meteorology, at the Lorenz equations is contained in his paper

*On the prevalence of aperiodicity in simple systems*delivered at the Biennial Seminar of the Canadian Mathematical Congress in Calgary, Canada, in 1978. In 1980 he published

*Attractor sets and quasigeostrophic equilibrium*:-

Later papers include:The primary purpose of this study is to find out what the attractor set looks like for some simple atmospheric model.

*A very narrow spectral band*(1984) which investigates the spectral properties of the Lorenz system

*; The local structure of a chaotic attractor in four dimensions*(1984);

*Lyapunov numbers and the local structure of attractors*(1985);

*On the existence of a slow manifold*(1986);

*Atmospheric models as dynamical systems*(1986);

*Computational chaos - a prelude to computational instability*(1989); and

*The slow manifold - what is it?*(1992).

In 1991 Lorenz gave the Jessie and John Danz Lectures at the University of Washington. He used this series of three lectures as a basis for his famous text

*The essence of chaos*(1993). Andrew Fowler writes in a review:-

Lorenz was married to Jane Logan (who died in 2001); they had three children, two daughters Nancy and Cheryl, and a son Edward. As to his hobbies, he [2]:-This is an excellent book, providing a popular exposition of the science of chaos by an undisputed international authority. ... In measured tones, Lorenz starts slowly and builds up gradually a complete understanding of chaos. The discussion is not technical, but does not shirk genuine explanation. The reader is assumed to be intelligent, and is expected to cope with the whole battery of chaotic phenomena. By the end, the dogged reader will find himself with an unsensational but genuine understanding of the subject.

His daughter Cheryl said [1]:-... was an active hiker and cross-country skier and made it a point to visit mountain trails near every scientific meeting he attended.

He died from cancer at his home in Cambridge, Massachusetts.He was out hiking two and one-half weeks ago, and he finished a paper a week ago with a colleague.

Lorenz received many awards and honours for his remarkable contributions. For example: elected a fellow of the American Academy of Arts and Sciences (1961); awarded the Clarence Leroy Meisinger Award and the Carl Gustaf Rossby Research Medal (1969) both from the American Meteorological Society; awarded the Symons Memorial Gold Medal of the Royal Meteorological Society (1973); elected a Fellow of the National Academy of Sciences (United States) (1975); elected a Member of the Indian Academy of Sciences; elected a Member of the Norwegian Academy of Science and Letters (1981); awarded the Crafoord Prize of the Royal Swedish Academy of Sciences (1983):-

He was also awarded the Elliott Creson Medal from the Franklin Institute. In 1984 he was elected an Honorary Member of the Royal Meteorological Society, then in 1991 he was awarded the Kyoto Prize of the Inamori Foundation (Japanese equivalent of the Nobel prize) for discovering deterministic chaos which has:-... for fundamental contributions to the field of geophysical hydrodynamics, which in a unique way have contributed to a deeper understanding of the large-scale motions of the atmosphere and the sea.

In addition he was awarded the Roger Revelle Medal from the American Geophysical Union and the Louis J Battan Author's Award from the American Meteorological Society. In 2004 he was awarded the Buys Ballot medal of the Royal Netherlands Academy of Arts and Sciences which is awarded approximately every ten years to an individual that has made significant contributions to meteorology.... profoundly influenced a wide range of basic sciences and brought about one of the most dramatic changes in mankind's view of nature since Sir Isaac Newton.

**Article by:** *J J O'Connor* and *E F Robertson*

**List of References** (2 books/articles)

**Mathematicians born in the same country**

**Other Web sites**