爱德华·诺顿·劳仑次Edward Norton Lorenz

爱德华·诺顿·劳仑次在新罕布什尔州西部汉诺威镇的达特茅斯学院学习数学。他于1938年从达特茅斯学院毕业，获得学士学位，然后前往哈佛大学攻读数学硕士学位。他于1940年获得A.M.学位，并提交了他的第一篇数学论文以供发表。论文A generalization of the Dirac equations于1941年发表在Proceeding of the National Academy of Sciences上。

获得硕士学位后，劳仑次在美国陆军航空队服役。他在陆军航空队的工作涉及将他的数学技能应用于天气预报，很快他就开始寻求进一步学习气象学。他在麻省理工学院攻读了第二个硕士学位，这次是气象学的S.M.，于1943年毕业。然后，在第二次世界大战结束后，他继续在麻省理工学院攻读博士学位，James Austin是他的论文导师。他在提交了学位论文A Method of Applying the Hydrodynamic and Thermodynamic Equations to Atmospheric Models后，于1948年获得了Sc.D.学位。

劳仑次 自1946年起在麻省理工学院担任助理气象学家，但当他于1948年获得博士学位时，被提升为气象学家。他担任此职位直到1954年，在这八年期间，他发表了一些重要著作。他于1949年6月将Dynamic Models Illustrating the Energy Balance of the Atmosphere提交给Journal of the Atmospheric Sciences，并于1950年2月发表。劳仑次为这篇论文所写的摘要如下：-

通过从包含摩擦的水平运动方程中消去气压，得到了二维球形地球的广义涡度方程。广义涡度方程由表示密度场和风场的正式无穷级数所满足。明确获得了一个特定级数解的前几项。该级数似乎在北极附近收敛，并确定了一个极地气团的模型。在气团内部，最冷的风是东北风，最暖的是西南风，而最冷的空气则在北极。加热发生在西北风中，冷却发生在东南风中，而除了摩擦效应外，整个气团是冷却的。研究了气团的能量平衡。有人提出，类似的加热和冷却分布可能有助于维持大气的一般环流。

论文Seasonal and Irregular Variations of the Northern Hemisphere Sea-Level Pressure Profile一年后出现在同一期刊上。劳仑次的摘要开头如下：-

对北半球选定纬度圈上平均的五天平均海平面气压变化，以及选定纬度对之间五天平均气压差的变化，进行了统计研究。

1952年，他发表了Flow of Angular Momentum as a Predictor for the Zonal Westerlies。摘要的前几句话如下：-

提出了一个近似微分方程，将纬向西风速度的变化与同期的纬向风速和绝对角动量的经向流动联系起来。利用从分析的北半球地图中提取的气压和高度数据，借助地转风近似计算了动量流和纬向风速的值，对该方程进行了统计检验。

劳仑次 的整个职业生涯都在麻省理工学院度过，1954年被任命为助理教授，随后晋升为副教授，1962年成为气象学教授。他于1977年至1981年担任系主任，1987年退休时被授予荣休教授称号。

1961年发生的一件简单事件，使劳仑次得出了令他举世闻名的结果。他当时正用计算机研究他设计的大气模型，其中包含十二个微分方程。从运行程序得到结果后，他决定把计算继续推进下去。他没有把整个程序重新从头开始（因为尽管他用的是当时先进的计算机，但按后来的标准看慢得令人痛苦），而是从计算的中间点启动程序，输入机器在那个中间点算出的数据。他去喝了杯咖啡，回来后想看看计算机算得怎么样了。他惊讶地发现，在之前已经算过的区间内，计算机给出了明显不同的结果。起初他以为一定是硬件问题，因为给定相同的输入数据，软件每次都应该得出相同的答案。最终他发现，他为第二次运行输入的数据，打印出来的小数位数与机器存储的不一样，所以第二次运行的初始数据略有不同（在小数点后第四位有差异）。于是他开始试图理解，初始数据中一个微小的变化怎么会对计算产生如此重大的影响。劳仑次发现了混沌。

劳仑次 并不是第一个发现混沌的人。儒勒·昂利·庞加莱 在19世纪80年代研究三体问题时发现了混沌。然而，儒勒·昂利·庞加莱 的发现并未引发任何重大进展。既然 劳仑次 理解了他发现的重要性，便将其写成论文 Deterministic Nonperiodic Flow，并于1963年发表在 Journal of the Atmospheric Sciences 上。摘要开头如下：-

确定性常微分非线性方程的有限系统可以被设计用来表示受迫耗散流体动力学流动。这些方程的解可以与相空间中的轨迹相对应。对于具有有界解的系统，发现非周期解通常在小扰动下是不稳定的，因此略微不同的初始状态可以演变成相当不同的状态。具有有界解的系统被证明拥有有界数值解。

这篇论文，如同80年前 儒勒·昂利·庞加莱 的工作一样，在发表后的一段时间内影响相对较小。然而，它后来成为有史以来被引用最多的论文之一。这组方程以及由这组方程描述的吸引子现在分别被称为“劳仑次 方程”和“劳仑次 吸引子”。可以公平地说，劳仑次 以这篇论文开启了一场科学革命，他和许多其他人在随后的几年里将其发展。1972年，他在美国科学促进会上发表演讲，题目为 Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?。通过将混沌理论的精髓概括在演讲标题中，劳仑次 成功抓住了公众的想象力，“蝴蝶效应”一词很快成为混沌的流行术语。

关于常微分方程和差分方程中非周期行为的另一篇论述，其中 劳仑次 描述了他如何从气象学中对流的描述出发，得出 劳仑次 方程，包含在他于1978年在加拿大卡尔加里举行的加拿大数学大会双年研讨会上提交的论文 On the prevalence of aperiodicity in simple systems 中。1980年，他发表了 Attractor sets and quasigeostrophic equilibrium：-

本研究的主要目的是找出某些简单大气模型的吸引子集是什么样子。

后来的论文包括：A very narrow spectral band（1984），该文研究了劳仑次系统; The local structure of a chaotic attractor in four dimensions（1984）的谱性质；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）；以及The slow manifold - what is it?（1992）。

1991年，劳仑次在华盛顿大学作了Jessie and 弗瑞兹·约翰 Danz讲座。他以这一系列的三次讲座为基础，写成了他的著名著作The essence of chaos（1993）。Andrew Fowler在一篇评论中写道：-

这是一本极好的书，由一位无可争议的国际权威对混沌科学作了通俗的阐述。……劳仑次以平稳的语调，缓慢起步，逐步建立起对混沌的完整理解。讨论并不技术化，但也不回避真正的解释。读者被假定为有才智的，并应能应对全部混沌现象。读到最后，坚持不懈的读者会发现自己对这一主题有了不浮夸但真正的理解。

劳仑次与简·洛根结婚（她于2001年去世）；他们有三个孩子，两个女儿南希和谢丽尔，以及一个儿子劳仑次。至于他的爱好，他[2]：-

……是一名活跃的徒步旅行者和越野滑雪者，并坚持在参加每一次科学会议时都去探访附近的山间小径。

他的女儿谢丽尔说[1]：-

他两周半前还在远足，一周前还与一位同事完成了一篇论文。

他在马萨诸塞州剑桥的家中因癌症去世。

劳仑次因其卓越贡献获得了许多奖项和荣誉。例如：当选为美国艺术与科学院会士（1961年）；获得美国气象学会颁发的克拉伦斯·勒罗伊·迈辛格奖和卡尔·古斯塔夫·罗斯比研究奖章（1969年）；获得皇家气象学会颁发的西蒙斯纪念金奖（1973年）；当选为国家科学院（美国）会士（1975年）；当选为印度科学院院士；当选为挪威科学与文学院院士（1981年）；获得瑞典皇家科学院颁发的克拉福德奖（1983年）：-

……因对地球物理流体动力学领域的根本性贡献，这些贡献以独特的方式促进了对大气和海洋大尺度运动的更深入理解。

他还获得了富兰克林研究所颁发的埃德温·贝利·埃利奥特 Creson奖章。1984年，他当选为皇家气象学会荣誉会员，随后在1991年，他因发现确定性混沌而获得了稻盛财团颁发的京都奖（相当于日本的诺贝尔奖），这一发现具有：-

……深刻影响了广泛的基础科学领域，并带来了自艾萨克·牛顿以来人类自然观中最剧烈的变化之一。

此外，他还获得了美国地球物理联合会的罗杰·雷维尔奖章和美国气象学会的路易斯·J·巴坦作者奖。2004年，他被授予荷兰皇家艺术与科学院的拜斯·巴洛特奖章，该奖章大约每十年颁发一次，授予对气象学做出重大贡献的个人。

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:-

A 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.

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

The 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.

In 1952 he published Flow of Angular Momentum as a Predictor for the Zonal Westerlies. The first sentences of the abstract read:-

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.

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.

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:-

Finite 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.

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 title 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:-

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

Later papers include: 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:-

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.

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]:-

... was an active hiker and cross-country skier and made it a point to visit mountain trails near every scientific meeting he attended.

His daughter Cheryl said [1]:-

He was out hiking two and one-half weeks ago, and he finished a paper a week ago with a colleague.

He died from cancer at his home in Cambridge, Massachusetts.

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):-

... 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.

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:-

... 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.

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.