A masterpiece of physics, buried in the unvisited literature of meteorology

Years later, physicists would speak longingly of Lorentz's paper on these equations—"that beautiful masterpiece." For the first time, Lorentz's images made clear what it meant to say "this is complicated." All the richness of chaos was there.

By James Gleick

Translation | Lou Weishan

In the 1950s and 1960s, there was an air of unrealistic optimism about weather forecasting. Newspapers and magazines were filled with hopes for the science of meteorology, not just forecasting but also weather modification and control. Two technologies were maturing: computers and satellites. And an international collaboration called the Global Atmospheric Research Program was preparing to take full advantage of them. The idea was that human society would be liberated from the vagaries of weather, from being its victim to its master. Geodesic domes would cover cornfields. Airplanes would drop catalysts directly into clouds. Scientists would learn how to make and stop rain.

The intellectual father of this trend was John von Neumann, who designed his first computer with one of its intended functions being to control the weather. He gathered a group of meteorologists and pitched his plan to the general scientific community. He had a specific mathematical reason for his optimism. He noticed that a complex dynamical system could have points of instability—critical points where a slight nudge could have major consequences, like a nudge on a ball at the top of a hill. And von Neumann imagined that, with the help of computers, scientists would be able to calculate how the equations of motion of the fluid would behave over the next few days. Then a central committee of meteorologists would send out planes to spread smoke or cloud seeding, thus nudging the weather in the desired direction. But von Neumann ignored the possibility of chaos, where every point would be unstable.

By the 1980s, there was a vast organization devoted to pursuing von Neumann's goal, at least the weather-forecasting part of it, at great expense. In a plain-looking boxy building in suburban Maryland, near the Washington Beltway, with radar and radio antennas dotting the roof, America's top weather forecasters gathered. Their supercomputers ran weather models similar only in spirit to Lorenz's. Where the Royal-McBee LGP-30 could do sixty multiplications per second, a CDC Cyber ​​205 mainframe was measured in millions of floating-point operations per second. Where Lorenz was content with a dozen equations, modern global weather models process systems of 500,000. Their models understand the way water vapor releases and absorbs heat as air contracts and expands. Digital winds are affected by digital mountains. And every hour, data from every country in the world, from airplanes, satellites, and ships, comes in. The National Weather Center produces the second-best weather forecast in the world.

The best weather forecasts come from Reading, England, a university town an hour outside of London. The European Centre for Medium-Range Weather Forecasts is housed in a tree-lined building, a modern brick-and-glass structure in the style of the United Nations, filled with gifts from all over the world. It was created in the heyday of the European Common Market, when most of Western Europe decided to pool its talents and resources to make better weather forecasts. The Europeans attribute their success to their rotation of young talent (there are no civil servants) and to their Cray supercomputers, which always seem to be one model ahead of the ones the Americans use.

Weather forecasting marked the beginning of the use of computers to model complex systems, but it certainly was not the end. The same techniques helped scientists and social scientists in many other fields make predictions, ranging from the small-scale fluid flows that concern propeller designers to the large-scale financial flows that concern economists. In fact, by the 1970s and 1980s, economic forecasting with computers had become a lot like global weather forecasting. Models weave through a complex but arbitrary web of equations that transforms measurements of initial conditions (whether atmospheric pressure or the money supply) into a simulation of future trends. The researchers hope that the results will not deviate too far from reality due to the many unavoidable simplifying assumptions. If a model does produce a result that is clearly off the mark (flooding in the Sahara, say, or a tripling of interest rates), the researchers tweak the equations to bring the result back on track. In practice, economic models have repeatedly proven difficult to reliably predict the future, yet many people who should know better behave as if they believe in the results. Forecasts of economic growth or unemployment are often offered with the implication that one has precision to two or three decimal places. Governments and financial institutions often pay for such forecasts and act on them, perhaps out of necessity or lack of better options. Perhaps they know that variables like consumer confidence are not as well measured as humidity, and that we have not yet found perfect differential equations to describe changes in politics and fashion. But few people realize how fragile the process of modeling various flows on computers is, even if the data are fairly reliable and the laws that govern them, as in weather forecasts, are purely physical .

Computer modeling has indeed succeeded in turning weather forecasting from an art into a science. The European Centre for Medium-Range Weather Forecasts estimates that the world saves billions of dollars in damages each year with these predictions, which are statistically better than nothing. But beyond two or three days, even the world’s best weather forecasts are little more than guesswork, and beyond six or seven days, they become worthless.

The butterfly effect is why. For small-scale weather phenomena (in the eyes of a global weather forecaster, "small-scale" might mean thunderstorms and snowstorms), any prediction quickly deteriorates and becomes useless. Errors and uncertainties accumulate, amplified across a range of turbulent phenomena large and small, from dust devils and squalls to giant vortices visible only from satellites.

Modern global weather models use data sampled from a grid with a hundred kilometers between points, and even then some of the initial data has to be guessed, because ground stations and satellites can't see everywhere. But imagine that the entire Earth could be covered with sensors, thirty centimeters apart horizontally and thirty centimeters apart vertically, all the way up to the top of the atmosphere. Imagine that each sensor gave a completely accurate reading of temperature, pressure, humidity, and whatever else a meteorologist wants to know. Then at noon, an infinitely powerful computer read all this data and calculated what the weather would be like every minute thereafter (12:01, 12:02, 12:03, ...).

The computer will still not be able to predict whether it will be sunny or rainy in Princeton, New Jersey, one month from now. At noon, there will be random fluctuations in the space between the sensors that the computer is not aware of, tiny deviations from the average. By 12:01, these fluctuations will create tiny errors thirty centimeters away. These errors will soon accumulate at the scale of three meters, and so on, until they lead to significant differences at the scale of the entire planet.

All of this was counterintuitive even to a veteran meteorologist. One of Lorenz’s old friends was Robert White, an MIT meteorologist who later became the first director of the National Oceanic and Atmospheric Administration. Lorenz explained the butterfly effect to him and what he thought it might mean for long-term forecasts. White gave von Neumann’s answer. “Forecasting, that doesn’t matter,” he said. “It’s weather control .” His idea was that small artificial influences within human reach would be able to induce the large-scale changes in the weather we want.

Lorenz thinks otherwise. Yes, you can change the weather. You can make it different from what it was. But if you do that, you never know what it was like. It's like taking a shuffled deck of cards and shuffling it again. You know it will change your luck, but you don't know if it will be better or worse .

Lorenz's discovery was an accident, one of countless since Archimedes and his bathtub. Lorenz was never the type to shout "Eureka." This unexpected discovery simply led him to a place he never left. He was ready to explore the implications of this discovery by finding out what it meant for the way science understands the flow of various fluids.

Had he stopped at the butterfly effect, an image that illustrates predictability giving way to total randomness, Lorenz might have revealed only very bad news. But Lorenz saw more than randomness in his weather models. He saw an elaborate geometric structure, an order masquerading as randomness. He was, after all, a mathematician masquerading as a meteorologist, and at this point he began to lead a double life. He would write papers on pure meteorology. But he would also write papers on pure mathematics, with a slightly misleading introduction to the topic of weather. Eventually, that introduction would disappear altogether.

He turned his attention more and more to the mathematics of systems that never quite find a steady state, that almost repeat themselves but never quite do. Everyone knows that the weather is such a system—nonperiodic. Other examples abound in nature: animal populations that rise and fall almost regularly, epidemics that break out and recede on a nearly regular schedule, and so on. If the weather did ever reach a state exactly like one it had experienced before, with every wind and cloud exactly the same, then it would probably continue to repeat itself forever, and the problem of weather forecasting would become mundane.

Lorenz realized that there must be a connection between the weather’s reluctance to repeat itself and forecasters’ inability to predict it— a connection between aperiodicity and unpredictability . Finding a simple system of equations that would generate the aperiodicity he was looking for was not easy. At first, his computer models tended to fall into cycles that repeated themselves over and over. But Lorenz tried a variety of slight complications and finally succeeded by adding an equation for how the temperature difference between the east and west directions (corresponding to the real-world difference in heating between, say, the east coast of North America and the Atlantic Ocean) varied over time. The repetitions disappeared.

The butterfly effect is not an accident, but a necessity . Suppose, Lorenz reasoned, that instead of small perturbations accumulating and growing in the system, they stayed small, so that when the weather got arbitrarily close to a state it had experienced before, it would stay that way, and keep getting arbitrarily close to that state. In fact, such a cycle would be predictable—and ultimately boring. You can hardly imagine anything better than the butterfly effect to generate the rich, ever-changing weather on Earth.

The butterfly effect is also known by another technical name: sensitive dependence on initial conditions . However, sensitive dependence on initial conditions is not a new concept. It is reflected in folk tales:

One nail missing is like losing a shoe.

I have lost a horse, a horse;

A horse is missing, a rider is lost;

A knight less, a battle lost;

A battle lost means a kingdom lost.

In science, as in life, it is well known that there can be a tipping point in a chain of events that amplifies small changes. But chaos means that such points are everywhere. They are everywhere. In systems like the weather, the sensitive dependence on initial conditions is an inevitable consequence of the way small scales are intertwined with large scales.

His colleagues were amazed that Lorenz had grasped both the aperiodicity and the sensitive dependence on initial conditions, using only a toy model of the weather: twelve equations, which he then calculated over and over again with mechanical efficiency. How could such richness, such unpredictability (such chaos), emerge from a simple deterministic system?

Lorenz put aside the weather and tried to find a simpler way to generate this complex behavior. He found it in a system of just three equations. These equations were nonlinear, that is, the relationships they represented were not strictly proportional. A linear relationship can be represented as a straight line on a graph. It was also easy to understand: the more the better. Systems of linear equations are solvable, which makes them suitable for textbooks. Linear systems also have an important advantage of being modular: you can take them apart and put them back together again - the parts are additive.

Nonlinear systems are generally not solvable and not additive. In both fluid and mechanical systems, nonlinear terms are often features that people want to ignore in an attempt to get a simple and clear understanding. Take friction, for example. Without friction, the energy required to accelerate a hockey puck can be expressed by a simple linear equation. With friction, the relationship becomes more complicated because the energy required depends on the speed the puck already has. Nonlinearity means that the very act of playing the game changes the rules of the game. You can't give friction a constant importance because its importance depends on speed. And speed, in turn, depends on friction. This interdependence makes nonlinearity difficult to calculate, but it also creates a rich variety of behaviors that are not seen in linear systems. In fluid dynamics, everything can be reduced to a single classical equation, the Navier-Stokes equations. It's a marvel of simplicity that relates velocity, pressure, density, and viscosity of a fluid, but it happens to be nonlinear. So the nature of these relationships often becomes impossible to determine explicitly. Analyzing the behavior of a nonlinear equation like the Navier-Stokes equations is like navigating a maze whose walls rearrange with every step you take. As von Neumann himself said: “The nature of the equations… changes simultaneously in all the relevant respects: in degree and in degree. Hence the intractable mathematical difficulties which must follow.” If the Navier–Stokes equations did not contain the devil of nonlinearity, the world would be a very different place, and science would have no need for chaos.

Lorenz's three equations were inspired by a particular kind of fluid motion: the rise of hot gases or liquids, or convection. In the atmosphere, air near the ground expands and rises; off hot asphalt and radiator surfaces, hot vapors rise in ghostly wisps. Lorenz also enjoyed talking about convection in a hot cup of coffee. According to him, this is just one of countless fluid dynamic processes whose future we might hope to predict. How can we calculate how fast a cup of coffee will cool? If the coffee is only warm, its heat will slowly dissipate without any fluid dynamic motion. The coffee remains in a steady state. But if it is hot enough, convection processes will carry the hot coffee from the bottom of the cup to the cooler surface. Convection in coffee becomes clearly visible by adding a little cream to the cup. The resulting white vortex can be very complex. But the long-term fate of such a system is obvious. As heat continues to dissipate and friction slows the fluid, the whole motion must inevitably stop. Lorenz then joked to a group of scientists in all seriousness: "It may be difficult to predict the temperature of the coffee in one minute, but it should not be difficult to predict its temperature in one hour." The equations of motion that describe a slowly cooling cup of coffee must be able to reflect this fate of the system. They must be dissipative. The temperature of the coffee must gradually approach room temperature, and the velocity must approach zero.

Lorenz took a set of equations that describe convection and simplified them to the point of being unrealistic, discarding anything that could be wrong. There was almost nothing left of the original model, but he did keep the nonlinearities. To a physicist, these equations look simple. You glance at them (and many scientists did later) and say, "I can solve that."

"Yes," Lorenz said quietly, "that's what you tend to think when you see them. There are some nonlinear terms in them, but you think there must be some way to get around them. But you just can't do it."

The simplest textbook example of convection occurs in a box filled with a fluid, where one smooth bottom surface can be heated and the other smooth top surface can be cooled. The temperature difference between the hot bottom and the cold top controls the motion of the fluid flow. If the temperature difference is small, then the whole system remains stationary. The heat then flows from the bottom to the top by conduction, as if flowing through a piece of metal, which is not enough to overcome the fluid's natural tendency to remain motionless on a macroscopic scale. Moreover, the whole system is stable. Any random motion (such as that caused by a graduate student knocking on the experimental equipment) will gradually die out, causing the system to return to its steady state.

©Adolph E. Brotman

Tumbling fluid: When a liquid or gas is heated at the bottom, the fluid tends to self-organize into cylindrical vortices (left). The hot fluid rises on one side, gradually loses heat, and then sinks on the other side - this is the process of convection. With further heating (right), an instability begins and the vortices begin to oscillate back and forth along the long axis of the cylinder. At higher temperatures, the fluid flow becomes arbitrary and turbulent.

But increase the intensity of the heat, and a new kind of behavior emerges. As the fluid at the bottom heats up, it expands. As it expands, it becomes less dense. And as it becomes less dense, it becomes relatively light enough to overcome friction and rise to the top. In a carefully designed box, cylindrical vortices emerge, with hot fluid rising on one side and cool fluid sinking to replenish it on the other. Viewed from the side, the entire motion forms a continuous circle. And outside the lab, nature often creates its own convective vortex cells. When the sun heats a desert landscape, for example, roiling air currents can create mysterious patterns in cumulus clouds above or in the sandbanks below.

Increase the intensity of the heat further, and the behavior of the fluid becomes more complicated. The vortices begin to twist and wobble. Lorenz's equations are too simplistic to model this kind of complexity. They abstract just one feature of real-world convection: hot fluid rises and cold fluid sinks, tumbling in a circle like a Ferris wheel. The equations take into account the speed of this motion and the transfer of heat, and these physical processes interact with each other. As the hot fluid rises along the circle, it comes into contact with other, cooler parts and begins to lose heat. If it moves fast enough, the bottom fluid won't lose all of its extra heat by the time it reaches the top and starts to sink down the other side of the vortex, so it actually starts to hinder the motion of other hot fluid behind it.

Although Lorenz’s system didn’t fully model convection, it turned out that it had some definite counterparts in real systems. For example, his equations accurately describe an old-fashioned dynamo. An ancestor of modern generators, disk dynamos generate electric currents by spinning disks in a magnetic field. Under certain conditions, a two-disk dynamo can reverse the current in its circuit. As Lorenz’s equations became more widely known, some scientists suggested that the behavior of such a dynamo might explain another bizarre reversal phenomenon: the Earth’s magnetic field. This “geomagnetic dynamo” was known to have reversed many times in Earth’s history, with seemingly random, inexplicable intervals between reversals. Faced with such irregularities, theorists usually try to find explanations outside the system, proposing things like meteorite impacts. But perhaps the geomagnetic dynamo has its own chaos.

Another system that can be described precisely by Lorenz's equations is a kind of waterwheel, a mechanical analogue of circular motion for convection. At the top, water flows at a constant speed into buckets hung on the edge of the waterwheel. Each bucket then leaks water at a constant speed through a small hole at the bottom. If the water is flowing slowly, the top bucket will never accumulate enough water to overcome friction; but if the water flows faster, the weight of the top bucket will cause the waterwheel to start turning. The rotation may continue in the same direction. Or if the water flows so fast that the heavy bucket passes over the lowest point and comes to the other side, the whole waterwheel may slow down, stop, and then turn in the opposite direction, first in one direction and then in the other.

Faced with such a simple mechanical system, a physicist's intuition (his pre-chaos intuition) would tell him that, in the long run, if the water keeps flowing at a constant speed, a steady state will evolve. Either the waterwheel turns at a constant speed, or it oscillates steadily back and forth, turning in one direction and then in the other direction at constant intervals. But Lorenz found that this was not the case.

©Adolph E. Brotman

Lorenz's Waterwheel: The first famous chaotic system, discovered by Edward Lorenz, corresponds to exactly one mechanical device: a waterwheel. This simple device proved capable of generating surprisingly complex behavior.

The rotation of a waterwheel has some properties similar to the tumbling cylinder formed by a fluid during convection. The waterwheel is like a cross section of a cylinder. Both systems are driven at a constant speed (by water or heat), and both dissipate energy (the fluid loses heat, and the bucket leaks water). In both systems, the long-term behavior depends on the strength of the driving energy.

Water flows in at a constant rate from the top. If the water flows slowly, the bucket at the top will never accumulate enough water to overcome friction, and the wheel will never start turning. (Similarly, in a fluid, if there is not enough heat to overcome viscosity, the fluid will never start moving.)

If the water flow becomes faster, the weight of the highest bucket will drive the waterwheel to start rotating (left picture). The whole waterwheel will enter a uniform rotation in the same direction (center picture).

But if the water flows faster (right), the rotation becomes chaotic due to the inherent nonlinear effects of the system. As the buckets pass under the flow, the amount of water they can pick up depends on how fast they are turning. On the one hand, if the wheel is turning very fast, the buckets won't have much time to catch water. (Similarly, a fluid in a rapidly tumbling convection current won't have much time to absorb heat.) On the other hand, if the wheel is turning very fast, the buckets will get to the other side before all the water has escaped. Therefore, the heavy buckets moving upward on the other side will cause the rotation to slow down or even reverse.

In fact, Lorenz found that over the long term the rotations would reverse many times and never develop a stable frequency or repeat themselves in any predictable pattern.

The three equations (with their three variables) completely describe the motion of the system. Lorenz’s computer outputs ever-changing values ​​for those three variables: 0–10–0, 4–12–0, 9–20–0, 16–36–2, 30–66–7, 54–115–24, 93–192–74. As time passes through the system, five time units, a hundred, a thousand, the numbers rise and fall.

In order to use the data to get an intuitive picture, Lorenz used each group of three numbers as coordinates to determine a point in three-dimensional space. Thus, the sequence of numbers generated a sequence of points, a continuous trajectory that records the behavior of the system. Such a trajectory may come to a place and then end, which means that the system eventually enters a steady state, when the variables related to speed and temperature will no longer change. Or the trajectory may form a loop, which goes back and forth, which means that the system eventually enters a behavior pattern that repeats itself periodically.

Lorenz's system was neither of these things. Instead, its graph showed an infinite complexity. It always stayed within certain boundaries, never crossed them, but it also never repeated itself. It produced a strange and unique shape - a kind of double spiral in three-dimensional space, like a butterfly with its wings outstretched. This shape revealed pure disorder, because no point or pattern of points repeated itself. But it also revealed a new kind of order.

Years later, physicists would speak longingly of Lorenz's paper on the equations—"that beautiful masterpiece." It would be spoken of as if it were an ancient scroll containing the secrets of eternity. Of the thousands of technical papers that have written about chaos, few have been cited more often than "Deterministic Aperiodic Flows." And for years, no single object would inspire more illustrations and even animations than the mysterious curve depicted in that paper, the double spiral that came to be known as the Lorenz attractor. For the first time, Lorenz's images made clear what it meant to say "this is complicated." All the richness of chaos was there.

©James P. Crutchfield/Adolph E. Brotman

Lorenz attractor: This magical image, resembling an owl mask or butterfly wings, became a symbol for early explorers of chaos. It reveals subtle structures hidden behind a chaotic stream of data. Traditionally, the changing values ​​of a variable are represented as a so-called time series (above left). But showing the changing relationship between three variables requires a different technique. At any given moment, the values ​​of the three variables determine the location of a point in three-dimensional space; the movement of the point as the system changes represents these changing variables.

Since the system never repeats itself exactly, its trajectory never intersects itself. Instead, it keeps going around and around. The motion of an attractor is abstract, but it conveys some of the characteristics of the motion of real systems. For example, a jump from one wing of an attractor to the other corresponds to a reversal of direction in the motion of a waterwheel or a convecting fluid.

At the time, however, few could see this. Lorenz described his discovery to William Marcus, an MIT professor of applied mathematics and a polite scientist with a remarkable appreciation for the work of his colleagues. Marcus laughed and said, “Ed, we know (we know) that convection in fluids doesn’t behave like that at all.” Marcus told him that the complexity would surely wane, and the system would eventually settle into a stable, regular motion.

“Of course, we totally missed the point,” Marcus said more than two decades later, years after he had actually built a Lorenz waterwheel in his basement lab to “preach” to nonbelievers. “Ed wasn’t thinking about our physics at all. He was thinking about some general or abstract model that exhibited behavior that he instinctively felt was typical of some aspect of the external world, but he couldn’t tell us that. Only afterward did we realize that he must have had these views.”

Few laymen realized how insular the scientific world had become; it was like a battleship with watertight bulkheads that sealed off the compartments from each other. Biologists didn’t need to look at the mathematical literature; there was already enough to read—indeed, molecular biologists didn’t need to look at population biology; there was already enough to read. Physicists had better ways to spend their precious time than browsing meteorological journals. Some mathematicians would have been excited to see Lorenz’s discovery; and over the next decade, many physicists, astronomers, and biologists had been looking for things like it, and sometimes rediscovered it themselves. But Lorenz was a meteorologist, and no one thought to look for chaos on page 103 of Volume 20 of the Journal of the Atmospheric Sciences.

This article is authorized to be excerpted from "Chaos" (Posts and Telecommunications Press, 2021 edition), and the title is added by the editor.

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