Abbe’s data compilations and forecasts, “probabilities” as he called them, became a government service available to anyone with a telegraph. By 1900, the United States also had 114 automatic stations in operation along with its cadre of human observers.
At the height of his career, Abbe understood that the behavior of the atmosphere was the same as a fluid, writing that meteorology is “essentially the application of hydrodynamics and thermodynamics to the atmosphere.” This insight turned out to be crucial, but the math necessary to elaborate his vision was beyond his capabilities. The probability torch was passed to the Norwegian scientist, Vilhelm Bjerknes.
Calculus of the Clouds
Bjerknes was a mathematical wunderkind born into a scientific dynasty. By age 15, he was a laboratory and research assistant to his father, Carl Anton Bjerknes, a physicist and the world’s leading authority on hydrodynamics. Carl had developed a new resonance theory that drew parallels between the behavior of fluids and electromagnetism. Carl and Vilhelm were definitely not British aristocratic autodidacts with a penchant for natural history. Carl was chair of the pure mathematics department at the University of Oslo, and father and son were squarely in the vanguard of modern professional scientists.
Vilhelm Bjerknes received his masters of mathematics and physics from the University of Kristiania (later named Oslo) in 1888. He began to sketch out some of his most original scientific insights while still at the university, but in the increasingly competitive world of professional science, his father had developed a fear of publishing his own work, much less Vilhelm’s. Painfully, Vilhelm realized he had to end his collaborative work with his father or be doomed to fall into obscurity. He set out on his own.
Vilhelm studied electromagnetics with Henri Poincaré in Paris, and then, in Germany, worked with the famous physicist Heinrich Hertz. In 1895, he took up a professorship at the University of Stockholm and began to wrestle with the logistics of atmospheric hydrodynamics. He had married, and in 1897 his son Jacob, the next heir to the scientific dynasty, was born. Meanwhile, in a sort of Oedipal atonement, he began collecting his father’s papers on hydrodynamics and published a two-volume set before his father’s death in 1903.
In 1904, the year after his father Carl passed away, Vilhelm published a paper on how weather could be predicted using mathematical formulae. He devised a two-step forecasting rationale: diagnostic (what is), followed by prognostic (what will be). It was reminiscent of a grand theory that had been proposed a century earlier by Pierre-Simon, marquis de Laplace (1749–1827), one of the great mathematicians of the Age of Enlightenment.
Laplace had conjectured that science and mathematics would eventually converge into a system capable of predicting the future. In his treatise Theorie Analytique des Probabilités, written in the early decades of the nineteenth century, he proposed,
Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.
Laplace was voicing the hubris of scientific rationalism at one of its most exciting periods. In this passage, he envisaged how a God-like intelligence might calculate the future given an omniscient knowledge of the present. It speaks to a profoundly Newtonian and Cartesian idealism, where elegant formulae paralleled the perfect harmonies of golden proportions, and it must have been at the back of Bjerknes’s mind as he drew up his prognostic equations.
To enable his forecasts, Bjerknes devised a mathematical model that, in its basic form, is still in use today. His formulae were a phenomenal achievement, resting, as they did, on seven independant equations. Four dealt with each atmospheric variable — temperature, humidity, pressure and density (the continuity equation, the equation of state and the first and second law of thermodynamics) — while the other three consisted of hydrodynamic equations for motion. But there was a problem: he could not gather enough data about the initial state of the weather. No high-alititude instruments existed yet, nor did enough ocean-going ones. Furthermore, the equations were so complex that they could not be solved numerically or analytically. In theory, he could predict future weather patterns, but there weren’t enough mathematicians to process the data.
Highs, Lows and Weather Fronts
In 1917, while the First World War was still raging, Bjerknes founded the Bergen Geophysical Institute at the Bergen Museum in Norway. There he was assisted by his son Jacob, now 20 and a brilliant physicist in his own right. Jacob’s friend Halvor Solberg and Swedish meteorologist Tor Bergeron joined them, and in less than a year, this extraordinary team had come up with the theory of “fronts,” a term they adopted from the First World War battlefronts. They then went on to model what happens as a mid-latitude cyclone (low-pressure cell) is born, matures and then decays, and they introduced the symbols — semicircular red symbols for warm fronts and blue triangles for cold fronts — that we still use on forecast maps today.
Vilhelm Bjerknes’s understanding of the atmosphere was a natural extension of Abbe’s original insight: that, in essence, the atmosphere is a rarefied fluid. Like any liquid, it is liable to turbulence — the differential heating of night and day, summer and winter; the friction from mountains and the effects of landmasses and oceans — all of which creates eddies, like stirring milk in a cup of coffee. Francis Galton had divided these eddies into highs and lows, defined by their relative barometric pressure. It was well-known by Bjerknes’s time that a high-pressure zone, or air mass, could either be warm or cold, as could a low-pressure air mass. Also well-known was the fact that warm air masses usually form in southern regions or over landmasses in summer, whereas cold-air masses form near the poles or over snow-covered landmasses in winter.
But regardless of their temperature, highs and lows have completely different origins. Bjerknes’s team discovered that a high-pressure cell develops when air has time to accumulate in one region. This surplus of air grows cool and heavy at its lofty heights, and the chilled air flows downward and outward. Low-pressure cells are different creatures altogether, spawned by the collision of two high-pressure cells of different temperatures. Of course, highs and lows, aside from their pressure profile and genesis, differ in one very critical characteristic: their spin. The Coriolis effect causes a low to rotate counterclockwise and a high to rotate clockwise. In the southern hemisphere, the opposite holds true. It was the interaction and evolution of these cells that Bjerknes’s team mapped out clearly for the first time.
A good example would be the birth of a low-pressure cell in North America. Here, lows often form when a warm, southern high-pressure zone bumps into a cool, northern high-pressure zone. Due to the fact that high-pressure systems rotate clockwise, the winds at the northern edge of the southern system blow in the opposite direction to the winds at the southern edge of the northern system. They are like cogs grinding against each other. The only “solution” to this problem is a swirl rotating in the opposite direction, counterclockwise. This is the beginning of the cyclone, which becomes a low-pressure area. As the air is drawn toward its center, the Coriolis effect sets it swirling in the opposite direction to the high-pressure region. Looking at the interaction of the two highs and the low, we cans see that the low acts like a lubricant between the two highs. It pulls an isthmus of warm air into and above the cool air mass. Now, as the southern high moves from west to east, a frontal system is created. The front is merely the leading edge, the boundary, between a mass of cold air and a mass of warm air.
Because highs and lows are generally circular, a front is usually curved, like the semicircles you see on weather maps. Their resemblance to military strategy maps was something that the Bjerknes’s team didn’t miss
when they drew their first frontal system maps. But there’s a lot more going on in three dimensions than a two-dimensional weather map reveals. Slicing through a front vertically, in cross section, you immediately notice that it is wedge shaped. Cool air sinks, hugs the ground, so that when a cold air mass is advancing and encounters a warm air mass, it wedges under the warmer air and pushes it up and over the cold front. The rising air carries water vapor through the dew point, which then condenses, first creating clouds and then rain. An advancing warm front is exactly the same, only in reverse — it pushes up and over cold air and squeezes it ahead of the front, creating almost the same wedge in cross section, although, as we’ll see, with a difference. The low-pressure cell sits at the center of the action, mediating the contact zone between warm and cool.
To extend Bjerknes’s battle metaphor, cold fronts might be called blitzkrieg attacks, and warm fronts fifth columns. Cold fronts have a steeper wedge and move faster than warm fronts, two reasons why their arrival is more abrupt. A cold front is always dramatic, creating thunderstorms in the summer and rain and snow in the winter. Warm fronts aren’t as aggressive; they are more gradual, and their approach is easy to read. Because they move slowly and have longer, more tapered wedge profiles, sometimes hundreds of miles long, it’s easier to predict a warm front’s arrival. The sequence of clouds that ride the approach of the warm front begin with high cirrus that gradually transition into altostratus, which are then replaced by nimbostratus as the clouds lower and thicken over the retreating wedge of cold air. Rain or snow often accompanies the passage of a front because in both cases — either an advancing warm front or an advancing cold front — air rises, cools and creates adiabatic precipitation. The only fly in the ointment, at least in terms of accurate and timely meteorological forecasting, were those overly complex, time-consuming mathematical equations of Bjerknes’s, which rendered practical meteorological calculations all but impossible.
But there was a new player about to arrive on the scene. Lewis Fry Richardson was another in the series of British Quaker polymath scientists. (The list of English scientists who were also Quakers is astonishing: the physicist Roger Penrose; his father, Lionel Penrose, the great geneticist; Luke Howard, who we’ve already met; Arthur Stanley Eddington, the world-renowned physicist; crystallographer Kathleen Lonsdale and Stephen Hawking’s collaborator George Ellis, just to name a few.)
Richardson was familiar with Bjerknes’s theories when, in 1913, he became director of a research laboratory for the United Kingdom’s Meteorological Office. As a Quaker, he was a conscientious objector to Britain’s involvement in the First World War, but he volunteered for the ambulance corps in 1916. (This was the same year that Cleveland Abbe was laid to rest, along with a copy of his beloved Smellie’s The Philosophy of Natural History.) On furloughs between battles, Richardson worked on atmospheric equations as he sat on a bale of hay in the ambulance corps quarters. He took Bjerknes’s calculations and revised them, replacing the finely graded analogs of calculus with measurements that sampled time in regular discrete portions, the way a strobe light breaks up movement into a series of stills. Each of these “moments” were approximations of change, but together, in sequence, they created accurate patterns. It was almost digital.
Now, he could do something no one had ever attempted: predict the weather mathematically. To do this, he needed a lot of meteorological information on a large landmass, and as it turned out the only weather charts detailed enough for his experiment were from the past. Here is an instance of “standing on the shoulders of giants,” because Richardson went back to Bjerknes’s very detailed weather charts for central Europe, 7 a.m., May 20, 1910.
He divided the map of the atmosphere into 25 modules, each measuring 125 square miles. These modules were further subdivided into five vertical layers of air. He plugged in all the variables and then ran the numbers that would, if his equations were correct, predict the state of the weather at 1 p.m., six hours later. It went disasterously wrong. He expected that the barometric pressure would be at 30.9 inches in six hours, but instead it remained “nearly steady.” The defeat must have been stunning, but he went on to publish all his findings, including the failed forecast, in his 1922 publication Weather Prediction by Numerical Process.
Many decades later, his experiment was vindicated. Apparently the problem wasn’t his math; it was the initial data, as Peter Lynch of the Irish Meteorological Service pointed out in 2006. Lynch repeated Richardson’s experiment with the original data “initialized,” as it would have been today, and the forecast became totally accurate. Thanks to Lynch, almost a century later, central Europe finally got an accurate forecast for the afternoon of May 20, 1910. Richardson had been afflicted with “scientific prematurity,” a term coined by the fractal mathematician Benoit Mandelbrot years later.
And afflicted he was indeed. His theories were marginalized for decades after his 1922 publication, his self-confessed forecast failure having done little to promote scientific interest in his extraordinary set of forecasting equations. Despite the fact that he had simplified Bjerknes’s math, his theorems still required an inordinate amount of calculations, and in the pre-computer era, it just wasn’t feasible. “Perhaps some day in the dim future it will be possible to advance the computations faster than the weather adances,” he wrote. “But that is a dream.”
And yet his dream did come true, and probably sooner than he thought it would, thanks to John von Neumann.
Born into a Jewish Hungarian aristocratic family in 1903, John von Neumann was a numerical whiz kid. By age eight, he was proficient in differential calculus, and by 19 he had published two major mathematical papers. He accepted a professorship at the University of Berlin at the unheard of age of 23. His intellectual output was astonishing: on average he published a major mathematical paper a month. But being Jewish in the darkening climate of Europe held little future for him, and he knew it. When he was offered a professorship at Princeton in 1931, years before the scientific exodus of Jewish physicists from Europe, he took it.
By the mid-1930s, von Neumann had a reputation for tackling immense scientific and mathematical enigmas and became the go-to guy for technical problem-solving. He even rescued the first atomic bomb test from failure. The Manhattan Project was a rushed affair, and the bomb slated for the famous Trinity test had a subcritical mass of plutonium and was in danger of being a dud. Von Neuman used shaped charges, whose exacting contours he had mathematically determined, to implode the plutonium symmetrically to produce a successful chain reaction.
(Von Neumann’s agile mind seemed to require stimulating environments. In the decades after the war, while working at the Princeton enclave, he often played German marching music very loudly as he worked on his theorems. So loudly, in fact, that a neighbor of his, a certain A. Einstein, complained that it was interfering with his concentration.)
Prior to his work on the atomic bomb, von Neumann was interested in hydrodynamical turbulence (the random eddies and currents within liquids) and the nonlinear equations that could describe it. These equations were essential to the ultimate mathematical descripton of the atmosphere, and von Neumann realized that some sort of electronic calculating device would be needed to undertake the vast computations necessary to solve the equations. So in 1946 at Princeton, he masterminded the construction of the world’s first truly progammable electronic computer, the electronic numerical integrator and computer (ENIAC). Using the ENIAC, with the help of another genius mathematician by the name of Jule Charney, von Neumann programmed the first computer-generated forecasts in 1950. They were completely accurate. Charney sent the results to Richardson, then 69, in England. It must have been a wonderful moment for Richardson. His theories were vindicated, and today Richardson’s numerical technique is the gold standard of weather forecasting.
A chapter in Richardson’s 1922 book, Weather Prediction by Numerical Process, was devoted to turbulence, one of the most difficult ph
enomena to model mathematically. He acknowledged this complexity with a whimsical verse: “Big whirls have little whirls that feed on their velocity; and little whirls have lesser whirls, and so on, to viscosity . . .” If ever there was a literary equivalent to the Mandelbrot set, the famous fractal equation that creates a self-referential world of infinitely repeating patterns within patterns, then this was it. Richardson had intuited the forthcoming science of chaos and with it, a monumental challenge to forecasting.
The Butterfly Effect
“The weather is always doing something there; always attending strictly to business; always getting up new designs and trying them on people to see how they will go. But it gets through more business in Spring than in any other season. In the Spring I have counted one hundred and thirty-six different kinds of weather inside of twenty-four hours.”
Mark Twain on New England weather
Edward Lorenz (1917–2008) had always been a weather buff. The changeable New England weather was sure to inspire a young meteorologist like Edward, and as a boy growing up in West Hartford, Connecticut, he built a small weather station in his parents’ backyard, not unlike Luke Howard’s some 50 years earlier. Its centerpiece was a special thermometer that kept an automatic record of the daily high and low temperatures with little sliding markers. He’d check the temperatures twice daily and record the numbers in a notebook. He also had a passion for math, although his twin interests seemed like two solitudes. He could measure the sensation of warmth and cold and translate those sensibilities into averages and means, but that’s as far as it went. They weren’t part of elegant, mathematical equations.
His mind craved logical challenges. On weekends and weeknights, he spent hours poring over problems in mathematical puzzle books, sometimes enlisting his father’s help. As Lorenz got older, he began to lean more toward mathematics. In fact, after graduating from Dartmouth College in 1938, he went on to get a masters of mathematics from Harvard. But then the Second World War intervened.
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