Invisible in the Storm is a welcome and authoritative account of the 20^{th} century contributions of mathematically sophisticated meteorologists such as Vilhelm Berknes (1862–1951), CarlGustav Rossby (1898–1957), Jule Charney (1917–1981), and Ed Lorenz (1917–2008). The main storyline begins with Bjerknes’s paper, “Das Problem der Wettervorhersage betrachtet vom Standpunkte der Mechanik und der Physik” published in 1904 in the Meteorologische Zeitschrift where he set out two criteria for preparing an objective forecast of the future state of the weather using the equations of hydrodynamics and thermodynamics:

A sufficiently accurate knowledge of the state of the atmosphere at the initial time; and

A sufficiently accurate knowledge of the laws according to which one state of the atmosphere develops from the other.
Just the year before, Henri Poincaré (1854–1912) had undermined the notion of a perfectly precise observation of nature in his essay “Science and Method,” by noting that “small differences in the initial conditions produce very great ones in the final phenomena.” This is now recognized as one of the cannons of chaos theory.
Nevertheless, Bjerknes’ program, played out with realtime weather data of “sufficient accuracy,” attracted considerable international attention and agreement. His model of atmospheric change and prediction, however, was limited by the availability only of surface observations in Norway during World War I. By 1920, however, new information on the vertical structure of the atmosphere allowed his colleagues Jacob Bjerknes (1897–1975) and Tor Bergeron (1891–1977) to extend the model into three dimensions and incorporate two new features: air masses and fronts. As Bergeron later pointed out, the model required an abundance of observations and objective smoothing techniques before it could yield usable predictions. Issuing a daily weather forecast in real time was a daunting and exhausting task, requiring graphical methods of charting and analyzing massive amounts of data.
Rossby, both a mathematicallyastute theoretician and a proficient organizer, studied with Bjerknes before moving to the United States in 1926, in part to promote the Norwegian methods of analysis and forecasting, but with a much bigger vision of his own: to create a mathematical model of the general circulation of the ocean and atmosphere. His list of accomplishments is nothing short of amazing, including establishing the first weather service for a commercial airline, founding the department of meteorology at MIT and opening up a reciprocal program on dynamic oceanography at Woods Hole, training thousands of weather cadets for the war effort, establishing the department of geophysics at the University of Chicago, founding the Journal of Meteorology, and, between 1947 and his death, starting the meteorology program at the University of Stockholm, preparing the first effective computerized weather forecast there, and establishing the noted environmental science journal Tellus. Rossby identified planetary long waves in the westerlies (called Rossby waves) that support and steer the movements of air masses at the surface. His “Rossby equation,” published in 1939, made it possible to model and predict the movement of these upperair waves. His advocacy of the term atmospheric science reflected his view that meteorology needed to expand its horizons as a major branch of geophysics.
Charney, a Rossby protégé, studied mathematics and meteorology at UCLA and developed the theory of baroclinic “long waves” in the upper atmosphere. He developed a set of equations for calculating the largescale motions of these waves known as the “quasigeostrophic approximation.” From 1948 to 1956, Charney was a member of the Institute for Advanced Study Meteorology Project in Princeton. He served as director of the theoretical meteorology group, which constructed a successful mathematical model of the atmosphere and demonstrated that numerical weather prediction was both feasible and practicable. In 1954, Charney helped establish a numerical weather prediction unit within the U.S. Weather Bureau. From 1968 to 1971, Charney was chair of the U.S. Committee for the Global Atmospheric Research Programme (GARP), a decadelong international experiment to measure the global circulation of the atmosphere, model its behavior, and improve predictions of its future state. He was instrumental in articulating the global goals and vision of GARP, and consistently argued that scientists needed to view the atmosphere as a single, global system.
Lorenz, like Poincaré so many years before, also made fundamental contributions to understanding the chaotic behavior of the atmosphere — in his case, at the dawn of the era of computer modeling. Lorenz noted that the nonlinear nature of the equations used in numerical weather prediction would lead to multiple solutions from essentially identical initial states. Small perturbations caused large departures. By “chaos,” Lorenz meant something very precise: the behavior and final state of a dynamical system described by a set of nonlinear equations is very sensitive to its initial state. In computer weather modeling this means there is no way to prepare the initial state of the system with such care as to allow long term predictions. A famous aphorism is attributed to Lorenz, but in fact was not authored by him: “Can a butterfly flapping its wings in Brazil cause a Tornado in Texas?” Due to viscosity, certainly not, or we would be in trouble every time someone sneezed or a door slammed, yet meteorologists, sensitized to the limits placed upon their predictions by chaos theory, began talking about what they called the “Butterfly Effect.”
Invisible in the Storm focuses on the mathematical history of meteorology using an engaging and accessible narrative style, with only a handful of equations scattered throughout the book. One equation, noticeably missing, is the Rossby Equation:
There are numerous headshots of famous scientists, an abundance of illustrative diagrams, and some photographs representing weather phenomena.
Atmospheric scientists and mathematicians interested in complexity and forecasting will enjoy this book. It might be used productively in topical mathematics courses and intermediate and advanced courses on meteorology. It introduces a number of topics such as symplectic geometry and climate modeling that would require an instructor to bring additional insights and resources into a pedagogical situation. There are no notes and only a short list of additional readings. Historians of science may look askance at the lack of proper source documentation, which is almost “invisible in the book.”
The text concludes with subversive messages that mathematicians and modelers should take as challenges. A diagram showing a projected 3 ^{o}C climate warming in future decades has a caption that reads in part: “These predictions are highly simplistic feedback models that are the subject of much current research and debate.” Another example comes from the authors’ reproduction of René Magritte’s surrealist painting, La condition humaine, with the caption: “The point is that models are not the same as reality but should capture what interests us.” A third example (there are more) depicts a computerized chessboard, noting: “Computers model weather evolution using lists of positions and weather states, which can lose sight of the original identity of the air masses and what they are doing.”
Clearly, this book is informative and inspirational, leaving plenty of room for innovations by future generations of mathematicians and modelers.
James Rodger Fleming is Professor and Chair of the program in Science, Technolog, and Society at Colby College in Waterville, ME.