Book Review Invisible in the Storm: The Role of Mathematics in Understanding Weather Reviewed by Peter Lynch Invisible in the Storm: The Role of Mathematics specifies how the circulation can in Understanding Weather change when baroclinicity is pres- Ian Roulstone and John Norbury ent. It enables us to calculate how Princeton University Press, 2013 vortices in the atmosphere and US$35.00, 346 pages oceans behave, giving a holistic, ISBN-13: 978-0691152721 but quantitative, description. Bjerknes’s circulation theorem ini- The development of mathematical models for tiated the study of geophysical weather prediction is one of the great scientific tri- fluid dynamics. umphs of the twentieth century. Accurate weather The basic mechanical and physi- forecasts are now available routinely, and quality cal laws governing the atmosphere has improved to the point where occasional fore- were in place by 1900. Bjerknes cast failures evoke surprise and strong reaction developed a vision of how weather among users. The story of how this came about is forecasting could be put on solid of great intrinsic interest. theoretical foundations and drew General readers, having no specialized math- up what amounted to a manifesto ematical knowledge beyond school level, will for scientific prediction. He considered how pre- warmly welcome an accessible description of how cise, long-range predictions of astronomical events weather forecasting and climate prediction are were possible and tried employing a similarly sys- done. There is huge interest in weather forecast- tematic approach in meteorology. He recognized ing and in climate change, as well as a demand the intractability of the governing equations and for a well-written account of these subjects. In realized that analytical solution of them was im- this book, the central ideas behind modeling and possible. But even an approximate solution, which the basic procedures undertaken in simulating the would have required many months to carry out, atmosphere are conveyed without resorting to any might eventually lead to more practical methods. difficult mathematics. The authors give an excellent account of devel- The first chapter gives a good picture of the opments over the first two decades of the twen- scientific background around 1900. It opens with tieth century. The discovery by Henri Poincaré of an account of the circulation theorem derived by sensitive dependence on initial conditions arose the Norwegian meteorologist Vilhelm Bjerknes. in the context of studying the three-body prob- This theorem follows from work of Helmholtz and lem but had much wider implications. It meant Kelvin but makes allowance for a crucial property that unavoidable errors in the specification of the of the atmosphere: that pressure and density initial state of the atmosphere would grow over surfaces do not usually coincide. This is what time, ultimately rendering the forecast useless. It is meant by the term baroclinicity. The theorem imposed an inherent limit on prediction of future Peter Lynch is professor of meteorology in the School of weather. A central aim of the book is to explain Mathemtical Sciences at University College Dublin and how rational prediction of weather is possible at he blogs at thatsmath.com. His email address is peter. all in the presence of chaos. [email protected]. Predicting the weather is vastly more complex DOI: http://dx.doi.org/10.1090/noti1036 than predicting the return of a comet. The authors SEPTEMBER 2013 NOTICES OF THE AMS 1051 show how the problem can be reduced to the In a chapter entitled “When the Wind Blows solution of a system of seven equations (coupled the Wind”, the authors attempt to convey the nonlinear partial differential equations) in seven ideas of nonlinearity, a phenomenon that “makes variables: pressure, temperature, density, humid- forecasting so difficult and weather so interest- ity, and three components of the wind. The math- ing.” This is a key idea, and I feel that the attempt ematical details are very sketchy, even in the “Tech can at best be described as a qualified success. Boxes” (boxes separate from the running text, with Some of the discussion is lacking in clarity and additional technical details), but the overall ideas may not provide readers with the desired level of are well conveyed. understanding. Most of the key meteorologists are recognized Part 2 opens with a chapter in which the bril- in the narrative. In particular, William Ferrel’s liant work of Carl Gustaf Rossby is described. work in formulating the equations on a rotating Rossby had the capacity to reduce a problem to Earth is given due prominence. But a description its essentials and to devise conceptual models of the important work of the scientists working in that elucidated the mechanism of atmospheric Vienna, specifically Max Margules and Felix Exner, phenomena, unencumbered by extraneous de- is omitted. Margules anticipated the problems tails. In a landmark paper published in 1939, he that would arise if the continuity equation were explained the basic dynamics of the large wavelike used for prediction, and Exner carried out several disturbances in the atmosphere by using a simple numerical forecasts using a highly simplified set model based on conservation of absolute vorticity. of equations. Linearizing this, he produced an expression for the The method of solving a complicated system phase speed of the waves, thereby explaining the of equations by reducing them to a manageable, mechanism of propagation and also providing a algebraic form is given good treatment. The con- means of predicting the propagation of wave dis- turbances. Some mathematical details of Rossby’s sequences of discretization are described by a nice model are presented in a Tech Box. analogy with pixelation of a painting, Constable’s Rossby’s model assumed a wave disturbance of Hay Wain. Unfortunately, the indifferent quality a fluid with uniform depth. When the fluid depth of the illustration on page 34 detracts somewhat varies, the conserved quantity is the ratio of abso- from the presentation. Overall, the diagrams in the lute vorticity to depth, the potential vorticity (PV) book are good, but some of the photographs are in its simplest form. This can be used to explain poorly reproduced. the effect of a mountain chain on the flow. The Bjerknes’s original idea was to use mathematical authors describe a flow over the Andes but do equations to forecast the weather. However, the not mention that in the Southern Hemisphere the complexity of this task convinced him and his team configuration of troughs and ridges is reversed. to follow a more empirical line, which turned out Thus, their account and their Figure 5.12 are likely to be enormously fruitful. The conceptual models to be a source of confusion to readers. of warm and cold fronts and of the life cycles of Rossby’s formula was of limited value in practi- frontal depressions that emerged from the Bergen cal forecasting. The atmosphere is complex, and School dominated synoptic meteorology for most its behavior cannot be reduced to a simple travel- of the twentieth century and were of great practical ling wave on a uniform background flow. A much benefit to humankind. More quantitative methods more complete understanding of how midlatitude had to await scientific and technical developments disturbances develop from small beginnings was in mid-century. provided by Jule Charney when he showed that During the First World War, an extraordinary they grow through baroclinic instability. Charney’s numerical experiment was carried out by Lewis Fry work is rightly given prominence in the book. Hav- Richardson, who, using the best data set he could ing explained the mechanism of wave growth, he find, calculated changes in pressure and wind went on to produce a system of equations that using the basic equations of motion. However, he could be used for practical numerical prediction was unaware that errors in the initial data could while avoiding the problems encountered by Rich- completely spoil the forecast, and his results were ardson. Charney then led the team that carried completely unrealistic. Richardson’s attempt at out the first successful prediction on the ENIAC practical forecasting by numbers was so unsuc- computer in 1950. This was the beginning of real cessful and so impractical at the time that it had numerical weather prediction. The story is very a deterrent effect on other meteorologists. But of well told in the book. course, Richardson’s approach was ultimately the The limitations on prediction imposed by the right one, and the causes of the error in his forecast chaotic nature of the atmosphere are then dis- are now well understood and quite avoidable. The cussed. The work of Edward Lorenz was crucial authors provide a clear description of what Rich- to our understanding of what can and cannot be ardson achieved and of the remarkable prescience achieved. With our growing appreciation of the of his work. inherent limitations on weather forecasting, the 1052 NOTICES OF THE AMS VOLUME 60, NUMBER 8 emphasis has shifted from deterministic to proba- bilistic prediction, and the method of ensemble forecasting is now at the forefront of operational practice. All this is well described, including the application of probability forecasts to loss/cost Institute for Computational and Experimental Research in Mathematics models that can be used for rational decision mak- ing with great economic benefits. APPLY TO BECOME AN ICERM POSTDOC The authors have an interest in symplectic ge- ometry, the mathematical framework underlying ICERM’s postdoctoral program brings early career mathematicians to the institute in order to support Hamiltonian mechanics. They include a descrip- and expand their research and to create lasting career tion of the main ideas of symplectic geometry, but collaborations and connections. this is as likely to mystify as to inspire readers, especially as the link with PV is not clarified.