Until very recently I knew precious little about applications of mathematics to weather forecasting. I had a notion that mathematics is bound to be involved, that there are difficulties inherent in processing vast amounts of data and sensitivity of solutions of multivariable differential equations due to their non-linearity, and that the current level of forecasting would not be possible without the tremendous technological progress achieved during the second half of the past century.
The new book Invisible in the Storm: The Role of Mathematics in Understanding Weather by Ian Roulstone and John Norbury (Princeton University Press, 2013) was an eye opener.
It's not only that the book shows how much modern weather forecasting depends on mathematics, the book outlines the many ways in which mathematics is applicable. Equations of motion, heat, fluids, vorticity, gravitational waves, laws of conservation and laws of change, numerical computations, theory of stability and chaos, and probability theory, are all being used in forecasting, often as competing utilities.
Indeed, mathematics employed in modern weather forecasting is formidable; the book lucidly describes its intuitive essentials as a thread in the story of the development of meteorology as a science in the past century. Formal equations, with two accidental exceptions (one pages 130 an 234), have been collected in clearly marked (and not too numerous) "tech boxes". The authors do an admirable job of translating if not the equations per se then at least their origins, significance, and utilization into plain language.
Not that it was always clear how - and even what - to predict, and which mathematics to use to this end.
Evolution of what is possible and what is not led eventually to a culture shift: Instead of trying to say exactly what the weather will be in the future, we instead try to predict what it will most probably be.
But naturally, mathematics was not evolving all by itself. The authors excel in presenting establishment of the science of meteorology as a human endeavor. The history of meteorology is rich in perseverance, sacrifice, enthusiasm, ingenuity, useful missteps, multinational collaboration ... and plain hard work. Authors' fluent recount makes the story all the more fascinating, even if math applications are only at the back of your mind. The book is a superior read.