The modern understanding of wave turbulence began in 1929 when Rudolf Peierls published his thesis work on heat conduction in crystals. Subsequent developments in the 1960s centered on the study of surface waves, and eventually, Vladimir Zakharov and his coworkers developed the theory into a sharply honed tool to describe weakly interacting waves. Their work focused on deriving an equation for the nonlinear evolution of the wave spectrum. It resembles the Boltzmann equation—commonly associated with gaseous physics—which led to the name: wave kinetic equation.

Today wave turbulence theory is applied to a diverse range of systems, including internal gravity waves, which are important for the long-term dynamics of the ocean. The variety of questions that the theory can answer—made possible by comparing the kinetic equation to simulations—has led to new experimental designs and testable predictions. At the same time, mathematical progress has placed the formal derivation of the kinetic equation onto much firmer ground, decisively so for idealized systems, such as the nonlinear Schrödinger equation.

Until now, students and researchers who wanted to learn about the field had to choose between *Kolmogorov Spectra of Turbulence I: Wave Turbulence*, by Zakharov, Victor S. L’vov, and Gregory Falkovich and published in 1992, or *Wave Turbulence*, by Sergey Nazarenko and published in 2011*.* The latest option is the welcome addition of *Physics of Wave Turbulence* by Sébastien Galtier*.* The author has a long track record in the field, including original applications to plasma systems, magnetohydrodynamics, rotating and stratified waves, compressible waves, and the solar wind. It is clear from reading the book that the subject is close to the author’s heart.

For a newcomer to the field of wave turbulence, the formal derivation of the kinetic equation may appear austere and rigid. The process starts from the governing equations in a particular form—typically that of a canonical Hamiltonian partial differential equation system—and then follows a long sequence of partially convincing, partially mystical steps to finally arrive at the kinetic equation. Galtier’s book offers a different take by stressing how multiple time scales can be used in the derivation. Although the idea is not new, and the outcome of the derivation is the same, the fresh perspective should help new readers.

*Physics of Wave Turbulence* starts with a general introduction to turbulent systems and then chronicles the basic theory of hydrodynamic turbulence, or turbulence in fluid systems. It is mostly standard material, but Galtier adds fresh touches here and there, such as a novel derivation of the famous energy spectra found in turbulent systems—the first derivation of which was accomplished by Andrei Kolmogorov in 1941.

Spectral cascades, which describe the nonlinear flow of energy from large forcing scales to small dissipation scales, are introduced, and the dramatic differences between 2D and 3D spectral cascades are described. In a nutshell, in 3D the energy flows naturally to very small scales, but in 2D, a counterintuitive inverse cascade moves energy to larger scales. For subtle reasons, the large-scale turbulence in the atmosphere and in oceans behaves much like a peculiar 2D system, so inverse cascades are, in fact, crucially important in practice.

After presenting that material, Galtier introduces wave turbulence, which he discusses through a sequence of increasingly complex physical models. The models simulate capillary waves, which travel at the interface between two fluids, and the so-called inertial wave turbulence, which is hydrodynamic turbulence in rapidly rotating containers. Interestingly, some of those systems exhibit inverse cascades as well.

Further applications include Alfvén waves relevant to incompressible magnetohydrodynamic systems, compressible plasma waves, and, finally, gravitational waves; a chapter on possible scenarios of the primordial universe discusses gravitational wave applications. The numerous exercises, integrated in the main text with solutions provided at the end of the book, are a welcome feature.

Like its two siblings, and despite the large amount of mathematics contained in it, *Physics of Wave Turbulence* is true to its name, so certain specific questions that may vex mathematicians are not addressed. I don't think that it a weakness of the new book, which is an excellent addition to the textbook literature on the subject. It simply means that a useful mathematical treatment of wave turbulence remains to be written.

**Oliver Bühler** is a professor of mathematics and atmosphere ocean science at New York University’s Courant Institute of Mathematical Sciences. His research focuses on asymptotic and stochastic methods applied to geophysical fluid dynamics.