Wave turbulence: a solvable problem applied to the Navier–Stokes equations

Wave turbulence and eddy turbulence are the two regimes that we may encounter in nature. The attention of fluid mechanics being mainly focused on incompressible hydrodynamics, it is usually the second regime that is treated in books, whereas waves are often present in geophysics and astrophysics. In this review, I present the theory of wave turbulence which is free from the closure problem encountered in eddy turbulence. Basically, the wave amplitude is introduced in a multiple time scale method as a small parameter to derive the so-called kinetic equations from which exact results can be obtained (power-law spectra, direction of the cascade, Kolmogorov’s constant) and compared with the data. Two hydrodynamic applications are considered with capillary waves and inertial waves, the first leading to isotropic turbulence and the second to anisotropic turbulence.