Cascade of energy in turbulent flows

A starting point for the conventional theory of turbulence 12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes 19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., 1,2,10,11,16,17,21], and 22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy 1,15]. In a forthcoming article 9] we will discuss conditions for the actual existence of the inertial range in dimension three.