%0 Journal Article %T How to Extract a Spectrum from Hydrodynamic Equations %+ Imperial College London %+ Université Côte d'Azur (UCA) %A Gibbon, John %A Vincenzi, Dario %< avec comité de lecture %@ 0938-8974 %J Journal of Nonlinear Science %I Springer Verlag %V 32 %N 6 %P 87 %8 2022-09-16 %D 2022 %R 10.1007/s00332-022-09830-9 %Z Nonlinear Sciences [physics] %Z Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn]Journal articles %X Practical results gained from statistical theories of turbulence usually appear in the form of an inertial range energy spectrum E(k) ∼ k^{−q} and a cutoff wavenumber k_c. For example, the values q = 5/3 and k_c ∼ Re 3/4 are intimately associated with Kolmogorov's 1941 theory. To extract such spectral information from the Navier-Stokes equations, Doering and Gibbon [2002] introduced the idea of forming a set of dynamic wave-numbers κ_n (t) from ratios of norms of solutions. The time averages of the κ_n (t) can be interpreted as the 2nth-moments of the energy spectrum. They found that 1 < q 8/3, thereby confirming the earlier work of Sulem and Frisch [1975] who showed that when spatial intermittency is included, no inertial range can exist in the limit of vanishing viscosity unless q 8/3. Since the κ_n (t) are based on Navier-Stokes weak solutions, this approach connects empirical predictions of the energy spectrum with the mathematical analysis of the Navier-Stokes equations. This method is developed to show how it can be applied to many hydrodynamic models such as the two dimensional Navier-Stokes equations (in both the direct-and inverse-cascade regimes), the forced Burgers equation and shell models. %G English %2 https://cnrs.hal.science/hal-03875524/document %2 https://cnrs.hal.science/hal-03875524/file/bdspec7.pdf %L hal-03875524 %U https://cnrs.hal.science/hal-03875524 %~ CNRS %~ UNIV-COTEDAZUR