How to Extract a Spectrum from Hydrodynamic Equations

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The practical results gained from statistical theories of turbulence usually appear in the form of an inertial range energy spectrum $${\mathcal {E}}(k)\sim k^{-q}$$ and a cutoff wavenumber $$k_{c}$$ . For example, the values $$q=5/3$$ and $$\ell k_{c}\sim \mathrm {Re}^{3/4}$$ are intimately associated with Kolmogorov’s 1941 theory. To extract such spectral information from the Navier–Stokes equations, Doering and Gibbon (Phys. D 165, 163–175, 2020) introduced the idea of forming a set of dynamic wavenumbers $$\kappa _n(t)$$ from ratios of norms of solutions. The time averages of the $$\kappa _n(t)$$ can be interpreted as the 2nth moments of the energy spectrum. They found that $$1 < q \leqslant 8/3$$ , thereby confirming the earlier work of Sulem and Frisch (J. Fluid Mech. 72, 417–423, 1975) who showed that when spatial intermittency is included, no inertial range can exist in the limit of vanishing viscosity unless $$q \leqslant 8/3$$ . Since the $$\kappa _n(t)$$ 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.