Abstract

Practical results gained from statistical theories of turbulence usually appear in the form of an inertial-range power-law energy spectrum and a cut-off wave-number. To extract such spectral information from the three-dimensional Navier–Stokes equations, Doering and Gibbon (2002) introduced the idea of forming a set of dynamic wave-numbers Kn(t) from ratios of norms of solutions. The time averages of Kn(t) can be interpreted as the moments of the energy spectrum. Since the κn(t) are based on Navier–Stokes weak solutions, the mathematical bounds for Kn(t) connect empirical predictions of the energy spectrum with the 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.