Forcing regimes in the two-dimensional Navier-Stokes equations

Abstract

In the standard theoretical setting of body-forced turbulence, the forcing that sustains the flow is concentrated in a narrow range of length scales. However, in experiments of fractal-grid turbulence and in numerical simulations inspired by the renormalization group approach, more general forcing functions have been considered. These studies have shown that the phenomenology of turbulence is sensitive to the regularity of the forcing, which raises the wider question of the sensitivity of all Navier--Stokes mathematical estimates to the regularity of body forces. To answer this question, it is necessary to convert the traditional estimates based on the Grashof number, a dimensionless measure of the magnitude of the forcing, to a form dependent on the Reynolds number, the usual dimensionless number in experimental measurements and statistical theories of turbulence. To investigate these issues we consider the two-dimensional case and employ the full range of forcing regularity allowed by the theory of weak solutions to extend available estimates not only for the energy and enstrophy dissipation rates, but also for the dimension of the global attractor. What emerges is the existence of three distinct regimes as a function of the regularity of the forcing.