Scaling properties of numerical two-dimensional turbulence.

Abstract

We consider two-dimensional (2D) incompressible turbulent flow in a statistically steady state in which energy and enstrophy inputs, localized around a forcing mode, are compensated by a linear frictional force at large scales and viscous dissipation at small scales. The scaling properties of the energy and enstrophy established cascades are studied via the velocity structure functions. The extended self-similarity is used to obtain a better estimate of the scaling exponents of the structure functions at any order. Three distinct scaling regimes are observed. At large scales, corresponding to the inverse cascade of energy, the scaling exponents are well defined and similar to those observed in 3D isotropic turbulence, with large deviations from the Kolmogorov 1941 scaling (strong intermittency). At intermediate scales, where coherent structures dominate, no scaling is present. At small scales, corresponding to the direct enstrophy cascade, a second scaling regime is obtained, with almost nonanomalous scaling exponents (weak intermittency). The intermittency obtained in the two scaling regimes is found to be consistent with the hierarchical intermittency model of turbulence of She and L\'ev\^eque [Phys. Rev Lett. 72, 336 (1994)], developed in the context of 3D turbulence. The model is characterized by two main parameters, \ensuremath{\Delta} and \ensuremath{\beta}, describing respectively, the smallest dissipative scales and the degree of intermittency of the energy transfers. These parameters are measured in the two regimes. In the inverse cascade, \ensuremath{\beta}=0.7 and \ensuremath{\Delta}=0.47, close to the values observed in 3D turbulence (\ensuremath{\beta}=\ensuremath{\Delta}=2/3). In the direct enstrophy cascade, \ensuremath{\beta} is close to 1, and \ensuremath{\Delta} close to 0, which explains the weak intermittency observed.