The enstrophy cascade in forced two-dimensional turbulence

Abstract

We carry out direct numerical simulations of two-dimensional turbulence with forcing at different wavenumbers and resolutions up to 32768 2 grid points. In the absence of large-scale drag, a state is reached where enstrophy is quasi-stationary while energy is growing. In the enstrophy cascade range the energy spectrum has the form E ( k ) = ε ω 2/3 k −3 , without any logarithmic correction, where ε ω is the enstrophy dissipation and is of the order of unity. However, is varying between different simulations and is thus not a perfect constant. This variation can be understood as a consequence of large-scale dissipation intermittency, following the argument by Landau (Landau & Lifshitz, Fluid Mechanics , 1959, Pergamon). In the presence of a large-scale drag, we obtain a slightly steeper spectrum. When forcing is applied at a scale which is somewhat smaller than the computational domain, no vortices are formed, and the statistics remain close to Gaussian in the enstrophy cascade range. When forcing is applied at a smaller scale, long-lived coherent vortices form at larger scales than the forcing scale, and intermittency measures become very large at all scales, including the scales of the enstrophy cascade. We conclude that the enstrophy cascade with a k −3 -spectrum is a robust feature of the two-dimensional Navier–Stokes equations. However, there is a complete lack of universality of higher-order statistics of vorticity increments in the enstrophy cascade range.