Reynolds-number dependency in homogeneous, stationary two-dimensional turbulence

Turbulent solutions of the Navier-Stokes equations for two-dimensional (2D) flow are a paradigm for the chaotic space-time patterns and equilibrium distributions of turbulent geophysical and astrophysical ``thin'' flows on large horizontal scales. Here we investigate how homogeneous, stationary 2D turbulence varies with the Reynolds number (Re) in stationary solutions with large-scale, random forcing and viscous diffusion, also including hypoviscous diffusion to limit the inverse energy cascade. This survey is made over the computationally feasible range in Re ≫ 1, approximately between 1.5 × 10³ and 5.6 × 10⁶. For increasing Re we witness the emergence of vorticity fine structure within the filaments and vortex cores. The energy spectrum shape approaches the forward-enstrophy inertial-range form k^-3 at large Re, and the velocity structure function is independent of Re. All other statistical measures investigated in this study exhibit power-law scaling with Re, including energy, enstrophy, dissipation rates, and the vorticity structure function. The scaling exponents depend on the forcing properties though their influences on large-scale coherent structures, whose particular distributions are non-universal. A striking result is the Re independence of the intermittency measures of the flow, in contrast with the known behavior for 3D homogeneous turbulence of asymptotically increasing intermittency. This is a consequence of the control of the tails of the distribution functions by large-scale coherent vortices. Our analysis allows extrapolation toward the asymptotic limit of Re -> ∞ , fundamental to geophysical and astrophysical regimes and their large-scale simulation models where turbulent transport and dissipation must be parameterized.