Robustness of the inverse cascade in two-dimensional turbulence

We study quasisteady inverse cascades in unbounded and bounded two-dimensional turbulence driven by time-independent injection and dissipated by molecular viscosity. It is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy-range energy spectrum to be steeper than $k^{-5}$ (ruling out a direct cascade) unless $1-r\ll 1.$ A direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales $(1\mathrm{}-r\ll 1).$ These facts underlie the robustness of the Kolmogorov-Kraichnan $k^{-5/3}$ inverse cascade, which is readily observable in numerical simulations without an accompanying direct enstrophy cascade. We numerically demonstrate an instance where the $k^{-5/3}$ inverse-cascading range is realizable with 79% of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region. As equilibrium is approached, the respective logarithmic slopes -α and -β of the ranges of wave numbers lower and higher than the forcing wave number satisfy α+β≈8. These results are consistent with recent theoretical predictions.