Energy spectra of steady two-dimensional turbulent flows

The power spectrum is measured in direct numerical simulations of the two-dimensional Navier-Stokes equation and other two-dimensional flows with white-in-time forcing at large scales. For the Navier-Stokes equation the energy spectrum in the inertial range approaches k^-3 with increasing Reynolds number, with possible logarithmic corrections. A family of two-dimensional flows, including the surface quasigeostrophic equation, allows us to vary the locality of the “enstrophy” transfer, where enstrophy is the mean square of the convected quantity. Dimensional analysis based on the enstrophy dissipation correctly predicts the energy spectrum, whenever the enstrophy transfer can be assumed to be spectrally local. Otherwise, the enstrophy spectrum is steeper than would be expected on the basis of local transfer. In this case the data suggest a k^-1 passive scalar spectrum.