A study of the Navier-Stokes-alpha model for two-dimensional turbulence

For wavenumbers k such that k * alpha > 1, corresponding to spatial scales smaller than alpha, there are three candidate power laws for the energy spectrum of the Navier-Stokes-alpha model, corresponding to three possible dynamical eddy turnover time scales in the model equations: one from the smoothed field, the second from the rough field and the third from a special combination of the two. Using two-dimensional turbulence as a test case, we measure the scaling of the spectra from high-resolution simulations of the Navier-Stokes-alpha model, in the limit as alpha goes to infinity. We show that the energy spectrum of the smoothed velocity field scales as k^{-7} in the direct enstrophy cascade regime, consistent with dynamics dominated by the time scale of the rough velocity field.