Spectral scaling of the two-dimensional Navier-Stokes-$\alpha$ and Leray-$\alpha$ models

The NS-$\alpha$ model of turbulence is a mollification of the Navier-Stokes equations, such that the vorticity is advected by a velocity field that is smoothed over spatial scales of size smaller than $\alpha$. The spectral properties of the smoothed velocity field match those of Navier-Stokes turbulence for wavenumbers $k$ such that $k\alpha\ll 1$. For $k\alpha \gg 1$ it is not possible to predict the scaling of the energy spectrum {\it a priori} since the smoothed and unsmoothed velocities provide several possible characteristic timescales for the problem. The same holds true for the other $\alpha$-models of turbulence. We measure the $k\alpha \gg 1$ scaling of the energy spectra from high-resolution simulations in two-dimensions, in the limit as $\alpha \rightarrow \infty$, for two models: the Navier-Stokes-$\alpha$ model and the Leray-$\alpha$ model. The spectrum of the smoothed velocity field scales as $k^{-7}$ in the former and as $k^{-5}$ in the latter. These scalings correspond to the direct cascade of the conserved enstrophy in each case, the governing time scales given by $(k |v_k|)^{-1}$ and $(k\sqrt{( u_k, v_k)}\;)^{-1}$ respectively, where $u_k$ and $v_k$ are the fourier components of the filtered (smoothed) velocity field $u$ and unfiltered velocity field $v$.