  We present data from high-resolution numerical simulations of the
Navier-Stokes-$\alpha$ and the Leray-$\alpha$ models for two-dimensional
turbulence. It was shown previously (Lunasin et al., J. Turbulence, 8, (2007),
751-778), that for wavenumbers $k$ such that $k\alpha\gg 1$, the energy
spectrum of the smoothed velocity field for the two-dimensional
Navier-Stokes-$\alpha$ (NS-$\alpha$) model scales as $k^{-7}$. This result is
in agreement with the scaling deduced by dimensional analysis of the flux of
the conserved enstrophy using its characteristic time scale. We therefore
hypothesize that the spectral scaling of any $\alpha$-model in the sub-$\alpha$
spatial scales must depend only on the characteristic time scale and dynamics
of the dominant cascading quantity in that regime of scales. The data presented
here, from simulations of the two-dimensional Leray-$\alpha$ model, confirm our
hypothesis. We show that for $k\alpha\gg 1$, the energy spectrum for the
two-dimensional Leray-$\alpha$ scales as $k^{-5}$, as expected by the
characteristic time scale for the flux of the conserved enstrophy of the
Leray-$\alpha$ model. These results lead to our conclusion that the dominant
directly cascading quantity of the model equations must determine the scaling
of the energy spectrum.
