  We show, by using direct numerical simulations and theory, how, by increasing
the order of dissipativity ($\alpha$) in equations of hydrodynamics, there is a
transition from a dissipative to a conservative system. This remarkable result,
already conjectured for the asymptotic case $\alpha \to \infty$ [U. Frisch et
al., Phys. Rev. Lett. {\bf 101}, 144501 (2008)], is now shown to be true for
any large, but finite, value of $\alpha$ greater than a crossover value
$\alpha_{\rm crossover}$. We thus provide a self-consistent picture of how
dissipative systems, under certain conditions, start behaving like conservative
systems and hence elucidate the subtle connection between equilibrium
statistical mechanics and out-of-equilibrium turbulent flows.
