Anomalous scaling and universality in turbulent systems with power-law forcing

The problem of the interplay between normal and anomalous scaling in turbulent systems stirred by a random forcing with a power law spectrum is addressed [1]. By means of direct numerical simulations, we consider both linear and nonlinear systems. As for the linear case, we study passive scalars advected by a $2d$ velocity field in the inverse cascade regime. For the nonlinear case, we review a recent investigation of $3d$ Navier-Stokes turbulence [2], and we present new quantitative results for shell models of turbulence.\\ We show [3] that at varying the forcing spectrum slope, small-scale turbulent fluctuations change from a {\it forcing independent} to a {\it forcing dominated} statistics. If $y_c$ is the critical value of the spectrum slope separating the two regimes, we find that when the statistics is forcing dominated, for $y<y_c$, small scale turbulent statistics behaves accordingly to the dimensional scaling, i.e., intermittency is vanishingly small. On the other hand, for $y>y_c$, we find the same anomalous scaling measured in flows forced at large scales.\\ We also show that to get firm statements it is necessary to reach considerably high resolutions due to the presence of unavoidable subleading terms affecting all correlation functions. All findings support universality of anomalous scaling for the small scale fluctuations.