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

L Biferale,M Sbragaglia,M Cencini,F Toschi,A S Lanotte

doi:10.1088/1367-2630/6/1/037

Abstract

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. 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, and we present new quantitative results for shell models of turbulence. We 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.

Highlights

We review a recent investigation of 3d Navier– Stokes turbulence, and we present new quantitative results for shell models of turbulence
Our results confirm the universality scenario originating from the zero-mode picture which predicts two distinct regimes. (i) Forcing-dominated regime: the scaling of low-order structure functions is non-anomalous, with exponents dimensionally related to the forcing spectrum; for the higherorder moments, scaling is anomalous and dominated by the zero modes. (ii) Forcing-subleading regime: the dimensional scaling related to the balance with the forcing is subleading, at any order, with respect to the anomalous one, similar to the case of a standard large-scale injection
We have discussed the problem of small-scale fluctuations in turbulent systems stirred at all scales by a power-law forcing

Summary

Linear dynamics: passive scalar transport

The anomalous scaling exponents have values in agreement with those obtained for a large-scale forcing, the interplay of different power laws, when the the forcing is as (4), makes the identification of the exponents very difficult even at considerably high resolution Due to these difficulties in measuring the exponents, we looked directly at the PDF of scalar increments P(δrθ). All differences associated with the two regimes should disappear in the high-order statistics, i.e. for p > pc when anomalous scaling exponents imposed by the zero modes should show up irrespective of the forcing In testing this point, the presence of saturation for large p comes into play; it entails that, for large excursions, 104 102 r-ζ∞ P(δrθ) θrms. The above results provide support to the validity of the zero-mode picture beyond the boundaries of the Kraichnan model and in agreement with the Lagrangian investigations [17]

Nonlinear systems

The 3d Navier–Stokes problem

Shell models for turbulence

Conclusions