Exact resummations in the theory of hydrodynamic turbulence. III. Scenarios for anomalous scaling and intermittency.

Abstract

Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws ${\mathit{S}}_{\mathit{n}}$(R)\ensuremath{\sim}${\mathit{R}}^{{\mathrm{\ensuremath{\zeta}}}_{\mathit{n}}}$, and the correlation of energy dissipation ${\mathit{K}}_{\mathrm{\ensuremath{\epsilon}}\mathrm{\ensuremath{\epsilon}}}$(R)\ensuremath{\sim}${\mathit{R}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}}$. The goal is to understand from first principles what is the mechanism that is responsible for changing the exponents ${\mathrm{\ensuremath{\zeta}}}_{\mathit{n}}$ and \ensuremath{\mu} from their classical Kolmogorov values. In paper II of this series [V. S. L'vov and I. Procaccia, Phys. Rev. E 52, 3858 (1995)] it was shown that the existence of an ultraviolet scale (the dissipation scale \ensuremath{\eta}) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted \ensuremath{\Delta}. The exact resummation of ladder diagrams resulted in a ``bridging relation,'' which determined \ensuremath{\Delta} in terms of ${\mathrm{\ensuremath{\zeta}}}_{2}$: \ensuremath{\Delta}=2-${\mathrm{\ensuremath{\zeta}}}_{2}$. In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e., ${\mathrm{\ensuremath{\zeta}}}_{\mathit{n}}$ not linear in n) with the renormalization scale being the infrared outer scale of turbulence L. It is shown that deviations from the classical Kolmogorov 1941 theory scaling of ${\mathit{S}}_{\mathit{n}}$(R) (${\mathrm{\ensuremath{\zeta}}}_{\mathit{n}}$\ensuremath{\ne}n/3) must appear if the correlation of dissipation is mixing (i.e., \ensuremath{\mu}\ensuremath{\gtrsim}0). We suggest possible scenarios for multiscaling, and discuss the implication of these scenarios on the values of the scaling exponents ${\mathrm{\ensuremath{\zeta}}}_{\mathit{n}}$ and their ``bridge'' with \ensuremath{\mu}. \textcopyright{} 1996 The American Physical Society.