Large-scale coherence and ‘‘anomalous scaling’’ of high-order moments of velocity differences in strong turbulence

The Hamiltonian formulation of hydrodynamics in Clebsch variables is used for construction of a statistical theory of turbulence. It is shown that the interaction of the random and large-scale coherent components of the Clebsch fields is responsible for generation of two energy spectra E(k)∝${\mathit{k}}^{\mathrm{-}7/3}$ and E(k)∝${\mathit{k}}^{\mathrm{-}2}$ at scales somewhat larger than those corresponding to the -5/3 inertial range. This interaction is also responsible for the experimentally observed Gaussian statistics of the velocity differences at large scales, and the nontrivial scaling behavior of their high-order moments for inertial-range values of the displacement r. The ‘‘anomalous scaling exponents’’ are derived and compared with experimental data.