Large deviations, singularity, and lognormality of energy dissipation in turbulence.

Abstract

We study implications of the assumption of power-law dependence of moments of energy dissipation in turbulence on the Reynolds number Re, holding due to intermittency. We demonstrate that at Re→∞ the dissipation's logarithm divided by lnRe converges with probability one to a negative constant. This implies that the dissipation is singular in the limit, as is known phenomenologically. The proof uses a large deviations function, whose existence is implied by the power-law assumption, and which provides the general asymptotic form of the dissipation's distribution. A similar function exists for vorticity and for velocity differences where it proves the moments representation of the multifractal model (MF). Then we observe that derivative of the scaling exponents of the dissipation, considered as a function of the order of the moment, is small at the origin. Thus the variation with the order is slow and can be described by a quadratic function. Indeed, the quadratic function, which corresponds to log-normal statistics, fits the data. Moreover, combining the lognormal scaling with the MF we derive a formula for the anomalous scaling exponents of turbulence which also fits the data. Thus lognormality, not to be confused with the Kolmogorov (1962) assumption of lognormal dissipation in the inertial range, when used in conjunction with the MF provides a concise way to get all scaling exponents of turbulence available at present.