Perturbative anomalous exponents from Kolmogorov multipliers

Abstract

We introduce a perturbative framework for anomalous scaling in turbulent transport based on multiplier statistics, rather than zero-mode calculations. We illustrate the approach using a shell model combining deterministic and Kraichnan-like stochastic components. The problem is reduced to the analysis of a stationary Fokker--Planck equation for Kolmogorov multipliers, defined as ratios of successive scalar amplitudes. Its solution yields the invariant measure through a perturbative expansion around a Gaussian distribution. Using the resulting multiplier statistics, we compute explicit anomalous scaling exponents for structure functions of arbitrary order, including odd, even, and non-integer moments. More broadly, the results suggest that multiplier statistics provide a viable route for computing anomalous exponents in turbulent transport, complementing recent hidden-symmetry approaches while circumventing the limitations of zero-mode methods based on a closed Hopf hierarchy.