First-principles perturbative theory of anomalous scaling in a stochastic shell model of turbulence

Abstract

Deriving anomalous scaling exponents from the equations of motion remains a central problem in the statistical theory of turbulence. Here we obtain a first-principles perturbative solution for a nonlinear stochastic dyadic shell model. The model preserves the conservative cascade structure and exact scaling symmetry of the deterministic dynamics, while stochastic transfer fluctuations provide a perturbative setting in which the leading-order rescaled dynamics is Gaussian. Using the statistically restored hidden scaling symmetry of the inertial-range equations, we determine the stationary statistics of the rescaled variables. We then formulate anomalous scaling as a Perron--Frobenius eigenvalue problem for the multiplier statistics. The resulting perturbative expansion yields explicit analytical expressions for the scaling exponents of structure functions of arbitrary order in the weak-noise regime. Direct numerical simulations provide an independent verification of the theoretical predictions. The results demonstrate that the hidden-symmetry perturbation framework extends from linear random models to a genuinely nonlinear cascade system.