Instanton Calculus in Shell Models of Turbulence

Abstract

It has been shown recently that intermittency of the Gledzer Ohkitani Yamada (GOY) shell model of turbulence has to be related to singular structures whose dynamics in the inertial range includes interactions with a background of fluctuations. In this paper we propose a statistical theory of these objects by modelling the incoherent background as a Gaussian white-noise forcing of small strength $\Gamma $. A general scheme is developed for constructing instantons in spatially discrete dynamical systems and the Cram\'er function governing the probability distribution of effective singularities of exponent $z$ is computed up to first order in a semiclassical expansion in powers of $\Gamma $. The resulting predictions are compared with the statistics of coherent structures deduced from full simulations of the GOY model at very high Reynolds numbers.