Intermittency from instanton calculus at the transition to turbulence and fusion rules

Abstract

Understanding intermittency of turbulent systems from the underlying differential equations is an outstanding problem in fluid dynamics. Here, in the example of Burgers turbulence as a stringent test, we introduce a method that yields high-order structure function exponents by combining instanton calculus, fusion rule predictions, and low-order statistical inputs from direct numerical simulations. We use instantons to evaluate high velocity gradient (VG) moments at the onset of intermittency, and then infer scaling exponents in fully developed turbulence via fusion rules. We show that the method captures the crossover at $\mathrm{Re}_\lambda \approx 1$ in the VG moment scaling, highlight the necessity of including fluctuations around instantons, and discuss future extensions.