Blowup as a driving mechanism of turbulence in shell models

Abstract

Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949 Onsager noticed that the scaling properties in inertial range imply non-differentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup indeed represents the driving mechanism of inertial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inertial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations.

26 pages, 11 figures

References in corpus (4)

• The large deviation approach to statistical mechanics
• Renormalization and universality of blowup in hydrodynamic flows
• Bifurcations of blowup in inviscid shell models of convective turbulence
• Computation of anomalous scaling exponents of turbulence from self-similar instanton dynamics

Cited by in corpus (12)

• Cascades and transitions in turbulent flows
• Spontaneously stochastic solutions in one-dimensional inviscid systems
• Development of high vorticity structures in incompressible 3D Euler equations
• Inverse energy cascade in nonlocal helical shellmodels of turbulence
• Spontaneous stochasticity of velocity in turbulence models
• Instanton based importance sampling for rare events in stochastic PDEs
• Continuous representation for shell models of turbulence
• A nested polyhedra model of isotropic MHD turbulence
• Chaotic and regular instantons in helical shell models of turbulence
• Singularities in Fully Developed Turbulence
• Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions
• Nonlinear phase synchronization and the role of spacing in shell models