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 the 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 the inertial range imply nondifferentiability 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 the 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.

Singularities in fully developed turbulence

“…Noting the dissipative anomaly for the compressible case (Shivamoggi [25]), (24) and hence 11 Mailybaev [23] suggests, by considering a shell model for FDT, that the coherent structure generation may also be induced by the FTS. 12 The vorticity evolution equation for a compressible, barotropic fluid is…”

Chaotic and regular instantons in helical shell models of turbulence

“…In particular, in [12] the issue of intermittency was studied in one popular shell model [7] and it was argued that anomalous scaling exponents of velocity moments can be related to the scaling and statistics of instantons. Instantons are particular solutions of the inviscid equations of motion, intimately connected to the finite time blowup of the model with an infinite number of shells [13,14].…”

“…In the turbulent velocity field they are represented by coherent structures that traverse the inertial range towards large wave numbers. In this work, we attribute the word instanton to a self-similar inviscid structure localized in both time and scale, which is different from the viscous instantonic solutions generated within the Martin-Siggia-Rose formalism and widely studied for the original threedimensional NSE and for Burgers equations [12][13][14][15][16][17][18][19][20][21].…”

Finite-time singularities and flow regularization in a hydromagnetic shell model at extreme magnetic Prandtl numbers

“…A well-known criterion to establish whether a singularity might develop in a finite-time T ⋆ is given by the BKM theorem [40] which for an MHD fluid requires that the magnitude of the vorticity and of the current density become infinite at least as fast as T t 1 ( ) − ⋆ . A corresponding criterion for the loss of regularity in a hydrodynamic (HD) shell model is derived in [41], which implies that the maximum shell vorticity must grow at least as k u T t 1 ( ) n n ∼ − ⋆ as t T → ⋆ [45]. Therefore in order to establish the existence of singularities in shell models it is important to check the behavior of: .…”