Multi-time scale-invariance of turbulence in a shell model

TL;DR

This paper explores the multi-time scale-invariance in turbulence using a shell model, revealing a hidden symmetry that explains multifractal intermittency and provides new self-similarity rules for turbulent fluctuations.

Contribution

It introduces a hypothesis of restored hidden symmetry to derive multi-time self-similarity laws in turbulence, supported by numerical verification.

Findings

Self-similarity rule linking observable homogeneity to H"older exponent

Scaling laws for decorrelation times of turbulent fluctuations

Verification of self-similarity rules for multi-time structure functions

Abstract

When time and velocities are dynamically rescaled relative to the instantaneous turnover time, the Sabra shell model acquires another (hidden) form of scaling symmetry. It has been previously shown that this symmetry is statistically restored in the inertial interval of developed turbulence, thereby establishing a self-similarity property derived from first principles and replacing the broken $1/3$-scaling of the K41 theory. Multifractal intermittency follows from the restored hidden symmetry, in which the anomalous scaling exponents $\zeta_p$ are identified as Perron-Frobenius eigenvalues. In this paper, we use the hypothesis of restored hidden symmetry to address the multi-time statistics of turbulent fluctuations. The central result is the self-similarity rule stating that any observable that is time-scale homogeneous of degree $p$ is self-similar with the H\"older exponent $h = \zeta_p/p$. As a particular case, it yields the scaling laws for decorrelation times of fluctuations obtained previously within the phenomenological multifractal approach. As further applications, we formulate self-similarity rules for multi-time structure functions and multi-time Kolmogorov multipliers and verify them numerically.