# Superfast amplification and superfast nonlinear saturation of perturbations as the mechanism of turbulence

## TL;DR

This paper demonstrates through large-scale numerical simulations that perturbations in turbulence amplify faster than exponential, leading to superfast nonlinear saturation, which underpins the persistence of fully developed turbulence.

## Contribution

It provides the first large-scale numerical verification of superfast perturbation amplification and saturation as the turbulence mechanism, extending previous theoretical predictions.

## Key findings

- Superfast amplification confirmed at high resolution and Reynolds number.
- Superfast nonlinear saturation observed following amplification.
- Supports the mechanism of turbulence persistence via superfast perturbation dynamics.

## Abstract

Ruelle predicted that the maximal amplification of perturbations in homogeneous isotropic turbulence is exponential $e^{\sigma \sqrt{Re} t}$ (where $\sigma \sqrt{Re}$ is the maximal Liapunov exponent). In our earlier works, we predicted that the maximal amplification of perturbations in fully developed turbulence is faster than exponential $e^{\sigma \sqrt{Re} \sqrt{t} +\sigma_1 t}$. That is, we predicted superfast initial amplification of perturbations. Built upon our earlier numerical verification of our prediction, here we conduct a large numerical verification with resolution up to $2048^3$ and Reynolds number up to $6210$. Our direct numerical simulation here confirms our analytical prediction. Our numerical simulation also demonstrates that such superfast amplification of perturbations leads to superfast nonlinear saturation. We conclude that such superfast amplification and superfast nonlinear saturation of ever existing perturbations serve as the mechanism for the generation, development and persistence of fully developed turbulence.