Response function of turbulence computed via fluctuation-response relation of a Langevin system with vanishing noise

TL;DR

This paper derives a response function expression for turbulence models using a fluctuation-response relation from Langevin systems, and verifies its convergence to the noiseless case as noise vanishes.

Contribution

It introduces a novel fluctuation-response relation for turbulence models based on Langevin dynamics and confirms its validity through numerical verification.

Findings

Derived response function expression in turbulence models

Numerical verification shows convergence as noise approaches zero

Establishes a link between fluctuation-response relation and turbulence response

Abstract

For a shell model of the fully developed turbulence and the incompressible Navier-Stokes equations in the Fourier space, when a Gaussian white noise is artificially added to the equation of each mode, an expression of the mean linear response function in terms of the velocity correlation functions is derived by applying the method developed for nonequilibrium Langevin systems [Harada and Sasa, Phys. Rev. Lett. 95, 130602 (2005)]. We verify numerically for the shell model case that the derived expression of the response function, as the noise tends to zero, converges to the response function of the noiseless shell model.