Theoretical Model for the Kramers-Moyal Description of Turbulence Cascades

TL;DR

This paper derives a theoretical Kramers-Moyal model for turbulence cascades, showing that at high Reynolds numbers, the velocity increments follow a Fokker-Planck equation, aligning with phenomenological models.

Contribution

It provides a theoretical derivation of the Kramers-Moyal equation for turbulence, demonstrating the vanishing of higher-order coefficients and confirming the Fokker-Planck description.

Findings

Higher order Kramers-Moyal coefficients tend to zero at high Reynolds numbers.

Velocity increments are governed by a Fokker-Planck operator.

Results align with phenomenological turbulence models.

Abstract

We derive the Kramers-Moyal equation for the conditional probability density of velocity increments from the theoretical model recently proposed by V.Yakhot [Phys.Rev.E {\bf 57}, 1737 (1998)] in the limit of high Reynolds number limit. We show that the higher order (n>=3) Kramers-Moyal coefficients tends to zero and the velocity increments are evolved by the Fokker-Planck operator. Our result is compatible with the phenomenological descriptions by R.Friedrich and J.Peinke [Phys.Rev.Lett. {\bf 78}, 863 (1997)].