# Master equation for She-Leveque scaling and its classification in terms of other Markov models of developed turbulence

## TL;DR

This paper derives a Markov process representation of She-Leveque turbulence scaling, compares it with Kolmogorov scaling, and classifies it among other turbulence models using a unified stochastic framework.

## Contribution

It introduces a Markov process model for She-Leveque scaling, connecting it with diffusion processes and other cascade models, and establishes a comprehensive classification framework.

## Key findings

- Derived a jump process for She-Leveque scaling
- Established a diffusion process including Kolmogorov scaling
- Formulated a general Markov scaling law combining diffusion and jumps

## Abstract

We derive the Markov process equivalent to She-Leveque scaling in homogeneous and isotropic turbulence. The Markov process is a jump process for velocity increments $u(r)$ in scale $r$ in which the jumps occur randomly but with deterministic width in $u$. From its master equation we establish a prescription to simulate the She-Leveque process and compare it with Kolmogorov scaling. To put the She-Leveque process into the context of other established turbulence models on the Markov level, we derive a diffusion process for $u(r)$ from two properties of the Navier-Stokes equation. This diffusion process already includes Kolmogorov scaling, extended self-similarity and a class of random cascade models. The fluctuation theorem of this Markov process implies a "second law" that puts a loose bound on the multipliers of the random cascade models. This bound explicitly allows for inverse cascades, which are necessary to satisfy the fluctuation theorem. By adding a jump process to the diffusion process, we go beyond Kolmogorov scaling and formulate the most general scaling law for the class of Markov processes having both diffusion and jump parts. This Markov scaling law includes She-Leveque scaling and a scaling law derived by Yakhot.