Title: Master equation for She-Leveque scaling and its classification in terms of other Markov models of developed turbulence,
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.