L\'evy flights on a comb and the plasma staircase

TL;DR

This paper models confined Le9vy flights on a sawtooth-like potential (comb) and identifies conditions for their localization, inspired by plasma transport phenomena in tokamaks.

Contribution

It introduces a theoretical framework for Le9vy flights confined by a sawtooth potential, linking confinement conditions to the potential's shape and the flights' fractal dimension.

Findings

Le9vy flights are confined if the potential's teeth are sufficiently broad and the shape parameter exceeds a critical value.

For Cauchy flights (bc=1), confinement occurs if the shape parameter n > 3.

The model explains localization phenomena observed in plasma transport in tokamaks.

Abstract

We formulate the problem of confined L\'evy flight on a comb. The comb represents a sawtooth-like potential field $V(x)$, with the asymmetric teeth favoring net transport in a preferred direction. The shape effect is modeled as a power-law dependence $V(x) \propto |\Delta x|^n$ within the sawtooth period, followed by an abrupt drop-off to zero, after which the initial power-law dependence is reset. It is found that the L\'evy flights will be confined in the sense of generalized central limit theorem if (i) the spacing between the teeth is sufficiently broad, and (ii) $n > 4-\mu$, where $\mu$ is the fractal dimension of the flights. In particular, for the Cauchy flights ($\mu = 1$), $n>3$. The study is motivated by recent observations of localization-delocalization of transport avalanches in banded flows in the Tore Supra tokamak and is intended to devise a theory basis to explain the observed phenomenology.