Passive advection of fractional Brownian motion by random layered flows

TL;DR

This paper investigates the statistical behavior of passive advection in layered flows driven by fractional Brownian motion, revealing anomalous super-diffusion and unusual spectral properties.

Contribution

It introduces a novel analysis of passive advection with fractional Brownian motion, highlighting unique growth rates and spectral characteristics.

Findings

Mean-squared displacement grows as t^{2 - H}

Wigner-Ville spectrum follows a 1/f^{3 - H} power law

Sample fluctuations of the spectrum are significant

Abstract

We study statistical properties of the process $Y(t)$ of a passive advection by quenched random layered flows in situations when the inter-layer transfer is governed by a fractional Brownian motion $X(t)$ with the Hurst index $H \in (0,1)$. We show that the disorder-averaged mean-squared displacement of the passive advection grows in the large time $t$ limit in proportion to $t^{2 - H}$, which defines a family of anomalous super-diffusions. We evaluate the disorder-averaged Wigner-Ville spectrum of the advection process $Y(t)$ and demonstrate that it has a rather unusual power-law form $1/f^{3 - H}$ with a characteristic exponent which exceed the value $2$. Our results also suggest that sample-to-sample fluctuations of the spectrum can be very important.