Passive advection of fractional Brownian motion by random layered flows

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 ∈ (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.

Keywords

• anomalous diffusion
• super-diffusion
• spectral analysis
• random advection