Passive advection of fractional Brownian motion by random layered flows

Abstract

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.