Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows.

Quentin Jacquet,Rainer Hollerbach,Eun-Jin Kim

doi:10.3390/e20080613

Abstract

We report the time-evolution of Probability Density Functions (PDFs) in a toy model of self-organised shear flows, where the formation of shear flows is induced by a finite memory time of a stochastic forcing, manifested by the emergence of a bimodal PDF with the two peaks representing non-zero mean values of a shear flow. Using theoretical analyses of limiting cases, as well as numerical solutions of the full Fokker–Planck equation, we present a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing. From time-dependent PDFs, we calculate the information length (), which is the total number of statistically different states that a system passes through in time and utilise it to understand the information geometry associated with the formation of bimodal or unimodal PDFs. We identify the difference between the relaxation and build-up of the shear gradient in view of information change and discuss the total information length () which maps out the underlying attractor structures, highlighting a unique property of which depends on the trajectory/history of a PDF’s evolution.

Highlights

Many systems in nature and laboratories are far from equilibrium, constantly changing in time and space and exhibiting very complex behaviour
Is unimodal, the mean shear will decrease to zero in the long time limit; when the stationary Probability Density Functions (PDFs) is bimodal with a peak of ± x∗, x0 > x∗ models the relaxation of an initial super-critical gradient (x0 )
We investigated the time evolution of PDFs in a toy model of self-organised shear flows using a unified coloured approximation, and utilised the information length to understand information changes and attractor structures

Summary

Introduction

Many systems in nature and laboratories are far from equilibrium, constantly changing in time and space and exhibiting very complex behaviour. Examples include turbulence in astrophysical and laboratory plasmas, the stock market, and biological ecosystems. Despite having apparently different manifestations of complexity, these systems have much in common and are often governed by similar nonlinear dynamics. An ‘ordered’ collective behaviour (e.g., in the form of coherent structures) emerges on the macroscale out of complexity as a novel consequence of self-organisation. In the laboratory, in geophysical and astrophysical systems, coherent structures such as large-scale shear flows (such as zonal flows and streamers in laboratory plasmas, in the atmosphere and oceans, and in giant planets) and differential rotations in the Sun and other stars emerge from small-scale turbulence. There is overwhelming evidence from laboratory experiments, observations, and computational studies that these coherent structures play an absolutely critical role in determining the level of transport in the flow

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