Statistical State Dynamics: a new perspective on turbulence in shear flow

TL;DR

This paper introduces the statistical state dynamics (SSD) approach as a powerful new perspective for analyzing turbulence in shear flows, emphasizing its advantages in understanding cooperative mechanisms across scales.

Contribution

The paper presents SSD analysis methods and demonstrates their utility in revealing turbulence mechanisms that are difficult to analyze through traditional realization-based methods.

Findings

SSD reveals cooperative mechanisms in turbulence.

SSD provides more accurate statistical quantities.

Application to planetary and laboratory flows illustrates utility.

Abstract

Traditionally, single realizations of the turbulent state have been the object of study in shear flow turbulence. When a statistical quantity was needed it was obtained from a spatial, temporal or ensemble average of sample realizations of the turbulence. However, there are important advantages to studying the dynamics of the statistical state (the SSD) directly. In highly chaotic systems statistical quantities are often the most useful and the advantage of obtaining these statistics directly from a state variable is obvious. Moreover, quantities such as the probability density function (pdf) are often difficult to obtain accurately by sampling state trajectories even if the pdf is stationary. In the event that the pdf is time dependent, solving directly for the pdf as a state variable is the only alternative. However, perhaps the greatest advantage of the SSD approach is conceptual: adopting this perspective reveals directly the essential cooperative mechanisms among the disparate spatial and temporal scales that underly the turbulent state. While these cooperative mechanisms have distinct manifestation in the dynamics of realizations of turbulence both these cooperative mechanisms and the phenomena associated with them are not amenable to analysis directly through study of realizations as they are through the study of the associated SSD. In this review a selection of example problems in the turbulence of planetary and laboratory flows is examined using recently developed SSD analysis methods in order to illustrate the utility of this approach to the study of turbulence in shear flow.