Zonal Flow as Pattern Formation

TL;DR

This paper explores how zonal flows emerge from turbulence as a pattern formation process, using statistical equations derived from the Hasegawa-Mima model to analyze bifurcations and instabilities.

Contribution

It demonstrates that the emergence of zonal flows can be understood as a pattern formation bifurcation within a statistical framework, providing a new perspective on turbulence structure growth.

Findings

Zonal flows arise from bifurcations in statistical turbulence equations.

The dispersion relation for modulational instability can be derived from these equations.

The statistical approach offers a general view on the growth of coherent structures.

Abstract

In this section, we examine the transition from statistically homogeneous turbulence to inhomogeneous turbulence with zonal flows. Statistical equations of motion can be derived from the quasilinear approximation to the Hasegawa-Mima equation. We review recent work that finds a bifurcation of these equations and shows that the emergence of zonal flows mathematically follows a standard type of pattern formation. We also show that the dispersion relation of modulational instability can be extracted from the statistical equations of motion in a certain limit. The statistical formulation can thus be thought to offer a more general perspective on growth of coherent structures, namely through instability of a full turbulent spectrum. Finally, we offer a physical perspective on the growth of large-scale structures.