Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics

TL;DR

This paper develops a geometrical-optics reduction of the CE2 model for zonal flow dynamics, demonstrating its conservation properties and comparing its predictions with quasilinear simulations, revealing both similarities and limitations.

Contribution

The paper introduces a numerically stable, conservation-preserving GO reduction of CE2 and compares its results to quasilinear dynamics, highlighting its advantages and missing physics.

Findings

CE2-GO preserves energy and enstrophy conservation.

CE2-GO qualitatively matches energy partitioning in zonal flows.

Differences in zonal flow length scales between models.

Abstract

Zonal flows have been observed to appear spontaneously from turbulence in a number of physical settings. A complete theory for their behavior is still lacking. Recently, a number of studies have investigated the dynamics of zonal flows using quasilinear theories and the statistical framework of a second-order cumulant expansion (CE2). A geometrical-optics (GO) reduction of CE2, derived under an assumption of separation of scales between the fluctuations and the zonal flow, is studied here numerically. The reduced model, CE2-GO, has a similar phase-space mathematical structure to the traditional wave-kinetic equation, but that wave-kinetic equation has been shown to fail to preserve enstrophy conservation and to exhibit an ultraviolet catastrophe. CE2-GO, in contrast, preserves nonlinear conservation of both energy and enstrophy. We show here how to retain these conservation properties in a pseudospectral simulation of CE2-GO. We then present nonlinear simulations of CE2-GO and compare with direct simulations of quasilinear (QL) dynamics. We find that CE2-GO retains some similarities to QL. The partitioning of energy that resides in the zonal flow is in good quantitative agreement between CE2-GO and QL. On the other hand, the length scale of the zonal flow does not follow the same qualitative trend in the two models. Overall, these simulations indicate that CE2-GO provides a simpler and more tractable statistical paradigm than CE2, but CE2-GO is missing important physics.