Ordering of timescales predicts applicability of quasi-linear theory in unstable flows

TL;DR

This paper investigates when quasilinear approximations are valid in turbulent rotating convection systems by analyzing the ordering of key timescales, showing that the Generalised Quasilinear approximation generally outperforms the standard QL.

Contribution

It introduces a timescale ordering framework to predict the applicability of quasilinear approximations in turbulent flows, demonstrating the superior performance of GQL over QL.

Findings

QL approximations work well with short shearing or correlation timescales.

GQL systematically outperforms QL across different parameter regimes.

Timescale ordering predicts the validity of quasilinear methods in turbulence.

Abstract

We discuss the applicability of quasilinear-type approximations for a turbulent system with a large range of spatial and temporal scales. We consider a paradigm fluid system of rotating convection with a vertical and horizontal temperature gradients. In particular, the interaction of rotating with the horizontal temperature gradient drives a ``thermal wind'' shear flow whose strength is controlled by a horizontal temperature gradient. Varying the parameters systematically alters the ordering of the shearing timescale, the convective timescale, and the correlation timescale. We demonstrate that quasilinear-type approximations work well when the shearing timescale or the correlation timescale is sufficiently short. In all cases, the Generalised Quasilinear approximation (GQL) systematically outperforms the Quasilinear approximation (QL). We discuss the consequences for statistical theories of turbulence interacting with mean gradients. We conclude with comments about the general applicability of these ideas across a wide variety of non-linear physical systems.