from the traditional paradigm that turbulence is controlled by the flow shear $U''$. Our analytic predictions are in agreement with direct numerical simulations of the modified Terry–Horton system.'
author:
- 'Hongxuan Zhu , Yao Zhou,'
- 'I. Y. Dodin'
bibliography:
- 'Ref.bib'
title: Analytic theory of the tertiary instability and the Dimits shift within a scalar model
---

Introduction {#sec:Introduction}
============

The Dimits shift in magnetized plasmas is the shift between the threshold of drift-wave (DW) “primary” instability and the actual onset of transport that follows the scaling laws of developed turbulence [@Dimits00]. The Dimits shift is observed in both fluid and gyrokinetic simulations [@Lin98; @Rogers00; @Ricci06; @Numata07; @Mikkelsen08; @Kobayashi12; @St-Onge17] and is generally attributed to turbulence suppression by zonal flows (ZFs), which are generated by the “secondary” instability [@Rogers00; @Diamond01]. However, the Dimits shift is finite, meaning that ZFs cannot completely suppress DW turbulent transport if the primary-instability threshold is exceeded by far. Because of the detrimental effect that turbulent transport has on plasma confinement, it is important to understand this effect in detail.

After the seminal work [@Biglari90], it is widely accepted that ZFs can significantly suppress turbulence by shearing turbulent eddies. Based on this paradigm, the predator–prey 