# On Nonlocal Energy Transfer via Zonal Flow in the Dimits Shift

## TL;DR

This paper demonstrates the existence of the Dimits shift in a modified 2D Terry-Horton equation, revealing a nonlocal energy transfer mechanism via zonal flows that persists across various simplifications and is relevant for plasma physics.

## Contribution

It introduces a modified 2D Terry-Horton model capturing the Dimits shift and elucidates a nonlocal energy transfer mechanism through zonal flows, applicable to more complex systems.

## Key findings

- The Dimits shift persists in simplified models including quasilinear and four-mode truncations.
- An adiabatic electron response enables efficient nonlocal energy transfer to sound-scale lengths.
- The shift size matches numerical solutions and is explained by coupling between unstable and stable modes.

## Abstract

The two-dimensional Terry-Horton equation is shown to exhibit the Dimits shift when suitably modified to capture both the nonlinear enhancement of zonal/drift-wave interactions and the existence of residual Rosenbluth-Hinton states. This phenomenon persists through numerous simplifications of the equation, including a quasilinear approximation as well as a four-mode truncation. It is shown that the use of an appropriate adiabatic electron response, for which the electrons are not affected by the flux-averaged potential, results in an $\boldsymbol{E}\boldsymbol{\times}\boldsymbol{B}$ nonlinearity that can efficiently transfer energy nonlocally to length scales on the order of the sound radius. The size of the shift for the nonlinear system is heuristically calculated and found to be in excellent agreement with numerical solutions. The existence of the Dimits shift for this system is then understood as an ability of the unstable primary modes to efficiently couple to stable modes at smaller scales, and the shift ends when these stable modes eventually destabilize as the density gradient is increased. This nonlocal mechanism of energy transfer is argued to be generically important even for more physically complete systems.