Theory of the tertiary instability and the Dimits shift from reduced drift-wave models

TL;DR

This paper models tertiary modes in drift-wave turbulence as quantum harmonic oscillators to explain the Dimits shift, deriving growth rates and instability thresholds within a reduced plasma model.

Contribution

It introduces a quantum harmonic oscillator analogy for tertiary modes and derives an explicit expression for the Dimits shift in a reduced drift-wave model.

Findings

Tertiary modes are localized near extrema of zonal velocity.

The growth rate of tertiary modes is derived and related to primary instability.

The Dimits shift is explicitly calculated in the Terry–Horton limit.

Abstract

Tertiary modes in electrostatic drift-wave turbulence are localized near extrema of the zonal velocity $U(x)$ with respect to the radial coordinate $x$. We argue that these modes can be described as quantum harmonic oscillators with complex frequencies, so their spectrum can be readily calculated. The corresponding growth rate $\gamma_{\rm TI}$ is derived within the modified Hasegawa--Wakatani model. We show that $\gamma_{\rm TI}$ equals the primary-instability growth rate plus a term that depends on the local $U''$; hence, the instability threshold is shifted compared to that in homogeneous turbulence. This provides a generic explanation of the well-known yet elusive Dimits shift, which we find explicitly in the Terry--Horton limit. Linearly unstable tertiary modes either saturate due to the evolution of the zonal density or generate radially propagating structures when the shear $|U'|$ is sufficiently weakened by viscosity. The Dimits regime ends when such structures are generated continuously.