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

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 ${\gamma }_{\mathrm{TI}}$ is derived within the modified Hasegawa--Wakatani model. We show that ${\gamma }_{\mathrm{TI}}$ equals the primary-instability growth rate plus a term that depends on the local $U^{\prime \prime }$; 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 $|U^{\prime }|$ is sufficiently weakened by viscosity. The Dimits regime ends when such structures are generated continuously.

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• Dimits transition in three-dimensional ion-temperature-gradient turbulence
• Theory of the tertiary instability and the Dimits shift within a scalar model
• Structure formation in turbulence as instability of effective quantum plasma
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• Detecting Shearless Phase-Space Transport Barriers in Global Gyrokinetic Turbulence Simulations with Test Particle Map Models
• Predicting the Dimits shift through reduced mode tertiary instability analysis in a strongly driven gyrokinetic fluid limit
• Statistical theory of the broadband two-plasmon decay instability