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.

References in corpus (7)

• Linearized model Fokker-Planck collision operators for gyrokinetic simulations. I. Theory
• Bifurcation in electrostatic resistive drift wave turbulence
• Transition to subcritical turbulence in a tokamak plasma
• Zonal-flow dynamics from a phase-space perspective
• Formation of solitary zonal structures via the modulational instability of drift waves
• Nonlinear saturation and oscillations of collisionless zonal flows
• Effects of zonal flows on transport crossphase in dissipative trapped-electron mode turbulence in edge plasmas

Cited by in corpus (1)

• Wave-kinetic approach to zonal-flow dynamics: recent advances