Phase-space theory of drift wave–zonal flow interactions and the Dimits shift

Abstract

Zonal flows (ZFs) play a crucial role in regulating drift-wave (DW) turbulence. ZFs can even completely suppress DWs near the linear-instability threshold, which is known as the Dimits shift. This thesis aims to improve understanding of DW–ZF interactions by studying them within reduced models, including the modified Hasegawa–Mima equation (MHME), the modified Terry–Horton equation (MTHE), and the modified Hasegawa–Wakatani equations (MHWE). The goals of this thesis are twofold.

The first goal is to improve basic understanding of inhomogeneous DWs in ZFs. In particular, the MHME is studied using the newly developed theoretical tools, including the Wigner–Moyal equation and the wave-kinetic equation, which model DWs as quantumlike particles (“driftons”). Specifically: (i) drifton phase-space dynamics is revised, along with the discovery of the “runaway” trajectories; (ii) nonlinear stage of the modulational instability is explained from the phase-space perspective; (iii) a first rigorous analytic theory of the Kelvin–Helmholtz instability of ZFs is presented.

The second goal is to propose a new understanding of the so-called tertiary instability and the Dimits shift. In particular, the MTHE and the MHWE are studied in detail. It is discovered that tertiary modes are localized at the extrema of the ZF velocity U(x), where x is the radial coordinate. The tertiary instability can be understood as the primary-DW instability modified by the ZF “curvature” U'', rather than the ZF shear |U'|. This leads to an explicit calculation of the Dimits shift within the MTHE and also to a qualitative understanding of this effect in the MHWE. In particular, the same approach also provides improved understanding of the original study of tertiary instability within a reduced model of ion-temperature-gradient mode.

In addition to these two goals, connections are also made with the physics of Rossby waves and ZFs that is of interest to the geophysics community.