On non-local energy transfer via zonal flow in the Dimits shift

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 E × B nonlinearity that can efficiently transfer energy non-locally to length scales of 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 non-local mechanism of energy transfer is argued to be generically important even for more physically complete systems.

Theory of the Tertiary Instability and the Dimits Shift from Reduced Drift-Wave Models

“…These results show that mode localization is a key feature of the TI. This feature is missed in some previous studies [15,19,20] where the TI was derived from the interaction of just four Fourier harmonics. Also, our findings challenge the popular idea that the TI is a Kelvin-Helmholtz instability (KHI) [3,16,20].…”

“…However, basic understanding and generic description of the TI and the Dimits shift have been elusive.Here, we propose a simple yet quantitative theory of the TI using the modified Hasegawa-Wakatani equation (mHWE) [2,16] as a base turbulence model. We clarify several misconceptions regarding the TI, and we explicitly derive the Dimits shift in the limit corresponding to the Terry-Horton model [15,21]. Our approach is also applicable to other DW models, such as ion-temperaturegradient (ITG) turbulence [5], as discussed towards the end.…”

“…This constitutes the so-called Dimits shift [12][13][14][15][16][17], which has been attracting attention for two decades. The finite value of the Dimits shift is commonly attributed to the "tertiary" instability (TI) [5], and some theories of the TI have been proposed [5,[15][16][17][18][19][20]. However, basic understanding and generic description of the TI and the Dimits shift have been elusive.Here, we propose a simple yet quantitative theory of the TI using the modified Hasegawa-Wakatani equation (mHWE) [2,16] as a base turbulence model.…”

Nonlinear saturation and oscillations of collisionless zonal flows

“…denotes the zonal average of f, and L x is the system length in the x-direction. (For more details, see, for example, [29,49].) Below, we consider both the oHME and mHME and treat them on the same footing by using generalâ .…”

“…We argue that at N=1, ZFs oscillate due to the presence of so-called passing drifton trajectories, and we derive an approximate formula for the ZF amplitude as a function of time in this regime. In doing so, we also extend the applicability of the popular 'four-mode truncation' (4MT) model [16][17][18][19][20][21][22][23][24][25][26][27][28][29], which is commonly used for the linear stage, to nonlinear ZF-DW interactions. We also show that at N1, the passing trajectories vanish and ZFs saturate monotonically, which can be attributed to phase mixing of higher-order sidebands when the 4MT ceases to be a reasonable approximation.…”