Residual zonal flows in tokamaks and stellarators at arbitrary wavelengths

Abstract

In the linear collisionless limit, a zonal potential perturbation in a toroidal plasma relaxes, in general, to a non-zero residual value. Expressions for the residual value in tokamak and stellarator geometries, and for arbitrary wavelengths, are derived. These expressions involve averages over the lowest order particle trajectories, that typically cannot be evaluated analytically. In this work, an efficient numerical method for the evaluation of such expressions is reported. It is shown that this method is faster than direct gyrokinetic simulations performed with the Gene and EUTERPE codes. Calculations of the residual value in stellarators are provided for much shorter wavelengths than previously available in the literature. Electrons must be treated kinetically in stellarators because, unlike in tokamaks, kinetic electrons modify the residual value even at long wavelengths. This effect, that had already been predicted theoretically, is confirmed by gyrokinetic simulations.

Oscillatory relaxation of zonal flows in a multi-species stellarator plasma

“…The spectra of these potential time traces, which clearly exhibit two peaks corresponding to the GAM (around 50 kHz) and LFO oscillations (around 6-9 kHz), are also shown in the same figure. As expected for a long wavelength perturbation [16] the residual level is close to zero. Figure 2.…”

“…A positive relationship between an increased zonal flow residual and reduced turbulent transport in LHD configurations is found in [21,22,23], while in W7-X the residual level appears not to play an important role, but the ZF oscillation frequency seems to be related to the turbulent transport level [24]. These linear properties could provide a way to characterize stellarator configurations in respect of turbulent transport and their evaluation is relatively inexpensive [16,17], which makes them appealing to be used in the search for stellarator optimized configurations.…”

“…This work was afterwards extended in [15] including the damping of the oscillation. The linear zonal flow relaxation was studied recently in general stellarator configurations in [16,17] and analytical expressions for the residual level and oscillation frequency in stellarators were derived, evaluated numerically and compared to results of gyrokinetic simulations in realistic relevant configurations.…”

The parallel boundary condition for turbulence simulations in low magnetic shear devices

“…For the figures in this section, the results are produced from flux tube lengths chosen such that the "Continuous Magnetic Drifts" (blue) and "Exact Periodic" (red) boundary condition options are applicable, which correspond to [B × ∇B · ∇ψ] z± = 0 and [∇ψ · ∇α] z± = 0, respectively. The flux tubes where [B × ∇B · ∇ψ] z± = 0 are of particular interest, as linear studies [24,25] reveal a dependence on the radial bounce-averaged magnetic drift of the zonal flow residual in stellarators, a quantity that vanishes in axisymmetry. The bounce-average of B × ∇B · ∇ψ will thus be performed between two points where this term vanishes, making it possible that such flux tube lengths could result in unique zonal flow behavior compared to other tube lengths.…”

Impurity effects on residual zonal flow in deuterium (D)-tritium (T) plasmas

“…Rosenbluth-Hinton (RH) [6] demonstrated that the level of large scale ( 1 r bi k  , 2 where r k is the radial wave vector of ZF, bi i q    is the trapped ion radial width with being the inverse aspect ratio, q being the safety factor and i  being the ion gyroradius) residual ZF driven by ion temperature gradient (ITG) turbulence is not damped by collisionless process, but modified by neoclassical polarization shielding. Its extensions to short wavelength electron-temperature-gradient (ETG) turbulence [7,8], arbitrary radial wavelengths [9,10], shaped tokamak geometry, collisional case [11,12] and stellarators [13,14] were reported.…”