Saturation of single toroidal number Alfvén modes

Abstract

The results of numerical simulations are presented to illustrate the saturation mechanism of a single toroidal number Alfvén mode, driven unstable, in a tokamak plasma, by the resonant interaction with energetic ions. The effects of equilibrium geometry non-uniformities and finite mode radial width on the wave-particle nonlinear dynamics are discussed. Saturation occurs as the fast-ion density flattening produced by the radial flux associated to the resonant particles captured in the potential well of the Alfvén wave extends over the whole region where mode-particle power exchange can take place. The occurrence of two different saturation regimes is shown. In the first regime, dubbed resonance detuning, that region is limited by the resonance radial width (that is, the width of the region where the fast-ion resonance frequency matches the mode frequency). In the second regime, called radial decoupling, the power exchange region is limited by the mode radial width. In the former regime, the mode saturation amplitude scales quadratically with the growth rate; in the latter, it scales linearly. The occurrence of one or the other regime can be predicted on the basis of linear dynamics: in particular, the radial profile of the fast-ion resonance frequency and the mode structure. Here, we discuss how such properties can depend on the considered toroidal number and compare simulation results with the predictions obtained from a simplified nonlinear pendulum model.Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. © EUROfusionunstable. As the EP pressure gradient exceeds a certain threshold, even strongly damped continuum oscillations, besides these normal modes of the background plasma, can be driven unstable; they are called energetic particle modes (EPM) [3]. Alvénic fluctuations of various types, excited by EPs, have been identified in several tokamak experiments [4][5][6][7][8][9].Linear properties of Alfvénic fluctuations driven by EPs can be investigated within the theoretical framework of the generalised fishbone-like dispersion relation [3,10,11]. On this basis, several successful comparison between theoretical predictions and experimental observations, as well as numerical simulation results, have been reported [12][13][14][15][16][17][18][19][20].Concerning the nonlinear dynamics of Alfvén modes (and its consequences on the overall performances of tokamak plasmas), it is determined by two main factors: the nonlinear wave-wave coupling and nonlinear waveparticle interactions. With reference to the former factor, various wave-wave interactions leading to the breaking of the Alfvénic state have been analyzed in [21,22]. Although these effects play a crucial role in multi-scale dynamics in burning plasmas, we will focus, in our current work, on the nonlinear wave-particle interactions. This issue has been first addressed withi...

Nonlinear dynamics of shear Alfvén fluctuations in divertor tokamak test facility plasmas

“…Simulation results suggest that fluctuations are further enhanced in this self-consistent non-perturbative process, and saturate at a higher amplitude than the predicted linear scaling, which assumes constant mode frequency and frequencyindependent mode structure. 28,29,31 From Fig. 10, it is also interesting to note that, on longer time scale of the strongly unstable case with clear non-perturbative wave-EP interactions, the frequency chirping shows non-adiabatic features, as the mode structure is strongly modified.…”

“…5,24 Thus, in a realistic plasma, the complex behavior underlying the nonlinear interplay between SAW fluctuation and EPs depends on the relative importance of the two mechanisms. As shown theoretically 5,24 and by recent numerical simulations, 23,[25][26][27][28][29][30][31] the saturation mechanism is determined by the relative ordering of nonlinear EP orbit excursion to the perpendicular (with respect to equilibrium magnetic field) fluctuation wavelength and/or equilibrium nonuniformity; and it can be reflected by the relative scale lengths of wave-EP power transfer, mode structure and effective resonance condition. 23,28,29,31 For two paradigmatic cases, typically in the marginally unstable limit, nonlinear EP orbit excursion is restricted by the effective resonance condition, and is much smaller than the perpendicular fluctuation wavelength; that is, the resonant EP response is similar to that of a uniform plasma.…”

“…As shown theoretically 5,24 and by recent numerical simulations, 23,25-31 the saturation mechanism is determined by the relative ordering of nonlinear EP orbit excursion to the perpendicular (with respect to equilibrium magnetic field) fluctuation wavelength and/or equilibrium nonuniformity; and it can be reflected by the relative scale lengths of wave-EP power transfer, mode structure and effective resonance condition. 23, 28,29,31 For two paradigmatic cases, typically in the marginally unstable limit, nonlinear EP orbit excursion is restricted by the effective resonance condition, and is much smaller than the perpendicular fluctuation wavelength; that is, the resonant EP response is similar to that of a uniform plasma. Hence, in this regime, resonance detuning outweighs radial decoupling and, when only resonance detuning is considered, the saturated fluctuation amplitude scales quadratically with respect to the linear growth rate of the mode, 32,33 consistent with that predicted by wave-particle trapping mechanism typical of a 1-D beam-plasma system.…”

Shear Alfvén fluctuation spectrum in divertor tokamak test facility plasmas

“…Numerical simulation results, presented here, assume a typical DTT reference scenario and a model "slowing-down" EP distribution function, making use of the hybrid magnetohydrodynamic(MHD)gyrokinetic simulation code (HMGC) 20,21 . HMGC has been extensively applied to study the resonant interactions between SAW and EPs as well as the corresponding nonlinear behaviors and ensuing EP confinement issues [22][23][24][25][26][27] . HMGC has also been used to investigate plasma scenarios of practical interest, such as ITER 28 , Japan Atomic Energy Research Institute Tokamak-60 Upgrade (JT-60U) [29][30][31] , Doublet III-D (DIII-D) 32,33 and Fusion Advanced Studies Torus (FAST) 8,34,35 .…”

Drift kinetic theory of alpha transport by tokamak perturbations

“…Toroidal Alfvén eigenmode saturation has been estimated using a variety of methods. Some works balance the linear growth rate of the instability with the rate at which particles get trapped by the wave and flatten the gradient driving the instability, sometimes including collisions that restore the original gradient (Wu & White 1994; Fu & Park 1995; Wang & Briguglio 2016; Zhou & White 2016; Todo 2019). Other works consider the effects of zonal flows and nonlinear mode couplings (Spong, Carreras & Hedrick 1994; Chen & Zonca 2012).…”