Nonlinear gyrokinetic simulations of reversed shear Alfven eigenmodes in DIII-D tokamak

Abstract

Global gyrokinetic simulations of reversed shear Alfven eigenmodes (RSAE) excited by energetic particles (EP) in the DIII-D discharge #159,243 find that self-generated zonal flows dominate the RSAE nonlinear saturation. Effects of Coulomb collisions are negligible since the effective EP collision frequency is much smaller than the RSAE linear growth rate. Nonlinear interactions of multiple unstable RSAEs lower the saturation amplitude and resulting EP transport, and nonlinearly generate toroidal Alfven eigenmodes (TAE) and ellipticity induced Alfven eigenmodes (EAE). In both single mode and multiple modes simulations, the initial RSAE saturation amplitude and associated EP transport are an order of magnitude higher than the experimental observations, but quickly diminish within 0.1 ms after nonlinear saturation due to the formation of coherent phase space structures that flatten the EP distribution function at resonances. Results from these simulations of the RSAE only show that additional dissipation such as background micro-turbulence is needed to sustain the quasi-steady state RSAE amplitude and EP transport observed in this DIII-D experiment.

1 Introduction

Confinement of energetic particles (EP)  (Fasoli et al. 2007; Breizman and Sharapov 2011; Lauber 2013; Gorelenkov et al. 2014; Todo 2019; Heidbrink and White 2020), including both fusion products ($\alpha$ particles) and fast ions produced by auxiliary heating through ion cyclotron radio frequencies (ICRF) waves or neutral beams injection (NBI), determines the performance of current and future fusion reactors. Meso-scale Alfven Eigenmodes (AE) (Heidbrink 2008; Chen and Zonca 2016), driven by strong EP pressure gradients, can cause an anomalous EP transport in the core region that reduces the heating efficiency. The AE turbulence and associated EP transport, thus, have drawn intense experimental and theoretical research.

Thanks to the rapid growth of computational power, numerical simulations of AE and EP physics play an increasingly more important role in the explanation and prediction of EP confinement in the last two decades. Early kinetic-MHD simulation efforts to understand the nonlinear AE evolution focus on nonlinear wave particle interaction leading to mode saturation, frequency chirping, and fast ion loss (Spong et al. 1994; Fu and Park 1995; Pinches et al. 1998; Todo et al. 2003). The types of nonlinear frequency chirping are found to be dependent on the EP drive and the mode damping in kinetic-MHD simulations (Wang et al. 2021). Beyond the conventional paradigm of a 1D nonlinear dynamical system for the wave particle interaction, nonlinear gyrokinetic simulations  (Zhang et al. 2012) find a fast (i.e., non-adiabatic) oscillation of AE frequency and amplitude due to the intrinsically 2D nonlinear dynamical system, where the radial variations of mode amplitude induce resonance detuning and radial decoupling in the evolution of coherent structures in phase space  (Zonca et al. 2015). Both hybrid and gyrokinetic simulations (Lang et al. 2010; Chen et al. 2013) find that Coulomb collisions of EP increase the AE saturation level due to the destruction of coherent structures (Berk et al. 1992). Nonlinear MHD effects and zonal flows are first found to be not important for weakly unstable mode near marginality in kinetic-MHD simulations (Todo et al. 2010). Subsequent nonlinear gyrokinetic simulations  (Chen et al. 2018) find that the RSAE saturation amplitude is reduced by the self-generated $n=0$ component in the electron density and thermal ion distribution (zonal structures) when $\gamma /\omega \ge 3\%$ ($\gamma$ is AE linear growth rate and $\omega$ is AE frequency), which indicates the strength of the RSAE instability when the dominant mechanism for the RSAE saturation changes from nonlinear wave particle trapping near marginality to zonal structure effects far away from marginality. Furthermore, a systematic comparison between single mode and multiple mode AE has been reported in kinetic-MHD simulations (Vlad et al. 2018), while the $n=0$ component is not considered. Recently, verification and validation studies of AE  (Spong et al. 2012; Taimourzadeh et al. 2019) have been carried out. Moreover, a full comparison of nonlinear results has been reported between gyrokinetic and hybrid codes (Cole et al. 2017). Nonetheless, nonlinear physics of AE has not been fully understood regarding the role of zonal flows, multiple mode interactions, and cross-scale couplings with realistic parameters.

Furthermore, the AE turbulence and EP transport may depend on nonlinear cross-scale coupling with micro-turbulence  (Horton 1999; Heidbrink et al. 2009) and macro-scale MHD instabilities. In this work, we study both AE nonlinear physics and cross-scale coupling through integrated simulations using the global gyrokinetic toroidal code GTC (Lin et al. 1998). Recent GTC simulations coupling micro–meso-scales find the regulation of RSAE by ion temperature gradient (ITG) micro-turbulence  (Liu et al. 2022). In the presence of the background micro-turbulence, RSAE amplitude and EP transport decrease drastically at the initial saturation but subsequently increase to the experimental levels in the quasi-steady state with burst dynamics, due to the scattering of nonlinearly trapped EP and the damping of zonal structures by the micro-turbulence. The RSAE amplitude in the quasi-steady-state ITG-RSAE turbulence from gyrokinetic simulations, for the first time, agrees very well with experimental observations.

As a follow-up to Liu et al. (2022), we report here the detailed nonlinear physics of the AE turbulence regarding the role zonal flows and multiple mode interactions from GTC simulations of the RSAE only. The detailed physics of the cross-scale coupling between RSAE and micro-turbulence will be reported in a future paper. The simulated DIII-D discharge #159,243 has been selected as the reference case for verification and validation (V&V) of linear gyrokinetic simulations (Taimourzadeh et al. 2019). RSAEs are found to be excited by beam ions near the ${q}_{min}$ flux surface using eight simulation codes (including gyrokineticLin et al. 1998; Candy and Waltz 2003; Chen and Parker 2003; Jolliet et al. 2007; Mishchenko et al. 2014), gyrokinetic-MHD hybrid  (Todo and Sato 1998; Varela et al. 2017), and eigenvalue  (Cheng and Chance 1986) codes) with good agreement on both real frequency and linear growth rate. Poloidal contour plot and radial mode structures of the $n=4$ RSAE also agree with the electron cyclotron emission (ECE) (Austin and Lohr 2003) and electron cyclotron emission imaging (ECEI) (Tobias et al. 2010) measurements. Subsequent linear global GTC simulations (Wang et al. 2021) find that RSAE are unstable including the most unstable modes with toroidal mode number $n=\left(\mathrm{4,5}\right)$, less unstable modes $n=\left(\mathrm{3,6},\mathrm{7,8}\right)$, and marginally unstable modes $n=\left[\mathrm{9,12}\right]$. Compressible magnetic perturbation slightly increases the RSAE linear growth rate, while effects of trapped electrons on the linear dispersion relation are insignificant.

Building on the comprehensive linear simulations, we perform nonlinear gyrokinetic simulations to study RSAE saturation and EP transport. We examine the effects of zonal flows, collisions of beam ions, and nonlinear toroidal mode coupling on the RSAE amplitude and the associated EP transport. In single toroidal mode simulations, self-generated zonal flows strongly affect both RSAE amplitude and EP transport for the most unstable modes. Effects of Coulomb collisions of beam ions are found to be negligible since the effective collision frequency of beam ions is much smaller than linear growth rate of the RSAE.

In multiple toroidal modes simulation, the most unstable mode saturates at lower amplitude, while the stable and marginally unstable modes can be nonlinearly generated by toroidal mode coupling between strongly unstable modes. In the multiple mode simulation, toroidicity-induced Alfven eigenmode (TAE) and ellipticity induced Alfven eigenmode (EAE) could be nonlinearly generated outside the ${q}_{min}$ flux surface after RSAE nonlinear saturation.

In global simulations of both single mode and multiple modes, RSAE mode amplitude and associated EP transport are an order of magnitude higher than the experimental observations (Collins et al. 2016; Heidbrink et al. 2017) at the nonlinear saturation, but quickly diminished within 0.1 ms after nonlinear saturation due to the flattening of EP distribution function at resonances. No steady state can be obtained in the absence of additional dissipation or background micro-turbulence in the simulations of this DIII-D experiment where only a few toroidal modes are unstable  (Zonca et al. 2015; Liu et al. 2022; Bass and Waltz 2010; Duarte et al. 2017; Zeeland et al. 2019; Gorelenkov and Duarte 2021; Biancalani et al. 2021). These GTC simulation results show that the perfect equilibrium (i.e., without ubiquitous background micro-turbulence) as the initial condition for the RSAE simulation can lead to the excessive overshoots of the RSAE amplitude in the nonlinear saturation and the diminished EP transport immediately after the nonlinear saturation. Such a perfect initial condition has been used in nearly all plasma simulations, but nonetheless is not realistic. In fact, Liu et al. (2022) shows that the presence of the micro-turbulence as the background for the RSAE simulation can greatly reduce the RSAE initial nonlinear saturation amplitude and maintain a quasi-steady state EP transport consistent with experimental observations.

This paper is organized as follows: The equilibrium profiles and simulation parameters are described in Sect. 2. In Sect. 3, we present the simulation results of single mode simulations and discuss the zonal flow effects and phase space dynamics. In Sect. 4, we discuss EP collisional effects on the RSAE nonlinear dynamics. In Sect. 5, we present the simulation results of multiple mode interactions. Finally, conclusions and discussion are presented in Sect. 6.

2 Equilibrium profiles and simulation parameters

In this work, the kinetic EFIT magnetic equilibrium geometry and plasma profiles including the classical fast ion density as shown in Figs. 2 and 3 of Taimourzadeh et al. (2019) are selected from DIII-D shot # 159,243 at $805ms$. The safety factor $q$ profile has a reversed shear with minimal value ${q}_{min}=2.94$ near $\rho =0.5$, where RSAE can be excited by beam ions. Here, $\rho$ is square root of toroidal flux normalized by its separatrix value.

GTC  (Lin et al. 1998) used in this work is a global toroidal code using gyrokinetic  (Brizard and Hahm 2007) ions and drift-kinetic electrons, which has been applied to driftwave micro-turbulence  (Lin et al. 1998, 2002; Zhang et al. 2008; Xiao and Lin 2009; Wang et al. 2020; Nicolau et al. 2021), magnetohydrodynamic (MHD) instabilities  (McClenaghan et al. 2014; Brochard et al. 2022) and AE  (Deng et al. 2012a , 2012; Wang et al. 2013; Choi et al. 2021). In this paper, all species (EP, thermal ions and electrons) are treated on the same footing using the low noise $\delta f$ scheme (Parker and Lee 1993). Compressible magnetic perturbation $\delta {B}_{||}$   (Dong et al. 2017) and equilibrium current (Deng et al. 2012b) are incorporated for a physically comprehensive simulation. The electron response is further expanded using the fluid-kinetic hybrid model (Deng et al. 2012b). Global equilibrium pressure profiles of each species are fixed using a model particle and heat source. We have verified that fixing the global profiles, which has been routinely applied in the global gyrokinetic simulations of the micro-turbulence to prevent profile relaxation (Lin et al. 2002), does not change the nonlinear RSAE physics including the role of zonal flows in the RSAE saturation and nonlinear evolution of RSAE mode amplitude and EP transport. Time step is set to be ${10}^{-8}s$ to resolve the high-frequency EAE ($142kHz$) and to resolve the electron orbital frequency as well as the frequency of unphysically high ${k}_{||}$ Alfven wave for numerical stability. GTC uses a global field-aligned mesh with 32 parallel grids which are sufficient because of the long parallel wavelength. In the nonlinear electromagnetic simulations of single toroidal mode RSAE, based on the convergence studies, $5\times {10}^{4}$ unstructured perpendicular grids with grid size $\sim 1.3{\rho }_{i}$ and 1000 particles per cell for each species with a local Maxwellian are used, where ${\rho }_{i}\sim 2.1mm$ is the thermal ion gyroradius. In the simulation of multiple modes RSAE, however, a larger marker number and denser mesh are used to reduce the noise and to provide a better spatial resolution, where $2\times {10}^{5}$ unstructured perpendicular grids and 6000 particles per cell for each species are used. In all simulations, all poloidal $m$ harmonics are kept when using Fourier filtering to select specific toroidal modes.

3 Nonlinear simulations of a single toroidal mode

3.1 Effects of zonal flows on RSAE saturation

Nonlinear simulations of a single mode RSAE for the most unstable mode $n=4$, less unstable mode $n=6$, and marginal unstable mode $n=10$ are performed to investigate the effects of zonal flows on the RSAE nonlinear saturation. Figure 1 shows the time evolution of the normalized electrostatic potential $e\delta \phi /{T}_{e}$ (here, $e$ is electron charge) on the ${q}_{min}$ flux surface from simulations self-consistently keeping zonal flows or artificially suppressing zonal flows. For the most unstable mode $n=4$ with a linear growth rate $\gamma$ of ${\gamma }_{4}/{\omega }_{4}\sim 8\%$ (here ${\omega }_{4}$ is the real frequency) shown in Fig. 1a, the RSAE saturates at a very high level with an electrostatic potential $\delta \phi \sim {T}_{e}/e$ but quickly drops by a factor of more than 20 to a very low level within $0.2ms$ when zonal flows are suppressed in the simulations. In the simulation keeping zonal flows, zonal flows are generated by mode coupling with a nonlinear growth rate ${\gamma }_{0}\sim 2{\gamma }_{4}$. Zonal flows, in turn, reduce the mode saturation amplitude by a factor of more than 2. After nonlinear saturation, the mode amplitude also quickly drops to a very low level. For the less unstable mode $n=6$ with ${\gamma }_{6}/{\omega }_{6}\sim 5\%$ as shown in Fig. 1b, zonal flows also play an important role in the RSAE saturation, and the saturation amplitude is smaller than that of $n=4$ RSAE. However, for marginally unstable mode $n=10$ with ${\gamma }_{10}/{\omega }_{10}\sim 2\%$ as shown in Fig. 1c, the effects of zonal flows are insignificant, and the mode saturate at very low amplitudes. These results are consistent with previous GEM simulations  (Chen et al. 2018) of the $n=4$ RSAE using the same DIII-D equilibrium but scanning fast ion density, which found that the effects of zonal structures are insignificant for weakly unstable mode $\gamma /\omega \le 3\%$.

Fig. 1

Time history of perturbed electrostatic potentials $\mathrm{e\delta \phi }/{\mathrm{T}}_{\mathrm{e}}$ on the ${\mathrm{q}}_{\mathrm{min}}$ flux surface from single mode simulations of $n=4$ [panel (a)] with (green) or without (red) zonal flows (black). Time histories of $\mathrm{e\delta \phi }/{\mathrm{T}}_{\mathrm{e}}$ from single mode simulations of $n=6$ and $10$ are shown in panels (b) and (c)

The mode structures of the $n=4$ RSAE from simulations with zonal flows are in Fig. 2 from the linear phase [panel (a)] to the nonlinear saturation [panel (b)] and late nonlinear stage [panel (c)]. The corresponding radial profiles of each poloidal harmonics are shown in Fig. 3. Besides the dominant harmonic $m=12$, there are two subdominant harmonics $m=11$ and $m=13$. In the linear phase, the mode peaks at the ${q}_{min}$ flux surface and exhibits triangular shape with a wide radial structure. At the nonlinear saturation as shown in Fig. 2b, besides the dominant peak at the ${q}_{min}$ surface, another peak appears inside the ${q}_{min}$ surface. Away from the ${q}_{min}$ flux surface, the outer part of RSAE mode structure, which is subdominant but still visible in the linear phase, exhibits ballooning structure and saturates at a level of about $25\%$ of the main peak’s amplitude near ${q}_{min}$ surface. And the corresponding radial structures show coupling of different poloidal harmonics in the major radial domain $R=[\mathrm{2.05,2.2}]m$ due to strong magnetic shear. But it is not TAE since no clear TAE frequency is found at the RSAE saturation. At the late nonlinear stage, besides the decrease of the mode amplitude compared with that at the nonlinear saturation, the mode amplitude near ${q}_{min}$ as shown in Fig. 3c, is suppressed after initial nonlinear saturation due to zonal flows and flattening of resonant EP distribution in phase space.

Fig. 2

Poloidal contour plots of perturbed electrostatic potential $\mathrm{e\delta \phi }/{\mathrm{T}}_{\mathrm{e}}$ in linear phase [panel (a)], nonlinear saturation [panel (b)] and late nonlinear stage [panel (c)] from single mode simulation of $n=4$ with zonal flows

Fig. 3

Radial profiles of poloidal harmonics in linear phase [panel (a)], nonlinear saturation [panel (b)] and late nonlinear stage [panel (c)] from single mode simulation of $n=4$ with zonal flows

The instantaneous frequencies of electrostatic potential $e\delta \phi /{T}_{e}$ of $m=12$ harmonic and zonal flows near the ${q}_{min}$ flux surface are shown in Fig. 4. In linear phase, $m=12$ harmonic grows exponentially with the linear frequency, and zonal flows grow with a zero frequency. After nonlinear saturation, the frequency of $m=12$ harmonic exhibits up and down chirping in the nonlinear phase. For the zonal flows, however, GAM oscillation $\sim 40kHz$ together with their second and third harmonics appears in the nonlinear phase.

Fig. 4

Frequency spectrum evolution of perturbed electrostatic potential $\mathrm{e\delta \phi }/{\mathrm{T}}_{\mathrm{e}}$ of $m=12$ harmonic [panel (a)] and zonal flows [panel(b)] near the ${\mathrm{q}}_{\mathrm{min}}$ flux surface from single mode simulation of $n=4$ with zonal flows

The time evolution of the resulting effective EP diffusivity ${D}_{f}\left(t\right)$ driven by the most unstable RSAE $n=4$ is shown in Fig. 5. Here, ${D}_{f}\left(t\right)$ are obtained by averaging over the major radial domain $R=\left[\mathrm{1.95,2.04}\right]m$ of fast ion diffusivity ${D}_{f}\left(R,t\right)$ as function of major radius $R$ and time $t$,

$${D}_{f}\left(R,t\right)=\int {v}_{dr}\left(R,t\right)\delta f\left(R,t\right)d{v}^{3}/\nabla {n}_{f},$$

where ${v}_{dr}$ is the radial component of gyrocenter drift velocity ${{\varvec{v}}}_{d}$ including ${\varvec{E}}\times {\varvec{B}}$ drift and magnetic-flutter term,

$${{\varvec{v}}}_{d}=\frac{c{{\varvec{B}}}_{0}\times \nabla \langle \delta \phi \rangle }{{B}_{0}^{2}}+{v}_{||}\frac{\delta {{\varvec{B}}}_{\perp }}{{B}_{0}}$$

with gyro-averaging $<>$, $\delta f$ is the perturbed beam ion distribution function, and ${n}_{f}$ is the equilibrium beam ion density. Without zonal flows, the averaged EP diffusivity reaches $40{m}^{2}/s$ at the RSAE saturation, but drops to a very low level ${D}_{f}\ll 1{m}^{2}/s$ within $0.2ms$. When zonal flows are kept, the averaged EP diffusivity is reduced to $15{m}^{2}/s$ by zonal flows near the RSAE saturation, which is still much larger than experimental value of EP diffusivity. It also quickly drops to almost 0 in $0.1ms$ after the nonlinear saturation, which is inconsistent with the quasi-steady state EP transport observed in the experiment.

Fig. 5

Time history of averaged effective fast ion diffusivity of single mode simulations of n = 4 with (green) and without (red) zonal flows

3.2 Phase space dynamics

Despite the differences in the saturation amplitudes among various unstable RSAE, a common phenomenon is that all mode amplitudes drop quickly to a very low level after the nonlinear saturation, presumably due to the formation of coherent phase space structures. The dynamics of EP phase space coherent structures associated with the precessional resonance with the beta-induced Alfven eigenmodes (BAE)  (Zhang et al. 2012, 2013; Cheng et al. 2017) has been studied previously by GTC simulations and illustrated in the phase space of canonical variables $\left(\zeta ,{P}_{\zeta }\right)$, where $\zeta$ is toroidal angle and ${P}_{\zeta }$ is the canonical angular momentum,

$${P}_{\zeta }=g\frac{{m}_{f}}{{Z}_{f}{B}_{0}}{v}_{f||}-\psi ,$$

with $g$ the covariant component of equilibrium magnetic field, ${B}_{0}$ the equilibrium magnetic field, $\psi$ the poloidal magnetic flux function, and ${m}_{f}$, ${Z}_{f}$, ${v}_{f||}$ the beam ion mass, charge, and parallel velocity, respectively. For the precessional resonance, the magnetic moment $\mu$ and longitudinal invariant ${J}_{||}$ are constants, and the nonlinear dynamics of the gyrocenters is reduced to one degree of freedom described by the pair of action-angle variables $\left(\zeta ,{P}_{\zeta }\right)$.

In the current simulations, the dominant resonances responsible for the RSAE excitation have been identified by analyzing EP phase space structures shown in Fig. 6. In the axisymmetric tokamaks, the resonant condition of Alfven eigenmodes is $\omega -n{\omega }_{p}-\left(l+n\overline{q }-m\right){\omega }_{b}=0$ for passing particles, and $\omega -n{\omega }_{p}-l{\omega }_{b}=0$ for trapped particles, where $l$ is an integer number, $\overline{q }=\oint qd\theta /\oint d\theta$ with the poloidal angle $\theta$, ${\omega }_{b}=2\pi {\left(\oint d\theta /\dot{\theta }\right)}^{-1}$ is the transit/bounce frequency for passing/trapped particles, and ${\omega }_{p}={\left(2\pi \right)}^{-1}{\omega }_{b}\oint \left(\dot{\zeta }-q\dot{\theta }\right)d\theta /\dot{\theta }$ is the precessional frequency  (Zonca and Chen 2014), respectively. The relative strength of resonances can be inferred from the intensity of fast ion entropy $\delta {f}^{2}$ in the phase space using constants of motions of linear orbits $\left(E,\lambda \right)$, where $E$ is the kinetic energy and $\lambda =\mu {B}_{a}/E$ is the pitch angle with on-axis equilibrium magnetic field ${B}_{a}$. The trap-passing boundary at ${q}_{min}$ surface is roughly $\lambda =1-{r}_{0}/{R}_{0}$, where ${r}_{0}=0.158{R}_{0}$ is the distance from the radial location of ${q}_{min}$ at outer midplane to the magnetic axis with the on-axis major radius ${R}_{0}=1.732m$. In Fig. 6, the colored lines represent various resonances of trapped particles estimated from analytic model (White 2006) using the local parameters on the ${q}_{min}$ flux surface. The resonances appear to be very complicated because of multiple resonances mostly contributed by trapped particles. The dominant resonances are drift bounce resonances ${\omega }_{4}={\omega }_{d}+{\omega }_{b}$ and ${\omega }_{4}={\omega }_{d}+{2\omega }_{b}$ in both linear and nonlinear phases.

Fig. 6

Phase space $\left(\mathrm{E},\uplambda \right)$ structures of $\updelta {\mathrm{f}}^{2}$ in linear phase from single mode (n = 4) simulation with zonal flows. The colored lines are dominant resonances. Here ${\upomega }_{\mathrm{d}}=4{\upomega }_{\mathrm{p}}$

The amplitudes of the two frequencies (${\omega }_{d}$ and ${\omega }_{b}$) along the linear resonance lines appear to be comparable as shown in Fig. 7, i.e., there is no time scale separation between the bounce and precessional motion. Therefore, the nonlinear EP dynamics have intrinsically two-degree of freedom and need to be described in the phase space defined by two pairs of action-angle variables $\left(\zeta ,{P}_{\zeta }\right)$ and $\left(\theta ,{J}_{||}\right)$. This 2D dynamics may be responsible for the lack of a simple clump-hole pair across the resonance lines, contrary to that expected by the analytic model  (Berk et al. 1992; Hezaveh et al. 2022) based on the 1D nonlinear wave particle interaction.

Fig. 7

Dependence of precessional and bounce frequencies on EP energy E along RSAE resonance lines in Fig. 6 for drift bounce resonances ${\omega }_{4}={\omega }_{\mathrm{d}}+{\omega }_{\mathrm{b}}$ [panel (a)] and ${\omega }_{4}={\omega }_{\mathrm{d}}+{2\omega }_{\mathrm{b}}$ [panel (b)]

Despite the intrinsic two-degree of freedom, the dynamics system still preserves the magnetic moment$\mu$. We now choose some values of $\mu$ and analyze the nonlinear evolution of $\delta f$ in the phase space using the remaining two constants of motion $\left({P}_{\zeta },\lambda \right)$. Here, we focus on the $n=4$ RSAE with zonal flows. We first choose a magnetic moment of $\mu {B}_{a}=80keV$ to examine the nonlinear dynamics of dominant resonant particles, trapped particles. The phase space structures are present in Fig. 8 from linear phase in panel (a) to nonlinear saturation in panel (b) and late nonlinear stage in panel (c) and panel(d), where ${P}_{\zeta }$ is normalized by the poloidal magnetic flux function on the wall${\psi }_{w}$. Selecting the maximal $\delta {f}^{2}$ along ${P}_{\zeta }$ in Fig. 8a, the black line represents the dominant resonance in the linear phase. In the nonlinear phase, the clump–hole pairs form across the resonance line, where the red regions represent the positive value and blue regions represent the negative value. The phase space structures peak at $\left(-\mathrm{0.2,1.1}\right)$ in $\left({P}_{\zeta },\lambda \right)$ phase space, which means the dominant resonant particles are deeply trapped particles with l = 1 (Brochard et al. 2020) near ${q}_{min}$ surface. Near the nonlinear saturation, the structures broaden along the resonance line and extend in energy and radial directions. This may explain the radial extension of the mode structure as shown in Fig. 2b. After nonlinear saturation, the structures extend perpendicular to the resonance line and the co-passing resonant particles with $l=-7$ near trapped-passing boundary $\left({P}_{\zeta },\lambda \right)=\left(\mathrm{0.1,0.89}\right)$ also play important role, as shown in Fig. 8c. The clump–hole pairs lead to the flattening of EP profile, which reduces the drive of EP distribution function and the RSAE mode amplitude. From $0.4ms$ to$0.8ms$, the phase space structures barely change. As a result, the RSAE mode amplitude persists a lower level and never increases again. Moreover, two dominant negative regions which separated by the positive region near $\left({P}_{\zeta },\lambda \right)=\left(-\mathrm{0.2,1.1}\right)$ in linear phase and nonlinear saturation merge to form one large negative structure after$0.4ms$. The gradient of EP distribution function perpendicular to the resonance line decreases to almost $0$ from linear to nonlinear phase.

Fig. 8

Perturbed beam ion distribution function $\mathrm{\delta f}$ in $\left({\mathrm{P}}_{\upzeta },\uplambda \right)$ phase space at $0.2\mathrm{ms}$ [panel (a)], $0.27\mathrm{ms}$ [panel (b)], $0.4\mathrm{ms}$ [panel (c)], and $0.8\mathrm{ms}$ [panel d] from single mode (n = 4) simulation with zonal flows for a magnetic moment $\upmu {\mathrm{B}}_{\mathrm{a}}=80\mathrm{keV}$. The black line represents the dominant resonance

We then choose a magnetic moment of $\mu {B}_{a}=25keV$ to examine the phase space structure contributed by both passing and trapped particles. The phase space structures are plotted in Fig. 9 from linear phase in panel (a) to nonlinear saturation in panel (b) and late nonlinear stage in panel (c) and panel(d). In the linear phase, the dominant resonant particles are passing particles, where the left dominant structure is contributed by counter-passing particles with $l=14$ near $\left({P}_{\zeta },\lambda \right)=\left(-\mathrm{0.66,0.38}\right)$ and right one by co-passing particles with $l=10$ near $\left({P}_{\zeta },\lambda \right)=\left(\mathrm{0.2,0.4}\right)$. Near the nonlinear saturation, passing particles still dominate the resonance, as shown in Fig. 9b. After nonlinear saturation, trapped particles also play important role. Again, the flattened EP profile reduces the drive of EP distribution perturbation and RSAE mode amplitude.

Fig. 9

Perturbed beam ion distribution function $\mathrm{\delta f}$ in $\left({\mathrm{P}}_{\upzeta },\uplambda \right)$ phase space at $0.2\mathrm{ms}$ [panel (a)], $0.27\mathrm{ms}$ [panel (b)], $0.4\mathrm{ms}$ [panel (c)], and $0.8\mathrm{ms}$ [panel (d)] from single mode (n = 4) simulation with zonal flows for a magnetic moment $\upmu {\mathrm{B}}_{\mathrm{a}}=25\mathrm{keV}$. The black line represents the dominant resonance

4 Collisional effects on the RSAE nonlinear dynamics

We now perform nonlinear simulations of the most unstable n = 4 RSAE to study effects of Coulomb collisions, which can destroy the EP coherent phase space structures. The effects of collisions on the RSAE amplitude and EP transport level depend on the ratio of the RSAE growth rate and effective collision frequencies. The slowing down rate of beam ions is ${\nu }_{fe}^{\epsilon }=19{s}^{-1}$ due to thermal electrons and ${\nu }_{fi}^{\epsilon }=26{s}^{-1}$ due to thermal ions, and the pitch angle scattering rate due to thermal ions is ${\nu }_{fi}^{scatt}=21{s}^{-1}$ for plasma parameters near ${q}_{min}$ flux surface, i.e., electron density ${n}_{e}=3.1\times {10}^{13}c{m}^{-3}$, beam ion density ${n}_{f}=1.95\times {10}^{12}c{m}^{-3}$, electron temperature ${T}_{e}=850eV$, thermal ion temperature ${T}_{i}=935eV$, beam ion temperature ${T}_{f}=20keV$, and effective charge ${Z}_{eff}=1.66$. The effective collision frequency  (Hinton and Hazeltine 1976) is much larger than the collision frequency for the process of collisional diffusion across the small phase space structure associated with the sharp resonance due to the fact that the RSAE $\gamma /\omega \ll 1$.

To compare with earlier RSAE simulations  (Lang et al. 2010; Chen et al. 2013), we first study collisional effects in GTC simulations without zonal flows by including pitch angle scattering of the EP due to thermal ions. GTC simulations find that collisional effects on both RSAE saturation amplitude and associated fast ion diffusivity are insignificant when using realistic collision frequency. When we use a collision frequency $10$ times or $100$ times of the realistic frequency, collisional effects enhances both RSAE saturation amplitude and associated EP diffusivity, but the effects are still very weak at the late nonlinear stage as shown in Fig. 10(a). If the collision frequency is further amplified to 1000 times, a quasi-steady state of AE amplitude and EP transport can finally be obtained. The results that quasi-steady state of AE turbulence can be maintained with collisions and that AE saturation amplitude increases with the collision frequency are qualitatively consistent with analytical theory  (Berk and Breizman 1990) and earlier kinetic-MHD M3D-K  (Lang et al. 2010) as well as gyrokinetic GEM simulations  (Chen et al. 2013). However, when zonal flows are kept in the GTC simulations, collisional effects on the AE amplitude and EP diffusivity are insignificant, even the collision frequency is amplified by $100$ times as shown in Fig. 10(b). The neoclassic transport coefficient of EP can be estimated by the analytic theory  (Lin et al. 1995) in the banana regime as $D=0.2{m}^{2}/s$ for realistic collision frequency ${\nu }_{fi}\sim 20{s}^{-1}$ and in the collisional regime as $D=15{m}^{2}/s$ for $\nu =1000{\nu }_{fi}$, indicates that the neoclassic EP diffusivity is much smaller than the EP diffusivity induced by the RSAE.

Fig.10

Time history of fast ion diffusivity from single mode simulations of n = 4 RSAE with [panel (b)] and without [panel (a)] zonal flows in GTC simulation with EP pitch angle scattering

We conclude that the EP collisional effects on the RSAE nonlinear dynamics are negligible for the realistic collision frequency in the DIII-D plasmas. This conclusion is further supported by the estimates of relevant time scales, which show that the RSAE linear growth rate is much larger than the Coulomb collision frequency for the EP. Since resonant beam ions only occupy a small region of the EP phase space, the effective collisional diffusion rate can be amplified using a heuristic factor of ${\left({\omega }_{4}/{\gamma }_{4}\right)}^{2}$. The effective collisional diffusion rate ${\left({\omega }_{4}/{\gamma }_{4}\right)}^{2}{\nu }_{f}\sim 0.09{\gamma }_{4}$ is still an order smaller than the RSAE linear growth rate. We note that the fast ion replacement rate (defined as inverse of the fast ion confinement time) is similar to the Coulomb collision frequency. Therefore, the NBI source has negligible effects on the RSAE amplitude and EP transport.

5 Nonlinear simulations of multiple toroidal modes

We now perform the nonlinear simulation where multiple toroidal modes n = [0,10] of the RSAE are kept and interact with each other. As shown in Fig. 11, the normalized electrostatic potentials $\delta \phi$ of the most unstable $n=4$ RSAE on the ${q}_{min}$ surface grow to large amplitudes and saturate by the self-generated zonal flows and zonal structures, which grow with a growth rate of $\gamma \sim {2\gamma }_{4}$. Before the saturation of the $n=4$ mode at about $0.24ms$, the linearly stable modes such as n = 2 and weakly unstable modes such as $n=10$ are first generated by numerical coupling of the $n=4$ mode with linear growth rate $\gamma \sim {\gamma }_{4}$. After $0.24ms$, $n=2$ and $n=10$ modes are then generated by nonlinear mode coupling of the strongly unstable modes with linear growth rate $\gamma \sim {2\gamma }_{4}$, similar to earlier kinetic-MHD HMGC simulations  (Vlad et al. 2018). Moreover, the displaced EPs could also drive more unstable modes in regions that are originally stable. Compared with the single toroidal mode simulations, the saturation amplitude of the $n=4$ mode is reduced by $20\%$, while the saturation amplitude of less unstable mode $n=6$ is reduced by $5$ times despite no change in the linear growth rate. At the late nonlinear stage, all mode amplitudes and EP transport drop to a very low level.

Fig. 11

Time histories of perturbed electrostatic potentials $\mathrm{e\delta \phi }/{\mathrm{T}}_{\mathrm{e}}$ on the ${\mathrm{q}}_{\mathrm{min}}$ flux surface [panel (a)] and averaged effective fast ion diffusivity [panel (b)] from multiple modes simulation of $n=[\mathrm{0,10}]$

The poloidal mode structures from GTC simulation of multiple modes n = [0,10] RSAE are shown in Fig. 12 for the linear phase [panel (a)], at the nonlinear saturation [panel (b)], and the late nonlinear stage [panel (c)]. The corresponding radial profiles of dominant poloidal harmonics are shown in Fig. 13. We find that the most unstable mode $n=4$ dominates in the linear phase and at the nonlinear saturation, and peaks near the ${q}_{min}$ flux surface. At the late nonlinear stage, the $n=4$, $m=12$ harmonic diminishes near the ${q}_{min}$ surface. However, a TAE-like mode of $n=4$ with strong ballooning structure outside the ${q}_{min}$ flux surface is generated nonlinearly, as shown in Figs.12c and 13c. This TAE is not generated by outward EP transport driven by the RSAE turbulence since the global pressure profiles of all species are fixed in all simulations.

Fig. 12

Poloidal contour plots of perturbed electrostatic potential $\mathrm{e\delta \phi }/{\mathrm{T}}_{\mathrm{e}}$ at linear phase [panel (a)], nonlinear saturation [panel (b)] and late nonlinear stage [panel (c)] from multiple modes simulation of $n=[\mathrm{0,10}]$

Fig. 13

Radial profiles of poloidal harmonics in linear phase [panel a], nonlinear saturation [panel b] and late nonlinear stage [panel c] from multiple modes simulation of $n=[\mathrm{0,10}]$

The instantaneous frequency and amplitude for the electrostatic potential $e\delta \phi /{T}_{e}$ of the $m=12$ harmonic near the $q=3$ flux surface, where the TAE amplitude peaks, are shown in Fig. 14. In linear phase, this harmonic grows exponentially with the linear growth rate and frequency of the $n=4$ RSAE near the ${q}_{min}$ surface. Right before the RSAE saturation, the frequency starts to increase, and then splits into two branches ($\sim 86kHz$ TAE and $\sim 142kHz$ EAE) at the nonlinear saturation ($0.23ms$). During the frequency chirping from RSAE to TAE, the mode growth rate does not change, which does not support the resonant mode coupling. It seems that the nonlinear frequency upshift changes the mode from RSAE to TAE because of frequency and mode structure overlap. The lower frequency TAE branch quickly develops two sidebands ($\sim 65kHz$ and $\sim 90kHz$) accompanied by the drop of mode amplitude due to the rotation of phase space coherent structures  (Fasoli et al. 1998). At$0.28ms$, the TAE amplitude drops to the lowest level, and the frequency differences of the two TAE sidebands reaches the largest value. Then the two TAE sidebands start to merge accompanied by the increase of the TAE amplitude. After the second frequency bifurcation, a unique TAE persists with a frequency of$\sim 86kHz$. Besides the TAE, EAE with frequency $\sim 142kHz$ is also nonlinearly generated after RSAE saturation.

Fig. 14

Time history of frequency spectrum (left) and perturbed electrostatic potential $\mathrm{e\delta \phi }/{\mathrm{T}}_{\mathrm{e}}$ of $m=12$ harmonic near the $q=3$ flux surface (right)

The identification of the TAE and EAE is supported by the Alfven continua (Deng et al. 2012a) as shown in Fig. 15, where the three frequencies for the n = 4 mode observed in the GTC simulation lie in the TAE and EAE gaps.

Fig. 15

Alfven continua of $n=4$. The colored lines represent various poloidal harmonics and the orange dashed line is $\mathrm{q}$ profile. Horizontal blue lines represent frequency, radial location and width for RSAE, TAE, and EAE observed in GTC simulation

Unlike the quickly diminished RSAE, the TAE and EAE modes sustain a finite level from $0.4ms$ to the end of simulation. The TAE and EAE amplitudes are smaller than the RSAE amplitude at the RSAE saturation but larger than that at the late nonlinear stage. The EP diffusivity shown in Fig. 11b exponentially grows with a growth rate of $2{\gamma }_{4}$ at linear phase. Compared with EP diffusivity from the single mode simulation of $n=4$ with zonal flows (green line), the averaged EP diffusivity in the multi-mode simulation (black line) is further reduced to $8{m}^{2}/s$ at the RSAE saturation, but also quickly drops to almost 0 in $0.05ms$. The EP diffusivity ${D}_{f}<0.2{m}^{2}/s$ near the TAE and EAE mode location is also smaller than the experimental observations.

6 Conclusion and discussion

Using equilibrium geometry and plasma profiles from DIII-D discharge #159,243, nonlinear global gyrokinetic GTC simulations of the RSAE excited by beam ions have been performed to investigate the nonlinear evolution of mode amplitude and associated EP transport. GTC simulations find:

• (a) Self-generated zonal flows dominate the RSAE nonlinear saturation by suppressing both mode amplitude and EP transport at the nonlinear saturation.

• (b) The flattening of EP profile at the resonance reduces the EP drive and quenches the RSAE after nonlinear saturation.

• (c) Effects of EP Coulomb collisions on the RSAE nonlinear dynamics are negligible when using the realistic collision frequency.

• (d) Nonlinear interactions between multiple toroidal modes reduce RSAE amplitudes and EP transport.

• (e) TAE and EAE in the outer edge can be nonlinearly generated by the RSAE.

• (f) Both single mode and multiple modes simulations find a huge initial burst followed by a quickly diminished RSAE amplitude and EP transport, which is inconsistent with experimental observations.

The most important finding from our simulation is that both mode saturation amplitude and associated EP transport are much higher than the experimental observations at the nonlinear saturation, and that diffusivity drops to almost $0$ within 0.05 ms after nonlinear saturation. Therefore, nonlinear physics of multiple toroidal modes cannot explain the RSAE amplitude and EP transport measured in the DIII-D experiments. Additional dissipation or background micro-turbulence are believed to affect the AE saturation and EP transport  (Zonca et al. 2015; Liu et al. 2022; Bass and Waltz 2010; Duarte et al. 2017; Zeeland et al. 2019; Gorelenkov and Duarte 2021; Biancalani et al. 2021), and integrated simulations incorporating multiple physical processes are required. It should be noted that the present conclusion could also be affected by additional effects, such as slow evolution of the equilibrium profile and EP distribution function, which requires the long time (on the collision time scale) full-f simulation.