Nonlinear excitation of a geodesic acoustic mode by reversed shear Alfvén eignemodes

1.   Introduction

Good confinement of energetic particles (EPs), generated by fusion reaction or auxiliary heating such as neutral beam injection, is crucial to the performance of future tokamak devices such as ITER [1, 2] and CFETR [3], since the EPs are needed to be sufficiently well confined to heat thermal plasmas and maintain them in optimal conditions for self-sustained fusion reactor. However, shear Alfvén wave (SAW) instabilities, including Alfvén eigenmodes [4] (AEs) and EP continuum modes [5], could be resonantly excited by the free energy of the nonuniform EP distribution. As finite amplitude SAW instabilities may cause significant anomalous EP transport and performance degradation, theoretical analysis on the SAW nonlinear dynamics has been a key topic of continuous research, see [6] for a recent in-depth review.

In this work, we focus on the typical parameter regime of future burning plasmas operated at steady-state, where a significant fraction of plasma current is generated non-inductively. Since the radial profile of the non-inductive (e.g. bootstrap) current generally peaks off-axis, the magnetic shear in the plasma core region is reversed in this scenario, i.e. the safety factor q profile contains an off-axis local minimum $q_{\min }$ . As first identified during the advanced operation in JET [7], a specific branch of SAW, namely, the reversed shear Alfvén eigenmode (RSAE, an alternative name is Alfvén cascades) exists close to the flux surface of $q_{\min }$ , and can be readily driven unstable by EPs [8–10]. Considering the fact that the EP pressure gradient also peaks in the core region, one may reasonably expect that, among all the branches of the SAW instabilities, RSAEs are firstly excited as the EP pressure profile ramps up in time. Recent nonlinear simulation analyses suggest that, the core-localized RSAEs could efficiently induce convective type of EP radial transport, along with further impact on the global EP confinement property [11, 12]. In particular, the modulation to the EP density profile by RSAE may enhance the drive to the toroidal Alfvén eigenmodes (TAEs) [4], which generally locate in the outer-core region with finite magnetic shear [13] and are a dangerous concern for direct EP loss and damage to the plasma facing components [14]. Thus, in order to predict the steady-state performance of a burning plasma device (such as ITER), it is of great importance to study the nonlinear evolution of RSAE, complementary to the research activities focusing on TAE [6].

As clarified in [12, 15], the dynamics of an AE activity could be subdivided into three stages. In the 'linear' stage where the AE is exponentially growing, the important physics include the wave-particle linear resonance condition and forced drive of secondary modes including axisymmetric zero frequency zonal structures (ZS) [16]. Wave-particle trapping takes place in the following 'saturation' stage [17]. Note that the first two stages generally occur in extremely short timescales, which are characterized by $\mathcal{O}\left(\gamma_{\mathrm{L}}^{-1}\right)$ and $\mathcal{O}\left(\omega_{\mathrm{B}}^{-1}\right)$, respectively. Here, γ_L is the linear AE growth rate, and is independent of the mode amplitude, is the wave-particle trapping frequency, with |δϕ| being the mode amplitude. After the initial saturation stage due to wave-EP nonlinearity, wave–wave nonlinearity dominates in the longer timescale 'saturated' stage with a time scale ∝ 1/|δϕ|, where the AE spectral energy could spontaneously decay into linearly stable domains [18]. This nonlinear spontaneous decay process is sometimes classified into the alternative AE saturation mechanism via wave–wave nonlinear coupling [19], to be considered on the same footing with wave-EP nonlinearity. Since the RSAEs are mostly observed with quasi-static amplitudes rather than bursts, the wave–wave couplings in the 'saturated' stage have significant implication to the long timescale, global performance of fusion plasmas, where one could consider the initial state as a RSAE pump wave with a prescribed mode structure and finite amplitude for a simplified analytical study [20]. Furthermore, note the lowest order RSAE dispersion relation, $\omega^2 \simeq k_{\|}^2 v_{\mathrm{A}}^2+\omega_{\mathrm{G}}^2$, with the wavenumber parallel to the equilibrium magnetic field, m/n the poloidal/toroidal mode numbers, respectively, R₀ the major radius, v_A the Alfvén speed, and ω_G the frequency of geodesic acoustic mode (GAM, see section 3.1) [21] corresponding to geodesic curvature induced SAW continuum upshift. Consequently, the RSAE spectra are sensitive to the underlying toroidal mode number n, and RSAEs with different n are generally discrete eigenstates with clear frequency separations [13]. Thus, besides direct coupling between RSAEs with different n giving rise to the generation of low frequency AEs, other mechanisms usually involve the spontaneous excitation of n = 0, $\mathit{m} \simeq 0$ ZS. Here, we have also taken into account the fact that magnetic shear vanishes around , which makes each RSAE dominated by only one or two poloidal mode numbers [8]. Indeed, it has been recently shown that a RSAE pump wave could generate both electrostatic zonal flow as well as electromagnetic zonal current via modulational instability, where the zonal current could play a potentially important role in the long timescale RSAE saturation mechanism [20]. Alternatively, as we show in the present paper, if a critical amplitude threshold is exceeded, a RSAE pump wave could parametrically decay into a finite frequency GAM and a lower sideband kinetic RSAE (KRSAE). Analogous process for TAE has been recently published in [22, 23], which suggests the important role played by the thermal ion β_i value. Here, β is ratio between kinetic and magnetic pressures. We consider the high β_i limit in this work, , so the GAM frequency is larger than the difference between pump RSAE frequency and continuum accumulation point frequency, consistent with the parameter regime of future burning plasmas as mentioned above. Here, ϵ₀ is the inverse aspect ratio.

The physics mechanism presented in this work could have significant implication to the operation of burning plasma devices. On one hand, the GAM generated in this process, as a finite frequency branch of electrostatic ZS, could regulate drift wave turbulence [24] and improve plasma confinement. On the other hand, the GAM mainly suffers from thermal ion Landau damping in the plasma core region [25], and thus, it provides a viable route to alpha channelling [26]. In fact, the present work is consistent with the recent observation of high thermal ion temperature and improved confinement regime in JET with the presence of ~MeV EPs and rich RSAE activity [27]. Thus, we could reasonably expect this mechanism to be at work in future burning plasmas.

The rest of the paper is organized as follows. In section 2, the basic equations of the theoretical model are given. In section 3, the parametric decay process of the pump RSAE into a GAM and a KRSAE is investigated. Finally, conclusions and discussions are given in section 4.

2.   Theoretical model

We investigate the process of a pump RSAE (Ω₀ ≡ (ω₀, k₀)) decaying into a GAM (Ω_G ≡ (ω_G, k_G)) and a KRSAE (Ω_K ≡ (ω_K, k_K)) with the same toroidal and poloidal mode numbers as the pump RSAE in high β_i limit, as shown in figure 1. For simplicity of discussion while without loss of generality, we considered the case with RSAE frequency above the local maximum of SAW continuum, which corresponds to ${\mathit{nq}}_{\min }-m<0$ , as discussed in [8]. Extension of the present analysis to more general cases is straightforward.

Figure  1.  Cartoon of the pump RSAE decay into a GAM and a KRSAE in the high β_i limit.

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For the present analysis of nonlinear interactions between RSAE and GAM, the scalar potential δϕ and parallel component of vector potential δA_‖ are adopted as the field variables, while the parallel magnetic perturbation is negligible in consistency with the typical low thermal to magnetic pressure ratio. Furthermore, for convenience, δψ ≡ ωδA_‖ /(ck_‖) is taken as an alternative field variable for RSAE and KRSAE, where the ideal MHD limit can be achieved by taking δψ = δϕ. The scalar potential perturbation contains δϕ = δϕ₀ + δϕ_G + δϕ_K, with the subscript 0, G and K denoting the pump RSAE, GAM and KRSAE, respectively. The frequency and wavenumber matching conditions for the resonant decay are adopted as Ω₀ = Ω_G + Ω_K. Considering single-n RSAE/KRSAE with one dominant poloidal mode m, the perturbations of RSAE and KRSAE in straight-field-line (r, $\vartheta$,φ) coordinates can be expressed as

where ${\widehat{k}}_{\mathrm{G}}$ is the radial structure modification due to GAM excitation, and ϕ is the amplitude of the corresponding modes.

On the other hand, the GAM is assumed to be electrostatic and the higher order electromagnetic corrections [28] are neglected, and the GAM scalar potential can be represented as

with ${\widehat{k}}_{\mathrm{G}}$ being the GAM radial wavenumber.

The kinetic effects, such as finite Larmor radius (FLR) and wave-particle resonances, may quantitatively and qualitatively change the KRSAE behaviours from the MHD limit, especially for k_⊥ρ_i ~ 1. Thus a gyrokinetic theory is needed to obtain the nonlinear nonadiabatic electron response to KRSAE due to the kinetic effects. Applying a nonlinear gyrokinetic approach is also crucial in that, the parametric decay process is dominated by the contribution of Reynolds stress in the radially fast varying inertial layer [29]. The governing equations, describing the parametric decay process, can be derived from the quasi-neutrality condition

and nonlinear gyrokinetic vorticity equation [30]

with $\langle\cdots\rangle$ denoting velocity space integration. δH_k is the nonadiabatic particle response, which can be derived from the nonlinear gyrokinetic equation [31]

Here, J_k ≡ J₀(k_⊥ρ) with J₀ being the Bessel function of zero index accounting for FLR effects, ρ = v_⊥ /Ω_c is the Larmor radius, Ω_c = eB/(mc) is the cyclotron frequency, e and c are the unite charge and the speed of light, respectively. Furthermore, F₀ is the equilibrium particle distribution function, $\omega_{\mathrm{d}}=\left(v_{\perp}^2+2 v_{\|}^2\right) /\left(2 \Omega R_0\right)\left(k_r \sin \vartheta+k_{\vartheta} \cos \vartheta\right)$ is the magnetic drift frequency associated with magnetic curvature and gradient, l is the arc length along the equilibrium magnetic field line, δL_k ≡ δϕ_k - k_‖v_‖δψ_k /ω_k is the scalar potential in the frame moving with the particle's guiding center along B₀ [29], and other notations are standard. The free energy associated with pressure gradient of thermal plasma is neglected in equation (6), assuming that the free energy destabilizing the pump RSAE is due to the EP pressure gradient, while nonlinear mode coupling is dominated by non-resonant thermal plasma contribution. For the cases with EPs playing important roles in nonlinear mode coupling, interested readers may refer to [16].

3.   Parametric decay dispersion relation

3.1   Nonlinear GAM equation

The dispersion relation of GAM can be derived from the nonlinear surface averaged vorticity equation as [23],

with $\mathcal{E}_{\mathrm{G}} \equiv\left\langle\left(1-J_{\mathrm{G}}^2\right) F_0 / n_0\right\rangle-\left(T_{\mathrm{i}} /\left(n_0 e^2\right)\right) \overline{\left\langle e \omega_{\mathrm{d}}\left(J_{\mathrm{G}} \delta H_{\mathrm{G}, \mathrm{i}}^{\mathrm{L}}-\delta H_{\mathrm{G}, \mathrm{e}}^{\mathrm{L}}\right) / \omega_{\mathrm{G}}\right\rangle} / \delta \phi_{\mathrm{G}}$ being the GAM dispersion relation [25], and to the lowest order, one has $\mathcal{E}_{\mathrm{G}} \sim b_{\mathrm{G}}\left(1-\omega_{\mathrm{G}}^2 / \omega^2\right)$ with $\omega_{\mathrm{G}}^2 \simeq(7 / 4+\tau) v_{\mathrm{i}}^2 / R_0^2$, τ ≡ T_e/T_i and . The nonlinear coupling coefficient, $A_{\mathrm{G}} \equiv \Gamma_{\mathrm{K}}-\Gamma_0-\left(b_0-b_{\mathrm{K}}\right) k_{\|}^2 v_{\mathrm{A}}^2 \sigma_0 \sigma_{\mathrm{K}^*} /\left(\omega_0 \omega_{\mathrm{K}}\right)$, is valid for arbitrary wavelengths [29], and contains the contribution from nonlinear Reynolds and Maxwell stresses, respectively [18]. Here, $\Gamma_k \equiv\left\langle J_k^2 F_0 / n_0\right\rangle, \sigma_k \equiv \delta \psi_k / \delta \phi_k=1+\tau-\tau \Gamma_k$, and σ_k ≠ 1 corresponds to deviation from ideal MHD limit due to FLR effects. In deriving equation (7), the leading order linear ion responses to RSAE and KRSAE are derived and substituted into the Reynolds stress term of equation (5). For the convenience of the readers, the lowest order particle response to RSAE/KRSAE is $\delta H_{\mathrm{R}, \mathrm{e}}^{\mathrm{L}} \simeq-\left(e / T_{\mathrm{e}}\right) F_0 \delta \psi_{\mathrm{R}}$, $\delta H_{\mathrm{R}, \mathrm{i}}^{\mathrm{L}} \simeq\left(e / T_{\mathrm{i}}\right) F_0 J_{\mathrm{R}} \delta \phi_{\mathrm{R}}$, by noting the |k_‖,Rv_{t, e}| ≫ |ω_R| ≫|k_‖,Rv_{t, i}|, |ω_d| ordering. For particle response to GAM with and k_‖,G = 0, the linear nonadiabatic response to GAM can be derived to the leading order as , [25], with $\overline{(\cdots)} \equiv \int \mathrm{d} \vartheta(\cdots) /(2 \pi)$ denoting surface averaging.

3.2   Nonlinear KRSAE equation

The nonlinear electron response to KRSAE can be derived from the nonlinear gyrokinetic equation by substituting the linear GAM and RSAE responses into the nonlinear term of equation (6), and taking the |k_‖,K v_t,e| ≫ |ω_K| ≫ |ω_d,e| ordering, and one has $\delta H_{\mathrm{K}, \mathrm{e}}^{\mathrm{NL}}=-\mathrm{i} e c k_{0, \vartheta} k_{\mathrm{G}} \delta \phi_{\mathrm{G}^*} \delta \psi_0 /\left(T_{\mathrm{e}} B_0 \omega_0\right)$. On the other hand, the nonlinear ion response to KRSAE can be derived as $\delta H_{\mathrm{K}, \mathrm{i}}^{\mathrm{NL}}=-\mathrm{i} e c J_0 J_{\mathrm{G}} F_0 k_{0, \vartheta} k_{\mathrm{G}} k_{\|, 0} v_{\|}$$\delta \phi_0 \delta \phi_{\mathrm{G}^*} /\left(T_{\mathrm{i}} B_0 \omega_{\mathrm{K}} \omega_0\right)$, noting the |ω_K| ≫ |k_‖,Kv_{t, i}| ≫ |ω_{d, i}| ordering. Substituting the linear and nonlinear particle responses to KRSAE into the quasi-neutrality condition, we can obtain

describing the deviation from ideal MHD condition due to FLR effects (denoted by σ_K ≠ 1), as well as nonlinear electron response to KRSAE. The nonlinear ion nonadiabatic response $\delta H_{\mathrm{K},\mathrm{i}}^{\mathrm{NL}}$ is an odd function of v_‖,and gives no contribution to the quasi-neutrality condition.

Substituting equation (8) into the nonlinear vorticity equation, we can obtain the dispersion relation of KRSAE

where $\mathcal{E}_{\mathrm{K}} \equiv 1-\Gamma_{\mathrm{K}}-k_{\mathrm{K}, \|}^2 v_{\mathrm{A}}^2 \sigma_{\mathrm{K}} b_{\mathrm{K}} / \omega_{\mathrm{K}}^2+\Delta_{\mathrm{T}}$ is the WKB dispersion relation of KRSAE with ∆_T accounting for kinetic compression of both thermal and EPs. One can derive the KRSAE eigenmode dispersion relation by transforming into Fourier space [8, 32]

Here, Γ(···) is the Euler Gamma function, $\hat{G} \equiv-\alpha^2\left(\Omega^2-\right.\left.\Omega_{\mathrm{AM}}^2\right) / \Theta_m$, $\alpha^4=(1 / 4) \Theta_m \rho_{\mathrm{i}}^{-2}$, Θ_m = Ω_AMS²/n, $\Omega_{\mathrm{AM}}=n q_{\min }-m$, $S \equiv \sqrt{q_{\min }^{\prime \prime} r_0^2 / q_{\min }^2}$, r₀ is the $\delta \hat{W}_{\mathrm{f}}$ is the fluid contribution to the potential energy [6]. The nonlinear coupling coefficient A_K = Γ₀ - Γ_G - σ₀ω_K(1 - Γ_K)/(σ_Kω₀), with the first two terms (Γ₀ - Γ_G) from ion nonlinearity through Reynold stress, and the third term from the nonlinear electron correction to ideal MHD constraint as described by equation (9).

3.3   Nonlinear parametric dispersion relation

The nonlinear dispersion relation can be derived from equations (7) and (9) as

where the nonlinear drive $\mathcal{D}$ is given as

Since GAM and KRSAE are both considered as normal modes of the system, in the local limit, $\mathcal{E}_{\mathrm{G}}$and $\hat{\mathcal{E}}_{\mathrm{K}}$ can be expanded along the characteristics of GAM and KRSAE as $\mathcal{E}_{\mathrm{G}} \simeq-2 \mathrm{i} b_{\mathrm{G}}\left(\gamma+\gamma_{\mathrm{G}}\right) / \omega_{\mathrm{G}}$, $\hat{\mathcal{E}}_{\mathrm{K}^*} \simeq \mathrm{i} \partial_{\omega_0} \hat{\mathcal{E}}_{\mathrm{K}, \mathcal{R}}\left(\gamma+\gamma_{\mathrm{K}}\right)$, where $\gamma_{\mathrm{G}} \equiv-\mathcal{E}_{\mathrm{G}, \mathcal{I}} /\left(\partial_{\omega_{\mathrm{G}}} \mathcal{E}_{\mathrm{G}, \mathcal{R}}\right)$ and $\gamma_{\mathrm{K}} \equiv-\mathcal{E}_{\mathrm{K}, I} /\left(\partial_{\omega_{\mathrm{K}}} \mathcal{E}_{\mathrm{K}, \mathcal{R}}\right)$ are the dissipation rates of GAM and KRSAE due to electron and ion contributions to Landau damping. The subscripts '$\mathcal{R}$' and '$\mathcal{I}$' denote the real and imaginary parts, respectively.

The parametric dispersion relation can be rewritten as

The threshold condition for the parametric decay process can be obtained by taking γ = 0 in equation (12), i.e.

For typical KRSAE with $\left|k_{\perp} \rho_{\mathrm{i}}\right| \simeq 1$, equation (12) is a complex integro-differential equation, which usually requires numerical investigation. However, it can be analyzed in the |k_⊥ρ_i| ≪ 1 limit to give some insights of the excitation condition of the present nonlinear process. With the |k_⊥ρ_i| ≪ 1 assumption, one obtains Γ_k ≈ 1 - b_k and $\sigma_k \simeq 1+\tau b_k$,which can be used to estimate $A_{\mathrm{K}} \simeq\left[\left(b_{\mathrm{G}}-b_0\right)-b_{\mathrm{K}}\left(\omega_{\mathrm{K}} / \omega_0\right)\left(1+\tau b_0\right) /\right.\left.\left(1+\tau b_{\mathrm{K}}\right)\right]>0$, $A_{\mathrm{G}} \simeq b_{\mathrm{G}}\left(1-\omega_{\mathrm{AM}}^2 /\left(\omega_{\mathrm{K}} \omega_0\right)\right)>0$, $\omega_K \partial_{\omega_K} \mathcal{E}_K>0$, where $\omega_{\mathrm{AM}}^2=v_{\mathrm{A}}^2\left(n q_{\min }-m\right)^2 /\left(q_{\min } R_0\right)^2$. It can then be shown that, $\hat{\mathcal{D}}$ is positive, i.e. the pump RSAE will spontaneously drive GAM and KRSAE, if its amplitude exceeds the threshold. Considering that $\left|\delta B_r\right| \simeq\left|k_{\vartheta} \delta A_{\|}\right| \simeq\left|c k_{\vartheta} k_{\|} \delta \phi / \omega\right|$ and the matching condition $\left|k_{\mathrm{G}}\right|=\left|k_{r, 0}+k_{r, \mathrm{~K}}\right| \simeq\left|k_{r, \mathrm{~K}}\right|$, it can be estimated as

Thus, the parametric decay process can be relevant for RSAE nonlinear saturation, with a comparable cross-section with other dominant nonlinear mode coupling channels, e.g. zero-frequency zonal structure generation [20]. In estimating the threshold condition in equation (14), $\left(1-{\omega }_{\mathrm{AM}}^2/({\omega }_{\mathrm{K}}{\omega }_0)\right)~\mathrm{∆}~{ϵ}_0$ is assumed, and other parameters are used as in typical tokamak, e.g. γ_K/ω₀ ~ γ_G/ω₀ ~ 10^-2, k_‖ρ_i ~ 10^-3 and k_Gρ_i ≤ 1 [33].

If the nonlinear drive is well above the threshold, the growth rate of the GAM is proportional to the pump RSAE amplitude, i.e. $\gamma \propto\left|\delta \phi_0\right|$. This is typical for ZF excitation via parametric decay insability; meanwhile, for the forced driven process [16], the nonlinear growth rate is characterized by $\gamma \simeq 2 \gamma_\mathrm{L}$, as typically observed in numerical simulations [34].

In addition to the pump RSAE amplitude threshold as a necessary condition, we note that several other conditions are required for the parametric decay process to occur. First, GAM and KRSAE should be weakly damped normal modes. For GAM dominated by the ion Landau damping due to the thermal ion transit resonance, it requires $q_{\min }\sqrt{7/4+\tau }>1$ , which is generally satisfied in the advanced scenarios. Second, the GAM frequency ω_G must be larger than the difference between the pump RSAE frequency and the continuum accumulation point frequency ω_AM, i.e. ${\omega }_{\mathrm{G}}>{\omega }_0-{\omega }_{\mathrm{AM}}~\widehat{d}{\omega }_{\mathrm{A}}$ with $\widehat{d}$ representing the shift of RSAE frequency from SAW continuum accumulation point due to, e.g. non-perturbative EP effects [13]. For an order of magnitude estimation [6, 13], one typically has, $\widehat{d}~{\gamma }_{\mathrm{L}}/{\omega }_0$ . Noting that ${\omega }_{\mathrm{G}}/{\omega }_0~q\sqrt{{\beta }_{\mathrm{i}}}$ , this condition thus, sets the lower limit of thermal ion pressure ratio β_i for this nonlinear process to occur.

4.   Conclusions and discussions

In this work, using nonlinear gyrokinetic theory, we have investigated the nonlinear decay mechanism of a pump RSAE into a GAM and a KRSAE with the same toroidal and poloidal mode numbers as the pump RSAE. This channel of RSAE spectral energy transfer is important for future fusion reactors due to the core localized fusion alpha particle distribution, which may preferentially drive RSAEs in the advanced reversed shear scenarios. The investigated parametric decay process can take place due to the dense KRSAE spectrum, which makes the frequency and wavenumber matching conditions for the parametric decay process possible.

For the process to be prevalent in fusion reactors, the threshold condition of the parametric instability is $\hat{\mathcal{D}}_{\mathrm{th}}=\gamma_{\mathrm{K}} \gamma_{\mathrm{G}}$, i.e. the pump RSAE drive should overcome the threshold due to KRSAE and GAM dissipation to drive the parametric decay instability, and the threshold of pump RSAE amplitude can be estimated as $\delta B_r / B_0 \sim \mathcal{O}\left(10^{-4}\right)$. In addition, the GAM and KRSAE are required to be weakly damped normal modes, and the GAM frequency should be larger than the difference between the pump RSAE frequency and the continuum accumulation point frequency, which is determined by the underlying equilibrium profiles as well as the non-perturbative EP effects [13]. The corresponding criteria are estimated as, respectively, and , which are generally satisfied in burning plasma scenarios.

Note that other mode coupling processes are also possible in fusion plasmas, e.g. a RSAE pump wave could generate zero frequency ZF via modulational instability, as investigated in [20]. The relative importance of these processes in a realistic scenario requires further analyses, before quantitative estimation of RSAE saturation level and the related heating/transport can be made. However, the investigated parametric decay process can provide a novel mechanism of alpha channelling [26] that may effectively heat core thermal plasmas in the future fusion device. The pump RSAE, excited by alpha particles, transfers energy to a GAM and a KRSAE through the parametric decay process. KRSAE and GAM mainly lose energy to electrons and ions, respectively [23], leading to core localized plasma heating. The heating rate can be obtained by estimating the saturation level of KRSAE and GAM, from the fixed point solution of the coupled nonlinear equations, e.g. equations (45)–(47) of [23], and will be investigated in a separated publication. On the other hand, the nonlinearly generated GAM as a finite frequency component of zonal flow [21], can regulate other types of drift waves, including drift Alfvén waves and the associated anomalous transport, by scattering drift waves into short wavelength stable region [35], leading to cross-scale couplings [36] and improvement of the thermal plasma confinement.

As a final remark, in the parameter regimes where the KRSAE or GAM is heavily damped and becomes a virtual mode, the parametric decay instability can still occur and becomes a non-resonant decay process (also called the nonlinear Landau damping in some literatures [37]), with the decay being a result of the corresponding Landau damping of the sideband. This process can be of relevance noting that GAM could be heavily damped [25] in the center of the tokamak with relatively small q, where RSAE tends to be localized. Extension of the present analysis to the non-resonant decay process is straightforward.