Nonlinear reversed shear Alfven eigenmode saturation due to spontaneous zonal current generation

General nonlinear equations describing reversed shear Alfven eigenmode (RSAE) self-modulation via zero frequency zonal structure (ZFZS) generation are derived using nonlinear gyrokinetic theory, which are then applied to study the spontaneous ZFZS excitation as well as RSAE nonlinear saturation. It is found that both electrostatic zonal flow (ZF) and electromagnetic zonal current (ZC) can be preferentially excited by finite amplitude RSAE, depending on specific plasma parameters. The modification to local shear Alfven wave continuum is evaluated using the derived saturation level of ZC, which is shown to play a comparable role in saturating RSAE with the ZFZS scattering.


Introduction
The future tokamak-based fusion power plants and experimental devices towards this goal (e.g., the ITER (Tomabechi et al. 1991)), are expected to operate at steady state with a substantial non-inductive current fraction (Gormezano et al. 2007), which generally renders a reversed shear scenario (Huang et al. 2020). In this circumstance, the reversed shear Alfvén eigenmode (RSAE, also known as Alfvén cascade) is frequently observed (Sharapov et al. 2001) as driven unstable by the energetic particles (EPs) (Zonca et al. 2002;Chen et al. 2014;Fasoli et al. 2007;Chen & Zonca 2016) in present day tokamaks, and is expected to play significant roles in future reactors (Wang et al. 2018), in transporting fusion alpha particles to tokamak edge (Wang et al. 2019), which has a deleterious effects on plasma self-heating, and may damage the plasma facing components (Ding et al. 2015). RSAE is a branch of shear Alfvén wave (SAW) eigenmode localized radially near the minimum of the safety factor q-profile (Sharapov et al. 2002), which is denoted as q min . The lowest order RSAE frequency in the incompressible limit, ω ≃ k v A ≃ |n − m/q min |v A /R, reflects the sensitive dependence on the instantaneous q min value for given toroidal/poloidal mode numbers n/m, and this feature can be used in q-profile measurement, i.e., MHD spectroscopy (Chen et al. 2014;Fasoli et al. 2002). Here, k is the wave number parallel to the equilibrium magnetic field B, v A is the Alfvén speed, and R is the major radius. RSAEs are typically dominated by one or two poloidal harmonic, with the radial width ∝ q/(r 2 0 q ′′ ) with r 0 being the radial location of q min , and q ′′ ≡ ∂ 2 r q. Despite the fairly good understanding of these linear physics, the nonlinear dynamics of RSAE still attracts recent research interest, especially in view of the EP as well as thermal particle transport induced by the associated electromagnetic field perturbations (Wang et al. 2019Shi et al. 2020;Wei et al. 2021). The transport rate is closely related with the perturbation amplitude (Chen 1999;Zonca et al. 2021); and in reactor relevant cases with many SAWs simultaneously driven unstable by EPs, the EP orbit could become chaotic and eventually lost in the presence of many low amplitude SAWs (threshold value δB/B ∼ O(10 −4 )) (White et al. 2010a,b). Thus, the assessment of the nonlinear RSAE saturation mechanism and amplitude plays a crucial role in evaluating the operation scenario and the EP confinement property.
In general, the channels of SAW nonlinear saturation can be classified into two routes, namely, the wave-particle nonlinear interaction and wave-wave nonlinear couplings (Chen & Zonca 2016, 2013Zonca et al. 2015b;Berk & Breizman 1990). The former focuses on the perturbation to the resonant EP phase space distribution function by finite amplitude SAWs (Zonca et al. 2015b;Berk & Breizman 1990), and is widely investigated by numerical simulations, as reviewed in Refs. (Chen & Zonca 2016;Lauber 2013). By contrast, the latter is relatively less explored. Most of the previous analytical works consider the toroidal Alfvén eigenmode (TAE) (Cheng et al. 1985) as a paradigm case, including the saturation via ion induced scattering (Hahm & Chen 1995;Qiu et al. 2019a), nonlinear modification to the SAW continuum structure (Zonca et al. 1995;Chen et al. 1998), the spontaneous generation of zero frequency zonal structures (ZFZS) (Chen & Zonca 2012;Qiu et al. 2016aQiu et al. , 2017 as well as geodesic acoustic mode (GAM) (Qiu et al. 2018a(Qiu et al. , 2019b. Since the RSAEs are expected to be prevalent in future steadystate burning plasmas, the RSAE saturation via wave-wave nonlinearity deserves special attention. In particular, the toroidally symmetric zonal field structures (Zonca et al. 2021), including the ZFZS, are well known to play important roles in regulating drift wave turbulences (Hahm et al. 1999;Lin et al. 1998;Chen et al. 2000;Diamond et al. 2005) including drift Alfvén waves, and thus, leading to cross-scale couplings (Zonca et al. 2015a) and nonlinear saturation via scattering them to short radial wavelength regime (or shearing in some literatures). In this work, spontaneous ZFZS excitation by RSAE modulational instability is analyzed using nonlinear gyrokinetic theory.
As noted above, the spontaneous excitation of ZFZS by TAE is first discussed in (Chen & Zonca 2012). It is shown that under certain conditions, the zonal current (ZC) is preferentially excited over the electrostatic zonal flow (ZF), with the branch ratio of ZF/ZC excitation determined by various geometry effects, including the breaking of pure Alfvénic state by toroidicity and neoclassical shielding of ZF (Rosenbluth & Hinton 1998). In contrast to the TAE case, it is shown in Ref. (Qiu et al. 2016b) that for betainduced Alfvén eigenmode (BAE), the excitation of ZF generally dominates, due to the |k v A /ω| ≪ 1 ordering. For the case of RSAE analyzed herein, its frequency is sensitively determined by the value of q min and the underlying values of toroidal/poloidal mode numbers n/m, and generally sweeps in-between the typical BAE to TAE frequency ranges. It is shown that depending on the specific plasma parameters including k , both ZC and ZF generation may dominate, and the previous conclusions on TAE (Chen & Zonca 2012) and BAE (Qiu et al. 2016b) can be recovered as limiting cases of the general nonlinear dispersion relation derived without assuming specific plasma parameters.
We note that, several nonlinear processes, with comparable cross-sections, may be comparably important in saturating AEs, as addressed in Ref. (Qiu et al. 2018b). In particular, a channel unique for RSAE saturation is proposed in this work. Due to the ZC and the associated perturbed poloidal magnetic field generation, the q-profile is modulated, which leads to the modification of the local SAW continuum in the vicinity of q min , and consequently RSAE saturation. The relevance of this channel on RSAE nonlinear saturation is analyzed and evaluated. This paper is arranged as follows. In section 2, the theoretical model is given. In section 3, the general nonlinear equations describing RSAE evolution and ZFZS excitation are derived. Section 4 is devoted to study the linear growth stage of the modulational instability; and the nonlinear saturation of RSAE via ZFZS scattering is investigated in section 5. Finally, brief conclusion and discussion are given in section 6.

Theoretical model
The nonlinear evolution of this system is studied using the standard nonlinear perturbation theory, considering a shifted circular tokamak equilibrium described by a set of field-aligned flux coordinates (r, θ, ϕ). The perturbed fields are represented by two field variables, namely, the electrostatic potential δφ and the parallel component of vector potential δA , while the parallel magnetic field fluctuation δB is suppressed, consistent with β ≪ 1 ordering of typical laboratory plasmas. Here, β is the ratio of thermal to magnetic pressures. For convenience, δA is replaced by δψ ≡ ωδA /(ck ), such that the ideal MHD limit, i.e., vanishing parallel electric field fluctuation δE corresponds to simply δφ = δψ. In this work, it is assumed that the RSAE is excited by a source outside this nonlinear system, such as EPs, and the nonlinear coupling is dominated by bulk plasma contribution. For the cases with EPs contributing significantly to the nonlinear ZFZS generation (Todo et al. 2010;Biancalani et al. 2020), interested readers may refer to Refs. (Qiu et al. 2016a(Qiu et al. , 2017 for more systematic discussions. To start with, we consider the two-field coupled system which consists of a RSAE (subscript 'R') and ZFZS (subscript 'Z'), i.e., δφ = δφ R + δφ Z with δφ R = δφ 0 + δφ 0 * . Here, δφ 0 represents RSAE with positive real frequency and δφ 0 * represents the counterpart with negative real frequency, of which there may be a rich spectrum of different radial eigen-states.
Considering the reactor relevant parameter regime with nq ≫ 1, the ballooning mode representation (Connor et al. 1978) for RSAE is adopted, Here, m = m 0 + j with m 0 being the reference poloidal mode number, x ≡ nq − m 0 , Φ 0 is the parallel mode structure with the typical radial extension comparable to distance between neighboring mode rational surfaces, A 0 is the mode envelope amplitude andk r,0 is the radial envelope wavenumber accounting for the slowly varying radial structures. Note that, RSAE is typically characterized by one dominant poloidal harmonic, while multiple sub-dominant poloidal harmonics exist due to toroidicity. Furthermore, |Φ 0 | 2 dx = 1 is used as normalization condition.
Consequently, the ZFZS is expected to have a fine radial structure in addition to the well-known meso-scale structure (Diamond et al. 2005), as a result of the RSAE fine radial mode structure highly localized around q min . For ZFZS dominated by n = 0, m = 0 scalar potential perturbation (Chen et al. 2000), we take with Φ Z accounting for the fine radial structure (Qiu et al. 2016b) due to nonlinear mode coupling and A Z exp i k Z dr being the well-known meso-scale structure. This general representation adopted here can be applied to recover the results obtained from linear growth stage of the modulational instability, by separating RSAE into pump and upper/lower sidebands, as often used in previous papers (Chen et al. 2000); and is also recovered in section 4, from the derived general nonlinear equations.
The governing equations describing the nonlinear processes can be derived from quasineutrality condition and nonlinear gyrokinetic vorticity equation derived from parallel Ampère's law (Chen & Zonca 2016) Here, J k ≡ J 0 (k ⊥ ρ) with J 0 being the Bessel function of zero index accounting for finite Larmor radius effects, ρ = v ⊥ /Ω c is the Larmor radius with Ω c being the cyclotron frequency, F 0 is the equilibrium particle distribution function, is the magnetic drift frequency, l is the length along the equilibrium magnetic field line, · · · means velocity space integration, s is the summation of different particle species with s = i, e representing ion and electron, and δL k ≡ δφ k − k v δψ k /ω k . The three terms on the left hand side of Eq. (2.2) are, respectively, the field line bending, inertial and curvature coupling terms, dominating the linear SAW physics. The two terms on the right hand side of Eq. (2.2) correspond to Maxwell (MX) and Reynolds stresses (RS) (Chen et al. 2001) that contribute to nonlinear mode couplings as MX and RS do not cancel each other, with their contribution dominating in the radially fast varying inertial layer (Chen et al. 2000), and k=k ′ +k ′′ indicates the wavenumber and frequency matching condition required for nonlinear mode coupling. δH k is the nonadiabatic particle response, which can be derived from nonlinear gyrokinetic equation (Frieman & Chen 1982): the linear ion/electron responses can be derived to the leading order as δH L k,i = eF 0 J k δφ k /T i and δH L k,e = −eF 0 δψ k /T e . Furthermore, one can have that, to the leading order, ideal MHD constraint is satisfied, i.e., δφ R = δψ R , by substituting the ion/electron responses of RSAE into quasi-neutrality condition.
On the other hand, considering such a nonlinear system dominated by SAW instabilities, we can also use the parallel component of the nonlinear ideal Ohm's law as an alternative to Eq. (2.1),

General nonlinear equations
In this section, the general nonlinear equations describing the self-consistent RSAE evolution are derived, including the generation of ZFZS and the feedback modulation of RSAE by ZFZS. Generally speaking, the nonlinear process can be divided into two stages, i.e., linear growth stage and strongly nonlinear stage, determined by whether the modulation to the pump wave is small. The governing equations derived in this section, without separating RSAE into pump wave and its sidebands, are general, and can be used for describing both stages as shown in later sections (Guo et al. 2009;Chen et al. 2021).
Considering the nonlinear coupling dominated by the radially fast varying inertial region, one can obtain the equation describing the electrostatic ZF excitation from surface averaged vorticity equation as Here,χ Z = χ Z /(k 2 Z ρ 2 i ) ≃ 1.6q 2 ǫ −1/2 with χ Z being the well-known neoclassical shielding of ZFZS (Rosenbluth & Hinton 1998) and ǫ ≡ r/R being the inverse aspect ratio of the torus. One can note that, being radial modulation with (k r,0 −k r,0 * ) denoting envelope modulation (Chen & Zonca 2012) and (∂ r ln Φ 0 − ∂ r ln Φ 0 * ) denoting parallel mode structure evolution (Qiu et al. 2016b), which gives the fine radial structure of ZFZS. For RSAE typically dominated by one or two poloidal harmonics, (∂ r ln Φ 0 −∂ r ln Φ 0 * ) is the dominant term, and determines the zonal structure radial wavenumber k Z = −i(∂ r ln Φ 0 − ∂ r ln Φ 0 * ), as addressed in Ref. (Qiu et al. 2017).
The equation describing the electromagnetic ZC excitation can be derived from Eq. (2.4), considering k ,Z = 0 and noting δψ Z ≡ ω 0 δA ,Z /(ck ,0 ) is defined using the frequency and parallel wavenumber of RSAE, as In deriving Eq.(3.2), ideal MHD condition for RSAE (δφ 0 = δψ 0 ) is used, and ∂ r ln δψ Z = ∂ r ln |δφ 0 | 2 is noted. One can also derive the corresponding equations describing RSAE from Eq. (2.4) as which describes the deviation from ideal MHD constraint due to nonlinear ZFZS modulation. The other equation for RSAE can be derived from nonlinear vorticity equation as with the term proportional to δφ Z on the right hand side corresponding to RS contribution and δψ Z term corresponding to MX contribution. The third term on the left hand side of Eq. (3.4) is the SAW continuum upshift due to geodesic curvature induced compression, and ω G is the frequency of GAM (Qiu et al. 2009 with E 0 being the RSAE dispersion relation, and α ≡ (k 2 (3.5) is general, and can be reduced to various limits depending on different plasma parameters, through the value of α dependence on k ,0 ; e.g., the mode dynamics described by Eq. (3.5) is similar to the TAE case with (k 2 ,0 v 2 A /ω 2 0 ) ∼ O(1), and |k ,0 | ≃ 1/(2qR), and thus α ≃ 1. On the other hand, the mode behavior gets close to a BAE with k ,0 ≃ 0, and thus α ≃ 0. For simplicity of investigation, the RSAE WKB dispersion relation can be adopted here as E 0 ≃ (1 − k 2 ,0 v 2 A /ω 2 0 − ω 2 G /ω 2 0 ), while the radial global dispersion relation (Zonca et al. 2002) can be applied for more quantitative analysis.
Furthermore, subtracting Eq. (3.1) by α×Eq. (3.2), one can obtain, which can be substituted into Eq. (3.5), and yield the general equation describing the self-modulation of RSAE, as (3.7) Both ZF and ZC generation by RSAE are systematically accounted for in Eq. (3.7) on the same footing, with the first term in square brackets corresponds to the ZF generation while the second term corresponds to the ZC. On the one hand, which one is preferentially excited is shown in the section 4. On the other hand, ZFZS generation can lead to RSAE saturation by scattering it to linearly stable radial eigen-states. The nonlinear saturation level, can be determined by self-consistently solving Eq. (3.7) as a nonlinear Schrodinger equation , while a rough estimation is given in, e.g., Ref. (Qiu et al. 2018a), by separating the AE into a pump and its sidebands, and deriving the fixed point solution of the coupled nonlinear equations. Since RSAE linear properties is very sensitive to q-profile, the modulation of q-profile caused by the nonlinearly generated ZC may have a great impact on RSAE nonlinear saturation. This is discussed in the section 5.

ZFZS spontaneous excitation by RSAE via modulational instability
To investigate the linear growth stage of the modulational instability, we follow the analysis of Ref. (Chen & Zonca 2012), and consider the fluctuation consists of a constant-amplitude pump wave Ω P ≡ Ω P (ω P , k P ) and its upper and lower sidebands Ω ± ≡ Ω ± (ω ± , k ± ) due to the modulation of the ZFZS Ω Z ≡ Ω Z (ω Z , k Z ) (Chen et al. 2000). Here, subscripts 'P ', '+' and '−' denote RSAE pump, upper and lower sidebands respectively, with δφ 0 = δφ P + δφ + , δφ 0 * = δφ P * + δφ − , and Then the general nonlinear Eqs. (3.1)-(3.4) can be reduced to equations describing δφ Z , δφ + and δφ − generation by the fixed amplitude pump RSAE, while the feedback of ZFZS and RSAE sidebands to the pump wave is neglected, focusing on the initial stage of the nonlinear process. Considering the frequency/wave number matching condition (ω ± = ±ω P + ω Z , k ± = ±k P + k Z ) imbedded in the above expressions, Eq. (3.1) can be reduced to representing the competition of Reynolds and Maxwell stresses to break the pure Alfvénic state (Chen & Zonca 2013, 2012. On the other hand, Eq. (3.2) describing ZC excitation can be reduced to The nonlinear equation describing RSAE sidebands generation through the ZFZS modulation to pump RSAE can be derived from Eq. (3.5) as Eqs. (4.1)-(4.3), are equivalent to Eqs. (34)-(36) for TAE cases as derived in Ref. (Qiu et al. 2017), with the coefficient α generalized to include a broader parameter regime (α ≃ 1 for TAE as discussed in Ref. (Qiu et al. 2017)). Note that k 2 ⊥,± = k 2 ⊥,P + k 2 Z and k 2 ⊥,+ is used in the derivation later. Similarly, subtracting Eq. (4.1) by α×Eq. (4.2), one can obtain The modulational instability dispersion relation can then be derived by substituting Eq. (4.3) into (4.4), as withF = (ck Z k θ,P /B) 2 (−k 2 Z + k 2 θ,P + k 2 ,P v 2 A k 2 ⊥,+ /ω 2 P )/(ω P k 2 ⊥,+ ) being a nonlinear coupling coefficient. Furthermore, considering that RSAE sidebands still obey the dispersion relation of RSAE by E ± = E 0 (ω Z ± ω P , k Z ), one can expand E ± along the RSAE characteristics, as E ± ≃ (∂E 0 /∂ω 0 )(±ω Z − ∆) with ∆ ≡ −k 2 Z (∂ 2 E 0 /∂k 2 r )/(2∂E 0 /∂ω 0 ) being the frequency mismatch, describing the frequency shift of RSAE sidebands from the pump RSAE dispersion relation due to the ZFZS modulation. Denoting γ ≡ −iω Z and noting ω ± ≃ ±ω P , the modulational instability dispersion relation can be shown as Here, the first term on the right side of Eq. (4.6) is the threshold condition due to frequency mismatch and the second term represents the nonlinear drive. Thus, the ZFZS can be spontaneously excited when the nonlinear drive overcomes the threshold condition due to frequency mismatch, as the RSAE amplitude is large enough, or the nonlinear coupling is strong enough, as we address in the following discussion. Furthermore, the first term in brackets corresponds to ZF contribution to the nonlinear coupling, and the second term corresponds to ZC. For ZF contribution, there are two restrictions due to, first, the partial cancelation of RS and MX with the 'residual' drive due to deviation from ideal MHD limit due to plasma nonuniformity (reversed q-profile here) that breaks the pure Alfvénic state (Zonca et al. 2002;Chen & Zonca 2013), and second, the neoclassical shielding of ZF as shown by theχ Z in the denominator, withχ Z ∼ q 2 /ǫ, which is typically much larger than unity (Chen & Zonca 2016;Rosenbluth & Hinton 1998). For ZC contribution, there are also two important factors that crucially determine the nonlinear process, with the first being the sign of ∆, which is typically determined by specific plasma parameters. The ZC term is a driving term for ∆ > 0; while for ∆ < 0, the ZC term becomes a damping term for this nonlinear process; and then the condition for modulational instability becomes very stringent, with additional requirement for the ZF term being dominant over the ZC term, which is not easy to satisfy due to the two restrictions as addressed in the paragraph above. The other important factor is the value of α. As noted before, if RSAE localizes near the rational surface, α ≃ 0 because of k ≃ 0, meaning ZC generation is very weak and is similar to BAE case investigated in Ref. (Qiu et al. 2016b); on the other hand, if RSAE localizes in the middle of two neighboring rational surfaces, α ≃ 1, and this case is similar to the TAE case (Chen & Zonca 2012) with ZC generation preferred. So for RSAE, with k and its frequency determined by q min and the corresponding mode numbers, both ZF and ZC generation could be dominant, depending on the specific plasma parameters, and should be investigated case by case.

Nonlinear RSAE saturation
In strongly nonlinear stage, the feedback of ZFZS and RSAE sidebands to the pump wave can not be neglected anymore, as the sideband amplitudes become comparable to that of the pump wave. It is indicated in Eq. (3.7) that ZS can play self-regulatory roles on RSAE nonlinear evolution by scattering RSAE into short radial wavelength stable domains, which may lead to RSAE saturation. The saturation level, can be derived from the coupled pump wave and sidebands equations, as is shown in, e.g., Ref. (Zonca & Chen 2008). In addition, another related channel unique for RSAE nonlinear saturation may exist, due to the modulation to SAW continuum (Zonca et al. 1995;Chen et al. 1998) by the nonlinearly generated ZS (among which ZC playing a dominant role), considering the sensitive dependence of RSAE on SAW continuum accumulation point (for a visualization of RSAE physics dependence on q min , interested readers may refer to Ref.  and Fig. 3 therein). The nonlinearly generated ZC is a toroidal current sharply localized around q min , which can generate a perturbed poloidal magnetic field and further modulate q-profile and thus the SAW continuum near q min . Thus, one can reasonably speculate that ZC plays an important role in RSAE saturation by modifying the equilibrium continuum, similar to the mechanism investigated in Ref. (Zonca et al. 1995) and Ref. (Chen et al. 1998) for TAE.
Here, in consistency with the above speculation on the crucial role played by ZC, we consider a simplified case that ZC generation is dominant. This assumption is natural for scenarios with (k 2 ,0 v 2 A /ω 2 0 ) ∼ O(1), however, may also be important for other parameter regimes for the reasons addressed above. Thus, Eq. (3.7) can be simplified as Taking a two scale analysis by assuming ω R = ω 0 + i∂ τ , and expanding E 0 ≃ (∂E 0 /∂ω 0 )(i∂ τ − iγ L R − ∆) with ∆ being the nonlinearity induced frequency shift and γ L R is the linear RSAE growth rate †, Eq. (5.1) becomes Eq. (5.2) describes the RSAE nonlinear evolution due to, scattering to different radial eigen-state (denoted by ∆) with different linear growth/damping rates (γ L R ) and nonlinear self-modulation by ZC generation, and the saturation level can be estimated by balancing the frequency shift (Max (|γ L R |, |∆|)) and the nonlinear RSAE modulation by ZC, then one has Eq. (5.3) describes the RSAE saturation level due to scattering by self-generated ZC to neighboring (linearly stabler) radial eigen-states, assuming ∆ ≫ γ L R . Cases with ∆ ≪ γ L R can be evaluated similarly. The RSAE saturation level can be estimated by substituting typical parameters into the expression. Furthermore, Eq. (5.3) can be substituted into Eq. (3.2), and the saturation level of the perturbed poloidal magnetic field δB θ can be estimated as resulting in a modulation of local q min by In deriving Eqs. (5.4) and (5.5), we have noted δψ = ωδA /(ck ) and δB = ∇ × δA b. Typical parameters can be adopted, i.e., q min ∼ O(1), B 0 /B θ ∼ qR/r 0 , k ∼ 1/(qR), † noting the linear RSAE growth/damping rate here, which is not included in Eq. (4.6) assuming RSAE sidebands damping rates are typically smaller than frequency mismatch k θ,0 ρ d ∼ O(1) with ρ d being the drift orbit radius and which can be reasonably assumed as ∼ O(0.1). Thus, one can roughly estimate that δq/q ∼ O(10 −3 ). Noting that the modification to local RSAE continuum frequency is ∼ O(nδqv A /(qR)), and that for reactor burning plasma with ρ d /a ∼ O(10 −2 ), most unstable RSAEs are characterized by n O(10) (Wang et al. 2018), the modification to local SAW continuum is comparable to the RSAE linear growth rate γ L R or frequency differences between different radial eigen-states (∼ ∆). Thus, one expects that the ZC induced SAW continuum modification in the vicinity of q min , plays an important role in RSAE nonlinear saturation, though the self-consistent study analogous to Refs. (Zonca et al. 1995;Chen et al. 1998) is not presented. The systematic investigation of RSAE saturation due to nonlinear modification of SAW continuum and the resulting enhanced continuum damping, will be presented in a separate publication.

Conclusion and Discussion
The general equations describing RSAE self-modulation via nonlinear excitation of ZFZS are derived using gyrokinetic theory, which is then applied to study the spontaneous ZFZS excitation via modulational instability as well as RSAE nonlinear saturation due to scattering to stabler radial eigen-states. It is found that, both ZF and ZC can be dominant in the spontaneous excitation by RSAE, depending on the specific plasma parameters, especially q min that determines the RSAE parallel wavenumber and frequency. The obtained general modulational instability dispersion relation for ZFZS excitation by RSAE, Eq. (4.6), can recover the results of TAE (Chen & Zonca 2012) and BAE (Qiu et al. 2016b) in the proper limits, i.e., by taking |k v A /ω| → 1 and 0, respectively. The properties of ZFZS generation by RSAE, noting that the typical RSAE parallel wavenumber and frequency are in between those of TAE and BAE, can be understood based on the knowledge obtained from TAE (Chen & Zonca 2012) and BAE (Qiu et al. 2016b).
An interesting step forward is, the saturation level of RSAE is qualitatively estimated by balancing the nonlinear scattering by ZFZS with the frequency differences between different radial eigen-states (∼ ∆), assuming ZC playing a dominant role in the RSAE scattering. The corresponding ZC saturation level as well as the modification to local q min , are also estimated. It is found that, the resulting modification to local SAW continuum accumulational point frequency, can be at least, comparable to the RSAE linear growth rate or frequency mismatch between different radial eigen-states for burning plasma scenarios with most unstable RSAEs characterized by n O(10) (Wang et al. 2018). Thus, the modification of local SAW continuum by ZC is expected to contribute significantly to RSAE saturation (Zonca et al. 1995;Chen et al. 1998).
The above estimation based on Eq. (5.1), assumes dominant ZC generation by taking |k v A /ω| ∼ 1, is valid for other parameter regimes due to the weak coupling coefficients of ZF generation, except for the cases that q min is localized very close to a low-order rational surface, such that the RSAE properties is close to BAE with predominantly ZF generation (Qiu et al. 2016b). The logic underlying the reasoning presented in section 5 is that, the RSAE and ZFZS saturation levels are estimated without accounting for the modification to SAW continuum, which is then used to estimate the modification to SAW continuum by the saturated ZC, and is found that the modification to SAW continuum could be comparable to or even more important than the ZFZS shearing. Thus, the obtained results indicate that, the modification to SAW continuum, will start to influence the RSAE nonlinear evolution, before it saturates due to self-modulation via ZFZS generation. Our work thus indicates that multiple processes may contribute comparably to the RSAE saturation, and should be accounted for on the same footing, based on the solid understanding of each individual process, to properly assess the saturation and thus EP transport by RSAE. This is of particular importance since RSAEs are expected to be strongly excited by core-localized fusion alphas in future reactors characterized by the advanced reversed shear scenarios.