2. 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 $(r, \theta , \varphi )$. The perturbed fields are represented by two field variables, namely the electrostatic potential $\delta \phi$ and the parallel component of vector potential $\delta A_{\parallel }$, while the parallel magnetic field fluctuation $\delta B_{\parallel }$ is suppressed, consistent with $\beta \ll 1$ ordering of typical laboratory plasmas. Here, $\beta$ is the ratio of thermal to magnetic pressures. For convenience, $\delta A_{\parallel }$ is replaced by $\delta \psi \equiv \omega \delta A_{\parallel }/(ck_{\parallel })$, such that the ideal MHD limit, i.e. vanishing parallel electric field fluctuation $\delta E_{\parallel }$, corresponds to simply $\delta \phi =\delta \psi$. 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 nonlinear ZFZS generation (Todo, Berk & Breizman Reference Todo, Berk and Breizman2010; Biancalani et al. Reference Biancalani, Bottino, Lauber, Mishchenko and Vannini2020), interested readers may refer to Qiu et al. (Reference Qiu, Chen and Zonca2016a,  Reference Qiu, Chen and Zonca2017) 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. $\delta \phi =\delta \phi _{R}+\delta \phi _{Z}$ with $\delta \phi _{R}=\delta \phi _{0}+\delta \phi _{0^{*}}$. Here, $\delta \phi _{0}$ represents the RSAE with positive real frequency and $\delta \phi _{0^{*}}$ represents the counterpart with negative real frequency, of which there may be a rich spectrum of different radial eigenstates.

Considering the reactor-relevant parameter regime with $nq\gg 1$, the ballooning mode representation (Connor, Hastie & Taylor Reference Connor, Hastie and Taylor1978) for theRSAE is adopted:

(2.1)\begin{equation} \delta\phi_{0} = A_{0}\, \textrm{e}^{\textrm{i}(n\phi-m_{0}\theta-\omega_{0}t)}\, \textrm{e}^{\textrm{i}\int\hat{k}_{r,0}\, \textrm{d} r}\sum_{j}\, \textrm{e}^{-\textrm{i}j\theta}\varPhi_{0}(x-j)+\textrm{c.c.} \end{equation}

Here, $m=m_{0}+j$ with $m_{0}$ being the reference poloidal mode number, $x\equiv nq-m_{0}$, $\varPhi _{0}$ is the parallel mode structure with typical radial extension comparable to the distance between neighbouring mode rational surfaces, $A_{0}$ is the mode envelope amplitude and $\hat {k}_{r,0}$ is the radial envelope wavenumber accounting for the slowly varying radial structures. Note that the RSAE is typically characterized by one dominant poloidal harmonic, while multiple sub-dominant poloidal harmonics exist due to toroidicity. Furthermore, $\int |\varPhi _{0}|^{2}{\textrm {d}\kern0.06em x}=1$ is used as normalization condition.

Consequently, the ZFZS is expected to have a fine radial structure in addition to the well-known mesoscale structure (Diamond et al. Reference Diamond, Itoh, Itoh and Hahm2005), 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. Reference Chen, Lin and White2000), we take

(2.2)\begin{equation} \delta\phi_{Z} = A_{Z}\, \textrm{e}^{\textrm{i}\int\hat{k}_{Z}\, \textrm{d} r-\textrm{i}\omega_{Z}t}\sum_{j}\varPhi_{Z}(x-j)+\textrm{c.c.}, \end{equation}

with $\varPhi _{Z}$ accounting for the fine radial structure (Qiu et al. Reference Qiu, Chen and Zonca2016b) due to nonlinear mode coupling and $A_{Z}\exp {\textrm {i}\int \hat {k}_{Z}\, \textrm {d} r}$ being the well-known mesoscale structure. This general representation adopted here can be applied to recover the results obtained from the linear growth stage of the modulational instability, by separating the RSAE into pump and upper/lower sidebands, as often used in previous work (Chen et al. Reference Chen, Lin and White2000); and is also recovered in § 4, from the derived general nonlinear equations.

The governing equations describing the nonlinear processes can be derived from the quasi-neutrality condition

(2.3)\begin{equation} \frac{n_{0}\, \textrm{e}^{2}}{T_{i}}\left(1+\frac{T_{i}}{T_{e}}\right)\delta\phi_{k} = \sum_{s}\langle qJ_{k}\delta H_{k}\rangle_{s}\end{equation}

and the nonlinear gyrokinetic vorticity equation derived from parallel Ampère's law (Chen & Zonca Reference Chen and Zonca2016)

(2.4)\begin{align} & \frac{c^{2}}{4{\rm \pi}\omega_{k}^{2}}B\frac{\partial}{\partial l}\frac{k_{{\perp}}^{2}}{B}\frac{\partial}{\partial l}\delta\psi_k + \frac{\textrm{e}^{2}}{T_{i}}\langle (1-J_{k}^{2})F_{0}\rangle_{s} \delta\phi_{k}-\sum_{s}\left\langle \frac{q}{\omega_{k}}J_{k}\omega_{d}\delta H_{k}\right\rangle\nonumber\\ & \quad ={-}\textrm{i}\frac{c}{B_{0}\omega_{k}}\sum_{\boldsymbol{k}=\boldsymbol{k'} +\boldsymbol{k''}}\hat{\boldsymbol{b}}\cdot\boldsymbol{k''}\times \boldsymbol{k'}\left[\frac{c^{2}}{4{\rm \pi}}k_{{\perp}}''^{2} \frac{\partial_{l}\delta\psi_{k'}\partial_{l}\delta\psi_{k''}}{\omega_{k'} \omega_{k''}}\right.\nonumber\\ & \qquad +\left.\vphantom{\frac{c^{2}}{4{\rm \pi}}}\langle e(J_{k}J_{k'}-J_{k''}) \delta L_{k'}\delta H_{k''}\rangle \right]. \end{align}

Here, $J_{k}\equiv J_{0}(k_{\perp }\rho )$ with $J_{0}$ being the Bessel function of zero index accounting for finite Larmor radius effects, $\rho =v_{\perp }/\varOmega _{c}$ is the Larmor radius with $\varOmega _{c}$ being the cyclotron frequency, $F_{0}$ is the equilibrium particle distribution function, $\omega _{d}=(v_{\perp }^{2}+2v_{\parallel }^{2})(k_{r}\sin \theta +k_{\theta }\cos \theta )/(2\varOmega _{c}R)$ is the magnetic drift frequency, $l$ is the length along the equilibrium magnetic field line, $\langle \cdots \rangle$ means velocity space integration, $\sum _{s}$ is the summation of different particle species with $s=i, e$ representing ion and electron, and $\delta L_{k}\equiv \delta \phi _{k}-k_{\parallel }v_{\parallel }\delta \psi _{k}/\omega _{k}$. The three terms on the left-hand side of (2.4) 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 (2.4) correspond to Maxwell and Reynolds stresses (Chen et al. Reference Chen, Lin, White and Zonca2001) that contribute to nonlinear mode couplings as Maxwell and Reynolds stresses do not cancel each other, with their contribution dominating in the radially fast-varying inertial layer (Chen et al. Reference Chen, Lin and White2000), and $\sum _{\boldsymbol {k}=\boldsymbol {k'}+\boldsymbol {k''}}$ indicates the wavenumber and frequency matching condition required for nonlinear mode coupling. Here $\delta H_{k}$ is the non-adiabatic particle response, which can be derived from the nonlinear gyrokinetic equation (Frieman & Chen Reference Frieman and Chen1982):

(2.5)\begin{align} & (-\textrm{i}\omega_{k}+v_{{\parallel}}\partial_{l}+\textrm{i}\omega_{d})\delta H_{k} ={-}\textrm{i}\omega_{k}\frac{q}{T}F_{0}J_{k}\delta L_{k}\nonumber\\ & \quad -\frac{c}{B_{0}}\sum_{\boldsymbol{k}=\boldsymbol{k'} +\boldsymbol{k''}}\hat{\boldsymbol{b}}\cdot\boldsymbol{k''}\times \boldsymbol{k'}J_{k'}\delta L_{k'}\delta H_{k''}. \end{align}

For RSAE with $|k_{\parallel }v_{e}|\gg |\omega _k|\gg |k_{\parallel }v_{i}|, |\omega _{d}|$, the linear ion/electron responses can be derived to leading order as $\delta H^{L}_{k, i}=eF_{0}J_k\delta \phi _ k/T_{i}$ and $\delta H^{L}_{k,e}=-eF_{0}\delta \psi _{k}/T_{e}$. Furthermore, one can have that, to leading order, the ideal MHD constraint is satisfied, i.e. $\delta \phi _{R}=\delta \psi _{R}$, by substituting the ion/electron responses of the RSAE into the 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 (2.3):

(2.6)\begin{equation} \delta E_{{\parallel},k} ={-}\sum_{\boldsymbol{k}=\boldsymbol{k'}+ \boldsymbol{k''}}\hat{\boldsymbol{b}}\cdot\delta\boldsymbol{u}_{k'} \times\delta\boldsymbol{B}_{k''}/c,\end{equation}

with $\delta \boldsymbol {u}$ being the $\boldsymbol {E}\times \boldsymbol {B}$ drift velocity. We note that (2.6) is equivalent to (2.3), ignoring the high-order ${O}(k_{\perp }^{2}\rho _{i}^{2})$ corrections.

3. 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 the RSAE into pump wave and its sidebands, are general, and can be used for describing both stages as shown in later sections (Guo, Chen & Zonca Reference Guo, Chen and Zonca2009; Chen et al. Reference Chen, Wei, Wei and Qiu2021).

Considering the nonlinear coupling dominated by the radially fast-varying inertial region, one can obtain the equation describing the electrostatic ZF excitation from the surface-averaged vorticity equation as

(3.1)\begin{equation} \omega_{Z}\hat{\chi}_{Z}\delta\phi_{Z} ={-}\textrm{i}\frac{c}{B}k_{\theta}\left(1-\frac{k_{{\parallel},0}^{2} v_{A}^{2}}{\omega_{0}^{2}}\right)(k_{r,0}-k_{r,0^{*}}) |\delta\phi_{0}|^{2}.\end{equation}

Here, $\hat {\chi }_{Z}=\chi _{Z}/(k_{Z}^{2}\rho _{i}^{2})\simeq 1.6q^{2}\epsilon ^{-1/2}$ with $\chi _{Z}$ being the well-known neoclassical shielding of ZFZS (Rosenbluth & Hinton Reference Rosenbluth and Hinton1998) and $\epsilon \equiv r/R$ being the inverse aspect ratio of the torus. One can note that $(k_{r,0}-k_{r,0^{*}})|\delta \varPhi _{0}|^{2}\equiv [(\hat {k}_{r,0}-\hat {k}_{r,0^{*}})-\textrm {i}(\partial _{r}\ln \varPhi _{0}- \partial _{r}\ln \varPhi _{0^{*}})]|\delta \varPhi _{0}|^{2}$ is radial modulation with $(\hat {k}_{r,0}-\hat {k}_{r,0^{*}})$ denoting envelope modulation (Chen & Zonca Reference Chen and Zonca2012) and $(\partial _{r}\ln \varPhi _{0}-\partial _{r}\ln \varPhi _{0^{*}})$ denoting parallel mode structure evolution (Qiu et al. Reference Qiu, Chen and Zonca2016b), which gives the fine radial structure of ZFZS. For RSAE typically dominated by one or two poloidal harmonics, $(\partial _{r}\ln \varPhi _{0}-\partial _{r}\ln \varPhi _{0^{*}})$ is the dominant term, and determines the zonal structure radial wavenumber $k_{Z}=-\textrm {i}(\partial _{r} \ln \varPhi _{0}-\partial _{r}\ln \varPhi _{0^{*}})$, as addressed in Qiu et al. (Reference Qiu, Chen and Zonca2017).

The equation describing the electromagnetic ZC excitation can be derived from (2.6), considering $k_{\parallel ,Z}=0$ and noting $\delta \psi _{Z}\equiv \omega _{0}\delta A_{\parallel ,Z}/(ck_{\parallel ,0})$ is defined using the frequency and parallel wavenumber of RSAE, as

(3.2)\begin{equation} \delta\psi_{Z} ={-}\textrm{i}\frac{c}{B}k_{\theta,0}k_{Z}\frac{1}{\omega_{0}} |\delta\phi_{0}|^{2}.\end{equation}

In deriving the (3.2), the ideal MHD condition for RSAE ($\delta \phi _{0}=\delta \psi _{0}$) is used, and $\partial _{r}\ln \delta \psi _{Z}=\partial _{r}\ln |\delta \phi _{0}|^{2}$ is noted.

One can also derive the corresponding equations describing RSAE from (2.6) as

(3.3)\begin{equation} \delta\phi_{0}-\delta\psi_{0} ={-}\textrm{i}\frac{c}{B}\frac{k_{Z} k_{\theta,0}}{\omega_{0}}\delta\phi_{0}(\delta\phi_{Z}- \delta\psi_{Z}),\end{equation}

which describes the deviation from the ideal MHD constraint due to nonlinear ZFZS modulation. The other equation for RSAE can be derived from the nonlinear vorticity equation as

(3.4)\begin{align} & k_{{\perp},0}^{2}\left(-\frac{k_{{\parallel},0}^{2}v_{A}^{2}}{\omega_{0}^{2}} \delta\psi_{0}+\delta\phi_{0}-\frac{\omega_{G}^{2}}{\omega_{0}^{2}} \delta\phi_{0}\right)\nonumber\\ & \quad ={-}\textrm{i}\frac{c}{B\omega_{0}}k_{Z}k_{\theta,0} (k_{Z}^{2}-k_{\theta,0}^{2})\delta\phi_{0} \left(\delta\phi_{Z}-\frac{k_{{\parallel},0}^{2}v_{A}^{2}}{\omega_{0}^{2}} \delta\psi_{Z}\right), \end{align}

with the term proportional to $\delta \phi _{Z}$ on the right-hand side corresponding to Reynolds stress contribution and the $\delta \psi _{Z}$ term corresponding to Maxwell contribution. The third term on the left-hand side of (3.4) is the SAW continuum upshift due to geodesic-curvature-induced compression, and $\omega _{G}$ is the frequency of geodesic acoustic mode (Qiu, Chen & Zonca Reference Qiu, Chen and Zonca2009). Note that (3.1)–(3.4) are the general equations for ZFZS generation by SAW instabilities in the Wentzel–Kramer–Brillouin (WKB) limit, and have expression similar to that describing ZFZS generation by TAE, as derived in Qiu et al. (Reference Qiu, Chen and Zonca2017), except the coefficient of the $\delta \psi$ term on the right-hand side of (3.4), where $k^2_{\parallel } v^2_{A}/\omega ^2\simeq 1$ is applied for TAE. However, the RSAE frequency is sensitively determined by $k_{\parallel }$ through $q_{\min }$, and is between the BAE and TAE frequency ranges, so a more general expression is derived here for ZFZS generation by RSAE. However, though with similar WKB expressions of dispersion relation, the physics underlying the ZFZS generation by RSAE is very different from that of the TAE case, as is embedded in the $k_{\parallel }$ term that determines the relative importance of various terms, for example, whether ZF or ZC generation should be dominated.

Substituting (3.3) into (3.4), one then obtains the equation describing the modulation of RSAE by ZFZS:

(3.5)\begin{equation} k_{{\perp},0}^{2}\mathscr{E}_{0}\delta\phi_{0} ={-}\textrm{i}\frac{c}{B\omega_{0}}\left(k_{Z}^{2}-k_{\theta,0}^{2}- \frac{k_{{\parallel},0}^{2}v_{A}^{2}}{\omega_{0}^{2}}k_{{\perp},0}^{2}\right) k_{Z}k_{\theta,0}\delta\phi_{0}(\delta\phi_{Z}-\alpha\delta\psi_{Z}),\end{equation}

with $\mathscr {E}_0$ being the RSAE dispersion relation and $\alpha \equiv (k_{\parallel ,0}^{2}v_{A}^{2}/\omega _{0}^{2})(-2k_{\theta , 0}^{2})/(k_{Z}^{2}-k_{\theta ,0}^{2}-k_{\parallel ,0}^{2}v_{A}^{2} k_{\perp ,0}^{2}/\omega _{0}^{2})$. Equation (3.5) is general, and can be reduced to various limits depending on different plasma parameters, through the value of $\alpha$ dependence on $k_{\parallel ,0}$; e.g. the mode dynamics described by (3.5) is similar to the TAE case with $(k_{\parallel ,0}^{2}v_{A}^{2}/\omega _{0}^{2})\sim {O}(1)$, and $|k_{\parallel ,0}|\simeq 1/(2qR)$, and thus $\alpha \simeq 1$. On the other hand, the mode behaviour gets close to a BAE with $k_{\parallel ,0}\simeq 0$, and thus $\alpha \simeq 0$. For simplicity of investigation, the RSAE WKB dispersion relation can be adopted here as $\mathscr {E}_{0}\simeq (1-k_{\parallel ,0}^{2}v_{A}^{2}/\omega _{0}^{2}- \omega _{G}^{2}/\omega _{0}^{2})$, while the radial global dispersion relation (Zonca et al. Reference Zonca, Briguglio, Chen, Dettrick, Fogaccia, Testa and Vlad2002) can be applied for more quantitative analysis.

Furthermore, subtracting (3.1) by $\alpha \times$ (3.2), one can obtain

(3.6)\begin{equation} \delta\phi_{Z}-\alpha\delta\psi_{Z} ={-}\textrm{i}\frac{c}{B}k_{Z}k_{\theta,0} \left[\frac{1-k_{{\parallel},0}^{2}v_{A}^{2}/\omega_{0}^{2}}{\omega_{Z} \hat{\chi}_{Z}k_{Z}}(k_{r,0}-k_{r,0^{*}})+\frac{\alpha}{\omega_{0}}\right] |\delta\phi_{0}|^{2},\end{equation}

which can be substituted into (3.5), and yield the general equation describing the self-modulation of RSAE as

(3.7)\begin{align} k_{{\perp},0}^{2}\mathscr{E}_{0}\delta\phi_{0} & ={-}\left(\frac{c}{B}k_{Z}k_{\theta,0}\right)^{2} \frac{1}{\omega_{0}}\left(k_{Z}^{2}-k_{\theta,0}^{2}- \frac{k_{\parallel0}^{2}v_{A}^{2}}{\omega_{0}^{2}}k_{\perp0}^{2}\right)\nonumber\\ & \quad \times \left[\frac{1-k_{{\parallel},0}^{2}v_{A}^{2}/\omega_{0}^{2}}{\omega_{Z} \hat{\chi}_{Z}k_{Z}}(k_{r,0}-k_{r,0^{*}})+ \frac{\alpha}{\omega_{0}}\right]\delta\phi_{0}|\delta\phi_{0}|^{2}. \end{align}

Both ZF and ZC generation by RSAE are systematically accounted for in (3.7) on the same footing, with the first term in square brackets corresponding to ZF generation and the second term to ZC. On the one hand, which one is preferentially excited is addressed in § 4. On the other hand, ZFZS generation can lead to RSAE saturation by scattering it to linearly stable radial eigenstates. The nonlinear saturation level can be determined by self-consistently solving (3.7) as a nonlinear Schrödinger equation (Chen et al. Reference Chen, Wei, Wei and Qiu2021), while a rough estimation is given in, for example, Qiu et al. (Reference Qiu, Chen, Zonca and Chen2018a), by separating the RSAE into a pump and its sidebands, and deriving the fixed-point solution of the coupled nonlinear equations. Since RSAE linear properties are very sensitive to the $q$-profile, the modulation of the$q$-profile caused by the nonlinearly generated ZC may have a great impact on RSAE nonlinear saturation. This is discussed in § 5.

4. Spontaneous excitation of ZFZS by RSAE via modulational instability

To investigate the linear growth stage of the modulational instability, we follow the analysis of Chen & Zonca (Reference Chen and Zonca2012), and consider the fluctuation consisting of a constant-amplitude pump wave $\varOmega _{P}\equiv \varOmega _{P}(\omega _{P}, \boldsymbol {k}_{P})$ and its upper and lower sidebands $\varOmega _{\pm }\equiv \varOmega _{\pm }(\omega _{\pm }, \boldsymbol {k}_{\pm })$ due to the modulation of the ZFZS $\varOmega _{Z}\equiv \varOmega _{Z}(\omega _{Z}, \boldsymbol {k}_{Z})$ (Chen et al. Reference Chen, Lin and White2000). Here, subscripts ‘$P$’, ‘$+$’ and ‘$-$’ denote RSAE pump, upper sideband and lower sideband, respectively, with $\delta \phi _{0}=\delta \phi _{P}+\delta \phi _{+}$, $\delta \phi _{0^{*}}=\delta \phi _{P^{*}}+\delta \phi _{-}$, and

(4.1)\begin{gather} \delta\phi_{P} = A_{P}\, \textrm{e}^{\textrm{i}\left(n\phi-m_{0}\theta-\omega_{P}t\right)}\sum_{j}\, \textrm{e}^{-\textrm{i}j\theta}\varPhi_{P}\left(x-j\right)+\textrm{c.c.}, \end{gather}

(4.2)\begin{gather}\delta\phi_{{\pm}} = A_{{\pm}}\, \textrm{e}^{{\pm} \textrm{i}\left(n\phi-m_{0}\theta-\omega_{P}t\right)}\, \textrm{e}^{\textrm{i}\left(\int\widehat{k_{Z}}\, \textrm{d} r-\omega_{Z}t\right)} \sum_{j}\, \textrm{e}^{{\mp} ij\theta}\left\{ \begin{array}{c} \varPhi_{P}\left(x-j\right)\\ \varPhi_{P^{*}}\left(x-j\right) \end{array}\right\} +\textrm{c.c.} \end{gather}

Then the general nonlinear equations (3.1)–(3.4) can be reduced to equations describing $\delta \phi _{Z}$, $\delta \phi _{+}$ and $\delta \phi _{-}$ 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/wavenumber matching condition ($\omega _{\pm }=\pm \omega _{P}+\omega _{Z}$, $\boldsymbol {k}_{\pm }=\pm \boldsymbol {k}_{P}+\boldsymbol {k}_{Z}$) embedded in the above expressions, equation (3.1) can be reduced to

(4.3)\begin{equation} \omega_{Z}\hat{\chi}_{Z}\delta\phi_{Z} ={-}\textrm{i}\frac{c}{B}k_{Z}k_{\theta,P}\left(1-\frac{k_{{\parallel},P}^{2} v_{A}^{2}}{\omega_{P}^{2}}\right) (\delta\phi_{+}\delta\phi_{P^{*}}-\delta\phi_{-} \delta\phi_{P}),\end{equation}

with $(1-k^{2}_{\parallel ,P}v^{2}_{A}/\omega ^{2}_{P})$ representing the competition of Reynolds and Maxwell stresses in breaking the pure Alfvénic state (Chen & Zonca Reference Chen and Zonca2012,  Reference Chen and Zonca2013). On the other hand, (3.2) describing ZC excitation can be reduced to

(4.4)\begin{equation} \delta\psi_{Z} ={-}\textrm{i}\frac{c}{B}k_{\theta,P}k_{Z} \frac{1}{\omega_{P}}(\delta\phi_{+}\delta\phi_{P^{*}}+ \delta\phi_{-}\delta\phi_{P}).\end{equation}

The nonlinear equation describing RSAE sideband generation through the ZFZS modulation to pump RSAE can be derived from (3.5) as

(4.5)\begin{align} k_{{\perp},\pm}^{2}\mathscr{E}_{{\pm}}\delta\phi_{{\pm}} & ={-}\textrm{i}\frac{c}{B\omega_{{\pm}}}\left(k_{Z}^{2}-k_{\theta,P}^{2}- \frac{k_{{\parallel},P}^{2}v_{A}^{2}}{\omega_{P}^{2}}k_{{\perp},\pm}^{2}\right)\nonumber\\ & \quad \times k_{Z}k_{\theta,P}\left\{ \begin{array}{c} \delta\phi_{P}\\ \delta\phi_{P^{*}} \end{array}\right\} (\delta\phi_{Z}-\alpha\delta\psi_{Z}). \end{align}

Equations (4.3)–(4.5) are equivalent to (34)–(36) for TAE cases as derived in Qiu et al. (Reference Qiu, Chen and Zonca2017), with the coefficient $\alpha$ generalized to include a broader parameter regime ($\alpha \simeq 1$ for TAE as discussed in Qiu et al. (Reference Qiu, Chen and Zonca2017)). Note that $k_{\perp ,\pm }^{2}=k_{\perp ,P}^{2}+k_{Z}^{2}$ and $k_{\perp ,+}^{2}$ are used in the derivation later. Similarly, subtracting (4.3) by $\alpha \times$ (4.4), one can obtain

(4.6)\begin{align} \delta\phi_{Z}-\alpha\delta\psi_{Z} & = \textrm{i}\frac{c}{B}k_{Z}k_{\theta,P} \left[\left(\frac{1-k_{{\parallel},P}^{2}v_{A}^{2}/\omega_{P}^{2}}{\omega_{Z} \hat{\chi}_{Z}}+\frac{\alpha}{\omega_{P}}\right)\delta\phi_{+} \delta\phi_{P^{*}}\right.\nonumber\\ & \quad \left.-\left(\frac{1-k_{{\parallel},P}^{2}v_{A}^{2}/\omega_{P}^{2}}{\omega_{Z} \hat{\chi}_{Z}}-\frac{\alpha}{\omega_{P}}\right)\delta\phi_{-}\delta\phi_{P}\right]. \end{align}

The modulational instability dispersion relation can then be derived by substituting (4.5) into (4.6) as

(4.7)\begin{align} 1 & ={-}\hat{F}\left|\delta\phi_{P}\right|^{2}\left[\left(\frac{1-k_{{\parallel},P}^{2} v_{A}^{2}/\omega_{P}^{2}}{\omega_{Z}\hat{\chi}_{Z}}+ \frac{\alpha}{\omega_{P}}\right)\frac{1}{\mathscr{E}_{+}}\right.\nonumber\\ & \quad -\left.\left(\frac{1-k_{{\parallel},P}^{2}v_{A}^{2}/\omega_{P}^{2}}{\omega_{Z} \hat{\chi}_{Z}}-\frac{\alpha}{\omega_{P}}\right) \frac{1}{\mathscr{E}_{-}}\right], \end{align}

with $\hat {F}=(ck_{Z}k_{\theta ,P}/B)^{2}(-k_{Z}^{2}+k_{\theta ,P}^{2} +k_{\parallel ,P}^{2}v_{A}^{2}k_{\perp ,+}^{2}/\omega _{P}^{2})/(\omega _{P}k_{\perp ,+}^{2})$ being a nonlinear coupling coefficient. Furthermore, considering that RSAE sidebands still obey the dispersion relation of RSAE by $\mathscr {E}_{\pm }=\mathscr {E}_{0}(\omega _Z\pm \omega _P, k_{Z})$, one can expand $\mathscr {E}_{\pm }$ along the RSAE characteristics, as $\mathscr {E}_{\pm }\simeq (\partial \mathscr {E}_{0}/\partial \omega _{0})(\pm \omega _{Z}-\varDelta )$ with $\varDelta \equiv -k_{Z}^{2}(\partial ^{2}\mathscr {E}_{0}/\partial k_{r}^{2})/(2\partial \mathscr {E}_{0}/\partial \omega _{0})$ being the frequency mismatch, describing the frequency shift of RSAE sidebands from the pump RSAE dispersion relation due to the ZFZS modulation. Denoting $\gamma \equiv -\textrm {i}\omega _{Z}$ and noting $\omega _{\pm }\simeq \pm \omega _{P}$, the modulational instability dispersion relation can be written as

(4.8)\begin{equation} \gamma^{2} ={-}\varDelta^{2}+\frac{2\hat{F} |\delta\phi_{P}|^{2}}{\partial\mathscr{E}_{0}/\partial\omega_{0}} \left(\frac{1-k_{{\parallel},P}^{2}v_{A}^{2}/\omega_{0}^{2}}{\hat{\chi}_{Z}} +\frac{\alpha}{\omega_{P}}\varDelta\right).\end{equation}

The first term on the right-hand side of (4.8) 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 parentheses corresponds to the ZF contribution to the nonlinear coupling, and the second term corresponds to ZC. For the ZF contribution, there are two restrictions due to, first, the partial cancellation of Reynolds and Maxwell stresses with the ‘residual’ drive due to deviation from the ideal MHD limit due to plasma non-uniformity (reversed $q$-profile here) that breaks the pure Alfvénic state (Zonca et al. Reference Zonca, Briguglio, Chen, Dettrick, Fogaccia, Testa and Vlad2002; Chen & Zonca Reference Chen and Zonca2013), and second, the neoclassical shielding of ZF as shown by $\hat {\chi }_{Z}$ in the denominator, with $\hat {\chi }_{Z}\sim q^{2}/\epsilon$, which is typically much larger than unity (Rosenbluth & Hinton Reference Rosenbluth and Hinton1998; Chen & Zonca Reference Chen and Zonca2016).

For the ZC contribution, there are also two important factors that crucially determine the nonlinear process, with the first being the sign of $\varDelta$, which is typically determined by specific plasma parameters. The ZC term is a driving term for $\varDelta >0$; while for $\varDelta <0$, 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 $\alpha$. As noted before, if RSAE localizes near the rational surface, $\alpha \simeq 0$ because of $k_{\parallel }\simeq 0$, meaning that ZC generation is very weak and is similar to the BAE case investigated in Qiu et al. (Reference Qiu, Chen and Zonca2016b); on the other hand, if RSAE localizes in the middle of two neighbouring rational surfaces, $\alpha \simeq 1$, and this case is similar to the TAE case (Chen & Zonca Reference Chen and Zonca2012) with ZC generation preferred. So for RSAE, with $k_{\parallel }$ 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. Furthermore, the threshold amplitude can be roughly estimated in two limit cases, e.g. as RSAE frequency approaches TAE and BAE frequencies, as $\delta B_{\theta }/B \sim {O}(10^{-4})$ (Chen & Zonca Reference Chen and Zonca2012; Qiu et al. Reference Qiu, Chen and Zonca2016b).

5. Nonlinear RSAE saturation

In strongly nonlinear stage, the feedback of ZFZS and RSAE sidebands to the pump wave can no longer be neglected, as the sideband amplitudes become comparable to that of the pump wave. It is indicated in (3.7) that ZS can play self-regulatory roles in RSAE nonlinear evolution by scattering the 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 sideband equations, as is shown in, for example, Zonca & Chen (Reference Zonca and Chen2008). In addition, another related channel unique for RSAE nonlinear saturation may exist, due to the modulation to the SAW continuum (Zonca et al. Reference Zonca, Romanelli, Vlad and Kar1995; Chen et al. Reference Chen, Zonca, Santoro and Hu1998) by the nonlinearly generated ZS (among which ZC plays 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 Wang et al. (Reference Wang, Qiu, Zonca, Briguglio and Vlad2020) and figure 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 the $q$-profile and thus the SAW continuum near $q_{\min }$. Thus, one can reasonably speculate that the ZC plays an important role in RSAE saturation by modifying the equilibrium continuum, similar to the mechanism investigated in Zonca et al. (Reference Zonca, Romanelli, Vlad and Kar1995) and Chen et al. (Reference Chen, Zonca, Santoro and Hu1998) for TAE.

Here, consistent with the above speculation on the crucial role played by the ZC, we consider a simplified case that ZC generation is dominant. This assumption is natural for scenarios with $(k_{\parallel ,0}^{2}v_{A}^{2}/\omega _{0}^{2})\sim {O}(1)$; however, it may also be important for other parameter regimes for the reasons addressed above. Thus, (3.7) can be simplified as

(5.1)\begin{equation} \mathscr{E}_{0}\delta\phi_{0} = 2\left(\frac{c}{B\omega_{0}}k_{Z}k_{\theta,0}\right)^{2}\delta\phi_{0} |\delta\phi_{0}|^{2}.\end{equation}

Taking a two-scale analysis by assuming $\omega _{R}=\omega _{0}+\textrm {i}\partial _{\tau }$, and expanding $\mathscr {E}_{0}\simeq (\partial \mathscr {E}_{0}/\partial \omega _{0})(\textrm {i}\partial _{\tau }-\textrm {i}\gamma ^{L}_{R}-\varDelta )$ with $\varDelta$ being the nonlinearity-induced frequency shift and $\gamma ^{L}_{R}$ the linear RSAE growth rate,Footnote ¹ (5.1) becomes

(5.2)\begin{equation} \left[\textrm{i}\partial_{\tau}-\textrm{i}\gamma^{L}_{R}-\varDelta-2\left(\frac{c}{B\omega_{0}}k_{Z}k_{\theta,0}\right)^{2} \frac{|\delta\phi_{0}|^{2}}{\partial\mathscr{E}_{0}/\partial\omega_{0}}\right]\delta\phi_{0} = 0.\end{equation}

Equation (5.2) describes the RSAE nonlinear evolution due to scattering to different radial eigenstates (denoted by $\varDelta$) with different linear growth/damping rates ($\gamma ^{L}_{R}$) and nonlinear self-modulation by ZC generation, and the saturation level can be estimated by balancing the frequency shift (Max($|\gamma ^{L}_{R}|$, $|\varDelta |$)) and the nonlinear RSAE modulation by the ZC. Then one has

(5.3)\begin{equation} |\delta\phi_{0}|^{2} = \frac{\partial^{2}\mathscr{E}_{0}}{\partial k_{r}^{2}} \left(\frac{B\omega_{0}}{2ck_{\theta,0}}\right)^{2}.\end{equation}

Equation (5.3) describes the RSAE saturation level due to scattering by self-generated ZC to neighbouring (linearly stabler) radial eigenstates, assuming $\varDelta \gg \gamma ^{L}_{R}$. Cases with $\varDelta \ll \gamma ^{L}_{R}$ can be evaluated similarly. The RSAE saturation level can be estimated by substituting typical parameters into the expression. Furthermore, (5.3) can be substituted into (3.2), and the saturation level of the perturbed poloidal magnetic field $\delta B_{\theta }$ can be estimated as

(5.4)\begin{equation} \delta B_{\theta} \sim \frac{B_{0}k_{{\parallel},0}k_{Z}^{2}}{4k_{\theta,0}} \left.\frac{\partial^{2}\mathscr{E}_{0}}{\partial k_{r}^{2}}\right|_{k_{r}=0},\end{equation}

resulting in a modulation of local $q_{\min }$ by

(5.5)\begin{equation} \delta q \sim{-}q_{\min}\frac{B_{0}k_{{\parallel},0}k_{Z}^{2}}{4B_{\theta}k_{\theta,0}} \left.\frac{\partial^{2}\mathscr{E}_{0}}{\partial k_{r}^{2}}\right|_{k_{r}=0}.\end{equation}

In deriving (5.4) and (5.5), we have noted $\delta \psi =\omega \delta A_{\parallel }/(ck_{\parallel })$ and $\boldsymbol {\delta B}=\nabla \times \delta A_{\parallel }\boldsymbol {b}$. Typical parameters can be adopted, i.e. $q_{\min }\sim {O}(1)$, $B_{0}/B_{\theta }\sim qR/r_{0}$, $k_{\parallel }\sim 1/(qR)$, $k_{\theta ,0}\rho _{d}\sim {O}(1)$ with $\rho _{d}$ being the EP drift orbit radius and

(5.6)\begin{equation} k_{Z}^{2}\left.\frac{\partial^{2}\mathscr{E}_{0}}{\partial k_{r}^{2}}\right|_{k_{r}=0} \sim \frac{4\varDelta}{\omega},\end{equation}

which can be reasonably assumed as $\sim {O}(0.1)$. Thus, one can roughly estimate that $\delta B_{\theta }/B \sim {O}(10^{-4})$ as saturation amplitude and $\delta q/q\sim {O}(10^{-3})$. Such RSAE nonlinear saturation amplitude is equivalent to the estimation from the fixed-point solution in the driven-dissipative system described by three coupled equations such as (45)–(47) in Qiu et al. (Reference Qiu, Chen and Zonca2019a), and is in fact the same as the threshold amplitude on ZFZS nonlinear excitation. However, we note this is a rough order-of-magnitude estimation, since the nonlinear driven-dissipative system does not necessarily show a stationary saturation as described by the fixed-point solution. In fact, depending on the parameter regimes such as driving/dissipative rate and the initial conditions, the system may exhibit limit cycle oscillation, period doubling as well as route to chaos (Russell, Hanson & Ott Reference Russell, Hanson and Ott1980). Noting that the modification to local RSAE continuum frequency is $\sim O(n\delta qv_{A}/(qR))$ and that for reactor burning plasma with $\rho _{d}/a\sim O(10^{-2})$ most unstable RSAEs are characterized by $n\gtrsim O(10)$ (Wang et al. Reference Wang, Qiu, Zonca, Briguglio, Fogaccia, Vlad and Wang2018), the modification to local SAW continuum is comparable to the RSAE linear growth rate $\gamma ^{L}_{R}$ or frequency differences between different radial eigenstates ($\sim \varDelta$). Thus, one expects the ZC-induced SAW continuum modification in the vicinity of $q_{\min }$ to play an important role in RSAE nonlinear saturation, though the self-consistent study analogous to Zonca et al. (Reference Zonca, Romanelli, Vlad and Kar1995) and Chen et al. (Reference Chen, Zonca, Santoro and Hu1998) 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.