Nonlinear dynamics of the reversed shear Alfvén eigenmode in burning plasmas

1.   Introduction

Energetic particles (EPs) including fusion alpha particles are of crucial importance in magnetically confined fusion plasmas due to their contribution to plasma heating and potentially current drive [1, 2]. A key aspect of EP confinement is related to the shear Alfvén wave (SAW) instabilities [3] resonantly excited by EPs [4–8]. In magnetic confinement devices, SAW can be excited as various EP continuum modes (EPMs) [7] or discrete Alfvén modes (AEs) [9–11] inside the frequency gaps of the SAW continuum induced by equilibrium magnetic geometry and plasma nonuniformity. These SAW instabilities can then induce significant EPs anomalous transport loss across the magnetic surfaces, leading to plasma performance degradation and even damage of plasma facing components [12, 13]. With the EPs transport rate determined by the saturation amplitude and spectrum of SAW instabilities [14, 15], it is crucial to understand the nonlinear dynamics resulting in their saturation. In the past decades, nonlinear saturation of SAW instabilities has been broadly investigated both numerically and theoretically [16–33], among which, one of the most important channel is nonlinear wave-wave coupling [34, 35], i.e. nonlinear spectrum evolution of SAW instabilities due to interacting with other collective electromagnetic oscillations.

In the advanced scenarios of future reactor burning plasmas, a large fraction of non-inductive (e.g. bootstrap) current will be maintained [36] off-axis, and the magnetic shear is reversed in the plasma core region, where large fraction of energetic fusion alpha particles are generated [37]. As a result, a specific Alfvén eigenmode, namely the reversed shear Alfvén eigenmode (RSAE, also known as Alfvén cascade due to its frequency sweeping character [38–41]) could be excited and play important roles in transport of fusion alpha particles. In particular, as multiple-$n$ RSAEs can be strongly driven unstable simultaneously in reactors with machine size being much larger than fusion alpha particle characteristic orbit width, RSAEs can lead to strong alpha particle re-distribution and transport [37, 42]. RSAE is a branch of Alfvén eigenmodes localized around the SAW continuum extremum induced by the local minimum of the safety factor $q$-profile (labeled as $q_{\mathrm{min}}$) to minimize the continuum damping, and is characterized by a radial width of $\sim \sqrt{q/(r_0^2q^{''})}$ [41], with $r_0$ being the radial location of $q_{\mathrm{min}}$ and $q^{''}\equiv \partial^2_r q$. RSAE was originally observed in the advanced operation experiments in JT-60U tokamak [43], and was then detected in numerous JET discharges [44]. In present day tokamaks, RSAEs are generally excited by large orbit EP during current ramp up stage where reversed shear $q$-profile is created by insufficient current penetration [45]. In most cases, with $q_{\mathrm{min}}$ decreasing from, e.g., a rational value $m/n$ to $(m-1/2)/n$, the RSAEs exhibit upward frequency sweeping from beta-induced Alfvén eigenmode (BAE) [46] to the toroidal Alfveń eigenmode (TAE) [9, 11] frequency ranges. Here, $m$ and $n$ stand for the poloidal and toroidal mode numbers, respectively.

Due to the increasing importance in reactor burning plasmas operating at advanced scenarios, RSAE has drawn much attention in recent investigations. For instance, the resonant decay of RSAE into a generic low frequency Alfvén mode (LFAM) was investigated in reference [47], based on which, a potential alpha channelling mechanism [48] via the LFAM Landau damping was also proposed and analyzed. The modulational instability of a finite amplitude RSAE and excitation of the zero-frequency zonal structures were investigated in reference [49], where RSAE was saturated due to the modulation of SAW continuum and scattered into short-radial-wavelength stable domain. In particular, it is pointed out that, the generation of zonal current around the $q_{\mathrm{min}}$ region can be of particular importance, due to the sensitive dependence of RSAE on reversed shear profiles. Further numerical investigations of RSAE nonlinear dynamics can also be found in, e.g., references [50, 51], where RSAE nonlinear saturation due to wave-particle radial decoupling and zonal flow generation were investigated, respectively. It is also noteworthy that, in reference [52], a nonlinear saturation channel of RSAEs via nonlinear harmonic generation was investigated, where quasi-modes with double and/or triple toroidal mode numbers of the primary linearly unstable RSAE were generated due to kinetic electron contribution via “magnetic fluttering”, and led to RSAE nonlinear saturation. The setting of the simulation seems though, to some extend “artificial”, as only few toroidal mode numbers are kept in the simulation, it provides the important information of RSAE dissipation via nonlinear harmonic generation.

Motivated by reference [52], in this work, we present a potential nonlinear saturation mechanism for RSAE via nonlinear quasi-mode generation. This nonlinear mode coupling is achieved through the non-adiabatic responses to electrons, corresponding to the magnetic fluttering nonlinearity as addressed in reference [52]. Meanwhile, the mode coupling is generalized from RSAE self-coupling in the simulation [52] to include also the interaction between two RSAEs with different toroidal mode numbers, for which the nonlinear coupling could be much stronger. Generally, this quasi-mode could experience significant continuum or radiative damping, and provide a channel for primary RSAE energy dissipation. Using nonlinear gyrokinetic theory, the parametric dispersion relation of this nonlinear mode coupling process is derived. Focusing on the continuum damping of the quasi-mode, the resultant nonlinear damping to RSAE is then analyzed and estimated.

The remainder of this paper is arranged as follows. In section 2, the theoretical model is given. In section 3, the nonlinear dispersion relation describing the RSAE nonlinear evolution due to interaction with another background RSAE is derived. It is then used in section 4 to investigate the continuum damping of the quasi-mode. An estimation of the resulting damping to RSAE is also presented.

2.   Theoretical model

Considering two co-propagating RSAEs $\Omega_0\equiv\Omega_0(\omega_0,{\boldsymbol{k}}_0)$ and $\Omega_1\equiv\Omega_1(\omega_1,{\boldsymbol{k}}_1)$ coupling and generating a beat wave $\Omega_\mathrm{b}\equiv\Omega_\mathrm{b}(\omega_\mathrm{b},{\boldsymbol{k}}_\mathrm{b})$, with the frequency and wavenumber of $\Omega_\mathrm{b}$ determined by the matching condition $\Omega_\mathrm{b} = \Omega_0 +\Omega_1$, the beat wave $\Omega_\mathrm{b}$ is likely a high-frequency quasi-mode bearing significant continuum or radiative damping, as it may not satisfy the global RSAE dispersion relation with corresponding toroidal mode number $n_\mathrm{b} = n_0+n_1$. Here, for “co-propagating”, we mean the two RSAEs propagate in the same direction along the magnetic field line. This nonlinear coupling provides the primary RSAEs an indirect damping mechanism, and may result in their saturation. Here, for simplicity of the discussion while focusing on the main physics picture, we focus on the continuum damping of the quasi-mode, and investigate the resultant nonlinear saturation of primary RSAEs. A sketched illustration of the proposed process is given in figure 1, where two RSAEs with $n=3,\ 4$ couple and generate an $n = 7$ high-frequency quasi-mode, which can be heavily damped due to coupling with the corresponding shear Alfvén continuum.

Figure  1.  Schematic plots of Alfvén continuum for $n=3,\ 4$ and 7 respectively. With the red bars standing for Alfvén modes, (a) and (b) represent RSAEs, and (c) represents the high-frequency quasi-mode generated by the nonlinear coupling of two RSAEs. Here, the horizontal axis is $r/a$ in arbitrary unit, the vertical axis is the normalized frequency to $\omega_{\mathrm{A}0}$, i.e. the Alfvén frequency on the major axis.

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The nonlinear coupling of the two RSAEs and the resultant damping are investigated in a uniform low-$\beta$ magnetized plasma using the standard nonlinear perturbation theory, with $\beta\ll1$ being the ratio between plasma and magnetic pressures. Here, for “uniform”, we mean the effects associated with diamagnetic effects are systematically neglected, while noting magnetically confined plasma is intrinsically nonuniform. Introducing the scalar potential $\delta\phi$ and parallel component of vector potential $\delta A_\parallel$ as the perturbed field variables, one then has $\delta\phi = \delta\phi_0+\delta\phi_1+\delta\phi_\mathrm{b}$ with the subscripts $0$, $1$ and $\mathrm{b}$ denoting $\Omega_0$, $\Omega_1$ and $\Omega_\mathrm{b}$, respectively. For convenience of investigation, $\delta A_\parallel$ is replaced by $\delta\psi\equiv \omega\delta A_\parallel/(ck_\parallel)$, such that $\delta\phi = \delta\psi$ can straightforwardly recover the ideal MHD limit, i.e. vanishing parallel electric field fluctuation $\delta E_\parallel$. Both electrons and ions are chacterised by Maxwellian equilibrium distributions $F_\mathrm{M}$.

For RSAEs typically dominated by single-$n$ and single-$m$ mode structures near $q_{\mathrm{min}}$, we take

with $A_k(t)$ being the slowly varying mode amplitude, $\Phi_k(x)$ being the parallel mode structure localized around $q_{\mathrm{min}}$ with $x\equiv nq-m$, and the normalization $\int |\Phi_k|^2\mathrm{d}x = 1$ can be adopted.

Nonlinear mode equations can be derived from charge quasi-neutrality condition

and nonlinear gyrokinetic vorticity equation [53]

Here, the terms on the left hand side of equation (2) are field line bending, inertia and curvature-pressure coupling terms, respectively, whereas the terms on the right hand side represent Reynolds and Maxwell stresses dominating in short wavelength limit. $N_0$ is the equilibrium particle density, $q_j$ is the electric charge, the angular brackets $\langle \dots\rangle$ denote velocity space integration, $\partial_l$ is the spatial derivative along the equilibrium magnetic field, $k_\bot = \sqrt{k^2_r+k^2_{\theta}}$ is the perpendicular wavenumber, $J_k\equiv J_0(k_\bot \rho)$ with $J_0$ being the Bessel function of zero index accounting for finite Larmor radius effects, and $\rho = v_\bot/\Omega_\mathrm{c}$ is the Larmor radius with $\Omega_\mathrm{c}$ being the cyclotron frequency. Furthermore, $\omega_\mathrm{d} = (v^2_\bot+2v^2_\parallel)/ (2\Omega_\mathrm{c}R_0)(k_r \sin\theta+k_\theta\cos\theta)$ is the magnetic drift frequency, $\Lambda_{k''}^{k'} = (c/B_0){\hat{\boldsymbol{b}}}\cdot{\boldsymbol{ k}}''\times {\boldsymbol{k}}'$ accounts for perpendicular coupling with the constraint of frequency and wavevector matching conditions, and $\delta L_k\equiv\delta \phi_k-k_\parallel v_\parallel\delta\psi_k/\omega_k$ is the scalar potential in the frame moving with guiding center. The non-adiabatic particle response $\delta H_k$ is derived from the nonlinear gyrokinetic equation [54]:

3.   Nonlinear mode equations

In this section, the coupled nonlinear equations for RSAE and the high-frequency quasi-mode are derived in sections 3.1 and 3.2, which are then combined and give the parametric dispersion relation in section 3.3.

3.1   Nonlinear quasi-mode $\mathit{\Omega_{\mathrm{\mathit{b}}}}$ genaration

The nonlinear equation for quasi-mode $\Omega_\mathrm{b}$ generation can be derived from the quasi-neutrality condition and the nonlinear gyrokinetic vorticity equation. The nonlinear non-adiabatic particle response of $\Omega_\mathrm{b}$ can be derived from the nonlinear component of equation (3) noting the $k_\parallel v_\mathrm{e}\gg\omega\gg k_\parallel v_\mathrm{i}\gtrsim\omega_\mathrm{d}$ ordering, and one obtains

In deriving equations (4) and (5), the linear particle responses $\delta H_{k\mathrm{i}}^\mathrm{L} = (e/T_\mathrm{i})F_{\mathrm{M}}J_k\delta\phi_k$ and $\delta H_{k\mathrm{e}}^\mathrm{L} = -(e/T_\mathrm{e})F_{\mathrm{M}}\delta\psi_k$ are used (interested readers may refer to the Appendix for the detailed derivation). With $\delta H_{\mathrm{be}}^{\mathrm{NL}}$ representing the coupling between $\Omega_0$ and $\Omega_1$ due to nonlinear electron contribution, it corresponds to the magnetic fluttering nonlinearity investigated in [52]. Substituting equations (4) and (5) into the quasi-neutrality condition, one obtains

i.e. breaking of ideal MHD constraint due to nonlinear mode coupling, while finite parallel electric field associated with linear FLR effects is not included here for simplicity [55]. This is also consistent with the $b_k\ll1$ ordering for linear unstable RSAEs with typically $k^{-1}_{\perp}$ comparable to EP characteristic orbit width. Substituting the particle responses into the nonlinear gyrokinetic vorticity equation, one obtains

with $b_k = k_\bot^2\rho_\mathrm{i}^2/2$, $v_\mathrm{A}$ being the Alfvén speed, $\omega_\mathrm{G}\equiv \sqrt{7/4+\tau}v_\mathrm{i}/R_0$ being the leading order geodesic acoustic mode frequency [56, 57] and $\tau\equiv T_\mathrm{e}/T_\mathrm{i}$. Combining equations (6) and (7), one obtains

Equation (8) is the desired nonlinear equation describing the high-frequency quasi-mode $\Omega_\mathrm{b}$ generation due to $\Omega_0$ and $\Omega_1$ coupling, with the $\Omega_\mathrm{b}$ dielectric function $\varepsilon_{\mathrm{Ab}}$ defined as

and the nonlinear coupling coefficient $\beta_\mathrm{b}$ given by

It is worth mentioning that, the $\Omega_\mathrm{b}$ dielectric function, $\varepsilon_{\mathrm{Ab}}$, may not satisfy the global linear RSAE dispersion relation for toroidal mode number $n_\mathrm{b}$, and $\Omega_\mathrm{b}$ could be a quasi-mode experiencing heavy damping, leading to the dissipation of both itself and the primary RSAEs, as shown later.

3.2   Nonlinear equation for $\mathit{\Omega_0}$

The nonlinear coupling equation for the test RSAE $\Omega_0$ can be derived following a similar procedure. However, noting that $\Omega_\mathrm{b}$ is a quasi-mode, one needs to keep both the linear and nonlinear particle responses since they could be of the same order. The resultant nonlinear non-adiabatic particle responses of $\Omega_0$ are respectively

Substituting equations (9) and (10) into the quasi-neutrality condition, one obtains

On the other hand, the nonlinear gyrokinetic vorticity equation yields

Combining equations (11) and (12), one obtains the nonlinear equation of $\Omega_0$

with the linear $\Omega_0$ dielectric function in the WKB limit given by

the nonlinear coupling coefficient $\beta_0$ given by

and $\varepsilon_{\mathrm{A}0}^{\mathrm{NL}}$ due to nonlinear particle contribution to $\Omega_\mathrm{b}$ being

Equation (13) is the nonlinear equation for the test RSAE $\Omega_0$ evolution due to the feedback of the quasi-mode $\Omega_\mathrm{b}$, and can be coupled with equation (8) to yield the nonlinear dispersion relation for $\Omega_0$ regulation via the high-frequency quasi-mode generation.

3.3   Parametric dispersion relation

Combining equations (8) and (13), one obtains

Equation (15) describes the evolution of the test RSAE $\Omega_0$ due to the nonlinear interaction with another RSAE $\Omega_1$, which can also be considered as the “parametric decay dispersion relation” of $\Omega_1$ decaying into $\Omega_0$ and $\Omega_\mathrm{b}$. Noting that $\varepsilon_{\mathrm{A}0}^{\mathrm{NL}}$ related term contributes only to the nonlinear frequency shift, re-organizing equation (15) and taking the imaginary part, one obtains

In deriving equation (16), we have expanded $\varepsilon_{\mathrm{A}0}\simeq \mathrm{i}\partial_{\omega_{0\mathrm{r}}}\varepsilon_{\mathrm{A0r}}(\partial_t-\gamma_{0})\simeq -(2\mathrm{i}/\omega_{0\mathrm{r}})\gamma_{\mathrm{ND}}$, with $\gamma_0$ being the linear growth rate of $\Omega_0$ and $\gamma_{\mathrm{ND}}$ being its damping rate due to scattering by $\Omega_\mathrm{b}$, respectively. It is also noteworthy that, as $\Omega_\mathrm{b}$ is a quasi-mode with the imaginary part of $\varepsilon_{\mathrm{Ab}}$ being comparable to the real part, no expansion is made to $\varepsilon_{\mathrm{Ab}}$. Meanwhile, for the continuum damping of interest, $\mathrm{Im} (1/\varepsilon_{\mathrm{Ab}}) = - {\text{π}}\delta(\varepsilon_{\mathrm{Ab}})$ is taken, corresponding to the absorption of the nonlinear generated quasi-mode $\Omega_\mathrm{b}$ near the SAW continuum resonance layer [55].

4.   Nonlinear damping to test RSAE

An estimation of the nonlinear damping rate is made to quantify the contribution of this nonlinear process. Multiplying equation (16) with $\Phi_0^{*}$ and averaging over radial mode structure, one obtains

Here, $\langle \dots\rangle_x\equiv \int\cdots \mathrm{d}x$ denotes the integration over $x$, with the weighting of $|\Phi_0|^2$. To make analytical progress, the parallel mode structures for RSAEs are taken as $\Phi_k\simeq \exp(-x^2/2\Delta_k^2)/( {\text{π}}^{1/4}\Delta_k^{1/2})$ with $\Delta_k$ being the characteristic radial width of the parallel mode structures and one typically has $\Delta_0\sim \Delta_1\lesssim{\cal{O}}(1)$.

Equation (17) gives the test RSAE $\Omega_0$ damping rate due to coupling to a “background” RSAE $\Omega_1$. As multiple-$n$ RSAEs could be driven unstable simultaneously [37] at the same location, all the background RSAEs interacting with $\Omega_0$ should be taken into account. Summation over all the RSAEs within strong or moderate coupling range to the test RSAE $\Omega_0$, and assuming that the integrated electromagnetic fluctuation amplitude induced by RSAEs is of the same order as the background RSAEs, the nonlinear damping rate can be estimated as

In estimating $\gamma_{\mathrm{ND}}$, $\delta(\varepsilon_{\mathrm{Ab}}) = \partial\varepsilon_{\mathrm{Ab}}/\partial x\sum\nolimits_{x_0}^{}\delta(x-x_0)\simeq -2x/\varpi_\mathrm{b}^2\sum\nolimits_{x_0}^{}\delta(x-x_0)$ is taken, with $\varpi\equiv \omega/\omega_\mathrm{A}$, $\omega_\mathrm{A}\equiv v_\mathrm{A}/(q_{\mathrm{min}}R_0)$ being the local Alfvén frequency and $x_0$ being the zero points of $\varepsilon_{\mathrm{Ab}}$. Other parameters are taken as $T_\mathrm{i}/T_\mathrm{E}\sim {\cal{O}}(10^{-2})$, $R_0/\rho_\mathrm{i}\sim {\cal{O}}(10^3)$, $|\delta B_{\mathrm{\mathit{r}}}/B_0|^2\sim{\cal{O}}(10^{-7})$, $b\sim k^2_\theta\rho_\mathrm{i}^2\sim(T_\mathrm{i}/T_\mathrm{E})/q^2$ for linearly unstable RSAEs with $T_\mathrm{E}$ being the EP characteristic energy, $\delta$B_r being the perturbed radial magnetic field, and B₀ being the equilibrium magnetic field amplitude.

Equation (18) shows an appreciable nonlinear damping to the test RSAE $\Omega_0$, which could make significant contribution to its nonlinear saturation. Note that in the present work, only the scattering to high-frequency quasi-mode is taken into account. Nevertheless, one can generalize the analysis to include other damping channels including other nonlinear mode coupling mechanism [47, 49, 58]. This is, however, beyond the scope of the present work, focusing on providing an interpretation to the simulation of reference [52], with the generalization to include coupling to background RSAEs with different toroidal mode numbers.

5.   Conclusion

Motivated by recent simulation study [52], a novel mechanism for RSAE nonlinear saturation is proposed and analyzed, which is achieved through generation of a high-frequency quasi-mode by the nonlinear mode coupling of two RSAEs. This high-frequency quasi-mode can be significantly damped due to coupling to the corresponding SAW continuum, thus leads to a nonlinear damping effect to the RSAEs, and promotes their nonlinear saturation. The nonlinear dispersion relation describing this nonlinear coupling process is derived based on the nonlinear gyrokinetic theory. To estimate the relevance of this nonlinear saturation mechanism to RSAE, an estimation of the nonlinear damping rate to the test RSAE is given by $\gamma_{\mathrm{ND}}/\omega_{0\mathrm{r}}\sim {\cal{O}}(10^{-3}-10^{-2})$ under typical parameters of the future burning plasmas. This result could be comparable with the typical RSAE linear growth rate excited by resonant EPs, and thus, demonstrate the significance of the nonlinear saturation mechanism proposed here.

The nonlinear coupling coefficient derived in this work is complicated, and depends on various conditions including the frequency, wavenumber, radial mode structure of the RSAEs and the structure of Alfvén continuum corresponding to the mode number of quasi-mode. This study seeks to estimate the relevance of the nonlinear saturation mechanism and has not done a thorough investigation on the optimised parameter regimes for this process to occur and dominate. For more detailed analysis, interested readers may refer to [47] with the nonlinear coupling coefficient having similar features.

As a final remark, the high-frequency quasi-mode discussed here, is damped due to the coupling to local Alfvén continuum only, whereas other damping effects (e.g. radiative damping and Landau damping due to frequency mismatch) are not included, and inclusion of which could yield an enhanced regulation effect to RSAE. Besides, the final nonlinear saturation of RSAE may require other channels including the self-consistent evolution of EPs distribution function [59], spontaneous decay into LFAM [47], zonal field generation [49] and geodesic acoustic mode (GAM) generation [58]. Further comprehensive and detailed investigations, particularly through large scale nonlinear gyrokinetic simulations, are required to assess the saturation level of RSAE and the energetic particle transport rate.

Appendix.   Derivation of nonlinear particle responses to $\mathbf{\mathbf{\Omega_{\mathrm{\mathbf{b}}}}}$

In this appendix, the nonlinear particle responses to $\Omega_\mathrm{b}$ are derived, based on the nonlinear components of equation (3). Specifically, for ion, noting $\omega\gg k_\parallel v_\mathrm{i}\gtrsim\omega_\mathrm{d}$, one has

Note in this derivation, $\delta L\equiv \delta\phi-(k_\parallel v_\parallel/\omega)\delta\psi\simeq \delta\phi$ for ion with $\omega\gg k_\parallel v_\mathrm{i}$, and $\delta H_{k\mathrm{i}}^\mathrm{L} = (e/T_\mathrm{i})F_\mathrm{M}J_k\delta\phi_k$ is used.

For electron with $k_\parallel v_\mathrm{e}\gg\omega \gg\omega_\mathrm{d}$ and negligible FLR effects, only the $k_\parallel v_\parallel$ term is kept, and the nonlinear gyrokinetic equation for electron can be written as

One obtains, the nonlinear electron response to $\Omega_\mathrm{b}$

Note in this derivation, $\delta L\equiv \delta\phi-(k_\parallel v_\parallel/\omega)\delta\psi\simeq -(k_\parallel v_\parallel/\omega)\delta\psi$ for electron with $\omega\ll k_\parallel v_\mathrm{e}$, and $\delta H_{k\mathrm{e}}^\mathrm{L} = -(e/T_\mathrm{e})F_\mathrm{M}\delta\psi_k$ is used. The nonlinear non-adiabatic responses to $\Omega_0$ can be derived similarly.