Low-frequency shear Alfvén waves at DIII-D: theoretical interpretation of experimental observations

The low-frequency Alfvén wave spectrum in the kinetic thermal-ion (KTI) gap frequency range Chen and Zonca (2007) has been of research interest since the first observations of beta-induced Alfvén eigenmodes (BAEs) Heidbrink et al. (1993); Turnbull et al. (1993). These modes are characterized with frequencies comparable to thermal ion transit and/or bounce frequencies, and can interact with both thermal and fast particles Zonca et al. (1996, 1999, 2010); Chavdarovski and Zonca (2009); Lauber et al. (2009); Chen and Zonca (2016), with possible (positive/negative) impact on the corresponding transport processes resulting from finite fluctuation and zonal field structures levels Chen and Zonca (2007, 2016); Zonca et al. (2021). The effects of energetic particles (EPs) on low-frequency shear Alfvén waves (SAWs) ranging from kinetic ballooning mode (KBM) Cheng (1982); Tang et al. (1980); Biglari and Chen (1991) to BAE are one of areas widely studied in the magnetic fusion literature. Recent papers on this topic cover the interpretation and modeling of experimental measurements by currently developed innovative diagnostics Sharapov et al. (2013); Gorelenkov et al. (2014); Heidbrink et al. (2021a, b, c), as well as latest progress in comparing numerical investigation and/or simulation results with observed phenomena Curran et al. (2012); Lauber (2013); Chavdarovski and Zonca (2014); Fasoli et al. (2016); Bierwage and Lauber (2017); Choi et al. (2021).

A series of dedicated experiments have been recently conducted on DIII-D to investigate the stability of the low-frequency SAWs Heidbrink et al. (2021a, b, c). The experiments show that the observed low-frequency mode¹¹1We will refer from now on only to the low frequency Alfvén mode (LFAM) which belongs to low-frequency SAWs predominantly Alfvénic polarization, keeping in mind that this terminology is the same as the low-frequency mode observed in recent DIII-D experiments Heidbrink et al. (2021a)., which was previously misidentified as ‘beta-induced Alfvén acoustic eigenmode (BAAE)’ Gorelenkov et al. (2007, 2009), is actually a lower-frequency reactive unstable KBM which favors high thermal electron temperature but almost has no coupling with energetic ions Heidbrink et al. (2021a); while the BAE is resonantly excited by energetic ions with its stability depending sensitively on the beam power and injection geometry Heidbrink et al. (2021b), consistent with earlier theoretical predictions Chen and Zonca (2017) based on the GFLDR theoretical framework Zonca and Chen (2014a, b). These instabilities are also found to occur when the minimum of the safety factor ($q_{min}$) approaches rational values and the modes in ascending pattern of higher frequency BAEs and LFAMs are separated by approximately the toroidal rotation frequency ($f_{rot}$). However, the subtle differences between them are that, for LFAMs, the maximum frequency appears at rational values of $q_{min}$ and the detected modes are radially localized near $q_{min}$, while BAEs occur at times near rational $q_{min}$ values but the timing of unstable modes is less precise than that for LFAMs. In addition, compared with the LFAMs, the BAE eigenfunction shows more deviation from the radial position of $q_{min}$ spatially. Although dedicated numerical simulations of the linear properties of the BAEs and LFAMs Varela et al. (2018); Choi et al. (2021) have been carried out, the above experimental phenomena have not been fully explained. Motivated by this, the present work aims to provide an in-depth theoretical understanding of the linear properties of low-frequency SAWs, with particular attention to the effects of energetic ions on their stability. The analysis is carried out based on the theoretical framework of the generalized fishbone-like dispersion relation (GFLDR) Chen et al. (1984); Chen (1994); Tsai and Chen (1993); Zonca and Chen (2006); Zonca et al. (2007); Zonca and Chen (2014a, b), and provides qualitative and quantitative interpretation of the main instability mechanisms underlying the numerical simulation results and experimental observations. As a result, our analysis provides yet another evidence of the predictive strength of the GFLDR theoretical framework and of its enhanced “interpretative capability for both experimental and numerical simulation results” Zonca and Chen (2014a, b).

In this work, unlike the previous paper not considering effects due to energetic particles (EPs) Ma et al. (2022), we focus on the BAE excitation via transit resonance with passing fast ions created by NBI heating Heidbrink et al. (2021b). In this case, the dynamics of various species enter the dispersion relation of low-frequency SAW, and affect its behavior linearly at different pressure gradient scale lengths. For DIII-D discharge #178631, Fig. 1 shows the radial dependence of different scale lengths of thermal and energetic particle pressure ($L_{P_{th}}$ and $L_{P_{E}}$), as well as the estimated radial mode width ($\Delta_{m}$) for weak and/or vanishing magnetic shear range, i.e., $|s|=|(r/q)(dq/dr)|\lesssim 0.05$. More specifically, the EP pressure profiles are given by the following two limits. One is the relaxed EP profile provided with EFIT reconstruction Lao et al. (1985), where the fast-ion pressure is the difference between the equilibrium pressure and the thermal pressure. The other is the “classical” EP profile obtained by TRANSP/NUBEAM Pankin et al. (2004) in the absence of fast-ion transport by instabilities. The pressure scale lengths of EPs are denoted by $L_{P_{E;rel}}$ and $L_{P_{E;cl}}$ for these two cases (respectively). The true EP profile when the modes are destabilized likely lies between these two limits. The actual pressure is closest to the EFIT-based one but this is measured after the unstable modes have (presumably) caused the gradients to flatten. Meanwhile, for the weak and/or vanishing magnetic shear region and given toroidal and poloidal mode numbers $(n,m)$, the normalized parallel wave vector is $\Omega_{A,m}=k_{\parallel n0}q_{min}R_{0}=nq_{min}-m$, and the radial width of the mode can then be estimated by $\Delta_{m}\simeq 1/|nq^{\prime\prime}|^{1/2}$ Zonca and Chen (2000); Zonca et al. (2002). Here, $k_{\parallel n0}$ represents the parallel wave-vector at $r_{0}$, where $q$ has a minimum given by $q_{min}$, $q^{\prime\prime}$ denotes the second derivative of $q$ in the radial direction, and $R_{0}$ is the torus major radius. It can be found that in this region, $L_{P_{th}}\gg\Delta_{m}$, which yields the usual local limit of the mode dispersion relation. This is the case for the reactive unstable LFAM in the absence of EPs already studied in Ref. Ma et al. (2022). However, for the energetic ion-driven BAEs, there are two distinct cases: the moderate EP pressure gradient case with $L_{P_{E;rel}}>\Delta_{m}$, which also approximately yields the usual local GFLDR Tsai and Chen (1993); Chen (1994); Zonca et al. (1996); Zonca and Chen (2000); Zonca et al. (2002, 2007); Zonca and Chen (2014a, b); and the strong EP pressure gradient case with $L_{P_{E;rel}}\simeq\Delta_{m}$, for which the global dispersion relation of low-frequency SAWs is needed and will be discussed in Sec. II.

Performing detailed numerical investigations of the two cases, it is found that the LFAMs and BAEs can both be driven unstable, however, due to different instability mechanisms, these modes yield different experimental observations. All these features can be, quantitatively and qualitatively, interpreted theoretically based on the GFLDR. Moreover, it is also confirmed that the stability of BAAE is not affected by EPs, even though it becomes weakly damped after coupling with KBM, consistent with theoretical predictions by Chen and Zonca Chen and Zonca (2017) as well as numerical simulation results reported in Refs. Choi et al. (2021); Lauber (2013); Bierwage and Lauber (2017).

The paper is structured as follows. Local and global dispersion relations for the low-frequency SAWs near weak and/or vanishing magnetic shear are introduced and discussed in Sec. II in different parameter regimes, depending on the relative magnitude of $L_{P_{E}}$ and $\Delta_{m}$. Detailed numerical investigations and theoretical analysis of the low-frequency SAWs in the presence of EPs are discussed in Sec. III, where comparisons between theory and experiments are also made. Finally, conclusions and further discussions are given in Sec. IV.

In this Section, we will present analytical dispersion relations for low-frequency SAW excitation in weakly reversed-shear DIII-D discharges. As stated in the previous Section, two cases determined by the relative magnitude of $L_{P_{E}}$ and $\Delta_{m}$ will be used to investigate the low-frequency SAW stability: case I, the local GFLDR model corresponding to $L_{P_{E}}>\Delta_{m}$; and case II, the global GFLDR corresponding to $L_{P_{E}}\simeq\Delta_{m}$.

Consider case I first. For $L_{P_{E;rel}}>\Delta_{m}$, the scales of $L_{P_{E}}$ and $\Delta_{m}$ can be separated, and the vorticity equation Chen (1994); Tsai and Chen (1993); Zonca et al. (1996); Zonca and Chen (2014a, b); Chen and Zonca (2016) which governs shear Alfvén waves (SAWs) can yield the low-frequency electromagnetic fluctuation dispersion relation in the usual local limit, as derived and discussed in great details in Refs. Tsai and Chen (1993); Chen (1994); Zonca et al. (2007); Zonca and Chen (2014a, b); Chen and Zonca (2016). We just note that, for DIII-D case of interest, the reversed magnetic shear configuration and thermal plasma compression effects should be accounted for properly Ma et al. (2022). Thus, for $s=0$ at $r_{0}$ but with finite $S\equiv(r/q)[q^{{}^{\prime\prime}}]^{1/2}$, the local GFLDR for low-frequency SAWs can be written as Zonca et al. (2002, 2007); Zonca and Chen (2014a, b); Chen and Zonca (2017)

where the generalized inertia term $\Lambda_{n}(\omega)$ here, including both diamagnetic effects as well as kinetic effects of circulating and trapped particle dynamics, has been derived explicitly in Ref. Chavdarovski and Zonca (2009) and the main results are summarized in Appendix A. The right hand side of Eq. (1) contains both “fluid” ($\delta{\hat{W}}_{nf}$) and “kinetic” ($\delta{\hat{W}}_{nk}$) contributions to the potential energy in the “regular” ideal region. In the low-frequency limits ($|\Lambda_{n}^{2}|\ll 1$), $\delta{\hat{W}}_{nf}$ is independent of the frequency and the explicit expression, specialized to the ($s$, $\alpha$) model equilibrium Connor et al. (1978) with circular flux surfaces, reads,

where $\alpha=\alpha_{c}+\alpha_{E}$, $\alpha_{c}=-R_{0}q_{min}^{2}d\beta/dr$ and $\alpha_{E}=-\frac{1}{2}R_{0}q_{min}^{2}d(\beta_{E\parallel}+\beta_{E\perp})/dr$. Note that Eq. (2) includes the contribution of the energetic particle adiabatic and convective responses as well Chen et al. (1984).

The term $\delta{\hat{W}}_{nk}$ is always a function of the mode frequency $\omega$, as it reflects resonant as well as non-resonant wave-particle interactions. For simplicity but still relevant to the DIII-D case, we take $F_{0E}$ to be a single pitch angle ($\lambda=\mu/\varepsilon$) slowing-down beam ion equilibrium distribution function; i.e., $F_{0E}=\frac{B_{0}\beta_{E}(r)}{2^{5}\sqrt{2}\pi^{2}m_{E}\varepsilon_{b}}\sqrt{(1-\lambda_{0}B_{0})}\varepsilon^{-3/2}\delta(\lambda-\lambda_{0})$. Here, $\beta_{E}(r)\equiv 8\pi P_{E}(r)/B_{0}^{2}$ is the ratio of EP kinetic and magnetic pressures and $B_{0}$ the on-axis equilibrium magnetic field, $\delta(x)$ is the Dirac function, $\mu$ is the magnetic moment and $\varepsilon=\upsilon^{2}/2\leq\varepsilon_{b}$ with $\varepsilon_{b}$ being the EP birth energy per unit mass. Then the explicit expression of non-adiabatic contribution $\delta{\hat{W}}_{nku}$ for the passing energetic ions is given by Tsai and Chen (1993); Chen (1994)

where ${\bar{\omega}}=\omega/\omega_{tEm}$ and $\omega_{tEm}\equiv\sqrt{2\varepsilon_{b}}/qR_{0}$ is the EP transit frequency at the maximum particle energy.

It is worthwhile emphasizing that the finite $k_{\parallel n0}q_{min}R_{0}$ in Eq. (1) plays an important stabilizing role since it represents the finite line bending effect at $r=r_{0}$ Zonca et al. (2007); Zonca and Chen (2014a, b). Furthermore, the expression of $\Lambda_{n}$ depends on the mode polarization via $S_{f}\equiv(i\delta E_{\parallel}/k_{\parallel})_{a.c.}\big{/}\delta\phi_{d.c.}$, where $a.c.$ and $d.c.$ refer to the sinusoidal and nearly constant (flute-like) components of the parallel electric field, wave vector, and scalar potential fluctuation Chavdarovski and Zonca (2014); Chen and Zonca (2017). The detailed expression of $S_{f}$, again, is given in the Appendix A. Here, we just note that $|S_{f}|$ is much smaller than unity for shear Alfvén wave and order of unity for ion acoustic wave Chavdarovski and Zonca (2009, 2014); Chen and Zonca (2017).

We remark here that, in the moderate pressure gradient case, the local GFLDR for the low-frequency SAWs is enough to delineate the underlying physics of the experimental and simulation results. However, the local GFLDR for the low-frequency SAWs, given by Eq. (1), will fail in the presence of strong EP pressure gradient, i.e., case II. In this case, two typical scale lengths $L_{P_{E,cl}}$ and $\Delta_{m}$ can not be separated anymore and, thus, a global dispersion relation is needed which can be derived from the vorticity equation, i.e., Eq. (1) of Ref. Zonca et al. (2002). Noting that the mode structure is dominated by single toroidal and poloidal mode numbers, $(n,m)$, the governing equation reads

where ${\bf k}_{\perp}/k_{\theta}=-({\bf e_{\theta}}-{\bf e}_{r}\xi)$ with ${\bf e}_{r}$ and ${\bf e}_{\theta}$ being, respectively, the radial and poloidal unit vectors, $x^{2}=nq_{min}^{\prime\prime}(r-r_{0})^{2}$, $\xi\equiv(i/n^{1/2})S(\partial/\partial x)$, and $\delta\phi_{m}$ is the $m$th poloidal harmonic of the scalar field perturbation. It is worth noting that, toroidal coupling among different poloidal harmonics is typically not important for modes in the reversed magnetic shear region, consistent with the mode being dominated by single $m$ and $n$. The terms $F$ and $K$ in Eq. (4) represent, respectively, the fluid-like particle and energetic ion contributions with their explicit form reading

where $D_{S}=S\sqrt{\Omega_{A,m}/n}$, $q_{E}$ and $m_{E}$ are the electric charge and mass of energetic ions, $\Omega_{dE}=(\upsilon_{E\perp}^{2}/2+\upsilon_{E\parallel}^{2})/\omega_{cE}R_{0}$, $\omega_{tE}=\upsilon_{E\parallel}/qR_{0}$, $QF_{0E}=(\omega\partial_{\varepsilon}+{\hat{\omega}}_{\ast E})F_{0E}$, ${\hat{\omega}}_{\ast E}F_{0E}=\omega^{-1}_{cE}({\bf k}\times{\bf b})\cdot\nabla F_{0E}$, $\omega_{cE}=q_{E}B/m_{E}c$, $\langle(...)\rangle_{\upsilon}=\int d^{3}\upsilon(...)$, and the subscripts $\parallel$ and $\perp$ represent the parallel and perpendicular components with respect to the equilibrium magnetic field ${\bf b}$.

Equation (4) is an ordinary differential equation and, generally, requires a numerical approach to be solved. However, for DIII-D case, the radial dependence of the normalized pressure gradient of energetic ions with the classical profile, as is shown by black curve in Fig. 2, can be well fitted by the analytic formula $\alpha_{E}(\rho)=c_{1}\left(1-(\rho-c_{2})^{2}/c_{3}^{2}\right)$, with $c_{1}=0.7099$, $c_{2}=0.3018$ and $c_{3}=0.2944$. This allows us to obtain simple analytical dispersion relations for low-frequency SAWs excitation. We just note that the maximum drive of energetic ions is located around $\rho=c_{2}=0.3018$, which deviates from the radial position of $q_{min}$. Then $\alpha_{E}(r)$ in Eq. (3) can be rewritten as

where $\delta_{a}=c_{1}/\alpha_{E0}$, $\delta_{b}=r_{0}-c_{2}a$ and $\delta_{c}=c_{3}a/L_{PE;cl}$, $a$ is the minor radius, $\alpha_{E0}$ and $L_{PE;cl}$ are evaluated at $r=r_{0}$.

Introducing the notation $x=r-r_{0}=\sigma z-\delta_{b}$, Eq. (4) is readily cast into the form

where $\epsilon_{A0}=\Lambda^{2}-\Omega^{2}_{A,m}$, $\delta{\hat{W}}_{nku0}=\frac{\pi\alpha_{E0}}{4\sqrt{2}}\left[2-{\bar{\omega}}\ln\left(\frac{{\bar{\omega}}+1}{{\bar{\omega}}-1}\right)\right]$. Then, Eq. (7) yields the following global dispersion relation for low-frequency SAWs,

Here, the integer $L$ is the radial eigenmode number. The corresponding eigenfunction reads

where $H_{L}(z)$ represents $L$th order Hermite polynomials and the causality constraints upon the discrete bound modes requiring ${\cal R}e(\sigma^{2})>0$, where $\sigma^{2}$ is solved for from the second of Eqs. (7) consistently with the dispersion relation, Eq. (8). The typical radial width, $w$, of $\delta\phi_{m}(r)$ is determined by $w^{2}=4\sigma^{2}$.

Equations (1) and (8) constitute the results of the present section, i.e., the local and global GFLDR for the low-frequency SAWs excited by energetic ions. With their explicit form, we can compute the individual terms involved in equations and investigate the linear properties of the experimentally observed low-frequency SAWs.

In this Section, we separately present numerical results for the local and global low-frequency SAW stability properties in the presence of energetic ions, for which the dispersion relation is given by Eqs. (1) and (8).

The numerical investigations use experimental equilibrium and profiles as shown in Fig. 3 for the DIII-D shot #178631 at the time $t=1200$ ms Heidbrink et al. (2021a), where the $q$-profile has a reversed shear configuration with $q_{min}=1.37$ at $r_{0}/a=0.28$ and $q_{min}$ decreases from 1.49 to 1.18 in the time window $1050$ ms $<t<1350$ ms, as shown in Fig. 6 (b) in Ref. Heidbrink et al., 2021a.

The present work has addressed linear properties of the low-frequency shear Alfvén waves (SAWs) with the consideration of energetic ions in DIII-D reversed magnetic shear tokamak experiments. By analyzing the experimental equilibrium profiles, the local and global models for low-frequency SAWs for weak and/or vanishing magnetic shear are discussed based on the unified theoretical framework of the generalize fishbone-like dispersion relation (GFLDR). Resorting to numerical and theoretical analyses, the dependences of mode frequency, growth rate and polarization on the minimum of the safety factor ($q_{min}$), as well as the instability mechanisms are delineated.

The main results of this work are that the LFAMs and BAEs observed in DIII-D experiments are, respectively, the reactive-type and dissipative-type unstable modes with predominantly Alfvénic polarization. Due to the different instability mechanisms, BAE peak occurs further away from the rational $q_{min}$ than LFAM peak does. The BAE eigenfunction is localized at the radial position with the strongest energetic-ion-drive spatially, which leads to deviation from the radial position of $q_{min}$.

The theoretical analysis explains many experimental observations.

1. 1.

   The theory successfully explains the temporal pattern of two bands of instability, the BAE band and the LFAM band, that both appear near rational values of $q_{min}$ but with distinctly different stability properties.

2. 2.

   The predicted values of KBM frequency are in excellent agreement with the experimental LFAM frequencies. The KBM can be unstable even in the absence of energetic particles (EPs).

3. 3.

   The predicted values of BAE frequency span the same range as the experimentally observed values.

4. 4.

   The theory also successfully explains the absence of a third branch of instability at BAAE frequencies, as that branch is predicted to be stable.

5. 5.

   Experimentally, an individual unstable BAE spans a much larger range of frequencies than an unstable LFAM, another feature successfully reproduced by theory.

6. 6.

   Experimentally, unstable LFAMs only persist for a few milliseconds. The short duration of the LFAM is consistent with the very strong $q_{min}$ dependence of the KBM growth rate.

7. 7.

   In experiment, unstable BAEs persist longer than LFAMs, which is consistent with the weaker dependence of the BAE growth rate on $q_{min}$ in theory.

8. 8.

   Temporally, in experiment, LFAMs occur at rational values of $q_{min}$; BAEs also occur near rational values but less precisely. This feature is also reproduced by the theoretical stability predictions: the KBM growth rate peaks sharply at rational $q_{min}$ values but the peak of the BAE growth rate deviates slightly.

9. 9.

   In experiment, for both the LFAM and the BAE, unstable modes with higher values of toroidal mode number $n$ are of shorter duration than lower values of $n$. The narrower growth rate curves as $n$ increases successfully explains this feature.

10. 10.

    Experimentally, the BAE radial eigenfunction has an approximately gaussian shape, consistent with the theoretical prediction that the $L=0$ radial harmonic is most unstable.

11. 11.

    Experimentally, the LFAM is more unstable in plasmas with hydrogen than in pure deuterium plasmas Heidbrink et al. (2021c), a feature explained by the higher value of $\omega_{A}$ in hydrogen plasmas. As Eq. (13) shows, a larger value of $\omega_{A}$ lowers the instability threshold.

On the other hand, there are three discrepancies between theory and experiment.

1. 1.

   Although the predicted KBM growth rate correctly peaks sharply for rational values of $q_{min}$, it remains positive for a much longer duration than the LFAMs are observed experimentally. Evidently, an additional damping mechanism is missing in the theory.

2. 2.

   Although the predicted KBM growth rate has changed significantly for the cases with and without EPs, there is no apparent dependence of LFAM stability on EPs experimentally. Therefore, a more precise theoretical model and more comprehensive experimental analysis are needed for meaningful comparison.

3. 3.

   Although the predicted BAE frequency spans the observed values, the predicted frequency has a parabolic shape with time, while the experimental frequency has a less regular shape. A likely explanation for this discrepancy is imprecise modeling of the fast-ion distribution function.

Finally, there is one theoretical prediction that is inconclusive experimentally: the mode polarization. Theory predicts predominately Alfvénic polarization for both the KBM and the BAE. In experiment, low toroidal mode number ($n\leq 3$) BAEs are usually observed on external magnetic coils; LFAMs are never detected, but the inferred toroidal mode numbers typically span a larger range than those normally detected for RSAEs or BAEs. DIII-D is equipped with one diagnostic that can detect internal magnetic fields, a radial interferometer-polarimeter (RIP) Chen et al. (2016) that measures the line integral of the density and radial magnetic field, $\int n_{e}B_{r}dl$. This diagnostic clearly detects RSAEs and BAEs, which is consistent with their expected shear-wave polarization. Fluctuations are observed by RIP for some LFAMs, indicating that there is at least some magnetic component, but the signal is weaker than for RSAEs and BAEs. It is not presently known if this difference is due to a line-integral effect associated with the mode structure or if the LFAM polarization is less Alfvénic than the other modes.

One of authors (R.R. Ma) would like to acknowledge Dr. Lei Yang and Dr. Yunpeng Zou for their useful discussions and the DIII-D team for providing the experimental data. The authors thank Dr. Xiaodi Du for helpful comments concerning the mode polarization. R.R Ma is also grateful to the Center for Nonlinear Plasma Science (CNPS) for its enlightening academic discussion, which provides a valuable sources of scientific stimuli.

This work has been supported in part by the National key R&D Program of China under Grant Nos. 2022YFE03040002 and 2018YFE0304103, by the National Science Foundation of China under Grant Nos. 12261131622 and 12175053 and Natural Science Foundation of Sichuan under Grant No. 2022NSFSC1814 and Sichuan Science and Technology Program under Grant No. 2022ZYD0019. This work has also been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 – EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Awards DE-FC02-04ER54698 and DE-SC0020337.

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