Perturbative analysis of low-frequency instabilities in high-field ST40 experiments

The control and avoidance of energetic-particle (EP)-driven MHD instabilities [1, 2] are essential for achieving burning plasmas relevant to economical, high-gain fusion reactors. Ignition is projected to be possible only if alpha particle losses are kept below 30% while losses of as little as 5% of the alpha particles will likely be already damaging for the ITER first wall [3]. This poses a serious unresolved problem, given that MHD instabilities have led to losses of over 50% of the injected beam ions in TFTR [4], DIII-D [5] and NSTX [6]. Due to their proven harmful impact, EP-driven instabilities and their associated transport have been identified as one of the main physics gaps that need to be closed to confidently design a low-capital-cost tokamak fusion pilot plant [7].

The understanding and prediction of the nature of Alfvénic oscillations in fusion devices are valuable tools to inform solutions for minimizing energetic particle transport. Typically, Alfvén waves exhibit either a steady frequency of oscillation, leading to diffusive losses as a result of resonance overlap, or a chirping frequency, leading to convective losses, though more complex behaviors, such as avalanching regimes, are also observed in experiments ³ . For the case of dominant diffusive losses, reduced modeling such as those based on quasilinear theory, are expected to be sufficient to capture the essence of the self-consistent evolution of the fast ion distribution function. For chirping, more comprehensive tools that capture its intrinsic nonlinear bounce-phase nature are needed for assessing fast ion transport.

Experimentally, a distinction has been observed in terms of the Alfvénic nonlinear response in tokamaks of different aspect ratios. Spherical tokamaks (STs) tend to exhibit frequent Alfvénic chirping and avalanching, accompanied by wave amplitude bursting, while conventional tokamaks tend to have Alfvénic waves oscillating with a nearly fixed frequency and a quasi-steady amplitude [8]. Employing nonlinear kinetic theory near threshold, [9, 10] predicted that collisions and injection speed (i.e. sub- vs supra-Alfvénic) alone cannot account for this difference and that the microturbulence scattering experienced by resonant fast ions was key to explain the observations ⁴ . In fact, [12] had predicted that turbulent scattering can exceed collisional scattering in typical scenarios of conventional tokamaks. It is known that STs naturally exhibit lower turbulent transport with respect to conventional tokamaks. For example, on NSTX the total thermal ion diffusivity has been found to be of order of its neoclassical level [13] in the mid-radius region of higher collisionality H-mode plasmas. These distinct turbulence features have been found to be consistent with the observation that chirping instabilities are rare in conventional tokamaks and common in STs [9] since the turbulence acts to effectively increase the scattering experienced by the resonant fast ions [12] and therefore to prevent the chirping and avalanching responses. Experiments with reduced turbulence in DIII-D (using negative triangularity) [14, 15] and in ASDEX-U (through flattening of thermal gradients due to impurity accumulation in the core) [16] observed more prevalence of bursting/chirping than in their usual operational configurations.

STs are a potentially transformative route to a more compact and possibly lower cost fusion power producing facility because of their fundamental properties of enhanced confinement and stability at low aspect ratio [17, 18], which have been established quantitatively in the first generation of high-powered STs, NSTX [19, 20] and MAST [21]. ST40, depicted in figure 1, is a high toroidal field and high beta ST [22] that was built and is owned and operated by Tokamak Energy Ltd UK, a private company whose mission it is to develop the ST for commercial power generation. ST40's most recent period of operation has achieved its goal of producing plasmas with central ion temperatures in excess of 100 million Kelvin (8.6 keV) [23], values considered to be near those required to produce net fusion power.

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The purpose of this paper is to further enhance the understanding of fast-ion-driven instabilities by performing the characterization of the linear and nonlinear Alfvénic behavior observed in a recent campaign of the high-field, high-temperature ST40 ST. We first describe basic characteristics of the ST40 machine, involved diagnostics and the equilibrium reconstruction. Then, we show MHD eigenmode analysis using the NOVA code [24] and its kinetic post-processor NOVA-K code [25, 26]. To better understand the underlying physics of oscillation bifurcation, we use two selected discharges in which clear transitions between constant frequency and chirping phases were observed. The results are then interpreted in terms of reduced nonlinear analytic theory of driven marginally-unstable waves in the presence of background dissipation [27, 28], specifically regarding the sustainment of phase space structures that are thought to be connected with chirping [27, 29, 30].

ST40 is a compact device with major radius $R_{0} = 0.4-0.6$ m and minor radius $a = 0.25-0.33$ m (with aspect ratio typically in the range $A = 1.6-1.8$). In the most recent campaign, typical parameters were plasma currents ranging from $I_{\mathrm {p}}\approx0.4-0.8$ MA and on-axis toroidal magnetics fields of $B_{\mathrm {T}}\approx1.5-2.3T$ produced by copper magnets. These toroidal field values are over twice those of any other high-powered ST. The plasma current flattop phase lasted up to 0.2 s. 2 MW of nominal neutral beam heating were available, with one source at 25 keV and the other at 55 keV. All beams are launched from the midplane and are tangential, co-injected with respect to the plasma current.

The ST40 discharges analyzed in this work were achieved with geometric axis radius $R_{\mathrm {geom}} = 0.45$, elongation κ = 1.5, aspect ratio A = 1.65, plasma current $I_{\mathrm {p}} = 0.6$ MA, and vacuum toroidal magnetic field $B_{\mathrm {T}} = 2.3T$ at the major radius. 1.8 MW of deuterium neutral beams were injected into deuterium plasmas with line average densities of $4\:10^{19}\textrm{m}^{-3}$. High temperatures were also achieved in hydrogen thermal plasmas with deuterium neutral beam injection. The pulse length was approximately 0.1 s in duration.

During the analysis performed by this work, however, measurements of the full kinetic profiles were not available. Temperatures were measured using line ratios from an x-ray crystal spectrometer measuring He-like Ar and a charge exchange diagnostic measuring C6+ at three radial positions on the low field side, from the plasma center to mid-radius. The line integrated electron density was obtained from interferometry. The plasma profiles were inferred using an integrated analysis workflow [23] that incorporated the line-of-sight integrated and profile measurements of the electron and ion temperatures and toroidal rotation, the line-of-sight integrated density measurements, as well as equilibrium constraints given by NUBEAM [31]. The TRANSP transport code [32] was then used to determine the temperature of the main hydrogenic species, as well as the fast ion density profile. Ranges of profiles for each discharge that were consistent with these constraints gave the uncertainty in the profile inferences. These uncertainties were typically around 20%. More details of the integrated analysis workflow can be found in [23]. These runs have time evolving temperatures and densities, along with evolving equilibria reconstruction from magnetics only.

We analyze waves in two ST40 shots that exhibit a clear transition in their behavior from the quasi-steady, fixed-frequency phase to the wave chirping phase, when they exhibit rapid frequency variations on a millisecond time scale, accompanied by bursts in amplitude. In shots 9831 and 9894, such a transition occurred approximately around 0.08 s and 0.092 s, respectively—see spectra of magnetic fluctuations shown in figures 2 and 3. These transitions are helpful in offering a view on the background plasma parameters whose changes correlate with a change of the nonlinear character. In this work, we focus on the effect of equilibrium, macroscopic profile changes over tens of milliseconds on determining favorable conditions for the emergence of chirping. However, as recently identified by Bland et al [33], sub-millisecond timescale changes can be relevant for determining the strength of an Alfvénic response in ST40. As Alfvénic waves grow in amplitude, they can eventually induce fast ion losses. Those losses have been linked to a reduction of plasma rotation in ST40, which in turn, was observed to mediate a H-L mode transition, as inferred by D-alpha measurements [33]. The modeling of the more extreme phenomenon and its relation to confinement mode transitions are beyond the scope of this study.

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The NUBEAM/TRANSP calculated fast ion distribution function for shot 9831 appears in figure 4. Kinetic and fast ion profiles from TRANSP were used as a basis for computing the linear Alfvénic continua by the ideal MHD code NOVA [24]. The most intense Alfvénic waves were identified with toroidal mode number n = 1 in the experiments. From the equilibrium reconstruction using the profiles inferred from the experiments, NOVA surveys did not find any reasonable mode of purely Alfvénic polarization. Once the acoustic branch was included in the computations, beta-induced Alfvén-acoustic eigenmode (BAAE) were identified with frequency plausible with experimental data. BAAEs [34] are mixed polarization modes that reside in the gap that arise from the coupling between the acoustic and the Alfvénic continua. The calculated eigenstructures and continua for discharges 9831 and 9894, at times near the transition into chirping, are presented in figures 5 and 6, respectively. Their eigenstructures are peaked at around 0.3 and 0.1 in units of normalized square root of poloidal flux, respectively. In identifying the mode frequency, toroidal rotation inferred from charge exchange measurements was taken into account. No further experimental characteristic could be inferred from internal measurements, due to limitations on diagnostics, which leaves open the possibility that other types of modes were excited instead. Our linear, perturbative MHD computations indicate BAAEs as a mode compatible with available measurements ⁵ . We note, however, that the validity of an ideal MHD description for modes with low frequency (as compared to the Alfvén frequency) is in general limited. Those modes are known to be damped at high $T_{i}/T_{e}$ [35] and can also be damped due to the inclusion of effects not present in NOVA analysis, such as thermal ion damping in a (gyro)kinetic framework, parallel electric fields and non-perturbative and diamagnetic effects [36–51] ⁶ .

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For the nonlinear analysis, we employ the theoretical understanding of the dynamics of marginally unstable eigenmodes, which, to leading order, can be described by an integro-differential evolution equation of cubic nonlinearity [52]. Such equation has been shown to indicate a chirping response in its explosive phase [28]. Building on that, a criterion for chirping likelihood ($\mathrm {Crt}$) was developed for eigenmodes in a tokamak [9, 53], where phase-space integration over the resonance surfaces in action space and realistic particle orbits are accounted for,

$$\begin{equation} \mathrm {Crt}=\frac{1}{N}\underset{j,\sigma_{\parallel}}{\sum}\int dP_{\varphi}\int d\mu\frac{\left|V_{j}\right|^{4}}{\omega_{\theta}\nu_{\mathrm {drag}}^{4}}\left|\frac{\partial\Omega_{j}}{\partial I}\right|\frac{\partial f}{\partial I}\mathrm {Int} < 0 \end{equation} \tag{ 1 }$$

where

$$\begin{equation} \mathrm {Int}\equiv Re\int_{0}^{\infty}dz\frac{z}{\frac{\nu_{\mathrm {stoch}}^{3}}{\nu_{\mathrm {drag}}^{3}}z-i}\exp\left[-\frac{2}{3}\frac{\nu_{\mathrm {stoch}}^{3}}{\nu_{\mathrm {drag}}^{3}}z^{3}+iz^{2}\right]. \end{equation} \tag{ 2 }$$

The chirping criterion $\mathrm {Crt}$ (equation (1)) is critically sensitive to the ratio between the effective frequencies experienced by resonant fast ions due to stochastic and coherent processes (represented by the effective rates $\nu_{\mathrm {stoch}}$ and $\nu_{\mathrm {drag}}$, respectively), comprising both collisional and micro-turbulent sources. The criterion, numerically implemented into NOVA-K [25, 26], provides a relatively simple prediction of the likelihood of chirping: if $\mathrm {Crt}\lt0$, the mode is expected to chirp; if $\mathrm {Crt}\gt0$, the mode is expected to exhibit a regular and quasi-steady frequency response. In equation (1), V_j is proportional to the interaction Hamiltonian, ω_θ is the poloidal bounce frequency, I is the relevant action for particle redistribution, f is the fast ion distribution function, j and $\sigma_{\parallel}$ are resonance labels, $\Omega_{j}$ denotes the resonance condition in action variables, P_ϕ is the canonical toroidal momentum, µ is the magnetic moment, and N is a normalization constant.

The distribution used in the numerical calculations for $\mathrm {Crt}$ involves a superposition of two fast ion components due to one neutral beam injector at 25 keV and another at 55 keV. They are each fit into an analytic form consisting of a Gaussian function in pitch angle and a slowing down form in energy, The NUBEAM reconstructed distribution is used to infer fitting parameters. More details of how the distribution is specified in terms of Legendre polynomials are given in [54].

Table 1 shows the chirping analysis of the two ST40 discharges, each at a time before and after the Alfvénic transition. It is found that the marked decrease of anomalous transport correlates with the mode entering the chirping phase in the experiments. In figures 7(a) and 8(a), it can be seen that the TRANSP inferred thermal ion heat conductivity χ_i , used as a proxy for the fast ion anomalous diffusivity in experiments [55] in the absence of direct turbulence measurements, decreases as the mode nonlinear character transitions. Following previous analyses [9, 56], the fast ion transport coefficients are calculated from the thermal ones using the scaling relations that resulted from gyrokinetic surveys [12, 57] in both passing and trapping regimes. The effective collisional coefficients are evaluated by NOVA-K, as described in [10, 58].

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As stated above, in both discharges 9831 and 9894, there occurs a considerable reduction of χ_i preceding the emergence of chirping, as inferred by TRANSP. Interestingly, however, the underlying reason for such reduction appears to be different in nature for the two discharges. For 9831, figures 7(b) and (c) indicates that the initially flat density profile at 0.04 s becomes much steeper at 0.09 s, which effectively decreases the ion temperature gradient (ITG) drive associated with $\eta_{i} = \left|\nabla\ln T_{i}\right|/\left|\nabla\ln n_{i}\right|$. As seen in figures 8(b) and (c), shot 9894 did not exhibit substantial changes in either the ion density, temperature gradients or η_i between the two times. The turbulent activity reduction in 9894 can be ascribed to the effect of ITG suppression by enhanced plasma beta [59–61]. As evidenced in figure 9, during the chirping, the total (thermal plus fast ion species) plasma beta locally at the peak of each chirping eigenstructure is twice as large for 9894 than it is for 9831. Besides, in 9894 the beta continues to grow throughout the chirping phase, in contrast to a flat beta during the chirping window of 9831 (roughly from 0.08 s to 0.09 s). In both discharges, the q profile did not present any considerable variation throughout the transition (it is known that if q gets more reversed, ITGs tend to be stabilized—see [62]).

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A detailed turbulence linear gyrokinetic stability analysis, which is beyond the scope of this work, has been recently performed [63] for similar shots (including #9831) as those analyzed in our paper. [63] found that ITGs and trapped-electron mode (TEM)s are dominant in the plasma core, which is the region of interest in our studies. The turbulence channel was found to be dominated by low k, ion-scale modes. Here, we are only interested in the transport coefficients, anomalous and collisional, at the ion scale, in order to infer fast ion scattering, since it is known to affect the character of the nonlinear response of Alfvén eigenmodes. The ion diffusivities have been calculated within TRANSP in interpretative mode (the measured temperature and density profiles were used as input) in the presence of sources and sinks due to atomic physics, charge exchange, impurities, convective, conductive, and ion-electron couplings. The heat deposition profiles were calculated by NUBEAM. TRANSP then computed diffusivities and conductivities via a power balance framework. The results show that, within the plasma profile uncertainties, the particle diffusivities and heat conductivities are well above the neoclassical levels [64].

We have also investigated the relative importance between neoclassical and anomalous contributions to χ_i . In one of the cases, 9831, the neoclassical ion conductivity is lower than the total ion conductivity by an order of magnitude, indicating that the transport is highly anomalous. However, in shot 9894 the neoclassical component appears to be of the same order as the total χ_i at the radii of interest around the mode location, indicating that turbulent and neoclassical transport are comparable. We note that in TRANSP χ_i is determined from the heat conduction term in the power balance, and is therefore dependent on the ITG. We note that here is an intrinsic uncertainty in the χ_i inference due to uncertainties in extrapolated T_i profile that are amplified by taking its gradient. The neoclassical χ_i component instead is computed from theoretical formulae.

Besides the already discussed limitations of the perturbative ideal MHD theory, the present analysis has neglected a few factors that might also affect the likelihood for each nonlinear scenario, for instance additional sources of fast ion scattering and changes in temperature and wave amplitude that can affect the damping. Theoretically, though, the likelihood for chirping does not depend on the exact value of the background damping rate, provided that the mode is sufficiently near its instability threshold [9, 53]. Beyond predicting the likely nonlinear scenario, a natural extension of the analysis presented here would be towards validating numerical quantification of losses using the kick model framework [65, 66]. However, lack of direct measurement of the neutron rate would make this task too speculative at this point.

Predicting the emergence of bursty chirping/avalanching fast ion driven instabilities offers key insights both on their fundamental understanding and also on their practical consequences for fusion reactor engineering, such as the quantification of their induced intermittent neutron fluxes towards the edge and how they relate to blanket design. The present study employed TRANSP and NOVA/NOVA-K analyses on two high-field, high-temperature ST40 discharges, with two approximately 1 MW tangential NBIs at 25 keV and 55 keV. An integrated data analysis approach was used to constrain several quantities based on the available measurements. Perturbative MHD analysis indicates that the modes appear to be n = 1 BAAEs, which involve mixed acoustic and Alfvénic polarization. The analyzed discharges 9831 and 9894 were chosen because they exhibited a clear transition between the fixed-frequency and the chirping phases, which offered a testbed for validating nonlinear models. Stability assessment suggests that the chirping behavior in ST40 can be inhibited by micro-turbulent scattering on fast ions, which is interpreted in terms of the suppression of coherent phase-space structures that support chirping [27]. The results suggest that fast ion anomalous diffusion likely mediates the transition, with the chirping phase observed to be achieved following a marked decrease in the turbulent transport coefficients, as inferred from temperature gradients of extrapolated plasma profiles. This is consistent with the theoretical expectation that the likelihood for chirping emergence near marginal stability is controlled by the ratio between diffusive and convective coefficients experienced by resonant fast ions [9, 53, 67], and with previous observations in NSTX [56], DIII-D [14] and ASDEX-U [16]. Interestingly, the underlying reason for the decrease of micro-turbulent scattering is found to be distinct in the two ST40 discharges analyzed—while in 9831 there is a marked decrease in the key ITG drive parameter $\eta_{i} = \left|\nabla\ln T_{i}\right|/\left|\nabla\ln n_{i}\right|$ mainly due to ion density profile steepening, in 9894 the turbulent reduction is interpreted in terms of a pronounced enhancement of total plasma beta at the mode location. Uncertainty in plasma profiles for the cases analyzed in this work, however, restricts any quantitative statement. In the next phase of ST40 operation, Thomson scattering measurements will be available and the charge exchange recombination spectroscopy capability will be enhanced. In addition, soft x-ray and bolometric cameras and new neutron and fast ion loss detectors will be installed. New data will provide further insight on the preliminary conclusions that were drawn from this first phase of operation.

V.N.D. thanks J. Parisi and G.J. Wilkie for pointing out to the effect of plasma beta on ITG suppression; A. Bierwage and X. Wang for pointing out limitations of the ideal perturbative MHD model; S. Sharapov for discussions on experimental techniques to suppress fast ion diffusivity, and J.W. Berkery for providing useful comments to the manuscript. This manuscript is based upon work supported by the US Department of Energy, Office of Science, Office of Fusion Energy Sciences, and has been authored by Princeton University under Contracts CRADA NFE-19-07769 and DE-AC02-09CH11466 with the US Department of Energy. The publisher, by accepting the article for publication acknowledges, that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.