Experimental investigation of kinetic instabilities driven by runaway electrons in the EXL-50 spherical torus

In plasmas related to nuclear fusion, energetic particles that are significantly hotter than the plasma bulk can be generated through various mechanisms, such as auxiliary heating (electron cyclotron resonance heating (ECRH), ion cyclotron resonance heating (ICRH) and neutral beam injection (NBI)), alpha particles[1] or runaway avalanche[2]. During plasma disruptions, the current quench (CQ) induces a large toroidal electric field, which accelerates runaway electrons (REs) and produces a relativistic beam. In the process of slowing down of energetic particles, kinetic instabilities driven by them can be generated through nonlinear wave-particle interactions (WPIs).

One of the primary threats to the stable operation of tokamak fusion reactors is the transfer of the plasma current from thermal to relativistic electrons during CQ[3]. REs can cause serious localized damage to the plasma-facing components (PFCs)[4]. Numerous theoretical and experimental research efforts have been dedicated to developing effective mitigation methods for these effects, particularly for future fusion reactors. A simple and fundamental mitigation strategy is to inject impurities into the post-disruption plasma to increase radiation, which is expected to be used in ITER. Other mitigation strategies under investigation include magnetic perturbations generated by external magnetic coils[5], as well as WPIs driven by either kinetic instabilities[6] or external wave injection[7].

The interaction of plasma waves or instabilities with energetic particles can cause the redistribution and loss of energetic particles. The resulting instabilities enhance the transport of energetic particles, and energetic particles escape from the confinement region, which causes additional particle energy loss. Therefore, appropriate instability may favor the mitigation of REs, as observed in the DIII-D tokamak[8].

Kinetic instabilities driven by energetic particles have two forms, one involves the evolution of mode frequency with the plasma equilibrium parameters, and the other is the frequency chirping. Because the instabilities that redistribute energetic particles are generated by the particles themselves, the process is inherently nonlinear. Relativistic REs, being high-energy beams, are capable of producing chirping instabilities.

In experiments, Alfvén eigenmodes and whistler waves excited by energetic particles are observed. The frequency of steady-frequency instabilities is quasi-monochromatic, and its parametric behavior in a cold plasma can be expressed as [9, 10, 11],

where $k$ represents the wavenumber, $v_{A}$ denotes the Alfvén velocity, $k_{\parallel}$ is the wavenumber parallel to the magnetic field, $c$ is the speed of light, and $\omega_{pi}$ represents the ion plasma frequency, respectively. The Alfvén velocity is given by $v_{A}=B/\sqrt{\mu_{0}\rho}$, where $B$ is the magnetic field strength, $\mu_{0}$ is the permeability of vacuum, and $\rho$ is the plasma mass density ($\rho\propto n_{e}$). Therefore, it scales inversely with the square root of plasma density ($v_{A}\propto n_{e}^{-1/2}$). From Eq. (1), the instability frequency scales with the plasma density, ranging from $n_{e}^{-1/2}$ ($k_{\parallel}^{2}c^{2}\ll\omega_{pi}^{2}$) to $n_{e}^{-1}$ ($k_{\parallel}^{2}c^{2}\gg\omega_{pi}^{2}$)[10].

A commonly used analytical theory for beam instability is the Berk-Breizman model, which demonstrates various nonlinear behaviors in steady-state dynamics. In a dissipative plasma with a large number of energetic particles, nonlinear WPIs produce self-trapped structures (holes and clumps) in the phase-space. The presence of a particle source can also affect the nonlinear behavior. Typically, chirping instabilities exhibit two branches of frequency, one increasing upward and the other decreasing downward. In the Berk-Breizman model, chirping instabilities spontaneously burst with a frequency shift and time relationship of $\delta\omega\propto\sqrt{t}$[12, 13]. The effect of various collision operators on the nonlinear frequency chirping of fast-ion-driven toroidicity-induced Alfvén eigenmodes (TAEs) has been studied in tokamak and Wendelstein 7-X (W7-X)[14]. The Krook operator is associated with periodic, well-separated chirping events that are routinely observed in experiments and numerical simulations. If the diffusion effect increases, the motion of hole-clump pairs can be suppressed, leading to the gradual disappearance of sub-branches[15]. Additionally, drag can generate asymmetric frequency chirping, which is necessary to enhance holes and suppress clumps.

Relaxation oscillations accompanied by quasi-periodic chirping instabilities burst have been observed in various devices with typical periods in the order of milliseconds[16, 17, 18, 19, 20, 21, 8]. In fusion experiments, chirping instabilities are widespread in various frequency bands and exhibit different frequency structures, such as chirping-up frequency, chirping-down frequency and chirping-bifurcating frequency.

These instabilities are excited when REs resonate with the wave. The wave phase seen by REs remains stationary when[10, 11],

where $\omega$ is the wave frequency, $v_{\parallel}$ is the parallel velocity, $k_{\perp}$ is the perpendicular component of the wavenumber, $v_{d}$ is the orbital drift, $\Omega_{ce}$ is the electron cyclotron frequency, and $\gamma$ is the relativistic factor, respectively. Here, $l$ is an integer, and the main excitation components are $l=0,\pm 1$. If $\Omega_{ce}>0$, then the cases of $l=-1$, $l=0$ and $l=+1$ correspond to the anomalous Doppler, Cherenkov and normal Doppler resonances, respectively. Theoretically, resonance may occur at any value of the integer $l$, but in tokamak experiments, the measured wave frequency is much smaller than the electron cyclotron frequency ($\omega\ll\Omega_{ce}$), so these resonant conditions are fulfilled when $l=-1$ or $l=0$.

The threshold for magnetosonic-whistler instability driven by relativistic REs is expressed as[9, 22],

where $Z$ represents the effective ion charge number, $B_{T}$ is the magnetic field strength in unit of T, $T_{eV}$ is the background plasma temperature in unit of eV, $n_{r}$ and $n_{e}$ refer to the RE density and background plasma density, respectively. Eq. (3) implies that collisional damping has less impact on high temperature plasmas, resulting in a lower density of REs required to drive the instability. This threshold is independent of plasma density and is only relevant for the fraction of REs at low magnetic field strengths. The temporal evolution of this threshold is highly sensitive to various plasma parameters.

This study presents the first observation of kinetic instabilities driven by REs in the XuanLong-50 (EXL-50) spherical torus[23]. This paper is organized as follows. In Section 2, the EXL-50 experiment and the diagnostics used in this study are introduced. The characteristics of the observed kinetic instabilities are described in Section 3, including dispersion relation, chirping characteristic, resonant condition and excitation threshold. Finally, the conclusion is given in Section 4.

The experiments are carried out on the EXL-50 spherical tokamak at the Energy iNNovation (ENN) science and technology development private limited company in Langfang, China. Hydrogen plasmas are used and ECRH is employed to heat plasma and drive the plasma current. The EXL-50 is a medium-sized spherical tokamak with a major radius of $R_{0}=0.58$ m and a minor radius of $a=0.41$ m, and it has been in operation since early 2020[23]. The toroidal field $B_{t}$ can reach a maximum of approximately 0.5 T at $r\approx 0.48$ m, and the aspect ratio $A$ is smaller than 2.

Three sets of ECRH systems (28 GHz) are used and can produce a large number of energetic electrons in the EXL-50 spherical tokamak. Discharges of the plasma current substantially above 100 kA can be obtained by 100 kW ECRH. In this study, only ECRH is used to heat and maintain the plasma, with the plasma current kept at $I_{p}$ = 50-100 kA and the line-integrated plasma density at $\int{n_{e}dl}$ = (1-4)$\times 10^{18}$ m^-2.

The diagnostic layout of the EXL-50 spherical tokamak used in this study is illustrated in Fig. 1. A microwave interferometer operated at 140 GHz is used to measure the line-integrated plasma density[24]. A distant hard X-ray (HXR) scintillating plastic detector is placed a few meters away from the EXL-50 machine to detect the volumetric bremsstrahlung emission resulting from the loss of energetic electrons to the machine wall. The interaction between energetic electrons lost to the wall and the wall material causes a rapid increase of this measured signal. A high-frequency magnetic pickup coil mounted at the low-field side (LFS) on the mid-plane of the vacuum vessel is used to diagnose the frequency chirping instabilities[25]. The coil has a frequency response from 1 MHz to 200 MHz and is digitized by a digital oscilloscope with a maximum sampling rate of 3 GS/s.

Fig. 2 shows the plasma parameters and raw signal measured by the magnetic pickup coil obtained from a typical ECRH discharge. The plasma is ignited and sustained using two sets of ECRH systems. The loop voltage is close to zero during the flat-top of the plasma current. For the normal discharges in EXL-50, the plasma current is carried by the energetic electrons produced by ECRH[23]. The velocity distribution of energetic electrons driven by ECRH is different to that of runaway electrons induced by a toroidal electric field. In the former case, the energetic electrons possess similar magnitudes of parallel and perpendicular velocities, while in the latter, the parallel velocities dominate. On the other hand, considerable toroidal electric field can be induced by internal reconnection event (IRE) or disruption in EXL-50. Part of energetic electrons is converted to runaway electrons during IRE or disruption phase which will excite special instabilities. When the second ECRH system is shut down, the plasma density drops sharply, and there are several relaxation oscillations in the line-integrated plasma density during 2.7-3 s. There are many high frequency fluctuations in the raw signal data from the magnetic pickup coil.

The frequency of chirping instabilities can be obtained from the spectrogram of the magnetic field measured by the high-frequency magnetic pickup coil. The time evolution of a detailed frequency of kinetic instabilities is shown in Fig. 2(d) during the time of 2.7-2.76 s as shown in the shadow of Fig. 2(a), and it consists of two components, $\omega\left(t\right)=\omega_{0}\left(t\right)+\delta\omega\left(t\right)$, where $\omega_{0}\left(t\right)$ and $\delta\omega\left(t\right)$ are the central frequency and the time-varying frequency part of each burst cycle, respectively. Obviously, the central frequency of chirping instabilities depends on the line-integrated plasma density, which will be described in detail later. The frequency of kinetic instabilities is driven by REs, and during this period, the instability frequency varies within a range of 14-18 MHz.

For the first time on the EXL-50 spherical tokamak, frequency chirping instabilities driven by energetic electrons have been observed. These instabilities are identified using a high-frequency magnetic pickup coil mounted on the vacuum vessel. They are detected in a plasma driven by the ECRH and associated with energetic electrons during the ECRH steady-state plasma.

The central frequency of kinetic instabilities is exponentially related to the line-integrated plasma density, $f\propto\left(\int{n_{e}dl}\right)^{-b}$, where the exponent $b$ is between 0.5 to 1, in agreement with Alfvénic scaling. Although the frequency chirps by 14.8-16.6 MHz on a timescale of 1 ms, there are numerous frequency bands in the EXL-50 spherical tokamak. The measured instability frequency structures are consistent with the hole-clump model for frequency chirping driven by energetic electrons. From the resonance condition, the chirping instabilities have negative frequencies, which has been experimentally observed. Theoretically, the excitation threshold of the whistler instability driven by REs is related to the ratio of the RE density to the background plasma density, and such a relationship is first demonstrated experimentally in this study. The relationship between the plasma current and plasma density during instability bursts is in agreement with that predicted by the theoretical model of the whistler instability threshold excited by REs. The instability can be stabilized by increasing the plasma density, which is consistent with the wave-particle resonance mechanism. This research demonstrates the controlled excitation of chirping instabilities in a tokamak plasma and describes several new features, which include the dispersion relation, the chirping characteristic and the resonance condition. It improves the understanding of the physical picture of RE control and mitigation.